Group #16845761 — subgroups

Subgroups — #16845761

Order 3968.

252 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 105 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	#14521033	C₂	1	2	yes	maximal
3	C₃₁⋊#338	C₂	1	2	yes	maximal, complemented
4	C₂×C₃₁×#131	C₂	1	1	yes	maximal, characteristic
5	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
6	C₃₁×#131	—	2	4	no	—
7	C₂×M₁₆×C₃₁	C₄	1	2	yes	—
8	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₃₁	C₄	1	2	yes	—
10	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×#92474	C₂²	1	4	yes	—
12	#11758424	—	2	2	no	—
13	#11758421	—	2	2	no	—
14	#11758407	—	2	4	no	—
15	#41×C₃₁	—	2	2	no	—
16	#41×C₃₁	—	2	2	no	—
17	C₂×Q₈×C₃₁	D₄	1	1	yes	characteristic
18	C₂×D₄×C₃₁	D₄	1	1	yes	characteristic
19	C₂×D₄×C₃₁	Q₈	1	2	yes	—
20	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
22	M₁₆×C₃₁	—	2	2	no	—
23	M₁₆×C₃₁	—	2	4	no	—
24	C₂×C₈×C₃₁	—	2	4	no	—
25	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₃₁	D₄	1	4	yes	—
29	#92474	—	2	4	no	—
30	#92474	—	2	4	no	—
31	C₂×C₃₁⋊C₈	—	2	4	no	—
32	C₂³×C₃₁	C₄⋊C₄	1	2	yes	—
33	C₂³×C₃₁	C₂×Q₈	1	1	yes	characteristic
34	Q₈×C₃₁	—	2	2	no	—
35	D₄×C₃₁	—	2	4	no	—
36	D₄×C₃₁	—	2	2	no	—
37	C₂×C₄×C₃₁	—	2	2	no	—
38	C₂×C₄×C₃₁	—	2	2	no	—
39	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₃₁	#41	1	2	yes	—
44	C₈×C₃₁	—	2	2	no	—
45	C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
46	C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
47	C₃₁⋊C₈	—	4	4	no	—
48	#1813	—	31	1	no	maximal, Sylow
49	C₂²×C₃₁	—	2	2	no	—
50	C₂²×C₃₁	—	4	1	no	—
51	C₂²×C₃₁	—	4	2	no	—
52	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₃₁	C₂²⋊Q₈	1	2	yes	—
54	C₄×C₃₁	—	2	2	no	—
55	C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₄×C₃₁	#117	1	1	yes	characteristic
57	C₄×C₃₁	#121	1	2	yes	—
58	C₂×#131	D₃₁	1	1	yes	characteristic
59	C₂×#108	—	31	2	no	—
60	#413	—	31	2	no	—
61	#338	—	31	2	no	—
62	C₂×C₃₁	—	2	1	no	—
63	C₂×C₃₁	—	2	2	no	—
64	C₂×C₃₁	—	4	2	no	—
65	C₂×C₃₁	#380	1	1	yes	characteristic
66	C₂×#41	C₂×D₃₁	1	1	yes	characteristic
67	#131	—	2	4	no	—
68	C₂×M₁₆	—	31	4	no	—
69	C₂×M₁₆	Dic₃₁	1	2	yes	—
70	C₂×M₁₆	C₂×D₃₁	1	1	yes	characteristic
71	C₂²×C₈	Dic₃₁	1	2	yes	—
72	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
73	#108	—	62	4	no	—
74	#101	—	62	2	no	—
75	#100	—	62	2	no	—
76	C₃₁	#1813	1	1	yes	characteristic, complemented, Sylow
77	#41	—	2	2	no	—
78	#41	—	2	2	no	—
79	C₂×Q₈	D₁₂₄	1	1	yes	characteristic
80	C₂×D₄	Dic₆₂	1	2	yes	—
81	C₂×D₄	D₁₂₄	1	1	yes	characteristic
82	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
83	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	—	2	4	no	—
86	M₁₆	—	62	4	no	—
87	M₁₆	—	62	4	no	—
88	C₂×C₈	—	2	4	no	—
89	C₂×C₈	—	62	4	no	—
90	C₂×C₈	C₂×Dic₃₁	1	2	yes	—
91	C₂×C₈	C₂×Dic₃₁	1	1	yes	characteristic
92	C₂×C₈	C₂×Dic₃₁	1	1	yes	characteristic
93	C₂×C₈	C₂²⋊D₃₁	1	4	yes	—
94	C₂³	C₄⋊Dic₃₁	1	2	yes	—
95	C₂³	C₂×Dic₆₂	1	1	yes	characteristic
96	Q₈	—	2	2	no	—
97	D₄	—	2	4	no	—
98	D₄	—	2	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	C₄⋊Dic₃₁	1	2	yes	—
102	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
103	C₂×C₄	#92495	1	2	yes	—
104	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
105	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
106	C₈	—	2	2	no	—
107	C₈	—	124	4	no	—
108	C₈	C₂²⋊Dic₃₁	1	2	yes	—
109	C₈	C₂²⋊Dic₃₁	1	2	yes	—
110	C₂²	—	2	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	4	2	no	—
113	C₂²	#11758501	1	2	yes	—
114	C₂²	C₂×C₄⋊Dic₃₁	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	#11758503	1	1	yes	characteristic
117	C₄	#11758507	1	2	yes	—
118	C₄	C₂×C₂²⋊Dic₃₁	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	2	no	—
121	C₂	—	4	2	no	—
122	C₂	#14520909	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758407

C₂

yes

maximal, complemented

#14521033

C₂

yes

maximal

C₃₁⋊#338

C₂

yes

maximal, complemented

C₂×C₃₁×#131

C₂

yes

maximal, characteristic

C₂×#41×C₃₁

C₂²

yes

characteristic

C₃₁×#131

—

C₂×M₁₆×C₃₁

C₄

yes

—

C₂×M₁₆×C₃₁

C₂²

yes

characteristic

C₂²×C₈×C₃₁

C₄

yes

—

C₂²×C₈×C₃₁

C₂²

yes

characteristic

C₂×#92474

C₂²

yes

—

#11758424

—

#11758421

—

#11758407

—

#41×C₃₁

—

#41×C₃₁

—

C₂×Q₈×C₃₁

D₄

yes

characteristic

C₂×D₄×C₃₁

D₄

yes

characteristic

C₂×D₄×C₃₁

Q₈

yes

—

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

M₁₆×C₃₁

—

M₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₂×C₈×C₃₁

C₂×C₄

yes

characteristic

C₂×C₈×C₃₁

C₂×C₄

yes

characteristic

C₂×C₈×C₃₁

D₄

yes

—

#92474

—

#92474

—

C₂×C₃₁⋊C₈

—

C₂³×C₃₁

C₄⋊C₄

yes

—

C₂³×C₃₁

C₂×Q₈

yes

characteristic

Q₈×C₃₁

—

D₄×C₃₁

—

D₄×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

—

C₈×C₃₁

C₂²⋊C₄

yes

—

C₃₁⋊C₈

—

#1813

—

maximal, Sylow

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

—

C₂×#131

D₃₁

yes

—

—

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

#380

yes

characteristic

C₂×#41

C₂×D₃₁

yes

characteristic

#131

—

C₂×M₁₆

—

C₂×M₁₆

Dic₃₁

yes

—

C₂×M₁₆

C₂×D₃₁

yes

characteristic

C₂²×C₈

Dic₃₁

yes

—

C₂²×C₈

C₂×D₃₁

yes

—

—

—

C₃₁

#1813

yes

characteristic, complemented, Sylow

#41

—

#41

—

C₂×Q₈

D₁₂₄

yes

characteristic

C₂×D₄

Dic₆₂

yes

—

C₂×D₄

D₁₂₄

yes

characteristic

C₂²×C₄

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₃₁

yes

—

C₂×C₈

C₂×Dic₃₁

yes

characteristic

C₂×C₈

C₂×Dic₃₁

yes

characteristic

C₂×C₈

C₂²⋊D₃₁

yes

—

C₂³

C₄⋊Dic₃₁

yes

—

C₂³

C₂×Dic₆₂

yes

characteristic

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₄⋊Dic₃₁

yes

—

102

C₂×C₄

C₂×D₁₂₄

yes

characteristic

103

C₂×C₄

#92495

yes

—

104

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

105

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

106

C₈

—

107

C₈

—

124

—

108

C₈

C₂²⋊Dic₃₁

yes

—

109

C₈

C₂²⋊Dic₃₁

yes

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

#11758501

yes

—

114

C₂²

C₂×C₄⋊Dic₃₁

yes

characteristic

115

C₄

—

116

C₄

#11758503

yes

characteristic

117

C₄

#11758507

yes

—

118

C₄

C₂×C₂²⋊Dic₃₁

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

—

122

C₂

#14520909

yes

characteristic

	subgroup	G/H
center	C₄	#11758503
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₂×C₃₁×#131	C₂
socle	C₂×C₃₁	#380
prime radical	C₂×#41×C₃₁	C₂²
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂×#131	D₃₁
hypocenter	C₃₁	#1813
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

subgroup

G/H

center

C₄

#11758503

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₂×C₃₁×#131

C₂

socle

C₂×C₃₁

#380

prime radical

C₂×#41×C₃₁

C₂²

cosocle

C₂²×C₄×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

N	H ≅ G/N	type
1. C₂×#11758407	C₂	semidirect
3. C₃₁⋊#338	C₂	semidirect
76. C₃₁	#1813	semidirect

H ≅ G/N

type

1. C₂×#11758407

C₂

semidirect

3. C₃₁⋊#338

C₂

semidirect

76. C₃₁

#1813

semidirect

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

subgroup	index
C₂×#11758407	2
#14521033	2
C₃₁⋊#338	2
C₂×C₃₁×#131	2
#1813	31

subgroup

index

C₂×#11758407

#14521033

C₃₁⋊#338

C₂×C₃₁×#131

#1813

Subgroups — #16845761

Conjugacy classes of subgroups

Complements

Maximal subgroups