Group #22586938 — subgroups

Subgroups — #22586938

Order 9088.

264 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 81 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	#17986056	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#11765849	C₂	1	1	yes	maximal, characteristic, complemented
3	#17985739	C₂	1	4	yes	maximal, complemented
4	C₂×#124×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#11765849	—	2	1	no	—
6	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
7	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
8	#14556039	C₂²	1	2	yes	complemented
9	#14556039	C₂²	1	2	yes	complemented
10	C₄×C₇₁⋊C₈	—	2	1	no	—
11	C₂²×Q₈×C₇₁	C₄	1	1	yes	characteristic
12	C₂²×D₄×C₇₁	C₄	1	1	yes	characteristic
13	#124×C₇₁	—	2	1	no	—
14	#124×C₇₁	—	2	1	no	—
15	#124×C₇₁	C₄	1	2	yes	—
16	#124×C₇₁	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₄×C₇₁	—	2	1	no	—
18	C₂×C₂²⋊C₄×C₇₁	—	2	1	no	—
19	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
20	#11765849	—	2	4	no	—
21	C₂×C₇₁⋊C₈	—	2	2	no	—
22	C₂×C₇₁⋊C₈	—	2	2	no	—
23	C₂⁴×C₇₁	—	2	1	no	—
24	C₂×Q₈×C₇₁	—	2	1	no	—
25	C₂×Q₈×C₇₁	—	4	2	no	—
26	C₂×Q₈×C₇₁	C₂×C₄	1	2	yes	—
27	C₂×D₄×C₇₁	—	2	1	no	—
28	C₂×D₄×C₇₁	—	4	2	no	—
29	C₂×D₄×C₇₁	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₇₁	—	2	1	no	—
31	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
34	C₂²⋊C₄×C₇₁	—	2	2	no	—
35	C₂²⋊C₄×C₇₁	—	2	2	no	—
36	C₂²⋊C₄×C₇₁	—	2	2	no	—
37	C₂²⋊C₄×C₇₁	—	2	2	no	—
38	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₇₁	D₄	1	1	yes	characteristic
40	C₄²×C₇₁	D₄	1	1	yes	characteristic
41	C₄²×C₇₁	C₂³	1	1	yes	characteristic
42	C₂³×C₇₁	—	2	2	no	—
43	C₂³×C₇₁	—	4	2	no	—
44	C₂³×C₇₁	—	4	2	no	—
45	C₂³×C₇₁	—	4	2	no	—
46	C₂³×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
47	Q₈×C₇₁	—	4	2	no	—
48	Q₈×C₇₁	—	4	2	no	—
49	D₄×C₇₁	—	4	2	no	—
50	D₄×C₇₁	—	4	2	no	—
51	C₂×C₄×C₇₁	—	2	2	no	—
52	C₂×C₄×C₇₁	—	2	2	no	—
53	C₂×C₄×C₇₁	—	2	2	no	—
54	C₂×C₄×C₇₁	—	4	1	no	—
55	C₂×C₄×C₇₁	—	4	2	no	—
56	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
62	C₇₁⋊C₈	—	2	4	no	—
63	C₇₁⋊C₈	—	4	2	no	—
64	C₂²×C₇₁	—	2	1	no	—
65	C₂²×C₇₁	—	2	1	no	—
66	C₂²×C₇₁	—	4	2	no	—
67	C₂²×C₇₁	—	4	2	no	—
68	C₂²×C₇₁	—	4	2	no	—
69	C₂²×C₇₁	—	8	4	no	—
70	C₂²×C₇₁	#104	1	2	yes	—
71	C₂²×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₄×C₇₁	—	2	1	no	—
73	C₄×C₇₁	—	2	1	no	—
74	C₄×C₇₁	—	2	1	no	—
75	C₄×C₇₁	—	2	1	no	—
76	C₄×C₇₁	—	2	1	no	—
77	C₄×C₇₁	—	4	2	no	—
78	C₄×C₇₁	#100	1	2	yes	—
79	C₂×C₇₁	—	2	2	no	—
80	C₂×C₇₁	—	8	2	no	—
81	C₂×C₇₁	C₂×#100	1	1	yes	characteristic
82	C₂×C₇₁	C₂×#104	1	1	yes	characteristic
83	C₂×C₇₁	#421	1	1	yes	characteristic
84	#1527	—	71	1	no	maximal, Sylow
85	C₇₁	#1527	1	1	yes	characteristic, complemented, Sylow
86	C₂×#124	D₇₁	1	1	yes	characteristic
87	#432	—	71	1	no	—
88	C₄×M₁₆	—	71	1	no	—
89	#329	—	71	4	no	—
90	C₂²×Q₈	Dic₇₁	1	1	yes	characteristic
91	C₂²×D₄	Dic₇₁	1	1	yes	characteristic
92	C₂×M₁₆	—	142	1	no	—
93	#124	—	2	1	no	—
94	#124	—	2	1	no	—
95	#124	Dic₇₁	1	2	yes	—
96	#124	C₂×D₇₁	1	2	yes	—
97	C₂×C₂²⋊C₄	—	2	1	no	—
98	C₂×C₂²⋊C₄	—	2	1	no	—
99	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
100	C₄⋊C₈	—	142	1	no	—
101	C₂²⋊C₈	—	142	1	no	—
102	#97	—	71	2	no	—
103	#97	—	71	2	no	—
104	C₄×C₈	—	142	1	no	—
105	C₂⁴	—	2	1	no	—
106	C₂×Q₈	—	2	1	no	—
107	C₂×Q₈	—	4	2	no	—
108	C₂×Q₈	C₂×Dic₇₁	1	2	yes	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	4	2	no	—
111	C₂×D₄	C₂×Dic₇₁	1	2	yes	—
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	C₂×Dic₇₁	1	1	yes	characteristic
114	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
115	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
116	M₁₆	—	142	4	no	—
117	C₂×C₈	—	142	2	no	—
118	C₂×C₈	—	142	2	no	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₂²⋊C₄	—	2	2	no	—
121	C₂²⋊C₄	—	2	2	no	—
122	C₂²⋊C₄	—	2	2	no	—
123	C₄²	C₂×Dic₇₁	1	1	yes	characteristic
124	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
125	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
126	C₄²	C₂²×D₇₁	1	1	yes	characteristic
127	C₂³	—	2	2	no	—
128	C₂³	—	4	2	no	—
129	C₂³	—	4	2	no	—
130	C₂³	—	4	2	no	—
131	C₂³	C₂²⋊Dic₇₁	1	1	yes	characteristic
132	Q₈	—	4	2	no	—
133	Q₈	—	4	2	no	—
134	D₄	—	4	2	no	—
135	D₄	—	4	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
142	C₂×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
143	C₂×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
144	C₂×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
145	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
146	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
147	C₈	—	284	2	no	—
148	C₈	—	142	4	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	4	2	no	—
152	C₂²	—	4	2	no	—
153	C₂²	—	4	2	no	—
154	C₂²	—	8	4	no	—
155	C₂²	#14556055	1	2	yes	—
156	C₂²	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	4	2	no	—
163	C₄	#14556045	1	2	yes	—
164	C₂	—	2	2	no	—
165	C₂	—	8	2	no	—
166	C₂	C₂×#14556045	1	1	yes	characteristic
167	C₂	C₂×#14556055	1	1	yes	characteristic
168	C₂	#17986016	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986056

C₂

yes

maximal, characteristic, complemented

C₄×#11765849

C₂

yes

maximal, characteristic, complemented

#17985739

C₂

yes

maximal, complemented

C₂×#124×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

—

C₄⋊(C₇₁⋊C₈)

—

C₂²⋊(C₇₁⋊C₈)

—

#14556039

C₂²

yes

complemented

#14556039

C₂²

yes

complemented

C₄×C₇₁⋊C₈

—

C₂²×Q₈×C₇₁

C₄

yes

characteristic

C₂²×D₄×C₇₁

C₄

yes

characteristic

#124×C₇₁

—

#124×C₇₁

—

#124×C₇₁

C₄

yes

—

#124×C₇₁

C₂²

yes

—

C₂×C₂²⋊C₄×C₇₁

—

C₂×C₂²⋊C₄×C₇₁

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂⁴×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

C₂×C₄

yes

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

C₂²⋊C₄

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#104

yes

—

C₂²×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#100

yes

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

C₂×#124

D₇₁

yes

characteristic

#432

—

C₄×M₁₆

—

#329

—

C₂²×Q₈

Dic₇₁

yes

characteristic

C₂²×D₄

Dic₇₁

yes

characteristic

C₂×M₁₆

—

142

—

#124

—

#124

—

#124

Dic₇₁

yes

—

#124

C₂×D₇₁

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

100

C₄⋊C₈

—

142

—

101

C₂²⋊C₈

—

142

—

102

#97

—

103

#97

—

104

C₄×C₈

—

142

—

105

C₂⁴

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

C₂×Dic₇₁

yes

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

C₂×Dic₇₁

yes

—

112

C₂²×C₄

—

113

C₂²×C₄

C₂×Dic₇₁

yes

characteristic

114

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

115

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

116

M₁₆

—

142

—

117

C₂×C₈

—

142

—

118

C₂×C₈

—

142

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

—

122

C₂²⋊C₄

—

123

C₄²

C₂×Dic₇₁

yes

characteristic

124

C₄²

C₂²⋊D₇₁

yes

characteristic

125

C₄²

C₂²⋊D₇₁

yes

characteristic

126

C₄²

C₂²×D₇₁

yes

characteristic

127

C₂³

—

128

C₂³

—

129

C₂³

—

130

C₂³

—

131

C₂³

C₂²⋊Dic₇₁

yes

characteristic

132

Q₈

—

133

Q₈

—

134

D₄

—

135

D₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

C₂²⋊Dic₇₁

yes

characteristic

142

C₂×C₄

C₂²⋊Dic₇₁

yes

characteristic

143

C₂×C₄

C₂²⋊Dic₇₁

yes

characteristic

144

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

145

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

146

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

147

C₈

—

284

—

148

C₈

—

142

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

#14556055

yes

—

156

C₂²

C₂×C₂²⋊Dic₇₁

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#14556045

yes

—

164

C₂

—

165

C₂

—

166

C₂

C₂×#14556045

yes

characteristic

167

C₂

C₂×#14556055

yes

characteristic

168

C₂

#17986016

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₄²	C₂²×D₇₁
Fitting	C₂×#124×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₂²⋊C₄
hypercenter	C₂×#124	D₇₁
hypocenter	C₇₁	#1527
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#124

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₄²

C₂²×D₇₁

Fitting

C₂×#124×C₇₁

C₂

socle

C₂²×C₇₁

C₂×C₂²⋊C₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#124

N	H ≅ G/N	type
1. #17986056	C₂	semidirect
2. C₄×#11765849	C₂	semidirect
3. #17985739	C₂	semidirect
8. #14556039	C₂²	semidirect
9. #14556039	C₂²	semidirect
85. C₇₁	#1527	semidirect

H ≅ G/N

type

1. #17986056

C₂

semidirect

2. C₄×#11765849

C₂

semidirect

3. #17985739

C₂

semidirect

8. #14556039

C₂²

semidirect

9. #14556039

C₂²

semidirect

85. C₇₁

#1527

semidirect

Subgroups — #22586938

Conjugacy classes of subgroups

Complements