Group #16846499 — subgroups

Subgroups — #16846499

Order 3968.

184 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 85 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	#14521097	C₂	1	2	yes	maximal
2	C₂×#11758407	C₂	1	1	yes	maximal, characteristic
3	#14521004	C₂	1	1	yes	maximal, characteristic
4	#14520820	C₂	1	2	yes	maximal
5	C₂×C₃₁×#108	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
7	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	#11758503	C₂²	1	1	yes	characteristic
9	C₂×#92474	C₂²	1	2	yes	—
10	C₂×#92474	C₂²	1	1	yes	characteristic
11	C₃₁×#108	—	2	1	no	—
12	C₃₁×#108	—	2	1	no	—
13	#11758410	—	2	4	no	—
14	#11758407	C₄	1	4	yes	complemented
15	#11758403	—	2	1	no	—
16	C₈×Dic₃₁	—	2	1	no	—
17	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
18	M₁₆×C₃₁	—	2	2	no	—
19	M₁₆×C₃₁	—	2	2	no	—
20	C₂×C₈×C₃₁	—	2	2	no	—
21	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₃₁	D₄	1	1	yes	characteristic
23	C₂×C₈×C₃₁	D₄	1	1	yes	characteristic
24	C₂×C₈×C₃₁	Q₈	1	2	yes	—
25	C₂²⋊Dic₃₁	Q₈	1	2	yes	—
26	C₄⋊Dic₃₁	—	2	1	no	—
27	C₄×Dic₃₁	—	2	1	no	—
28	#92474	—	2	2	no	—
29	#92474	—	2	1	no	—
30	#92474	C₂×C₄	1	4	yes	—
31	#92474	D₄	1	2	yes	—
32	C₂×C₃₁⋊C₈	—	2	1	no	—
33	C₂×C₃₁⋊C₈	—	2	2	no	—
34	C₂³×C₃₁	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
40	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
41	C₂×Dic₃₁	—	2	2	no	—
42	C₈×C₃₁	—	4	2	no	—
43	C₈×C₃₁	C₄⋊C₄	1	2	yes	—
44	C₈×C₃₁	C₄⋊C₄	1	2	yes	—
45	C₃₁⋊C₈	—	2	4	no	—
46	C₃₁⋊C₈	—	2	2	no	—
47	#1953	—	31	1	no	maximal, Sylow
48	C₂²×C₃₁	—	4	1	no	—
49	C₂²×C₃₁	C₄×Q₈	1	1	yes	characteristic
50	C₂²×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
51	C₂²×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
52	C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
54	C₄×C₃₁	#121	1	1	yes	characteristic
55	C₄×C₃₁	#125	1	1	yes	characteristic
56	Dic₃₁	—	4	2	no	—
57	#430	—	31	2	no	—
58	C₂×#108	—	31	1	no	—
59	C₂×#108	D₃₁	1	1	yes	characteristic
60	#405	—	31	1	no	—
61	#344	—	31	2	no	—
62	C₂×C₃₁	—	2	1	no	—
63	C₂×C₃₁	—	2	1	no	—
64	C₂×C₃₁	—	2	1	no	—
65	C₂×C₃₁	#389	1	1	yes	characteristic
66	C₂×M₁₆	—	31	2	no	—
67	C₂×M₁₆	—	31	1	no	—
68	C₂×M₁₆	C₂×D₃₁	1	2	yes	—
69	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
70	#117	—	31	1	no	—
71	#108	—	2	1	no	—
72	#108	—	2	1	no	—
73	#108	—	31	4	no	—
74	#108	—	62	4	no	—
75	#97	—	62	1	no	—
76	C₄×C₈	—	62	1	no	—
77	C₃₁	#1953	1	1	yes	characteristic, complemented, Sylow
78	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
79	M₁₆	—	2	2	no	—
80	M₁₆	—	2	2	no	—
81	M₁₆	—	31	4	no	—
82	M₁₆	—	31	2	no	—
83	M₁₆	—	62	1	no	—
84	M₁₆	—	62	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	62	2	no	—
87	C₂×C₈	—	62	1	no	—
88	C₂×C₈	Dic₆₂	1	2	yes	—
89	C₂×C₈	C₄×D₃₁	1	2	yes	—
90	C₂×C₈	C₂²⋊D₃₁	1	1	yes	characteristic
91	C₂×C₈	C₂²⋊D₃₁	1	1	yes	characteristic
92	C₄⋊C₄	—	62	1	no	—
93	C₂²⋊C₄	—	31	2	no	—
94	C₄²	—	62	1	no	—
95	C₂³	Q₈×D₃₁	1	1	yes	characteristic
96	C₂×C₄	—	62	2	no	—
97	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
98	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
99	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
100	C₂×C₄	#92497	1	1	yes	characteristic
101	C₂×C₄	#92499	1	1	yes	characteristic
102	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
103	C₈	—	4	2	no	—
104	C₈	—	62	4	no	—
105	C₈	—	62	2	no	—
106	C₈	Dic₃₁⋊C₄	1	2	yes	—
107	C₈	Dic₃₁⋊C₄	1	2	yes	—
108	C₂²	—	4	1	no	—
109	C₂²	#11758464	1	1	yes	characteristic
110	C₂²	#11758465	1	1	yes	characteristic
111	C₂²	#11758523	1	1	yes	characteristic
112	C₄	—	124	2	no	—
113	C₄	#11758467	1	1	yes	characteristic
114	C₄	#11758470	1	1	yes	characteristic
115	C₄	C₂×Dic₃₁⋊C₄	1	1	yes	characteristic
116	C₄	#11758515	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	2	1	no	—
119	C₂	—	2	1	no	—
120	C₂	#14520945	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521097

C₂

yes

maximal

C₂×#11758407

C₂

yes

maximal, characteristic

#14521004

C₂

yes

maximal, characteristic

#14520820

C₂

yes

maximal

C₂×C₃₁×#108

C₂

yes

maximal, characteristic

C₂×M₁₆×C₃₁

C₂²

yes

—

C₂²×C₈×C₃₁

C₂²

yes

characteristic

#11758503

C₂²

yes

characteristic

C₂×#92474

C₂²

yes

—

C₂×#92474

C₂²

yes

characteristic

C₃₁×#108

—

C₃₁×#108

—

#11758410

—

#11758407

C₄

yes

complemented

#11758403

—

C₈×Dic₃₁

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

M₁₆×C₃₁

—

M₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₂×C₈×C₃₁

D₄

yes

characteristic

C₂×C₈×C₃₁

D₄

yes

characteristic

C₂×C₈×C₃₁

Q₈

yes

—

C₂²⋊Dic₃₁

Q₈

yes

—

C₄⋊Dic₃₁

—

C₄×Dic₃₁

—

#92474

—

#92474

—

#92474

C₂×C₄

yes

—

#92474

D₄

yes

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

C₂×C₄×C₃₁

yes

C₂×C₄×C₃₁

yes

C₂×Dic₃₁

—

C₈×C₃₁

—

C₈×C₃₁

C₄⋊C₄

yes

—

C₈×C₃₁

C₄⋊C₄

yes

—

C₃₁⋊C₈

—

C₃₁⋊C₈

—

#1953

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

C₄×Q₈

yes

characteristic

C₂²×C₃₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₃₁

C₄⋊Q₈

yes

characteristic

C₄×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₃₁

C₄×D₄

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

Dic₃₁

—

#430

—

C₂×#108

—

C₂×#108

D₃₁

yes

characteristic

#405

—

#344

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

#389

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₁

yes

—

C₂²×C₈

C₂×D₃₁

yes

—

—

—

—

—

—

C₄×C₈

—

C₃₁

#1953

yes

characteristic, complemented, Sylow

C₂²×C₄

C₂²×D₃₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₆₂

yes

—

C₂×C₈

C₄×D₃₁

yes

—

C₂×C₈

C₂²⋊D₃₁

yes

characteristic

C₂×C₈

C₂²⋊D₃₁

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

Q₈×D₃₁

yes

characteristic

C₂×C₄

—

C₂×C₄

C₂×Dic₆₂

yes

characteristic

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

C₂×C₄

D₄×D₃₁

yes

characteristic

100

C₂×C₄

#92497

yes

characteristic

101

C₂×C₄

#92499

yes

characteristic

102

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

103

C₈

—

104

C₈

—

105

C₈

—

106

C₈

Dic₃₁⋊C₄

yes

—

107

C₈

Dic₃₁⋊C₄

yes

—

108

C₂²

—

109

C₂²

#11758464

yes

characteristic

110

C₂²

#11758465

yes

characteristic

111

C₂²

#11758523

yes

characteristic

112

C₄

—

124

—

113

C₄

#11758467

yes

characteristic

114

C₄

#11758470

yes

characteristic

115

C₄

C₂×Dic₃₁⋊C₄

yes

characteristic

116

C₄

#11758515

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

120

C₂

#14520945

yes

characteristic

	subgroup	G/H
center	C₄	#11758515
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₂×C₃₁×#108	C₂
socle	C₂×C₃₁	#389
prime radical	C₂³×C₃₁	C₂×Q₈
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂×#108	D₃₁
hypocenter	C₃₁	#1953
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₄

#11758515

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₂×C₃₁×#108

C₂

socle

C₂×C₃₁

#389

prime radical

C₂³×C₃₁

C₂×Q₈

cosocle

C₂²×C₄×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
14. #11758407	C₄	semidirect
77. C₃₁	#1953	semidirect

H ≅ G/N

type

14. #11758407

C₄

semidirect

77. C₃₁

#1953

semidirect

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

subgroup	index
#14521097	2
C₂×#11758407	2
#14521004	2
#14520820	2
C₂×C₃₁×#108	2
#1953	31

subgroup

index

C₂×C₃₁×#108

Subgroups — #16846499

Conjugacy classes of subgroups

Complements

Maximal subgroups