Subgroups — #37582380

Order 24448.

376 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27133291	C₂	1	1	yes	maximal, characteristic, complemented
2	#27133146	C₂	1	1	yes	maximal, characteristic
3	#27132968	C₂	1	2	yes	maximal, complemented
4	#27132965	C₂	1	2	yes	maximal, complemented
5	#27132951	C₂	1	2	yes	maximal, complemented
6	#27132946	C₂	1	2	yes	maximal, complemented
7	#27132706	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#20259254	C₂	1	1	yes	maximal, characteristic, complemented
9	#27132689	C₂	1	1	yes	maximal, characteristic, complemented
10	#27132664	C₂	1	1	yes	maximal, characteristic
11	#416×C₁₉₁	C₂	1	1	yes	maximal, characteristic
12	C₂×#15690271	C₂²	1	1	yes	characteristic
13	#20259322	—	2	1	no	—
14	#20259321	C₂²	1	2	yes	—
15	#20259314	—	2	1	no	—
16	#20259313	C₂²	1	2	yes	—
17	#20259299	—	2	2	no	—
18	Q₈×Dic₁₉₁	C₂²	1	2	yes	—
19	D₄×Dic₁₉₁	—	2	2	no	—
20	D₄×Dic₁₉₁	C₂²	1	2	yes	—
21	#20259281	—	2	1	no	—
22	#20259281	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₉₁	—	2	1	no	—
24	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
26	#20259255	C₂²	1	2	yes	—
27	#20259254	C₂²	1	2	yes	—
28	#20259254	C₂²	1	2	yes	—
29	#20259253	C₂²	1	2	yes	—
30	Dic₁₉₁⋊C₈	C₂²	1	2	yes	—
31	Dic₁₉₁⋊C₈	C₂²	1	2	yes	—
32	#20259246	C₂²	1	2	yes	—
33	#20259243	C₂²	1	2	yes	—
34	C₂×#41×C₁₉₁	C₂²	1	1	yes	characteristic
35	C₂²×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
36	#117×C₁₉₁	C₂²	1	1	yes	characteristic
37	#103×C₁₉₁	C₂²	1	1	yes	characteristic
38	#103×C₁₉₁	C₂²	1	1	yes	characteristic
39	#102×C₁₉₁	C₂²	1	1	yes	characteristic
40	#102×C₁₉₁	C₂²	1	1	yes	characteristic
41	C₂×Dic₃₈₂	—	2	2	no	—
42	C₂²×Dic₁₉₁	—	2	2	no	—
43	C₂²×Dic₁₉₁	—	2	1	no	—
44	C₂²×Dic₁₉₁	—	2	1	no	—
45	#15690271	—	2	2	no	—
46	C₂×C₁₉₁⋊C₈	C₂³	1	2	yes	—
47	C₄⋊Dic₁₉₁	—	2	2	no	—
48	C₄⋊Dic₁₉₁	—	2	2	no	—
49	C₄⋊Dic₁₉₁	—	2	1	no	—
50	C₄⋊Dic₁₉₁	—	2	1	no	—
51	C₄⋊Dic₁₉₁	D₄	1	1	yes	characteristic
52	C₄⋊Dic₁₉₁	D₄	1	1	yes	characteristic
53	C₄⋊Dic₁₉₁	C₂³	1	2	yes	—
54	C₄⋊Dic₁₉₁	C₂³	1	2	yes	—
55	Dic₁₉₁⋊C₄	—	2	2	no	—
56	Dic₁₉₁⋊C₄	—	2	2	no	—
57	C₂²⋊Dic₁₉₁	—	2	2	no	—
58	C₂²⋊Dic₁₉₁	D₄	1	2	yes	—
59	C₄×Dic₁₉₁	—	2	2	no	—
60	C₄×Dic₁₉₁	C₂³	1	2	yes	—
61	#41×C₁₉₁	—	2	2	no	—
62	#41×C₁₉₁	Q₈	1	4	yes	complemented
63	C₂×Q₈×C₁₉₁	C₂³	1	1	yes	characteristic
64	C₂×D₄×C₁₉₁	—	2	1	no	—
65	C₂×D₄×C₁₉₁	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₁₉₁	—	2	1	no	—
67	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₁₉₁	—	2	2	no	—
69	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₁₉₁	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₁₉₁	C₂³	1	1	yes	characteristic
73	C₂²⋊C₄×C₁₉₁	—	2	1	no	—
74	C₄²×C₁₉₁	—	2	1	no	—
75	Dic₃₈₂	—	8	2	no	—
76	C₂×Dic₁₉₁	—	2	2	no	—
77	C₂×Dic₁₉₁	—	2	2	no	—
78	C₂×Dic₁₉₁	—	8	2	no	—
79	C₂×Dic₁₉₁	—	8	1	no	—
80	C₂×Dic₁₉₁	—	8	1	no	—
81	C₂×Dic₁₉₁	C₂×D₄	1	2	yes	—
82	C₂×Dic₁₉₁	C₂×D₄	1	2	yes	—
83	C₁₉₁⋊C₈	—	2	2	no	—
84	C₂³×C₁₉₁	—	2	1	no	—
85	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
86	Q₈×C₁₉₁	—	2	1	no	—
87	Q₈×C₁₉₁	C₂×Q₈	1	2	yes	—
88	D₄×C₁₉₁	—	2	4	no	—
89	D₄×C₁₉₁	—	2	1	no	—
90	D₄×C₁₉₁	C₂×Q₈	1	2	yes	—
91	C₂×C₄×C₁₉₁	—	2	1	no	—
92	C₂×C₄×C₁₉₁	—	2	4	no	—
93	C₂×C₄×C₁₉₁	—	2	1	no	—
94	C₂×C₄×C₁₉₁	—	2	1	no	—
95	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₉₁	C₂×Q₈	1	2	yes	—
97	C₂×C₄×C₁₉₁	#41	1	2	yes	—
98	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
99	C₈×C₁₉₁	—	2	1	no	—
100	C₈×C₁₉₁	—	2	1	no	—
101	C₂²×C₁₉₁	—	2	1	no	—
102	C₂²×C₁₉₁	—	2	1	no	—
103	C₂²×C₁₉₁	—	2	2	no	—
104	C₂²×C₁₉₁	—	4	1	no	—
105	C₂²×C₁₉₁	—	4	1	no	—
106	C₂²×C₁₉₁	C₂²⋊Q₈	1	2	yes	—
107	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
108	C₄×C₁₉₁	—	2	2	no	—
109	C₄×C₁₉₁	—	4	1	no	—
110	C₄×C₁₉₁	—	4	1	no	—
111	C₄×C₁₉₁	C₂²⋊Q₈	1	2	yes	—
112	C₄×C₁₉₁	C₂²×Q₈	1	1	yes	characteristic
113	C₄×C₁₉₁	C₂×#41	1	1	yes	characteristic
114	Dic₁₉₁	—	2	2	no	—
115	Dic₁₉₁	—	2	2	no	—
116	Dic₁₉₁	—	4	2	no	—
117	Dic₁₉₁	—	4	2	no	—
118	C₂×C₁₉₁	—	2	2	no	—
119	C₂×C₁₉₁	—	4	2	no	—
120	C₂×C₁₉₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₁₉₁	#577	1	1	yes	characteristic
122	C₂×C₁₉₁	#577	1	1	yes	characteristic
123	C₁₉₁	#3084	1	1	yes	characteristic, complemented, Sylow
124	#3084	—	191	1	no	maximal, Sylow
125	#544	—	191	1	no	—
126	#520	—	191	1	no	—
127	#479	—	191	2	no	—
128	#478	—	191	2	no	—
129	#476	—	191	2	no	—
130	#475	—	191	2	no	—
131	#428	—	191	1	no	—
132	C₂×#106	—	191	1	no	—
133	#424	—	191	1	no	—
134	#417	—	191	1	no	—
135	#416	D₁₉₁	1	1	yes	characteristic
136	C₂×#41	C₂×D₁₉₁	1	1	yes	characteristic
137	C₂×M₁₆	—	191	1	no	—
138	C₂²×C₈	C₂×D₁₉₁	1	1	yes	characteristic
139	C₄⋊Q₈	—	382	1	no	—
140	C₄⋊Q₈	—	191	2	no	—
141	#125	—	382	1	no	—
142	#125	—	191	2	no	—
143	C₂²⋊Q₈	—	382	2	no	—
144	C₄×Q₈	—	191	2	no	—
145	C₄×D₄	—	382	2	no	—
146	C₄×D₄	—	191	2	no	—
147	#117	—	382	1	no	—
148	#117	—	191	1	no	—
149	#117	C₂×D₁₉₁	1	1	yes	characteristic
150	C₂×C₄⋊C₄	—	382	1	no	—
151	C₂×C₄⋊C₄	—	191	1	no	—
152	C₂×C₄⋊C₄	—	191	1	no	—
153	#107	—	191	2	no	—
154	#106	—	191	2	no	—
155	#106	—	191	2	no	—
156	#106	—	191	2	no	—
157	C₄⋊C₈	—	191	2	no	—
158	C₄⋊C₈	—	191	2	no	—
159	#103	—	191	2	no	—
160	#103	C₂×D₁₉₁	1	1	yes	characteristic
161	#103	C₂×D₁₉₁	1	1	yes	characteristic
162	#102	—	191	2	no	—
163	#102	C₂×D₁₉₁	1	1	yes	characteristic
164	#102	C₂×D₁₉₁	1	1	yes	characteristic
165	#41	—	2	2	no	—
166	#41	Dic₃₈₂	1	4	yes	complemented
167	C₂×Q₈	—	382	2	no	—
168	C₂×Q₈	C₂²×D₁₉₁	1	1	yes	characteristic
169	C₂×D₄	—	2	1	no	—
170	C₂×D₄	C₂²×D₁₉₁	1	1	yes	characteristic
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	382	1	no	—
173	C₂²×C₄	—	382	1	no	—
174	C₂²×C₄	—	382	2	no	—
175	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
176	M₁₆	—	382	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	191	2	no	—
179	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
180	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
181	C₄⋊C₄	—	382	2	no	—
182	C₄⋊C₄	—	382	2	no	—
183	C₄⋊C₄	—	382	2	no	—
184	C₄⋊C₄	—	382	2	no	—
185	C₄⋊C₄	—	382	1	no	—
186	C₄⋊C₄	—	382	1	no	—
187	C₄⋊C₄	—	191	2	no	—
188	C₄⋊C₄	—	191	2	no	—
189	C₄⋊C₄	—	191	1	no	—
190	C₄⋊C₄	—	191	1	no	—
191	C₄⋊C₄	C₂²×D₁₉₁	1	1	yes	characteristic
192	C₄⋊C₄	C₂²×D₁₉₁	1	1	yes	characteristic
193	C₂²⋊C₄	—	2	1	no	—
194	C₂²⋊C₄	—	382	2	no	—
195	C₂²⋊C₄	—	191	2	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	382	2	no	—
198	C₄²	—	191	2	no	—
199	C₂³	—	2	1	no	—
200	C₂³	D₄×D₁₉₁	1	1	yes	characteristic
201	Q₈	—	2	1	no	—
202	Q₈	—	1528	2	no	—
203	Q₈	C₂×Dic₃₈₂	1	2	yes	—
204	D₄	—	2	4	no	—
205	D₄	—	2	1	no	—
206	D₄	C₂×Dic₃₈₂	1	2	yes	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	382	2	no	—
212	C₂×C₄	—	382	2	no	—
213	C₂×C₄	—	191	2	no	—
214	C₂×C₄	—	191	2	no	—
215	C₂×C₄	—	1528	2	no	—
216	C₂×C₄	—	1528	1	no	—
217	C₂×C₄	—	1528	1	no	—
218	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
219	C₂×C₄	C₂×Dic₃₈₂	1	2	yes	—
220	C₂×C₄	#15690286	1	2	yes	—
221	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
222	C₈	—	2	1	no	—
223	C₈	—	2	1	no	—
224	C₈	—	382	2	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	4	1	no	—
230	C₂²	#20259299	1	2	yes	—
231	C₂²	C₂×D₄×D₁₉₁	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	4	1	no	—
234	C₄	—	4	1	no	—
235	C₄	—	382	2	no	—
236	C₄	—	382	2	no	—
237	C₄	—	764	2	no	—
238	C₄	—	764	2	no	—
239	C₄	#20259299	1	2	yes	—
240	C₄	C₂²×Dic₃₈₂	1	1	yes	characteristic
241	C₄	C₂×#15690286	1	1	yes	characteristic
242	C₂	—	2	2	no	—
243	C₂	—	4	2	no	—
244	C₂	C₂×#20259299	1	1	yes	characteristic
245	C₂	#27133470	1	1	yes	characteristic
246	C₂	#27133470	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	#416×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
prime radical	C₂×#41×C₁₉₁	C₂²
hypercenter	#416	D₁₉₁
hypocenter	C₁₉₁	#3084
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
1. #27133291	C₂	semidirect
3. #27132968	C₂	semidirect
4. #27132965	C₂	semidirect
5. #27132951	C₂	semidirect
6. #27132946	C₂	semidirect
7. #27132706	C₂	semidirect
8. C₂×#20259254	C₂	semidirect
9. #27132689	C₂	semidirect
62. #41×C₁₉₁	Q₈	semidirect
123. C₁₉₁	#3084	semidirect
166. #41	Dic₃₈₂	semidirect

Maximal subgroups

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

subgroup	index
#27133291	2
#27133146	2
#27132968	2
#27132965	2
#27132951	2
#27132946	2
#27132706	2
C₂×#20259254	2
#27132689	2
#27132664	2
#416×C₁₉₁	2
#3084	191