Group #31917988 — subgroups

Subgroups — #31917988

Order 18176.

170 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 75 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	M₃₂×Dic₇₁	C₂	1	1	yes	maximal, characteristic
2	#22586537	C₂	1	2	yes	maximal, complemented
3	#22586480	C₂	1	1	yes	maximal, characteristic
4	#22586475	C₂	1	2	yes	maximal, complemented
5	C₇₁×#1400	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556068	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₇₁	C₄	1	1	yes	characteristic
8	#17985833	C₂²	1	2	yes	—
9	#17985774	C₄	1	1	yes	characteristic
10	#17985774	C₂²	1	1	yes	characteristic
11	C₁₆×Dic₇₁	C₄	1	1	yes	characteristic
12	C₁₆×Dic₇₁	C₂²	1	1	yes	characteristic
13	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
14	M₆₄×C₇₁	C₂²	1	2	yes	—
15	C₂²×C₇₁⋊C₈	C₈	1	1	yes	characteristic
16	C₂×C₄×Dic₇₁	C₈	1	1	yes	characteristic
17	#14556068	—	2	2	no	—
18	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₈×Dic₇₁	C₈	1	1	yes	characteristic
21	C₈×Dic₇₁	C₈	1	1	yes	characteristic
22	C₈×Dic₇₁	C₂×C₄	1	1	yes	characteristic
23	C₈×Dic₇₁	C₂×C₄	1	1	yes	characteristic
24	C₇₁⋊C₃₂	—	2	2	no	—
25	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
26	M₃₂×C₇₁	—	2	2	no	—
27	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
29	C₃₂×C₇₁	—	2	2	no	—
30	C₂²×Dic₇₁	—	2	1	no	—
31	C₂×C₇₁⋊C₈	—	2	2	no	—
32	C₂×C₇₁⋊C₈	C₂×C₈	1	1	yes	characteristic
33	C₂×C₇₁⋊C₈	C₂×C₈	1	1	yes	characteristic
34	C₄×Dic₇₁	—	2	1	no	—
35	C₄×Dic₇₁	C₂×C₈	1	1	yes	characteristic
36	C₄×Dic₇₁	C₂×C₈	1	1	yes	characteristic
37	C₇₁⋊C₁₆	—	2	1	no	—
38	C₇₁⋊C₁₆	M₁₆	1	2	yes	—
39	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
40	C₂×C₈×C₇₁	—	2	1	no	—
41	C₂×C₈×C₇₁	—	2	1	no	—
42	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
44	C₁₆×C₇₁	—	2	1	no	—
45	C₁₆×C₇₁	#41	1	2	yes	—
46	C₂³×C₇₁	M₃₂	1	1	yes	characteristic
47	C₂×C₄×C₇₁	—	2	1	no	—
48	C₂×C₄×C₇₁	—	2	1	no	—
49	C₂×C₄×C₇₁	M₃₂	1	1	yes	characteristic
50	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
51	C₂×Dic₇₁	—	2	1	no	—
52	C₂×Dic₇₁	—	2	1	no	—
53	C₂×Dic₇₁	—	4	2	no	—
54	C₈×C₇₁	—	2	1	no	—
55	C₈×C₇₁	#117	1	1	yes	characteristic
56	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
57	C₇₁⋊C₈	—	2	1	no	—
58	C₇₁⋊C₈	M₃₂	1	2	yes	—
59	C₂²×C₇₁	—	2	1	no	—
60	C₂²×C₇₁	—	4	1	no	—
61	C₂²×C₇₁	C₂×M₃₂	1	1	yes	characteristic
62	C₄×C₇₁	—	2	1	no	—
63	C₄×C₇₁	#431	1	1	yes	characteristic
64	C₄×C₇₁	C₂×M₃₂	1	1	yes	characteristic
65	Dic₇₁	—	2	2	no	—
66	Dic₇₁	—	4	1	no	—
67	#13642	—	71	1	no	maximal, Sylow
68	C₂×C₇₁	—	2	1	no	—
69	C₂×C₇₁	—	4	1	no	—
70	C₂×C₇₁	#2163	1	1	yes	characteristic
71	C₄×M₃₂	—	71	1	no	—
72	#1422	—	71	2	no	—
73	#1400	—	71	1	no	—
74	#1400	D₇₁	1	1	yes	characteristic
75	#1398	—	71	2	no	—
76	C₇₁	#13642	1	1	yes	characteristic, complemented, Sylow
77	C₂×M₃₂	—	71	1	no	—
78	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
79	C₂×C₄×C₈	—	71	1	no	—
80	M₆₄	—	71	2	no	—
81	M₆₄	C₂×D₇₁	1	2	yes	—
82	#346	—	71	1	no	—
83	#346	—	71	1	no	—
84	C₄×C₁₆	—	71	1	no	—
85	C₄×C₁₆	—	71	1	no	—
86	C₂²×C₈	—	71	1	no	—
87	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
88	C₂×C₄²	—	71	1	no	—
89	M₃₂	—	2	2	no	—
90	M₃₂	—	142	2	no	—
91	C₂×C₁₆	—	71	1	no	—
92	C₂×C₁₆	—	71	1	no	—
93	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
94	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
95	C₄×C₈	—	71	1	no	—
96	C₄×C₈	—	71	1	no	—
97	C₄×C₈	—	71	1	no	—
98	C₄×C₈	—	71	1	no	—
99	C₃₂	—	2	2	no	—
100	C₃₂	—	142	2	no	—
101	C₂²×C₄	—	142	1	no	—
102	C₂²×C₄	C₈×D₇₁	1	1	yes	characteristic
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	71	1	no	—
106	C₂×C₈	—	71	1	no	—
107	C₂×C₈	—	142	2	no	—
108	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
109	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
110	C₄²	—	71	1	no	—
111	C₄²	—	71	1	no	—
112	C₄²	—	142	1	no	—
113	C₁₆	—	2	1	no	—
114	C₁₆	—	71	2	no	—
115	C₁₆	—	142	1	no	—
116	C₁₆	#11765864	1	2	yes	—
117	C₂³	#14556067	1	1	yes	characteristic
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	284	2	no	—
121	C₂×C₄	—	142	1	no	—
122	C₂×C₄	—	142	1	no	—
123	C₂×C₄	#14556067	1	1	yes	characteristic
124	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	71	2	no	—
127	C₈	—	142	1	no	—
128	C₈	#14556084	1	1	yes	characteristic
129	C₈	M₁₆×D₇₁	1	1	yes	characteristic
130	C₂²	—	2	1	no	—
131	C₂²	—	4	1	no	—
132	C₂²	C₂×#14556067	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	284	1	no	—
135	C₄	—	142	2	no	—
136	C₄	#17986049	1	1	yes	characteristic
137	C₄	M₃₂×D₇₁	1	1	yes	characteristic
138	C₂	—	2	1	no	—
139	C₂	—	4	1	no	—
140	C₂	#22590140	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

M₃₂×Dic₇₁

C₂

yes

maximal, characteristic

#22586537

C₂

yes

maximal, complemented

#22586480

C₂

yes

maximal, characteristic

#22586475

C₂

yes

maximal, complemented

C₇₁×#1400

C₂

yes

maximal, characteristic

C₂×#14556068

C₄

yes

characteristic

C₂×C₈×Dic₇₁

C₄

yes

characteristic

#17985833

C₂²

yes

—

#17985774

C₄

yes

characteristic

#17985774

C₂²

yes

characteristic

C₁₆×Dic₇₁

C₄

yes

characteristic

C₁₆×Dic₇₁

C₂²

yes

characteristic

C₂×M₃₂×C₇₁

C₂²

yes

characteristic

M₆₄×C₇₁

C₂²

yes

—

C₂²×C₇₁⋊C₈

C₈

yes

characteristic

C₂×C₄×Dic₇₁

C₈

yes

characteristic

#14556068

—

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

characteristic

C₈×Dic₇₁

C₈

yes

characteristic

C₈×Dic₇₁

C₈

yes

characteristic

C₈×Dic₇₁

C₂×C₄

yes

characteristic

C₈×Dic₇₁

C₂×C₄

yes

characteristic

C₇₁⋊C₃₂

—

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂³

yes

characteristic

C₃₂×C₇₁

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂×C₈

yes

characteristic

C₂×C₇₁⋊C₈

C₂×C₈

yes

characteristic

C₄×Dic₇₁

—

C₄×Dic₇₁

C₂×C₈

yes

characteristic

C₄×Dic₇₁

C₂×C₈

yes

characteristic

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

M₁₆

yes

—

C₂²×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

#41

yes

—

C₂³×C₇₁

M₃₂

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

M₃₂

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₈

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₈×C₇₁

#117

yes

characteristic

C₈×C₇₁

C₂×M₁₆

yes

characteristic

C₇₁⋊C₈

—

C₇₁⋊C₈

M₃₂

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×M₃₂

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

#431

yes

characteristic

C₄×C₇₁

C₂×M₃₂

yes

characteristic

Dic₇₁

—

Dic₇₁

—

#13642

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#2163

yes

characteristic

C₄×M₃₂

—

#1422

—

#1400

—

#1400

D₇₁

yes

characteristic

#1398

—

C₇₁

#13642

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

C₂×D₇₁

yes

characteristic

C₂×C₄×C₈

—

M₆₄

—

M₆₄

C₂×D₇₁

yes

—

#346

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

C₂²×C₈

—

C₂²×C₈

C₄×D₇₁

yes

characteristic

C₂×C₄²

—

M₃₂

—

M₃₂

—

142

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₇₁

yes

characteristic

C₂×C₁₆

C₂²×D₇₁

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₃₂

—

100

C₃₂

—

142

—

101

C₂²×C₄

—

142

—

102

C₂²×C₄

C₈×D₇₁

yes

characteristic

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

142

—

108

C₂×C₈

C₈×D₇₁

yes

characteristic

109

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

110

C₄²

—

111

C₄²

—

112

C₄²

—

142

—

113

C₁₆

—

114

C₁₆

—

115

C₁₆

—

142

—

116

C₁₆

#11765864

yes

—

117

C₂³

#14556067

yes

characteristic

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

284

—

121

C₂×C₄

—

142

—

122

C₂×C₄

—

142

—

123

C₂×C₄

#14556067

yes

characteristic

124

C₂×C₄

C₂×C₈×D₇₁

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

142

—

128

C₈

#14556084

yes

characteristic

129

C₈

M₁₆×D₇₁

yes

characteristic

130

C₂²

—

131

C₂²

—

132

C₂²

C₂×#14556067

yes

characteristic

133

C₄

—

134

C₄

—

284

—

135

C₄

—

142

—

136

C₄

#17986049

yes

characteristic

137

C₄

M₃₂×D₇₁

yes

characteristic

138

C₂

—

139

C₂

—

140

C₂

#22590140

yes

characteristic

	subgroup	G/H
center	C₈	#14556084
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂×C₁₆	C₂²×D₇₁
Fitting	C₇₁×#1400	C₂
socle	C₂×C₇₁	#2163
prime radical	C₂³×C₇₁	M₃₂
cosocle	C₂×C₁₆×C₇₁	C₂³
hypercenter	#1400	D₇₁
hypocenter	C₇₁	#13642
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ #1400

subgroup

G/H

center

C₈

#14556084

commutator

C₂×C₄×C₇₁

C₂²×C₈

Frattini

C₂×C₁₆

C₂²×D₇₁

Fitting

C₇₁×#1400

C₂

socle

C₂×C₇₁

#2163

prime radical

C₂³×C₇₁

M₃₂

cosocle

C₂×C₁₆×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ #1400

N	H ≅ G/N	type
2. #22586537	C₂	semidirect
4. #22586475	C₂	semidirect
76. C₇₁	#13642	semidirect

H ≅ G/N

type

2. #22586537

C₂

semidirect

4. #22586475

C₂

semidirect

76. C₇₁

#13642

semidirect

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

subgroup	index
M₃₂×Dic₇₁	2
#22586537	2
#22586480	2
#22586475	2
C₇₁×#1400	2
#13642	71

subgroup

index

M₃₂×Dic₇₁

C₇₁×#1400

Subgroups — #31917988

Conjugacy classes of subgroups

Complements

Maximal subgroups