Group #13101782 — subgroups

Subgroups — #13101782

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	#11717267	C₂	1	1	yes	maximal, characteristic
2	#11716991	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3967	C₂	1	2	yes	maximal, complemented
4	#11716605	C₂	1	2	yes	maximal
5	#11716594	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
8	#87808	—	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	#65089	—	5	2	no	maximal
14	C₂²×C₅⋊D₅	—	2	1	no	—
15	C₅²×Q₈	—	2	2	no	—
16	C₂×C₄×C₅²	—	2	1	no	—
17	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
18	C₂×#1039	D₄	1	2	yes	—
19	C₄⋊(C₅⋊D₅)	—	2	2	no	—
20	C₂×C₅×Dic₅	—	2	2	no	—
21	C₅×#168	—	2	2	no	—
22	#4093	—	5	2	no	—
23	#4093	—	10	2	no	—
24	#4036	—	5	2	no	—
25	C₅×#103	—	5	2	no	—
26	#3967	D₅	1	2	yes	complemented
27	#3953	—	5	2	no	—
28	C₈×Dic₅	—	5	2	no	—
29	#3941	—	5	2	no	—
30	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
31	C₂×C₅⋊D₅	—	4	1	no	—
32	C₂×C₅⋊D₅	—	4	2	no	—
33	C₄×C₅²	—	4	1	no	—
34	C₄×C₅²	#41	1	2	yes	—
35	#1039	—	2	2	no	—
36	C₅×Dic₅	—	4	2	no	—
37	C₂×C₅×Q₈	—	2	2	no	—
38	C₂×C₅×Q₈	C₂×D₅	1	2	yes	—
39	C₂×D₂₀	—	5	2	no	—
40	C₂×D₂₀	—	10	2	no	—
41	C₂×C₅×C₈	—	5	2	no	—
42	C₅×C₄⋊C₄	—	5	2	no	—
43	#685	—	10	2	no	—
44	#685	—	10	2	no	—
45	#685	—	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	#124	1	1	yes	characteristic
54	C₂×C₅²	#135	1	2	yes	—
55	C₅⋊D₅	—	8	1	no	—
56	C₂²×D₅	—	10	4	no	—
57	C₂²×D₅	—	10	2	no	—
58	C₅×Q₈	—	2	2	no	—
59	C₅×Q₈	—	2	2	no	—
60	C₅×Q₈	—	4	4	no	—
61	C₂×C₄×C₅	—	2	4	no	—
62	C₂×C₄×C₅	—	2	2	no	—
63	C₂×C₄×C₅	—	2	2	no	—
64	C₂×C₄×C₅	—	10	2	no	—
65	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
66	C₂×Dic₅	—	2	2	no	—
67	C₂×Dic₅	—	5	2	no	—
68	C₂×Dic₅	—	5	2	no	—
69	C₂×Dic₅	—	10	4	no	—
70	D₂₀	—	10	2	no	—
71	D₂₀	—	10	2	no	—
72	D₂₀	—	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	—	5	2	no	—
78	#102	—	25	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	D₄×D₅	1	2	yes	—
83	C₂×D₅	—	20	4	no	—
84	C₂×D₅	—	20	8	no	—
85	C₂×D₅	—	20	2	no	—
86	C₂×D₅	—	20	2	no	—
87	C₂×D₅	—	20	2	no	—
88	C₄×C₅	—	2	4	no	—
89	C₄×C₅	—	4	4	no	—
90	C₄×C₅	—	4	2	no	—
91	C₄×C₅	—	20	2	no	—
92	C₄×C₅	#709	1	2	yes	—
93	C₄×C₅	#711	1	2	yes	—
94	Dic₅	—	4	2	no	—
95	Dic₅	—	10	2	no	—
96	Dic₅	—	10	2	no	—
97	Dic₅	—	20	4	no	—
98	C₂×Q₈	D₅²	1	1	yes	characteristic
99	C₂×D₄	—	25	1	no	—
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	#4031	1	2	yes	—
108	C₂×C₅	#4062	1	2	yes	—
109	C₂×C₅	#4065	1	2	yes	—
110	D₅	—	40	4	no	—
111	D₅	—	40	2	no	—
112	C₂³	—	50	1	no	—
113	Q₈	—	2	2	no	—
114	D₄	—	50	2	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	10	2	no	—
117	C₂×C₄	—	25	2	no	—
118	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
119	C₈	—	10	2	no	—
120	C₅	—	2	2	no	—
121	C₅	#65089	1	2	yes	complemented
122	C₂²	—	100	2	no	—
123	C₂²	—	100	1	no	—
124	C₂²	#87886	1	1	yes	characteristic
125	C₄	—	4	1	no	—
126	C₄	—	20	2	no	—
127	C₄	—	50	2	no	—
128	C₄	#87879	1	2	yes	—
129	C₂	—	200	1	no	—
130	C₂	#11717068	1	2	yes	—
131	C₂	#11717071	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11717267

C₂

yes

maximal, characteristic

#11716991

C₂

yes

maximal, characteristic, complemented

C₅×#3967

C₂

yes

maximal, complemented

#11716605

C₂

yes

maximal

#11716594

C₂

yes

maximal, characteristic, complemented

C₂×C₅²×Q₈

C₂²

yes

characteristic

C₂×C₄⋊(C₅⋊D₅)

C₂²

yes

characteristic

#87808

—

C₄×#1039

C₂²

yes

characteristic

C₅×C₄⋊Dic₅

C₂²

yes

—

C₂×C₅×#168

C₂²

yes

—

#87780

—

#65089

—

maximal

C₂²×C₅⋊D₅

—

C₅²×Q₈

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

D₄

yes

—

C₄⋊(C₅⋊D₅)

—

C₂×C₅×Dic₅

—

—

—

—

—

—

D₅

yes

complemented

#3953

—

C₈×Dic₅

—

#3941

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₄×C₅²

—

C₄×C₅²

#41

yes

—

#1039

—

C₅×Dic₅

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

C₂×D₅

yes

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×C₅×C₈

—

C₅×C₄⋊C₄

—

#685

—

#685

—

#685

—

C₄⋊Dic₅

—

C₄⋊Dic₅

C₂×D₅

yes

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

—

C₂×#168

C₂×D₅

yes

—

—

C₂×C₅²

yes

C₂×C₅²

yes

—

C₅⋊D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

D₂₀

—

D₂₀

—

D₂₀

—

C₅×C₈

—

—

—

—

—

—

C₄×C₈

—

C₅²

#488

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

D₄×D₅

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#709

yes

—

C₄×C₅

#711

yes

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

C₂×Q₈

D₅²

yes

characteristic

C₂×D₄

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₂×C₅

—

106

C₂×C₅

—

107

C₂×C₅

#4031

yes

—

108

C₂×C₅

#4062

yes

—

109

C₂×C₅

#4065

yes

—

110

D₅

—

111

D₅

—

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

#65089

yes

complemented

122

C₂²

—

100

—

123

C₂²

—

100

—

124

C₂²

#87886

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

#87879

yes

—

129

C₂

—

200

—

130

C₂

#11717068

yes

—

131

C₂

#11717071

yes

characteristic

	subgroup	G/H
center	C₂²	#87886
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂×D₅²
Fitting	C₂×C₅²×Q₈	C₂²
socle	C₂²×C₅²	C₂×D₄
prime radical	C₂×C₄⋊(C₅⋊D₅)	C₂²
hypercenter	C₂×Q₈	D₅²
hypocenter	C₅²	#488
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₈

subgroup

G/H

center

C₂²

#87886

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂×D₅²

Fitting

C₂×C₅²×Q₈

C₂²

socle

C₂²×C₅²

C₂×D₄

prime radical

C₂×C₄⋊(C₅⋊D₅)

C₂²

hypercenter

C₂×Q₈

D₅²

hypocenter

C₅²

#488

LCS

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₈

N	H ≅ G/N	type
2. #11716991	C₂	semidirect
3. C₅×#3967	C₂	semidirect
5. #11716594	C₂	semidirect
26. #3967	D₅	semidirect
80. C₅²	#488	semidirect
121. C₅	#65089	semidirect

H ≅ G/N

type

2. #11716991

C₂

semidirect

3. C₅×#3967

C₂

semidirect

5. #11716594

C₂

semidirect

26. #3967

D₅

semidirect

80. C₅²

#488

semidirect

121. C₅

#65089

semidirect

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

subgroup	index
#11717267	2
#11716991	2
C₅×#3967	2
#11716605	2
#11716594	2
#65089	5

subgroup

index

Subgroups — #13101782

Conjugacy classes of subgroups

Complements

Maximal subgroups