Group #22588215 — subgroups

Subgroups — #22588215

Order 9088.

272 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14556039	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556039	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#17985754	C₂	1	1	yes	maximal, characteristic
6	#17985754	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊C₄×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	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊(C₇₁⋊C₈)	C₄	1	4	yes	complemented
13	#14556039	—	2	2	no	—
14	#14556039	C₄	1	4	yes	complemented
15	C₄×C₇₁⋊C₈	—	2	2	no	—
16	C₂×C₄⋊C₄×C₇₁	C₄	1	2	yes	—
17	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
20	#95×C₇₁	C₄	1	2	yes	—
21	C₂×C₇₁⋊C₈	—	2	1	no	—
22	C₂×C₇₁⋊C₈	—	2	1	no	—
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	C₂×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
30	C₂×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
31	C₂×C₇₁⋊C₈	D₄	1	2	yes	—
32	C₂×C₇₁⋊C₈	Q₈	1	2	yes	—
33	C₂²×C₄×C₇₁	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₇₁	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₇₁	—	2	4	no	—
39	C₄²×C₇₁	—	2	1	no	—
40	C₄²×C₇₁	—	2	1	no	—
41	C₄²×C₇₁	—	2	1	no	—
42	C₄²×C₇₁	—	2	1	no	—
43	C₄²×C₇₁	C₂×C₄	1	2	yes	—
44	C₄²×C₇₁	C₂×C₄	1	2	yes	—
45	C₂³×C₇₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₇₁	—	2	2	no	—
47	C₂×C₄×C₇₁	—	2	2	no	—
48	C₂×C₄×C₇₁	—	2	2	no	—
49	C₂×C₄×C₇₁	—	2	2	no	—
50	C₂×C₄×C₇₁	—	2	2	no	—
51	C₂×C₄×C₇₁	—	2	2	no	—
52	C₂×C₄×C₇₁	—	2	2	no	—
53	C₂×C₄×C₇₁	C₄²	1	2	yes	—
54	C₂×C₄×C₇₁	C₄²	1	2	yes	—
55	C₂×C₄×C₇₁	M₁₆	1	4	yes	—
56	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
61	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
62	C₇₁⋊C₈	—	2	2	no	—
63	C₇₁⋊C₈	—	2	2	no	—
64	C₇₁⋊C₈	—	2	2	no	—
65	C₇₁⋊C₈	C₄⋊C₄	1	4	yes	complemented
66	C₂²×C₇₁	C₂×C₄²	1	1	yes	characteristic
67	C₂²×C₇₁	C₄×D₄	1	1	yes	characteristic
68	C₂²×C₇₁	C₄×Q₈	1	1	yes	characteristic
69	C₂²×C₇₁	C₂×M₁₆	1	2	yes	—
70	C₂²×C₇₁	#131	1	2	yes	—
71	C₄×C₇₁	—	2	2	no	—
72	C₄×C₇₁	—	2	2	no	—
73	C₄×C₇₁	—	2	4	no	—
74	C₄×C₇₁	—	4	2	no	—
75	C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₄×C₇₁	#117	1	1	yes	characteristic
77	C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
78	C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₇₁	C₄×M₁₆	1	1	yes	characteristic
81	C₂×C₇₁	#405	1	1	yes	characteristic
82	C₂×C₇₁	#435	1	2	yes	—
83	C₂×C₇₁	#446	1	2	yes	—
84	#1771	—	71	1	no	maximal, Sylow
85	C₇₁	#1771	1	1	yes	characteristic, complemented, Sylow
86	C₂×C₄⋊C₈	—	71	1	no	—
87	C₂×#97	—	71	1	no	—
88	C₂×#97	—	71	1	no	—
89	C₂×C₄×C₈	—	71	1	no	—
90	C₄×C₄⋊C₄	D₇₁	1	1	yes	characteristic
91	#336	—	71	1	no	—
92	#336	—	71	1	no	—
93	C₂²×C₈	—	71	1	no	—
94	C₂²×C₈	—	71	1	no	—
95	C₂²×C₈	—	71	1	no	—
96	C₂²×C₈	—	71	1	no	—
97	C₂×C₄⋊C₄	Dic₇₁	1	2	yes	—
98	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
99	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
100	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
101	C₄⋊C₈	—	71	4	no	—
102	#97	—	71	4	no	—
103	#97	—	142	2	no	—
104	C₄×C₈	—	142	2	no	—
105	#95	Dic₇₁	1	2	yes	—
106	C₂²×C₄	C₂×Dic₇₁	1	2	yes	—
107	C₂²×C₄	C₂×Dic₇₁	1	1	yes	characteristic
108	C₂²×C₄	C₂×Dic₇₁	1	1	yes	characteristic
109	C₂²×C₄	C₂×Dic₇₁	1	2	yes	—
110	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
111	C₂×C₈	—	71	2	no	—
112	C₂×C₈	—	71	2	no	—
113	C₂×C₈	—	71	2	no	—
114	C₂×C₈	—	71	2	no	—
115	C₂×C₈	—	71	2	no	—
116	C₂×C₈	—	71	2	no	—
117	C₂×C₈	—	142	1	no	—
118	C₂×C₈	—	142	1	no	—
119	C₂×C₈	—	142	1	no	—
120	C₂×C₈	—	142	1	no	—
121	C₂×C₈	—	142	1	no	—
122	C₂×C₈	—	142	1	no	—
123	C₄⋊C₄	—	2	4	no	—
124	C₄²	—	2	1	no	—
125	C₄²	—	2	1	no	—
126	C₄²	—	2	1	no	—
127	C₄²	—	2	1	no	—
128	C₄²	C₄×D₇₁	1	2	yes	—
129	C₄²	C₄×D₇₁	1	2	yes	—
130	C₂³	C₂²×Dic₇₁	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	C₄×Dic₇₁	1	2	yes	—
139	C₂×C₄	C₄×Dic₇₁	1	2	yes	—
140	C₂×C₄	#11765849	1	4	yes	—
141	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
143	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
144	C₂×C₄	Q₈×D₇₁	1	1	yes	characteristic
145	C₂×C₄	#11765864	1	1	yes	characteristic
146	C₂×C₄	#11765866	1	1	yes	characteristic
147	C₈	—	71	4	no	—
148	C₈	—	142	2	no	—
149	C₈	—	142	2	no	—
150	C₈	—	142	2	no	—
151	C₂²	C₂×C₄×Dic₇₁	1	1	yes	characteristic
152	C₂²	D₄×Dic₇₁	1	1	yes	characteristic
153	C₂²	Q₈×Dic₇₁	1	1	yes	characteristic
154	C₂²	C₂×#11765849	1	2	yes	—
155	C₂²	#14556135	1	2	yes	—
156	C₄	—	2	4	no	—
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	4	2	no	—
160	C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
161	C₄	#14556086	1	1	yes	characteristic
162	C₄	#14556090	1	1	yes	characteristic
163	C₄	#14556093	1	1	yes	characteristic
164	C₂	C₄⋊C₄×Dic₇₁	1	1	yes	characteristic
165	C₂	C₄×#11765849	1	1	yes	characteristic
166	C₂	#17985955	1	1	yes	characteristic
167	C₂	#17986071	1	2	yes	—
168	C₂	#17986130	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₇₁⋊C₈)

C₂

yes

maximal, characteristic

C₂×#14556039

C₂

yes

maximal, characteristic

C₂×#14556039

C₂

yes

maximal, characteristic

C₂×C₄×C₇₁⋊C₈

C₂

yes

maximal, characteristic

#17985754

C₂

yes

maximal, characteristic

#17985754

C₂

yes

maximal, characteristic

C₄×C₄⋊C₄×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₄

yes

complemented

#14556039

—

#14556039

C₄

yes

complemented

C₄×C₇₁⋊C₈

—

C₂×C₄⋊C₄×C₇₁

C₄

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₂×C₄²×C₇₁

C₂²

yes

characteristic

#95×C₇₁

C₄

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

D₄

yes

—

C₂×C₇₁⋊C₈

Q₈

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

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

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄²

yes

—

C₂×C₄×C₇₁

C₄²

yes

—

C₂×C₄×C₇₁

M₁₆

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

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

C₄⋊C₄

yes

complemented

C₂²×C₇₁

C₂×C₄²

yes

characteristic

C₂²×C₇₁

C₄×D₄

yes

characteristic

C₂²×C₇₁

C₄×Q₈

yes

characteristic

C₂²×C₇₁

C₂×M₁₆

yes

—

C₂²×C₇₁

#131

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₁

#117

yes

characteristic

C₄×C₇₁

C₄×D₄

yes

characteristic

C₄×C₇₁

C₄×Q₈

yes

characteristic

C₂×C₇₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₇₁

C₄×M₁₆

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₂×C₇₁

#446

yes

—

#1771

—

maximal, Sylow

C₇₁

#1771

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₄×C₄⋊C₄

D₇₁

yes

characteristic

#336

—

#336

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

Dic₇₁

yes

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₂×C₄²

C₂×D₇₁

yes

characteristic

100

C₂×C₄²

C₂×D₇₁

yes

characteristic

101

C₄⋊C₈

—

102

#97

—

103

#97

—

142

—

104

C₄×C₈

—

142

—

105

#95

Dic₇₁

yes

—

106

C₂²×C₄

C₂×Dic₇₁

yes

—

107

C₂²×C₄

C₂×Dic₇₁

yes

characteristic

108

C₂²×C₄

C₂×Dic₇₁

yes

characteristic

109

C₂²×C₄

C₂×Dic₇₁

yes

—

110

C₂²×C₄

C₂²×D₇₁

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

142

—

118

C₂×C₈

—

142

—

119

C₂×C₈

—

142

—

120

C₂×C₈

—

142

—

121

C₂×C₈

—

142

—

122

C₂×C₈

—

142

—

123

C₄⋊C₄

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

C₄×D₇₁

yes

—

129

C₄²

C₄×D₇₁

yes

—

130

C₂³

C₂²×Dic₇₁

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

C₄×Dic₇₁

yes

—

139

C₂×C₄

C₄×Dic₇₁

yes

—

140

C₂×C₄

#11765849

yes

—

141

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

142

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

143

C₂×C₄

D₄×D₇₁

yes

characteristic

144

C₂×C₄

Q₈×D₇₁

yes

characteristic

145

C₂×C₄

#11765864

yes

characteristic

146

C₂×C₄

#11765866

yes

characteristic

147

C₈

—

148

C₈

—

142

—

149

C₈

—

142

—

150

C₈

—

142

—

151

C₂²

C₂×C₄×Dic₇₁

yes

characteristic

152

C₂²

D₄×Dic₇₁

yes

characteristic

153

C₂²

Q₈×Dic₇₁

yes

characteristic

154

C₂²

C₂×#11765849

yes

—

155

C₂²

#14556135

yes

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

C₄⋊C₄×D₇₁

yes

characteristic

161

C₄

#14556086

yes

characteristic

162

C₄

#14556090

yes

characteristic

163

C₄

#14556093

yes

characteristic

164

C₂

C₄⋊C₄×Dic₇₁

yes

characteristic

165

C₂

C₄×#11765849

yes

characteristic

166

C₂

#17985955

yes

characteristic

167

C₂

#17986071

yes

—

168

C₂

#17986130

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₇₁
commutator	C₂²×C₇₁	C₂×C₄²
Fitting	C₄×C₄⋊C₄×C₇₁	C₂
socle	C₂³×C₇₁	C₂²×C₄
hypercenter	C₄×C₄⋊C₄	D₇₁
hypocenter	C₇₁	#1771
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₇₁

commutator

C₂²×C₇₁

C₂×C₄²

Fitting

C₄×C₄⋊C₄×C₇₁

C₂

socle

C₂³×C₇₁

C₂²×C₄

hypercenter

C₄×C₄⋊C₄

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
12. C₄⋊(C₇₁⋊C₈)	C₄	semidirect
14. #14556039	C₄	semidirect
65. C₇₁⋊C₈	C₄⋊C₄	semidirect
85. C₇₁	#1771	semidirect

H ≅ G/N

type

12. C₄⋊(C₇₁⋊C₈)

C₄

semidirect

14. #14556039

C₄

semidirect

65. C₇₁⋊C₈

C₄⋊C₄

semidirect

85. C₇₁

#1771

semidirect

Subgroups — #22588215

Conjugacy classes of subgroups

Complements