Group #31913813 — subgroups

Subgroups — #31913813

Order 18176.

342 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 163 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	#22590096	C₂	1	2	yes	maximal
2	#22586422	C₂	1	4	yes	maximal
3	C₄²×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556068	C₂²	1	4	yes	—
5	C₄⋊(C₇₁⋊C₁₆)	—	2	4	no	—
6	C₂×C₄×C₈×C₇₁	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₇₁	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
10	C₄³×C₇₁	C₄	1	1	yes	characteristic
11	#14556068	—	2	8	no	—
12	C₂×C₇₁⋊C₁₆	—	2	4	no	—
13	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
15	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₇₁	—	2	2	no	—
17	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
19	C₄×C₈×C₇₁	—	2	4	no	—
20	C₄×C₈×C₇₁	—	2	4	no	—
21	C₄×C₈×C₇₁	C₈	1	2	yes	—
22	C₄×C₈×C₇₁	C₈	1	2	yes	—
23	C₄×C₈×C₇₁	D₄	1	2	yes	—
24	C₄×C₈×C₇₁	Q₈	1	2	yes	—
25	C₇₁⋊C₁₆	—	4	4	no	—
26	C₂²×C₄×C₇₁	—	2	2	no	—
27	C₂²×C₄×C₇₁	M₁₆	1	2	yes	—
28	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₈×C₇₁	—	2	2	no	—
30	C₂×C₈×C₇₁	—	2	2	no	—
31	C₂×C₈×C₇₁	—	2	2	no	—
32	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
33	C₂×C₈×C₇₁	C₂×C₈	1	4	yes	—
34	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₈×C₇₁	#41	1	4	yes	—
37	C₄²×C₇₁	—	2	8	no	—
38	C₄²×C₇₁	—	2	2	no	—
39	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
40	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
41	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
42	C₄²×C₇₁	M₁₆	1	4	yes	—
43	C₂³×C₇₁	C₂×M₁₆	1	1	yes	characteristic
44	C₂×C₄×C₇₁	—	2	8	no	—
45	C₂×C₄×C₇₁	—	2	4	no	—
46	C₂×C₄×C₇₁	—	2	4	no	—
47	C₂×C₄×C₇₁	C₄⋊C₈	1	2	yes	—
48	C₂×C₄×C₇₁	C₄⋊C₈	1	2	yes	—
49	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₇₁	#117	1	2	yes	—
51	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
52	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
53	C₈×C₇₁	—	2	2	no	—
54	C₈×C₇₁	—	2	4	no	—
55	C₈×C₇₁	#124	1	2	yes	—
56	C₈×C₇₁	#125	1	2	yes	—
57	C₂²×C₇₁	—	2	2	no	—
58	C₂²×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
59	C₂²×C₇₁	#432	1	2	yes	—
60	C₄×C₇₁	—	2	8	no	—
61	C₄×C₇₁	—	2	2	no	—
62	C₄×C₇₁	#364	1	4	yes	—
63	C₄×C₇₁	#382	1	1	yes	characteristic
64	C₄×C₇₁	#423	1	1	yes	characteristic
65	C₄×C₇₁	#431	1	2	yes	—
66	#12883	—	71	1	no	maximal, Sylow
67	C₂×C₇₁	—	2	2	no	—
68	C₂×C₇₁	#1831	1	1	yes	characteristic
69	C₂×C₇₁	C₂×#364	1	2	yes	—
70	#2150	—	71	2	no	—
71	C₄²×C₈	D₇₁	1	1	yes	characteristic
72	#1376	—	71	4	no	—
73	C₇₁	#12883	1	1	yes	characteristic, complemented, Sylow
74	C₂×M₃₂	—	71	4	no	—
75	C₂×C₄×C₈	Dic₇₁	1	2	yes	—
76	C₂×C₄×C₈	Dic₇₁	1	1	yes	characteristic
77	C₂×C₄×C₈	C₂×D₇₁	1	2	yes	—
78	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
79	C₄³	Dic₇₁	1	1	yes	characteristic
80	C₄⋊C₁₆	—	142	4	no	—
81	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
82	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
83	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
84	C₂×C₄²	—	2	2	no	—
85	C₂×C₄²	C₂×Dic₇₁	1	2	yes	—
86	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
87	M₃₂	—	142	8	no	—
88	C₂×C₁₆	—	142	4	no	—
89	C₄×C₈	—	2	4	no	—
90	C₄×C₈	—	2	4	no	—
91	C₄×C₈	C₇₁⋊C₈	1	2	yes	—
92	C₄×C₈	C₇₁⋊C₈	1	2	yes	—
93	C₄×C₈	Dic₁₄₂	1	2	yes	—
94	C₄×C₈	D₂₈₄	1	2	yes	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	#11765849	1	2	yes	—
97	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
102	C₂×C₈	C₂×C₇₁⋊C₈	1	4	yes	—
103	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
104	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
105	C₂×C₈	#11765865	1	4	yes	—
106	C₄²	—	2	8	no	—
107	C₄²	—	2	2	no	—
108	C₄²	C₄⋊Dic₇₁	1	2	yes	—
109	C₄²	C₂×C₇₁⋊C₈	1	1	yes	characteristic
110	C₄²	C₂×C₇₁⋊C₈	1	1	yes	characteristic
111	C₄²	#11765849	1	4	yes	—
112	C₁₆	—	284	4	no	—
113	C₂³	C₂×#11765849	1	1	yes	characteristic
114	C₂×C₄	—	2	8	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
118	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
119	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
120	C₂×C₄	#14556087	1	2	yes	—
121	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
122	C₂×C₄	C₂×#11765849	1	2	yes	—
123	C₈	—	2	4	no	—
124	C₈	—	2	2	no	—
125	C₈	#14556116	1	2	yes	—
126	C₈	#14556120	1	2	yes	—
127	C₂²	—	2	2	no	—
128	C₂²	C₂×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
129	C₂²	#17986056	1	2	yes	—
130	C₄	—	2	8	no	—
131	C₄	—	2	2	no	—
132	C₄	#17985820	1	4	yes	—
133	C₄	#17985865	1	1	yes	characteristic
134	C₄	#17986023	1	1	yes	characteristic
135	C₄	#17986052	1	2	yes	—
136	C₂	—	2	2	no	—
137	C₂	#22588457	1	1	yes	characteristic
138	C₂	C₂×#17985820	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22590096

C₂

yes

maximal

#22586422

C₂

yes

maximal

C₄²×C₈×C₇₁

C₂

yes

maximal, characteristic

C₂×#14556068

C₂²

yes

—

C₄⋊(C₇₁⋊C₁₆)

—

C₂×C₄×C₈×C₇₁

C₄

yes

—

C₂×C₄×C₈×C₇₁

C₄

yes

characteristic

C₂×C₄×C₈×C₇₁

C₂²

yes

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

C₄³×C₇₁

C₄

yes

characteristic

#14556068

—

C₂×C₇₁⋊C₁₆

—

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₁

C₂×C₄

yes

—

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄²×C₇₁

—

C₂×C₄²×C₇₁

C₂×C₄

yes

—

C₂×C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄×C₈×C₇₁

—

C₄×C₈×C₇₁

—

C₄×C₈×C₇₁

C₈

yes

—

C₄×C₈×C₇₁

C₈

yes

—

C₄×C₈×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

Q₈

yes

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

M₁₆

yes

—

C₂²×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

C₂×C₈

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₁

#41

yes

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

C₄⋊C₄

yes

—

C₄²×C₇₁

C₂×C₈

yes

characteristic

C₄²×C₇₁

C₂×C₈

yes

characteristic

C₄²×C₇₁

M₁₆

yes

—

C₂³×C₇₁

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄⋊C₈

yes

—

C₂×C₄×C₇₁

C₄⋊C₈

yes

—

C₂×C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

#117

yes

—

C₂×C₄×C₇₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₁

C₂×M₁₆

yes

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

#124

yes

—

C₈×C₇₁

#125

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₇₁

#432

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#364

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

#12883

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

#1831

yes

characteristic

C₂×C₇₁

C₂×#364

yes

—

#2150

—

C₄²×C₈

D₇₁

yes

characteristic

#1376

—

C₇₁

#12883

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×C₄×C₈

Dic₇₁

yes

—

C₂×C₄×C₈

Dic₇₁

yes

characteristic

C₂×C₄×C₈

C₂×D₇₁

yes

—

C₂×C₄×C₈

C₂×D₇₁

yes

characteristic

C₄³

Dic₇₁

yes

characteristic

C₄⋊C₁₆

—

142

—

C₂²×C₈

C₂×Dic₇₁

yes

characteristic

C₂²×C₈

C₂×Dic₇₁

yes

—

C₂²×C₈

C₂²×D₇₁

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×Dic₇₁

yes

—

C₂×C₄²

C₂×Dic₇₁

yes

characteristic

M₃₂

—

142

—

C₂×C₁₆

—

142

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₇₁⋊C₈

yes

—

C₄×C₈

C₇₁⋊C₈

yes

—

C₄×C₈

Dic₁₄₂

yes

—

C₄×C₈

D₂₈₄

yes

—

C₂²×C₄

—

C₂²×C₄

#11765849

yes

—

C₂²×C₄

C₂²×Dic₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₄⋊Dic₇₁

yes

—

102

C₂×C₈

C₂×C₇₁⋊C₈

yes

—

103

C₂×C₈

C₂×D₂₈₄

yes

characteristic

104

C₂×C₈

C₂×Dic₁₄₂

yes

characteristic

105

C₂×C₈

#11765865

yes

—

106

C₄²

—

107

C₄²

—

108

C₄²

C₄⋊Dic₇₁

yes

—

109

C₄²

C₂×C₇₁⋊C₈

yes

characteristic

110

C₄²

C₂×C₇₁⋊C₈

yes

characteristic

111

C₄²

#11765849

yes

—

112

C₁₆

—

284

—

113

C₂³

C₂×#11765849

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₄⋊(C₇₁⋊C₈)

yes

—

118

C₂×C₄

C₄⋊(C₇₁⋊C₈)

yes

—

119

C₂×C₄

C₂×C₄⋊Dic₇₁

yes

characteristic

120

C₂×C₄

#14556087

yes

—

121

C₂×C₄

C₂²×C₇₁⋊C₈

yes

characteristic

122

C₂×C₄

C₂×#11765849

yes

—

123

C₈

—

124

C₈

—

125

C₈

#14556116

yes

—

126

C₈

#14556120

yes

—

127

C₂²

—

128

C₂²

C₂×C₄⋊(C₇₁⋊C₈)

yes

characteristic

129

C₂²

#17986056

yes

—

130

C₄

—

131

C₄

—

132

C₄

#17985820

yes

—

133

C₄

#17985865

yes

characteristic

134

C₄

#17986023

yes

characteristic

135

C₄

#17986052

yes

—

136

C₂

—

137

C₂

#22588457

yes

characteristic

138

C₂

C₂×#17985820

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₄²×C₈×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₄⋊C₈
prime radical	C₂³×C₇₁	C₂×M₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₄²×C₈	D₇₁
hypocenter	C₇₁	#12883
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

subgroup

G/H

center

C₂×C₈

C₂×D₂₈₄

commutator

C₂×C₄×C₇₁

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₄²×C₈×C₇₁

C₂

socle

C₂²×C₇₁

C₂×C₄⋊C₈

prime radical

C₂³×C₇₁

C₂×M₁₆

cosocle

C₂²×C₈×C₇₁

C₂³

hypercenter

C₄²×C₈

D₇₁

hypocenter

C₇₁

#12883

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

N	H ≅ G/N	type
73. C₇₁	#12883	semidirect

H ≅ G/N

type

73. C₇₁

#12883

semidirect

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

subgroup	index
#22590096	2
#22586422	2
C₄²×C₈×C₇₁	2
#12883	71

subgroup

index

#22590096

#22586422

C₄²×C₈×C₇₁

#12883

Subgroups — #31913813

Conjugacy classes of subgroups

Complements

Maximal subgroups