Subgroups — #37546565

Order 23168.

332 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26042007	C₂	1	1	yes	maximal, characteristic, complemented
2	#26042002	C₂	1	1	yes	maximal, characteristic
3	#26041648	C₂	1	1	yes	maximal, characteristic, complemented
4	#26041639	C₂	1	1	yes	maximal, characteristic
5	#26041634	C₂	1	1	yes	maximal, characteristic
6	#26041587	C₂	1	1	yes	maximal, characteristic, complemented
7	#26041580	C₂	1	1	yes	maximal, characteristic
8	#26041575	C₂	1	1	yes	maximal, characteristic, complemented
9	#26041502	C₂	1	1	yes	maximal, characteristic
10	#26041502	C₂	1	1	yes	maximal, characteristic
11	#26041490	C₂	1	1	yes	maximal, characteristic, complemented
12	#26041387	C₂	1	1	yes	maximal, characteristic, complemented
13	#26041381	C₂	1	1	yes	maximal, characteristic
14	C₄×Dic₇₂₄	C₂	1	1	yes	maximal, characteristic, complemented
15	#446×C₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Dic₇₂₄	C₂²	1	1	yes	characteristic
17	C₂×Dic₇₂₄	C₂²	1	1	yes	characteristic
18	C₂×#15668438	C₂²	1	1	yes	characteristic
19	#19153816	C₂²	1	1	yes	characteristic
20	#19153816	C₂²	1	1	yes	characteristic
21	#19153814	—	2	1	no	—
22	#19153811	—	2	1	no	—
23	#19153811	—	2	1	no	—
24	#19153808	C₂²	1	1	yes	characteristic
25	#19153808	C₂²	1	1	yes	characteristic
26	#19153806	—	2	1	no	—
27	#19153806	—	2	1	no	—
28	#19153804	C₂²	1	1	yes	characteristic
29	#19153804	C₂²	1	1	yes	characteristic
30	#19153792	—	2	1	no	—
31	C₄×Dic₃₆₂	C₂²	1	1	yes	characteristic
32	C₄×Dic₃₆₂	C₂²	1	1	yes	characteristic
33	C₄×Dic₃₆₂	C₂²	1	1	yes	characteristic
34	C₄×Dic₃₆₂	C₂²	1	1	yes	characteristic
35	C₄×Dic₃₆₂	C₂²	1	1	yes	characteristic
36	#19153781	—	2	1	no	—
37	C₄×D₇₂₄	C₂²	1	1	yes	characteristic
38	#19153774	—	2	1	no	—
39	#19153750	C₂²	1	1	yes	characteristic
40	#19153750	C₂²	1	1	yes	characteristic
41	#19153748	C₂²	1	1	yes	characteristic
42	#19153741	C₂²	1	1	yes	characteristic
43	#19153741	C₂²	1	1	yes	characteristic
44	#19153741	C₂²	1	1	yes	characteristic
45	#19153741	C₂²	1	1	yes	characteristic
46	#19153741	C₂²	1	1	yes	characteristic
47	#19153741	C₂²	1	1	yes	characteristic
48	#19153741	C₂²	1	1	yes	characteristic
49	#19153738	C₂²	1	1	yes	characteristic
50	#19153738	C₂²	1	1	yes	characteristic
51	#19153738	C₂²	1	1	yes	characteristic
52	C₄×Q₈×C₁₈₁	C₂²	1	1	yes	characteristic
53	C₄⋊C₈×C₁₈₁	C₂²	1	1	yes	characteristic
54	C₄⋊C₈×C₁₈₁	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₁₈₁	C₂²	1	1	yes	characteristic
56	#97×C₁₈₁	C₂²	1	1	yes	characteristic
57	#97×C₁₈₁	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₈₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₃₆₂	—	2	1	no	—
60	C₂×Dic₃₆₂	C₂³	1	1	yes	characteristic
61	C₂×Dic₃₆₂	C₂³	1	1	yes	characteristic
62	C₂×Dic₃₆₂	C₂³	1	1	yes	characteristic
63	C₂×D₇₂₄	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₁₈₁	—	2	1	no	—
65	Dic₇₂₄	—	2	2	no	—
66	Dic₇₂₄	—	2	2	no	—
67	#15668438	—	2	2	no	—
68	C₄⋊Dic₁₈₁	—	2	1	no	—
69	C₄⋊Dic₁₈₁	—	2	1	no	—
70	C₄⋊Dic₁₈₁	—	2	1	no	—
71	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
75	#15668428	—	2	1	no	—
76	#15668428	—	2	1	no	—
77	#15668428	—	2	1	no	—
78	#15668428	—	2	1	no	—
79	#15668428	—	2	1	no	—
80	#15668428	—	2	1	no	—
81	#15668428	—	2	1	no	—
82	#15668428	—	2	1	no	—
83	#15668428	—	2	1	no	—
84	#15668426	—	2	1	no	—
85	#15668426	—	2	1	no	—
86	#15668426	—	2	1	no	—
87	#15668426	—	2	1	no	—
88	#15668426	—	2	1	no	—
89	C₄×Dic₁₈₁	—	2	1	no	—
90	C₄×Dic₁₈₁	—	2	1	no	—
91	C₄×Dic₁₈₁	—	2	1	no	—
92	C₄×Dic₁₈₁	—	2	1	no	—
93	C₄×Dic₁₈₁	—	2	1	no	—
94	C₂×Q₈×C₁₈₁	D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
96	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₁₈₁	D₄	1	1	yes	characteristic
100	C₄⋊C₄×C₁₈₁	D₄	1	1	yes	characteristic
101	C₄⋊C₄×C₁₈₁	D₄	1	1	yes	characteristic
102	C₄²×C₁₈₁	C₂³	1	1	yes	characteristic
103	C₄²×C₁₈₁	C₂³	1	1	yes	characteristic
104	C₄²×C₁₈₁	C₂³	1	1	yes	characteristic
105	C₂²×D₁₈₁	—	2	1	no	—
106	Dic₃₆₂	—	2	1	no	—
107	Dic₃₆₂	—	2	1	no	—
108	Dic₃₆₂	—	2	1	no	—
109	Dic₃₆₂	—	2	1	no	—
110	Dic₃₆₂	—	2	1	no	—
111	Dic₃₆₂	—	4	2	no	—
112	Dic₃₆₂	#41	1	2	yes	—
113	D₇₂₄	—	2	1	no	—
114	D₇₂₄	—	2	1	no	—
115	C₂×Dic₁₈₁	—	2	1	no	—
116	C₂×Dic₁₈₁	—	2	1	no	—
117	C₂×Dic₁₈₁	—	2	1	no	—
118	C₂×Dic₁₈₁	—	2	1	no	—
119	C₂×Dic₁₈₁	—	2	1	no	—
120	C₂×Dic₁₈₁	—	2	1	no	—
121	C₂×Dic₁₈₁	—	2	1	no	—
122	C₄×D₁₈₁	—	4	2	no	—
123	Q₈×C₁₈₁	—	4	1	no	—
124	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₈₁	C₂⁴	1	1	yes	characteristic
131	C₈×C₁₈₁	—	2	1	no	—
132	C₈×C₁₈₁	—	2	1	no	—
133	C₈×C₁₈₁	—	2	1	no	—
134	C₈×C₁₈₁	—	2	1	no	—
135	C₂²×C₁₈₁	C₂²×D₄	1	1	yes	characteristic
136	C₂×D₁₈₁	—	4	1	no	—
137	C₂×D₁₈₁	—	4	1	no	—
138	C₂×D₁₈₁	—	4	1	no	—
139	C₄×C₁₈₁	—	2	1	no	—
140	C₄×C₁₈₁	—	2	1	no	—
141	C₄×C₁₈₁	—	2	1	no	—
142	C₄×C₁₈₁	—	2	1	no	—
143	C₄×C₁₈₁	—	2	1	no	—
144	C₄×C₁₈₁	—	2	1	no	—
145	C₄×C₁₈₁	C₂×#41	1	1	yes	characteristic
146	C₄×C₁₈₁	2₋¹⁺⁴	1	1	yes	characteristic
147	Dic₁₈₁	—	2	2	no	—
148	Dic₁₈₁	—	4	1	no	—
149	Dic₁₈₁	—	4	1	no	—
150	Dic₁₈₁	—	4	1	no	—
151	Dic₁₈₁	—	4	1	no	—
152	Dic₁₈₁	—	4	1	no	—
153	Dic₁₈₁	—	4	1	no	—
154	C₂×C₁₈₁	#548	1	1	yes	characteristic
155	C₂×C₁₈₁	#575	1	1	yes	characteristic
156	C₂×C₁₈₁	#578	1	1	yes	characteristic
157	D₁₈₁	—	8	1	no	—
158	C₁₈₁	#3377	1	1	yes	characteristic, complemented, Sylow
159	#3377	—	181	1	no	maximal, Sylow
160	#557	—	181	1	no	—
161	#556	—	181	1	no	—
162	#491	—	181	1	no	—
163	#489	—	181	1	no	—
164	#488	—	181	1	no	—
165	#479	—	181	1	no	—
166	#478	—	181	1	no	—
167	#477	—	181	1	no	—
168	#464	—	181	1	no	—
169	#464	—	181	1	no	—
170	#462	—	181	1	no	—
171	#446	D₁₈₁	1	1	yes	characteristic, complemented
172	#441	—	181	1	no	—
173	#440	—	181	1	no	—
174	C₄×Q₁₆	—	181	1	no	—
175	C₂×Q₁₆	—	181	1	no	—
176	C₂×Q₁₆	—	181	1	no	—
177	C₂×SD₁₆	—	181	1	no	—
178	C₄⋊Q₈	—	362	1	no	—
179	C₄⋊Q₈	—	181	1	no	—
180	C₄⋊Q₈	—	181	1	no	—
181	#126	—	362	1	no	—
182	#126	—	362	1	no	—
183	#125	—	362	1	no	—
184	#125	—	362	1	no	—
185	#125	—	181	1	no	—
186	#125	—	181	1	no	—
187	#124	—	181	1	no	—
188	#124	—	181	1	no	—
189	C₂²⋊Q₈	—	362	1	no	—
190	C₄×Q₈	—	362	1	no	—
191	C₄×Q₈	—	181	1	no	—
192	C₄×Q₈	—	181	1	no	—
193	C₄×Q₈	—	181	1	no	—
194	C₄×Q₈	—	181	1	no	—
195	C₄×Q₈	—	181	1	no	—
196	C₄×Q₈	C₂×D₁₈₁	1	1	yes	characteristic
197	C₄×D₄	—	181	1	no	—
198	#117	—	362	1	no	—
199	#107	—	181	1	no	—
200	#107	—	181	1	no	—
201	#106	—	181	1	no	—
202	C₄⋊C₈	C₂×D₁₈₁	1	1	yes	characteristic
203	C₄⋊C₈	C₂×D₁₈₁	1	1	yes	characteristic
204	C₄⋊C₈	C₂×D₁₈₁	1	1	yes	characteristic
205	#103	—	181	1	no	—
206	#103	—	181	1	no	—
207	#103	—	181	1	no	—
208	#103	—	181	1	no	—
209	#103	—	181	1	no	—
210	#103	—	181	1	no	—
211	#103	—	181	1	no	—
212	#102	—	181	1	no	—
213	#102	—	181	1	no	—
214	#102	—	181	1	no	—
215	#97	C₂×D₁₈₁	1	1	yes	characteristic
216	#97	C₂×D₁₈₁	1	1	yes	characteristic
217	C₄×C₈	C₂×D₁₈₁	1	1	yes	characteristic
218	C₂×Q₈	—	362	1	no	—
219	C₂×Q₈	—	181	1	no	—
220	C₂×Q₈	—	181	1	no	—
221	C₂×Q₈	—	181	1	no	—
222	C₂×Q₈	D₇₂₄	1	1	yes	characteristic
223	C₂×D₄	—	181	1	no	—
224	C₂²×C₄	—	362	1	no	—
225	Q₁₆	—	362	2	no	—
226	Q₁₆	—	362	2	no	—
227	SD₁₆	—	362	2	no	—
228	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
229	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
230	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
231	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
232	C₄⋊C₄	—	362	1	no	—
233	C₄⋊C₄	—	362	1	no	—
234	C₄⋊C₄	—	362	1	no	—
235	C₄⋊C₄	—	362	1	no	—
236	C₄⋊C₄	—	362	1	no	—
237	C₄⋊C₄	—	362	1	no	—
238	C₄⋊C₄	—	362	1	no	—
239	C₄⋊C₄	—	362	1	no	—
240	C₄⋊C₄	—	362	1	no	—
241	C₄⋊C₄	—	362	1	no	—
242	C₄⋊C₄	—	362	1	no	—
243	C₄⋊C₄	—	362	1	no	—
244	C₄⋊C₄	—	181	1	no	—
245	C₄⋊C₄	—	181	1	no	—
246	C₄⋊C₄	—	181	1	no	—
247	C₄⋊C₄	—	181	1	no	—
248	C₄⋊C₄	D₇₂₄	1	1	yes	characteristic
249	C₄⋊C₄	D₇₂₄	1	1	yes	characteristic
250	C₄⋊C₄	D₇₂₄	1	1	yes	characteristic
251	C₂²⋊C₄	—	362	1	no	—
252	C₂²⋊C₄	—	362	1	no	—
253	C₂²⋊C₄	—	362	1	no	—
254	C₂²⋊C₄	—	362	1	no	—
255	C₂²⋊C₄	—	362	1	no	—
256	C₄²	—	362	1	no	—
257	C₄²	—	362	1	no	—
258	C₄²	—	362	1	no	—
259	C₄²	—	362	1	no	—
260	C₄²	—	362	1	no	—
261	C₄²	C₂²×D₁₈₁	1	1	yes	characteristic
262	C₄²	C₂²×D₁₈₁	1	1	yes	characteristic
263	C₄²	C₂²×D₁₈₁	1	1	yes	characteristic
264	C₂³	—	362	1	no	—
265	Q₈	—	4	1	no	—
266	Q₈	—	362	1	no	—
267	Q₈	—	362	1	no	—
268	Q₈	—	362	1	no	—
269	Q₈	—	362	1	no	—
270	Q₈	—	362	1	no	—
271	Q₈	—	181	2	no	—
272	Q₈	—	724	2	no	—
273	D₄	—	362	1	no	—
274	D₄	—	362	1	no	—
275	C₂×C₄	—	362	1	no	—
276	C₂×C₄	—	362	1	no	—
277	C₂×C₄	—	362	1	no	—
278	C₂×C₄	—	362	1	no	—
279	C₂×C₄	—	362	1	no	—
280	C₂×C₄	—	362	1	no	—
281	C₂×C₄	—	362	1	no	—
282	C₂×C₄	—	724	2	no	—
283	C₂×C₄	C₂×D₇₂₄	1	1	yes	characteristic
284	C₂×C₄	C₂×D₇₂₄	1	1	yes	characteristic
285	C₂×C₄	C₂×D₇₂₄	1	1	yes	characteristic
286	C₂×C₄	C₂×D₇₂₄	1	1	yes	characteristic
287	C₂×C₄	C₂×D₇₂₄	1	1	yes	characteristic
288	C₂×C₄	C₂×D₇₂₄	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₁₈₁	1	1	yes	characteristic
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₂²	—	724	1	no	—
295	C₂²	—	724	1	no	—
296	C₂²	—	724	1	no	—
297	C₂²	C₂²×D₇₂₄	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	362	2	no	—
305	C₄	—	724	1	no	—
306	C₄	—	724	1	no	—
307	C₄	—	724	1	no	—
308	C₄	—	724	1	no	—
309	C₄	—	724	1	no	—
310	C₄	—	724	1	no	—
311	C₄	#41×D₁₈₁	1	1	yes	characteristic
312	C₄	#19153864	1	1	yes	characteristic
313	C₂	—	1448	1	no	—
314	C₂	#26041941	1	1	yes	characteristic
315	C₂	#26042079	1	1	yes	characteristic
316	C₂	#26042096	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₂₄
commutator, cosocle	C₂×C₄×C₁₈₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₈₁
Fitting	#446×C₁₈₁	C₂
socle	C₂²×C₁₈₁	C₂²×D₄
prime radical	C₂×D₇₂₄	C₂³
hypercenter	#446	D₁₈₁
hypocenter	C₁₈₁	#3377
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

Complements

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

N	H ≅ G/N	type
1. #26042007	C₂	semidirect
3. #26041648	C₂	semidirect
6. #26041587	C₂	semidirect
8. #26041575	C₂	semidirect
11. #26041490	C₂	semidirect
12. #26041387	C₂	semidirect
14. C₄×Dic₇₂₄	C₂	semidirect
15. #446×C₁₈₁	C₂	semidirect
158. C₁₈₁	#3377	semidirect
171. #446	D₁₈₁	semidirect

Maximal subgroups

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

subgroup	index
#26042007	2
#26042002	2
#26041648	2
#26041639	2
#26041634	2
#26041587	2
#26041580	2
#26041575	2
#26041502	2
#26041490	2
#26041387	2
#26041381	2
C₄×Dic₇₂₄	2
#446×C₁₈₁	2
#3377	181