Group #37573361 — subgroups

Subgroups — #37573361

Order 24448.

396 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 105 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	1	yes	maximal, characteristic, complemented
2	C₂×#20259239	C₂	1	4	yes	maximal, complemented
3	C₄×#15690271	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#127×C₁₉₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#15690271	C₂²	1	2	yes	—
6	C₂×#15690271	C₂²	1	2	yes	—
7	C₄⋊(C₁₉₁⋊C₈)	—	2	2	no	—
8	#20259239	—	2	8	no	—
9	#20259233	—	2	1	no	—
10	C₄×C₁₉₁⋊C₈	—	2	1	no	—
11	C₂²×D₄×C₁₉₁	C₄	1	4	yes	—
12	C₂²×D₄×C₁₉₁	C₂²	1	2	yes	—
13	#127×C₁₉₁	—	2	1	no	—
14	#127×C₁₉₁	—	2	1	no	—
15	#127×C₁₉₁	—	2	2	no	—
16	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
17	#15690271	—	2	4	no	—
18	#15690271	—	2	2	no	—
19	#15690271	D₄	1	4	yes	complemented
20	C₂×C₁₉₁⋊C₈	—	2	2	no	—
21	C₂×C₁₉₁⋊C₈	—	2	2	no	—
22	C₂⁴×C₁₉₁	C₂×C₄	1	4	yes	—
23	C₂×D₄×C₁₉₁	—	2	4	no	—
24	C₂×D₄×C₁₉₁	—	2	4	no	—
25	C₂×D₄×C₁₉₁	—	2	2	no	—
26	C₂×D₄×C₁₉₁	—	2	2	no	—
27	C₂×D₄×C₁₉₁	—	2	4	no	—
28	C₂×D₄×C₁₉₁	—	2	4	no	—
29	C₂×D₄×C₁₉₁	—	4	8	no	—
30	C₂²×C₄×C₁₉₁	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
32	C₄²×C₁₉₁	D₄	1	1	yes	characteristic
33	C₄²×C₁₉₁	D₄	1	1	yes	characteristic
34	C₄²×C₁₉₁	D₄	1	2	yes	—
35	C₁₉₁⋊C₈	—	2	4	no	—
36	C₁₉₁⋊C₈	—	4	2	no	—
37	C₂³×C₁₉₁	—	2	4	no	—
38	C₂³×C₁₉₁	—	2	8	no	—
39	C₂³×C₁₉₁	—	4	8	no	—
40	C₂³×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
41	D₄×C₁₉₁	—	4	8	no	—
42	D₄×C₁₉₁	—	4	8	no	—
43	D₄×C₁₉₁	—	4	4	no	—
44	D₄×C₁₉₁	—	4	2	no	—
45	D₄×C₁₉₁	—	4	2	no	—
46	C₂×C₄×C₁₉₁	—	2	4	no	—
47	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	2	yes	—
49	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₉₁	#41	1	2	yes	—
53	C₂²×C₁₉₁	—	2	2	no	—
54	C₂²×C₁₉₁	—	4	4	no	—
55	C₂²×C₁₉₁	—	4	8	no	—
56	C₂²×C₁₉₁	—	8	8	no	—
57	C₂²×C₁₉₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₁₉₁	C₄×D₄	1	2	yes	—
59	C₄×C₁₉₁	—	2	2	no	—
60	C₄×C₁₉₁	#100	1	4	yes	complemented
61	C₄×C₁₉₁	#121	1	2	yes	—
62	C₄×C₁₉₁	#124	1	1	yes	characteristic
63	C₄×C₁₉₁	#127	1	1	yes	characteristic
64	C₂×C₁₉₁	—	2	2	no	—
65	C₂×C₁₉₁	—	8	4	no	—
66	C₂×C₁₉₁	#390	1	1	yes	characteristic
67	C₂×C₁₉₁	C₂×#100	1	2	yes	—
68	C₁₉₁	#1965	1	1	yes	characteristic, complemented, Sylow
69	#1965	—	191	1	no	maximal, Sylow
70	C₂×#127	D₁₉₁	1	1	yes	characteristic
71	#423	—	191	1	no	—
72	C₂×#100	—	191	4	no	—
73	C₄×M₁₆	—	191	1	no	—
74	C₂²×D₄	Dic₁₉₁	1	4	yes	—
75	C₂²×D₄	C₂×D₁₉₁	1	2	yes	—
76	C₂×M₁₆	—	191	2	no	—
77	C₂×M₁₆	—	191	2	no	—
78	#127	—	2	1	no	—
79	#127	—	2	1	no	—
80	#127	—	2	2	no	—
81	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
82	C₄⋊C₈	—	382	2	no	—
83	#100	—	382	8	no	—
84	#97	—	382	1	no	—
85	C₄×C₈	—	382	1	no	—
86	C₂⁴	C₂×Dic₁₉₁	1	4	yes	—
87	C₂×D₄	—	2	4	no	—
88	C₂×D₄	—	2	4	no	—
89	C₂×D₄	—	2	4	no	—
90	C₂×D₄	—	2	2	no	—
91	C₂×D₄	—	2	2	no	—
92	C₂×D₄	—	2	4	no	—
93	C₂×D₄	—	4	8	no	—
94	C₂²×C₄	C₂×Dic₁₉₁	1	2	yes	—
95	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
96	M₁₆	—	382	2	no	—
97	M₁₆	—	382	4	no	—
98	M₁₆	—	191	4	no	—
99	C₂×C₈	—	382	2	no	—
100	C₂×C₈	—	382	2	no	—
101	C₄²	C₂²⋊D₁₉₁	1	1	yes	characteristic
102	C₄²	C₂²⋊D₁₉₁	1	1	yes	characteristic
103	C₄²	C₂²⋊D₁₉₁	1	2	yes	—
104	C₂³	—	2	4	no	—
105	C₂³	—	2	8	no	—
106	C₂³	—	4	8	no	—
107	C₂³	C₂²×Dic₁₉₁	1	1	yes	characteristic
108	D₄	—	4	8	no	—
109	D₄	—	4	8	no	—
110	D₄	—	4	4	no	—
111	D₄	—	4	2	no	—
112	D₄	—	4	2	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	C₂²⋊Dic₁₉₁	1	2	yes	—
115	C₂×C₄	C₂²⋊Dic₁₉₁	1	2	yes	—
116	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
117	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
119	C₂×C₄	#15690286	1	2	yes	—
120	C₈	—	382	4	no	—
121	C₈	—	764	2	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	4	4	no	—
124	C₂²	—	4	8	no	—
125	C₂²	—	8	8	no	—
126	C₂²	C₂×C₂²⋊Dic₁₉₁	1	1	yes	characteristic
127	C₂²	D₄×Dic₁₉₁	1	2	yes	—
128	C₄	—	2	2	no	—
129	C₄	#20259239	1	4	yes	complemented
130	C₄	#20259293	1	2	yes	—
131	C₄	#20259309	1	1	yes	characteristic
132	C₄	#20259318	1	1	yes	characteristic
133	C₂	—	2	2	no	—
134	C₂	—	8	4	no	—
135	C₂	#27132566	1	1	yes	characteristic
136	C₂	C₂×#20259239	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27132687

C₂

yes

maximal, characteristic, complemented

C₂×#20259239

C₂

yes

maximal, complemented

C₄×#15690271

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₁₉₁

C₂

yes

maximal, characteristic

C₂×#15690271

C₂²

yes

—

C₂×#15690271

C₂²

yes

—

C₄⋊(C₁₉₁⋊C₈)

—

#20259239

—

#20259233

—

C₄×C₁₉₁⋊C₈

—

C₂²×D₄×C₁₉₁

C₄

yes

—

C₂²×D₄×C₁₉₁

C₂²

yes

—

#127×C₁₉₁

—

#127×C₁₉₁

—

#127×C₁₉₁

—

C₂×C₄²×C₁₉₁

C₂²

yes

—

—

D₄

yes

complemented

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₂⁴×C₁₉₁

C₂×C₄

yes

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂²×C₄×C₁₉₁

C₂×C₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₄²×C₁₉₁

D₄

yes

characteristic

C₄²×C₁₉₁

D₄

yes

characteristic

C₄²×C₁₉₁

D₄

yes

—

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

C₂²×C₄

yes

characteristic

D₄×C₁₉₁

—

D₄×C₁₉₁

—

D₄×C₁₉₁

—

D₄×C₁₉₁

—

D₄×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

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₁

C₄×D₄

yes

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#100

yes

complemented

C₄×C₁₉₁

#121

yes

—

C₄×C₁₉₁

yes

C₄×C₁₉₁

yes

C₂×C₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

#390

yes

characteristic

C₂×C₁₉₁

C₂×#100

yes

—

C₁₉₁

#1965

yes

characteristic, complemented, Sylow

#1965

—

191

maximal, Sylow

C₂×#127

D₁₉₁

yes

characteristic

#423

—

191

—

C₂×#100

—

191

—

C₄×M₁₆

—

191

—

C₂²×D₄

Dic₁₉₁

yes

—

C₂²×D₄

C₂×D₁₉₁

yes

—

C₂×M₁₆

—

191

—

C₂×M₁₆

—

191

—

#127

—

#127

—

#127

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

C₄⋊C₈

—

382

—

#100

—

382

—

#97

—

382

—

C₄×C₈

—

382

—

C₂⁴

C₂×Dic₁₉₁

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

C₂×Dic₁₉₁

yes

—

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

M₁₆

—

382

—

M₁₆

—

382

—

M₁₆

—

191

—

C₂×C₈

—

382

—

100

C₂×C₈

—

382

—

101

C₄²

C₂²⋊D₁₉₁

yes

characteristic

102

C₄²

C₂²⋊D₁₉₁

yes

characteristic

103

C₄²

C₂²⋊D₁₉₁

yes

—

104

C₂³

—

105

C₂³

—

106

C₂³

—

107

C₂³

C₂²×Dic₁₉₁

yes

characteristic

108

D₄

—

109

D₄

—

110

D₄

—

111

D₄

—

112

D₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂²⋊Dic₁₉₁

yes

—

115

C₂×C₄

C₂²⋊Dic₁₉₁

yes

—

116

C₂×C₄

D₄×D₁₉₁

yes

—

117

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

119

C₂×C₄

#15690286

yes

—

120

C₈

—

382

—

121

C₈

—

764

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

C₂×C₂²⋊Dic₁₉₁

yes

characteristic

127

C₂²

D₄×Dic₁₉₁

yes

—

128

C₄

—

129

C₄

#20259239

yes

complemented

130

C₄

#20259293

yes

—

131

C₄

#20259309

yes

characteristic

132

C₄

#20259318

yes

characteristic

133

C₂

—

134

C₂

—

135

C₂

#27132566

yes

characteristic

136

C₂

C₂×#20259239

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₉₁
commutator	C₂³×C₁₉₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting, prime radical	C₂×#127×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₉₁	C₂³
hypercenter	C₂×#127	D₁₉₁
hypocenter	C₁₉₁	#1965
LCS	G ▷ C₂³×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₉₁

commutator

C₂³×C₁₉₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₁

Fitting, prime radical

C₂×#127×C₁₉₁

C₂

socle

C₂²×C₁₉₁

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₉₁

C₂³

D₁₉₁

C₁₉₁

G ▷ C₂³×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. #27132687	C₂	semidirect
2. C₂×#20259239	C₂	semidirect
3. C₄×#15690271	C₂	semidirect
19. #15690271	D₄	semidirect
60. C₄×C₁₉₁	#100	semidirect
68. C₁₉₁	#1965	semidirect
129. C₄	#20259239	semidirect

H ≅ G/N

type

1. #27132687

C₂

semidirect

2. C₂×#20259239

C₂

semidirect

3. C₄×#15690271

C₂

semidirect

19. #15690271

D₄

semidirect

60. C₄×C₁₉₁

#100

semidirect

68. C₁₉₁

#1965

semidirect

129. C₄

#20259239

semidirect

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

subgroup	index
#27132687	2
C₂×#20259239	2
C₄×#15690271	2
C₂×#127×C₁₉₁	2
#1965	191

subgroup

index

C₂×#127×C₁₉₁

191

Subgroups — #37573361

Conjugacy classes of subgroups

Complements

Maximal subgroups