Group #37583974 — subgroups

Subgroups — #37583974

Order 24448.

322 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 117 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	#27133275	C₂	1	2	yes	maximal
2	#27132971	C₂	1	4	yes	maximal, complemented
3	#27132921	C₂	1	2	yes	maximal
4	#27132904	C₂	1	2	yes	maximal
5	#27132874	C₂	1	4	yes	maximal, complemented
6	#424×C₁₉₁	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₇₆₄	C₂²	1	2	yes	—
8	C₂×Dic₇₆₄	C₂²	1	2	yes	—
9	#20259322	C₂²	1	2	yes	—
10	#20259315	—	2	4	no	—
11	#20259314	C₂²	1	2	yes	—
12	#20259312	—	2	4	no	—
13	#20259301	C₂²	1	4	yes	—
14	C₄×Dic₃₈₂	C₂²	1	4	yes	—
15	#20259256	C₂²	1	2	yes	—
16	#20259254	C₂²	1	2	yes	—
17	#20259247	C₂²	1	4	yes	—
18	#20259247	C₂²	1	4	yes	—
19	C₂×M₁₆×C₁₉₁	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
21	#117×C₁₉₁	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₉₁	C₂²	1	4	yes	—
23	C₂×Dic₃₈₂	C₂³	1	4	yes	—
24	Dic₇₆₄	—	4	2	no	—
25	Dic₇₆₄	—	4	2	no	—
26	C₄⋊Dic₁₉₁	—	2	4	no	—
27	C₄⋊Dic₁₉₁	C₂³	1	4	yes	—
28	Dic₁₉₁⋊C₄	—	2	4	no	—
29	Dic₁₉₁⋊C₄	—	2	4	no	—
30	Dic₁₉₁⋊C₄	—	2	4	no	—
31	C₂²⋊Dic₁₉₁	—	2	4	no	—
32	C₄×Dic₁₉₁	—	2	4	no	—
33	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
34	M₁₆×C₁₉₁	—	4	1	no	—
35	C₂×C₈×C₁₉₁	—	4	1	no	—
36	C₂×C₈×C₁₉₁	C₂³	1	2	yes	—
37	C₂×C₈×C₁₉₁	C₂³	1	2	yes	—
38	C₄⋊C₄×C₁₉₁	D₄	1	2	yes	—
39	C₂²⋊C₄×C₁₉₁	D₄	1	2	yes	—
40	C₄²×C₁₉₁	C₂³	1	2	yes	—
41	Dic₃₈₂	—	2	4	no	—
42	Dic₃₈₂	—	2	4	no	—
43	C₂×Dic₁₉₁	—	2	4	no	—
44	C₂×Dic₁₉₁	—	2	4	no	—
45	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₉₁	—	2	1	no	—
47	C₂×C₄×C₁₉₁	—	2	1	no	—
48	C₂×C₄×C₁₉₁	C₂×D₄	1	4	yes	—
49	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
51	C₈×C₁₉₁	—	2	2	no	—
52	C₈×C₁₉₁	—	2	2	no	—
53	C₂²×C₁₉₁	—	2	1	no	—
54	C₂²×C₁₉₁	—	2	2	no	—
55	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
56	C₄×C₁₉₁	—	2	1	no	—
57	C₄×C₁₉₁	—	2	4	no	—
58	C₄×C₁₉₁	H(5,𝔽₂)	1	1	yes	characteristic
59	C₄×C₁₉₁	2₋¹⁺⁴	1	1	yes	characteristic
60	Dic₁₉₁	—	4	4	no	—
61	Dic₁₉₁	—	4	4	no	—
62	C₂×C₁₉₁	—	4	1	no	—
63	C₂×C₁₉₁	#537	1	1	yes	characteristic
64	C₂×C₁₉₁	#578	1	2	yes	—
65	C₁₉₁	#3261	1	1	yes	characteristic, complemented, Sylow
66	#3261	—	191	1	no	maximal, Sylow
67	#541	—	191	2	no	—
68	#479	—	191	4	no	—
69	#470	—	191	2	no	—
70	#467	—	191	2	no	—
71	#462	—	191	4	no	—
72	#424	D₁₉₁	1	1	yes	characteristic
73	C₂×Q₁₆	—	191	2	no	—
74	C₂×Q₁₆	—	191	2	no	—
75	C₂×M₁₆	C₂×D₁₉₁	1	1	yes	characteristic
76	C₂²×C₈	C₂×D₁₉₁	1	1	yes	characteristic
77	C₄⋊Q₈	—	191	2	no	—
78	#126	—	382	4	no	—
79	#125	—	382	4	no	—
80	#125	—	191	2	no	—
81	C₂²⋊Q₈	—	191	4	no	—
82	C₄×Q₈	—	191	4	no	—
83	#117	C₂×D₁₉₁	1	1	yes	characteristic
84	#107	—	191	2	no	—
85	#106	—	191	2	no	—
86	C₄⋊C₈	C₂×D₁₉₁	1	4	yes	—
87	#103	—	191	4	no	—
88	#103	—	191	4	no	—
89	C₂×Q₈	—	191	4	no	—
90	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
91	Q₁₆	—	764	2	no	—
92	Q₁₆	—	764	2	no	—
93	M₁₆	—	4	1	no	—
94	C₂×C₈	—	4	1	no	—
95	C₂×C₈	C₂²×D₁₉₁	1	2	yes	—
96	C₂×C₈	C₂²×D₁₉₁	1	2	yes	—
97	C₄⋊C₄	—	382	4	no	—
98	C₄⋊C₄	—	382	4	no	—
99	C₄⋊C₄	—	382	4	no	—
100	C₄⋊C₄	—	382	4	no	—
101	C₄⋊C₄	—	191	4	no	—
102	C₄⋊C₄	D₇₆₄	1	2	yes	—
103	C₂²⋊C₄	—	382	4	no	—
104	C₂²⋊C₄	D₇₆₄	1	2	yes	—
105	C₄²	—	382	4	no	—
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	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	382	4	no	—
113	C₂×C₄	—	382	4	no	—
114	C₂×C₄	C₂×D₇₆₄	1	4	yes	—
115	C₂×C₄	C₂×D₇₆₄	1	1	yes	characteristic
116	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
117	C₈	—	2	2	no	—
118	C₈	—	2	2	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	1	no	—
121	C₂²	C₂²×D₇₆₄	1	1	yes	characteristic
122	C₄	—	2	4	no	—
123	C₄	—	2	1	no	—
124	C₄	—	764	4	no	—
125	C₄	—	764	4	no	—
126	C₄	#20259367	1	1	yes	characteristic
127	C₄	#20259370	1	1	yes	characteristic
128	C₂	—	4	1	no	—
129	C₂	#27133244	1	1	yes	characteristic
130	C₂	#27133480	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27133275

C₂

yes

maximal

#27132971

C₂

yes

maximal, complemented

#27132921

C₂

yes

maximal

#27132904

C₂

yes

maximal

#27132874

C₂

yes

maximal, complemented

#424×C₁₉₁

C₂

yes

maximal, characteristic

C₂×Dic₇₆₄

C₂²

yes

—

C₂×Dic₇₆₄

C₂²

yes

—

#20259322

C₂²

yes

—

#20259315

—

#20259314

C₂²

yes

—

#20259312

—

#20259301

C₂²

yes

—

C₄×Dic₃₈₂

C₂²

yes

—

#20259256

C₂²

yes

—

#20259254

C₂²

yes

—

#20259247

C₂²

yes

—

#20259247

C₂²

yes

—

C₂×M₁₆×C₁₉₁

C₂²

yes

characteristic

C₂²×C₈×C₁₉₁

C₂²

yes

characteristic

#117×C₁₉₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₁

C₂²

yes

—

C₂×Dic₃₈₂

C₂³

yes

—

Dic₇₆₄

—

Dic₇₆₄

—

C₄⋊Dic₁₉₁

—

C₄⋊Dic₁₉₁

C₂³

yes

—

Dic₁₉₁⋊C₄

—

Dic₁₉₁⋊C₄

—

Dic₁₉₁⋊C₄

—

C₂²⋊Dic₁₉₁

—

C₄×Dic₁₉₁

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

M₁₆×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

C₂³

yes

—

C₂×C₈×C₁₉₁

C₂³

yes

—

C₄⋊C₄×C₁₉₁

D₄

yes

—

C₂²⋊C₄×C₁₉₁

D₄

yes

—

C₄²×C₁₉₁

C₂³

yes

—

Dic₃₈₂

—

Dic₃₈₂

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂³×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂⁴

yes

characteristic

C₈×C₁₉₁

—

C₈×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂²×D₄

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

—

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

#578

yes

—

C₁₉₁

#3261

yes

characteristic, complemented, Sylow

#3261

—

191

maximal, Sylow

#541

—

191

—

#479

—

191

—

#470

—

191

—

#467

—

191

—

#462

—

191

—

#424

D₁₉₁

yes

characteristic

C₂×Q₁₆

—

191

—

C₂×Q₁₆

—

191

—

C₂×M₁₆

C₂×D₁₉₁

yes

characteristic

C₂²×C₈

C₂×D₁₉₁

yes

characteristic

C₄⋊Q₈

—

191

—

#126

—

382

—

#125

—

382

—

#125

—

191

—

C₂²⋊Q₈

—

191

—

C₄×Q₈

—

191

—

#117

C₂×D₁₉₁

yes

characteristic

#107

—

191

—

#106

—

191

—

C₄⋊C₈

C₂×D₁₉₁

yes

—

#103

—

191

—

#103

—

191

—

C₂×Q₈

—

191

—

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

Q₁₆

—

764

—

Q₁₆

—

764

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²×D₁₉₁

yes

—

C₂×C₈

C₂²×D₁₉₁

yes

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

382

—

100

C₄⋊C₄

—

382

—

101

C₄⋊C₄

—

191

—

102

C₄⋊C₄

D₇₆₄

yes

—

103

C₂²⋊C₄

—

382

—

104

C₂²⋊C₄

D₇₆₄

yes

—

105

C₄²

—

382

—

106

C₄²

C₂²×D₁₉₁

yes

—

107

C₂³

C₂×D₇₆₄

yes

characteristic

108

Q₈

—

382

—

109

Q₈

—

382

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

382

—

113

C₂×C₄

—

382

—

114

C₂×C₄

C₂×D₇₆₄

yes

—

115

C₂×C₄

C₂×D₇₆₄

yes

characteristic

116

C₂×C₄

C₂³×D₁₉₁

yes

characteristic

117

C₈

—

118

C₈

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂²×D₇₆₄

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

764

—

125

C₄

—

764

—

126

C₄

#20259367

yes

characteristic

127

C₄

#20259370

yes

characteristic

128

C₂

—

129

C₂

#27133244

yes

characteristic

130

C₂

#27133480

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₇₆₄
commutator, cosocle	C₂×C₄×C₁₉₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	#424×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
prime radical	C₂³×C₁₉₁	C₂×D₄
hypercenter	#424	D₁₉₁
hypocenter	C₁₉₁	#3261
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #424

subgroup

G/H

center

C₂²

C₂²×D₇₆₄

commutator, cosocle

C₂×C₄×C₁₉₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉₁

Fitting

#424×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 ◁ #117 ◁ #424

N	H ≅ G/N	type
2. #27132971	C₂	semidirect
5. #27132874	C₂	semidirect
65. C₁₉₁	#3261	semidirect

H ≅ G/N

type

2. #27132971

C₂

semidirect

5. #27132874

C₂

semidirect

65. C₁₉₁

#3261

semidirect

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

subgroup	index
#27133275	2
#27132971	2
#27132921	2
#27132904	2
#27132874	2
#424×C₁₉₁	2
#3261	191

subgroup

index

#424×C₁₉₁

191

Subgroups — #37583974

Conjugacy classes of subgroups

Complements

Maximal subgroups