Group #22588116 — subgroups

Subgroups — #22588116

Order 9088.

360 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 157 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	C₂×#14556038	C₂	1	2	yes	maximal, complemented
3	C₄×C₂²⋊Dic₇₁	C₂	1	1	yes	maximal, characteristic
4	#17985753	C₂	1	2	yes	maximal, complemented
5	C₂³×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₈	C₂²	1	2	yes	—
7	C₂×C₂²⋊Dic₇₁	C₄	1	2	yes	—
8	C₂×C₄×Dic₇₁	C₂²	1	2	yes	—
9	C₂²⋊(C₇₁⋊C₈)	C₄	1	4	yes	complemented
10	#14556038	—	2	4	no	—
11	C₇₁⋊#95	C₄	1	2	yes	—
12	C₂³×C₄×C₇₁	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₇₁	—	2	4	no	—
14	C₂²×C₈×C₇₁	C₄	1	4	yes	complemented
15	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₂²×Dic₇₁	C₂×C₄	1	4	yes	—
18	C₂×C₇₁⋊C₈	—	2	2	no	—
19	C₂×C₇₁⋊C₈	—	2	2	no	—
20	C₂×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
21	C₂²⋊Dic₇₁	—	2	4	no	—
22	C₄×Dic₇₁	—	2	2	no	—
23	C₄×Dic₇₁	—	2	2	no	—
24	C₂⁴×C₇₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₇₁	—	2	2	no	—
26	C₂²×C₄×C₇₁	—	2	2	no	—
27	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₇₁	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₇₁	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₇₁	—	2	2	no	—
32	C₂×C₈×C₇₁	—	2	8	no	—
33	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
34	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
35	C₂×C₈×C₇₁	D₄	1	4	yes	—
36	C₂³×C₇₁	—	2	2	no	—
37	C₂³×C₇₁	—	2	2	no	—
38	C₂³×C₇₁	C₄²	1	2	yes	—
39	C₂³×C₇₁	M₁₆	1	4	yes	—
40	C₂³×C₇₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₇₁	—	2	2	no	—
42	C₂×C₄×C₇₁	—	2	2	no	—
43	C₂×C₄×C₇₁	—	2	2	no	—
44	C₂×C₄×C₇₁	—	2	4	no	—
45	C₂×C₄×C₇₁	C₄²	1	2	yes	—
46	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₇₁	#41	1	2	yes	—
50	C₂×Dic₇₁	—	2	4	no	—
51	C₂×Dic₇₁	—	2	4	no	—
52	C₂×Dic₇₁	—	2	4	no	—
53	C₈×C₇₁	—	2	2	no	—
54	C₈×C₇₁	C₂²⋊C₄	1	4	yes	complemented
55	C₇₁⋊C₈	—	2	4	no	—
56	C₂²×C₇₁	—	2	2	no	—
57	C₂²×C₇₁	—	2	2	no	—
58	C₂²×C₇₁	—	2	8	no	—
59	C₂²×C₇₁	#97	1	4	yes	complemented
60	C₂²×C₇₁	C₂×C₄²	1	1	yes	characteristic
61	C₂²×C₇₁	C₄×D₄	1	2	yes	—
62	C₂²×C₇₁	C₂×M₁₆	1	2	yes	—
63	C₂²×C₇₁	#131	1	2	yes	—
64	C₄×C₇₁	—	2	2	no	—
65	C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₄×C₇₁	#117	1	1	yes	characteristic
67	C₄×C₇₁	C₄×D₄	1	2	yes	—
68	Dic₇₁	—	4	4	no	—
69	C₂×C₇₁	—	2	4	no	—
70	C₂×C₇₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₇₁	C₂×#97	1	1	yes	characteristic
72	C₂×C₇₁	#405	1	1	yes	characteristic
73	C₂×C₇₁	#435	1	4	yes	—
74	#1752	—	71	1	no	maximal, Sylow
75	C₇₁	#1752	1	1	yes	characteristic, complemented, Sylow
76	C₂³×C₈	D₇₁	1	1	yes	characteristic
77	C₂×C₂²⋊C₈	—	71	1	no	—
78	C₂×#97	—	71	2	no	—
79	C₄×C₂²⋊C₄	—	71	1	no	—
80	#336	—	71	2	no	—
81	C₂³×C₄	C₂×D₇₁	1	1	yes	characteristic
82	C₂²×C₈	—	2	4	no	—
83	C₂²×C₈	—	71	2	no	—
84	C₂²×C₈	Dic₇₁	1	4	yes	complemented
85	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
86	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
87	C₂×C₂²⋊C₄	—	71	2	no	—
88	C₂×C₄²	—	71	2	no	—
89	C₂²⋊C₈	—	71	4	no	—
90	#97	—	142	4	no	—
91	#95	—	71	2	no	—
92	C₂⁴	C₄×D₇₁	1	1	yes	characteristic
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	71	4	no	—
96	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
97	C₂²×C₄	C₄×D₇₁	1	2	yes	—
98	C₂²×C₄	C₂×Dic₇₁	1	2	yes	—
99	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	8	no	—
102	C₂×C₈	—	71	4	no	—
103	C₂×C₈	—	142	2	no	—
104	C₂×C₈	—	142	2	no	—
105	C₂×C₈	C₂×Dic₇₁	1	2	yes	—
106	C₂×C₈	C₂×Dic₇₁	1	2	yes	—
107	C₂×C₈	C₂²⋊D₇₁	1	4	yes	—
108	C₂²⋊C₄	—	142	4	no	—
109	C₄²	—	142	2	no	—
110	C₄²	—	142	2	no	—
111	C₂³	—	2	2	no	—
112	C₂³	—	2	2	no	—
113	C₂³	C₄×Dic₇₁	1	2	yes	—
114	C₂³	#11765848	1	4	yes	—
115	C₂³	C₂×C₄×D₇₁	1	1	yes	characteristic
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	142	4	no	—
121	C₂×C₄	—	142	4	no	—
122	C₂×C₄	—	142	4	no	—
123	C₂×C₄	C₄×Dic₇₁	1	2	yes	—
124	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
125	C₂×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
126	C₂×C₄	C₂×C₂²⋊D₇₁	1	2	yes	—
127	C₂×C₄	#11765865	1	2	yes	—
128	C₈	—	2	2	no	—
129	C₈	—	142	4	no	—
130	C₈	C₂²⋊Dic₇₁	1	4	yes	complemented
131	C₂²	—	2	8	no	—
132	C₂²	—	2	2	no	—
133	C₂²	—	2	2	no	—
134	C₂²	#14556038	1	4	yes	complemented
135	C₂²	C₂×C₄×Dic₇₁	1	1	yes	characteristic
136	C₂²	C₄×C₂²⋊D₇₁	1	2	yes	—
137	C₂²	C₂×#11765848	1	2	yes	—
138	C₂²	#14556134	1	2	yes	—
139	C₄	—	2	2	no	—
140	C₄	—	284	4	no	—
141	C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
142	C₄	#14556087	1	1	yes	characteristic
143	C₄	C₄×C₂²⋊D₇₁	1	2	yes	—
144	C₂	—	2	4	no	—
145	C₂	C₄×C₂²⋊Dic₇₁	1	1	yes	characteristic
146	C₂	C₂×#14556038	1	1	yes	characteristic
147	C₂	#17985954	1	1	yes	characteristic
148	C₂	#17986070	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊(C₇₁⋊C₈)

C₂

yes

maximal, characteristic

C₂×#14556038

C₂

yes

maximal, complemented

C₄×C₂²⋊Dic₇₁

C₂

yes

maximal, characteristic

#17985753

C₂

yes

maximal, complemented

C₂³×C₈×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

—

C₂×C₂²⋊Dic₇₁

C₄

yes

—

C₂×C₄×Dic₇₁

C₂²

yes

—

C₂²⋊(C₇₁⋊C₈)

C₄

yes

complemented

#14556038

—

C₇₁⋊#95

C₄

yes

—

C₂³×C₄×C₇₁

C₂²

yes

characteristic

C₂²×C₈×C₇₁

—

C₂²×C₈×C₇₁

C₄

yes

complemented

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂²×Dic₇₁

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₂⁴×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₈×C₇₁

D₄

yes

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

C₄²

yes

—

C₂³×C₇₁

M₁₆

yes

—

C₂³×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

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

—

C₂×C₄×C₇₁

#41

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₈×C₇₁

C₂²⋊C₄

yes

complemented

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#97

yes

complemented

C₂²×C₇₁

C₂×C₄²

yes

characteristic

C₂²×C₇₁

C₄×D₄

yes

—

C₂²×C₇₁

C₂×M₁₆

yes

—

C₂²×C₇₁

#131

yes

—

C₄×C₇₁

—

C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₁

#117

yes

characteristic

C₄×C₇₁

C₄×D₄

yes

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

C₄×C₂²⋊C₄

yes

characteristic

C₂×C₇₁

C₂×#97

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

#1752

—

maximal, Sylow

C₇₁

#1752

yes

characteristic, complemented, Sylow

C₂³×C₈

D₇₁

yes

characteristic

C₂×C₂²⋊C₈

—

C₂×#97

—

C₄×C₂²⋊C₄

—

#336

—

C₂³×C₄

C₂×D₇₁

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

Dic₇₁

yes

complemented

C₂²×C₈

C₂×D₇₁

yes

characteristic

C₂²×C₈

C₂×D₇₁

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂²⋊C₈

—

#97

—

142

—

#95

—

C₂⁴

C₄×D₇₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

C₄×D₇₁

yes

—

C₂²×C₄

C₂×Dic₇₁

yes

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

142

—

104

C₂×C₈

—

142

—

105

C₂×C₈

C₂×Dic₇₁

yes

—

106

C₂×C₈

C₂×Dic₇₁

yes

—

107

C₂×C₈

C₂²⋊D₇₁

yes

—

108

C₂²⋊C₄

—

142

—

109

C₄²

—

142

—

110

C₄²

—

142

—

111

C₂³

—

112

C₂³

—

113

C₂³

C₄×Dic₇₁

yes

—

114

C₂³

#11765848

yes

—

115

C₂³

C₂×C₄×D₇₁

yes

characteristic

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

142

—

121

C₂×C₄

—

142

—

122

C₂×C₄

—

142

—

123

C₂×C₄

C₄×Dic₇₁

yes

—

124

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

125

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

126

C₂×C₄

C₂×C₂²⋊D₇₁

yes

—

127

C₂×C₄

#11765865

yes

—

128

C₈

—

129

C₈

—

142

—

130

C₈

C₂²⋊Dic₇₁

yes

complemented

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

#14556038

yes

complemented

135

C₂²

C₂×C₄×Dic₇₁

yes

characteristic

136

C₂²

C₄×C₂²⋊D₇₁

yes

—

137

C₂²

C₂×#11765848

yes

—

138

C₂²

#14556134

yes

—

139

C₄

—

140

C₄

—

284

—

141

C₄

C₂×C₂²⋊Dic₇₁

yes

characteristic

142

C₄

#14556087

yes

characteristic

143

C₄

C₄×C₂²⋊D₇₁

yes

—

144

C₂

—

145

C₂

C₄×C₂²⋊Dic₇₁

yes

characteristic

146

C₂

C₂×#14556038

yes

characteristic

147

C₂

#17985954

yes

characteristic

148

C₂

#17986070

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₇₁
commutator	C₂²×C₇₁	C₂×C₄²
Fitting	C₂³×C₈×C₇₁	C₂
socle	C₂³×C₇₁	C₂²×C₄
hypercenter	C₂³×C₈	D₇₁
hypocenter	C₇₁	#1752
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₇₁

commutator

C₂²×C₇₁

C₂×C₄²

Fitting

C₂³×C₈×C₇₁

C₂

socle

C₂³×C₇₁

C₂²×C₄

hypercenter

C₂³×C₈

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂²×C₄ ◁ C₂³×C₈

N	H ≅ G/N	type
2. C₂×#14556038	C₂	semidirect
4. #17985753	C₂	semidirect
9. C₂²⋊(C₇₁⋊C₈)	C₄	semidirect
14. C₂²×C₈×C₇₁	C₄	semidirect
54. C₈×C₇₁	C₂²⋊C₄	semidirect
59. C₂²×C₇₁	#97	semidirect
75. C₇₁	#1752	semidirect
84. C₂²×C₈	Dic₇₁	semidirect
130. C₈	C₂²⋊Dic₇₁	semidirect
134. C₂²	#14556038	semidirect

H ≅ G/N

type

2. C₂×#14556038

C₂

semidirect

4. #17985753

C₂

semidirect

9. C₂²⋊(C₇₁⋊C₈)

C₄

semidirect

14. C₂²×C₈×C₇₁

C₄

semidirect

54. C₈×C₇₁

C₂²⋊C₄

semidirect

59. C₂²×C₇₁

#97

semidirect

75. C₇₁

#1752

semidirect

84. C₂²×C₈

Dic₇₁

semidirect

130. C₈

C₂²⋊Dic₇₁

semidirect

134. C₂²

#14556038

semidirect

Subgroups — #22588116

Conjugacy classes of subgroups

Complements