Group #31918087 — subgroups

Subgroups — #31918087

Order 18176.

158 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 91 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	C₁₆×Dic₁₄₂	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₇₁⋊C₃₂)	C₂	1	1	yes	maximal, characteristic
3	Dic₇₁⋊C₃₂	C₂	1	2	yes	maximal
4	#22586473	C₂	1	2	yes	maximal
5	C₄×C₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₁₄₂	C₄	1	1	yes	characteristic
7	C₂×C₇₁⋊C₃₂	C₂²	1	2	yes	—
8	C₄⋊(C₇₁⋊C₁₆)	C₄	1	1	yes	characteristic
9	Dic₇₁⋊C₁₆	C₄	1	2	yes	—
10	C₁₆×Dic₇₁	C₂²	1	2	yes	—
11	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
12	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
14	C₄×Dic₁₄₂	C₈	1	1	yes	characteristic
15	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
16	C₄⋊(C₇₁⋊C₈)	C₈	1	1	yes	characteristic
17	Dic₇₁⋊C₈	C₈	1	2	yes	—
18	C₈×Dic₇₁	C₂×C₄	1	2	yes	—
19	C₇₁⋊C₃₂	—	2	2	no	—
20	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
22	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₃₂×C₇₁	—	2	1	no	—
24	C₃₂×C₇₁	Q₈	1	2	yes	complemented
25	C₂×Dic₁₄₂	C₁₆	1	1	yes	characteristic
26	C₂×C₇₁⋊C₈	C₂×C₈	1	2	yes	—
27	C₄⋊Dic₇₁	C₁₆	1	1	yes	characteristic
28	Dic₇₁⋊C₄	C₁₆	1	2	yes	—
29	C₄×Dic₇₁	C₂×C₈	1	2	yes	—
30	C₇₁⋊C₁₆	—	2	2	no	—
31	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
33	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
34	C₁₆×C₇₁	—	2	1	no	—
35	C₁₆×C₇₁	C₂×Q₈	1	1	yes	characteristic
36	C₁₆×C₇₁	#41	1	1	yes	characteristic
37	C₂×C₄×C₇₁	C₂×C₁₆	1	1	yes	characteristic
38	C₂×C₄×C₇₁	C₂×C₁₆	1	1	yes	characteristic
39	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
40	C₂×Dic₇₁	C₂×C₁₆	1	2	yes	—
41	C₂×Dic₇₁	C₂×C₁₆	1	2	yes	—
42	Dic₁₄₂	—	2	2	no	—
43	C₈×C₇₁	—	2	1	no	—
44	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
45	C₈×C₇₁	#131	1	1	yes	characteristic
46	C₇₁⋊C₈	—	2	2	no	—
47	C₂²×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
48	C₄×C₇₁	—	2	1	no	—
49	C₄×C₇₁	M₆₄	1	2	yes	—
50	C₄×C₇₁	C₈×Q₈	1	1	yes	characteristic
51	C₄×C₇₁	#504	1	1	yes	characteristic
52	Dic₇₁	—	2	2	no	—
53	Dic₇₁	—	2	2	no	—
54	#13660	—	71	1	no	maximal, Sylow
55	C₂×C₇₁	Q₈×C₁₆	1	1	yes	characteristic
56	C₂×C₇₁	C₂×M₆₄	1	1	yes	characteristic
57	C₂×C₇₁	#2258	1	1	yes	characteristic
58	Q₈×C₁₆	—	71	1	no	—
59	C₄⋊C₃₂	—	71	2	no	—
60	C₄⋊C₃₂	—	71	1	no	—
61	#1397	—	71	2	no	—
62	C₄×C₃₂	D₇₁	1	1	yes	characteristic
63	C₇₁	#13660	1	1	yes	characteristic, complemented, Sylow
64	C₈×Q₈	—	71	1	no	—
65	C₂×C₃₂	—	71	2	no	—
66	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
67	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
68	C₄⋊C₁₆	—	71	2	no	—
69	C₄⋊C₁₆	—	71	1	no	—
70	C₄×C₁₆	—	71	2	no	—
71	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
72	C₄×Q₈	—	71	1	no	—
73	C₂×C₁₆	—	71	2	no	—
74	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
75	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
76	C₄⋊C₈	—	71	2	no	—
77	C₄⋊C₈	—	71	1	no	—
78	C₄×C₈	—	71	2	no	—
79	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
80	C₃₂	—	2	1	no	—
81	C₃₂	—	142	2	no	—
82	C₃₂	Dic₁₄₂	1	2	yes	complemented
83	C₂×Q₈	—	71	1	no	—
84	C₂×C₈	—	71	2	no	—
85	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
86	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
87	C₄⋊C₄	—	71	2	no	—
88	C₄⋊C₄	—	71	1	no	—
89	C₄²	—	71	2	no	—
90	C₄²	C₈×D₇₁	1	1	yes	characteristic
91	C₁₆	—	2	1	no	—
92	C₁₆	—	142	2	no	—
93	C₁₆	C₂×Dic₁₄₂	1	1	yes	characteristic
94	C₁₆	#11765865	1	1	yes	characteristic
95	Q₈	—	142	2	no	—
96	C₂×C₄	—	71	2	no	—
97	C₂×C₄	—	71	2	no	—
98	C₂×C₄	C₁₆×D₇₁	1	1	yes	characteristic
99	C₂×C₄	C₁₆×D₇₁	1	1	yes	characteristic
100	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	142	2	no	—
103	C₈	C₄×Dic₁₄₂	1	1	yes	characteristic
104	C₈	#14556134	1	1	yes	characteristic
105	C₂²	C₂×C₁₆×D₇₁	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	142	2	no	—
108	C₄	—	142	2	no	—
109	C₄	#17985832	1	2	yes	—
110	C₄	C₈×Dic₁₄₂	1	1	yes	characteristic
111	C₄	#17986415	1	1	yes	characteristic
112	C₂	C₁₆×Dic₁₄₂	1	1	yes	characteristic
113	C₂	C₂×#17985832	1	1	yes	characteristic
114	C₂	#22590633	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×Dic₁₄₂

C₂

yes

maximal, characteristic

C₄⋊(C₇₁⋊C₃₂)

C₂

yes

maximal, characteristic

Dic₇₁⋊C₃₂

C₂

yes

maximal

#22586473

C₂

yes

maximal

C₄×C₃₂×C₇₁

C₂

yes

maximal, characteristic

C₈×Dic₁₄₂

C₄

yes

characteristic

C₂×C₇₁⋊C₃₂

C₂²

yes

—

C₄⋊(C₇₁⋊C₁₆)

C₄

yes

characteristic

Dic₇₁⋊C₁₆

C₄

yes

—

C₁₆×Dic₇₁

C₂²

yes

—

C₂×C₃₂×C₇₁

C₂²

yes

characteristic

C₂×C₃₂×C₇₁

C₂²

yes

characteristic

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₄×Dic₁₄₂

C₈

yes

characteristic

C₂×C₇₁⋊C₁₆

C₂×C₄

yes

—

C₄⋊(C₇₁⋊C₈)

C₈

yes

characteristic

Dic₇₁⋊C₈

C₈

yes

—

C₈×Dic₇₁

C₂×C₄

yes

—

C₇₁⋊C₃₂

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₁

C₂³

yes

characteristic

C₄×C₈×C₇₁

C₂×C₄

yes

characteristic

C₃₂×C₇₁

—

C₃₂×C₇₁

Q₈

yes

complemented

C₂×Dic₁₄₂

C₁₆

yes

characteristic

C₂×C₇₁⋊C₈

C₂×C₈

yes

—

C₄⋊Dic₇₁

C₁₆

yes

characteristic

Dic₇₁⋊C₄

C₁₆

yes

—

C₄×Dic₇₁

C₂×C₈

yes

—

C₇₁⋊C₁₆

—

C₂×C₈×C₇₁

C₂×C₈

yes

characteristic

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₄²×C₇₁

C₂×C₈

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

C₂×Q₈

yes

characteristic

C₁₆×C₇₁

#41

yes

characteristic

C₂×C₄×C₇₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₈

yes

characteristic

C₂×Dic₇₁

C₂×C₁₆

yes

—

C₂×Dic₇₁

C₂×C₁₆

yes

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

C₄×Q₈

yes

characteristic

C₈×C₇₁

#131

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

C₂²×C₁₆

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

M₆₄

yes

—

C₄×C₇₁

C₈×Q₈

yes

characteristic

C₄×C₇₁

#504

yes

characteristic

Dic₇₁

—

Dic₇₁

—

#13660

—

maximal, Sylow

C₂×C₇₁

Q₈×C₁₆

yes

characteristic

C₂×C₇₁

C₂×M₆₄

yes

characteristic

C₂×C₇₁

#2258

yes

characteristic

Q₈×C₁₆

—

C₄⋊C₃₂

—

C₄⋊C₃₂

—

#1397

—

C₄×C₃₂

D₇₁

yes

characteristic

C₇₁

#13660

yes

characteristic, complemented, Sylow

C₈×Q₈

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₇₁

yes

characteristic

C₂×C₃₂

C₂×D₇₁

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₇₁

yes

characteristic

C₂×C₁₆

C₂²×D₇₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₇₁

yes

characteristic

C₃₂

—

C₃₂

—

142

—

C₃₂

Dic₁₄₂

yes

complemented

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

C₈×D₇₁

yes

characteristic

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₈×D₇₁

yes

characteristic

C₁₆

—

C₁₆

—

142

—

C₁₆

C₂×Dic₁₄₂

yes

characteristic

C₁₆

#11765865

yes

characteristic

Q₈

—

142

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₁₆×D₇₁

yes

characteristic

C₂×C₄

C₁₆×D₇₁

yes

characteristic

100

C₂×C₄

C₂×C₈×D₇₁

yes

characteristic

101

C₈

—

102

C₈

—

142

—

103

C₈

C₄×Dic₁₄₂

yes

characteristic

104

C₈

#14556134

yes

characteristic

105

C₂²

C₂×C₁₆×D₇₁

yes

characteristic

106

C₄

—

107

C₄

—

142

—

108

C₄

—

142

—

109

C₄

#17985832

yes

—

110

C₄

C₈×Dic₁₄₂

yes

characteristic

111

C₄

#17986415

yes

characteristic

112

C₂

C₁₆×Dic₁₄₂

yes

characteristic

113

C₂

C₂×#17985832

yes

characteristic

114

C₂

#22590633

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₇₁
commutator, socle, prime radical	C₂²×C₇₁	C₂²×C₁₆
Fitting	C₄×C₃₂×C₇₁	C₂
cosocle	C₂×C₁₆×C₇₁	C₂³
hypercenter	C₄×C₃₂	D₇₁
hypocenter	C₇₁	#13660
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₃₂

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₇₁

commutator, socle, prime radical

C₂²×C₇₁

C₂²×C₁₆

Fitting

C₄×C₃₂×C₇₁

C₂

cosocle

C₂×C₁₆×C₇₁

C₂³

hypercenter

C₄×C₃₂

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂×C₁₆ ◁ C₄×C₃₂

N	H ≅ G/N	type
24. C₃₂×C₇₁	Q₈	semidirect
63. C₇₁	#13660	semidirect
82. C₃₂	Dic₁₄₂	semidirect

H ≅ G/N

type

24. C₃₂×C₇₁

Q₈

semidirect

63. C₇₁

#13660

semidirect

82. C₃₂

Dic₁₄₂

semidirect

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

subgroup	index
C₁₆×Dic₁₄₂	2
C₄⋊(C₇₁⋊C₃₂)	2
Dic₇₁⋊C₃₂	2
#22586473	2
C₄×C₃₂×C₇₁	2
#13660	71

subgroup

index

C₁₆×Dic₁₄₂

C₄⋊(C₇₁⋊C₃₂)

Dic₇₁⋊C₃₂

#22586473

C₄×C₃₂×C₇₁

#13660

Subgroups — #31918087

Conjugacy classes of subgroups

Complements

Maximal subgroups