Group #37573663 — subgroups

Subgroups — #37573663

Order 24448.

294 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 87 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×#20259242	C₂	1	2	yes	maximal
2	#27132565	C₂	1	1	yes	maximal, characteristic
3	#27132425	C₂	1	2	yes	maximal, complemented
4	#27132418	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#125×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₇₆₄	C₂²	1	1	yes	characteristic
7	C₂²×C₁₉₁⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
9	C₂×(C₂×D₁₉₁)⋊C₄	—	2	2	no	—
10	#20259242	—	2	4	no	—
11	#125×C₁₉₁	—	2	4	no	—
12	C₂×C₄⋊C₄×C₁₉₁	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
15	C₂³×D₁₉₁	—	2	1	no	—
16	C₂×D₇₆₄	—	2	2	no	—
17	C₂×D₇₆₄	—	2	2	no	—
18	C₂×D₇₆₄	—	2	1	no	—
19	C₂×D₇₆₄	Q₈	1	2	yes	—
20	C₂²×Dic₁₉₁	—	2	1	no	—
21	C₂×C₁₉₁⋊C₈	—	2	2	no	—
22	C₂×C₁₉₁⋊C₈	—	2	2	no	—
23	C₂×C₁₉₁⋊C₈	—	2	2	no	—
24	C₄⋊Dic₁₉₁	—	2	2	no	—
25	(C₂×D₁₉₁)⋊C₄	—	4	4	no	—
26	C₂²×C₄×C₁₉₁	—	2	2	no	—
27	C₂²×C₄×C₁₉₁	D₄	1	2	yes	—
28	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₁₉₁	—	2	8	no	—
30	C₄⋊C₄×C₁₉₁	—	2	2	no	—
31	C₄⋊C₄×C₁₉₁	D₄	1	4	yes	—
32	C₄²×C₁₉₁	—	2	2	no	—
33	C₂²×D₁₉₁	—	2	2	no	—
34	C₂²×D₁₉₁	—	4	1	no	—
35	C₂²×D₁₉₁	—	4	1	no	—
36	C₂²×D₁₉₁	—	4	2	no	—
37	C₂²×D₁₉₁	—	4	2	no	—
38	D₇₆₄	—	2	4	no	—
39	D₇₆₄	—	4	2	no	—
40	C₂×Dic₁₉₁	—	4	1	no	—
41	C₂×Dic₁₉₁	—	4	2	no	—
42	C₁₉₁⋊C₈	—	4	2	no	—
43	C₂³×C₁₉₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₉₁	—	2	2	no	—
45	C₂×C₄×C₁₉₁	—	2	4	no	—
46	C₂×C₄×C₁₉₁	—	2	2	no	—
47	C₂×C₄×C₁₉₁	—	2	2	no	—
48	C₂×C₄×C₁₉₁	—	4	2	no	—
49	C₂×C₄×C₁₉₁	—	4	2	no	—
50	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₉₁	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₁₉₁	#41	1	2	yes	—
53	C₂×C₄×C₁₉₁	#41	1	1	yes	characteristic
54	C₂²×C₁₉₁	#120	1	1	yes	characteristic
55	C₂²×C₁₉₁	C₂²⋊Q₈	1	1	yes	characteristic
56	C₂²×C₁₉₁	#124	1	1	yes	characteristic
57	C₂²×C₁₉₁	#135	1	2	yes	—
58	C₂²×C₁₉₁	AGL(1,ℤ/8)	1	2	yes	—
59	C₂×D₁₉₁	—	4	2	no	—
60	C₂×D₁₉₁	—	4	4	no	—
61	C₂×D₁₉₁	—	8	2	no	—
62	C₂×D₁₉₁	—	8	1	no	—
63	C₂×D₁₉₁	—	8	1	no	—
64	C₄×C₁₉₁	—	4	2	no	—
65	C₄×C₁₉₁	—	4	4	no	—
66	C₄×C₁₉₁	C₂²⋊Q₈	1	2	yes	—
67	C₄×C₁₉₁	#124	1	2	yes	—
68	Dic₁₉₁	—	8	1	no	—
69	C₂×C₁₉₁	#393	1	1	yes	characteristic
70	C₂×C₁₉₁	#449	1	2	yes	—
71	C₂×C₁₉₁	#478	1	2	yes	—
72	C₂×C₁₉₁	#488	1	1	yes	characteristic
73	C₂×C₁₉₁	#490	1	1	yes	characteristic
74	D₁₉₁	—	8	2	no	—
75	C₁₉₁	#2025	1	1	yes	characteristic, complemented, Sylow
76	#2025	—	191	1	no	maximal, Sylow
77	C₂×#125	D₁₉₁	1	1	yes	characteristic, complemented
78	C₂×#102	—	191	2	no	—
79	#390	—	191	1	no	—
80	#340	—	191	2	no	—
81	#336	—	191	1	no	—
82	C₂²×D₄	—	191	1	no	—
83	C₂²×C₈	—	191	2	no	—
84	#125	—	2	4	no	—
85	C₂×C₄⋊C₄	—	2	2	no	—
86	C₂×C₄⋊C₄	—	191	1	no	—
87	C₂×C₄⋊C₄	C₂×D₁₉₁	1	2	yes	—
88	C₂×C₂²⋊C₄	—	382	2	no	—
89	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
90	#102	—	382	4	no	—
91	C₂⁴	—	382	1	no	—
92	C₂×D₄	—	382	2	no	—
93	C₂×D₄	—	382	2	no	—
94	C₂×D₄	—	382	1	no	—
95	C₂×D₄	—	191	2	no	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	—	382	1	no	—
98	C₂²×C₄	C₂²⋊D₁₉₁	1	2	yes	—
99	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
100	C₂×C₈	—	382	2	no	—
101	C₂×C₈	—	382	2	no	—
102	C₂×C₈	—	382	2	no	—
103	C₄⋊C₄	—	2	8	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	382	2	no	—
106	C₄⋊C₄	C₂²⋊D₁₉₁	1	4	yes	—
107	C₂²⋊C₄	—	764	4	no	—
108	C₄²	—	2	2	no	—
109	C₂³	—	382	2	no	—
110	C₂³	—	764	2	no	—
111	C₂³	—	764	2	no	—
112	C₂³	—	764	1	no	—
113	C₂³	—	764	1	no	—
114	C₂³	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
115	D₄	—	382	4	no	—
116	D₄	—	764	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	764	2	no	—
124	C₂×C₄	—	764	1	no	—
125	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
126	C₂×C₄	Q₈×D₁₉₁	1	1	yes	characteristic
127	C₂×C₄	#15690288	1	2	yes	—
128	C₂×C₄	#15690288	1	1	yes	characteristic
129	C₈	—	764	2	no	—
130	C₂²	—	1528	1	no	—
131	C₂²	—	1528	1	no	—
132	C₂²	—	1528	2	no	—
133	C₂²	—	764	4	no	—
134	C₂²	—	764	2	no	—
135	C₂²	#20259291	1	1	yes	characteristic
136	C₂²	#20259298	1	1	yes	characteristic
137	C₂²	#20259311	1	1	yes	characteristic
138	C₂²	#20259342	1	2	yes	—
139	C₂²	#20259347	1	2	yes	—
140	C₄	—	4	2	no	—
141	C₄	—	4	4	no	—
142	C₄	—	1528	1	no	—
143	C₄	#20259298	1	2	yes	—
144	C₄	#20259311	1	2	yes	—
145	C₂	—	1528	2	no	—
146	C₂	#27132574	1	1	yes	characteristic
147	C₂	#27132808	1	2	yes	—
148	C₂	#27132960	1	2	yes	—
149	C₂	#27133020	1	1	yes	characteristic
150	C₂	#27133029	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20259242

C₂

yes

maximal

#27132565

C₂

yes

maximal, characteristic

#27132425

C₂

yes

maximal, complemented

#27132418

C₂

yes

maximal, characteristic, complemented

C₂×#125×C₁₉₁

C₂

yes

maximal, characteristic, complemented

C₂²×D₇₆₄

C₂²

yes

characteristic

C₂²×C₁₉₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₉₁

C₂²

yes

characteristic

C₂×(C₂×D₁₉₁)⋊C₄

—

#20259242

—

#125×C₁₉₁

—

C₂×C₄⋊C₄×C₁₉₁

—

C₂×C₄⋊C₄×C₁₉₁

C₂²

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₂³×D₁₉₁

—

C₂×D₇₆₄

—

C₂×D₇₆₄

—

C₂×D₇₆₄

—

C₂×D₇₆₄

Q₈

yes

—

C₂²×Dic₁₉₁

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₄⋊Dic₁₉₁

—

(C₂×D₁₉₁)⋊C₄

—

C₂²×C₄×C₁₉₁

—

C₂²×C₄×C₁₉₁

D₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₄⋊C₄×C₁₉₁

D₄

yes

—

C₄²×C₁₉₁

—

C₂²×D₁₉₁

—

C₂²×D₁₉₁

—

C₂²×D₁₉₁

—

C₂²×D₁₉₁

—

C₂²×D₁₉₁

—

D₇₆₄

—

D₇₆₄

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

—

C₂×C₄×C₁₉₁

yes

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

C₂²⋊Q₈

yes

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

yes

—

C₂²×C₁₉₁

AGL(1,ℤ/8)

yes

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

C₂²⋊Q₈

yes

—

C₄×C₁₉₁

#124

yes

—

Dic₁₉₁

—

C₂×C₁₉₁

#393

yes

characteristic

C₂×C₁₉₁

#449

yes

—

C₂×C₁₉₁

#478

yes

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

D₁₉₁

—

C₁₉₁

#2025

yes

characteristic, complemented, Sylow

#2025

—

191

maximal, Sylow

C₂×#125

D₁₉₁

yes

characteristic, complemented

C₂×#102

—

191

—

#390

—

191

—

#340

—

191

—

#336

—

191

—

C₂²×D₄

—

191

—

C₂²×C₈

—

191

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

191

—

C₂×C₄⋊C₄

C₂×D₁₉₁

yes

—

C₂×C₂²⋊C₄

—

382

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

#102

—

382

—

C₂⁴

—

382

—

C₂×D₄

—

382

—

C₂×D₄

—

382

—

C₂×D₄

—

382

—

C₂×D₄

—

191

—

C₂²×C₄

—

C₂²×C₄

—

382

—

C₂²×C₄

C₂²⋊D₁₉₁

yes

—

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

100

C₂×C₈

—

382

—

101

C₂×C₈

—

382

—

102

C₂×C₈

—

382

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

382

—

106

C₄⋊C₄

C₂²⋊D₁₉₁

yes

—

107

C₂²⋊C₄

—

764

—

108

C₄²

—

109

C₂³

—

382

—

110

C₂³

—

764

—

111

C₂³

—

764

—

112

C₂³

—

764

—

113

C₂³

—

764

—

114

C₂³

C₂×C₂²⋊D₁₉₁

yes

characteristic

115

D₄

—

382

—

116

D₄

—

764

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

764

—

124

C₂×C₄

—

764

—

125

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

—

126

C₂×C₄

Q₈×D₁₉₁

yes

characteristic

127

C₂×C₄

#15690288

yes

—

128

C₂×C₄

#15690288

yes

characteristic

129

C₈

—

764

—

130

C₂²

—

1528

—

131

C₂²

—

1528

—

132

C₂²

—

1528

—

133

C₂²

—

764

—

134

C₂²

—

764

—

135

C₂²

#20259291

yes

characteristic

136

C₂²

#20259298

yes

characteristic

137

C₂²

#20259311

yes

characteristic

138

C₂²

#20259342

yes

—

139

C₂²

#20259347

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

1528

—

143

C₄

#20259298

yes

—

144

C₄

#20259311

yes

—

145

C₂

—

1528

—

146

C₂

#27132574

yes

characteristic

147

C₂

#27132808

yes

—

148

C₂

#27132960

yes

—

149

C₂

#27133020

yes

characteristic

150

C₂

#27133029

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉₁
commutator, cosocle	C₂²×C₄×C₁₉₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting	C₂×#125×C₁₉₁	C₂
socle	C₂³×C₁₉₁	C₂×D₄
prime radical	C₂²×D₇₆₄	C₂²
hypercenter	C₂×#125	D₁₉₁
hypocenter	C₁₉₁	#2025
LCS	G ▷ C₂²×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉₁

commutator, cosocle

C₂²×C₄×C₁₉₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉₁

Fitting

C₂×#125×C₁₉₁

C₂

socle

C₂³×C₁₉₁

C₂×D₄

prime radical

C₂²×D₇₆₄

C₂²

D₁₉₁

C₁₉₁

G ▷ C₂²×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
3. #27132425	C₂	semidirect
4. #27132418	C₂	semidirect
5. C₂×#125×C₁₉₁	C₂	semidirect
75. C₁₉₁	#2025	semidirect
77. C₂×#125	D₁₉₁	semidirect

H ≅ G/N

type

3. #27132425

C₂

semidirect

4. #27132418

C₂

semidirect

5. C₂×#125×C₁₉₁

C₂

semidirect

75. C₁₉₁

#2025

semidirect

77. C₂×#125

D₁₉₁

semidirect

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

subgroup	index
C₂×#20259242	2
#27132565	2
#27132425	2
#27132418	2
C₂×#125×C₁₉₁	2
#2025	191

subgroup

index

C₂×#125×C₁₉₁

191

Subgroups — #37573663

Conjugacy classes of subgroups

Complements

Maximal subgroups