Group #37583865 — subgroups

Subgroups — #37583865

Order 24448.

348 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 137 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	#27133220	C₂	1	2	yes	maximal
2	#27132959	C₂	1	4	yes	maximal, complemented
3	#27132904	C₂	1	4	yes	maximal
4	#27132874	C₂	1	4	yes	maximal, complemented
5	C₂×C₄⋊C₈×C₁₉₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₆₄	C₂²	1	4	yes	—
7	#20259322	C₂²	1	2	yes	—
8	#20259322	C₂²	1	2	yes	—
9	#20259314	—	2	2	no	—
10	#20259301	—	2	2	no	—
11	#20259301	C₂²	1	4	yes	—
12	C₄×Dic₃₈₂	—	2	2	no	—
13	C₄×Dic₃₈₂	C₂²	1	4	yes	—
14	#20259281	—	2	2	no	—
15	#20259256	C₂²	1	4	yes	—
16	#20259247	C₂²	1	8	yes	—
17	C₂²×C₈×C₁₉₁	C₂²	1	2	yes	—
18	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₁₉₁	C₂²	1	4	yes	—
20	C₂×Dic₃₈₂	—	2	2	no	—
21	C₂×Dic₃₈₂	C₂³	1	4	yes	—
22	Dic₇₆₄	—	4	4	no	—
23	C₄⋊Dic₁₉₁	—	2	2	no	—
24	C₄⋊Dic₁₉₁	—	2	2	no	—
25	C₄⋊Dic₁₉₁	—	2	2	no	—
26	C₄⋊Dic₁₉₁	C₂³	1	4	yes	—
27	Dic₁₉₁⋊C₄	—	2	4	no	—
28	Dic₁₉₁⋊C₄	—	2	4	no	—
29	C₂²⋊Dic₁₉₁	—	2	4	no	—
30	C₄×Dic₁₉₁	—	2	4	no	—
31	C₂²×C₄×C₁₉₁	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₉₁	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₁₉₁	—	4	2	no	—
35	C₂×C₈×C₁₉₁	C₂³	1	4	yes	—
36	C₄²×C₁₉₁	D₄	1	2	yes	—
37	C₄²×C₁₉₁	C₂³	1	2	yes	—
38	Dic₃₈₂	—	2	4	no	—
39	Dic₃₈₂	—	2	4	no	—
40	Dic₃₈₂	—	4	4	no	—
41	C₂×Dic₁₉₁	—	2	4	no	—
42	C₂×Dic₁₉₁	—	2	4	no	—
43	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₉₁	—	2	1	no	—
45	C₂×C₄×C₁₉₁	—	2	1	no	—
46	C₂×C₄×C₁₉₁	—	4	1	no	—
47	C₂×C₄×C₁₉₁	Q₁₆	1	4	yes	complemented
48	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
52	C₈×C₁₉₁	—	2	4	no	—
53	C₂²×C₁₉₁	—	2	1	no	—
54	C₂²×C₁₉₁	—	2	1	no	—
55	C₂²×C₁₉₁	C₂×Q₁₆	1	2	yes	—
56	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
57	C₄×C₁₉₁	—	2	1	no	—
58	C₄×C₁₉₁	—	2	2	no	—
59	C₄×C₁₉₁	C₂×Q₁₆	1	4	yes	—
60	C₄×C₁₉₁	H(5,𝔽₂)	1	1	yes	characteristic
61	C₄×C₁₉₁	2₋¹⁺⁴	1	1	yes	characteristic
62	Dic₁₉₁	—	4	4	no	—
63	Dic₁₉₁	—	4	4	no	—
64	C₂×C₁₉₁	—	2	2	no	—
65	C₂×C₁₉₁	#537	1	1	yes	characteristic
66	C₂×C₁₉₁	C₂²×Q₁₆	1	1	yes	characteristic
67	C₂×C₁₉₁	#575	1	1	yes	characteristic
68	C₁₉₁	#3249	1	1	yes	characteristic, complemented, Sylow
69	#3249	—	191	1	no	maximal, Sylow
70	#533	—	191	2	no	—
71	#477	—	191	4	no	—
72	#467	—	191	4	no	—
73	#462	—	191	4	no	—
74	C₂×C₄⋊C₈	D₁₉₁	1	1	yes	characteristic
75	C₂×Q₁₆	—	191	4	no	—
76	C₂²×C₈	C₂×D₁₉₁	1	2	yes	—
77	C₄⋊Q₈	—	191	2	no	—
78	C₄⋊Q₈	—	191	2	no	—
79	#125	—	382	2	no	—
80	C₂²⋊Q₈	—	382	2	no	—
81	C₂²⋊Q₈	—	191	4	no	—
82	C₄×Q₈	—	382	2	no	—
83	C₄×Q₈	—	191	4	no	—
84	#117	—	382	2	no	—
85	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
86	#107	—	191	4	no	—
87	C₄⋊C₈	C₂×D₁₉₁	1	4	yes	—
88	#103	—	191	8	no	—
89	C₂×Q₈	—	382	2	no	—
90	C₂×Q₈	—	191	4	no	—
91	C₂²×C₄	D₇₆₄	1	1	yes	characteristic
92	C₂²×C₄	D₇₆₄	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
94	Q₁₆	—	764	4	no	—
95	C₂×C₈	—	4	2	no	—
96	C₂×C₈	C₂²×D₁₉₁	1	4	yes	—
97	C₄⋊C₄	—	382	4	no	—
98	C₄⋊C₄	—	382	4	no	—
99	C₄⋊C₄	—	382	2	no	—
100	C₄⋊C₄	—	382	2	no	—
101	C₄⋊C₄	—	382	2	no	—
102	C₄⋊C₄	—	191	4	no	—
103	C₂²⋊C₄	—	382	4	no	—
104	C₄²	—	382	4	no	—
105	C₄²	D₇₆₄	1	2	yes	—
106	C₄²	C₂²×D₁₉₁	1	2	yes	—
107	C₂³	C₂×D₇₆₄	1	1	yes	characteristic
108	Q₈	—	382	4	no	—
109	Q₈	—	382	4	no	—
110	Q₈	—	764	4	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	—	382	4	no	—
115	C₂×C₄	—	382	4	no	—
116	C₂×C₄	Dic₇₆₄	1	4	yes	complemented
117	C₂×C₄	C₂×D₇₆₄	1	1	yes	characteristic
118	C₂×C₄	C₂×D₇₆₄	1	2	yes	—
119	C₂×C₄	C₂×D₇₆₄	1	2	yes	—
120	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
121	C₈	—	2	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	C₂×Dic₇₆₄	1	2	yes	—
125	C₂²	C₂²×D₇₆₄	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	2	1	no	—
128	C₄	—	764	4	no	—
129	C₄	—	764	4	no	—
130	C₄	C₂×Dic₇₆₄	1	4	yes	—
131	C₄	#20259367	1	1	yes	characteristic
132	C₄	#20259370	1	1	yes	characteristic
133	C₂	—	2	2	no	—
134	C₂	#27133244	1	1	yes	characteristic
135	C₂	C₂²×Dic₇₆₄	1	1	yes	characteristic
136	C₂	#27133463	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27133220

C₂

yes

maximal

#27132959

C₂

yes

maximal, complemented

#27132904

C₂

yes

maximal

#27132874

C₂

yes

maximal, complemented

C₂×C₄⋊C₈×C₁₉₁

C₂

yes

maximal, characteristic

C₂×Dic₇₆₄

C₂²

yes

—

#20259322

C₂²

yes

—

#20259322

C₂²

yes

—

#20259314

—

#20259301

—

#20259301

C₂²

yes

—

C₄×Dic₃₈₂

—

C₄×Dic₃₈₂

C₂²

yes

—

#20259281

—

#20259256

C₂²

yes

—

#20259247

C₂²

yes

—

C₂²×C₈×C₁₉₁

C₂²

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₁

C₂²

yes

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

C₂³

yes

—

Dic₇₆₄

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

C₂³

yes

—

Dic₁₉₁⋊C₄

—

Dic₁₉₁⋊C₄

—

C₂²⋊Dic₁₉₁

—

C₄×Dic₁₉₁

—

C₂²×C₄×C₁₉₁

D₄

yes

characteristic

C₂²×C₄×C₁₉₁

D₄

yes

characteristic

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

C₂³

yes

—

C₄²×C₁₉₁

D₄

yes

—

C₄²×C₁₉₁

C₂³

yes

—

Dic₃₈₂

—

Dic₃₈₂

—

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂³×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

Q₁₆

yes

complemented

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂⁴

yes

characteristic

C₈×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂×Q₁₆

yes

—

C₂²×C₁₉₁

C₂²×D₄

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂×Q₁₆

yes

—

C₄×C₁₉₁

H(5,𝔽₂)

yes

characteristic

C₄×C₁₉₁

2₋¹⁺⁴

yes

characteristic

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

#537

yes

characteristic

C₂×C₁₉₁

C₂²×Q₁₆

yes

C₂×C₁₉₁

yes

C₁₉₁

yes

characteristic, complemented, Sylow

#3249

—

191

maximal, Sylow

#533

—

191

—

#477

—

191

—

#467

—

191

—

#462

—

191

—

C₂×C₄⋊C₈

D₁₉₁

yes

characteristic

C₂×Q₁₆

—

191

—

C₂²×C₈

C₂×D₁₉₁

yes

—

C₄⋊Q₈

—

191

—

C₄⋊Q₈

—

191

—

#125

—

382

—

C₂²⋊Q₈

—

382

—

C₂²⋊Q₈

—

191

—

C₄×Q₈

—

382

—

C₄×Q₈

—

191

—

#117

—

382

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

#107

—

191

—

C₄⋊C₈

C₂×D₁₉₁

yes

—

#103

—

191

—

C₂×Q₈

—

382

—

C₂×Q₈

—

191

—

C₂²×C₄

D₇₆₄

yes

characteristic

C₂²×C₄

D₇₆₄

yes

characteristic

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

Q₁₆

—

764

—

C₂×C₈

—

C₂×C₈

C₂²×D₁₉₁

yes

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

382

—

100

C₄⋊C₄

—

382

—

101

C₄⋊C₄

—

382

—

102

C₄⋊C₄

—

191

—

103

C₂²⋊C₄

—

382

—

104

C₄²

—

382

—

105

C₄²

D₇₆₄

yes

—

106

C₄²

C₂²×D₁₉₁

yes

—

107

C₂³

C₂×D₇₆₄

yes

characteristic

108

Q₈

—

382

—

109

Q₈

—

382

—

110

Q₈

—

764

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

382

—

115

C₂×C₄

—

382

—

116

C₂×C₄

Dic₇₆₄

yes

complemented

117

C₂×C₄

C₂×D₇₆₄

yes

characteristic

118

C₂×C₄

C₂×D₇₆₄

yes

—

119

C₂×C₄

C₂×D₇₆₄

yes

—

120

C₂×C₄

C₂³×D₁₉₁

yes

characteristic

121

C₈

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂×Dic₇₆₄

yes

—

125

C₂²

C₂²×D₇₆₄

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

764

—

129

C₄

—

764

—

130

C₄

C₂×Dic₇₆₄

yes

—

131

C₄

#20259367

yes

characteristic

132

C₄

#20259370

yes

characteristic

133

C₂

—

134

C₂

#27133244

yes

characteristic

135

C₂

C₂²×Dic₇₆₄

yes

characteristic

136

C₂

#27133463

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₆₄
commutator, cosocle	C₂×C₄×C₁₉₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	C₂×C₄⋊C₈×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
prime radical	C₂³×C₁₉₁	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₁₉₁
hypocenter	C₁₉₁	#3249
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂²

C₂²×D₇₆₄

commutator, cosocle

C₂×C₄×C₁₉₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉₁

Fitting

C₂×C₄⋊C₈×C₁₉₁

C₂

socle

C₂²×C₁₉₁

C₂²×D₄

prime radical

C₂³×C₁₉₁

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₁₉₁

hypocenter

C₁₉₁

#3249

LCS

G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
2. #27132959	C₂	semidirect
4. #27132874	C₂	semidirect
47. C₂×C₄×C₁₉₁	Q₁₆	semidirect
68. C₁₉₁	#3249	semidirect
116. C₂×C₄	Dic₇₆₄	semidirect

H ≅ G/N

type

2. #27132959

C₂

semidirect

4. #27132874

C₂

semidirect

47. C₂×C₄×C₁₉₁

Q₁₆

semidirect

68. C₁₉₁

#3249

semidirect

116. C₂×C₄

Dic₇₆₄

semidirect

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

subgroup	index
#27133220	2
#27132959	2
#27132904	2
#27132874	2
C₂×C₄⋊C₈×C₁₉₁	2
#3249	191

subgroup

index

C₂×C₄⋊C₈×C₁₉₁

191

Subgroups — #37583865

Conjugacy classes of subgroups

Complements

Maximal subgroups