Group #13102115 — subgroups

Subgroups — C₅×#65144 #13102115

Order 1600.

232 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 94 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	C₅×#4034	C₂	1	2	yes	maximal
2	C₅²×#106	C₂	1	1	yes	maximal, characteristic
3	C₅×#3949	C₂	1	1	yes	maximal, characteristic
4	C₅×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
5	C₅×#3940	C₂	1	2	yes	maximal
6	C₂×C₅×Dic₁₀	—	2	2	no	—
7	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
8	C₅²×C₄⋊C₄	C₂²	1	2	yes	—
9	C₅×C₄⋊Dic₅	C₂²	1	2	yes	—
10	C₅×#683	—	2	2	no	—
11	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
12	C₂×C₅×#168	C₂²	1	1	yes	characteristic
13	C₅×#498	—	5	1	no	maximal
14	#65144	C₅	1	1	yes	maximal, characteristic, direct product factor
15	C₂×C₄×C₅²	—	2	2	no	—
16	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
17	C₂×C₅×Dic₅	—	2	2	no	—
18	C₂×C₅×Dic₅	D₄	1	2	yes	—
19	C₅×Dic₁₀	—	4	4	no	—
20	C₅²×C₈	Q₈	1	2	yes	complemented
21	C₅×#168	Q₈	1	2	yes	complemented
22	C₅×C₄⋊Q₈	—	5	2	no	—
23	#4034	C₂×C₅	1	2	yes	—
24	C₅×#106	—	2	2	no	—
25	C₅×#106	—	5	1	no	—
26	C₅×#106	—	5	2	no	—
27	C₅×#106	D₅	1	1	yes	characteristic
28	C₅×#106	C₂×C₅	1	1	yes	characteristic
29	C₄×C₅×C₈	—	5	1	no	—
30	#3949	C₂×C₅	1	1	yes	characteristic
31	C₈×Dic₅	C₂×C₅	1	1	yes	characteristic
32	#3940	C₂×C₅	1	2	yes	—
33	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
34	C₄×C₅²	—	4	2	no	—
35	C₄×C₅²	C₂×Q₈	1	1	yes	characteristic
36	C₄×C₅²	C₂×Q₈	1	1	yes	characteristic
37	C₅×Dic₅	—	4	2	no	—
38	C₅×Dic₅	SD₁₆	1	4	yes	—
39	C₂×C₅×Q₈	—	10	2	no	—
40	C₂×Dic₁₀	—	2	2	no	—
41	C₂×C₅×C₈	—	2	2	no	—
42	C₂×C₅×C₈	—	5	1	no	—
43	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic
44	C₂×C₅×C₈	C₂²×C₅	1	1	yes	characteristic
45	C₅×C₄⋊C₄	—	2	4	no	—
46	C₅×C₄⋊C₄	—	5	2	no	—
47	C₅×C₄⋊C₄	—	10	2	no	—
48	C₅×C₄⋊C₄	C₂×D₅	1	2	yes	—
49	C₅×C₄⋊C₄	C₂²×C₅	1	2	yes	—
50	C₄²×C₅	—	5	1	no	—
51	C₄⋊Dic₅	C₂²×C₅	1	2	yes	—
52	#683	—	2	2	no	—
53	C₄×Dic₅	C₂²×C₅	1	1	yes	characteristic
54	C₂×#168	C₂²×C₅	1	1	yes	characteristic
55	#498	—	5	1	no	Sylow
56	C₂×C₅²	C₄⋊Q₈	1	1	yes	characteristic
57	C₂×C₅²	C₂×SD₁₆	1	2	yes	—
58	C₅×Q₈	—	20	4	no	—
59	C₂×C₄×C₅	—	2	2	no	—
60	C₂×C₄×C₅	—	2	2	no	—
61	C₂×C₄×C₅	—	2	2	no	—
62	C₂×C₄×C₅	—	4	4	no	—
63	C₂×C₄×C₅	—	5	2	no	—
64	C₂×C₄×C₅	—	10	2	no	—
65	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
66	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
67	C₂×Dic₅	—	2	2	no	—
68	C₂×Dic₅	C₅×D₄	1	2	yes	—
69	Dic₁₀	—	4	4	no	—
70	C₅×C₈	—	2	4	no	—
71	C₅×C₈	—	5	2	no	—
72	C₅×C₈	Dic₁₀	1	2	yes	complemented
73	C₅×C₈	C₅×Q₈	1	2	yes	complemented
74	#168	C₅×Q₈	1	2	yes	complemented
75	C₄⋊Q₈	—	5	2	no	—
76	#106	—	5	1	no	—
77	#106	—	5	2	no	—
78	#106	C₅×D₅	1	1	yes	characteristic
79	C₄×C₈	—	5	1	no	—
80	C₅²	#498	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₅	—	2	2	no	—
82	C₂²×C₅	D₄×D₅	1	1	yes	characteristic
83	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
84	C₄×C₅	—	2	2	no	—
85	C₄×C₅	—	2	2	no	—
86	C₄×C₅	—	4	2	no	—
87	C₄×C₅	—	4	2	no	—
88	C₄×C₅	—	5	4	no	—
89	C₄×C₅	—	8	4	no	—
90	C₄×C₅	—	20	2	no	—
91	C₄×C₅	C₂×Dic₁₀	1	1	yes	characteristic
92	C₄×C₅	Q₈×D₅	1	1	yes	characteristic
93	C₄×C₅	C₂×C₅×Q₈	1	1	yes	characteristic
94	C₄×C₅	C₂×C₅×Q₈	1	1	yes	characteristic
95	Dic₅	—	4	2	no	—
96	Dic₅	C₅×SD₁₆	1	4	yes	—
97	C₂×Q₈	—	10	2	no	—
98	C₂×C₈	—	5	1	no	—
99	C₂×C₈	C₂×C₅×D₅	1	1	yes	characteristic
100	C₄⋊C₄	—	5	2	no	—
101	C₄⋊C₄	—	10	2	no	—
102	C₄⋊C₄	C₂×C₅×D₅	1	2	yes	—
103	C₄²	—	5	1	no	—
104	C₂×C₅	—	2	2	no	—
105	C₂×C₅	—	2	4	no	—
106	C₂×C₅	#4034	1	1	yes	characteristic
107	C₂×C₅	D₅×SD₁₆	1	2	yes	—
108	C₂×C₅	C₅×C₄⋊Q₈	1	1	yes	characteristic
109	C₂×C₅	C₂×C₅×SD₁₆	1	2	yes	—
110	Q₈	—	20	4	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	5	2	no	—
113	C₂×C₄	—	10	2	no	—
114	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
115	C₈	—	5	2	no	—
116	C₈	C₅×Dic₁₀	1	2	yes	complemented
117	C₅	—	2	2	no	—
118	C₅	#65144	1	1	yes	characteristic, direct product factor
119	C₅	C₅×#498	1	1	yes	characteristic, complemented
120	C₂²	C₅×D₄×D₅	1	1	yes	characteristic
121	C₄	—	4	2	no	—
122	C₄	—	5	4	no	—
123	C₄	—	20	2	no	—
124	C₄	C₂×C₅×Dic₁₀	1	1	yes	characteristic
125	C₄	C₅×Q₈×D₅	1	1	yes	characteristic
126	C₂	C₅×#4034	1	1	yes	characteristic
127	C₂	C₅×D₅×SD₁₆	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#4034

C₂

yes

maximal

C₅²×#106

C₂

yes

maximal, characteristic

C₅×#3949

C₂

yes

maximal, characteristic

C₅×C₈×Dic₅

C₂

yes

maximal, characteristic

C₅×#3940

C₂

yes

maximal

C₂×C₅×Dic₁₀

—

C₂×C₅²×C₈

C₂²

yes

characteristic

C₅²×C₄⋊C₄

C₂²

yes

—

C₅×C₄⋊Dic₅

C₂²

yes

—

C₅×#683

—

C₄×C₅×Dic₅

C₂²

yes

characteristic

C₂×C₅×#168

C₂²

yes

—

C₅

yes

maximal, characteristic, direct product factor

C₂×C₄×C₅²

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×C₅×Dic₅

—

C₂×C₅×Dic₅

D₄

yes

—

C₅×Dic₁₀

—

C₅²×C₈

Q₈

yes

complemented

C₅×#168

Q₈

yes

complemented

C₅×C₄⋊Q₈

—

#4034

C₂×C₅

yes

—

—

—

—

D₅

yes

characteristic

C₅×#106

C₂×C₅

yes

characteristic

C₄×C₅×C₈

—

#3949

C₂×C₅

yes

characteristic

C₈×Dic₅

C₂×C₅

yes

characteristic

#3940

C₂×C₅

yes

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

—

C₄×C₅²

C₂×Q₈

yes

characteristic

C₄×C₅²

C₂×Q₈

yes

characteristic

C₅×Dic₅

—

C₅×Dic₅

SD₁₆

yes

—

C₂×C₅×Q₈

—

C₂×Dic₁₀

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂×D₅

yes

characteristic

C₂×C₅×C₈

C₂²×C₅

yes

characteristic

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

C₂×D₅

yes

—

C₅×C₄⋊C₄

C₂²×C₅

yes

—

C₄²×C₅

—

C₄⋊Dic₅

C₂²×C₅

yes

—

#683

—

C₄×Dic₅

C₂²×C₅

yes

characteristic

C₂×#168

C₂²×C₅

yes

characteristic

#498

—

Sylow

C₂×C₅²

C₄⋊Q₈

yes

characteristic

C₂×C₅²

C₂×SD₁₆

yes

—

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

characteristic

C₂×C₄×C₅

C₂³×C₅

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

C₅×D₄

yes

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

Dic₁₀

yes

complemented

C₅×C₈

C₅×Q₈

yes

complemented

#168

C₅×Q₈

yes

complemented

C₄⋊Q₈

—

#106

—

#106

—

#106

C₅×D₅

yes

characteristic

C₄×C₈

—

C₅²

#498

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

D₄×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

C₂×Dic₁₀

yes

characteristic

C₄×C₅

Q₈×D₅

yes

characteristic

C₄×C₅

C₂×C₅×Q₈

yes

characteristic

C₄×C₅

C₂×C₅×Q₈

yes

characteristic

Dic₅

—

Dic₅

C₅×SD₁₆

yes

—

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

C₂×C₅×D₅

yes

characteristic

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

C₂×C₅×D₅

yes

—

103

C₄²

—

104

C₂×C₅

—

105

C₂×C₅

—

106

C₂×C₅

#4034

yes

characteristic

107

C₂×C₅

D₅×SD₁₆

yes

—

108

C₂×C₅

C₅×C₄⋊Q₈

yes

characteristic

109

C₂×C₅

C₂×C₅×SD₁₆

yes

—

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂²×C₅×D₅

yes

characteristic

115

C₈

—

116

C₈

C₅×Dic₁₀

yes

complemented

117

C₅

—

118

C₅

#65144

yes

characteristic, direct product factor

119

C₅

C₅×#498

yes

characteristic, complemented

120

C₂²

C₅×D₄×D₅

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

C₂×C₅×Dic₁₀

yes

characteristic

125

C₄

C₅×Q₈×D₅

yes

characteristic

126

C₂

C₅×#4034

yes

characteristic

127

C₂

C₅×D₅×SD₁₆

yes

—

	subgroup	G/H
center	C₂²×C₅	D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂³×C₅
Frattini	C₂×C₄	C₂²×C₅×D₅
Fitting	C₅²×#106	C₂
socle, prime radical	C₂²×C₅²	C₂×D₄
hypercenter	C₅×#106	D₅
hypocenter	C₅	C₅×#498
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#106

subgroup

G/H

center

C₂²×C₅

D₄×D₅

commutator, cosocle

C₂×C₄×C₅

C₂³×C₅

Frattini

C₂×C₄

C₂²×C₅×D₅

Fitting

C₅²×#106

C₂

socle, prime radical

C₂²×C₅²

C₂×D₄

D₅

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#106

N	H ≅ G/N	type
14. #65144	C₅	direct (N × H)
20. C₅²×C₈	Q₈	semidirect
21. C₅×#168	Q₈	semidirect
72. C₅×C₈	Dic₁₀	semidirect
73. C₅×C₈	C₅×Q₈	semidirect
74. #168	C₅×Q₈	semidirect
80. C₅²	#498	semidirect
116. C₈	C₅×Dic₁₀	semidirect
118. C₅	#65144	direct (N × H)
119. C₅	C₅×#498	semidirect

H ≅ G/N

type

14. #65144

C₅

direct (N × H)

20. C₅²×C₈

Q₈

semidirect

21. C₅×#168

Q₈

semidirect

72. C₅×C₈

Dic₁₀

semidirect

73. C₅×C₈

C₅×Q₈

semidirect

74. #168

C₅×Q₈

semidirect

80. C₅²

#498

semidirect

116. C₈

C₅×Dic₁₀

semidirect

118. C₅

#65144

direct (N × H)

119. C₅

C₅×#498

semidirect

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

subgroup	index
C₅×#4034	2
C₅²×#106	2
C₅×#3949	2
C₅×C₈×Dic₅	2
C₅×#3940	2
C₅×#498	5
#65144	5

subgroup

index

C₅²×#106

C₅×C₈×Dic₅

Subgroups — C₅×#65144 #13102115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#65144 #13102115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65144 #13102115