Group #37573367 — subgroups

Subgroups — #37573367

Order 24448.

312 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 89 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×#20259310	C₂	1	1	yes	maximal, characteristic
2	C₂×#20259240	C₂	1	2	yes	maximal, complemented
3	C₂×#20259238	C₂	1	2	yes	maximal
4	C₄×#15690271	C₂	1	1	yes	maximal, characteristic, complemented
5	#423×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Dic₃₈₂	C₂²	1	1	yes	characteristic
7	C₂²×D₇₆₄	C₂²	1	1	yes	characteristic
8	C₂×#15690271	C₂²	1	2	yes	—
9	#20259310	—	2	2	no	—
10	#20259310	—	2	1	no	—
11	#20259310	—	2	1	no	—
12	C₂×(C₂×D₁₉₁)⋊C₄	C₄	1	4	yes	—
13	#20259240	—	2	4	no	—
14	#20259238	—	2	4	no	—
15	#20259233	—	2	1	no	—
16	C₄×C₁₉₁⋊C₈	—	2	1	no	—
17	C₂×M₁₆×C₁₉₁	C₂²	1	2	yes	—
18	C₂×C₄²×C₁₉₁	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₁₉₁	—	2	2	no	—
20	C₂³×D₁₉₁	C₂×C₄	1	2	yes	—
21	C₂×Dic₃₈₂	—	2	2	no	—
22	C₂×Dic₃₈₂	—	2	1	no	—
23	C₂×Dic₃₈₂	—	2	1	no	—
24	C₂×Dic₃₈₂	—	2	2	no	—
25	C₂×D₇₆₄	—	2	2	no	—
26	C₂×D₇₆₄	—	2	1	no	—
27	C₂×D₇₆₄	—	2	1	no	—
28	C₂×D₇₆₄	—	2	2	no	—
29	C₂²×Dic₁₉₁	C₂×C₄	1	2	yes	—
30	#15690271	—	2	2	no	—
31	#15690271	D₄	1	4	yes	—
32	C₂×C₁₉₁⋊C₈	—	2	2	no	—
33	(C₂×D₁₉₁)⋊C₄	—	2	4	no	—
34	(C₂×D₁₉₁)⋊C₄	—	2	4	no	—
35	C₂²×C₄×C₁₉₁	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
37	M₁₆×C₁₉₁	—	2	2	no	—
38	M₁₆×C₁₉₁	—	2	2	no	—
39	C₂×C₈×C₁₉₁	—	2	2	no	—
40	C₄²×C₁₉₁	D₄	1	2	yes	—
41	C₄²×C₁₉₁	D₄	1	1	yes	characteristic
42	C₄²×C₁₉₁	D₄	1	1	yes	characteristic
43	C₂²×D₁₉₁	—	2	2	no	—
44	C₂²×D₁₉₁	—	2	4	no	—
45	C₂²×D₁₉₁	—	4	4	no	—
46	Dic₃₈₂	—	4	2	no	—
47	Dic₃₈₂	—	4	1	no	—
48	Dic₃₈₂	—	4	1	no	—
49	D₇₆₄	—	4	2	no	—
50	D₇₆₄	—	4	1	no	—
51	D₇₆₄	—	4	1	no	—
52	C₂×Dic₁₉₁	—	2	2	no	—
53	C₂×Dic₁₉₁	—	2	4	no	—
54	C₁₉₁⋊C₈	—	2	4	no	—
55	C₂³×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₉₁	—	4	2	no	—
57	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×C₁₉₁	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₉₁	#41	1	2	yes	—
63	C₈×C₁₉₁	—	4	2	no	—
64	C₂²×C₁₉₁	—	2	2	no	—
65	C₂²×C₁₉₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₁₉₁	C₄×D₄	1	2	yes	—
67	C₂×D₁₉₁	—	4	2	no	—
68	C₂×D₁₉₁	—	4	4	no	—
69	C₂×D₁₉₁	—	8	4	no	—
70	C₄×C₁₉₁	—	2	2	no	—
71	C₄×C₁₉₁	—	2	2	no	—
72	C₄×C₁₉₁	#121	1	2	yes	—
73	C₄×C₁₉₁	#124	1	1	yes	characteristic
74	C₄×C₁₉₁	#127	1	1	yes	characteristic
75	Dic₁₉₁	—	4	2	no	—
76	C₂×C₁₉₁	—	2	2	no	—
77	C₂×C₁₉₁	#390	1	1	yes	characteristic
78	C₂×C₁₉₁	#413	1	2	yes	—
79	D₁₉₁	—	8	2	no	—
80	C₁₉₁	#1967	1	1	yes	characteristic, complemented, Sylow
81	#1967	—	191	1	no	maximal, Sylow
82	C₂×#124	—	191	1	no	—
83	#423	D₁₉₁	1	1	yes	characteristic, complemented
84	C₂×#101	—	191	2	no	—
85	C₂×#100	—	191	2	no	—
86	C₄×M₁₆	—	191	1	no	—
87	C₂²×Q₈	—	191	1	no	—
88	C₂²×D₄	—	191	1	no	—
89	C₂×M₁₆	—	191	2	no	—
90	C₂×M₁₆	C₂×D₁₉₁	1	2	yes	—
91	#124	—	382	2	no	—
92	#124	—	382	1	no	—
93	#124	—	382	1	no	—
94	C₂×C₂²⋊C₄	—	191	4	no	—
95	C₂×C₄²	C₂×D₁₉₁	1	1	yes	characteristic
96	C₄⋊C₈	—	2	2	no	—
97	#101	—	382	4	no	—
98	#100	—	382	4	no	—
99	#97	—	382	1	no	—
100	C₄×C₈	—	382	1	no	—
101	C₂⁴	—	191	2	no	—
102	C₂×Q₈	—	382	2	no	—
103	C₂×Q₈	—	382	1	no	—
104	C₂×Q₈	—	382	1	no	—
105	C₂×Q₈	—	382	2	no	—
106	C₂×D₄	—	382	2	no	—
107	C₂×D₄	—	382	1	no	—
108	C₂×D₄	—	382	1	no	—
109	C₂×D₄	—	382	2	no	—
110	C₂²×C₄	—	191	2	no	—
111	C₂²×C₄	C₄×D₁₉₁	1	2	yes	—
112	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
113	M₁₆	—	2	2	no	—
114	M₁₆	—	2	2	no	—
115	M₁₆	—	382	2	no	—
116	M₁₆	—	191	4	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	382	2	no	—
119	C₂²⋊C₄	—	382	4	no	—
120	C₂²⋊C₄	—	382	4	no	—
121	C₄²	C₂²⋊D₁₉₁	1	2	yes	—
122	C₄²	D₇₆₄	1	1	yes	characteristic
123	C₄²	D₇₆₄	1	1	yes	characteristic
124	C₂³	—	382	4	no	—
125	C₂³	—	382	2	no	—
126	C₂³	—	764	4	no	—
127	C₂³	C₂×C₄×D₁₉₁	1	1	yes	characteristic
128	Q₈	—	764	2	no	—
129	Q₈	—	764	1	no	—
130	Q₈	—	764	1	no	—
131	D₄	—	764	2	no	—
132	D₄	—	764	1	no	—
133	D₄	—	764	1	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	382	2	no	—
136	C₂×C₄	—	382	4	no	—
137	C₂×C₄	(C₂×D₁₉₁)⋊C₄	1	2	yes	—
138	C₂×C₄	(C₂×D₁₉₁)⋊C₄	1	2	yes	—
139	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
140	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
141	C₂×C₄	C₂×D₇₆₄	1	1	yes	characteristic
142	C₂×C₄	#15690288	1	2	yes	—
143	C₈	—	4	2	no	—
144	C₈	—	382	4	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	1528	4	no	—
147	C₂²	—	764	2	no	—
148	C₂²	—	764	4	no	—
149	C₂²	C₂×(C₂×D₁₉₁)⋊C₄	1	1	yes	characteristic
150	C₂²	#20259284	1	2	yes	—
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	—	764	2	no	—
154	C₄	#20259297	1	2	yes	—
155	C₄	#20259311	1	1	yes	characteristic
156	C₄	#20259318	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	—	1528	2	no	—
159	C₂	#27132564	1	1	yes	characteristic
160	C₂	#27132645	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20259310

C₂

yes

maximal, characteristic

C₂×#20259240

C₂

yes

maximal, complemented

C₂×#20259238

C₂

yes

maximal

C₄×#15690271

C₂

yes

maximal, characteristic, complemented

#423×C₁₉₁

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₃₈₂

C₂²

yes

characteristic

C₂²×D₇₆₄

C₂²

yes

characteristic

C₂×#15690271

C₂²

yes

—

#20259310

—

#20259310

—

#20259310

—

C₂×(C₂×D₁₉₁)⋊C₄

C₄

yes

—

#20259240

—

#20259238

—

#20259233

—

C₄×C₁₉₁⋊C₈

—

C₂×M₁₆×C₁₉₁

C₂²

yes

—

C₂×C₄²×C₁₉₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₁

—

C₂³×D₁₉₁

C₂×C₄

yes

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×Dic₃₈₂

—

C₂×D₇₆₄

—

C₂×D₇₆₄

—

C₂×D₇₆₄

—

C₂×D₇₆₄

—

C₂²×Dic₁₉₁

C₂×C₄

yes

—

#15690271

—

#15690271

D₄

yes

—

C₂×C₁₉₁⋊C₈

—

(C₂×D₁₉₁)⋊C₄

—

(C₂×D₁₉₁)⋊C₄

—

C₂²×C₄×C₁₉₁

C₂×C₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

M₁₆×C₁₉₁

—

M₁₆×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₄²×C₁₉₁

D₄

yes

—

C₄²×C₁₉₁

D₄

yes

characteristic

C₄²×C₁₉₁

D₄

yes

characteristic

C₂²×D₁₉₁

—

C₂²×D₁₉₁

—

C₂²×D₁₉₁

—

Dic₃₈₂

—

Dic₃₈₂

—

Dic₃₈₂

—

D₇₆₄

—

D₇₆₄

—

D₇₆₄

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

—

C₈×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₁

C₄×D₄

yes

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#121

yes

—

C₄×C₁₉₁

yes

C₄×C₁₉₁

yes

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

#390

yes

characteristic

C₂×C₁₉₁

#413

yes

—

D₁₉₁

—

C₁₉₁

#1967

yes

characteristic, complemented, Sylow

#1967

—

191

maximal, Sylow

C₂×#124

—

191

—

#423

D₁₉₁

yes

characteristic, complemented

C₂×#101

—

191

—

C₂×#100

—

191

—

C₄×M₁₆

—

191

—

C₂²×Q₈

—

191

—

C₂²×D₄

—

191

—

C₂×M₁₆

—

191

—

C₂×M₁₆

C₂×D₁₉₁

yes

—

#124

—

382

—

#124

—

382

—

#124

—

382

—

C₂×C₂²⋊C₄

—

191

—

C₂×C₄²

C₂×D₁₉₁

yes

characteristic

C₄⋊C₈

—

#101

—

382

—

#100

—

382

—

#97

—

382

—

100

C₄×C₈

—

382

—

101

C₂⁴

—

191

—

102

C₂×Q₈

—

382

—

103

C₂×Q₈

—

382

—

104

C₂×Q₈

—

382

—

105

C₂×Q₈

—

382

—

106

C₂×D₄

—

382

—

107

C₂×D₄

—

382

—

108

C₂×D₄

—

382

—

109

C₂×D₄

—

382

—

110

C₂²×C₄

—

191

—

111

C₂²×C₄

C₄×D₁₉₁

yes

—

112

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

382

—

116

M₁₆

—

191

—

117

C₂×C₈

—

118

C₂×C₈

—

382

—

119

C₂²⋊C₄

—

382

—

120

C₂²⋊C₄

—

382

—

121

C₄²

C₂²⋊D₁₉₁

yes

—

122

C₄²

D₇₆₄

yes

characteristic

123

C₄²

D₇₆₄

yes

characteristic

124

C₂³

—

382

—

125

C₂³

—

382

—

126

C₂³

—

764

—

127

C₂³

C₂×C₄×D₁₉₁

yes

characteristic

128

Q₈

—

764

—

129

Q₈

—

764

—

130

Q₈

—

764

—

131

D₄

—

764

—

132

D₄

—

764

—

133

D₄

—

764

—

134

C₂×C₄

—

135

C₂×C₄

—

382

—

136

C₂×C₄

—

382

—

137

C₂×C₄

(C₂×D₁₉₁)⋊C₄

yes

—

138

C₂×C₄

(C₂×D₁₉₁)⋊C₄

yes

—

139

C₂×C₄

D₄×D₁₉₁

yes

—

140

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

141

C₂×C₄

C₂×D₇₆₄

yes

characteristic

142

C₂×C₄

#15690288

yes

—

143

C₈

—

144

C₈

—

382

—

145

C₂²

—

146

C₂²

—

1528

—

147

C₂²

—

764

—

148

C₂²

—

764

—

149

C₂²

C₂×(C₂×D₁₉₁)⋊C₄

yes

characteristic

150

C₂²

#20259284

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

764

—

154

C₄

#20259297

yes

—

155

C₄

#20259311

yes

characteristic

156

C₄

#20259318

yes

characteristic

157

C₂

—

158

C₂

—

1528

—

159

C₂

#27132564

yes

characteristic

160

C₂

#27132645

yes

—

	subgroup	G/H
center	C₂²	C₂×(C₂×D₁₉₁)⋊C₄
commutator	C₂³×C₁₉₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting	#423×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂×C₂²⋊C₄
prime radical	C₂²×D₇₆₄	C₂²
cosocle	C₂²×C₄×C₁₉₁	C₂³
hypercenter	#423	D₁₉₁
hypocenter	C₁₉₁	#1967
LCS	G ▷ C₂³×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×(C₂×D₁₉₁)⋊C₄

commutator

C₂³×C₁₉₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₁

Fitting

#423×C₁₉₁

C₂

socle

C₂²×C₁₉₁

C₂×C₂²⋊C₄

prime radical

C₂²×D₇₆₄

C₂²

cosocle

C₂²×C₄×C₁₉₁

C₂³

D₁₉₁

C₁₉₁

G ▷ C₂³×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
2. C₂×#20259240	C₂	semidirect
4. C₄×#15690271	C₂	semidirect
5. #423×C₁₉₁	C₂	semidirect
80. C₁₉₁	#1967	semidirect
83. #423	D₁₉₁	semidirect

H ≅ G/N

type

2. C₂×#20259240

C₂

semidirect

4. C₄×#15690271

C₂

semidirect

5. #423×C₁₉₁

C₂

semidirect

80. C₁₉₁

#1967

semidirect

83. #423

D₁₉₁

semidirect

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

subgroup	index
C₂×#20259310	2
C₂×#20259240	2
C₂×#20259238	2
C₄×#15690271	2
#423×C₁₉₁	2
#1967	191

subgroup

index

#423×C₁₉₁

191

Subgroups — #37573367

Conjugacy classes of subgroups

Complements

Maximal subgroups