Subgroups — #37543952

Order 23168.

370 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26041900	C₂	1	1	yes	maximal, characteristic
2	#26041756	C₂	1	1	yes	maximal, characteristic, complemented
3	#26041621	C₂	1	1	yes	maximal, characteristic, complemented
4	#26041604	C₂	1	1	yes	maximal, characteristic
5	#26041575	C₂	1	1	yes	maximal, characteristic, complemented
6	#26041565	C₂	1	1	yes	maximal, characteristic, complemented
7	#26041502	C₂	1	1	yes	maximal, characteristic, complemented
8	#26041502	C₂	1	1	yes	maximal, characteristic, complemented
9	#26041437	C₂	1	1	yes	maximal, characteristic, complemented
10	#26041420	C₂	1	1	yes	maximal, characteristic
11	#26041387	C₂	1	1	yes	maximal, characteristic, complemented
12	#26041381	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×Dic₇₂₄	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#15668438	C₂	1	1	yes	maximal, characteristic, complemented
15	#433×C₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Dic₃₆₂	C₂²	1	1	yes	characteristic
17	C₂×Dic₇₂₄	C₂²	1	1	yes	characteristic
18	C₂×Dic₇₂₄	C₂²	1	1	yes	characteristic
19	C₂×#15668438	C₂²	1	1	yes	characteristic
20	C₂×#15668438	C₂²	1	1	yes	characteristic
21	#19153816	C₂²	1	1	yes	characteristic
22	#19153811	—	2	1	no	—
23	#19153804	C₂²	1	1	yes	characteristic
24	#19153802	—	2	1	no	—
25	#19153798	—	2	1	no	—
26	#19153795	C₂²	1	1	yes	characteristic
27	#19153794	—	2	1	no	—
28	#19153790	C₂²	1	1	yes	characteristic
29	C₄×Dic₃₆₂	C₂²	1	1	yes	characteristic
30	C₄×Dic₃₆₂	C₂²	1	1	yes	characteristic
31	C₄×Dic₃₆₂	C₂²	1	1	yes	characteristic
32	C₄×Dic₃₆₂	C₂²	1	1	yes	characteristic
33	C₄×Dic₃₆₂	C₂²	1	1	yes	characteristic
34	#19153781	—	2	1	no	—
35	#19153781	—	2	1	no	—
36	C₄×D₇₂₄	C₂²	1	1	yes	characteristic
37	#19153775	C₂²	1	1	yes	characteristic
38	#19153772	—	2	1	no	—
39	#19153772	—	2	1	no	—
40	#19153750	C₂²	1	1	yes	characteristic
41	#19153750	C₂²	1	1	yes	characteristic
42	#19153748	C₂²	1	1	yes	characteristic
43	#19153748	C₂²	1	1	yes	characteristic
44	#19153741	C₂²	1	1	yes	characteristic
45	#19153741	C₂²	1	1	yes	characteristic
46	#19153741	C₂²	1	1	yes	characteristic
47	#19153741	C₂²	1	1	yes	characteristic
48	#19153741	C₂²	1	1	yes	characteristic
49	#19153741	C₂²	1	1	yes	characteristic
50	#19153738	C₂²	1	1	yes	characteristic
51	#19153738	C₂²	1	1	yes	characteristic
52	#117×C₁₈₁	C₂²	1	1	yes	characteristic
53	C₄⋊C₈×C₁₈₁	C₂²	1	1	yes	characteristic
54	C₄⋊C₈×C₁₈₁	C₂²	1	1	yes	characteristic
55	C₂²⋊C₈×C₁₈₁	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₁₈₁	C₂²	1	1	yes	characteristic
57	#97×C₁₈₁	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₈₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₃₆₂	—	2	2	no	—
60	C₂×Dic₃₆₂	—	2	2	no	—
61	C₂×Dic₃₆₂	—	2	1	no	—
62	C₂×Dic₃₆₂	—	2	1	no	—
63	C₂×Dic₃₆₂	C₂³	1	1	yes	characteristic
64	C₂×Dic₃₆₂	C₂³	1	1	yes	characteristic
65	C₂×Dic₃₆₂	C₂³	1	1	yes	characteristic
66	C₂×D₇₂₄	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₁₈₁	—	2	1	no	—
68	C₂²×Dic₁₈₁	—	2	1	no	—
69	C₂×C₄×D₁₈₁	—	2	1	no	—
70	Dic₇₂₄	—	2	2	no	—
71	Dic₇₂₄	—	2	2	no	—
72	#15668438	—	2	2	no	—
73	#15668438	—	2	2	no	—
74	C₄⋊Dic₁₈₁	—	2	1	no	—
75	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
79	#15668428	—	2	1	no	—
80	#15668428	—	2	1	no	—
81	#15668428	—	2	1	no	—
82	#15668428	—	2	1	no	—
83	#15668428	—	2	1	no	—
84	#15668428	—	2	1	no	—
85	C₂²⋊Dic₁₈₁	—	2	1	no	—
86	C₂²⋊Dic₁₈₁	—	2	1	no	—
87	#15668426	—	2	1	no	—
88	#15668426	—	2	1	no	—
89	#15668426	—	2	1	no	—
90	#15668426	—	2	1	no	—
91	C₄×Dic₁₈₁	—	2	1	no	—
92	C₄×Dic₁₈₁	—	2	1	no	—
93	C₄×Dic₁₈₁	—	2	1	no	—
94	C₄×Dic₁₈₁	—	2	1	no	—
95	C₄×Dic₁₈₁	—	2	1	no	—
96	C₄×Dic₁₈₁	—	2	1	no	—
97	C₂²×C₄×C₁₈₁	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₈₁	D₄	1	1	yes	characteristic
103	C₄⋊C₄×C₁₈₁	D₄	1	1	yes	characteristic
104	C₂²⋊C₄×C₁₈₁	D₄	1	1	yes	characteristic
105	C₂²⋊C₄×C₁₈₁	D₄	1	1	yes	characteristic
106	C₄²×C₁₈₁	C₂³	1	1	yes	characteristic
107	C₄²×C₁₈₁	C₂³	1	1	yes	characteristic
108	C₂²×D₁₈₁	—	2	1	no	—
109	Dic₃₆₂	—	2	4	no	—
110	Dic₃₆₂	—	2	1	no	—
111	Dic₃₆₂	—	2	1	no	—
112	Dic₃₆₂	—	2	1	no	—
113	Dic₃₆₂	—	2	1	no	—
114	Dic₃₆₂	—	8	1	no	—
115	Dic₃₆₂	#41	1	2	yes	—
116	Dic₃₆₂	#41	1	2	yes	—
117	D₇₂₄	—	2	1	no	—
118	D₇₂₄	—	2	1	no	—
119	C₂²⋊D₁₈₁	—	8	1	no	—
120	C₂×Dic₁₈₁	—	2	4	no	—
121	C₂×Dic₁₈₁	—	2	1	no	—
122	C₂×Dic₁₈₁	—	2	1	no	—
123	C₂×Dic₁₈₁	—	2	1	no	—
124	C₂×Dic₁₈₁	—	2	1	no	—
125	C₂×Dic₁₈₁	—	2	1	no	—
126	C₂×Dic₁₈₁	—	2	1	no	—
127	C₂×Dic₁₈₁	—	2	1	no	—
128	C₄×D₁₈₁	—	8	1	no	—
129	C₂³×C₁₈₁	C₂×D₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₈₁	C₂⁴	1	1	yes	characteristic
138	C₈×C₁₈₁	—	2	1	no	—
139	C₈×C₁₈₁	—	2	1	no	—
140	C₈×C₁₈₁	—	2	1	no	—
141	C₈×C₁₈₁	—	2	1	no	—
142	C₂²×C₁₈₁	—	2	1	no	—
143	C₂²×C₁₈₁	—	2	1	no	—
144	C₂²×C₁₈₁	—	2	1	no	—
145	C₂²×C₁₈₁	C₂²×D₄	1	1	yes	characteristic
146	C₂×D₁₈₁	—	4	1	no	—
147	C₂×D₁₈₁	—	4	1	no	—
148	C₂×D₁₈₁	—	4	1	no	—
149	C₄×C₁₈₁	—	2	1	no	—
150	C₄×C₁₈₁	—	2	1	no	—
151	C₄×C₁₈₁	—	2	1	no	—
152	C₄×C₁₈₁	—	2	1	no	—
153	C₄×C₁₈₁	—	2	1	no	—
154	C₄×C₁₈₁	C₂×#41	1	1	yes	characteristic
155	C₄×C₁₈₁	C₂×#41	1	1	yes	characteristic
156	Dic₁₈₁	—	2	2	no	—
157	Dic₁₈₁	—	2	2	no	—
158	Dic₁₈₁	—	4	1	no	—
159	Dic₁₈₁	—	4	1	no	—
160	Dic₁₈₁	—	4	1	no	—
161	Dic₁₈₁	—	4	1	no	—
162	Dic₁₈₁	—	4	1	no	—
163	C₂×C₁₈₁	—	4	1	no	—
164	C₂×C₁₈₁	#525	1	1	yes	characteristic
165	C₂×C₁₈₁	#577	1	1	yes	characteristic
166	C₂×C₁₈₁	#578	1	1	yes	characteristic
167	D₁₈₁	—	8	1	no	—
168	C₁₈₁	#3121	1	1	yes	characteristic, complemented, Sylow
169	#3121	—	181	1	no	maximal, Sylow
170	#542	—	181	1	no	—
171	#519	—	181	1	no	—
172	#485	—	181	1	no	—
173	#482	—	181	1	no	—
174	#477	—	181	1	no	—
175	#475	—	181	1	no	—
176	#464	—	181	1	no	—
177	#464	—	181	1	no	—
178	#451	—	181	1	no	—
179	#448	—	181	1	no	—
180	#441	—	181	1	no	—
181	#440	—	181	1	no	—
182	C₄×Q₁₆	—	181	1	no	—
183	C₄×SD₁₆	—	181	1	no	—
184	#433	D₁₈₁	1	1	yes	characteristic, complemented
185	C₂²×Q₈	—	181	1	no	—
186	C₂×Q₁₆	—	181	1	no	—
187	C₂×Q₁₆	—	181	1	no	—
188	C₂×SD₁₆	—	181	1	no	—
189	C₂×SD₁₆	—	181	1	no	—
190	C₄⋊Q₈	—	181	1	no	—
191	#126	—	362	1	no	—
192	#124	—	362	1	no	—
193	#124	—	181	1	no	—
194	#123	—	362	1	no	—
195	C₂²⋊Q₈	—	362	1	no	—
196	C₂²⋊Q₈	—	181	1	no	—
197	#121	—	181	1	no	—
198	C₄×Q₈	—	362	1	no	—
199	C₄×Q₈	—	362	1	no	—
200	C₄×Q₈	—	181	1	no	—
201	C₄×Q₈	—	181	1	no	—
202	C₄×Q₈	—	181	1	no	—
203	C₄×Q₈	—	181	1	no	—
204	C₄×Q₈	—	181	1	no	—
205	C₄×D₄	—	181	1	no	—
206	#117	—	362	1	no	—
207	#117	—	362	1	no	—
208	#117	—	181	1	no	—
209	#117	C₂×D₁₈₁	1	1	yes	characteristic
210	#107	—	181	1	no	—
211	#107	—	181	1	no	—
212	#106	—	181	1	no	—
213	#106	—	181	1	no	—
214	C₄⋊C₈	C₂×D₁₈₁	1	1	yes	characteristic
215	C₄⋊C₈	C₂×D₁₈₁	1	1	yes	characteristic
216	#103	—	181	1	no	—
217	#103	—	181	1	no	—
218	#103	—	181	1	no	—
219	#103	—	181	1	no	—
220	#103	—	181	1	no	—
221	#103	—	181	1	no	—
222	#102	—	181	1	no	—
223	#102	—	181	1	no	—
224	C₂²⋊C₈	C₂×D₁₈₁	1	1	yes	characteristic
225	C₂²⋊C₈	C₂×D₁₈₁	1	1	yes	characteristic
226	#97	C₂×D₁₈₁	1	1	yes	characteristic
227	C₄×C₈	C₂×D₁₈₁	1	1	yes	characteristic
228	C₂×Q₈	—	362	2	no	—
229	C₂×Q₈	—	362	2	no	—
230	C₂×Q₈	—	362	1	no	—
231	C₂×Q₈	—	362	1	no	—
232	C₂×Q₈	—	181	1	no	—
233	C₂×Q₈	—	181	1	no	—
234	C₂×Q₈	—	181	1	no	—
235	C₂×D₄	—	362	1	no	—
236	C₂×D₄	—	181	1	no	—
237	C₂²×C₄	—	362	1	no	—
238	C₂²×C₄	—	362	1	no	—
239	C₂²×C₄	C₂²×D₁₈₁	1	1	yes	characteristic
240	Q₁₆	—	362	2	no	—
241	Q₁₆	—	362	2	no	—
242	SD₁₆	—	362	2	no	—
243	SD₁₆	—	362	2	no	—
244	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
245	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
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	—	181	1	no	—
256	C₄⋊C₄	—	181	1	no	—
257	C₄⋊C₄	—	181	1	no	—
258	C₄⋊C₄	—	181	1	no	—
259	C₄⋊C₄	D₇₂₄	1	1	yes	characteristic
260	C₄⋊C₄	D₇₂₄	1	1	yes	characteristic
261	C₂²⋊C₄	—	362	1	no	—
262	C₂²⋊C₄	—	362	1	no	—
263	C₂²⋊C₄	—	362	1	no	—
264	C₂²⋊C₄	—	362	1	no	—
265	C₂²⋊C₄	—	362	1	no	—
266	C₂²⋊C₄	—	362	1	no	—
267	C₂²⋊C₄	D₇₂₄	1	1	yes	characteristic
268	C₂²⋊C₄	D₇₂₄	1	1	yes	characteristic
269	C₄²	—	362	1	no	—
270	C₄²	—	362	1	no	—
271	C₄²	—	362	1	no	—
272	C₄²	—	362	1	no	—
273	C₄²	—	362	1	no	—
274	C₄²	—	362	1	no	—
275	C₄²	C₂²×D₁₈₁	1	1	yes	characteristic
276	C₄²	C₂²×D₁₈₁	1	1	yes	characteristic
277	C₂³	—	362	1	no	—
278	C₂³	C₂×D₇₂₄	1	1	yes	characteristic
279	Q₈	—	362	4	no	—
280	Q₈	—	362	1	no	—
281	Q₈	—	362	1	no	—
282	Q₈	—	362	1	no	—
283	Q₈	—	362	1	no	—
284	Q₈	—	1448	1	no	—
285	Q₈	—	181	2	no	—
286	Q₈	—	181	2	no	—
287	D₄	—	362	1	no	—
288	D₄	—	362	1	no	—
289	D₄	—	1448	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	362	4	no	—
293	C₂×C₄	—	362	1	no	—
294	C₂×C₄	—	362	1	no	—
295	C₂×C₄	—	362	1	no	—
296	C₂×C₄	—	362	1	no	—
297	C₂×C₄	—	362	1	no	—
298	C₂×C₄	—	362	1	no	—
299	C₂×C₄	—	362	1	no	—
300	C₂×C₄	—	1448	1	no	—
301	C₂×C₄	C₂×D₇₂₄	1	1	yes	characteristic
302	C₂×C₄	C₂×D₇₂₄	1	1	yes	characteristic
303	C₂×C₄	C₂×D₇₂₄	1	1	yes	characteristic
304	C₂×C₄	C₂×D₇₂₄	1	1	yes	characteristic
305	C₂×C₄	C₂×D₇₂₄	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₁₈₁	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	724	1	no	—
315	C₂²	—	724	1	no	—
316	C₂²	—	724	1	no	—
317	C₂²	C₂²×D₇₂₄	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	362	2	no	—
324	C₄	—	362	2	no	—
325	C₄	—	724	1	no	—
326	C₄	—	724	1	no	—
327	C₄	—	724	1	no	—
328	C₄	—	724	1	no	—
329	C₄	—	724	1	no	—
330	C₄	#41×D₁₈₁	1	1	yes	characteristic
331	C₄	#41×D₁₈₁	1	1	yes	characteristic
332	C₂	—	4	1	no	—
333	C₂	—	1448	1	no	—
334	C₂	#26041791	1	1	yes	characteristic
335	C₂	#26042091	1	1	yes	characteristic
336	C₂	#26042096	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₂₄
commutator, cosocle	C₂×C₄×C₁₈₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₈₁
Fitting	#433×C₁₈₁	C₂
socle	C₂²×C₁₈₁	C₂²×D₄
prime radical	#19153790	C₂²
hypercenter	#433	D₁₈₁
hypocenter	C₁₈₁	#3121
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #433

Complements

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

N	H ≅ G/N	type
2. #26041756	C₂	semidirect
3. #26041621	C₂	semidirect
5. #26041575	C₂	semidirect
6. #26041565	C₂	semidirect
7. #26041502	C₂	semidirect
8. #26041502	C₂	semidirect
9. #26041437	C₂	semidirect
11. #26041387	C₂	semidirect
12. #26041381	C₂	semidirect
13. C₄×Dic₇₂₄	C₂	semidirect
14. C₄×#15668438	C₂	semidirect
15. #433×C₁₈₁	C₂	semidirect
168. C₁₈₁	#3121	semidirect
184. #433	D₁₈₁	semidirect

Maximal subgroups

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

subgroup	index
#26041900	2
#26041756	2
#26041621	2
#26041604	2
#26041575	2
#26041565	2
#26041502	2
#26041437	2
#26041420	2
#26041387	2
#26041381	2
C₄×Dic₇₂₄	2
C₄×#15668438	2
#433×C₁₈₁	2
#3121	181