Subgroups — C₁₀₉×#24829 #42142823

Order 27904.

956 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#476×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#442×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#2542	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₉×#2349	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₀₉×#2084	C₂	1	2	yes	maximal, complemented
6	C₁₀₉×#2065	C₂	1	2	yes	maximal, complemented
7	C₁₀₉×#1993	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₀₉×#1940	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₀₉×#1879	C₂	1	2	yes	maximal
10	C₁₀₉×#1804	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	2	no	—
15	C₂×C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
17	C₂×C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
18	#476×C₁₀₉	—	2	4	no	—
19	#442×C₁₀₉	—	2	4	no	—
20	C₂×#106×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂×#106×C₁₀₉	C₂²	1	2	yes	—
22	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
24	C₂×#102×C₁₀₉	C₂²	1	1	yes	characteristic
25	C₂×#102×C₁₀₉	C₂²	1	1	yes	characteristic
26	C₂×#102×C₁₀₉	C₂²	1	2	yes	—
27	C₂×#102×C₁₀₉	C₂²	1	2	yes	—
28	C₂×#97×C₁₀₉	C₂²	1	1	yes	characteristic
29	#399×C₁₀₉	—	2	1	no	—
30	#399×C₁₀₉	—	2	1	no	—
31	#390×C₁₀₉	C₂²	1	2	yes	—
32	#389×C₁₀₉	C₂²	1	2	yes	—
33	#385×C₁₀₉	—	2	2	no	—
34	#384×C₁₀₉	C₂²	1	2	yes	—
35	#382×C₁₀₉	C₂²	1	1	yes	characteristic
36	#380×C₁₀₉	—	2	1	no	—
37	C₄×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
38	C₂×#95×C₁₀₉	—	2	1	no	—
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	8	no	—
58	C₄×D₄×C₁₀₉	—	2	4	no	—
59	C₄×D₄×C₁₀₉	—	2	2	no	—
60	C₄×D₄×C₁₀₉	—	2	2	no	—
61	C₄×D₄×C₁₀₉	—	2	2	no	—
62	C₄×D₄×C₁₀₉	—	4	2	no	—
63	C₄×D₄×C₁₀₉	D₄	1	4	yes	—
64	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
65	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
66	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
67	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
68	C₂×C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
69	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
70	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
73	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
74	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
75	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
76	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
77	C₂×C₂²⋊C₄×C₁₀₉	—	2	2	no	—
78	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
79	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
80	#106×C₁₀₉	—	2	2	no	—
81	#106×C₁₀₉	—	2	4	no	—
82	C₄⋊C₈×C₁₀₉	—	2	2	no	—
83	C₄⋊C₈×C₁₀₉	—	4	2	no	—
84	#102×C₁₀₉	—	2	2	no	—
85	#102×C₁₀₉	—	2	4	no	—
86	#102×C₁₀₉	—	2	2	no	—
87	#102×C₁₀₉	—	4	2	no	—
88	#97×C₁₀₉	—	2	2	no	—
89	#95×C₁₀₉	—	2	1	no	—
90	#95×C₁₀₉	—	2	2	no	—
91	#95×C₁₀₉	—	2	1	no	—
92	#95×C₁₀₉	D₄	1	1	yes	characteristic
93	#95×C₁₀₉	D₄	1	1	yes	characteristic
94	C₂⁴×C₁₀₉	—	2	1	no	—
95	C₂⁴×C₁₀₉	—	2	1	no	—
96	C₂×Q₈×C₁₀₉	—	4	4	no	—
97	C₂×Q₈×C₁₀₉	—	4	2	no	—
98	C₂×D₄×C₁₀₉	—	2	1	no	—
99	C₂×D₄×C₁₀₉	—	2	1	no	—
100	C₂×D₄×C₁₀₉	—	2	1	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	C₂×D₄	1	2	yes	—
107	C₂×D₄×C₁₀₉	C₂×Q₈	1	2	yes	—
108	C₂×D₄×C₁₀₉	C₂×Q₈	1	2	yes	—
109	C₂×D₄×C₁₀₉	#41	1	2	yes	—
110	C₂²×C₄×C₁₀₉	—	2	1	no	—
111	C₂²×C₄×C₁₀₉	—	2	4	no	—
112	C₂²×C₄×C₁₀₉	—	2	2	no	—
113	C₂²×C₄×C₁₀₉	—	2	2	no	—
114	C₂²×C₄×C₁₀₉	—	2	1	no	—
115	C₂²×C₄×C₁₀₉	—	4	4	no	—
116	C₂²×C₄×C₁₀₉	—	4	4	no	—
117	C₂²×C₄×C₁₀₉	—	4	2	no	—
118	C₂²×C₄×C₁₀₉	—	4	4	no	—
119	C₂²×C₄×C₁₀₉	—	8	2	no	—
120	C₂²×C₄×C₁₀₉	—	8	2	no	—
121	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
125	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
126	D₈×C₁₀₉	—	2	8	no	—
127	C₂×C₈×C₁₀₉	—	2	1	no	—
128	C₂×C₈×C₁₀₉	—	2	1	no	—
129	C₂×C₈×C₁₀₉	—	2	1	no	—
130	C₂×C₈×C₁₀₉	—	2	2	no	—
131	C₂×C₈×C₁₀₉	—	2	2	no	—
132	C₂×C₈×C₁₀₉	—	2	1	no	—
133	C₂×C₈×C₁₀₉	—	2	2	no	—
134	C₂×C₈×C₁₀₉	—	2	1	no	—
135	C₂×C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
136	C₄⋊C₄×C₁₀₉	—	2	4	no	—
137	C₄⋊C₄×C₁₀₉	—	2	1	no	—
138	C₄⋊C₄×C₁₀₉	—	2	1	no	—
139	C₄⋊C₄×C₁₀₉	—	2	2	no	—
140	C₄⋊C₄×C₁₀₉	—	2	1	no	—
141	C₄⋊C₄×C₁₀₉	—	2	2	no	—
142	C₄⋊C₄×C₁₀₉	—	2	2	no	—
143	C₄⋊C₄×C₁₀₉	—	2	2	no	—
144	C₄⋊C₄×C₁₀₉	—	4	4	no	—
145	C₄⋊C₄×C₁₀₉	—	4	2	no	—
146	C₄⋊C₄×C₁₀₉	—	4	4	no	—
147	C₄⋊C₄×C₁₀₉	—	8	2	no	—
148	C₄⋊C₄×C₁₀₉	C₂×D₄	1	2	yes	—
149	C₂²⋊C₄×C₁₀₉	—	2	4	no	—
150	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
151	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
152	C₂²⋊C₄×C₁₀₉	—	4	2	no	—
153	C₂²⋊C₄×C₁₀₉	—	4	4	no	—
154	C₄²×C₁₀₉	—	2	2	no	—
155	C₄²×C₁₀₉	—	2	2	no	—
156	C₄²×C₁₀₉	C₂×D₄	1	2	yes	—
157	C₄²×C₁₀₉	#41	1	2	yes	—
158	C₂³×C₁₀₉	—	2	2	no	—
159	C₂³×C₁₀₉	—	2	2	no	—
160	C₂³×C₁₀₉	—	2	2	no	—
161	C₂³×C₁₀₉	—	2	2	no	—
162	C₂³×C₁₀₉	—	4	2	no	—
163	C₂³×C₁₀₉	—	4	2	no	—
164	C₂³×C₁₀₉	—	4	1	no	—
165	C₂³×C₁₀₉	—	4	1	no	—
166	C₂³×C₁₀₉	—	4	2	no	—
167	C₂³×C₁₀₉	—	4	2	no	—
168	C₂³×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
169	Q₈×C₁₀₉	—	8	4	no	—
170	D₄×C₁₀₉	—	2	2	no	—
171	D₄×C₁₀₉	—	2	2	no	—
172	D₄×C₁₀₉	—	2	2	no	—
173	D₄×C₁₀₉	—	2	2	no	—
174	D₄×C₁₀₉	—	2	2	no	—
175	D₄×C₁₀₉	—	4	1	no	—
176	D₄×C₁₀₉	—	4	1	no	—
177	D₄×C₁₀₉	C₂²⋊Q₈	1	4	yes	complemented
178	C₂×C₄×C₁₀₉	—	2	2	no	—
179	C₂×C₄×C₁₀₉	—	2	2	no	—
180	C₂×C₄×C₁₀₉	—	2	2	no	—
181	C₂×C₄×C₁₀₉	—	2	2	no	—
182	C₂×C₄×C₁₀₉	—	2	2	no	—
183	C₂×C₄×C₁₀₉	—	2	2	no	—
184	C₂×C₄×C₁₀₉	—	2	2	no	—
185	C₂×C₄×C₁₀₉	—	2	2	no	—
186	C₂×C₄×C₁₀₉	—	4	2	no	—
187	C₂×C₄×C₁₀₉	—	4	1	no	—
188	C₂×C₄×C₁₀₉	—	4	2	no	—
189	C₂×C₄×C₁₀₉	—	4	2	no	—
190	C₂×C₄×C₁₀₉	—	4	2	no	—
191	C₂×C₄×C₁₀₉	—	4	2	no	—
192	C₂×C₄×C₁₀₉	—	4	2	no	—
193	C₂×C₄×C₁₀₉	—	4	2	no	—
194	C₂×C₄×C₁₀₉	—	4	2	no	—
195	C₂×C₄×C₁₀₉	—	8	4	no	—
196	C₂×C₄×C₁₀₉	—	8	8	no	—
197	C₂×C₄×C₁₀₉	—	8	4	no	—
198	C₂×C₄×C₁₀₉	#121	1	2	yes	—
199	C₂×C₄×C₁₀₉	#121	1	2	yes	—
200	C₂×C₄×C₁₀₉	AGL(1,ℤ/8)	1	4	yes	—
201	C₂×C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
202	C₂×C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
203	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
204	C₂×C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
205	C₂×C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
206	C₂×C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
207	C₈×C₁₀₉	—	2	2	no	—
208	C₈×C₁₀₉	—	4	1	no	—
209	C₈×C₁₀₉	—	4	2	no	—
210	C₂²×C₁₀₉	—	2	4	no	—
211	C₂²×C₁₀₉	—	4	4	no	—
212	C₂²×C₁₀₉	—	4	2	no	—
213	C₂²×C₁₀₉	—	4	2	no	—
214	C₂²×C₁₀₉	—	4	4	no	—
215	C₂²×C₁₀₉	—	4	4	no	—
216	C₂²×C₁₀₉	—	4	2	no	—
217	C₂²×C₁₀₉	—	8	1	no	—
218	C₂²×C₁₀₉	—	8	1	no	—
219	C₂²×C₁₀₉	—	8	2	no	—
220	C₂²×C₁₀₉	C₂×#121	1	1	yes	characteristic
221	C₂²×C₁₀₉	#546	1	1	yes	characteristic
222	C₂²×C₁₀₉	D₄×Q₈	1	1	yes	characteristic
223	C₂²×C₁₀₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
224	C₂²×C₁₀₉	#577	1	2	yes	—
225	C₄×C₁₀₉	—	2	2	no	—
226	C₄×C₁₀₉	—	2	2	no	—
227	C₄×C₁₀₉	—	2	4	no	—
228	C₄×C₁₀₉	—	4	2	no	—
229	C₄×C₁₀₉	—	4	2	no	—
230	C₄×C₁₀₉	—	8	1	no	—
231	C₄×C₁₀₉	—	8	2	no	—
232	C₄×C₁₀₉	—	8	2	no	—
233	C₄×C₁₀₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
234	C₄×C₁₀₉	#537	1	1	yes	characteristic
235	C₄×C₁₀₉	#552	1	1	yes	characteristic
236	C₄×C₁₀₉	#554	1	1	yes	characteristic
237	#24829	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
238	C₂×C₁₀₉	—	4	4	no	—
239	C₂×C₁₀₉	—	8	2	no	—
240	C₂×C₁₀₉	#2417	1	1	yes	characteristic
241	C₂×C₁₀₉	#3038	1	1	yes	characteristic
242	C₂×C₁₀₉	#3042	1	1	yes	characteristic
243	C₂×C₁₀₉	#3306	1	2	yes	—
244	C₂×C₁₀₉	#3384	1	2	yes	—
245	C₂×#476	C₂×C₁₀₉	1	1	yes	characteristic, complemented
246	C₂×#442	C₂×C₁₀₉	1	1	yes	characteristic
247	#2542	C₂×C₁₀₉	1	1	yes	characteristic, complemented
248	#2349	C₂×C₁₀₉	1	1	yes	characteristic, complemented
249	#2084	C₂×C₁₀₉	1	2	yes	complemented
250	#2065	C₂×C₁₀₉	1	2	yes	complemented
251	#1993	C₂×C₁₀₉	1	1	yes	characteristic, complemented
252	#1940	C₂×C₁₀₉	1	1	yes	characteristic, complemented
253	#1879	C₂×C₁₀₉	1	2	yes	—
254	#1804	C₂×C₁₀₉	1	1	yes	characteristic, complemented
255	#1802	C₂×C₁₀₉	1	2	yes	complemented
256	C₁₀₉	#24829	1	1	yes	characteristic, direct product factor, Sylow
257	C₂²×D₈	C₂²×C₁₀₉	1	1	yes	characteristic
258	C₂×C₄⋊Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
259	C₂×C₂²⋊Q₈	—	2	2	no	—
260	C₂×C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
261	C₂×C₄×D₄	C₂²×C₁₀₉	1	2	yes	—
262	C₂×C₄×D₄	C₂²×C₁₀₉	1	1	yes	characteristic
263	#476	—	2	4	no	—
264	#442	—	2	4	no	—
265	C₂×#106	C₂²×C₁₀₉	1	2	yes	—
266	C₂×#106	C₂²×C₁₀₉	1	1	yes	characteristic
267	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	1	yes	characteristic
268	C₂×C₄⋊C₈	C₂²×C₁₀₉	1	2	yes	—
269	C₂×#102	C₂²×C₁₀₉	1	1	yes	characteristic
270	C₂×#102	C₂²×C₁₀₉	1	2	yes	—
271	C₂×#102	C₂²×C₁₀₉	1	1	yes	characteristic
272	C₂×#102	C₂²×C₁₀₉	1	2	yes	—
273	C₂×#97	C₂²×C₁₀₉	1	1	yes	characteristic
274	#399	—	2	1	no	—
275	#399	—	2	1	no	—
276	#390	C₂²×C₁₀₉	1	2	yes	—
277	#389	C₂²×C₁₀₉	1	2	yes	—
278	#385	—	2	2	no	—
279	#384	C₂²×C₁₀₉	1	2	yes	—
280	#382	C₂²×C₁₀₉	1	1	yes	characteristic
281	#380	—	2	1	no	—
282	C₄×C₄⋊C₄	C₂²×C₁₀₉	1	1	yes	characteristic
283	C₂×#95	—	2	1	no	—
284	#340	C₂²×C₁₀₉	1	2	yes	—
285	#340	C₂²×C₁₀₉	1	2	yes	—
286	#340	C₂²×C₁₀₉	1	2	yes	—
287	#336	C₂²×C₁₀₉	1	2	yes	—
288	C₂²×Q₈	—	2	1	no	—
289	C₂²×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
290	C₂²×D₄	C₂³×C₁₀₉	1	1	yes	characteristic
291	C₂³×C₄	—	2	1	no	—
292	C₂³×C₄	—	2	1	no	—
293	C₂³×C₄	—	2	2	no	—
294	C₂×D₈	—	2	2	no	—
295	C₂×D₈	—	2	2	no	—
296	C₂×D₈	Q₈×C₁₀₉	1	4	yes	complemented
297	C₂²×C₈	C₂³×C₁₀₉	1	2	yes	—
298	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
299	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
300	C₄⋊Q₈	—	2	2	no	—
301	C₄⋊Q₈	—	2	2	no	—
302	C₂²⋊Q₈	—	4	8	no	—
303	C₄×D₄	—	2	4	no	—
304	C₄×D₄	—	2	2	no	—
305	C₄×D₄	—	2	2	no	—
306	C₄×D₄	—	2	2	no	—
307	C₄×D₄	—	4	2	no	—
308	C₄×D₄	D₄×C₁₀₉	1	4	yes	—
309	C₂×C₄⋊C₄	—	2	2	no	—
310	C₂×C₄⋊C₄	—	2	2	no	—
311	C₂×C₄⋊C₄	—	2	1	no	—
312	C₂×C₄⋊C₄	—	2	2	no	—
313	C₂×C₄⋊C₄	D₄×C₁₀₉	1	2	yes	—
314	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
315	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
316	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	2	yes	—
317	C₂×C₄⋊C₄	C₂³×C₁₀₉	1	1	yes	characteristic
318	C₂×C₂²⋊C₄	—	2	1	no	—
319	C₂×C₂²⋊C₄	—	2	2	no	—
320	C₂×C₂²⋊C₄	—	2	1	no	—
321	C₂×C₂²⋊C₄	—	2	2	no	—
322	C₂×C₂²⋊C₄	—	2	2	no	—
323	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
324	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
325	#106	—	2	2	no	—
326	#106	—	2	4	no	—
327	C₄⋊C₈	—	2	2	no	—
328	C₄⋊C₈	—	4	2	no	—
329	#102	—	2	2	no	—
330	#102	—	2	4	no	—
331	#102	—	2	2	no	—
332	#102	—	4	2	no	—
333	#97	—	2	2	no	—
334	#95	—	2	1	no	—
335	#95	—	2	2	no	—
336	#95	—	2	1	no	—
337	#95	D₄×C₁₀₉	1	1	yes	characteristic
338	#95	D₄×C₁₀₉	1	1	yes	characteristic
339	C₂⁴	—	2	1	no	—
340	C₂⁴	—	2	1	no	—
341	C₂×Q₈	—	4	4	no	—
342	C₂×Q₈	—	4	2	no	—
343	C₂×D₄	—	2	1	no	—
344	C₂×D₄	—	2	1	no	—
345	C₂×D₄	—	2	1	no	—
346	C₂×D₄	—	2	1	no	—
347	C₂×D₄	—	2	1	no	—
348	C₂×D₄	—	2	1	no	—
349	C₂×D₄	—	2	1	no	—
350	C₂×D₄	—	2	1	no	—
351	C₂×D₄	C₂×D₄×C₁₀₉	1	2	yes	—
352	C₂×D₄	C₂×Q₈×C₁₀₉	1	2	yes	—
353	C₂×D₄	C₂×Q₈×C₁₀₉	1	2	yes	—
354	C₂×D₄	#41×C₁₀₉	1	2	yes	—
355	C₂²×C₄	—	2	1	no	—
356	C₂²×C₄	—	2	4	no	—
357	C₂²×C₄	—	2	2	no	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	2	1	no	—
360	C₂²×C₄	—	4	4	no	—
361	C₂²×C₄	—	4	4	no	—
362	C₂²×C₄	—	4	2	no	—
363	C₂²×C₄	—	4	4	no	—
364	C₂²×C₄	—	8	2	no	—
365	C₂²×C₄	—	8	2	no	—
366	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
367	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
368	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
369	C₂²×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
370	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
371	D₈	—	2	8	no	—
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	—	2	1	no	—
375	C₂×C₈	—	2	2	no	—
376	C₂×C₈	—	2	2	no	—
377	C₂×C₈	—	2	1	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	2	1	no	—
380	C₂×C₈	C₂×Q₈×C₁₀₉	1	2	yes	—
381	C₄⋊C₄	—	2	4	no	—
382	C₄⋊C₄	—	2	1	no	—
383	C₄⋊C₄	—	2	1	no	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	—	2	1	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	2	2	no	—
388	C₄⋊C₄	—	2	2	no	—
389	C₄⋊C₄	—	4	4	no	—
390	C₄⋊C₄	—	4	2	no	—
391	C₄⋊C₄	—	4	4	no	—
392	C₄⋊C₄	—	8	2	no	—
393	C₄⋊C₄	C₂×D₄×C₁₀₉	1	2	yes	—
394	C₂²⋊C₄	—	2	4	no	—
395	C₂²⋊C₄	—	4	4	no	—
396	C₂²⋊C₄	—	4	4	no	—
397	C₂²⋊C₄	—	4	2	no	—
398	C₂²⋊C₄	—	4	4	no	—
399	C₄²	—	2	2	no	—
400	C₄²	—	2	2	no	—
401	C₄²	C₂×D₄×C₁₀₉	1	2	yes	—
402	C₄²	#41×C₁₀₉	1	2	yes	—
403	C₂³	—	2	2	no	—
404	C₂³	—	2	2	no	—
405	C₂³	—	2	2	no	—
406	C₂³	—	2	2	no	—
407	C₂³	—	4	2	no	—
408	C₂³	—	4	2	no	—
409	C₂³	—	4	1	no	—
410	C₂³	—	4	1	no	—
411	C₂³	—	4	2	no	—
412	C₂³	—	4	2	no	—
413	C₂³	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
414	Q₈	—	8	4	no	—
415	D₄	—	2	2	no	—
416	D₄	—	2	2	no	—
417	D₄	—	2	2	no	—
418	D₄	—	2	2	no	—
419	D₄	—	2	2	no	—
420	D₄	—	4	1	no	—
421	D₄	—	4	1	no	—
422	D₄	C₂²⋊Q₈×C₁₀₉	1	4	yes	complemented
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	4	2	no	—
437	C₂×C₄	—	4	2	no	—
438	C₂×C₄	—	4	2	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	8	4	no	—
441	C₂×C₄	—	8	8	no	—
442	C₂×C₄	—	8	4	no	—
443	C₂×C₄	#121×C₁₀₉	1	2	yes	—
444	C₂×C₄	#121×C₁₀₉	1	2	yes	—
445	C₂×C₄	AGL(1,ℤ/8)×C₁₀₉	1	4	yes	—
446	C₂×C₄	C₂²×D₄×C₁₀₉	1	1	yes	characteristic
447	C₂×C₄	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
448	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
449	C₂×C₄	C₂×#41×C₁₀₉	1	1	yes	characteristic
450	C₂×C₄	H(5,𝔽₂)×C₁₀₉	1	1	yes	characteristic
451	C₂×C₄	2₋¹⁺⁴×C₁₀₉	1	1	yes	characteristic
452	C₈	—	2	2	no	—
453	C₈	—	4	1	no	—
454	C₈	—	4	2	no	—
455	C₂²	—	2	4	no	—
456	C₂²	—	4	2	no	—
457	C₂²	—	4	2	no	—
458	C₂²	—	4	4	no	—
459	C₂²	—	4	4	no	—
460	C₂²	—	4	4	no	—
461	C₂²	—	4	2	no	—
462	C₂²	—	8	1	no	—
463	C₂²	—	8	1	no	—
464	C₂²	—	8	2	no	—
465	C₂²	C₂×#121×C₁₀₉	1	1	yes	characteristic
466	C₂²	#546×C₁₀₉	1	1	yes	characteristic
467	C₂²	D₄×Q₈×C₁₀₉	1	1	yes	characteristic
468	C₂²	C₂×AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
469	C₂²	#577×C₁₀₉	1	2	yes	—
470	C₄	—	2	2	no	—
471	C₄	—	2	2	no	—
472	C₄	—	2	4	no	—
473	C₄	—	4	2	no	—
474	C₄	—	4	2	no	—
475	C₄	—	8	1	no	—
476	C₄	—	8	2	no	—
477	C₄	—	8	2	no	—
478	C₄	C₂×C₂²⋊Q₈×C₁₀₉	1	1	yes	characteristic
479	C₄	#537×C₁₀₉	1	1	yes	characteristic
480	C₄	#552×C₁₀₉	1	1	yes	characteristic
481	C₄	#554×C₁₀₉	1	1	yes	characteristic
482	C₂	—	4	4	no	—
483	C₂	—	8	2	no	—
484	C₂	C₁₀₉×#2417	1	1	yes	characteristic
485	C₂	C₁₀₉×#3038	1	1	yes	characteristic
486	C₂	C₁₀₉×#3042	1	1	yes	characteristic
487	C₂	C₁₀₉×#3306	1	2	yes	—
488	C₂	C₁₀₉×#3384	1	2	yes	—

	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×#476×C₁₀₉	C₂	semidirect
3. C₁₀₉×#2542	C₂	semidirect
4. C₁₀₉×#2349	C₂	semidirect
5. C₁₀₉×#2084	C₂	semidirect
6. C₁₀₉×#2065	C₂	semidirect
7. C₁₀₉×#1993	C₂	semidirect
8. C₁₀₉×#1940	C₂	semidirect
10. C₁₀₉×#1804	C₂	semidirect
11. C₁₀₉×#1802	C₂	semidirect
51. C₂×D₈×C₁₀₉	Q₈	semidirect
177. D₄×C₁₀₉	C₂²⋊Q₈	semidirect
237. #24829	C₁₀₉	direct (N × H)
245. C₂×#476	C₂×C₁₀₉	semidirect
247. #2542	C₂×C₁₀₉	semidirect
248. #2349	C₂×C₁₀₉	semidirect
249. #2084	C₂×C₁₀₉	semidirect
250. #2065	C₂×C₁₀₉	semidirect
251. #1993	C₂×C₁₀₉	semidirect
252. #1940	C₂×C₁₀₉	semidirect
254. #1804	C₂×C₁₀₉	semidirect
255. #1802	C₂×C₁₀₉	semidirect
256. C₁₀₉	#24829	direct (N × H)
296. C₂×D₈	Q₈×C₁₀₉	semidirect
422. 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×#476×C₁₀₉	2
C₂×#442×C₁₀₉	2
C₁₀₉×#2542	2
C₁₀₉×#2349	2
C₁₀₉×#2084	2
C₁₀₉×#2065	2
C₁₀₉×#1993	2
C₁₀₉×#1940	2
C₁₀₉×#1879	2
C₁₀₉×#1804	2
C₁₀₉×#1802	2
#24829	109

Subgroups — C109×#24829 #42142823

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#24829 #42142823