Subgroups — #37543540

Order 23168.

384 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#125×D₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#15668450	C₂	1	1	yes	maximal, characteristic
3	#26041588	C₂	1	1	yes	maximal, characteristic, complemented
4	#26041586	C₂	1	1	yes	maximal, characteristic, complemented
5	#26041584	C₂	1	1	yes	maximal, characteristic, complemented
6	#26041582	C₂	1	1	yes	maximal, characteristic, complemented
7	#26041580	C₂	1	1	yes	maximal, characteristic, complemented
8	#26041578	C₂	1	1	yes	maximal, characteristic, complemented
9	#26041576	C₂	1	1	yes	maximal, characteristic, complemented
10	#26041320	C₂	1	1	yes	maximal, characteristic, complemented
11	#26041318	C₂	1	1	yes	maximal, characteristic, complemented
12	#26041301	C₂	1	1	yes	maximal, characteristic, complemented
13	#26041279	C₂	1	1	yes	maximal, characteristic
14	#26041279	C₂	1	1	yes	maximal, characteristic
15	#479×C₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#15668450	C₂²	1	1	yes	characteristic
17	C₂×#15668432	C₂²	1	1	yes	characteristic
18	C₂×#15668432	C₂²	1	1	yes	characteristic
19	#19153808	C₂²	1	1	yes	characteristic
20	#19153807	C₂²	1	1	yes	characteristic
21	#19153807	C₂²	1	1	yes	characteristic
22	#19153806	—	2	1	no	—
23	#19153806	—	2	1	no	—
24	Q₈×Dic₁₈₁	C₂²	1	1	yes	characteristic
25	C₄×D₇₂₄	—	2	1	no	—
26	C₄×D₇₂₄	C₂²	1	1	yes	characteristic
27	#19153778	—	2	1	no	—
28	#19153778	C₂²	1	1	yes	characteristic
29	#19153774	—	2	1	no	—
30	#19153774	C₂²	1	1	yes	characteristic
31	C₄⋊C₄×D₁₈₁	—	2	1	no	—
32	C₄⋊C₄×D₁₈₁	—	2	1	no	—
33	C₄⋊C₄×D₁₈₁	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₁₈₁	C₂²	1	1	yes	characteristic
35	C₄²×D₁₈₁	—	2	1	no	—
36	C₄²×D₁₈₁	C₂²	1	1	yes	characteristic
37	#19153750	C₂²	1	1	yes	characteristic
38	#19153749	C₂²	1	1	yes	characteristic
39	#19153749	C₂²	1	1	yes	characteristic
40	#19153748	C₂²	1	1	yes	characteristic
41	#19153747	C₂²	1	1	yes	characteristic
42	#19153747	C₂²	1	1	yes	characteristic
43	C₄⋊#13089678	C₂²	1	1	yes	characteristic
44	#19153745	C₂²	1	1	yes	characteristic
45	#19153745	C₂²	1	1	yes	characteristic
46	#19153740	C₂²	1	1	yes	characteristic
47	#19153740	C₂²	1	1	yes	characteristic
48	#19153738	C₂²	1	1	yes	characteristic
49	#19153738	C₂²	1	1	yes	characteristic
50	#19153736	C₂²	1	1	yes	characteristic
51	#19153736	C₂²	1	1	yes	characteristic
52	#125×C₁₈₁	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₈₁	C₂²	1	1	yes	characteristic
54	#107×C₁₈₁	C₂²	1	1	yes	characteristic
55	#106×C₁₈₁	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₈₁	C₂²	1	1	yes	characteristic
57	#103×C₁₈₁	C₂²	1	1	yes	characteristic
58	#103×C₁₈₁	C₂²	1	1	yes	characteristic
59	#15668450	—	2	2	no	—
60	#15668450	Q₈	1	4	yes	—
61	C₂×D₇₂₄	—	2	1	no	—
62	C₂×D₇₂₄	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₁₈₁	—	2	1	no	—
64	C₂×C₄×D₁₈₁	—	2	1	no	—
65	C₂×C₄×D₁₈₁	—	2	1	no	—
66	C₂×C₄×D₁₈₁	—	2	1	no	—
67	C₂×C₄×D₁₈₁	—	2	1	no	—
68	C₂×C₄×D₁₈₁	D₄	1	1	yes	characteristic
69	C₂×C₄×D₁₈₁	D₄	1	1	yes	characteristic
70	C₂×C₄×D₁₈₁	C₂³	1	1	yes	characteristic
71	#15668432	—	2	2	no	—
72	#15668432	—	2	2	no	—
73	C₂×#13089678	C₂³	1	1	yes	characteristic
74	C₂×#13089678	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₈₁	—	2	1	no	—
76	C₄⋊Dic₁₈₁	—	2	1	no	—
77	C₄⋊Dic₁₈₁	—	2	1	no	—
78	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
81	#15668428	—	2	1	no	—
82	#15668428	—	2	1	no	—
83	#15668428	—	2	1	no	—
84	#15668428	—	2	1	no	—
85	#15668428	—	2	1	no	—
86	#15668428	—	2	1	no	—
87	#15668426	—	2	1	no	—
88	#15668426	—	2	1	no	—
89	#15668426	—	2	1	no	—
90	C₄×Dic₁₈₁	—	2	1	no	—
91	C₄×Dic₁₈₁	—	2	1	no	—
92	C₄×Dic₁₈₁	D₄	1	1	yes	characteristic
93	C₄×Dic₁₈₁	D₄	1	1	yes	characteristic
94	C₄×Dic₁₈₁	C₂³	1	1	yes	characteristic
95	C₂×Q₈×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	C₄⋊C₄×C₁₈₁	—	2	1	no	—
99	C₄⋊C₄×C₁₈₁	—	2	1	no	—
100	C₄⋊C₄×C₁₈₁	—	2	1	no	—
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	—	2	1	no	—
105	C₄²×C₁₈₁	C₂³	1	1	yes	characteristic
106	C₂²×D₁₈₁	—	2	1	no	—
107	C₂²×D₁₈₁	C₂×D₄	1	1	yes	characteristic
108	D₇₂₄	—	2	4	no	—
109	D₇₂₄	—	2	1	no	—
110	D₇₂₄	C₂×Q₈	1	2	yes	—
111	C₂×Dic₁₈₁	—	2	1	no	—
112	C₂×Dic₁₈₁	—	2	1	no	—
113	C₂×Dic₁₈₁	—	2	1	no	—
114	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
115	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
116	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
117	C₄×D₁₈₁	—	2	4	no	—
118	C₄×D₁₈₁	—	4	1	no	—
119	C₄×D₁₈₁	—	4	1	no	—
120	C₄×D₁₈₁	—	4	2	no	—
121	C₄×D₁₈₁	—	4	2	no	—
122	C₄×D₁₈₁	—	4	2	no	—
123	C₄×D₁₈₁	—	4	2	no	—
124	C₄×D₁₈₁	C₂×Q₈	1	2	yes	—
125	C₄×D₁₈₁	#41	1	2	yes	—
126	#13089678	—	2	1	no	—
127	#13089678	—	2	1	no	—
128	Q₈×C₁₈₁	—	2	1	no	—
129	Q₈×C₁₈₁	C₂×Q₈	1	2	yes	—
130	C₂×C₄×C₁₈₁	—	2	1	no	—
131	C₂×C₄×C₁₈₁	—	2	1	no	—
132	C₂×C₄×C₁₈₁	—	2	1	no	—
133	C₂×C₄×C₁₈₁	—	2	1	no	—
134	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₈₁	C₂⁴	1	1	yes	characteristic
137	C₈×C₁₈₁	—	2	1	no	—
138	C₈×C₁₈₁	—	2	1	no	—
139	C₂²×C₁₈₁	C₂²×D₄	1	1	yes	characteristic
140	C₂×D₁₈₁	—	2	1	no	—
141	C₂×D₁₈₁	—	2	1	no	—
142	C₂×D₁₈₁	—	2	2	no	—
143	C₂×D₁₈₁	—	4	1	no	—
144	C₂×D₁₈₁	—	4	1	no	—
145	C₂×D₁₈₁	C₂²⋊Q₈	1	2	yes	—
146	C₄×C₁₈₁	—	2	2	no	—
147	C₄×C₁₈₁	—	2	1	no	—
148	C₄×C₁₈₁	—	2	1	no	—
149	C₄×C₁₈₁	—	4	1	no	—
150	C₄×C₁₈₁	—	4	1	no	—
151	C₄×C₁₈₁	—	4	1	no	—
152	C₄×C₁₈₁	C₂²×Q₈	1	1	yes	characteristic
153	C₄×C₁₈₁	C₂×#41	1	1	yes	characteristic
154	Dic₁₈₁	—	2	1	no	—
155	Dic₁₈₁	—	2	1	no	—
156	Dic₁₈₁	—	4	1	no	—
157	Dic₁₈₁	—	4	1	no	—
158	Dic₁₈₁	—	4	1	no	—
159	Dic₁₈₁	C₂²⋊Q₈	1	2	yes	—
160	C₂×C₁₈₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
161	C₂×C₁₈₁	#575	1	1	yes	characteristic
162	C₂×C₁₈₁	#575	1	1	yes	characteristic
163	D₁₈₁	—	2	2	no	—
164	D₁₈₁	—	4	2	no	—
165	C₁₈₁	#3080	1	1	yes	characteristic, complemented, Sylow
166	#3080	—	181	1	no	maximal, Sylow
167	C₂×#125	—	181	1	no	—
168	C₄×#41	—	181	1	no	—
169	#479	—	181	1	no	—
170	#479	—	181	1	no	—
171	#479	—	181	1	no	—
172	#479	D₁₈₁	1	1	yes	characteristic, complemented
173	#478	—	181	1	no	—
174	#478	—	181	1	no	—
175	#478	—	181	1	no	—
176	#478	—	181	1	no	—
177	#428	—	181	1	no	—
178	#428	—	181	1	no	—
179	#423	—	181	1	no	—
180	#417	—	181	1	no	—
181	#417	—	181	1	no	—
182	C₂×#41	—	181	1	no	—
183	C₂×M₁₆	—	181	1	no	—
184	C₂×M₁₆	—	181	1	no	—
185	#125	—	362	1	no	—
186	#125	—	362	1	no	—
187	#125	—	181	1	no	—
188	#125	—	181	1	no	—
189	#125	—	181	1	no	—
190	#125	C₂×D₁₈₁	1	1	yes	characteristic
191	C₄×Q₈	—	181	1	no	—
192	C₄×Q₈	C₂×D₁₈₁	1	1	yes	characteristic
193	C₄×D₄	—	362	1	no	—
194	C₄×D₄	—	362	1	no	—
195	C₄×D₄	—	181	1	no	—
196	C₄×D₄	—	181	1	no	—
197	#117	—	362	1	no	—
198	#117	—	181	1	no	—
199	C₂×C₄⋊C₄	—	362	1	no	—
200	C₂×C₄⋊C₄	—	362	1	no	—
201	C₂×C₄⋊C₄	—	181	1	no	—
202	C₂×C₄⋊C₄	—	181	1	no	—
203	C₂×C₄²	—	362	1	no	—
204	C₂×C₄²	—	181	1	no	—
205	#107	—	181	1	no	—
206	#107	—	181	1	no	—
207	#107	—	181	1	no	—
208	#107	C₂×D₁₈₁	1	1	yes	characteristic
209	#106	—	181	1	no	—
210	#106	—	181	1	no	—
211	#106	—	181	1	no	—
212	#106	C₂×D₁₈₁	1	1	yes	characteristic
213	C₄⋊C₈	—	181	1	no	—
214	C₄⋊C₈	—	181	1	no	—
215	C₄⋊C₈	—	181	1	no	—
216	C₄⋊C₈	C₂×D₁₈₁	1	1	yes	characteristic
217	#103	—	181	1	no	—
218	#103	—	181	1	no	—
219	#103	C₂×D₁₈₁	1	1	yes	characteristic
220	#103	C₂×D₁₈₁	1	1	yes	characteristic
221	#102	—	181	1	no	—
222	#102	—	181	1	no	—
223	#102	—	181	1	no	—
224	#102	—	181	1	no	—
225	#41	—	362	2	no	—
226	#41	—	181	4	no	—
227	C₂×Q₈	C₂²×D₁₈₁	1	1	yes	characteristic
228	C₂×D₄	—	362	1	no	—
229	C₂×D₄	—	181	1	no	—
230	C₂²×C₄	—	362	1	no	—
231	C₂²×C₄	—	362	1	no	—
232	C₂²×C₄	—	362	1	no	—
233	C₂²×C₄	—	362	1	no	—
234	C₂²×C₄	—	362	1	no	—
235	C₂²×C₄	—	181	1	no	—
236	C₂²×C₄	—	181	1	no	—
237	C₂²×C₄	—	181	1	no	—
238	M₁₆	—	362	2	no	—
239	M₁₆	—	362	2	no	—
240	C₂×C₈	—	181	1	no	—
241	C₂×C₈	—	181	1	no	—
242	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	362	1	no	—
248	C₄⋊C₄	—	362	1	no	—
249	C₄⋊C₄	—	362	1	no	—
250	C₄⋊C₄	—	362	1	no	—
251	C₄⋊C₄	—	362	1	no	—
252	C₄⋊C₄	—	362	1	no	—
253	C₄⋊C₄	—	362	1	no	—
254	C₄⋊C₄	—	362	1	no	—
255	C₄⋊C₄	—	362	1	no	—
256	C₄⋊C₄	—	181	1	no	—
257	C₄⋊C₄	—	181	1	no	—
258	C₄⋊C₄	—	181	1	no	—
259	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
260	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
261	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
262	C₂²⋊C₄	—	362	1	no	—
263	C₂²⋊C₄	—	362	1	no	—
264	C₂²⋊C₄	—	362	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	362	1	no	—
267	C₄²	—	362	1	no	—
268	C₄²	—	181	1	no	—
269	C₄²	—	181	1	no	—
270	C₄²	—	181	1	no	—
271	C₄²	C₂²×D₁₈₁	1	1	yes	characteristic
272	C₂³	—	362	1	no	—
273	C₂³	—	181	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	Q₈×D₁₈₁	1	2	yes	—
276	D₄	—	362	4	no	—
277	D₄	—	362	1	no	—
278	D₄	—	181	2	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	362	4	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	—	181	1	no	—
290	C₂×C₄	—	181	2	no	—
291	C₂×C₄	—	181	2	no	—
292	C₂×C₄	—	724	1	no	—
293	C₂×C₄	—	724	1	no	—
294	C₂×C₄	—	724	2	no	—
295	C₂×C₄	—	724	2	no	—
296	C₂×C₄	—	724	2	no	—
297	C₂×C₄	—	724	2	no	—
298	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
299	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
300	C₂×C₄	C₂³×D₁₈₁	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	362	1	no	—
304	C₈	—	362	1	no	—
305	C₂²	—	362	1	no	—
306	C₂²	—	362	1	no	—
307	C₂²	—	362	2	no	—
308	C₂²	—	181	2	no	—
309	C₂²	—	724	1	no	—
310	C₂²	—	724	1	no	—
311	C₂²	C₂×D₄×D₁₈₁	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	362	1	no	—
319	C₄	—	362	1	no	—
320	C₄	—	181	2	no	—
321	C₄	—	724	1	no	—
322	C₄	—	724	1	no	—
323	C₄	—	724	1	no	—
324	C₄	C₂×Q₈×D₁₈₁	1	1	yes	characteristic
325	C₄	#41×D₁₈₁	1	1	yes	characteristic
326	C₂	—	362	2	no	—
327	C₂	—	724	2	no	—
328	C₂	C₂²⋊Q₈×D₁₈₁	1	1	yes	characteristic
329	C₂	#26042076	1	1	yes	characteristic
330	C₂	#26042078	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₈₁
commutator, cosocle	C₂×C₄×C₁₈₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₈₁
Fitting	#479×C₁₈₁	C₂
socle	C₂²×C₁₈₁	C₂²×D₄
prime radical	C₂×#15668450	C₂²
hypercenter	#479	D₁₈₁
hypocenter	C₁₈₁	#3080
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
1. #125×D₁₈₁	C₂	semidirect
3. #26041588	C₂	semidirect
4. #26041586	C₂	semidirect
5. #26041584	C₂	semidirect
6. #26041582	C₂	semidirect
7. #26041580	C₂	semidirect
8. #26041578	C₂	semidirect
9. #26041576	C₂	semidirect
10. #26041320	C₂	semidirect
11. #26041318	C₂	semidirect
12. #26041301	C₂	semidirect
15. #479×C₁₈₁	C₂	semidirect
165. C₁₈₁	#3080	semidirect
172. #479	D₁₈₁	semidirect

Maximal subgroups

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

subgroup	index
#125×D₁₈₁	2
C₄×#15668450	2
#26041588	2
#26041586	2
#26041584	2
#26041582	2
#26041580	2
#26041578	2
#26041576	2
#26041320	2
#26041318	2
#26041301	2
#26041279	2
#479×C₁₈₁	2
#3080	181