Group #13027845 — subgroups

Subgroups — #13027845

Order 1280.

186 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 92 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	#10604774	C₂	1	2	yes	maximal, complemented
2	#10601716	C₂	1	1	yes	maximal, characteristic
3	#10598798	C₂	1	2	yes	maximal, complemented
4	#10598538	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#3928	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#443	C₄	1	1	yes	characteristic
7	#65429	C₂²	1	2	yes	—
8	#64877	—	2	2	no	—
9	#64877	C₂²	1	2	yes	—
10	C₂×#3928	—	2	2	no	—
11	C₂×#3928	C₂²	1	1	yes	characteristic
12	C₂×#3928	C₂²	1	1	yes	characteristic
13	#64781	C₄	1	2	yes	—
14	#64683	C₄	1	1	yes	characteristic
15	C₄×#672	C₂²	1	1	yes	characteristic
16	#13647	—	5	1	no	maximal, Sylow
17	C₅×#135	C₈	1	1	yes	characteristic
18	C₅×#131	C₂×C₄	1	2	yes	—
19	C₅×#108	C₈	1	1	yes	characteristic
20	C₅×#104	C₈	1	2	yes	—
21	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
22	#3944	—	2	2	no	—
23	#3944	C₂×C₄	1	2	yes	—
24	C₂×#672	—	2	2	no	—
25	C₂×#672	C₂×C₄	1	1	yes	characteristic
26	C₂×#672	C₂³	1	1	yes	characteristic
27	#3928	—	2	2	no	—
28	#3928	—	2	1	no	—
29	#3928	D₄	1	1	yes	characteristic
30	#3928	D₄	1	1	yes	characteristic
31	#2258	—	5	2	no	—
32	#2170	—	5	1	no	—
33	#1422	—	5	1	no	—
34	#1401	—	5	2	no	—
35	C₄×C₃₂	—	5	1	no	—
36	C₅×#41	C₂×C₈	1	2	yes	—
37	C₅×Q₁₆	C₁₆	1	1	yes	characteristic
38	C₅×SD₁₆	C₁₆	1	2	yes	—
39	C₅×D₈	C₁₆	1	1	yes	characteristic
40	C₅×M₁₆	—	2	2	no	—
41	C₅×M₁₆	C₂×C₈	1	2	yes	—
42	C₂×C₅×C₈	—	2	2	no	—
43	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
44	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
45	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
46	#672	—	2	2	no	—
47	#672	—	2	1	no	—
48	#672	C₂×D₄	1	1	yes	characteristic
49	#672	#41	1	1	yes	characteristic
50	#504	—	5	2	no	—
51	#443	AGL(1,𝔽₅)	1	1	yes	characteristic
52	M₆₄	—	5	2	no	—
53	M₆₄	—	10	2	no	—
54	C₂×C₃₂	—	5	1	no	—
55	C₂×C₃₂	—	5	1	no	—
56	C₂×C₃₂	—	10	2	no	—
57	#364	—	5	1	no	—
58	#350	—	5	2	no	—
59	C₄×C₁₆	—	5	1	no	—
60	C₅×Q₈	C₂×C₁₆	1	2	yes	—
61	C₅×D₄	—	2	2	no	—
62	C₅×D₄	C₂×C₁₆	1	2	yes	—
63	C₂×C₄×C₅	—	2	1	no	—
64	C₂×C₄×C₅	—	2	2	no	—
65	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
66	C₅×C₈	—	2	2	no	—
67	C₅×C₈	C₂×C₁₆	1	1	yes	characteristic
68	C₅×C₈	C₂×C₁₆	1	1	yes	characteristic
69	C₅×C₈	C₄×D₄	1	1	yes	characteristic
70	C₅×C₈	#131	1	1	yes	characteristic
71	#135	#170	1	1	yes	characteristic
72	#131	C₂×AGL(1,𝔽₅)	1	2	yes	—
73	M₃₂	—	5	2	no	—
74	M₃₂	—	10	2	no	—
75	C₂×C₁₆	—	5	1	no	—
76	C₂×C₁₆	—	5	1	no	—
77	C₂×C₁₆	—	10	2	no	—
78	#108	#170	1	1	yes	characteristic
79	#104	#170	1	2	yes	—
80	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
81	C₃₂	—	5	1	no	—
82	C₃₂	—	5	1	no	—
83	C₃₂	—	10	1	no	—
84	C₃₂	—	10	2	no	—
85	C₂²×C₅	—	2	2	no	—
86	C₂²×C₅	#504	1	1	yes	characteristic
87	C₄×C₅	—	2	2	no	—
88	C₄×C₅	—	2	2	no	—
89	C₄×C₅	C₈×D₄	1	1	yes	characteristic
90	C₄×C₅	C₂²×C₁₆	1	1	yes	characteristic
91	#41	C₂×#170	1	2	yes	—
92	Q₁₆	#674	1	1	yes	characteristic
93	SD₁₆	#674	1	2	yes	—
94	D₈	#674	1	1	yes	characteristic
95	M₁₆	—	2	2	no	—
96	M₁₆	C₂×#170	1	2	yes	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	C₂×#170	1	1	yes	characteristic
99	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
100	C₄²	C₂×#170	1	1	yes	characteristic
101	C₁₆	—	5	1	no	—
102	C₁₆	—	5	1	no	—
103	C₁₆	—	10	1	no	—
104	C₁₆	—	10	2	no	—
105	C₂×C₅	—	2	1	no	—
106	C₂×C₅	—	4	2	no	—
107	C₂×C₅	D₄×C₁₆	1	1	yes	characteristic
108	Q₈	C₂×#674	1	2	yes	—
109	D₄	—	2	2	no	—
110	D₄	C₂×#674	1	2	yes	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	C₂²×#170	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	C₂×#674	1	1	yes	characteristic
116	C₈	C₂×#674	1	1	yes	characteristic
117	C₈	#4131	1	1	yes	characteristic
118	C₈	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
119	C₅	#13647	1	1	yes	characteristic, complemented, Sylow
120	C₂²	—	2	2	no	—
121	C₂²	#65791	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	C₂²×#674	1	1	yes	characteristic
125	C₄	D₄×#170	1	1	yes	characteristic
126	C₂	—	2	1	no	—
127	C₂	—	4	2	no	—
128	C₂	D₄×#674	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10604774

C₂

yes

maximal, complemented

#10601716

C₂

yes

maximal, characteristic

#10598798

C₂

yes

maximal, complemented

#10598538

C₂

yes

maximal, characteristic, complemented

C₄×#3928

C₂

yes

maximal, characteristic, complemented

C₅×#443

C₄

yes

characteristic

#65429

C₂²

yes

—

#64877

—

#64877

C₂²

yes

—

C₂×#3928

—

C₂×#3928

C₂²

yes

characteristic

C₂×#3928

C₂²

yes

characteristic

#64781

C₄

yes

—

#64683

C₄

yes

characteristic

C₄×#672

C₂²

yes

—

C₈

yes

characteristic

C₅×#131

C₂×C₄

yes

—

C₅×#108

C₈

yes

characteristic

C₅×#104

C₈

yes

—

C₄×C₅×C₈

C₂×C₄

yes

characteristic

#3944

—

#3944

C₂×C₄

yes

—

C₂×#672

—

C₂×#672

C₂×C₄

yes

characteristic

C₂×#672

C₂³

yes

—

—

D₄

yes

characteristic

#3928

D₄

yes

—

—

—

—

C₄×C₃₂

—

C₅×#41

C₂×C₈

yes

—

C₅×Q₁₆

C₁₆

yes

characteristic

C₅×SD₁₆

C₁₆

yes

—

C₅×D₈

C₁₆

yes

characteristic

C₅×M₁₆

—

C₅×M₁₆

C₂×C₈

yes

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂×C₈

yes

characteristic

C₂×C₅×C₈

C₂²×C₄

yes

characteristic

C₄²×C₅

C₂×C₈

yes

—

—

C₂×D₄

yes

yes

—

yes

M₆₄

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

#364

—

#350

—

C₄×C₁₆

—

C₅×Q₈

C₂×C₁₆

yes

—

C₅×D₄

—

C₅×D₄

C₂×C₁₆

yes

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×C₈

yes

characteristic

C₅×C₈

—

C₅×C₈

C₂×C₁₆

yes

characteristic

C₅×C₈

C₂×C₁₆

yes

characteristic

C₅×C₈

C₄×D₄

yes

C₅×C₈

yes

yes

yes

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

yes

yes

—

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

C₃₂

—

C₃₂

—

C₃₂

—

C₃₂

—

C₂²×C₅

—

C₂²×C₅

#504

yes

characteristic

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₈×D₄

yes

characteristic

C₄×C₅

C₂²×C₁₆

yes

characteristic

#41

C₂×#170

yes

—

Q₁₆

#674

yes

characteristic

SD₁₆

#674

yes

—

D₈

#674

yes

characteristic

M₁₆

—

M₁₆

C₂×#170

yes

—

C₂×C₈

—

C₂×C₈

C₂×#170

yes

characteristic

C₂×C₈

C₂²×AGL(1,𝔽₅)

yes

characteristic

100

C₄²

C₂×#170

yes

characteristic

101

C₁₆

—

102

C₁₆

—

103

C₁₆

—

104

C₁₆

—

105

C₂×C₅

—

106

C₂×C₅

—

107

C₂×C₅

D₄×C₁₆

yes

characteristic

108

Q₈

C₂×#674

yes

—

109

D₄

—

110

D₄

C₂×#674

yes

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²×#170

yes

characteristic

114

C₈

—

115

C₈

C₂×#674

yes

characteristic

116

C₈

C₂×#674

yes

characteristic

117

C₈

#4131

yes

characteristic

118

C₈

D₄×AGL(1,𝔽₅)

yes

characteristic

119

C₅

#13647

yes

characteristic, complemented, Sylow

120

C₂²

—

121

C₂²

#65791

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

C₂²×#674

yes

characteristic

125

C₄

D₄×#170

yes

characteristic

126

C₂

—

127

C₂

—

128

C₂

D₄×#674

yes

characteristic

	subgroup	G/H
center	C₈	D₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#443	C₄
socle	C₂×C₅	D₄×C₁₆
prime radical	C₅×#135	C₈
cosocle	C₂×#672	C₂³
hypercenter	#443	AGL(1,𝔽₅)
hypocenter	C₅	#13647
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #443

subgroup

G/H

center

C₈

D₄×AGL(1,𝔽₅)

commutator

C₄×C₅

C₂²×C₁₆

Frattini

C₂×C₈

C₂²×AGL(1,𝔽₅)

Fitting

C₅×#443

C₄

socle

C₂×C₅

D₄×C₁₆

C₈

C₂³

C₅

G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ #443

N	H ≅ G/N	type
1. #10604774	C₂	semidirect
3. #10598798	C₂	semidirect
4. #10598538	C₂	semidirect
5. C₄×#3928	C₂	semidirect
119. C₅	#13647	semidirect

H ≅ G/N

type

1. #10604774

C₂

semidirect

3. #10598798

C₂

semidirect

4. #10598538

C₂

semidirect

5. C₄×#3928

C₂

semidirect

119. C₅

#13647

semidirect

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

subgroup	index
#10604774	2
#10601716	2
#10598798	2
#10598538	2
C₄×#3928	2
#13647	5

subgroup

index

Subgroups — #13027845

Conjugacy classes of subgroups

Complements

Maximal subgroups