Group #13104822 — subgroups

Subgroups — #13104822

Order 1600.

328 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 64 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	C₄²×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
2	#11716897	C₂	1	2	yes	maximal, complemented
3	#11716896	C₂	1	2	yes	maximal
4	#11716895	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₅⋊D₅	C₄	1	1	yes	characteristic
6	C₂×C₄×C₅⋊D₅	C₄	1	1	yes	characteristic
7	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
8	C₂×#6034	C₂²	1	4	yes	—
9	C₄²×C₅²	C₄	1	1	yes	characteristic
10	C₄×#1039	C₄	1	1	yes	characteristic
11	C₄×#1039	C₂²	1	1	yes	characteristic
12	C₄×#1039	C₂²	1	1	yes	characteristic
13	#65684	—	5	2	no	maximal
14	C₂²×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
18	C₂×#1039	C₂×C₄	1	1	yes	characteristic
19	C₂×#1039	C₂×C₄	1	1	yes	characteristic
20	C₂×#1039	C₂³	1	1	yes	characteristic
21	C₄×C₅⋊D₅	—	2	2	no	—
22	C₄×C₅⋊D₅	—	2	2	no	—
23	C₄×C₅⋊D₅	—	2	2	no	—
24	#6034	—	2	4	no	—
25	C₄²×D₅	—	5	2	no	—
26	C₄²×D₅	—	10	2	no	—
27	#4004	—	5	4	no	—
28	#4003	—	5	4	no	—
29	#4002	—	5	4	no	—
30	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₅⋊D₅	—	2	1	no	—
32	C₂×C₅⋊D₅	—	2	1	no	—
33	C₂×C₅⋊D₅	M₁₆	1	2	yes	—
34	C₄×C₅²	—	2	1	no	—
35	C₄×C₅²	—	2	1	no	—
36	C₄×C₅²	—	2	1	no	—
37	#1039	—	2	1	no	—
38	#1039	M₁₆	1	2	yes	—
39	#1039	#41	1	2	yes	—
40	C₂×C₄×D₅	—	5	2	no	—
41	C₂×C₄×D₅	—	5	2	no	—
42	C₂×C₄×D₅	—	5	2	no	—
43	C₂×C₄×D₅	—	10	2	no	—
44	C₂×C₄×D₅	—	10	2	no	—
45	C₂×C₄×D₅	—	10	2	no	—
46	C₂×#170	—	5	8	no	—
47	C₄²×C₅	—	2	2	no	—
48	C₄²×C₅	AGL(1,𝔽₅)	1	2	yes	—
49	C₄×Dic₅	—	5	2	no	—
50	C₄×Dic₅	—	5	2	no	—
51	C₄×Dic₅	—	5	2	no	—
52	C₄×Dic₅	—	10	2	no	—
53	C₄×Dic₅	—	10	2	no	—
54	C₄×Dic₅	—	10	2	no	—
55	#432	—	25	1	no	Sylow
56	C₂×C₅²	#117	1	1	yes	characteristic
57	C₂×C₅²	C₂×M₁₆	1	1	yes	characteristic
58	C₂×C₅²	C₂×M₁₆	1	1	yes	characteristic
59	C₅⋊D₅	—	2	2	no	—
60	C₂²×D₅	—	5	2	no	—
61	C₂²×D₅	—	10	2	no	—
62	C₂×C₄×C₅	—	2	2	no	—
63	C₂×C₄×C₅	—	2	2	no	—
64	C₂×C₄×C₅	—	2	2	no	—
65	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
66	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
67	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
68	C₂×Dic₅	—	5	2	no	—
69	C₂×Dic₅	—	5	2	no	—
70	C₂×Dic₅	—	5	2	no	—
71	C₂×Dic₅	—	10	2	no	—
72	C₂×Dic₅	—	10	2	no	—
73	C₂×Dic₅	—	10	2	no	—
74	C₄×D₅	—	10	4	no	—
75	C₄×D₅	—	10	4	no	—
76	C₄×D₅	—	10	4	no	—
77	C₄×D₅	—	10	8	no	—
78	C₄×D₅	—	10	8	no	—
79	C₄×D₅	—	10	8	no	—
80	#170	—	10	8	no	—
81	C₂×C₄²	—	25	1	no	—
82	C₄⋊C₈	—	25	2	no	—
83	C₂²⋊C₈	—	25	2	no	—
84	#97	—	25	2	no	—
85	C₅²	#432	1	1	yes	characteristic, complemented, Sylow
86	C₂²×C₅	—	2	2	no	—
87	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	2	yes	—
88	C₂×D₅	—	5	4	no	—
89	C₂×D₅	—	10	4	no	—
90	C₂×D₅	—	10	4	no	—
91	C₂×D₅	—	10	4	no	—
92	C₂×D₅	—	10	2	no	—
93	C₂×D₅	—	10	2	no	—
94	C₄×C₅	—	2	4	no	—
95	C₄×C₅	—	2	4	no	—
96	C₄×C₅	—	2	4	no	—
97	C₄×C₅	—	2	2	no	—
98	C₄×C₅	—	2	2	no	—
99	C₄×C₅	—	2	2	no	—
100	Dic₅	—	5	4	no	—
101	Dic₅	—	5	4	no	—
102	Dic₅	—	10	4	no	—
103	Dic₅	—	10	4	no	—
104	Dic₅	—	10	4	no	—
105	Dic₅	—	10	2	no	—
106	C₂²×C₄	—	25	1	no	—
107	C₂²×C₄	—	25	1	no	—
108	C₂²×C₄	—	25	1	no	—
109	C₂×C₈	—	25	4	no	—
110	C₄²	—	25	1	no	—
111	C₄²	—	25	1	no	—
112	C₄²	—	25	1	no	—
113	C₄²	#1044	1	1	yes	characteristic
114	C₂×C₅	—	2	2	no	—
115	C₂×C₅	—	2	2	no	—
116	C₂×C₅	—	2	2	no	—
117	C₂×C₅	D₅⋊M₁₆	1	2	yes	—
118	C₂×C₅	D₅⋊M₁₆	1	2	yes	—
119	C₂×C₅	#4130	1	2	yes	—
120	D₅	—	10	8	no	—
121	D₅	—	10	4	no	—
122	C₂³	—	25	1	no	—
123	C₂×C₄	—	25	1	no	—
124	C₂×C₄	—	25	1	no	—
125	C₂×C₄	—	25	1	no	—
126	C₂×C₄	—	50	2	no	—
127	C₂×C₄	—	50	2	no	—
128	C₂×C₄	—	50	2	no	—
129	C₂×C₄	C₂×#1044	1	1	yes	characteristic
130	C₂×C₄	C₂×#1044	1	1	yes	characteristic
131	C₂×C₄	C₂×#1044	1	1	yes	characteristic
132	C₈	—	50	4	no	—
133	C₅	—	2	2	no	—
134	C₅	#65684	1	2	yes	complemented
135	C₂²	—	25	2	no	—
136	C₂²	—	50	1	no	—
137	C₂²	—	50	1	no	—
138	C₂²	C₂²×#1044	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	25	2	no	—
143	C₄	—	25	2	no	—
144	C₄	—	50	1	no	—
145	C₂	—	50	2	no	—
146	C₂	#11717448	1	1	yes	characteristic
147	C₂	#11717448	1	1	yes	characteristic
148	C₂	#11717451	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄²×C₅⋊D₅

C₂

yes

maximal, characteristic

#11716897

C₂

yes

maximal, complemented

#11716896

C₂

yes

maximal

#11716895

C₂

yes

maximal, complemented

C₂×C₄×C₅⋊D₅

C₄

yes

characteristic

C₂×C₄×C₅⋊D₅

C₄

yes

characteristic

C₂×C₄×C₅⋊D₅

C₂²

yes

characteristic

C₂×#6034

C₂²

yes

—

C₄²×C₅²

C₄

yes

characteristic

C₄×#1039

C₄

yes

characteristic

C₄×#1039

C₂²

yes

characteristic

C₄×#1039

C₂²

yes

characteristic

#65684

—

maximal

C₂²×C₅⋊D₅

C₂×C₄

yes

characteristic

C₂×C₄×C₅²

C₂×C₄

yes

characteristic

C₂×C₄×C₅²

C₂×C₄

yes

characteristic

C₂×C₄×C₅²

C₂×C₄

yes

characteristic

C₂×#1039

C₂×C₄

yes

characteristic

C₂×#1039

C₂×C₄

yes

characteristic

C₂×#1039

C₂³

yes

characteristic

C₄×C₅⋊D₅

—

C₄×C₅⋊D₅

—

C₄×C₅⋊D₅

—

#6034

—

C₄²×D₅

—

C₄²×D₅

—

#4004

—

#4003

—

#4002

—

C₂²×C₅²

C₂²×C₄

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

M₁₆

yes

—

C₄×C₅²

—

C₄×C₅²

—

C₄×C₅²

—

#1039

—

#1039

M₁₆

yes

—

#1039

#41

yes

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×#170

—

C₄²×C₅

—

C₄²×C₅

AGL(1,𝔽₅)

yes

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

—

C₂×C₅²

yes

C₂×C₅²

C₂×M₁₆

yes

characteristic

C₂×C₅²

C₂×M₁₆

yes

characteristic

C₅⋊D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×AGL(1,𝔽₅)

yes

—

C₂×C₄×C₅

C₂×AGL(1,𝔽₅)

yes

—

C₂×C₄×C₅

C₂×AGL(1,𝔽₅)

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

#170

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₅²

#432

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

C₂²×AGL(1,𝔽₅)

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

100

Dic₅

—

101

Dic₅

—

102

Dic₅

—

103

Dic₅

—

104

Dic₅

—

105

Dic₅

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂×C₈

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₄²

#1044

yes

characteristic

114

C₂×C₅

—

115

C₂×C₅

—

116

C₂×C₅

—

117

C₂×C₅

D₅⋊M₁₆

yes

—

118

C₂×C₅

D₅⋊M₁₆

yes

—

119

C₂×C₅

#4130

yes

—

120

D₅

—

121

D₅

—

122

C₂³

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×#1044

yes

characteristic

130

C₂×C₄

C₂×#1044

yes

characteristic

131

C₂×C₄

C₂×#1044

yes

characteristic

132

C₈

—

133

C₅

—

134

C₅

#65684

yes

complemented

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

C₂²×#1044

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₂

—

146

C₂

#11717448

yes

characteristic

147

C₂

#11717448

yes

characteristic

148

C₂

#11717451

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#1044
commutator, socle	C₂²×C₅²	C₂²×C₄
Fitting	C₄²×C₅²	C₄
prime radical	C₂²×C₅⋊D₅	C₂×C₄
cosocle	C₂×#1039	C₂³
hypercenter	C₄²	#1044
hypocenter	C₅²	#432
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄²

subgroup

G/H

center, Frattini

C₂²

C₂²×#1044

commutator, socle

C₂²×C₅²

C₂²×C₄

Fitting

C₄²×C₅²

C₄

prime radical

C₂²×C₅⋊D₅

C₂×C₄

C₂³

C₄²

C₅²

G ▷ C₂²×C₅² ▷ C₅²

C₁ ◁ C₂² ◁ C₄²

N	H ≅ G/N	type
2. #11716897	C₂	semidirect
4. #11716895	C₂	semidirect
85. C₅²	#432	semidirect
134. C₅	#65684	semidirect

H ≅ G/N

type

2. #11716897

C₂

semidirect

4. #11716895

C₂

semidirect

85. C₅²

#432

semidirect

134. C₅

#65684

semidirect

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

subgroup	index
C₄²×C₅⋊D₅	2
#11716897	2
#11716896	2
#11716895	2
#65684	5

subgroup

index

C₄²×C₅⋊D₅

Subgroups — #13104822

Conjugacy classes of subgroups

Complements

Maximal subgroups