Group #37572516 — subgroups

Subgroups — #37572516

Order 24448.

272 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 113 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	#27132687	C₂	1	2	yes	maximal
2	C₂×#20259241	C₂	1	2	yes	maximal
3	#27132432	C₂	1	2	yes	maximal
4	C₂×C₄×Q₈×C₁₉₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#15690271	C₂²	1	4	yes	—
6	C₄⋊(C₁₉₁⋊C₈)	—	2	2	no	—
7	C₄⋊(C₁₉₁⋊C₈)	—	2	2	no	—
8	#20259241	—	2	4	no	—
9	C₂²×Q₈×C₁₉₁	C₂²	1	1	yes	characteristic
10	C₄×Q₈×C₁₉₁	—	2	2	no	—
11	C₄×Q₈×C₁₉₁	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₉₁	C₄	1	2	yes	—
13	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₉₁	C₄	1	2	yes	—
15	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
16	#15690271	—	2	4	no	—
17	#15690271	—	2	4	no	—
18	C₂×C₁₉₁⋊C₈	—	2	4	no	—
19	C₂×Q₈×C₁₉₁	—	2	2	no	—
20	C₂×Q₈×C₁₉₁	—	2	2	no	—
21	C₂×Q₈×C₁₉₁	D₄	1	2	yes	—
22	C₂×Q₈×C₁₉₁	Q₈	1	2	yes	—
23	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₉₁	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₉₁	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₁₉₁	—	2	4	no	—
29	C₄⋊C₄×C₁₉₁	—	2	2	no	—
30	C₄²×C₁₉₁	—	2	2	no	—
31	C₄²×C₁₉₁	—	2	2	no	—
32	C₄²×C₁₉₁	D₄	1	2	yes	—
33	C₄²×C₁₉₁	D₄	1	2	yes	—
34	C₁₉₁⋊C₈	—	4	4	no	—
35	C₂³×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
36	Q₈×C₁₉₁	—	2	4	no	—
37	Q₈×C₁₉₁	—	2	4	no	—
38	C₂×C₄×C₁₉₁	—	2	2	no	—
39	C₂×C₄×C₁₉₁	—	2	2	no	—
40	C₂×C₄×C₁₉₁	—	2	2	no	—
41	C₂×C₄×C₁₉₁	—	2	2	no	—
42	C₂×C₄×C₁₉₁	—	4	1	no	—
43	C₂×C₄×C₁₉₁	—	4	2	no	—
44	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₉₁	C₄⋊C₄	1	4	yes	—
48	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₁₉₁	#41	1	2	yes	—
52	C₂²×C₁₉₁	—	2	1	no	—
53	C₂²×C₁₉₁	—	2	1	no	—
54	C₂²×C₁₉₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₁₉₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₉₁	#117	1	1	yes	characteristic
57	C₄×C₁₉₁	—	2	4	no	—
58	C₄×C₁₉₁	—	2	1	no	—
59	C₄×C₁₉₁	—	2	2	no	—
60	C₄×C₁₉₁	—	4	2	no	—
61	C₄×C₁₉₁	#101	1	2	yes	complemented
62	C₄×C₁₉₁	#121	1	2	yes	—
63	C₄×C₁₉₁	C₂²⋊Q₈	1	2	yes	—
64	C₂×C₁₉₁	—	2	1	no	—
65	C₂×C₁₉₁	—	2	1	no	—
66	C₂×C₁₉₁	#380	1	1	yes	characteristic
67	C₂×C₁₉₁	C₂×#101	1	1	yes	characteristic
68	C₂×C₁₉₁	#413	1	1	yes	characteristic
69	C₁₉₁	#1801	1	1	yes	characteristic, complemented, Sylow
70	#1801	—	191	1	no	maximal, Sylow
71	C₂×C₄×Q₈	D₁₉₁	1	1	yes	characteristic
72	#423	—	191	2	no	—
73	C₂×#101	—	191	2	no	—
74	#343	—	191	2	no	—
75	C₂²×Q₈	C₂×D₁₉₁	1	1	yes	characteristic
76	C₂×M₁₆	—	191	4	no	—
77	C₄×Q₈	—	2	2	no	—
78	C₄×Q₈	—	2	2	no	—
79	C₂×C₄⋊C₄	Dic₁₉₁	1	2	yes	—
80	C₂×C₄⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
81	C₂×C₄²	Dic₁₉₁	1	2	yes	—
82	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
83	C₄⋊C₈	—	382	2	no	—
84	C₄⋊C₈	—	382	2	no	—
85	#101	—	382	4	no	—
86	C₂×Q₈	—	2	2	no	—
87	C₂×Q₈	—	2	2	no	—
88	C₂×Q₈	D₇₆₄	1	2	yes	—
89	C₂×Q₈	Dic₃₈₂	1	2	yes	—
90	C₂²×C₄	C₂×Dic₁₉₁	1	1	yes	characteristic
91	C₂²×C₄	C₂×Dic₁₉₁	1	2	yes	—
92	C₂²×C₄	C₂×Dic₁₉₁	1	2	yes	—
93	C₂²×C₄	C₂×Dic₁₉₁	1	1	yes	characteristic
94	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
95	M₁₆	—	382	4	no	—
96	M₁₆	—	382	4	no	—
97	C₂×C₈	—	382	4	no	—
98	C₄⋊C₄	—	2	4	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄²	—	2	2	no	—
101	C₄²	—	2	2	no	—
102	C₄²	C₂²⋊D₁₉₁	1	2	yes	—
103	C₄²	C₂²⋊D₁₉₁	1	2	yes	—
104	C₂³	C₂²×Dic₁₉₁	1	1	yes	characteristic
105	Q₈	—	2	4	no	—
106	Q₈	—	2	4	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	C₂²⋊Dic₁₉₁	1	1	yes	characteristic
114	C₂×C₄	C₂²⋊Dic₁₉₁	1	2	yes	—
115	C₂×C₄	C₂²⋊Dic₁₉₁	1	1	yes	characteristic
116	C₂×C₄	C₄⋊Dic₁₉₁	1	4	yes	—
117	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
118	C₂×C₄	C₂×D₇₆₄	1	1	yes	characteristic
119	C₂×C₄	C₂×Dic₃₈₂	1	1	yes	characteristic
120	C₂×C₄	#15690287	1	2	yes	—
121	C₈	—	764	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	C₂×C₂²⋊Dic₁₉₁	1	1	yes	characteristic
125	C₂²	C₂×C₄⋊Dic₁₉₁	1	1	yes	characteristic
126	C₂²	#20259281	1	1	yes	characteristic
127	C₄	—	2	4	no	—
128	C₄	—	2	1	no	—
129	C₄	—	2	2	no	—
130	C₄	—	4	2	no	—
131	C₄	#20259241	1	2	yes	complemented
132	C₄	#20259296	1	2	yes	—
133	C₄	#20259301	1	2	yes	—
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	#27132523	1	1	yes	characteristic
137	C₂	C₂×#20259241	1	1	yes	characteristic
138	C₂	#27132647	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27132687

C₂

yes

maximal

C₂×#20259241

C₂

yes

maximal

#27132432

C₂

yes

maximal

C₂×C₄×Q₈×C₁₉₁

C₂

yes

maximal, characteristic

C₂×#15690271

C₂²

yes

—

C₄⋊(C₁₉₁⋊C₈)

—

C₄⋊(C₁₉₁⋊C₈)

—

#20259241

—

C₂²×Q₈×C₁₉₁

C₂²

yes

characteristic

C₄×Q₈×C₁₉₁

—

C₄×Q₈×C₁₉₁

—

C₂×C₄⋊C₄×C₁₉₁

C₄

yes

—

C₂×C₄⋊C₄×C₁₉₁

C₂²

yes

characteristic

C₂×C₄²×C₁₉₁

C₄

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

#15690271

—

#15690271

—

C₂×C₁₉₁⋊C₈

—

C₂×Q₈×C₁₉₁

—

C₂×Q₈×C₁₉₁

—

C₂×Q₈×C₁₉₁

D₄

yes

—

C₂×Q₈×C₁₉₁

Q₈

yes

—

C₂²×C₄×C₁₉₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₁

C₂×C₄

yes

—

C₂²×C₄×C₁₉₁

C₂×C₄

yes

—

C₂²×C₄×C₁₉₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄²×C₁₉₁

—

C₄²×C₁₉₁

—

C₄²×C₁₉₁

D₄

yes

—

C₄²×C₁₉₁

D₄

yes

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

C₂²×C₄

yes

characteristic

Q₈×C₁₉₁

—

Q₈×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₉₁

#117

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#101

yes

complemented

C₄×C₁₉₁

#121

yes

—

C₄×C₁₉₁

C₂²⋊Q₈

yes

—

C₂×C₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#1801

—

191

maximal, Sylow

C₂×C₄×Q₈

D₁₉₁

yes

characteristic

#423

—

191

—

C₂×#101

—

191

—

#343

—

191

—

C₂²×Q₈

C₂×D₁₉₁

yes

characteristic

C₂×M₁₆

—

191

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

Dic₁₉₁

yes

—

C₂×C₄⋊C₄

C₂×D₁₉₁

yes

characteristic

C₂×C₄²

Dic₁₉₁

yes

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

C₄⋊C₈

—

382

—

C₄⋊C₈

—

382

—

#101

—

382

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₇₆₄

yes

—

C₂×Q₈

Dic₃₈₂

yes

—

C₂²×C₄

C₂×Dic₁₉₁

yes

characteristic

C₂²×C₄

C₂×Dic₁₉₁

yes

—

C₂²×C₄

C₂×Dic₁₉₁

yes

—

C₂²×C₄

C₂×Dic₁₉₁

yes

characteristic

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

M₁₆

—

382

—

M₁₆

—

382

—

C₂×C₈

—

382

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₂²⋊D₁₉₁

yes

—

103

C₄²

C₂²⋊D₁₉₁

yes

—

104

C₂³

C₂²×Dic₁₉₁

yes

characteristic

105

Q₈

—

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²⋊Dic₁₉₁

yes

characteristic

114

C₂×C₄

C₂²⋊Dic₁₉₁

yes

—

115

C₂×C₄

C₂²⋊Dic₁₉₁

yes

characteristic

116

C₂×C₄

C₄⋊Dic₁₉₁

yes

—

117

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

—

118

C₂×C₄

C₂×D₇₆₄

yes

characteristic

119

C₂×C₄

C₂×Dic₃₈₂

yes

characteristic

120

C₂×C₄

#15690287

yes

—

121

C₈

—

764

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂×C₂²⋊Dic₁₉₁

yes

characteristic

125

C₂²

C₂×C₄⋊Dic₁₉₁

yes

characteristic

126

C₂²

#20259281

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

#20259241

yes

complemented

132

C₄

#20259296

yes

—

133

C₄

#20259301

yes

—

134

C₂

—

135

C₂

—

136

C₂

#27132523

yes

characteristic

137

C₂

C₂×#20259241

yes

characteristic

138

C₂

#27132647

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₉₁
commutator, prime radical	C₂³×C₁₉₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting	C₂×C₄×Q₈×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₉₁	C₂³
hypercenter	C₂×C₄×Q₈	D₁₉₁
hypocenter	C₁₉₁	#1801
LCS	G ▷ C₂³×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₉₁

commutator, prime radical

C₂³×C₁₉₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₁

Fitting

C₂×C₄×Q₈×C₁₉₁

C₂

socle

C₂²×C₁₉₁

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₉₁

C₂³

hypercenter

C₂×C₄×Q₈

D₁₉₁

hypocenter

C₁₉₁

#1801

LCS

G ▷ C₂³×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
61. C₄×C₁₉₁	#101	semidirect
69. C₁₉₁	#1801	semidirect
131. C₄	#20259241	semidirect

H ≅ G/N

type

61. C₄×C₁₉₁

#101

semidirect

69. C₁₉₁

#1801

semidirect

131. C₄

#20259241

semidirect

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

subgroup	index
#27132687	2
C₂×#20259241	2
#27132432	2
C₂×C₄×Q₈×C₁₉₁	2
#1801	191

subgroup

index

C₂×C₄×Q₈×C₁₉₁

191

Subgroups — #37572516

Conjugacy classes of subgroups

Complements

Maximal subgroups