Group #31917938 — subgroups

Subgroups — #31917938

Order 18176.

206 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 147 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	C₂²×C₇₁⋊C₃₂	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₇₁⋊C₃₂)	C₂	1	4	yes	maximal, complemented
4	C₇₁×#2108	C₂	1	1	yes	maximal, characteristic
5	#17985833	—	2	2	no	—
6	C₂×C₇₁⋊C₃₂	—	2	2	no	—
7	C₂×C₇₁⋊C₃₂	C₂²	1	2	yes	—
8	C₂×C₇₁⋊C₃₂	C₂²	1	2	yes	—
9	C₂×M₃₂×C₇₁	C₄	1	1	yes	characteristic
10	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
11	#405×C₇₁	C₄	1	1	yes	characteristic
12	#346×C₇₁	C₄	1	2	yes	—
13	C₄×C₁₆×C₇₁	C₂²	1	2	yes	—
14	C₇₁⋊C₃₂	—	2	2	no	—
15	C₇₁⋊C₃₂	—	2	2	no	—
16	C₂×M₁₆×C₇₁	C₈	1	1	yes	characteristic
17	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
18	#117×C₇₁	C₈	1	1	yes	characteristic
19	M₃₂×C₇₁	—	2	2	no	—
20	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
21	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₇₁	D₄	1	2	yes	—
23	C₂×C₁₆×C₇₁	Q₈	1	2	yes	—
24	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
25	#97×C₇₁	C₈	1	2	yes	—
26	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
28	M₁₆×C₇₁	—	2	2	no	—
29	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
30	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
31	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
33	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
34	C₄⋊C₄×C₇₁	C₁₆	1	2	yes	—
35	C₂²⋊C₄×C₇₁	C₁₆	1	2	yes	—
36	C₄²×C₇₁	C₂×C₈	1	2	yes	—
37	C₁₆×C₇₁	—	2	2	no	—
38	C₁₆×C₇₁	C₄⋊C₄	1	2	yes	—
39	C₁₆×C₇₁	C₂×D₄	1	1	yes	characteristic
40	C₁₆×C₇₁	C₂×Q₈	1	1	yes	characteristic
41	C₂³×C₇₁	C₂×C₁₆	1	1	yes	characteristic
42	C₂×C₄×C₇₁	C₄⋊C₈	1	2	yes	—
43	C₂×C₄×C₇₁	C₂×C₁₆	1	4	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	C₈×C₇₁	—	2	2	no	—
48	C₈×C₇₁	C₄⋊C₈	1	2	yes	—
49	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
51	C₂²×C₇₁	—	2	2	no	—
52	C₂²×C₇₁	C₄⋊C₁₆	1	2	yes	—
53	C₂²×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
54	C₄×C₇₁	—	2	4	no	—
55	C₄×C₇₁	C₄⋊C₁₆	1	2	yes	—
56	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
57	C₄×C₇₁	C₂×M₃₂	1	1	yes	characteristic
58	#13628	—	71	1	no	maximal, Sylow
59	C₂×C₇₁	—	2	2	no	—
60	C₂×C₇₁	C₂×C₄⋊C₁₆	1	1	yes	characteristic
61	C₂×C₇₁	#2258	1	2	yes	—
62	C₂×M₆₄	—	71	1	no	—
63	C₂²×C₃₂	—	71	1	no	—
64	#2108	D₇₁	1	1	yes	characteristic
65	C₄⋊C₃₂	—	71	4	no	—
66	C₇₁	#13628	1	1	yes	characteristic, complemented, Sylow
67	C₂×M₃₂	Dic₇₁	1	1	yes	characteristic
68	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
69	#405	Dic₇₁	1	1	yes	characteristic
70	M₆₄	—	142	2	no	—
71	C₂×C₃₂	—	71	2	no	—
72	C₂×C₃₂	—	71	2	no	—
73	C₂×C₃₂	—	142	2	no	—
74	#346	Dic₇₁	1	2	yes	—
75	C₄×C₁₆	C₂×D₇₁	1	2	yes	—
76	C₂×M₁₆	C₇₁⋊C₈	1	1	yes	characteristic
77	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
78	#117	C₇₁⋊C₈	1	1	yes	characteristic
79	M₃₂	—	2	2	no	—
80	C₂×C₁₆	Dic₁₄₂	1	2	yes	—
81	C₂×C₁₆	D₂₈₄	1	2	yes	—
82	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
83	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
84	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
85	#97	C₇₁⋊C₈	1	2	yes	—
86	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
87	C₃₂	—	142	2	no	—
88	C₃₂	—	142	2	no	—
89	C₂²×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
92	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
93	C₂×C₈	C₂×C₇₁⋊C₈	1	1	yes	characteristic
94	C₂×C₈	#11765849	1	2	yes	—
95	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
96	C₄⋊C₄	C₇₁⋊C₁₆	1	2	yes	—
97	C₂²⋊C₄	C₇₁⋊C₁₆	1	2	yes	—
98	C₄²	C₂×C₇₁⋊C₈	1	2	yes	—
99	C₁₆	—	2	2	no	—
100	C₁₆	C₄⋊Dic₇₁	1	2	yes	—
101	C₁₆	C₂×D₂₈₄	1	1	yes	characteristic
102	C₁₆	C₂×Dic₁₄₂	1	1	yes	characteristic
103	C₂³	C₂×C₇₁⋊C₁₆	1	1	yes	characteristic
104	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
105	C₂×C₄	C₂×C₇₁⋊C₁₆	1	4	yes	—
106	C₂×C₄	C₂×C₇₁⋊C₁₆	1	1	yes	characteristic
107	C₂×C₄	#14556068	1	2	yes	—
108	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
111	C₈	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
112	C₈	C₂×#11765849	1	1	yes	characteristic
113	C₂²	—	2	2	no	—
114	C₂²	C₄⋊(C₇₁⋊C₁₆)	1	2	yes	—
115	C₂²	C₂²×C₇₁⋊C₁₆	1	1	yes	characteristic
116	C₄	—	2	4	no	—
117	C₄	C₄⋊(C₇₁⋊C₁₆)	1	2	yes	—
118	C₄	C₂×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
119	C₄	C₂×#14556068	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	C₂×C₄⋊(C₇₁⋊C₁₆)	1	1	yes	characteristic
122	C₂	#22590634	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17985833

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₃₂

C₂

yes

maximal, characteristic

C₄⋊(C₇₁⋊C₃₂)

C₂

yes

maximal, complemented

C₇₁×#2108

C₂

yes

maximal, characteristic

#17985833

—

C₂×C₇₁⋊C₃₂

—

C₂×C₇₁⋊C₃₂

C₂²

yes

—

C₂×C₇₁⋊C₃₂

C₂²

yes

—

C₂×M₃₂×C₇₁

C₄

yes

characteristic

C₂²×C₁₆×C₇₁

C₂²

yes

characteristic

#405×C₇₁

C₄

yes

characteristic

#346×C₇₁

C₄

yes

—

C₄×C₁₆×C₇₁

C₂²

yes

—

C₇₁⋊C₃₂

—

C₇₁⋊C₃₂

—

C₂×M₁₆×C₇₁

C₈

yes

characteristic

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

#117×C₇₁

C₈

yes

characteristic

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

D₄

yes

—

C₂×C₁₆×C₇₁

Q₈

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

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

C₂×C₈

yes

—

C₂×C₈×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

M₁₆

yes

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₇₁

C₁₆

yes

—

C₂²⋊C₄×C₇₁

C₁₆

yes

—

C₄²×C₇₁

C₂×C₈

yes

—

C₁₆×C₇₁

—

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

—

C₂×C₄×C₇₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₁

M₃₂

yes

—

C₂×C₄×C₇₁

C₂²×C₈

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

C₄⋊C₈

yes

—

C₈×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₈×C₇₁

C₂×M₁₆

yes

characteristic

C₂²×C₇₁

—

C₂²×C₇₁

C₄⋊C₁₆

yes

—

C₂²×C₇₁

C₂²×C₁₆

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

C₄⋊C₁₆

yes

—

C₄×C₇₁

C₂×C₄⋊C₈

yes

characteristic

C₄×C₇₁

C₂×M₃₂

yes

characteristic

#13628

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

C₂×C₄⋊C₁₆

yes

characteristic

C₂×C₇₁

#2258

yes

—

C₂×M₆₄

—

C₂²×C₃₂

—

#2108

D₇₁

yes

characteristic

C₄⋊C₃₂

—

C₇₁

#13628

yes

characteristic, complemented, Sylow

C₂×M₃₂

Dic₇₁

yes

characteristic

C₂²×C₁₆

C₂×D₇₁

yes

characteristic

#405

Dic₇₁

yes

characteristic

M₆₄

—

142

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

142

—

#346

Dic₇₁

yes

—

C₄×C₁₆

C₂×D₇₁

yes

—

C₂×M₁₆

C₇₁⋊C₈

yes

characteristic

C₂²×C₈

C₂×Dic₇₁

yes

characteristic

#117

C₇₁⋊C₈

yes

characteristic

M₃₂

—

C₂×C₁₆

Dic₁₄₂

yes

—

C₂×C₁₆

D₂₈₄

yes

—

C₂×C₁₆

C₂×Dic₇₁

yes

—

C₂×C₁₆

C₂×Dic₇₁

yes

characteristic

C₂×C₁₆

C₂²×D₇₁

yes

characteristic

#97

C₇₁⋊C₈

yes

—

C₄×C₈

C₂×Dic₇₁

yes

—

C₃₂

—

142

—

C₃₂

—

142

—

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

#11765849

yes

—

C₂×C₈

C₂²×Dic₇₁

yes

characteristic

C₄⋊C₄

C₇₁⋊C₁₆

yes

—

C₂²⋊C₄

C₇₁⋊C₁₆

yes

—

C₄²

C₂×C₇₁⋊C₈

yes

—

C₁₆

—

100

C₁₆

C₄⋊Dic₇₁

yes

—

101

C₁₆

C₂×D₂₈₄

yes

characteristic

102

C₁₆

C₂×Dic₁₄₂

yes

characteristic

103

C₂³

C₂×C₇₁⋊C₁₆

yes

characteristic

104

C₂×C₄

C₄⋊(C₇₁⋊C₈)

yes

—

105

C₂×C₄

C₂×C₇₁⋊C₁₆

yes

—

106

C₂×C₄

C₂×C₇₁⋊C₁₆

yes

characteristic

107

C₂×C₄

#14556068

yes

—

108

C₂×C₄

C₂²×C₇₁⋊C₈

yes

characteristic

109

C₈

—

110

C₈

C₄⋊(C₇₁⋊C₈)

yes

—

111

C₈

C₂×C₄⋊Dic₇₁

yes

characteristic

112

C₈

C₂×#11765849

yes

characteristic

113

C₂²

—

114

C₂²

C₄⋊(C₇₁⋊C₁₆)

yes

—

115

C₂²

C₂²×C₇₁⋊C₁₆

yes

characteristic

116

C₄

—

117

C₄

C₄⋊(C₇₁⋊C₁₆)

yes

—

118

C₄

C₂×C₄⋊(C₇₁⋊C₈)

yes

characteristic

119

C₄

C₂×#14556068

yes

characteristic

120

C₂

—

121

C₂

C₂×C₄⋊(C₇₁⋊C₁₆)

yes

characteristic

122

C₂

#22590634

yes

—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₇₁
commutator, socle	C₂²×C₇₁	C₂²×C₁₆
Fitting	C₇₁×#2108	C₂
prime radical	C₂³×C₇₁	C₂×C₁₆
cosocle	C₂×C₁₆×C₇₁	C₂³
hypercenter	#2108	D₇₁
hypocenter	C₇₁	#13628
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₁₆ ◁ #2108

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₇₁

commutator, socle

C₂²×C₇₁

C₂²×C₁₆

Fitting

C₇₁×#2108

C₂

prime radical

C₂³×C₇₁

C₂×C₁₆

cosocle

C₂×C₁₆×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂×C₁₆ ◁ #2108

N	H ≅ G/N	type
3. C₄⋊(C₇₁⋊C₃₂)	C₂	semidirect
66. C₇₁	#13628	semidirect

H ≅ G/N

type

3. C₄⋊(C₇₁⋊C₃₂)

C₂

semidirect

66. C₇₁

#13628

semidirect

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

subgroup	index
C₂×#17985833	2
C₂²×C₇₁⋊C₃₂	2
C₄⋊(C₇₁⋊C₃₂)	2
C₇₁×#2108	2
#13628	71

subgroup

index

C₂×#17985833

C₂²×C₇₁⋊C₃₂

C₄⋊(C₇₁⋊C₃₂)

C₇₁×#2108

#13628

Subgroups — #31917938

Conjugacy classes of subgroups

Complements

Maximal subgroups