Group #37573627 — subgroups

Subgroups — #37573627

Order 24448.

382 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 75 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×#20259348	C₂	1	2	yes	maximal, complemented
2	C₂×#20259239	C₂	1	2	yes	maximal, complemented
3	#27132629	C₂	1	1	yes	maximal, characteristic, complemented
4	#27132428	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₈×C₁₉₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15690287	C₂²	1	1	yes	characteristic
7	#20259348	—	2	4	no	—
8	#20259348	—	2	4	no	—
9	C₂×#15690275	—	2	2	no	—
10	C₂×#15690272	—	2	2	no	—
11	C₂×#15690271	C₂²	1	1	yes	characteristic
12	C₂×#15690271	C₂²	1	2	yes	—
13	#20259239	—	2	4	no	—
14	(C₂×D₁₉₁)⋊C₈	—	2	2	no	—
15	C₂²×D₄×C₁₉₁	C₂²	1	2	yes	—
16	C₂×D₈×C₁₉₁	—	2	4	no	—
17	C₂×D₈×C₁₉₁	—	2	2	no	—
18	C₂²×C₈×C₁₉₁	C₂²	1	1	yes	characteristic
19	#15690287	—	2	2	no	—
20	#15690287	—	2	2	no	—
21	C₂×Dic₃₈₂	D₄	1	1	yes	characteristic
22	C₂×D₇₆₄	D₄	1	1	yes	characteristic
23	C₂×C₂²⋊D₁₉₁	—	2	1	no	—
24	C₂×C₄×D₁₉₁	—	2	1	no	—
25	#15690275	—	2	8	no	—
26	#15690272	—	2	8	no	—
27	#15690271	—	2	2	no	—
28	#15690271	—	2	2	no	—
29	#15690271	D₄	1	4	yes	—
30	C₂×C₁₉₁⋊C₈	—	2	2	no	—
31	C₂×C₁₉₁⋊C₈	—	2	1	no	—
32	C₂⁴×C₁₉₁	—	2	2	no	—
33	C₂×D₄×C₁₉₁	—	2	2	no	—
34	C₂×D₄×C₁₉₁	—	2	2	no	—
35	C₂×D₄×C₁₉₁	—	2	2	no	—
36	C₂×D₄×C₁₉₁	—	2	2	no	—
37	C₂×D₄×C₁₉₁	—	2	2	no	—
38	C₂×D₄×C₁₉₁	D₄	1	4	yes	—
39	C₂²×C₄×C₁₉₁	C₂³	1	1	yes	characteristic
40	D₈×C₁₉₁	—	2	8	no	—
41	C₂×C₈×C₁₉₁	—	2	2	no	—
42	C₂×C₈×C₁₉₁	D₄	1	2	yes	—
43	C₂²×D₁₉₁	—	2	1	no	—
44	Dic₃₈₂	—	2	2	no	—
45	D₇₆₄	—	2	2	no	—
46	C₂²⋊D₁₉₁	—	4	2	no	—
47	C₂×Dic₁₉₁	—	2	1	no	—
48	C₄×D₁₉₁	—	4	1	no	—
49	C₄×D₁₉₁	—	4	1	no	—
50	C₁₉₁⋊C₈	—	2	4	no	—
51	C₁₉₁⋊C₈	—	4	1	no	—
52	C₂³×C₁₉₁	—	2	4	no	—
53	C₂³×C₁₉₁	—	4	2	no	—
54	C₂³×C₁₉₁	—	4	2	no	—
55	C₂³×C₁₉₁	—	8	4	no	—
56	C₂³×C₁₉₁	#41	1	1	yes	characteristic
57	D₄×C₁₉₁	—	2	4	no	—
58	D₄×C₁₉₁	—	2	4	no	—
59	D₄×C₁₉₁	—	4	2	no	—
60	D₄×C₁₉₁	—	4	2	no	—
61	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₁₉₁	C₂×D₄	1	2	yes	—
64	C₂×C₄×C₁₉₁	C₂×D₄	1	1	yes	characteristic
65	C₈×C₁₉₁	—	2	2	no	—
66	C₂²×C₁₉₁	—	2	2	no	—
67	C₂²×C₁₉₁	—	4	4	no	—
68	C₂²×C₁₉₁	—	8	4	no	—
69	C₂²×C₁₉₁	—	8	4	no	—
70	C₂²×C₁₉₁	—	8	4	no	—
71	C₂²×C₁₉₁	#121	1	1	yes	characteristic
72	C₂²×C₁₉₁	#121	1	2	yes	—
73	C₂×D₁₉₁	—	4	1	no	—
74	C₂×D₁₉₁	—	4	2	no	—
75	C₄×C₁₉₁	#120	1	2	yes	—
76	C₄×C₁₉₁	#120	1	1	yes	characteristic
77	C₄×C₁₉₁	#127	1	1	yes	characteristic
78	Dic₁₉₁	—	4	1	no	—
79	C₂×C₁₉₁	—	2	2	no	—
80	C₂×C₁₉₁	—	8	4	no	—
81	C₂×C₁₉₁	#392	1	1	yes	characteristic
82	C₂×C₁₉₁	#472	1	2	yes	—
83	D₁₉₁	—	8	1	no	—
84	C₁₉₁	#2018	1	1	yes	characteristic, complemented, Sylow
85	#2018	—	191	1	no	maximal, Sylow
86	C₂×AGL(1,ℤ/8)	—	191	2	no	—
87	C₂²×D₈	D₁₉₁	1	1	yes	characteristic, complemented
88	C₂×#100	—	191	2	no	—
89	#408	—	191	1	no	—
90	#341	—	191	1	no	—
91	C₂×#41	—	191	1	no	—
92	C₂²×D₄	C₂×D₁₉₁	1	2	yes	—
93	AGL(1,ℤ/8)	—	382	4	no	—
94	AGL(1,ℤ/8)	—	382	4	no	—
95	C₂×SD₁₆	—	382	2	no	—
96	C₂×D₈	—	2	4	no	—
97	C₂×D₈	—	2	2	no	—
98	C₂×D₈	—	382	2	no	—
99	C₂×M₁₆	—	191	1	no	—
100	C₂×M₁₆	—	191	2	no	—
101	C₂²×C₈	C₂×D₁₉₁	1	1	yes	characteristic
102	#100	—	382	4	no	—
103	C₂²⋊C₈	—	382	2	no	—
104	C₂⁴	—	2	2	no	—
105	#41	—	382	2	no	—
106	#41	—	382	2	no	—
107	C₂×Q₈	—	191	1	no	—
108	C₂×D₄	—	2	2	no	—
109	C₂×D₄	—	2	2	no	—
110	C₂×D₄	—	2	2	no	—
111	C₂×D₄	—	2	2	no	—
112	C₂×D₄	—	2	2	no	—
113	C₂×D₄	—	382	1	no	—
114	C₂×D₄	—	191	1	no	—
115	C₂×D₄	C₂²⋊D₁₉₁	1	4	yes	—
116	C₂²×C₄	—	382	1	no	—
117	C₂²×C₄	C₂²×D₁₉₁	1	1	yes	characteristic
118	SD₁₆	—	382	8	no	—
119	D₈	—	2	8	no	—
120	D₈	—	382	8	no	—
121	M₁₆	—	382	2	no	—
122	M₁₆	—	382	2	no	—
123	M₁₆	—	191	4	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	382	2	no	—
126	C₂×C₈	—	382	1	no	—
127	C₂×C₈	C₂²⋊D₁₉₁	1	2	yes	—
128	C₂³	—	2	4	no	—
129	C₂³	—	4	2	no	—
130	C₂³	—	4	2	no	—
131	C₂³	—	8	4	no	—
132	C₂³	—	382	1	no	—
133	C₂³	#15690286	1	1	yes	characteristic
134	Q₈	—	382	2	no	—
135	D₄	—	2	4	no	—
136	D₄	—	2	4	no	—
137	D₄	—	4	2	no	—
138	D₄	—	4	2	no	—
139	D₄	—	382	2	no	—
140	D₄	—	764	2	no	—
141	C₂×C₄	—	382	1	no	—
142	C₂×C₄	—	764	1	no	—
143	C₂×C₄	—	764	1	no	—
144	C₂×C₄	D₄×D₁₉₁	1	1	yes	characteristic
145	C₂×C₄	D₄×D₁₉₁	1	2	yes	—
146	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊D₁₉₁	1	1	yes	characteristic
148	C₈	—	2	2	no	—
149	C₈	—	382	4	no	—
150	C₈	—	764	1	no	—
151	C₂²	—	2	2	no	—
152	C₂²	—	4	4	no	—
153	C₂²	—	8	4	no	—
154	C₂²	—	8	4	no	—
155	C₂²	—	8	4	no	—
156	C₂²	—	764	2	no	—
157	C₂²	—	764	1	no	—
158	C₂²	#20259293	1	1	yes	characteristic
159	C₂²	#20259294	1	2	yes	—
160	C₄	—	764	1	no	—
161	C₄	#20259290	1	2	yes	—
162	C₄	#20259291	1	1	yes	characteristic
163	C₄	#20259318	1	1	yes	characteristic
164	C₂	—	2	2	no	—
165	C₂	—	8	4	no	—
166	C₂	—	1528	1	no	—
167	C₂	#27132571	1	1	yes	characteristic
168	C₂	#27132930	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20259348

C₂

yes

maximal, complemented

C₂×#20259239

C₂

yes

maximal, complemented

#27132629

C₂

yes

maximal, characteristic, complemented

#27132428

C₂

yes

maximal, characteristic, complemented

C₂²×D₈×C₁₉₁

C₂

yes

maximal, characteristic, complemented

C₂×#15690287

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×#15690271

C₂²

yes

—

#20259239

—

(C₂×D₁₉₁)⋊C₈

—

C₂²×D₄×C₁₉₁

C₂²

yes

—

C₂×D₈×C₁₉₁

—

C₂×D₈×C₁₉₁

—

C₂²×C₈×C₁₉₁

C₂²

yes

characteristic

#15690287

—

#15690287

—

C₂×Dic₃₈₂

D₄

yes

characteristic

C₂×D₇₆₄

D₄

yes

characteristic

C₂×C₂²⋊D₁₉₁

—

C₂×C₄×D₁₉₁

—

—

—

—

—

D₄

yes

—

C₂×C₁₉₁⋊C₈

—

C₂×C₁₉₁⋊C₈

—

C₂⁴×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

—

C₂×D₄×C₁₉₁

D₄

yes

—

C₂²×C₄×C₁₉₁

C₂³

yes

characteristic

D₈×C₁₉₁

—

C₂×C₈×C₁₉₁

—

C₂×C₈×C₁₉₁

D₄

yes

—

C₂²×D₁₉₁

—

Dic₃₈₂

—

D₇₆₄

—

C₂²⋊D₁₉₁

—

C₂×Dic₁₉₁

—

C₄×D₁₉₁

—

C₄×D₁₉₁

—

C₁₉₁⋊C₈

—

C₁₉₁⋊C₈

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

—

C₂³×C₁₉₁

#41

yes

characteristic

D₄×C₁₉₁

—

D₄×C₁₉₁

—

D₄×C₁₉₁

—

D₄×C₁₉₁

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

—

C₂×C₄×C₁₉₁

C₂×D₄

yes

characteristic

C₈×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

—

C₂²×C₁₉₁

#121

yes

characteristic

C₂²×C₁₉₁

#121

yes

—

C₂×D₁₉₁

—

C₂×D₁₉₁

—

C₄×C₁₉₁

#120

yes

—

C₄×C₁₉₁

yes

C₄×C₁₉₁

yes

Dic₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

—

C₂×C₁₉₁

#392

yes

characteristic

C₂×C₁₉₁

#472

yes

—

D₁₉₁

—

C₁₉₁

#2018

yes

characteristic, complemented, Sylow

#2018

—

191

maximal, Sylow

C₂×AGL(1,ℤ/8)

—

191

—

C₂²×D₈

D₁₉₁

yes

characteristic, complemented

C₂×#100

—

191

—

#408

—

191

—

#341

—

191

—

C₂×#41

—

191

—

C₂²×D₄

C₂×D₁₉₁

yes

—

AGL(1,ℤ/8)

—

382

—

AGL(1,ℤ/8)

—

382

—

C₂×SD₁₆

—

382

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

382

—

C₂×M₁₆

—

191

—

100

C₂×M₁₆

—

191

—

101

C₂²×C₈

C₂×D₁₉₁

yes

characteristic

102

#100

—

382

—

103

C₂²⋊C₈

—

382

—

104

C₂⁴

—

105

#41

—

382

—

106

#41

—

382

—

107

C₂×Q₈

—

191

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

382

—

114

C₂×D₄

—

191

—

115

C₂×D₄

C₂²⋊D₁₉₁

yes

—

116

C₂²×C₄

—

382

—

117

C₂²×C₄

C₂²×D₁₉₁

yes

characteristic

118

SD₁₆

—

382

—

119

D₈

—

120

D₈

—

382

—

121

M₁₆

—

382

—

122

M₁₆

—

382

—

123

M₁₆

—

191

—

124

C₂×C₈

—

125

C₂×C₈

—

382

—

126

C₂×C₈

—

382

—

127

C₂×C₈

C₂²⋊D₁₉₁

yes

—

128

C₂³

—

129

C₂³

—

130

C₂³

—

131

C₂³

—

132

C₂³

—

382

—

133

C₂³

#15690286

yes

characteristic

134

Q₈

—

382

—

135

D₄

—

136

D₄

—

137

D₄

—

138

D₄

—

139

D₄

—

382

—

140

D₄

—

764

—

141

C₂×C₄

—

382

—

142

C₂×C₄

—

764

—

143

C₂×C₄

—

764

—

144

C₂×C₄

D₄×D₁₉₁

yes

characteristic

145

C₂×C₄

D₄×D₁₉₁

yes

—

146

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

—

147

C₂×C₄

C₂×C₂²⋊D₁₉₁

yes

characteristic

148

C₈

—

149

C₈

—

382

—

150

C₈

—

764

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

764

—

157

C₂²

—

764

—

158

C₂²

#20259293

yes

characteristic

159

C₂²

#20259294

yes

—

160

C₄

—

764

—

161

C₄

#20259290

yes

—

162

C₄

#20259291

yes

characteristic

163

C₄

#20259318

yes

characteristic

164

C₂

—

165

C₂

—

166

C₂

—

1528

—

167

C₂

#27132571

yes

characteristic

168

C₂

#27132930

yes

—

	subgroup	G/H
center	C₂²	#20259293
commutator, cosocle	C₂²×C₄×C₁₉₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉₁
Fitting	C₂²×D₈×C₁₉₁	C₂
socle	C₂²×C₁₉₁	#121
hypercenter	C₂²×D₈	D₁₉₁
hypocenter	C₁₉₁	#2018
LCS	G ▷ C₂²×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₈

subgroup

G/H

center

C₂²

#20259293

commutator, cosocle

C₂²×C₄×C₁₉₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉₁

Fitting

C₂²×D₈×C₁₉₁

C₂

socle

C₂²×C₁₉₁

#121

hypercenter

C₂²×D₈

D₁₉₁

hypocenter

C₁₉₁

#2018

LCS

G ▷ C₂²×C₄×C₁₉₁ ▷ C₂×C₁₉₁ ▷ C₁₉₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₈

N	H ≅ G/N	type
1. C₂×#20259348	C₂	semidirect
2. C₂×#20259239	C₂	semidirect
3. #27132629	C₂	semidirect
4. #27132428	C₂	semidirect
5. C₂²×D₈×C₁₉₁	C₂	semidirect
84. C₁₉₁	#2018	semidirect
87. C₂²×D₈	D₁₉₁	semidirect

H ≅ G/N

type

1. C₂×#20259348

C₂

semidirect

2. C₂×#20259239

C₂

semidirect

3. #27132629

C₂

semidirect

4. #27132428

C₂

semidirect

5. C₂²×D₈×C₁₉₁

C₂

semidirect

84. C₁₉₁

#2018

semidirect

87. C₂²×D₈

D₁₉₁

semidirect

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

subgroup	index
C₂×#20259348	2
C₂×#20259239	2
#27132629	2
#27132428	2
C₂²×D₈×C₁₉₁	2
#2018	191

subgroup

index

C₂²×D₈×C₁₉₁

191

Subgroups — #37573627

Conjugacy classes of subgroups

Complements

Maximal subgroups