Group #13102525 — subgroups

Subgroups — #13102525

Order 1600.

600 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 221 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×C₄⋊#1039	C₂	1	2	yes	maximal
3	C₂×#87806	C₂	1	1	yes	maximal, characteristic
4	C₅²×#95	C₂	1	1	yes	maximal, characteristic
5	#11716698	C₂	1	2	yes	maximal
6	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₅²	C₂²	1	2	yes	—
8	C₂²×#1039	C₂²	1	1	yes	characteristic
9	C₂²×#1039	C₂²	1	2	yes	—
10	C₂²×#1039	C₂²	1	1	yes	characteristic
11	C₄⋊#1039	—	2	2	no	—
12	C₄⋊#1039	—	2	4	no	—
13	#87806	—	2	2	no	—
14	#64950	—	5	6	no	maximal
15	C₂³×C₅²	C₂³	1	1	yes	characteristic
16	C₂×C₄×C₅²	—	2	2	no	—
17	C₂×C₄×C₅²	—	2	2	no	—
18	C₂×C₄×C₅²	—	2	2	no	—
19	C₂×C₄×C₅²	D₄	1	2	yes	—
20	C₂×C₄×C₅²	Q₈	1	4	yes	—
21	C₂×#1039	—	2	1	no	—
22	C₂×#1039	—	2	1	no	—
23	C₂×#1039	—	2	1	no	—
24	C₂×#1039	—	2	2	no	—
25	C₂×#1039	—	2	2	no	—
26	C₂×#1039	—	2	2	no	—
27	C₂×#1039	—	2	2	no	—
28	C₂×#1039	D₄	1	2	yes	—
29	C₂×C₄⋊Dic₅	—	5	12	no	—
30	C₂×C₄⋊Dic₅	—	5	6	no	—
31	C₂×#683	—	5	6	no	—
32	C₅×#95	D₅	1	6	yes	—
33	#3963	—	5	12	no	—
34	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₅²	C₂×Q₈	1	2	yes	—
37	C₂²×C₅²	#41	1	1	yes	characteristic
38	C₂²×C₅²	#41	1	2	yes	—
39	C₄×C₅²	—	2	4	no	—
40	C₄×C₅²	—	2	2	no	—
41	#1039	—	2	2	no	—
42	#1039	—	4	2	no	—
43	#1039	—	4	1	no	—
44	C₂²×C₄×C₅	C₂×D₅	1	12	yes	—
45	C₂²×C₄×C₅	C₂×D₅	1	6	yes	—
46	C₂²×Dic₅	—	5	6	no	—
47	C₂²×Dic₅	—	5	6	no	—
48	C₂²×Dic₅	—	5	12	no	—
49	C₄⋊Dic₅	—	10	24	no	—
50	C₄⋊Dic₅	—	10	12	no	—
51	#683	—	10	12	no	—
52	#398	—	25	1	no	Sylow
53	C₂×C₅²	#121	1	1	yes	characteristic
54	C₂×C₅²	C₂²⋊Q₈	1	2	yes	—
55	C₂×C₅²	#123	1	1	yes	characteristic
56	C₂×C₅²	#125	1	2	yes	—
57	C₂×C₅²	C₄⋊Q₈	1	1	yes	characteristic
58	C₂³×C₅	C₂²×D₅	1	6	yes	—
59	C₂×C₄×C₅	—	2	12	no	—
60	C₂×C₄×C₅	—	2	12	no	—
61	C₂×C₄×C₅	—	2	12	no	—
62	C₂×C₄×C₅	Dic₁₀	1	24	yes	—
63	C₂×C₄×C₅	D₂₀	1	12	yes	—
64	C₂×Dic₅	—	5	12	no	—
65	C₂×Dic₅	—	10	6	no	—
66	C₂×Dic₅	—	10	6	no	—
67	C₂×Dic₅	—	10	6	no	—
68	C₂×Dic₅	—	10	12	no	—
69	C₂×Dic₅	—	10	12	no	—
70	C₂×Dic₅	—	10	12	no	—
71	C₂×Dic₅	—	10	12	no	—
72	C₂×C₄⋊C₄	—	25	1	no	—
73	C₂×C₄⋊C₄	—	25	2	no	—
74	C₂×C₄⋊C₄	—	25	1	no	—
75	#95	—	25	2	no	—
76	#95	C₅⋊D₅	1	1	yes	characteristic
77	C₅²	#398	1	1	yes	characteristic, complemented, Sylow
78	C₂²×C₅	C₂×Dic₁₀	1	12	yes	—
79	C₂²×C₅	C₂×D₂₀	1	6	yes	—
80	C₂²×C₅	D₄×D₅	1	6	yes	—
81	C₂²×C₅	#711	1	12	yes	—
82	C₂²×C₅	#713	1	6	yes	—
83	C₄×C₅	—	2	24	no	—
84	C₄×C₅	—	2	12	no	—
85	Dic₅	—	10	12	no	—
86	Dic₅	—	20	6	no	—
87	Dic₅	—	20	12	no	—
88	C₂²×C₄	—	25	1	no	—
89	C₂²×C₄	—	25	1	no	—
90	C₂²×C₄	—	25	2	no	—
91	C₂²×C₄	C₂×C₅⋊D₅	1	2	yes	—
92	C₂²×C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
93	C₄⋊C₄	—	50	4	no	—
94	C₄⋊C₄	—	50	2	no	—
95	C₄⋊C₄	—	50	2	no	—
96	C₂×C₅	#4015	1	6	yes	—
97	C₂×C₅	#4024	1	12	yes	—
98	C₂×C₅	#4032	1	6	yes	—
99	C₂×C₅	#4036	1	12	yes	—
100	C₂×C₅	#4041	1	6	yes	—
101	C₂³	C₂²×C₅⋊D₅	1	1	yes	characteristic
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	25	2	no	—
106	C₂×C₄	—	50	1	no	—
107	C₂×C₄	—	50	1	no	—
108	C₂×C₄	—	50	1	no	—
109	C₂×C₄	—	50	2	no	—
110	C₂×C₄	—	50	2	no	—
111	C₂×C₄	—	50	2	no	—
112	C₂×C₄	—	50	2	no	—
113	C₂×C₄	#6045	1	4	yes	—
114	C₂×C₄	C₄⋊(C₅⋊D₅)	1	2	yes	—
115	C₅	#64950	1	6	yes	complemented
116	C₂²	C₂×#6045	1	2	yes	—
117	C₂²	C₂×C₄⋊(C₅⋊D₅)	1	1	yes	characteristic
118	C₂²	D₄×C₅⋊D₅	1	1	yes	characteristic
119	C₂²	#87902	1	2	yes	—
120	C₂²	#87904	1	1	yes	characteristic
121	C₄	—	2	4	no	—
122	C₄	—	2	2	no	—
123	C₄	—	50	2	no	—
124	C₄	—	100	1	no	—
125	C₄	—	100	2	no	—
126	C₂	#11717189	1	1	yes	characteristic
127	C₂	#11717198	1	2	yes	—
128	C₂	#11717206	1	1	yes	characteristic
129	C₂	#11717210	1	2	yes	—
130	C₂	#11717215	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#1039

C₂

yes

maximal, characteristic

C₂×C₄⋊#1039

C₂

yes

maximal

C₂×#87806

C₂

yes

maximal, characteristic

C₅²×#95

C₂

yes

maximal, characteristic

#11716698

C₂

yes

maximal

C₂²×C₄×C₅²

C₂²

yes

characteristic

C₂²×C₄×C₅²

C₂²

yes

—

C₂²×#1039

C₂²

yes

characteristic

C₂²×#1039

C₂²

yes

—

C₂²×#1039

C₂²

yes

—

—

—

—

C₂³×C₅²

C₂³

yes

characteristic

C₂×C₄×C₅²

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

D₄

yes

—

C₂×C₄×C₅²

Q₈

yes

—

—

—

—

—

—

—

—

D₄

yes

—

C₂×C₄⋊Dic₅

—

C₂×C₄⋊Dic₅

—

C₂×#683

—

C₅×#95

D₅

yes

—

#3963

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₂²×C₅²

C₂×D₄

yes

characteristic

C₂²×C₅²

C₂×Q₈

yes

—

C₂²×C₅²

#41

yes

characteristic

C₂²×C₅²

#41

yes

—

C₄×C₅²

—

C₄×C₅²

—

#1039

—

#1039

—

#1039

—

C₂²×C₄×C₅

C₂×D₅

yes

—

C₂²×C₄×C₅

C₂×D₅

yes

—

C₂²×Dic₅

—

C₂²×Dic₅

—

C₂²×Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

—

—

C₂×C₅²

yes

C₂×C₅²

C₂²⋊Q₈

yes

—

C₂×C₅²

#123

yes

characteristic

C₂×C₅²

#125

yes

—

C₂×C₅²

C₄⋊Q₈

yes

characteristic

C₂³×C₅

C₂²×D₅

yes

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

Dic₁₀

yes

—

C₂×C₄×C₅

D₂₀

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

#95

—

#95

C₅⋊D₅

yes

characteristic

C₅²

#398

yes

characteristic, complemented, Sylow

C₂²×C₅

C₂×Dic₁₀

yes

—

C₂²×C₅

C₂×D₂₀

yes

—

C₂²×C₅

D₄×D₅

yes

—

C₂²×C₅

#711

yes

—

C₂²×C₅

#713

yes

—

C₄×C₅

—

C₄×C₅

—

Dic₅

—

Dic₅

—

Dic₅

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₅⋊D₅

yes

—

C₂²×C₄

C₂×C₅⋊D₅

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂×C₅

#4015

yes

—

C₂×C₅

#4024

yes

—

C₂×C₅

#4032

yes

—

C₂×C₅

#4036

yes

—

100

C₂×C₅

#4041

yes

—

101

C₂³

C₂²×C₅⋊D₅

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

#6045

yes

—

114

C₂×C₄

C₄⋊(C₅⋊D₅)

yes

—

115

C₅

#64950

yes

complemented

116

C₂²

C₂×#6045

yes

—

117

C₂²

C₂×C₄⋊(C₅⋊D₅)

yes

characteristic

118

C₂²

D₄×C₅⋊D₅

yes

characteristic

119

C₂²

#87902

yes

—

120

C₂²

#87904

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

100

—

125

C₄

—

100

—

126

C₂

#11717189

yes

characteristic

127

C₂

#11717198

yes

—

128

C₂

#11717206

yes

characteristic

129

C₂

#11717210

yes

—

130

C₂

#11717215

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×C₅⋊D₅
commutator, socle, prime radical, cosocle	C₂³×C₅²	C₂³
Fitting	C₅²×#95	C₂
hypercenter	#95	C₅⋊D₅
hypocenter	C₅²	#398
LCS	G ▷ C₂³×C₅² ▷ C₅²
UCS	C₁ ◁ C₂³ ◁ #95

subgroup

G/H

center, Frattini

C₂³

C₂²×C₅⋊D₅

commutator, socle, prime radical, cosocle

C₂³×C₅²

C₂³

Fitting

C₅²×#95

C₂

C₅⋊D₅

C₅²

G ▷ C₂³×C₅² ▷ C₅²

C₁ ◁ C₂³ ◁ #95

N	H ≅ G/N	type
77. C₅²	#398	semidirect
115. C₅	#64950	semidirect

H ≅ G/N

type

77. C₅²

#398

semidirect

115. C₅

#64950

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×#87806	2
C₅²×#95	2
#11716698	2
#64950	5

subgroup

index

C₂×C₄⋊#1039

C₂×#87806

C₅²×#95

#11716698

#64950

Subgroups — #13102525

Conjugacy classes of subgroups

Complements

Maximal subgroups