Subgroups — #37544856

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	#26041835	C₂	1	1	yes	maximal, characteristic, complemented
2	#26041826	C₂	1	1	yes	maximal, characteristic
3	#26041683	C₂	1	1	yes	maximal, characteristic, complemented
4	#26041676	C₂	1	1	yes	maximal, characteristic, complemented
5	#26041624	C₂	1	1	yes	maximal, characteristic, complemented
6	#26041624	C₂	1	1	yes	maximal, characteristic, complemented
7	#26041606	C₂	1	1	yes	maximal, characteristic, complemented
8	#26041605	C₂	1	1	yes	maximal, characteristic, complemented
9	#26041602	C₂	1	1	yes	maximal, characteristic, complemented
10	#26041601	C₂	1	1	yes	maximal, characteristic, complemented
11	#26041580	C₂	1	1	yes	maximal, characteristic, complemented
12	#26041579	C₂	1	1	yes	maximal, characteristic, complemented
13	#26041576	C₂	1	1	yes	maximal, characteristic, complemented
14	#26041331	C₂	1	1	yes	maximal, characteristic, complemented
15	#478×C₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#15668449	—	2	1	no	—
17	#19153815	C₂²	1	1	yes	characteristic
18	#19153815	C₂²	1	1	yes	characteristic
19	#19153815	C₂²	1	1	yes	characteristic
20	#19153812	C₂²	1	1	yes	characteristic
21	#19153808	C₂²	1	1	yes	characteristic
22	#19153802	—	2	1	no	—
23	#19153794	—	2	1	no	—
24	#19153794	—	2	1	no	—
25	#19153791	C₂²	1	1	yes	characteristic
26	#19153789	—	2	1	no	—
27	#19153787	C₂²	1	1	yes	characteristic
28	#19153781	—	2	1	no	—
29	#19153781	—	2	1	no	—
30	C₄×D₇₂₄	C₂²	1	1	yes	characteristic
31	C₄×C₂²⋊D₁₈₁	—	2	1	no	—
32	#19153774	C₂²	1	1	yes	characteristic
33	#19153774	C₂²	1	1	yes	characteristic
34	C₄²×D₁₈₁	C₂²	1	1	yes	characteristic
35	#19153750	C₂²	1	1	yes	characteristic
36	#19153749	C₂²	1	1	yes	characteristic
37	#19153749	C₂²	1	1	yes	characteristic
38	#19153748	C₂²	1	1	yes	characteristic
39	#19153747	C₂²	1	1	yes	characteristic
40	#19153747	C₂²	1	1	yes	characteristic
41	C₄⋊#13089678	C₂²	1	1	yes	characteristic
42	#19153738	C₂²	1	1	yes	characteristic
43	#19153738	C₂²	1	1	yes	characteristic
44	#19153737	C₂²	1	1	yes	characteristic
45	#19153737	C₂²	1	1	yes	characteristic
46	#19153736	C₂²	1	1	yes	characteristic
47	#19153736	C₂²	1	1	yes	characteristic
48	#19153728	C₂²	1	1	yes	characteristic
49	#19153728	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	#15668449	—	4	4	no	—
60	C₂×Dic₃₆₂	—	2	1	no	—
61	C₂×Dic₃₆₂	—	2	1	no	—
62	C₂×Dic₃₆₂	—	2	1	no	—
63	C₂×D₇₂₄	—	2	1	no	—
64	C₂×D₇₂₄	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊D₁₈₁	—	2	1	no	—
66	C₂×C₂²⋊D₁₈₁	—	2	1	no	—
67	C₂×C₂²⋊D₁₈₁	—	2	1	no	—
68	C₂×C₄×D₁₈₁	—	2	1	no	—
69	C₂×C₄×D₁₈₁	D₄	1	1	yes	characteristic
70	C₂×C₄×D₁₈₁	D₄	1	1	yes	characteristic
71	C₂×C₄×D₁₈₁	C₂³	1	1	yes	characteristic
72	C₂×#13089678	C₂³	1	1	yes	characteristic
73	C₂×#13089678	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
77	#15668428	—	2	1	no	—
78	#15668428	—	2	1	no	—
79	#15668428	—	2	1	no	—
80	#15668428	—	2	1	no	—
81	#15668428	—	2	1	no	—
82	C₂²⋊Dic₁₈₁	—	2	1	no	—
83	#15668426	—	2	1	no	—
84	#15668426	—	2	1	no	—
85	#15668426	—	2	1	no	—
86	#15668426	—	2	1	no	—
87	C₄×Dic₁₈₁	—	2	1	no	—
88	C₄×Dic₁₈₁	—	2	1	no	—
89	C₄×Dic₁₈₁	D₄	1	1	yes	characteristic
90	C₄×Dic₁₈₁	D₄	1	1	yes	characteristic
91	C₄×Dic₁₈₁	C₂³	1	1	yes	characteristic
92	C₂×D₄×C₁₈₁	C₂³	1	1	yes	characteristic
93	C₂²×C₄×C₁₈₁	—	2	1	no	—
94	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
95	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
96	C₄⋊C₄×C₁₈₁	—	2	1	no	—
97	C₄⋊C₄×C₁₈₁	—	2	1	no	—
98	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
101	C₂²⋊C₄×C₁₈₁	—	2	1	no	—
102	C₄²×C₁₈₁	C₂³	1	1	yes	characteristic
103	C₂²×D₁₈₁	—	2	1	no	—
104	C₂²×D₁₈₁	C₂×D₄	1	1	yes	characteristic
105	Dic₃₆₂	—	4	2	no	—
106	Dic₃₆₂	—	4	2	no	—
107	Dic₃₆₂	—	4	2	no	—
108	D₇₂₄	—	2	1	no	—
109	D₇₂₄	—	2	1	no	—
110	D₇₂₄	—	4	2	no	—
111	C₂²⋊D₁₈₁	—	4	2	no	—
112	C₂²⋊D₁₈₁	—	4	2	no	—
113	C₂²⋊D₁₈₁	—	4	2	no	—
114	C₂×Dic₁₈₁	—	2	1	no	—
115	C₂×Dic₁₈₁	—	2	1	no	—
116	C₂×Dic₁₈₁	—	2	1	no	—
117	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
118	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
120	C₄×D₁₈₁	—	2	1	no	—
121	C₄×D₁₈₁	—	2	1	no	—
122	C₄×D₁₈₁	—	2	2	no	—
123	C₄×D₁₈₁	—	2	2	no	—
124	C₄×D₁₈₁	—	4	2	no	—
125	#13089678	—	2	1	no	—
126	#13089678	—	2	1	no	—
127	C₂³×C₁₈₁	—	2	1	no	—
128	D₄×C₁₈₁	—	2	1	no	—
129	D₄×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	—	4	2	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	—	4	1	no	—
140	C₂²×C₁₈₁	—	4	1	no	—
141	C₂²×C₁₈₁	—	4	1	no	—
142	C₂²×C₁₈₁	C₂²×D₄	1	1	yes	characteristic
143	C₂×D₁₈₁	—	2	1	no	—
144	C₂×D₁₈₁	—	2	1	no	—
145	C₂×D₁₈₁	—	2	1	no	—
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	—	4	1	no	—
152	C₄×C₁₈₁	—	4	1	no	—
153	C₄×C₁₈₁	—	4	1	no	—
154	C₄×C₁₈₁	2₋¹⁺⁴	1	1	yes	characteristic
155	C₄×C₁₈₁	2₋¹⁺⁴	1	1	yes	characteristic
156	Dic₁₈₁	—	2	1	no	—
157	Dic₁₈₁	—	4	1	no	—
158	Dic₁₈₁	—	4	1	no	—
159	Dic₁₈₁	—	4	1	no	—
160	Dic₁₈₁	#135	1	2	yes	—
161	Dic₁₈₁	AGL(1,ℤ/8)	1	2	yes	—
162	C₂×C₁₈₁	—	8	1	no	—
163	C₂×C₁₈₁	#536	1	1	yes	characteristic
164	C₂×C₁₈₁	C₂×#135	1	1	yes	characteristic
165	C₂×C₁₈₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
166	D₁₈₁	—	4	1	no	—
167	D₁₈₁	—	8	1	no	—
168	C₁₈₁	#3228	1	1	yes	characteristic, complemented, Sylow
169	#3228	—	181	1	no	maximal, Sylow
170	#533	—	181	1	no	—
171	#532	—	181	1	no	—
172	#500	—	181	1	no	—
173	#498	—	181	1	no	—
174	#486	—	181	1	no	—
175	#486	—	181	1	no	—
176	#482	—	181	1	no	—
177	#482	—	181	1	no	—
178	#482	—	181	1	no	—
179	#482	—	181	1	no	—
180	#478	—	181	1	no	—
181	#478	—	181	1	no	—
182	#478	—	181	1	no	—
183	#478	D₁₈₁	1	1	yes	characteristic, complemented
184	#431	—	181	1	no	—
185	C₂×#41	—	362	1	no	—
186	C₄⋊Q₈	—	181	1	no	—
187	C₄⋊Q₈	—	181	1	no	—
188	C₄⋊Q₈	—	181	1	no	—
189	#127	—	181	1	no	—
190	#125	—	181	1	no	—
191	#125	C₂×D₁₈₁	1	1	yes	characteristic
192	#124	—	362	1	no	—
193	C₂²⋊Q₈	—	362	1	no	—
194	C₂²⋊Q₈	—	362	1	no	—
195	#121	—	362	1	no	—
196	#121	—	181	1	no	—
197	#121	—	181	1	no	—
198	C₄×Q₈	—	362	1	no	—
199	C₄×Q₈	—	362	1	no	—
200	C₄×D₄	—	362	1	no	—
201	C₄×D₄	—	181	1	no	—
202	C₄×D₄	C₂×D₁₈₁	1	1	yes	characteristic
203	#117	—	181	1	no	—
204	#117	—	181	1	no	—
205	C₂×C₄²	—	181	1	no	—
206	#107	—	181	1	no	—
207	#107	—	181	1	no	—
208	#107	—	181	1	no	—
209	#107	C₂×D₁₈₁	1	1	yes	characteristic
210	#106	—	181	1	no	—
211	#106	—	181	1	no	—
212	#106	—	181	1	no	—
213	#106	C₂×D₁₈₁	1	1	yes	characteristic
214	C₄⋊C₈	—	181	1	no	—
215	C₄⋊C₈	C₂×D₁₈₁	1	1	yes	characteristic
216	#102	—	181	1	no	—
217	#102	—	181	1	no	—
218	#102	—	181	1	no	—
219	#102	—	181	1	no	—
220	#102	—	181	1	no	—
221	#102	—	181	1	no	—
222	#102	C₂×D₁₈₁	1	1	yes	characteristic
223	#102	C₂×D₁₈₁	1	1	yes	characteristic
224	C₂²⋊C₈	—	181	1	no	—
225	C₂²⋊C₈	—	181	1	no	—
226	C₄×C₈	—	181	1	no	—
227	C₄×C₈	—	181	1	no	—
228	#41	—	724	4	no	—
229	C₂×Q₈	—	362	1	no	—
230	C₂×Q₈	—	362	1	no	—
231	C₂×Q₈	—	362	1	no	—
232	C₂×D₄	—	362	1	no	—
233	C₂×D₄	—	362	1	no	—
234	C₂×D₄	—	362	1	no	—
235	C₂×D₄	—	362	1	no	—
236	C₂×D₄	—	181	1	no	—
237	C₂×D₄	C₂²×D₁₈₁	1	1	yes	characteristic
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	362	1	no	—
240	C₂²×C₄	—	181	1	no	—
241	C₂²×C₄	—	181	1	no	—
242	C₂²×C₄	—	181	1	no	—
243	C₂×C₈	—	181	1	no	—
244	C₂×C₈	—	181	1	no	—
245	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	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	—	181	1	no	—
255	C₄⋊C₄	—	181	1	no	—
256	C₄⋊C₄	—	181	1	no	—
257	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
258	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
259	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
260	C₂²⋊C₄	—	2	1	no	—
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	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	—	2	1	no	—
273	C₂³	—	362	1	no	—
274	C₂³	—	181	1	no	—
275	Q₈	—	724	2	no	—
276	Q₈	—	724	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	—	724	2	no	—
283	D₄	—	724	2	no	—
284	D₄	—	724	2	no	—
285	D₄	—	724	2	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	362	2	no	—
291	C₂×C₄	—	362	2	no	—
292	C₂×C₄	—	362	1	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	—	181	1	no	—
298	C₂×C₄	—	181	1	no	—
299	C₂×C₄	—	181	1	no	—
300	C₂×C₄	—	724	2	no	—
301	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
302	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
303	C₂×C₄	C₂³×D₁₈₁	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	362	1	no	—
307	C₈	—	362	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	362	1	no	—
312	C₂²	—	362	1	no	—
313	C₂²	—	362	1	no	—
314	C₂²	—	724	1	no	—
315	C₂²	—	724	1	no	—
316	C₂²	—	724	1	no	—
317	C₂²	C₂×D₄×D₁₈₁	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	362	1	no	—
324	C₄	—	181	2	no	—
325	C₄	—	181	2	no	—
326	C₄	—	724	1	no	—
327	C₄	—	724	1	no	—
328	C₄	—	724	1	no	—
329	C₄	#19153863	1	1	yes	characteristic
330	C₄	#19153864	1	1	yes	characteristic
331	C₂	—	8	1	no	—
332	C₂	—	1448	1	no	—
333	C₂	—	724	1	no	—
334	C₂	#26041852	1	1	yes	characteristic
335	C₂	#135×D₁₈₁	1	1	yes	characteristic
336	C₂	AGL(1,ℤ/8)×D₁₈₁	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	#26041826	C₂
hypercenter	#478	D₁₈₁
hypocenter	C₁₈₁	#3228
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
1. #26041835	C₂	semidirect
3. #26041683	C₂	semidirect
4. #26041676	C₂	semidirect
5. #26041624	C₂	semidirect
6. #26041624	C₂	semidirect
7. #26041606	C₂	semidirect
8. #26041605	C₂	semidirect
9. #26041602	C₂	semidirect
10. #26041601	C₂	semidirect
11. #26041580	C₂	semidirect
12. #26041579	C₂	semidirect
13. #26041576	C₂	semidirect
14. #26041331	C₂	semidirect
15. #478×C₁₈₁	C₂	semidirect
168. C₁₈₁	#3228	semidirect
183. #478	D₁₈₁	semidirect

Maximal subgroups

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

subgroup	index
#26041835	2
#26041826	2
#26041683	2
#26041676	2
#26041624	2
#26041606	2
#26041605	2
#26041602	2
#26041601	2
#26041580	2
#26041579	2
#26041576	2
#26041331	2
#478×C₁₈₁	2
#3228	181