Group #13027547 — subgroups

Subgroups — #13027547

Order 1280.

330 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 116 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×D₅×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#64877	C₂	1	2	yes	maximal, complemented
3	#10598654	C₂	1	4	yes	maximal, complemented
4	C₂²×C₈×D₅	C₄	1	1	yes	characteristic
5	C₂²×C₅×C₁₆	C₄	1	1	yes	characteristic
6	C₂²×#672	C₂²	1	1	yes	characteristic
7	C₂×D₅×C₁₆	—	2	4	no	—
8	C₂×D₅×C₁₆	C₄	1	2	yes	—
9	C₂×D₅×C₁₆	C₂²	1	2	yes	—
10	#64877	—	2	4	no	—
11	C₂×#3928	C₂²	1	4	yes	complemented
12	#13553	—	5	1	no	maximal, Sylow
13	C₂²×C₄×D₅	C₈	1	1	yes	characteristic
14	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
15	C₂²×#168	C₈	1	1	yes	characteristic
16	C₂×C₈×D₅	—	2	2	no	—
17	C₂×C₈×D₅	—	2	2	no	—
18	C₂×C₈×D₅	C₈	1	2	yes	—
19	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
20	C₂×C₅×C₁₆	—	2	2	no	—
21	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
22	C₂×#672	C₂×C₄	1	1	yes	characteristic
23	C₂×#672	D₄	1	4	yes	—
24	C₂×#672	C₂³	1	1	yes	characteristic
25	D₅×C₁₆	—	2	4	no	—
26	D₅×C₁₆	—	2	4	no	—
27	#3928	—	2	4	no	—
28	C₂³×C₁₆	—	5	1	no	—
29	C₂×M₆₄	—	5	2	no	—
30	C₂²⋊C₃₂	—	5	4	no	—
31	C₂³×D₅	C₁₆	1	1	yes	characteristic
32	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
33	C₂²×Dic₅	C₁₆	1	1	yes	characteristic
34	C₂×C₄×D₅	—	2	2	no	—
35	C₂×C₄×D₅	—	2	2	no	—
36	C₂×C₄×D₅	C₁₆	1	2	yes	—
37	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
38	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
39	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
40	C₂×C₅×C₈	M₁₆	1	2	yes	—
41	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
42	C₂×#168	—	2	2	no	—
43	C₂×#168	C₂×C₈	1	2	yes	—
44	C₈×D₅	—	2	4	no	—
45	C₈×D₅	—	2	2	no	—
46	C₈×D₅	—	2	2	no	—
47	C₅×C₁₆	—	2	2	no	—
48	#672	C₂²⋊C₄	1	2	yes	—
49	#672	C₂×D₄	1	2	yes	—
50	C₂³×C₈	—	5	1	no	—
51	C₂²×C₁₆	—	5	1	no	—
52	C₂²×C₁₆	—	5	2	no	—
53	C₂²×C₁₆	—	5	2	no	—
54	C₂²×C₁₆	—	10	4	no	—
55	C₂²×C₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic
56	M₆₄	—	10	4	no	—
57	C₂×C₃₂	—	5	4	no	—
58	C₂³×C₅	C₂×C₁₆	1	1	yes	characteristic
59	C₂²×D₅	—	2	2	no	—
60	C₂²×D₅	—	2	4	no	—
61	C₂²×D₅	C₂×C₁₆	1	2	yes	—
62	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
63	C₂×C₄×C₅	C₂×C₁₆	1	1	yes	characteristic
64	C₂×C₄×C₅	M₃₂	1	2	yes	—
65	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
66	C₂×Dic₅	—	2	2	no	—
67	C₂×Dic₅	C₂×C₁₆	1	2	yes	—
68	C₄×D₅	—	2	4	no	—
69	C₄×D₅	—	2	2	no	—
70	C₄×D₅	—	2	2	no	—
71	C₅×C₈	C₂²⋊C₈	1	2	yes	—
72	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₅×C₈	C₂×M₁₆	1	1	yes	characteristic
74	#168	—	2	2	no	—
75	C₂³×C₄	—	5	1	no	—
76	C₂²×C₈	—	5	1	no	—
77	C₂²×C₈	—	5	2	no	—
78	C₂²×C₈	—	5	2	no	—
79	C₂²×C₈	—	10	2	no	—
80	C₂²×C₈	—	10	2	no	—
81	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
82	C₂×C₁₆	—	2	2	no	—
83	C₂×C₁₆	—	5	4	no	—
84	C₂×C₁₆	—	5	1	no	—
85	C₂×C₁₆	—	5	1	no	—
86	C₂×C₁₆	—	10	4	no	—
87	C₂×C₁₆	—	10	4	no	—
88	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	—
89	C₃₂	—	10	4	no	—
90	C₂²×C₅	—	2	2	no	—
91	C₂²×C₅	C₂²⋊C₁₆	1	2	yes	—
92	C₂²×C₅	C₂²×C₁₆	1	1	yes	characteristic
93	C₂×D₅	—	2	8	no	—
94	C₂×D₅	—	2	2	no	—
95	C₂×D₅	—	2	2	no	—
96	C₂×D₅	M₆₄	1	4	yes	complemented
97	C₄×C₅	C₂²⋊C₁₆	1	2	yes	—
98	C₄×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
99	C₄×C₅	C₂×M₃₂	1	1	yes	characteristic
100	Dic₅	—	2	2	no	—
101	C₂⁴	—	5	1	no	—
102	C₂²×C₄	—	5	1	no	—
103	C₂²×C₄	—	5	2	no	—
104	C₂²×C₄	—	5	2	no	—
105	C₂²×C₄	—	10	2	no	—
106	C₂²×C₄	—	10	2	no	—
107	C₂²×C₄	D₅⋊C₈	1	1	yes	characteristic
108	C₂×C₈	—	5	2	no	—
109	C₂×C₈	—	10	2	no	—
110	C₂×C₈	—	10	2	no	—
111	C₂×C₈	—	10	2	no	—
112	C₂×C₈	—	10	4	no	—
113	C₂×C₈	D₅⋊C₈	1	1	yes	characteristic
114	C₂×C₈	#700	1	2	yes	—
115	C₂×C₈	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
116	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
117	C₁₆	—	2	2	no	—
118	C₁₆	—	5	2	no	—
119	C₁₆	—	5	2	no	—
120	C₂×C₅	—	2	2	no	—
121	C₂×C₅	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
122	C₂×C₅	C₂×M₆₄	1	2	yes	—
123	D₅	—	2	4	no	—
124	C₂³	—	5	2	no	—
125	C₂³	—	10	4	no	—
126	C₂³	—	10	2	no	—
127	C₂³	D₅⋊C₁₆	1	1	yes	characteristic
128	C₂×C₄	—	5	2	no	—
129	C₂×C₄	—	10	2	no	—
130	C₂×C₄	—	10	4	no	—
131	C₂×C₄	—	10	2	no	—
132	C₂×C₄	—	10	2	no	—
133	C₂×C₄	D₅⋊C₁₆	1	1	yes	characteristic
134	C₂×C₄	#3990	1	2	yes	—
135	C₂×C₄	#4003	1	2	yes	—
136	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
137	C₈	—	10	2	no	—
138	C₈	#4003	1	2	yes	—
139	C₈	C₂×#700	1	1	yes	characteristic
140	C₈	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
141	C₅	#13553	1	1	yes	characteristic, complemented, Sylow
142	C₂²	—	2	2	no	—
143	C₂²	—	5	4	no	—
144	C₂²	—	10	8	no	—
145	C₂²	—	10	2	no	—
146	C₂²	—	10	2	no	—
147	C₂²	#64887	1	2	yes	—
148	C₂²	C₂×D₅⋊C₁₆	1	1	yes	characteristic
149	C₄	—	10	2	no	—
150	C₄	#64887	1	2	yes	—
151	C₄	C₂×#3990	1	1	yes	characteristic
152	C₄	C₂×#4003	1	1	yes	characteristic
153	C₂	—	2	2	no	—
154	C₂	—	10	4	no	—
155	C₂	D₅⋊M₆₄	1	2	yes	—
156	C₂	C₂×#64887	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₅×C₁₆

C₂

yes

maximal, characteristic

C₂×#64877

C₂

yes

maximal, complemented

#10598654

C₂

yes

maximal, complemented

C₂²×C₈×D₅

C₄

yes

characteristic

C₂²×C₅×C₁₆

C₄

yes

characteristic

C₂²×#672

C₂²

yes

characteristic

C₂×D₅×C₁₆

—

C₂×D₅×C₁₆

C₄

yes

—

C₂×D₅×C₁₆

C₂²

yes

—

#64877

—

C₂×#3928

C₂²

yes

complemented

#13553

—

maximal, Sylow

C₂²×C₄×D₅

C₈

yes

characteristic

C₂²×C₅×C₈

C₂×C₄

yes

characteristic

C₂²×#168

C₈

yes

characteristic

C₂×C₈×D₅

—

C₂×C₈×D₅

—

C₂×C₈×D₅

C₈

yes

—

C₂×C₈×D₅

C₂×C₄

yes

—

C₂×C₅×C₁₆

—

C₂×C₅×C₁₆

C₂×C₄

yes

—

C₂×#672

C₂×C₄

yes

characteristic

C₂×#672

D₄

yes

—

C₂×#672

C₂³

yes

characteristic

D₅×C₁₆

—

D₅×C₁₆

—

#3928

—

C₂³×C₁₆

—

C₂×M₆₄

—

C₂²⋊C₃₂

—

C₂³×D₅

C₁₆

yes

characteristic

C₂²×C₄×C₅

C₂×C₈

yes

characteristic

C₂²×Dic₅

C₁₆

yes

characteristic

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

C₁₆

yes

—

C₂×C₄×D₅

C₂×C₈

yes

—

C₂×C₅×C₈

C₂²⋊C₄

yes

—

C₂×C₅×C₈

C₂×C₈

yes

characteristic

C₂×C₅×C₈

M₁₆

yes

—

C₂×C₅×C₈

C₂²×C₄

yes

characteristic

C₂×#168

—

C₂×#168

C₂×C₈

yes

—

C₈×D₅

—

C₈×D₅

—

C₈×D₅

—

C₅×C₁₆

—

#672

C₂²⋊C₄

yes

—

#672

C₂×D₄

yes

—

C₂³×C₈

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

AGL(1,𝔽₅)

yes

characteristic

M₆₄

—

C₂×C₃₂

—

C₂³×C₅

C₂×C₁₆

yes

characteristic

C₂²×D₅

—

C₂²×D₅

—

C₂²×D₅

C₂×C₁₆

yes

—

C₂×C₄×C₅

C₂²⋊C₈

yes

—

C₂×C₄×C₅

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅

M₃₂

yes

—

C₂×C₄×C₅

C₂²×C₈

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

C₂×C₁₆

yes

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₅×C₈

C₂²⋊C₈

yes

—

C₅×C₈

C₂×C₂²⋊C₄

yes

characteristic

C₅×C₈

C₂×M₁₆

yes

characteristic

#168

—

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×AGL(1,𝔽₅)

yes

—

C₃₂

—

C₂²×C₅

—

C₂²×C₅

C₂²⋊C₁₆

yes

—

C₂²×C₅

C₂²×C₁₆

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

M₆₄

yes

complemented

C₄×C₅

C₂²⋊C₁₆

yes

—

C₄×C₅

C₂×C₂²⋊C₈

yes

characteristic

C₄×C₅

C₂×M₃₂

yes

characteristic

100

Dic₅

—

101

C₂⁴

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

D₅⋊C₈

yes

characteristic

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

D₅⋊C₈

yes

characteristic

114

C₂×C₈

#700

yes

—

115

C₂×C₈

C₂²⋊AGL(1,𝔽₅)

yes

—

116

C₂×C₈

C₂²×AGL(1,𝔽₅)

yes

characteristic

117

C₁₆

—

118

C₁₆

—

119

C₁₆

—

120

C₂×C₅

—

121

C₂×C₅

C₂×C₂²⋊C₁₆

yes

characteristic

122

C₂×C₅

C₂×M₆₄

yes

—

123

D₅

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

D₅⋊C₁₆

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

D₅⋊C₁₆

yes

characteristic

134

C₂×C₄

#3990

yes

—

135

C₂×C₄

#4003

yes

—

136

C₂×C₄

C₂×D₅⋊C₈

yes

characteristic

137

C₈

—

138

C₈

#4003

yes

—

139

C₈

C₂×#700

yes

characteristic

140

C₈

C₂×C₂²⋊AGL(1,𝔽₅)

yes

characteristic

141

C₅

#13553

yes

characteristic, complemented, Sylow

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

#64887

yes

—

148

C₂²

C₂×D₅⋊C₁₆

yes

characteristic

149

C₄

—

150

C₄

#64887

yes

—

151

C₄

C₂×#3990

yes

characteristic

152

C₄

C₂×#4003

yes

characteristic

153

C₂

—

154

C₂

—

155

C₂

D₅⋊M₆₄

yes

—

156

C₂

C₂×#64887

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂²×C₁₆
Fitting	C₂²×C₅×C₁₆	C₄
prime radical	C₂³×D₅	C₁₆
cosocle	C₂×#672	C₂³
hypercenter	C₂²×C₁₆	AGL(1,𝔽₅)
hypocenter	C₅	#13553
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×AGL(1,𝔽₅)

commutator, socle

C₂²×C₅

C₂²×C₁₆

Fitting

C₂²×C₅×C₁₆

C₄

prime radical

C₂³×D₅

C₁₆

C₂³

C₂²×C₁₆

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
2. C₂×#64877	C₂	semidirect
3. #10598654	C₂	semidirect
11. C₂×#3928	C₂²	semidirect
96. C₂×D₅	M₆₄	semidirect
141. C₅	#13553	semidirect

H ≅ G/N

type

2. C₂×#64877

C₂

semidirect

3. #10598654

C₂

semidirect

11. C₂×#3928

C₂²

semidirect

96. C₂×D₅

M₆₄

semidirect

141. C₅

#13553

semidirect

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

subgroup	index
C₂²×D₅×C₁₆	2
C₂×#64877	2
#10598654	2
#13553	5

subgroup

index

C₂²×D₅×C₁₆

C₂×#64877

#10598654

#13553

Subgroups — #13027547

Conjugacy classes of subgroups

Complements

Maximal subgroups