Group #13027800 — subgroups

Subgroups — #13027800

Order 1280.

198 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 96 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	D₅⋊M₆₄	C₂	1	2	yes	maximal
2	#10599865	C₂	1	1	yes	maximal, characteristic
3	#10598658	C₂	1	2	yes	maximal, complemented
4	#10598538	C₂	1	2	yes	maximal, complemented
5	C₂×#3930	C₄	1	1	yes	characteristic
6	C₂×D₅×C₁₆	C₂²	1	1	yes	characteristic
7	#64978	C₄	1	1	yes	characteristic
8	#64877	C₂²	1	4	yes	—
9	#64842	—	2	2	no	—
10	D₅⋊C₃₂	—	2	2	no	—
11	C₅×#346	C₄	1	1	yes	characteristic
12	#64722	C₄	1	1	yes	characteristic
13	C₁₆×Dic₅	C₂²	1	1	yes	characteristic
14	C₄×#672	C₂²	1	1	yes	characteristic
15	#13630	—	5	1	no	maximal, Sylow
16	C₂×#676	C₈	1	1	yes	characteristic
17	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
18	#4018	C₈	1	1	yes	characteristic
19	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
22	#3947	C₈	1	1	yes	characteristic
23	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
24	C₂×#672	C₂×C₄	1	1	yes	characteristic
25	C₂×#672	C₂³	1	1	yes	characteristic
26	#3935	C₈	1	1	yes	characteristic
27	#3930	—	2	2	no	—
28	D₅×C₁₆	D₄	1	2	yes	—
29	D₅×C₁₆	Q₈	1	2	yes	—
30	#3928	—	2	4	no	—
31	C₂×M₆₄	—	5	2	no	—
32	#2108	—	5	1	no	—
33	#1422	—	5	2	no	—
34	#1422	—	5	2	no	—
35	C₂×C₄×D₅	C₂×C₈	1	1	yes	characteristic
36	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
37	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
38	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
39	#685	—	2	1	no	—
40	#683	—	2	1	no	—
41	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
42	C₂×#168	C₂×C₈	1	1	yes	characteristic
43	C₂×#168	C₂×C₈	1	1	yes	characteristic
44	#676	C₁₆	1	4	yes	—
45	C₈×D₅	C₄⋊C₄	1	2	yes	—
46	C₈×D₅	M₁₆	1	2	yes	—
47	C₅×C₁₆	—	2	1	no	—
48	C₅×C₁₆	C₄⋊C₄	1	2	yes	—
49	#672	—	2	1	no	—
50	#672	C₂×D₄	1	1	yes	characteristic
51	#672	C₂×Q₈	1	1	yes	characteristic
52	C₂×M₃₂	—	5	1	no	—
53	C₂²×C₁₆	—	5	1	no	—
54	#405	—	5	1	no	—
55	M₆₄	—	5	4	no	—
56	M₆₄	—	10	2	no	—
57	C₂×C₃₂	—	10	2	no	—
58	#346	—	5	1	no	—
59	#346	AGL(1,𝔽₅)	1	1	yes	characteristic
60	C₄×C₁₆	—	5	1	no	—
61	C₄×C₁₆	—	5	1	no	—
62	C₂²×D₅	M₃₂	1	1	yes	characteristic
63	C₂×C₄×C₅	—	2	1	no	—
64	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
65	C₂×Dic₅	—	2	1	no	—
66	C₂×Dic₅	M₃₂	1	1	yes	characteristic
67	C₄×D₅	C₄⋊C₈	1	2	yes	—
68	C₄×D₅	C₂×C₁₆	1	2	yes	—
69	C₅×C₈	C₂×C₁₆	1	2	yes	—
70	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₅×C₈	C₂×M₁₆	1	1	yes	characteristic
72	#168	C₄⋊C₈	1	2	yes	—
73	#168	C₂×C₁₆	1	2	yes	—
74	C₂×M₁₆	—	5	1	no	—
75	C₂²×C₈	—	5	1	no	—
76	#117	—	5	1	no	—
77	M₃₂	—	10	2	no	—
78	C₂×C₁₆	—	5	1	no	—
79	C₂×C₁₆	—	5	2	no	—
80	C₂×C₁₆	—	5	2	no	—
81	C₂×C₁₆	—	5	1	no	—
82	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
83	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
84	#97	—	5	1	no	—
85	#97	—	5	1	no	—
86	C₄×C₈	—	5	1	no	—
87	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
88	C₃₂	—	10	4	no	—
89	C₂²×C₅	C₂×M₃₂	1	1	yes	characteristic
90	C₂×D₅	—	4	1	no	—
91	C₂×D₅	C₄⋊C₁₆	1	2	yes	—
92	C₄×C₅	—	4	1	no	—
93	C₄×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
94	C₄×C₅	C₂²×C₁₆	1	1	yes	characteristic
95	Dic₅	—	4	1	no	—
96	Dic₅	C₄⋊C₁₆	1	2	yes	—
97	C₂²×C₄	—	5	1	no	—
98	M₁₆	—	5	4	no	—
99	C₂×C₈	—	5	1	no	—
100	C₂×C₈	—	5	1	no	—
101	C₂×C₈	—	5	2	no	—
102	C₂×C₈	—	5	2	no	—
103	C₂×C₈	D₅⋊C₈	1	1	yes	characteristic
104	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
105	C₄⋊C₄	—	10	1	no	—
106	C₂²⋊C₄	—	10	1	no	—
107	C₄²	—	5	1	no	—
108	C₄²	D₅⋊C₈	1	1	yes	characteristic
109	C₁₆	—	2	1	no	—
110	C₁₆	—	5	1	no	—
111	C₁₆	—	5	1	no	—
112	C₁₆	—	10	1	no	—
113	C₁₆	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
114	C₂×C₅	—	2	1	no	—
115	C₂×C₅	C₂×C₄⋊C₁₆	1	1	yes	characteristic
116	D₅	—	2	2	no	—
117	C₂³	—	5	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	5	1	no	—
120	C₂×C₄	—	5	2	no	—
121	C₂×C₄	—	5	2	no	—
122	C₂×C₄	—	10	1	no	—
123	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
124	C₈	—	5	2	no	—
125	C₈	—	5	2	no	—
126	C₈	D₅⋊C₁₆	1	2	yes	—
127	C₈	D₅⋊M₁₆	1	1	yes	characteristic
128	C₈	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
129	C₅	#13630	1	1	yes	characteristic, complemented, Sylow
130	C₂²	—	5	2	no	—
131	C₂²	—	20	1	no	—
132	C₂²	D₅⋊M₃₂	1	1	yes	characteristic
133	C₄	—	4	1	no	—
134	C₄	—	5	2	no	—
135	C₄	—	20	1	no	—
136	C₄	D₅⋊(C₄⋊C₈)	1	1	yes	characteristic
137	C₄	C₂×D₅⋊C₁₆	1	1	yes	characteristic
138	C₂	—	2	1	no	—
139	C₂	—	10	2	no	—
140	C₂	D₅⋊(C₄⋊C₁₆)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₅⋊M₆₄

C₂

yes

maximal

#10599865

C₂

yes

maximal, characteristic

#10598658

C₂

yes

maximal, complemented

#10598538

C₂

yes

maximal, complemented

C₂×#3930

C₄

yes

characteristic

C₂×D₅×C₁₆

C₂²

yes

characteristic

#64978

C₄

yes

characteristic

#64877

C₂²

yes

—

#64842

—

D₅⋊C₃₂

—

C₅×#346

C₄

yes

characteristic

#64722

C₄

yes

characteristic

C₁₆×Dic₅

C₂²

yes

characteristic

C₄×#672

C₂²

yes

—

C₈

yes

characteristic

C₂×C₈×D₅

C₂×C₄

yes

characteristic

#4018

C₈

yes

characteristic

C₂×C₅×C₁₆

C₂×C₄

yes

characteristic

C₂×C₅×C₁₆

C₂×C₄

yes

characteristic

C₄×C₅×C₈

C₂×C₄

yes

characteristic

#3947

C₈

yes

characteristic

C₈×Dic₅

C₂×C₄

yes

characteristic

C₂×#672

C₂×C₄

yes

characteristic

C₂×#672

C₂³

yes

characteristic

#3935

C₈

yes

characteristic

#3930

—

D₅×C₁₆

D₄

yes

—

D₅×C₁₆

Q₈

yes

—

#3928

—

C₂×M₆₄

—

#2108

—

#1422

—

#1422

—

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

#685

—

#683

—

C₄×Dic₅

C₂×C₈

yes

characteristic

C₂×#168

C₂×C₈

yes

characteristic

C₂×#168

C₂×C₈

yes

characteristic

#676

C₁₆

yes

—

C₈×D₅

C₄⋊C₄

yes

—

C₈×D₅

M₁₆

yes

—

C₅×C₁₆

—

C₅×C₁₆

C₄⋊C₄

yes

—

#672

—

#672

C₂×D₄

yes

characteristic

#672

C₂×Q₈

yes

characteristic

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

M₆₄

—

M₆₄

—

C₂×C₃₂

—

—

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⋊C₈

yes

—

C₄×D₅

C₂×C₁₆

yes

—

C₅×C₈

C₂×C₁₆

yes

—

C₅×C₈

C₂×C₄⋊C₄

yes

characteristic

C₅×C₈

C₂×M₁₆

yes

characteristic

#168

C₄⋊C₈

yes

—

#168

C₂×C₁₆

yes

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

C₂×C₁₆

yes

—

—

C₄×C₈

—

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

C₃₂

—

C₂²×C₅

C₂×M₃₂

yes

characteristic

C₂×D₅

—

C₂×D₅

C₄⋊C₁₆

yes

—

C₄×C₅

—

C₄×C₅

C₂×C₄⋊C₈

yes

characteristic

C₄×C₅

C₂²×C₁₆

yes

characteristic

Dic₅

—

Dic₅

C₄⋊C₁₆

yes

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

D₅⋊C₈

yes

characteristic

104

C₂×C₈

C₂²×AGL(1,𝔽₅)

yes

characteristic

105

C₄⋊C₄

—

106

C₂²⋊C₄

—

107

C₄²

—

108

C₄²

D₅⋊C₈

yes

characteristic

109

C₁₆

—

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

113

C₁₆

C₄⋊AGL(1,𝔽₅)

yes

—

114

C₂×C₅

—

115

C₂×C₅

C₂×C₄⋊C₁₆

yes

characteristic

116

D₅

—

117

C₂³

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

C₂×D₅⋊C₈

yes

characteristic

124

C₈

—

125

C₈

—

126

C₈

D₅⋊C₁₆

yes

—

127

C₈

D₅⋊M₁₆

yes

characteristic

128

C₈

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

129

C₅

#13630

yes

characteristic, complemented, Sylow

130

C₂²

—

131

C₂²

—

132

C₂²

D₅⋊M₃₂

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

D₅⋊(C₄⋊C₈)

yes

characteristic

137

C₄

C₂×D₅⋊C₁₆

yes

characteristic

138

C₂

—

139

C₂

—

140

C₂

D₅⋊(C₄⋊C₁₆)

yes

characteristic

	subgroup	G/H
center	C₈	C₂×C₄⋊AGL(1,𝔽₅)
commutator	C₄×C₅	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#346	C₄
socle	C₂×C₅	C₂×C₄⋊C₁₆
prime radical	C₂²×D₅	M₃₂
cosocle	C₂×#672	C₂³
hypercenter	#346	AGL(1,𝔽₅)
hypocenter	C₅	#13630
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ #346

subgroup

G/H

center

C₈

C₂×C₄⋊AGL(1,𝔽₅)

commutator

C₄×C₅

C₂²×C₁₆

Frattini

C₂×C₈

C₂²×AGL(1,𝔽₅)

Fitting

C₅×#346

C₄

socle

C₂×C₅

C₂×C₄⋊C₁₆

prime radical

C₂²×D₅

M₃₂

C₂³

C₅

G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ #346

N	H ≅ G/N	type
3. #10598658	C₂	semidirect
4. #10598538	C₂	semidirect
129. C₅	#13630	semidirect

H ≅ G/N

type

3. #10598658

C₂

semidirect

4. #10598538

C₂

semidirect

129. C₅

#13630

semidirect

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

subgroup	index
D₅⋊M₆₄	2
#10599865	2
#10598658	2
#10598538	2
#13630	5

subgroup

index

D₅⋊M₆₄

Subgroups — #13027800

Conjugacy classes of subgroups

Complements

Maximal subgroups