Group #16845747 — subgroups

Subgroups — #16845747

Order 3968.

292 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 101 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	C₂³×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#11758410	C₂	1	2	yes	maximal, complemented
3	#14521013	C₂	1	1	yes	maximal, characteristic
4	#14520813	C₂	1	2	yes	maximal, complemented
5	C₃₁×#407	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
8	C₂×#92474	C₂²	1	2	yes	—
9	C₂²×C₃₁⋊C₈	—	2	4	no	—
10	C₂²×C₃₁⋊C₈	C₄	1	4	yes	—
11	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₂²×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
13	C₃₁×C₂²⋊C₈	—	2	2	no	—
14	C₂²⋊(C₃₁⋊C₈)	—	2	2	no	—
15	#11758410	—	2	4	no	—
16	C₂⁴×C₃₁	Q₈	1	1	yes	characteristic
17	C₂²×C₄×C₃₁	—	2	2	no	—
18	C₂²×C₄×C₃₁	—	2	2	no	—
19	C₂²×C₄×C₃₁	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₃₁	D₄	1	2	yes	—
21	C₂²×C₄×C₃₁	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
23	M₁₆×C₃₁	—	2	2	no	—
24	M₁₆×C₃₁	—	2	2	no	—
25	C₂×C₈×C₃₁	—	2	2	no	—
26	#92474	—	2	2	no	—
27	#92474	—	2	2	no	—
28	C₂×C₃₁⋊C₈	—	2	2	no	—
29	C₂×C₃₁⋊C₈	—	2	8	no	—
30	C₂×C₃₁⋊C₈	—	2	2	no	—
31	C₂×C₃₁⋊C₈	C₂×C₄	1	2	yes	—
32	C₂×C₃₁⋊C₈	C₂×C₄	1	2	yes	—
33	C₂×C₃₁⋊C₈	D₄	1	4	yes	—
34	C₂³×C₃₁	—	2	4	no	—
35	C₂³×C₃₁	—	2	2	no	—
36	C₂³×C₃₁	C₄⋊C₄	1	2	yes	—
37	C₂³×C₃₁	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₃₁	—	2	2	no	—
39	C₂×C₄×C₃₁	—	2	2	no	—
40	C₂×C₄×C₃₁	—	2	2	no	—
41	C₂×C₄×C₃₁	—	2	4	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×D₄	1	1	yes	characteristic
46	C₂×C₄×C₃₁	#41	1	2	yes	—
47	C₈×C₃₁	—	4	2	no	—
48	C₃₁⋊C₈	—	2	2	no	—
49	C₃₁⋊C₈	—	4	2	no	—
50	C₃₁⋊C₈	C₂²⋊C₄	1	4	yes	—
51	#1810	—	31	1	no	maximal, Sylow
52	C₂²×C₃₁	—	2	8	no	—
53	C₂²×C₃₁	—	2	2	no	—
54	C₂²×C₃₁	—	2	2	no	—
55	C₂²×C₃₁	—	2	2	no	—
56	C₂²×C₃₁	#108	1	4	yes	complemented
57	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₃₁	C₂²⋊Q₈	1	2	yes	—
59	C₄×C₃₁	—	2	2	no	—
60	C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₄×C₃₁	#117	1	1	yes	characteristic
62	C₄×C₃₁	#121	1	2	yes	—
63	C₂³×C₈	—	31	1	no	—
64	C₂×#108	—	31	2	no	—
65	#407	—	31	1	no	—
66	#407	D₃₁	1	1	yes	characteristic
67	#341	—	31	2	no	—
68	C₂×C₃₁	—	2	4	no	—
69	C₂×C₃₁	—	2	2	no	—
70	C₂×C₃₁	#380	1	1	yes	characteristic
71	C₂×C₃₁	C₂×#108	1	2	yes	—
72	C₂³×C₄	C₂×D₃₁	1	1	yes	characteristic
73	C₂×M₁₆	—	31	2	no	—
74	C₂×M₁₆	C₂×D₃₁	1	2	yes	—
75	C₂²×C₈	—	31	1	no	—
76	C₂²×C₈	—	31	4	no	—
77	C₂²×C₈	—	31	1	no	—
78	C₂²×C₈	—	62	4	no	—
79	#108	—	62	4	no	—
80	C₂²⋊C₈	—	2	2	no	—
81	C₂²⋊C₈	—	62	2	no	—
82	C₃₁	#1810	1	1	yes	characteristic, complemented, Sylow
83	C₂⁴	Dic₆₂	1	1	yes	characteristic
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	Dic₆₂	1	1	yes	characteristic
87	C₂²×C₄	C₄×D₃₁	1	2	yes	—
88	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
89	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	62	2	no	—
93	M₁₆	—	62	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	31	2	no	—
96	C₂×C₈	—	31	4	no	—
97	C₂×C₈	—	31	2	no	—
98	C₂×C₈	—	62	8	no	—
99	C₂×C₈	—	62	2	no	—
100	C₂×C₈	—	62	2	no	—
101	C₂³	—	2	4	no	—
102	C₂³	—	2	2	no	—
103	C₂³	Dic₃₁⋊C₄	1	2	yes	—
104	C₂³	C₂×Dic₆₂	1	1	yes	characteristic
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	Dic₃₁⋊C₄	1	2	yes	—
110	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
111	C₂×C₄	D₄×D₃₁	1	2	yes	—
112	C₂×C₄	#92497	1	2	yes	—
113	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
114	C₈	—	4	2	no	—
115	C₈	—	31	4	no	—
116	C₈	—	62	2	no	—
117	C₈	—	124	2	no	—
118	C₂²	—	2	8	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	2	no	—
121	C₂²	—	2	2	no	—
122	C₂²	#11758410	1	4	yes	complemented
123	C₂²	#11758455	1	2	yes	—
124	C₂²	C₂×Dic₃₁⋊C₄	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	#11758454	1	1	yes	characteristic
127	C₄	C₂²⋊C₄×D₃₁	1	1	yes	characteristic
128	C₄	#11758516	1	2	yes	—
129	C₂	—	2	4	no	—
130	C₂	—	2	2	no	—
131	C₂	#14520907	1	1	yes	characteristic
132	C₂	C₂×#11758410	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₃₁⋊C₈

C₂

yes

maximal, characteristic

C₂×#11758410

C₂

yes

maximal, complemented

#14521013

C₂

yes

maximal, characteristic

#14520813

C₂

yes

maximal, complemented

C₃₁×#407

C₂

yes

maximal, characteristic

C₂³×C₄×C₃₁

C₂²

yes

characteristic

C₂×M₁₆×C₃₁

C₂²

yes

—

C₂×#92474

C₂²

yes

—

C₂²×C₃₁⋊C₈

—

C₂²×C₃₁⋊C₈

C₄

yes

—

C₂²×C₃₁⋊C₈

C₂²

yes

characteristic

C₂²×C₃₁⋊C₈

C₂²

yes

characteristic

C₃₁×C₂²⋊C₈

—

C₂²⋊(C₃₁⋊C₈)

—

#11758410

—

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

—

M₁₆×C₃₁

—

C₂×C₈×C₃₁

—

#92474

—

#92474

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

C₂×C₄

yes

—

C₂×C₃₁⋊C₈

C₂×C₄

yes

—

C₂×C₃₁⋊C₈

D₄

yes

—

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

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

#41

yes

—

C₈×C₃₁

—

C₃₁⋊C₈

—

C₃₁⋊C₈

—

C₃₁⋊C₈

C₂²⋊C₄

yes

—

#1810

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#108

yes

complemented

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

—

C₂³×C₈

—

C₂×#108

—

#407

—

#407

D₃₁

yes

characteristic

#341

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

#380

yes

characteristic

C₂×C₃₁

C₂×#108

yes

—

C₂³×C₄

C₂×D₃₁

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₁

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

#108

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₃₁

#1810

yes

characteristic, complemented, Sylow

C₂⁴

Dic₆₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₆₂

yes

characteristic

C₂²×C₄

C₄×D₃₁

yes

—

C₂²×C₄

C₂²⋊D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂³

—

102

C₂³

—

103

C₂³

Dic₃₁⋊C₄

yes

—

104

C₂³

C₂×Dic₆₂

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

Dic₃₁⋊C₄

yes

—

110

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

111

C₂×C₄

D₄×D₃₁

yes

—

112

C₂×C₄

#92497

yes

—

113

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

124

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

#11758410

yes

complemented

123

C₂²

#11758455

yes

—

124

C₂²

C₂×Dic₃₁⋊C₄

yes

characteristic

125

C₄

—

126

C₄

#11758454

yes

characteristic

127

C₄

C₂²⋊C₄×D₃₁

yes

characteristic

128

C₄

#11758516

yes

—

129

C₂

—

130

C₂

—

131

C₂

#14520907

yes

characteristic

132

C₂

C₂×#11758410

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₃₁
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₃₁×#407	C₂
socle	C₂²×C₃₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₃₁	Q₈
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	#407	D₃₁
hypocenter	C₃₁	#1810
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₃₁

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₃₁×#407

C₂

socle

C₂²×C₃₁

C₂×C₄⋊C₄

prime radical

C₂⁴×C₃₁

Q₈

cosocle

C₂²×C₄×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
2. C₂×#11758410	C₂	semidirect
4. #14520813	C₂	semidirect
56. C₂²×C₃₁	#108	semidirect
82. C₃₁	#1810	semidirect
122. C₂²	#11758410	semidirect

H ≅ G/N

type

2. C₂×#11758410

C₂

semidirect

4. #14520813

C₂

semidirect

56. C₂²×C₃₁

#108

semidirect

82. C₃₁

#1810

semidirect

122. C₂²

#11758410

semidirect

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

subgroup	index
C₂³×C₃₁⋊C₈	2
C₂×#11758410	2
#14521013	2
#14520813	2
C₃₁×#407	2
#1810	31

subgroup

index

C₂³×C₃₁⋊C₈

C₃₁×#407

Subgroups — #16845747

Conjugacy classes of subgroups

Complements

Maximal subgroups