Group #37571604 — subgroups

Subgroups — #37571604

Order 24448.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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	#27133379	C₂	1	1	yes	maximal, characteristic, complemented
2	#27133037	C₂	1	1	yes	maximal, characteristic, complemented
3	#27132859	C₂	1	2	yes	maximal, complemented
4	#27132390	C₂	1	2	yes	maximal, complemented
5	#333×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15690274	—	2	2	no	—
7	C₂×D₁₅₂₈	—	2	2	no	—
8	#20259319	C₂²	1	1	yes	characteristic
9	#20259311	—	2	2	no	—
10	#20259297	—	2	2	no	—
11	C₄×D₇₆₄	C₂²	1	2	yes	—
12	#20259280	—	2	2	no	—
13	#20259244	—	2	2	no	—
14	C₄×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₉₁	C₂²	1	2	yes	—
17	C₂×D₇₆₄	—	2	2	no	—
18	C₂×D₇₆₄	—	2	2	no	—
19	C₂×D₇₆₄	—	2	1	no	—
20	C₂×D₇₆₄	D₄	1	2	yes	—
21	C₂×C₄×D₁₉₁	—	2	2	no	—
22	#15690274	—	4	4	no	—
23	D₁₅₂₈	—	4	4	no	—
24	C₂×C₁₉₁⋊C₈	—	2	1	no	—
25	C₄⋊Dic₁₉₁	D₄	1	2	yes	—
26	(C₂×D₁₉₁)⋊C₄	—	2	4	no	—
27	(C₂×D₁₉₁)⋊C₄	—	2	2	no	—
28	C₄×Dic₁₉₁	—	2	2	no	—
29	C₂×Q₈×C₁₉₁	D₄	1	2	yes	—
30	C₂×C₈×C₁₉₁	—	2	2	no	—
31	C₄⋊C₄×C₁₉₁	—	2	2	no	—
32	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
33	C₂²×D₁₉₁	—	2	2	no	—
34	C₂²×D₁₉₁	—	4	1	no	—
35	D₇₆₄	—	2	2	no	—
36	D₇₆₄	—	2	2	no	—
37	D₇₆₄	—	4	2	no	—
38	D₇₆₄	—	4	2	no	—
39	D₇₆₄	—	4	2	no	—
40	D₇₆₄	—	8	2	no	—
41	C₂×Dic₁₉₁	—	2	2	no	—
42	C₄×D₁₉₁	—	4	2	no	—
43	C₄×D₁₉₁	—	4	2	no	—
44	C₁₉₁⋊C₈	—	2	2	no	—
45	Q₈×C₁₉₁	—	4	2	no	—
46	C₂×C₄×C₁₉₁	—	2	2	no	—
47	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
49	C₈×C₁₉₁	—	4	2	no	—
50	C₂²×C₁₉₁	#120	1	1	yes	characteristic
51	C₂×D₁₉₁	—	4	2	no	—
52	C₂×D₁₉₁	—	4	2	no	—
53	C₂×D₁₉₁	—	4	2	no	—
54	C₂×D₁₉₁	—	8	2	no	—
55	C₂×D₁₉₁	—	8	1	no	—
56	C₄×C₁₉₁	—	2	1	no	—
57	C₄×C₁₉₁	—	4	2	no	—
58	C₄×C₁₉₁	#135	1	2	yes	—
59	C₄×C₁₉₁	AGL(1,ℤ/8)	1	2	yes	—
60	Dic₁₉₁	—	4	2	no	—
61	C₂×C₁₉₁	#449	1	2	yes	—
62	C₂×C₁₉₁	#453	1	1	yes	characteristic
63	D₁₉₁	—	8	2	no	—
64	D₁₉₁	—	16	1	no	—
65	C₁₉₁	#1624	1	1	yes	characteristic, complemented, Sylow
66	#1624	—	191	1	no	maximal, Sylow
67	#555	—	191	1	no	—
68	#492	—	191	1	no	—
69	#459	—	191	2	no	—
70	#333	D₁₉₁	1	1	yes	characteristic, complemented
71	D₄⋊C₈	—	191	2	no	—
72	C₂×SD₁₆	—	382	2	no	—
73	C₂×D₈	—	382	2	no	—
74	C₄⋊Q₈	C₂×D₁₉₁	1	1	yes	characteristic
75	#127	—	191	1	no	—
76	#124	—	382	2	no	—
77	#121	—	382	2	no	—
78	C₄×D₄	—	191	2	no	—
79	#117	—	382	2	no	—
80	C₄⋊C₈	C₂×D₁₉₁	1	2	yes	—
81	#102	—	382	2	no	—
82	C₄×C₈	—	191	1	no	—
83	C₂×Q₈	D₇₆₄	1	2	yes	—
84	C₂×D₄	—	382	2	no	—
85	C₂×D₄	—	382	2	no	—
86	C₂×D₄	—	382	1	no	—
87	C₂×D₄	—	191	2	no	—
88	C₂²×C₄	—	382	2	no	—
89	SD₁₆	—	764	4	no	—
90	D₈	—	764	4	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	382	1	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	191	2	no	—
95	C₂²⋊C₄	—	382	4	no	—
96	C₂²⋊C₄	—	382	2	no	—
97	C₄²	—	382	2	no	—
98	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
99	C₂³	—	382	2	no	—
100	C₂³	—	764	1	no	—
101	Q₈	—	4	2	no	—
102	D₄	—	382	2	no	—
103	D₄	—	382	2	no	—
104	D₄	—	1528	2	no	—
105	D₄	—	764	2	no	—
106	D₄	—	764	2	no	—
107	D₄	—	764	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	382	2	no	—
110	C₂×C₄	—	764	2	no	—
111	C₂×C₄	—	764	2	no	—
112	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
113	C₂×C₄	C₂×D₇₆₄	1	1	yes	characteristic
114	C₈	—	4	2	no	—
115	C₈	—	382	2	no	—
116	C₂²	—	1528	2	no	—
117	C₂²	—	1528	1	no	—
118	C₂²	—	764	2	no	—
119	C₂²	—	764	2	no	—
120	C₂²	—	764	2	no	—
121	C₂²	#20259292	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	4	2	no	—
124	C₄	—	764	2	no	—
125	C₄	#20259345	1	2	yes	—
126	C₄	#20259349	1	2	yes	—
127	C₂	—	3056	1	no	—
128	C₂	—	1528	2	no	—
129	C₂	#27132811	1	2	yes	—
130	C₂	#27132833	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27133379

C₂

yes

maximal, characteristic, complemented

#27133037

C₂

yes

maximal, characteristic, complemented

#27132859

C₂

yes

maximal, complemented

#27132390

C₂

yes

maximal, complemented

#333×C₁₉₁

C₂

yes

maximal, characteristic, complemented

C₂×#15690274

—

C₂×D₁₅₂₈

—

#20259319

C₂²

yes

characteristic

#20259311

—

#20259297

—

C₄×D₇₆₄

C₂²

yes

—

#20259280

—

#20259244

—

C₄×C₁₉₁⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₉₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₁

C₂²

yes

—

C₂×D₇₆₄

—

C₂×D₇₆₄

—

C₂×D₇₆₄

—

C₂×D₇₆₄

D₄

yes

—

C₂×C₄×D₁₉₁

—

#15690274

—

D₁₅₂₈

—

C₂×C₁₉₁⋊C₈

—

C₄⋊Dic₁₉₁

D₄

yes

—

(C₂×D₁₉₁)⋊C₄

—

(C₂×D₁₉₁)⋊C₄

—

C₄×Dic₁₉₁

—

C₂×Q₈×C₁₉₁

D₄

yes

—

C₂×C₈×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄²×C₁₉₁

C₂³

yes

characteristic

C₂²×D₁₉₁

—

C₂²×D₁₉₁

—

D₇₆₄

—

D₇₆₄

—

D₇₆₄

—

D₇₆₄

—

D₇₆₄

—

D₇₆₄

—

C₂×Dic₁₉₁

—

C₄×D₁₉₁

—

C₄×D₁₉₁

—

C₁₉₁⋊C₈

—

Q₈×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₈×C₁₉₁

—

C₂²×C₁₉₁

#120

yes

characteristic

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#135

yes

—

C₄×C₁₉₁

AGL(1,ℤ/8)

yes

—

Dic₁₉₁

—

C₂×C₁₉₁

#449

yes

—

C₂×C₁₉₁

#453

yes

characteristic

D₁₉₁

—

D₁₉₁

—

C₁₉₁

#1624

yes

characteristic, complemented, Sylow

#1624

—

191

maximal, Sylow

#555

—

191

—

#492

—

191

—

#459

—

191

—

#333

D₁₉₁

yes

characteristic, complemented

D₄⋊C₈

—

191

—

C₂×SD₁₆

—

382

—

C₂×D₈

—

382

—

C₄⋊Q₈

C₂×D₁₉₁

yes

characteristic

#127

—

191

—

#124

—

382

—

#121

—

382

—

C₄×D₄

—

191

—

#117

—

382

—

C₄⋊C₈

C₂×D₁₉₁

yes

—

#102

—

382

—

C₄×C₈

—

191

—

C₂×Q₈

D₇₆₄

yes

—

C₂×D₄

—

382

—

C₂×D₄

—

382

—

C₂×D₄

—

382

—

C₂×D₄

—

191

—

C₂²×C₄

—

382

—

SD₁₆

—

764

—

D₈

—

764

—

C₂×C₈

—

C₂×C₈

—

382

—

C₄⋊C₄

—

C₄⋊C₄

—

191

—

C₂²⋊C₄

—

382

—

C₂²⋊C₄

—

382

—

C₄²

—

382

—

C₄²

C₂²×D₁₉₁

yes

characteristic

C₂³

—

382

—

100

C₂³

—

764

—

101

Q₈

—

102

D₄

—

382

—

103

D₄

—

382

—

104

D₄

—

1528

—

105

D₄

—

764

—

106

D₄

—

764

—

107

D₄

—

764

—

108

C₂×C₄

—

109

C₂×C₄

—

382

—

110

C₂×C₄

—

764

—

111

C₂×C₄

—

764

—

112

C₂×C₄

D₄×D₁₉₁

yes

—

113

C₂×C₄

C₂×D₇₆₄

yes

characteristic

114

C₈

—

115

C₈

—

382

—

116

C₂²

—

1528

—

117

C₂²

—

1528

—

118

C₂²

—

764

—

119

C₂²

—

764

—

120

C₂²

—

764

—

121

C₂²

#20259292

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

764

—

125

C₄

#20259345

yes

—

126

C₄

#20259349

yes

—

127

C₂

—

3056

—

128

C₂

—

1528

—

129

C₂

#27132811

yes

—

130

C₂

#27132833

yes

characteristic

	subgroup	G/H
center	C₂²	#20259292
commutator, cosocle	C₄²×C₁₉₁	C₂³
Frattini	C₄²	C₂²×D₁₉₁
Fitting	#333×C₁₉₁	C₂
socle	C₂²×C₁₉₁	#120
hypercenter	#333	D₁₉₁
hypocenter	C₁₉₁	#1624
LCS	G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#20259292

commutator, cosocle

C₄²×C₁₉₁

C₂³

Frattini

C₄²

C₂²×D₁₉₁

Fitting

#333×C₁₉₁

C₂

C₂²×C₁₉₁

D₁₉₁

C₁₉₁

G ▷ C₄²×C₁₉₁ ▷ C₂²×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
1. #27133379	C₂	semidirect
2. #27133037	C₂	semidirect
3. #27132859	C₂	semidirect
4. #27132390	C₂	semidirect
5. #333×C₁₉₁	C₂	semidirect
65. C₁₉₁	#1624	semidirect
70. #333	D₁₉₁	semidirect

H ≅ G/N

type

1. #27133379

C₂

semidirect

2. #27133037

C₂

semidirect

3. #27132859

C₂

semidirect

4. #27132390

C₂

semidirect

5. #333×C₁₉₁

C₂

semidirect

65. C₁₉₁

#1624

semidirect

70. #333

D₁₉₁

semidirect

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

subgroup	index
#27133379	2
#27133037	2
#27132859	2
#27132390	2
#333×C₁₉₁	2
#1624	191

subgroup

index

#333×C₁₉₁

191

Subgroups — #37571604

Conjugacy classes of subgroups

Complements

Maximal subgroups