Group #13104575 — subgroups

Subgroups — #13104575

Order 1600.

323 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 323 conjugacy classes of subgroups — too many to draw. Only its 109 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×C₄⋊#1042	C₂	1	2	yes	maximal
2	C₄²×C₅×D₅	C₂	1	1	yes	maximal, characteristic
3	#11716849	C₂	1	4	yes	maximal
4	C₂²×#1042	C₂²	1	4	yes	—
5	C₂×C₄×C₅×D₅	C₂²	1	2	yes	—
6	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
7	C₄⋊#1042	—	2	4	no	—
8	C₄²×C₅²	C₄	1	1	yes	characteristic, complemented
9	C₄×C₅×Dic₅	C₄	1	1	yes	characteristic, complemented
10	C₄×C₅×Dic₅	C₄	1	2	yes	complemented
11	#65689	—	5	1	no	maximal
12	#65225	—	5	1	no	maximal
13	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
14	C₂×#1042	—	2	4	no	—
15	C₂×#1042	—	2	8	no	—
16	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
18	C₂×C₅×Dic₅	C₂×C₄	1	2	yes	—
19	C₂×C₅×Dic₅	C₂×C₄	1	1	yes	characteristic
20	C₄×C₅×D₅	—	2	4	no	—
21	C₄×C₅×D₅	D₄	1	2	yes	—
22	C₄×C₅×D₅	Q₈	1	2	yes	—
23	C₂×C₄⋊AGL(1,𝔽₅)	—	5	2	no	—
24	C₂×C₄²×C₅	—	5	1	no	—
25	C₂×C₄⋊Dic₅	—	5	2	no	—
26	C₄²×D₅	D₅	1	1	yes	characteristic
27	(C₂×AGL(1,𝔽₅))⋊C₄	—	5	4	no	—
28	#3963	—	5	4	no	—
29	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₅×D₅	C₂×D₄	1	1	yes	characteristic
31	C₂×C₅×D₅	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₅×D₅	#41	1	4	yes	—
33	#1042	—	4	4	no	—
34	C₄×C₅²	—	2	2	no	—
35	C₄×C₅²	C₄⋊C₄	1	2	yes	—
36	C₅×Dic₅	—	2	2	no	—
37	C₅×Dic₅	C₄⋊C₄	1	2	yes	—
38	C₂²×AGL(1,𝔽₅)	—	5	4	no	—
39	C₂²×C₄×C₅	—	5	1	no	—
40	C₂²×C₄×C₅	—	5	2	no	—
41	C₂²×Dic₅	—	5	4	no	—
42	C₂×C₄×D₅	C₂×D₅	1	2	yes	—
43	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
44	C₄⋊AGL(1,𝔽₅)	—	10	4	no	—
45	C₄²×C₅	—	4	1	no	—
46	C₄²×C₅	—	5	2	no	—
47	C₄²×C₅	—	5	1	no	—
48	C₄²×C₅	Dic₅	1	1	yes	characteristic, complemented
49	C₄²×C₅	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
50	C₄⋊Dic₅	—	10	4	no	—
51	C₄×Dic₅	Dic₅	1	1	yes	characteristic, complemented
52	C₄×Dic₅	Dic₅	1	2	yes	complemented
53	#382	—	25	1	no	Sylow
54	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₅²	#117	1	2	yes	—
56	C₅×D₅	#124	1	2	yes	—
57	C₅×D₅	#125	1	2	yes	—
58	C₂³×C₅	—	5	1	no	—
59	C₂²×D₅	C₂²×D₅	1	1	yes	characteristic
60	C₂×AGL(1,𝔽₅)	—	10	4	no	—
61	C₂×AGL(1,𝔽₅)	—	10	8	no	—
62	C₂×C₄×C₅	—	4	2	no	—
63	C₂×C₄×C₅	—	4	1	no	—
64	C₂×C₄×C₅	—	5	1	no	—
65	C₂×C₄×C₅	—	5	2	no	—
66	C₂×C₄×C₅	—	5	2	no	—
67	C₂×C₄×C₅	—	5	2	no	—
68	C₂×C₄×C₅	—	10	4	no	—
69	C₂×C₄×C₅	C₂×Dic₅	1	2	yes	—
70	C₂×C₄×C₅	C₂×Dic₅	1	1	yes	characteristic
71	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
72	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
73	C₂×Dic₅	—	10	4	no	—
74	C₂×Dic₅	—	10	8	no	—
75	C₂×Dic₅	C₂×Dic₅	1	2	yes	—
76	C₂×Dic₅	C₂×Dic₅	1	1	yes	characteristic
77	C₄×D₅	—	2	4	no	—
78	C₄×D₅	Dic₁₀	1	2	yes	—
79	C₄×D₅	D₂₀	1	2	yes	—
80	C₂×C₄⋊C₄	—	25	2	no	—
81	C₂×C₄²	—	5	1	no	—
82	#95	—	25	4	no	—
83	C₅²	#382	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₅	—	4	1	no	—
85	C₂²×C₅	—	5	1	no	—
86	C₂²×C₅	—	5	1	no	—
87	C₂²×C₅	—	5	4	no	—
88	C₂²×C₅	C₂²×Dic₅	1	1	yes	characteristic
89	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
90	C₂×D₅	C₂×Dic₁₀	1	1	yes	characteristic
91	C₂×D₅	C₂×D₂₀	1	1	yes	characteristic
92	C₂×D₅	#709	1	4	yes	—
93	AGL(1,𝔽₅)	—	20	4	no	—
94	C₄×C₅	—	2	2	no	—
95	C₄×C₅	—	2	2	no	—
96	C₄×C₅	—	4	4	no	—
97	C₄×C₅	—	4	2	no	—
98	C₄×C₅	—	5	2	no	—
99	C₄×C₅	—	10	2	no	—
100	C₄×C₅	C₄⋊Dic₅	1	2	yes	—
101	C₄×C₅	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
102	Dic₅	—	2	2	no	—
103	Dic₅	—	20	4	no	—
104	Dic₅	C₄⋊Dic₅	1	2	yes	—
105	C₂²×C₄	—	5	1	no	—
106	C₂²×C₄	—	5	2	no	—
107	C₂²×C₄	—	25	4	no	—
108	C₄⋊C₄	—	50	4	no	—
109	C₄²	—	5	2	no	—
110	C₄²	—	5	1	no	—
111	C₄²	#1042	1	1	yes	characteristic, complemented
112	C₂×C₅	—	4	1	no	—
113	C₂×C₅	—	4	2	no	—
114	C₂×C₅	—	5	2	no	—
115	C₂×C₅	—	5	2	no	—
116	C₂×C₅	C₂×C₄⋊Dic₅	1	1	yes	characteristic
117	C₂×C₅	#4072	1	2	yes	—
118	C₂×C₅	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
119	C₂×C₅	#4130	1	2	yes	—
120	D₅	#4016	1	2	yes	—
121	D₅	#4021	1	2	yes	—
122	C₂³	—	5	1	no	—
123	C₂×C₄	—	5	1	no	—
124	C₂×C₄	—	5	2	no	—
125	C₂×C₄	—	5	2	no	—
126	C₂×C₄	—	5	2	no	—
127	C₂×C₄	—	10	4	no	—
128	C₂×C₄	—	50	4	no	—
129	C₂×C₄	—	50	8	no	—
130	C₂×C₄	C₂×#1042	1	2	yes	—
131	C₂×C₄	C₂×#1042	1	1	yes	characteristic
132	C₅	—	4	1	no	—
133	C₅	#65225	1	1	yes	characteristic, complemented
134	C₅	#65689	1	1	yes	characteristic, complemented
135	C₂²	—	5	1	no	—
136	C₂²	—	5	1	no	—
137	C₂²	—	5	4	no	—
138	C₂²	C₂²×#1042	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	5	2	no	—
141	C₄	—	10	2	no	—
142	C₄	—	100	4	no	—
143	C₄	C₄⋊#1042	1	2	yes	—
144	C₂	—	5	2	no	—
145	C₂	—	5	2	no	—
146	C₂	C₂×C₄⋊#1042	1	1	yes	characteristic
147	C₂	#11717425	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#1042

C₂

yes

maximal

C₄²×C₅×D₅

C₂

yes

maximal, characteristic

#11716849

C₂

yes

maximal

C₂²×#1042

C₂²

yes

—

C₂×C₄×C₅×D₅

C₂²

yes

—

C₂×C₄×C₅×D₅

C₂²

yes

characteristic

C₄⋊#1042

—

C₄²×C₅²

C₄

yes

characteristic, complemented

C₄×C₅×Dic₅

C₄

yes

characteristic, complemented

C₄×C₅×Dic₅

C₄

yes

—

—

C₂²×C₅×D₅

C₂³

yes

characteristic

C₂×#1042

—

C₂×#1042

—

C₂×C₄×C₅²

C₂×C₄

yes

—

C₂×C₄×C₅²

C₂×C₄

yes

characteristic

C₂×C₅×Dic₅

C₂×C₄

yes

—

C₂×C₅×Dic₅

C₂×C₄

yes

characteristic

C₄×C₅×D₅

—

C₄×C₅×D₅

D₄

yes

—

C₄×C₅×D₅

Q₈

yes

—

C₂×C₄⋊AGL(1,𝔽₅)

—

C₂×C₄²×C₅

—

C₂×C₄⋊Dic₅

—

C₄²×D₅

D₅

yes

characteristic

(C₂×AGL(1,𝔽₅))⋊C₄

—

#3963

—

C₂²×C₅²

C₂²×C₄

yes

characteristic

C₂×C₅×D₅

C₂×D₄

yes

characteristic

C₂×C₅×D₅

C₂×Q₈

yes

characteristic

C₂×C₅×D₅

#41

yes

—

#1042

—

C₄×C₅²

—

C₄×C₅²

C₄⋊C₄

yes

—

C₅×Dic₅

—

C₅×Dic₅

C₄⋊C₄

yes

—

C₂²×AGL(1,𝔽₅)

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

—

C₂²×Dic₅

—

C₂×C₄×D₅

C₂×D₅

yes

—

C₂×C₄×D₅

C₂×D₅

yes

characteristic

C₄⋊AGL(1,𝔽₅)

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

Dic₅

yes

characteristic, complemented

C₄²×C₅

AGL(1,𝔽₅)

yes

characteristic, complemented

C₄⋊Dic₅

—

C₄×Dic₅

Dic₅

yes

characteristic, complemented

C₄×Dic₅

Dic₅

yes

complemented

#382

—

Sylow

C₂×C₅²

C₂×C₄⋊C₄

yes

characteristic

C₂×C₅²

#117

yes

—

C₅×D₅

#124

yes

—

C₅×D₅

#125

yes

—

C₂³×C₅

—

C₂²×D₅

yes

characteristic

C₂×AGL(1,𝔽₅)

—

C₂×AGL(1,𝔽₅)

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×Dic₅

yes

—

C₂×C₄×C₅

C₂×Dic₅

yes

characteristic

C₂×C₄×C₅

C₂×AGL(1,𝔽₅)

yes

—

C₂×C₄×C₅

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

yes

—

C₂×Dic₅

yes

characteristic

C₄×D₅

—

C₄×D₅

Dic₁₀

yes

—

C₄×D₅

D₂₀

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

#95

—

C₅²

#382

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂²×Dic₅

yes

characteristic

C₂²×C₅

C₂²×AGL(1,𝔽₅)

yes

characteristic

C₂×D₅

C₂×Dic₁₀

yes

characteristic

C₂×D₅

C₂×D₂₀

yes

characteristic

C₂×D₅

#709

yes

—

AGL(1,𝔽₅)

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

100

C₄×C₅

C₄⋊Dic₅

yes

—

101

C₄×C₅

C₄⋊AGL(1,𝔽₅)

yes

—

102

Dic₅

—

103

Dic₅

—

104

Dic₅

C₄⋊Dic₅

yes

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₄⋊C₄

—

109

C₄²

—

110

C₄²

—

111

C₄²

#1042

yes

characteristic, complemented

112

C₂×C₅

—

113

C₂×C₅

—

114

C₂×C₅

—

115

C₂×C₅

—

116

C₂×C₅

C₂×C₄⋊Dic₅

yes

characteristic

117

C₂×C₅

#4072

yes

—

118

C₂×C₅

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

119

C₂×C₅

#4130

yes

—

120

D₅

#4016

yes

—

121

D₅

#4021

yes

—

122

C₂³

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₂×#1042

yes

—

131

C₂×C₄

C₂×#1042

yes

characteristic

132

C₅

—

133

C₅

#65225

yes

characteristic, complemented

134

C₅

#65689

yes

characteristic, complemented

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

C₂²×#1042

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

100

—

143

C₄

C₄⋊#1042

yes

—

144

C₂

—

145

C₂

—

146

C₂

C₂×C₄⋊#1042

yes

characteristic

147

C₂

#11717425

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#1042
commutator, socle	C₂²×C₅²	C₂²×C₄
Fitting	C₄²×C₅²	C₄
prime radical, cosocle	C₂²×C₅×D₅	C₂³
hypercenter	C₄²	#1042
hypocenter	C₅²	#382
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄²

subgroup

G/H

center, Frattini

C₂²

C₂²×#1042

commutator, socle

C₂²×C₅²

C₂²×C₄

Fitting

C₄²×C₅²

C₄

prime radical, cosocle

C₂²×C₅×D₅

C₂³

C₄²

C₅²

G ▷ C₂²×C₅² ▷ C₅²

C₁ ◁ C₂² ◁ C₄²

N	H ≅ G/N	type
8. C₄²×C₅²	C₄	semidirect
9. C₄×C₅×Dic₅	C₄	semidirect
10. C₄×C₅×Dic₅	C₄	semidirect
48. C₄²×C₅	Dic₅	semidirect
49. C₄²×C₅	AGL(1,𝔽₅)	semidirect
51. C₄×Dic₅	Dic₅	semidirect
52. C₄×Dic₅	Dic₅	semidirect
83. C₅²	#382	semidirect
111. C₄²	#1042	semidirect
133. C₅	#65225	semidirect
134. C₅	#65689	semidirect

H ≅ G/N

type

8. C₄²×C₅²

C₄

semidirect

9. C₄×C₅×Dic₅

C₄

semidirect

10. C₄×C₅×Dic₅

C₄

semidirect

48. C₄²×C₅

Dic₅

semidirect

49. C₄²×C₅

AGL(1,𝔽₅)

semidirect

51. C₄×Dic₅

Dic₅

semidirect

52. C₄×Dic₅

Dic₅

semidirect

83. C₅²

#382

semidirect

111. C₄²

#1042

semidirect

133. C₅

#65225

semidirect

134. C₅

#65689

semidirect

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

subgroup	index
C₂×C₄⋊#1042	2
C₄²×C₅×D₅	2
#11716849	2
#65689	5
#65225	5

subgroup

index

C₂×C₄⋊#1042

C₄²×C₅×D₅

#11716849

#65689

#65225

Subgroups — #13104575

Conjugacy classes of subgroups

Complements

Maximal subgroups