Group #12013996 — subgroups

Subgroups — #12013996

Order 1280.

260 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 79 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	C₅×#1710	C₂	1	1	yes	maximal, characteristic, complemented
2	#10602175	C₂	1	2	yes	maximal, complemented
3	#10602174	C₂	1	2	yes	maximal
4	#10598853	C₂	1	1	yes	maximal, characteristic
5	C₈×#672	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#500	C₂²	1	2	yes	—
7	C₅×#487	—	2	2	no	—
8	#64988	C₂²	1	1	yes	characteristic
9	#64984	C₂²	1	1	yes	characteristic
10	#64983	—	2	2	no	—
11	C₅×C₈²	C₂²	1	1	yes	characteristic
12	#64712	—	2	4	no	—
13	#64711	—	2	4	no	—
14	C₄×#672	C₂²	1	2	yes	—
15	#12511	—	5	1	no	maximal, Sylow
16	C₅×C₄⋊Q₈	—	2	2	no	—
17	C₂×Dic₂₀	—	2	2	no	—
18	C₂×D₄₀	—	2	2	no	—
19	#4020	—	2	1	no	—
20	#4015	—	2	1	no	—
21	C₅×#107	—	2	2	no	—
22	C₅×#107	—	2	2	no	—
23	C₅×#103	—	4	4	no	—
24	C₄×C₅×C₈	D₄	1	2	yes	—
25	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
26	#3953	—	4	4	no	—
27	C₂×#672	D₄	1	4	yes	—
28	#2241	—	5	2	no	—
29	#2240	—	5	2	no	—
30	#1710	D₅	1	1	yes	characteristic, complemented
31	#1709	—	5	1	no	—
32	C₈×C₁₆	—	5	1	no	—
33	C₂×C₅×Q₈	—	4	2	no	—
34	C₂×D₂₀	—	4	1	no	—
35	C₂×D₂₀	—	4	2	no	—
36	C₂×Dic₁₀	—	4	1	no	—
37	C₂×C₅×C₈	D₈	1	4	yes	—
38	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
39	C₅×C₄⋊C₄	—	4	2	no	—
40	C₅×C₄⋊C₄	—	4	2	no	—
41	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
42	C₄⋊Dic₅	—	4	2	no	—
43	Dic₂₀	—	4	4	no	—
44	D₄₀	—	4	4	no	—
45	#672	—	2	4	no	—
46	#500	C₂×D₅	1	2	yes	—
47	#494	—	5	1	no	—
48	#493	—	5	1	no	—
49	#487	—	2	2	no	—
50	#486	—	10	2	no	—
51	#358	—	10	4	no	—
52	#357	—	10	4	no	—
53	C₄×C₁₆	—	5	2	no	—
54	C₈²	C₂×D₅	1	1	yes	characteristic
55	C₂²×D₅	—	8	1	no	—
56	C₅×Q₈	—	8	4	no	—
57	C₂×C₄×C₅	—	8	2	no	—
58	C₂×C₄×C₅	#127	1	1	yes	characteristic
59	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
60	C₂×Dic₅	—	8	1	no	—
61	D₂₀	—	8	2	no	—
62	D₂₀	—	8	4	no	—
63	Dic₁₀	—	8	2	no	—
64	C₅×C₈	—	2	4	no	—
65	C₅×C₈	#135	1	4	yes	—
66	C₂×Q₁₆	—	10	2	no	—
67	C₂×D₈	—	10	2	no	—
68	C₄⋊Q₈	—	2	2	no	—
69	C₄⋊Q₈	—	10	1	no	—
70	#127	—	10	1	no	—
71	C₂×C₁₆	—	5	4	no	—
72	#107	—	2	2	no	—
73	#107	—	2	2	no	—
74	#103	—	4	4	no	—
75	#102	—	20	4	no	—
76	C₄×C₈	C₂²⋊D₅	1	2	yes	—
77	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
78	C₂²×C₅	#493	1	1	yes	characteristic
79	C₂×D₅	—	16	1	no	—
80	C₂×D₅	—	16	2	no	—
81	C₄×C₅	—	16	2	no	—
82	C₄×C₅	#495	1	2	yes	—
83	C₄×C₅	#508	1	4	yes	—
84	Dic₅	—	16	1	no	—
85	C₂×Q₈	—	4	2	no	—
86	C₂×Q₈	—	20	1	no	—
87	C₂×D₄	—	20	2	no	—
88	C₂×D₄	—	20	1	no	—
89	Q₁₆	—	20	4	no	—
90	D₈	—	20	4	no	—
91	C₂×C₈	#686	1	4	yes	—
92	C₂×C₈	D₄×D₅	1	2	yes	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	4	2	no	—
95	C₄⋊C₄	—	20	2	no	—
96	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
97	C₁₆	—	10	4	no	—
98	C₂×C₅	#1709	1	1	yes	characteristic
99	C₂×C₅	#2249	1	2	yes	—
100	D₅	—	32	1	no	—
101	C₂³	—	40	1	no	—
102	Q₈	—	8	4	no	—
103	Q₈	—	40	2	no	—
104	D₄	—	40	2	no	—
105	D₄	—	40	4	no	—
106	C₂×C₄	—	8	2	no	—
107	C₂×C₄	—	40	1	no	—
108	C₂×C₄	C₂×#686	1	2	yes	—
109	C₂×C₄	#4086	1	1	yes	characteristic
110	C₈	—	2	4	no	—
111	C₈	#4065	1	4	yes	—
112	C₅	#12511	1	1	yes	characteristic, complemented, Sylow
113	C₂²	—	80	1	no	—
114	C₂²	—	80	2	no	—
115	C₂²	#65362	1	1	yes	characteristic
116	C₄	—	16	2	no	—
117	C₄	—	80	1	no	—
118	C₄	#65480	1	2	yes	—
119	C₄	#65483	1	4	yes	—
120	C₂	—	160	1	no	—
121	C₂	#10602090	1	1	yes	characteristic
122	C₂	#10602113	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#1710

C₂

yes

maximal, characteristic, complemented

#10602175

C₂

yes

maximal, complemented

#10602174

C₂

yes

maximal

#10598853

C₂

yes

maximal, characteristic

C₈×#672

C₂

yes

maximal, characteristic, complemented

C₅×#500

C₂²

yes

—

C₅×#487

—

#64988

C₂²

yes

characteristic

#64984

C₂²

yes

characteristic

#64983

—

C₅×C₈²

C₂²

yes

—

—

C₂²

yes

—

#12511

—

maximal, Sylow

C₅×C₄⋊Q₈

—

C₂×Dic₂₀

—

C₂×D₄₀

—

—

—

—

—

—

C₄×C₅×C₈

D₄

yes

—

C₄×C₅×C₈

C₂³

yes

characteristic

#3953

—

C₂×#672

D₄

yes

—

#2241

—

#2240

—

#1710

D₅

yes

characteristic, complemented

#1709

—

C₈×C₁₆

—

C₂×C₅×Q₈

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

D₈

yes

—

C₂×C₅×C₈

C₂×D₄

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂×D₄

yes

characteristic

C₄⋊Dic₅

—

Dic₂₀

—

D₄₀

—

#672

—

#500

C₂×D₅

yes

—

—

—

—

—

—

—

C₄×C₁₆

—

C₈²

C₂×D₅

yes

characteristic

C₂²×D₅

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#127

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

—

C₂×Dic₅

—

D₂₀

—

D₂₀

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

#135

yes

—

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

—

—

—

—

—

C₄×C₈

C₂²⋊D₅

yes

—

C₄×C₈

C₂²×D₅

yes

characteristic

C₂²×C₅

#493

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

#495

yes

—

C₄×C₅

#508

yes

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

D₈

—

C₂×C₈

#686

yes

—

C₂×C₈

D₄×D₅

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₂²⋊D₅

yes

characteristic

C₁₆

—

C₂×C₅

#1709

yes

characteristic

C₂×C₅

#2249

yes

—

100

D₅

—

101

C₂³

—

102

Q₈

—

103

Q₈

—

104

D₄

—

105

D₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂×#686

yes

—

109

C₂×C₄

#4086

yes

characteristic

110

C₈

—

111

C₈

#4065

yes

—

112

C₅

#12511

yes

characteristic, complemented, Sylow

113

C₂²

—

114

C₂²

—

115

C₂²

#65362

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

#65480

yes

—

119

C₄

#65483

yes

—

120

C₂

—

160

—

121

C₂

#10602090

yes

characteristic

122

C₂

#10602113

yes

—

	subgroup	G/H
center	C₂²	#65362
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#1710	C₂
socle	C₂²×C₅	#493
prime radical	#64984	C₂²
hypercenter	#1710	D₅
hypocenter	C₅	#12511
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

subgroup

G/H

center

C₂²

#65362

commutator, cosocle

C₄×C₅×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₅

C₂

C₂²×C₅

C₂²

D₅

C₅

G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

N	H ≅ G/N	type
1. C₅×#1710	C₂	semidirect
2. #10602175	C₂	semidirect
5. C₈×#672	C₂	semidirect
30. #1710	D₅	semidirect
112. C₅	#12511	semidirect

H ≅ G/N

type

1. C₅×#1710

C₂

semidirect

2. #10602175

C₂

semidirect

5. C₈×#672

C₂

semidirect

30. #1710

D₅

semidirect

112. C₅

#12511

semidirect

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

subgroup	index
C₅×#1710	2
#10602175	2
#10602174	2
#10598853	2
C₈×#672	2
#12511	5

subgroup

index

Subgroups — #12013996

Conjugacy classes of subgroups

Complements

Maximal subgroups