Group #31918011 — subgroups

Subgroups — #31918011

Order 18176.

202 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 113 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×#17985833	C₂	1	2	yes	maximal, complemented
2	C₄⋊(C₇₁⋊C₃₂)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊(C₇₁⋊C₃₂)	C₂	1	2	yes	maximal, complemented
4	C₄×C₇₁⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
6	#17985833	—	2	4	no	—
7	C₂×C₇₁⋊C₃₂	C₂²	1	1	yes	characteristic
8	C₂×C₇₁⋊C₃₂	C₂²	1	2	yes	—
9	C₂×C₇₁⋊C₃₂	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
11	C₈×D₄×C₇₁	C₄	1	1	yes	characteristic
12	C₄⋊C₁₆×C₇₁	C₄	1	1	yes	characteristic
13	C₂²⋊C₁₆×C₇₁	C₄	1	2	yes	—
14	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
15	C₇₁⋊C₃₂	—	2	2	no	—
16	C₇₁⋊C₃₂	—	2	1	no	—
17	C₇₁⋊C₃₂	D₄	1	2	yes	complemented
18	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
19	C₄×D₄×C₇₁	C₈	1	1	yes	characteristic
20	C₂×C₁₆×C₇₁	—	2	2	no	—
21	C₂×C₁₆×C₇₁	—	2	2	no	—
22	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
25	C₄⋊C₈×C₇₁	C₈	1	1	yes	characteristic
26	C₂²⋊C₈×C₇₁	C₈	1	2	yes	—
27	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₂×D₄×C₇₁	C₁₆	1	1	yes	characteristic
29	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
30	C₂×C₈×C₇₁	—	2	2	no	—
31	C₂×C₈×C₇₁	—	2	2	no	—
32	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
34	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
35	C₄⋊C₄×C₇₁	C₁₆	1	1	yes	characteristic
36	C₂²⋊C₄×C₇₁	C₁₆	1	2	yes	—
37	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
38	C₁₆×C₇₁	—	2	2	no	—
39	C₁₆×C₇₁	—	2	1	no	—
40	C₁₆×C₇₁	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₇₁	#41	1	1	yes	characteristic
42	C₂³×C₇₁	C₂×C₁₆	1	2	yes	—
43	D₄×C₇₁	—	2	2	no	—
44	C₂×C₄×C₇₁	—	2	2	no	—
45	C₂×C₄×C₇₁	—	2	2	no	—
46	C₂×C₄×C₇₁	C₂×C₁₆	1	1	yes	characteristic
47	C₂×C₄×C₇₁	C₂×C₁₆	1	2	yes	—
48	C₂×C₄×C₇₁	C₂×C₁₆	1	1	yes	characteristic
49	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
50	C₈×C₇₁	—	2	2	no	—
51	C₈×C₇₁	—	2	1	no	—
52	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
53	C₈×C₇₁	#131	1	1	yes	characteristic
54	C₂²×C₇₁	—	2	2	no	—
55	C₂²×C₇₁	—	2	2	no	—
56	C₂²×C₇₁	—	2	2	no	—
57	C₂²×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
58	C₄×C₇₁	—	2	2	no	—
59	C₄×C₇₁	—	2	1	no	—
60	C₄×C₇₁	M₆₄	1	2	yes	complemented
61	C₄×C₇₁	C₈×D₄	1	1	yes	characteristic
62	C₄×C₇₁	#504	1	1	yes	characteristic
63	#13646	—	71	1	no	maximal, Sylow
64	C₂×C₇₁	—	4	2	no	—
65	C₂×C₇₁	D₄×C₁₆	1	1	yes	characteristic
66	C₂×C₇₁	C₂×M₆₄	1	1	yes	characteristic
67	C₂×C₇₁	#2258	1	1	yes	characteristic
68	C₂×M₆₄	—	71	2	no	—
69	D₄×C₁₆	D₇₁	1	1	yes	characteristic
70	C₄⋊C₃₂	—	71	1	no	—
71	C₂²⋊C₃₂	—	71	2	no	—
72	C₄×C₃₂	—	71	1	no	—
73	C₇₁	#13646	1	1	yes	characteristic, complemented, Sylow
74	C₂²×C₁₆	C₂×D₇₁	1	2	yes	—
75	C₈×D₄	Dic₇₁	1	1	yes	characteristic
76	M₆₄	—	142	4	no	—
77	C₂×C₃₂	—	71	1	no	—
78	C₂×C₃₂	—	71	1	no	—
79	C₂×C₃₂	—	71	2	no	—
80	C₄⋊C₁₆	Dic₇₁	1	1	yes	characteristic
81	C₂²⋊C₁₆	Dic₇₁	1	2	yes	—
82	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
83	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
84	C₄×D₄	C₇₁⋊C₈	1	1	yes	characteristic
85	C₂×C₁₆	—	2	2	no	—
86	C₂×C₁₆	—	2	2	no	—
87	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
88	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
89	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
90	C₄⋊C₈	C₇₁⋊C₈	1	1	yes	characteristic
91	C₂²⋊C₈	C₇₁⋊C₈	1	2	yes	—
92	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
93	C₃₂	—	71	2	no	—
94	C₃₂	—	142	2	no	—
95	C₃₂	—	142	1	no	—
96	C₂×D₄	C₇₁⋊C₁₆	1	1	yes	characteristic
97	C₂²×C₄	C₂×C₇₁⋊C₈	1	2	yes	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	C₂×C₇₁⋊C₈	1	1	yes	characteristic
101	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
102	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
103	C₄⋊C₄	C₇₁⋊C₁₆	1	1	yes	characteristic
104	C₂²⋊C₄	C₇₁⋊C₁₆	1	2	yes	—
105	C₄²	C₂×C₇₁⋊C₈	1	1	yes	characteristic
106	C₁₆	—	2	2	no	—
107	C₁₆	—	2	1	no	—
108	C₁₆	D₄×D₇₁	1	1	yes	characteristic
109	C₁₆	#11765864	1	1	yes	characteristic
110	C₂³	C₂×C₇₁⋊C₁₆	1	2	yes	—
111	D₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	C₂×C₇₁⋊C₁₆	1	1	yes	characteristic
115	C₂×C₄	C₂×C₇₁⋊C₁₆	1	2	yes	—
116	C₂×C₄	C₂×C₇₁⋊C₁₆	1	1	yes	characteristic
117	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
118	C₈	—	2	2	no	—
119	C₈	—	2	1	no	—
120	C₈	D₄×Dic₇₁	1	1	yes	characteristic
121	C₈	#14556135	1	1	yes	characteristic
122	C₂²	—	2	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	—	2	2	no	—
125	C₂²	C₂²×C₇₁⋊C₁₆	1	1	yes	characteristic
126	C₄	—	2	1	no	—
127	C₄	—	2	2	no	—
128	C₄	#17985833	1	2	yes	complemented
129	C₄	D₄×C₇₁⋊C₈	1	1	yes	characteristic
130	C₄	#17986416	1	1	yes	characteristic
131	C₂	—	4	2	no	—
132	C₂	D₄×C₇₁⋊C₁₆	1	1	yes	characteristic
133	C₂	C₂×#17985833	1	1	yes	characteristic
134	C₂	#22590634	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17985833

C₂

yes

maximal, complemented

C₄⋊(C₇₁⋊C₃₂)

C₂

yes

maximal, characteristic, complemented

C₂²⋊(C₇₁⋊C₃₂)

C₂

yes

maximal, complemented

C₄×C₇₁⋊C₃₂

C₂

yes

maximal, characteristic, complemented

D₄×C₁₆×C₇₁

C₂

yes

maximal, characteristic

#17985833

—

C₂×C₇₁⋊C₃₂

C₂²

yes

characteristic

C₂×C₇₁⋊C₃₂

C₂²

yes

—

C₂×C₇₁⋊C₃₂

C₂²

yes

characteristic

C₂²×C₁₆×C₇₁

C₂²

yes

—

C₈×D₄×C₇₁

C₄

yes

characteristic

C₄⋊C₁₆×C₇₁

C₄

yes

characteristic

C₂²⋊C₁₆×C₇₁

C₄

yes

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₇₁⋊C₃₂

—

C₇₁⋊C₃₂

—

C₇₁⋊C₃₂

D₄

yes

complemented

C₂²×C₈×C₇₁

C₂×C₄

yes

—

C₄×D₄×C₇₁

C₈

yes

characteristic

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂³

yes

characteristic

C₄⋊C₈×C₇₁

C₈

yes

characteristic

C₂²⋊C₈×C₇₁

C₈

yes

—

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×D₄×C₇₁

C₁₆

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₈

yes

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

C₂×C₈

yes

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₇₁

C₁₆

yes

characteristic

C₂²⋊C₄×C₇₁

C₁₆

yes

—

C₄²×C₇₁

C₂×C₈

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

—

C₁₆×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

#41

yes

characteristic

C₂³×C₇₁

C₂×C₁₆

yes

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×C₁₆

yes

—

C₂×C₄×C₇₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₈

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

C₄×D₄

yes

characteristic

C₈×C₇₁

#131

yes

characteristic

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂²×C₁₆

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

M₆₄

yes

complemented

C₄×C₇₁

C₈×D₄

yes

C₄×C₇₁

yes

—

C₂×C₇₁

—

C₂×C₇₁

D₄×C₁₆

yes

characteristic

C₂×C₇₁

C₂×M₆₄

yes

characteristic

C₂×C₇₁

#2258

yes

characteristic

C₂×M₆₄

—

D₄×C₁₆

D₇₁

yes

characteristic

C₄⋊C₃₂

—

C₂²⋊C₃₂

—

C₄×C₃₂

—

C₇₁

#13646

yes

characteristic, complemented, Sylow

C₂²×C₁₆

C₂×D₇₁

yes

—

C₈×D₄

Dic₇₁

yes

characteristic

M₆₄

—

142

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₄⋊C₁₆

Dic₇₁

yes

characteristic

C₂²⋊C₁₆

Dic₇₁

yes

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

C₂²×C₈

C₂×Dic₇₁

yes

—

C₄×D₄

C₇₁⋊C₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₇₁

yes

characteristic

C₂×C₁₆

C₂×Dic₇₁

yes

—

C₂×C₁₆

C₂²×D₇₁

yes

characteristic

C₄⋊C₈

C₇₁⋊C₈

yes

characteristic

C₂²⋊C₈

C₇₁⋊C₈

yes

—

C₄×C₈

C₂×Dic₇₁

yes

characteristic

C₃₂

—

C₃₂

—

142

—

C₃₂

—

142

—

C₂×D₄

C₇₁⋊C₁₆

yes

characteristic

C₂²×C₄

C₂×C₇₁⋊C₈

yes

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

C₂×C₇₁⋊C₈

yes

characteristic

101

C₂×C₈

C₂×C₇₁⋊C₈

yes

—

102

C₂×C₈

C₂²×Dic₇₁

yes

characteristic

103

C₄⋊C₄

C₇₁⋊C₁₆

yes

characteristic

104

C₂²⋊C₄

C₇₁⋊C₁₆

yes

—

105

C₄²

C₂×C₇₁⋊C₈

yes

characteristic

106

C₁₆

—

107

C₁₆

—

108

C₁₆

D₄×D₇₁

yes

characteristic

109

C₁₆

#11765864

yes

characteristic

110

C₂³

C₂×C₇₁⋊C₁₆

yes

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂×C₇₁⋊C₁₆

yes

characteristic

115

C₂×C₄

C₂×C₇₁⋊C₁₆

yes

—

116

C₂×C₄

C₂×C₇₁⋊C₁₆

yes

characteristic

117

C₂×C₄

C₂²×C₇₁⋊C₈

yes

characteristic

118

C₈

—

119

C₈

—

120

C₈

D₄×Dic₇₁

yes

characteristic

121

C₈

#14556135

yes

characteristic

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

C₂²×C₇₁⋊C₁₆

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

#17985833

yes

complemented

129

C₄

D₄×C₇₁⋊C₈

yes

characteristic

130

C₄

#17986416

yes

characteristic

131

C₂

—

132

C₂

D₄×C₇₁⋊C₁₆

yes

characteristic

133

C₂

C₂×#17985833

yes

characteristic

134

C₂

#22590634

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₇₁
commutator, socle	C₂²×C₇₁	C₂²×C₁₆
Fitting	D₄×C₁₆×C₇₁	C₂
prime radical	C₂×D₄×C₇₁	C₁₆
cosocle	C₂×C₁₆×C₇₁	C₂³
hypercenter	D₄×C₁₆	D₇₁
hypocenter	C₇₁	#13646
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₁₆ ◁ D₄×C₁₆

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₇₁

commutator, socle

C₂²×C₇₁

C₂²×C₁₆

Fitting

D₄×C₁₆×C₇₁

C₂

prime radical

C₂×D₄×C₇₁

C₁₆

cosocle

C₂×C₁₆×C₇₁

C₂³

hypercenter

D₄×C₁₆

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂×C₁₆ ◁ D₄×C₁₆

N	H ≅ G/N	type
1. C₂×#17985833	C₂	semidirect
2. C₄⋊(C₇₁⋊C₃₂)	C₂	semidirect
3. C₂²⋊(C₇₁⋊C₃₂)	C₂	semidirect
4. C₄×C₇₁⋊C₃₂	C₂	semidirect
17. C₇₁⋊C₃₂	D₄	semidirect
60. C₄×C₇₁	M₆₄	semidirect
73. C₇₁	#13646	semidirect
128. C₄	#17985833	semidirect

H ≅ G/N

type

1. C₂×#17985833

C₂

semidirect

2. C₄⋊(C₇₁⋊C₃₂)

C₂

semidirect

3. C₂²⋊(C₇₁⋊C₃₂)

C₂

semidirect

4. C₄×C₇₁⋊C₃₂

C₂

semidirect

17. C₇₁⋊C₃₂

D₄

semidirect

60. C₄×C₇₁

M₆₄

semidirect

73. C₇₁

#13646

semidirect

128. C₄

#17985833

semidirect

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

subgroup	index
C₂×#17985833	2
C₄⋊(C₇₁⋊C₃₂)	2
C₂²⋊(C₇₁⋊C₃₂)	2
C₄×C₇₁⋊C₃₂	2
D₄×C₁₆×C₇₁	2
#13646	71

subgroup

index

C₂×#17985833

C₄⋊(C₇₁⋊C₃₂)

C₂²⋊(C₇₁⋊C₃₂)

C₄×C₇₁⋊C₃₂

D₄×C₁₆×C₇₁

#13646

Subgroups — #31918011

Conjugacy classes of subgroups

Complements

Maximal subgroups