Group #13102502 — subgroups

Subgroups — C₅×#65532 #13102502

Order 1600.

486 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 86 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×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	C₅×#4081	C₂	1	3	yes	maximal, complemented
3	C₅×#3966	C₂	1	3	yes	maximal, complemented
4	C₅²×#41	—	2	3	no	—
5	C₂×C₅²×D₄	—	2	3	no	—
6	C₂×C₅²×D₄	C₂²	1	3	yes	—
7	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic, complemented
8	C₅×C₂²⋊Dic₅	—	2	3	no	—
9	C₅×C₂²⋊Dic₅	C₂²	1	3	yes	complemented
10	C₅×#683	—	2	3	no	—
11	C₅×#458	—	5	1	no	maximal
12	#65532	C₅	1	1	yes	maximal, characteristic, direct product factor
13	C₂³×C₅²	—	2	1	no	—
14	C₂³×C₅²	D₄	1	3	yes	—
15	C₂³×C₅²	C₂³	1	1	yes	characteristic
16	C₅²×Q₈	—	2	1	no	—
17	C₅²×D₄	—	2	3	no	—
18	C₅²×D₄	—	2	6	no	—
19	C₂×C₄×C₅²	—	2	3	no	—
20	C₂×C₄×C₅²	D₄	1	3	yes	—
21	C₂×C₅×Dic₅	—	2	3	no	—
22	C₂×C₅×Dic₅	—	2	3	no	—
23	C₅×H(5,𝔽₂)	—	2	2	no	—
24	C₅×H(5,𝔽₂)	D₅	1	1	yes	characteristic
25	C₅×H(5,𝔽₂)	C₂×C₅	1	1	yes	characteristic
26	C₅×#123	—	5	3	no	—
27	#4081	C₂×C₅	1	3	yes	complemented
28	C₅×#99	—	5	3	no	—
29	#3966	C₂×C₅	1	3	yes	complemented
30	C₂²×C₅²	—	2	3	no	—
31	C₂²×C₅²	—	4	1	no	—
32	C₂²×C₅²	—	4	3	no	—
33	C₂²×C₅²	—	8	1	no	—
34	C₂²×C₅²	C₂×D₄	1	3	yes	—
35	C₄×C₅²	—	2	3	no	—
36	C₅×Dic₅	—	4	3	no	—
37	C₅×Dic₅	—	4	1	no	—
38	C₅×#41	—	2	12	no	—
39	C₅×#41	—	2	3	no	—
40	C₅×#41	—	2	3	no	—
41	C₂×C₅×D₄	—	2	12	no	—
42	C₂×C₅×D₄	—	2	3	no	—
43	C₂×C₅×D₄	—	2	3	no	—
44	C₂×C₅×D₄	—	2	6	no	—
45	C₂×C₅×D₄	C₂×D₅	1	3	yes	—
46	C₂×C₅×D₄	C₂²×C₅	1	3	yes	—
47	C₂²×C₄×C₅	—	5	1	no	—
48	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic, complemented
49	C₅×C₄⋊C₄	—	10	3	no	—
50	C₅×C₂²⋊C₄	—	5	3	no	—
51	C₅×C₂²⋊C₄	—	10	3	no	—
52	C₂²⋊Dic₅	—	2	3	no	—
53	C₂²⋊Dic₅	C₂²×C₅	1	3	yes	complemented
54	#683	—	2	3	no	—
55	#458	—	5	1	no	Sylow
56	C₂×C₅²	—	2	3	no	—
57	C₂×C₅²	—	4	3	no	—
58	C₂×C₅²	#120	1	1	yes	characteristic
59	C₂³×C₅	—	2	4	no	—
60	C₂³×C₅	—	2	1	no	—
61	C₂³×C₅	—	2	1	no	—
62	C₂³×C₅	—	2	6	no	—
63	C₂³×C₅	—	2	2	no	—
64	C₂³×C₅	C₂²⋊D₅	1	3	yes	—
65	C₂³×C₅	C₅×D₄	1	3	yes	—
66	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
67	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
68	C₅×Q₈	—	2	4	no	—
69	C₅×Q₈	—	2	1	no	—
70	C₅×Q₈	—	2	1	no	—
71	C₅×D₄	—	2	12	no	—
72	C₅×D₄	—	2	3	no	—
73	C₅×D₄	—	2	3	no	—
74	C₅×D₄	—	2	24	no	—
75	C₅×D₄	—	2	6	no	—
76	C₅×D₄	—	2	6	no	—
77	C₂×C₄×C₅	—	2	12	no	—
78	C₂×C₄×C₅	—	2	3	no	—
79	C₂×C₄×C₅	—	2	3	no	—
80	C₂×C₄×C₅	—	2	6	no	—
81	C₂×C₄×C₅	—	10	3	no	—
82	C₂×C₄×C₅	—	10	3	no	—
83	C₂×C₄×C₅	C₂²⋊D₅	1	3	yes	—
84	C₂×C₄×C₅	C₅×D₄	1	3	yes	—
85	C₂×Dic₅	—	2	3	no	—
86	C₂×Dic₅	—	2	3	no	—
87	H(5,𝔽₂)	C₅×D₅	1	1	yes	characteristic
88	#123	—	5	3	no	—
89	#99	—	5	3	no	—
90	C₅²	#458	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₅	—	2	12	no	—
92	C₂²×C₅	—	2	3	no	—
93	C₂²×C₅	—	2	3	no	—
94	C₂²×C₅	—	2	6	no	—
95	C₂²×C₅	—	4	1	no	—
96	C₂²×C₅	—	4	1	no	—
97	C₂²×C₅	—	4	3	no	—
98	C₂²×C₅	—	4	3	no	—
99	C₂²×C₅	—	8	2	no	—
100	C₂²×C₅	—	8	6	no	—
101	C₂²×C₅	—	8	4	no	—
102	C₂²×C₅	—	8	1	no	—
103	C₂²×C₅	—	8	1	no	—
104	C₂²×C₅	C₂×C₂²⋊D₅	1	3	yes	—
105	C₂²×C₅	C₂×C₅×D₄	1	3	yes	—
106	C₄×C₅	—	2	12	no	—
107	C₄×C₅	—	2	3	no	—
108	C₄×C₅	—	2	3	no	—
109	C₄×C₅	—	20	3	no	—
110	C₄×C₅	—	20	1	no	—
111	Dic₅	—	4	3	no	—
112	Dic₅	—	4	1	no	—
113	#41	—	2	3	no	—
114	C₂×D₄	—	2	3	no	—
115	C₂×D₄	C₂×C₅×D₅	1	3	yes	—
116	C₂²×C₄	—	5	1	no	—
117	C₄⋊C₄	—	10	3	no	—
118	C₂²⋊C₄	—	5	3	no	—
119	C₂²⋊C₄	—	10	3	no	—
120	C₂×C₅	—	2	3	no	—
121	C₂×C₅	—	2	3	no	—
122	C₂×C₅	—	2	2	no	—
123	C₂×C₅	—	4	6	no	—
124	C₂×C₅	—	4	12	no	—
125	C₂×C₅	—	4	3	no	—
126	C₂×C₅	—	4	3	no	—
127	C₂×C₅	#4099	1	1	yes	characteristic
128	C₂×C₅	C₅×#120	1	1	yes	characteristic
129	C₂³	—	2	1	no	—
130	C₂³	C₅×C₂²⋊D₅	1	3	yes	—
131	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
132	Q₈	—	2	1	no	—
133	D₄	—	2	3	no	—
134	D₄	—	2	6	no	—
135	C₂×C₄	—	2	3	no	—
136	C₂×C₄	—	10	3	no	—
137	C₂×C₄	—	10	3	no	—
138	C₂×C₄	C₅×C₂²⋊D₅	1	3	yes	—
139	C₅	—	2	2	no	—
140	C₅	#65532	1	1	yes	characteristic, direct product factor
141	C₅	C₅×#458	1	1	yes	characteristic, complemented
142	C₂²	—	2	3	no	—
143	C₂²	—	4	1	no	—
144	C₂²	—	4	3	no	—
145	C₂²	—	8	1	no	—
146	C₂²	C₂×C₅×C₂²⋊D₅	1	3	yes	—
147	C₄	—	2	3	no	—
148	C₄	—	20	3	no	—
149	C₄	—	20	1	no	—
150	C₂	—	2	3	no	—
151	C₂	—	4	3	no	—
152	C₂	C₅×#4099	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²×H(5,𝔽₂)

C₂

yes

maximal, characteristic

C₅×#4081

C₂

yes

maximal, complemented

C₅×#3966

C₂

yes

maximal, complemented

C₅²×#41

—

C₂×C₅²×D₄

—

C₂×C₅²×D₄

C₂²

yes

—

C₂²×C₅×Dic₅

C₂²

yes

characteristic, complemented

C₅×C₂²⋊Dic₅

—

C₅×C₂²⋊Dic₅

C₂²

yes

—

—

C₅

yes

maximal, characteristic, direct product factor

C₂³×C₅²

—

C₂³×C₅²

D₄

yes

—

C₂³×C₅²

C₂³

yes

characteristic

C₅²×Q₈

—

C₅²×D₄

—

C₅²×D₄

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

D₄

yes

—

C₂×C₅×Dic₅

—

C₂×C₅×Dic₅

—

C₅×H(5,𝔽₂)

—

C₅×H(5,𝔽₂)

D₅

yes

characteristic

C₅×H(5,𝔽₂)

C₂×C₅

yes

characteristic

C₅×#123

—

#4081

C₂×C₅

yes

complemented

C₅×#99

—

#3966

C₂×C₅

yes

complemented

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

C₂×D₄

yes

—

C₄×C₅²

—

C₅×Dic₅

—

C₅×Dic₅

—

C₅×#41

—

C₅×#41

—

C₅×#41

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

C₂×D₅

yes

—

C₂×C₅×D₄

C₂²×C₅

yes

—

C₂²×C₄×C₅

—

C₂²×Dic₅

C₂²×C₅

yes

characteristic, complemented

C₅×C₄⋊C₄

—

C₅×C₂²⋊C₄

—

C₅×C₂²⋊C₄

—

C₂²⋊Dic₅

—

C₂²⋊Dic₅

C₂²×C₅

yes

—

—

C₂×C₅²

—

C₂×C₅²

—

C₂×C₅²

#120

yes

characteristic

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

C₂²⋊D₅

yes

—

C₂³×C₅

C₅×D₄

yes

—

C₂³×C₅

C₂²×D₅

yes

characteristic

C₂³×C₅

yes

characteristic

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₅×D₄

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

H(5,𝔽₂)

C₅×D₅

yes

characteristic

#123

—

#99

—

C₅²

#458

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

100

C₂²×C₅

—

101

C₂²×C₅

—

102

C₂²×C₅

—

103

C₂²×C₅

—

104

C₂²×C₅

C₂×C₂²⋊D₅

yes

—

105

C₂²×C₅

C₂×C₅×D₄

yes

—

106

C₄×C₅

—

107

C₄×C₅

—

108

C₄×C₅

—

109

C₄×C₅

—

110

C₄×C₅

—

111

Dic₅

—

112

Dic₅

—

113

#41

—

114

C₂×D₄

—

115

C₂×D₄

C₂×C₅×D₅

yes

—

116

C₂²×C₄

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

119

C₂²⋊C₄

—

120

C₂×C₅

—

121

C₂×C₅

—

122

C₂×C₅

—

123

C₂×C₅

—

124

C₂×C₅

—

125

C₂×C₅

—

126

C₂×C₅

—

127

C₂×C₅

#4099

yes

characteristic

128

C₂×C₅

C₅×#120

yes

characteristic

129

C₂³

—

130

C₂³

C₅×C₂²⋊D₅

yes

—

131

C₂³

C₂²×C₅×D₅

yes

characteristic

132

Q₈

—

133

D₄

—

134

D₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

C₅×C₂²⋊D₅

yes

—

139

C₅

—

140

C₅

#65532

yes

characteristic, direct product factor

141

C₅

C₅×#458

yes

characteristic, complemented

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂×C₅×C₂²⋊D₅

yes

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₂

—

151

C₂

—

152

C₂

C₅×#4099

yes

characteristic

	subgroup	G/H
center	C₂×C₅	#4099
commutator, cosocle	C₂³×C₅	C₂³×C₅
Frattini	C₂³	C₂²×C₅×D₅
Fitting, prime radical	C₅²×H(5,𝔽₂)	C₂
socle	C₂×C₅²	#120
hypercenter	C₅×H(5,𝔽₂)	D₅
hypocenter	C₅	C₅×#458
LCS	G ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₅ ◁ C₂³×C₅ ◁ C₅×H(5,𝔽₂)

subgroup

G/H

C₂×C₅

C₂³×C₅

C₂³

C₂²×C₅×D₅

Fitting, prime radical

C₅²×H(5,𝔽₂)

C₂

socle

C₂×C₅²

#120

hypercenter

C₅×H(5,𝔽₂)

D₅

hypocenter

C₅

C₅×#458

LCS

G ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂×C₅ ◁ C₂³×C₅ ◁ C₅×H(5,𝔽₂)

N	H ≅ G/N	type
2. C₅×#4081	C₂	semidirect
3. C₅×#3966	C₂	semidirect
7. C₂²×C₅×Dic₅	C₂²	semidirect
9. C₅×C₂²⋊Dic₅	C₂²	semidirect
12. #65532	C₅	direct (N × H)
27. #4081	C₂×C₅	semidirect
29. #3966	C₂×C₅	semidirect
48. C₂²×Dic₅	C₂²×C₅	semidirect
53. C₂²⋊Dic₅	C₂²×C₅	semidirect
90. C₅²	#458	semidirect
140. C₅	#65532	direct (N × H)
141. C₅	C₅×#458	semidirect

H ≅ G/N

type

2. C₅×#4081

C₂

semidirect

3. C₅×#3966

C₂

semidirect

7. C₂²×C₅×Dic₅

C₂²

semidirect

9. C₅×C₂²⋊Dic₅

C₂²

semidirect

12. #65532

C₅

direct (N × H)

27. #4081

C₂×C₅

semidirect

29. #3966

C₂×C₅

semidirect

48. C₂²×Dic₅

C₂²×C₅

semidirect

53. C₂²⋊Dic₅

C₂²×C₅

semidirect

90. C₅²

#458

semidirect

140. C₅

#65532

direct (N × H)

141. C₅

C₅×#458

semidirect

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

subgroup	index
C₅²×H(5,𝔽₂)	2
C₅×#4081	2
C₅×#3966	2
C₅×#458	5
#65532	5

subgroup

index

C₅²×H(5,𝔽₂)

Subgroups — C₅×#65532 #13102502

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#65532 #13102502

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65532 #13102502