Group #37583073 — subgroups

Subgroups — #37583073

Order 24448.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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×#20259314	C₂	1	1	yes	maximal, characteristic
2	#27133069	C₂	1	4	yes	maximal, complemented
3	#27133062	C₂	1	4	yes	maximal, complemented
4	#27132706	C₂	1	4	yes	maximal
5	C₄×#15690271	C₂	1	1	yes	maximal, characteristic
6	#533×C₁₉₁	C₂	1	1	yes	maximal, characteristic
7	C₂×#15690271	C₂²	1	2	yes	—
8	#20259314	—	2	2	no	—
9	#20259314	C₂²	1	2	yes	—
10	#20259314	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₁₉₁	C₂²	1	2	yes	—
12	C₂×Dic₁₉₁⋊C₄	—	2	2	no	—
13	#20259255	C₂²	1	8	yes	—
14	#20259253	C₂²	1	8	yes	—
15	#20259233	C₂²	1	2	yes	—
16	C₄×C₁₉₁⋊C₈	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₁₉₁	C₂²	1	2	yes	—
18	#125×C₁₉₁	C₂²	1	2	yes	—
19	C₂²⋊Q₈×C₁₉₁	—	2	2	no	—
20	C₄×Q₈×C₁₉₁	—	2	2	no	—
21	#117×C₁₉₁	C₂²	1	2	yes	—
22	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
23	C₂²×Dic₁₉₁	—	2	2	no	—
24	#15690271	Q₈	1	8	yes	complemented
25	C₂×C₁₉₁⋊C₈	C₂³	1	4	yes	—
26	C₄⋊Dic₁₉₁	—	2	2	no	—
27	C₄⋊Dic₁₉₁	C₂³	1	4	yes	—
28	Dic₁₉₁⋊C₄	—	2	4	no	—
29	Dic₁₉₁⋊C₄	—	2	4	no	—
30	C₂×Q₈×C₁₉₁	—	2	2	no	—
31	C₂²×C₄×C₁₉₁	D₄	1	2	yes	—
32	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₁₉₁	—	2	4	no	—
34	C₄⋊C₄×C₁₉₁	—	2	2	no	—
35	C₄⋊C₄×C₁₉₁	C₂³	1	4	yes	—
36	C₂²⋊C₄×C₁₉₁	—	2	2	no	—
37	C₄²×C₁₉₁	—	2	2	no	—
38	C₄²×C₁₉₁	D₄	1	2	yes	—
39	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
40	C₄²×C₁₉₁	C₂³	1	1	yes	characteristic
41	C₂×Dic₁₉₁	—	2	4	no	—
42	C₂×Dic₁₉₁	—	4	4	no	—
43	C₁₉₁⋊C₈	C₂×Q₈	1	8	yes	—
44	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
45	Q₈×C₁₉₁	—	4	4	no	—
46	C₂×C₄×C₁₉₁	—	2	4	no	—
47	C₂×C₄×C₁₉₁	—	4	2	no	—
48	C₂×C₄×C₁₉₁	C₂×D₄	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×Q₈	1	4	yes	—
52	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
53	C₂²×C₁₉₁	—	2	2	no	—
54	C₂²×C₁₉₁	C₄⋊Q₈	1	2	yes	—
55	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
56	C₄×C₁₉₁	—	2	2	no	—
57	C₄×C₁₉₁	—	2	2	no	—
58	C₄×C₁₉₁	—	4	4	no	—
59	C₄×C₁₉₁	C₄⋊Q₈	1	2	yes	—
60	C₄×C₁₉₁	C₂²×Q₈	1	2	yes	—
61	Dic₁₉₁	—	4	4	no	—
62	C₂×C₁₉₁	—	2	2	no	—
63	C₂×C₁₉₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₁₉₁	#575	1	2	yes	—
65	C₁₉₁	#3166	1	1	yes	characteristic, complemented, Sylow
66	#3166	—	191	1	no	maximal, Sylow
67	#533	D₁₉₁	1	1	yes	characteristic
68	C₂×#125	—	191	1	no	—
69	#501	—	191	4	no	—
70	#499	—	191	4	no	—
71	#428	—	191	4	no	—
72	C₄×M₁₆	—	191	1	no	—
73	C₂×M₁₆	—	191	2	no	—
74	C₄⋊Q₈	C₂×D₁₉₁	1	2	yes	—
75	#125	—	382	2	no	—
76	#125	—	191	2	no	—
77	#125	—	191	2	no	—
78	#125	C₂×D₁₉₁	1	2	yes	—
79	C₂²⋊Q₈	—	2	2	no	—
80	C₄×Q₈	—	2	2	no	—
81	#117	C₂×D₁₉₁	1	2	yes	—
82	C₂×C₄⋊C₄	—	382	2	no	—
83	C₂×C₄⋊C₄	—	191	2	no	—
84	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
85	#107	—	191	8	no	—
86	#106	—	191	8	no	—
87	#97	—	191	2	no	—
88	C₄×C₈	—	191	2	no	—
89	C₂×Q₈	—	2	2	no	—
90	C₂²×C₄	—	382	2	no	—
91	C₂²×C₄	C₂²⋊D₁₉₁	1	2	yes	—
92	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
93	M₁₆	—	191	8	no	—
94	C₂×C₈	—	191	4	no	—
95	C₄⋊C₄	—	2	4	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	382	4	no	—
98	C₄⋊C₄	—	382	4	no	—
99	C₄⋊C₄	—	382	2	no	—
100	C₄⋊C₄	—	191	4	no	—
101	C₄⋊C₄	C₂²×D₁₉₁	1	4	yes	—
102	C₂²⋊C₄	—	2	2	no	—
103	C₄²	—	2	2	no	—
104	C₄²	C₂²⋊D₁₉₁	1	2	yes	—
105	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
106	C₄²	C₂²×D₁₉₁	1	1	yes	characteristic
107	C₂³	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
108	Q₈	—	4	4	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	382	4	no	—
112	C₂×C₄	—	764	4	no	—
113	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
114	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
115	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
116	C₂×C₄	Q₈×D₁₉₁	1	4	yes	—
117	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
118	C₈	—	191	8	no	—
119	C₂²	—	2	2	no	—
120	C₂²	#20259320	1	2	yes	—
121	C₂²	C₂²×C₂²⋊D₁₉₁	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	4	4	no	—
125	C₄	—	764	4	no	—
126	C₄	#20259320	1	2	yes	—
127	C₄	C₂×Q₈×D₁₉₁	1	2	yes	—
128	C₂	—	2	2	no	—
129	C₂	C₂×#20259320	1	1	yes	characteristic
130	C₂	#27133461	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20259314

C₂

yes

maximal, characteristic

#27133069

C₂

yes

maximal, complemented

#27133062

C₂

yes

maximal, complemented

#27132706

C₂

yes

maximal

C₄×#15690271

C₂

yes

maximal, characteristic

#533×C₁₉₁

C₂

yes

maximal, characteristic

C₂×#15690271

C₂²

yes

—

#20259314

—

#20259314

C₂²

yes

—

#20259314

C₂²

yes

—

C₂×C₄⋊Dic₁₉₁

C₂²

yes

—

C₂×Dic₁₉₁⋊C₄

—

#20259255

C₂²

yes

—

#20259253

C₂²

yes

—

#20259233

C₂²

yes

—

C₄×C₁₉₁⋊C₈

C₂²

yes

—

C₄⋊Q₈×C₁₉₁

C₂²

yes

—

#125×C₁₉₁

C₂²

yes

—

C₂²⋊Q₈×C₁₉₁

—

C₄×Q₈×C₁₉₁

—

#117×C₁₉₁

C₂²

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₂²×Dic₁₉₁

—

#15690271

Q₈

yes

complemented

C₂×C₁₉₁⋊C₈

C₂³

yes

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

C₂³

yes

—

Dic₁₉₁⋊C₄

—

Dic₁₉₁⋊C₄

—

C₂×Q₈×C₁₉₁

—

C₂²×C₄×C₁₉₁

D₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

C₂³

yes

—

C₂²⋊C₄×C₁₉₁

—

C₄²×C₁₉₁

—

C₄²×C₁₉₁

D₄

yes

—

C₄²×C₁₉₁

C₂³

yes

characteristic

C₄²×C₁₉₁

C₂³

yes

characteristic

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₁₉₁⋊C₈

C₂×Q₈

yes

—

C₂³×C₁₉₁

C₂×D₄

yes

characteristic

Q₈×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×Q₈

yes

—

C₂×C₄×C₁₉₁

C₂⁴

yes

characteristic

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₄⋊Q₈

yes

—

C₂²×C₁₉₁

C₂²×D₄

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

C₄⋊Q₈

yes

—

C₄×C₁₉₁

C₂²×Q₈

yes

—

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₉₁

#575

yes

—

C₁₉₁

#3166

yes

characteristic, complemented, Sylow

#3166

—

191

maximal, Sylow

#533

D₁₉₁

yes

characteristic

C₂×#125

—

191

—

#501

—

191

—

#499

—

191

—

#428

—

191

—

C₄×M₁₆

—

191

—

C₂×M₁₆

—

191

—

C₄⋊Q₈

C₂×D₁₉₁

yes

—

#125

—

382

—

#125

—

191

—

#125

—

191

—

#125

C₂×D₁₉₁

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₁₉₁

yes

—

C₂×C₄⋊C₄

—

382

—

C₂×C₄⋊C₄

—

191

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

#107

—

191

—

#106

—

191

—

#97

—

191

—

C₄×C₈

—

191

—

C₂×Q₈

—

C₂²×C₄

—

382

—

C₂²×C₄

C₂²⋊D₁₉₁

yes

—

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

M₁₆

—

191

—

C₂×C₈

—

191

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

382

—

100

C₄⋊C₄

—

191

—

101

C₄⋊C₄

C₂²×D₁₉₁

yes

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

C₂²⋊D₁₉₁

yes

—

105

C₄²

C₂²×D₁₉₁

yes

characteristic

106

C₄²

C₂²×D₁₉₁

yes

characteristic

107

C₂³

C₂×C₂²⋊D₁₉₁

yes

characteristic

108

Q₈

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

382

—

112

C₂×C₄

—

764

—

113

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

—

114

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

—

115

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

116

C₂×C₄

Q₈×D₁₉₁

yes

—

117

C₂×C₄

C₂³×D₁₉₁

yes

characteristic

118

C₈

—

191

—

119

C₂²

—

120

C₂²

#20259320

yes

—

121

C₂²

C₂²×C₂²⋊D₁₉₁

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

764

—

126

C₄

#20259320

yes

—

127

C₄

C₂×Q₈×D₁₉₁

yes

—

128

C₂

—

129

C₂

C₂×#20259320

yes

characteristic

130

C₂

#27133461

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₉₁
commutator, cosocle	C₂×C₄×C₁₉₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	#533×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
prime radical	C₂³×C₁₉₁	C₂×D₄
hypercenter	#533	D₁₉₁
hypocenter	C₁₉₁	#3166
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₉₁

commutator, cosocle

C₂×C₄×C₁₉₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉₁

Fitting

#533×C₁₉₁

C₂

socle

C₂²×C₁₉₁

C₂²×D₄

prime radical

C₂³×C₁₉₁

C₂×D₄

D₁₉₁

C₁₉₁

G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #27133069	C₂	semidirect
3. #27133062	C₂	semidirect
24. #15690271	Q₈	semidirect
65. C₁₉₁	#3166	semidirect

H ≅ G/N

type

2. #27133069

C₂

semidirect

3. #27133062

C₂

semidirect

24. #15690271

Q₈

semidirect

65. C₁₉₁

#3166

semidirect

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

subgroup	index
C₂×#20259314	2
#27133069	2
#27133062	2
#27132706	2
C₄×#15690271	2
#533×C₁₉₁	2
#3166	191

subgroup

index

#533×C₁₉₁

191

Subgroups — #37583073

Conjugacy classes of subgroups

Complements

Maximal subgroups