Group #37575239 — subgroups

Subgroups — #37575239

Order 24448.

516 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 205 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	C₂×#20259322	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₃₈₂	C₂	1	2	yes	maximal
3	#27132569	C₂	1	2	yes	maximal
4	#27132548	C₂	1	8	yes	maximal
5	#27132529	C₂	1	1	yes	maximal, characteristic
6	#376×C₁₉₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₃₈₂	C₂²	1	2	yes	—
8	#20259322	C₄	1	8	yes	complemented
9	C₄×Dic₃₈₂	—	2	8	no	—
10	C₂×C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₁₉₁	C₂²	1	4	yes	—
12	C₂×Dic₁₉₁⋊C₄	C₂²	1	4	yes	—
13	C₂×C₄×Dic₁₉₁	C₂²	1	4	yes	—
14	C₁₉₁⋊#95	C₂²	1	4	yes	—
15	C₁₉₁⋊#95	C₂²	1	8	yes	—
16	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
19	#95×C₁₉₁	C₂²	1	4	yes	—
20	C₂×Dic₃₈₂	—	2	4	no	—
21	C₂×Dic₃₈₂	—	2	4	no	—
22	C₂×Dic₃₈₂	C₂×C₄	1	8	yes	—
23	C₂²×Dic₁₉₁	C₂³	1	4	yes	—
24	C₂²×Dic₁₉₁	C₂³	1	4	yes	—
25	C₄⋊Dic₁₉₁	—	2	4	no	—
26	C₄⋊Dic₁₉₁	C₂×C₄	1	16	yes	—
27	Dic₁₉₁⋊C₄	—	2	8	no	—
28	C₄×Dic₁₉₁	—	2	8	no	—
29	C₂²×C₄×C₁₉₁	C₂³	1	2	yes	—
30	C₂²×C₄×C₁₉₁	C₂³	1	2	yes	—
31	C₂²×C₄×C₁₉₁	C₂³	1	2	yes	—
32	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
33	C₄²×C₁₉₁	—	2	2	no	—
34	C₄²×C₁₉₁	C₂×C₄	1	4	yes	—
35	C₄²×C₁₉₁	D₄	1	4	yes	—
36	Dic₃₈₂	—	2	16	no	—
37	C₂×Dic₁₉₁	—	2	4	no	—
38	C₂×Dic₁₉₁	—	2	4	no	—
39	C₂×Dic₁₉₁	—	2	4	no	—
40	C₂×Dic₁₉₁	—	2	8	no	—
41	C₂×Dic₁₉₁	C₂²×C₄	1	8	yes	—
42	C₂³×C₁₉₁	C₂⁴	1	1	yes	characteristic
43	C₂×C₄×C₁₉₁	—	2	4	no	—
44	C₂×C₄×C₁₉₁	—	2	4	no	—
45	C₂×C₄×C₁₉₁	—	2	2	no	—
46	C₂×C₄×C₁₉₁	—	2	4	no	—
47	C₂×C₄×C₁₉₁	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₁₉₁	C₂×D₄	1	4	yes	—
50	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₉₁	#41	1	2	yes	—
52	C₂²×C₁₉₁	C₂³×C₄	1	1	yes	characteristic
53	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
54	C₂²×C₁₉₁	C₂×#41	1	1	yes	characteristic
55	C₂²×C₁₉₁	2₋¹⁺⁴	1	4	yes	—
56	C₄×C₁₉₁	—	2	4	no	—
57	C₄×C₁₉₁	—	2	4	no	—
58	C₄×C₁₉₁	—	4	2	no	—
59	C₄×C₁₉₁	C₄×D₄	1	4	yes	—
60	Dic₁₉₁	—	2	8	no	—
61	Dic₁₉₁	—	4	4	no	—
62	C₂×C₁₉₁	C₂×C₄×D₄	1	1	yes	characteristic
63	C₂×C₁₉₁	#519	1	2	yes	—
64	C₂×C₁₉₁	#536	1	2	yes	—
65	C₂×C₁₉₁	#541	1	2	yes	—
66	C₁₉₁	#2327	1	1	yes	characteristic, complemented, Sylow
67	#2327	—	191	1	no	maximal, Sylow
68	C₂×C₄⋊Q₈	—	191	1	no	—
69	C₂×C₄×Q₈	—	191	2	no	—
70	#391	—	191	2	no	—
71	#387	—	191	8	no	—
72	#382	—	191	1	no	—
73	#376	D₁₉₁	1	1	yes	characteristic
74	C₂²×Q₈	—	191	2	no	—
75	C₄⋊Q₈	—	191	8	no	—
76	C₄×Q₈	—	382	8	no	—
77	C₂×C₄⋊C₄	—	191	2	no	—
78	C₂×C₄⋊C₄	—	191	4	no	—
79	C₂×C₄⋊C₄	—	191	4	no	—
80	C₂×C₄²	—	191	4	no	—
81	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
82	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
83	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
84	#95	—	191	4	no	—
85	#95	—	191	8	no	—
86	#95	C₂×D₁₉₁	1	4	yes	—
87	C₂×Q₈	—	382	4	no	—
88	C₂×Q₈	—	382	4	no	—
89	C₂×Q₈	—	191	8	no	—
90	C₂²×C₄	—	191	4	no	—
91	C₂²×C₄	—	191	4	no	—
92	C₂²×C₄	C₂²×D₁₉₁	1	2	yes	—
93	C₂²×C₄	C₂²×D₁₉₁	1	2	yes	—
94	C₂²×C₄	C₂²×D₁₉₁	1	2	yes	—
95	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
96	C₄⋊C₄	—	382	8	no	—
97	C₄⋊C₄	—	382	4	no	—
98	C₄⋊C₄	—	191	16	no	—
99	C₄²	—	2	2	no	—
100	C₄²	—	382	8	no	—
101	C₄²	C₄×D₁₉₁	1	4	yes	—
102	C₄²	D₇₆₄	1	4	yes	—
103	C₂³	C₂³×D₁₉₁	1	1	yes	characteristic
104	Q₈	—	382	16	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	382	4	no	—
110	C₂×C₄	—	382	4	no	—
111	C₂×C₄	—	382	4	no	—
112	C₂×C₄	—	382	8	no	—
113	C₂×C₄	—	191	8	no	—
114	C₂×C₄	C₂×C₄×D₁₉₁	1	4	yes	—
115	C₂×C₄	C₂×C₄×D₁₉₁	1	2	yes	—
116	C₂×C₄	C₂×D₇₆₄	1	4	yes	—
117	C₂×C₄	C₂×D₇₆₄	1	2	yes	—
118	C₂×C₄	#15690287	1	2	yes	—
119	C₂²	C₂²×C₄×D₁₉₁	1	1	yes	characteristic
120	C₂²	C₂²×D₇₆₄	1	1	yes	characteristic
121	C₂²	C₂×#15690287	1	1	yes	characteristic
122	C₂²	#20259369	1	4	yes	—
123	C₄	—	2	4	no	—
124	C₄	—	2	4	no	—
125	C₄	—	4	2	no	—
126	C₄	—	382	8	no	—
127	C₄	—	764	4	no	—
128	C₄	C₄×D₇₆₄	1	4	yes	—
129	C₂	C₂×C₄×D₇₆₄	1	1	yes	characteristic
130	C₂	#27133140	1	2	yes	—
131	C₂	#27133239	1	2	yes	—
132	C₂	#27133275	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20259322

C₂

yes

maximal, characteristic

C₂×C₄×Dic₃₈₂

C₂

yes

maximal

#27132569

C₂

yes

maximal

#27132548

C₂

yes

maximal

#27132529

C₂

yes

maximal, characteristic

#376×C₁₉₁

C₂

yes

maximal, characteristic

C₂²×Dic₃₈₂

C₂²

yes

—

#20259322

C₄

yes

complemented

C₄×Dic₃₈₂

—

C₂×C₄⋊Dic₁₉₁

C₂²

yes

—

C₂×C₄⋊Dic₁₉₁

C₂²

yes

—

C₂×Dic₁₉₁⋊C₄

C₂²

yes

—

C₂×C₄×Dic₁₉₁

C₂²

yes

—

C₁₉₁⋊#95

C₂²

yes

—

C₁₉₁⋊#95

C₂²

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

#95×C₁₉₁

C₂²

yes

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

C₂×C₄

yes

—

C₂²×Dic₁₉₁

C₂³

yes

—

C₂²×Dic₁₉₁

C₂³

yes

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

C₂×C₄

yes

—

Dic₁₉₁⋊C₄

—

C₄×Dic₁₉₁

—

C₂²×C₄×C₁₉₁

C₂³

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₄²×C₁₉₁

—

C₄²×C₁₉₁

C₂×C₄

yes

—

C₄²×C₁₉₁

D₄

yes

—

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

C₂²×C₄

yes

—

C₂³×C₁₉₁

C₂⁴

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

—

C₂×C₄×C₁₉₁

C₂²×C₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

#41

yes

—

C₂²×C₁₉₁

C₂³×C₄

yes

characteristic

C₂²×C₁₉₁

C₂²×D₄

yes

characteristic

C₂²×C₁₉₁

C₂×#41

yes

characteristic

C₂²×C₁₉₁

2₋¹⁺⁴

yes

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

C₄×D₄

yes

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

C₂×C₄×D₄

yes

characteristic

C₂×C₁₉₁

#519

yes

—

C₂×C₁₉₁

#536

yes

—

C₂×C₁₉₁

#541

yes

—

C₁₉₁

#2327

yes

characteristic, complemented, Sylow

#2327

—

191

maximal, Sylow

C₂×C₄⋊Q₈

—

191

—

C₂×C₄×Q₈

—

191

—

#391

—

191

—

#387

—

191

—

#382

—

191

—

#376

D₁₉₁

yes

characteristic

C₂²×Q₈

—

191

—

C₄⋊Q₈

—

191

—

C₄×Q₈

—

382

—

C₂×C₄⋊C₄

—

191

—

C₂×C₄⋊C₄

—

191

—

C₂×C₄⋊C₄

—

191

—

C₂×C₄²

—

191

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

#95

—

191

—

#95

—

191

—

#95

C₂×D₁₉₁

yes

—

C₂×Q₈

—

382

—

C₂×Q₈

—

382

—

C₂×Q₈

—

191

—

C₂²×C₄

—

191

—

C₂²×C₄

—

191

—

C₂²×C₄

C₂²×D₁₉₁

yes

—

C₂²×C₄

C₂²×D₁₉₁

yes

—

C₂²×C₄

C₂²×D₁₉₁

yes

—

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

C₄⋊C₄

—

382

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

191

—

C₄²

—

100

C₄²

—

382

—

101

C₄²

C₄×D₁₉₁

yes

—

102

C₄²

D₇₆₄

yes

—

103

C₂³

C₂³×D₁₉₁

yes

characteristic

104

Q₈

—

382

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

382

—

110

C₂×C₄

—

382

—

111

C₂×C₄

—

382

—

112

C₂×C₄

—

382

—

113

C₂×C₄

—

191

—

114

C₂×C₄

C₂×C₄×D₁₉₁

yes

—

115

C₂×C₄

C₂×C₄×D₁₉₁

yes

—

116

C₂×C₄

C₂×D₇₆₄

yes

—

117

C₂×C₄

C₂×D₇₆₄

yes

—

118

C₂×C₄

#15690287

yes

—

119

C₂²

C₂²×C₄×D₁₉₁

yes

characteristic

120

C₂²

C₂²×D₇₆₄

yes

characteristic

121

C₂²

C₂×#15690287

yes

characteristic

122

C₂²

#20259369

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

382

—

127

C₄

—

764

—

128

C₄

C₄×D₇₆₄

yes

—

129

C₂

C₂×C₄×D₇₆₄

yes

characteristic

130

C₂

#27133140

yes

—

131

C₂

#27133239

yes

—

132

C₂

#27133275

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉₁
commutator	C₂²×C₁₉₁	C₂³×C₄
Fitting	#376×C₁₉₁	C₂
socle, prime radical, cosocle	C₂³×C₁₉₁	C₂⁴
hypercenter	#376	D₁₉₁
hypocenter	C₁₉₁	#2327
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ #376

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₉₁

commutator

C₂²×C₁₉₁

C₂³×C₄

Fitting

#376×C₁₉₁

C₂

socle, prime radical, cosocle

C₂³×C₁₉₁

C₂⁴

D₁₉₁

C₁₉₁

G ▷ C₂²×C₁₉₁ ▷ C₁₉₁

C₁ ◁ C₂³ ◁ #376

N	H ≅ G/N	type
8. #20259322	C₄	semidirect
66. C₁₉₁	#2327	semidirect

H ≅ G/N

type

8. #20259322

C₄

semidirect

66. C₁₉₁

#2327

semidirect

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

subgroup	index
C₂×#20259322	2
C₂×C₄×Dic₃₈₂	2
#27132569	2
#27132548	2
#27132529	2
#376×C₁₉₁	2
#2327	191

subgroup

index

C₂×C₄×Dic₃₈₂

#376×C₁₉₁

191

Subgroups — #37575239

Conjugacy classes of subgroups

Complements

Maximal subgroups