Group #13099338 — subgroups

Subgroups — #13099338

Order 1600.

288 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 92 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	C₂×C₄⋊#1039	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#681	C₂	1	2	yes	maximal
3	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
4	C₂²×#1039	C₄	1	2	yes	—
5	C₄⋊#1039	—	2	2	no	—
6	C₅×#681	—	2	2	no	—
7	C₅×#681	C₄	1	4	yes	complemented
8	C₂×C₅×#168	—	2	2	no	—
9	#64774	—	5	2	no	maximal
10	C₂³×C₅²	C₂×C₄	1	1	yes	characteristic
11	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
12	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
13	C₂×C₄×C₅²	D₄	1	2	yes	—
14	C₂×C₄×C₅²	Q₈	1	1	yes	characteristic
15	C₂×#1039	—	2	2	no	—
16	C₂×#1039	—	4	2	no	—
17	C₅×#168	—	2	4	no	—
18	C₂×C₅×M₁₆	—	5	2	no	—
19	C₂×C₄⋊Dic₅	—	5	2	no	—
20	C₂×C₄⋊Dic₅	—	10	2	no	—
21	C₂×#681	D₅	1	2	yes	—
22	C₂²×C₅²	—	2	1	no	—
23	C₂²×C₅²	—	2	1	no	—
24	C₂²×C₅²	C₄²	1	1	yes	characteristic
25	C₂²×C₅²	C₂²⋊C₄	1	1	yes	characteristic
26	C₂²×C₅²	C₄⋊C₄	1	1	yes	characteristic
27	C₄×C₅²	C₂²⋊C₄	1	2	yes	—
28	C₄×C₅²	C₄⋊C₄	1	2	yes	—
29	#1039	—	4	2	no	—
30	C₂²×C₄×C₅	—	2	2	no	—
31	C₂²×C₄×C₅	C₂×D₅	1	2	yes	—
32	C₂²×Dic₅	—	5	4	no	—
33	C₂²×Dic₅	—	10	4	no	—
34	C₅×M₁₆	—	5	4	no	—
35	C₅×M₁₆	—	10	2	no	—
36	C₂×C₅×C₈	—	10	2	no	—
37	C₄⋊Dic₅	—	10	8	no	—
38	C₄⋊Dic₅	—	10	2	no	—
39	C₄⋊Dic₅	—	10	2	no	—
40	#681	—	2	2	no	—
41	#681	Dic₅	1	4	yes	complemented
42	C₂×#168	—	2	2	no	—
43	#343	—	25	1	no	Sylow
44	C₂×C₅²	—	2	1	no	—
45	C₂×C₅²	—	2	1	no	—
46	C₂×C₅²	#95	1	1	yes	characteristic
47	C₂×C₅²	#100	1	1	yes	characteristic
48	C₂×C₅²	#101	1	1	yes	characteristic
49	C₂³×C₅	—	2	2	no	—
50	C₂³×C₅	C₄×D₅	1	2	yes	—
51	C₂×C₄×C₅	—	2	4	no	—
52	C₂×C₄×C₅	—	2	2	no	—
53	C₂×C₄×C₅	—	2	4	no	—
54	C₂×C₄×C₅	—	2	2	no	—
55	C₂×C₄×C₅	Dic₁₀	1	2	yes	—
56	C₂×C₄×C₅	C₄×D₅	1	2	yes	—
57	C₂×C₄×C₅	D₂₀	1	2	yes	—
58	C₂×C₄×C₅	C₂×Dic₅	1	2	yes	—
59	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
60	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
61	C₂×Dic₅	—	10	8	no	—
62	C₂×Dic₅	—	10	4	no	—
63	C₂×Dic₅	—	20	8	no	—
64	C₂×Dic₅	—	20	4	no	—
65	C₅×C₈	—	10	4	no	—
66	#168	—	2	4	no	—
67	C₂×M₁₆	—	5	2	no	—
68	C₂×C₄⋊C₄	—	25	1	no	—
69	C₅²	#343	1	1	yes	characteristic, complemented, Sylow
70	C₂²×C₅	—	2	4	no	—
71	C₂²×C₅	—	2	4	no	—
72	C₂²×C₅	—	2	2	no	—
73	C₂²×C₅	—	2	2	no	—
74	C₂²×C₅	—	2	2	no	—
75	C₂²×C₅	—	2	2	no	—
76	C₂²×C₅	—	2	2	no	—
77	C₂²×C₅	C₄×Dic₅	1	2	yes	—
78	C₂²×C₅	#683	1	2	yes	—
79	C₂²×C₅	#685	1	2	yes	—
80	C₄×C₅	—	2	4	no	—
81	C₄×C₅	—	2	4	no	—
82	C₄×C₅	#683	1	2	yes	—
83	C₄×C₅	C₄⋊Dic₅	1	2	yes	—
84	C₄×C₅	#685	1	2	yes	—
85	C₄×C₅	C₂²⋊Dic₅	1	2	yes	—
86	Dic₅	—	20	8	no	—
87	Dic₅	—	20	4	no	—
88	C₂²×C₄	—	25	2	no	—
89	C₂²×C₄	D₅²	1	1	yes	characteristic
90	M₁₆	—	5	4	no	—
91	M₁₆	—	10	2	no	—
92	C₂×C₈	—	10	2	no	—
93	C₄⋊C₄	—	50	2	no	—
94	C₂×C₅	—	2	4	no	—
95	C₂×C₅	—	2	4	no	—
96	C₂×C₅	—	2	2	no	—
97	C₂×C₅	—	2	2	no	—
98	C₂×C₅	—	2	2	no	—
99	C₂×C₅	—	2	2	no	—
100	C₂×C₅	—	2	2	no	—
101	C₂×C₅	#3955	1	2	yes	—
102	C₂×C₅	#3956	1	2	yes	—
103	C₂×C₅	#3963	1	2	yes	—
104	C₂³	#6036	1	1	yes	characteristic
105	C₂×C₄	—	50	2	no	—
106	C₂×C₄	—	100	2	no	—
107	C₂×C₄	D₅×Dic₅	1	2	yes	—
108	C₂×C₄	#6037	1	1	yes	characteristic
109	C₂×C₄	#6038	1	2	yes	—
110	C₂×C₄	#6039	1	1	yes	characteristic
111	C₈	—	10	4	no	—
112	C₅	—	2	2	no	—
113	C₅	#64774	1	2	yes	complemented
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	Dic₅²	1	1	yes	characteristic
117	C₂²	#87779	1	1	yes	characteristic
118	C₂²	#87781	1	1	yes	characteristic
119	C₄	—	100	2	no	—
120	C₄	#87778	1	2	yes	—
121	C₄	#87780	1	2	yes	—
122	C₂	—	2	1	no	—
123	C₂	—	2	1	no	—
124	C₂	#11716600	1	1	yes	characteristic
125	C₂	#11716602	1	1	yes	characteristic
126	C₂	#11716620	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#1039

C₂

yes

maximal, characteristic

C₂×C₅×#681

C₂

yes

maximal

C₂²×C₄×C₅²

C₂²

yes

characteristic

C₂²×#1039

C₄

yes

—

C₄⋊#1039

—

C₅×#681

—

C₅×#681

C₄

yes

complemented

C₂×C₅×#168

—

#64774

—

maximal

C₂³×C₅²

C₂×C₄

yes

characteristic

C₂×C₄×C₅²

C₂×C₄

yes

—

C₂×C₄×C₅²

D₄

yes

characteristic

C₂×C₄×C₅²

D₄

yes

—

C₂×C₄×C₅²

Q₈

yes

—

—

—

C₂×C₅×M₁₆

—

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

—

C₂×#681

D₅

yes

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

C₄²

yes

characteristic

C₂²×C₅²

C₂²⋊C₄

yes

characteristic

C₂²×C₅²

C₄⋊C₄

yes

characteristic

C₄×C₅²

C₂²⋊C₄

yes

—

C₄×C₅²

C₄⋊C₄

yes

—

#1039

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₅

yes

—

C₂²×Dic₅

—

C₂²×Dic₅

—

C₅×M₁₆

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

#681

—

#681

Dic₅

yes

—

—

C₂×C₅²

—

C₂×C₅²

—

C₂×C₅²

yes

C₂×C₅²

yes

C₂×C₅²

yes

C₂³×C₅

—

C₂³×C₅

C₄×D₅

yes

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

Dic₁₀

yes

—

C₂×C₄×C₅

C₄×D₅

yes

—

C₂×C₄×C₅

D₂₀

yes

—

C₂×C₄×C₅

C₂×Dic₅

yes

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₅×C₈

—

#168

—

C₂×M₁₆

—

C₂×C₄⋊C₄

—

C₅²

#343

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₄×Dic₅

yes

—

C₂²×C₅

#683

yes

—

C₂²×C₅

#685

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#683

yes

—

C₄×C₅

C₄⋊Dic₅

yes

—

C₄×C₅

#685

yes

—

C₄×C₅

C₂²⋊Dic₅

yes

—

Dic₅

—

Dic₅

—

C₂²×C₄

—

C₂²×C₄

D₅²

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

—

100

C₂×C₅

—

101

C₂×C₅

#3955

yes

—

102

C₂×C₅

#3956

yes

—

103

C₂×C₅

#3963

yes

—

104

C₂³

#6036

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

—

100

—

107

C₂×C₄

D₅×Dic₅

yes

—

108

C₂×C₄

#6037

yes

characteristic

109

C₂×C₄

#6038

yes

—

110

C₂×C₄

#6039

yes

characteristic

111

C₈

—

112

C₅

—

113

C₅

#64774

yes

complemented

114

C₂²

—

115

C₂²

—

116

C₂²

Dic₅²

yes

characteristic

117

C₂²

#87779

yes

characteristic

118

C₂²

#87781

yes

characteristic

119

C₄

—

100

—

120

C₄

#87778

yes

—

121

C₄

#87780

yes

—

122

C₂

—

123

C₂

—

124

C₂

#11716600

yes

characteristic

125

C₂

#11716602

yes

characteristic

126

C₂

#11716620

yes

characteristic

	subgroup	G/H
center	C₂²	#87779
commutator	C₂²×C₅²	C₄²
Frattini, hypercenter	C₂²×C₄	D₅²
Fitting, cosocle	C₂²×C₄×C₅²	C₂²
socle	C₂²×C₅²	C₂²⋊C₄
prime radical	C₂³×C₅²	C₂×C₄
hypocenter	C₅²	#343
LCS	G ▷ C₂²×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

subgroup

G/H

center

C₂²

#87779

commutator

C₂²×C₅²

C₄²

Frattini, hypercenter

C₂²×C₄

D₅²

Fitting, cosocle

C₂²×C₄×C₅²

C₂²

socle

C₂²×C₅²

C₂²⋊C₄

prime radical

C₂³×C₅²

C₂×C₄

hypocenter

C₅²

#343

LCS

G ▷ C₂²×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₂²×C₄

N	H ≅ G/N	type
7. C₅×#681	C₄	semidirect
41. #681	Dic₅	semidirect
69. C₅²	#343	semidirect
113. C₅	#64774	semidirect

H ≅ G/N

type

7. C₅×#681

C₄

semidirect

41. #681

Dic₅

semidirect

69. C₅²

#343

semidirect

113. C₅

#64774

semidirect

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

subgroup	index
C₂×C₄⋊#1039	2
C₂×C₅×#681	2
#64774	5

subgroup

index

C₂×C₄⋊#1039

C₂×C₅×#681

#64774

Subgroups — #13099338

Conjugacy classes of subgroups

Complements

Maximal subgroups