Group #13103581 — subgroups

Subgroups — #13103581

Order 1600.

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

normal subgroups only

This group has 267 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	C₂×#87849	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#700	C₂	1	1	yes	maximal, characteristic
3	C₄×D₅×Dic₅	C₂	1	1	yes	maximal, characteristic
4	#11716760	C₂	1	4	yes	maximal, complemented
5	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
6	C₂×D₅×Dic₅	C₄	1	2	yes	—
7	C₂×#6032	C₂²	1	2	yes	—
8	#87849	—	2	2	no	—
9	C₂×C₅×#170	C₂²	1	2	yes	—
10	C₅×#700	—	2	2	no	—
11	C₄×#1039	C₄	1	1	yes	characteristic
12	C₄×C₅×Dic₅	C₄	1	1	yes	characteristic
13	Dic₅²	C₂²	1	2	yes	—
14	#65682	—	5	1	no	maximal
15	#65407	—	5	1	no	maximal
16	C₂²×C₅×D₅	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
18	C₂×#1039	C₂×C₄	1	2	yes	—
19	C₂×C₅×Dic₅	C₂×C₄	1	2	yes	—
20	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
21	C₄×C₅×D₅	D₄	1	2	yes	—
22	C₄×C₅×D₅	Q₈	1	2	yes	—
23	D₅×Dic₅	—	2	4	no	—
24	#6032	—	2	2	no	—
25	C₅×#170	—	2	2	no	—
26	C₂×#700	—	5	1	no	—
27	C₂×#700	D₅	1	1	yes	characteristic
28	C₂×C₅×M₁₆	—	5	1	no	—
29	C₂×C₄×Dic₅	—	5	1	no	—
30	C₂×#681	—	5	1	no	—
31	C₄²×D₅	—	5	1	no	—
32	#4004	—	5	4	no	—
33	#3946	—	5	4	no	—
34	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₅×D₅	—	2	2	no	—
36	C₂×C₅×D₅	C₄⋊C₄	1	2	yes	—
37	C₄×C₅²	C₄⋊C₄	1	2	yes	—
38	#1039	M₁₆	1	4	yes	complemented
39	C₅×Dic₅	—	2	2	no	—
40	C₅×Dic₅	C₂×D₄	1	1	yes	characteristic
41	C₅×Dic₅	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₅	—	5	1	no	—
43	C₂²×Dic₅	—	5	2	no	—
44	C₂×C₄×D₅	—	5	2	no	—
45	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
46	C₂×#170	—	5	2	no	—
47	C₂×#170	C₂×D₅	1	2	yes	—
48	#700	—	2	2	no	—
49	#700	—	10	2	no	—
50	C₅×M₁₆	—	10	2	no	—
51	C₂×C₅×C₈	—	5	2	no	—
52	C₄²×C₅	—	5	1	no	—
53	C₄×Dic₅	—	5	1	no	—
54	C₄×Dic₅	—	5	1	no	—
55	C₄×Dic₅	—	5	2	no	—
56	C₄×Dic₅	—	5	2	no	—
57	C₄×Dic₅	—	20	1	no	—
58	C₄×Dic₅	AGL(1,𝔽₅)	1	1	yes	characteristic
59	#681	—	10	2	no	—
60	C₂×#168	—	5	2	no	—
61	#423	—	25	1	no	Sylow
62	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₅²	C₂×M₁₆	1	2	yes	—
64	C₅×D₅	—	2	2	no	—
65	C₂³×C₅	—	5	1	no	—
66	C₂²×D₅	C₄×D₅	1	1	yes	characteristic
67	C₂×C₄×C₅	—	4	1	no	—
68	C₂×C₄×C₅	—	5	1	no	—
69	C₂×C₄×C₅	—	5	2	no	—
70	C₂×C₄×C₅	—	5	2	no	—
71	C₂×C₄×C₅	—	5	2	no	—
72	C₂×C₄×C₅	C₄×D₅	1	1	yes	characteristic
73	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
74	C₂×Dic₅	—	5	2	no	—
75	C₂×Dic₅	—	5	2	no	—
76	C₂×Dic₅	—	10	4	no	—
77	C₂×Dic₅	—	20	2	no	—
78	C₂×Dic₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
79	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
80	C₄×D₅	—	10	4	no	—
81	C₄×D₅	Dic₁₀	1	2	yes	—
82	C₄×D₅	C₂²⋊D₅	1	2	yes	—
83	#170	—	2	2	no	—
84	#170	—	10	2	no	—
85	C₅×C₈	—	10	2	no	—
86	#168	—	10	2	no	—
87	C₂×M₁₆	—	5	1	no	—
88	C₂×M₁₆	—	25	1	no	—
89	C₂×C₄²	—	25	1	no	—
90	C₄⋊C₈	—	25	4	no	—
91	C₅²	#423	1	1	yes	characteristic, complemented, Sylow
92	C₂²×C₅	—	4	1	no	—
93	C₂²×C₅	—	5	2	no	—
94	C₂²×C₅	—	10	2	no	—
95	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
96	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
97	C₂×D₅	—	2	2	no	—
98	C₂×D₅	#683	1	2	yes	—
99	C₄×C₅	—	4	2	no	—
100	C₄×C₅	—	5	1	no	—
101	C₄×C₅	—	5	1	no	—
102	C₄×C₅	—	10	2	no	—
103	C₄×C₅	#683	1	2	yes	—
104	C₄×C₅	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
105	Dic₅	—	2	2	no	—
106	Dic₅	—	5	4	no	—
107	Dic₅	—	5	4	no	—
108	Dic₅	—	20	4	no	—
109	Dic₅	C₂×Dic₁₀	1	1	yes	characteristic
110	Dic₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
111	C₂²×C₄	—	5	1	no	—
112	C₂²×C₄	—	25	2	no	—
113	M₁₆	—	10	2	no	—
114	M₁₆	—	50	2	no	—
115	C₂×C₈	—	5	2	no	—
116	C₂×C₈	—	25	2	no	—
117	C₄²	—	5	1	no	—
118	C₄²	—	25	2	no	—
119	C₄²	—	25	1	no	—
120	C₂×C₅	—	4	2	no	—
121	C₂×C₅	—	4	1	no	—
122	C₂×C₅	—	10	2	no	—
123	C₂×C₅	D₅×M₁₆	1	2	yes	—
124	C₂×C₅	C₂×#683	1	1	yes	characteristic
125	C₂×C₅	D₅⋊M₁₆	1	2	yes	—
126	C₂×C₅	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
127	D₅	—	2	2	no	—
128	C₂³	—	5	1	no	—
129	C₂×C₄	—	5	1	no	—
130	C₂×C₄	—	5	2	no	—
131	C₂×C₄	—	5	2	no	—
132	C₂×C₄	—	5	2	no	—
133	C₂×C₄	—	25	2	no	—
134	C₂×C₄	—	50	4	no	—
135	C₂×C₄	D₅×AGL(1,𝔽₅)	1	1	yes	characteristic
136	C₈	—	10	2	no	—
137	C₈	—	50	2	no	—
138	C₅	—	4	1	no	—
139	C₅	#65407	1	1	yes	characteristic, complemented
140	C₅	#65682	1	1	yes	characteristic, complemented
141	C₂²	—	5	2	no	—
142	C₂²	—	10	2	no	—
143	C₂²	C₂×D₅×AGL(1,𝔽₅)	1	1	yes	characteristic
144	C₄	—	5	1	no	—
145	C₄	—	5	1	no	—
146	C₄	—	10	2	no	—
147	C₄	—	25	4	no	—
148	C₄	#87825	1	2	yes	—
149	C₂	—	10	2	no	—
150	C₂	#11717331	1	2	yes	—
151	C₂	C₂×#87825	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#87849

C₂

yes

maximal, characteristic

C₂×C₅×#700

C₂

yes

maximal, characteristic

C₄×D₅×Dic₅

C₂

yes

maximal, characteristic

#11716760

C₂

yes

maximal, complemented

C₂×C₄×C₅×D₅

C₂²

yes

characteristic

C₂×D₅×Dic₅

C₄

yes

—

C₂×#6032

C₂²

yes

—

#87849

—

C₂×C₅×#170

C₂²

yes

—

C₅×#700

—

C₄×#1039

C₄

yes

characteristic

C₄×C₅×Dic₅

C₄

yes

characteristic

Dic₅²

C₂²

yes

—

—

—

C₂²×C₅×D₅

C₂×C₄

yes

characteristic

C₂×C₄×C₅²

C₂×C₄

yes

characteristic

C₂×#1039

C₂×C₄

yes

—

C₂×C₅×Dic₅

C₂×C₄

yes

—

C₂×C₅×Dic₅

C₂³

yes

characteristic

C₄×C₅×D₅

D₄

yes

—

C₄×C₅×D₅

Q₈

yes

—

D₅×Dic₅

—

—

—

—

D₅

yes

characteristic

C₂×C₅×M₁₆

—

C₂×C₄×Dic₅

—

C₂×#681

—

C₄²×D₅

—

#4004

—

#3946

—

C₂²×C₅²

C₂²×C₄

yes

characteristic

C₂×C₅×D₅

—

C₂×C₅×D₅

C₄⋊C₄

yes

—

C₄×C₅²

C₄⋊C₄

yes

—

#1039

M₁₆

yes

complemented

C₅×Dic₅

—

C₅×Dic₅

C₂×D₄

yes

characteristic

C₅×Dic₅

C₂×Q₈

yes

characteristic

C₂²×C₄×C₅

—

C₂²×Dic₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

C₂×D₅

yes

characteristic

C₂×#170

—

C₂×#170

C₂×D₅

yes

—

#700

—

#700

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₄²×C₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

yes

—

—

—

C₂×C₅²

C₂×C₄⋊C₄

yes

characteristic

C₂×C₅²

C₂×M₁₆

yes

—

C₅×D₅

—

C₂³×C₅

—

C₂²×D₅

C₄×D₅

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₄×D₅

yes

characteristic

C₂×C₄×C₅

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

C₂×AGL(1,𝔽₅)

yes

—

C₂×Dic₅

C₂²×D₅

yes

characteristic

C₄×D₅

—

C₄×D₅

Dic₁₀

yes

—

C₄×D₅

C₂²⋊D₅

yes

—

#170

—

#170

—

C₅×C₈

—

#168

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×C₄²

—

C₄⋊C₈

—

C₅²

#423

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂×C₄×D₅

yes

characteristic

C₂²×C₅

C₂²×AGL(1,𝔽₅)

yes

characteristic

C₂×D₅

—

C₂×D₅

#683

yes

—

C₄×C₅

—

100

C₄×C₅

—

101

C₄×C₅

—

102

C₄×C₅

—

103

C₄×C₅

#683

yes

—

104

C₄×C₅

C₄⋊AGL(1,𝔽₅)

yes

—

105

Dic₅

—

106

Dic₅

—

107

Dic₅

—

108

Dic₅

—

109

Dic₅

C₂×Dic₁₀

yes

characteristic

110

Dic₅

C₂×C₂²⋊D₅

yes

characteristic

111

C₂²×C₄

—

112

C₂²×C₄

—

113

M₁₆

—

114

M₁₆

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

120

C₂×C₅

—

121

C₂×C₅

—

122

C₂×C₅

—

123

C₂×C₅

D₅×M₁₆

yes

—

124

C₂×C₅

C₂×#683

yes

characteristic

125

C₂×C₅

D₅⋊M₁₆

yes

—

126

C₂×C₅

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

127

D₅

—

128

C₂³

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

D₅×AGL(1,𝔽₅)

yes

characteristic

136

C₈

—

137

C₈

—

138

C₅

—

139

C₅

#65407

yes

characteristic, complemented

140

C₅

#65682

yes

characteristic, complemented

141

C₂²

—

142

C₂²

—

143

C₂²

C₂×D₅×AGL(1,𝔽₅)

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

#87825

yes

—

149

C₂

—

150

C₂

#11717331

yes

—

151

C₂

C₂×#87825

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×D₅×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅²	C₂²×C₄
Fitting	C₂×C₄×C₅²	C₂×C₄
prime radical	C₂²×C₅×D₅	C₂×C₄
cosocle	C₂×C₅×Dic₅	C₂³
hypercenter	C₂×C₄	D₅×AGL(1,𝔽₅)
hypocenter	C₅²	#423
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄

subgroup

G/H

center, Frattini

C₂²

C₂×D₅×AGL(1,𝔽₅)

commutator, socle

C₂²×C₅²

C₂²×C₄

Fitting

C₂×C₄×C₅²

C₂×C₄

prime radical

C₂²×C₅×D₅

C₂×C₄

cosocle

C₂×C₅×Dic₅

C₂³

C₂×C₄

C₅²

G ▷ C₂²×C₅² ▷ C₅²

C₁ ◁ C₂² ◁ C₂×C₄

N	H ≅ G/N	type
4. #11716760	C₂	semidirect
38. #1039	M₁₆	semidirect
91. C₅²	#423	semidirect
139. C₅	#65407	semidirect
140. C₅	#65682	semidirect

H ≅ G/N

type

4. #11716760

C₂

semidirect

38. #1039

M₁₆

semidirect

91. C₅²

#423

semidirect

139. C₅

#65407

semidirect

140. C₅

#65682

semidirect

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

subgroup	index
C₂×#87849	2
C₂×C₅×#700	2
C₄×D₅×Dic₅	2
#11716760	2
#65682	5
#65407	5

subgroup

index

C₂×#87849

C₂×C₅×#700

C₄×D₅×Dic₅

#11716760

#65682

#65407

Subgroups — #13103581

Conjugacy classes of subgroups

Complements

Maximal subgroups