Group #13104566 — subgroups

Subgroups — #13104566

Order 1600.

267 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 267 conjugacy classes of subgroups — too many to draw. Only its 101 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	C₂×#87849	C₂	1	2	yes	maximal
2	C₄²×C₅×D₅	C₂	1	1	yes	maximal, characteristic
3	#11716847	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₅×D₅	C₄	1	2	yes	—
5	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
6	C₂×#6032	C₂²	1	4	yes	—
7	#87849	—	2	4	no	—
8	C₄²×C₅²	C₄	1	1	yes	characteristic
9	C₄×C₅×Dic₅	C₄	1	1	yes	characteristic
10	C₄×C₅×Dic₅	C₂²	1	2	yes	—
11	#65682	—	5	1	no	maximal
12	#65213	—	5	1	no	maximal
13	C₂²×C₅×D₅	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
15	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
16	C₂×C₅×Dic₅	C₂×C₄	1	2	yes	—
17	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
18	C₄×C₅×D₅	—	2	4	no	—
19	C₄×C₅×D₅	D₄	1	2	yes	—
20	C₄×C₅×D₅	Q₈	1	2	yes	—
21	#6032	—	2	4	no	—
22	C₂×#700	—	5	2	no	—
23	C₂×C₄²×C₅	—	5	1	no	—
24	C₂×#681	—	5	2	no	—
25	C₄²×D₅	D₅	1	1	yes	characteristic
26	#4004	—	5	4	no	—
27	C₄⋊#168	—	5	4	no	—
28	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₅×D₅	—	2	2	no	—
30	C₂×C₅×D₅	C₄⋊C₄	1	2	yes	—
31	C₄×C₅²	—	2	2	no	—
32	C₄×C₅²	C₄⋊C₄	1	2	yes	—
33	C₅×Dic₅	M₁₆	1	4	yes	complemented
34	C₅×Dic₅	C₂×D₄	1	1	yes	characteristic
35	C₅×Dic₅	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₅	—	5	1	no	—
37	C₂²×C₄×C₅	—	5	2	no	—
38	C₂×C₄×D₅	Dic₅	1	2	yes	—
39	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
40	C₂×#170	—	5	4	no	—
41	#700	—	10	4	no	—
42	C₄²×C₅	—	4	1	no	—
43	C₄²×C₅	—	5	2	no	—
44	C₄²×C₅	—	5	1	no	—
45	C₄²×C₅	Dic₅	1	1	yes	characteristic
46	C₄²×C₅	AGL(1,𝔽₅)	1	1	yes	characteristic
47	C₄×Dic₅	Dic₅	1	1	yes	characteristic
48	C₄×Dic₅	C₂×D₅	1	2	yes	—
49	#681	—	10	4	no	—
50	C₂×#168	—	5	4	no	—
51	#423	—	25	1	no	Sylow
52	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₅²	C₂×M₁₆	1	2	yes	—
54	C₅×D₅	—	2	2	no	—
55	C₂³×C₅	—	5	1	no	—
56	C₂²×D₅	C₂×Dic₅	1	1	yes	characteristic
57	C₂×C₄×C₅	—	4	2	no	—
58	C₂×C₄×C₅	—	4	1	no	—
59	C₂×C₄×C₅	—	5	2	no	—
60	C₂×C₄×C₅	—	5	1	no	—
61	C₂×C₄×C₅	—	5	2	no	—
62	C₂×C₄×C₅	—	5	2	no	—
63	C₂×C₄×C₅	—	10	4	no	—
64	C₂×C₄×C₅	C₂×Dic₅	1	2	yes	—
65	C₂×C₄×C₅	C₂×Dic₅	1	1	yes	characteristic
66	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
67	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
68	C₂×Dic₅	C₂×Dic₅	1	2	yes	—
69	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
70	C₄×D₅	—	2	4	no	—
71	C₄×D₅	Dic₁₀	1	2	yes	—
72	C₄×D₅	D₂₀	1	2	yes	—
73	#170	—	10	4	no	—
74	#168	—	10	4	no	—
75	C₂×M₁₆	—	25	2	no	—
76	C₂×C₄²	—	5	1	no	—
77	C₄⋊C₈	—	25	4	no	—
78	C₅²	#423	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₅	—	4	1	no	—
80	C₂²×C₅	—	5	2	no	—
81	C₂²×C₅	—	10	2	no	—
82	C₂²×C₅	C₂²×Dic₅	1	1	yes	characteristic
83	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
84	C₂×D₅	—	2	2	no	—
85	C₂×D₅	C₄⋊Dic₅	1	2	yes	—
86	C₄×C₅	—	2	2	no	—
87	C₄×C₅	—	2	2	no	—
88	C₄×C₅	—	4	4	no	—
89	C₄×C₅	—	4	2	no	—
90	C₄×C₅	—	5	4	no	—
91	C₄×C₅	—	5	1	no	—
92	C₄×C₅	—	5	1	no	—
93	C₄×C₅	C₄⋊Dic₅	1	2	yes	—
94	C₄×C₅	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
95	Dic₅	#681	1	4	yes	complemented
96	Dic₅	C₂×Dic₁₀	1	1	yes	characteristic
97	Dic₅	C₂×D₂₀	1	1	yes	characteristic
98	C₂²×C₄	—	5	1	no	—
99	C₂²×C₄	—	5	2	no	—
100	M₁₆	—	50	4	no	—
101	C₂×C₈	—	25	4	no	—
102	C₄²	—	5	2	no	—
103	C₄²	—	5	1	no	—
104	C₄²	#1042	1	1	yes	characteristic
105	C₂×C₅	—	4	2	no	—
106	C₂×C₅	—	4	1	no	—
107	C₂×C₅	—	10	2	no	—
108	C₂×C₅	C₂×#681	1	2	yes	—
109	C₂×C₅	C₂×C₄⋊Dic₅	1	1	yes	characteristic
110	C₂×C₅	D₅⋊M₁₆	1	2	yes	—
111	C₂×C₅	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
112	D₅	—	2	2	no	—
113	C₂³	—	5	1	no	—
114	C₂×C₄	—	5	1	no	—
115	C₂×C₄	—	5	2	no	—
116	C₂×C₄	—	5	2	no	—
117	C₂×C₄	—	5	2	no	—
118	C₂×C₄	—	10	4	no	—
119	C₂×C₄	C₂×#1042	1	2	yes	—
120	C₂×C₄	C₂×#1042	1	1	yes	characteristic
121	C₈	—	50	4	no	—
122	C₅	—	4	1	no	—
123	C₅	#65213	1	1	yes	characteristic, complemented
124	C₅	#65682	1	1	yes	characteristic, complemented
125	C₂²	—	5	2	no	—
126	C₂²	—	10	2	no	—
127	C₂²	C₂²×#1042	1	1	yes	characteristic
128	C₄	—	2	2	no	—
129	C₄	—	5	4	no	—
130	C₄	—	5	1	no	—
131	C₄	—	5	1	no	—
132	C₄	C₄⋊#1042	1	2	yes	—
133	C₂	—	10	2	no	—
134	C₂	D₅⋊#681	1	2	yes	—
135	C₂	C₂×C₄⋊#1042	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#87849

C₂

yes

maximal

C₄²×C₅×D₅

C₂

yes

maximal, characteristic

#11716847

C₂

yes

maximal, complemented

C₂×C₄×C₅×D₅

C₄

yes

—

C₂×C₄×C₅×D₅

C₂²

yes

characteristic

C₂×#6032

C₂²

yes

—

#87849

—

C₄²×C₅²

C₄

yes

characteristic

C₄×C₅×Dic₅

C₄

yes

characteristic

C₄×C₅×Dic₅

C₂²

yes

—

—

—

C₂²×C₅×D₅

C₂×C₄

yes

characteristic

C₂×C₄×C₅²

C₂×C₄

yes

—

C₂×C₄×C₅²

C₂×C₄

yes

characteristic

C₂×C₅×Dic₅

C₂×C₄

yes

—

C₂×C₅×Dic₅

C₂³

yes

characteristic

C₄×C₅×D₅

—

C₄×C₅×D₅

D₄

yes

—

C₄×C₅×D₅

Q₈

yes

—

#6032

—

C₂×#700

—

C₂×C₄²×C₅

—

C₂×#681

—

C₄²×D₅

D₅

yes

characteristic

#4004

—

C₄⋊#168

—

C₂²×C₅²

C₂²×C₄

yes

characteristic

C₂×C₅×D₅

—

C₂×C₅×D₅

C₄⋊C₄

yes

—

C₄×C₅²

—

C₄×C₅²

C₄⋊C₄

yes

—

C₅×Dic₅

M₁₆

yes

complemented

C₅×Dic₅

C₂×D₄

yes

characteristic

C₅×Dic₅

C₂×Q₈

yes

characteristic

C₂²×C₄×C₅

—

C₂²×C₄×C₅

—

C₂×C₄×D₅

Dic₅

yes

—

C₂×C₄×D₅

C₂×D₅

yes

characteristic

C₂×#170

—

#700

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

—

C₄²×C₅

Dic₅

yes

characteristic

C₄²×C₅

AGL(1,𝔽₅)

yes

characteristic

C₄×Dic₅

Dic₅

yes

characteristic

C₄×Dic₅

C₂×D₅

yes

—

—

—

—

C₂×C₅²

C₂×C₄⋊C₄

yes

characteristic

C₂×C₅²

C₂×M₁₆

yes

—

C₅×D₅

—

C₂³×C₅

—

C₂²×D₅

C₂×Dic₅

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×Dic₅

yes

—

C₂×C₄×C₅

C₂×Dic₅

yes

characteristic

C₂×C₄×C₅

C₂×AGL(1,𝔽₅)

yes

—

C₂×C₄×C₅

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂×Dic₅

yes

—

C₂×Dic₅

C₂²×D₅

yes

characteristic

C₄×D₅

—

C₄×D₅

Dic₁₀

yes

—

C₄×D₅

D₂₀

yes

—

#170

—

#168

—

C₂×M₁₆

—

C₂×C₄²

—

C₄⋊C₈

—

C₅²

#423

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂²×Dic₅

yes

characteristic

C₂²×C₅

C₂²×AGL(1,𝔽₅)

yes

characteristic

C₂×D₅

—

C₂×D₅

C₄⋊Dic₅

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₄⋊Dic₅

yes

—

C₄×C₅

C₄⋊AGL(1,𝔽₅)

yes

—

Dic₅

#681

yes

complemented

Dic₅

C₂×Dic₁₀

yes

characteristic

Dic₅

C₂×D₂₀

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

100

M₁₆

—

101

C₂×C₈

—

102

C₄²

—

103

C₄²

—

104

C₄²

#1042

yes

characteristic

105

C₂×C₅

—

106

C₂×C₅

—

107

C₂×C₅

—

108

C₂×C₅

C₂×#681

yes

—

109

C₂×C₅

C₂×C₄⋊Dic₅

yes

characteristic

110

C₂×C₅

D₅⋊M₁₆

yes

—

111

C₂×C₅

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

112

D₅

—

113

C₂³

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂×#1042

yes

—

120

C₂×C₄

C₂×#1042

yes

characteristic

121

C₈

—

122

C₅

—

123

C₅

#65213

yes

characteristic, complemented

124

C₅

#65682

yes

characteristic, complemented

125

C₂²

—

126

C₂²

—

127

C₂²

C₂²×#1042

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

C₄⋊#1042

yes

—

133

C₂

—

134

C₂

D₅⋊#681

yes

—

135

C₂

C₂×C₄⋊#1042

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#1042
commutator, socle	C₂²×C₅²	C₂²×C₄
Fitting	C₄²×C₅²	C₄
prime radical	C₂²×C₅×D₅	C₂×C₄
cosocle	C₂×C₅×Dic₅	C₂³
hypercenter	C₄²	#1042
hypocenter	C₅²	#423
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄²

subgroup

G/H

center, Frattini

C₂²

C₂²×#1042

commutator, socle

C₂²×C₅²

C₂²×C₄

Fitting

C₄²×C₅²

C₄

prime radical

C₂²×C₅×D₅

C₂×C₄

cosocle

C₂×C₅×Dic₅

C₂³

C₄²

C₅²

G ▷ C₂²×C₅² ▷ C₅²

C₁ ◁ C₂² ◁ C₄²

N	H ≅ G/N	type
3. #11716847	C₂	semidirect
33. C₅×Dic₅	M₁₆	semidirect
78. C₅²	#423	semidirect
95. Dic₅	#681	semidirect
123. C₅	#65213	semidirect
124. C₅	#65682	semidirect

H ≅ G/N

type

3. #11716847

C₂

semidirect

33. C₅×Dic₅

M₁₆

semidirect

78. C₅²

#423

semidirect

95. Dic₅

#681

semidirect

123. C₅

#65213

semidirect

124. C₅

#65682

semidirect

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

subgroup	index
C₂×#87849	2
C₄²×C₅×D₅	2
#11716847	2
#65682	5
#65213	5

subgroup

index

C₄²×C₅×D₅

Subgroups — #13104566

Conjugacy classes of subgroups

Complements

Maximal subgroups