Group #13103333 — subgroups

Subgroups — #13103333

Order 1600.

167 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 167 conjugacy classes of subgroups — too many to draw. Only its 32 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	C₄⋊#6057	C₂	1	1	yes	maximal, characteristic
2	#11717355	C₂	1	1	yes	maximal, characteristic
3	#11716790	C₂	1	1	yes	maximal, characteristic
4	#11716788	C₂	1	2	yes	maximal
5	#11716771	C₂	1	2	yes	maximal
6	C₂×#6057	—	2	1	no	—
7	#87872	C₂²	1	1	yes	characteristic
8	C₄⋊#1044	C₂²	1	2	yes	—
9	C₄×#1044	C₂²	1	1	yes	characteristic
10	C₄⋊#1043	C₂²	1	1	yes	characteristic
11	#87840	—	2	1	no	—
12	#87832	—	2	1	no	—
13	#87830	—	2	1	no	—
14	#87767	C₂²	1	2	yes	—
15	C₂×#1044	—	2	2	no	—
16	C₂×#1044	D₄	1	1	yes	characteristic
17	C₂×#1044	D₄	1	1	yes	characteristic
18	C₂×#1043	—	2	1	no	—
19	#6057	—	4	2	no	—
20	C₄×C₅⋊D₅	C₂³	1	1	yes	characteristic
21	C₅×Dic₁₀	—	2	1	no	—
22	#6039	Q₈	1	2	yes	complemented
23	#6036	—	2	1	no	—
24	C₅×#168	—	2	2	no	—
25	Q₈×AGL(1,𝔽₅)	—	10	1	no	—
26	#3993	—	10	2	no	—
27	C₂×C₅⋊D₅	C₂×D₄	1	1	yes	characteristic
28	#1044	—	2	1	no	—
29	#1044	—	4	2	no	—
30	#1044	SD₁₆	1	2	yes	—
31	#1043	—	4	1	no	—
32	C₄×C₅²	#41	1	1	yes	characteristic
33	#1039	C₂×Q₈	1	1	yes	characteristic
34	C₅×Dic₅	—	2	2	no	—
35	Q₈×D₅	—	10	1	no	—
36	C₄⋊AGL(1,𝔽₅)	—	10	1	no	—
37	C₄⋊AGL(1,𝔽₅)	—	10	2	no	—
38	C₄⋊AGL(1,𝔽₅)	—	10	2	no	—
39	C₄⋊AGL(1,𝔽₅)	—	10	2	no	—
40	C₄×AGL(1,𝔽₅)	—	10	1	no	—
41	C₄×AGL(1,𝔽₅)	—	10	2	no	—
42	C₈×D₅	—	10	2	no	—
43	#475	—	25	1	no	maximal, Sylow
44	C₂×C₅²	C₂²⋊Q₈	1	1	yes	characteristic
45	C₅⋊D₅	C₂×SD₁₆	1	1	yes	characteristic
46	C₅⋊D₅	#137	1	1	yes	characteristic
47	C₂×AGL(1,𝔽₅)	—	10	1	no	—
48	C₂×AGL(1,𝔽₅)	—	10	1	no	—
49	C₂×AGL(1,𝔽₅)	—	10	2	no	—
50	C₂×AGL(1,𝔽₅)	—	20	2	no	—
51	C₂×AGL(1,𝔽₅)	—	20	2	no	—
52	C₅×Q₈	—	10	1	no	—
53	C₄×D₅	—	10	2	no	—
54	C₄×D₅	—	10	1	no	—
55	C₄×D₅	—	10	2	no	—
56	Dic₁₀	—	2	1	no	—
57	Dic₁₀	—	10	2	no	—
58	C₅×C₈	—	10	2	no	—
59	#168	—	2	2	no	—
60	C₄⋊Q₈	—	25	1	no	—
61	C₄×Q₈	—	25	1	no	—
62	#106	—	25	2	no	—
63	C₄⋊C₈	—	25	1	no	—
64	#103	—	25	2	no	—
65	C₅²	#475	1	1	yes	characteristic, complemented, Sylow
66	C₂×D₅	—	10	1	no	—
67	C₂×D₅	—	10	2	no	—
68	AGL(1,𝔽₅)	—	10	2	no	—
69	AGL(1,𝔽₅)	—	20	1	no	—
70	AGL(1,𝔽₅)	—	20	2	no	—
71	AGL(1,𝔽₅)	—	40	2	no	—
72	AGL(1,𝔽₅)	—	40	2	no	—
73	C₄×C₅	—	2	2	no	—
74	C₄×C₅	—	2	1	no	—
75	C₄×C₅	—	10	2	no	—
76	Dic₅	—	2	2	no	—
77	Dic₅	—	10	1	no	—
78	Dic₅	—	10	2	no	—
79	C₂×Q₈	—	25	1	no	—
80	C₂×Q₈	—	50	1	no	—
81	C₂×C₈	—	25	2	no	—
82	C₄⋊C₄	—	25	2	no	—
83	C₄⋊C₄	—	25	1	no	—
84	C₄⋊C₄	—	50	1	no	—
85	C₄⋊C₄	—	50	1	no	—
86	C₄²	—	25	1	no	—
87	C₄²	—	50	1	no	—
88	C₂×C₅	—	2	1	no	—
89	C₂×C₅	—	2	2	no	—
90	D₅	—	10	1	no	—
91	D₅	—	10	2	no	—
92	D₅	—	10	1	no	—
93	D₅	—	10	2	no	—
94	Q₈	—	10	1	no	—
95	Q₈	—	25	2	no	—
96	Q₈	—	100	2	no	—
97	C₂×C₄	—	25	1	no	—
98	C₂×C₄	—	25	1	no	—
99	C₂×C₄	—	25	1	no	—
100	C₂×C₄	—	50	1	no	—
101	C₂×C₄	—	50	1	no	—
102	C₂×C₄	—	50	2	no	—
103	C₈	—	10	2	no	—
104	C₅	—	2	1	no	—
105	C₅	—	2	2	no	—
106	C₂²	—	25	1	no	—
107	C₄	—	10	2	no	—
108	C₄	—	25	2	no	—
109	C₄	—	25	1	no	—
110	C₄	—	50	1	no	—
111	C₄	—	100	2	no	—
112	C₄	—	100	1	no	—
113	C₄	(D₅⋊AGL(1,𝔽₅))⋊C₂	1	1	yes	characteristic
114	C₂	—	25	1	no	—
115	C₂	—	25	1	no	—
116	C₂	#11717310	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊#6057

C₂

yes

maximal, characteristic

#11717355

C₂

yes

maximal, characteristic

#11716790

C₂

yes

maximal, characteristic

#11716788

C₂

yes

maximal

#11716771

C₂

yes

maximal

C₂×#6057

—

#87872

C₂²

yes

characteristic

C₄⋊#1044

C₂²

yes

—

C₄×#1044

C₂²

yes

characteristic

C₄⋊#1043

C₂²

yes

—

—

—

C₂²

yes

—

C₂×#1044

—

C₂×#1044

D₄

yes

characteristic

C₂×#1044

D₄

yes

characteristic

C₂×#1043

—

#6057

—

C₄×C₅⋊D₅

C₂³

yes

characteristic

C₅×Dic₁₀

—

#6039

Q₈

yes

—

—

—

—

C₂×C₅⋊D₅

C₂×D₄

yes

—

—

SD₁₆

yes

—

#1043

—

C₄×C₅²

#41

yes

characteristic

#1039

C₂×Q₈

yes

characteristic

C₅×Dic₅

—

Q₈×D₅

—

—

—

—

—

—

—

C₈×D₅

—

#475

—

maximal, Sylow

C₂×C₅²

C₂²⋊Q₈

yes

characteristic

C₅⋊D₅

C₂×SD₁₆

yes

C₅⋊D₅

yes

—

—

—

—

—

C₅×Q₈

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

#168

—

C₄⋊Q₈

—

C₄×Q₈

—

#106

—

C₄⋊C₈

—

#103

—

C₅²

#475

yes

characteristic, complemented, Sylow

C₂×D₅

—

C₂×D₅

—

—

—

—

—

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

Dic₅

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₂×C₅

—

C₂×C₅

—

D₅

—

D₅

—

D₅

—

D₅

—

Q₈

—

Q₈

—

Q₈

—

100

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₈

—

104

C₅

—

105

C₅

—

106

C₂²

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

100

—

112

C₄

—

100

—

113

C₄

(D₅⋊AGL(1,𝔽₅))⋊C₂

yes

characteristic

114

C₂

—

115

C₂

—

116

C₂

#11717310

yes

characteristic

	subgroup	G/H
center	C₂	#11717310
commutator, cosocle	C₄×C₅⋊D₅	C₂³
Frattini, hypercenter	C₄	(D₅⋊AGL(1,𝔽₅))⋊C₂
Fitting	C₄×C₅²	#41
socle	C₂×C₅²	C₂²⋊Q₈
prime radical	C₂×C₅⋊D₅	C₂×D₄
hypocenter	C₅²	#475
LCS	G ▷ C₄×C₅⋊D₅ ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₄

subgroup

G/H

center

C₂

#11717310

commutator, cosocle

C₄×C₅⋊D₅

C₂³

Frattini, hypercenter

C₄

C₄×C₅²

C₂×C₅²

C₂²⋊Q₈

prime radical

C₂×C₅⋊D₅

C₂×D₄

hypocenter

C₅²

#475

LCS

G ▷ C₄×C₅⋊D₅ ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂ ◁ C₄

N	H ≅ G/N	type
22. #6039	Q₈	semidirect
65. C₅²	#475	semidirect

H ≅ G/N

type

22. #6039

Q₈

semidirect

65. C₅²

#475

semidirect

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

subgroup	index
C₄⋊#6057	2
#11717355	2
#11716790	2
#11716788	2
#11716771	2
#475	25

subgroup

index

Subgroups — #13103333

Conjugacy classes of subgroups

Complements

Maximal subgroups