Group #31917634 — subgroups

Subgroups — #31917634

Order 18176.

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

normal subgroups only

This group has 238 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	C₂²×C₇₁⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	#22589936	C₂	1	1	yes	maximal, characteristic
3	#22586473	C₂	1	2	yes	maximal, complemented
4	C₃₂×Dic₇₁	C₂	1	2	yes	maximal, complemented
5	C₂×M₆₄×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556068	C₄	1	1	yes	characteristic
7	#17985954	C₄	1	1	yes	characteristic
8	C₂×C₇₁⋊C₃₂	C₄	1	4	yes	complemented
9	C₂×C₇₁⋊C₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₇₁⋊C₃₂	C₂²	1	1	yes	characteristic
11	#17985774	C₄	1	2	yes	—
12	C₁₆×Dic₇₁	C₂²	1	2	yes	—
13	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
14	M₆₄×C₇₁	C₄	1	4	yes	complemented
15	C₂×C₃₂×C₇₁	C₂²	1	2	yes	—
16	C₂×#11765849	C₈	1	1	yes	characteristic
17	#14556087	C₈	1	1	yes	characteristic
18	#14556068	C₈	1	4	yes	—
19	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
20	#14556038	C₈	1	2	yes	—
21	C₈×Dic₇₁	C₂×C₄	1	2	yes	—
22	C₇₁⋊C₃₂	C₂×C₄	1	2	yes	—
23	C₇₁⋊C₃₂	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
27	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
28	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
29	C₃₂×C₇₁	C₂×C₄	1	4	yes	—
30	#11765849	C₁₆	1	4	yes	—
31	C₂×C₇₁⋊C₈	C₂×C₈	1	2	yes	—
32	C₄⋊Dic₇₁	C₁₆	1	2	yes	—
33	C₂²⋊Dic₇₁	C₁₆	1	2	yes	—
34	C₄×Dic₇₁	C₂×C₈	1	2	yes	—
35	C₇₁⋊C₁₆	C₂×C₈	1	4	yes	—
36	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
37	C₂×C₈×C₇₁	C₄²	1	2	yes	—
38	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
39	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
40	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
41	C₁₆×C₇₁	C₄²	1	2	yes	—
42	C₁₆×C₇₁	C₂²×C₄	1	1	yes	characteristic
43	C₁₆×C₇₁	C₂²×C₄	1	1	yes	characteristic
44	C₂³×C₇₁	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₇₁	C₄×C₈	1	2	yes	—
46	C₂×C₄×C₇₁	C₂×C₁₆	1	1	yes	characteristic
47	C₂×C₄×C₇₁	C₂×C₁₆	1	2	yes	—
48	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
49	C₂×Dic₇₁	C₂×C₁₆	1	4	yes	—
50	C₈×C₇₁	C₄×C₈	1	2	yes	—
51	C₈×C₇₁	C₂×C₄²	1	1	yes	characteristic
52	C₈×C₇₁	C₂²×C₈	1	1	yes	characteristic
53	C₇₁⋊C₈	C₂×C₁₆	1	4	yes	—
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	C₂²×C₁₆	1	1	yes	characteristic
60	Dic₇₁	—	2	4	no	—
61	#13549	—	71	1	no	maximal, Sylow
62	C₂×C₇₁	—	2	2	no	—
63	C₂×C₇₁	C₂×C₄×C₁₆	1	1	yes	characteristic
64	C₂×C₇₁	#2258	1	2	yes	—
65	C₂×M₆₄	D₇₁	1	1	yes	characteristic
66	C₂²×C₃₂	—	71	1	no	—
67	#2108	—	71	1	no	—
68	#1397	—	71	2	no	—
69	C₄×C₃₂	—	71	2	no	—
70	C₇₁	#13549	1	1	yes	characteristic, complemented, Sylow
71	C₂×M₃₂	—	71	1	no	—
72	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
73	#405	—	71	1	no	—
74	M₆₄	Dic₇₁	1	4	yes	complemented
75	C₂×C₃₂	—	71	4	no	—
76	C₂×C₃₂	—	71	1	no	—
77	C₂×C₃₂	—	71	1	no	—
78	C₂×C₃₂	C₂×D₇₁	1	2	yes	—
79	#346	—	71	2	no	—
80	C₄×C₁₆	—	71	2	no	—
81	C₂×M₁₆	—	71	1	no	—
82	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
83	#117	—	71	1	no	—
84	M₃₂	—	71	4	no	—
85	C₂×C₁₆	—	71	2	no	—
86	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
87	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
88	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
89	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
90	#97	—	71	2	no	—
91	C₄×C₈	—	71	2	no	—
92	C₃₂	—	71	2	no	—
93	C₃₂	—	71	2	no	—
94	C₃₂	C₂×Dic₇₁	1	4	yes	—
95	C₂²×C₄	C₈×D₇₁	1	1	yes	characteristic
96	M₁₆	—	71	4	no	—
97	C₂×C₈	—	71	2	no	—
98	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
99	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
100	C₂×C₈	C₈×D₇₁	1	2	yes	—
101	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
102	C₄⋊C₄	—	71	2	no	—
103	C₂²⋊C₄	—	71	2	no	—
104	C₄²	—	71	2	no	—
105	C₁₆	—	71	4	no	—
106	C₁₆	C₄×Dic₇₁	1	2	yes	—
107	C₁₆	C₂×C₄×D₇₁	1	1	yes	characteristic
108	C₁₆	C₂²×Dic₇₁	1	1	yes	characteristic
109	C₂³	C₁₆×D₇₁	1	1	yes	characteristic
110	C₂×C₄	—	71	4	no	—
111	C₂×C₄	C₈×Dic₇₁	1	2	yes	—
112	C₂×C₄	C₁₆×D₇₁	1	2	yes	—
113	C₂×C₄	C₁₆×D₇₁	1	1	yes	characteristic
114	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
115	C₈	—	71	4	no	—
116	C₈	C₈×Dic₇₁	1	2	yes	—
117	C₈	C₂×C₄×Dic₇₁	1	1	yes	characteristic
118	C₈	C₂×C₈×D₇₁	1	1	yes	characteristic
119	C₂²	—	2	2	no	—
120	C₂²	C₁₆×Dic₇₁	1	2	yes	—
121	C₂²	C₂×C₁₆×D₇₁	1	1	yes	characteristic
122	C₄	—	142	4	no	—
123	C₄	C₁₆×Dic₇₁	1	2	yes	—
124	C₄	C₂×C₈×Dic₇₁	1	1	yes	characteristic
125	C₄	C₂×C₁₆×D₇₁	1	1	yes	characteristic
126	C₂	—	2	2	no	—
127	C₂	C₂×C₁₆×Dic₇₁	1	1	yes	characteristic
128	C₂	#22590635	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₇₁⋊C₃₂

C₂

yes

maximal, characteristic

#22589936

C₂

yes

maximal, characteristic

#22586473

C₂

yes

maximal, complemented

C₃₂×Dic₇₁

C₂

yes

maximal, complemented

C₂×M₆₄×C₇₁

C₂

yes

maximal, characteristic

C₂×#14556068

C₄

yes

characteristic

#17985954

C₄

yes

characteristic

C₂×C₇₁⋊C₃₂

C₄

yes

complemented

C₂×C₇₁⋊C₃₂

C₂²

yes

characteristic

C₂×C₇₁⋊C₃₂

C₂²

yes

characteristic

#17985774

C₄

yes

—

C₁₆×Dic₇₁

C₂²

yes

—

C₂²×C₁₆×C₇₁

C₂²

yes

characteristic

M₆₄×C₇₁

C₄

yes

complemented

C₂×C₃₂×C₇₁

C₂²

yes

—

C₂×#11765849

C₈

yes

characteristic

#14556087

C₈

yes

characteristic

#14556068

C₈

yes

—

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

—

#14556038

C₈

yes

—

C₈×Dic₇₁

C₂×C₄

yes

—

C₇₁⋊C₃₂

C₂×C₄

yes

—

C₇₁⋊C₃₂

C₂×C₄

yes

—

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

C₂×C₁₆×C₇₁

C₂³

yes

characteristic

C₃₂×C₇₁

C₂×C₄

yes

—

#11765849

C₁₆

yes

—

C₂×C₇₁⋊C₈

C₂×C₈

yes

—

C₄⋊Dic₇₁

C₁₆

yes

—

C₂²⋊Dic₇₁

C₁₆

yes

—

C₄×Dic₇₁

C₂×C₈

yes

—

C₇₁⋊C₁₆

C₂×C₈

yes

—

C₂²×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

C₄²

yes

—

C₂×C₈×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

C₂×C₈

yes

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₁₆×C₇₁

C₄²

yes

—

C₁₆×C₇₁

C₂²×C₄

yes

characteristic

C₁₆×C₇₁

C₂²×C₄

yes

characteristic

C₂³×C₇₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₁

C₄×C₈

yes

—

C₂×C₄×C₇₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×C₁₆

yes

—

C₂×C₄×C₇₁

C₂²×C₈

yes

characteristic

C₂×Dic₇₁

C₂×C₁₆

yes

—

C₈×C₇₁

C₄×C₈

yes

—

C₈×C₇₁

C₂×C₄²

yes

characteristic

C₈×C₇₁

C₂²×C₈

yes

characteristic

C₇₁⋊C₈

C₂×C₁₆

yes

—

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

C₂²×C₁₆

yes

characteristic

Dic₇₁

—

#13549

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

C₂×C₄×C₁₆

yes

characteristic

C₂×C₇₁

#2258

yes

—

C₂×M₆₄

D₇₁

yes

characteristic

C₂²×C₃₂

—

#2108

—

#1397

—

C₄×C₃₂

—

C₇₁

#13549

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₇₁

yes

characteristic

#405

—

M₆₄

Dic₇₁

yes

complemented

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₇₁

yes

—

#346

—

C₄×C₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₄×D₇₁

yes

characteristic

#117

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₇₁

yes

characteristic

C₂×C₁₆

C₄×D₇₁

yes

—

C₂×C₁₆

C₂×Dic₇₁

yes

—

C₂×C₁₆

C₂²×D₇₁

yes

characteristic

#97

—

C₄×C₈

—

C₃₂

—

C₃₂

—

C₃₂

C₂×Dic₇₁

yes

—

C₂²×C₄

C₈×D₇₁

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

C₄×Dic₇₁

yes

—

C₂×C₈

C₈×D₇₁

yes

characteristic

100

C₂×C₈

C₈×D₇₁

yes

—

101

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

102

C₄⋊C₄

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₁₆

—

106

C₁₆

C₄×Dic₇₁

yes

—

107

C₁₆

C₂×C₄×D₇₁

yes

characteristic

108

C₁₆

C₂²×Dic₇₁

yes

characteristic

109

C₂³

C₁₆×D₇₁

yes

characteristic

110

C₂×C₄

—

111

C₂×C₄

C₈×Dic₇₁

yes

—

112

C₂×C₄

C₁₆×D₇₁

yes

—

113

C₂×C₄

C₁₆×D₇₁

yes

characteristic

114

C₂×C₄

C₂×C₈×D₇₁

yes

characteristic

115

C₈

—

116

C₈

C₈×Dic₇₁

yes

—

117

C₈

C₂×C₄×Dic₇₁

yes

characteristic

118

C₈

C₂×C₈×D₇₁

yes

characteristic

119

C₂²

—

120

C₂²

C₁₆×Dic₇₁

yes

—

121

C₂²

C₂×C₁₆×D₇₁

yes

characteristic

122

C₄

—

142

—

123

C₄

C₁₆×Dic₇₁

yes

—

124

C₄

C₂×C₈×Dic₇₁

yes

characteristic

125

C₄

C₂×C₁₆×D₇₁

yes

characteristic

126

C₂

—

127

C₂

C₂×C₁₆×Dic₇₁

yes

characteristic

128

C₂

#22590635

yes

—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₇₁
commutator	C₂×C₇₁	C₂×C₄×C₁₆
Fitting	C₂×M₆₄×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×C₁₆
prime radical	C₂³×C₇₁	C₂×C₁₆
cosocle	C₂×C₁₆×C₇₁	C₂³
hypercenter	C₂×M₆₄	D₇₁
hypocenter	C₇₁	#13549
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₇₁

commutator

C₂×C₇₁

C₂×C₄×C₁₆

Fitting

C₂×M₆₄×C₇₁

C₂

socle

C₂²×C₇₁

C₂²×C₁₆

prime radical

C₂³×C₇₁

C₂×C₁₆

cosocle

C₂×C₁₆×C₇₁

C₂³

hypercenter

C₂×M₆₄

D₇₁

C₇₁

G ▷ C₂×C₇₁ ▷ C₇₁

C₁ ◁ C₂×C₁₆ ◁ C₂×M₆₄

N	H ≅ G/N	type
3. #22586473	C₂	semidirect
4. C₃₂×Dic₇₁	C₂	semidirect
8. C₂×C₇₁⋊C₃₂	C₄	semidirect
14. M₆₄×C₇₁	C₄	semidirect
70. C₇₁	#13549	semidirect
74. M₆₄	Dic₇₁	semidirect

H ≅ G/N

type

3. #22586473

C₂

semidirect

4. C₃₂×Dic₇₁

C₂

semidirect

8. C₂×C₇₁⋊C₃₂

C₄

semidirect

14. M₆₄×C₇₁

C₄

semidirect

70. C₇₁

#13549

semidirect

74. M₆₄

Dic₇₁

semidirect

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

subgroup	index
C₂²×C₇₁⋊C₃₂	2
#22589936	2
#22586473	2
C₃₂×Dic₇₁	2
C₂×M₆₄×C₇₁	2
#13549	71

subgroup

index

C₂²×C₇₁⋊C₃₂

#22589936

#22586473

C₃₂×Dic₇₁

C₂×M₆₄×C₇₁

#13549

Subgroups — #31917634

Conjugacy classes of subgroups

Complements

Maximal subgroups