Group #13103108 — subgroups

Subgroups — #13103108

Order 1600.

584 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 139 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	C₅²×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	#11717269	C₂	1	2	yes	maximal, complemented
3	#11717267	C₂	1	1	yes	maximal, characteristic, complemented
4	#11716704	C₂	1	2	yes	maximal, complemented
5	#11716703	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅²×#41	—	2	2	no	—
7	C₅²×#41	—	2	2	no	—
8	C₅²×#41	C₂²	1	2	yes	—
9	C₂×C₅²×Q₈	—	2	2	no	—
10	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
12	#87811	—	2	2	no	—
13	#87809	—	2	2	no	—
14	#87808	—	2	2	no	—
15	C₄×#1039	C₂²	1	1	yes	characteristic
16	#87804	C₂²	1	2	yes	—
17	#65534	—	5	6	no	maximal
18	C₂²×C₅⋊D₅	—	2	1	no	—
19	C₅²×Q₈	—	2	2	no	—
20	C₅²×Q₈	—	2	2	no	—
21	C₅²×Q₈	D₄	1	2	yes	—
22	C₅²×D₄	—	2	2	no	—
23	C₅²×D₄	—	2	2	no	—
24	C₅²×D₄	D₄	1	2	yes	—
25	C₂×C₄×C₅²	—	2	4	no	—
26	C₂×C₄×C₅²	—	2	2	no	—
27	C₂×C₄×C₅²	D₄	1	2	yes	—
28	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
29	C₂×#1039	—	2	1	no	—
30	C₄⋊(C₅⋊D₅)	—	2	2	no	—
31	#6029	—	2	2	no	—
32	C₅×2₋¹⁺⁴	D₅	1	6	yes	complemented
33	#4095	—	5	12	no	—
34	#4093	—	5	6	no	—
35	#3969	—	5	12	no	—
36	#3968	—	5	6	no	—
37	C₂²×C₅²	—	2	2	no	—
38	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
39	C₂×C₅⋊D₅	—	4	2	no	—
40	C₂×C₅⋊D₅	—	4	1	no	—
41	C₄×C₅²	—	2	2	no	—
42	C₄×C₅²	—	2	2	no	—
43	C₄×C₅²	C₂×D₄	1	2	yes	—
44	#1039	—	4	1	no	—
45	C₅×#41	—	2	12	no	—
46	C₅×#41	—	2	12	no	—
47	C₅×#41	C₂×D₅	1	12	yes	—
48	C₂×C₅×Q₈	—	2	12	no	—
49	C₂×C₅×Q₈	C₂×D₅	1	6	yes	—
50	C₂×D₂₀	—	5	6	no	—
51	#688	—	10	12	no	—
52	#686	—	10	12	no	—
53	#685	—	10	12	no	—
54	C₄×Dic₅	—	5	6	no	—
55	#681	—	5	12	no	—
56	#454	—	25	1	no	Sylow
57	C₂×C₅²	—	2	1	no	—
58	C₂×C₅²	—	4	2	no	—
59	C₂×C₅²	#120	1	1	yes	characteristic
60	C₅⋊D₅	—	8	1	no	—
61	C₂²×D₅	—	10	6	no	—
62	C₅×Q₈	—	2	12	no	—
63	C₅×Q₈	—	2	12	no	—
64	C₅×Q₈	C₂²⋊D₅	1	12	yes	—
65	C₅×D₄	—	2	12	no	—
66	C₅×D₄	—	2	12	no	—
67	C₅×D₄	C₂²⋊D₅	1	12	yes	—
68	C₂×C₄×C₅	—	2	24	no	—
69	C₂×C₄×C₅	—	2	12	no	—
70	C₂×C₄×C₅	C₂²⋊D₅	1	12	yes	—
71	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
72	C₂×Dic₅	—	10	6	no	—
73	D₂₀	—	10	12	no	—
74	#168	—	10	12	no	—
75	2₋¹⁺⁴	C₅⋊D₅	1	1	yes	characteristic, complemented
76	AGL(1,ℤ/8)	—	25	2	no	—
77	#124	—	25	1	no	—
78	#104	—	25	2	no	—
79	#101	—	25	1	no	—
80	C₅²	#454	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₅	—	2	12	no	—
82	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
83	C₂×D₅	—	20	12	no	—
84	C₂×D₅	—	20	6	no	—
85	C₄×C₅	—	2	12	no	—
86	C₄×C₅	—	2	12	no	—
87	C₄×C₅	C₂×C₂²⋊D₅	1	12	yes	—
88	Dic₅	—	20	6	no	—
89	#41	—	2	2	no	—
90	#41	—	2	2	no	—
91	#41	C₂×C₅⋊D₅	1	2	yes	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	C₂×C₅⋊D₅	1	1	yes	characteristic
94	C₂×D₄	—	25	1	no	—
95	SD₁₆	—	50	2	no	—
96	D₈	—	50	2	no	—
97	M₁₆	—	25	2	no	—
98	C₂²⋊C₄	—	50	2	no	—
99	C₄²	—	25	1	no	—
100	C₂×C₅	—	2	6	no	—
101	C₂×C₅	—	4	12	no	—
102	C₂×C₅	#4099	1	6	yes	—
103	D₅	—	40	6	no	—
104	C₂³	—	50	1	no	—
105	Q₈	—	2	2	no	—
106	Q₈	—	2	2	no	—
107	Q₈	C₂²⋊(C₅⋊D₅)	1	2	yes	—
108	D₄	—	2	2	no	—
109	D₄	—	2	2	no	—
110	D₄	—	50	2	no	—
111	D₄	C₂²⋊(C₅⋊D₅)	1	2	yes	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	50	1	no	—
115	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	2	yes	—
116	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
117	C₈	—	50	2	no	—
118	C₅	#65534	1	6	yes	complemented
119	C₂²	—	2	2	no	—
120	C₂²	—	100	2	no	—
121	C₂²	—	100	1	no	—
122	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	2	no	—
125	C₄	—	100	1	no	—
126	C₄	C₂×C₂²⋊(C₅⋊D₅)	1	2	yes	—
127	C₂	—	2	1	no	—
128	C₂	—	4	2	no	—
129	C₂	—	200	1	no	—
130	C₂	#11717273	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²×2₋¹⁺⁴

C₂

yes

maximal, characteristic, complemented

#11717269

C₂

yes

maximal, complemented

#11717267

C₂

yes

maximal, characteristic, complemented

#11716704

C₂

yes

maximal, complemented

#11716703

C₂

yes

maximal, characteristic, complemented

C₅²×#41

—

C₅²×#41

—

C₅²×#41

C₂²

yes

—

C₂×C₅²×Q₈

—

C₂×C₅²×Q₈

C₂²

yes

characteristic

C₂×C₄⋊(C₅⋊D₅)

C₂²

yes

—

—

—

C₂²

yes

characteristic

#87804

C₂²

yes

—

#65534

—

maximal

C₂²×C₅⋊D₅

—

C₅²×Q₈

—

C₅²×Q₈

—

C₅²×Q₈

D₄

yes

—

C₅²×D₄

—

C₅²×D₄

—

C₅²×D₄

D₄

yes

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

D₄

yes

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

—

C₄⋊(C₅⋊D₅)

—

#6029

—

C₅×2₋¹⁺⁴

D₅

yes

—

—

—

—

C₂²×C₅²

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₄×C₅²

—

C₄×C₅²

—

C₄×C₅²

C₂×D₄

yes

—

#1039

—

C₅×#41

—

C₅×#41

—

C₅×#41

C₂×D₅

yes

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

C₂×D₅

yes

—

C₂×D₂₀

—

#688

—

#686

—

#685

—

C₄×Dic₅

—

#681

—

#454

—

Sylow

C₂×C₅²

—

C₂×C₅²

—

C₂×C₅²

#120

yes

characteristic

C₅⋊D₅

—

C₂²×D₅

—

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

C₂²⋊D₅

yes

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

C₂²⋊D₅

yes

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

D₂₀

—

#168

—

2₋¹⁺⁴

C₅⋊D₅

yes

characteristic, complemented

—

—

—

—

C₅²

#454

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

C₂×C₂²⋊D₅

yes

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₂×C₂²⋊D₅

yes

—

Dic₅

—

#41

—

#41

—

#41

C₂×C₅⋊D₅

yes

—

C₂×Q₈

—

C₂×Q₈

C₂×C₅⋊D₅

yes

characteristic

C₂×D₄

—

SD₁₆

—

D₈

—

M₁₆

—

C₂²⋊C₄

—

C₄²

—

100

C₂×C₅

—

101

C₂×C₅

—

102

C₂×C₅

#4099

yes

—

103

D₅

—

104

C₂³

—

105

Q₈

—

106

Q₈

—

107

Q₈

C₂²⋊(C₅⋊D₅)

yes

—

108

D₄

—

109

D₄

—

110

D₄

—

111

D₄

C₂²⋊(C₅⋊D₅)

yes

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

—

116

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

117

C₈

—

118

C₅

#65534

yes

complemented

119

C₂²

—

120

C₂²

—

100

—

121

C₂²

—

100

—

122

C₂²

C₂×C₂²⋊(C₅⋊D₅)

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

100

—

126

C₄

C₂×C₂²⋊(C₅⋊D₅)

yes

—

127

C₂

—

128

C₂

—

129

C₂

—

200

—

130

C₂

#11717273

yes

characteristic

	subgroup	G/H
center	C₂	#11717273
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₅²×2₋¹⁺⁴	C₂
socle	C₂×C₅²	#120
hypercenter	2₋¹⁺⁴	C₅⋊D₅
hypocenter	C₅²	#454
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ 2₋¹⁺⁴

subgroup

G/H

center

C₂

#11717273

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂²×C₅⋊D₅

Fitting

C₅²×2₋¹⁺⁴

C₂

socle

C₂×C₅²

#120

hypercenter

2₋¹⁺⁴

C₅⋊D₅

hypocenter

C₅²

#454

LCS

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ 2₋¹⁺⁴

N	H ≅ G/N	type
1. C₅²×2₋¹⁺⁴	C₂	semidirect
2. #11717269	C₂	semidirect
3. #11717267	C₂	semidirect
4. #11716704	C₂	semidirect
5. #11716703	C₂	semidirect
32. C₅×2₋¹⁺⁴	D₅	semidirect
75. 2₋¹⁺⁴	C₅⋊D₅	semidirect
80. C₅²	#454	semidirect
118. C₅	#65534	semidirect

H ≅ G/N

type

1. C₅²×2₋¹⁺⁴

C₂

semidirect

2. #11717269

C₂

semidirect

3. #11717267

C₂

semidirect

4. #11716704

C₂

semidirect

5. #11716703

C₂

semidirect

32. C₅×2₋¹⁺⁴

D₅

semidirect

75. 2₋¹⁺⁴

C₅⋊D₅

semidirect

80. C₅²

#454

semidirect

118. C₅

#65534

semidirect

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

subgroup	index
C₅²×2₋¹⁺⁴	2
#11717269	2
#11717267	2
#11716704	2
#11716703	2
#65534	5

subgroup

index

C₅²×2₋¹⁺⁴

Subgroups — #13103108

Conjugacy classes of subgroups

Complements

Maximal subgroups