Group #31906802 — subgroups

Subgroups — #31906802

Order 18176.

288 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 119 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	#22590143	C₂	1	2	yes	maximal
2	#22586282	C₂	1	4	yes	maximal
3	C₇₁×#2960	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₇₁⋊C₁₆)	—	2	2	no	—
5	C₂²⋊(C₇₁⋊C₁₆)	C₂²	1	4	yes	—
6	#17985775	—	2	2	no	—
7	#533×C₇₁	C₄	1	1	yes	characteristic
8	#446×C₇₁	—	2	2	no	—
9	C₈×Q₈×C₇₁	—	2	2	no	—
10	#424×C₇₁	C₂²	1	2	yes	—
11	#423×C₇₁	C₄	1	1	yes	characteristic
12	#405×C₇₁	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₇₁⋊C₁₆	—	2	4	no	—
15	C₂×M₁₆×C₇₁	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
18	C₄⋊Q₈×C₇₁	C₈	1	2	yes	—
19	#125×C₇₁	C₈	1	2	yes	—
20	C₂²⋊Q₈×C₇₁	—	2	2	no	—
21	C₄×Q₈×C₇₁	—	2	2	no	—
22	#117×C₇₁	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
24	C₄⋊C₈×C₇₁	—	2	4	no	—
25	C₄⋊C₈×C₇₁	—	2	2	no	—
26	#97×C₇₁	—	2	2	no	—
27	C₄×C₈×C₇₁	—	2	2	no	—
28	C₄×C₈×C₇₁	D₄	1	2	yes	—
29	C₄×C₈×C₇₁	D₄	1	2	yes	—
30	C₇₁⋊C₁₆	—	4	4	no	—
31	C₂×Q₈×C₇₁	—	2	2	no	—
32	C₂²×C₄×C₇₁	M₁₆	1	2	yes	—
33	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
34	M₁₆×C₇₁	—	4	2	no	—
35	C₂×C₈×C₇₁	—	2	2	no	—
36	C₂×C₈×C₇₁	—	2	2	no	—
37	C₂×C₈×C₇₁	—	4	1	no	—
38	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
41	C₄⋊C₄×C₇₁	—	2	4	no	—
42	C₄⋊C₄×C₇₁	—	2	2	no	—
43	C₄⋊C₄×C₇₁	C₂×C₈	1	4	yes	—
44	C₂²⋊C₄×C₇₁	—	2	2	no	—
45	C₄²×C₇₁	—	2	2	no	—
46	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
47	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
48	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
49	C₂³×C₇₁	C₂×M₁₆	1	1	yes	characteristic
50	Q₈×C₇₁	—	4	4	no	—
51	C₂×C₄×C₇₁	—	2	1	no	—
52	C₂×C₄×C₇₁	—	2	1	no	—
53	C₂×C₄×C₇₁	—	2	4	no	—
54	C₂×C₄×C₇₁	—	4	2	no	—
55	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
56	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
57	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
59	C₈×C₇₁	—	2	2	no	—
60	C₈×C₇₁	—	2	1	no	—
61	C₈×C₇₁	—	4	2	no	—
62	C₈×C₇₁	#101	1	2	yes	—
63	C₂²×C₇₁	—	2	2	no	—
64	C₂²×C₇₁	—	2	1	no	—
65	C₂²×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
66	C₄×C₇₁	—	2	2	no	—
67	C₄×C₇₁	—	2	2	no	—
68	C₄×C₇₁	—	2	1	no	—
69	C₄×C₇₁	—	4	4	no	—
70	C₄×C₇₁	#410	1	1	yes	characteristic
71	C₄×C₇₁	C₂×#101	1	1	yes	characteristic
72	#11497	—	71	1	no	maximal, Sylow
73	C₂×C₇₁	—	4	1	no	—
74	C₂×C₇₁	#1461	1	1	yes	characteristic
75	C₂×C₇₁	#2117	1	2	yes	—
76	#2960	D₇₁	1	1	yes	characteristic
77	#2163	—	71	2	no	—
78	#1323	—	71	4	no	—
79	C₇₁	#11497	1	1	yes	characteristic, complemented, Sylow
80	#533	Dic₇₁	1	1	yes	characteristic
81	#446	—	2	2	no	—
82	C₈×Q₈	—	2	2	no	—
83	#424	C₂×D₇₁	1	2	yes	—
84	#423	Dic₇₁	1	1	yes	characteristic
85	#405	Dic₇₁	1	2	yes	—
86	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
87	C₄⋊C₁₆	—	142	2	no	—
88	C₂²⋊C₁₆	—	71	4	no	—
89	#346	—	142	2	no	—
90	C₂×M₁₆	C₂×Dic₇₁	1	2	yes	—
91	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
92	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
93	C₄⋊Q₈	C₇₁⋊C₈	1	2	yes	—
94	#125	C₇₁⋊C₈	1	2	yes	—
95	C₂²⋊Q₈	—	2	2	no	—
96	C₄×Q₈	—	2	2	no	—
97	#117	C₂×Dic₇₁	1	2	yes	—
98	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
99	C₂×C₁₆	—	142	4	no	—
100	C₄⋊C₈	—	2	4	no	—
101	C₄⋊C₈	—	2	2	no	—
102	#97	—	2	2	no	—
103	C₄×C₈	—	2	2	no	—
104	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
105	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
106	C₂×Q₈	—	2	2	no	—
107	C₂²×C₄	#11765849	1	2	yes	—
108	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
109	M₁₆	—	4	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	4	1	no	—
113	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
114	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
115	C₂×C₈	C₂×C₂²⋊D₇₁	1	2	yes	—
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	C₂×C₇₁⋊C₈	1	4	yes	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₄²	—	2	2	no	—
121	C₄²	C₂²⋊Dic₇₁	1	2	yes	—
122	C₄²	C₂×C₇₁⋊C₈	1	1	yes	characteristic
123	C₄²	C₂×C₇₁⋊C₈	1	1	yes	characteristic
124	C₁₆	—	284	4	no	—
125	C₂³	C₂×#11765849	1	1	yes	characteristic
126	Q₈	—	4	4	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	2	yes	—
132	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	2	yes	—
133	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
134	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	—	2	1	no	—
137	C₈	—	4	2	no	—
138	C₈	#14556047	1	2	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	C₂×C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
142	C₄	—	2	2	no	—
143	C₄	—	2	2	no	—
144	C₄	—	2	1	no	—
145	C₄	—	4	4	no	—
146	C₄	#17985973	1	1	yes	characteristic
147	C₄	C₂×#14556047	1	1	yes	characteristic
148	C₂	—	4	1	no	—
149	C₂	#22586664	1	1	yes	characteristic
150	C₂	#22589968	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22590143

C₂

yes

maximal

#22586282

C₂

yes

maximal

C₇₁×#2960

C₂

yes

maximal, characteristic

C₄⋊(C₇₁⋊C₁₆)

—

C₂²⋊(C₇₁⋊C₁₆)

C₂²

yes

—

#17985775

—

#533×C₇₁

C₄

yes

characteristic

#446×C₇₁

—

C₈×Q₈×C₇₁

—

#424×C₇₁

C₂²

yes

—

#423×C₇₁

C₄

yes

characteristic

#405×C₇₁

C₄

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₇₁⋊C₁₆

—

C₂×M₁₆×C₇₁

C₂×C₄

yes

—

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₄⋊Q₈×C₇₁

C₈

yes

—

#125×C₇₁

C₈

yes

—

C₂²⋊Q₈×C₇₁

—

C₄×Q₈×C₇₁

—

#117×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

—

#97×C₇₁

—

C₄×C₈×C₇₁

—

C₄×C₈×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

D₄

yes

—

C₇₁⋊C₁₆

—

C₂×Q₈×C₇₁

—

C₂²×C₄×C₇₁

M₁₆

yes

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

C₂×C₈

yes

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂²⋊C₄

yes

—

C₄²×C₇₁

C₂×C₈

yes

characteristic

C₄²×C₇₁

C₂×C₈

yes

characteristic

C₂³×C₇₁

C₂×M₁₆

yes

characteristic

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₈

yes

—

C₂×C₄×C₇₁

C₂²⋊C₈

yes

—

C₂×C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₈

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

#101

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×C₂²⋊C₈

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

—

C₂×C₇₁

—

C₂×C₇₁

#1461

yes

characteristic

C₂×C₇₁

#2117

yes

—

#2960

D₇₁

yes

characteristic

#2163

—

#1323

—

C₇₁

#11497

yes

characteristic, complemented, Sylow

#533

Dic₇₁

yes

characteristic

#446

—

C₈×Q₈

—

#424

C₂×D₇₁

yes

—

#423

Dic₇₁

yes

characteristic

#405

Dic₇₁

yes

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

C₄⋊C₁₆

—

142

—

C₂²⋊C₁₆

—

#346

—

142

—

C₂×M₁₆

C₂×Dic₇₁

yes

—

C₂²×C₈

C₂×Dic₇₁

yes

characteristic

C₂²×C₈

C₂²×D₇₁

yes

characteristic

C₄⋊Q₈

C₇₁⋊C₈

yes

—

#125

C₇₁⋊C₈

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×Dic₇₁

yes

—

C₂×C₄²

C₂×Dic₇₁

yes

characteristic

C₂×C₁₆

—

142

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

#97

—

103

C₄×C₈

—

104

C₄×C₈

C₂²⋊D₇₁

yes

—

105

C₄×C₈

C₂²⋊D₇₁

yes

—

106

C₂×Q₈

—

107

C₂²×C₄

#11765849

yes

—

108

C₂²×C₄

C₂²×Dic₇₁

yes

characteristic

109

M₁₆

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₂²⋊Dic₇₁

yes

characteristic

114

C₂×C₈

C₂²⋊Dic₇₁

yes

characteristic

115

C₂×C₈

C₂×C₂²⋊D₇₁

yes

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

C₂×C₇₁⋊C₈

yes

—

119

C₂²⋊C₄

—

120

C₄²

—

121

C₄²

C₂²⋊Dic₇₁

yes

—

122

C₄²

C₂×C₇₁⋊C₈

yes

characteristic

123

C₄²

C₂×C₇₁⋊C₈

yes

characteristic

124

C₁₆

—

284

—

125

C₂³

C₂×#11765849

yes

characteristic

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₂²⋊(C₇₁⋊C₈)

yes

—

132

C₂×C₄

C₂²⋊(C₇₁⋊C₈)

yes

—

133

C₂×C₄

C₂×C₂²⋊Dic₇₁

yes

characteristic

134

C₂×C₄

C₂²×C₇₁⋊C₈

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

#14556047

yes

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂×C₂²⋊(C₇₁⋊C₈)

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

#17985973

yes

characteristic

147

C₄

C₂×#14556047

yes

characteristic

148

C₂

—

149

C₂

#22586664

yes

characteristic

150

C₂

#22589968

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#2960	C₂
socle	C₂²×C₇₁	C₂×C₂²⋊C₈
prime radical	C₂³×C₇₁	C₂×M₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#2960	D₇₁
hypocenter	C₇₁	#11497
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2960

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊Dic₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₇₁×#2960

C₂

socle

C₂²×C₇₁

C₂×C₂²⋊C₈

prime radical

C₂³×C₇₁

C₂×M₁₆

cosocle

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2960

N	H ≅ G/N	type
79. C₇₁	#11497	semidirect

H ≅ G/N

type

79. C₇₁

#11497

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22590143	2
#22586282	2
C₇₁×#2960	2
#11497	71

subgroup

index

#22590143

#22586282

C₇₁×#2960

#11497

Subgroups — #31906802

Conjugacy classes of subgroups

Complements

Maximal subgroups