Subgroups — #37546420

Order 23168.

348 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26042012	C₂	1	1	yes	maximal, characteristic
2	#26042006	C₂	1	1	yes	maximal, characteristic, complemented
3	#26041636	C₂	1	1	yes	maximal, characteristic, complemented
4	#26041632	C₂	1	1	yes	maximal, characteristic, complemented
5	#26041624	C₂	1	1	yes	maximal, characteristic
6	#26041582	C₂	1	1	yes	maximal, characteristic, complemented
7	#26041563	C₂	1	1	yes	maximal, characteristic, complemented
8	#26041502	C₂	1	1	yes	maximal, characteristic, complemented
9	#26041500	C₂	1	1	yes	maximal, characteristic, complemented
10	#26041478	C₂	1	1	yes	maximal, characteristic
11	#26041403	C₂	1	1	yes	maximal, characteristic, complemented
12	#26041373	C₂	1	1	yes	maximal, characteristic, complemented
13	#26041366	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#15668437	C₂	1	1	yes	maximal, characteristic, complemented
15	#478×C₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Dic₇₂₄	C₂²	1	1	yes	characteristic
17	C₂×#15668438	C₂²	1	1	yes	characteristic
18	C₂×#15668437	C₂²	1	1	yes	characteristic
19	#19153815	—	2	1	no	—
20	#19153815	C₂²	1	1	yes	characteristic
21	#19153812	C₂²	1	1	yes	characteristic
22	#19153807	C₂²	1	1	yes	characteristic
23	#19153806	—	2	1	no	—
24	#19153806	—	2	1	no	—
25	#19153804	C₂²	1	1	yes	characteristic
26	#19153803	C₂²	1	1	yes	characteristic
27	#19153802	—	2	1	no	—
28	#19153799	—	2	1	no	—
29	#19153798	—	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	C₄×C₂²⋊D₁₈₁	—	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	#19153739	C₂²	1	1	yes	characteristic
43	#19153739	C₂²	1	1	yes	characteristic
44	#19153739	C₂²	1	1	yes	characteristic
45	#19153738	C₂²	1	1	yes	characteristic
46	#19153737	C₂²	1	1	yes	characteristic
47	#19153736	C₂²	1	1	yes	characteristic
48	#19153736	C₂²	1	1	yes	characteristic
49	C₄×#13089678	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₈₁	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₈₁	C₂²	1	1	yes	characteristic
52	#125×C₁₈₁	C₂²	1	1	yes	characteristic
53	C₄×D₄×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	#102×C₁₈₁	C₂²	1	1	yes	characteristic
58	#102×C₁₈₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₃₆₂	—	2	1	no	—
60	C₂×Dic₃₆₂	D₄	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×C₂²⋊D₁₈₁	—	2	1	no	—
65	C₂×C₂²⋊D₁₈₁	—	2	1	no	—
66	C₂×C₄×D₁₈₁	—	2	1	no	—
67	Dic₇₂₄	—	2	2	no	—
68	#15668438	—	2	2	no	—
69	#15668437	—	2	2	no	—
70	C₂×#13089678	C₂³	1	1	yes	characteristic
71	C₂×#13089678	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₁₈₁	—	2	1	no	—
73	C₄⋊Dic₁₈₁	D₄	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	D₄	1	1	yes	characteristic
82	#15668428	D₄	1	1	yes	characteristic
83	C₂²⋊Dic₁₈₁	—	2	1	no	—
84	C₂²⋊Dic₁₈₁	—	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×D₄×C₁₈₁	C₂³	1	1	yes	characteristic
95	C₂²×C₄×C₁₈₁	—	2	1	no	—
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	C₂²⋊C₄×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	—	4	2	no	—
111	Dic₃₆₂	#41	1	2	yes	—
112	D₇₂₄	—	2	1	no	—
113	D₇₂₄	—	2	1	no	—
114	C₂²⋊D₁₈₁	—	4	2	no	—
115	C₂²⋊D₁₈₁	—	8	1	no	—
116	C₂×Dic₁₈₁	—	2	1	no	—
117	C₂×Dic₁₈₁	—	2	1	no	—
118	C₂×Dic₁₈₁	—	2	1	no	—
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	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
123	C₄×D₁₈₁	—	8	1	no	—
124	#13089678	—	2	1	no	—
125	#13089678	—	2	1	no	—
126	C₂³×C₁₈₁	—	2	1	no	—
127	D₄×C₁₈₁	—	2	1	no	—
128	D₄×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	—	8	1	no	—
133	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₈₁	C₂⁴	1	1	yes	characteristic
136	C₈×C₁₈₁	—	2	1	no	—
137	C₈×C₁₈₁	—	2	1	no	—
138	C₂²×C₁₈₁	—	4	1	no	—
139	C₂²×C₁₈₁	—	4	1	no	—
140	C₂²×C₁₈₁	—	4	1	no	—
141	C₂²×C₁₈₁	C₂²×D₄	1	1	yes	characteristic
142	C₂×D₁₈₁	—	4	1	no	—
143	C₂×D₁₈₁	—	4	1	no	—
144	C₂×D₁₈₁	—	4	1	no	—
145	C₄×C₁₈₁	—	2	1	no	—
146	C₄×C₁₈₁	—	2	1	no	—
147	C₄×C₁₈₁	—	4	1	no	—
148	C₄×C₁₈₁	—	4	1	no	—
149	C₄×C₁₈₁	—	4	1	no	—
150	C₄×C₁₈₁	C₂×#41	1	1	yes	characteristic
151	C₄×C₁₈₁	2₋¹⁺⁴	1	1	yes	characteristic
152	Dic₁₈₁	—	2	2	no	—
153	Dic₁₈₁	—	2	1	no	—
154	Dic₁₈₁	—	2	1	no	—
155	Dic₁₈₁	—	2	1	no	—
156	Dic₁₈₁	—	4	1	no	—
157	Dic₁₈₁	—	4	1	no	—
158	Dic₁₈₁	#135	1	2	yes	—
159	C₂×C₁₈₁	—	8	1	no	—
160	C₂×C₁₈₁	#548	1	1	yes	characteristic
161	C₂×C₁₈₁	C₂×#135	1	1	yes	characteristic
162	C₂×C₁₈₁	#577	1	1	yes	characteristic
163	D₁₈₁	—	8	1	no	—
164	C₁₈₁	#3365	1	1	yes	characteristic, complemented, Sylow
165	#3365	—	181	1	no	maximal, Sylow
166	#559	—	181	1	no	—
167	#557	—	181	1	no	—
168	#488	—	181	1	no	—
169	#488	—	181	1	no	—
170	#486	—	181	1	no	—
171	#478	—	181	1	no	—
172	#478	D₁₈₁	1	1	yes	characteristic, complemented
173	#475	—	181	1	no	—
174	#464	—	181	1	no	—
175	#464	—	181	1	no	—
176	#460	—	181	1	no	—
177	C₈×Q₈	—	181	1	no	—
178	C₄×Q₁₆	—	181	1	no	—
179	C₄×SD₁₆	—	181	1	no	—
180	C₄×SD₁₆	—	181	1	no	—
181	C₂×Q₁₆	—	181	1	no	—
182	C₂×SD₁₆	—	181	1	no	—
183	C₂×SD₁₆	—	181	1	no	—
184	C₄⋊Q₈	—	362	1	no	—
185	C₄⋊Q₈	—	181	1	no	—
186	#127	—	181	1	no	—
187	#125	—	362	1	no	—
188	#125	—	362	1	no	—
189	#125	—	181	1	no	—
190	#125	C₂×D₁₈₁	1	1	yes	characteristic
191	#124	—	362	1	no	—
192	#124	—	181	1	no	—
193	#124	—	181	1	no	—
194	#123	—	362	1	no	—
195	#123	—	362	1	no	—
196	C₄×Q₈	—	362	1	no	—
197	C₄×Q₈	—	181	1	no	—
198	C₄×Q₈	—	181	1	no	—
199	C₄×Q₈	—	181	1	no	—
200	C₄×Q₈	—	181	1	no	—
201	C₄×Q₈	—	181	1	no	—
202	C₄×D₄	—	362	1	no	—
203	C₄×D₄	—	181	1	no	—
204	C₄×D₄	C₂×D₁₈₁	1	1	yes	characteristic
205	#107	C₂×D₁₈₁	1	1	yes	characteristic
206	#106	—	181	1	no	—
207	#106	C₂×D₁₈₁	1	1	yes	characteristic
208	C₄⋊C₈	—	181	1	no	—
209	C₄⋊C₈	—	181	1	no	—
210	C₄⋊C₈	C₂×D₁₈₁	1	1	yes	characteristic
211	#103	—	181	1	no	—
212	#103	—	181	1	no	—
213	#103	—	181	1	no	—
214	#103	—	181	1	no	—
215	#102	—	181	1	no	—
216	#102	—	181	1	no	—
217	#102	—	181	1	no	—
218	#102	—	181	1	no	—
219	#102	C₂×D₁₈₁	1	1	yes	characteristic
220	#102	C₂×D₁₈₁	1	1	yes	characteristic
221	C₄×C₈	—	181	1	no	—
222	C₄×C₈	—	181	1	no	—
223	C₄×C₈	—	181	1	no	—
224	C₂×Q₈	—	362	1	no	—
225	C₂×Q₈	—	181	1	no	—
226	C₂×Q₈	—	181	1	no	—
227	C₂×Q₈	—	181	1	no	—
228	C₂×D₄	—	362	1	no	—
229	C₂×D₄	—	362	1	no	—
230	C₂×D₄	—	181	1	no	—
231	C₂×D₄	C₂²×D₁₈₁	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	362	1	no	—
234	Q₁₆	—	362	2	no	—
235	SD₁₆	—	362	2	no	—
236	SD₁₆	—	362	2	no	—
237	C₂×C₈	—	181	1	no	—
238	C₂×C₈	—	181	1	no	—
239	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	2	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	—	362	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	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
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	—	362	1	no	—
263	C₄²	—	362	1	no	—
264	C₄²	—	362	1	no	—
265	C₄²	—	362	1	no	—
266	C₄²	—	362	1	no	—
267	C₄²	—	181	1	no	—
268	C₄²	—	181	1	no	—
269	C₄²	C₂²×D₁₈₁	1	1	yes	characteristic
270	C₂³	—	2	1	no	—
271	C₂³	—	362	1	no	—
272	Q₈	—	362	2	no	—
273	Q₈	—	362	1	no	—
274	Q₈	—	362	1	no	—
275	Q₈	—	362	1	no	—
276	Q₈	—	181	2	no	—
277	Q₈	—	724	2	no	—
278	D₄	—	2	1	no	—
279	D₄	—	2	1	no	—
280	D₄	—	362	1	no	—
281	D₄	—	362	1	no	—
282	D₄	—	1448	1	no	—
283	D₄	—	724	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	8	1	no	—
288	C₂×C₄	—	362	1	no	—
289	C₂×C₄	—	362	1	no	—
290	C₂×C₄	—	362	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	—	181	1	no	—
296	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
297	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₁₈₁	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	362	1	no	—
302	C₈	—	362	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	724	1	no	—
307	C₂²	—	724	1	no	—
308	C₂²	—	724	1	no	—
309	C₂²	C₂×D₄×D₁₈₁	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	362	2	no	—
316	C₄	—	362	1	no	—
317	C₄	—	362	1	no	—
318	C₄	—	362	1	no	—
319	C₄	—	181	2	no	—
320	C₄	—	724	1	no	—
321	C₄	—	724	1	no	—
322	C₄	#41×D₁₈₁	1	1	yes	characteristic
323	C₄	#19153864	1	1	yes	characteristic
324	C₂	—	8	1	no	—
325	C₂	—	1448	1	no	—
326	C₂	#26041939	1	1	yes	characteristic
327	C₂	#135×D₁₈₁	1	1	yes	characteristic
328	C₂	#26042089	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₈₁
commutator, cosocle	C₂×C₄×C₁₈₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₈₁
Fitting	#478×C₁₈₁	C₂
socle	C₂²×C₁₈₁	C₂²×D₄
prime radical	#19153812	C₂²
hypercenter	#478	D₁₈₁
hypocenter	C₁₈₁	#3365
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
2. #26042006	C₂	semidirect
3. #26041636	C₂	semidirect
4. #26041632	C₂	semidirect
6. #26041582	C₂	semidirect
7. #26041563	C₂	semidirect
8. #26041502	C₂	semidirect
9. #26041500	C₂	semidirect
11. #26041403	C₂	semidirect
12. #26041373	C₂	semidirect
13. #26041366	C₂	semidirect
14. C₄×#15668437	C₂	semidirect
15. #478×C₁₈₁	C₂	semidirect
164. C₁₈₁	#3365	semidirect
172. #478	D₁₈₁	semidirect

Maximal subgroups

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

subgroup	index
#26042012	2
#26042006	2
#26041636	2
#26041632	2
#26041624	2
#26041582	2
#26041563	2
#26041502	2
#26041500	2
#26041478	2
#26041403	2
#26041373	2
#26041366	2
C₄×#15668437	2
#478×C₁₈₁	2
#3365	181