Group #13027829 — subgroups

Subgroups — #13027829

Order 1280.

170 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 68 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	#10599864	C₂	1	1	yes	maximal, characteristic
2	#10598658	C₂	1	2	yes	maximal, complemented
3	#10598657	C₂	1	2	yes	maximal
4	#10598537	C₂	1	2	yes	maximal, complemented
5	C₂×#3930	C₄	1	1	yes	characteristic
6	C₂×#3930	C₂²	1	1	yes	characteristic
7	C₄×C₈×D₅	C₄	1	1	yes	characteristic
8	#64877	C₂²	1	4	yes	—
9	C₅×#346	C₄	1	1	yes	characteristic
10	C₁₆×Dic₅	C₄	1	1	yes	characteristic
11	C₁₆×Dic₅	C₂²	1	1	yes	characteristic
12	#64681	C₂²	1	1	yes	characteristic
13	#13642	—	5	1	no	maximal, Sylow
14	C₂×C₈×D₅	C₈	1	1	yes	characteristic
15	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
16	C₄²×D₅	C₈	1	1	yes	characteristic
17	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
18	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
20	C₈×Dic₅	C₈	1	1	yes	characteristic
21	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
22	C₂×#672	C₂×C₄	1	1	yes	characteristic
23	C₂×#672	C₂³	1	1	yes	characteristic
24	C₄×#168	C₈	1	1	yes	characteristic
25	#3930	—	2	2	no	—
26	#3930	—	2	2	no	—
27	#3928	—	2	4	no	—
28	C₄×M₃₂	—	5	1	no	—
29	#1422	—	5	2	no	—
30	#1400	—	5	2	no	—
31	#1398	—	5	2	no	—
32	C₂×C₄×D₅	—	2	1	no	—
33	C₂×C₄×D₅	C₂×C₈	1	1	yes	characteristic
34	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
35	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
37	C₄×Dic₅	—	2	1	no	—
38	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
39	C₂×#168	C₂×C₈	1	1	yes	characteristic
40	C₂×#168	C₂×C₈	1	1	yes	characteristic
41	C₈×D₅	—	2	1	no	—
42	C₈×D₅	—	2	1	no	—
43	C₈×D₅	—	2	2	no	—
44	C₅×C₁₆	—	2	1	no	—
45	C₅×C₁₆	—	2	1	no	—
46	#672	M₁₆	1	2	yes	—
47	#672	#41	1	2	yes	—
48	C₂×M₃₂	—	5	1	no	—
49	C₂×M₃₂	—	5	1	no	—
50	C₂×C₄×C₈	—	5	1	no	—
51	M₆₄	—	5	4	no	—
52	#346	—	5	1	no	—
53	#346	AGL(1,𝔽₅)	1	1	yes	characteristic
54	C₄×C₁₆	—	5	1	no	—
55	C₄×C₁₆	—	5	1	no	—
56	C₂²×D₅	M₃₂	1	1	yes	characteristic
57	C₂×C₄×C₅	—	2	1	no	—
58	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
59	C₂×Dic₅	—	2	1	no	—
60	C₂×Dic₅	M₃₂	1	1	yes	characteristic
61	C₄×D₅	—	2	1	no	—
62	C₄×D₅	—	2	1	no	—
63	C₄×D₅	—	4	2	no	—
64	C₅×C₈	—	2	1	no	—
65	C₅×C₈	#117	1	1	yes	characteristic
66	C₅×C₈	C₂×M₁₆	1	1	yes	characteristic
67	#168	—	2	1	no	—
68	#168	M₃₂	1	2	yes	—
69	C₂²×C₈	—	5	1	no	—
70	C₂²×C₈	—	5	1	no	—
71	C₂×C₄²	—	5	1	no	—
72	M₃₂	—	10	2	no	—
73	M₃₂	—	10	2	no	—
74	C₂×C₁₆	—	5	1	no	—
75	C₂×C₁₆	—	5	1	no	—
76	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
77	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
78	C₄×C₈	—	5	1	no	—
79	C₄×C₈	—	5	1	no	—
80	C₄×C₈	—	5	1	no	—
81	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
82	C₃₂	—	10	4	no	—
83	C₂²×C₅	C₂×M₃₂	1	1	yes	characteristic
84	C₂×D₅	—	2	1	no	—
85	C₂×D₅	—	4	1	no	—
86	C₄×C₅	—	4	1	no	—
87	C₄×C₅	#431	1	1	yes	characteristic
88	C₄×C₅	C₂×M₃₂	1	1	yes	characteristic
89	Dic₅	—	2	2	no	—
90	Dic₅	—	2	1	no	—
91	C₂²×C₄	—	5	1	no	—
92	C₂²×C₄	—	10	1	no	—
93	C₂×C₈	—	5	1	no	—
94	C₂×C₈	—	5	1	no	—
95	C₂×C₈	—	10	2	no	—
96	C₂×C₈	—	10	1	no	—
97	C₂×C₈	—	10	1	no	—
98	C₂×C₈	D₅⋊C₈	1	1	yes	characteristic
99	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
100	C₄²	—	5	1	no	—
101	C₄²	—	10	1	no	—
102	C₄²	D₅⋊C₈	1	1	yes	characteristic
103	C₁₆	—	2	1	no	—
104	C₁₆	—	2	1	no	—
105	C₁₆	—	5	2	no	—
106	C₁₆	—	5	2	no	—
107	C₂×C₅	—	2	1	no	—
108	C₂×C₅	#2163	1	1	yes	characteristic
109	D₅	—	4	1	no	—
110	C₂³	—	5	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	5	1	no	—
113	C₂×C₄	—	10	1	no	—
114	C₂×C₄	—	10	1	no	—
115	C₂×C₄	—	10	1	no	—
116	C₂×C₄	—	20	2	no	—
117	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
118	C₈	—	2	1	no	—
119	C₈	—	5	2	no	—
120	C₈	—	10	1	no	—
121	C₈	D₅⋊M₁₆	1	1	yes	characteristic
122	C₈	#4130	1	1	yes	characteristic
123	C₅	#13642	1	1	yes	characteristic, complemented, Sylow
124	C₂²	—	10	1	no	—
125	C₂²	—	20	1	no	—
126	C₂²	D₅⋊M₃₂	1	1	yes	characteristic
127	C₄	—	4	1	no	—
128	C₄	—	10	2	no	—
129	C₄	—	10	1	no	—
130	C₄	#65678	1	1	yes	characteristic
131	C₄	D₅⋊M₃₂	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	20	1	no	—
134	C₂	#10603830	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10599864

C₂

yes

maximal, characteristic

#10598658

C₂

yes

maximal, complemented

#10598657

C₂

yes

maximal

#10598537

C₂

yes

maximal, complemented

C₂×#3930

C₄

yes

characteristic

C₂×#3930

C₂²

yes

characteristic

C₄×C₈×D₅

C₄

yes

characteristic

#64877

C₂²

yes

—

C₅×#346

C₄

yes

characteristic

C₁₆×Dic₅

C₄

yes

characteristic

C₁₆×Dic₅

C₂²

yes

characteristic

#64681

C₂²

yes

characteristic

#13642

—

maximal, Sylow

C₂×C₈×D₅

C₈

yes

characteristic

C₂×C₈×D₅

C₂×C₄

yes

characteristic

C₄²×D₅

C₈

yes

characteristic

C₂×C₅×C₁₆

C₂×C₄

yes

characteristic

C₂×C₅×C₁₆

C₂×C₄

yes

characteristic

C₄×C₅×C₈

C₂×C₄

yes

characteristic

C₈×Dic₅

C₈

yes

characteristic

C₈×Dic₅

C₂×C₄

yes

characteristic

C₂×#672

C₂×C₄

yes

characteristic

C₂×#672

C₂³

yes

characteristic

C₄×#168

C₈

yes

—

—

—

C₄×M₃₂

—

#1422

—

#1400

—

#1398

—

C₂×C₄×D₅

—

C₂×C₄×D₅

C₂×C₈

yes

characteristic

C₂×C₅×C₈

C₂×C₈

yes

characteristic

C₂×C₅×C₈

C₂²×C₄

yes

characteristic

C₄²×C₅

C₂×C₈

yes

characteristic

C₄×Dic₅

—

C₄×Dic₅

C₂×C₈

yes

characteristic

C₂×#168

C₂×C₈

yes

characteristic

C₂×#168

C₂×C₈

yes

characteristic

C₈×D₅

—

C₈×D₅

—

C₈×D₅

—

C₅×C₁₆

—

C₅×C₁₆

—

#672

M₁₆

yes

—

#672

#41

yes

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×C₄×C₈

—

M₆₄

—

—

yes

C₄×C₁₆

—

C₄×C₁₆

—

C₂²×D₅

M₃₂

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×C₈

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

M₃₂

yes

characteristic

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₅×C₈

—

C₅×C₈

#117

yes

characteristic

C₅×C₈

C₂×M₁₆

yes

characteristic

#168

—

#168

M₃₂

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

C₂×C₁₆

yes

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

C₃₂

—

C₂²×C₅

C₂×M₃₂

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

#431

yes

characteristic

C₄×C₅

C₂×M₃₂

yes

characteristic

Dic₅

—

Dic₅

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₅⋊C₈

yes

characteristic

C₂×C₈

C₂²×AGL(1,𝔽₅)

yes

characteristic

100

C₄²

—

101

C₄²

—

102

C₄²

D₅⋊C₈

yes

characteristic

103

C₁₆

—

104

C₁₆

—

105

C₁₆

—

106

C₁₆

—

107

C₂×C₅

—

108

C₂×C₅

#2163

yes

characteristic

109

D₅

—

110

C₂³

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂×D₅⋊C₈

yes

characteristic

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

D₅⋊M₁₆

yes

characteristic

122

C₈

#4130

yes

characteristic

123

C₅

#13642

yes

characteristic, complemented, Sylow

124

C₂²

—

125

C₂²

—

126

C₂²

D₅⋊M₃₂

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#65678

yes

characteristic

131

C₄

D₅⋊M₃₂

yes

characteristic

132

C₂

—

133

C₂

—

134

C₂

#10603830

yes

characteristic

	subgroup	G/H
center	C₈	#4130
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₂×C₈	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#346	C₄
socle	C₂×C₅	#2163
prime radical	C₂²×D₅	M₃₂
cosocle	C₂×#672	C₂³
hypercenter	#346	AGL(1,𝔽₅)
hypocenter	C₅	#13642
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₈ ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₈

#4130

commutator

C₂×C₄×C₅

C₂²×C₈

Frattini

C₂×C₈

C₂²×AGL(1,𝔽₅)

C₄

C₂×C₅

C₂²×D₅

M₃₂

C₂³

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₈ ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
2. #10598658	C₂	semidirect
4. #10598537	C₂	semidirect
123. C₅	#13642	semidirect

H ≅ G/N

type

2. #10598658

C₂

semidirect

4. #10598537

C₂

semidirect

123. C₅

#13642

semidirect

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

subgroup	index
#10599864	2
#10598658	2
#10598657	2
#10598537	2
#13642	5

subgroup

index

Subgroups — #13027829

Conjugacy classes of subgroups

Complements

Maximal subgroups