Group #37575240 — subgroups

Subgroups — #37575240

Order 24448.

516 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 237 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×Q₈×Dic₁₉₁	C₂	1	2	yes	maximal
2	#27132570	C₂	1	2	yes	maximal
3	#27132549	C₂	1	8	yes	maximal
4	#27132529	C₂	1	1	yes	maximal, characteristic
5	#27132509	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₁₉₁	C₂	1	1	yes	maximal, characteristic
7	Q₈×Dic₁₉₁	—	2	8	no	—
8	C₂×C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₁₉₁	C₂²	1	4	yes	—
10	C₂×C₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₁₉₁	C₂²	1	4	yes	—
12	C₂×C₄×Dic₁₉₁	C₂²	1	1	yes	characteristic
13	C₁₉₁⋊#95	C₂²	1	4	yes	—
14	C₁₉₁⋊#95	C₂²	1	4	yes	—
15	C₁₉₁⋊#95	C₂²	1	8	yes	—
16	C₂²×Q₈×C₁₉₁	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₁₉₁	C₄	1	8	yes	complemented
18	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	4	yes	—
19	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
20	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
21	C₂²×Dic₁₉₁	C₂³	1	4	yes	—
22	C₂²×Dic₁₉₁	C₂³	1	2	yes	—
23	C₄⋊Dic₁₉₁	—	2	8	no	—
24	C₄⋊Dic₁₉₁	—	2	4	no	—
25	C₄×Dic₁₉₁	—	2	8	no	—
26	C₄×Dic₁₉₁	—	2	2	no	—
27	C₄×Dic₁₉₁	D₄	1	4	yes	—
28	C₂×Q₈×C₁₉₁	—	2	4	no	—
29	C₂×Q₈×C₁₉₁	—	2	4	no	—
30	C₂×Q₈×C₁₉₁	C₂×C₄	1	8	yes	—
31	C₂²×C₄×C₁₉₁	C₂³	1	2	yes	—
32	C₂²×C₄×C₁₉₁	C₂³	1	4	yes	—
33	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₁₉₁	C₂×C₄	1	16	yes	—
35	C₄²×C₁₉₁	C₂×C₄	1	4	yes	—
36	C₂×Dic₁₉₁	—	2	4	no	—
37	C₂×Dic₁₉₁	—	2	4	no	—
38	C₂×Dic₁₉₁	—	2	2	no	—
39	C₂×Dic₁₉₁	—	2	8	no	—
40	C₂×Dic₁₉₁	—	2	4	no	—
41	C₂×Dic₁₉₁	C₂×D₄	1	4	yes	—
42	C₂³×C₁₉₁	C₂⁴	1	1	yes	characteristic
43	Q₈×C₁₉₁	—	2	16	no	—
44	C₂×C₄×C₁₉₁	—	2	4	no	—
45	C₂×C₄×C₁₉₁	—	2	4	no	—
46	C₂×C₄×C₁₉₁	—	2	4	no	—
47	C₂×C₄×C₁₉₁	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₁₉₁	C₂²×C₄	1	8	yes	—
49	C₂×C₄×C₁₉₁	C₂²×C₄	1	2	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	8	no	—
57	C₄×C₁₉₁	—	2	4	no	—
58	C₄×C₁₉₁	C₄×D₄	1	4	yes	—
59	Dic₁₉₁	—	2	4	no	—
60	Dic₁₉₁	—	4	4	no	—
61	Dic₁₉₁	—	4	2	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	D₁₉₁	1	1	yes	characteristic
69	C₂×C₄×Q₈	—	191	2	no	—
70	#391	—	191	2	no	—
71	#387	—	191	8	no	—
72	#382	—	191	1	no	—
73	#376	—	191	1	no	—
74	C₂²×Q₈	C₂×D₁₉₁	1	2	yes	—
75	C₄⋊Q₈	Dic₁₉₁	1	8	yes	complemented
76	C₄×Q₈	—	382	8	no	—
77	C₂×C₄⋊C₄	—	191	2	no	—
78	C₂×C₄⋊C₄	—	191	4	no	—
79	C₂×C₄⋊C₄	C₂×D₁₉₁	1	4	yes	—
80	C₂×C₄²	—	191	4	no	—
81	C₂×C₄²	—	191	1	no	—
82	C₂×C₄²	—	191	1	no	—
83	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
84	#95	—	191	4	no	—
85	#95	—	191	4	no	—
86	#95	—	191	8	no	—
87	C₂×Q₈	—	2	4	no	—
88	C₂×Q₈	—	2	4	no	—
89	C₂×Q₈	C₂×Dic₁₉₁	1	8	yes	—
90	C₂²×C₄	—	191	2	no	—
91	C₂²×C₄	—	191	4	no	—
92	C₂²×C₄	—	191	2	no	—
93	C₂²×C₄	C₂²×D₁₉₁	1	2	yes	—
94	C₂²×C₄	C₂²×D₁₉₁	1	4	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	C₂×Dic₁₉₁	1	16	yes	—
99	C₄²	—	382	8	no	—
100	C₄²	—	382	2	no	—
101	C₄²	—	191	4	no	—
102	C₄²	C₂×Dic₁₉₁	1	4	yes	—
103	C₂³	C₂³×D₁₉₁	1	1	yes	characteristic
104	Q₈	—	2	16	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	382	4	no	—
109	C₂×C₄	—	382	4	no	—
110	C₂×C₄	—	382	2	no	—
111	C₂×C₄	—	382	8	no	—
112	C₂×C₄	—	382	4	no	—
113	C₂×C₄	—	191	4	no	—
114	C₂×C₄	C₂²×Dic₁₉₁	1	4	yes	—
115	C₂×C₄	C₂²×Dic₁₉₁	1	8	yes	—
116	C₂×C₄	C₂²×Dic₁₉₁	1	2	yes	—
117	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
118	C₂×C₄	#15690286	1	2	yes	—
119	C₂²	C₂³×Dic₁₉₁	1	1	yes	characteristic
120	C₂²	C₂×D₄×D₁₉₁	1	1	yes	characteristic
121	C₂²	C₂×#15690286	1	1	yes	characteristic
122	C₂²	#20259370	1	4	yes	—
123	C₄	—	2	8	no	—
124	C₄	—	2	4	no	—
125	C₄	—	382	4	no	—
126	C₄	—	764	4	no	—
127	C₄	—	764	2	no	—
128	C₄	D₄×Dic₁₉₁	1	4	yes	—
129	C₂	C₂×D₄×Dic₁₉₁	1	1	yes	characteristic
130	C₂	#27133141	1	2	yes	—
131	C₂	#27133238	1	2	yes	—
132	C₂	#27133274	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₁₉₁

C₂

yes

maximal

#27132570

C₂

yes

maximal

#27132549

C₂

yes

maximal

#27132529

C₂

yes

maximal, characteristic

#27132509

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₉₁

C₂

yes

maximal, characteristic

Q₈×Dic₁₉₁

—

C₂×C₄⋊Dic₁₉₁

C₂²

yes

—

C₂×C₄⋊Dic₁₉₁

C₂²

yes

—

C₂×C₄×Dic₁₉₁

C₂²

yes

characteristic

C₂×C₄×Dic₁₉₁

C₂²

yes

—

C₂×C₄×Dic₁₉₁

C₂²

yes

characteristic

C₁₉₁⋊#95

C₂²

yes

—

C₁₉₁⋊#95

C₂²

yes

—

C₁₉₁⋊#95

C₂²

yes

—

C₂²×Q₈×C₁₉₁

C₂²

yes

—

C₄⋊Q₈×C₁₉₁

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₉₁

C₂²

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₂²×Dic₁₉₁

C₂³

yes

—

C₂²×Dic₁₉₁

C₂³

yes

—

C₂²×Dic₁₉₁

C₂³

yes

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₄×Dic₁₉₁

—

C₄×Dic₁₉₁

—

C₄×Dic₁₉₁

D₄

yes

—

C₂×Q₈×C₁₉₁

—

C₂×Q₈×C₁₉₁

—

C₂×Q₈×C₁₉₁

C₂×C₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₁

C₂×C₄

yes

—

C₄²×C₁₉₁

C₂×C₄

yes

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

C₂×D₄

yes

—

C₂³×C₁₉₁

C₂⁴

yes

characteristic

Q₈×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

C₂²×C₄

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×D₄

yes

—

Dic₁₉₁

—

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

D₁₉₁

yes

characteristic

C₂×C₄×Q₈

—

191

—

#391

—

191

—

#387

—

191

—

#382

—

191

—

#376

—

191

—

C₂²×Q₈

C₂×D₁₉₁

yes

—

C₄⋊Q₈

Dic₁₉₁

yes

complemented

C₄×Q₈

—

382

—

C₂×C₄⋊C₄

—

191

—

C₂×C₄⋊C₄

—

191

—

C₂×C₄⋊C₄

C₂×D₁₉₁

yes

—

C₂×C₄²

—

191

—

C₂×C₄²

—

191

—

C₂×C₄²

—

191

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

#95

—

191

—

#95

—

191

—

#95

—

191

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₉₁

yes

—

C₂²×C₄

—

191

—

C₂²×C₄

—

191

—

C₂²×C₄

—

191

—

C₂²×C₄

C₂²×D₁₉₁

yes

—

C₂²×C₄

C₂²×D₁₉₁

yes

—

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

C₄⋊C₄

—

382

—

C₄⋊C₄

—

382

—

C₄⋊C₄

C₂×Dic₁₉₁

yes

—

C₄²

—

382

—

100

C₄²

—

382

—

101

C₄²

—

191

—

102

C₄²

C₂×Dic₁₉₁

yes

—

103

C₂³

C₂³×D₁₉₁

yes

characteristic

104

Q₈

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

382

—

109

C₂×C₄

—

382

—

110

C₂×C₄

—

382

—

111

C₂×C₄

—

382

—

112

C₂×C₄

—

382

—

113

C₂×C₄

—

191

—

114

C₂×C₄

C₂²×Dic₁₉₁

yes

—

115

C₂×C₄

C₂²×Dic₁₉₁

yes

—

116

C₂×C₄

C₂²×Dic₁₉₁

yes

—

117

C₂×C₄

D₄×D₁₉₁

yes

—

118

C₂×C₄

#15690286

yes

—

119

C₂²

C₂³×Dic₁₉₁

yes

characteristic

120

C₂²

C₂×D₄×D₁₉₁

yes

characteristic

121

C₂²

C₂×#15690286

yes

characteristic

122

C₂²

#20259370

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

382

—

126

C₄

—

764

—

127

C₄

—

764

—

128

C₄

D₄×Dic₁₉₁

yes

—

129

C₂

C₂×D₄×Dic₁₉₁

yes

characteristic

130

C₂

#27133141

yes

—

131

C₂

#27133238

yes

—

132

C₂

#27133274

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉₁
commutator	C₂²×C₁₉₁	C₂³×C₄
Fitting	C₂×C₄⋊Q₈×C₁₉₁	C₂
socle, prime radical, cosocle	C₂³×C₁₉₁	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₁₉₁
hypocenter	C₁₉₁	#2327
LCS	G ▷ C₂²×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₉₁

commutator

C₂²×C₁₉₁

C₂³×C₄

Fitting

C₂×C₄⋊Q₈×C₁₉₁

C₂

socle, prime radical, cosocle

C₂³×C₁₉₁

C₂⁴

hypercenter

C₂×C₄⋊Q₈

D₁₉₁

C₁₉₁

G ▷ C₂²×C₁₉₁ ▷ C₁₉₁

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
17. C₄⋊Q₈×C₁₉₁	C₄	semidirect
66. C₁₉₁	#2327	semidirect
75. C₄⋊Q₈	Dic₁₉₁	semidirect

H ≅ G/N

type

17. C₄⋊Q₈×C₁₉₁

C₄

semidirect

66. C₁₉₁

#2327

semidirect

75. C₄⋊Q₈

Dic₁₉₁

semidirect

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

subgroup	index
C₂×Q₈×Dic₁₉₁	2
#27132570	2
#27132549	2
#27132529	2
#27132509	2
C₂×C₄⋊Q₈×C₁₉₁	2
#2327	191

subgroup

index

C₂×Q₈×Dic₁₉₁

C₂×C₄⋊Q₈×C₁₉₁

191

Subgroups — #37575240

Conjugacy classes of subgroups

Complements

Maximal subgroups