Subgroups — #37546331

Order 23168.

342 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26042000	C₂	1	1	yes	maximal, characteristic
2	#26041987	C₂	1	1	yes	maximal, characteristic
3	#26041676	C₂	1	1	yes	maximal, characteristic, complemented
4	#26041619	C₂	1	1	yes	maximal, characteristic, complemented
5	#26041617	C₂	1	1	yes	maximal, characteristic, complemented
6	#26041608	C₂	1	1	yes	maximal, characteristic, complemented
7	#26041607	C₂	1	1	yes	maximal, characteristic, complemented
8	#26041572	C₂	1	1	yes	maximal, characteristic, complemented
9	#26041562	C₂	1	1	yes	maximal, characteristic, complemented
10	#26041534	C₂	1	1	yes	maximal, characteristic, complemented
11	Q₁₆×Dic₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#26041355	C₂	1	1	yes	maximal, characteristic
13	#26041321	C₂	1	1	yes	maximal, characteristic, complemented
14	#26041318	C₂	1	1	yes	maximal, characteristic, complemented
15	#470×C₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#15668432	C₂²	1	1	yes	characteristic
17	#19153815	C₂²	1	1	yes	characteristic
18	#19153815	C₂²	1	1	yes	characteristic
19	#19153811	—	2	1	no	—
20	#19153809	—	2	1	no	—
21	#19153806	—	2	1	no	—
22	#19153802	—	2	1	no	—
23	#19153796	—	2	1	no	—
24	#19153793	—	2	1	no	—
25	#19153792	C₂²	1	1	yes	characteristic
26	#19153792	C₂²	1	1	yes	characteristic
27	Q₈×Dic₁₈₁	C₂²	1	1	yes	characteristic
28	Q₈×Dic₁₈₁	C₂²	1	1	yes	characteristic
29	#19153781	—	2	1	no	—
30	C₄×C₂²⋊D₁₈₁	C₂²	1	1	yes	characteristic
31	#19153776	—	2	1	no	—
32	#19153775	C₂²	1	1	yes	characteristic
33	#19153774	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊Dic₁₈₁	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₁₈₁	C₂²	1	1	yes	characteristic
36	#19153750	C₂²	1	1	yes	characteristic
37	#19153750	C₂²	1	1	yes	characteristic
38	#19153749	C₂²	1	1	yes	characteristic
39	#19153748	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	#19153747	C₂²	1	1	yes	characteristic
44	#19153745	C₂²	1	1	yes	characteristic
45	#19153740	C₂²	1	1	yes	characteristic
46	#19153740	C₂²	1	1	yes	characteristic
47	#19153740	C₂²	1	1	yes	characteristic
48	#19153740	C₂²	1	1	yes	characteristic
49	C₂²⋊#13089678	C₂²	1	1	yes	characteristic
50	#19153728	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₈₁	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₁₈₁	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₁₈₁	C₂²	1	1	yes	characteristic
54	C₂²⋊Q₈×C₁₈₁	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×C₁₈₁	C₂²	1	1	yes	characteristic
56	#106×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×C₂²⋊D₁₈₁	D₄	1	1	yes	characteristic
62	C₂²×Dic₁₈₁	—	2	1	no	—
63	C₂×C₄×D₁₈₁	—	2	1	no	—
64	C₂×C₄×D₁₈₁	C₂³	1	1	yes	characteristic
65	#15668432	—	2	2	no	—
66	C₂×#13089678	C₂³	1	1	yes	characteristic
67	C₂×#13089678	C₂³	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
81	C₂²⋊Dic₁₈₁	—	2	1	no	—
82	C₂²⋊Dic₁₈₁	D₄	1	1	yes	characteristic
83	#15668426	—	2	1	no	—
84	#15668426	—	2	1	no	—
85	#15668426	—	2	1	no	—
86	#15668426	D₄	1	1	yes	characteristic
87	C₄×Dic₁₈₁	—	2	1	no	—
88	C₄×Dic₁₈₁	—	2	1	no	—
89	C₄×Dic₁₈₁	—	2	1	no	—
90	C₄×Dic₁₈₁	—	2	1	no	—
91	C₄×Dic₁₈₁	C₂³	1	1	yes	characteristic
92	C₂×Q₈×C₁₈₁	C₂³	1	1	yes	characteristic
93	C₂×Q₈×C₁₈₁	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₁₈₁	C₂³	1	1	yes	characteristic
95	Q₁₆×C₁₈₁	—	2	2	no	—
96	M₁₆×C₁₈₁	—	2	2	no	—
97	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
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	C₂²⋊C₄×C₁₈₁	—	2	1	no	—
104	C₂²⋊C₄×C₁₈₁	—	2	1	no	—
105	C₂²×D₁₈₁	C₂×D₄	1	1	yes	characteristic
106	Dic₃₆₂	—	4	2	no	—
107	Dic₃₆₂	—	8	1	no	—
108	C₂²⋊D₁₈₁	—	2	2	no	—
109	C₂×Dic₁₈₁	—	2	1	no	—
110	C₂×Dic₁₈₁	—	2	1	no	—
111	C₂×Dic₁₈₁	—	2	1	no	—
112	C₂×Dic₁₈₁	—	2	1	no	—
113	C₂×Dic₁₈₁	—	8	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	1	no	—
118	C₄×D₁₈₁	—	2	1	no	—
119	C₄×D₁₈₁	—	8	1	no	—
120	#13089678	—	2	1	no	—
121	#13089678	—	2	1	no	—
122	C₂³×C₁₈₁	C₂×D₄	1	1	yes	characteristic
123	Q₈×C₁₈₁	—	2	1	no	—
124	Q₈×C₁₈₁	—	2	1	no	—
125	Q₈×C₁₈₁	—	2	1	no	—
126	Q₈×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	—	2	1	no	—
131	C₂×C₄×C₁₈₁	—	2	1	no	—
132	C₂×C₄×C₁₈₁	—	2	1	no	—
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	#41	1	2	yes	—
137	C₂²×C₁₈₁	—	2	1	no	—
138	C₂²×C₁₈₁	—	2	1	no	—
139	C₂²×C₁₈₁	—	2	1	no	—
140	C₂²×C₁₈₁	C₂²×D₄	1	1	yes	characteristic
141	C₂×D₁₈₁	—	2	1	no	—
142	C₂×D₁₈₁	—	2	1	no	—
143	C₂×D₁₈₁	—	2	1	no	—
144	C₄×C₁₈₁	—	2	1	no	—
145	C₄×C₁₈₁	—	4	1	no	—
146	C₄×C₁₈₁	—	4	1	no	—
147	C₄×C₁₈₁	—	4	1	no	—
148	C₄×C₁₈₁	—	4	1	no	—
149	C₄×C₁₈₁	C₂×#41	1	1	yes	characteristic
150	C₄×C₁₈₁	2₋¹⁺⁴	1	1	yes	characteristic
151	Dic₁₈₁	—	2	1	no	—
152	Dic₁₈₁	—	2	1	no	—
153	Dic₁₈₁	—	4	1	no	—
154	Dic₁₈₁	—	4	1	no	—
155	Dic₁₈₁	—	4	1	no	—
156	Dic₁₈₁	—	4	1	no	—
157	Dic₁₈₁	#137	1	2	yes	—
158	C₂×C₁₈₁	—	4	1	no	—
159	C₂×C₁₈₁	#547	1	1	yes	characteristic
160	C₂×C₁₈₁	C₂×#137	1	1	yes	characteristic
161	C₂×C₁₈₁	#577	1	1	yes	characteristic
162	D₁₈₁	—	4	1	no	—
163	C₁₈₁	#3356	1	1	yes	characteristic, complemented, Sylow
164	#3356	—	181	1	no	maximal, Sylow
165	#556	—	181	1	no	—
166	#554	—	181	1	no	—
167	#498	—	181	1	no	—
168	#485	—	181	1	no	—
169	#485	—	181	1	no	—
170	#483	—	181	1	no	—
171	#483	—	181	1	no	—
172	#477	—	181	1	no	—
173	#475	—	181	1	no	—
174	#470	—	181	1	no	—
175	#470	D₁₈₁	1	1	yes	characteristic, complemented
176	C₄×Q₁₆	—	181	1	no	—
177	#436	—	181	1	no	—
178	#428	—	181	1	no	—
179	#428	—	181	1	no	—
180	C₂×Q₁₆	C₂×D₁₈₁	1	1	yes	characteristic
181	C₂×M₁₆	—	181	1	no	—
182	C₂×M₁₆	C₂×D₁₈₁	1	1	yes	characteristic
183	C₄⋊Q₈	—	181	1	no	—
184	C₄⋊Q₈	—	181	1	no	—
185	#126	—	362	1	no	—
186	#126	—	362	1	no	—
187	#125	—	362	1	no	—
188	#124	—	362	1	no	—
189	C₂²⋊Q₈	—	362	1	no	—
190	C₂²⋊Q₈	—	362	1	no	—
191	C₂²⋊Q₈	—	181	1	no	—
192	C₂²⋊Q₈	—	181	1	no	—
193	C₂²⋊Q₈	C₂×D₁₈₁	1	1	yes	characteristic
194	C₂²⋊Q₈	C₂×D₁₈₁	1	1	yes	characteristic
195	C₄×Q₈	—	362	1	no	—
196	C₄×Q₈	—	181	1	no	—
197	C₄×Q₈	—	181	1	no	—
198	C₄×D₄	—	362	1	no	—
199	C₄×D₄	—	181	1	no	—
200	#117	—	181	1	no	—
201	#117	—	181	1	no	—
202	C₂×C₄⋊C₄	—	181	1	no	—
203	C₂×C₄⋊C₄	—	181	1	no	—
204	#107	—	181	1	no	—
205	#107	—	181	1	no	—
206	#107	—	181	1	no	—
207	#106	—	181	1	no	—
208	#106	—	181	1	no	—
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	#103	—	181	1	no	—
215	#103	—	181	1	no	—
216	#103	—	181	1	no	—
217	#103	—	181	1	no	—
218	#103	C₂×D₁₈₁	1	1	yes	characteristic
219	#103	C₂×D₁₈₁	1	1	yes	characteristic
220	C₂²⋊C₈	—	181	1	no	—
221	C₂²⋊C₈	—	181	1	no	—
222	C₄×C₈	—	181	1	no	—
223	C₂×Q₈	—	362	1	no	—
224	C₂×Q₈	—	362	1	no	—
225	C₂×Q₈	C₂²×D₁₈₁	1	1	yes	characteristic
226	C₂×Q₈	C₂²×D₁₈₁	1	1	yes	characteristic
227	C₂×D₄	—	181	1	no	—
228	C₂²×C₄	—	362	1	no	—
229	C₂²×C₄	—	362	1	no	—
230	C₂²×C₄	—	181	1	no	—
231	C₂²×C₄	C₂²×D₁₈₁	1	1	yes	characteristic
232	Q₁₆	—	2	2	no	—
233	M₁₆	—	2	2	no	—
234	M₁₆	—	362	2	no	—
235	C₂×C₈	—	181	1	no	—
236	C₂×C₈	—	181	1	no	—
237	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	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	—	362	1	no	—
247	C₄⋊C₄	—	362	1	no	—
248	C₄⋊C₄	—	362	1	no	—
249	C₄⋊C₄	—	181	1	no	—
250	C₄⋊C₄	—	181	1	no	—
251	C₄⋊C₄	—	181	1	no	—
252	C₄⋊C₄	—	181	1	no	—
253	C₄⋊C₄	—	181	1	no	—
254	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
256	C₂²⋊C₄	—	2	1	no	—
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	362	1	no	—
259	C₂²⋊C₄	—	362	1	no	—
260	C₂²⋊C₄	—	362	1	no	—
261	C₂²⋊C₄	—	362	1	no	—
262	C₂²⋊C₄	—	181	1	no	—
263	C₂²⋊C₄	—	181	1	no	—
264	C₄²	—	362	1	no	—
265	C₄²	—	362	1	no	—
266	C₄²	—	362	1	no	—
267	C₄²	—	362	1	no	—
268	C₄²	—	181	1	no	—
269	C₂³	—	181	1	no	—
270	C₂³	D₄×D₁₈₁	1	1	yes	characteristic
271	Q₈	—	2	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	1448	1	no	—
276	Q₈	—	724	2	no	—
277	D₄	—	362	2	no	—
278	C₂×C₄	—	2	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	—	2	1	no	—
284	C₂×C₄	—	362	1	no	—
285	C₂×C₄	—	362	1	no	—
286	C₂×C₄	—	362	1	no	—
287	C₂×C₄	—	362	1	no	—
288	C₂×C₄	—	362	1	no	—
289	C₂×C₄	—	362	1	no	—
290	C₂×C₄	—	1448	1	no	—
291	C₂×C₄	—	1448	1	no	—
292	C₂×C₄	—	181	1	no	—
293	C₂×C₄	—	181	1	no	—
294	C₂×C₄	—	181	1	no	—
295	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
296	C₂×C₄	C₂³×D₁₈₁	1	1	yes	characteristic
297	C₈	—	2	1	no	—
298	C₈	—	362	1	no	—
299	C₈	—	362	1	no	—
300	C₈	#15668448	1	2	yes	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	362	1	no	—
305	C₂²	—	362	1	no	—
306	C₂²	—	362	1	no	—
307	C₂²	C₂×D₄×D₁₈₁	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	362	1	no	—
314	C₄	—	362	1	no	—
315	C₄	—	181	2	no	—
316	C₄	—	724	1	no	—
317	C₄	—	724	1	no	—
318	C₄	—	724	1	no	—
319	C₄	—	724	1	no	—
320	C₄	C₂×#15668448	1	1	yes	characteristic
321	C₄	#19153863	1	1	yes	characteristic
322	C₂	—	4	1	no	—
323	C₂	—	724	1	no	—
324	C₂	#26041929	1	1	yes	characteristic
325	C₂	#137×D₁₈₁	1	1	yes	characteristic
326	C₂	#26042088	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₈₁
commutator, cosocle	C₂×C₄×C₁₈₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₈₁
Fitting	#470×C₁₈₁	C₂
socle	C₂²×C₁₈₁	C₂²×D₄
prime radical	C₂×C₂²⋊D₁₈₁	D₄
hypercenter	#470	D₁₈₁
hypocenter	C₁₈₁	#3356
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #470

Complements

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

N	H ≅ G/N	type
3. #26041676	C₂	semidirect
4. #26041619	C₂	semidirect
5. #26041617	C₂	semidirect
6. #26041608	C₂	semidirect
7. #26041607	C₂	semidirect
8. #26041572	C₂	semidirect
9. #26041562	C₂	semidirect
10. #26041534	C₂	semidirect
11. Q₁₆×Dic₁₈₁	C₂	semidirect
13. #26041321	C₂	semidirect
14. #26041318	C₂	semidirect
15. #470×C₁₈₁	C₂	semidirect
163. C₁₈₁	#3356	semidirect
175. #470	D₁₈₁	semidirect

Maximal subgroups

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

subgroup	index
#26042000	2
#26041987	2
#26041676	2
#26041619	2
#26041617	2
#26041608	2
#26041607	2
#26041572	2
#26041562	2
#26041534	2
Q₁₆×Dic₁₈₁	2
#26041355	2
#26041321	2
#26041318	2
#470×C₁₈₁	2
#3356	181