Group #13102354 — subgroups

Subgroups — C₅×#65382 #13102354

Order 1600.

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

normal subgroups only

This group has 264 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	C₅²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#4015	C₂	1	1	yes	maximal, characteristic
3	C₅×#3942	C₂	1	4	yes	maximal
4	C₄×C₅×#168	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅²×Q₈	—	2	1	no	—
6	C₂×C₅×Dic₁₀	C₂²	1	2	yes	—
7	C₅²×C₄⋊C₄	—	2	1	no	—
8	C₅²×C₄⋊C₄	C₂²	1	2	yes	—
9	C₄²×C₅²	C₂²	1	1	yes	characteristic
10	C₅×C₄⋊Dic₅	—	2	1	no	—
11	C₅×C₄⋊Dic₅	—	2	1	no	—
12	C₂×C₅×#168	C₂²	1	2	yes	—
13	C₅×#487	—	5	1	no	maximal
14	#65382	C₅	1	1	yes	maximal, characteristic, direct product factor
15	C₅²×Q₈	—	4	2	no	—
16	C₂×C₄×C₅²	—	2	2	no	—
17	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
18	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
19	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
20	C₂×C₅×Dic₅	—	2	2	no	—
21	C₅×Dic₁₀	—	2	4	no	—
22	C₅×#168	—	2	2	no	—
23	C₅×C₄⋊Q₈	—	2	2	no	—
24	C₅×C₄⋊Q₈	—	5	1	no	—
25	C₅×C₄⋊Q₈	D₅	1	1	yes	characteristic
26	C₅×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
27	#4015	C₂×C₅	1	1	yes	characteristic
28	C₅×#103	—	5	4	no	—
29	C₄×C₅×C₈	—	5	1	no	—
30	#3942	C₂×C₅	1	4	yes	—
31	C₄×#168	C₂×C₅	1	1	yes	characteristic
32	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
33	C₄×C₅²	—	4	2	no	—
34	C₄×C₅²	SD₁₆	1	2	yes	—
35	C₄×C₅²	Q₁₆	1	2	yes	—
36	C₄×C₅²	#41	1	2	yes	—
37	C₅×Dic₅	—	4	2	no	—
38	C₂×C₅×Q₈	—	2	4	no	—
39	C₂×C₅×Q₈	—	2	1	no	—
40	C₂×C₅×Q₈	—	2	1	no	—
41	C₂×C₅×Q₈	—	5	2	no	—
42	C₂×Dic₁₀	C₂²×C₅	1	2	yes	—
43	C₂×C₅×C₈	—	5	2	no	—
44	C₅×C₄⋊C₄	—	2	4	no	—
45	C₅×C₄⋊C₄	—	2	1	no	—
46	C₅×C₄⋊C₄	—	2	1	no	—
47	C₅×C₄⋊C₄	—	2	4	no	—
48	C₅×C₄⋊C₄	—	10	1	no	—
49	C₅×C₄⋊C₄	—	10	1	no	—
50	C₅×C₄⋊C₄	C₂×D₅	1	2	yes	—
51	C₅×C₄⋊C₄	C₂²×C₅	1	2	yes	—
52	C₄²×C₅	—	2	2	no	—
53	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
54	C₄²×C₅	C₂²×C₅	1	1	yes	characteristic
55	C₄⋊Dic₅	—	2	1	no	—
56	C₄⋊Dic₅	—	2	1	no	—
57	C₂×#168	C₂²×C₅	1	2	yes	—
58	#487	—	5	1	no	Sylow
59	C₂×C₅²	#124	1	1	yes	characteristic
60	C₂×C₅²	C₂×SD₁₆	1	1	yes	characteristic
61	C₂×C₅²	C₂×Q₁₆	1	1	yes	characteristic
62	C₅×Q₈	—	4	8	no	—
63	C₅×Q₈	—	4	2	no	—
64	C₅×Q₈	—	4	2	no	—
65	C₅×Q₈	—	10	4	no	—
66	C₂×C₄×C₅	—	2	8	no	—
67	C₂×C₄×C₅	—	2	2	no	—
68	C₂×C₄×C₅	—	2	2	no	—
69	C₂×C₄×C₅	—	2	2	no	—
70	C₂×C₄×C₅	—	2	2	no	—
71	C₂×C₄×C₅	—	2	2	no	—
72	C₂×C₄×C₅	—	10	2	no	—
73	C₂×C₄×C₅	C₂²⋊D₅	1	1	yes	characteristic
74	C₂×C₄×C₅	C₂²⋊D₅	1	1	yes	characteristic
75	C₂×C₄×C₅	C₅×D₄	1	1	yes	characteristic
76	C₂×C₄×C₅	C₅×D₄	1	1	yes	characteristic
77	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
78	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
79	C₂×Dic₅	—	2	2	no	—
80	Dic₁₀	—	2	4	no	—
81	C₅×C₈	—	10	2	no	—
82	#168	—	2	2	no	—
83	C₄⋊Q₈	—	5	1	no	—
84	C₄⋊Q₈	C₅×D₅	1	1	yes	characteristic
85	#103	—	5	4	no	—
86	C₄×C₈	—	5	1	no	—
87	C₅²	#487	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₅	—	2	2	no	—
89	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
90	C₂²×C₅	C₂×C₅×D₄	1	1	yes	characteristic
91	C₄×C₅	—	2	4	no	—
92	C₄×C₅	—	2	4	no	—
93	C₄×C₅	—	2	4	no	—
94	C₄×C₅	—	4	8	no	—
95	C₄×C₅	—	4	2	no	—
96	C₄×C₅	—	4	2	no	—
97	C₄×C₅	—	20	2	no	—
98	C₄×C₅	#687	1	2	yes	—
99	C₄×C₅	#689	1	2	yes	—
100	C₄×C₅	C₅×SD₁₆	1	2	yes	—
101	C₄×C₅	C₅×Q₁₆	1	2	yes	—
102	C₄×C₅	#713	1	2	yes	—
103	C₄×C₅	C₅×#41	1	2	yes	—
104	Dic₅	—	4	2	no	—
105	C₂×Q₈	—	2	1	no	—
106	C₂×Q₈	—	5	2	no	—
107	C₂×C₈	—	5	2	no	—
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	10	1	no	—
110	C₄⋊C₄	—	10	1	no	—
111	C₄⋊C₄	C₂×C₅×D₅	1	2	yes	—
112	C₄²	C₂×C₅×D₅	1	1	yes	characteristic
113	C₂×C₅	—	2	2	no	—
114	C₂×C₅	—	2	2	no	—
115	C₂×C₅	—	2	2	no	—
116	C₂×C₅	C₂×#687	1	1	yes	characteristic
117	C₂×C₅	C₂×#689	1	1	yes	characteristic
118	C₂×C₅	#4093	1	1	yes	characteristic
119	C₂×C₅	C₅×#124	1	1	yes	characteristic
120	C₂×C₅	C₂×C₅×SD₁₆	1	1	yes	characteristic
121	C₂×C₅	C₂×C₅×Q₁₆	1	1	yes	characteristic
122	Q₈	—	4	2	no	—
123	Q₈	—	10	4	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	10	2	no	—
126	C₂×C₄	C₅×C₂²⋊D₅	1	1	yes	characteristic
127	C₂×C₄	C₅×C₂²⋊D₅	1	1	yes	characteristic
128	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
129	C₈	—	10	2	no	—
130	C₅	—	2	2	no	—
131	C₅	#65382	1	1	yes	characteristic, direct product factor
132	C₅	C₅×#487	1	1	yes	characteristic, complemented
133	C₂²	C₂×C₅×C₂²⋊D₅	1	1	yes	characteristic
134	C₄	—	4	2	no	—
135	C₄	—	20	2	no	—
136	C₄	C₅×#687	1	2	yes	—
137	C₄	C₅×#689	1	2	yes	—
138	C₄	C₅×#713	1	2	yes	—
139	C₂	C₂×C₅×#687	1	1	yes	characteristic
140	C₂	C₂×C₅×#689	1	1	yes	characteristic
141	C₂	C₅×#4093	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₅×#4015

C₂

yes

maximal, characteristic

C₅×#3942

C₂

yes

maximal

C₄×C₅×#168

C₂

yes

maximal, characteristic

C₂×C₅²×Q₈

—

C₂×C₅×Dic₁₀

C₂²

yes

—

C₅²×C₄⋊C₄

—

C₅²×C₄⋊C₄

C₂²

yes

—

C₄²×C₅²

C₂²

yes

characteristic

C₅×C₄⋊Dic₅

—

C₅×C₄⋊Dic₅

—

C₂×C₅×#168

C₂²

yes

—

C₅×#487

—

maximal

#65382

C₅

yes

maximal, characteristic, direct product factor

C₅²×Q₈

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

D₄

yes

characteristic

C₂×C₄×C₅²

D₄

yes

characteristic

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×C₅×Dic₅

—

C₅×Dic₁₀

—

C₅×#168

—

C₅×C₄⋊Q₈

—

C₅×C₄⋊Q₈

—

C₅×C₄⋊Q₈

D₅

yes

characteristic

C₅×C₄⋊Q₈

C₂×C₅

yes

characteristic

#4015

C₂×C₅

yes

characteristic

C₅×#103

—

C₄×C₅×C₈

—

#3942

C₂×C₅

yes

—

C₄×#168

C₂×C₅

yes

characteristic

C₂²×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

—

C₄×C₅²

SD₁₆

yes

—

C₄×C₅²

Q₁₆

yes

—

C₄×C₅²

#41

yes

—

C₅×Dic₅

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×Dic₁₀

C₂²×C₅

yes

—

C₂×C₅×C₈

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

C₂×D₅

yes

—

C₅×C₄⋊C₄

C₂²×C₅

yes

—

C₄²×C₅

—

C₄²×C₅

C₂×D₅

yes

characteristic

C₄²×C₅

C₂²×C₅

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₂×#168

C₂²×C₅

yes

—

—

C₂×C₅²

yes

C₂×C₅²

C₂×SD₁₆

yes

characteristic

C₂×C₅²

C₂×Q₁₆

yes

characteristic

C₅×Q₈

—

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

—

C₂×C₄×C₅

C₂²⋊D₅

yes

characteristic

C₂×C₄×C₅

C₂²⋊D₅

yes

characteristic

C₂×C₄×C₅

C₅×D₄

yes

characteristic

C₂×C₄×C₅

C₅×D₄

yes

characteristic

C₂×C₄×C₅

C₂²×D₅

yes

characteristic

C₂×C₄×C₅

C₂³×C₅

yes

characteristic

C₂×Dic₅

—

Dic₁₀

—

C₅×C₈

—

#168

—

C₄⋊Q₈

—

C₄⋊Q₈

C₅×D₅

yes

characteristic

#103

—

C₄×C₈

—

C₅²

#487

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

#687

yes

—

C₄×C₅

#689

yes

—

100

C₄×C₅

C₅×SD₁₆

yes

—

101

C₄×C₅

C₅×Q₁₆

yes

—

102

C₄×C₅

#713

yes

—

103

C₄×C₅

C₅×#41

yes

—

104

Dic₅

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×C₈

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

C₂×C₅×D₅

yes

—

112

C₄²

C₂×C₅×D₅

yes

characteristic

113

C₂×C₅

—

114

C₂×C₅

—

115

C₂×C₅

—

116

C₂×C₅

C₂×#687

yes

characteristic

117

C₂×C₅

C₂×#689

yes

characteristic

118

C₂×C₅

#4093

yes

characteristic

119

C₂×C₅

C₅×#124

yes

characteristic

120

C₂×C₅

C₂×C₅×SD₁₆

yes

characteristic

121

C₂×C₅

C₂×C₅×Q₁₆

yes

characteristic

122

Q₈

—

123

Q₈

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

C₅×C₂²⋊D₅

yes

characteristic

127

C₂×C₄

C₅×C₂²⋊D₅

yes

characteristic

128

C₂×C₄

C₂²×C₅×D₅

yes

characteristic

129

C₈

—

130

C₅

—

131

C₅

#65382

yes

characteristic, direct product factor

132

C₅

C₅×#487

yes

characteristic, complemented

133

C₂²

C₂×C₅×C₂²⋊D₅

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

C₅×#687

yes

—

137

C₄

C₅×#689

yes

—

138

C₄

C₅×#713

yes

—

139

C₂

C₂×C₅×#687

yes

characteristic

140

C₂

C₂×C₅×#689

yes

characteristic

141

C₂

C₅×#4093

yes

characteristic

	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×#487
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×#487

LCS

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×C₄⋊Q₈

N	H ≅ G/N	type
14. #65382	C₅	direct (N × H)
87. C₅²	#487	semidirect
131. C₅	#65382	direct (N × H)
132. C₅	C₅×#487	semidirect

H ≅ G/N

type

14. #65382

C₅

direct (N × H)

87. C₅²

#487

semidirect

131. C₅

#65382

direct (N × H)

132. C₅

C₅×#487

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×#4015	2
C₅×#3942	2
C₄×C₅×#168	2
C₅×#487	5
#65382	5

subgroup

index

C₅²×C₄⋊Q₈

C₄×C₅×#168

Subgroups — C₅×#65382 #13102354

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#65382 #13102354

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65382 #13102354