Subgroups — #27132768

Order 12224.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15690277	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₃₈₂	C₂	1	1	yes	maximal, characteristic
3	#20259253	C₂	1	1	yes	maximal, characteristic
4	#20259246	C₂	1	1	yes	maximal, characteristic
5	#20259245	C₂	1	1	yes	maximal, characteristic
6	#20259233	C₂	1	1	yes	maximal, characteristic
7	C₄×Q₈×C₁₉₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₃₈₂	C₂²	1	1	yes	characteristic
9	#15690277	C₄	1	4	yes	complemented
10	C₂×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
13	Dic₁₉₁⋊C₄	—	2	1	no	—
14	C₄×Dic₁₉₁	—	2	1	no	—
15	C₂×Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₁₉₁	—	2	1	no	—
17	C₄⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
18	C₄²×C₁₉₁	—	2	1	no	—
19	C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
20	Dic₃₈₂	—	2	1	no	—
21	Dic₃₈₂	C₂×C₄	1	2	yes	—
22	C₂×Dic₁₉₁	—	2	1	no	—
23	C₂×Dic₁₉₁	—	2	1	no	—
24	C₁₉₁⋊C₈	—	2	1	no	—
25	C₁₉₁⋊C₈	C₂×C₄	1	2	yes	—
26	Q₈×C₁₉₁	—	2	1	no	—
27	Q₈×C₁₉₁	C₂×C₄	1	2	yes	—
28	C₂×C₄×C₁₉₁	—	2	1	no	—
29	C₂×C₄×C₁₉₁	—	2	1	no	—
30	C₂×C₄×C₁₉₁	D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₉₁	D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
33	C₂²×C₁₉₁	C₂×D₄	1	1	yes	characteristic
34	C₄×C₁₉₁	—	2	2	no	—
35	C₄×C₁₉₁	—	2	1	no	—
36	C₄×C₁₉₁	—	2	1	no	—
37	C₄×C₁₉₁	—	4	1	no	—
38	C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
39	C₄×C₁₉₁	#41	1	1	yes	characteristic
40	Dic₁₉₁	—	2	2	no	—
41	Dic₁₉₁	—	4	1	no	—
42	C₂×C₁₉₁	C₄×D₄	1	1	yes	characteristic
43	C₂×C₁₉₁	#137	1	1	yes	characteristic
44	C₂×C₁₉₁	#137	1	1	yes	characteristic
45	C₁₉₁	#441	1	1	yes	characteristic, complemented, Sylow
46	#441	—	191	1	no	maximal, Sylow
47	C₂×Q₁₆	—	191	1	no	—
48	C₄×Q₈	—	191	1	no	—
49	C₄×Q₈	D₁₉₁	1	1	yes	characteristic
50	#106	—	191	1	no	—
51	#103	—	191	1	no	—
52	#103	—	191	1	no	—
53	#97	—	191	1	no	—
54	C₂×Q₈	—	191	1	no	—
55	C₂×Q₈	C₂×D₁₉₁	1	1	yes	characteristic
56	Q₁₆	—	191	4	no	—
57	C₂×C₈	—	191	1	no	—
58	C₂×C₈	—	191	1	no	—
59	C₄⋊C₄	—	2	1	no	—
60	C₄⋊C₄	—	382	1	no	—
61	C₄⋊C₄	—	191	1	no	—
62	C₄⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
63	C₄²	—	2	1	no	—
64	C₄²	—	382	1	no	—
65	C₄²	C₂×D₁₉₁	1	1	yes	characteristic
66	Q₈	—	2	1	no	—
67	Q₈	—	382	1	no	—
68	Q₈	—	191	2	no	—
69	Q₈	C₄×D₁₉₁	1	2	yes	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	382	1	no	—
73	C₂×C₄	—	382	1	no	—
74	C₂×C₄	C₂²⋊D₁₉₁	1	1	yes	characteristic
75	C₂×C₄	C₂²⋊D₁₉₁	1	1	yes	characteristic
76	C₂×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
77	C₈	—	382	1	no	—
78	C₈	—	191	2	no	—
79	C₂²	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
80	C₄	—	2	1	no	—
81	C₄	—	2	1	no	—
82	C₄	—	2	2	no	—
83	C₄	—	4	1	no	—
84	C₄	—	382	2	no	—
85	C₄	—	764	1	no	—
86	C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
87	C₄	#15690287	1	1	yes	characteristic
88	C₂	C₄×C₂²⋊D₁₉₁	1	1	yes	characteristic
89	C₂	#20259352	1	1	yes	characteristic
90	C₂	#20259353	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₉₁
commutator	C₄×C₁₉₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₉₁
Fitting	C₄×Q₈×C₁₉₁	C₂
socle, prime radical	C₂²×C₁₉₁	C₂×D₄
cosocle	C₂×C₄×C₁₉₁	C₂³
hypercenter	C₄×Q₈	D₁₉₁
hypocenter	C₁₉₁	#441
LCS	G ▷ C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
9. #15690277	C₄	semidirect
45. C₁₉₁	#441	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690277	2
C₄×Dic₃₈₂	2
#20259253	2
#20259246	2
#20259245	2
#20259233	2
C₄×Q₈×C₁₉₁	2
#441	191