Group #16845753 — subgroups

Subgroups — #16845753

Order 3968.

272 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 109 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×#11758407	C₂	1	2	yes	maximal, complemented
2	#14521013	C₂	1	2	yes	maximal
3	#14520814	C₂	1	2	yes	maximal, complemented
4	C₂²×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
6	C₂×M₁₆×C₃₁	—	2	4	no	—
7	C₂×M₁₆×C₃₁	C₄	1	4	yes	—
8	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×#92474	C₂²	1	4	yes	—
11	C₂²⋊(C₃₁⋊C₈)	—	2	4	no	—
12	#11758407	—	2	4	no	—
13	C₂⁴×C₃₁	Q₈	1	1	yes	characteristic
14	C₂²×C₄×C₃₁	—	2	2	no	—
15	C₂²×C₄×C₃₁	—	2	2	no	—
16	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₃₁	D₄	1	2	yes	—
18	C₂²×C₄×C₃₁	Q₈	1	1	yes	characteristic
19	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
20	M₁₆×C₃₁	—	2	8	no	—
21	C₂×C₈×C₃₁	—	2	2	no	—
22	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₃₁	D₄	1	4	yes	—
25	#92474	—	2	4	no	—
26	#92474	—	2	4	no	—
27	C₂×C₃₁⋊C₈	—	2	4	no	—
28	C₂³×C₃₁	—	2	2	no	—
29	C₂³×C₃₁	—	4	2	no	—
30	C₂³×C₃₁	C₄⋊C₄	1	2	yes	—
31	C₂³×C₃₁	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄×C₃₁	—	2	2	no	—
33	C₂×C₄×C₃₁	—	2	2	no	—
34	C₂×C₄×C₃₁	—	2	2	no	—
35	C₂×C₄×C₃₁	—	2	4	no	—
36	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
37	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₃₁	#41	1	2	yes	—
41	C₈×C₃₁	—	2	2	no	—
42	C₈×C₃₁	C₂²⋊C₄	1	4	yes	—
43	C₃₁⋊C₈	—	4	4	no	—
44	#1811	—	31	1	no	maximal, Sylow
45	C₂²×C₃₁	—	2	2	no	—
46	C₂²×C₃₁	—	2	2	no	—
47	C₂²×C₃₁	—	2	4	no	—
48	C₂²×C₃₁	—	4	4	no	—
49	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₃₁	C₂²⋊Q₈	1	2	yes	—
51	C₄×C₃₁	—	2	2	no	—
52	C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₃₁	#117	1	1	yes	characteristic
54	C₄×C₃₁	#121	1	2	yes	—
55	C₂²×M₁₆	D₃₁	1	1	yes	characteristic
56	C₂×#108	—	31	2	no	—
57	#407	—	31	2	no	—
58	#341	—	31	2	no	—
59	C₂×C₃₁	—	2	2	no	—
60	C₂×C₃₁	—	4	2	no	—
61	C₂×C₃₁	#380	1	1	yes	characteristic
62	C₂×C₃₁	#430	1	2	yes	—
63	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
64	C₂×M₁₆	—	2	4	no	—
65	C₂×M₁₆	—	31	4	no	—
66	C₂×M₁₆	Dic₃₁	1	4	yes	—
67	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
68	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
69	#108	—	62	4	no	—
70	C₂²⋊C₈	—	62	4	no	—
71	C₃₁	#1811	1	1	yes	characteristic, complemented, Sylow
72	C₂⁴	Dic₆₂	1	1	yes	characteristic
73	C₂²×C₄	—	2	2	no	—
74	C₂²×C₄	—	2	2	no	—
75	C₂²×C₄	Dic₆₂	1	1	yes	characteristic
76	C₂²×C₄	D₁₂₄	1	2	yes	—
77	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
78	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
79	M₁₆	—	2	8	no	—
80	M₁₆	—	62	4	no	—
81	M₁₆	—	62	4	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	62	4	no	—
84	C₂×C₈	C₂×Dic₃₁	1	2	yes	—
85	C₂×C₈	C₂×Dic₃₁	1	2	yes	—
86	C₂×C₈	C₂²⋊D₃₁	1	4	yes	—
87	C₂³	—	2	2	no	—
88	C₂³	—	4	2	no	—
89	C₂³	C₄⋊Dic₃₁	1	2	yes	—
90	C₂³	C₂×Dic₆₂	1	1	yes	characteristic
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	C₄⋊Dic₃₁	1	2	yes	—
96	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
97	C₂×C₄	#92495	1	2	yes	—
98	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
99	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
100	C₈	—	2	2	no	—
101	C₈	—	124	4	no	—
102	C₈	C₂²⋊Dic₃₁	1	4	yes	—
103	C₂²	—	2	2	no	—
104	C₂²	—	2	4	no	—
105	C₂²	—	2	2	no	—
106	C₂²	—	4	4	no	—
107	C₂²	#11758501	1	2	yes	—
108	C₂²	C₂×C₄⋊Dic₃₁	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	#11758503	1	1	yes	characteristic
111	C₄	#11758507	1	2	yes	—
112	C₄	C₂×C₂²⋊Dic₃₁	1	1	yes	characteristic
113	C₂	—	2	2	no	—
114	C₂	—	4	2	no	—
115	C₂	#14520909	1	1	yes	characteristic
116	C₂	#14521098	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758407

C₂

yes

maximal, complemented

#14521013

C₂

yes

maximal

#14520814

C₂

yes

maximal, complemented

C₂²×M₁₆×C₃₁

C₂

yes

maximal, characteristic

C₂³×C₄×C₃₁

C₂²

yes

characteristic

C₂×M₁₆×C₃₁

—

C₂×M₁₆×C₃₁

C₄

yes

—

C₂²×C₈×C₃₁

C₂²

yes

characteristic

C₂²×C₈×C₃₁

C₂²

yes

characteristic

C₂×#92474

C₂²

yes

—

C₂²⋊(C₃₁⋊C₈)

—

#11758407

—

C₂⁴×C₃₁

Q₈

yes

characteristic

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

Q₈

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

characteristic

M₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₂×C₈×C₃₁

D₄

yes

—

#92474

—

#92474

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

C₄⋊C₄

yes

—

C₂³×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₄⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₈×C₃₁

—

C₈×C₃₁

C₂²⋊C₄

yes

—

C₃₁⋊C₈

—

#1811

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₁

C₂²⋊Q₈

yes

—

C₄×C₃₁

—

C₄×C₃₁

C₂×C₂²⋊C₄

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

—

C₂²×M₁₆

D₃₁

yes

—

—

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

#380

yes

characteristic

C₂×C₃₁

#430

yes

—

C₂³×C₄

C₂×D₃₁

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

Dic₃₁

yes

—

C₂²×C₈

C₂×D₃₁

yes

characteristic

C₂²×C₈

C₂×D₃₁

yes

characteristic

#108

—

C₂²⋊C₈

—

C₃₁

#1811

yes

characteristic, complemented, Sylow

C₂⁴

Dic₆₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₆₂

yes

characteristic

C₂²×C₄

D₁₂₄

yes

—

C₂²×C₄

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₃₁

yes

—

C₂×C₈

C₂×Dic₃₁

yes

—

C₂×C₈

C₂²⋊D₃₁

yes

—

C₂³

—

C₂³

—

C₂³

C₄⋊Dic₃₁

yes

—

C₂³

C₂×Dic₆₂

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₃₁

yes

—

C₂×C₄

C₂×D₁₂₄

yes

characteristic

C₂×C₄

#92495

yes

—

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

100

C₈

—

101

C₈

—

124

—

102

C₈

C₂²⋊Dic₃₁

yes

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

#11758501

yes

—

108

C₂²

C₂×C₄⋊Dic₃₁

yes

characteristic

109

C₄

—

110

C₄

#11758503

yes

characteristic

111

C₄

#11758507

yes

—

112

C₄

C₂×C₂²⋊Dic₃₁

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

#14520909

yes

characteristic

116

C₂

#14521098

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₂₄
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₂²×M₁₆×C₃₁	C₂
socle	C₂²×C₃₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₃₁	Q₈
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂²×M₁₆	D₃₁
hypocenter	C₃₁	#1811
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×D₁₂₄

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₂²×M₁₆×C₃₁

C₂

socle

C₂²×C₃₁

C₂×C₄⋊C₄

prime radical

C₂⁴×C₃₁

Q₈

cosocle

C₂²×C₄×C₃₁

C₂³

hypercenter

C₂²×M₁₆

D₃₁

hypocenter

C₃₁

#1811

LCS

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

N	H ≅ G/N	type
1. C₂×#11758407	C₂	semidirect
3. #14520814	C₂	semidirect
71. C₃₁	#1811	semidirect

H ≅ G/N

type

1. C₂×#11758407

C₂

semidirect

3. #14520814

C₂

semidirect

71. C₃₁

#1811

semidirect

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

subgroup	index
C₂×#11758407	2
#14521013	2
#14520814	2
C₂²×M₁₆×C₃₁	2
#1811	31

subgroup

index

C₂²×M₁₆×C₃₁

Subgroups — #16845753

Conjugacy classes of subgroups

Complements

Maximal subgroups