Group #13018817 — subgroups

Subgroups — #13018817

Order 1280.

354 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 144 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	#10603859	C₂	1	2	yes	maximal
2	#10603818	C₂	1	4	yes	maximal
3	C₈²×D₅	C₂	1	1	yes	maximal, characteristic
4	C₂×#3996	—	2	4	no	—
5	#65681	C₂²	1	4	yes	—
6	C₄×C₈×D₅	C₂²	1	2	yes	—
7	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
8	#64882	—	2	8	no	—
9	C₅×C₈²	C₄	1	1	yes	characteristic
10	C₈×#168	C₄	1	1	yes	characteristic
11	C₈×#168	C₄	1	2	yes	—
12	#8823	—	5	1	no	maximal, Sylow
13	C₂×#700	—	2	4	no	—
14	C₂×C₈×D₅	D₄	1	2	yes	—
15	C₂×C₈×D₅	Q₈	1	4	yes	—
16	C₄²×D₅	C₂³	1	1	yes	characteristic
17	C₄⋊#170	—	2	8	no	—
18	#3996	—	4	8	no	—
19	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
20	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
21	C₈×Dic₅	D₄	1	4	yes	—
22	C₈×Dic₅	Q₈	1	2	yes	—
23	C₄×#168	C₂×C₄	1	2	yes	—
24	C₄×#168	C₂×C₄	1	1	yes	characteristic
25	#1848	—	5	2	no	—
26	#1567	—	5	4	no	—
27	C₂×C₈²	—	5	1	no	—
28	C₂×C₄×D₅	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×D₅	C₂×Q₈	1	2	yes	—
30	C₂×#170	—	4	4	no	—
31	#700	—	4	8	no	—
32	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
33	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
34	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
35	C₄×Dic₅	C₂×D₄	1	2	yes	—
36	C₄×Dic₅	C₂×Q₈	1	1	yes	characteristic
37	C₂×#168	C₄⋊C₄	1	4	yes	—
38	C₂×#168	C₄⋊C₄	1	2	yes	—
39	C₈×D₅	—	2	8	no	—
40	C₈×D₅	D₈	1	4	yes	—
41	C₈×D₅	Q₁₆	1	4	yes	—
42	C₂×#108	—	10	4	no	—
43	#423	—	5	4	no	—
44	C₂×C₄×C₈	—	5	2	no	—
45	C₂×C₄×C₈	—	5	1	no	—
46	#335	—	10	8	no	—
47	C₈²	—	5	2	no	—
48	C₈²	—	5	1	no	—
49	C₈²	AGL(1,𝔽₅)	1	1	yes	characteristic
50	C₂²×D₅	C₄⋊Q₈	1	1	yes	characteristic
51	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
53	C₂×Dic₅	#127	1	1	yes	characteristic
54	C₂×Dic₅	C₄⋊Q₈	1	2	yes	—
55	C₄×D₅	—	2	4	no	—
56	C₄×D₅	C₂×D₈	1	2	yes	—
57	C₄×D₅	C₂×Q₁₆	1	2	yes	—
58	#170	—	8	4	no	—
59	C₅×C₈	#107	1	4	yes	—
60	C₅×C₈	#108	1	8	yes	complemented
61	#168	—	2	4	no	—
62	#168	#107	1	4	yes	—
63	C₂×M₁₆	—	10	4	no	—
64	C₂²×C₈	—	5	2	no	—
65	C₂²×C₈	—	5	4	no	—
66	C₂×C₄²	—	5	1	no	—
67	#108	—	20	8	no	—
68	C₄⋊C₈	—	10	8	no	—
69	C₄×C₈	—	5	4	no	—
70	C₄×C₈	—	5	2	no	—
71	C₄×C₈	—	5	1	no	—
72	C₄×C₈	—	5	2	no	—
73	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
74	C₄×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
75	C₂²×C₅	#384	1	1	yes	characteristic
76	C₂×D₅	—	2	2	no	—
77	C₂×D₅	#500	1	2	yes	—
78	C₄×C₅	C₂×#107	1	2	yes	—
79	C₄×C₅	C₂×#108	1	4	yes	—
80	Dic₅	—	2	2	no	—
81	Dic₅	#493	1	1	yes	characteristic
82	Dic₅	#494	1	1	yes	characteristic
83	C₂²×C₄	—	5	1	no	—
84	C₂²×C₄	—	5	2	no	—
85	M₁₆	—	20	8	no	—
86	C₂×C₈	—	5	2	no	—
87	C₂×C₈	—	5	4	no	—
88	C₂×C₈	—	5	4	no	—
89	C₂×C₈	—	5	4	no	—
90	C₂×C₈	—	10	8	no	—
91	C₂×C₈	—	20	4	no	—
92	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
93	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	4	yes	—
94	C₄²	—	5	2	no	—
95	C₄²	—	5	1	no	—
96	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
97	C₂×C₅	#1845	1	1	yes	characteristic
98	C₂×C₅	#1848	1	2	yes	—
99	D₅	—	2	2	no	—
100	C₂³	—	5	1	no	—
101	C₂×C₄	—	5	1	no	—
102	C₂×C₄	—	5	2	no	—
103	C₂×C₄	—	5	2	no	—
104	C₂×C₄	—	5	2	no	—
105	C₂×C₄	—	10	4	no	—
106	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
107	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	2	yes	—
108	C₈	—	5	4	no	—
109	C₈	—	10	4	no	—
110	C₈	—	40	4	no	—
111	C₈	#3994	1	4	yes	—
112	C₈	#3996	1	8	yes	complemented
113	C₅	#8823	1	1	yes	characteristic, complemented, Sylow
114	C₂²	—	5	2	no	—
115	C₂²	—	10	2	no	—
116	C₂²	#65688	1	1	yes	characteristic
117	C₄	—	5	1	no	—
118	C₄	—	5	1	no	—
119	C₄	—	10	2	no	—
120	C₄	C₂×#3994	1	2	yes	—
121	C₄	C₂×#3996	1	4	yes	—
122	C₂	—	10	2	no	—
123	C₂	#10603848	1	1	yes	characteristic
124	C₂	#10603859	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10603859

C₂

yes

maximal

#10603818

C₂

yes

maximal

C₈²×D₅

C₂

yes

maximal, characteristic

C₂×#3996

—

#65681

C₂²

yes

—

C₄×C₈×D₅

C₂²

yes

—

C₄×C₈×D₅

C₂²

yes

characteristic

#64882

—

C₅×C₈²

C₄

yes

characteristic

C₈×#168

C₄

yes

characteristic

C₈×#168

C₄

yes

—

#8823

—

maximal, Sylow

C₂×#700

—

C₂×C₈×D₅

D₄

yes

—

C₂×C₈×D₅

Q₈

yes

—

C₄²×D₅

C₂³

yes

characteristic

C₄⋊#170

—

#3996

—

C₄×C₅×C₈

C₂×C₄

yes

characteristic

C₄×C₅×C₈

C₂×C₄

yes

—

C₈×Dic₅

D₄

yes

—

C₈×Dic₅

Q₈

yes

—

C₄×#168

C₂×C₄

yes

—

C₄×#168

C₂×C₄

yes

characteristic

#1848

—

#1567

—

C₂×C₈²

—

C₂×C₄×D₅

C₂×D₄

yes

characteristic

C₂×C₄×D₅

C₂×Q₈

yes

—

C₂×#170

—

#700

—

C₂×C₅×C₈

C₄⋊C₄

yes

—

C₂×C₅×C₈

C₄⋊C₄

yes

—

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄×Dic₅

C₂×D₄

yes

—

C₄×Dic₅

C₂×Q₈

yes

characteristic

C₂×#168

C₄⋊C₄

yes

—

C₂×#168

C₄⋊C₄

yes

—

C₈×D₅

—

C₈×D₅

D₈

yes

—

C₈×D₅

Q₁₆

yes

—

C₂×#108

—

#423

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#335

—

C₈²

—

C₈²

—

C₈²

AGL(1,𝔽₅)

yes

characteristic

C₂²×D₅

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

—

C₂×Dic₅

#127

yes

characteristic

C₂×Dic₅

C₄⋊Q₈

yes

—

C₄×D₅

—

C₄×D₅

C₂×D₈

yes

—

C₄×D₅

C₂×Q₁₆

yes

—

#170

—

C₅×C₈

#107

yes

—

C₅×C₈

yes

—

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

#108

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

—

C₂²×C₅

#384

yes

characteristic

C₂×D₅

—

C₂×D₅

#500

yes

—

C₄×C₅

C₂×#107

yes

—

C₄×C₅

C₂×#108

yes

—

Dic₅

—

Dic₅

yes

Dic₅

yes

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊AGL(1,𝔽₅)

yes

—

C₂×C₈

C₄⋊AGL(1,𝔽₅)

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×AGL(1,𝔽₅)

yes

C₂×C₅

yes

C₂×C₅

yes

—

D₅

—

100

C₂³

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

107

C₂×C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

#3994

yes

—

112

C₈

#3996

yes

complemented

113

C₅

#8823

yes

characteristic, complemented, Sylow

114

C₂²

—

115

C₂²

—

116

C₂²

#65688

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

C₂×#3994

yes

—

121

C₄

C₂×#3996

yes

—

122

C₂

—

123

C₂

#10603848

yes

characteristic

124

C₂

#10603859

yes

—

	subgroup	G/H
center	C₂²	#65688
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₄²	C₂²×AGL(1,𝔽₅)
Fitting	C₅×C₈²	C₄
socle	C₂²×C₅	#384
prime radical	C₂²×D₅	C₄⋊Q₈
cosocle	C₄²×D₅	C₂³
hypercenter	C₈²	AGL(1,𝔽₅)
hypocenter	C₅	#8823
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

subgroup

G/H

center

C₂²

#65688

commutator

C₄²×C₅

C₂²×C₄

Frattini

C₄²

C₂²×AGL(1,𝔽₅)

Fitting

C₅×C₈²

C₄

socle

C₂²×C₅

#384

prime radical

C₂²×D₅

C₄⋊Q₈

cosocle

C₄²×D₅

C₂³

C₈²

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
60. C₅×C₈	#108	semidirect
112. C₈	#3996	semidirect
113. C₅	#8823	semidirect

H ≅ G/N

type

60. C₅×C₈

#108

semidirect

112. C₈

#3996

semidirect

113. C₅

#8823

semidirect

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

subgroup	index
#10603859	2
#10603818	2
C₈²×D₅	2
#8823	5

subgroup

index

#10603859

#10603818

C₈²×D₅

#8823

Subgroups — #13018817

Conjugacy classes of subgroups

Complements

Maximal subgroups