Subgroups — #37546250

Order 23168.

336 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26042003	C₂	1	1	yes	maximal, characteristic
2	#26041967	C₂	1	1	yes	maximal, characteristic
3	#26041623	C₂	1	1	yes	maximal, characteristic, complemented
4	#26041623	C₂	1	1	yes	maximal, characteristic, complemented
5	#26041622	C₂	1	1	yes	maximal, characteristic, complemented
6	#26041616	C₂	1	1	yes	maximal, characteristic, complemented
7	#26041583	C₂	1	1	yes	maximal, characteristic, complemented
8	#26041571	C₂	1	1	yes	maximal, characteristic, complemented
9	#26041538	C₂	1	1	yes	maximal, characteristic, complemented
10	#26041521	C₂	1	1	yes	maximal, characteristic, complemented
11	#26041387	C₂	1	1	yes	maximal, characteristic, complemented
12	#26041352	C₂	1	1	yes	maximal, characteristic
13	#26041320	C₂	1	1	yes	maximal, characteristic, complemented
14	#26041314	C₂	1	1	yes	maximal, characteristic, complemented
15	#501×C₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Dic₇₂₄	C₂²	1	1	yes	characteristic
17	C₂×#15668432	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₁₈₁	C₂²	1	1	yes	characteristic
19	#19153811	—	2	1	no	—
20	#19153811	—	2	1	no	—
21	#19153811	—	2	1	no	—
22	#19153807	—	2	1	no	—
23	#19153805	—	2	1	no	—
24	#19153801	—	2	1	no	—
25	#19153796	C₂²	1	1	yes	characteristic
26	#19153796	C₂²	1	1	yes	characteristic
27	#19153796	C₂²	1	1	yes	characteristic
28	#19153796	C₂²	1	1	yes	characteristic
29	Q₈×Dic₁₈₁	—	2	1	no	—
30	C₄×Dic₃₆₂	C₂²	1	1	yes	characteristic
31	C₄×Dic₃₆₂	C₂²	1	1	yes	characteristic
32	C₄×D₇₂₄	C₂²	1	1	yes	characteristic
33	#19153774	—	2	1	no	—
34	#19153774	C₂²	1	1	yes	characteristic
35	#19153774	C₂²	1	1	yes	characteristic
36	#19153774	C₂²	1	1	yes	characteristic
37	C₄⋊C₄×D₁₈₁	C₂²	1	1	yes	characteristic
38	#19153749	C₂²	1	1	yes	characteristic
39	#19153749	C₂²	1	1	yes	characteristic
40	#19153749	C₂²	1	1	yes	characteristic
41	#19153748	C₂²	1	1	yes	characteristic
42	#19153747	C₂²	1	1	yes	characteristic
43	C₄⋊#13089678	C₂²	1	1	yes	characteristic
44	#19153741	C₂²	1	1	yes	characteristic
45	#19153741	C₂²	1	1	yes	characteristic
46	#19153739	C₂²	1	1	yes	characteristic
47	#19153739	C₂²	1	1	yes	characteristic
48	#19153739	C₂²	1	1	yes	characteristic
49	#19153739	C₂²	1	1	yes	characteristic
50	#19153728	C₂²	1	1	yes	characteristic
51	#19153728	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₁₈₁	C₂²	1	1	yes	characteristic
53	#125×C₁₈₁	C₂²	1	1	yes	characteristic
54	#107×C₁₈₁	C₂²	1	1	yes	characteristic
55	#107×C₁₈₁	C₂²	1	1	yes	characteristic
56	#106×C₁₈₁	C₂²	1	1	yes	characteristic
57	#106×C₁₈₁	C₂²	1	1	yes	characteristic
58	#97×C₁₈₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₃₆₂	C₂³	1	1	yes	characteristic
60	C₂×Dic₃₆₂	C₂³	1	1	yes	characteristic
61	C₂×D₇₂₄	D₄	1	1	yes	characteristic
62	C₂×C₄×D₁₈₁	—	2	1	no	—
63	C₂×C₄×D₁₈₁	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₁₈₁	C₂³	1	1	yes	characteristic
65	Dic₇₂₄	—	2	2	no	—
66	#15668432	—	2	2	no	—
67	C₂×#13089678	C₂³	1	1	yes	characteristic
68	C₂×#13089678	C₂³	1	1	yes	characteristic
69	C₈×D₁₈₁	—	2	2	no	—
70	C₄⋊Dic₁₈₁	—	2	1	no	—
71	C₄⋊Dic₁₈₁	—	2	1	no	—
72	C₄⋊Dic₁₈₁	—	2	1	no	—
73	C₄⋊Dic₁₈₁	—	2	1	no	—
74	C₄⋊Dic₁₈₁	D₄	1	1	yes	characteristic
75	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
77	#15668428	—	2	1	no	—
78	#15668428	—	2	1	no	—
79	#15668428	—	2	1	no	—
80	#15668426	—	2	1	no	—
81	#15668426	—	2	1	no	—
82	#15668426	—	2	1	no	—
83	#15668426	—	2	1	no	—
84	#15668426	—	2	1	no	—
85	#15668426	—	2	1	no	—
86	#15668426	—	2	1	no	—
87	#15668426	D₄	1	1	yes	characteristic
88	#15668426	D₄	1	1	yes	characteristic
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	—	2	1	no	—
95	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
96	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
97	C₄⋊C₄×C₁₈₁	—	2	1	no	—
98	C₄⋊C₄×C₁₈₁	—	2	1	no	—
99	C₄⋊C₄×C₁₈₁	—	2	1	no	—
100	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
104	C₄²×C₁₈₁	C₂³	1	1	yes	characteristic
105	C₂²×D₁₈₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×D₁₈₁	C₂×D₄	1	1	yes	characteristic
107	Dic₃₆₂	—	2	1	no	—
108	Dic₃₆₂	—	2	1	no	—
109	Dic₃₆₂	—	2	1	no	—
110	Dic₃₆₂	—	2	1	no	—
111	D₇₂₄	—	4	1	no	—
112	C₂×Dic₁₈₁	—	2	1	no	—
113	C₂×Dic₁₈₁	—	2	1	no	—
114	C₂×Dic₁₈₁	—	2	1	no	—
115	C₂×Dic₁₈₁	—	2	1	no	—
116	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
117	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
118	C₄×D₁₈₁	—	2	1	no	—
119	C₄×D₁₈₁	—	2	1	no	—
120	C₄×D₁₈₁	—	2	1	no	—
121	C₄×D₁₈₁	—	2	1	no	—
122	C₄×D₁₈₁	—	4	2	no	—
123	#13089678	—	2	1	no	—
124	#13089678	—	2	1	no	—
125	Q₈×C₁₈₁	—	4	2	no	—
126	C₂×C₄×C₁₈₁	—	2	1	no	—
127	C₂×C₄×C₁₈₁	—	2	1	no	—
128	C₂×C₄×C₁₈₁	—	2	1	no	—
129	C₂×C₄×C₁₈₁	—	2	1	no	—
130	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₈₁	C₂⁴	1	1	yes	characteristic
133	C₈×C₁₈₁	—	2	1	no	—
134	C₈×C₁₈₁	#41	1	2	yes	—
135	C₂²×C₁₈₁	C₂²×D₄	1	1	yes	characteristic
136	C₂×D₁₈₁	—	2	1	no	—
137	C₂×D₁₈₁	—	2	1	no	—
138	C₂×D₁₈₁	—	2	1	no	—
139	C₂×D₁₈₁	—	2	1	no	—
140	C₂×D₁₈₁	—	2	1	no	—
141	C₂×D₁₈₁	—	2	1	no	—
142	C₄×C₁₈₁	—	2	1	no	—
143	C₄×C₁₈₁	—	2	1	no	—
144	C₄×C₁₈₁	—	4	1	no	—
145	C₄×C₁₈₁	—	4	1	no	—
146	C₄×C₁₈₁	—	4	1	no	—
147	C₄×C₁₈₁	—	4	1	no	—
148	C₄×C₁₈₁	C₂×#41	1	1	yes	characteristic
149	C₄×C₁₈₁	2₋¹⁺⁴	1	1	yes	characteristic
150	Dic₁₈₁	—	2	1	no	—
151	Dic₁₈₁	—	2	1	no	—
152	Dic₁₈₁	—	4	1	no	—
153	Dic₁₈₁	—	4	1	no	—
154	Dic₁₈₁	—	4	1	no	—
155	Dic₁₈₁	—	4	1	no	—
156	C₂×C₁₈₁	#547	1	1	yes	characteristic
157	C₂×C₁₈₁	#575	1	1	yes	characteristic
158	C₂×C₁₈₁	#578	1	1	yes	characteristic
159	D₁₈₁	—	4	1	no	—
160	D₁₈₁	—	4	1	no	—
161	C₁₈₁	#3349	1	1	yes	characteristic, complemented, Sylow
162	#3349	—	181	1	no	maximal, Sylow
163	#556	—	181	1	no	—
164	#552	—	181	1	no	—
165	#501	D₁₈₁	1	1	yes	characteristic, complemented
166	#485	—	181	1	no	—
167	#485	—	181	1	no	—
168	#485	—	181	1	no	—
169	#484	—	181	1	no	—
170	#479	—	181	1	no	—
171	#477	—	181	1	no	—
172	#470	—	181	1	no	—
173	#467	—	181	1	no	—
174	#441	—	181	1	no	—
175	#435	—	181	1	no	—
176	#428	—	181	1	no	—
177	#427	—	181	1	no	—
178	C₂×Q₁₆	—	181	1	no	—
179	C₂×M₁₆	—	181	1	no	—
180	C₂²×C₈	—	181	1	no	—
181	C₄⋊Q₈	C₂×D₁₈₁	1	1	yes	characteristic
182	#126	—	362	1	no	—
183	#126	—	362	1	no	—
184	#126	—	362	1	no	—
185	#125	—	362	1	no	—
186	#125	C₂×D₁₈₁	1	1	yes	characteristic
187	#124	—	362	1	no	—
188	#123	—	362	1	no	—
189	C₂²⋊Q₈	—	181	1	no	—
190	C₂²⋊Q₈	—	181	1	no	—
191	C₂²⋊Q₈	—	181	1	no	—
192	C₂²⋊Q₈	—	181	1	no	—
193	C₄×Q₈	—	362	1	no	—
194	C₄×Q₈	—	181	1	no	—
195	C₄×Q₈	—	181	1	no	—
196	C₄×D₄	—	181	1	no	—
197	#117	—	362	1	no	—
198	#117	—	181	1	no	—
199	#117	—	181	1	no	—
200	#117	—	181	1	no	—
201	C₂×C₄⋊C₄	—	181	1	no	—
202	#107	—	181	1	no	—
203	#107	—	181	1	no	—
204	#107	—	181	1	no	—
205	#107	C₂×D₁₈₁	1	1	yes	characteristic
206	#107	C₂×D₁₈₁	1	1	yes	characteristic
207	#106	—	181	1	no	—
208	#106	—	181	1	no	—
209	#106	C₂×D₁₈₁	1	1	yes	characteristic
210	#106	C₂×D₁₈₁	1	1	yes	characteristic
211	C₄⋊C₈	—	181	1	no	—
212	#103	—	181	1	no	—
213	#103	—	181	1	no	—
214	#103	—	181	1	no	—
215	#103	—	181	1	no	—
216	#103	—	181	1	no	—
217	#103	—	181	1	no	—
218	C₂²⋊C₈	—	181	1	no	—
219	C₂²⋊C₈	—	181	1	no	—
220	#97	C₂×D₁₈₁	1	1	yes	characteristic
221	C₂×Q₈	—	2	1	no	—
222	C₂×Q₈	—	181	1	no	—
223	C₂×Q₈	—	181	1	no	—
224	C₂×D₄	—	181	1	no	—
225	C₂²×C₄	—	362	1	no	—
226	C₂²×C₄	—	181	1	no	—
227	C₂²×C₄	—	181	1	no	—
228	Q₁₆	—	362	2	no	—
229	M₁₆	—	362	2	no	—
230	C₂×C₈	—	362	2	no	—
231	C₂×C₈	—	181	1	no	—
232	C₂×C₈	—	181	1	no	—
233	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
234	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	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	—	362	1	no	—
245	C₄⋊C₄	—	181	1	no	—
246	C₄⋊C₄	—	181	1	no	—
247	C₄⋊C₄	—	181	1	no	—
248	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
249	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
250	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
251	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
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	C₂²⋊C₄	—	362	1	no	—
257	C₂²⋊C₄	—	362	1	no	—
258	C₂²⋊C₄	—	362	1	no	—
259	C₂²⋊C₄	—	181	1	no	—
260	C₂²⋊C₄	—	181	1	no	—
261	C₄²	—	362	1	no	—
262	C₄²	—	362	1	no	—
263	C₄²	—	362	1	no	—
264	C₄²	—	362	1	no	—
265	C₄²	—	362	1	no	—
266	C₄²	C₂²×D₁₈₁	1	1	yes	characteristic
267	C₂³	—	181	1	no	—
268	C₂³	—	181	1	no	—
269	Q₈	—	4	2	no	—
270	Q₈	—	362	1	no	—
271	Q₈	—	362	1	no	—
272	Q₈	—	362	1	no	—
273	Q₈	—	362	1	no	—
274	D₄	—	724	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	362	1	no	—
280	C₂×C₄	—	362	1	no	—
281	C₂×C₄	—	362	1	no	—
282	C₂×C₄	—	362	1	no	—
283	C₂×C₄	—	362	1	no	—
284	C₂×C₄	—	362	1	no	—
285	C₂×C₄	—	362	1	no	—
286	C₂×C₄	—	362	1	no	—
287	C₂×C₄	—	181	1	no	—
288	C₂×C₄	—	181	1	no	—
289	C₂×C₄	—	724	2	no	—
290	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
291	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
292	C₂×C₄	C₂³×D₁₈₁	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	362	1	no	—
295	C₈	—	362	1	no	—
296	C₈	#15668450	1	2	yes	—
297	C₂²	—	362	1	no	—
298	C₂²	—	362	1	no	—
299	C₂²	—	362	1	no	—
300	C₂²	—	362	1	no	—
301	C₂²	—	362	1	no	—
302	C₂²	—	362	1	no	—
303	C₂²	C₂×D₄×D₁₈₁	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	362	1	no	—
311	C₄	—	362	1	no	—
312	C₄	—	724	1	no	—
313	C₄	—	724	1	no	—
314	C₄	—	724	1	no	—
315	C₄	—	724	1	no	—
316	C₄	C₂×#15668450	1	1	yes	characteristic
317	C₄	#19153864	1	1	yes	characteristic
318	C₂	—	724	1	no	—
319	C₂	—	724	1	no	—
320	C₂	#26041932	1	1	yes	characteristic
321	C₂	#26042075	1	1	yes	characteristic
322	C₂	#26042095	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₈₁
commutator, cosocle	C₂×C₄×C₁₈₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₈₁
Fitting	#501×C₁₈₁	C₂
socle	C₂²×C₁₈₁	C₂²×D₄
prime radical	C₂×D₇₂₄	D₄
hypercenter	#501	D₁₈₁
hypocenter	C₁₈₁	#3349
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #501

Complements

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

N	H ≅ G/N	type
3. #26041623	C₂	semidirect
4. #26041623	C₂	semidirect
5. #26041622	C₂	semidirect
6. #26041616	C₂	semidirect
7. #26041583	C₂	semidirect
8. #26041571	C₂	semidirect
9. #26041538	C₂	semidirect
10. #26041521	C₂	semidirect
11. #26041387	C₂	semidirect
13. #26041320	C₂	semidirect
14. #26041314	C₂	semidirect
15. #501×C₁₈₁	C₂	semidirect
161. C₁₈₁	#3349	semidirect
165. #501	D₁₈₁	semidirect

Maximal subgroups

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

subgroup	index
#26042003	2
#26041967	2
#26041623	2
#26041622	2
#26041616	2
#26041583	2
#26041571	2
#26041538	2
#26041521	2
#26041387	2
#26041352	2
#26041320	2
#26041314	2
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