Group #22586187 — subgroups

Subgroups — #22586187

Order 9088.

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

normal subgroups only

This group has 228 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	C₂×C₂²⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₇₁⋊C₈	—	2	1	no	—
5	C₂×C₂²⋊Dic₇₁	C₄	1	2	yes	—
6	C₂×C₄×Dic₇₁	—	2	1	no	—
7	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
8	C₂²⋊(C₇₁⋊C₈)	C₄	1	2	yes	complemented
9	C₇₁⋊#95	—	2	1	no	—
10	C₂³×C₄×C₇₁	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₇₁	C₄	1	2	yes	complemented
12	C₂²⋊C₈×C₇₁	—	2	1	no	—
13	C₂²⋊C₈×C₇₁	—	2	1	no	—
14	C₂²×Dic₇₁	—	2	2	no	—
15	C₂×C₇₁⋊C₈	—	2	2	no	—
16	C₂×C₇₁⋊C₈	—	4	2	no	—
17	C₂²⋊Dic₇₁	—	2	2	no	—
18	C₂²⋊Dic₇₁	C₈	1	4	yes	complemented
19	C₄×Dic₇₁	—	4	2	no	—
20	C₂⁴×C₇₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₇₁	—	4	2	no	—
22	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₇₁	Q₈	1	1	yes	characteristic
28	C₂×C₈×C₇₁	—	2	2	no	—
29	C₂×C₈×C₇₁	—	4	2	no	—
30	C₂³×C₇₁	—	4	1	no	—
31	C₂³×C₇₁	—	4	1	no	—
32	C₂³×C₇₁	C₄²	1	1	yes	characteristic
33	C₂³×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂³×C₇₁	C₄⋊C₄	1	1	yes	characteristic
35	C₂³×C₇₁	C₂×C₈	1	2	yes	—
36	C₂³×C₇₁	M₁₆	1	2	yes	—
37	C₂×C₄×C₇₁	—	2	2	no	—
38	C₂×C₄×C₇₁	—	2	2	no	—
39	C₂×C₄×C₇₁	—	2	2	no	—
40	C₂×C₄×C₇₁	—	4	1	no	—
41	C₂×C₄×C₇₁	—	4	1	no	—
42	C₂×C₄×C₇₁	—	4	1	no	—
43	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
47	C₂×Dic₇₁	—	2	4	no	—
48	C₂×Dic₇₁	—	4	2	no	—
49	C₂×Dic₇₁	—	4	2	no	—
50	C₈×C₇₁	—	4	2	no	—
51	C₇₁⋊C₈	—	4	2	no	—
52	C₂²×C₇₁	—	2	1	no	—
53	C₂²×C₇₁	—	2	1	no	—
54	C₂²×C₇₁	—	2	1	no	—
55	C₂²×C₇₁	—	2	1	no	—
56	C₂²×C₇₁	—	2	1	no	—
57	C₂²×C₇₁	—	2	1	no	—
58	C₂²×C₇₁	—	4	2	no	—
59	C₂²×C₇₁	—	4	2	no	—
60	C₂²×C₇₁	#95	1	1	yes	characteristic
61	C₂²×C₇₁	C₄×C₈	1	1	yes	characteristic
62	C₂²×C₇₁	#97	1	1	yes	characteristic
63	C₂²×C₇₁	C₂²⋊C₈	1	2	yes	—
64	C₂²×C₇₁	C₄⋊C₈	1	2	yes	—
65	C₄×C₇₁	—	2	1	no	—
66	C₄×C₇₁	—	2	1	no	—
67	C₄×C₇₁	—	2	1	no	—
68	C₄×C₇₁	#99	1	2	yes	—
69	Dic₇₁	—	4	4	no	—
70	C₂×C₇₁	—	2	2	no	—
71	C₂×C₇₁	—	2	1	no	—
72	C₂×C₇₁	—	2	1	no	—
73	C₂×C₇₁	—	2	2	no	—
74	C₂×C₇₁	#336	1	1	yes	characteristic
75	C₂×C₇₁	#342	1	1	yes	characteristic
76	C₂×C₇₁	#344	1	1	yes	characteristic
77	#1282	—	71	1	no	maximal, Sylow
78	C₇₁	#1282	1	1	yes	characteristic, complemented, Sylow
79	C₂×C₂²⋊C₈	—	71	1	no	—
80	C₂×C₂²⋊C₈	D₇₁	1	1	yes	characteristic
81	C₄×C₂²⋊C₄	—	71	1	no	—
82	C₂³×C₄	C₂×D₇₁	1	1	yes	characteristic
83	C₂²×C₈	—	142	1	no	—
84	C₂²×C₈	Dic₇₁	1	2	yes	complemented
85	C₂×C₂²⋊C₄	—	71	2	no	—
86	C₂×C₄²	—	142	1	no	—
87	C₂²⋊C₈	—	2	1	no	—
88	C₂²⋊C₈	—	2	1	no	—
89	C₂²⋊C₈	—	71	2	no	—
90	C₂²⋊C₈	—	142	1	no	—
91	#95	—	142	1	no	—
92	C₂⁴	C₄×D₇₁	1	1	yes	characteristic
93	C₂²×C₄	—	4	2	no	—
94	C₂²×C₄	—	142	2	no	—
95	C₂²×C₄	Dic₁₄₂	1	1	yes	characteristic
96	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
97	C₂²×C₄	D₂₈₄	1	1	yes	characteristic
98	C₂²×C₄	C₂×Dic₇₁	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
100	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	4	2	no	—
103	C₂×C₈	—	284	2	no	—
104	C₂×C₈	—	142	2	no	—
105	C₂²⋊C₄	—	71	4	no	—
106	C₂²⋊C₄	—	142	2	no	—
107	C₄²	—	284	2	no	—
108	C₂³	—	4	1	no	—
109	C₂³	—	4	1	no	—
110	C₂³	C₄×Dic₇₁	1	1	yes	characteristic
111	C₂³	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
112	C₂³	Dic₇₁⋊C₄	1	1	yes	characteristic
113	C₂³	C₈×D₇₁	1	2	yes	—
114	C₂³	#11765848	1	2	yes	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	284	2	no	—
122	C₂×C₄	—	284	2	no	—
123	C₂×C₄	—	142	4	no	—
124	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
125	C₂×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
126	C₂×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
127	C₂×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
128	C₈	—	4	2	no	—
129	C₈	—	284	2	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	—	4	2	no	—
137	C₂²	—	4	2	no	—
138	C₂²	C₇₁⋊#95	1	1	yes	characteristic
139	C₂²	C₈×Dic₇₁	1	1	yes	characteristic
140	C₂²	#14556038	1	1	yes	characteristic
141	C₂²	(C₂×D₇₁)⋊C₈	1	2	yes	—
142	C₂²	Dic₇₁⋊C₈	1	2	yes	—
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	284	4	no	—
147	C₄	#14556043	1	2	yes	—
148	C₂	—	2	2	no	—
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	—	2	2	no	—
152	C₂	#17985753	1	1	yes	characteristic
153	C₂	#17985766	1	1	yes	characteristic
154	C₂	#17985770	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊(C₇₁⋊C₈)

C₂

yes

maximal, characteristic

C₄×C₂²⋊Dic₇₁

C₂

yes

maximal, characteristic

C₂×C₂²⋊C₈×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

—

C₂×C₂²⋊Dic₇₁

C₄

yes

—

C₂×C₄×Dic₇₁

—

C₂²⋊(C₇₁⋊C₈)

—

C₂²⋊(C₇₁⋊C₈)

C₄

yes

complemented

C₇₁⋊#95

—

C₂³×C₄×C₇₁

C₂²

yes

characteristic

C₂²×C₈×C₇₁

C₄

yes

complemented

C₂²⋊C₈×C₇₁

—

C₂²⋊C₈×C₇₁

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂²⋊Dic₇₁

—

C₂²⋊Dic₇₁

C₈

yes

complemented

C₄×Dic₇₁

—

C₂⁴×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

Q₈

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

C₄²

yes

characteristic

C₂³×C₇₁

C₂²⋊C₄

yes

characteristic

C₂³×C₇₁

C₄⋊C₄

yes

characteristic

C₂³×C₇₁

C₂×C₈

yes

—

C₂³×C₇₁

M₁₆

yes

—

C₂×C₄×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

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#95

yes

characteristic

C₂²×C₇₁

C₄×C₈

yes

characteristic

C₂²×C₇₁

#97

yes

characteristic

C₂²×C₇₁

C₂²⋊C₈

yes

—

C₂²×C₇₁

C₄⋊C₈

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#99

yes

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

C₂×C₂²⋊C₈

—

C₂×C₂²⋊C₈

D₇₁

yes

characteristic

C₄×C₂²⋊C₄

—

C₂³×C₄

C₂×D₇₁

yes

characteristic

C₂²×C₈

—

142

—

C₂²×C₈

Dic₇₁

yes

complemented

C₂×C₂²⋊C₄

—

C₂×C₄²

—

142

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

142

—

#95

—

142

—

C₂⁴

C₄×D₇₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

142

—

C₂²×C₄

Dic₁₄₂

yes

characteristic

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

D₂₈₄

yes

characteristic

C₂²×C₄

C₂×Dic₇₁

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

100

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

284

—

104

C₂×C₈

—

142

—

105

C₂²⋊C₄

—

106

C₂²⋊C₄

—

142

—

107

C₄²

—

284

—

108

C₂³

—

109

C₂³

—

110

C₂³

C₄×Dic₇₁

yes

characteristic

111

C₂³

(C₂×D₇₁)⋊C₄

yes

characteristic

112

C₂³

Dic₇₁⋊C₄

yes

characteristic

113

C₂³

C₈×D₇₁

yes

—

114

C₂³

#11765848

yes

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

284

—

122

C₂×C₄

—

284

—

123

C₂×C₄

—

142

—

124

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

125

C₂×C₄

C₂²⋊Dic₇₁

yes

characteristic

126

C₂×C₄

Dic₇₁⋊C₄

yes

characteristic

127

C₂×C₄

C₄⋊Dic₇₁

yes

characteristic

128

C₈

—

129

C₈

—

284

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

C₇₁⋊#95

yes

characteristic

139

C₂²

C₈×Dic₇₁

yes

characteristic

140

C₂²

#14556038

yes

characteristic

141

C₂²

(C₂×D₇₁)⋊C₈

yes

—

142

C₂²

Dic₇₁⋊C₈

yes

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

284

—

147

C₄

#14556043

yes

—

148

C₂

—

149

C₂

—

150

C₂

—

151

C₂

—

152

C₂

#17985753

yes

characteristic

153

C₂

#17985766

yes

characteristic

154

C₂

#17985770

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₇₁
commutator	C₂²×C₇₁	C₄×C₈
Frattini	C₂³×C₄	C₂×D₇₁
Fitting	C₂×C₂²⋊C₈×C₇₁	C₂
socle	C₂²×C₇₁	#95
hypercenter	C₂×C₂²⋊C₈	D₇₁
hypocenter	C₇₁	#1282
LCS	G ▷ C₂²×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center

C₂×C₄

C₂²⋊Dic₇₁

commutator

C₂²×C₇₁

C₄×C₈

Frattini

C₂³×C₄

C₂×D₇₁

Fitting

C₂×C₂²⋊C₈×C₇₁

C₂

socle

C₂²×C₇₁

#95

hypercenter

C₂×C₂²⋊C₈

D₇₁

hypocenter

C₇₁

#1282

LCS

G ▷ C₂²×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
8. C₂²⋊(C₇₁⋊C₈)	C₄	semidirect
11. C₂²×C₈×C₇₁	C₄	semidirect
18. C₂²⋊Dic₇₁	C₈	semidirect
78. C₇₁	#1282	semidirect
84. C₂²×C₈	Dic₇₁	semidirect

H ≅ G/N

type

8. C₂²⋊(C₇₁⋊C₈)

C₄

semidirect

11. C₂²×C₈×C₇₁

C₄

semidirect

18. C₂²⋊Dic₇₁

C₈

semidirect

78. C₇₁

#1282

semidirect

84. C₂²×C₈

Dic₇₁

semidirect

Subgroups — #22586187

Conjugacy classes of subgroups

Complements