Group #37583914 — subgroups

Subgroups — #37583914

Order 24448.

342 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 125 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	#27132949	C₂	1	4	yes	maximal, complemented
3	#27132921	C₂	1	4	yes	maximal
4	#27132874	C₂	1	4	yes	maximal, complemented
5	#423×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	#20259254	C₂²	1	4	yes	—
16	#20259247	C₂²	1	8	yes	—
17	C₂×M₁₆×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	2	yes	—
32	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
33	M₁₆×C₁₉₁	—	4	2	no	—
34	C₂×C₈×C₁₉₁	C₂³	1	4	yes	—
35	C₄²×C₁₉₁	D₄	1	1	yes	characteristic
36	C₄²×C₁₉₁	D₄	1	1	yes	characteristic
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	4	no	—
45	C₂×C₄×C₁₉₁	—	2	1	no	—
46	C₂×C₄×C₁₉₁	—	2	1	no	—
47	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₉₁	C₂⁴	1	1	yes	characteristic
51	C₈×C₁₉₁	—	2	4	no	—
52	C₂²×C₁₉₁	—	2	1	no	—
53	C₂²×C₁₉₁	—	2	2	no	—
54	C₂²×C₁₉₁	C₂²×D₄	1	1	yes	characteristic
55	C₄×C₁₉₁	—	2	1	no	—
56	C₄×C₁₉₁	—	2	2	no	—
57	C₄×C₁₉₁	#137	1	4	yes	—
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	C₂×#137	1	2	yes	—
65	C₁₉₁	#3255	1	1	yes	characteristic, complemented, Sylow
66	#3255	—	191	1	no	maximal, Sylow
67	#533	—	191	2	no	—
68	#475	—	191	4	no	—
69	#470	—	191	4	no	—
70	#462	—	191	4	no	—
71	#423	D₁₉₁	1	1	yes	characteristic
72	C₂×Q₁₆	—	191	4	no	—
73	C₂×M₁₆	C₂×D₁₉₁	1	2	yes	—
74	C₄⋊Q₈	—	191	2	no	—
75	C₄⋊Q₈	—	191	2	no	—
76	#125	—	382	2	no	—
77	C₂²⋊Q₈	—	382	2	no	—
78	C₂²⋊Q₈	—	191	4	no	—
79	C₄×Q₈	—	382	2	no	—
80	C₄×Q₈	—	191	4	no	—
81	#117	—	382	2	no	—
82	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
83	#106	—	191	4	no	—
84	C₄⋊C₈	C₂×D₁₉₁	1	4	yes	—
85	#103	—	191	8	no	—
86	C₂×Q₈	—	382	2	no	—
87	C₂×Q₈	—	191	4	no	—
88	C₂²×C₄	D₇₆₄	1	2	yes	—
89	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
90	Q₁₆	—	764	4	no	—
91	M₁₆	—	4	2	no	—
92	C₂×C₈	C₂²×D₁₉₁	1	4	yes	—
93	C₄⋊C₄	—	382	4	no	—
94	C₄⋊C₄	—	382	4	no	—
95	C₄⋊C₄	—	382	2	no	—
96	C₄⋊C₄	—	382	2	no	—
97	C₄⋊C₄	—	382	2	no	—
98	C₄⋊C₄	—	191	4	no	—
99	C₂²⋊C₄	—	382	4	no	—
100	C₄²	—	382	4	no	—
101	C₄²	D₇₆₄	1	1	yes	characteristic
102	C₄²	D₇₆₄	1	1	yes	characteristic
103	C₄²	C₂²×D₁₉₁	1	2	yes	—
104	C₂³	C₂×D₇₆₄	1	1	yes	characteristic
105	Q₈	—	382	4	no	—
106	Q₈	—	382	4	no	—
107	Q₈	—	764	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	382	4	no	—
112	C₂×C₄	—	382	4	no	—
113	C₂×C₄	C₂×D₇₆₄	1	1	yes	characteristic
114	C₂×C₄	C₂×D₇₆₄	1	2	yes	—
115	C₂×C₄	C₂×D₇₆₄	1	2	yes	—
116	C₂×C₄	C₂³×D₁₉₁	1	1	yes	characteristic
117	C₈	—	2	4	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₂²×D₇₆₄	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	1	no	—
123	C₄	—	764	4	no	—
124	C₄	—	764	4	no	—
125	C₄	#20259355	1	4	yes	—
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	C₂×#20259355	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27133220

C₂

yes

maximal

#27132949

C₂

yes

maximal, complemented

#27132921

C₂

yes

maximal

#27132874

C₂

yes

maximal, complemented

#423×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

—

#20259254

C₂²

yes

—

#20259247

C₂²

yes

—

C₂×M₁₆×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

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

M₁₆×C₁₉₁

—

C₂×C₈×C₁₉₁

C₂³

yes

—

C₄²×C₁₉₁

D₄

yes

characteristic

C₄²×C₁₉₁

D₄

yes

characteristic

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

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

C₂²×D₄

yes

characteristic

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#137

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

C₂×#137

yes

—

C₁₉₁

#3255

yes

characteristic, complemented, Sylow

#3255

—

191

maximal, Sylow

#533

—

191

—

#475

—

191

—

#470

—

191

—

#462

—

191

—

#423

D₁₉₁

yes

characteristic

C₂×Q₁₆

—

191

—

C₂×M₁₆

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

#106

—

191

—

C₄⋊C₈

C₂×D₁₉₁

yes

—

#103

—

191

—

C₂×Q₈

—

382

—

C₂×Q₈

—

191

—

C₂²×C₄

D₇₆₄

yes

—

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

Q₁₆

—

764

—

M₁₆

—

C₂×C₈

C₂²×D₁₉₁

yes

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

382

—

C₄⋊C₄

—

191

—

C₂²⋊C₄

—

382

—

100

C₄²

—

382

—

101

C₄²

D₇₆₄

yes

characteristic

102

C₄²

D₇₆₄

yes

characteristic

103

C₄²

C₂²×D₁₉₁

yes

—

104

C₂³

C₂×D₇₆₄

yes

characteristic

105

Q₈

—

382

—

106

Q₈

—

382

—

107

Q₈

—

764

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

382

—

112

C₂×C₄

—

382

—

113

C₂×C₄

C₂×D₇₆₄

yes

characteristic

114

C₂×C₄

C₂×D₇₆₄

yes

—

115

C₂×C₄

C₂×D₇₆₄

yes

—

116

C₂×C₄

C₂³×D₁₉₁

yes

characteristic

117

C₈

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂²×D₇₆₄

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

764

—

124

C₄

—

764

—

125

C₄

#20259355

yes

—

126

C₄

#20259367

yes

characteristic

127

C₄

#20259370

yes

characteristic

128

C₂

—

129

C₂

#27133244

yes

characteristic

130

C₂

C₂×#20259355

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₇₆₄
commutator, cosocle	C₂×C₄×C₁₉₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₁
Fitting	#423×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂²×D₄
prime radical	C₂³×C₁₉₁	C₂×D₄
hypercenter	#423	D₁₉₁
hypocenter	C₁₉₁	#3255
LCS	G ▷ C₂×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂²×D₇₆₄

commutator, cosocle

C₂×C₄×C₁₉₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉₁

Fitting

#423×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 ◁ #423

N	H ≅ G/N	type
2. #27132949	C₂	semidirect
4. #27132874	C₂	semidirect
65. C₁₉₁	#3255	semidirect

H ≅ G/N

type

2. #27132949

C₂

semidirect

4. #27132874

C₂

semidirect

65. C₁₉₁

#3255

semidirect

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

subgroup	index
#27133220	2
#27132949	2
#27132921	2
#27132874	2
#423×C₁₉₁	2
#3255	191

subgroup

index

#423×C₁₉₁

191

Subgroups — #37583914

Conjugacy classes of subgroups

Complements

Maximal subgroups