Subgroups — C₁₀₉×#24284 #42142278

Order 27904.

1432 conjugacy classes of subgroups.

The subgroup lattice has 792 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×AGL(1,ℤ/8)×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#472×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#471×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₉×#3005	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₉×#3003	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₀₉×#2930	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₀₉×#2046	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₀₉×#2045	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₀₉×#2019	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₀₉×#2018	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₀₉×#1978	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₀₉×#1977	C₂	1	1	yes	maximal, characteristic, complemented
13	C₁₀₉×#1811	C₂	1	1	yes	maximal, characteristic, complemented
14	C₁₀₉×#1785	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₀₉×#1783	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×#41×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂³×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×#137×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
19	C₂×AGL(1,ℤ/8)×C₁₀₉	—	2	4	no	—
20	C₂×AGL(1,ℤ/8)×C₁₀₉	—	2	4	no	—
21	C₂×AGL(1,ℤ/8)×C₁₀₉	—	2	2	no	—
22	C₂×AGL(1,ℤ/8)×C₁₀₉	—	2	2	no	—
23	C₂×AGL(1,ℤ/8)×C₁₀₉	C₂²	1	1	yes	characteristic
24	C₂×AGL(1,ℤ/8)×C₁₀₉	C₂²	1	1	yes	characteristic
25	C₂×AGL(1,ℤ/8)×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
26	C₂²×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
27	C₂²×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₂²×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
29	C₂²×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
30	C₂×#131×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
31	C₂²×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
32	#525×C₁₀₉	C₂²	1	1	yes	characteristic
33	#472×C₁₀₉	—	2	4	no	—
34	#471×C₁₀₉	—	2	4	no	—
35	#452×C₁₀₉	—	2	2	no	—
36	#450×C₁₀₉	—	2	2	no	—
37	#449×C₁₀₉	—	2	2	no	—
38	#447×C₁₀₉	—	2	2	no	—
39	#436×C₁₀₉	—	2	2	no	—
40	#435×C₁₀₉	—	2	2	no	—
41	C₂×#108×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
42	C₂×#108×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
43	#424×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
44	#419×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
45	#418×C₁₀₉	C₂²	1	1	yes	characteristic
46	#418×C₁₀₉	C₂²	1	1	yes	characteristic
47	#418×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
48	C₂×#101×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
49	C₂×#101×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
50	C₂×#100×C₁₀₉	C₂²	1	1	yes	characteristic
51	C₂×#100×C₁₀₉	C₂²	1	1	yes	characteristic
52	C₂×#100×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₂×#100×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₂×#100×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
55	C₂×#100×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
56	#408×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
57	#407×C₁₀₉	C₂²	1	1	yes	characteristic
58	#407×C₁₀₉	C₂²	1	1	yes	characteristic
59	#407×C₁₀₉	C₂²	1	1	yes	characteristic
60	#405×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
61	#341×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
62	#341×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
63	C₂⁵×C₁₀₉	—	2	1	no	—
64	C₂×#41×C₁₀₉	—	2	2	no	—
65	C₂×#41×C₁₀₉	—	2	2	no	—
66	C₂×#41×C₁₀₉	—	2	2	no	—
67	C₂×#41×C₁₀₉	—	2	2	no	—
68	C₂×#41×C₁₀₉	—	4	1	no	—
69	C₂×#41×C₁₀₉	—	4	1	no	—
70	C₂×#41×C₁₀₉	C₂³	1	1	yes	characteristic
71	C₂²×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
72	C₂²×D₄×C₁₀₉	—	2	2	no	—
73	C₂²×D₄×C₁₀₉	—	2	2	no	—
74	C₂²×D₄×C₁₀₉	—	2	1	no	—
75	C₂²×D₄×C₁₀₉	—	2	2	no	—
76	C₂²×D₄×C₁₀₉	—	2	2	no	—
77	C₂²×D₄×C₁₀₉	—	2	1	no	—
78	C₂²×D₄×C₁₀₉	—	4	1	no	—
79	C₂²×D₄×C₁₀₉	—	4	1	no	—
80	C₂²×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
81	C₂²×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
82	C₂²×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₁₀₉	—	2	1	no	—
84	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
85	#137×C₁₀₉	—	2	2	no	—
86	#137×C₁₀₉	—	4	1	no	—
87	AGL(1,ℤ/8)×C₁₀₉	—	2	16	no	—
88	AGL(1,ℤ/8)×C₁₀₉	—	2	16	no	—
89	AGL(1,ℤ/8)×C₁₀₉	—	2	2	no	—
90	AGL(1,ℤ/8)×C₁₀₉	—	2	2	no	—
91	AGL(1,ℤ/8)×C₁₀₉	—	2	2	no	—
92	AGL(1,ℤ/8)×C₁₀₉	—	4	1	no	—
93	AGL(1,ℤ/8)×C₁₀₉	—	4	1	no	—
94	AGL(1,ℤ/8)×C₁₀₉	—	4	1	no	—
95	C₂×Q₁₆×C₁₀₉	—	2	1	no	—
96	C₂×SD₁₆×C₁₀₉	—	2	2	no	—
97	C₂×SD₁₆×C₁₀₉	—	2	2	no	—
98	C₂×SD₁₆×C₁₀₉	—	2	1	no	—
99	C₂×SD₁₆×C₁₀₉	—	2	1	no	—
100	C₂×SD₁₆×C₁₀₉	—	2	1	no	—
101	C₂×SD₁₆×C₁₀₉	—	2	1	no	—
102	C₂×SD₁₆×C₁₀₉	—	2	2	no	—
103	C₂×SD₁₆×C₁₀₉	—	2	1	no	—
104	C₂×SD₁₆×C₁₀₉	—	2	1	no	—
105	C₂×SD₁₆×C₁₀₉	—	2	2	no	—
106	C₂×SD₁₆×C₁₀₉	D₄	1	4	yes	complemented
107	C₂×D₈×C₁₀₉	—	2	2	no	—
108	C₂×D₈×C₁₀₉	—	2	2	no	—
109	C₂×D₈×C₁₀₉	—	2	1	no	—
110	C₂×D₈×C₁₀₉	—	2	1	no	—
111	C₂×D₈×C₁₀₉	—	2	1	no	—
112	C₂×D₈×C₁₀₉	—	2	2	no	—
113	C₂×D₈×C₁₀₉	—	2	1	no	—
114	C₂×D₈×C₁₀₉	—	2	1	no	—
115	C₂×D₈×C₁₀₉	—	2	2	no	—
116	C₂×D₈×C₁₀₉	D₄	1	4	yes	complemented
117	#131×C₁₀₉	—	2	2	no	—
118	#131×C₁₀₉	—	2	2	no	—
119	C₂×M₁₆×C₁₀₉	—	2	2	no	—
120	C₂×M₁₆×C₁₀₉	—	2	2	no	—
121	C₂×M₁₆×C₁₀₉	—	2	1	no	—
122	C₂×M₁₆×C₁₀₉	D₄	1	2	yes	—
123	C₂×M₁₆×C₁₀₉	D₄	1	2	yes	—
124	C₂×M₁₆×C₁₀₉	C₂³	1	1	yes	characteristic
125	C₂×M₁₆×C₁₀₉	C₂³	1	1	yes	characteristic
126	C₂×M₁₆×C₁₀₉	C₂³	1	1	yes	characteristic
127	C₂×M₁₆×C₁₀₉	C₂³	1	1	yes	characteristic
128	C₂×M₁₆×C₁₀₉	C₂³	1	1	yes	characteristic
129	C₂×M₁₆×C₁₀₉	C₂³	1	1	yes	characteristic
130	C₂²×C₈×C₁₀₉	—	2	1	no	—
131	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
132	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
133	#123×C₁₀₉	—	2	1	no	—
134	C₂²⋊Q₈×C₁₀₉	D₄	1	2	yes	—
135	#121×C₁₀₉	D₄	1	2	yes	—
136	#120×C₁₀₉	—	2	1	no	—
137	C₄×D₄×C₁₀₉	—	2	2	no	—
138	#117×C₁₀₉	C₂³	1	1	yes	characteristic
139	#108×C₁₀₉	—	2	2	no	—
140	#108×C₁₀₉	—	2	2	no	—
141	C₄⋊C₈×C₁₀₉	—	2	2	no	—
142	#103×C₁₀₉	—	2	2	no	—
143	#102×C₁₀₉	—	2	2	no	—
144	#102×C₁₀₉	—	2	2	no	—
145	#102×C₁₀₉	—	2	2	no	—
146	#101×C₁₀₉	—	2	2	no	—
147	#101×C₁₀₉	—	4	1	no	—
148	#100×C₁₀₉	—	2	2	no	—
149	#100×C₁₀₉	—	2	2	no	—
150	#100×C₁₀₉	—	2	2	no	—
151	#100×C₁₀₉	—	4	1	no	—
152	#100×C₁₀₉	—	4	1	no	—
153	#100×C₁₀₉	—	4	1	no	—
154	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
155	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
156	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
157	C₂²⋊C₈×C₁₀₉	—	2	2	no	—
158	#97×C₁₀₉	—	2	1	no	—
159	C₄×C₈×C₁₀₉	—	2	1	no	—
160	C₂⁴×C₁₀₉	—	2	1	no	—
161	C₂⁴×C₁₀₉	—	2	1	no	—
162	C₂⁴×C₁₀₉	—	2	1	no	—
163	C₂⁴×C₁₀₉	—	4	2	no	—
164	C₂⁴×C₁₀₉	—	4	2	no	—
165	C₂⁴×C₁₀₉	—	4	2	no	—
166	C₂⁴×C₁₀₉	—	8	4	no	—
167	C₂⁴×C₁₀₉	#41	1	1	yes	characteristic
168	#41×C₁₀₉	—	2	8	no	—
169	#41×C₁₀₉	—	2	1	no	—
170	#41×C₁₀₉	—	2	1	no	—
171	#41×C₁₀₉	—	2	1	no	—
172	#41×C₁₀₉	—	2	1	no	—
173	#41×C₁₀₉	—	4	4	no	—
174	#41×C₁₀₉	—	4	4	no	—
175	#41×C₁₀₉	—	4	4	no	—
176	C₂×Q₈×C₁₀₉	—	2	1	no	—
177	C₂×Q₈×C₁₀₉	—	2	1	no	—
178	C₂×Q₈×C₁₀₉	—	2	1	no	—
179	C₂×Q₈×C₁₀₉	—	2	1	no	—
180	C₂×Q₈×C₁₀₉	—	2	1	no	—
181	C₂×Q₈×C₁₀₉	C₂×D₄	1	2	yes	—
182	C₂×Q₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
183	C₂×D₄×C₁₀₉	—	2	8	no	—
184	C₂×D₄×C₁₀₉	—	2	1	no	—
185	C₂×D₄×C₁₀₉	—	2	2	no	—
186	C₂×D₄×C₁₀₉	—	2	1	no	—
187	C₂×D₄×C₁₀₉	—	2	1	no	—
188	C₂×D₄×C₁₀₉	—	2	1	no	—
189	C₂×D₄×C₁₀₉	—	2	1	no	—
190	C₂×D₄×C₁₀₉	—	2	1	no	—
191	C₂×D₄×C₁₀₉	—	2	1	no	—
192	C₂×D₄×C₁₀₉	—	2	1	no	—
193	C₂×D₄×C₁₀₉	—	2	1	no	—
194	C₂×D₄×C₁₀₉	—	2	1	no	—
195	C₂×D₄×C₁₀₉	—	2	1	no	—
196	C₂×D₄×C₁₀₉	—	2	1	no	—
197	C₂×D₄×C₁₀₉	—	2	1	no	—
198	C₂×D₄×C₁₀₉	—	2	1	no	—
199	C₂×D₄×C₁₀₉	—	2	1	no	—
200	C₂×D₄×C₁₀₉	—	2	1	no	—
201	C₂×D₄×C₁₀₉	—	4	4	no	—
202	C₂×D₄×C₁₀₉	—	4	4	no	—
203	C₂×D₄×C₁₀₉	—	4	2	no	—
204	C₂×D₄×C₁₀₉	—	4	2	no	—
205	C₂×D₄×C₁₀₉	—	4	2	no	—
206	C₂×D₄×C₁₀₉	—	4	4	no	—
207	C₂×D₄×C₁₀₉	—	4	2	no	—
208	C₂×D₄×C₁₀₉	—	4	2	no	—
209	C₂×D₄×C₁₀₉	—	4	2	no	—
210	C₂×D₄×C₁₀₉	C₂×D₄	1	2	yes	—
211	C₂×D₄×C₁₀₉	C₂×D₄	1	2	yes	—
212	C₂×D₄×C₁₀₉	C₂×D₄	1	2	yes	—
213	C₂×D₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
214	C₂²×C₄×C₁₀₉	—	2	1	no	—
215	C₂²×C₄×C₁₀₉	—	2	1	no	—
216	C₂²×C₄×C₁₀₉	—	2	1	no	—
217	C₂²×C₄×C₁₀₉	—	2	1	no	—
218	C₂²×C₄×C₁₀₉	—	2	1	no	—
219	C₂²×C₄×C₁₀₉	—	2	1	no	—
220	C₂²×C₄×C₁₀₉	—	4	2	no	—
221	C₂²×C₄×C₁₀₉	—	4	2	no	—
222	C₂²×C₄×C₁₀₉	—	4	2	no	—
223	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
224	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
225	C₂²×C₄×C₁₀₉	#41	1	1	yes	characteristic
226	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
227	Q₁₆×C₁₀₉	—	4	2	no	—
228	SD₁₆×C₁₀₉	—	2	8	no	—
229	SD₁₆×C₁₀₉	—	2	4	no	—
230	SD₁₆×C₁₀₉	—	2	4	no	—
231	SD₁₆×C₁₀₉	—	4	2	no	—
232	SD₁₆×C₁₀₉	—	4	2	no	—
233	SD₁₆×C₁₀₉	—	4	2	no	—
234	SD₁₆×C₁₀₉	—	4	2	no	—
235	D₈×C₁₀₉	—	2	8	no	—
236	D₈×C₁₀₉	—	2	4	no	—
237	D₈×C₁₀₉	—	2	4	no	—
238	D₈×C₁₀₉	—	4	2	no	—
239	D₈×C₁₀₉	—	4	2	no	—
240	D₈×C₁₀₉	—	4	2	no	—
241	M₁₆×C₁₀₉	—	2	4	no	—
242	M₁₆×C₁₀₉	—	2	4	no	—
243	M₁₆×C₁₀₉	—	2	1	no	—
244	M₁₆×C₁₀₉	—	2	1	no	—
245	M₁₆×C₁₀₉	—	2	1	no	—
246	M₁₆×C₁₀₉	—	2	1	no	—
247	M₁₆×C₁₀₉	—	2	1	no	—
248	M₁₆×C₁₀₉	—	2	1	no	—
249	M₁₆×C₁₀₉	—	2	1	no	—
250	M₁₆×C₁₀₉	—	2	1	no	—
251	M₁₆×C₁₀₉	—	2	1	no	—
252	M₁₆×C₁₀₉	—	2	1	no	—
253	M₁₆×C₁₀₉	—	2	1	no	—
254	M₁₆×C₁₀₉	—	2	1	no	—
255	M₁₆×C₁₀₉	—	4	2	no	—
256	C₂×C₈×C₁₀₉	—	2	1	no	—
257	C₂×C₈×C₁₀₉	—	2	1	no	—
258	C₂×C₈×C₁₀₉	—	2	1	no	—
259	C₂×C₈×C₁₀₉	—	2	1	no	—
260	C₂×C₈×C₁₀₉	—	2	1	no	—
261	C₂×C₈×C₁₀₉	—	2	1	no	—
262	C₂×C₈×C₁₀₉	—	2	1	no	—
263	C₂×C₈×C₁₀₉	—	2	1	no	—
264	C₂×C₈×C₁₀₉	—	4	2	no	—
265	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
266	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
267	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
268	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
269	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
270	C₄⋊C₄×C₁₀₉	—	2	2	no	—
271	C₄⋊C₄×C₁₀₉	C₂×D₄	1	2	yes	—
272	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
273	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
274	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
275	C₄²×C₁₀₉	—	2	1	no	—
276	C₂³×C₁₀₉	—	2	1	no	—
277	C₂³×C₁₀₉	—	2	1	no	—
278	C₂³×C₁₀₉	—	2	1	no	—
279	C₂³×C₁₀₉	—	2	2	no	—
280	C₂³×C₁₀₉	—	2	2	no	—
281	C₂³×C₁₀₉	—	2	2	no	—
282	C₂³×C₁₀₉	—	4	2	no	—
283	C₂³×C₁₀₉	—	4	8	no	—
284	C₂³×C₁₀₉	—	4	1	no	—
285	C₂³×C₁₀₉	—	4	1	no	—
286	C₂³×C₁₀₉	—	4	1	no	—
287	C₂³×C₁₀₉	—	4	1	no	—
288	C₂³×C₁₀₉	—	4	1	no	—
289	C₂³×C₁₀₉	—	4	1	no	—
290	C₂³×C₁₀₉	—	8	2	no	—
291	C₂³×C₁₀₉	—	8	2	no	—
292	C₂³×C₁₀₉	—	8	2	no	—
293	C₂³×C₁₀₉	—	8	4	no	—
294	C₂³×C₁₀₉	—	8	4	no	—
295	C₂³×C₁₀₉	—	8	8	no	—
296	C₂³×C₁₀₉	—	8	8	no	—
297	C₂³×C₁₀₉	#121	1	2	yes	—
298	C₂³×C₁₀₉	C₂×#41	1	1	yes	characteristic
299	Q₈×C₁₀₉	—	2	2	no	—
300	Q₈×C₁₀₉	—	2	1	no	—
301	Q₈×C₁₀₉	—	2	1	no	—
302	Q₈×C₁₀₉	—	2	2	no	—
303	Q₈×C₁₀₉	—	4	1	no	—
304	Q₈×C₁₀₉	—	4	1	no	—
305	D₄×C₁₀₉	—	2	2	no	—
306	D₄×C₁₀₉	—	2	2	no	—
307	D₄×C₁₀₉	—	2	2	no	—
308	D₄×C₁₀₉	—	2	1	no	—
309	D₄×C₁₀₉	—	2	1	no	—
310	D₄×C₁₀₉	—	2	2	no	—
311	D₄×C₁₀₉	—	2	2	no	—
312	D₄×C₁₀₉	—	2	2	no	—
313	D₄×C₁₀₉	—	4	8	no	—
314	D₄×C₁₀₉	—	4	8	no	—
315	D₄×C₁₀₉	—	4	1	no	—
316	D₄×C₁₀₉	—	4	1	no	—
317	D₄×C₁₀₉	—	4	1	no	—
318	D₄×C₁₀₉	—	4	1	no	—
319	D₄×C₁₀₉	—	4	1	no	—
320	D₄×C₁₀₉	—	4	1	no	—
321	D₄×C₁₀₉	—	4	1	no	—
322	D₄×C₁₀₉	—	4	1	no	—
323	D₄×C₁₀₉	—	8	2	no	—
324	C₂×C₄×C₁₀₉	—	2	1	no	—
325	C₂×C₄×C₁₀₉	—	2	1	no	—
326	C₂×C₄×C₁₀₉	—	2	2	no	—
327	C₂×C₄×C₁₀₉	—	2	2	no	—
328	C₂×C₄×C₁₀₉	—	2	2	no	—
329	C₂×C₄×C₁₀₉	—	2	2	no	—
330	C₂×C₄×C₁₀₉	—	2	2	no	—
331	C₂×C₄×C₁₀₉	—	2	1	no	—
332	C₂×C₄×C₁₀₉	—	2	2	no	—
333	C₂×C₄×C₁₀₉	—	4	8	no	—
334	C₂×C₄×C₁₀₉	—	4	1	no	—
335	C₂×C₄×C₁₀₉	—	4	1	no	—
336	C₂×C₄×C₁₀₉	—	4	1	no	—
337	C₂×C₄×C₁₀₉	—	4	1	no	—
338	C₂×C₄×C₁₀₉	#121	1	2	yes	—
339	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
340	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
341	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
342	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
343	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
344	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
345	C₈×C₁₀₉	—	2	2	no	—
346	C₈×C₁₀₉	—	4	1	no	—
347	C₈×C₁₀₉	—	4	1	no	—
348	C₈×C₁₀₉	—	4	1	no	—
349	C₈×C₁₀₉	—	4	1	no	—
350	C₈×C₁₀₉	—	4	1	no	—
351	C₈×C₁₀₉	—	4	1	no	—
352	C₈×C₁₀₉	#120	1	2	yes	—
353	C₈×C₁₀₉	#120	1	2	yes	—
354	C₂²×C₁₀₉	—	2	2	no	—
355	C₂²×C₁₀₉	—	2	2	no	—
356	C₂²×C₁₀₉	—	2	2	no	—
357	C₂²×C₁₀₉	—	2	2	no	—
358	C₂²×C₁₀₉	—	4	2	no	—
359	C₂²×C₁₀₉	—	4	2	no	—
360	C₂²×C₁₀₉	—	4	2	no	—
361	C₂²×C₁₀₉	—	4	1	no	—
362	C₂²×C₁₀₉	—	4	2	no	—
363	C₂²×C₁₀₉	—	4	2	no	—
364	C₂²×C₁₀₉	—	4	2	no	—
365	C₂²×C₁₀₉	—	8	2	no	—
366	C₂²×C₁₀₉	—	8	2	no	—
367	C₂²×C₁₀₉	—	8	2	no	—
368	C₂²×C₁₀₉	—	8	2	no	—
369	C₂²×C₁₀₉	—	8	2	no	—
370	C₂²×C₁₀₉	—	8	2	no	—
371	C₂²×C₁₀₉	—	8	16	no	—
372	C₂²×C₁₀₉	—	8	2	no	—
373	C₂²×C₁₀₉	—	8	2	no	—
374	C₂²×C₁₀₉	—	8	2	no	—
375	C₂²×C₁₀₉	C₂×#121	1	1	yes	characteristic
376	C₂²×C₁₀₉	#546	1	1	yes	characteristic
377	C₂²×C₁₀₉	#546	1	1	yes	characteristic
378	C₄×C₁₀₉	—	2	2	no	—
379	C₄×C₁₀₉	—	4	2	no	—
380	C₄×C₁₀₉	—	4	1	no	—
381	C₄×C₁₀₉	—	4	2	no	—
382	C₄×C₁₀₉	C₂×#120	1	1	yes	characteristic
383	C₄×C₁₀₉	#535	1	1	yes	characteristic
384	C₄×C₁₀₉	D₄²	1	1	yes	characteristic
385	C₄×C₁₀₉	D₄²	1	1	yes	characteristic
386	#24284	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
387	C₂×C₁₀₉	—	2	1	no	—
388	C₂×C₁₀₉	—	2	1	no	—
389	C₂×C₁₀₉	—	4	2	no	—
390	C₂×C₁₀₉	—	8	1	no	—
391	C₂×C₁₀₉	—	8	2	no	—
392	C₂×C₁₀₉	—	8	2	no	—
393	C₂×C₁₀₉	—	8	2	no	—
394	C₂×C₁₀₉	#2408	1	1	yes	characteristic
395	C₂×C₁₀₉	#3068	1	2	yes	—
396	C₂²×AGL(1,ℤ/8)	C₂×C₁₀₉	1	1	yes	characteristic, complemented
397	C₂×#472	C₂×C₁₀₉	1	1	yes	characteristic, complemented
398	C₂×#471	C₂×C₁₀₉	1	1	yes	characteristic, complemented
399	#3005	C₂×C₁₀₉	1	1	yes	characteristic, complemented
400	#3003	C₂×C₁₀₉	1	1	yes	characteristic, complemented
401	#2930	C₂×C₁₀₉	1	1	yes	characteristic, complemented
402	#2046	C₂×C₁₀₉	1	1	yes	characteristic, complemented
403	#2045	C₂×C₁₀₉	1	1	yes	characteristic, complemented
404	#2019	C₂×C₁₀₉	1	1	yes	characteristic, complemented
405	#2018	C₂×C₁₀₉	1	1	yes	characteristic, complemented
406	#1978	C₂×C₁₀₉	1	1	yes	characteristic, complemented
407	#1977	C₂×C₁₀₉	1	1	yes	characteristic, complemented
408	#1811	C₂×C₁₀₉	1	1	yes	characteristic, complemented
409	#1785	C₂×C₁₀₉	1	1	yes	characteristic, complemented
410	#1783	C₂×C₁₀₉	1	1	yes	characteristic, complemented
411	C₁₀₉	#24284	1	1	yes	characteristic, direct product factor, Sylow
412	C₂²×#41	C₂²×C₁₀₉	1	1	yes	characteristic
413	C₂³×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
414	C₂×#137	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
415	C₂×AGL(1,ℤ/8)	—	2	4	no	—
416	C₂×AGL(1,ℤ/8)	—	2	4	no	—
417	C₂×AGL(1,ℤ/8)	—	2	2	no	—
418	C₂×AGL(1,ℤ/8)	—	2	2	no	—
419	C₂×AGL(1,ℤ/8)	C₂²×C₁₀₉	1	1	yes	characteristic
420	C₂×AGL(1,ℤ/8)	C₂²×C₁₀₉	1	1	yes	characteristic
421	C₂×AGL(1,ℤ/8)	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
422	C₂²×SD₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
423	C₂²×SD₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
424	C₂²×D₈	C₂²×C₁₀₉	1	1	yes	characteristic
425	C₂²×D₈	C₂²×C₁₀₉	1	1	yes	characteristic
426	C₂×#131	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
427	C₂²×M₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
428	#525	C₂²×C₁₀₉	1	1	yes	characteristic
429	#472	—	2	4	no	—
430	#471	—	2	4	no	—
431	#452	—	2	2	no	—
432	#450	—	2	2	no	—
433	#449	—	2	2	no	—
434	#447	—	2	2	no	—
435	#436	—	2	2	no	—
436	#435	—	2	2	no	—
437	C₂×#108	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
438	C₂×#108	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
439	#424	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
440	#419	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
441	#418	C₂²×C₁₀₉	1	1	yes	characteristic
442	#418	C₂²×C₁₀₉	1	1	yes	characteristic
443	#418	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
444	C₂×#101	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
445	C₂×#101	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
446	C₂×#100	C₂²×C₁₀₉	1	1	yes	characteristic
447	C₂×#100	C₂²×C₁₀₉	1	1	yes	characteristic
448	C₂×#100	C₂²×C₁₀₉	1	1	yes	characteristic
449	C₂×#100	C₂²×C₁₀₉	1	1	yes	characteristic
450	C₂×#100	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
451	C₂×#100	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
452	#408	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
453	#407	C₂²×C₁₀₉	1	1	yes	characteristic
454	#407	C₂²×C₁₀₉	1	1	yes	characteristic
455	#407	C₂²×C₁₀₉	1	1	yes	characteristic
456	#405	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
457	#341	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
458	#341	C₂²×C₁₀₉	1	1	yes	characteristic, complemented
459	C₂⁵	—	2	1	no	—
460	C₂×#41	—	2	2	no	—
461	C₂×#41	—	2	2	no	—
462	C₂×#41	—	2	2	no	—
463	C₂×#41	—	2	2	no	—
464	C₂×#41	—	4	1	no	—
465	C₂×#41	—	4	1	no	—
466	C₂×#41	C₂³×C₁₀₉	1	1	yes	characteristic
467	C₂²×Q₈	C₂³×C₁₀₉	1	1	yes	characteristic
468	C₂²×D₄	—	2	2	no	—
469	C₂²×D₄	—	2	2	no	—
470	C₂²×D₄	—	2	1	no	—
471	C₂²×D₄	—	2	2	no	—
472	C₂²×D₄	—	2	2	no	—
473	C₂²×D₄	—	2	1	no	—
474	C₂²×D₄	—	4	1	no	—
475	C₂²×D₄	—	4	1	no	—
476	C₂²×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
477	C₂²×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
478	C₂²×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
479	C₂³×C₄	—	2	1	no	—
480	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
481	#137	—	2	2	no	—
482	#137	—	4	1	no	—
483	AGL(1,ℤ/8)	—	2	16	no	—
484	AGL(1,ℤ/8)	—	2	16	no	—
485	AGL(1,ℤ/8)	—	2	2	no	—
486	AGL(1,ℤ/8)	—	2	2	no	—
487	AGL(1,ℤ/8)	—	2	2	no	—
488	AGL(1,ℤ/8)	—	4	1	no	—
489	AGL(1,ℤ/8)	—	4	1	no	—
490	AGL(1,ℤ/8)	—	4	1	no	—
491	C₂×Q₁₆	—	2	1	no	—
492	C₂×SD₁₆	—	2	2	no	—
493	C₂×SD₁₆	—	2	2	no	—
494	C₂×SD₁₆	—	2	1	no	—
495	C₂×SD₁₆	—	2	1	no	—
496	C₂×SD₁₆	—	2	1	no	—
497	C₂×SD₁₆	—	2	1	no	—
498	C₂×SD₁₆	—	2	2	no	—
499	C₂×SD₁₆	—	2	1	no	—
500	C₂×SD₁₆	—	2	1	no	—
501	C₂×SD₁₆	—	2	2	no	—
502	C₂×SD₁₆	D₄×C₁₀₉	1	4	yes	complemented
503	C₂×D₈	—	2	2	no	—
504	C₂×D₈	—	2	2	no	—
505	C₂×D₈	—	2	1	no	—
506	C₂×D₈	—	2	1	no	—
507	C₂×D₈	—	2	1	no	—
508	C₂×D₈	—	2	2	no	—
509	C₂×D₈	—	2	1	no	—
510	C₂×D₈	—	2	1	no	—
511	C₂×D₈	—	2	2	no	—
512	C₂×D₈	D₄×C₁₀₉	1	4	yes	complemented
513	#131	—	2	2	no	—
514	#131	—	2	2	no	—
515	C₂×M₁₆	—	2	2	no	—
516	C₂×M₁₆	—	2	2	no	—
517	C₂×M₁₆	—	2	1	no	—
518	C₂×M₁₆	D₄×C₁₀₉	1	2	yes	—
519	C₂×M₁₆	D₄×C₁₀₉	1	2	yes	—
520	C₂×M₁₆	C₂³×C₁₀₉	1	1	yes	characteristic
521	C₂×M₁₆	C₂³×C₁₀₉	1	1	yes	characteristic
522	C₂×M₁₆	C₂³×C₁₀₉	1	1	yes	characteristic
523	C₂×M₁₆	C₂³×C₁₀₉	1	1	yes	characteristic
524	C₂×M₁₆	C₂³×C₁₀₉	1	1	yes	characteristic
525	C₂×M₁₆	C₂³×C₁₀₉	1	1	yes	characteristic
526	C₂²×C₈	—	2	1	no	—
527	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
528	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
529	#123	—	2	1	no	—
530	C₂²⋊Q₈	D₄×C₁₀₉	1	2	yes	—
531	#121	D₄×C₁₀₉	1	2	yes	—
532	#120	—	2	1	no	—
533	C₄×D₄	—	2	2	no	—
534	#117	C₂³×C₁₀₉	1	1	yes	characteristic
535	#108	—	2	2	no	—
536	#108	—	2	2	no	—
537	C₄⋊C₈	—	2	2	no	—
538	#103	—	2	2	no	—
539	#102	—	2	2	no	—
540	#102	—	2	2	no	—
541	#102	—	2	2	no	—
542	#101	—	2	2	no	—
543	#101	—	4	1	no	—
544	#100	—	2	2	no	—
545	#100	—	2	2	no	—
546	#100	—	2	2	no	—
547	#100	—	4	1	no	—
548	#100	—	4	1	no	—
549	#100	—	4	1	no	—
550	C₂²⋊C₈	—	2	2	no	—
551	C₂²⋊C₈	—	2	2	no	—
552	C₂²⋊C₈	—	2	2	no	—
553	C₂²⋊C₈	—	2	2	no	—
554	#97	—	2	1	no	—
555	C₄×C₈	—	2	1	no	—
556	C₂⁴	—	2	1	no	—
557	C₂⁴	—	2	1	no	—
558	C₂⁴	—	2	1	no	—
559	C₂⁴	—	4	2	no	—
560	C₂⁴	—	4	2	no	—
561	C₂⁴	—	4	2	no	—
562	C₂⁴	—	8	4	no	—
563	C₂⁴	#41×C₁₀₉	1	1	yes	characteristic
564	#41	—	2	8	no	—
565	#41	—	2	1	no	—
566	#41	—	2	1	no	—
567	#41	—	2	1	no	—
568	#41	—	2	1	no	—
569	#41	—	4	4	no	—
570	#41	—	4	4	no	—
571	#41	—	4	4	no	—
572	C₂×Q₈	—	2	1	no	—
573	C₂×Q₈	—	2	1	no	—
574	C₂×Q₈	—	2	1	no	—
575	C₂×Q₈	—	2	1	no	—
576	C₂×Q₈	—	2	1	no	—
577	C₂×Q₈	C₂×D₄×C₁₀₉	1	2	yes	—
578	C₂×Q₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
579	C₂×D₄	—	2	8	no	—
580	C₂×D₄	—	2	1	no	—
581	C₂×D₄	—	2	2	no	—
582	C₂×D₄	—	2	1	no	—
583	C₂×D₄	—	2	1	no	—
584	C₂×D₄	—	2	1	no	—
585	C₂×D₄	—	2	1	no	—
586	C₂×D₄	—	2	1	no	—
587	C₂×D₄	—	2	1	no	—
588	C₂×D₄	—	2	1	no	—
589	C₂×D₄	—	2	1	no	—
590	C₂×D₄	—	2	1	no	—
591	C₂×D₄	—	2	1	no	—
592	C₂×D₄	—	2	1	no	—
593	C₂×D₄	—	2	1	no	—
594	C₂×D₄	—	2	1	no	—
595	C₂×D₄	—	2	1	no	—
596	C₂×D₄	—	2	1	no	—
597	C₂×D₄	—	4	4	no	—
598	C₂×D₄	—	4	4	no	—
599	C₂×D₄	—	4	2	no	—
600	C₂×D₄	—	4	2	no	—
601	C₂×D₄	—	4	2	no	—
602	C₂×D₄	—	4	4	no	—
603	C₂×D₄	—	4	2	no	—
604	C₂×D₄	—	4	2	no	—
605	C₂×D₄	—	4	2	no	—
606	C₂×D₄	C₂×D₄×C₁₀₉	1	2	yes	—
607	C₂×D₄	C₂×D₄×C₁₀₉	1	2	yes	—
608	C₂×D₄	C₂×D₄×C₁₀₉	1	2	yes	—
609	C₂×D₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
610	C₂²×C₄	—	2	1	no	—
611	C₂²×C₄	—	2	1	no	—
612	C₂²×C₄	—	2	1	no	—
613	C₂²×C₄	—	2	1	no	—
614	C₂²×C₄	—	2	1	no	—
615	C₂²×C₄	—	2	1	no	—
616	C₂²×C₄	—	4	2	no	—
617	C₂²×C₄	—	4	2	no	—
618	C₂²×C₄	—	4	2	no	—
619	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
620	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
621	C₂²×C₄	#41×C₁₀₉	1	1	yes	characteristic
622	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
623	Q₁₆	—	4	2	no	—
624	SD₁₆	—	2	8	no	—
625	SD₁₆	—	2	4	no	—
626	SD₁₆	—	2	4	no	—
627	SD₁₆	—	4	2	no	—
628	SD₁₆	—	4	2	no	—
629	SD₁₆	—	4	2	no	—
630	SD₁₆	—	4	2	no	—
631	D₈	—	2	8	no	—
632	D₈	—	2	4	no	—
633	D₈	—	2	4	no	—
634	D₈	—	4	2	no	—
635	D₈	—	4	2	no	—
636	D₈	—	4	2	no	—
637	M₁₆	—	2	4	no	—
638	M₁₆	—	2	4	no	—
639	M₁₆	—	2	1	no	—
640	M₁₆	—	2	1	no	—
641	M₁₆	—	2	1	no	—
642	M₁₆	—	2	1	no	—
643	M₁₆	—	2	1	no	—
644	M₁₆	—	2	1	no	—
645	M₁₆	—	2	1	no	—
646	M₁₆	—	2	1	no	—
647	M₁₆	—	2	1	no	—
648	M₁₆	—	2	1	no	—
649	M₁₆	—	2	1	no	—
650	M₁₆	—	2	1	no	—
651	M₁₆	—	4	2	no	—
652	C₂×C₈	—	2	1	no	—
653	C₂×C₈	—	2	1	no	—
654	C₂×C₈	—	2	1	no	—
655	C₂×C₈	—	2	1	no	—
656	C₂×C₈	—	2	1	no	—
657	C₂×C₈	—	2	1	no	—
658	C₂×C₈	—	2	1	no	—
659	C₂×C₈	—	2	1	no	—
660	C₂×C₈	—	4	2	no	—
661	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
662	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
663	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
664	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
665	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
666	C₄⋊C₄	—	2	2	no	—
667	C₄⋊C₄	C₂×D₄×C₁₀₉	1	2	yes	—
668	C₂²⋊C₄	—	2	2	no	—
669	C₂²⋊C₄	—	2	1	no	—
670	C₂²⋊C₄	—	2	2	no	—
671	C₄²	—	2	1	no	—
672	C₂³	—	2	1	no	—
673	C₂³	—	2	1	no	—
674	C₂³	—	2	1	no	—
675	C₂³	—	2	2	no	—
676	C₂³	—	2	2	no	—
677	C₂³	—	2	2	no	—
678	C₂³	—	4	2	no	—
679	C₂³	—	4	8	no	—
680	C₂³	—	4	1	no	—
681	C₂³	—	4	1	no	—
682	C₂³	—	4	1	no	—
683	C₂³	—	4	1	no	—
684	C₂³	—	4	1	no	—
685	C₂³	—	4	1	no	—
686	C₂³	—	8	2	no	—
687	C₂³	—	8	2	no	—
688	C₂³	—	8	2	no	—
689	C₂³	—	8	4	no	—
690	C₂³	—	8	4	no	—
691	C₂³	—	8	8	no	—
692	C₂³	—	8	8	no	—
693	C₂³	#121×C₁₀₉	1	2	yes	—
694	C₂³	C₂×#41×C₁₀₉	1	1	yes	characteristic
695	Q₈	—	2	2	no	—
696	Q₈	—	2	1	no	—
697	Q₈	—	2	1	no	—
698	Q₈	—	2	2	no	—
699	Q₈	—	4	1	no	—
700	Q₈	—	4	1	no	—
701	D₄	—	2	2	no	—
702	D₄	—	2	2	no	—
703	D₄	—	2	2	no	—
704	D₄	—	2	1	no	—
705	D₄	—	2	1	no	—
706	D₄	—	2	2	no	—
707	D₄	—	2	2	no	—
708	D₄	—	2	2	no	—
709	D₄	—	4	8	no	—
710	D₄	—	4	8	no	—
711	D₄	—	4	1	no	—
712	D₄	—	4	1	no	—
713	D₄	—	4	1	no	—
714	D₄	—	4	1	no	—
715	D₄	—	4	1	no	—
716	D₄	—	4	1	no	—
717	D₄	—	4	1	no	—
718	D₄	—	4	1	no	—
719	D₄	—	8	2	no	—
720	C₂×C₄	—	2	1	no	—
721	C₂×C₄	—	2	1	no	—
722	C₂×C₄	—	2	2	no	—
723	C₂×C₄	—	2	2	no	—
724	C₂×C₄	—	2	2	no	—
725	C₂×C₄	—	2	2	no	—
726	C₂×C₄	—	2	2	no	—
727	C₂×C₄	—	2	1	no	—
728	C₂×C₄	—	2	2	no	—
729	C₂×C₄	—	4	8	no	—
730	C₂×C₄	—	4	1	no	—
731	C₂×C₄	—	4	1	no	—
732	C₂×C₄	—	4	1	no	—
733	C₂×C₄	—	4	1	no	—
734	C₂×C₄	#121×C₁₀₉	1	2	yes	—
735	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
736	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
737	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
738	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
739	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
740	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
741	C₈	—	2	2	no	—
742	C₈	—	4	1	no	—
743	C₈	—	4	1	no	—
744	C₈	—	4	1	no	—
745	C₈	—	4	1	no	—
746	C₈	—	4	1	no	—
747	C₈	—	4	1	no	—
748	C₈	#120×C₁₀₉	1	2	yes	—
749	C₈	#120×C₁₀₉	1	2	yes	—
750	C₂²	—	2	2	no	—
751	C₂²	—	2	2	no	—
752	C₂²	—	2	2	no	—
753	C₂²	—	2	2	no	—
754	C₂²	—	4	2	no	—
755	C₂²	—	4	2	no	—
756	C₂²	—	4	2	no	—
757	C₂²	—	4	2	no	—
758	C₂²	—	4	1	no	—
759	C₂²	—	4	2	no	—
760	C₂²	—	4	2	no	—
761	C₂²	—	8	2	no	—
762	C₂²	—	8	2	no	—
763	C₂²	—	8	2	no	—
764	C₂²	—	8	2	no	—
765	C₂²	—	8	2	no	—
766	C₂²	—	8	2	no	—
767	C₂²	—	8	16	no	—
768	C₂²	—	8	2	no	—
769	C₂²	—	8	2	no	—
770	C₂²	—	8	2	no	—
771	C₂²	C₂×#121×C₁₀₉	1	1	yes	characteristic
772	C₂²	#546×C₁₀₉	1	1	yes	characteristic
773	C₂²	#546×C₁₀₉	1	1	yes	characteristic
774	C₄	—	2	2	no	—
775	C₄	—	4	2	no	—
776	C₄	—	4	2	no	—
777	C₄	—	4	1	no	—
778	C₄	C₂×#120×C₁₀₉	1	1	yes	characteristic
779	C₄	#535×C₁₀₉	1	1	yes	characteristic
780	C₄	D₄²×C₁₀₉	1	1	yes	characteristic
781	C₄	D₄²×C₁₀₉	1	1	yes	characteristic
782	C₂	—	2	1	no	—
783	C₂	—	2	1	no	—
784	C₂	—	4	2	no	—
785	C₂	—	8	1	no	—
786	C₂	—	8	2	no	—
787	C₂	—	8	2	no	—
788	C₂	—	8	2	no	—
789	C₂	C₁₀₉×#2408	1	1	yes	characteristic
790	C₂	C₁₀₉×#3068	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₀₉	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂³×C₄×C₁₀₉ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×AGL(1,ℤ/8)×C₁₀₉	C₂	semidirect
2. C₂×#472×C₁₀₉	C₂	semidirect
3. C₂×#471×C₁₀₉	C₂	semidirect
4. C₁₀₉×#3005	C₂	semidirect
5. C₁₀₉×#3003	C₂	semidirect
6. C₁₀₉×#2930	C₂	semidirect
7. C₁₀₉×#2046	C₂	semidirect
8. C₁₀₉×#2045	C₂	semidirect
9. C₁₀₉×#2019	C₂	semidirect
10. C₁₀₉×#2018	C₂	semidirect
11. C₁₀₉×#1978	C₂	semidirect
12. C₁₀₉×#1977	C₂	semidirect
13. C₁₀₉×#1811	C₂	semidirect
14. C₁₀₉×#1785	C₂	semidirect
15. C₁₀₉×#1783	C₂	semidirect
18. C₂×#137×C₁₀₉	C₂²	semidirect
25. C₂×AGL(1,ℤ/8)×C₁₀₉	C₂²	semidirect
30. C₂×#131×C₁₀₉	C₂²	semidirect
41. C₂×#108×C₁₀₉	C₂²	semidirect
42. C₂×#108×C₁₀₉	C₂²	semidirect
43. #424×C₁₀₉	C₂²	semidirect
44. #419×C₁₀₉	C₂²	semidirect
47. #418×C₁₀₉	C₂²	semidirect
48. C₂×#101×C₁₀₉	C₂²	semidirect
49. C₂×#101×C₁₀₉	C₂²	semidirect
54. C₂×#100×C₁₀₉	C₂²	semidirect
55. C₂×#100×C₁₀₉	C₂²	semidirect
56. #408×C₁₀₉	C₂²	semidirect
60. #405×C₁₀₉	C₂²	semidirect
61. #341×C₁₀₉	C₂²	semidirect
62. #341×C₁₀₉	C₂²	semidirect
106. C₂×SD₁₆×C₁₀₉	D₄	semidirect
116. C₂×D₈×C₁₀₉	D₄	semidirect
386. #24284	C₁₀₉	direct (N × H)
396. C₂²×AGL(1,ℤ/8)	C₂×C₁₀₉	semidirect
397. C₂×#472	C₂×C₁₀₉	semidirect
398. C₂×#471	C₂×C₁₀₉	semidirect
399. #3005	C₂×C₁₀₉	semidirect
400. #3003	C₂×C₁₀₉	semidirect
401. #2930	C₂×C₁₀₉	semidirect
402. #2046	C₂×C₁₀₉	semidirect
403. #2045	C₂×C₁₀₉	semidirect
404. #2019	C₂×C₁₀₉	semidirect
405. #2018	C₂×C₁₀₉	semidirect
406. #1978	C₂×C₁₀₉	semidirect
407. #1977	C₂×C₁₀₉	semidirect
408. #1811	C₂×C₁₀₉	semidirect
409. #1785	C₂×C₁₀₉	semidirect
410. #1783	C₂×C₁₀₉	semidirect
411. C₁₀₉	#24284	direct (N × H)
414. C₂×#137	C₂²×C₁₀₉	semidirect
421. C₂×AGL(1,ℤ/8)	C₂²×C₁₀₉	semidirect
426. C₂×#131	C₂²×C₁₀₉	semidirect
437. C₂×#108	C₂²×C₁₀₉	semidirect
438. C₂×#108	C₂²×C₁₀₉	semidirect
439. #424	C₂²×C₁₀₉	semidirect
440. #419	C₂²×C₁₀₉	semidirect
443. #418	C₂²×C₁₀₉	semidirect
444. C₂×#101	C₂²×C₁₀₉	semidirect
445. C₂×#101	C₂²×C₁₀₉	semidirect
450. C₂×#100	C₂²×C₁₀₉	semidirect
451. C₂×#100	C₂²×C₁₀₉	semidirect
452. #408	C₂²×C₁₀₉	semidirect
456. #405	C₂²×C₁₀₉	semidirect
457. #341	C₂²×C₁₀₉	semidirect
458. #341	C₂²×C₁₀₉	semidirect
502. C₂×SD₁₆	D₄×C₁₀₉	semidirect
512. C₂×D₈	D₄×C₁₀₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×AGL(1,ℤ/8)×C₁₀₉	2
C₂×#472×C₁₀₉	2
C₂×#471×C₁₀₉	2
C₁₀₉×#3005	2
C₁₀₉×#3003	2
C₁₀₉×#2930	2
C₁₀₉×#2046	2
C₁₀₉×#2045	2
C₁₀₉×#2019	2
C₁₀₉×#2018	2
C₁₀₉×#1978	2
C₁₀₉×#1977	2
C₁₀₉×#1811	2
C₁₀₉×#1785	2
C₁₀₉×#1783	2
#24284	109

Subgroups — C109×#24284 #42142278

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#24284 #42142278