Group #13025381 — subgroups

Subgroups — #13025381

Order 1280.

198 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 62 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	#10604069	C₂	1	2	yes	maximal
2	#10600131	C₂	1	1	yes	maximal, characteristic
3	#10598641	C₂	1	1	yes	maximal, characteristic
4	#10598639	C₂	1	1	yes	maximal, characteristic
5	#10598608	C₂	1	2	yes	maximal
6	#65709	—	2	2	no	—
7	C₂×Dic₄₀	C₄	1	1	yes	characteristic
8	#65162	C₂²	1	2	yes	—
9	#64886	C₂²	1	1	yes	characteristic
10	#64881	C₂²	1	2	yes	—
11	#64879	C₂²	1	1	yes	characteristic
12	#64872	—	2	2	no	—
13	C₅×#367	C₄	1	1	yes	characteristic
14	#64722	C₂²	1	1	yes	characteristic
15	#64712	C₄	1	2	yes	—
16	#12088	—	5	1	no	maximal, Sylow
17	C₂×Dic₂₀	C₂×C₄	1	2	yes	—
18	#4033	—	2	2	no	—
19	#4002	—	4	2	no	—
20	C₄⋊#170	—	2	2	no	—
21	C₄⋊#170	—	4	2	no	—
22	C₄×#170	D₄	1	2	yes	—
23	C₂×#674	—	2	1	no	—
24	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₅×#107	C₂×C₄	1	2	yes	—
26	C₈×Dic₅	C₂³	1	1	yes	characteristic
27	C₂×#672	C₂×C₄	1	1	yes	characteristic
28	#3942	—	2	2	no	—
29	Dic₄₀	—	2	2	no	—
30	#2176	—	5	1	no	—
31	#1593	—	5	2	no	—
32	#1368	—	5	1	no	—
33	#1334	—	5	2	no	—
34	#1313	—	5	1	no	—
35	C₂×Dic₁₀	—	2	2	no	—
36	C₂×#170	—	2	2	no	—
37	C₂×#170	—	4	2	no	—
38	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
39	C₅×C₄⋊C₄	—	2	2	no	—
40	#683	—	4	2	no	—
41	C₄×Dic₅	—	4	2	no	—
42	C₄×Dic₅	C₂×D₄	1	1	yes	characteristic
43	C₂×#168	#41	1	1	yes	characteristic
44	Dic₂₀	—	2	2	no	—
45	Dic₂₀	—	2	2	no	—
46	#674	—	4	1	no	—
47	C₅×C₁₆	M₁₆	1	2	yes	—
48	#672	—	2	1	no	—
49	C₂×Q₃₂	—	5	1	no	—
50	#446	—	10	2	no	—
51	C₄×Q₁₆	—	5	2	no	—
52	#367	AGL(1,𝔽₅)	1	1	yes	characteristic
53	#358	—	5	2	no	—
54	#346	—	5	1	no	—
55	#346	—	5	1	no	—
56	#334	—	5	2	no	—
57	Q₈⋊C₈	—	10	2	no	—
58	#322	—	5	1	no	—
59	C₂×C₄×C₅	—	4	2	no	—
60	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
61	C₂×Dic₅	—	4	2	no	—
62	C₂×Dic₅	AGL(1,ℤ/8)	1	2	yes	—
63	Dic₁₀	—	4	2	no	—
64	Dic₁₀	—	4	2	no	—
65	#170	—	4	2	no	—
66	#170	—	8	2	no	—
67	C₅×C₈	C₂×M₁₆	1	1	yes	characteristic
68	C₅×C₈	#131	1	1	yes	characteristic
69	#168	—	2	1	no	—
70	C₂×Q₁₆	—	5	2	no	—
71	C₄×Q₈	—	10	2	no	—
72	Q₃₂	—	10	2	no	—
73	C₂×C₁₆	—	5	1	no	—
74	C₂×C₁₆	—	10	1	no	—
75	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
76	#107	C₂×AGL(1,𝔽₅)	1	2	yes	—
77	C₄⋊C₈	—	10	2	no	—
78	C₄⋊C₈	—	20	2	no	—
79	#103	—	10	2	no	—
80	#97	—	20	2	no	—
81	C₄×C₈	—	5	1	no	—
82	C₄×C₈	—	5	2	no	—
83	C₂²×C₅	#442	1	1	yes	characteristic
84	C₄×C₅	—	8	2	no	—
85	C₄×C₅	#436	1	1	yes	characteristic
86	C₄×C₅	#444	1	1	yes	characteristic
87	Dic₅	—	2	2	no	—
88	Dic₅	—	8	2	no	—
89	C₂×Q₈	—	10	2	no	—
90	Q₁₆	—	10	2	no	—
91	Q₁₆	—	10	2	no	—
92	C₂×C₈	—	5	1	no	—
93	C₂×C₈	—	10	2	no	—
94	C₂×C₈	—	20	2	no	—
95	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	20	2	no	—
98	C₄²	—	5	1	no	—
99	C₄²	—	20	2	no	—
100	C₁₆	—	10	1	no	—
101	C₁₆	—	20	1	no	—
102	C₁₆	#700	1	2	yes	—
103	C₂×C₅	#1594	1	1	yes	characteristic
104	C₂×C₅	#2180	1	2	yes	—
105	Q₈	—	20	2	no	—
106	Q₈	—	20	2	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	5	2	no	—
109	C₂×C₄	—	20	2	no	—
110	C₂×C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
111	C₈	—	10	1	no	—
112	C₈	—	20	2	no	—
113	C₈	—	40	2	no	—
114	C₈	C₂×#700	1	1	yes	characteristic
115	C₈	#4133	1	1	yes	characteristic
116	C₅	#12088	1	1	yes	characteristic, complemented, Sylow
117	C₂²	#65731	1	1	yes	characteristic
118	C₄	—	8	2	no	—
119	C₄	—	10	2	no	—
120	C₄	—	40	2	no	—
121	C₄	#65705	1	1	yes	characteristic
122	C₄	#65741	1	1	yes	characteristic
123	C₂	#10604070	1	1	yes	characteristic
124	C₂	#10604130	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10604069

C₂

yes

maximal

#10600131

C₂

yes

maximal, characteristic

#10598641

C₂

yes

maximal, characteristic

#10598639

C₂

yes

maximal, characteristic

#10598608

C₂

yes

maximal

#65709

—

C₂×Dic₄₀

C₄

yes

characteristic

#65162

C₂²

yes

—

#64886

C₂²

yes

characteristic

#64881

C₂²

yes

—

#64879

C₂²

yes

characteristic

#64872

—

C₅×#367

C₄

yes

characteristic

#64722

C₂²

yes

characteristic

#64712

C₄

yes

—

#12088

—

maximal, Sylow

C₂×Dic₂₀

C₂×C₄

yes

—

—

—

—

—

D₄

yes

—

C₂×#674

—

C₂×C₅×C₁₆

C₂×C₄

yes

characteristic

C₅×#107

C₂×C₄

yes

—

C₈×Dic₅

C₂³

yes

characteristic

C₂×#672

C₂×C₄

yes

characteristic

#3942

—

Dic₄₀

—

—

—

—

—

—

C₂×Dic₁₀

—

C₂×#170

—

C₂×#170

—

C₂×C₅×C₈

C₂²×C₄

yes

characteristic

C₅×C₄⋊C₄

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

C₂×D₄

yes

yes

Dic₂₀

—

Dic₂₀

—

#674

—

C₅×C₁₆

M₁₆

yes

—

#672

—

C₂×Q₃₂

—

#446

—

C₄×Q₁₆

—

yes

—

—

—

—

Q₈⋊C₈

—

#322

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₄×D₄

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

AGL(1,ℤ/8)

yes

—

Dic₁₀

—

Dic₁₀

—

#170

—

#170

—

C₅×C₈

C₂×M₁₆

yes

C₅×C₈

yes

—

C₂×Q₁₆

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

yes

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₅

#442

yes

characteristic

C₄×C₅

—

C₄×C₅

yes

C₄×C₅

yes

Dic₅

—

Dic₅

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×AGL(1,𝔽₅)

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

100

C₁₆

—

101

C₁₆

—

102

C₁₆

#700

yes

—

103

C₂×C₅

#1594

yes

characteristic

104

C₂×C₅

#2180

yes

—

105

Q₈

—

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

D₄×AGL(1,𝔽₅)

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

C₂×#700

yes

characteristic

115

C₈

#4133

yes

characteristic

116

C₅

#12088

yes

characteristic, complemented, Sylow

117

C₂²

#65731

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#65705

yes

characteristic

122

C₄

#65741

yes

characteristic

123

C₂

#10604070

yes

characteristic

124

C₂

#10604130

yes

—

	subgroup	G/H
center	C₂²	#65731
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#367	C₄
socle, prime radical	C₂²×C₅	#442
cosocle	C₈×Dic₅	C₂³
hypercenter	#367	AGL(1,𝔽₅)
hypocenter	C₅	#12088
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

subgroup

G/H

center

C₂²

#65731

commutator

C₂×C₅×C₈

C₂²×C₄

Frattini

C₂×C₈

C₂²×AGL(1,𝔽₅)

C₄

C₂²×C₅

C₈×Dic₅

C₂³

C₅

G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

N	H ≅ G/N	type
116. C₅	#12088	semidirect

H ≅ G/N

type

116. C₅

#12088

semidirect

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

subgroup	index
#10604069	2
#10600131	2
#10598641	2
#10598639	2
#10598608	2
#12088	5

subgroup

index

Subgroups — #13025381

Conjugacy classes of subgroups

Complements

Maximal subgroups