Group #13102363 — subgroups

Subgroups — C₅×#65391 #13102363

Order 1600.

304 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 126 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	C₂²×C₅²×C₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#4072	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#681	C₂	1	1	yes	maximal, characteristic
4	C₅×#3946	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
6	C₂×C₅²×C₈	—	2	2	no	—
7	C₂×C₅²×C₈	C₂²	1	2	yes	—
8	C₅×C₂²⋊Dic₅	C₄	1	2	yes	—
9	C₅×C₄⋊Dic₅	C₄	1	2	yes	—
10	C₄×C₅×Dic₅	C₂²	1	2	yes	—
11	C₅×#681	—	2	2	no	—
12	C₂×C₅×#168	C₂²	1	2	yes	—
13	C₅×#424	—	5	1	no	maximal
14	#65391	C₅	1	1	yes	maximal, characteristic, direct product factor
15	C₂³×C₅²	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₅²	D₄	1	2	yes	—
18	C₂×C₄×C₅²	Q₈	1	2	yes	—
19	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
20	C₂×C₅×Dic₅	C₂×C₄	1	4	yes	—
21	C₅²×C₈	—	2	2	no	—
22	C₅×#168	—	2	2	no	—
23	C₂×C₅×M₁₆	—	5	1	no	—
24	C₂²×C₅×C₈	—	2	2	no	—
25	C₂²×C₅×C₈	D₅	1	1	yes	characteristic
26	C₂²×C₅×C₈	C₂×C₅	1	1	yes	characteristic
27	C₅×#117	—	5	1	no	—
28	#4072	C₂×C₅	1	1	yes	characteristic
29	C₂×#681	C₂×C₅	1	1	yes	characteristic
30	C₅×C₄⋊C₈	—	5	4	no	—
31	#3946	C₂×C₅	1	4	yes	complemented
32	C₂²×C₅²	—	2	2	no	—
33	C₂²×C₅²	C₄⋊C₄	1	2	yes	—
34	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
35	C₄×C₅²	C₄⋊C₄	1	2	yes	—
36	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
37	C₄×C₅²	C₂×Q₈	1	1	yes	characteristic
38	C₅×Dic₅	—	2	4	no	—
39	C₂²×C₄×C₅	—	2	2	no	—
40	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
41	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
42	C₅×M₁₆	—	10	2	no	—
43	C₂×C₅×C₈	—	2	8	no	—
44	C₂×C₅×C₈	—	2	2	no	—
45	C₂×C₅×C₈	—	2	2	no	—
46	C₂×C₅×C₈	—	2	4	no	—
47	C₂×C₅×C₈	—	5	2	no	—
48	C₂×C₅×C₈	C₂×D₅	1	2	yes	—
49	C₂×C₅×C₈	C₂²×C₅	1	2	yes	—
50	C₅×C₄⋊C₄	—	5	2	no	—
51	C₅×C₂²⋊C₄	—	5	2	no	—
52	C₄²×C₅	—	5	2	no	—
53	C₂²⋊Dic₅	C₄×C₅	1	2	yes	—
54	C₄⋊Dic₅	C₄×C₅	1	2	yes	—
55	C₄×Dic₅	C₂²×C₅	1	2	yes	—
56	#681	—	2	2	no	—
57	C₂×#168	C₂²×C₅	1	2	yes	—
58	#424	—	5	1	no	Sylow
59	C₂×C₅²	—	2	2	no	—
60	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₅²	#131	1	2	yes	—
62	C₂³×C₅	—	2	2	no	—
63	C₂³×C₅	C₄×D₅	1	1	yes	characteristic
64	C₂³×C₅	C₂×C₄×C₅	1	1	yes	characteristic
65	C₂×C₄×C₅	—	2	4	no	—
66	C₂×C₄×C₅	—	2	2	no	—
67	C₂×C₄×C₅	—	2	4	no	—
68	C₂×C₄×C₅	—	2	2	no	—
69	C₂×C₄×C₅	—	5	4	no	—
70	C₂×C₄×C₅	Dic₁₀	1	2	yes	—
71	C₂×C₄×C₅	C₄×D₅	1	1	yes	characteristic
72	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
73	C₂×C₄×C₅	C₂×C₄×C₅	1	1	yes	characteristic
74	C₂×C₄×C₅	C₅×D₄	1	2	yes	—
75	C₂×C₄×C₅	C₅×Q₈	1	2	yes	—
76	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
77	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
78	C₂×Dic₅	C₂×C₄×C₅	1	4	yes	—
79	C₅×C₈	—	2	8	no	—
80	C₅×C₈	—	2	2	no	—
81	C₅×C₈	—	2	2	no	—
82	C₅×C₈	—	10	2	no	—
83	#168	—	2	2	no	—
84	C₂×M₁₆	—	5	1	no	—
85	C₂²×C₈	C₅×D₅	1	1	yes	characteristic
86	#117	—	5	1	no	—
87	C₄⋊C₈	—	5	4	no	—
88	C₅²	#424	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₅	—	2	8	no	—
90	C₂²×C₅	—	2	2	no	—
91	C₂²×C₅	—	2	2	no	—
92	C₂²×C₅	—	2	2	no	—
93	C₂²×C₅	—	2	4	no	—
94	C₂²×C₅	#683	1	2	yes	—
95	C₂²×C₅	C₅×C₄⋊C₄	1	2	yes	—
96	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
97	C₂²×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
98	C₄×C₅	—	2	2	no	—
99	C₄×C₅	—	2	2	no	—
100	C₄×C₅	—	2	4	no	—
101	C₄×C₅	—	10	4	no	—
102	C₄×C₅	#683	1	2	yes	—
103	C₄×C₅	C₅×C₄⋊C₄	1	2	yes	—
104	C₄×C₅	C₂×Dic₁₀	1	1	yes	characteristic
105	C₄×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
106	C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
107	C₄×C₅	C₂×C₅×Q₈	1	1	yes	characteristic
108	Dic₅	—	2	4	no	—
109	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
110	M₁₆	—	10	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	5	2	no	—
113	C₂×C₈	C₂×C₅×D₅	1	2	yes	—
114	C₄⋊C₄	—	5	2	no	—
115	C₂²⋊C₄	—	5	2	no	—
116	C₄²	—	5	2	no	—
117	C₂×C₅	—	2	8	no	—
118	C₂×C₅	—	2	2	no	—
119	C₂×C₅	—	2	2	no	—
120	C₂×C₅	—	2	4	no	—
121	C₂×C₅	—	2	2	no	—
122	C₂×C₅	#4047	1	2	yes	—
123	C₂×C₅	C₂×#683	1	1	yes	characteristic
124	C₂×C₅	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₅	C₅×#131	1	2	yes	—
126	C₂³	C₄×C₅×D₅	1	1	yes	characteristic
127	C₂×C₄	—	5	4	no	—
128	C₂×C₄	C₅×Dic₁₀	1	2	yes	—
129	C₂×C₄	C₄×C₅×D₅	1	1	yes	characteristic
130	C₂×C₄	C₅×C₂²⋊D₅	1	2	yes	—
131	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
132	C₈	—	2	2	no	—
133	C₈	—	10	2	no	—
134	C₅	—	2	2	no	—
135	C₅	#65391	1	1	yes	characteristic, direct product factor
136	C₅	C₅×#424	1	1	yes	characteristic, complemented
137	C₂²	—	2	2	no	—
138	C₂²	C₅×#683	1	2	yes	—
139	C₂²	C₂×C₄×C₅×D₅	1	1	yes	characteristic
140	C₄	—	10	4	no	—
141	C₄	C₅×#683	1	2	yes	—
142	C₄	C₂×C₅×Dic₁₀	1	1	yes	characteristic
143	C₄	C₂×C₅×C₂²⋊D₅	1	1	yes	characteristic
144	C₂	—	2	2	no	—
145	C₂	C₅×#4047	1	2	yes	—
146	C₂	C₂×C₅×#683	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₅²×C₈

C₂

yes

maximal, characteristic

C₅×#4072

C₂

yes

maximal, characteristic

C₂×C₅×#681

C₂

yes

maximal, characteristic

C₅×#3946

C₂

yes

maximal, complemented

C₂²×C₄×C₅²

C₂²

yes

characteristic

C₂×C₅²×C₈

—

C₂×C₅²×C₈

C₂²

yes

—

C₅×C₂²⋊Dic₅

C₄

yes

—

C₅×C₄⋊Dic₅

C₄

yes

—

C₄×C₅×Dic₅

C₂²

yes

—

C₅×#681

—

C₂×C₅×#168

C₂²

yes

—

C₅×#424

—

maximal

#65391

C₅

yes

maximal, characteristic, direct product factor

C₂³×C₅²

C₂×C₄

yes

characteristic

C₂×C₄×C₅²

C₂×C₄

yes

characteristic

C₂×C₄×C₅²

D₄

yes

—

C₂×C₄×C₅²

Q₈

yes

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×C₅×Dic₅

C₂×C₄

yes

—

C₅²×C₈

—

C₅×#168

—

C₂×C₅×M₁₆

—

C₂²×C₅×C₈

—

C₂²×C₅×C₈

D₅

yes

characteristic

C₂²×C₅×C₈

C₂×C₅

yes

characteristic

C₅×#117

—

#4072

C₂×C₅

yes

characteristic

C₂×#681

C₂×C₅

yes

characteristic

C₅×C₄⋊C₈

—

#3946

C₂×C₅

yes

complemented

C₂²×C₅²

—

C₂²×C₅²

C₄⋊C₄

yes

—

C₂²×C₅²

C₂²×C₄

yes

characteristic

C₄×C₅²

C₄⋊C₄

yes

—

C₄×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

C₂×Q₈

yes

characteristic

C₅×Dic₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₅

yes

characteristic

C₂²×C₄×C₅

C₂²×C₅

yes

characteristic

C₅×M₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂×D₅

yes

—

C₂×C₅×C₈

C₂²×C₅

yes

—

C₅×C₄⋊C₄

—

C₅×C₂²⋊C₄

—

C₄²×C₅

—

C₂²⋊Dic₅

C₄×C₅

yes

—

C₄⋊Dic₅

C₄×C₅

yes

—

C₄×Dic₅

C₂²×C₅

yes

—

#681

—

C₂×#168

C₂²×C₅

yes

—

#424

—

Sylow

C₂×C₅²

—

C₂×C₅²

C₂×C₄⋊C₄

yes

characteristic

C₂×C₅²

#131

yes

—

C₂³×C₅

—

C₂³×C₅

C₄×D₅

yes

characteristic

C₂³×C₅

C₂×C₄×C₅

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

Dic₁₀

yes

—

C₂×C₄×C₅

C₄×D₅

yes

characteristic

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

yes

characteristic

C₂×C₄×C₅

C₅×D₄

yes

—

C₂×C₄×C₅

C₅×Q₈

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

characteristic

C₂×C₄×C₅

C₂³×C₅

yes

characteristic

C₂×Dic₅

C₂×C₄×C₅

yes

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

#168

—

C₂×M₁₆

—

C₂²×C₈

C₅×D₅

yes

characteristic

#117

—

C₄⋊C₈

—

C₅²

#424

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

#683

yes

—

C₂²×C₅

C₅×C₄⋊C₄

yes

—

C₂²×C₅

C₂×C₄×D₅

yes

characteristic

C₂²×C₅

C₂²×C₄×C₅

yes

characteristic

C₄×C₅

—

C₄×C₅

—

100

C₄×C₅

—

101

C₄×C₅

—

102

C₄×C₅

#683

yes

—

103

C₄×C₅

C₅×C₄⋊C₄

yes

—

104

C₄×C₅

C₂×Dic₁₀

yes

characteristic

105

C₄×C₅

C₂×C₂²⋊D₅

yes

characteristic

106

C₄×C₅

C₂×C₅×D₄

yes

characteristic

107

C₄×C₅

C₂×C₅×Q₈

yes

characteristic

108

Dic₅

—

109

C₂²×C₄

C₂×C₅×D₅

yes

characteristic

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₂×C₅×D₅

yes

—

114

C₄⋊C₄

—

115

C₂²⋊C₄

—

116

C₄²

—

117

C₂×C₅

—

118

C₂×C₅

—

119

C₂×C₅

—

120

C₂×C₅

—

121

C₂×C₅

—

122

C₂×C₅

#4047

yes

—

123

C₂×C₅

C₂×#683

yes

characteristic

124

C₂×C₅

C₂×C₅×C₄⋊C₄

yes

characteristic

125

C₂×C₅

C₅×#131

yes

—

126

C₂³

C₄×C₅×D₅

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

C₅×Dic₁₀

yes

—

129

C₂×C₄

C₄×C₅×D₅

yes

characteristic

130

C₂×C₄

C₅×C₂²⋊D₅

yes

—

131

C₂×C₄

C₂²×C₅×D₅

yes

characteristic

132

C₈

—

133

C₈

—

134

C₅

—

135

C₅

#65391

yes

characteristic, direct product factor

136

C₅

C₅×#424

yes

characteristic, complemented

137

C₂²

—

138

C₂²

C₅×#683

yes

—

139

C₂²

C₂×C₄×C₅×D₅

yes

characteristic

140

C₄

—

141

C₄

C₅×#683

yes

—

142

C₄

C₂×C₅×Dic₁₀

yes

characteristic

143

C₄

C₂×C₅×C₂²⋊D₅

yes

characteristic

144

C₂

—

145

C₂

C₅×#4047

yes

—

146

C₂

C₂×C₅×#683

yes

characteristic

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

subgroup

G/H

center

C₂×C₄×C₅

C₂²×D₅

commutator

C₂²×C₅

C₂²×C₄×C₅

Frattini

C₂×C₄

C₂²×C₅×D₅

Fitting

C₂²×C₅²×C₈

C₂

socle

C₂²×C₅²

C₂²×C₄

prime radical

C₂³×C₅²

C₂×C₄

cosocle

C₂×C₄×C₅

C₂³×C₅

hypercenter

C₂²×C₅×C₈

D₅

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₅×C₈

N	H ≅ G/N	type
4. C₅×#3946	C₂	semidirect
14. #65391	C₅	direct (N × H)
31. #3946	C₂×C₅	semidirect
88. C₅²	#424	semidirect
135. C₅	#65391	direct (N × H)
136. C₅	C₅×#424	semidirect

H ≅ G/N

type

4. C₅×#3946

C₂

semidirect

14. #65391

C₅

direct (N × H)

31. #3946

C₂×C₅

semidirect

88. C₅²

#424

semidirect

135. C₅

#65391

direct (N × H)

136. C₅

C₅×#424

semidirect

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

subgroup	index
C₂²×C₅²×C₈	2
C₅×#4072	2
C₂×C₅×#681	2
C₅×#3946	2
C₅×#424	5
#65391	5

subgroup

index

C₂²×C₅²×C₈

C₂×C₅×#681

Subgroups — C₅×#65391 #13102363

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#65391 #13102363

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65391 #13102363