Subgroups — #37543970

Order 23168.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26041887	C₂	1	1	yes	maximal, characteristic, complemented
2	#26041751	C₂	1	1	yes	maximal, characteristic
3	#26041623	C₂	1	1	yes	maximal, characteristic, complemented
4	#26041618	C₂	1	1	yes	maximal, characteristic, complemented
5	#26041581	C₂	1	1	yes	maximal, characteristic, complemented
6	#26041578	C₂	1	1	yes	maximal, characteristic, complemented
7	#26041483	C₂	1	1	yes	maximal, characteristic, complemented
8	#26041482	C₂	1	1	yes	maximal, characteristic, complemented
9	#26041431	C₂	1	1	yes	maximal, characteristic
10	#26041428	C₂	1	1	yes	maximal, characteristic
11	#26041379	C₂	1	1	yes	maximal, characteristic, complemented
12	#26041377	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#15668438	C₂	1	1	yes	maximal, characteristic, complemented
14	#26041338	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×SD₁₆×C₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₁₈₁	C₂²	1	1	yes	characteristic
17	C₂×#15668438	C₂²	1	1	yes	characteristic
18	C₂×#15668437	C₂²	1	1	yes	characteristic
19	C₂×#15668436	C₂²	1	1	yes	characteristic
20	#19153814	C₂²	1	1	yes	characteristic, complemented
21	#19153811	—	2	1	no	—
22	#19153810	—	2	1	no	—
23	#19153808	—	2	1	no	—
24	#19153807	C₂²	1	1	yes	characteristic, complemented
25	#19153806	—	2	1	no	—
26	#19153796	C₂²	1	1	yes	characteristic
27	#19153792	C₂²	1	1	yes	characteristic
28	C₄×Dic₃₆₂	C₂²	1	1	yes	characteristic, complemented
29	C₄×D₇₂₄	C₂²	1	1	yes	characteristic
30	D₄×Dic₁₈₁	C₂²	1	1	yes	characteristic
31	#19153778	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊D₁₈₁	—	2	1	no	—
33	#19153776	—	2	1	no	—
34	#19153774	C₂²	1	1	yes	characteristic
35	#19153773	C₂²	1	1	yes	characteristic
36	C₂×#15668426	—	2	1	no	—
37	C₂²⋊C₄×D₁₈₁	—	2	1	no	—
38	#19153749	C₂²	1	1	yes	characteristic, complemented
39	#19153749	C₂²	1	1	yes	characteristic, complemented
40	#19153748	C₂²	1	1	yes	characteristic, complemented
41	#19153745	C₂²	1	1	yes	characteristic, complemented
42	#19153745	C₂²	1	1	yes	characteristic, complemented
43	#19153741	C₂²	1	1	yes	characteristic, complemented
44	#19153740	C₂²	1	1	yes	characteristic, complemented
45	#19153739	C₂²	1	1	yes	characteristic, complemented
46	#19153738	C₂²	1	1	yes	characteristic
47	#19153737	C₂²	1	1	yes	characteristic
48	#19153736	C₂²	1	1	yes	characteristic
49	#19153728	C₂²	1	1	yes	characteristic
50	#19153728	C₂²	1	1	yes	characteristic
51	#19153727	C₂²	1	1	yes	characteristic, complemented
52	C₂×SD₁₆×C₁₈₁	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₈₁	C₂²	1	1	yes	characteristic, complemented
54	C₄×D₄×C₁₈₁	C₂²	1	1	yes	characteristic
55	#106×C₁₈₁	C₂²	1	1	yes	characteristic, complemented
56	#103×C₁₈₁	C₂²	1	1	yes	characteristic, complemented
57	#102×C₁₈₁	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₈₁	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₁₈₁	—	2	1	no	—
60	C₂×Dic₃₆₂	C₂³	1	1	yes	characteristic
61	C₂×D₇₂₄	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₈₁	—	2	1	no	—
63	D₄×D₁₈₁	—	2	2	no	—
64	D₄×D₁₈₁	—	2	2	no	—
65	C₂²×Dic₁₈₁	—	2	1	no	—
66	C₂×C₄×D₁₈₁	—	2	1	no	—
67	C₂×C₄×D₁₈₁	—	2	1	no	—
68	C₂×C₄×D₁₈₁	C₂³	1	1	yes	characteristic
69	#15668438	—	2	2	no	—
70	#15668437	—	2	2	no	—
71	#15668436	—	2	2	no	—
72	C₂×#13089678	C₂³	1	1	yes	characteristic
73	C₂×#13089678	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₈₁	—	2	1	no	—
75	C₄⋊Dic₁₈₁	—	2	1	no	—
76	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₈₁	C₂³	1	1	yes	characteristic
78	#15668428	—	2	1	no	—
79	#15668428	—	2	1	no	—
80	#15668428	—	2	1	no	—
81	#15668428	—	2	1	no	—
82	#15668428	—	2	1	no	—
83	#15668428	D₄	1	1	yes	characteristic
84	#15668428	D₄	1	1	yes	characteristic
85	C₂²⋊Dic₁₈₁	—	2	1	no	—
86	#15668426	—	2	1	no	—
87	#15668426	—	2	1	no	—
88	#15668426	—	2	1	no	—
89	#15668426	—	2	1	no	—
90	#15668426	—	2	1	no	—
91	#15668426	D₄	1	1	yes	characteristic
92	#15668426	D₄	1	1	yes	characteristic
93	C₄×Dic₁₈₁	—	2	1	no	—
94	C₄×Dic₁₈₁	—	2	1	no	—
95	C₄×Dic₁₈₁	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₁₈₁	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₁₈₁	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₁₈₁	—	2	1	no	—
99	SD₁₆×C₁₈₁	—	2	2	no	—
100	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₈₁	—	2	1	no	—
103	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₈₁	C₂³	1	1	yes	characteristic
105	C₂²⋊C₄×C₁₈₁	—	2	1	no	—
106	C₄²×C₁₈₁	—	2	1	no	—
107	C₄²×C₁₈₁	C₂³	1	1	yes	characteristic
108	C₂²×D₁₈₁	—	2	1	no	—
109	C₂²×D₁₈₁	—	2	4	no	—
110	C₂²×D₁₈₁	—	8	1	no	—
111	C₂²×D₁₈₁	—	8	1	no	—
112	C₂²×D₁₈₁	C₂×D₄	1	1	yes	characteristic
113	Dic₃₆₂	—	2	1	no	—
114	Dic₃₆₂	—	2	1	no	—
115	D₇₂₄	—	2	1	no	—
116	D₇₂₄	#41	1	2	yes	—
117	C₂²⋊D₁₈₁	—	2	4	no	—
118	C₂×Dic₁₈₁	—	2	1	no	—
119	C₂×Dic₁₈₁	—	2	1	no	—
120	C₂×Dic₁₈₁	—	2	1	no	—
121	C₂×Dic₁₈₁	—	8	1	no	—
122	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
125	C₄×D₁₈₁	—	2	1	no	—
126	C₄×D₁₈₁	—	2	1	no	—
127	C₄×D₁₈₁	—	8	1	no	—
128	C₄×D₁₈₁	—	8	1	no	—
129	#13089678	—	2	1	no	—
130	#13089678	—	2	1	no	—
131	C₂³×C₁₈₁	—	2	1	no	—
132	Q₈×C₁₈₁	—	2	1	no	—
133	Q₈×C₁₈₁	—	2	1	no	—
134	D₄×C₁₈₁	—	2	1	no	—
135	D₄×C₁₈₁	#41	1	2	yes	—
136	C₂×C₄×C₁₈₁	—	2	1	no	—
137	C₂×C₄×C₁₈₁	—	2	1	no	—
138	C₂×C₄×C₁₈₁	—	2	1	no	—
139	C₂×C₄×C₁₈₁	—	8	1	no	—
140	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₈₁	C₂⁴	1	1	yes	characteristic
143	C₈×C₁₈₁	—	2	1	no	—
144	C₈×C₁₈₁	—	2	1	no	—
145	C₂²×C₁₈₁	—	2	2	no	—
146	C₂²×C₁₈₁	—	4	1	no	—
147	C₂²×C₁₈₁	—	4	1	no	—
148	C₂²×C₁₈₁	C₂²×D₄	1	1	yes	characteristic
149	C₂×D₁₈₁	—	2	1	no	—
150	C₂×D₁₈₁	—	2	2	no	—
151	C₂×D₁₈₁	—	2	1	no	—
152	C₂×D₁₈₁	—	2	1	no	—
153	C₂×D₁₈₁	—	4	1	no	—
154	C₂×D₁₈₁	—	4	1	no	—
155	C₂×D₁₈₁	—	8	4	no	—
156	C₄×C₁₈₁	—	2	1	no	—
157	C₄×C₁₈₁	—	2	1	no	—
158	C₄×C₁₈₁	—	4	1	no	—
159	C₄×C₁₈₁	—	4	1	no	—
160	C₄×C₁₈₁	—	4	1	no	—
161	C₄×C₁₈₁	C₂×#41	1	1	yes	characteristic
162	C₄×C₁₈₁	C₂×#41	1	1	yes	characteristic
163	Dic₁₈₁	—	2	1	no	—
164	Dic₁₈₁	—	2	1	no	—
165	Dic₁₈₁	—	2	1	no	—
166	Dic₁₈₁	—	4	1	no	—
167	Dic₁₈₁	—	4	1	no	—
168	Dic₁₈₁	—	4	1	no	—
169	C₂×C₁₈₁	—	4	2	no	—
170	C₂×C₁₈₁	#525	1	1	yes	characteristic
171	C₂×C₁₈₁	#577	1	1	yes	characteristic
172	C₂×C₁₈₁	#577	1	1	yes	characteristic
173	D₁₈₁	—	4	1	no	—
174	D₁₈₁	—	4	2	no	—
175	C₁₈₁	#3123	1	1	yes	characteristic, complemented, Sylow
176	#3123	—	181	1	no	maximal, Sylow
177	#541	—	181	1	no	—
178	#518	—	181	1	no	—
179	#485	—	181	1	no	—
180	#485	—	181	1	no	—
181	#478	—	181	1	no	—
182	#478	—	181	1	no	—
183	#461	—	181	1	no	—
184	#461	—	181	1	no	—
185	#450	—	181	1	no	—
186	#450	—	181	1	no	—
187	#440	—	181	1	no	—
188	#440	—	181	1	no	—
189	C₄×SD₁₆	—	181	1	no	—
190	C₄×SD₁₆	D₁₈₁	1	1	yes	characteristic, complemented
191	#433	—	181	1	no	—
192	C₂²×D₄	—	181	1	no	—
193	C₂×SD₁₆	—	181	1	no	—
194	C₂×SD₁₆	—	181	1	no	—
195	C₂×SD₁₆	—	181	1	no	—
196	C₂×SD₁₆	C₂×D₁₈₁	1	1	yes	characteristic
197	C₄⋊Q₈	—	181	1	no	—
198	#126	—	362	1	no	—
199	#126	—	362	1	no	—
200	#125	—	362	1	no	—
201	#125	—	362	1	no	—
202	#125	—	181	1	no	—
203	C₂²⋊Q₈	—	181	1	no	—
204	C₂²⋊Q₈	—	181	1	no	—
205	C₄×Q₈	—	181	1	no	—
206	C₄×Q₈	C₂×D₁₈₁	1	1	yes	characteristic, complemented
207	C₄×D₄	—	362	1	no	—
208	C₄×D₄	—	362	1	no	—
209	C₄×D₄	—	181	1	no	—
210	C₄×D₄	—	181	1	no	—
211	C₄×D₄	—	181	1	no	—
212	C₄×D₄	C₂×D₁₈₁	1	1	yes	characteristic
213	#117	—	181	1	no	—
214	#117	—	181	1	no	—
215	C₂×C₂²⋊C₄	—	362	1	no	—
216	C₂×C₂²⋊C₄	—	362	1	no	—
217	#107	—	181	1	no	—
218	#107	—	181	1	no	—
219	#106	—	181	1	no	—
220	#106	C₂×D₁₈₁	1	1	yes	characteristic, complemented
221	C₄⋊C₈	—	181	1	no	—
222	C₄⋊C₈	—	181	1	no	—
223	#103	—	181	1	no	—
224	#103	—	181	1	no	—
225	#103	—	181	1	no	—
226	#103	C₂×D₁₈₁	1	1	yes	characteristic, complemented
227	#102	—	181	1	no	—
228	#102	—	181	1	no	—
229	#102	—	181	1	no	—
230	#102	C₂×D₁₈₁	1	1	yes	characteristic
231	C₂²⋊C₈	—	181	1	no	—
232	C₂²⋊C₈	—	181	1	no	—
233	#97	—	181	1	no	—
234	C₄×C₈	C₂×D₁₈₁	1	1	yes	characteristic, complemented
235	C₂⁴	—	362	1	no	—
236	C₂×Q₈	—	181	1	no	—
237	C₂×Q₈	C₂²×D₁₈₁	1	1	yes	characteristic
238	C₂×D₄	—	362	2	no	—
239	C₂×D₄	—	362	2	no	—
240	C₂×D₄	—	362	1	no	—
241	C₂×D₄	—	181	1	no	—
242	C₂×D₄	C₂²×D₁₈₁	1	1	yes	characteristic
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	362	1	no	—
245	C₂²×C₄	—	362	1	no	—
246	C₂²×C₄	—	362	1	no	—
247	C₂²×C₄	—	181	1	no	—
248	SD₁₆	—	2	2	no	—
249	SD₁₆	—	362	2	no	—
250	SD₁₆	—	362	2	no	—
251	SD₁₆	—	362	2	no	—
252	C₂×C₈	—	181	1	no	—
253	C₂×C₈	—	181	1	no	—
254	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
255	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	362	1	no	—
258	C₄⋊C₄	—	362	1	no	—
259	C₄⋊C₄	—	362	1	no	—
260	C₄⋊C₄	—	362	1	no	—
261	C₄⋊C₄	—	362	1	no	—
262	C₄⋊C₄	—	362	1	no	—
263	C₄⋊C₄	—	362	1	no	—
264	C₄⋊C₄	—	181	1	no	—
265	C₄⋊C₄	—	181	1	no	—
266	C₄⋊C₄	—	181	1	no	—
267	C₄⋊C₄	—	181	1	no	—
268	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
269	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	362	1	no	—
272	C₂²⋊C₄	—	362	1	no	—
273	C₂²⋊C₄	—	362	1	no	—
274	C₂²⋊C₄	—	362	1	no	—
275	C₂²⋊C₄	—	362	1	no	—
276	C₂²⋊C₄	—	362	1	no	—
277	C₂²⋊C₄	—	181	1	no	—
278	C₂²⋊C₄	—	181	1	no	—
279	C₄²	—	2	1	no	—
280	C₄²	—	362	1	no	—
281	C₄²	—	362	1	no	—
282	C₄²	—	181	1	no	—
283	C₄²	C₂²×D₁₈₁	1	1	yes	characteristic
284	C₂³	—	2	1	no	—
285	C₂³	—	362	4	no	—
286	C₂³	—	362	1	no	—
287	C₂³	—	1448	1	no	—
288	C₂³	—	1448	1	no	—
289	C₂³	—	181	1	no	—
290	Q₈	—	2	1	no	—
291	Q₈	—	2	1	no	—
292	Q₈	—	362	1	no	—
293	Q₈	—	362	1	no	—
294	D₄	—	2	1	no	—
295	D₄	—	362	4	no	—
296	D₄	—	362	1	no	—
297	D₄	—	181	2	no	—
298	D₄	#15668449	1	2	yes	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	8	1	no	—
303	C₂×C₄	—	362	1	no	—
304	C₂×C₄	—	362	1	no	—
305	C₂×C₄	—	362	1	no	—
306	C₂×C₄	—	362	1	no	—
307	C₂×C₄	—	362	1	no	—
308	C₂×C₄	—	1448	1	no	—
309	C₂×C₄	—	1448	1	no	—
310	C₂×C₄	—	1448	1	no	—
311	C₂×C₄	—	181	1	no	—
312	C₂×C₄	—	181	1	no	—
313	C₂×C₄	—	181	1	no	—
314	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
315	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₁₈₁	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	362	1	no	—
320	C₈	—	362	1	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	362	1	no	—
325	C₂²	—	362	2	no	—
326	C₂²	—	362	1	no	—
327	C₂²	—	362	1	no	—
328	C₂²	—	1448	4	no	—
329	C₂²	—	724	1	no	—
330	C₂²	—	724	1	no	—
331	C₂²	C₂×D₄×D₁₈₁	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	362	1	no	—
338	C₄	—	362	1	no	—
339	C₄	—	362	1	no	—
340	C₄	—	724	1	no	—
341	C₄	—	724	1	no	—
342	C₄	—	724	1	no	—
343	C₄	#41×D₁₈₁	1	1	yes	characteristic
344	C₄	C₂×#15668449	1	1	yes	characteristic
345	C₂	—	4	2	no	—
346	C₂	—	724	1	no	—
347	C₂	—	724	2	no	—
348	C₂	#26041788	1	1	yes	characteristic
349	C₂	#26042085	1	1	yes	characteristic
350	C₂	#26042086	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₈₁
commutator, cosocle	C₂×C₄×C₁₈₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₈₁
Fitting	C₄×SD₁₆×C₁₈₁	C₂
socle	C₂²×C₁₈₁	C₂²×D₄
prime radical	C₂×D₄×D₁₈₁	C₂²
hypercenter	C₄×SD₁₆	D₁₈₁
hypocenter	C₁₈₁	#3123
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #26041887	C₂	semidirect
3. #26041623	C₂	semidirect
4. #26041618	C₂	semidirect
5. #26041581	C₂	semidirect
6. #26041578	C₂	semidirect
7. #26041483	C₂	semidirect
8. #26041482	C₂	semidirect
11. #26041379	C₂	semidirect
12. #26041377	C₂	semidirect
13. C₄×#15668438	C₂	semidirect
14. #26041338	C₂	semidirect
15. C₄×SD₁₆×C₁₈₁	C₂	semidirect
20. #19153814	C₂²	semidirect
24. #19153807	C₂²	semidirect
28. C₄×Dic₃₆₂	C₂²	semidirect
38. #19153749	C₂²	semidirect
39. #19153749	C₂²	semidirect
40. #19153748	C₂²	semidirect
41. #19153745	C₂²	semidirect
42. #19153745	C₂²	semidirect
43. #19153741	C₂²	semidirect
44. #19153740	C₂²	semidirect
45. #19153739	C₂²	semidirect
51. #19153727	C₂²	semidirect
53. C₄×Q₈×C₁₈₁	C₂²	semidirect
55. #106×C₁₈₁	C₂²	semidirect
56. #103×C₁₈₁	C₂²	semidirect
58. C₄×C₈×C₁₈₁	C₂²	semidirect
175. C₁₈₁	#3123	semidirect
190. C₄×SD₁₆	D₁₈₁	semidirect
206. C₄×Q₈	C₂×D₁₈₁	semidirect
220. #106	C₂×D₁₈₁	semidirect
226. #103	C₂×D₁₈₁	semidirect
234. C₄×C₈	C₂×D₁₈₁	semidirect

Maximal subgroups

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

subgroup	index
#26041887	2
#26041751	2
#26041623	2
#26041618	2
#26041581	2
#26041578	2
#26041483	2
#26041482	2
#26041431	2
#26041428	2
#26041379	2
#26041377	2
C₄×#15668438	2
#26041338	2
C₄×SD₁₆×C₁₈₁	2
#3123	181