Group #37571634 — subgroups

Subgroups — #37571634

Order 24448.

314 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 57 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	#27133298	C₂	1	1	yes	maximal, characteristic
2	#27132950	C₂	1	2	yes	maximal, complemented
3	#27132407	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132392	C₂	1	2	yes	maximal, complemented
5	#492×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄×D₁₉₁	—	2	2	no	—
7	#20259293	—	2	2	no	—
8	#20259290	—	2	2	no	—
9	C₄×D₇₆₄	C₂²	1	2	yes	—
10	#20259253	—	2	2	no	—
11	C₄⋊(C₁₉₁⋊C₈)	C₂²	1	2	yes	complemented
12	#20259242	—	2	2	no	—
13	C₂×SD₁₆×C₁₉₁	—	2	2	no	—
14	C₄⋊Q₈×C₁₉₁	C₂²	1	1	yes	characteristic, complemented
15	#127×C₁₉₁	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic, complemented
17	C₂³×D₁₉₁	—	2	2	no	—
18	C₂×D₇₆₄	D₄	1	2	yes	—
19	C₂×C₂²⋊D₁₉₁	—	2	4	no	—
20	D₄×D₁₉₁	—	4	4	no	—
21	D₄×D₁₉₁	—	4	4	no	—
22	C₂×C₄×D₁₉₁	—	2	2	no	—
23	C₂×C₁₉₁⋊C₈	—	2	2	no	—
24	C₄⋊Dic₁₉₁	D₄	1	2	yes	—
25	C₂²⋊Dic₁₉₁	—	2	2	no	—
26	(C₂×D₁₉₁)⋊C₄	—	2	2	no	—
27	C₂×Q₈×C₁₉₁	—	2	1	no	—
28	C₂×D₄×C₁₉₁	—	2	2	no	—
29	C₂×D₄×C₁₉₁	D₄	1	2	yes	—
30	SD₁₆×C₁₉₁	—	4	4	no	—
31	C₂×C₈×C₁₉₁	—	2	1	no	—
32	C₄⋊C₄×C₁₉₁	—	2	2	no	—
33	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
34	C₂²×D₁₉₁	—	2	2	no	—
35	C₂²×D₁₉₁	—	4	8	no	—
36	C₂²×D₁₉₁	—	8	2	no	—
37	D₇₆₄	—	2	2	no	—
38	D₇₆₄	SD₁₆	1	4	yes	complemented
39	C₂²⋊D₁₉₁	—	4	8	no	—
40	C₂×Dic₁₉₁	—	2	2	no	—
41	C₄×D₁₉₁	—	4	2	no	—
42	C₄×D₁₉₁	—	4	2	no	—
43	C₁₉₁⋊C₈	—	4	2	no	—
44	C₂³×C₁₉₁	—	2	2	no	—
45	Q₈×C₁₉₁	—	4	2	no	—
46	D₄×C₁₉₁	—	4	2	no	—
47	D₄×C₁₉₁	—	4	2	no	—
48	D₄×C₁₉₁	—	4	2	no	—
49	C₂×C₄×C₁₉₁	—	4	1	no	—
50	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
52	C₈×C₁₉₁	—	2	2	no	—
53	C₂²×C₁₉₁	—	4	4	no	—
54	C₂²×C₁₉₁	—	4	2	no	—
55	C₂²×C₁₉₁	#120	1	1	yes	characteristic
56	C₂×D₁₉₁	—	2	4	no	—
57	C₂×D₁₉₁	—	4	2	no	—
58	C₂×D₁₉₁	—	4	2	no	—
59	C₂×D₁₉₁	—	8	8	no	—
60	C₄×C₁₉₁	—	2	1	no	—
61	C₄×C₁₉₁	—	8	1	no	—
62	C₄×C₁₉₁	C₂×SD₁₆	1	2	yes	—
63	C₄×C₁₉₁	AGL(1,ℤ/8)	1	2	yes	—
64	Dic₁₉₁	—	4	2	no	—
65	C₂×C₁₉₁	—	8	2	no	—
66	C₂×C₁₉₁	#450	1	2	yes	—
67	C₂×C₁₉₁	#453	1	1	yes	characteristic
68	D₁₉₁	—	4	4	no	—
69	C₁₉₁	#1629	1	1	yes	characteristic, complemented, Sylow
70	#1629	—	191	1	no	maximal, Sylow
71	D₄²	—	191	1	no	—
72	#492	D₁₉₁	1	1	yes	characteristic, complemented
73	#476	—	191	2	no	—
74	#331	—	191	1	no	—
75	D₄⋊C₈	—	191	2	no	—
76	C₂²×D₄	—	382	2	no	—
77	C₂×SD₁₆	—	2	2	no	—
78	C₄⋊Q₈	C₂×D₁₉₁	1	1	yes	characteristic, complemented
79	#127	C₂×D₁₉₁	1	1	yes	characteristic
80	#121	—	382	2	no	—
81	#120	—	382	2	no	—
82	C₄×D₄	—	191	2	no	—
83	#106	—	382	2	no	—
84	C₄⋊C₈	—	191	2	no	—
85	#102	—	382	2	no	—
86	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic, complemented
87	C₂⁴	—	382	2	no	—
88	C₂×Q₈	—	2	1	no	—
89	C₂×D₄	—	2	2	no	—
90	C₂×D₄	—	382	4	no	—
91	C₂×D₄	—	191	2	no	—
92	C₂×D₄	—	764	4	no	—
93	C₂×D₄	—	764	4	no	—
94	C₂×D₄	C₂²⋊D₁₉₁	1	2	yes	—
95	C₂²×C₄	—	382	2	no	—
96	SD₁₆	—	4	4	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	382	2	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	191	2	no	—
101	C₂²⋊C₄	—	382	2	no	—
102	C₂²⋊C₄	—	382	2	no	—
103	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
104	C₂³	—	2	2	no	—
105	C₂³	—	382	2	no	—
106	C₂³	—	1528	2	no	—
107	C₂³	—	764	8	no	—
108	Q₈	—	4	2	no	—
109	D₄	—	4	2	no	—
110	D₄	—	4	2	no	—
111	D₄	—	4	2	no	—
112	D₄	—	382	2	no	—
113	D₄	—	191	4	no	—
114	D₄	—	764	8	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	382	2	no	—
117	C₂×C₄	—	764	2	no	—
118	C₂×C₄	—	764	2	no	—
119	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
120	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₈	—	764	2	no	—
123	C₂²	—	4	4	no	—
124	C₂²	—	4	2	no	—
125	C₂²	—	382	4	no	—
126	C₂²	—	1528	8	no	—
127	C₂²	—	764	2	no	—
128	C₂²	—	764	2	no	—
129	C₂²	#20259290	1	1	yes	characteristic
130	C₄	—	2	1	no	—
131	C₄	—	8	1	no	—
132	C₄	—	764	2	no	—
133	C₄	SD₁₆×D₁₉₁	1	2	yes	—
134	C₄	#20259349	1	2	yes	—
135	C₂	—	8	2	no	—
136	C₂	—	764	4	no	—
137	C₂	#27132812	1	2	yes	—
138	C₂	#27132830	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27133298

C₂

yes

maximal, characteristic

#27132950

C₂

yes

maximal, complemented

#27132407

C₂

yes

maximal, characteristic, complemented

#27132392

C₂

yes

maximal, complemented

#492×C₁₉₁

C₂

yes

maximal, characteristic, complemented

C₂×D₄×D₁₉₁

—

#20259293

—

#20259290

—

C₄×D₇₆₄

C₂²

yes

—

#20259253

—

C₄⋊(C₁₉₁⋊C₈)

C₂²

yes

complemented

#20259242

—

C₂×SD₁₆×C₁₉₁

—

C₄⋊Q₈×C₁₉₁

C₂²

yes

characteristic, complemented

#127×C₁₉₁

C₂²

yes

characteristic

C₄×C₈×C₁₉₁

C₂²

yes

characteristic, complemented

C₂³×D₁₉₁

—

C₂×D₇₆₄

D₄

yes

—

C₂×C₂²⋊D₁₉₁

—

D₄×D₁₉₁

—

D₄×D₁₉₁

—

C₂×C₄×D₁₉₁

—

C₂×C₁₉₁⋊C₈

—

C₄⋊Dic₁₉₁

D₄

yes

—

C₂²⋊Dic₁₉₁

—

(C₂×D₁₉₁)⋊C₄

—

C₂×Q₈×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

D₄

yes

—

SD₁₆×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄²×C₁₉₁

C₂³

yes

characteristic

C₂²×D₁₉₁

—

C₂²×D₁₉₁

—

C₂²×D₁₉₁

—

D₇₆₄

—

D₇₆₄

SD₁₆

yes

complemented

C₂²⋊D₁₉₁

—

C₂×Dic₁₉₁

—

C₄×D₁₉₁

—

C₄×D₁₉₁

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

—

Q₈×C₁₉₁

—

D₄×C₁₉₁

—

D₄×C₁₉₁

—

D₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₈×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

#120

yes

characteristic

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂×SD₁₆

yes

—

C₄×C₁₉₁

AGL(1,ℤ/8)

yes

—

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

#450

yes

—

C₂×C₁₉₁

#453

yes

characteristic

D₁₉₁

—

C₁₉₁

#1629

yes

characteristic, complemented, Sylow

#1629

—

191

maximal, Sylow

D₄²

—

191

—

#492

D₁₉₁

yes

characteristic, complemented

#476

—

191

—

#331

—

191

—

D₄⋊C₈

—

191

—

C₂²×D₄

—

382

—

C₂×SD₁₆

—

C₄⋊Q₈

C₂×D₁₉₁

yes

characteristic, complemented

#127

C₂×D₁₉₁

yes

characteristic

#121

—

382

—

#120

—

382

—

C₄×D₄

—

191

—

#106

—

382

—

C₄⋊C₈

—

191

—

#102

—

382

—

C₄×C₈

C₂×D₁₉₁

yes

characteristic, complemented

C₂⁴

—

382

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

382

—

C₂×D₄

—

191

—

C₂×D₄

—

764

—

C₂×D₄

—

764

—

C₂×D₄

C₂²⋊D₁₉₁

yes

—

C₂²×C₄

—

382

—

SD₁₆

—

C₂×C₈

—

C₂×C₈

—

382

—

C₄⋊C₄

—

100

C₄⋊C₄

—

191

—

101

C₂²⋊C₄

—

382

—

102

C₂²⋊C₄

—

382

—

103

C₄²

C₂²×D₁₉₁

yes

characteristic

104

C₂³

—

105

C₂³

—

382

—

106

C₂³

—

1528

—

107

C₂³

—

764

—

108

Q₈

—

109

D₄

—

110

D₄

—

111

D₄

—

112

D₄

—

382

—

113

D₄

—

191

—

114

D₄

—

764

—

115

C₂×C₄

—

116

C₂×C₄

—

382

—

117

C₂×C₄

—

764

—

118

C₂×C₄

—

764

—

119

C₂×C₄

D₄×D₁₉₁

yes

—

120

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

121

C₈

—

122

C₈

—

764

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

382

—

126

C₂²

—

1528

—

127

C₂²

—

764

—

128

C₂²

—

764

—

129

C₂²

#20259290

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

764

—

133

C₄

SD₁₆×D₁₉₁

yes

—

134

C₄

#20259349

yes

—

135

C₂

—

136

C₂

—

764

—

137

C₂

#27132812

yes

—

138

C₂

#27132830

yes

characteristic

	subgroup	G/H
center	C₂²	#20259290
commutator, cosocle	C₄²×C₁₉₁	C₂³
Frattini	C₄²	C₂²×D₁₉₁
Fitting	#492×C₁₉₁	C₂
socle	C₂²×C₁₉₁	#120
prime radical	#27133298	C₂
hypercenter	#492	D₁₉₁
hypocenter	C₁₉₁	#1629
LCS	G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#20259290

commutator, cosocle

C₄²×C₁₉₁

C₂³

Frattini

C₄²

C₂²×D₁₉₁

Fitting

#492×C₁₉₁

C₂

C₂²×C₁₉₁

C₂

D₁₉₁

C₁₉₁

G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
2. #27132950	C₂	semidirect
3. #27132407	C₂	semidirect
4. #27132392	C₂	semidirect
5. #492×C₁₉₁	C₂	semidirect
11. C₄⋊(C₁₉₁⋊C₈)	C₂²	semidirect
14. C₄⋊Q₈×C₁₉₁	C₂²	semidirect
16. C₄×C₈×C₁₉₁	C₂²	semidirect
38. D₇₆₄	SD₁₆	semidirect
69. C₁₉₁	#1629	semidirect
72. #492	D₁₉₁	semidirect
78. C₄⋊Q₈	C₂×D₁₉₁	semidirect
86. C₄×C₈	C₂×D₁₉₁	semidirect

H ≅ G/N

type

2. #27132950

C₂

semidirect

3. #27132407

C₂

semidirect

4. #27132392

C₂

semidirect

5. #492×C₁₉₁

C₂

semidirect

11. C₄⋊(C₁₉₁⋊C₈)

C₂²

semidirect

14. C₄⋊Q₈×C₁₉₁

C₂²

semidirect

16. C₄×C₈×C₁₉₁

C₂²

semidirect

38. D₇₆₄

SD₁₆

semidirect

69. C₁₉₁

#1629

semidirect

72. #492

D₁₉₁

semidirect

78. C₄⋊Q₈

C₂×D₁₉₁

semidirect

86. C₄×C₈

C₂×D₁₉₁

semidirect

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

subgroup	index
#27133298	2
#27132950	2
#27132407	2
#27132392	2
#492×C₁₉₁	2
#1629	191

subgroup

index

#492×C₁₉₁

191

Subgroups — #37571634

Conjugacy classes of subgroups

Complements

Maximal subgroups