Group #37574764 — subgroups

Subgroups — #37574764

Order 24448.

132 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 47 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	#27133064	C₂	1	1	yes	maximal, characteristic
2	#27133041	C₂	1	1	yes	maximal, characteristic
3	#27132468	C₂	1	1	yes	maximal, characteristic
4	#27132467	C₂	1	1	yes	maximal, characteristic
5	#27132466	C₂	1	1	yes	maximal, characteristic
6	#27132438	C₂	1	1	yes	maximal, characteristic
7	#358×C₁₉₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₇₆₄	C₂²	1	1	yes	characteristic
9	C₂×#15690277	—	2	1	no	—
10	#20259320	—	2	1	no	—
11	#20259313	—	2	1	no	—
12	C₂×C₁₉₁⋊C₁₆	C₂²	1	1	yes	characteristic
13	#20259256	C₂²	1	1	yes	characteristic
14	#20259253	—	2	1	no	—
15	C₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₁₉₁	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₁₉₁	C₂²	1	1	yes	characteristic
18	#107×C₁₉₁	C₂²	1	1	yes	characteristic
19	C₂×Dic₃₈₂	—	2	1	no	—
20	Dic₇₆₄	—	2	1	no	—
21	Dic₇₆₄	—	2	1	no	—
22	#15690277	—	4	2	no	—
23	C₂×C₁₉₁⋊C₈	D₄	1	1	yes	characteristic
24	C₄⋊Dic₁₉₁	—	2	1	no	—
25	C₄⋊Dic₁₉₁	—	4	1	no	—
26	Dic₁₉₁⋊C₄	—	4	1	no	—
27	Dic₁₉₁⋊C₄	—	4	1	no	—
28	Dic₁₉₁⋊C₄	—	4	1	no	—
29	C₄×Dic₁₉₁	D₄	1	1	yes	characteristic
30	C₁₉₁⋊C₁₆	—	2	1	no	—
31	C₂×Q₈×C₁₉₁	—	2	1	no	—
32	Q₁₆×C₁₉₁	—	2	1	no	—
33	Q₁₆×C₁₉₁	—	2	1	no	—
34	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₁₉₁	—	2	1	no	—
36	C₁₆×C₁₉₁	—	2	1	no	—
37	Dic₃₈₂	—	4	1	no	—
38	Dic₃₈₂	—	4	1	no	—
39	C₂×Dic₁₉₁	—	4	1	no	—
40	C₂×Dic₁₉₁	—	4	1	no	—
41	C₂×Dic₁₉₁	D₈	1	1	yes	characteristic
42	C₂×Dic₁₉₁	D₈	1	1	yes	characteristic
43	C₁₉₁⋊C₈	SD₁₆	1	2	yes	—
44	Q₈×C₁₉₁	—	4	1	no	—
45	Q₈×C₁₉₁	—	4	1	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	C₈×C₁₉₁	#41	1	1	yes	characteristic
50	C₈×C₁₉₁	#41	1	1	yes	characteristic
51	C₂²×C₁₉₁	C₂×D₈	1	1	yes	characteristic
52	C₄×C₁₉₁	—	8	1	no	—
53	C₄×C₁₉₁	—	8	1	no	—
54	C₄×C₁₉₁	#124	1	1	yes	characteristic
55	C₄×C₁₉₁	C₂×SD₁₆	1	1	yes	characteristic
56	Dic₁₉₁	—	2	1	no	—
57	Dic₁₉₁	—	2	1	no	—
58	Dic₁₉₁	—	8	1	no	—
59	Dic₁₉₁	—	8	1	no	—
60	C₂×C₁₉₁	#486	1	1	yes	characteristic
61	C₂×C₁₉₁	#510	1	1	yes	characteristic
62	C₂×C₁₉₁	#510	1	1	yes	characteristic
63	C₁₉₁	#2245	1	1	yes	characteristic, complemented, Sylow
64	#2245	—	191	1	no	maximal, Sylow
65	#499	—	191	1	no	—
66	#494	—	191	1	no	—
67	#358	—	191	1	no	—
68	#358	—	191	1	no	—
69	#358	—	191	1	no	—
70	#358	D₁₉₁	1	1	yes	characteristic
71	#346	—	191	1	no	—
72	C₂×Q₁₆	—	382	1	no	—
73	C₂×Q₁₆	—	191	1	no	—
74	C₂×Q₁₆	C₂×D₁₉₁	1	1	yes	characteristic
75	C₄⋊Q₈	—	382	1	no	—
76	#125	—	382	1	no	—
77	C₂×C₁₆	—	191	1	no	—
78	C₂×C₁₆	C₂×D₁₉₁	1	1	yes	characteristic
79	#107	—	191	1	no	—
80	#107	C₂×D₁₉₁	1	1	yes	characteristic
81	#106	—	382	1	no	—
82	C₄×C₈	—	191	1	no	—
83	C₂×Q₈	—	2	1	no	—
84	C₂×Q₈	—	382	1	no	—
85	Q₁₆	—	2	1	no	—
86	Q₁₆	—	2	1	no	—
87	Q₁₆	—	382	1	no	—
88	Q₁₆	—	382	1	no	—
89	Q₁₆	—	764	2	no	—
90	C₂×C₈	—	191	1	no	—
91	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
92	C₄⋊C₄	—	2	1	no	—
93	C₄⋊C₄	—	382	1	no	—
94	C₄⋊C₄	—	764	1	no	—
95	C₄⋊C₄	—	764	1	no	—
96	C₄⋊C₄	—	764	1	no	—
97	C₄⋊C₄	—	764	1	no	—
98	C₄²	—	191	1	no	—
99	C₁₆	—	2	1	no	—
100	C₁₆	—	382	1	no	—
101	Q₈	—	4	1	no	—
102	Q₈	—	4	1	no	—
103	Q₈	—	764	1	no	—
104	Q₈	—	764	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	191	1	no	—
108	C₂×C₄	—	191	1	no	—
109	C₂×C₄	—	764	1	no	—
110	C₂×C₄	—	764	1	no	—
111	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
112	C₈	—	191	2	no	—
113	C₈	#15690286	1	1	yes	characteristic
114	C₈	#15690287	1	1	yes	characteristic
115	C₂²	D₈×D₁₉₁	1	1	yes	characteristic
116	C₄	—	8	1	no	—
117	C₄	—	8	1	no	—
118	C₄	—	382	1	no	—
119	C₄	—	382	1	no	—
120	C₄	—	1528	1	no	—
121	C₄	—	1528	1	no	—
122	C₄	#20259308	1	1	yes	characteristic
123	C₄	SD₁₆×D₁₉₁	1	1	yes	characteristic
124	C₂	#27133008	1	1	yes	characteristic
125	C₂	#27133102	1	1	yes	characteristic
126	C₂	#27133105	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27133064

C₂

yes

maximal, characteristic

#27133041

C₂

yes

maximal, characteristic

#27132468

C₂

yes

maximal, characteristic

#27132467

C₂

yes

maximal, characteristic

#27132466

C₂

yes

maximal, characteristic

#27132438

C₂

yes

maximal, characteristic

#358×C₁₉₁

C₂

yes

maximal, characteristic

C₂×Dic₇₆₄

C₂²

yes

—

—

—

C₂×C₁₉₁⋊C₁₆

C₂²

yes

characteristic

#20259256

C₂²

yes

characteristic

#20259253

—

C₈×Dic₁₉₁

C₂²

yes

characteristic

C₂×Q₁₆×C₁₉₁

C₂²

yes

characteristic

C₂×C₁₆×C₁₉₁

C₂²

yes

characteristic

#107×C₁₉₁

C₂²

yes

characteristic

C₂×Dic₃₈₂

—

Dic₇₆₄

—

Dic₇₆₄

—

#15690277

—

C₂×C₁₉₁⋊C₈

D₄

yes

characteristic

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

Dic₁₉₁⋊C₄

—

Dic₁₉₁⋊C₄

—

Dic₁₉₁⋊C₄

—

C₄×Dic₁₉₁

D₄

yes

characteristic

C₁₉₁⋊C₁₆

—

C₂×Q₈×C₁₉₁

—

Q₁₆×C₁₉₁

—

Q₁₆×C₁₉₁

—

C₂×C₈×C₁₉₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₁

—

C₁₆×C₁₉₁

—

Dic₃₈₂

—

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

D₈

yes

characteristic

C₂×Dic₁₉₁

D₈

yes

characteristic

C₁₉₁⋊C₈

SD₁₆

yes

—

Q₈×C₁₉₁

—

Q₈×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

C₈×C₁₉₁

yes

C₈×C₁₉₁

yes

C₂²×C₁₉₁

C₂×D₈

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#124

yes

characteristic

C₄×C₁₉₁

C₂×SD₁₆

yes

characteristic

Dic₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#2245

—

191

maximal, Sylow

#499

—

191

—

#494

—

191

—

#358

—

191

—

#358

—

191

—

#358

—

191

—

#358

D₁₉₁

yes

characteristic

#346

—

191

—

C₂×Q₁₆

—

382

—

C₂×Q₁₆

—

191

—

C₂×Q₁₆

C₂×D₁₉₁

yes

characteristic

C₄⋊Q₈

—

382

—

#125

—

382

—

C₂×C₁₆

—

191

—

C₂×C₁₆

C₂×D₁₉₁

yes

characteristic

#107

—

191

—

#107

C₂×D₁₉₁

yes

characteristic

#106

—

382

—

C₄×C₈

—

191

—

C₂×Q₈

—

C₂×Q₈

—

382

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

382

—

Q₁₆

—

382

—

Q₁₆

—

764

—

C₂×C₈

—

191

—

C₂×C₈

C₂²×D₁₉₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

764

—

C₄⋊C₄

—

764

—

C₄⋊C₄

—

764

—

C₄⋊C₄

—

764

—

C₄²

—

191

—

C₁₆

—

100

C₁₆

—

382

—

101

Q₈

—

102

Q₈

—

103

Q₈

—

764

—

104

Q₈

—

764

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

191

—

108

C₂×C₄

—

191

—

109

C₂×C₄

—

764

—

110

C₂×C₄

—

764

—

111

C₂×C₄

D₄×D₁₉₁

yes

characteristic

112

C₈

—

191

—

113

C₈

#15690286

yes

characteristic

114

C₈

#15690287

yes

characteristic

115

C₂²

D₈×D₁₉₁

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

382

—

119

C₄

—

382

—

120

C₄

—

1528

—

121

C₄

—

1528

—

122

C₄

#20259308

yes

characteristic

123

C₄

SD₁₆×D₁₉₁

yes

characteristic

124

C₂

#27133008

yes

characteristic

125

C₂

#27133102

yes

characteristic

126

C₂

#27133105

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₉₁
commutator, cosocle	C₂×C₈×C₁₉₁	C₂³
Frattini	C₂×C₈	C₂²×D₁₉₁
Fitting	#358×C₁₉₁	C₂
socle, prime radical	C₂²×C₁₉₁	C₂×D₈
hypercenter	#358	D₁₉₁
hypocenter	C₁₉₁	#2245
LCS	G ▷ C₂×C₈×C₁₉₁ ▷ C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₁₉₁

commutator, cosocle

C₂×C₈×C₁₉₁

C₂³

Frattini

C₂×C₈

C₂²×D₁₉₁

Fitting

#358×C₁₉₁

C₂

socle, prime radical

C₂²×C₁₉₁

C₂×D₈

D₁₉₁

C₁₉₁

G ▷ C₂×C₈×C₁₉₁ ▷ C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
63. C₁₉₁	#2245	semidirect

H ≅ G/N

type

63. C₁₉₁

#2245

semidirect

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

subgroup	index
#27133064	2
#27133041	2
#27132468	2
#27132467	2
#27132466	2
#27132438	2
#358×C₁₉₁	2
#2245	191

subgroup

index

#358×C₁₉₁

191

Subgroups — #37574764

Conjugacy classes of subgroups

Complements

Maximal subgroups