Group #31917934 — subgroups

Subgroups — #31917934

Order 18176.

218 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 127 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	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	Dic₇₁⋊C₃₂	C₂	1	4	yes	maximal, complemented
4	C₂×M₆₄×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₁⋊C₁₆	C₄	1	1	yes	characteristic
6	C₂×C₈×Dic₇₁	C₄	1	1	yes	characteristic
7	#17985833	—	2	2	no	—
8	C₂×C₇₁⋊C₃₂	C₂²	1	2	yes	—
9	C₁₆×Dic₇₁	C₄	1	2	yes	—
10	C₁₆×Dic₇₁	C₂²	1	2	yes	—
11	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
12	M₆₄×C₇₁	—	2	2	no	—
13	C₂×C₃₂×C₇₁	C₂²	1	2	yes	—
14	C₂²×C₇₁⋊C₈	C₈	1	1	yes	characteristic
15	C₂×C₄×Dic₇₁	C₈	1	1	yes	characteristic
16	C₂×C₇₁⋊C₁₆	—	2	2	no	—
17	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
18	C₈×Dic₇₁	C₈	1	2	yes	—
19	C₈×Dic₇₁	C₂×C₄	1	2	yes	—
20	C₇₁⋊C₃₂	—	2	2	no	—
21	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₇₁	D₄	1	2	yes	—
24	C₂×C₁₆×C₇₁	Q₈	1	2	yes	—
25	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
26	C₃₂×C₇₁	—	2	2	no	—
27	C₂²×Dic₇₁	C₁₆	1	2	yes	—
28	C₂×C₇₁⋊C₈	—	2	2	no	—
29	C₂×C₇₁⋊C₈	C₂×C₈	1	2	yes	—
30	C₄×Dic₇₁	C₁₆	1	1	yes	characteristic
31	C₄×Dic₇₁	C₁₆	1	1	yes	characteristic
32	C₄×Dic₇₁	C₂×C₈	1	2	yes	—
33	C₇₁⋊C₁₆	—	2	2	no	—
34	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
36	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
37	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
38	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
39	C₁₆×C₇₁	C₄⋊C₄	1	2	yes	—
40	C₁₆×C₇₁	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₇₁	C₂×Q₈	1	1	yes	characteristic
42	C₂³×C₇₁	C₂×C₁₆	1	1	yes	characteristic
43	C₂×C₄×C₇₁	C₄⋊C₈	1	2	yes	—
44	C₂×C₄×C₇₁	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₇₁	M₃₂	1	2	yes	—
46	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
47	C₂×Dic₇₁	—	2	4	no	—
48	C₂×Dic₇₁	C₂×C₁₆	1	2	yes	—
49	C₂×Dic₇₁	C₂×C₁₆	1	2	yes	—
50	C₈×C₇₁	C₄⋊C₈	1	2	yes	—
51	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
53	C₇₁⋊C₈	—	2	2	no	—
54	C₂²×C₇₁	—	2	2	no	—
55	C₂²×C₇₁	C₄⋊C₁₆	1	2	yes	—
56	C₂²×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
57	C₄×C₇₁	C₄⋊C₁₆	1	2	yes	—
58	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
59	C₄×C₇₁	C₂×M₃₂	1	1	yes	characteristic
60	Dic₇₁	—	2	2	no	—
61	Dic₇₁	M₆₄	1	4	yes	complemented
62	#13627	—	71	1	no	maximal, Sylow
63	C₂×C₇₁	—	2	2	no	—
64	C₂×C₇₁	C₂×C₄⋊C₁₆	1	1	yes	characteristic
65	C₂×C₇₁	C₂×M₆₄	1	2	yes	—
66	C₂×M₆₄	—	71	1	no	—
67	C₂×M₆₄	D₇₁	1	1	yes	characteristic
68	C₂×C₄×C₁₆	—	71	1	no	—
69	C₄⋊C₃₂	—	71	4	no	—
70	C₇₁	#13627	1	1	yes	characteristic, complemented, Sylow
71	C₂²×C₁₆	—	71	1	no	—
72	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
73	C₂×C₄×C₈	—	71	1	no	—
74	M₆₄	—	2	2	no	—
75	M₆₄	—	142	2	no	—
76	C₂×C₃₂	—	71	2	no	—
77	C₂×C₃₂	C₂×D₇₁	1	2	yes	—
78	C₄×C₁₆	—	71	2	no	—
79	C₄×C₁₆	—	71	2	no	—
80	C₂²×C₈	—	71	1	no	—
81	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
82	C₂×C₄²	—	71	1	no	—
83	C₂×C₁₆	—	71	2	no	—
84	C₂×C₁₆	—	142	2	no	—
85	C₂×C₁₆	Dic₁₄₂	1	2	yes	—
86	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
87	C₂×C₁₆	C₂²⋊D₇₁	1	2	yes	—
88	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
89	C₄×C₈	—	71	2	no	—
90	C₄×C₈	—	71	2	no	—
91	C₃₂	—	2	2	no	—
92	C₃₂	—	142	2	no	—
93	C₂²×C₄	—	71	2	no	—
94	C₂²×C₄	C₈×D₇₁	1	1	yes	characteristic
95	C₂×C₈	—	71	2	no	—
96	C₂×C₈	—	142	2	no	—
97	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
98	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
99	C₂×C₈	#11765848	1	2	yes	—
100	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
101	C₄²	—	71	1	no	—
102	C₄²	—	71	1	no	—
103	C₄²	—	71	2	no	—
104	C₁₆	—	142	2	no	—
105	C₁₆	Dic₇₁⋊C₄	1	2	yes	—
106	C₁₆	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
107	C₁₆	C₂×Dic₁₄₂	1	1	yes	characteristic
108	C₂³	C₁₆×D₇₁	1	1	yes	characteristic
109	C₂×C₄	—	71	2	no	—
110	C₂×C₄	—	71	2	no	—
111	C₂×C₄	—	142	4	no	—
112	C₂×C₄	Dic₇₁⋊C₈	1	2	yes	—
113	C₂×C₄	C₁₆×D₇₁	1	1	yes	characteristic
114	C₂×C₄	#14556067	1	2	yes	—
115	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
116	C₈	—	142	2	no	—
117	C₈	Dic₇₁⋊C₈	1	2	yes	—
118	C₈	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
119	C₈	C₂×#11765848	1	1	yes	characteristic
120	C₂²	—	2	2	no	—
121	C₂²	Dic₇₁⋊C₁₆	1	2	yes	—
122	C₂²	C₂×C₁₆×D₇₁	1	1	yes	characteristic
123	C₄	—	71	4	no	—
124	C₄	—	142	2	no	—
125	C₄	Dic₇₁⋊C₁₆	1	2	yes	—
126	C₄	C₂×Dic₇₁⋊C₈	1	1	yes	characteristic
127	C₄	C₂×#14556067	1	1	yes	characteristic
128	C₂	—	2	2	no	—
129	C₂	C₂×Dic₇₁⋊C₁₆	1	1	yes	characteristic
130	C₂	M₆₄×D₇₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17985833

C₂

yes

maximal, characteristic

C₂×C₁₆×Dic₇₁

C₂

yes

maximal, characteristic

Dic₇₁⋊C₃₂

C₂

yes

maximal, complemented

C₂×M₆₄×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₁₆

C₄

yes

characteristic

C₂×C₈×Dic₇₁

C₄

yes

characteristic

#17985833

—

C₂×C₇₁⋊C₃₂

C₂²

yes

—

C₁₆×Dic₇₁

C₄

yes

—

C₁₆×Dic₇₁

C₂²

yes

—

C₂²×C₁₆×C₇₁

C₂²

yes

characteristic

M₆₄×C₇₁

—

C₂×C₃₂×C₇₁

C₂²

yes

—

C₂²×C₇₁⋊C₈

C₈

yes

characteristic

C₂×C₄×Dic₇₁

C₈

yes

characteristic

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

—

C₈×Dic₇₁

C₈

yes

—

C₈×Dic₇₁

C₂×C₄

yes

—

C₇₁⋊C₃₂

—

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

D₄

yes

—

C₂×C₁₆×C₇₁

Q₈

yes

—

C₂×C₁₆×C₇₁

C₂³

yes

characteristic

C₃₂×C₇₁

—

C₂²×Dic₇₁

C₁₆

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂×C₈

yes

—

C₄×Dic₇₁

C₁₆

yes

characteristic

C₄×Dic₇₁

C₁₆

yes

characteristic

C₄×Dic₇₁

C₂×C₈

yes

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

M₁₆

yes

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₁₆×C₇₁

C₄⋊C₄

yes

—

C₁₆×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

C₂×Q₈

yes

characteristic

C₂³×C₇₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₈

yes

—

C₂×C₄×C₇₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₁

M₃₂

yes

—

C₂×C₄×C₇₁

C₂²×C₈

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

C₂×C₁₆

yes

—

C₂×Dic₇₁

C₂×C₁₆

yes

—

C₈×C₇₁

C₄⋊C₈

yes

—

C₈×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₈×C₇₁

C₂×M₁₆

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

C₄⋊C₁₆

yes

—

C₂²×C₇₁

C₂²×C₁₆

yes

characteristic

C₄×C₇₁

C₄⋊C₁₆

yes

—

C₄×C₇₁

C₂×C₄⋊C₈

yes

characteristic

C₄×C₇₁

C₂×M₃₂

yes

characteristic

Dic₇₁

—

Dic₇₁

M₆₄

yes

complemented

#13627

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

C₂×C₄⋊C₁₆

yes

characteristic

C₂×C₇₁

C₂×M₆₄

yes

—

C₂×M₆₄

—

C₂×M₆₄

D₇₁

yes

characteristic

C₂×C₄×C₁₆

—

C₄⋊C₃₂

—

C₇₁

#13627

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₇₁

yes

characteristic

C₂×C₄×C₈

—

M₆₄

—

M₆₄

—

142

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₇₁

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

C₂²×C₈

—

C₂²×C₈

C₄×D₇₁

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

142

—

C₂×C₁₆

Dic₁₄₂

yes

—

C₂×C₁₆

C₄×D₇₁

yes

characteristic

C₂×C₁₆

C₂²⋊D₇₁

yes

—

C₂×C₁₆

C₂²×D₇₁

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₃₂

—

C₃₂

—

142

—

C₂²×C₄

—

C₂²×C₄

C₈×D₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

Dic₇₁⋊C₄

yes

—

C₂×C₈

C₈×D₇₁

yes

characteristic

C₂×C₈

#11765848

yes

—

100

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

101

C₄²

—

102

C₄²

—

103

C₄²

—

104

C₁₆

—

142

—

105

C₁₆

Dic₇₁⋊C₄

yes

—

106

C₁₆

C₂×C₂²⋊D₇₁

yes

characteristic

107

C₁₆

C₂×Dic₁₄₂

yes

characteristic

108

C₂³

C₁₆×D₇₁

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

142

—

112

C₂×C₄

Dic₇₁⋊C₈

yes

—

113

C₂×C₄

C₁₆×D₇₁

yes

characteristic

114

C₂×C₄

#14556067

yes

—

115

C₂×C₄

C₂×C₈×D₇₁

yes

characteristic

116

C₈

—

142

—

117

C₈

Dic₇₁⋊C₈

yes

—

118

C₈

C₂×Dic₇₁⋊C₄

yes

characteristic

119

C₈

C₂×#11765848

yes

characteristic

120

C₂²

—

121

C₂²

Dic₇₁⋊C₁₆

yes

—

122

C₂²

C₂×C₁₆×D₇₁

yes

characteristic

123

C₄

—

124

C₄

—

142

—

125

C₄

Dic₇₁⋊C₁₆

yes

—

126

C₄

C₂×Dic₇₁⋊C₈

yes

characteristic

127

C₄

C₂×#14556067

yes

characteristic

128

C₂

—

129

C₂

C₂×Dic₇₁⋊C₁₆

yes

characteristic

130

C₂

M₆₄×D₇₁

yes

—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₇₁
commutator, socle	C₂²×C₇₁	C₂²×C₁₆
Fitting	C₂×M₆₄×C₇₁	C₂
prime radical	C₂³×C₇₁	C₂×C₁₆
cosocle	C₂×C₁₆×C₇₁	C₂³
hypercenter	C₂×M₆₄	D₇₁
hypocenter	C₇₁	#13627
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₇₁

commutator, socle

C₂²×C₇₁

C₂²×C₁₆

Fitting

C₂×M₆₄×C₇₁

C₂

prime radical

C₂³×C₇₁

C₂×C₁₆

cosocle

C₂×C₁₆×C₇₁

C₂³

hypercenter

C₂×M₆₄

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

N	H ≅ G/N	type
3. Dic₇₁⋊C₃₂	C₂	semidirect
61. Dic₇₁	M₆₄	semidirect
70. C₇₁	#13627	semidirect

H ≅ G/N

type

3. Dic₇₁⋊C₃₂

C₂

semidirect

61. Dic₇₁

M₆₄

semidirect

70. C₇₁

#13627

semidirect

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

subgroup	index
C₂×#17985833	2
C₂×C₁₆×Dic₇₁	2
Dic₇₁⋊C₃₂	2
C₂×M₆₄×C₇₁	2
#13627	71

subgroup

index

C₂×#17985833

C₂×C₁₆×Dic₇₁

Dic₇₁⋊C₃₂

C₂×M₆₄×C₇₁

#13627

Subgroups — #31917934

Conjugacy classes of subgroups

Complements

Maximal subgroups