Group #16845757 — subgroups

Subgroups — #16845757

Order 3968.

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

normal subgroups only

This group has 260 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	#14521087	C₂	1	2	yes	maximal, complemented
2	#14521023	C₂	1	2	yes	maximal
3	#14520806	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	2	no	—
7	C₃₁×#131	—	2	2	no	—
8	C₂×M₁₆×C₃₁	C₄	1	2	yes	complemented
9	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₃₁	C₄	1	2	yes	complemented
11	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	#11758503	C₂²	1	4	yes	—
13	#11758422	—	2	4	no	—
14	#11758406	—	2	2	no	—
15	#11758405	—	2	2	no	—
16	#41×C₃₁	—	2	2	no	—
17	#41×C₃₁	—	2	2	no	—
18	C₂×Q₈×C₃₁	Q₈	1	1	yes	characteristic
19	C₂×D₄×C₃₁	D₄	1	2	yes	—
20	C₂×D₄×C₃₁	Q₈	1	1	yes	characteristic
21	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
23	M₁₆×C₃₁	—	2	2	no	—
24	M₁₆×C₃₁	—	2	4	no	—
25	C₂×C₈×C₃₁	—	2	2	no	—
26	C₂×C₈×C₃₁	—	2	2	no	—
27	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
28	C₂×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₃₁	D₄	1	2	yes	—
31	C₂×C₈×C₃₁	D₄	1	2	yes	—
32	C₂²⋊Dic₃₁	—	2	4	no	—
33	C₄⋊Dic₃₁	—	2	4	no	—
34	C₄×Dic₃₁	—	2	4	no	—
35	C₂³×C₃₁	C₄⋊C₄	1	2	yes	—
36	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
37	Q₈×C₃₁	—	2	2	no	—
38	D₄×C₃₁	—	2	4	no	—
39	D₄×C₃₁	—	2	2	no	—
40	C₂×C₄×C₃₁	—	2	2	no	—
41	C₂×C₄×C₃₁	—	2	2	no	—
42	C₂×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₃₁	#41	1	2	yes	—
47	C₂×Dic₃₁	—	4	4	no	—
48	C₈×C₃₁	—	2	2	no	—
49	C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
52	#1812	—	31	1	no	maximal, Sylow
53	C₂²×C₃₁	—	2	2	no	—
54	C₂²×C₃₁	—	4	1	no	—
55	C₂²×C₃₁	—	4	2	no	—
56	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₃₁	#121	1	2	yes	—
58	C₄×C₃₁	—	2	2	no	—
59	C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₃₁	#117	1	1	yes	characteristic
61	C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
62	Dic₃₁	—	8	4	no	—
63	C₂×#131	D₃₁	1	1	yes	characteristic
64	#427	—	31	2	no	—
65	#410	—	31	2	no	—
66	H(3,ℤ/4)	—	31	2	no	—
67	C₂×C₃₁	—	2	2	no	—
68	C₂×C₃₁	—	2	1	no	—
69	C₂×C₃₁	—	4	2	no	—
70	C₂×C₃₁	#380	1	1	yes	characteristic
71	C₂×#41	C₂×D₃₁	1	1	yes	characteristic
72	#131	—	2	2	no	—
73	#131	—	2	2	no	—
74	C₂×M₁₆	Dic₃₁	1	2	yes	complemented
75	C₂×M₁₆	C₂×D₃₁	1	1	yes	characteristic
76	C₂²×C₈	Dic₃₁	1	2	yes	complemented
77	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
78	#117	—	31	4	no	—
79	#107	—	62	2	no	—
80	#106	—	62	2	no	—
81	#99	—	62	4	no	—
82	C₃₁	#1812	1	1	yes	characteristic, complemented, Sylow
83	#41	—	2	2	no	—
84	#41	—	2	2	no	—
85	C₂×Q₈	Dic₆₂	1	1	yes	characteristic
86	C₂×D₄	Dic₆₂	1	1	yes	characteristic
87	C₂×D₄	D₁₂₄	1	2	yes	—
88	C₂²×C₄	C₂×Dic₃₁	1	2	yes	—
89	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	4	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	C₂×Dic₃₁	1	2	yes	—
95	C₂×C₈	C₂×Dic₃₁	1	1	yes	characteristic
96	C₂×C₈	C₂×Dic₃₁	1	1	yes	characteristic
97	C₂×C₈	C₂²⋊D₃₁	1	2	yes	—
98	C₂×C₈	C₂²⋊D₃₁	1	2	yes	—
99	C₄⋊C₄	—	62	4	no	—
100	C₂²⋊C₄	—	62	4	no	—
101	C₄²	—	62	4	no	—
102	C₂³	C₄⋊Dic₃₁	1	2	yes	—
103	C₂³	C₂×D₁₂₄	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	D₄	—	2	4	no	—
106	D₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	124	4	no	—
110	C₂×C₄	C₄⋊Dic₃₁	1	2	yes	—
111	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
112	C₂×C₄	#92495	1	2	yes	—
113	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
114	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
115	C₈	—	2	2	no	—
116	C₈	C₂²⋊Dic₃₁	1	1	yes	characteristic
117	C₈	C₂²⋊Dic₃₁	1	1	yes	characteristic
118	C₈	C₂²⋊Dic₃₁	1	2	yes	—
119	C₂²	—	2	2	no	—
120	C₂²	—	4	1	no	—
121	C₂²	—	4	2	no	—
122	C₂²	C₂×C₄⋊Dic₃₁	1	1	yes	characteristic
123	C₂²	#11758507	1	2	yes	—
124	C₄	—	2	2	no	—
125	C₄	—	248	4	no	—
126	C₄	#11758501	1	2	yes	—
127	C₄	#11758503	1	1	yes	characteristic
128	C₄	C₂×C₂²⋊Dic₃₁	1	1	yes	characteristic
129	C₂	—	2	2	no	—
130	C₂	—	2	1	no	—
131	C₂	—	4	2	no	—
132	C₂	#14520909	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521087

C₂

yes

maximal, complemented

#14521023

C₂

yes

maximal

#14520806

C₂

yes

maximal, complemented

C₂×C₃₁×#131

C₂

yes

maximal, characteristic

C₂×#41×C₃₁

C₂²

yes

characteristic

C₃₁×#131

—

C₃₁×#131

—

C₂×M₁₆×C₃₁

C₄

yes

complemented

C₂×M₁₆×C₃₁

C₂²

yes

characteristic

C₂²×C₈×C₃₁

C₄

yes

complemented

C₂²×C₈×C₃₁

C₂²

yes

characteristic

#11758503

C₂²

yes

—

#11758422

—

#11758406

—

#11758405

—

#41×C₃₁

—

#41×C₃₁

—

C₂×Q₈×C₃₁

Q₈

yes

characteristic

C₂×D₄×C₃₁

D₄

yes

—

C₂×D₄×C₃₁

Q₈

yes

characteristic

C₂²×C₄×C₃₁

C₂×C₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

M₁₆×C₃₁

—

M₁₆×C₃₁

—

C₂×C₈×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

—

C₂×C₈×C₃₁

D₄

yes

—

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄×Dic₃₁

—

C₂³×C₃₁

C₄⋊C₄

yes

—

C₂³×C₃₁

C₂×D₄

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

—

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₈×C₃₁

—

C₈×C₃₁

C₂²⋊C₄

yes

characteristic

C₈×C₃₁

C₂²⋊C₄

yes

characteristic

C₈×C₃₁

C₂²⋊C₄

yes

—

#1812

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₁

#121

yes

—

C₄×C₃₁

—

C₄×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₃₁

#117

yes

characteristic

C₄×C₃₁

C₂²⋊Q₈

yes

—

Dic₃₁

—

C₂×#131

D₃₁

yes

—

—

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

#380

yes

characteristic

C₂×#41

C₂×D₃₁

yes

characteristic

#131

—

#131

—

C₂×M₁₆

Dic₃₁

yes

complemented

C₂×M₁₆

C₂×D₃₁

yes

characteristic

C₂²×C₈

Dic₃₁

yes

complemented

C₂²×C₈

C₂×D₃₁

yes

—

—

—

—

C₃₁

#1812

yes

characteristic, complemented, Sylow

#41

—

#41

—

C₂×Q₈

Dic₆₂

yes

characteristic

C₂×D₄

Dic₆₂

yes

characteristic

C₂×D₄

D₁₂₄

yes

—

C₂²×C₄

C₂×Dic₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

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

—

C₂×C₈

C₂²⋊D₃₁

yes

—

C₄⋊C₄

—

100

C₂²⋊C₄

—

101

C₄²

—

102

C₂³

C₄⋊Dic₃₁

yes

—

103

C₂³

C₂×D₁₂₄

yes

characteristic

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

124

—

110

C₂×C₄

C₄⋊Dic₃₁

yes

—

111

C₂×C₄

C₂×Dic₆₂

yes

characteristic

112

C₂×C₄

#92495

yes

—

113

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

114

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

115

C₈

—

116

C₈

C₂²⋊Dic₃₁

yes

characteristic

117

C₈

C₂²⋊Dic₃₁

yes

characteristic

118

C₈

C₂²⋊Dic₃₁

yes

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₂×C₄⋊Dic₃₁

yes

characteristic

123

C₂²

#11758507

yes

—

124

C₄

—

125

C₄

—

248

—

126

C₄

#11758501

yes

—

127

C₄

#11758503

yes

characteristic

128

C₄

C₂×C₂²⋊Dic₃₁

yes

characteristic

129

C₂

—

130

C₂

—

131

C₂

—

132

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	#1812
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. #14521087	C₂	semidirect
3. #14520806	C₂	semidirect
8. C₂×M₁₆×C₃₁	C₄	semidirect
10. C₂²×C₈×C₃₁	C₄	semidirect
74. C₂×M₁₆	Dic₃₁	semidirect
76. C₂²×C₈	Dic₃₁	semidirect
82. C₃₁	#1812	semidirect

H ≅ G/N

type

1. #14521087

C₂

semidirect

3. #14520806

C₂

semidirect

8. C₂×M₁₆×C₃₁

C₄

semidirect

10. C₂²×C₈×C₃₁

C₄

semidirect

74. C₂×M₁₆

Dic₃₁

semidirect

76. C₂²×C₈

Dic₃₁

semidirect

82. C₃₁

#1812

semidirect

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

subgroup	index
#14521087	2
#14521023	2
#14520806	2
C₂×C₃₁×#131	2
#1812	31

subgroup

index

C₂×C₃₁×#131

Subgroups — #16845757

Conjugacy classes of subgroups

Complements

Maximal subgroups