Subgroups — #37546597

Order 23168.

338 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26042005	C₂	1	1	yes	maximal, characteristic, complemented
2	#26041988	C₂	1	1	yes	maximal, characteristic
3	#26041681	C₂	1	1	yes	maximal, characteristic, complemented
4	#26041679	C₂	1	1	yes	maximal, characteristic, complemented
5	#26041676	C₂	1	1	yes	maximal, characteristic, complemented
6	#26041588	C₂	1	1	yes	maximal, characteristic, complemented
7	#26041586	C₂	1	1	yes	maximal, characteristic, complemented
8	#26041579	C₂	1	1	yes	maximal, characteristic
9	#26041577	C₂	1	1	yes	maximal, characteristic
10	#26041572	C₂	1	1	yes	maximal, characteristic, complemented
11	#26041557	C₂	1	1	yes	maximal, characteristic
12	C₈×Dic₃₆₂	C₂	1	1	yes	maximal, characteristic, complemented
13	SD₁₆×Dic₁₈₁	C₂	1	1	yes	maximal, characteristic
14	SD₁₆×Dic₁₈₁	C₂	1	1	yes	maximal, characteristic
15	C₄×SD₁₆×C₁₈₁	C₂	1	1	yes	maximal, characteristic
16	#19153815	—	2	1	no	—
17	#19153815	C₂²	1	1	yes	characteristic
18	#19153815	C₂²	1	1	yes	characteristic
19	#19153808	—	2	1	no	—
20	#19153808	C₂²	1	1	yes	characteristic
21	#19153808	C₂²	1	1	yes	characteristic
22	#19153807	—	2	1	no	—
23	#19153807	C₂²	1	1	yes	characteristic
24	#19153807	C₂²	1	1	yes	characteristic
25	#19153795	—	2	1	no	—
26	#19153793	—	2	1	no	—
27	#19153793	—	2	1	no	—
28	Q₈×Dic₁₈₁	C₂²	1	1	yes	characteristic
29	Q₈×Dic₁₈₁	C₂²	1	1	yes	characteristic
30	C₄×Dic₃₆₂	—	2	1	no	—
31	C₄×Dic₃₆₂	C₂²	1	1	yes	characteristic
32	D₄×Dic₁₈₁	C₂²	1	1	yes	characteristic
33	D₄×Dic₁₈₁	C₂²	1	1	yes	characteristic
34	C₂×#15668428	—	2	1	no	—
35	#19153750	C₂²	1	1	yes	characteristic
36	#19153750	C₂²	1	1	yes	characteristic
37	#19153749	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	C₄⋊#13089678	C₂²	1	1	yes	characteristic
44	#19153745	C₂²	1	1	yes	characteristic
45	#19153745	C₂²	1	1	yes	characteristic
46	#19153740	C₂²	1	1	yes	characteristic
47	#19153740	C₂²	1	1	yes	characteristic
48	#19153737	C₂²	1	1	yes	characteristic
49	#19153737	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₈₁	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₈₁	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₁₈₁	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₈₁	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₁₈₁	C₂²	1	1	yes	characteristic
55	#106×C₁₈₁	C₂²	1	1	yes	characteristic
56	#103×C₁₈₁	C₂²	1	1	yes	characteristic
57	#102×C₁₈₁	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₈₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₃₆₂	—	2	1	no	—
60	C₂×Dic₃₆₂	—	2	1	no	—
61	C₂×Dic₃₆₂	D₄	1	1	yes	characteristic
62	C₂²×Dic₁₈₁	—	2	1	no	—
63	C₂²×Dic₁₈₁	—	2	1	no	—
64	C₂×#13089678	C₂³	1	1	yes	characteristic
65	C₂×#13089678	C₂³	1	1	yes	characteristic
66	C₄⋊Dic₁₈₁	—	2	1	no	—
67	C₄⋊Dic₁₈₁	—	2	1	no	—
68	C₄⋊Dic₁₈₁	—	2	1	no	—
69	C₄⋊Dic₁₈₁	—	2	1	no	—
70	C₄⋊Dic₁₈₁	D₄	1	1	yes	characteristic
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	—	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	C₂²⋊Dic₁₈₁	—	2	1	no	—
87	C₄×Dic₁₈₁	—	2	1	no	—
88	C₄×Dic₁₈₁	—	2	1	no	—
89	C₄×Dic₁₈₁	C₂³	1	1	yes	characteristic
90	C₄×Dic₁₈₁	C₂³	1	1	yes	characteristic
91	C₂×Q₈×C₁₈₁	C₂³	1	1	yes	characteristic
92	C₂×D₄×C₁₈₁	C₂³	1	1	yes	characteristic
93	C₂²×C₄×C₁₈₁	—	2	1	no	—
94	SD₁₆×C₁₈₁	Q₈	1	4	yes	complemented
95	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
96	C₂×C₈×C₁₈₁	C₂³	1	1	yes	characteristic
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	C₂²⋊C₄×C₁₈₁	—	2	1	no	—
101	C₄²×C₁₈₁	—	2	1	no	—
102	C₄²×C₁₈₁	C₂³	1	1	yes	characteristic
103	Dic₃₆₂	—	2	2	no	—
104	Dic₃₆₂	—	4	2	no	—
105	Dic₃₆₂	—	8	1	no	—
106	C₂×Dic₁₈₁	—	2	1	no	—
107	C₂×Dic₁₈₁	—	2	1	no	—
108	C₂×Dic₁₈₁	—	2	1	no	—
109	C₂×Dic₁₈₁	—	2	1	no	—
110	C₂×Dic₁₈₁	—	4	2	no	—
111	C₂×Dic₁₈₁	—	8	1	no	—
112	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
113	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
114	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
115	C₂×Dic₁₈₁	C₂×D₄	1	1	yes	characteristic
116	#13089678	—	2	1	no	—
117	#13089678	—	2	1	no	—
118	C₂³×C₁₈₁	—	2	1	no	—
119	Q₈×C₁₈₁	—	2	1	no	—
120	Q₈×C₁₈₁	C₂×Q₈	1	2	yes	—
121	D₄×C₁₈₁	—	2	1	no	—
122	D₄×C₁₈₁	C₂×Q₈	1	2	yes	—
123	C₂×C₄×C₁₈₁	—	2	1	no	—
124	C₂×C₄×C₁₈₁	—	2	1	no	—
125	C₂×C₄×C₁₈₁	—	2	1	no	—
126	C₂×C₄×C₁₈₁	—	8	1	no	—
127	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₈₁	C₂⁴	1	1	yes	characteristic
130	C₈×C₁₈₁	—	2	1	no	—
131	C₈×C₁₈₁	C₂×Q₈	1	2	yes	—
132	C₂²×C₁₈₁	—	2	2	no	—
133	C₂²×C₁₈₁	—	4	1	no	—
134	C₂²×C₁₈₁	—	4	1	no	—
135	C₂²×C₁₈₁	C₂²×D₄	1	1	yes	characteristic
136	C₄×C₁₈₁	—	2	2	no	—
137	C₄×C₁₈₁	—	2	1	no	—
138	C₄×C₁₈₁	—	2	1	no	—
139	C₄×C₁₈₁	—	4	1	no	—
140	C₄×C₁₈₁	—	4	1	no	—
141	C₄×C₁₈₁	C₂²×Q₈	1	1	yes	characteristic
142	C₄×C₁₈₁	2₋¹⁺⁴	1	1	yes	characteristic
143	Dic₁₈₁	—	2	1	no	—
144	Dic₁₈₁	—	2	1	no	—
145	Dic₁₈₁	—	2	1	no	—
146	Dic₁₈₁	—	4	1	no	—
147	Dic₁₈₁	—	4	1	no	—
148	Dic₁₈₁	—	4	1	no	—
149	Dic₁₈₁	—	4	1	no	—
150	Dic₁₈₁	#135	1	2	yes	—
151	C₂×C₁₈₁	—	4	2	no	—
152	C₂×C₁₈₁	D₄×Q₈	1	1	yes	characteristic
153	C₂×C₁₈₁	C₂×#135	1	1	yes	characteristic
154	C₂×C₁₈₁	#577	1	1	yes	characteristic
155	C₁₈₁	#3381	1	1	yes	characteristic, complemented, Sylow
156	#3381	—	181	1	no	maximal, Sylow
157	#557	—	181	1	no	—
158	#554	—	181	1	no	—
159	#499	—	181	1	no	—
160	#499	—	181	1	no	—
161	#498	—	181	1	no	—
162	#479	—	181	1	no	—
163	#479	—	181	1	no	—
164	#478	—	181	1	no	—
165	#478	—	181	1	no	—
166	#477	—	181	1	no	—
167	#474	—	181	1	no	—
168	C₈×Q₈	—	181	1	no	—
169	C₄×SD₁₆	—	181	1	no	—
170	C₄×SD₁₆	—	181	1	no	—
171	C₄×SD₁₆	D₁₈₁	1	1	yes	characteristic
172	C₂×SD₁₆	C₂×D₁₈₁	1	1	yes	characteristic
173	C₄⋊Q₈	—	362	1	no	—
174	C₄⋊Q₈	—	181	1	no	—
175	C₄⋊Q₈	—	181	1	no	—
176	#125	—	362	1	no	—
177	#125	—	362	1	no	—
178	#125	—	181	1	no	—
179	#125	—	181	1	no	—
180	#125	—	181	1	no	—
181	#125	—	181	1	no	—
182	C₂²⋊Q₈	—	362	1	no	—
183	C₂²⋊Q₈	—	362	1	no	—
184	C₂²⋊Q₈	—	362	1	no	—
185	C₄×Q₈	—	362	1	no	—
186	C₄×Q₈	—	181	1	no	—
187	C₄×Q₈	—	181	1	no	—
188	C₄×Q₈	—	181	1	no	—
189	C₄×Q₈	C₂×D₁₈₁	1	1	yes	characteristic
190	C₄×D₄	—	181	1	no	—
191	C₄×D₄	—	181	1	no	—
192	C₄×D₄	C₂×D₁₈₁	1	1	yes	characteristic
193	C₂×C₄⋊C₄	—	362	1	no	—
194	#107	—	181	1	no	—
195	#107	—	181	1	no	—
196	#107	—	181	1	no	—
197	#107	—	181	1	no	—
198	#106	—	181	1	no	—
199	#106	—	181	1	no	—
200	#106	—	181	1	no	—
201	#106	—	181	1	no	—
202	#106	C₂×D₁₈₁	1	1	yes	characteristic
203	C₄⋊C₈	—	181	1	no	—
204	C₄⋊C₈	—	181	1	no	—
205	C₄⋊C₈	—	181	1	no	—
206	#103	—	181	1	no	—
207	#103	—	181	1	no	—
208	#103	C₂×D₁₈₁	1	1	yes	characteristic
209	#102	—	181	1	no	—
210	#102	—	181	1	no	—
211	#102	C₂×D₁₈₁	1	1	yes	characteristic
212	C₄×C₈	—	181	1	no	—
213	C₄×C₈	—	181	1	no	—
214	C₄×C₈	C₂×D₁₈₁	1	1	yes	characteristic
215	C₂×Q₈	—	362	1	no	—
216	C₂×Q₈	—	362	1	no	—
217	C₂×Q₈	—	181	1	no	—
218	C₂×Q₈	C₂²×D₁₈₁	1	1	yes	characteristic
219	C₂×D₄	C₂²×D₁₈₁	1	1	yes	characteristic
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	362	1	no	—
222	C₂²×C₄	—	362	1	no	—
223	SD₁₆	Dic₃₆₂	1	4	yes	complemented
224	C₂×C₈	—	181	1	no	—
225	C₂×C₈	—	181	1	no	—
226	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
227	C₂×C₈	C₂²×D₁₈₁	1	1	yes	characteristic
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	362	1	no	—
230	C₄⋊C₄	—	362	1	no	—
231	C₄⋊C₄	—	362	1	no	—
232	C₄⋊C₄	—	362	1	no	—
233	C₄⋊C₄	—	362	1	no	—
234	C₄⋊C₄	—	362	1	no	—
235	C₄⋊C₄	—	362	1	no	—
236	C₄⋊C₄	—	362	1	no	—
237	C₄⋊C₄	—	362	1	no	—
238	C₄⋊C₄	—	362	1	no	—
239	C₄⋊C₄	—	362	1	no	—
240	C₄⋊C₄	—	362	1	no	—
241	C₄⋊C₄	—	181	1	no	—
242	C₄⋊C₄	—	181	1	no	—
243	C₄⋊C₄	—	181	1	no	—
244	C₄⋊C₄	—	181	1	no	—
245	C₄⋊C₄	—	181	1	no	—
246	C₄⋊C₄	—	181	1	no	—
247	C₄⋊C₄	—	181	1	no	—
248	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
249	C₄⋊C₄	C₂²×D₁₈₁	1	1	yes	characteristic
250	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	362	1	no	—
252	C₂²⋊C₄	—	362	1	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	362	1	no	—
255	C₄²	—	362	1	no	—
256	C₄²	—	181	1	no	—
257	C₄²	—	181	1	no	—
258	C₄²	C₂²×D₁₈₁	1	1	yes	characteristic
259	C₂³	—	2	1	no	—
260	Q₈	—	2	1	no	—
261	Q₈	—	362	2	no	—
262	Q₈	—	1448	1	no	—
263	Q₈	—	724	2	no	—
264	Q₈	C₂×Dic₃₆₂	1	2	yes	—
265	D₄	—	2	1	no	—
266	D₄	C₂×Dic₃₆₂	1	2	yes	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	8	1	no	—
271	C₂×C₄	—	362	1	no	—
272	C₂×C₄	—	362	1	no	—
273	C₂×C₄	—	362	1	no	—
274	C₂×C₄	—	362	1	no	—
275	C₂×C₄	—	1448	1	no	—
276	C₂×C₄	—	181	1	no	—
277	C₂×C₄	—	181	1	no	—
278	C₂×C₄	—	181	1	no	—
279	C₂×C₄	—	181	1	no	—
280	C₂×C₄	—	724	2	no	—
281	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
282	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
283	C₂×C₄	C₂³×D₁₈₁	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	362	1	no	—
286	C₈	—	362	1	no	—
287	C₈	C₂×Dic₃₆₂	1	2	yes	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	C₂×D₄×D₁₈₁	1	1	yes	characteristic
292	C₄	—	2	2	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	1	no	—
297	C₄	—	362	1	no	—
298	C₄	—	362	1	no	—
299	C₄	—	362	1	no	—
300	C₄	—	181	2	no	—
301	C₄	—	724	1	no	—
302	C₄	—	724	1	no	—
303	C₄	—	724	1	no	—
304	C₄	—	724	1	no	—
305	C₄	C₂²×Dic₃₆₂	1	1	yes	characteristic
306	C₄	#19153863	1	1	yes	characteristic
307	C₂	—	4	2	no	—
308	C₂	D₄×Dic₃₆₂	1	1	yes	characteristic
309	C₂	#135×D₁₈₁	1	1	yes	characteristic
310	C₂	#26042085	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×C₁₈₁	C₂³
hypercenter	C₄×SD₁₆	D₁₈₁
hypocenter	C₁₈₁	#3381
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. #26042005	C₂	semidirect
3. #26041681	C₂	semidirect
4. #26041679	C₂	semidirect
5. #26041676	C₂	semidirect
6. #26041588	C₂	semidirect
7. #26041586	C₂	semidirect
10. #26041572	C₂	semidirect
12. C₈×Dic₃₆₂	C₂	semidirect
94. SD₁₆×C₁₈₁	Q₈	semidirect
155. C₁₈₁	#3381	semidirect
223. SD₁₆	Dic₃₆₂	semidirect

Maximal subgroups

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

subgroup	index
#26042005	2
#26041988	2
#26041681	2
#26041679	2
#26041676	2
#26041588	2
#26041586	2
#26041579	2
#26041577	2
#26041572	2
#26041557	2
C₈×Dic₃₆₂	2
SD₁₆×Dic₁₈₁	2
C₄×SD₁₆×C₁₈₁	2
#3381	181