Group #13102616 — subgroups

Subgroups — #13102616

Order 1600.

584 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 111 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	#11717520	C₂	1	1	yes	maximal, characteristic, complemented
2	#11717267	C₂	1	1	yes	maximal, characteristic, complemented
3	#11717228	C₂	1	2	yes	maximal, complemented
4	C₅²×#101	C₂	1	1	yes	maximal, characteristic, complemented
5	#11716692	C₂	1	2	yes	maximal, complemented
6	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
7	#87904	—	2	2	no	—
8	Q₈×C₅⋊D₅	—	2	2	no	—
9	#87900	—	2	2	no	—
10	#87900	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
12	C₅²×M₁₆	C₂²	1	2	yes	—
13	#87808	—	2	2	no	—
14	C₄×#1039	C₂²	1	1	yes	characteristic
15	#87801	—	2	2	no	—
16	#87800	—	2	2	no	—
17	#65042	—	5	6	no	maximal
18	C₂²×C₅⋊D₅	—	2	1	no	—
19	C₅²×Q₈	—	2	2	no	—
20	C₂×C₄×C₅²	D₄	1	2	yes	—
21	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
22	C₂²⋊(C₅⋊D₅)	—	2	2	no	—
23	C₂×#1039	—	2	1	no	—
24	C₄⋊(C₅⋊D₅)	—	2	2	no	—
25	C₄⋊(C₅⋊D₅)	—	2	2	no	—
26	C₄⋊(C₅⋊D₅)	D₄	1	2	yes	—
27	C₄×C₅⋊D₅	—	2	4	no	—
28	C₄×C₅⋊D₅	—	2	2	no	—
29	#6045	—	2	2	no	—
30	#6045	D₄	1	2	yes	—
31	C₅²×C₈	—	2	2	no	—
32	#4147	—	5	6	no	—
33	#4093	—	5	6	no	—
34	#4054	—	5	12	no	—
35	C₅×#101	D₅	1	6	yes	complemented
36	#3957	—	5	12	no	—
37	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
38	C₂×C₅⋊D₅	—	2	2	no	—
39	C₂×C₅⋊D₅	—	4	2	no	—
40	C₂×C₅⋊D₅	—	4	1	no	—
41	C₄×C₅²	—	2	2	no	—
42	C₄×C₅²	C₂×D₄	1	2	yes	—
43	#1039	—	2	2	no	—
44	#1039	—	4	1	no	—
45	C₂×C₅×Q₈	C₂×D₅	1	6	yes	—
46	#713	—	10	12	no	—
47	Q₈×D₅	—	10	12	no	—
48	#709	—	5	12	no	—
49	#709	—	10	12	no	—
50	C₂×D₂₀	—	5	6	no	—
51	C₅×M₁₆	C₂×D₅	1	12	yes	—
52	#685	—	10	12	no	—
53	C₄×Dic₅	—	5	6	no	—
54	D₄₀	—	10	12	no	—
55	#677	—	10	12	no	—
56	#454	—	25	1	no	Sylow
57	C₂×C₅²	—	2	1	no	—
58	C₂×C₅²	#120	1	1	yes	characteristic
59	C₅⋊D₅	—	4	2	no	—
60	C₅⋊D₅	—	8	1	no	—
61	C₂²×D₅	—	10	6	no	—
62	C₅×Q₈	—	2	12	no	—
63	C₂×C₄×C₅	D₂₀	1	12	yes	—
64	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
65	C₂²⋊D₅	—	10	12	no	—
66	C₂×Dic₅	—	10	6	no	—
67	D₂₀	—	5	12	no	—
68	D₂₀	—	10	12	no	—
69	D₂₀	—	10	12	no	—
70	C₄×D₅	—	10	24	no	—
71	C₄×D₅	—	10	12	no	—
72	Dic₁₀	—	5	12	no	—
73	Dic₁₀	—	10	12	no	—
74	C₅×C₈	—	2	12	no	—
75	2₋¹⁺⁴	—	25	1	no	—
76	AGL(1,ℤ/8)	—	25	2	no	—
77	#124	—	25	1	no	—
78	#104	—	25	2	no	—
79	#101	C₅⋊D₅	1	1	yes	characteristic, complemented
80	C₅²	#454	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₅	C₂×D₂₀	1	6	yes	—
82	C₂×D₅	—	10	12	no	—
83	C₂×D₅	—	20	12	no	—
84	C₂×D₅	—	20	6	no	—
85	C₄×C₅	—	2	12	no	—
86	C₄×C₅	D₄×D₅	1	12	yes	—
87	Dic₅	—	10	12	no	—
88	Dic₅	—	20	6	no	—
89	#41	—	25	2	no	—
90	#41	—	50	2	no	—
91	#41	—	50	2	no	—
92	C₂×Q₈	—	50	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	C₂×C₅⋊D₅	1	2	yes	—
98	C₂²⋊C₄	—	50	2	no	—
99	C₄²	—	25	1	no	—
100	C₂×C₅	—	2	6	no	—
101	C₂×C₅	#4028	1	6	yes	—
102	D₅	—	20	12	no	—
103	D₅	—	40	6	no	—
104	C₂³	—	50	1	no	—
105	Q₈	—	2	2	no	—
106	Q₈	—	25	2	no	—
107	Q₈	—	50	2	no	—
108	D₄	—	25	2	no	—
109	D₄	—	50	2	no	—
110	D₄	—	50	2	no	—
111	D₄	—	50	2	no	—
112	C₂×C₄	—	50	4	no	—
113	C₂×C₄	—	50	2	no	—
114	C₂×C₄	—	50	1	no	—
115	C₂×C₄	C₄⋊(C₅⋊D₅)	1	2	yes	—
116	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
117	C₈	—	2	2	no	—
118	C₅	#65042	1	6	yes	complemented
119	C₂²	—	50	2	no	—
120	C₂²	—	100	2	no	—
121	C₂²	—	100	1	no	—
122	C₂²	C₂×C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	50	2	no	—
125	C₄	—	100	1	no	—
126	C₄	D₄×C₅⋊D₅	1	2	yes	—
127	C₂	—	2	1	no	—
128	C₂	—	100	2	no	—
129	C₂	—	200	1	no	—
130	C₂	#11717202	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11717520

C₂

yes

maximal, characteristic, complemented

#11717267

C₂

yes

maximal, characteristic, complemented

#11717228

C₂

yes

maximal, complemented

C₅²×#101

C₂

yes

maximal, characteristic, complemented

#11716692

C₂

yes

maximal, complemented

C₂×C₅²×Q₈

C₂²

yes

characteristic

#87904

—

Q₈×C₅⋊D₅

—

#87900

—

#87900

C₂²

yes

—

C₂×C₄⋊(C₅⋊D₅)

C₂²

yes

characteristic

C₅²×M₁₆

C₂²

yes

—

#87808

—

C₄×#1039

C₂²

yes

—

—

—

C₂²×C₅⋊D₅

—

C₅²×Q₈

—

C₂×C₄×C₅²

D₄

yes

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂²⋊(C₅⋊D₅)

—

C₂×#1039

—

C₄⋊(C₅⋊D₅)

—

C₄⋊(C₅⋊D₅)

—

C₄⋊(C₅⋊D₅)

D₄

yes

—

C₄×C₅⋊D₅

—

C₄×C₅⋊D₅

—

#6045

—

#6045

D₄

yes

—

C₅²×C₈

—

—

—

—

D₅

yes

complemented

#3957

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₄×C₅²

—

C₄×C₅²

C₂×D₄

yes

—

#1039

—

#1039

—

C₂×C₅×Q₈

C₂×D₅

yes

—

#713

—

Q₈×D₅

—

#709

—

#709

—

C₂×D₂₀

—

C₅×M₁₆

C₂×D₅

yes

—

#685

—

C₄×Dic₅

—

D₄₀

—

#677

—

#454

—

Sylow

C₂×C₅²

—

C₂×C₅²

#120

yes

characteristic

C₅⋊D₅

—

C₅⋊D₅

—

C₂²×D₅

—

C₅×Q₈

—

C₂×C₄×C₅

D₂₀

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂²⋊D₅

—

C₂×Dic₅

—

D₂₀

—

D₂₀

—

D₂₀

—

C₄×D₅

—

C₄×D₅

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

2₋¹⁺⁴

—

—

—

—

C₅⋊D₅

yes

characteristic, complemented

C₅²

#454

yes

characteristic, complemented, Sylow

C₂²×C₅

C₂×D₂₀

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

D₄×D₅

yes

—

Dic₅

—

Dic₅

—

#41

—

#41

—

#41

—

C₂×Q₈

—

C₂×Q₈

C₂×C₅⋊D₅

yes

characteristic

C₂×D₄

—

SD₁₆

—

D₈

—

M₁₆

C₂×C₅⋊D₅

yes

—

C₂²⋊C₄

—

C₄²

—

100

C₂×C₅

—

101

C₂×C₅

#4028

yes

—

102

D₅

—

103

D₅

—

104

C₂³

—

105

Q₈

—

106

Q₈

—

107

Q₈

—

108

D₄

—

109

D₄

—

110

D₄

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

C₄⋊(C₅⋊D₅)

yes

—

116

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

117

C₈

—

118

C₅

#65042

yes

complemented

119

C₂²

—

120

C₂²

—

100

—

121

C₂²

—

100

—

122

C₂²

C₂×C₄⋊(C₅⋊D₅)

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

100

—

126

C₄

D₄×C₅⋊D₅

yes

—

127

C₂

—

128

C₂

—

100

—

129

C₂

—

200

—

130

C₂

#11717202

yes

characteristic

	subgroup	G/H
center	C₂	#11717202
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₅²×#101	C₂
socle	C₂×C₅²	#120
hypercenter	#101	C₅⋊D₅
hypocenter	C₅²	#454
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

subgroup

G/H

center

C₂

#11717202

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂²×C₅⋊D₅

Fitting

C₅²×#101

C₂

C₂×C₅²

C₅⋊D₅

C₅²

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

N	H ≅ G/N	type
1. #11717520	C₂	semidirect
2. #11717267	C₂	semidirect
3. #11717228	C₂	semidirect
4. C₅²×#101	C₂	semidirect
5. #11716692	C₂	semidirect
35. C₅×#101	D₅	semidirect
79. #101	C₅⋊D₅	semidirect
80. C₅²	#454	semidirect
118. C₅	#65042	semidirect

H ≅ G/N

type

1. #11717520

C₂

semidirect

2. #11717267

C₂

semidirect

3. #11717228

C₂

semidirect

4. C₅²×#101

C₂

semidirect

5. #11716692

C₂

semidirect

35. C₅×#101

D₅

semidirect

79. #101

C₅⋊D₅

semidirect

80. C₅²

#454

semidirect

118. C₅

#65042

semidirect

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

subgroup	index
#11717520	2
#11717267	2
#11717228	2
C₅²×#101	2
#11716692	2
#65042	5

subgroup

index

C₅²×#101

Subgroups — #13102616

Conjugacy classes of subgroups

Complements

Maximal subgroups