Group #37574321 — subgroups

Subgroups — #37574321

Order 24448.

152 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 73 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	C₄×C₈×D₁₉₁	C₂	1	1	yes	maximal, characteristic
2	Dic₁₉₁⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	(C₂×D₁₉₁)⋊C₁₆	C₂	1	2	yes	maximal
4	#27132439	C₂	1	1	yes	maximal, characteristic, complemented
5	#346×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₁₉₁	C₄	1	1	yes	characteristic
7	C₂×C₈×D₁₉₁	C₂²	1	1	yes	characteristic
8	C₄²×D₁₉₁	C₄	1	1	yes	characteristic
9	C₂×C₁₉₁⋊C₁₆	C₂²	1	2	yes	—
10	C₄×C₁₉₁⋊C₈	C₄	1	1	yes	characteristic
11	C₈×Dic₁₉₁	C₄	1	1	yes	characteristic
12	C₈×Dic₁₉₁	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₁₉₁	C₂²	1	2	yes	—
14	C₄×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₁₉₁	C₈	1	1	yes	characteristic
16	C₂×C₄×D₁₉₁	C₈	1	1	yes	characteristic
17	C₂×C₄×D₁₉₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₉₁⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₉₁⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₈×D₁₉₁	—	2	2	no	—
21	C₈×D₁₉₁	—	2	2	no	—
22	C₄×Dic₁₉₁	C₈	1	1	yes	characteristic
23	C₄×Dic₁₉₁	C₈	1	1	yes	characteristic
24	C₄×Dic₁₉₁	C₂×C₄	1	1	yes	characteristic
25	C₁₉₁⋊C₁₆	—	2	2	no	—
26	C₂×C₈×C₁₉₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₁₉₁	C₂³	1	1	yes	characteristic
28	C₄²×C₁₉₁	C₂×C₄	1	1	yes	characteristic
29	C₁₆×C₁₉₁	—	2	2	no	—
30	C₂²×D₁₉₁	C₂×C₈	1	1	yes	characteristic
31	C₂×Dic₁₉₁	C₂×C₈	1	1	yes	characteristic
32	C₂×Dic₁₉₁	C₂×C₈	1	1	yes	characteristic
33	C₂×Dic₁₉₁	C₂×C₈	1	1	yes	characteristic
34	C₄×D₁₉₁	—	2	2	no	—
35	C₄×D₁₉₁	—	2	1	no	—
36	C₄×D₁₉₁	—	2	1	no	—
37	C₄×D₁₉₁	—	2	2	no	—
38	C₁₉₁⋊C₈	—	2	1	no	—
39	C₁₉₁⋊C₈	—	2	1	no	—
40	C₂×C₄×C₁₉₁	C₂×C₈	1	1	yes	characteristic
41	C₂×C₄×C₁₉₁	C₂×C₈	1	1	yes	characteristic
42	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
43	C₈×C₁₉₁	M₁₆	1	2	yes	—
44	C₈×C₁₉₁	#41	1	2	yes	—
45	C₂²×C₁₉₁	C₂²×C₈	1	1	yes	characteristic
46	C₂×D₁₉₁	—	2	1	no	—
47	C₂×D₁₉₁	—	2	1	no	—
48	C₂×D₁₉₁	M₃₂	1	2	yes	—
49	C₄×C₁₉₁	—	2	1	no	—
50	C₄×C₁₉₁	—	2	1	no	—
51	C₄×C₁₉₁	#117	1	1	yes	characteristic
52	C₄×C₁₉₁	C₂×M₁₆	1	1	yes	characteristic
53	Dic₁₉₁	—	2	1	no	—
54	Dic₁₉₁	—	2	1	no	—
55	Dic₁₉₁	M₃₂	1	2	yes	—
56	C₂×C₁₉₁	#431	1	1	yes	characteristic
57	C₂×C₁₉₁	C₂×M₃₂	1	1	yes	characteristic
58	C₂×C₁₉₁	C₂×M₃₂	1	1	yes	characteristic
59	D₁₉₁	—	2	2	no	—
60	C₁₉₁	#2163	1	1	yes	characteristic, complemented, Sylow
61	#2163	—	191	1	no	maximal, Sylow
62	C₂×C₄×C₈	—	191	1	no	—
63	C₄⋊C₁₆	—	191	2	no	—
64	C₂²⋊C₁₆	—	191	2	no	—
65	#346	—	191	1	no	—
66	#346	D₁₉₁	1	1	yes	characteristic, complemented
67	C₂²×C₈	—	191	1	no	—
68	C₂²×C₈	—	191	1	no	—
69	C₂×C₄²	—	191	1	no	—
70	C₂×C₁₆	—	191	2	no	—
71	C₂×C₁₆	C₂×D₁₉₁	1	2	yes	—
72	C₄×C₈	—	191	1	no	—
73	C₄×C₈	—	191	1	no	—
74	C₄×C₈	—	191	1	no	—
75	C₄×C₈	C₂×D₁₉₁	1	1	yes	characteristic
76	C₂²×C₄	—	191	1	no	—
77	C₂²×C₄	—	191	1	no	—
78	C₂²×C₄	—	191	1	no	—
79	C₂×C₈	—	382	2	no	—
80	C₂×C₈	—	382	2	no	—
81	C₂×C₈	—	191	1	no	—
82	C₂×C₈	—	191	1	no	—
83	C₂×C₈	C₄×D₁₉₁	1	1	yes	characteristic
84	C₂×C₈	C₂²×D₁₉₁	1	1	yes	characteristic
85	C₄²	—	191	1	no	—
86	C₄²	—	191	1	no	—
87	C₄²	—	191	1	no	—
88	C₄²	C₄×D₁₉₁	1	1	yes	characteristic
89	C₁₆	—	2	2	no	—
90	C₁₆	—	382	2	no	—
91	C₂³	—	191	1	no	—
92	C₂×C₄	—	382	2	no	—
93	C₂×C₄	—	382	2	no	—
94	C₂×C₄	—	382	1	no	—
95	C₂×C₄	—	382	1	no	—
96	C₂×C₄	—	191	1	no	—
97	C₂×C₄	—	191	1	no	—
98	C₂×C₄	—	191	1	no	—
99	C₂×C₄	C₈×D₁₉₁	1	1	yes	characteristic
100	C₂×C₄	C₈×D₁₉₁	1	1	yes	characteristic
101	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
102	C₈	—	382	1	no	—
103	C₈	—	382	1	no	—
104	C₈	#15690270	1	2	yes	—
105	C₈	#15690287	1	2	yes	—
106	C₂²	—	382	1	no	—
107	C₂²	—	382	1	no	—
108	C₂²	—	191	2	no	—
109	C₂²	C₂×C₈×D₁₉₁	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	382	1	no	—
113	C₄	—	382	1	no	—
114	C₄	—	191	2	no	—
115	C₄	#20259279	1	1	yes	characteristic
116	C₄	C₂×#15690270	1	1	yes	characteristic
117	C₂	—	382	2	no	—
118	C₂	#27132714	1	1	yes	characteristic
119	C₂	M₃₂×D₁₉₁	1	1	yes	characteristic
120	C₂	M₃₂×D₁₉₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₈×D₁₉₁

C₂

yes

maximal, characteristic

Dic₁₉₁⋊C₁₆

C₂

yes

maximal, complemented

(C₂×D₁₉₁)⋊C₁₆

C₂

yes

maximal

#27132439

C₂

yes

maximal, characteristic, complemented

#346×C₁₉₁

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₉₁

C₄

yes

characteristic

C₂×C₈×D₁₉₁

C₂²

yes

characteristic

C₄²×D₁₉₁

C₄

yes

characteristic

C₂×C₁₉₁⋊C₁₆

C₂²

yes

—

C₄×C₁₉₁⋊C₈

C₄

yes

characteristic

C₈×Dic₁₉₁

C₄

yes

characteristic

C₈×Dic₁₉₁

C₂²

yes

characteristic

C₂×C₁₆×C₁₉₁

C₂²

yes

—

C₄×C₈×C₁₉₁

C₂²

yes

characteristic

C₂×C₄×D₁₉₁

C₈

yes

characteristic

C₂×C₄×D₁₉₁

C₈

yes

characteristic

C₂×C₄×D₁₉₁

C₂×C₄

yes

characteristic

C₂×C₁₉₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₉₁⋊C₈

C₂×C₄

yes

characteristic

C₈×D₁₉₁

—

C₈×D₁₉₁

—

C₄×Dic₁₉₁

C₈

yes

characteristic

C₄×Dic₁₉₁

C₈

yes

characteristic

C₄×Dic₁₉₁

C₂×C₄

yes

characteristic

C₁₉₁⋊C₁₆

—

C₂×C₈×C₁₉₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₉₁

C₂³

yes

characteristic

C₄²×C₁₉₁

C₂×C₄

yes

characteristic

C₁₆×C₁₉₁

—

C₂²×D₁₉₁

C₂×C₈

yes

characteristic

C₂×Dic₁₉₁

C₂×C₈

yes

characteristic

C₂×Dic₁₉₁

C₂×C₈

yes

characteristic

C₂×Dic₁₉₁

C₂×C₈

yes

characteristic

C₄×D₁₉₁

—

C₄×D₁₉₁

—

C₄×D₁₉₁

—

C₄×D₁₉₁

—

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

—

C₂×C₄×C₁₉₁

C₂×C₈

yes

characteristic

C₂×C₄×C₁₉₁

C₂×C₈

yes

characteristic

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₈×C₁₉₁

M₁₆

yes

—

C₈×C₁₉₁

#41

yes

—

C₂²×C₁₉₁

C₂²×C₈

yes

characteristic

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

M₃₂

yes

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#117

yes

characteristic

C₄×C₁₉₁

C₂×M₁₆

yes

characteristic

Dic₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

M₃₂

yes

—

C₂×C₁₉₁

#431

yes

characteristic

C₂×C₁₉₁

C₂×M₃₂

yes

characteristic

C₂×C₁₉₁

C₂×M₃₂

yes

characteristic

D₁₉₁

—

C₁₉₁

#2163

yes

characteristic, complemented, Sylow

#2163

—

191

maximal, Sylow

C₂×C₄×C₈

—

191

—

C₄⋊C₁₆

—

191

—

C₂²⋊C₁₆

—

191

—

#346

—

191

—

#346

D₁₉₁

yes

characteristic, complemented

C₂²×C₈

—

191

—

C₂²×C₈

—

191

—

C₂×C₄²

—

191

—

C₂×C₁₆

—

191

—

C₂×C₁₆

C₂×D₁₉₁

yes

—

C₄×C₈

—

191

—

C₄×C₈

—

191

—

C₄×C₈

—

191

—

C₄×C₈

C₂×D₁₉₁

yes

characteristic

C₂²×C₄

—

191

—

C₂²×C₄

—

191

—

C₂²×C₄

—

191

—

C₂×C₈

—

382

—

C₂×C₈

—

382

—

C₂×C₈

—

191

—

C₂×C₈

—

191

—

C₂×C₈

C₄×D₁₉₁

yes

characteristic

C₂×C₈

C₂²×D₁₉₁

yes

characteristic

C₄²

—

191

—

C₄²

—

191

—

C₄²

—

191

—

C₄²

C₄×D₁₉₁

yes

characteristic

C₁₆

—

C₁₆

—

382

—

C₂³

—

191

—

C₂×C₄

—

382

—

C₂×C₄

—

382

—

C₂×C₄

—

382

—

C₂×C₄

—

382

—

C₂×C₄

—

191

—

C₂×C₄

—

191

—

C₂×C₄

—

191

—

C₂×C₄

C₈×D₁₉₁

yes

characteristic

100

C₂×C₄

C₈×D₁₉₁

yes

characteristic

101

C₂×C₄

C₂×C₄×D₁₉₁

yes

characteristic

102

C₈

—

382

—

103

C₈

—

382

—

104

C₈

#15690270

yes

—

105

C₈

#15690287

yes

—

106

C₂²

—

382

—

107

C₂²

—

382

—

108

C₂²

—

191

—

109

C₂²

C₂×C₈×D₁₉₁

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

382

—

113

C₄

—

382

—

114

C₄

—

191

—

115

C₄

#20259279

yes

characteristic

116

C₄

C₂×#15690270

yes

characteristic

117

C₂

—

382

—

118

C₂

#27132714

yes

characteristic

119

C₂

M₃₂×D₁₉₁

yes

characteristic

120

C₂

M₃₂×D₁₉₁

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₉₁
commutator, socle	C₂²×C₁₉₁	C₂²×C₈
Fitting	#346×C₁₉₁	C₂
prime radical	C₂²×D₁₉₁	C₂×C₈
cosocle	C₂×C₈×C₁₉₁	C₂³
hypercenter	#346	D₁₉₁
hypocenter	C₁₉₁	#2163
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂×C₈ ◁ #346

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₉₁

commutator, socle

C₂²×C₁₉₁

C₂²×C₈

Fitting

#346×C₁₉₁

C₂

prime radical

C₂²×D₁₉₁

C₂×C₈

cosocle

C₂×C₈×C₁₉₁

C₂³

D₁₉₁

C₁₉₁

G ▷ C₂²×C₁₉₁ ▷ C₁₉₁

C₁ ◁ C₂×C₈ ◁ #346

N	H ≅ G/N	type
2. Dic₁₉₁⋊C₁₆	C₂	semidirect
4. #27132439	C₂	semidirect
5. #346×C₁₉₁	C₂	semidirect
60. C₁₉₁	#2163	semidirect
66. #346	D₁₉₁	semidirect

H ≅ G/N

type

2. Dic₁₉₁⋊C₁₆

C₂

semidirect

4. #27132439

C₂

semidirect

5. #346×C₁₉₁

C₂

semidirect

60. C₁₉₁

#2163

semidirect

66. #346

D₁₉₁

semidirect

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

subgroup	index
C₄×C₈×D₁₉₁	2
Dic₁₉₁⋊C₁₆	2
(C₂×D₁₉₁)⋊C₁₆	2
#27132439	2
#346×C₁₉₁	2
#2163	191

subgroup

index

C₄×C₈×D₁₉₁

Dic₁₉₁⋊C₁₆

(C₂×D₁₉₁)⋊C₁₆

#27132439

#346×C₁₉₁

#2163

191

Subgroups — #37574321

Conjugacy classes of subgroups

Complements

Maximal subgroups