Group #13100869 — subgroups

Subgroups — #13100869

Order 1600.

296 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 78 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×#87799	C₂	1	1	yes	maximal, characteristic
2	#11716992	C₂	1	1	yes	maximal, characteristic
3	C₂×#87767	C₂	1	1	yes	maximal, characteristic
4	C₅×#3946	C₂	1	2	yes	maximal, complemented
5	#11716612	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
7	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
8	C₂×#6029	C₂²	1	1	yes	characteristic
9	#87799	—	2	2	no	—
10	C₄×C₅×Dic₅	C₂²	1	2	yes	—
11	C₂×C₅×#168	C₂²	1	2	yes	—
12	#87781	C₄	1	2	yes	—
13	#87779	C₄	1	1	yes	characteristic
14	#87779	C₄	1	1	yes	characteristic
15	#87767	—	2	2	no	—
16	#65128	—	5	2	no	maximal
17	C₂²×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
19	C₂×#1039	C₂×C₄	1	1	yes	characteristic
20	C₄×C₅⋊D₅	D₄	1	2	yes	—
21	C₄×C₅⋊D₅	Q₈	1	2	yes	—
22	C₂×C₅×Dic₅	C₂×C₄	1	2	yes	—
23	C₂×C₅×Dic₅	C₂×C₄	1	2	yes	—
24	C₅²×C₈	—	2	1	no	—
25	#6029	—	2	1	no	—
26	C₅×#168	—	2	2	no	—
27	C₂×#676	—	5	2	no	—
28	C₂×#676	—	10	2	no	—
29	C₂×C₈×D₅	—	5	2	no	—
30	#4018	—	5	2	no	—
31	C₅×C₄⋊C₈	—	5	2	no	—
32	#3946	—	5	2	no	—
33	#3946	D₅	1	2	yes	complemented
34	C₄⋊#168	—	5	2	no	—
35	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₅⋊D₅	—	2	2	no	—
37	C₂×C₅⋊D₅	C₄⋊C₄	1	2	yes	—
38	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
39	C₄×C₅²	C₂×Q₈	1	1	yes	characteristic
40	#1039	C₄⋊C₄	1	2	yes	—
41	C₅×Dic₅	—	2	2	no	—
42	C₅×Dic₅	—	2	2	no	—
43	C₂×C₄×D₅	—	5	2	no	—
44	C₂×C₄×D₅	—	10	2	no	—
45	C₂×C₅×C₈	—	2	2	no	—
46	C₂×C₅×C₈	—	5	2	no	—
47	C₂×C₅×C₈	C₂×D₅	1	2	yes	—
48	C₄²×C₅	—	5	2	no	—
49	#685	—	5	2	no	—
50	#685	—	5	2	no	—
51	#683	—	5	2	no	—
52	#683	—	5	2	no	—
53	C₄×Dic₅	C₂×D₅	1	2	yes	—
54	C₂×#168	—	5	2	no	—
55	C₂×#168	—	10	2	no	—
56	C₂×#168	C₂×D₅	1	2	yes	—
57	#676	—	10	4	no	—
58	#676	—	10	8	no	—
59	C₈×D₅	—	10	4	no	—
60	#424	—	25	1	no	Sylow
61	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₅²	#131	1	2	yes	—
63	C₅⋊D₅	—	2	2	no	—
64	C₂²×D₅	—	5	2	no	—
65	C₂²×D₅	—	10	2	no	—
66	C₂×C₄×C₅	—	2	2	no	—
67	C₂×C₄×C₅	—	5	2	no	—
68	C₂×C₄×C₅	—	5	2	no	—
69	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
70	C₂×Dic₅	—	5	2	no	—
71	C₂×Dic₅	—	10	2	no	—
72	C₂×Dic₅	C₄×D₅	1	2	yes	—
73	C₂×Dic₅	C₄×D₅	1	2	yes	—
74	C₄×D₅	—	5	4	no	—
75	C₄×D₅	—	5	4	no	—
76	C₄×D₅	—	10	4	no	—
77	C₄×D₅	—	10	4	no	—
78	C₅×C₈	—	2	4	no	—
79	C₅×C₈	—	2	2	no	—
80	C₅×C₈	—	10	2	no	—
81	#168	—	2	2	no	—
82	#168	—	10	4	no	—
83	#168	—	10	2	no	—
84	C₂×M₁₆	—	25	1	no	—
85	C₂²×C₈	—	25	1	no	—
86	#117	—	25	1	no	—
87	C₄⋊C₈	—	5	2	no	—
88	C₄⋊C₈	—	25	2	no	—
89	C₅²	#424	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₅	—	2	2	no	—
91	C₂²×C₅	C₂×C₄×D₅	1	2	yes	—
92	C₂×D₅	—	5	4	no	—
93	C₂×D₅	—	10	4	no	—
94	C₂×D₅	—	10	2	no	—
95	C₂×D₅	—	10	2	no	—
96	C₂×D₅	—	20	4	no	—
97	C₄×C₅	—	2	2	no	—
98	C₄×C₅	—	2	2	no	—
99	C₄×C₅	—	10	2	no	—
100	C₄×C₅	—	10	2	no	—
101	C₄×C₅	D₄×D₅	1	2	yes	—
102	C₄×C₅	Q₈×D₅	1	2	yes	—
103	Dic₅	—	2	2	no	—
104	Dic₅	—	2	2	no	—
105	Dic₅	—	5	4	no	—
106	Dic₅	—	10	4	no	—
107	C₂²×C₄	—	25	1	no	—
108	M₁₆	—	50	2	no	—
109	C₂×C₈	—	5	2	no	—
110	C₂×C₈	—	25	1	no	—
111	C₂×C₈	—	50	2	no	—
112	C₂×C₈	D₅²	1	1	yes	characteristic
113	C₄⋊C₄	—	25	2	no	—
114	C₂²⋊C₄	—	25	1	no	—
115	C₂²⋊C₄	—	25	1	no	—
116	C₄²	—	5	2	no	—
117	C₂×C₅	—	2	4	no	—
118	C₂×C₅	—	2	2	no	—
119	C₂×C₅	D₅×C₄⋊C₄	1	2	yes	—
120	C₂×C₅	#4047	1	2	yes	—
121	C₂×C₅	#4053	1	2	yes	—
122	D₅	—	10	4	no	—
123	D₅	—	20	4	no	—
124	C₂³	—	25	1	no	—
125	C₂×C₄	—	5	2	no	—
126	C₂×C₄	—	5	2	no	—
127	C₂×C₄	—	25	1	no	—
128	C₂×C₄	—	25	2	no	—
129	C₂×C₄	—	25	2	no	—
130	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
131	C₈	—	2	1	no	—
132	C₈	—	10	2	no	—
133	C₈	—	50	1	no	—
134	C₅	—	2	2	no	—
135	C₅	#65128	1	2	yes	complemented
136	C₂²	—	25	2	no	—
137	C₂²	—	50	2	no	—
138	C₂²	C₄×D₅²	1	1	yes	characteristic
139	C₄	—	10	2	no	—
140	C₄	—	10	2	no	—
141	C₄	—	25	2	no	—
142	C₄	#87872	1	1	yes	characteristic
143	C₄	#87886	1	1	yes	characteristic
144	C₂	—	50	2	no	—
145	C₂	#11716922	1	2	yes	—
146	C₂	#11717001	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#87799

C₂

yes

maximal, characteristic

#11716992

C₂

yes

maximal, characteristic

C₂×#87767

C₂

yes

maximal, characteristic

C₅×#3946

C₂

yes

maximal, complemented

#11716612

C₂

yes

maximal, complemented

C₂×C₄×C₅⋊D₅

C₂²

yes

characteristic

C₂×C₅²×C₈

C₂²

yes

characteristic

C₂×#6029

C₂²

yes

characteristic

#87799

—

C₄×C₅×Dic₅

C₂²

yes

—

C₂×C₅×#168

C₂²

yes

—

#87781

C₄

yes

—

#87779

C₄

yes

characteristic

#87779

C₄

yes

—

—

C₂²×C₅⋊D₅

C₂×C₄

yes

characteristic

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

C₂×C₄

yes

characteristic

C₄×C₅⋊D₅

D₄

yes

—

C₄×C₅⋊D₅

Q₈

yes

—

C₂×C₅×Dic₅

C₂×C₄

yes

—

C₂×C₅×Dic₅

C₂×C₄

yes

—

C₅²×C₈

—

—

—

—

—

C₂×C₈×D₅

—

#4018

—

C₅×C₄⋊C₈

—

#3946

—

#3946

D₅

yes

complemented

C₄⋊#168

—

C₂²×C₅²

C₂²×C₄

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

C₄⋊C₄

yes

—

C₄×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

C₂×Q₈

yes

characteristic

#1039

C₄⋊C₄

yes

—

C₅×Dic₅

—

C₅×Dic₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂×D₅

yes

—

C₄²×C₅

—

—

—

—

—

C₄×Dic₅

C₂×D₅

yes

—

C₂×#168

—

C₂×#168

—

C₂×#168

C₂×D₅

yes

—

#676

—

#676

—

C₈×D₅

—

#424

—

Sylow

C₂×C₅²

C₂×C₄⋊C₄

yes

characteristic

C₂×C₅²

#131

yes

—

C₅⋊D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

C₄×D₅

yes

—

C₂×Dic₅

C₄×D₅

yes

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

#168

—

#168

—

#168

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

C₄⋊C₈

—

C₄⋊C₈

—

C₅²

#424

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

C₂×C₄×D₅

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

100

C₄×C₅

—

101

C₄×C₅

D₄×D₅

yes

—

102

C₄×C₅

Q₈×D₅

yes

—

103

Dic₅

—

104

Dic₅

—

105

Dic₅

—

106

Dic₅

—

107

C₂²×C₄

—

108

M₁₆

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

D₅²

yes

characteristic

113

C₄⋊C₄

—

114

C₂²⋊C₄

—

115

C₂²⋊C₄

—

116

C₄²

—

117

C₂×C₅

—

118

C₂×C₅

—

119

C₂×C₅

D₅×C₄⋊C₄

yes

—

120

C₂×C₅

#4047

yes

—

121

C₂×C₅

#4053

yes

—

122

D₅

—

123

D₅

—

124

C₂³

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₂×D₅²

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

—

134

C₅

—

135

C₅

#65128

yes

complemented

136

C₂²

—

137

C₂²

—

138

C₂²

C₄×D₅²

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

#87872

yes

characteristic

143

C₄

#87886

yes

characteristic

144

C₂

—

145

C₂

#11716922

yes

—

146

C₂

#11717001

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×D₅²
commutator, socle	C₂²×C₅²	C₂²×C₄
Fitting	C₂×C₅²×C₈	C₂²
prime radical	C₂²×C₅⋊D₅	C₂×C₄
cosocle	C₂×C₄×C₅²	C₂³
hypercenter	C₂×C₈	D₅²
hypocenter	C₅²	#424
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂×D₅²

commutator, socle

C₂²×C₅²

C₂²×C₄

Fitting

C₂×C₅²×C₈

C₂²

prime radical

C₂²×C₅⋊D₅

C₂×C₄

cosocle

C₂×C₄×C₅²

C₂³

hypercenter

C₂×C₈

D₅²

C₅²

G ▷ C₂²×C₅² ▷ C₅²

C₁ ◁ C₂×C₄ ◁ C₂×C₈

N	H ≅ G/N	type
4. C₅×#3946	C₂	semidirect
5. #11716612	C₂	semidirect
33. #3946	D₅	semidirect
89. C₅²	#424	semidirect
135. C₅	#65128	semidirect

H ≅ G/N

type

4. C₅×#3946

C₂

semidirect

5. #11716612

C₂

semidirect

33. #3946

D₅

semidirect

89. C₅²

#424

semidirect

135. C₅

#65128

semidirect

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

subgroup	index
C₂×#87799	2
#11716992	2
C₂×#87767	2
C₅×#3946	2
#11716612	2
#65128	5

subgroup

index

Subgroups — #13100869

Conjugacy classes of subgroups

Complements

Maximal subgroups