Group #37574569 — subgroups

Subgroups — #37574569

Order 24448.

154 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 49 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	C₂×Dic₁₅₂₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#20259268	C₂	1	1	yes	maximal, characteristic
3	#27133041	C₂	1	1	yes	maximal, characteristic
4	#27132757	C₂	1	1	yes	maximal, characteristic
5	Dic₁₉₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	#27132466	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×Dic₇₆₄	C₂²	1	1	yes	characteristic
10	C₂×#15690277	—	2	1	no	—
11	#20259320	—	2	1	no	—
12	#20259287	—	2	1	no	—
13	Dic₁₅₂₈	—	2	2	no	—
14	#20259268	—	2	2	no	—
15	C₂×C₁₉₁⋊C₁₆	C₂²	1	1	yes	characteristic
16	#20259245	—	2	1	no	—
17	C₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₁₉₁	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×C₁₉₁	C₂²	1	1	yes	characteristic
20	#107×C₁₉₁	C₂²	1	1	yes	characteristic
21	C₂×Dic₃₈₂	—	2	1	no	—
22	C₂×Dic₃₈₂	—	2	1	no	—
23	Dic₇₆₄	—	2	1	no	—
24	Dic₇₆₄	—	2	1	no	—
25	Dic₇₆₄	—	2	1	no	—
26	Dic₇₆₄	D₄	1	2	yes	—
27	#15690277	—	4	2	no	—
28	C₂×C₁₉₁⋊C₈	D₄	1	1	yes	characteristic
29	Dic₁₉₁⋊C₄	—	4	1	no	—
30	Dic₁₉₁⋊C₄	—	4	1	no	—
31	Dic₁₉₁⋊C₄	—	4	1	no	—
32	C₄×Dic₁₉₁	—	4	1	no	—
33	C₄×Dic₁₉₁	D₄	1	1	yes	characteristic
34	C₁₉₁⋊C₁₆	—	2	1	no	—
35	C₂×Q₈×C₁₉₁	—	2	1	no	—
36	Q₁₆×C₁₉₁	—	2	1	no	—
37	Q₁₆×C₁₉₁	—	2	1	no	—
38	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₁₉₁	—	2	1	no	—
40	C₁₆×C₁₉₁	—	2	1	no	—
41	Dic₃₈₂	—	2	2	no	—
42	Dic₃₈₂	—	4	1	no	—
43	Dic₃₈₂	—	4	1	no	—
44	Dic₃₈₂	—	4	1	no	—
45	C₂×Dic₁₉₁	—	4	1	no	—
46	C₂×Dic₁₉₁	—	4	1	no	—
47	C₂×Dic₁₉₁	D₈	1	1	yes	characteristic
48	C₂×Dic₁₉₁	D₈	1	1	yes	characteristic
49	C₁₉₁⋊C₈	—	2	1	no	—
50	Q₈×C₁₉₁	—	4	1	no	—
51	Q₈×C₁₉₁	—	4	1	no	—
52	C₂×C₄×C₁₉₁	—	4	1	no	—
53	C₂×C₄×C₁₉₁	—	4	1	no	—
54	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
55	C₈×C₁₉₁	C₂×D₄	1	1	yes	characteristic
56	C₈×C₁₉₁	#41	1	1	yes	characteristic
57	C₂²×C₁₉₁	C₂×D₈	1	1	yes	characteristic
58	C₄×C₁₉₁	—	8	1	no	—
59	C₄×C₁₉₁	—	8	1	no	—
60	C₄×C₁₉₁	#121	1	1	yes	characteristic
61	C₄×C₁₉₁	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Dic₁₉₁	—	2	1	no	—
63	Dic₁₉₁	—	4	2	no	—
64	Dic₁₉₁	—	8	1	no	—
65	Dic₁₉₁	Q₃₂	1	2	yes	complemented
66	C₂×C₁₉₁	#459	1	1	yes	characteristic
67	C₂×C₁₉₁	C₂×Q₃₂	1	1	yes	characteristic
68	C₂×C₁₉₁	#510	1	1	yes	characteristic
69	C₁₉₁	#2209	1	1	yes	characteristic, complemented, Sylow
70	#2209	—	191	1	no	maximal, Sylow
71	C₂×Q₃₂	—	191	1	no	—
72	C₂×Q₃₂	—	191	1	no	—
73	#494	—	191	1	no	—
74	C₄×Q₁₆	—	191	1	no	—
75	C₄⋊C₁₆	—	191	1	no	—
76	#358	—	191	1	no	—
77	#358	D₁₉₁	1	1	yes	characteristic
78	C₂×Q₁₆	—	382	1	no	—
79	C₂×Q₁₆	—	191	1	no	—
80	C₂×Q₁₆	—	191	1	no	—
81	C₂×Q₁₆	C₂×D₁₉₁	1	1	yes	characteristic
82	C₄⋊Q₈	—	382	1	no	—
83	C₄×Q₈	—	382	1	no	—
84	Q₃₂	—	382	2	no	—
85	Q₃₂	—	382	2	no	—
86	C₂×C₁₆	—	191	1	no	—
87	C₂×C₁₆	C₂×D₁₉₁	1	1	yes	characteristic
88	#107	C₂×D₁₉₁	1	1	yes	characteristic
89	#103	—	382	1	no	—
90	C₄×C₈	—	191	1	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	382	1	no	—
93	C₂×Q₈	—	382	1	no	—
94	Q₁₆	—	2	1	no	—
95	Q₁₆	—	2	1	no	—
96	Q₁₆	—	382	1	no	—
97	Q₁₆	—	382	1	no	—
98	Q₁₆	—	382	1	no	—
99	Q₁₆	—	191	2	no	—
100	Q₁₆	—	764	2	no	—
101	C₂×C₈	—	191	1	no	—
102	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	764	1	no	—
105	C₄⋊C₄	—	764	1	no	—
106	C₄⋊C₄	—	764	1	no	—
107	C₄²	—	191	1	no	—
108	C₄²	—	764	1	no	—
109	C₁₆	—	2	1	no	—
110	C₁₆	—	382	1	no	—
111	Q₈	—	4	1	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	382	2	no	—
114	Q₈	—	764	1	no	—
115	Q₈	—	764	1	no	—
116	Q₈	—	764	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	191	1	no	—
120	C₂×C₄	—	191	1	no	—
121	C₂×C₄	—	764	1	no	—
122	C₂×C₄	—	764	1	no	—
123	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
124	C₈	—	382	1	no	—
125	C₈	D₄×D₁₉₁	1	1	yes	characteristic
126	C₈	#15690287	1	1	yes	characteristic
127	C₂²	D₈×D₁₉₁	1	1	yes	characteristic
128	C₄	—	8	1	no	—
129	C₄	—	8	1	no	—
130	C₄	—	382	1	no	—
131	C₄	—	191	2	no	—
132	C₄	—	1528	1	no	—
133	C₄	—	764	2	no	—
134	C₄	#20259295	1	1	yes	characteristic
135	C₄	#20259349	1	1	yes	characteristic
136	C₂	#27132857	1	1	yes	characteristic
137	C₂	Q₃₂×D₁₉₁	1	1	yes	characteristic
138	C₂	#27133102	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₅₂₈

C₂

yes

maximal, characteristic

C₂×#20259268

C₂

yes

maximal, characteristic

#27133041

C₂

yes

maximal, characteristic

#27132757

C₂

yes

maximal, characteristic

Dic₁₉₁⋊C₁₆

C₂

yes

maximal, characteristic

#27132466

C₂

yes

maximal, characteristic

#358×C₁₉₁

C₂

yes

maximal, characteristic

C₂×Dic₇₆₄

C₂²

yes

characteristic

C₂×Dic₇₆₄

C₂²

yes

—

—

—

Dic₁₅₂₈

—

#20259268

—

C₂×C₁₉₁⋊C₁₆

C₂²

yes

characteristic

#20259245

—

C₈×Dic₁₉₁

C₂²

yes

characteristic

C₂×Q₁₆×C₁₉₁

C₂²

yes

characteristic

C₂×C₁₆×C₁₉₁

C₂²

yes

characteristic

#107×C₁₉₁

C₂²

yes

characteristic

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

Dic₇₆₄

—

Dic₇₆₄

—

Dic₇₆₄

—

Dic₇₆₄

D₄

yes

—

#15690277

—

C₂×C₁₉₁⋊C₈

D₄

yes

characteristic

Dic₁₉₁⋊C₄

—

Dic₁₉₁⋊C₄

—

Dic₁₉₁⋊C₄

—

C₄×Dic₁₉₁

—

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

—

Dic₃₈₂

—

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

D₈

yes

characteristic

C₂×Dic₁₉₁

D₈

yes

characteristic

C₁₉₁⋊C₈

—

Q₈×C₁₉₁

—

Q₈×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₈×C₁₉₁

C₂×D₄

yes

characteristic

C₈×C₁₉₁

#41

yes

characteristic

C₂²×C₁₉₁

C₂×D₈

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

yes

C₄×C₁₉₁

yes

Dic₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

Q₃₂

yes

complemented

C₂×C₁₉₁

#459

yes

characteristic

C₂×C₁₉₁

C₂×Q₃₂

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#2209

—

191

maximal, Sylow

C₂×Q₃₂

—

191

—

C₂×Q₃₂

—

191

—

#494

—

191

—

C₄×Q₁₆

—

191

—

C₄⋊C₁₆

—

191

—

#358

—

191

—

#358

D₁₉₁

yes

characteristic

C₂×Q₁₆

—

382

—

C₂×Q₁₆

—

191

—

C₂×Q₁₆

—

191

—

C₂×Q₁₆

C₂×D₁₉₁

yes

characteristic

C₄⋊Q₈

—

382

—

C₄×Q₈

—

382

—

Q₃₂

—

382

—

Q₃₂

—

382

—

C₂×C₁₆

—

191

—

C₂×C₁₆

C₂×D₁₉₁

yes

characteristic

#107

C₂×D₁₉₁

yes

characteristic

#103

—

382

—

C₄×C₈

—

191

—

C₂×Q₈

—

C₂×Q₈

—

382

—

C₂×Q₈

—

382

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

382

—

Q₁₆

—

382

—

Q₁₆

—

382

—

Q₁₆

—

191

—

100

Q₁₆

—

764

—

101

C₂×C₈

—

191

—

102

C₂×C₈

C₂²×D₁₉₁

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

764

—

105

C₄⋊C₄

—

764

—

106

C₄⋊C₄

—

764

—

107

C₄²

—

191

—

108

C₄²

—

764

—

109

C₁₆

—

110

C₁₆

—

382

—

111

Q₈

—

112

Q₈

—

113

Q₈

—

382

—

114

Q₈

—

764

—

115

Q₈

—

764

—

116

Q₈

—

764

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

191

—

120

C₂×C₄

—

191

—

121

C₂×C₄

—

764

—

122

C₂×C₄

—

764

—

123

C₂×C₄

D₄×D₁₉₁

yes

characteristic

124

C₈

—

382

—

125

C₈

D₄×D₁₉₁

yes

characteristic

126

C₈

#15690287

yes

characteristic

127

C₂²

D₈×D₁₉₁

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

382

—

131

C₄

—

191

—

132

C₄

—

1528

—

133

C₄

—

764

—

134

C₄

#20259295

yes

characteristic

135

C₄

#20259349

yes

characteristic

136

C₂

#27132857

yes

characteristic

137

C₂

Q₃₂×D₁₉₁

yes

characteristic

138

C₂

#27133102

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	#2209
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
65. Dic₁₉₁	Q₃₂	semidirect
69. C₁₉₁	#2209	semidirect

H ≅ G/N

type

65. Dic₁₉₁

Q₃₂

semidirect

69. C₁₉₁

#2209

semidirect

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

subgroup	index
C₂×Dic₁₅₂₈	2
C₂×#20259268	2
#27133041	2
#27132757	2
Dic₁₉₁⋊C₁₆	2
#27132466	2
#358×C₁₉₁	2
#2209	191

subgroup

index

C₂×Dic₁₅₂₈

Dic₁₉₁⋊C₁₆

#358×C₁₉₁

191

Subgroups — #37574569

Conjugacy classes of subgroups

Complements

Maximal subgroups