Group #37574057 — subgroups

Subgroups — #37574057

Order 24448.

254 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 95 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×#20259256	C₂	1	2	yes	maximal
2	C₂×#20259247	C₂	1	2	yes	maximal
3	#27132569	C₂	1	1	yes	maximal, characteristic
4	#27132533	C₂	1	1	yes	maximal, characteristic
5	#336×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₈₂	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₉₁	—	2	2	no	—
8	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
10	#20259256	—	2	4	no	—
11	#20259247	—	2	4	no	—
12	C₁₉₁⋊#95	—	2	2	no	—
13	C₂²×C₈×C₁₉₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₈₂	—	2	2	no	—
16	C₂×Dic₃₈₂	—	2	1	no	—
17	C₂×Dic₃₈₂	—	2	2	no	—
18	C₂×Dic₃₈₂	Q₈	1	2	yes	—
19	C₂²×Dic₁₉₁	—	2	1	no	—
20	C₂²×Dic₁₉₁	—	2	1	no	—
21	C₂²×Dic₁₉₁	—	2	2	no	—
22	C₄⋊Dic₁₉₁	—	2	2	no	—
23	C₄⋊Dic₁₉₁	—	2	2	no	—
24	C₄⋊Dic₁₉₁	—	2	2	no	—
25	C₄⋊Dic₁₉₁	—	4	4	no	—
26	C₂²×C₄×C₁₉₁	D₄	1	2	yes	—
27	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₁₉₁	—	2	2	no	—
29	C₂×C₈×C₁₉₁	—	2	2	no	—
30	C₂×C₈×C₁₉₁	D₄	1	4	yes	—
31	C₄²×C₁₉₁	—	2	2	no	—
32	Dic₃₈₂	—	2	4	no	—
33	Dic₃₈₂	—	4	2	no	—
34	C₂×Dic₁₉₁	—	2	2	no	—
35	C₂×Dic₁₉₁	—	4	1	no	—
36	C₂×Dic₁₉₁	—	4	1	no	—
37	C₂×Dic₁₉₁	—	4	2	no	—
38	C₂×Dic₁₉₁	—	4	1	no	—
39	C₂×Dic₁₉₁	—	4	4	no	—
40	C₂×Dic₁₉₁	—	4	2	no	—
41	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₉₁	—	2	2	no	—
43	C₂×C₄×C₁₉₁	—	2	2	no	—
44	C₂×C₄×C₁₉₁	Q₁₆	1	4	yes	—
45	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₁₉₁	#41	1	2	yes	—
48	C₂×C₄×C₁₉₁	#41	1	1	yes	characteristic
49	C₈×C₁₉₁	—	2	4	no	—
50	C₂²×C₁₉₁	C₂²⋊Q₈	1	1	yes	characteristic
51	C₂²×C₁₉₁	#123	1	1	yes	characteristic
52	C₂²×C₁₉₁	#127	1	1	yes	characteristic
53	C₂²×C₁₉₁	C₂×Q₁₆	1	2	yes	—
54	C₂²×C₁₉₁	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₁₉₁	—	2	4	no	—
56	C₄×C₁₉₁	C₂²⋊Q₈	1	2	yes	—
57	C₄×C₁₉₁	#123	1	2	yes	—
58	Dic₁₉₁	—	4	2	no	—
59	Dic₁₉₁	—	8	2	no	—
60	Dic₁₉₁	—	8	1	no	—
61	C₂×C₁₉₁	#399	1	1	yes	characteristic
62	C₂×C₁₉₁	#477	1	2	yes	—
63	C₂×C₁₉₁	#484	1	2	yes	—
64	C₂×C₁₉₁	#494	1	1	yes	characteristic
65	C₂×C₁₉₁	#496	1	1	yes	characteristic
66	C₁₉₁	#2095	1	1	yes	characteristic, complemented, Sylow
67	#2095	—	191	1	no	maximal, Sylow
68	C₂×#107	—	191	2	no	—
69	C₂×#103	—	191	2	no	—
70	#391	—	191	1	no	—
71	#384	—	191	1	no	—
72	#336	D₁₉₁	1	1	yes	characteristic
73	C₂²×Q₈	—	191	1	no	—
74	C₂²×C₈	C₂×D₁₉₁	1	2	yes	—
75	C₂×C₄⋊C₄	—	382	2	no	—
76	C₂×C₄⋊C₄	—	191	2	no	—
77	C₂×C₄⋊C₄	—	191	1	no	—
78	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
79	#107	—	382	4	no	—
80	#103	—	382	4	no	—
81	#95	—	382	2	no	—
82	C₂×Q₈	—	382	1	no	—
83	C₂×Q₈	—	382	2	no	—
84	C₂×Q₈	—	382	2	no	—
85	C₂×Q₈	—	191	2	no	—
86	C₂²×C₄	—	382	1	no	—
87	C₂²×C₄	—	382	1	no	—
88	C₂²×C₄	—	382	2	no	—
89	C₂²×C₄	D₇₆₄	1	2	yes	—
90	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	D₇₆₄	1	4	yes	—
94	C₄⋊C₄	—	382	2	no	—
95	C₄⋊C₄	—	382	2	no	—
96	C₄⋊C₄	—	382	2	no	—
97	C₄⋊C₄	—	764	4	no	—
98	C₄²	—	2	2	no	—
99	C₂³	C₂×D₇₆₄	1	1	yes	characteristic
100	Q₈	—	382	4	no	—
101	Q₈	—	764	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	382	2	no	—
105	C₂×C₄	—	764	4	no	—
106	C₂×C₄	—	764	2	no	—
107	C₂×C₄	—	764	2	no	—
108	C₂×C₄	—	764	1	no	—
109	C₂×C₄	—	764	1	no	—
110	C₂×C₄	—	764	1	no	—
111	C₂×C₄	Dic₇₆₄	1	4	yes	—
112	C₂×C₄	C₂×D₇₆₄	1	2	yes	—
113	C₂×C₄	Q₈×D₁₉₁	1	1	yes	characteristic
114	C₂×C₄	#15690286	1	2	yes	—
115	C₂×C₄	#15690286	1	1	yes	characteristic
116	C₈	—	2	4	no	—
117	C₂²	#20259302	1	1	yes	characteristic
118	C₂²	#20259306	1	1	yes	characteristic
119	C₂²	#20259319	1	1	yes	characteristic
120	C₂²	C₂×Dic₇₆₄	1	2	yes	—
121	C₂²	#20259350	1	2	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	1528	2	no	—
124	C₄	—	1528	1	no	—
125	C₄	—	764	2	no	—
126	C₄	#20259302	1	2	yes	—
127	C₄	#20259306	1	2	yes	—
128	C₂	#27132600	1	1	yes	characteristic
129	C₂	#27132959	1	2	yes	—
130	C₂	#27132999	1	2	yes	—
131	C₂	#27133043	1	1	yes	characteristic
132	C₂	#27133052	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20259256

C₂

yes

maximal

C₂×#20259247

C₂

yes

maximal

#27132569

C₂

yes

maximal, characteristic

#27132533

C₂

yes

maximal, characteristic

#336×C₁₉₁

C₂

yes

maximal, characteristic

C₂²×Dic₃₈₂

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₉₁

—

C₂×C₄⋊Dic₁₉₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₉₁

C₂²

yes

—

#20259256

—

#20259247

—

C₁₉₁⋊#95

—

C₂²×C₈×C₁₉₁

C₂²

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

Q₈

yes

—

C₂²×Dic₁₉₁

—

C₂²×Dic₁₉₁

—

C₂²×Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₂²×C₄×C₁₉₁

D₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

D₄

yes

—

C₄²×C₁₉₁

—

Dic₃₈₂

—

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂³×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

Q₁₆

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

—

C₂×C₄×C₁₉₁

#41

yes

characteristic

C₈×C₁₉₁

—

C₂²×C₁₉₁

C₂²⋊Q₈

yes

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

C₂×Q₁₆

yes

—

C₂²×C₁₉₁

AGL(1,ℤ/8)

yes

—

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂²⋊Q₈

yes

—

C₄×C₁₉₁

#123

yes

—

Dic₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

#399

yes

characteristic

C₂×C₁₉₁

#477

yes

—

C₂×C₁₉₁

#484

yes

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#2095

—

191

maximal, Sylow

C₂×#107

—

191

—

C₂×#103

—

191

—

#391

—

191

—

#384

—

191

—

#336

D₁₉₁

yes

characteristic

C₂²×Q₈

—

191

—

C₂²×C₈

C₂×D₁₉₁

yes

—

C₂×C₄⋊C₄

—

382

—

C₂×C₄⋊C₄

—

191

—

C₂×C₄⋊C₄

—

191

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

#107

—

382

—

#103

—

382

—

#95

—

382

—

C₂×Q₈

—

382

—

C₂×Q₈

—

382

—

C₂×Q₈

—

382

—

C₂×Q₈

—

191

—

C₂²×C₄

—

382

—

C₂²×C₄

—

382

—

C₂²×C₄

—

382

—

C₂²×C₄

D₇₆₄

yes

—

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₇₆₄

yes

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

764

—

C₄²

—

C₂³

C₂×D₇₆₄

yes

characteristic

100

Q₈

—

382

—

101

Q₈

—

764

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

382

—

105

C₂×C₄

—

764

—

106

C₂×C₄

—

764

—

107

C₂×C₄

—

764

—

108

C₂×C₄

—

764

—

109

C₂×C₄

—

764

—

110

C₂×C₄

—

764

—

111

C₂×C₄

Dic₇₆₄

yes

—

112

C₂×C₄

C₂×D₇₆₄

yes

—

113

C₂×C₄

Q₈×D₁₉₁

yes

characteristic

114

C₂×C₄

#15690286

yes

—

115

C₂×C₄

#15690286

yes

characteristic

116

C₈

—

117

C₂²

#20259302

yes

characteristic

118

C₂²

#20259306

yes

characteristic

119

C₂²

#20259319

yes

characteristic

120

C₂²

C₂×Dic₇₆₄

yes

—

121

C₂²

#20259350

yes

—

122

C₄

—

123

C₄

—

1528

—

124

C₄

—

1528

—

125

C₄

—

764

—

126

C₄

#20259302

yes

—

127

C₄

#20259306

yes

—

128

C₂

#27132600

yes

characteristic

129

C₂

#27132959

yes

—

130

C₂

#27132999

yes

—

131

C₂

#27133043

yes

characteristic

132

C₂

#27133052

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₇₆₄
commutator, cosocle	C₂²×C₄×C₁₉₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting	#336×C₁₉₁	C₂
socle, prime radical	C₂³×C₁₉₁	C₂×D₄
hypercenter	#336	D₁₉₁
hypocenter	C₁₉₁	#2095
LCS	G ▷ C₂²×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₇₆₄

commutator, cosocle

C₂²×C₄×C₁₉₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉₁

Fitting

#336×C₁₉₁

C₂

socle, prime radical

C₂³×C₁₉₁

C₂×D₄

D₁₉₁

C₁₉₁

G ▷ C₂²×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
66. C₁₉₁	#2095	semidirect

H ≅ G/N

type

66. C₁₉₁

#2095

semidirect

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

subgroup	index
C₂×#20259256	2
C₂×#20259247	2
#27132569	2
#27132533	2
#336×C₁₉₁	2
#2095	191

subgroup

index

#336×C₁₉₁

191

Subgroups — #37574057

Conjugacy classes of subgroups

Complements

Maximal subgroups