Group #22588199 — subgroups

Subgroups — #22588199

Order 9088.

260 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 133 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×C₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556038	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic
4	#17985845	C₂	1	1	yes	maximal, characteristic
5	#17985753	C₂	1	1	yes	maximal, characteristic
6	#17985753	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₇₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₇₁	C₂²	1	1	yes	characteristic
12	C₄⋊(C₇₁⋊C₈)	C₄	1	4	yes	complemented
13	#14556038	—	2	2	no	—
14	C₈×Dic₇₁	—	2	2	no	—
15	C₇₁⋊#95	C₄	1	2	yes	—
16	C₇₁⋊#95	C₄	1	2	yes	—
17	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₇₁	C₄	1	4	yes	complemented
21	C₂²×Dic₇₁	C₂×C₄	1	2	yes	—
22	C₂²×Dic₇₁	C₂×C₄	1	2	yes	—
23	C₂×C₇₁⋊C₈	—	2	1	no	—
24	C₂×C₇₁⋊C₈	—	2	1	no	—
25	C₂×C₇₁⋊C₈	—	2	1	no	—
26	C₂×C₇₁⋊C₈	—	2	1	no	—
27	C₂×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
28	C₂×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
29	C₄×Dic₇₁	—	2	1	no	—
30	C₄×Dic₇₁	—	2	1	no	—
31	C₄×Dic₇₁	D₄	1	2	yes	—
32	C₄×Dic₇₁	Q₈	1	2	yes	—
33	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₇₁	—	2	1	no	—
37	C₂×C₈×C₇₁	—	2	1	no	—
38	C₂×C₈×C₇₁	—	2	1	no	—
39	C₂×C₈×C₇₁	—	2	1	no	—
40	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
41	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
42	C₄²×C₇₁	C₂×C₄	1	2	yes	—
43	C₄²×C₇₁	C₂×C₄	1	2	yes	—
44	C₂³×C₇₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₇₁	—	2	2	no	—
46	C₂×C₄×C₇₁	—	2	2	no	—
47	C₂×C₄×C₇₁	C₄²	1	2	yes	—
48	C₂×C₄×C₇₁	C₄²	1	2	yes	—
49	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
54	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
55	C₂×Dic₇₁	—	2	2	no	—
56	C₂×Dic₇₁	—	2	2	no	—
57	C₂×Dic₇₁	—	2	2	no	—
58	C₂×Dic₇₁	—	2	2	no	—
59	C₂×Dic₇₁	—	2	2	no	—
60	C₂×Dic₇₁	C₄⋊C₄	1	4	yes	—
61	C₈×C₇₁	—	2	2	no	—
62	C₈×C₇₁	—	2	2	no	—
63	C₇₁⋊C₈	—	2	2	no	—
64	C₇₁⋊C₈	—	2	2	no	—
65	C₂²×C₇₁	C₂×C₄²	1	1	yes	characteristic
66	C₂²×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₇₁	#117	1	1	yes	characteristic
68	C₂²×C₇₁	#131	1	2	yes	—
69	C₂²×C₇₁	#131	1	2	yes	—
70	C₄×C₇₁	—	2	2	no	—
71	C₄×C₇₁	—	2	2	no	—
72	C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
73	C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
74	C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
75	C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
76	Dic₇₁	—	2	4	no	—
77	Dic₇₁	—	4	2	no	—
78	C₂×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₇₁	#405	1	1	yes	characteristic
80	C₂×C₇₁	#405	1	1	yes	characteristic
81	C₂×C₇₁	#424	1	2	yes	—
82	C₂×C₇₁	#433	1	2	yes	—
83	#1768	—	71	1	no	maximal, Sylow
84	C₇₁	#1768	1	1	yes	characteristic, complemented, Sylow
85	C₂×C₄⋊C₈	—	71	1	no	—
86	C₂×C₄⋊C₈	D₇₁	1	1	yes	characteristic
87	C₂×#97	—	71	1	no	—
88	C₂×C₄×C₈	—	71	1	no	—
89	#376	—	71	1	no	—
90	#336	—	71	1	no	—
91	#336	—	71	1	no	—
92	C₂²×C₈	—	71	1	no	—
93	C₂²×C₈	—	71	1	no	—
94	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
95	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
96	C₂×C₄²	—	71	1	no	—
97	C₂×C₄²	—	71	1	no	—
98	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
99	C₄⋊C₈	—	71	4	no	—
100	C₄⋊C₈	Dic₇₁	1	4	yes	complemented
101	#97	—	142	2	no	—
102	C₄×C₈	—	142	2	no	—
103	#95	—	71	2	no	—
104	#95	—	71	2	no	—
105	C₂²×C₄	—	71	2	no	—
106	C₂²×C₄	—	71	2	no	—
107	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
108	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
109	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	71	2	no	—
115	C₂×C₈	—	71	2	no	—
116	C₂×C₈	—	142	1	no	—
117	C₂×C₈	—	142	1	no	—
118	C₂×C₈	—	142	1	no	—
119	C₂×C₈	—	142	1	no	—
120	C₂×C₈	C₂×Dic₇₁	1	2	yes	—
121	C₂×C₈	C₂×Dic₇₁	1	2	yes	—
122	C₄²	—	71	2	no	—
123	C₄²	—	71	2	no	—
124	C₄²	—	142	1	no	—
125	C₄²	—	142	1	no	—
126	C₄²	C₄×D₇₁	1	2	yes	—
127	C₄²	C₂×Dic₇₁	1	2	yes	—
128	C₂³	C₂×C₄×D₇₁	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	71	4	no	—
132	C₂×C₄	—	142	2	no	—
133	C₂×C₄	—	142	2	no	—
134	C₂×C₄	—	142	2	no	—
135	C₂×C₄	—	142	2	no	—
136	C₂×C₄	—	142	2	no	—
137	C₂×C₄	C₄×Dic₇₁	1	2	yes	—
138	C₂×C₄	C₄×Dic₇₁	1	2	yes	—
139	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
140	C₂×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
141	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
142	C₂×C₄	Q₈×D₇₁	1	1	yes	characteristic
143	C₂×C₄	#11765864	1	1	yes	characteristic
144	C₂×C₄	#11765866	1	1	yes	characteristic
145	C₈	—	2	2	no	—
146	C₈	—	2	2	no	—
147	C₈	—	142	2	no	—
148	C₈	—	142	2	no	—
149	C₂²	C₂×C₄×Dic₇₁	1	1	yes	characteristic
150	C₂²	C₄⋊C₄×D₇₁	1	1	yes	characteristic
151	C₂²	#14556086	1	1	yes	characteristic
152	C₂²	#14556134	1	2	yes	—
153	C₂²	#14556136	1	2	yes	—
154	C₄	—	2	2	no	—
155	C₄	—	2	2	no	—
156	C₄	—	284	2	no	—
157	C₄	—	142	4	no	—
158	C₄	#14556090	1	1	yes	characteristic
159	C₄	D₄×Dic₇₁	1	1	yes	characteristic
160	C₄	#14556093	1	1	yes	characteristic
161	C₄	Q₈×Dic₇₁	1	1	yes	characteristic
162	C₂	C₄⋊C₄×Dic₇₁	1	1	yes	characteristic
163	C₂	#17985954	1	1	yes	characteristic
164	C₂	#17985957	1	1	yes	characteristic
165	C₂	#17986024	1	2	yes	—
166	C₂	#17986059	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₇₁⋊C₈)

C₂

yes

maximal, characteristic

C₂×#14556038

C₂

yes

maximal, characteristic

C₂×C₈×Dic₇₁

C₂

yes

maximal, characteristic

#17985845

C₂

yes

maximal, characteristic

#17985753

C₂

yes

maximal, characteristic

#17985753

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₇₁

C₂²

yes

characteristic

C₂×C₄×Dic₇₁

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₄

yes

complemented

#14556038

—

C₈×Dic₇₁

—

C₇₁⋊#95

C₄

yes

—

C₇₁⋊#95

C₄

yes

—

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₄

yes

complemented

C₂²×Dic₇₁

C₂×C₄

yes

—

C₂²×Dic₇₁

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

D₄

yes

—

C₄×Dic₇₁

Q₈

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

C₂×C₄

yes

—

C₂³×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄²

yes

—

C₂×C₄×C₇₁

C₄²

yes

—

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×Q₈

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

C₄⋊C₄

yes

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₂×C₄²

yes

characteristic

C₂²×C₇₁

C₂×C₄⋊C₄

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

—

C₂²×C₇₁

#131

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₄×D₄

yes

characteristic

C₄×C₇₁

C₄×D₄

yes

characteristic

C₄×C₇₁

C₄×Q₈

yes

characteristic

C₄×C₇₁

C₄×Q₈

yes

characteristic

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

C₄×C₄⋊C₄

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₂×C₇₁

#433

yes

—

#1768

—

maximal, Sylow

C₇₁

#1768

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₇₁

yes

characteristic

C₂×#97

—

C₂×C₄×C₈

—

#376

—

#336

—

#336

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇₁

yes

characteristic

C₂²×C₈

C₂×D₇₁

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

Dic₇₁

yes

complemented

101

#97

—

142

—

102

C₄×C₈

—

142

—

103

#95

—

104

#95

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

C₄×D₇₁

yes

characteristic

108

C₂²×C₄

C₄×D₇₁

yes

characteristic

109

C₂²×C₄

C₂²×D₇₁

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

142

—

117

C₂×C₈

—

142

—

118

C₂×C₈

—

142

—

119

C₂×C₈

—

142

—

120

C₂×C₈

C₂×Dic₇₁

yes

—

121

C₂×C₈

C₂×Dic₇₁

yes

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

142

—

125

C₄²

—

142

—

126

C₄²

C₄×D₇₁

yes

—

127

C₄²

C₂×Dic₇₁

yes

—

128

C₂³

C₂×C₄×D₇₁

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

142

—

133

C₂×C₄

—

142

—

134

C₂×C₄

—

142

—

135

C₂×C₄

—

142

—

136

C₂×C₄

—

142

—

137

C₂×C₄

C₄×Dic₇₁

yes

—

138

C₂×C₄

C₄×Dic₇₁

yes

—

139

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

140

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

141

C₂×C₄

D₄×D₇₁

yes

characteristic

142

C₂×C₄

Q₈×D₇₁

yes

characteristic

143

C₂×C₄

#11765864

yes

characteristic

144

C₂×C₄

#11765866

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

142

—

148

C₈

—

142

—

149

C₂²

C₂×C₄×Dic₇₁

yes

characteristic

150

C₂²

C₄⋊C₄×D₇₁

yes

characteristic

151

C₂²

#14556086

yes

characteristic

152

C₂²

#14556134

yes

—

153

C₂²

#14556136

yes

—

154

C₄

—

155

C₄

—

156

C₄

—

284

—

157

C₄

—

142

—

158

C₄

#14556090

yes

characteristic

159

C₄

D₄×Dic₇₁

yes

characteristic

160

C₄

#14556093

yes

characteristic

161

C₄

Q₈×Dic₇₁

yes

characteristic

162

C₂

C₄⋊C₄×Dic₇₁

yes

characteristic

163

C₂

#17985954

yes

characteristic

164

C₂

#17985957

yes

characteristic

165

C₂

#17986024

yes

—

166

C₂

#17986059

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₇₁
commutator	C₂²×C₇₁	C₂×C₄²
Fitting	C₂×C₄⋊C₈×C₇₁	C₂
socle	C₂³×C₇₁	C₂²×C₄
hypercenter	C₂×C₄⋊C₈	D₇₁
hypocenter	C₇₁	#1768
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₇₁

commutator

C₂²×C₇₁

C₂×C₄²

Fitting

C₂×C₄⋊C₈×C₇₁

C₂

socle

C₂³×C₇₁

C₂²×C₄

hypercenter

C₂×C₄⋊C₈

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
12. C₄⋊(C₇₁⋊C₈)	C₄	semidirect
20. C₄⋊C₈×C₇₁	C₄	semidirect
84. C₇₁	#1768	semidirect
100. C₄⋊C₈	Dic₇₁	semidirect

H ≅ G/N

type

12. C₄⋊(C₇₁⋊C₈)

C₄

semidirect

20. C₄⋊C₈×C₇₁

C₄

semidirect

84. C₇₁

#1768

semidirect

100. C₄⋊C₈

Dic₇₁

semidirect

Subgroups — #22588199

Conjugacy classes of subgroups

Complements