Group #37574593 — subgroups

Subgroups — #37574593

Order 24448.

170 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 47 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	#27133099	C₂	1	2	yes	maximal, complemented
2	#27132781	C₂	1	1	yes	maximal, characteristic, complemented
3	#27132484	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132476	C₂	1	2	yes	maximal, complemented
5	#493×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#20259344	C₂²	1	1	yes	characteristic
7	#20259328	—	2	1	no	—
8	#20259266	—	2	2	no	—
9	#20259263	—	2	2	no	—
10	#20259262	C₂²	1	2	yes	—
11	#20259258	C₂²	1	1	yes	characteristic
12	#20259250	—	2	1	no	—
13	C₂×D₈×C₁₉₁	—	2	1	no	—
14	C₂×D₈×C₁₉₁	C₂²	1	2	yes	—
15	#127×C₁₉₁	—	2	1	no	—
16	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
17	#15690287	—	2	1	no	—
18	Dic₇₆₄	D₄	1	1	yes	characteristic, complemented
19	#15690276	—	2	1	no	—
20	D₁₅₂₈	D₄	1	1	yes	characteristic, complemented
21	#15690271	—	2	1	no	—
22	#15690270	—	4	1	no	—
23	C₈×D₁₉₁	—	4	1	no	—
24	C₁₉₁⋊C₁₆	—	2	2	no	—
25	C₂×D₄×C₁₉₁	—	2	2	no	—
26	C₂×D₄×C₁₉₁	—	4	2	no	—
27	D₈×C₁₉₁	—	2	2	no	—
28	D₈×C₁₉₁	—	2	2	no	—
29	D₈×C₁₉₁	—	4	1	no	—
30	D₈×C₁₉₁	—	4	1	no	—
31	C₂×C₈×C₁₉₁	D₄	1	1	yes	characteristic
32	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
33	C₄²×C₁₉₁	D₄	1	1	yes	characteristic
34	Dic₃₈₂	—	2	1	no	—
35	D₇₆₄	—	2	1	no	—
36	C₂²⋊D₁₉₁	—	4	1	no	—
37	C₄×D₁₉₁	—	4	1	no	—
38	C₁₉₁⋊C₈	—	4	1	no	—
39	C₂³×C₁₉₁	—	4	2	no	—
40	D₄×C₁₉₁	—	4	2	no	—
41	D₄×C₁₉₁	—	4	2	no	—
42	D₄×C₁₉₁	—	8	4	no	—
43	C₂×C₄×C₁₉₁	—	2	1	no	—
44	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
45	C₈×C₁₉₁	D₈	1	1	yes	characteristic
46	C₈×C₁₉₁	D₈	1	1	yes	characteristic
47	C₈×C₁₉₁	C₂×D₄	1	1	yes	characteristic
48	C₈×C₁₉₁	#41	1	1	yes	characteristic
49	C₂²×C₁₉₁	—	8	4	no	—
50	C₂²×C₁₉₁	—	8	2	no	—
51	C₂²×C₁₉₁	AGL(1,ℤ/8)	1	1	yes	characteristic
52	C₂×D₁₉₁	—	4	1	no	—
53	C₄×C₁₉₁	—	2	2	no	—
54	C₄×C₁₉₁	#121	1	1	yes	characteristic
55	C₄×C₁₉₁	C₂×D₈	1	1	yes	characteristic
56	Dic₁₉₁	—	4	1	no	—
57	C₂×C₁₉₁	—	2	1	no	—
58	C₂×C₁₉₁	—	16	2	no	—
59	C₂×C₁₉₁	#459	1	1	yes	characteristic
60	D₁₉₁	—	8	1	no	—
61	C₁₉₁	#2213	1	1	yes	characteristic, complemented, Sylow
62	#2213	—	191	1	no	maximal, Sylow
63	#509	—	191	2	no	—
64	#493	D₁₉₁	1	1	yes	characteristic, complemented
65	#443	—	191	1	no	—
66	#364	—	191	1	no	—
67	#361	—	191	2	no	—
68	#135	—	191	1	no	—
69	C₂×D₈	—	2	1	no	—
70	C₂×D₈	C₂×D₁₉₁	1	2	yes	—
71	#131	—	382	1	no	—
72	#127	—	2	1	no	—
73	SD₃₂	—	382	2	no	—
74	D₁₆	—	382	2	no	—
75	M₃₂	—	191	2	no	—
76	#108	—	191	1	no	—
77	#104	—	382	1	no	—
78	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
79	#41	—	382	1	no	—
80	C₂×D₄	—	2	2	no	—
81	C₂×D₄	—	4	2	no	—
82	Q₁₆	—	191	1	no	—
83	SD₁₆	—	382	1	no	—
84	D₈	—	2	2	no	—
85	D₈	—	2	2	no	—
86	D₈	—	4	1	no	—
87	D₈	—	4	1	no	—
88	D₈	—	191	1	no	—
89	M₁₆	—	382	1	no	—
90	M₁₆	—	764	1	no	—
91	C₂×C₈	—	764	1	no	—
92	C₂×C₈	C₂²⋊D₁₉₁	1	1	yes	characteristic
93	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
94	C₄²	C₂²⋊D₁₉₁	1	1	yes	characteristic
95	C₁₆	—	382	2	no	—
96	C₂³	—	4	2	no	—
97	Q₈	—	382	1	no	—
98	D₄	—	4	2	no	—
99	D₄	—	4	2	no	—
100	D₄	—	8	4	no	—
101	D₄	—	382	1	no	—
102	D₄	—	764	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	764	1	no	—
105	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
106	C₈	—	764	1	no	—
107	C₈	#15690272	1	1	yes	characteristic
108	C₈	#15690272	1	1	yes	characteristic
109	C₈	D₄×D₁₉₁	1	1	yes	characteristic
110	C₈	#15690286	1	1	yes	characteristic
111	C₂²	—	8	4	no	—
112	C₂²	—	8	2	no	—
113	C₂²	—	764	1	no	—
114	C₂²	#20259348	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	764	1	no	—
117	C₄	#20259293	1	1	yes	characteristic
118	C₄	C₂×#15690272	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	16	2	no	—
121	C₂	—	1528	1	no	—
122	C₂	#27132855	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27133099

C₂

yes

maximal, complemented

#27132781

C₂

yes

maximal, characteristic, complemented

#27132484

C₂

yes

maximal, characteristic, complemented

#27132476

C₂

yes

maximal, complemented

#493×C₁₉₁

C₂

yes

maximal, characteristic, complemented

#20259344

C₂²

yes

—

—

—

C₂²

yes

—

#20259258

C₂²

yes

characteristic

#20259250

—

C₂×D₈×C₁₉₁

—

C₂×D₈×C₁₉₁

C₂²

yes

—

#127×C₁₉₁

—

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

#15690287

—

Dic₇₆₄

D₄

yes

characteristic, complemented

#15690276

—

D₁₅₂₈

D₄

yes

characteristic, complemented

#15690271

—

#15690270

—

C₈×D₁₉₁

—

C₁₉₁⋊C₁₆

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

—

D₈×C₁₉₁

—

D₈×C₁₉₁

—

D₈×C₁₉₁

—

D₈×C₁₉₁

—

C₂×C₈×C₁₉₁

D₄

yes

characteristic

C₂×C₈×C₁₉₁

C₂³

yes

characteristic

C₄²×C₁₉₁

D₄

yes

characteristic

Dic₃₈₂

—

D₇₆₄

—

C₂²⋊D₁₉₁

—

C₄×D₁₉₁

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

—

D₄×C₁₉₁

—

D₄×C₁₉₁

—

D₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₈×C₁₉₁

D₈

yes

characteristic

C₈×C₁₉₁

D₈

yes

characteristic

C₈×C₁₉₁

C₂×D₄

yes

characteristic

C₈×C₁₉₁

#41

yes

characteristic

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

AGL(1,ℤ/8)

yes

characteristic

C₂×D₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#121

yes

characteristic

C₄×C₁₉₁

C₂×D₈

yes

characteristic

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

#459

yes

characteristic

D₁₉₁

—

C₁₉₁

#2213

yes

characteristic, complemented, Sylow

#2213

—

191

maximal, Sylow

#509

—

191

—

#493

D₁₉₁

yes

characteristic, complemented

#443

—

191

—

#364

—

191

—

#361

—

191

—

#135

—

191

—

C₂×D₈

—

C₂×D₈

C₂×D₁₉₁

yes

—

#131

—

382

—

#127

—

SD₃₂

—

382

—

D₁₆

—

382

—

M₃₂

—

191

—

#108

—

191

—

#104

—

382

—

C₄×C₈

C₂×D₁₉₁

yes

characteristic

#41

—

382

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

191

—

SD₁₆

—

382

—

D₈

—

D₈

—

D₈

—

D₈

—

D₈

—

191

—

M₁₆

—

382

—

M₁₆

—

764

—

C₂×C₈

—

764

—

C₂×C₈

C₂²⋊D₁₉₁

yes

characteristic

C₂×C₈

C₂²×D₁₉₁

yes

characteristic

C₄²

C₂²⋊D₁₉₁

yes

characteristic

C₁₆

—

382

—

C₂³

—

Q₈

—

382

—

D₄

—

D₄

—

100

D₄

—

101

D₄

—

382

—

102

D₄

—

764

—

103

C₂×C₄

—

104

C₂×C₄

—

764

—

105

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

106

C₈

—

764

—

107

C₈

#15690272

yes

characteristic

108

C₈

#15690272

yes

characteristic

109

C₈

D₄×D₁₉₁

yes

characteristic

110

C₈

#15690286

yes

characteristic

111

C₂²

—

112

C₂²

—

113

C₂²

—

764

—

114

C₂²

#20259348

yes

characteristic

115

C₄

—

116

C₄

—

764

—

117

C₄

#20259293

yes

characteristic

118

C₄

C₂×#15690272

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

—

1528

—

122

C₂

#27132855

yes

characteristic

	subgroup	G/H
center	C₂	#27132855
commutator, cosocle	C₂×C₈×C₁₉₁	C₂³
Frattini	C₂×C₈	C₂²×D₁₉₁
Fitting	#493×C₁₉₁	C₂
socle	C₂×C₁₉₁	#459
hypercenter	#493	D₁₉₁
hypocenter	C₁₉₁	#2213
LCS	G ▷ C₂×C₈×C₁₉₁ ▷ C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #493

subgroup

G/H

center

C₂

#27132855

commutator, cosocle

C₂×C₈×C₁₉₁

C₂³

Frattini

C₂×C₈

C₂²×D₁₉₁

Fitting

#493×C₁₉₁

C₂

C₂×C₁₉₁

D₁₉₁

C₁₉₁

G ▷ C₂×C₈×C₁₉₁ ▷ C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #493

N	H ≅ G/N	type
1. #27133099	C₂	semidirect
2. #27132781	C₂	semidirect
3. #27132484	C₂	semidirect
4. #27132476	C₂	semidirect
5. #493×C₁₉₁	C₂	semidirect
18. Dic₇₆₄	D₄	semidirect
20. D₁₅₂₈	D₄	semidirect
61. C₁₉₁	#2213	semidirect
64. #493	D₁₉₁	semidirect

H ≅ G/N

type

1. #27133099

C₂

semidirect

2. #27132781

C₂

semidirect

3. #27132484

C₂

semidirect

4. #27132476

C₂

semidirect

5. #493×C₁₉₁

C₂

semidirect

18. Dic₇₆₄

D₄

semidirect

20. D₁₅₂₈

D₄

semidirect

61. C₁₉₁

#2213

semidirect

64. #493

D₁₉₁

semidirect

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

subgroup	index
#27133099	2
#27132781	2
#27132484	2
#27132476	2
#493×C₁₉₁	2
#2213	191

subgroup

index

#493×C₁₉₁

191

Subgroups — #37574593

Conjugacy classes of subgroups

Complements

Maximal subgroups