Group #31917640 — subgroups

Subgroups — #31917640

Order 18176.

214 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 159 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
2	#22586476	C₂	1	4	yes	maximal, complemented
3	C₇₁×#2108	C₂	1	1	yes	maximal, characteristic
4	#17985833	C₄	1	4	yes	complemented
5	#17985833	C₂²	1	4	yes	—
6	C₂×C₇₁⋊C₃₂	C₄	1	4	yes	complemented
7	C₂×M₃₂×C₇₁	C₄	1	1	yes	characteristic
8	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
9	#405×C₇₁	C₄	1	1	yes	characteristic
10	#346×C₇₁	C₂²	1	2	yes	—
11	C₄×C₁₆×C₇₁	C₄	1	2	yes	—
12	C₇₁⋊C₃₂	C₂×C₄	1	8	yes	—
13	C₂×M₁₆×C₇₁	C₈	1	1	yes	characteristic
14	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
15	#117×C₇₁	C₈	1	1	yes	characteristic
16	M₃₂×C₇₁	C₈	1	4	yes	—
17	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
19	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
20	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
21	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
22	#97×C₇₁	C₈	1	2	yes	—
23	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
25	M₁₆×C₇₁	—	2	2	no	—
26	C₂×C₈×C₇₁	C₄²	1	2	yes	—
27	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
28	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
29	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
30	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
31	C₄⋊C₄×C₇₁	—	2	1	no	—
32	C₂²⋊C₄×C₇₁	—	2	1	no	—
33	C₄²×C₇₁	C₂×C₈	1	2	yes	—
34	C₁₆×C₇₁	C₄²	1	1	yes	characteristic
35	C₁₆×C₇₁	C₄²	1	1	yes	characteristic
36	C₁₆×C₇₁	C₂×C₈	1	4	yes	—
37	C₁₆×C₇₁	C₂²×C₄	1	2	yes	—
38	C₂³×C₇₁	M₃₂	1	1	yes	characteristic
39	C₂×C₄×C₇₁	—	2	2	no	—
40	C₂×C₄×C₇₁	C₄×C₈	1	2	yes	—
41	C₂×C₄×C₇₁	M₃₂	1	1	yes	characteristic
42	C₂×C₄×C₇₁	M₃₂	1	2	yes	—
43	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
44	C₈×C₇₁	—	2	2	no	—
45	C₈×C₇₁	C₄×C₈	1	2	yes	—
46	C₈×C₇₁	C₂×C₄²	1	1	yes	characteristic
47	C₈×C₇₁	C₂²×C₈	1	1	yes	characteristic
48	C₂²×C₇₁	—	4	1	no	—
49	C₂²×C₇₁	#346	1	2	yes	—
50	C₂²×C₇₁	C₂×M₃₂	1	1	yes	characteristic
51	C₄×C₇₁	—	4	2	no	—
52	C₄×C₇₁	#346	1	2	yes	—
53	C₄×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
54	C₄×C₇₁	C₂×M₃₂	1	1	yes	characteristic
55	#13551	—	71	1	no	maximal, Sylow
56	C₂×C₇₁	—	2	1	no	—
57	C₂×C₇₁	—	2	2	no	—
58	C₂×C₇₁	C₂×#346	1	1	yes	characteristic
59	C₂×M₆₄	—	71	2	no	—
60	#2108	D₇₁	1	1	yes	characteristic
61	#1398	—	71	4	no	—
62	C₇₁	#13551	1	1	yes	characteristic, complemented, Sylow
63	C₂×M₃₂	Dic₇₁	1	1	yes	characteristic
64	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
65	#405	Dic₇₁	1	1	yes	characteristic
66	M₆₄	—	71	4	no	—
67	M₆₄	—	71	4	no	—
68	C₂×C₃₂	—	71	4	no	—
69	#346	C₂×D₇₁	1	2	yes	—
70	C₄×C₁₆	Dic₇₁	1	2	yes	—
71	C₂×M₁₆	C₇₁⋊C₈	1	1	yes	characteristic
72	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
73	#117	C₇₁⋊C₈	1	1	yes	characteristic
74	M₃₂	C₇₁⋊C₈	1	4	yes	—
75	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
76	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
77	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
78	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
79	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
80	#97	C₇₁⋊C₈	1	2	yes	—
81	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
82	C₃₂	—	71	8	no	—
83	C₂²×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
86	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
87	C₂×C₈	C₂×C₇₁⋊C₈	1	1	yes	characteristic
88	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
89	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
90	C₄⋊C₄	—	2	1	no	—
91	C₂²⋊C₄	—	2	1	no	—
92	C₄²	C₂×C₇₁⋊C₈	1	2	yes	—
93	C₁₆	C₄×Dic₇₁	1	1	yes	characteristic
94	C₁₆	C₄×Dic₇₁	1	1	yes	characteristic
95	C₁₆	C₂×C₇₁⋊C₈	1	4	yes	—
96	C₁₆	C₂×C₄×D₇₁	1	2	yes	—
97	C₂³	#14556068	1	1	yes	characteristic
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	C₄×C₇₁⋊C₈	1	2	yes	—
100	C₂×C₄	#14556068	1	2	yes	—
101	C₂×C₄	#14556068	1	1	yes	characteristic
102	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
103	C₈	—	2	2	no	—
104	C₈	C₄×C₇₁⋊C₈	1	2	yes	—
105	C₈	C₂×C₄×Dic₇₁	1	1	yes	characteristic
106	C₈	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
107	C₂²	—	4	1	no	—
108	C₂²	#17985775	1	2	yes	—
109	C₂²	C₂×#14556068	1	1	yes	characteristic
110	C₄	—	4	2	no	—
111	C₄	#17985775	1	2	yes	—
112	C₄	C₂×C₄×C₇₁⋊C₈	1	1	yes	characteristic
113	C₄	C₂×#14556068	1	1	yes	characteristic
114	C₂	—	2	1	no	—
115	C₂	—	2	2	no	—
116	C₂	C₂×#17985775	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17985833

C₂

yes

maximal

#22586476

C₂

yes

maximal, complemented

C₇₁×#2108

C₂

yes

maximal, characteristic

#17985833

C₄

yes

complemented

#17985833

C₂²

yes

—

C₂×C₇₁⋊C₃₂

C₄

yes

complemented

C₂×M₃₂×C₇₁

C₄

yes

characteristic

C₂²×C₁₆×C₇₁

C₂²

yes

characteristic

#405×C₇₁

C₄

yes

characteristic

#346×C₇₁

C₂²

yes

—

C₄×C₁₆×C₇₁

C₄

yes

—

C₇₁⋊C₃₂

C₂×C₄

yes

—

C₂×M₁₆×C₇₁

C₈

yes

characteristic

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

#117×C₇₁

C₈

yes

characteristic

M₃₂×C₇₁

C₈

yes

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

C₂×C₁₆×C₇₁

C₂³

yes

characteristic

#97×C₇₁

C₈

yes

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂×C₈

yes

characteristic

M₁₆×C₇₁

—

C₂×C₈×C₇₁

C₄²

yes

—

C₂×C₈×C₇₁

C₂×C₈

yes

—

C₂×C₈×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

C₂×C₈

yes

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

C₂×C₈

yes

—

C₁₆×C₇₁

C₄²

yes

characteristic

C₁₆×C₇₁

C₄²

yes

characteristic

C₁₆×C₇₁

C₂×C₈

yes

—

C₁₆×C₇₁

C₂²×C₄

yes

—

C₂³×C₇₁

M₃₂

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄×C₈

yes

—

C₂×C₄×C₇₁

M₃₂

yes

characteristic

C₂×C₄×C₇₁

M₃₂

yes

—

C₂×C₄×C₇₁

C₂²×C₈

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

C₄×C₈

yes

—

C₈×C₇₁

C₂×C₄²

yes

characteristic

C₈×C₇₁

C₂²×C₈

yes

characteristic

C₂²×C₇₁

—

C₂²×C₇₁

#346

yes

—

C₂²×C₇₁

C₂×M₃₂

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

#346

yes

—

C₄×C₇₁

C₂×C₄×C₈

yes

characteristic

C₄×C₇₁

C₂×M₃₂

yes

characteristic

#13551

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

C₂×#346

yes

characteristic

C₂×M₆₄

—

#2108

D₇₁

yes

characteristic

#1398

—

C₇₁

#13551

yes

characteristic, complemented, Sylow

C₂×M₃₂

Dic₇₁

yes

characteristic

C₂²×C₁₆

C₂×D₇₁

yes

characteristic

#405

Dic₇₁

yes

characteristic

M₆₄

—

M₆₄

—

C₂×C₃₂

—

#346

C₂×D₇₁

yes

—

C₄×C₁₆

Dic₇₁

yes

—

C₂×M₁₆

C₇₁⋊C₈

yes

characteristic

C₂²×C₈

C₂×Dic₇₁

yes

characteristic

#117

C₇₁⋊C₈

yes

characteristic

M₃₂

C₇₁⋊C₈

yes

—

C₂×C₁₆

C₄×D₇₁

yes

—

C₂×C₁₆

C₄×D₇₁

yes

—

C₂×C₁₆

C₂×Dic₇₁

yes

characteristic

C₂×C₁₆

C₂×Dic₇₁

yes

—

C₂×C₁₆

C₂²×D₇₁

yes

characteristic

#97

C₇₁⋊C₈

yes

—

C₄×C₈

C₂×Dic₇₁

yes

—

C₃₂

—

C₂²×C₄

C₂×C₇₁⋊C₈

yes

characteristic

M₁₆

—

C₂×C₈

C₄×Dic₇₁

yes

—

C₂×C₈

C₂×C₇₁⋊C₈

yes

—

C₂×C₈

C₂×C₇₁⋊C₈

yes

characteristic

C₂×C₈

C₂×C₇₁⋊C₈

yes

—

C₂×C₈

C₂²×Dic₇₁

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×C₇₁⋊C₈

yes

—

C₁₆

C₄×Dic₇₁

yes

characteristic

C₁₆

C₄×Dic₇₁

yes

characteristic

C₁₆

C₂×C₇₁⋊C₈

yes

—

C₁₆

C₂×C₄×D₇₁

yes

—

C₂³

#14556068

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄×C₇₁⋊C₈

yes

—

100

C₂×C₄

#14556068

yes

—

101

C₂×C₄

#14556068

yes

characteristic

102

C₂×C₄

C₂²×C₇₁⋊C₈

yes

characteristic

103

C₈

—

104

C₈

C₄×C₇₁⋊C₈

yes

—

105

C₈

C₂×C₄×Dic₇₁

yes

characteristic

106

C₈

C₂²×C₇₁⋊C₈

yes

characteristic

107

C₂²

—

108

C₂²

#17985775

yes

—

109

C₂²

C₂×#14556068

yes

characteristic

110

C₄

—

111

C₄

#17985775

yes

—

112

C₄

C₂×C₄×C₇₁⋊C₈

yes

characteristic

113

C₄

C₂×#14556068

yes

characteristic

114

C₂

—

115

C₂

—

116

C₂

C₂×#17985775

yes

characteristic

	subgroup	G/H
center	C₁₆	C₄×Dic₇₁
commutator	C₄×C₇₁	C₂×C₄×C₈
Frattini	C₂×C₁₆	C₂²×D₇₁
Fitting	C₇₁×#2108	C₂
socle	C₂×C₇₁	C₂×#346
prime radical	C₂³×C₇₁	M₃₂
cosocle	C₂×C₁₆×C₇₁	C₂³
hypercenter	#2108	D₇₁
hypocenter	C₇₁	#13551
LCS	G ▷ C₄×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₁₆ ◁ #2108

subgroup

G/H

center

C₁₆

C₄×Dic₇₁

commutator

C₄×C₇₁

C₂×C₄×C₈

Frattini

C₂×C₁₆

C₂²×D₇₁

Fitting

C₇₁×#2108

C₂

socle

C₂×C₇₁

C₂×#346

prime radical

C₂³×C₇₁

M₃₂

cosocle

C₂×C₁₆×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₄×C₇₁ ▷ C₇₁

C₁ ◁ C₁₆ ◁ #2108

N	H ≅ G/N	type
2. #22586476	C₂	semidirect
4. #17985833	C₄	semidirect
6. C₂×C₇₁⋊C₃₂	C₄	semidirect
62. C₇₁	#13551	semidirect

H ≅ G/N

type

2. #22586476

C₂

semidirect

4. #17985833

C₄

semidirect

6. C₂×C₇₁⋊C₃₂

C₄

semidirect

62. C₇₁

#13551

semidirect

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

subgroup	index
C₂×#17985833	2
#22586476	2
C₇₁×#2108	2
#13551	71

subgroup

index

C₂×#17985833

#22586476

C₇₁×#2108

#13551

Subgroups — #31917640

Conjugacy classes of subgroups

Complements

Maximal subgroups