Group #37572569 — subgroups

Subgroups — #37572569

Order 24448.

252 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 81 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×#20259330	C₂	1	1	yes	maximal, characteristic
2	C₂×#20259257	C₂	1	2	yes	maximal, complemented
3	#27132645	C₂	1	2	yes	maximal
4	C₁₉₁⋊#338	C₂	1	1	yes	maximal, characteristic, complemented
5	#338×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15690287	C₂²	1	1	yes	characteristic
7	#20259330	—	2	4	no	—
8	C₂×#15690271	C₂²	1	2	yes	—
9	C₂×#15690270	C₄	1	2	yes	—
10	C₂²×C₁₉₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₁₉₁	C₄	1	2	yes	—
12	#20259257	—	2	4	no	—
13	#20259240	—	2	2	no	—
14	#20259238	—	2	2	no	—
15	C₂×M₁₆×C₁₉₁	C₂²	1	2	yes	—
16	C₂×M₁₆×C₁₉₁	C₂²	1	1	yes	characteristic
17	#15690287	—	2	2	no	—
18	#15690287	—	2	2	no	—
19	C₂×Dic₃₈₂	D₄	1	1	yes	characteristic
20	C₂×D₇₆₄	D₄	1	1	yes	characteristic
21	C₂×C₂²⋊D₁₉₁	Q₈	1	2	yes	—
22	C₂×C₄×D₁₉₁	C₂×C₄	1	2	yes	—
23	#15690271	—	2	2	no	—
24	#15690271	—	2	2	no	—
25	#15690270	—	2	4	no	—
26	C₂×C₁₉₁⋊C₈	—	2	2	no	—
27	C₂×C₁₉₁⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₉₁⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₉₁⋊C₈	D₄	1	4	yes	—
30	C₈×D₁₉₁	—	2	4	no	—
31	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
32	M₁₆×C₁₉₁	—	2	2	no	—
33	M₁₆×C₁₉₁	—	2	2	no	—
34	M₁₆×C₁₉₁	—	2	2	no	—
35	C₂×C₈×C₁₉₁	—	2	2	no	—
36	C₂×C₈×C₁₉₁	C₂×C₄	1	2	yes	—
37	C₂²×D₁₉₁	C₄⋊C₄	1	2	yes	—
38	Dic₃₈₂	—	2	2	no	—
39	D₇₆₄	—	2	2	no	—
40	C₂²⋊D₁₉₁	—	2	4	no	—
41	C₂×Dic₁₉₁	C₄⋊C₄	1	2	yes	—
42	C₄×D₁₉₁	—	2	2	no	—
43	C₄×D₁₉₁	—	2	2	no	—
44	C₁₉₁⋊C₈	—	4	2	no	—
45	C₁₉₁⋊C₈	C₂²⋊C₄	1	2	yes	—
46	C₁₉₁⋊C₈	C₂²⋊C₄	1	2	yes	—
47	C₂³×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₉₁	#41	1	2	yes	—
52	C₈×C₁₉₁	—	2	2	no	—
53	C₈×C₁₉₁	—	4	2	no	—
54	C₂²×C₁₉₁	—	4	1	no	—
55	C₂²×C₁₉₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₉₁	C₂²⋊Q₈	1	2	yes	—
57	C₂×D₁₉₁	—	2	2	no	—
58	C₂×D₁₉₁	—	4	2	no	—
59	C₄×C₁₉₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₁₉₁	#117	1	1	yes	characteristic
61	C₄×C₁₉₁	#121	1	2	yes	—
62	Dic₁₉₁	—	2	2	no	—
63	C₂×C₁₉₁	—	2	1	no	—
64	C₂×C₁₉₁	—	2	2	no	—
65	C₂×C₁₉₁	#380	1	1	yes	characteristic
66	D₁₉₁	—	4	2	no	—
67	C₁₉₁	#1813	1	1	yes	characteristic, complemented, Sylow
68	#1813	—	191	1	no	maximal, Sylow
69	C₂×#131	—	191	1	no	—
70	C₂×#108	—	191	2	no	—
71	#413	—	191	2	no	—
72	#338	—	191	1	no	—
73	#338	D₁₉₁	1	1	yes	characteristic, complemented
74	C₂×#41	—	191	1	no	—
75	#131	—	382	4	no	—
76	C₂×M₁₆	—	191	2	no	—
77	C₂×M₁₆	—	191	2	no	—
78	C₂×M₁₆	C₂×D₁₉₁	1	2	yes	—
79	C₂×M₁₆	C₂×D₁₉₁	1	1	yes	characteristic
80	C₂²×C₈	—	191	2	no	—
81	C₂²×C₈	—	191	1	no	—
82	#108	—	382	4	no	—
83	#101	—	382	2	no	—
84	#100	—	382	2	no	—
85	#41	—	382	2	no	—
86	#41	—	382	2	no	—
87	C₂×Q₈	—	191	1	no	—
88	C₂×D₄	—	191	2	no	—
89	C₂×D₄	—	191	1	no	—
90	C₂²×C₄	—	191	2	no	—
91	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
92	M₁₆	—	2	2	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	—	382	4	no	—
96	M₁₆	—	382	2	no	—
97	M₁₆	—	382	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	382	4	no	—
100	C₂×C₈	—	382	2	no	—
101	C₂×C₈	—	191	1	no	—
102	C₂×C₈	—	191	1	no	—
103	C₂×C₈	—	191	4	no	—
104	C₂×C₈	C₄×D₁₉₁	1	2	yes	—
105	C₂³	—	191	2	no	—
106	C₂³	Q₈×D₁₉₁	1	1	yes	characteristic
107	Q₈	—	382	2	no	—
108	D₄	—	382	4	no	—
109	D₄	—	382	2	no	—
110	C₂×C₄	—	382	2	no	—
111	C₂×C₄	—	382	2	no	—
112	C₂×C₄	—	191	2	no	—
113	C₂×C₄	C₂×C₄×D₁₉₁	1	1	yes	characteristic
114	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
115	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
116	C₂×C₄	#15690287	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	191	2	no	—
120	C₈	—	191	2	no	—
121	C₈	—	764	2	no	—
122	C₂²	—	4	1	no	—
123	C₂²	—	382	2	no	—
124	C₂²	—	764	2	no	—
125	C₂²	C₄⋊C₄×D₁₉₁	1	1	yes	characteristic
126	C₂²	#20259300	1	2	yes	—
127	C₄	—	382	2	no	—
128	C₄	C₂²⋊C₄×D₁₉₁	1	1	yes	characteristic
129	C₄	#20259279	1	1	yes	characteristic
130	C₄	#20259295	1	2	yes	—
131	C₂	—	2	1	no	—
132	C₂	—	2	2	no	—
133	C₂	—	764	2	no	—
134	C₂	#27132522	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20259330

C₂

yes

maximal, characteristic

C₂×#20259257

C₂

yes

maximal, complemented

#27132645

C₂

yes

maximal

C₁₉₁⋊#338

C₂

yes

maximal, characteristic, complemented

#338×C₁₉₁

C₂

yes

maximal, characteristic, complemented

C₂×#15690287

C₂²

yes

characteristic

#20259330

—

C₂×#15690271

C₂²

yes

—

C₂×#15690270

C₄

yes

—

C₂²×C₁₉₁⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₁₉₁

C₄

yes

—

#20259257

—

#20259240

—

#20259238

—

C₂×M₁₆×C₁₉₁

C₂²

yes

—

C₂×M₁₆×C₁₉₁

C₂²

yes

characteristic

#15690287

—

#15690287

—

C₂×Dic₃₈₂

D₄

yes

characteristic

C₂×D₇₆₄

D₄

yes

characteristic

C₂×C₂²⋊D₁₉₁

Q₈

yes

—

C₂×C₄×D₁₉₁

C₂×C₄

yes

—

#15690271

—

#15690271

—

#15690270

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₉₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₉₁⋊C₈

D₄

yes

—

C₈×D₁₉₁

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

M₁₆×C₁₉₁

—

M₁₆×C₁₉₁

—

M₁₆×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

C₂×C₄

yes

—

C₂²×D₁₉₁

C₄⋊C₄

yes

—

Dic₃₈₂

—

D₇₆₄

—

C₂²⋊D₁₉₁

—

C₂×Dic₁₉₁

C₄⋊C₄

yes

—

C₄×D₁₉₁

—

C₄×D₁₉₁

—

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

C₂²⋊C₄

yes

—

C₁₉₁⋊C₈

C₂²⋊C₄

yes

—

C₂³×C₁₉₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

—

C₈×C₁₉₁

—

C₈×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₉₁

C₂²⋊Q₈

yes

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₄×C₁₉₁

C₂×C₂²⋊C₄

yes

C₄×C₁₉₁

yes

C₄×C₁₉₁

yes

—

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

#380

yes

characteristic

D₁₉₁

—

C₁₉₁

#1813

yes

characteristic, complemented, Sylow

#1813

—

191

maximal, Sylow

C₂×#131

—

191

—

C₂×#108

—

191

—

#413

—

191

—

#338

—

191

—

#338

D₁₉₁

yes

characteristic, complemented

C₂×#41

—

191

—

#131

—

382

—

C₂×M₁₆

—

191

—

C₂×M₁₆

—

191

—

C₂×M₁₆

C₂×D₁₉₁

yes

—

C₂×M₁₆

C₂×D₁₉₁

yes

characteristic

C₂²×C₈

—

191

—

C₂²×C₈

—

191

—

#108

—

382

—

#101

—

382

—

#100

—

382

—

#41

—

382

—

#41

—

382

—

C₂×Q₈

—

191

—

C₂×D₄

—

191

—

C₂×D₄

—

191

—

C₂²×C₄

—

191

—

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

382

—

M₁₆

—

382

—

M₁₆

—

382

—

C₂×C₈

—

C₂×C₈

—

382

—

100

C₂×C₈

—

382

—

101

C₂×C₈

—

191

—

102

C₂×C₈

—

191

—

103

C₂×C₈

—

191

—

104

C₂×C₈

C₄×D₁₉₁

yes

—

105

C₂³

—

191

—

106

C₂³

Q₈×D₁₉₁

yes

characteristic

107

Q₈

—

382

—

108

D₄

—

382

—

109

D₄

—

382

—

110

C₂×C₄

—

382

—

111

C₂×C₄

—

382

—

112

C₂×C₄

—

191

—

113

C₂×C₄

C₂×C₄×D₁₉₁

yes

characteristic

114

C₂×C₄

D₄×D₁₉₁

yes

—

115

C₂×C₄

D₄×D₁₉₁

yes

characteristic

116

C₂×C₄

#15690287

yes

—

117

C₈

—

118

C₈

—

119

C₈

—

191

—

120

C₈

—

191

—

121

C₈

—

764

—

122

C₂²

—

123

C₂²

—

382

—

124

C₂²

—

764

—

125

C₂²

C₄⋊C₄×D₁₉₁

yes

characteristic

126

C₂²

#20259300

yes

—

127

C₄

—

382

—

128

C₄

C₂²⋊C₄×D₁₉₁

yes

characteristic

129

C₄

#20259279

yes

characteristic

130

C₄

#20259295

yes

—

131

C₂

—

132

C₂

—

133

C₂

—

764

—

134

C₂

#27132522

yes

characteristic

	subgroup	G/H
center	C₄	#20259279
commutator	C₂×C₄×C₁₉₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting	#338×C₁₉₁	C₂
socle	C₂×C₁₉₁	#380
prime radical	C₂×#15690287	C₂²
cosocle	C₂²×C₄×C₁₉₁	C₂³
hypercenter	#338	D₁₉₁
hypocenter	C₁₉₁	#1813
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#20259279

commutator

C₂×C₄×C₁₉₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₁

Fitting

#338×C₁₉₁

C₂

C₂×C₁₉₁

C₂²

C₂²×C₄×C₁₉₁

C₂³

D₁₉₁

C₁₉₁

G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
2. C₂×#20259257	C₂	semidirect
4. C₁₉₁⋊#338	C₂	semidirect
5. #338×C₁₉₁	C₂	semidirect
67. C₁₉₁	#1813	semidirect
73. #338	D₁₉₁	semidirect

H ≅ G/N

type

2. C₂×#20259257

C₂

semidirect

4. C₁₉₁⋊#338

C₂

semidirect

5. #338×C₁₉₁

C₂

semidirect

67. C₁₉₁

#1813

semidirect

73. #338

D₁₉₁

semidirect

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

subgroup	index
C₂×#20259330	2
C₂×#20259257	2
#27132645	2
C₁₉₁⋊#338	2
#338×C₁₉₁	2
#1813	191

subgroup

index

C₂×#20259330

C₂×#20259257

#27132645

C₁₉₁⋊#338

#338×C₁₉₁

#1813

191

Subgroups — #37572569

Conjugacy classes of subgroups

Complements

Maximal subgroups