Subgroups — C₁₀₉×#24811 #42142805

Order 27904.

998 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#474×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#2542	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₉×#2083	C₂	1	2	yes	maximal, complemented
6	C₁₀₉×#2067	C₂	1	2	yes	maximal, complemented
7	C₁₀₉×#1990	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₀₉×#1939	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₀₉×#1879	C₂	1	2	yes	maximal
10	C₁₀₉×#1802	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₀₉×#1802	C₂	1	2	yes	maximal, complemented
12	C₂²×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
15	C₂×C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
16	C₂×C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
18	C₂×C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₁₀₉	—	2	1	no	—
20	#474×C₁₀₉	—	2	4	no	—
21	C₄×D₈×C₁₀₉	—	2	4	no	—
22	C₂×#107×C₁₀₉	C₂²	1	2	yes	—
23	C₂×#107×C₁₀₉	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
26	C₂×#102×C₁₀₉	C₂²	1	2	yes	—
27	C₂×#102×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₂×#102×C₁₀₉	C₂²	1	1	yes	characteristic
29	C₂×#102×C₁₀₉	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
31	#399×C₁₀₉	—	2	2	no	—
32	#390×C₁₀₉	C₂²	1	2	yes	—
33	#389×C₁₀₉	C₂²	1	2	yes	—
34	#385×C₁₀₉	—	2	2	no	—
35	#384×C₁₀₉	C₂²	1	1	yes	characteristic
36	#384×C₁₀₉	C₂²	1	2	yes	—
37	#380×C₁₀₉	—	2	1	no	—
38	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
39	#340×C₁₀₉	C₂²	1	2	yes	—
40	#340×C₁₀₉	C₂²	1	2	yes	—
41	#340×C₁₀₉	C₂²	1	2	yes	—
42	#336×C₁₀₉	C₂²	1	2	yes	—
43	C₂²×Q₈×C₁₀₉	—	2	1	no	—
44	C₂²×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₁₀₉	—	2	1	no	—
47	C₂³×C₄×C₁₀₉	—	2	1	no	—
48	C₂³×C₄×C₁₀₉	—	2	2	no	—
49	C₂×D₈×C₁₀₉	—	2	2	no	—
50	C₂×D₈×C₁₀₉	—	2	2	no	—
51	C₂×D₈×C₁₀₉	Q₈	1	4	yes	complemented
52	C₂²×C₈×C₁₀₉	C₂³	1	2	yes	—
53	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
55	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
56	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
57	C₂²⋊Q₈×C₁₀₉	—	4	4	no	—
58	C₂²⋊Q₈×C₁₀₉	—	4	4	no	—
59	C₄×D₄×C₁₀₉	—	2	4	no	—
60	C₄×D₄×C₁₀₉	—	2	2	no	—
61	C₄×D₄×C₁₀₉	—	2	2	no	—
62	C₄×D₄×C₁₀₉	—	2	2	no	—
63	C₄×D₄×C₁₀₉	—	4	2	no	—
64	C₄×D₄×C₁₀₉	D₄	1	4	yes	—
65	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
66	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
67	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
68	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
69	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
70	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
71	C₂×C₄⋊C₄×C₁₀₉	—	4	4	no	—
72	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
75	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
77	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
78	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
79	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
80	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
81	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
82	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
83	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
84	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
85	#107×C₁₀₉	—	2	2	no	—
86	#107×C₁₀₉	—	2	4	no	—
87	C₄⋊C₈×C₁₀₉	—	2	2	no	—
88	C₄⋊C₈×C₁₀₉	—	4	2	no	—
89	#102×C₁₀₉	—	2	2	no	—
90	#102×C₁₀₉	—	2	4	no	—
91	#102×C₁₀₉	—	2	2	no	—
92	#102×C₁₀₉	—	4	2	no	—
93	C₄×C₈×C₁₀₉	—	2	2	no	—
94	#95×C₁₀₉	—	2	2	no	—
95	#95×C₁₀₉	D₄	1	2	yes	—
96	C₂⁴×C₁₀₉	—	2	1	no	—
97	C₂⁴×C₁₀₉	—	2	1	no	—
98	C₂×Q₈×C₁₀₉	—	4	2	no	—
99	C₂×Q₈×C₁₀₉	—	4	2	no	—
100	C₂×Q₈×C₁₀₉	—	4	2	no	—
101	C₂×D₄×C₁₀₉	—	2	1	no	—
102	C₂×D₄×C₁₀₉	—	2	1	no	—
103	C₂×D₄×C₁₀₉	—	2	1	no	—
104	C₂×D₄×C₁₀₉	—	2	1	no	—
105	C₂×D₄×C₁₀₉	—	2	1	no	—
106	C₂×D₄×C₁₀₉	—	2	1	no	—
107	C₂×D₄×C₁₀₉	—	2	1	no	—
108	C₂×D₄×C₁₀₉	—	2	1	no	—
109	C₂×D₄×C₁₀₉	C₂×D₄	1	2	yes	—
110	C₂×D₄×C₁₀₉	C₂×Q₈	1	2	yes	—
111	C₂×D₄×C₁₀₉	C₂×Q₈	1	2	yes	—
112	C₂×D₄×C₁₀₉	#41	1	2	yes	—
113	C₂²×C₄×C₁₀₉	—	2	1	no	—
114	C₂²×C₄×C₁₀₉	—	2	4	no	—
115	C₂²×C₄×C₁₀₉	—	2	2	no	—
116	C₂²×C₄×C₁₀₉	—	2	2	no	—
117	C₂²×C₄×C₁₀₉	—	2	1	no	—
118	C₂²×C₄×C₁₀₉	—	4	4	no	—
119	C₂²×C₄×C₁₀₉	—	4	2	no	—
120	C₂²×C₄×C₁₀₉	—	4	2	no	—
121	C₂²×C₄×C₁₀₉	—	4	2	no	—
122	C₂²×C₄×C₁₀₉	—	4	4	no	—
123	C₂²×C₄×C₁₀₉	—	4	2	no	—
124	C₂²×C₄×C₁₀₉	—	4	2	no	—
125	C₂²×C₄×C₁₀₉	—	8	2	no	—
126	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
129	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
131	D₈×C₁₀₉	—	2	8	no	—
132	C₂×C₈×C₁₀₉	—	2	1	no	—
133	C₂×C₈×C₁₀₉	—	2	1	no	—
134	C₂×C₈×C₁₀₉	—	2	1	no	—
135	C₂×C₈×C₁₀₉	—	2	2	no	—
136	C₂×C₈×C₁₀₉	—	2	2	no	—
137	C₂×C₈×C₁₀₉	—	2	1	no	—
138	C₂×C₈×C₁₀₉	—	2	2	no	—
139	C₂×C₈×C₁₀₉	—	2	1	no	—
140	C₂×C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
141	C₄⋊C₄×C₁₀₉	—	2	2	no	—
142	C₄⋊C₄×C₁₀₉	—	2	1	no	—
143	C₄⋊C₄×C₁₀₉	—	2	1	no	—
144	C₄⋊C₄×C₁₀₉	—	2	2	no	—
145	C₄⋊C₄×C₁₀₉	—	2	1	no	—
146	C₄⋊C₄×C₁₀₉	—	2	2	no	—
147	C₄⋊C₄×C₁₀₉	—	2	2	no	—
148	C₄⋊C₄×C₁₀₉	—	2	2	no	—
149	C₄⋊C₄×C₁₀₉	—	4	4	no	—
150	C₄⋊C₄×C₁₀₉	—	4	2	no	—
151	C₄⋊C₄×C₁₀₉	—	4	4	no	—
152	C₄⋊C₄×C₁₀₉	—	4	2	no	—
153	C₄⋊C₄×C₁₀₉	—	4	4	no	—
154	C₄⋊C₄×C₁₀₉	—	8	1	no	—
155	C₄⋊C₄×C₁₀₉	D₈	1	4	yes	complemented
156	C₄⋊C₄×C₁₀₉	C₂×D₄	1	2	yes	—
157	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
158	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
159	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
160	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
161	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
162	C₄²×C₁₀₉	—	2	2	no	—
163	C₄²×C₁₀₉	—	2	2	no	—
164	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
165	C₄²×C₁₀₉	#41	1	2	yes	—
166	C₂³×C₁₀₉	—	2	2	no	—
167	C₂³×C₁₀₉	—	2	2	no	—
168	C₂³×C₁₀₉	—	2	2	no	—
169	C₂³×C₁₀₉	—	2	2	no	—
170	C₂³×C₁₀₉	—	4	1	no	—
171	C₂³×C₁₀₉	—	4	1	no	—
172	C₂³×C₁₀₉	—	4	1	no	—
173	C₂³×C₁₀₉	—	4	1	no	—
174	C₂³×C₁₀₉	—	4	1	no	—
175	C₂³×C₁₀₉	—	4	1	no	—
176	C₂³×C₁₀₉	—	4	1	no	—
177	C₂³×C₁₀₉	—	4	1	no	—
178	C₂³×C₁₀₉	—	4	1	no	—
179	C₂³×C₁₀₉	—	4	1	no	—
180	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
181	Q₈×C₁₀₉	—	8	4	no	—
182	D₄×C₁₀₉	—	2	2	no	—
183	D₄×C₁₀₉	—	2	2	no	—
184	D₄×C₁₀₉	—	2	2	no	—
185	D₄×C₁₀₉	—	2	2	no	—
186	D₄×C₁₀₉	—	2	2	no	—
187	D₄×C₁₀₉	—	4	1	no	—
188	D₄×C₁₀₉	—	4	1	no	—
189	D₄×C₁₀₉	C₂²⋊Q₈	1	4	yes	complemented
190	C₂×C₄×C₁₀₉	—	2	2	no	—
191	C₂×C₄×C₁₀₉	—	2	1	no	—
192	C₂×C₄×C₁₀₉	—	2	2	no	—
193	C₂×C₄×C₁₀₉	—	2	1	no	—
194	C₂×C₄×C₁₀₉	—	2	2	no	—
195	C₂×C₄×C₁₀₉	—	2	2	no	—
196	C₂×C₄×C₁₀₉	—	2	2	no	—
197	C₂×C₄×C₁₀₉	—	2	2	no	—
198	C₂×C₄×C₁₀₉	—	4	1	no	—
199	C₂×C₄×C₁₀₉	—	4	1	no	—
200	C₂×C₄×C₁₀₉	—	4	8	no	—
201	C₂×C₄×C₁₀₉	—	4	1	no	—
202	C₂×C₄×C₁₀₉	—	4	2	no	—
203	C₂×C₄×C₁₀₉	—	4	2	no	—
204	C₂×C₄×C₁₀₉	—	4	2	no	—
205	C₂×C₄×C₁₀₉	—	4	1	no	—
206	C₂×C₄×C₁₀₉	—	4	2	no	—
207	C₂×C₄×C₁₀₉	—	4	2	no	—
208	C₂×C₄×C₁₀₉	—	4	1	no	—
209	C₂×C₄×C₁₀₉	—	4	2	no	—
210	C₂×C₄×C₁₀₉	—	8	8	no	—
211	C₂×C₄×C₁₀₉	—	8	4	no	—
212	C₂×C₄×C₁₀₉	#121	1	2	yes	—
213	C₂×C₄×C₁₀₉	#121	1	2	yes	—
214	C₂×C₄×C₁₀₉	C₂×D₈	1	4	yes	—
215	C₂×C₄×C₁₀₉	C₂×D₈	1	2	yes	—
216	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
217	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
219	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
220	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
221	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
222	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
223	C₈×C₁₀₉	—	2	2	no	—
224	C₈×C₁₀₉	—	4	1	no	—
225	C₈×C₁₀₉	—	4	2	no	—
226	C₂²×C₁₀₉	—	2	4	no	—
227	C₂²×C₁₀₉	—	4	2	no	—
228	C₂²×C₁₀₉	—	4	2	no	—
229	C₂²×C₁₀₉	—	4	2	no	—
230	C₂²×C₁₀₉	—	4	2	no	—
231	C₂²×C₁₀₉	—	4	2	no	—
232	C₂²×C₁₀₉	—	4	2	no	—
233	C₂²×C₁₀₉	—	4	2	no	—
234	C₂²×C₁₀₉	—	4	2	no	—
235	C₂²×C₁₀₉	—	4	2	no	—
236	C₂²×C₁₀₉	—	8	1	no	—
237	C₂²×C₁₀₉	—	8	1	no	—
238	C₂²×C₁₀₉	—	8	1	no	—
239	C₂²×C₁₀₉	—	8	1	no	—
240	C₂²×C₁₀₉	C₂×#121	1	1	yes	characteristic
241	C₂²×C₁₀₉	#546	1	1	yes	characteristic
242	C₂²×C₁₀₉	D₄×Q₈	1	1	yes	characteristic
243	C₂²×C₁₀₉	C₂²×D₈	1	1	yes	characteristic
244	C₂²×C₁₀₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
245	C₂²×C₁₀₉	#577	1	1	yes	characteristic
246	C₂²×C₁₀₉	#578	1	1	yes	characteristic
247	C₄×C₁₀₉	—	2	4	no	—
248	C₄×C₁₀₉	—	2	2	no	—
249	C₄×C₁₀₉	—	4	2	no	—
250	C₄×C₁₀₉	—	4	2	no	—
251	C₄×C₁₀₉	—	8	1	no	—
252	C₄×C₁₀₉	—	8	2	no	—
253	C₄×C₁₀₉	—	8	2	no	—
254	C₄×C₁₀₉	#459	1	4	yes	—
255	C₄×C₁₀₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
256	C₄×C₁₀₉	#537	1	1	yes	characteristic
257	C₄×C₁₀₉	#552	1	1	yes	characteristic
258	C₄×C₁₀₉	#554	1	1	yes	characteristic
259	#24811	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
260	C₂×C₁₀₉	—	4	4	no	—
261	C₂×C₁₀₉	—	8	2	no	—
262	C₂×C₁₀₉	#2417	1	1	yes	characteristic
263	C₂×C₁₀₉	C₂×#459	1	1	yes	characteristic
264	C₂×C₁₀₉	#3046	1	1	yes	characteristic
265	C₂×C₁₀₉	#3294	1	1	yes	characteristic
266	C₂×C₁₀₉	#3330	1	1	yes	characteristic
267	C₂×C₁₀₉	Q₈×D₈	1	1	yes	characteristic
268	C₂×C₁₀₉	#3384	1	1	yes	characteristic
269	C₂×#474	C₂×C₁₀₉	1	1	yes	characteristic, complemented
270	C₂×C₄×D₈	C₂×C₁₀₉	1	1	yes	characteristic
271	#2542	C₂×C₁₀₉	1	1	yes	characteristic, complemented
272	D₄×C₄⋊C₄	C₂×C₁₀₉	1	1	yes	characteristic, complemented
273	#2083	C₂×C₁₀₉	1	2	yes	complemented
274	#2067	C₂×C₁₀₉	1	2	yes	complemented
275	#1990	C₂×C₁₀₉	1	1	yes	characteristic, complemented
276	#1939	C₂×C₁₀₉	1	1	yes	characteristic, complemented
277	#1879	C₂×C₁₀₉	1	2	yes	—
278	#1802	C₂×C₁₀₉	1	1	yes	characteristic, complemented
279	#1802	C₂×C₁₀₉	1	2	yes	complemented
280	C₁₀₉	#24811	1	1	yes	characteristic, direct product factor, Sylow
281	C₂²×D₈	C₂²×C₁₀₉	1	1	yes	characteristic
282	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
283	C₂×C₂²⋊Q₈	—	2	1	no	—
284	C₂×C₂²⋊Q₈	—	2	1	no	—
285	C₂×C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
286	C₂×C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
287	C₂×C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
288	C₂²×C₄⋊C₄	—	2	1	no	—
289	#474	—	2	4	no	—
290	C₄×D₈	—	2	4	no	—
291	C₂×#107	C₂²×C₁₀₉	1	2	yes	—
292	C₂×#107	C₂²×C₁₀₉	1	1	yes	characteristic
293	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
294	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
295	C₂×#102	C₂²×C₁₀₉	1	1	yes	characteristic
296	C₂×#102	C₂²×C₁₀₉	1	2	yes	—
297	C₂×#102	C₂²×C₁₀₉	1	1	yes	characteristic
298	C₂×#102	C₂²×C₁₀₉	1	2	yes	—
299	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
300	#399	—	2	2	no	—
301	#390	C₂²×C₁₀₉	1	2	yes	—
302	#389	C₂²×C₁₀₉	1	2	yes	—
303	#385	—	2	2	no	—
304	#384	C₂²×C₁₀₉	1	2	yes	—
305	#384	C₂²×C₁₀₉	1	1	yes	characteristic
306	#380	—	2	1	no	—
307	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
308	#340	C₂²×C₁₀₉	1	2	yes	—
309	#340	C₂²×C₁₀₉	1	2	yes	—
310	#340	C₂²×C₁₀₉	1	2	yes	—
311	#336	C₂²×C₁₀₉	1	2	yes	—
312	C₂²×Q₈	—	2	1	no	—
313	C₂²×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
314	C₂²×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
315	C₂³×C₄	—	2	1	no	—
316	C₂³×C₄	—	2	1	no	—
317	C₂³×C₄	—	2	2	no	—
318	C₂×D₈	—	2	2	no	—
319	C₂×D₈	—	2	2	no	—
320	C₂×D₈	Q₈×C₁₀₉	1	4	yes	complemented
321	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
322	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
323	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
324	C₄⋊Q₈	—	2	2	no	—
325	C₄⋊Q₈	—	2	2	no	—
326	C₂²⋊Q₈	—	4	4	no	—
327	C₂²⋊Q₈	—	4	4	no	—
328	C₄×D₄	—	2	4	no	—
329	C₄×D₄	—	2	2	no	—
330	C₄×D₄	—	2	2	no	—
331	C₄×D₄	—	2	2	no	—
332	C₄×D₄	—	4	2	no	—
333	C₄×D₄	D₄×C₁₀₉	1	4	yes	—
334	C₂×C₄⋊C₄	—	2	1	no	—
335	C₂×C₄⋊C₄	—	2	2	no	—
336	C₂×C₄⋊C₄	—	2	2	no	—
337	C₂×C₄⋊C₄	—	2	2	no	—
338	C₂×C₄⋊C₄	—	2	1	no	—
339	C₂×C₄⋊C₄	—	2	1	no	—
340	C₂×C₄⋊C₄	—	4	4	no	—
341	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
342	C₂×C₄⋊C₄	D₄×C₁₀₉	1	1	yes	characteristic
343	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
344	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
345	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
346	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
347	C₂×C₂²⋊C₄	—	2	1	no	—
348	C₂×C₂²⋊C₄	—	2	2	no	—
349	C₂×C₂²⋊C₄	—	2	1	no	—
350	C₂×C₂²⋊C₄	—	2	2	no	—
351	C₂×C₂²⋊C₄	—	2	2	no	—
352	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
353	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
354	#107	—	2	2	no	—
355	#107	—	2	4	no	—
356	C₄⋊C₈	—	2	2	no	—
357	C₄⋊C₈	—	4	2	no	—
358	#102	—	2	2	no	—
359	#102	—	2	4	no	—
360	#102	—	2	2	no	—
361	#102	—	4	2	no	—
362	C₄×C₈	—	2	2	no	—
363	#95	—	2	2	no	—
364	#95	D₄×C₁₀₉	1	2	yes	—
365	C₂⁴	—	2	1	no	—
366	C₂⁴	—	2	1	no	—
367	C₂×Q₈	—	4	2	no	—
368	C₂×Q₈	—	4	2	no	—
369	C₂×Q₈	—	4	2	no	—
370	C₂×D₄	—	2	1	no	—
371	C₂×D₄	—	2	1	no	—
372	C₂×D₄	—	2	1	no	—
373	C₂×D₄	—	2	1	no	—
374	C₂×D₄	—	2	1	no	—
375	C₂×D₄	—	2	1	no	—
376	C₂×D₄	—	2	1	no	—
377	C₂×D₄	—	2	1	no	—
378	C₂×D₄	C₂×D₄×C₁₀₉	1	2	yes	—
379	C₂×D₄	C₂×Q₈×C₁₀₉	1	2	yes	—
380	C₂×D₄	C₂×Q₈×C₁₀₉	1	2	yes	—
381	C₂×D₄	#41×C₁₀₉	1	2	yes	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	2	4	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	1	no	—
387	C₂²×C₄	—	4	4	no	—
388	C₂²×C₄	—	4	2	no	—
389	C₂²×C₄	—	4	2	no	—
390	C₂²×C₄	—	4	2	no	—
391	C₂²×C₄	—	4	4	no	—
392	C₂²×C₄	—	4	2	no	—
393	C₂²×C₄	—	4	2	no	—
394	C₂²×C₄	—	8	2	no	—
395	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
396	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
397	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
398	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
399	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
400	D₈	—	2	8	no	—
401	C₂×C₈	—	2	1	no	—
402	C₂×C₈	—	2	1	no	—
403	C₂×C₈	—	2	1	no	—
404	C₂×C₈	—	2	2	no	—
405	C₂×C₈	—	2	2	no	—
406	C₂×C₈	—	2	1	no	—
407	C₂×C₈	—	2	2	no	—
408	C₂×C₈	—	2	1	no	—
409	C₂×C₈	C₂×Q₈×C₁₀₉	1	2	yes	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	2	1	no	—
412	C₄⋊C₄	—	2	1	no	—
413	C₄⋊C₄	—	2	2	no	—
414	C₄⋊C₄	—	2	1	no	—
415	C₄⋊C₄	—	2	2	no	—
416	C₄⋊C₄	—	2	2	no	—
417	C₄⋊C₄	—	2	2	no	—
418	C₄⋊C₄	—	4	4	no	—
419	C₄⋊C₄	—	4	2	no	—
420	C₄⋊C₄	—	4	4	no	—
421	C₄⋊C₄	—	4	2	no	—
422	C₄⋊C₄	—	4	4	no	—
423	C₄⋊C₄	—	8	1	no	—
424	C₄⋊C₄	D₈×C₁₀₉	1	4	yes	complemented
425	C₄⋊C₄	C₂×D₄×C₁₀₉	1	2	yes	—
426	C₂²⋊C₄	—	2	4	no	—
427	C₂²⋊C₄	—	4	4	no	—
428	C₂²⋊C₄	—	4	4	no	—
429	C₂²⋊C₄	—	4	2	no	—
430	C₂²⋊C₄	—	4	4	no	—
431	C₄²	—	2	2	no	—
432	C₄²	—	2	2	no	—
433	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
434	C₄²	#41×C₁₀₉	1	2	yes	—
435	C₂³	—	2	2	no	—
436	C₂³	—	2	2	no	—
437	C₂³	—	2	2	no	—
438	C₂³	—	2	2	no	—
439	C₂³	—	4	1	no	—
440	C₂³	—	4	1	no	—
441	C₂³	—	4	1	no	—
442	C₂³	—	4	1	no	—
443	C₂³	—	4	1	no	—
444	C₂³	—	4	1	no	—
445	C₂³	—	4	1	no	—
446	C₂³	—	4	1	no	—
447	C₂³	—	4	1	no	—
448	C₂³	—	4	1	no	—
449	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
450	Q₈	—	8	4	no	—
451	D₄	—	2	2	no	—
452	D₄	—	2	2	no	—
453	D₄	—	2	2	no	—
454	D₄	—	2	2	no	—
455	D₄	—	2	2	no	—
456	D₄	—	4	1	no	—
457	D₄	—	4	1	no	—
458	D₄	C₂²⋊Q₈×C₁₀₉	1	4	yes	complemented
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	4	1	no	—
468	C₂×C₄	—	4	1	no	—
469	C₂×C₄	—	4	8	no	—
470	C₂×C₄	—	4	1	no	—
471	C₂×C₄	—	4	2	no	—
472	C₂×C₄	—	4	2	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	4	1	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	4	2	no	—
477	C₂×C₄	—	4	1	no	—
478	C₂×C₄	—	4	2	no	—
479	C₂×C₄	—	8	8	no	—
480	C₂×C₄	—	8	4	no	—
481	C₂×C₄	#121×C₁₀₉	1	2	yes	—
482	C₂×C₄	#121×C₁₀₉	1	2	yes	—
483	C₂×C₄	C₂×D₈×C₁₀₉	1	4	yes	—
484	C₂×C₄	C₂×D₈×C₁₀₉	1	2	yes	—
485	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
486	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
487	C₂×C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
488	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
489	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
490	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
491	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
492	C₈	—	2	2	no	—
493	C₈	—	4	1	no	—
494	C₈	—	4	2	no	—
495	C₂²	—	2	4	no	—
496	C₂²	—	4	2	no	—
497	C₂²	—	4	2	no	—
498	C₂²	—	4	2	no	—
499	C₂²	—	4	2	no	—
500	C₂²	—	4	2	no	—
501	C₂²	—	4	2	no	—
502	C₂²	—	4	2	no	—
503	C₂²	—	4	2	no	—
504	C₂²	—	4	2	no	—
505	C₂²	—	8	1	no	—
506	C₂²	—	8	1	no	—
507	C₂²	—	8	1	no	—
508	C₂²	—	8	1	no	—
509	C₂²	C₂×#121×C₁₀₉	1	1	yes	characteristic
510	C₂²	#546×C₁₀₉	1	1	yes	characteristic
511	C₂²	D₄×Q₈×C₁₀₉	1	1	yes	characteristic
512	C₂²	C₂²×D₈×C₁₀₉	1	1	yes	characteristic
513	C₂²	C₂×AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
514	C₂²	#577×C₁₀₉	1	1	yes	characteristic
515	C₂²	#578×C₁₀₉	1	1	yes	characteristic
516	C₄	—	2	4	no	—
517	C₄	—	2	2	no	—
518	C₄	—	4	2	no	—
519	C₄	—	4	2	no	—
520	C₄	—	8	1	no	—
521	C₄	—	8	2	no	—
522	C₄	—	8	2	no	—
523	C₄	#459×C₁₀₉	1	4	yes	—
524	C₄	C₂×C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
525	C₄	#537×C₁₀₉	1	1	yes	characteristic
526	C₄	#552×C₁₀₉	1	1	yes	characteristic
527	C₄	#554×C₁₀₉	1	1	yes	characteristic
528	C₂	—	4	4	no	—
529	C₂	—	8	2	no	—
530	C₂	C₁₀₉×#2417	1	1	yes	characteristic
531	C₂	C₂×#459×C₁₀₉	1	1	yes	characteristic
532	C₂	C₁₀₉×#3046	1	1	yes	characteristic
533	C₂	C₁₀₉×#3294	1	1	yes	characteristic
534	C₂	C₁₀₉×#3330	1	1	yes	characteristic
535	C₂	Q₈×D₈×C₁₀₉	1	1	yes	characteristic
536	C₂	C₁₀₉×#3384	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₀₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
prime radical	C₂²×D₈×C₁₀₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₄×C₄⋊C₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#474×C₁₀₉	C₂	semidirect
3. C₁₀₉×#2542	C₂	semidirect
4. D₄×C₄⋊C₄×C₁₀₉	C₂	semidirect
5. C₁₀₉×#2083	C₂	semidirect
6. C₁₀₉×#2067	C₂	semidirect
7. C₁₀₉×#1990	C₂	semidirect
8. C₁₀₉×#1939	C₂	semidirect
10. C₁₀₉×#1802	C₂	semidirect
11. C₁₀₉×#1802	C₂	semidirect
51. C₂×D₈×C₁₀₉	Q₈	semidirect
155. C₄⋊C₄×C₁₀₉	D₈	semidirect
189. D₄×C₁₀₉	C₂²⋊Q₈	semidirect
259. #24811	C₁₀₉	direct (N × H)
269. C₂×#474	C₂×C₁₀₉	semidirect
271. #2542	C₂×C₁₀₉	semidirect
272. D₄×C₄⋊C₄	C₂×C₁₀₉	semidirect
273. #2083	C₂×C₁₀₉	semidirect
274. #2067	C₂×C₁₀₉	semidirect
275. #1990	C₂×C₁₀₉	semidirect
276. #1939	C₂×C₁₀₉	semidirect
278. #1802	C₂×C₁₀₉	semidirect
279. #1802	C₂×C₁₀₉	semidirect
280. C₁₀₉	#24811	direct (N × H)
320. C₂×D₈	Q₈×C₁₀₉	semidirect
424. C₄⋊C₄	D₈×C₁₀₉	semidirect
458. D₄	C₂²⋊Q₈×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#474×C₁₀₉	2
C₂×C₄×D₈×C₁₀₉	2
C₁₀₉×#2542	2
D₄×C₄⋊C₄×C₁₀₉	2
C₁₀₉×#2083	2
C₁₀₉×#2067	2
C₁₀₉×#1990	2
C₁₀₉×#1939	2
C₁₀₉×#1879	2
C₁₀₉×#1802	2
#24811	109

Subgroups — C109×#24811 #42142805

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#24811 #42142805