Group #37572152 — subgroups

Subgroups — #37572152

Order 24448.

176 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27133059	C₂	1	1	yes	maximal, characteristic
2	#27133059	C₂	1	1	yes	maximal, characteristic
3	#27133042	C₂	1	1	yes	maximal, characteristic
4	#27133014	C₂	1	1	yes	maximal, characteristic
5	#27133013	C₂	1	1	yes	maximal, characteristic
6	#27132385	C₂	1	1	yes	maximal, characteristic
7	#498×C₁₉₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#15690277	—	2	2	no	—
9	#20259322	C₂²	1	1	yes	characteristic
10	#20259322	C₂²	1	1	yes	characteristic
11	#20259253	—	2	2	no	—
12	#20259253	—	2	2	no	—
13	#20259247	—	2	2	no	—
14	#20259246	—	2	2	no	—
15	C₄×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
19	#106×C₁₉₁	—	2	2	no	—
20	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
21	C₂×Dic₃₈₂	—	2	1	no	—
22	C₂×Dic₃₈₂	—	2	1	no	—
23	#15690277	—	4	4	no	—
24	C₂×C₁₉₁⋊C₈	D₄	1	2	yes	—
25	C₂×C₁₉₁⋊C₈	D₄	1	2	yes	—
26	C₄⋊Dic₁₉₁	—	2	1	no	—
27	C₄⋊Dic₁₉₁	—	2	1	no	—
28	C₄⋊Dic₁₉₁	—	2	1	no	—
29	C₄⋊Dic₁₉₁	—	2	1	no	—
30	C₂×Q₈×C₁₉₁	—	2	1	no	—
31	C₂×Q₈×C₁₉₁	—	2	1	no	—
32	C₂×C₈×C₁₉₁	D₄	1	2	yes	—
33	C₄⋊C₄×C₁₉₁	—	2	1	no	—
34	C₄⋊C₄×C₁₉₁	—	2	1	no	—
35	C₄⋊C₄×C₁₉₁	—	2	1	no	—
36	C₄⋊C₄×C₁₉₁	—	2	1	no	—
37	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
38	Dic₃₈₂	—	4	2	no	—
39	Dic₃₈₂	—	4	2	no	—
40	C₂×Dic₁₉₁	—	4	1	no	—
41	C₂×Dic₁₉₁	—	4	1	no	—
42	C₁₉₁⋊C₈	—	2	2	no	—
43	C₁₉₁⋊C₈	SD₁₆	1	4	yes	—
44	Q₈×C₁₉₁	—	4	2	no	—
45	Q₈×C₁₉₁	—	4	2	no	—
46	C₂×C₄×C₁₉₁	—	4	1	no	—
47	C₂×C₄×C₁₉₁	—	4	1	no	—
48	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
51	C₈×C₁₉₁	—	2	2	no	—
52	C₂²×C₁₉₁	#127	1	1	yes	characteristic
53	C₄×C₁₉₁	—	8	1	no	—
54	C₄×C₁₉₁	—	8	1	no	—
55	C₄×C₁₉₁	C₂×SD₁₆	1	2	yes	—
56	C₄×C₁₉₁	#137	1	2	yes	—
57	C₄×C₁₉₁	#137	1	2	yes	—
58	Dic₁₉₁	—	8	1	no	—
59	Dic₁₉₁	—	8	1	no	—
60	C₂×C₁₉₁	#492	1	1	yes	characteristic
61	C₂×C₁₉₁	#497	1	1	yes	characteristic
62	C₂×C₁₉₁	#497	1	1	yes	characteristic
63	C₁₉₁	#1716	1	1	yes	characteristic, complemented, Sylow
64	#1716	—	191	1	no	maximal, Sylow
65	#498	—	191	1	no	—
66	#498	—	191	1	no	—
67	#498	D₁₉₁	1	1	yes	characteristic
68	#494	—	191	1	no	—
69	#487	—	191	1	no	—
70	#487	—	191	1	no	—
71	#322	—	191	1	no	—
72	C₂×Q₁₆	—	382	2	no	—
73	C₄⋊Q₈	—	191	1	no	—
74	C₄⋊Q₈	—	191	1	no	—
75	C₄⋊Q₈	C₂×D₁₉₁	1	1	yes	characteristic
76	C₄⋊Q₈	C₂×D₁₉₁	1	1	yes	characteristic
77	#106	—	2	2	no	—
78	#106	—	382	2	no	—
79	#106	—	382	2	no	—
80	#103	—	382	2	no	—
81	#103	—	382	2	no	—
82	C₄×C₈	—	191	1	no	—
83	C₄×C₈	—	191	1	no	—
84	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	2	1	no	—
87	C₂×Q₈	—	382	1	no	—
88	C₂×Q₈	—	382	1	no	—
89	Q₁₆	—	764	4	no	—
90	C₂×C₈	—	191	2	no	—
91	C₂×C₈	—	191	2	no	—
92	C₂×C₈	C₂²⋊D₁₉₁	1	2	yes	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	382	1	no	—
98	C₄⋊C₄	—	382	1	no	—
99	C₄⋊C₄	—	382	1	no	—
100	C₄⋊C₄	—	382	1	no	—
101	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
102	Q₈	—	4	2	no	—
103	Q₈	—	4	2	no	—
104	Q₈	—	764	2	no	—
105	Q₈	—	764	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	764	1	no	—
109	C₂×C₄	—	764	1	no	—
110	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
111	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	382	2	no	—
115	C₈	—	191	4	no	—
116	C₂²	#20259318	1	1	yes	characteristic
117	C₄	—	8	1	no	—
118	C₄	—	8	1	no	—
119	C₄	—	1528	1	no	—
120	C₄	—	1528	1	no	—
121	C₄	SD₁₆×D₁₉₁	1	2	yes	—
122	C₄	#20259351	1	2	yes	—
123	C₄	#20259353	1	2	yes	—
124	C₂	#27133034	1	1	yes	characteristic
125	C₂	#27133054	1	1	yes	characteristic
126	C₂	#27133055	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27133059

C₂

yes

maximal, characteristic

#27133059

C₂

yes

maximal, characteristic

#27133042

C₂

yes

maximal, characteristic

#27133014

C₂

yes

maximal, characteristic

#27133013

C₂

yes

maximal, characteristic

#27132385

C₂

yes

maximal, characteristic

#498×C₁₉₁

C₂

yes

maximal, characteristic

C₂×#15690277

—

#20259322

C₂²

yes

characteristic

#20259322

C₂²

yes

—

—

—

—

C₄×C₁₉₁⋊C₈

C₂²

yes

characteristic

C₄×C₁₉₁⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₉₁

C₂²

yes

characteristic

C₄⋊Q₈×C₁₉₁

C₂²

yes

characteristic

#106×C₁₉₁

—

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

#15690277

—

C₂×C₁₉₁⋊C₈

D₄

yes

—

C₂×C₁₉₁⋊C₈

D₄

yes

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₂×Q₈×C₁₉₁

—

C₂×Q₈×C₁₉₁

—

C₂×C₈×C₁₉₁

D₄

yes

—

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄²×C₁₉₁

C₂³

yes

characteristic

Dic₃₈₂

—

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

SD₁₆

yes

—

Q₈×C₁₉₁

—

Q₈×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₈×C₁₉₁

—

C₂²×C₁₉₁

#127

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂×SD₁₆

yes

—

C₄×C₁₉₁

#137

yes

—

C₄×C₁₉₁

#137

yes

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#1716

—

191

maximal, Sylow

#498

—

191

—

#498

—

191

—

#498

D₁₉₁

yes

characteristic

#494

—

191

—

#487

—

191

—

#487

—

191

—

#322

—

191

—

C₂×Q₁₆

—

382

—

C₄⋊Q₈

—

191

—

C₄⋊Q₈

—

191

—

C₄⋊Q₈

C₂×D₁₉₁

yes

characteristic

C₄⋊Q₈

C₂×D₁₉₁

yes

characteristic

#106

—

#106

—

382

—

#106

—

382

—

#103

—

382

—

#103

—

382

—

C₄×C₈

—

191

—

C₄×C₈

—

191

—

C₄×C₈

C₂×D₁₉₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

382

—

C₂×Q₈

—

382

—

Q₁₆

—

764

—

C₂×C₈

—

191

—

C₂×C₈

—

191

—

C₂×C₈

C₂²⋊D₁₉₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

382

—

100

C₄⋊C₄

—

382

—

101

C₄²

C₂²×D₁₉₁

yes

characteristic

102

Q₈

—

103

Q₈

—

104

Q₈

—

764

—

105

Q₈

—

764

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

764

—

109

C₂×C₄

—

764

—

110

C₂×C₄

D₄×D₁₉₁

yes

characteristic

111

C₂×C₄

D₄×D₁₉₁

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

113

C₈

—

114

C₈

—

382

—

115

C₈

—

191

—

116

C₂²

#20259318

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

1528

—

120

C₄

—

1528

—

121

C₄

SD₁₆×D₁₉₁

yes

—

122

C₄

#20259351

yes

—

123

C₄

#20259353

yes

—

124

C₂

#27133034

yes

characteristic

125

C₂

#27133054

yes

characteristic

126

C₂

#27133055

yes

characteristic

	subgroup	G/H
center	C₂²	#20259318
commutator, cosocle	C₄²×C₁₉₁	C₂³
Frattini	C₄²	C₂²×D₁₉₁
Fitting	#498×C₁₉₁	C₂
socle, prime radical	C₂²×C₁₉₁	#127
hypercenter	#498	D₁₉₁
hypocenter	C₁₉₁	#1716
LCS	G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#20259318

commutator, cosocle

C₄²×C₁₉₁

C₂³

Frattini

C₄²

C₂²×D₁₉₁

Fitting

#498×C₁₉₁

C₂

C₂²×C₁₉₁

D₁₉₁

C₁₉₁

G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
63. C₁₉₁	#1716	semidirect

H ≅ G/N

type

63. C₁₉₁

#1716

semidirect

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

subgroup	index
#27133059	2
#27133042	2
#27133014	2
#27133013	2
#27132385	2
#498×C₁₉₁	2
#1716	191

subgroup

index

#498×C₁₉₁

191

Subgroups — #37572152

Conjugacy classes of subgroups

Complements

Maximal subgroups