Group #13102938 — subgroups

Subgroups — #13102938

Order 1600.

560 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 141 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	C₅²×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#87809	C₂	1	2	yes	maximal, complemented
3	C₄×C₄⋊(C₅⋊D₅)	C₂	1	1	yes	maximal, characteristic, complemented
4	#11716699	C₂	1	2	yes	maximal, complemented
5	C₄⋊#6029	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅²×D₄	—	2	1	no	—
7	C₂×C₅²×D₄	—	2	1	no	—
8	C₂×C₅²×D₄	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₅⋊D₅	—	2	1	no	—
11	C₄²×C₅²	C₂²	1	1	yes	characteristic
12	#87809	—	2	4	no	—
13	#87808	—	2	1	no	—
14	C₄⋊#1039	C₂²	1	1	yes	characteristic
15	C₂×#6029	C₂²	1	2	yes	—
16	#65361	—	5	6	no	maximal
17	C₂³×C₅²	—	2	2	no	—
18	C₂²×C₅⋊D₅	—	2	1	no	—
19	C₅²×D₄	—	2	2	no	—
20	C₅²×D₄	—	2	2	no	—
21	C₅²×D₄	—	4	2	no	—
22	C₅²×D₄	—	4	2	no	—
23	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
24	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
25	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
26	C₂×#1039	—	2	1	no	—
27	C₄⋊(C₅⋊D₅)	—	2	1	no	—
28	C₄⋊(C₅⋊D₅)	D₄	1	2	yes	complemented
29	C₄×C₅⋊D₅	—	4	2	no	—
30	#6029	—	2	2	no	—
31	C₅×#127	D₅	1	6	yes	complemented
32	C₂×#686	—	5	12	no	—
33	C₄×D₂₀	—	5	6	no	—
34	#3964	—	5	12	no	—
35	C₄⋊#168	—	5	6	no	—
36	C₂²×C₅²	—	4	2	no	—
37	C₂²×C₅²	—	4	2	no	—
38	C₂²×C₅²	—	4	2	no	—
39	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
40	C₂×C₅⋊D₅	—	2	2	no	—
41	C₂×C₅⋊D₅	—	4	1	no	—
42	C₂×C₅⋊D₅	—	4	1	no	—
43	C₄×C₅²	—	2	1	no	—
44	C₄×C₅²	D₈	1	2	yes	complemented
45	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
46	C₄×C₅²	#41	1	1	yes	characteristic
47	#1039	—	4	1	no	—
48	C₂×C₅×D₄	—	2	6	no	—
49	C₂×C₅×D₄	—	2	6	no	—
50	C₂×C₅×D₄	C₂×D₅	1	12	yes	—
51	C₂×D₂₀	—	5	6	no	—
52	C₂×C₄×D₅	—	10	6	no	—
53	C₄²×C₅	C₂×D₅	1	6	yes	—
54	#686	—	10	24	no	—
55	#685	—	10	6	no	—
56	C₄⋊Dic₅	—	5	6	no	—
57	C₂×#168	—	5	12	no	—
58	#459	—	25	1	no	Sylow
59	C₂×C₅²	—	8	2	no	—
60	C₂×C₅²	#121	1	1	yes	characteristic
61	C₂×C₅²	C₂×D₈	1	1	yes	characteristic
62	C₂×C₅²	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₅⋊D₅	—	4	2	no	—
64	C₂³×C₅	—	2	12	no	—
65	C₂²×D₅	—	10	6	no	—
66	C₅×D₄	—	2	12	no	—
67	C₅×D₄	—	2	12	no	—
68	C₅×D₄	—	4	12	no	—
69	C₅×D₄	—	4	12	no	—
70	C₂×C₄×C₅	C₂²⋊D₅	1	6	yes	—
71	C₂×C₄×C₅	C₂²⋊D₅	1	6	yes	—
72	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
73	C₂×Dic₅	—	10	6	no	—
74	D₂₀	—	5	12	no	—
75	D₂₀	—	10	6	no	—
76	C₄×D₅	—	20	12	no	—
77	#168	—	10	12	no	—
78	C₂×D₈	—	25	2	no	—
79	#127	C₅⋊D₅	1	1	yes	characteristic, complemented
80	C₄×D₄	—	25	1	no	—
81	C₄⋊C₈	—	25	1	no	—
82	#102	—	25	2	no	—
83	C₅²	#459	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₅	—	4	12	no	—
85	C₂²×C₅	—	4	12	no	—
86	C₂²×C₅	—	4	12	no	—
87	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
88	C₂×D₅	—	10	12	no	—
89	C₂×D₅	—	20	6	no	—
90	C₂×D₅	—	20	6	no	—
91	C₄×C₅	—	2	6	no	—
92	C₄×C₅	#686	1	12	yes	complemented
93	C₄×C₅	D₄×D₅	1	6	yes	—
94	C₄×C₅	#711	1	6	yes	—
95	Dic₅	—	20	6	no	—
96	C₂×D₄	—	2	1	no	—
97	C₂×D₄	—	2	1	no	—
98	C₂×D₄	—	25	1	no	—
99	C₂×D₄	C₂×C₅⋊D₅	1	2	yes	—
100	C₂²×C₄	—	50	1	no	—
101	D₈	—	50	4	no	—
102	C₂×C₈	—	25	2	no	—
103	C₄⋊C₄	—	25	1	no	—
104	C₂²⋊C₄	—	50	1	no	—
105	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
106	C₂×C₅	—	8	12	no	—
107	C₂×C₅	C₂×#686	1	6	yes	—
108	C₂×C₅	#4078	1	6	yes	—
109	C₂×C₅	#4084	1	6	yes	—
110	D₅	—	20	12	no	—
111	C₂³	—	2	2	no	—
112	C₂³	—	50	1	no	—
113	D₄	—	2	2	no	—
114	D₄	—	2	2	no	—
115	D₄	—	4	2	no	—
116	D₄	—	4	2	no	—
117	D₄	—	25	2	no	—
118	D₄	—	50	1	no	—
119	C₂×C₄	—	50	1	no	—
120	C₂×C₄	—	100	2	no	—
121	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
122	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
123	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
124	C₈	—	50	2	no	—
125	C₅	#65361	1	6	yes	complemented
126	C₂²	—	4	2	no	—
127	C₂²	—	4	2	no	—
128	C₂²	—	4	2	no	—
129	C₂²	—	50	2	no	—
130	C₂²	—	100	1	no	—
131	C₂²	—	100	1	no	—
132	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	100	1	no	—
135	C₄	#87809	1	2	yes	complemented
136	C₄	D₄×C₅⋊D₅	1	1	yes	characteristic
137	C₄	#87902	1	1	yes	characteristic
138	C₂	—	8	2	no	—
139	C₂	—	100	2	no	—
140	C₂	C₂×#87809	1	1	yes	characteristic
141	C₂	#11717252	1	1	yes	characteristic
142	C₂	#11717258	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²×#127

C₂

yes

maximal, characteristic, complemented

C₂×#87809

C₂

yes

maximal, complemented

C₄×C₄⋊(C₅⋊D₅)

C₂

yes

maximal, characteristic, complemented

#11716699

C₂

yes

maximal, complemented

C₄⋊#6029

C₂

yes

maximal, characteristic, complemented

C₂×C₅²×D₄

—

C₂×C₅²×D₄

—

C₂×C₅²×D₄

C₂²

yes

—

C₂×C₄⋊(C₅⋊D₅)

C₂²

yes

characteristic

C₂×C₄×C₅⋊D₅

—

C₄²×C₅²

C₂²

yes

—

—

C₂²

yes

characteristic

C₂×#6029

C₂²

yes

—

#65361

—

maximal

C₂³×C₅²

—

C₂²×C₅⋊D₅

—

C₅²×D₄

—

C₅²×D₄

—

C₅²×D₄

—

C₅²×D₄

—

C₂×C₄×C₅²

D₄

yes

characteristic

C₂×C₄×C₅²

D₄

yes

characteristic

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

—

C₄⋊(C₅⋊D₅)

—

C₄⋊(C₅⋊D₅)

D₄

yes

complemented

C₄×C₅⋊D₅

—

#6029

—

C₅×#127

D₅

yes

complemented

C₂×#686

—

C₄×D₂₀

—

#3964

—

C₄⋊#168

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₄×C₅²

—

C₄×C₅²

D₈

yes

complemented

C₄×C₅²

C₂×D₄

yes

C₄×C₅²

yes

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

C₂×D₅

yes

—

C₂×D₂₀

—

C₂×C₄×D₅

—

C₄²×C₅

C₂×D₅

yes

—

#686

—

#685

—

C₄⋊Dic₅

—

C₂×#168

—

#459

—

Sylow

C₂×C₅²

—

C₂×C₅²

#121

yes

characteristic

C₂×C₅²

C₂×D₈

yes

characteristic

C₂×C₅²

AGL(1,ℤ/8)

yes

characteristic

C₅⋊D₅

—

C₂³×C₅

—

C₂²×D₅

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

D₂₀

—

D₂₀

—

C₄×D₅

—

#168

—

C₂×D₈

—

#127

C₅⋊D₅

yes

characteristic, complemented

C₄×D₄

—

C₄⋊C₈

—

#102

—

C₅²

#459

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂×C₂²⋊D₅

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

#686

yes

complemented

C₄×C₅

D₄×D₅

yes

—

C₄×C₅

#711

yes

—

Dic₅

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×C₅⋊D₅

yes

—

100

C₂²×C₄

—

101

D₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

C₂×C₅⋊D₅

yes

characteristic

106

C₂×C₅

—

107

C₂×C₅

C₂×#686

yes

—

108

C₂×C₅

#4078

yes

—

109

C₂×C₅

#4084

yes

—

110

D₅

—

111

C₂³

—

112

C₂³

—

113

D₄

—

114

D₄

—

115

D₄

—

116

D₄

—

117

D₄

—

118

D₄

—

119

C₂×C₄

—

120

C₂×C₄

—

100

—

121

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

characteristic

122

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

characteristic

123

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

124

C₈

—

125

C₅

#65361

yes

complemented

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

100

—

131

C₂²

—

100

—

132

C₂²

C₂×C₂²⋊(C₅⋊D₅)

yes

characteristic

133

C₄

—

134

C₄

—

100

—

135

C₄

#87809

yes

complemented

136

C₄

D₄×C₅⋊D₅

yes

characteristic

137

C₄

#87902

yes

characteristic

138

C₂

—

139

C₂

—

100

—

140

C₂

C₂×#87809

yes

characteristic

141

C₂

#11717252

yes

characteristic

142

C₂

#11717258

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊(C₅⋊D₅)
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₅²×#127	C₂
socle	C₂²×C₅²	C₂×D₄
hypercenter	#127	C₅⋊D₅
hypocenter	C₅²	#459
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

subgroup

G/H

center

C₂²

C₂×C₂²⋊(C₅⋊D₅)

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂²×C₅⋊D₅

Fitting

C₅²×#127

C₂

socle

C₂²×C₅²

C₂×D₄

C₅⋊D₅

C₅²

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #127

N	H ≅ G/N	type
1. C₅²×#127	C₂	semidirect
2. C₂×#87809	C₂	semidirect
3. C₄×C₄⋊(C₅⋊D₅)	C₂	semidirect
4. #11716699	C₂	semidirect
5. C₄⋊#6029	C₂	semidirect
28. C₄⋊(C₅⋊D₅)	D₄	semidirect
31. C₅×#127	D₅	semidirect
44. C₄×C₅²	D₈	semidirect
79. #127	C₅⋊D₅	semidirect
83. C₅²	#459	semidirect
92. C₄×C₅	#686	semidirect
125. C₅	#65361	semidirect
135. C₄	#87809	semidirect

H ≅ G/N

type

1. C₅²×#127

C₂

semidirect

2. C₂×#87809

C₂

semidirect

3. C₄×C₄⋊(C₅⋊D₅)

C₂

semidirect

4. #11716699

C₂

semidirect

5. C₄⋊#6029

C₂

semidirect

28. C₄⋊(C₅⋊D₅)

D₄

semidirect

31. C₅×#127

D₅

semidirect

44. C₄×C₅²

D₈

semidirect

79. #127

C₅⋊D₅

semidirect

83. C₅²

#459

semidirect

92. C₄×C₅

#686

semidirect

125. C₅

#65361

semidirect

135. C₄

#87809

semidirect

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

subgroup	index
C₅²×#127	2
C₂×#87809	2
C₄×C₄⋊(C₅⋊D₅)	2
#11716699	2
C₄⋊#6029	2
#65361	5

subgroup

index

C₅²×#127

C₄×C₄⋊(C₅⋊D₅)

Subgroups — #13102938

Conjugacy classes of subgroups

Complements

Maximal subgroups