Subgroups — #37546464

Order 23168.

338 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26042009	C₂	1	1	yes	maximal, characteristic, complemented
2	#26042001	C₂	1	1	yes	maximal, characteristic
3	#26041641	C₂	1	1	yes	maximal, characteristic
4	#26041640	C₂	1	1	yes	maximal, characteristic
5	#26041628	C₂	1	1	yes	maximal, characteristic, complemented
6	#26041568	C₂	1	1	yes	maximal, characteristic
7	#26041563	C₂	1	1	yes	maximal, characteristic, complemented
8	#26041504	C₂	1	1	yes	maximal, characteristic
9	#26041502	C₂	1	1	yes	maximal, characteristic
10	#26041488	C₂	1	1	yes	maximal, characteristic, complemented
11	#26041408	C₂	1	1	yes	maximal, characteristic, complemented
12	#26041385	C₂	1	1	yes	maximal, characteristic, complemented
13	#26041384	C₂	1	1	yes	maximal, characteristic, complemented
14	#26041366	C₂	1	1	yes	maximal, characteristic
15	#475×C₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Dic₇₂₄	C₂²	1	1	yes	characteristic
17	C₂×#15668439	C₂²	1	1	yes	characteristic
18	C₂×#15668438	C₂²	1	1	yes	characteristic
19	#19153815	—	2	1	no	—
20	#19153815	C₂²	1	1	yes	characteristic
21	#19153815	C₂²	1	1	yes	characteristic
22	#19153814	—	2	1	no	—
23	#19153814	—	2	1	no	—
24	#19153814	C₂²	1	1	yes	characteristic
25	#19153811	—	2	1	no	—
26	#19153810	—	2	1	no	—
27	#19153805	C₂²	1	1	yes	characteristic
28	#19153804	C₂²	1	1	yes	characteristic
29	#19153794	—	2	1	no	—
30	C₄×Dic₃₆₂	C₂²	1	1	yes	characteristic
31	C₄×Dic₃₆₂	C₂²	1	1	yes	characteristic
32	#19153781	—	2	1	no	—
33	#19153781	C₂²	1	1	yes	characteristic
34	#19153781	C₂²	1	1	yes	characteristic
35	#19153781	C₂²	1	1	yes	characteristic
36	#19153778	C₂²	1	1	yes	characteristic
37	#19153773	—	2	1	no	—
38	#19153747	C₂²	1	1	yes	characteristic
39	#19153745	C₂²	1	1	yes	characteristic
40	#19153745	C₂²	1	1	yes	characteristic
41	#19153741	C₂²	1	1	yes	characteristic
42	#19153740	C₂²	1	1	yes	characteristic
43	#19153739	C₂²	1	1	yes	characteristic
44	#19153739	C₂²	1	1	yes	characteristic
45	#19153739	C₂²	1	1	yes	characteristic
46	#19153738	C₂²	1	1	yes	characteristic
47	#19153736	C₂²	1	1	yes	characteristic
48	#19153736	C₂²	1	1	yes	characteristic
49	#19153727	C₂²	1	1	yes	characteristic
50	#19153726	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₈₁	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₁₈₁	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₈₁	C₂²	1	1	yes	characteristic
54	#106×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	C₂×Dic₃₆₂	—	2	1	no	—
60	C₂×Dic₃₆₂	—	2	1	no	—
61	C₂×Dic₃₆₂	D₄	1	1	yes	characteristic
62	C₂×Dic₃₆₂	C₂³	1	1	yes	characteristic
63	C₂×Dic₃₆₂	C₂³	1	1	yes	characteristic
64	C₂×D₇₂₄	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₁₈₁	—	2	1	no	—
66	Dic₇₂₄	—	2	2	no	—
67	#15668439	—	2	2	no	—
68	#15668438	—	2	2	no	—
69	C₂×#13089678	C₂³	1	1	yes	characteristic
70	C₂×#13089678	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₁₈₁	D₄	1	1	yes	characteristic
72	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
73	#15668428	—	2	1	no	—
74	#15668428	—	2	1	no	—
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	D₄	1	1	yes	characteristic
82	#15668428	D₄	1	1	yes	characteristic
83	#15668426	—	2	1	no	—
84	#15668426	—	2	1	no	—
85	#15668426	—	2	1	no	—
86	#15668426	—	2	1	no	—
87	#15668426	—	2	1	no	—
88	C₄×Dic₁₈₁	—	2	1	no	—
89	C₄×Dic₁₈₁	—	2	1	no	—
90	C₄×Dic₁₈₁	—	2	1	no	—
91	C₄×Dic₁₈₁	—	2	1	no	—
92	C₄×Dic₁₈₁	C₂³	1	1	yes	characteristic
93	C₄×Dic₁₈₁	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₁₈₁	—	2	1	no	—
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	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
103	C₄²×C₁₈₁	—	2	1	no	—
104	C₄²×C₁₈₁	C₂³	1	1	yes	characteristic
105	C₂²×D₁₈₁	—	2	1	no	—
106	Dic₃₆₂	—	2	1	no	—
107	Dic₃₆₂	—	2	2	no	—
108	Dic₃₆₂	—	2	1	no	—
109	Dic₃₆₂	—	2	1	no	—
110	Dic₃₆₂	—	8	1	no	—
111	Dic₃₆₂	—	8	1	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	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
122	C₄×D₁₈₁	—	4	2	no	—
123	#13089678	—	2	1	no	—
124	#13089678	—	2	1	no	—
125	Q₈×C₁₈₁	—	2	1	no	—
126	Q₈×C₁₈₁	—	2	1	no	—
127	Q₈×C₁₈₁	—	8	1	no	—
128	C₂×C₄×C₁₈₁	—	2	1	no	—
129	C₂×C₄×C₁₈₁	—	2	1	no	—
130	C₂×C₄×C₁₈₁	—	2	1	no	—
131	C₂×C₄×C₁₈₁	—	2	1	no	—
132	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₈₁	C₂⁴	1	1	yes	characteristic
135	C₈×C₁₈₁	—	2	1	no	—
136	C₈×C₁₈₁	—	2	1	no	—
137	C₂²×C₁₈₁	C₂²×D₄	1	1	yes	characteristic
138	C₂×D₁₈₁	—	4	1	no	—
139	C₂×D₁₈₁	—	4	1	no	—
140	C₂×D₁₈₁	—	4	1	no	—
141	C₄×C₁₈₁	—	2	1	no	—
142	C₄×C₁₈₁	—	2	1	no	—
143	C₄×C₁₈₁	—	4	1	no	—
144	C₄×C₁₈₁	—	4	1	no	—
145	C₄×C₁₈₁	—	4	1	no	—
146	C₄×C₁₈₁	—	4	1	no	—
147	C₄×C₁₈₁	C₂×#41	1	1	yes	characteristic
148	C₄×C₁₈₁	2₋¹⁺⁴	1	1	yes	characteristic
149	Dic₁₈₁	—	2	2	no	—
150	Dic₁₈₁	—	2	1	no	—
151	Dic₁₈₁	—	2	1	no	—
152	Dic₁₈₁	—	2	1	no	—
153	Dic₁₈₁	—	4	1	no	—
154	Dic₁₈₁	—	4	1	no	—
155	Dic₁₈₁	#137	1	2	yes	—
156	C₂×C₁₈₁	#548	1	1	yes	characteristic
157	C₂×C₁₈₁	C₂×#137	1	1	yes	characteristic
158	C₂×C₁₈₁	#577	1	1	yes	characteristic
159	D₁₈₁	—	8	1	no	—
160	C₁₈₁	#3369	1	1	yes	characteristic, complemented, Sylow
161	#3369	—	181	1	no	maximal, Sylow
162	Q₈²	—	181	1	no	—
163	#556	—	181	1	no	—
164	#489	—	181	1	no	—
165	#489	—	181	1	no	—
166	#487	—	181	1	no	—
167	#476	—	181	1	no	—
168	#475	—	181	1	no	—
169	#475	D₁₈₁	1	1	yes	characteristic, complemented
170	#464	—	181	1	no	—
171	#464	—	181	1	no	—
172	#462	—	181	1	no	—
173	#446	—	181	1	no	—
174	#441	—	181	1	no	—
175	#441	—	181	1	no	—
176	C₄×SD₁₆	—	181	1	no	—
177	C₂×Q₁₆	—	181	1	no	—
178	C₂×Q₁₆	—	181	1	no	—
179	C₂×SD₁₆	—	181	1	no	—
180	C₄⋊Q₈	—	362	1	no	—
181	C₄⋊Q₈	—	362	1	no	—
182	C₄⋊Q₈	—	362	1	no	—
183	C₄⋊Q₈	—	181	1	no	—
184	C₄⋊Q₈	—	181	1	no	—
185	C₄⋊Q₈	—	181	1	no	—
186	C₄⋊Q₈	C₂×D₁₈₁	1	1	yes	characteristic
187	#126	—	362	1	no	—
188	#126	—	362	1	no	—
189	#124	—	181	1	no	—
190	#124	—	181	1	no	—
191	C₂²⋊Q₈	—	362	1	no	—
192	C₄×Q₈	—	362	1	no	—
193	C₄×Q₈	—	181	1	no	—
194	C₄×Q₈	—	181	1	no	—
195	C₄×Q₈	—	181	1	no	—
196	C₄×Q₈	—	181	1	no	—
197	C₄×Q₈	—	181	1	no	—
198	C₄×Q₈	C₂×D₁₈₁	1	1	yes	characteristic
199	C₄×D₄	—	181	1	no	—
200	#117	—	362	1	no	—
201	#106	—	181	1	no	—
202	#106	C₂×D₁₈₁	1	1	yes	characteristic
203	#106	C₂×D₁₈₁	1	1	yes	characteristic
204	C₄⋊C₈	—	181	1	no	—
205	C₄⋊C₈	—	181	1	no	—
206	C₄⋊C₈	C₂×D₁₈₁	1	1	yes	characteristic
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	#103	C₂×D₁₈₁	1	1	yes	characteristic
213	#103	C₂×D₁₈₁	1	1	yes	characteristic
214	#102	—	181	1	no	—
215	#102	—	181	1	no	—
216	#102	—	181	1	no	—
217	#97	—	181	1	no	—
218	#97	—	181	1	no	—
219	C₄×C₈	—	181	1	no	—
220	C₂×Q₈	—	2	1	no	—
221	C₂×Q₈	—	362	1	no	—
222	C₂×Q₈	—	362	1	no	—
223	C₂×Q₈	—	181	1	no	—
224	C₂×Q₈	—	181	1	no	—
225	C₂×Q₈	—	181	1	no	—
226	C₂×Q₈	C₂²×D₁₈₁	1	1	yes	characteristic
227	C₂×D₄	—	181	1	no	—
228	C₂²×C₄	—	362	1	no	—
229	Q₁₆	—	362	2	no	—
230	Q₁₆	—	362	2	no	—
231	SD₁₆	—	362	2	no	—
232	C₂×C₈	—	181	1	no	—
233	C₂×C₈	—	181	1	no	—
234	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
235	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
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	—	362	1	no	—
246	C₄⋊C₄	—	181	1	no	—
247	C₄⋊C₄	—	181	1	no	—
248	C₄⋊C₄	—	181	1	no	—
249	C₄⋊C₄	—	181	1	no	—
250	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
251	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
252	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
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	C₄²	—	2	1	no	—
259	C₄²	—	362	1	no	—
260	C₄²	—	362	1	no	—
261	C₄²	—	362	1	no	—
262	C₄²	—	362	1	no	—
263	C₄²	—	181	1	no	—
264	C₄²	—	181	1	no	—
265	C₄²	C₂²×D₁₈₁	1	1	yes	characteristic
266	C₂³	—	362	1	no	—
267	Q₈	—	2	1	no	—
268	Q₈	—	2	1	no	—
269	Q₈	—	8	1	no	—
270	Q₈	—	362	2	no	—
271	Q₈	—	362	1	no	—
272	Q₈	—	362	1	no	—
273	Q₈	—	362	1	no	—
274	Q₈	—	1448	1	no	—
275	Q₈	—	1448	1	no	—
276	Q₈	—	181	2	no	—
277	D₄	—	362	1	no	—
278	D₄	—	362	1	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	1	no	—
284	C₂×C₄	—	362	1	no	—
285	C₂×C₄	—	362	1	no	—
286	C₂×C₄	—	181	1	no	—
287	C₂×C₄	—	181	1	no	—
288	C₂×C₄	—	181	1	no	—
289	C₂×C₄	—	181	1	no	—
290	C₂×C₄	—	724	2	no	—
291	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
292	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
293	C₂×C₄	C₂³×D₁₈₁	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	362	1	no	—
297	C₈	—	362	1	no	—
298	C₂²	—	724	1	no	—
299	C₂²	—	724	1	no	—
300	C₂²	—	724	1	no	—
301	C₂²	C₂×D₄×D₁₈₁	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	362	2	no	—
309	C₄	—	362	1	no	—
310	C₄	—	362	1	no	—
311	C₄	—	362	1	no	—
312	C₄	—	181	2	no	—
313	C₄	—	724	1	no	—
314	C₄	—	724	1	no	—
315	C₄	#41×D₁₈₁	1	1	yes	characteristic
316	C₄	#19153864	1	1	yes	characteristic
317	C₂	—	1448	1	no	—
318	C₂	#26041939	1	1	yes	characteristic
319	C₂	#137×D₁₈₁	1	1	yes	characteristic
320	C₂	#26042085	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₈₁
commutator, cosocle	C₂×C₄×C₁₈₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₈₁
Fitting	#475×C₁₈₁	C₂
socle	C₂²×C₁₈₁	C₂²×D₄
prime radical	C₂×D₇₂₄	C₂³
hypercenter	#475	D₁₈₁
hypocenter	C₁₈₁	#3369
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #475

Complements

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

N	H ≅ G/N	type
1. #26042009	C₂	semidirect
5. #26041628	C₂	semidirect
7. #26041563	C₂	semidirect
10. #26041488	C₂	semidirect
11. #26041408	C₂	semidirect
12. #26041385	C₂	semidirect
13. #26041384	C₂	semidirect
15. #475×C₁₈₁	C₂	semidirect
160. C₁₈₁	#3369	semidirect
169. #475	D₁₈₁	semidirect

Maximal subgroups

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

subgroup	index
#26042009	2
#26042001	2
#26041641	2
#26041640	2
#26041628	2
#26041568	2
#26041563	2
#26041504	2
#26041502	2
#26041488	2
#26041408	2
#26041385	2
#26041384	2
#26041366	2
#475×C₁₈₁	2
#3369	181