Group #37573473 — subgroups

Subgroups — #37573473

Order 24448.

292 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 129 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×C₄⋊(C₁₉₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₁₉₁	C₂	1	1	yes	maximal, characteristic
3	#27132425	C₂	1	4	yes	maximal
4	C₂×C₄⋊Q₈×C₁₉₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₉₁⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
7	C₂×C₄×Dic₁₉₁	—	2	1	no	—
8	C₄⋊(C₁₉₁⋊C₈)	—	2	2	no	—
9	C₁₉₁⋊#95	—	2	1	no	—
10	C₂²×Q₈×C₁₉₁	—	2	1	no	—
11	C₄⋊Q₈×C₁₉₁	—	2	2	no	—
12	C₄⋊Q₈×C₁₉₁	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₁₉₁	—	2	1	no	—
14	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
16	C₂²×Dic₁₉₁	—	2	2	no	—
17	C₂×C₁₉₁⋊C₈	—	2	2	no	—
18	C₂×C₁₉₁⋊C₈	—	2	4	no	—
19	C₄⋊Dic₁₉₁	—	2	2	no	—
20	C₄⋊Dic₁₉₁	Q₈	1	4	yes	complemented
21	C₄×Dic₁₉₁	—	4	2	no	—
22	C₂×Q₈×C₁₉₁	—	2	4	no	—
23	C₂×Q₈×C₁₉₁	—	4	2	no	—
24	C₂×Q₈×C₁₉₁	—	4	2	no	—
25	C₂²×C₄×C₁₉₁	—	2	2	no	—
26	C₂²×C₄×C₁₉₁	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₉₁	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₁₉₁	—	2	4	no	—
30	C₄⋊C₄×C₁₉₁	—	2	2	no	—
31	C₄⋊C₄×C₁₉₁	C₂×C₄	1	4	yes	—
32	C₄²×C₁₉₁	C₂×C₄	1	2	yes	—
33	C₄²×C₁₉₁	D₄	1	2	yes	—
34	C₂×Dic₁₉₁	—	2	4	no	—
35	C₂×Dic₁₉₁	—	4	2	no	—
36	C₂×Dic₁₉₁	—	4	2	no	—
37	C₁₉₁⋊C₈	—	4	2	no	—
38	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₁₉₁	—	4	8	no	—
40	C₂×C₄×C₁₉₁	—	2	2	no	—
41	C₂×C₄×C₁₉₁	—	2	4	no	—
42	C₂×C₄×C₁₉₁	—	4	2	no	—
43	C₂×C₄×C₁₉₁	—	4	2	no	—
44	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₁₉₁	D₈	1	2	yes	—
47	C₂×C₄×C₁₉₁	SD₁₆	1	2	yes	—
48	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₉₁	C₂×Q₈	1	2	yes	—
51	C₂×C₄×C₁₉₁	#41	1	2	yes	—
52	C₂²×C₁₉₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₁₉₁	C₂²⋊Q₈	1	2	yes	—
54	C₂²×C₁₉₁	C₂×D₈	1	1	yes	characteristic
55	C₂²×C₁₉₁	C₂×SD₁₆	1	1	yes	characteristic
56	C₂²×C₁₉₁	#137	1	2	yes	—
57	C₄×C₁₉₁	—	2	2	no	—
58	C₄×C₁₉₁	—	4	4	no	—
59	C₄×C₁₉₁	#102	1	4	yes	—
60	C₄×C₁₉₁	C₄×Q₈	1	2	yes	—
61	C₄×C₁₉₁	#124	1	1	yes	characteristic
62	C₄×C₁₉₁	C₄⋊Q₈	1	1	yes	characteristic
63	Dic₁₉₁	—	4	4	no	—
64	C₂×C₁₉₁	#391	1	1	yes	characteristic
65	C₂×C₁₉₁	C₂×#102	1	1	yes	characteristic
66	C₂×C₁₉₁	#419	1	1	yes	characteristic
67	C₂×C₁₉₁	#474	1	2	yes	—
68	C₂×C₁₉₁	#475	1	2	yes	—
69	C₁₉₁	#1988	1	1	yes	characteristic, complemented, Sylow
70	#1988	—	191	1	no	maximal, Sylow
71	C₂×C₄⋊Q₈	D₁₉₁	1	1	yes	characteristic
72	C₂×C₄⋊C₈	—	191	1	no	—
73	C₄×C₄⋊C₄	—	191	1	no	—
74	#340	—	191	4	no	—
75	C₂²×Q₈	—	2	1	no	—
76	C₂²×C₈	—	191	2	no	—
77	C₄⋊Q₈	—	2	2	no	—
78	C₄⋊Q₈	Dic₁₉₁	1	4	yes	complemented
79	C₂×C₄⋊C₄	—	2	1	no	—
80	C₂×C₄⋊C₄	—	191	2	no	—
81	C₂×C₄⋊C₄	C₂×D₁₉₁	1	2	yes	—
82	C₂×C₄²	—	382	1	no	—
83	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
84	C₄⋊C₈	—	382	2	no	—
85	#95	—	382	1	no	—
86	C₂×Q₈	—	2	4	no	—
87	C₂×Q₈	—	4	2	no	—
88	C₂×Q₈	—	4	2	no	—
89	C₂²×C₄	—	2	2	no	—
90	C₂²×C₄	—	382	2	no	—
91	C₂²×C₄	C₂²⋊D₁₉₁	1	1	yes	characteristic
92	C₂²×C₄	C₂²⋊D₁₉₁	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
94	C₂×C₈	—	382	2	no	—
95	C₂×C₈	—	382	4	no	—
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	382	2	no	—
99	C₄⋊C₄	—	191	4	no	—
100	C₄⋊C₄	C₂×Dic₁₉₁	1	4	yes	—
101	C₄²	—	764	2	no	—
102	C₄²	C₂×Dic₁₉₁	1	2	yes	—
103	C₄²	C₂²⋊D₁₉₁	1	2	yes	—
104	C₂³	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
105	Q₈	—	4	8	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	382	4	no	—
111	C₂×C₄	—	764	2	no	—
112	C₂×C₄	—	764	2	no	—
113	C₂×C₄	C₂²⋊Dic₁₉₁	1	2	yes	—
114	C₂×C₄	C₂²⋊Dic₁₉₁	1	2	yes	—
115	C₂×C₄	#15690272	1	2	yes	—
116	C₂×C₄	#15690275	1	2	yes	—
117	C₂×C₄	C₂²×Dic₁₉₁	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
119	C₂×C₄	Q₈×D₁₉₁	1	2	yes	—
120	C₂×C₄	#15690288	1	2	yes	—
121	C₈	—	764	2	no	—
122	C₂²	C₂×C₂²⋊Dic₁₉₁	1	1	yes	characteristic
123	C₂²	#20259298	1	2	yes	—
124	C₂²	C₂×#15690272	1	1	yes	characteristic
125	C₂²	C₂×#15690275	1	1	yes	characteristic
126	C₂²	#20259353	1	2	yes	—
127	C₄	—	2	2	no	—
128	C₄	—	4	4	no	—
129	C₄	—	764	4	no	—
130	C₄	#20259243	1	4	yes	—
131	C₄	Q₈×Dic₁₉₁	1	2	yes	—
132	C₄	#20259311	1	1	yes	characteristic
133	C₄	#20259320	1	1	yes	characteristic
134	C₂	#27132570	1	1	yes	characteristic
135	C₂	C₂×#20259243	1	1	yes	characteristic
136	C₂	#27132672	1	1	yes	characteristic
137	C₂	#27132940	1	2	yes	—
138	C₂	#27132945	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₁₉₁⋊C₈)

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₁₉₁

C₂

yes

maximal, characteristic

#27132425

C₂

yes

maximal

C₂×C₄⋊Q₈×C₁₉₁

C₂

yes

maximal, characteristic

C₂²×C₁₉₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₉₁

C₂²

yes

—

C₂×C₄×Dic₁₉₁

—

C₄⋊(C₁₉₁⋊C₈)

—

C₁₉₁⋊#95

—

C₂²×Q₈×C₁₉₁

—

C₄⋊Q₈×C₁₉₁

—

C₄⋊Q₈×C₁₉₁

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₉₁

—

C₂×C₄⋊C₄×C₁₉₁

C₂²

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₂²×Dic₁₉₁

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

Q₈

yes

complemented

C₄×Dic₁₉₁

—

C₂×Q₈×C₁₉₁

—

C₂×Q₈×C₁₉₁

—

C₂×Q₈×C₁₉₁

—

C₂²×C₄×C₁₉₁

—

C₂²×C₄×C₁₉₁

D₄

yes

characteristic

C₂²×C₄×C₁₉₁

D₄

yes

characteristic

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

C₂×C₄

yes

—

C₄²×C₁₉₁

C₂×C₄

yes

—

C₄²×C₁₉₁

D₄

yes

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

C₂×D₄

yes

characteristic

Q₈×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₁

D₈

yes

—

C₂×C₄×C₁₉₁

SD₁₆

yes

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×Q₈

yes

—

C₂×C₄×C₁₉₁

#41

yes

—

C₂²×C₁₉₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₁

C₂²⋊Q₈

yes

—

C₂²×C₁₉₁

C₂×D₈

yes

characteristic

C₂²×C₁₉₁

C₂×SD₁₆

yes

characteristic

C₂²×C₁₉₁

#137

yes

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#102

yes

—

C₄×C₁₉₁

C₄×Q₈

yes

—

C₄×C₁₉₁

#124

yes

characteristic

C₄×C₁₉₁

C₄⋊Q₈

yes

characteristic

Dic₁₉₁

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

—

C₂×C₁₉₁

#475

yes

—

C₁₉₁

#1988

yes

characteristic, complemented, Sylow

#1988

—

191

maximal, Sylow

C₂×C₄⋊Q₈

D₁₉₁

yes

characteristic

C₂×C₄⋊C₈

—

191

—

C₄×C₄⋊C₄

—

191

—

#340

—

191

—

C₂²×Q₈

—

C₂²×C₈

—

191

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₁₉₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

191

—

C₂×C₄⋊C₄

C₂×D₁₉₁

yes

—

C₂×C₄²

—

382

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

C₄⋊C₈

—

382

—

#95

—

382

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

382

—

C₂²×C₄

C₂²⋊D₁₉₁

yes

characteristic

C₂²×C₄

C₂²⋊D₁₉₁

yes

characteristic

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

C₂×C₈

—

382

—

C₂×C₈

—

382

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

191

—

100

C₄⋊C₄

C₂×Dic₁₉₁

yes

—

101

C₄²

—

764

—

102

C₄²

C₂×Dic₁₉₁

yes

—

103

C₄²

C₂²⋊D₁₉₁

yes

—

104

C₂³

C₂×C₂²⋊D₁₉₁

yes

characteristic

105

Q₈

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

382

—

111

C₂×C₄

—

764

—

112

C₂×C₄

—

764

—

113

C₂×C₄

C₂²⋊Dic₁₉₁

yes

—

114

C₂×C₄

C₂²⋊Dic₁₉₁

yes

—

115

C₂×C₄

#15690272

yes

—

116

C₂×C₄

#15690275

yes

—

117

C₂×C₄

C₂²×Dic₁₉₁

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

119

C₂×C₄

Q₈×D₁₉₁

yes

—

120

C₂×C₄

#15690288

yes

—

121

C₈

—

764

—

122

C₂²

C₂×C₂²⋊Dic₁₉₁

yes

characteristic

123

C₂²

#20259298

yes

—

124

C₂²

C₂×#15690272

yes

characteristic

125

C₂²

C₂×#15690275

yes

characteristic

126

C₂²

#20259353

yes

—

127

C₄

—

128

C₄

—

129

C₄

—

764

—

130

C₄

#20259243

yes

—

131

C₄

Q₈×Dic₁₉₁

yes

—

132

C₄

#20259311

yes

characteristic

133

C₄

#20259320

yes

characteristic

134

C₂

#27132570

yes

characteristic

135

C₂

C₂×#20259243

yes

characteristic

136

C₂

#27132672

yes

characteristic

137

C₂

#27132940

yes

—

138

C₂

#27132945

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉₁
commutator	C₂×C₄×C₁₉₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting	C₂×C₄⋊Q₈×C₁₉₁	C₂
socle, prime radical	C₂³×C₁₉₁	C₂×D₄
cosocle	C₂²×C₄×C₁₉₁	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₉₁
hypocenter	C₁₉₁	#1988
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉₁

commutator

C₂×C₄×C₁₉₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₁

Fitting

C₂×C₄⋊Q₈×C₁₉₁

C₂

socle, prime radical

C₂³×C₁₉₁

C₂×D₄

cosocle

C₂²×C₄×C₁₉₁

C₂³

hypercenter

C₂×C₄⋊Q₈

D₁₉₁

hypocenter

C₁₉₁

#1988

LCS

G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₁₉₁	C₄	semidirect
20. C₄⋊Dic₁₉₁	Q₈	semidirect
69. C₁₉₁	#1988	semidirect
78. C₄⋊Q₈	Dic₁₉₁	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₁₉₁

C₄

semidirect

20. C₄⋊Dic₁₉₁

Q₈

semidirect

69. C₁₉₁

#1988

semidirect

78. C₄⋊Q₈

Dic₁₉₁

semidirect

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

subgroup	index
C₂×C₄⋊(C₁₉₁⋊C₈)	2
C₄×C₄⋊Dic₁₉₁	2
#27132425	2
C₂×C₄⋊Q₈×C₁₉₁	2
#1988	191

subgroup

index

C₂×C₄⋊(C₁₉₁⋊C₈)

C₄×C₄⋊Dic₁₉₁

#27132425

C₂×C₄⋊Q₈×C₁₉₁

#1988

191

Subgroups — #37573473

Conjugacy classes of subgroups

Complements

Maximal subgroups