Group #13101656 — subgroups

Subgroups — #13101656

Order 1600.

268 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 58 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	#11717256	C₂	1	1	yes	maximal, characteristic
2	#11716991	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3964	C₂	1	2	yes	maximal
4	#11716608	C₂	1	2	yes	maximal, complemented
5	#11716594	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
7	C₂×#6045	C₂²	1	1	yes	characteristic
8	C₂²⋊#1039	—	2	2	no	—
9	C₄×#1039	C₂²	1	1	yes	characteristic
10	C₅×C₄⋊Dic₅	C₂²	1	2	yes	—
11	C₂×C₅×#168	C₂²	1	2	yes	—
12	#87780	—	2	2	no	—
13	#65057	—	5	2	no	maximal
14	C₂³×C₅²	—	2	1	no	—
15	C₅²×D₄	—	2	2	no	—
16	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
17	C₂×#1039	—	2	1	no	—
18	C₂×#1039	D₄	1	2	yes	—
19	#6045	—	2	2	no	—
20	C₂×C₅×Dic₅	—	2	2	no	—
21	C₅×#168	—	2	2	no	—
22	#4082	—	5	2	no	—
23	#4082	—	10	2	no	—
24	#4036	—	5	2	no	—
25	C₅×#102	—	5	2	no	—
26	#3964	D₅	1	2	yes	—
27	#3948	—	5	2	no	—
28	C₈×Dic₅	—	5	2	no	—
29	#3942	—	5	2	no	—
30	C₂²×C₅²	—	4	1	no	—
31	C₂²×C₅²	—	4	2	no	—
32	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
33	C₄×C₅²	#41	1	2	yes	—
34	#1039	—	2	2	no	—
35	#1039	—	4	1	no	—
36	C₅×Dic₅	—	4	2	no	—
37	C₂×C₅×D₄	—	2	2	no	—
38	C₂×C₅×D₄	C₂×D₅	1	2	yes	—
39	C₂×Dic₁₀	—	5	2	no	—
40	C₂×Dic₁₀	—	10	2	no	—
41	C₂×C₅×C₈	—	5	2	no	—
42	C₅×C₄⋊C₄	—	5	2	no	—
43	C₂²⋊Dic₅	—	10	2	no	—
44	C₂²⋊Dic₅	—	10	2	no	—
45	C₂²⋊Dic₅	—	10	8	no	—
46	C₄⋊Dic₅	—	10	2	no	—
47	C₄⋊Dic₅	C₂×D₅	1	2	yes	—
48	#683	—	10	2	no	—
49	C₄×Dic₅	—	5	2	no	—
50	C₄×Dic₅	—	10	2	no	—
51	C₂×#168	C₂×D₅	1	2	yes	—
52	#488	—	25	1	no	Sylow
53	C₂×C₅²	—	8	1	no	—
54	C₂×C₅²	#124	1	1	yes	characteristic
55	C₂×C₅²	#135	1	2	yes	—
56	C₂³×C₅	—	2	4	no	—
57	C₂³×C₅	—	2	2	no	—
58	C₅×D₄	—	2	2	no	—
59	C₅×D₄	—	2	2	no	—
60	C₅×D₄	—	4	4	no	—
61	C₂×C₄×C₅	—	2	2	no	—
62	C₂×C₄×C₅	—	10	2	no	—
63	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
64	C₂×Dic₅	—	2	2	no	—
65	C₂×Dic₅	—	5	2	no	—
66	C₂×Dic₅	—	5	2	no	—
67	C₂×Dic₅	—	10	4	no	—
68	C₂×Dic₅	—	10	4	no	—
69	C₂×Dic₅	—	10	2	no	—
70	Dic₁₀	—	10	2	no	—
71	Dic₁₀	—	10	2	no	—
72	Dic₁₀	—	20	4	no	—
73	C₅×C₈	—	10	2	no	—
74	#168	—	2	2	no	—
75	#125	—	25	1	no	—
76	#124	—	25	1	no	—
77	#103	—	25	2	no	—
78	#102	—	5	2	no	—
79	C₄×C₈	—	25	1	no	—
80	C₅²	#488	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₅	—	2	2	no	—
82	C₂²×C₅	—	4	4	no	—
83	C₂²×C₅	—	4	8	no	—
84	C₂²×C₅	—	4	2	no	—
85	C₂²×C₅	—	4	2	no	—
86	C₂²×C₅	—	4	2	no	—
87	C₂²×C₅	D₄×D₅	1	2	yes	—
88	C₄×C₅	—	2	4	no	—
89	C₄×C₅	—	20	2	no	—
90	C₄×C₅	#709	1	2	yes	—
91	C₄×C₅	#711	1	2	yes	—
92	Dic₅	—	4	2	no	—
93	Dic₅	—	10	2	no	—
94	Dic₅	—	10	2	no	—
95	Dic₅	—	20	4	no	—
96	Dic₅	—	20	4	no	—
97	Dic₅	—	20	2	no	—
98	C₂×Q₈	—	25	1	no	—
99	C₂×D₄	D₅²	1	1	yes	characteristic
100	C₂×C₈	—	5	2	no	—
101	C₄⋊C₄	—	5	2	no	—
102	C₄⋊C₄	—	50	2	no	—
103	C₂²⋊C₄	—	50	2	no	—
104	C₄²	—	25	1	no	—
105	C₂×C₅	—	2	4	no	—
106	C₂×C₅	—	2	2	no	—
107	C₂×C₅	—	8	4	no	—
108	C₂×C₅	—	8	2	no	—
109	C₂×C₅	#4031	1	2	yes	—
110	C₂×C₅	#4058	1	2	yes	—
111	C₂×C₅	#4062	1	2	yes	—
112	C₂³	—	2	1	no	—
113	Q₈	—	50	2	no	—
114	D₄	—	2	2	no	—
115	C₂×C₄	—	10	2	no	—
116	C₂×C₄	—	25	2	no	—
117	C₂×C₄	—	50	1	no	—
118	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
119	C₈	—	10	2	no	—
120	C₅	—	2	2	no	—
121	C₅	#65057	1	2	yes	complemented
122	C₂²	—	4	2	no	—
123	C₂²	—	4	1	no	—
124	C₂²	#87886	1	1	yes	characteristic
125	C₄	—	20	2	no	—
126	C₄	—	50	2	no	—
127	C₄	—	100	1	no	—
128	C₄	#87879	1	2	yes	—
129	C₂	—	8	1	no	—
130	C₂	#11717054	1	2	yes	—
131	C₂	#11717071	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11717256

C₂

yes

maximal, characteristic

#11716991

C₂

yes

maximal, characteristic, complemented

C₅×#3964

C₂

yes

maximal

#11716608

C₂

yes

maximal, complemented

#11716594

C₂

yes

maximal, characteristic, complemented

C₂×C₅²×D₄

C₂²

yes

characteristic

C₂×#6045

C₂²

yes

characteristic

C₂²⋊#1039

—

C₄×#1039

C₂²

yes

characteristic

C₅×C₄⋊Dic₅

C₂²

yes

—

C₂×C₅×#168

C₂²

yes

—

#87780

—

#65057

—

maximal

C₂³×C₅²

—

C₅²×D₄

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

—

C₂×#1039

D₄

yes

—

#6045

—

C₂×C₅×Dic₅

—

—

—

—

—

—

D₅

yes

—

#3948

—

C₈×Dic₅

—

#3942

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

#41

yes

—

#1039

—

#1039

—

C₅×Dic₅

—

C₂×C₅×D₄

—

C₂×C₅×D₄

C₂×D₅

yes

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

—

C₅×C₄⋊C₄

—

C₂²⋊Dic₅

—

C₂²⋊Dic₅

—

C₂²⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

C₂×D₅

yes

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

—

C₂×#168

C₂×D₅

yes

—

#488

—

Sylow

C₂×C₅²

—

C₂×C₅²

#124

yes

characteristic

C₂×C₅²

#135

yes

—

C₂³×C₅

—

C₂³×C₅

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

—

—

—

—

—

C₄×C₈

—

C₅²

#488

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

D₄×D₅

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#709

yes

—

C₄×C₅

#711

yes

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×D₄

D₅²

yes

characteristic

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₂×C₅

—

106

C₂×C₅

—

107

C₂×C₅

—

108

C₂×C₅

—

109

C₂×C₅

#4031

yes

—

110

C₂×C₅

#4058

yes

—

111

C₂×C₅

#4062

yes

—

112

C₂³

—

113

Q₈

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₂×D₅²

yes

characteristic

119

C₈

—

120

C₅

—

121

C₅

#65057

yes

complemented

122

C₂²

—

123

C₂²

—

124

C₂²

#87886

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

100

—

128

C₄

#87879

yes

—

129

C₂

—

130

C₂

#11717054

yes

—

131

C₂

#11717071

yes

characteristic

	subgroup	G/H
center	C₂²	#87886
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂×D₅²
Fitting, prime radical	C₂×C₅²×D₄	C₂²
socle	C₂²×C₅²	C₂×D₄
hypercenter	C₂×D₄	D₅²
hypocenter	C₅²	#488
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₄

subgroup

G/H

center

C₂²

#87886

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂×D₅²

Fitting, prime radical

C₂×C₅²×D₄

C₂²

socle

C₂²×C₅²

C₂×D₄

hypercenter

C₂×D₄

D₅²

hypocenter

C₅²

#488

LCS

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₄

N	H ≅ G/N	type
2. #11716991	C₂	semidirect
4. #11716608	C₂	semidirect
5. #11716594	C₂	semidirect
80. C₅²	#488	semidirect
121. C₅	#65057	semidirect

H ≅ G/N

type

2. #11716991

C₂

semidirect

4. #11716608

C₂

semidirect

5. #11716594

C₂

semidirect

80. C₅²

#488

semidirect

121. C₅

#65057

semidirect

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

subgroup	index
#11717256	2
#11716991	2
C₅×#3964	2
#11716608	2
#11716594	2
#65057	5

subgroup

index

Subgroups — #13101656

Conjugacy classes of subgroups

Complements

Maximal subgroups