Group #37572602 — subgroups

Subgroups — #37572602

Order 24448.

272 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 77 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×#20259280	C₂	1	1	yes	maximal, characteristic
2	#27132626	C₂	1	2	yes	maximal
3	#27132432	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132431	C₂	1	2	yes	maximal, complemented
5	#343×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₁₉₁	C₂²	1	1	yes	characteristic
7	C₂×#15690271	C₂²	1	2	yes	—
8	#20259280	—	2	4	no	—
9	C₂×C₄⋊Dic₁₉₁	C₂²	1	1	yes	characteristic
10	C₂×(C₂×D₁₉₁)⋊C₄	C₄	1	2	yes	—
11	C₂×C₄×Dic₁₉₁	C₄	1	2	yes	—
12	(C₂×D₁₉₁)⋊C₈	—	2	4	no	—
13	C₂×M₁₆×C₁₉₁	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₁₉₁	C₂²	1	1	yes	characteristic
15	C₂³×D₁₉₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₁₉₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×Dic₁₉₁	C₂×C₄	1	2	yes	—
18	C₂×C₄×D₁₉₁	—	2	1	no	—
19	C₂×C₄×D₁₉₁	—	2	2	no	—
20	C₂×C₄×D₁₉₁	—	2	1	no	—
21	C₂×C₄×D₁₉₁	D₄	1	4	yes	—
22	#15690271	—	2	2	no	—
23	#15690271	—	2	2	no	—
24	C₂×C₁₉₁⋊C₈	—	2	2	no	—
25	C₄⋊Dic₁₉₁	—	2	2	no	—
26	(C₂×D₁₉₁)⋊C₄	—	2	4	no	—
27	C₄×Dic₁₉₁	—	2	4	no	—
28	C₂²×C₄×C₁₉₁	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
30	M₁₆×C₁₉₁	—	2	2	no	—
31	M₁₆×C₁₉₁	—	2	2	no	—
32	C₂×C₈×C₁₉₁	—	2	2	no	—
33	C₄⋊C₄×C₁₉₁	—	2	2	no	—
34	C₂²×D₁₉₁	—	2	1	no	—
35	C₂²×D₁₉₁	—	2	1	no	—
36	C₂²×D₁₉₁	—	4	1	no	—
37	C₂²×D₁₉₁	—	4	1	no	—
38	C₂²×D₁₉₁	C₂²⋊C₄	1	2	yes	—
39	C₂×Dic₁₉₁	—	2	1	no	—
40	C₂×Dic₁₉₁	—	2	1	no	—
41	C₂×Dic₁₉₁	—	2	2	no	—
42	C₂×Dic₁₉₁	—	4	2	no	—
43	C₂×Dic₁₉₁	C₂²⋊C₄	1	2	yes	—
44	C₄×D₁₉₁	—	2	4	no	—
45	C₄×D₁₉₁	—	2	4	no	—
46	C₁₉₁⋊C₈	—	4	2	no	—
47	C₂³×C₁₉₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₉₁	—	2	2	no	—
49	C₂×C₄×C₁₉₁	—	4	2	no	—
50	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₉₁	#41	1	1	yes	characteristic
52	C₂×C₄×C₁₉₁	#41	1	2	yes	—
53	C₂×C₄×C₁₉₁	#41	1	1	yes	characteristic
54	C₈×C₁₉₁	—	4	2	no	—
55	C₂²×C₁₉₁	—	2	1	no	—
56	C₂²×C₁₉₁	—	2	1	no	—
57	C₂²×C₁₉₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₁₉₁	#117	1	1	yes	characteristic
59	C₂²×C₁₉₁	#117	1	1	yes	characteristic
60	C₂×D₁₉₁	—	2	2	no	—
61	C₂×D₁₉₁	—	2	2	no	—
62	C₂×D₁₉₁	—	2	2	no	—
63	C₂×D₁₉₁	—	4	2	no	—
64	C₂×D₁₉₁	—	4	2	no	—
65	C₄×C₁₉₁	—	4	2	no	—
66	C₄×C₁₉₁	#123	1	2	yes	—
67	C₄×C₁₉₁	#123	1	2	yes	—
68	Dic₁₉₁	—	2	2	no	—
69	Dic₁₉₁	—	4	2	no	—
70	C₂×C₁₉₁	—	2	1	no	—
71	C₂×C₁₉₁	—	2	1	no	—
72	C₂×C₁₉₁	#381	1	1	yes	characteristic
73	C₂×C₁₉₁	#413	1	1	yes	characteristic
74	C₂×C₁₉₁	#413	1	1	yes	characteristic
75	D₁₉₁	—	4	2	no	—
76	C₁₉₁	#1822	1	1	yes	characteristic, complemented, Sylow
77	#1822	—	191	1	no	maximal, Sylow
78	C₂×#117	—	191	1	no	—
79	#407	—	191	2	no	—
80	#343	—	191	2	no	—
81	#343	—	191	1	no	—
82	#343	D₁₉₁	1	1	yes	characteristic, complemented
83	C₂³×C₄	—	191	1	no	—
84	C₂×M₁₆	—	191	2	no	—
85	C₂×M₁₆	C₂×D₁₉₁	1	2	yes	—
86	#117	—	382	4	no	—
87	C₂×C₄⋊C₄	—	191	1	no	—
88	C₂×C₄⋊C₄	C₂×D₁₉₁	1	1	yes	characteristic
89	C₂×C₂²⋊C₄	—	191	2	no	—
90	C₂×C₄²	—	191	2	no	—
91	C₂²⋊C₈	—	382	4	no	—
92	C₂⁴	—	191	1	no	—
93	C₂²×C₄	—	382	1	no	—
94	C₂²×C₄	—	382	2	no	—
95	C₂²×C₄	—	382	1	no	—
96	C₂²×C₄	—	191	1	no	—
97	C₂²×C₄	—	191	4	no	—
98	C₂²×C₄	—	191	2	no	—
99	C₂²×C₄	C₄×D₁₉₁	1	2	yes	—
100	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
101	M₁₆	—	2	2	no	—
102	M₁₆	—	2	2	no	—
103	M₁₆	—	382	2	no	—
104	M₁₆	—	382	2	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	382	2	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	382	2	no	—
109	C₂²⋊C₄	—	382	4	no	—
110	C₄²	—	382	4	no	—
111	C₂³	—	382	1	no	—
112	C₂³	—	382	1	no	—
113	C₂³	—	191	2	no	—
114	C₂³	—	764	1	no	—
115	C₂³	—	764	1	no	—
116	C₂³	C₂×C₄×D₁₉₁	1	1	yes	characteristic
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	382	1	no	—
120	C₂×C₄	—	382	1	no	—
121	C₂×C₄	—	382	4	no	—
122	C₂×C₄	—	382	4	no	—
123	C₂×C₄	—	382	2	no	—
124	C₂×C₄	—	191	2	no	—
125	C₂×C₄	—	764	2	no	—
126	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
127	C₂×C₄	#15690286	1	1	yes	characteristic
128	C₂×C₄	#15690287	1	2	yes	—
129	C₂×C₄	#15690288	1	1	yes	characteristic
130	C₈	—	4	2	no	—
131	C₈	—	764	2	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	—	382	2	no	—
135	C₂²	—	382	2	no	—
136	C₂²	—	382	2	no	—
137	C₂²	—	764	2	no	—
138	C₂²	—	764	2	no	—
139	C₂²	C₂²⋊C₄×D₁₉₁	1	1	yes	characteristic
140	C₂²	#20259279	1	1	yes	characteristic
141	C₂²	#20259280	1	1	yes	characteristic
142	C₄	—	4	2	no	—
143	C₄	—	382	2	no	—
144	C₄	—	764	2	no	—
145	C₄	#20259304	1	2	yes	—
146	C₄	#20259307	1	2	yes	—
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	—	764	2	no	—
150	C₂	#27132525	1	1	yes	characteristic
151	C₂	#27132644	1	1	yes	characteristic
152	C₂	#27132646	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20259280

C₂

yes

maximal, characteristic

#27132626

C₂

yes

maximal

#27132432

C₂

yes

maximal, characteristic, complemented

#27132431

C₂

yes

maximal, complemented

#343×C₁₉₁

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₁₉₁

C₂²

yes

characteristic

C₂×#15690271

C₂²

yes

—

#20259280

—

C₂×C₄⋊Dic₁₉₁

C₂²

yes

characteristic

C₂×(C₂×D₁₉₁)⋊C₄

C₄

yes

—

C₂×C₄×Dic₁₉₁

C₄

yes

—

(C₂×D₁₉₁)⋊C₈

—

C₂×M₁₆×C₁₉₁

C₂²

yes

—

C₂×C₄⋊C₄×C₁₉₁

C₂²

yes

characteristic

C₂³×D₁₉₁

C₂×C₄

yes

characteristic

C₂²×Dic₁₉₁

C₂×C₄

yes

characteristic

C₂²×Dic₁₉₁

C₂×C₄

yes

—

C₂×C₄×D₁₉₁

—

C₂×C₄×D₁₉₁

—

C₂×C₄×D₁₉₁

—

C₂×C₄×D₁₉₁

D₄

yes

—

#15690271

—

#15690271

—

C₂×C₁₉₁⋊C₈

—

C₄⋊Dic₁₉₁

—

(C₂×D₁₉₁)⋊C₄

—

C₄×Dic₁₉₁

—

C₂²×C₄×C₁₉₁

C₂×C₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

M₁₆×C₁₉₁

—

M₁₆×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₄⋊C₄×C₁₉₁

—

C₂²×D₁₉₁

—

C₂²×D₁₉₁

—

C₂²×D₁₉₁

—

C₂²×D₁₉₁

—

C₂²×D₁₉₁

C₂²⋊C₄

yes

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

—

C₂×Dic₁₉₁

C₂²⋊C₄

yes

—

C₄×D₁₉₁

—

C₄×D₁₉₁

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

#41

yes

characteristic

C₂×C₄×C₁₉₁

#41

yes

—

C₂×C₄×C₁₉₁

#41

yes

characteristic

C₈×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

C₂×C₂²⋊C₄

yes

C₂²×C₁₉₁

yes

C₂²×C₁₉₁

yes

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₄×C₁₉₁

—

C₄×C₁₉₁

#123

yes

—

C₄×C₁₉₁

#123

yes

—

Dic₁₉₁

—

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

C₂×C₁₉₁

yes

D₁₉₁

—

C₁₉₁

#1822

yes

characteristic, complemented, Sylow

#1822

—

191

maximal, Sylow

C₂×#117

—

191

—

#407

—

191

—

#343

—

191

—

#343

—

191

—

#343

D₁₉₁

yes

characteristic, complemented

C₂³×C₄

—

191

—

C₂×M₁₆

—

191

—

C₂×M₁₆

C₂×D₁₉₁

yes

—

#117

—

382

—

C₂×C₄⋊C₄

—

191

—

C₂×C₄⋊C₄

C₂×D₁₉₁

yes

characteristic

C₂×C₂²⋊C₄

—

191

—

C₂×C₄²

—

191

—

C₂²⋊C₈

—

382

—

C₂⁴

—

191

—

C₂²×C₄

—

382

—

C₂²×C₄

—

382

—

C₂²×C₄

—

382

—

C₂²×C₄

—

191

—

C₂²×C₄

—

191

—

C₂²×C₄

—

191

—

C₂²×C₄

C₄×D₁₉₁

yes

—

100

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

101

M₁₆

—

102

M₁₆

—

103

M₁₆

—

382

—

104

M₁₆

—

382

—

105

C₂×C₈

—

106

C₂×C₈

—

382

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

382

—

109

C₂²⋊C₄

—

382

—

110

C₄²

—

382

—

111

C₂³

—

382

—

112

C₂³

—

382

—

113

C₂³

—

191

—

114

C₂³

—

764

—

115

C₂³

—

764

—

116

C₂³

C₂×C₄×D₁₉₁

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

382

—

120

C₂×C₄

—

382

—

121

C₂×C₄

—

382

—

122

C₂×C₄

—

382

—

123

C₂×C₄

—

382

—

124

C₂×C₄

—

191

—

125

C₂×C₄

—

764

—

126

C₂×C₄

D₄×D₁₉₁

yes

—

127

C₂×C₄

#15690286

yes

characteristic

128

C₂×C₄

#15690287

yes

—

129

C₂×C₄

#15690288

yes

characteristic

130

C₈

—

131

C₈

—

764

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

382

—

135

C₂²

—

382

—

136

C₂²

—

382

—

137

C₂²

—

764

—

138

C₂²

—

764

—

139

C₂²

C₂²⋊C₄×D₁₉₁

yes

characteristic

140

C₂²

#20259279

yes

characteristic

141

C₂²

#20259280

yes

characteristic

142

C₄

—

143

C₄

—

382

—

144

C₄

—

764

—

145

C₄

#20259304

yes

—

146

C₄

#20259307

yes

—

147

C₂

—

148

C₂

—

149

C₂

—

764

—

150

C₂

#27132525

yes

characteristic

151

C₂

#27132644

yes

characteristic

152

C₂

#27132646

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₉₁
commutator	C₂³×C₁₉₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting	#343×C₁₉₁	C₂
socle	C₂²×C₁₉₁	C₂×C₂²⋊C₄
prime radical	C₂³×D₁₉₁	C₂×C₄
cosocle	C₂²×C₄×C₁₉₁	C₂³
hypercenter	#343	D₁₉₁
hypocenter	C₁₉₁	#1822
LCS	G ▷ C₂³×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₉₁

commutator

C₂³×C₁₉₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₁

Fitting

#343×C₁₉₁

C₂

socle

C₂²×C₁₉₁

C₂×C₂²⋊C₄

prime radical

C₂³×D₁₉₁

C₂×C₄

cosocle

C₂²×C₄×C₁₉₁

C₂³

D₁₉₁

C₁₉₁

G ▷ C₂³×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
3. #27132432	C₂	semidirect
4. #27132431	C₂	semidirect
5. #343×C₁₉₁	C₂	semidirect
76. C₁₉₁	#1822	semidirect
82. #343	D₁₉₁	semidirect

H ≅ G/N

type

3. #27132432

C₂

semidirect

4. #27132431

C₂

semidirect

5. #343×C₁₉₁

C₂

semidirect

76. C₁₉₁

#1822

semidirect

82. #343

D₁₉₁

semidirect

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

subgroup	index
C₂×#20259280	2
#27132626	2
#27132432	2
#27132431	2
#343×C₁₉₁	2
#1822	191

subgroup

index

#343×C₁₉₁

191

Subgroups — #37572602

Conjugacy classes of subgroups

Complements

Maximal subgroups