Group #37574690 — subgroups

Subgroups — #37574690

Order 24448.

126 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 43 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	#27133060	C₂	1	1	yes	maximal, characteristic, complemented
2	#27132779	C₂	1	1	yes	maximal, characteristic
3	#27132487	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132483	C₂	1	1	yes	maximal, characteristic, complemented
5	#27132474	C₂	1	1	yes	maximal, characteristic
6	#27132472	C₂	1	1	yes	maximal, characteristic
7	#365×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#20259344	C₂²	1	1	yes	characteristic
9	#20259330	—	2	1	no	—
10	#20259321	—	2	1	no	—
11	#20259262	C₂²	1	1	yes	characteristic
12	#20259254	C₂²	1	1	yes	characteristic
13	#20259253	—	2	1	no	—
14	#20259248	—	2	1	no	—
15	C₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
16	M₃₂×C₁₉₁	C₂²	1	1	yes	characteristic
17	#108×C₁₉₁	C₂²	1	1	yes	characteristic
18	#106×C₁₉₁	C₂²	1	1	yes	characteristic
19	#15690287	—	2	1	no	—
20	C₂×Dic₃₈₂	—	4	1	no	—
21	Dic₇₆₄	Q₈	1	1	yes	characteristic, complemented
22	#15690276	—	2	1	no	—
23	D₁₅₂₈	Q₈	1	1	yes	characteristic, complemented
24	#15690270	—	4	1	no	—
25	C₂×C₁₉₁⋊C₈	D₄	1	1	yes	characteristic
26	C₈×D₁₉₁	—	4	1	no	—
27	C₄⋊Dic₁₉₁	—	2	1	no	—
28	Dic₁₉₁⋊C₄	—	4	1	no	—
29	C₄×Dic₁₉₁	D₄	1	1	yes	characteristic
30	C₁₉₁⋊C₁₆	—	2	1	no	—
31	M₁₆×C₁₉₁	—	2	1	no	—
32	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₁₉₁	—	2	1	no	—
34	C₁₆×C₁₉₁	—	2	1	no	—
35	Dic₃₈₂	—	2	1	no	—
36	Dic₃₈₂	—	8	2	no	—
37	D₇₆₄	—	2	1	no	—
38	C₂²⋊D₁₉₁	—	4	1	no	—
39	C₂×Dic₁₉₁	—	2	1	no	—
40	C₂×Dic₁₉₁	—	4	1	no	—
41	C₄×D₁₉₁	—	4	1	no	—
42	C₁₉₁⋊C₈	SD₁₆	1	1	yes	characteristic
43	C₁₉₁⋊C₈	SD₁₆	1	1	yes	characteristic
44	C₂×C₄×C₁₉₁	—	4	1	no	—
45	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
46	C₈×C₁₉₁	—	4	1	no	—
47	C₈×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
48	C₈×C₁₉₁	#41	1	1	yes	characteristic
49	C₂²×C₁₉₁	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₂×D₁₉₁	—	4	1	no	—
51	C₄×C₁₉₁	—	8	1	no	—
52	C₄×C₁₉₁	C₂²⋊Q₈	1	1	yes	characteristic
53	C₄×C₁₉₁	C₂×SD₁₆	1	1	yes	characteristic
54	Dic₁₉₁	—	2	2	no	—
55	Dic₁₉₁	—	4	1	no	—
56	Dic₁₉₁	—	8	1	no	—
57	C₂×C₁₉₁	—	2	1	no	—
58	C₂×C₁₉₁	#476	1	1	yes	characteristic
59	D₁₉₁	—	8	1	no	—
60	C₁₉₁	#2230	1	1	yes	characteristic, complemented, Sylow
61	#2230	—	191	1	no	maximal, Sylow
62	#498	—	191	1	no	—
63	#443	—	191	1	no	—
64	#365	—	191	1	no	—
65	#365	D₁₉₁	1	1	yes	characteristic, complemented
66	#364	—	191	1	no	—
67	#360	—	191	1	no	—
68	#360	—	191	1	no	—
69	#135	—	191	1	no	—
70	#131	—	382	1	no	—
71	C₄⋊Q₈	—	382	1	no	—
72	M₃₂	—	191	1	no	—
73	M₃₂	C₂×D₁₉₁	1	1	yes	characteristic
74	#108	C₂×D₁₉₁	1	1	yes	characteristic
75	#106	—	382	1	no	—
76	#106	—	191	1	no	—
77	#106	C₂×D₁₉₁	1	1	yes	characteristic
78	#104	—	382	1	no	—
79	C₄×C₈	—	191	1	no	—
80	#41	—	382	1	no	—
81	C₂×Q₈	—	764	1	no	—
82	Q₁₆	—	191	1	no	—
83	SD₁₆	—	382	1	no	—
84	D₈	—	191	1	no	—
85	M₁₆	—	2	1	no	—
86	M₁₆	—	764	1	no	—
87	C₂×C₈	—	191	1	no	—
88	C₂×C₈	—	764	1	no	—
89	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
90	C₄⋊C₄	—	2	1	no	—
91	C₄⋊C₄	—	382	1	no	—
92	C₄⋊C₄	—	764	1	no	—
93	C₄²	—	191	1	no	—
94	C₁₆	—	2	1	no	—
95	C₁₆	—	382	1	no	—
96	Q₈	—	382	1	no	—
97	Q₈	—	1528	2	no	—
98	D₄	—	382	1	no	—
99	D₄	—	764	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	382	1	no	—
102	C₂×C₄	—	764	1	no	—
103	C₂×C₄	—	764	1	no	—
104	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
105	C₈	—	4	1	no	—
106	C₈	—	191	1	no	—
107	C₈	—	191	1	no	—
108	C₈	Q₈×D₁₉₁	1	1	yes	characteristic
109	C₈	#15690287	1	1	yes	characteristic
110	C₂²	—	764	1	no	—
111	C₂²	#20259349	1	1	yes	characteristic
112	C₄	—	8	1	no	—
113	C₄	—	382	2	no	—
114	C₄	—	1528	1	no	—
115	C₄	—	764	1	no	—
116	C₄	#20259300	1	1	yes	characteristic
117	C₄	SD₁₆×D₁₉₁	1	1	yes	characteristic
118	C₂	—	2	1	no	—
119	C₂	—	1528	1	no	—
120	C₂	#27132952	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27133060

C₂

yes

maximal, characteristic, complemented

#27132779

C₂

yes

maximal, characteristic

#27132487

C₂

yes

maximal, characteristic, complemented

#27132483

C₂

yes

maximal, characteristic, complemented

#27132474

C₂

yes

maximal, characteristic

#27132472

C₂

yes

maximal, characteristic

#365×C₁₉₁

C₂

yes

maximal, characteristic, complemented

#20259344

C₂²

yes

—

—

C₂²

yes

characteristic

#20259254

C₂²

yes

characteristic

#20259253

—

#20259248

—

C₈×Dic₁₉₁

C₂²

yes

characteristic

M₃₂×C₁₉₁

C₂²

yes

characteristic

#108×C₁₉₁

C₂²

yes

characteristic

#106×C₁₉₁

C₂²

yes

characteristic

#15690287

—

C₂×Dic₃₈₂

—

Dic₇₆₄

Q₈

yes

characteristic, complemented

#15690276

—

D₁₅₂₈

Q₈

yes

characteristic, complemented

#15690270

—

C₂×C₁₉₁⋊C₈

D₄

yes

characteristic

C₈×D₁₉₁

—

C₄⋊Dic₁₉₁

—

Dic₁₉₁⋊C₄

—

C₄×Dic₁₉₁

D₄

yes

characteristic

C₁₉₁⋊C₁₆

—

M₁₆×C₁₉₁

—

C₂×C₈×C₁₉₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₁

—

C₁₆×C₁₉₁

—

Dic₃₈₂

—

Dic₃₈₂

—

D₇₆₄

—

C₂²⋊D₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₄×D₁₉₁

—

C₁₉₁⋊C₈

SD₁₆

yes

characteristic

C₁₉₁⋊C₈

SD₁₆

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₈×C₁₉₁

—

C₈×C₁₉₁

C₂×Q₈

yes

C₈×C₁₉₁

yes

C₂²×C₁₉₁

yes

C₂×D₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂²⋊Q₈

yes

characteristic

C₄×C₁₉₁

C₂×SD₁₆

yes

characteristic

Dic₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

#476

yes

characteristic

D₁₉₁

—

C₁₉₁

#2230

yes

characteristic, complemented, Sylow

#2230

—

191

maximal, Sylow

#498

—

191

—

#443

—

191

—

#365

—

191

—

#365

D₁₉₁

yes

characteristic, complemented

#364

—

191

—

#360

—

191

—

#360

—

191

—

#135

—

191

—

#131

—

382

—

C₄⋊Q₈

—

382

—

M₃₂

—

191

—

M₃₂

C₂×D₁₉₁

yes

characteristic

#108

C₂×D₁₉₁

yes

characteristic

#106

—

382

—

#106

—

191

—

#106

C₂×D₁₉₁

yes

characteristic

#104

—

382

—

C₄×C₈

—

191

—

#41

—

382

—

C₂×Q₈

—

764

—

Q₁₆

—

191

—

SD₁₆

—

382

—

D₈

—

191

—

M₁₆

—

M₁₆

—

764

—

C₂×C₈

—

191

—

C₂×C₈

—

764

—

C₂×C₈

C₂²×D₁₉₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

764

—

C₄²

—

191

—

C₁₆

—

C₁₆

—

382

—

Q₈

—

382

—

Q₈

—

1528

—

D₄

—

382

—

D₄

—

764

—

100

C₂×C₄

—

101

C₂×C₄

—

382

—

102

C₂×C₄

—

764

—

103

C₂×C₄

—

764

—

104

C₂×C₄

D₄×D₁₉₁

yes

characteristic

105

C₈

—

106

C₈

—

191

—

107

C₈

—

191

—

108

C₈

Q₈×D₁₉₁

yes

characteristic

109

C₈

#15690287

yes

characteristic

110

C₂²

—

764

—

111

C₂²

#20259349

yes

characteristic

112

C₄

—

113

C₄

—

382

—

114

C₄

—

1528

—

115

C₄

—

764

—

116

C₄

#20259300

yes

characteristic

117

C₄

SD₁₆×D₁₉₁

yes

characteristic

118

C₂

—

119

C₂

—

1528

—

120

C₂

#27132952

yes

characteristic

	subgroup	G/H
center	C₂	#27132952
commutator, cosocle	C₂×C₈×C₁₉₁	C₂³
Frattini	C₂×C₈	C₂²×D₁₉₁
Fitting	#365×C₁₉₁	C₂
socle	C₂×C₁₉₁	#476
prime radical	#20259344	C₂²
hypercenter	#365	D₁₉₁
hypocenter	C₁₉₁	#2230
LCS	G ▷ C₂×C₈×C₁₉₁ ▷ C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

subgroup

G/H

center

C₂

#27132952

commutator, cosocle

C₂×C₈×C₁₉₁

C₂³

Frattini

C₂×C₈

C₂²×D₁₉₁

Fitting

#365×C₁₉₁

C₂

C₂×C₁₉₁

C₂²

D₁₉₁

C₁₉₁

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

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

N	H ≅ G/N	type
1. #27133060	C₂	semidirect
3. #27132487	C₂	semidirect
4. #27132483	C₂	semidirect
7. #365×C₁₉₁	C₂	semidirect
21. Dic₇₆₄	Q₈	semidirect
23. D₁₅₂₈	Q₈	semidirect
60. C₁₉₁	#2230	semidirect
65. #365	D₁₉₁	semidirect

H ≅ G/N

type

1. #27133060

C₂

semidirect

3. #27132487

C₂

semidirect

4. #27132483

C₂

semidirect

7. #365×C₁₉₁

C₂

semidirect

21. Dic₇₆₄

Q₈

semidirect

23. D₁₅₂₈

Q₈

semidirect

60. C₁₉₁

#2230

semidirect

65. #365

D₁₉₁

semidirect

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

subgroup	index
#27133060	2
#27132779	2
#27132487	2
#27132483	2
#27132474	2
#27132472	2
#365×C₁₉₁	2
#2230	191

subgroup

index

#365×C₁₉₁

191

Subgroups — #37574690

Conjugacy classes of subgroups

Complements

Maximal subgroups