Group #13102343 — subgroups

Subgroups — C₅×#65371 #13102343

Order 1600.

248 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 78 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	C₅²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#4016	C₂	1	1	yes	maximal, characteristic
3	C₅×#3967	C₂	1	4	yes	maximal
4	C₅×#3935	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅²×Q₈	C₂²	1	2	yes	—
6	C₅²×C₄⋊C₄	—	2	2	no	—
7	C₄²×C₅²	C₂²	1	1	yes	characteristic
8	C₅×C₄⋊Dic₅	C₂²	1	2	yes	—
9	C₅×#683	—	2	2	no	—
10	C₂×C₅×#168	C₂²	1	2	yes	—
11	C₅×#491	—	5	1	no	maximal
12	#65371	C₅	1	1	yes	maximal, characteristic, direct product factor
13	C₅²×Q₈	—	2	4	no	—
14	C₂×C₄×C₅²	—	2	2	no	—
15	C₂×C₄×C₅²	D₄	1	2	yes	—
16	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
17	C₂×C₅×Dic₅	—	2	2	no	—
18	C₅×#168	—	2	2	no	—
19	C₅×C₄⋊Q₈	—	2	2	no	—
20	C₅×C₄⋊Q₈	D₅	1	1	yes	characteristic
21	C₅×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
22	C₅×#125	—	5	1	no	—
23	#4016	C₂×C₅	1	1	yes	characteristic
24	C₅×#103	—	5	4	no	—
25	C₅×#97	—	5	1	no	—
26	#3967	C₂×C₅	1	4	yes	—
27	#3935	C₂×C₅	1	1	yes	characteristic
28	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
29	C₄×C₅²	—	2	2	no	—
30	C₄×C₅²	—	4	2	no	—
31	C₄×C₅²	#41	1	2	yes	—
32	C₅×Dic₅	—	4	2	no	—
33	C₂×C₅×Q₈	—	2	4	no	—
34	C₂×C₅×Q₈	C₂×D₅	1	2	yes	—
35	C₂×C₅×Q₈	C₂²×C₅	1	2	yes	—
36	C₂×C₅×C₈	—	5	2	no	—
37	C₅×C₄⋊C₄	—	2	8	no	—
38	C₅×C₄⋊C₄	—	2	2	no	—
39	C₅×C₄⋊C₄	—	2	2	no	—
40	C₅×C₄⋊C₄	—	5	2	no	—
41	C₅×C₄⋊C₄	—	10	2	no	—
42	C₄²×C₅	—	2	2	no	—
43	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
44	C₄²×C₅	C₂²×C₅	1	1	yes	characteristic
45	C₄⋊Dic₅	C₂²×C₅	1	2	yes	—
46	#683	—	2	2	no	—
47	C₂×#168	C₂²×C₅	1	2	yes	—
48	#491	—	5	1	no	Sylow
49	C₂×C₅²	#124	1	1	yes	characteristic
50	C₂×C₅²	#137	1	2	yes	—
51	C₅×Q₈	—	2	4	no	—
52	C₅×Q₈	—	2	4	no	—
53	C₅×Q₈	—	4	8	no	—
54	C₂×C₄×C₅	—	2	8	no	—
55	C₂×C₄×C₅	—	2	2	no	—
56	C₂×C₄×C₅	—	2	2	no	—
57	C₂×C₄×C₅	—	2	4	no	—
58	C₂×C₄×C₅	—	2	2	no	—
59	C₂×C₄×C₅	—	10	2	no	—
60	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
61	C₂×C₄×C₅	C₅×D₄	1	2	yes	—
62	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
63	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
64	C₂×Dic₅	—	2	2	no	—
65	C₅×C₈	—	10	2	no	—
66	#168	—	2	2	no	—
67	C₄⋊Q₈	C₅×D₅	1	1	yes	characteristic
68	#125	—	5	1	no	—
69	#103	—	5	4	no	—
70	#97	—	5	1	no	—
71	C₅²	#491	1	1	yes	characteristic, complemented, Sylow
72	C₂²×C₅	—	2	2	no	—
73	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
74	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
75	C₄×C₅	—	2	8	no	—
76	C₄×C₅	—	2	2	no	—
77	C₄×C₅	—	2	2	no	—
78	C₄×C₅	—	2	4	no	—
79	C₄×C₅	—	4	8	no	—
80	C₄×C₅	—	4	2	no	—
81	C₄×C₅	—	4	2	no	—
82	C₄×C₅	—	20	2	no	—
83	C₄×C₅	#711	1	2	yes	—
84	C₄×C₅	C₅×#41	1	2	yes	—
85	Dic₅	—	4	2	no	—
86	C₂×Q₈	C₂×C₅×D₅	1	2	yes	—
87	C₂×C₈	—	5	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	5	2	no	—
90	C₄⋊C₄	—	10	2	no	—
91	C₄²	C₂×C₅×D₅	1	1	yes	characteristic
92	C₂×C₅	—	2	2	no	—
93	C₂×C₅	—	2	4	no	—
94	C₂×C₅	#4082	1	1	yes	characteristic
95	C₂×C₅	#4088	1	2	yes	—
96	C₂×C₅	C₅×#124	1	1	yes	characteristic
97	C₂×C₅	C₅×#137	1	2	yes	—
98	Q₈	—	2	4	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	10	2	no	—
101	C₂×C₄	C₅×C₂²⋊D₅	1	2	yes	—
102	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
103	C₈	—	10	2	no	—
104	C₅	—	2	2	no	—
105	C₅	#65371	1	1	yes	characteristic, direct product factor
106	C₅	C₅×#491	1	1	yes	characteristic, complemented
107	C₂²	C₂×C₅×C₂²⋊D₅	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	4	2	no	—
110	C₄	—	20	2	no	—
111	C₄	C₅×#711	1	2	yes	—
112	C₂	C₅×#4082	1	1	yes	characteristic
113	C₂	C₅×#4088	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₅×#4016

C₂

yes

maximal, characteristic

C₅×#3967

C₂

yes

maximal

C₅×#3935

C₂

yes

maximal, characteristic

C₂×C₅²×Q₈

C₂²

yes

—

C₅²×C₄⋊C₄

—

C₄²×C₅²

C₂²

yes

characteristic

C₅×C₄⋊Dic₅

C₂²

yes

—

C₅×#683

—

C₂×C₅×#168

C₂²

yes

—

C₅×#491

—

maximal

#65371

C₅

yes

maximal, characteristic, direct product factor

C₅²×Q₈

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

D₄

yes

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×C₅×Dic₅

—

C₅×#168

—

C₅×C₄⋊Q₈

—

C₅×C₄⋊Q₈

D₅

yes

characteristic

C₅×C₄⋊Q₈

C₂×C₅

yes

characteristic

C₅×#125

—

#4016

C₂×C₅

yes

characteristic

C₅×#103

—

C₅×#97

—

#3967

C₂×C₅

yes

—

#3935

C₂×C₅

yes

characteristic

C₂²×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

—

C₄×C₅²

—

C₄×C₅²

#41

yes

—

C₅×Dic₅

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

C₂×D₅

yes

—

C₂×C₅×Q₈

C₂²×C₅

yes

—

C₂×C₅×C₈

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

—

C₄²×C₅

C₂×D₅

yes

characteristic

C₄²×C₅

C₂²×C₅

yes

characteristic

C₄⋊Dic₅

C₂²×C₅

yes

—

#683

—

C₂×#168

C₂²×C₅

yes

—

—

C₂×C₅²

yes

C₂×C₅²

yes

—

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₅×D₄

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

characteristic

C₂×C₄×C₅

C₂³×C₅

yes

characteristic

C₂×Dic₅

—

C₅×C₈

—

#168

—

C₄⋊Q₈

C₅×D₅

yes

—

—

—

C₅²

#491

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

C₂×C₂²⋊D₅

yes

characteristic

C₂²×C₅

C₂×C₅×D₄

yes

characteristic

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#711

yes

—

C₄×C₅

C₅×#41

yes

—

Dic₅

—

C₂×Q₈

C₂×C₅×D₅

yes

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₅×D₅

yes

characteristic

C₂×C₅

—

C₂×C₅

—

C₂×C₅

#4082

yes

characteristic

C₂×C₅

#4088

yes

—

C₂×C₅

C₅×#124

yes

characteristic

C₂×C₅

C₅×#137

yes

—

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₅×C₂²⋊D₅

yes

—

102

C₂×C₄

C₂²×C₅×D₅

yes

characteristic

103

C₈

—

104

C₅

—

105

C₅

#65371

yes

characteristic, direct product factor

106

C₅

C₅×#491

yes

characteristic, complemented

107

C₂²

C₂×C₅×C₂²⋊D₅

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

C₅×#711

yes

—

112

C₂

C₅×#4082

yes

characteristic

113

C₂

C₅×#4088

yes

—

	subgroup	G/H
center	C₂²×C₅	C₂×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂³×C₅
Frattini	C₂×C₄	C₂²×C₅×D₅
Fitting	C₅²×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₅²	C₂×D₄
hypercenter	C₅×C₄⋊Q₈	D₅
hypocenter	C₅	C₅×#491
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×C₄⋊Q₈

subgroup

G/H

center

C₂²×C₅

C₂×C₂²⋊D₅

commutator, cosocle

C₂×C₄×C₅

C₂³×C₅

Frattini

C₂×C₄

C₂²×C₅×D₅

Fitting

C₅²×C₄⋊Q₈

C₂

socle, prime radical

C₂²×C₅²

C₂×D₄

hypercenter

C₅×C₄⋊Q₈

D₅

hypocenter

C₅

C₅×#491

LCS

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×C₄⋊Q₈

N	H ≅ G/N	type
12. #65371	C₅	direct (N × H)
71. C₅²	#491	semidirect
105. C₅	#65371	direct (N × H)
106. C₅	C₅×#491	semidirect

H ≅ G/N

type

12. #65371

C₅

direct (N × H)

71. C₅²

#491

semidirect

105. C₅

#65371

direct (N × H)

106. C₅

C₅×#491

semidirect

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

subgroup	index
C₅²×C₄⋊Q₈	2
C₅×#4016	2
C₅×#3967	2
C₅×#3935	2
C₅×#491	5
#65371	5

subgroup

index

C₅²×C₄⋊Q₈

Subgroups — C₅×#65371 #13102343

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#65371 #13102343

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65371 #13102343