Group #37572595 — subgroups

Subgroups — #37572595

Order 24448.

300 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 85 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×#20259281	C₂	1	1	yes	maximal, characteristic
2	#27132526	C₂	1	1	yes	maximal, characteristic
3	#27132429	C₂	1	4	yes	maximal
4	#381×C₁₉₁	C₂	1	1	yes	maximal, characteristic
5	#20259281	—	2	2	no	—
6	#20259281	—	2	2	no	—
7	C₂×C₄⋊Dic₁₉₁	C₄	1	2	yes	complemented
8	C₂×C₂²⋊Dic₁₉₁	C₂²	1	2	yes	—
9	C₂×C₂²⋊Dic₁₉₁	C₂²	1	2	yes	—
10	C₂×C₄×Dic₁₉₁	C₄	1	2	yes	complemented
11	C₁₉₁⋊#95	—	2	2	no	—
12	C₂³×C₄×C₁₉₁	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊C₄×C₁₉₁	C₂²	1	2	yes	—
14	#95×C₁₉₁	—	2	2	no	—
15	C₂²×Dic₁₉₁	—	2	2	no	—
16	C₂²×Dic₁₉₁	C₂×C₄	1	4	yes	—
17	C₄⋊Dic₁₉₁	—	2	4	no	—
18	C₂²⋊Dic₁₉₁	—	2	2	no	—
19	C₂²⋊Dic₁₉₁	—	2	2	no	—
20	C₂²⋊Dic₁₉₁	—	2	4	no	—
21	C₄×Dic₁₉₁	—	2	4	no	—
22	C₂⁴×C₁₉₁	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₁₉₁	—	2	4	no	—
24	C₂²×C₄×C₁₉₁	—	2	2	no	—
25	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₉₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₉₁	D₄	1	2	yes	—
28	C₂²×C₄×C₁₉₁	D₄	1	2	yes	—
29	C₂²⋊C₄×C₁₉₁	—	2	4	no	—
30	C₂×Dic₁₉₁	—	2	4	no	—
31	C₂×Dic₁₉₁	—	4	4	no	—
32	C₂×Dic₁₉₁	—	4	4	no	—
33	C₂×Dic₁₉₁	—	4	2	no	—
34	C₂³×C₁₉₁	—	4	2	no	—
35	C₂³×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
36	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
37	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
38	C₂³×C₁₉₁	#41	1	4	yes	—
39	C₂×C₄×C₁₉₁	—	2	2	no	—
40	C₂×C₄×C₁₉₁	—	2	2	no	—
41	C₂×C₄×C₁₉₁	—	2	4	no	—
42	C₂×C₄×C₁₉₁	—	2	4	no	—
43	C₂×C₄×C₁₉₁	—	4	4	no	—
44	C₂×C₄×C₁₉₁	—	4	2	no	—
45	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₁₉₁	—	2	2	no	—
48	C₂²×C₁₉₁	—	2	2	no	—
49	C₂²×C₁₉₁	—	2	2	no	—
50	C₂²×C₁₉₁	—	4	4	no	—
51	C₂²×C₁₉₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₁₉₁	#117	1	2	yes	—
53	C₂²×C₁₉₁	#123	1	2	yes	—
54	C₂²×C₁₉₁	#123	1	2	yes	—
55	C₄×C₁₉₁	—	2	2	no	—
56	C₄×C₁₉₁	—	2	2	no	—
57	C₄×C₁₉₁	—	8	2	no	—
58	Dic₁₉₁	—	4	4	no	—
59	Dic₁₉₁	—	8	2	no	—
60	C₂×C₁₉₁	—	2	2	no	—
61	C₂×C₁₉₁	—	2	2	no	—
62	C₂×C₁₉₁	—	2	2	no	—
63	C₂×C₁₉₁	#381	1	1	yes	characteristic
64	C₂×C₁₉₁	#410	1	2	yes	—
65	C₁₉₁	#1820	1	1	yes	characteristic, complemented, Sylow
66	#1820	—	191	1	no	maximal, Sylow
67	C₂×#117	—	191	1	no	—
68	#381	—	191	1	no	—
69	#381	D₁₉₁	1	1	yes	characteristic
70	#342	—	191	4	no	—
71	C₂³×C₄	C₂×D₁₉₁	1	1	yes	characteristic
72	#117	—	382	2	no	—
73	#117	—	382	2	no	—
74	C₂×C₄⋊C₄	—	191	2	no	—
75	C₂×C₂²⋊C₄	—	191	2	no	—
76	C₂×C₂²⋊C₄	—	191	2	no	—
77	C₂×C₂²⋊C₄	C₂×D₁₉₁	1	2	yes	—
78	C₂×C₄²	—	191	2	no	—
79	#95	—	2	2	no	—
80	#95	—	382	2	no	—
81	C₂⁴	C₂²×D₁₉₁	1	1	yes	characteristic
82	C₂²×C₄	—	2	4	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	382	2	no	—
85	C₂²×C₄	—	191	4	no	—
86	C₂²×C₄	C₄×D₁₉₁	1	1	yes	characteristic
87	C₂²×C₄	C₄×D₁₉₁	1	1	yes	characteristic
88	C₂²×C₄	C₂²⋊D₁₉₁	1	2	yes	—
89	C₂²×C₄	D₇₆₄	1	2	yes	—
90	C₄⋊C₄	—	382	4	no	—
91	C₂²⋊C₄	—	2	4	no	—
92	C₂²⋊C₄	—	382	2	no	—
93	C₂²⋊C₄	—	382	2	no	—
94	C₂²⋊C₄	—	382	4	no	—
95	C₄²	—	382	4	no	—
96	C₂³	—	4	2	no	—
97	C₂³	C₂×C₄×D₁₉₁	1	1	yes	characteristic
98	C₂³	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
99	C₂³	C₂×D₇₆₄	1	1	yes	characteristic
100	C₂³	#15690286	1	4	yes	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	4	4	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	382	4	no	—
108	C₂×C₄	—	764	4	no	—
109	C₂×C₄	—	764	2	no	—
110	C₂×C₄	—	764	4	no	—
111	C₂×C₄	(C₂×D₁₉₁)⋊C₄	1	2	yes	—
112	C₂×C₄	(C₂×D₁₉₁)⋊C₄	1	2	yes	—
113	C₂²	—	2	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	—	4	4	no	—
117	C₂²	C₂×(C₂×D₁₉₁)⋊C₄	1	1	yes	characteristic
118	C₂²	#20259278	1	2	yes	—
119	C₂²	#20259303	1	2	yes	—
120	C₂²	#20259306	1	2	yes	—
121	C₄	—	2	2	no	—
122	C₄	—	2	2	no	—
123	C₄	—	8	2	no	—
124	C₄	—	1528	2	no	—
125	C₄	—	764	4	no	—
126	C₂	—	2	2	no	—
127	C₂	—	2	2	no	—
128	C₂	—	2	2	no	—
129	C₂	#27132524	1	1	yes	characteristic
130	C₂	#27132636	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20259281

C₂

yes

maximal, characteristic

#27132526

C₂

yes

maximal, characteristic

#27132429

C₂

yes

maximal

#381×C₁₉₁

C₂

yes

maximal, characteristic

#20259281

—

#20259281

—

C₂×C₄⋊Dic₁₉₁

C₄

yes

complemented

C₂×C₂²⋊Dic₁₉₁

C₂²

yes

—

C₂×C₂²⋊Dic₁₉₁

C₂²

yes

—

C₂×C₄×Dic₁₉₁

C₄

yes

complemented

C₁₉₁⋊#95

—

C₂³×C₄×C₁₉₁

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₁₉₁

C₂²

yes

—

#95×C₁₉₁

—

C₂²×Dic₁₉₁

—

C₂²×Dic₁₉₁

C₂×C₄

yes

—

C₄⋊Dic₁₉₁

—

C₂²⋊Dic₁₉₁

—

C₂²⋊Dic₁₉₁

—

C₂²⋊Dic₁₉₁

—

C₄×Dic₁₉₁

—

C₂⁴×C₁₉₁

C₂³

yes

characteristic

C₂²×C₄×C₁₉₁

—

C₂²×C₄×C₁₉₁

—

C₂²×C₄×C₁₉₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₁

D₄

yes

—

C₂²×C₄×C₁₉₁

D₄

yes

—

C₂²⋊C₄×C₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

C₂²×C₄

yes

characteristic

C₂³×C₁₉₁

C₂×D₄

yes

characteristic

C₂³×C₁₉₁

C₂×D₄

yes

characteristic

C₂³×C₁₉₁

#41

yes

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₁

C₂²⋊C₄

yes

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₁

#117

yes

—

C₂²×C₁₉₁

#123

yes

—

C₂²×C₁₉₁

#123

yes

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

#381

yes

characteristic

C₂×C₁₉₁

#410

yes

—

C₁₉₁

#1820

yes

characteristic, complemented, Sylow

#1820

—

191

maximal, Sylow

C₂×#117

—

191

—

#381

—

191

—

#381

D₁₉₁

yes

characteristic

#342

—

191

—

C₂³×C₄

C₂×D₁₉₁

yes

characteristic

#117

—

382

—

#117

—

382

—

C₂×C₄⋊C₄

—

191

—

C₂×C₂²⋊C₄

—

191

—

C₂×C₂²⋊C₄

—

191

—

C₂×C₂²⋊C₄

C₂×D₁₉₁

yes

—

C₂×C₄²

—

191

—

#95

—

#95

—

382

—

C₂⁴

C₂²×D₁₉₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

382

—

C₂²×C₄

—

191

—

C₂²×C₄

C₄×D₁₉₁

yes

characteristic

C₂²×C₄

C₄×D₁₉₁

yes

characteristic

C₂²×C₄

C₂²⋊D₁₉₁

yes

—

C₂²×C₄

D₇₆₄

yes

—

C₄⋊C₄

—

382

—

C₂²⋊C₄

—

C₂²⋊C₄

—

382

—

C₂²⋊C₄

—

382

—

C₂²⋊C₄

—

382

—

C₄²

—

382

—

C₂³

—

C₂³

C₂×C₄×D₁₉₁

yes

characteristic

C₂³

C₂×C₂²⋊D₁₉₁

yes

characteristic

C₂³

C₂×D₇₆₄

yes

characteristic

100

C₂³

#15690286

yes

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

382

—

108

C₂×C₄

—

764

—

109

C₂×C₄

—

764

—

110

C₂×C₄

—

764

—

111

C₂×C₄

(C₂×D₁₉₁)⋊C₄

yes

—

112

C₂×C₄

(C₂×D₁₉₁)⋊C₄

yes

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

C₂×(C₂×D₁₉₁)⋊C₄

yes

characteristic

118

C₂²

#20259278

yes

—

119

C₂²

#20259303

yes

—

120

C₂²

#20259306

yes

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

1528

—

125

C₄

—

764

—

126

C₂

—

127

C₂

—

128

C₂

—

129

C₂

#27132524

yes

characteristic

130

C₂

#27132636

yes

—

	subgroup	G/H
center	C₂²	C₂×(C₂×D₁₉₁)⋊C₄
commutator	C₂³×C₁₉₁	C₂²×C₄
Frattini	C₂⁴	C₂²×D₁₉₁
Fitting	#381×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂×C₂²⋊C₄
prime radical, cosocle	C₂⁴×C₁₉₁	C₂³
hypercenter	#381	D₁₉₁
hypocenter	C₁₉₁	#1820
LCS	G ▷ C₂³×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #381

subgroup

G/H

center

C₂²

C₂×(C₂×D₁₉₁)⋊C₄

commutator

C₂³×C₁₉₁

C₂²×C₄

Frattini

C₂⁴

C₂²×D₁₉₁

Fitting

#381×C₁₉₁

C₂

socle

C₂²×C₁₉₁

C₂×C₂²⋊C₄

prime radical, cosocle

C₂⁴×C₁₉₁

C₂³

D₁₉₁

C₁₉₁

G ▷ C₂³×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #381

N	H ≅ G/N	type
7. C₂×C₄⋊Dic₁₉₁	C₄	semidirect
10. C₂×C₄×Dic₁₉₁	C₄	semidirect
65. C₁₉₁	#1820	semidirect

H ≅ G/N

type

7. C₂×C₄⋊Dic₁₉₁

C₄

semidirect

10. C₂×C₄×Dic₁₉₁

C₄

semidirect

65. C₁₉₁

#1820

semidirect

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

subgroup	index
C₂×#20259281	2
#27132526	2
#27132429	2
#381×C₁₉₁	2
#1820	191

subgroup

index

#381×C₁₉₁

191

Subgroups — #37572595

Conjugacy classes of subgroups

Complements

Maximal subgroups