Group #12013958 — subgroups

Subgroups — #12013958

Order 1280.

262 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 99 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	#10599859	C₂	1	1	yes	maximal, characteristic
2	#10599858	C₂	1	2	yes	maximal
3	#10599837	C₂	1	1	yes	maximal, characteristic
4	#10598856	C₂	1	2	yes	maximal
5	C₅×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₀	—	2	2	no	—
7	#64988	C₂²	1	2	yes	—
8	#64971	C₂²	1	2	yes	—
9	#64970	—	2	2	no	—
10	#64969	—	2	2	no	—
11	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
12	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
13	C₅×C₈²	C₂²	1	1	yes	characteristic
14	#64724	—	2	2	no	—
15	#64723	—	2	4	no	—
16	#12504	—	5	1	no	maximal, Sylow
17	C₂×Dic₂₀	—	2	2	no	—
18	C₂×Dic₂₀	—	2	2	no	—
19	#4015	—	2	2	no	—
20	#4015	—	2	2	no	—
21	C₂×C₅×C₁₆	D₄	1	2	yes	—
22	C₂×C₅×C₁₆	D₄	1	2	yes	—
23	C₄×C₅×C₈	D₄	1	1	yes	characteristic
24	C₄×C₅×C₈	D₄	1	1	yes	characteristic
25	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
26	#3950	—	2	2	no	—
27	#3950	—	2	2	no	—
28	#3949	—	4	4	no	—
29	#3948	—	4	4	no	—
30	Dic₄₀	—	4	4	no	—
31	#2252	—	5	1	no	—
32	#2247	—	5	1	no	—
33	#2241	—	5	2	no	—
34	#1707	—	5	2	no	—
35	C₈×C₁₆	D₅	1	1	yes	characteristic
36	C₂×Dic₁₀	—	4	2	no	—
37	C₂×Dic₁₀	—	4	2	no	—
38	C₂×C₅×C₈	D₈	1	2	yes	—
39	C₂×C₅×C₈	D₈	1	2	yes	—
40	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
41	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
42	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
43	C₄⋊Dic₅	—	4	2	no	—
44	C₄⋊Dic₅	—	4	2	no	—
45	C₄⋊Dic₅	—	4	2	no	—
46	C₄⋊Dic₅	—	4	2	no	—
47	Dic₂₀	—	4	4	no	—
48	Dic₂₀	—	4	4	no	—
49	C₅×C₁₆	—	2	2	no	—
50	C₅×C₁₆	SD₁₆	1	4	yes	—
51	C₂×Q₃₂	—	10	2	no	—
52	#500	—	5	2	no	—
53	#498	—	10	2	no	—
54	#494	—	5	2	no	—
55	#487	—	10	2	no	—
56	#366	—	10	2	no	—
57	#358	—	10	4	no	—
58	C₄×C₁₆	C₂×D₅	1	1	yes	characteristic
59	C₄×C₁₆	C₂×D₅	1	1	yes	characteristic
60	C₈²	C₂×D₅	1	1	yes	characteristic
61	C₂×C₄×C₅	#127	1	1	yes	characteristic
62	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
64	C₂×Dic₅	—	8	2	no	—
65	C₂×Dic₅	—	8	2	no	—
66	Dic₁₀	—	8	2	no	—
67	Dic₁₀	—	8	2	no	—
68	Dic₁₀	—	8	4	no	—
69	C₅×C₈	—	2	2	no	—
70	C₅×C₈	Q₃₂	1	4	yes	complemented
71	C₅×C₈	C₂×SD₁₆	1	2	yes	—
72	C₅×C₈	#135	1	2	yes	—
73	C₂×Q₁₆	—	10	2	no	—
74	C₂×Q₁₆	—	10	2	no	—
75	C₄⋊Q₈	—	10	2	no	—
76	C₄⋊Q₈	—	10	2	no	—
77	Q₃₂	—	20	4	no	—
78	C₂×C₁₆	D₂₀	1	2	yes	—
79	C₂×C₁₆	D₂₀	1	2	yes	—
80	#107	—	10	2	no	—
81	#107	—	10	2	no	—
82	#106	—	20	4	no	—
83	#103	—	20	4	no	—
84	C₄×C₈	D₂₀	1	1	yes	characteristic
85	C₄×C₈	D₂₀	1	1	yes	characteristic
86	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
87	C₂²×C₅	#493	1	1	yes	characteristic
88	C₄×C₅	#492	1	1	yes	characteristic
89	C₄×C₅	#495	1	1	yes	characteristic
90	C₄×C₅	C₂×Q₃₂	1	2	yes	—
91	C₄×C₅	#508	1	2	yes	—
92	Dic₅	—	16	2	no	—
93	Dic₅	—	16	2	no	—
94	C₂×Q₈	—	20	2	no	—
95	C₂×Q₈	—	20	2	no	—
96	Q₁₆	—	20	4	no	—
97	Q₁₆	—	20	4	no	—
98	C₂×C₈	D₄₀	1	2	yes	—
99	C₂×C₈	D₄₀	1	2	yes	—
100	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
101	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
102	C₄⋊C₄	—	20	2	no	—
103	C₄⋊C₄	—	20	2	no	—
104	C₄⋊C₄	—	20	2	no	—
105	C₄⋊C₄	—	20	2	no	—
106	C₄²	C₂×D₂₀	1	1	yes	characteristic
107	C₁₆	—	2	2	no	—
108	C₁₆	#677	1	4	yes	—
109	C₂×C₅	#1706	1	1	yes	characteristic
110	C₂×C₅	#2247	1	1	yes	characteristic
111	C₂×C₅	#2249	1	1	yes	characteristic
112	Q₈	—	40	4	no	—
113	Q₈	—	40	2	no	—
114	Q₈	—	40	2	no	—
115	C₂×C₄	—	40	2	no	—
116	C₂×C₄	—	40	2	no	—
117	C₂×C₄	#4020	1	1	yes	characteristic
118	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
119	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
120	C₈	—	2	2	no	—
121	C₈	Dic₄₀	1	4	yes	complemented
122	C₈	C₂×#677	1	2	yes	—
123	C₈	#4050	1	2	yes	—
124	C₅	#12504	1	1	yes	characteristic, complemented, Sylow
125	C₂²	#64984	1	1	yes	characteristic
126	C₄	—	80	2	no	—
127	C₄	—	80	2	no	—
128	C₄	#64982	1	1	yes	characteristic
129	C₄	#64985	1	1	yes	characteristic
130	C₄	#65193	1	2	yes	—
131	C₄	C₂×Dic₄₀	1	2	yes	—
132	C₂	#10598851	1	1	yes	characteristic
133	C₂	#10599855	1	1	yes	characteristic
134	C₂	#10599859	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10599859

C₂

yes

maximal, characteristic

#10599858

C₂

yes

maximal

#10599837

C₂

yes

maximal, characteristic

#10598856

C₂

yes

maximal

C₅×C₈×C₁₆

C₂

yes

maximal, characteristic

C₂×Dic₄₀

—

#64988

C₂²

yes

—

#64971

C₂²

yes

—

#64970

—

#64969

—

C₄×C₅×C₁₆

C₂²

yes

characteristic

C₄×C₅×C₁₆

C₂²

yes

characteristic

C₅×C₈²

C₂²

yes

—

—

—

C₂×Dic₂₀

—

C₂×Dic₂₀

—

#4015

—

#4015

—

C₂×C₅×C₁₆

D₄

yes

—

C₂×C₅×C₁₆

D₄

yes

—

C₄×C₅×C₈

D₄

yes

characteristic

C₄×C₅×C₈

D₄

yes

characteristic

C₄×C₅×C₈

C₂³

yes

—

—

—

—

Dic₄₀

—

—

—

—

—

C₈×C₁₆

D₅

yes

characteristic

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

D₈

yes

—

C₂×C₅×C₈

D₈

yes

—

C₂×C₅×C₈

C₂×D₄

yes

characteristic

C₂×C₅×C₈

C₂×D₄

yes

characteristic

C₄²×C₅

C₂×D₄

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

Dic₂₀

—

Dic₂₀

—

C₅×C₁₆

—

C₅×C₁₆

SD₁₆

yes

—

C₂×Q₃₂

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₅

yes

characteristic

C₄×C₁₆

C₂×D₅

yes

characteristic

C₈²

C₂×D₅

yes

characteristic

C₂×C₄×C₅

#127

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

Q₃₂

yes

complemented

C₅×C₈

C₂×SD₁₆

yes

—

C₅×C₈

#135

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

C₂×C₁₆

D₂₀

yes

—

C₂×C₁₆

D₂₀

yes

—

—

—

—

—

C₄×C₈

D₂₀

yes

characteristic

C₄×C₈

D₂₀

yes

characteristic

C₄×C₈

C₂²×D₅

yes

C₂²×C₅

yes

C₄×C₅

yes

C₄×C₅

yes

C₄×C₅

C₂×Q₃₂

yes

—

C₄×C₅

#508

yes

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

D₄₀

yes

—

C₂×C₈

D₄₀

yes

—

100

C₂×C₈

C₂×D₂₀

yes

characteristic

101

C₂×C₈

C₂×D₂₀

yes

characteristic

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄²

C₂×D₂₀

yes

characteristic

107

C₁₆

—

108

C₁₆

#677

yes

—

109

C₂×C₅

#1706

yes

characteristic

110

C₂×C₅

#2247

yes

characteristic

111

C₂×C₅

#2249

yes

characteristic

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

#4020

yes

characteristic

118

C₂×C₄

C₂×D₄₀

yes

characteristic

119

C₂×C₄

C₂×D₄₀

yes

characteristic

120

C₈

—

121

C₈

Dic₄₀

yes

complemented

122

C₈

C₂×#677

yes

—

123

C₈

#4050

yes

—

124

C₅

#12504

yes

characteristic, complemented, Sylow

125

C₂²

#64984

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

#64982

yes

characteristic

129

C₄

#64985

yes

characteristic

130

C₄

#65193

yes

—

131

C₄

C₂×Dic₄₀

yes

—

132

C₂

#10598851

yes

characteristic

133

C₂

#10599855

yes

characteristic

134

C₂

#10599859

yes

characteristic

	subgroup	G/H
center	C₂²	#64984
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×C₈×C₁₆	C₂
socle, prime radical	C₂²×C₅	#493
hypercenter	C₈×C₁₆	D₅
hypocenter	C₅	#12504
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#64984

commutator, cosocle

C₄×C₅×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₅

Fitting

C₅×C₈×C₁₆

C₂

socle, prime radical

C₂²×C₅

#493

hypercenter

C₈×C₁₆

D₅

hypocenter

C₅

#12504

LCS

G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
70. C₅×C₈	Q₃₂	semidirect
121. C₈	Dic₄₀	semidirect
124. C₅	#12504	semidirect

H ≅ G/N

type

70. C₅×C₈

Q₃₂

semidirect

121. C₈

Dic₄₀

semidirect

124. C₅

#12504

semidirect

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

subgroup	index
#10599859	2
#10599858	2
#10599837	2
#10598856	2
C₅×C₈×C₁₆	2
#12504	5

subgroup

index

C₅×C₈×C₁₆

Subgroups — #12013958

Conjugacy classes of subgroups

Complements

Maximal subgroups