Group #13104108 — subgroups

Subgroups — #13104108

Order 1600.

320 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 79 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×#87867	C₂	1	1	yes	maximal, characteristic
2	C₂×#87860	C₂	1	1	yes	maximal, characteristic
3	C₂×Dic₅²	C₂	1	1	yes	maximal, characteristic
4	#11716783	C₂	1	4	yes	maximal, complemented
5	C₂²×#1039	C₂²	1	1	yes	characteristic
6	C₂²×C₅×Dic₅	C₄	1	2	yes	—
7	#87867	—	2	2	no	—
8	C₂×#6034	C₂²	1	2	yes	—
9	#87860	—	2	2	no	—
10	C₂×#6033	C₂²	1	2	yes	—
11	Dic₅²	C₄	1	1	yes	characteristic
12	Dic₅²	C₄	1	1	yes	characteristic
13	Dic₅²	C₂²	1	2	yes	—
14	#65758	—	5	2	no	maximal
15	C₂³×C₅²	C₂×C₄	1	1	yes	characteristic
16	C₂×#1039	C₂×C₄	1	1	yes	characteristic
17	C₂×#1039	D₄	1	2	yes	—
18	C₂×#1039	Q₈	1	2	yes	—
19	C₂×#1039	C₂³	1	1	yes	characteristic
20	C₂×C₅×Dic₅	—	2	4	no	—
21	C₂×C₅×Dic₅	C₂×C₄	1	2	yes	—
22	C₂×C₅×Dic₅	C₂×C₄	1	2	yes	—
23	#6034	—	2	2	no	—
24	#6033	—	2	2	no	—
25	C₂×#704	—	5	2	no	—
26	C₂×#704	—	5	2	no	—
27	C₂×#704	—	10	2	no	—
28	C₂×C₄×Dic₅	—	5	2	no	—
29	#4004	—	5	4	no	—
30	C₄⋊#170	—	5	4	no	—
31	C₂²×C₅²	—	2	2	no	—
32	C₂²×C₅²	C₄⋊C₄	1	2	yes	—
33	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
34	#1039	C₄⋊C₄	1	2	yes	—
35	#1039	C₂×D₄	1	1	yes	characteristic
36	#1039	C₂×Q₈	1	1	yes	characteristic
37	C₅×Dic₅	—	2	2	no	—
38	C₅×Dic₅	M₁₆	1	4	yes	complemented
39	C₂²×C₄×C₅	—	5	2	no	—
40	C₂²×Dic₅	—	5	2	no	—
41	C₂²×Dic₅	—	10	2	no	—
42	C₂²×Dic₅	AGL(1,𝔽₅)	1	2	yes	—
43	#704	—	10	4	no	—
44	#704	—	10	4	no	—
45	#704	—	10	8	no	—
46	C₂×#170	—	5	4	no	—
47	C₂×#170	—	5	4	no	—
48	C₂×#170	—	10	4	no	—
49	C₄×Dic₅	—	5	2	no	—
50	C₄×Dic₅	—	5	2	no	—
51	C₄×Dic₅	—	5	2	no	—
52	C₄×Dic₅	—	5	2	no	—
53	#423	—	25	1	no	Sylow
54	C₂×C₅²	—	2	2	no	—
55	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₅²	C₂×M₁₆	1	2	yes	—
57	C₂³×C₅	—	2	2	no	—
58	C₂³×C₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
59	C₂×C₄×C₅	—	5	2	no	—
60	C₂×C₄×C₅	—	5	2	no	—
61	C₂×C₄×C₅	—	10	4	no	—
62	C₂×Dic₅	—	2	4	no	—
63	C₂×Dic₅	—	5	4	no	—
64	C₂×Dic₅	—	5	4	no	—
65	C₂×Dic₅	—	5	2	no	—
66	C₂×Dic₅	—	5	2	no	—
67	C₂×Dic₅	—	10	2	no	—
68	C₂×Dic₅	—	10	4	no	—
69	C₂×Dic₅	—	10	4	no	—
70	C₂×Dic₅	—	10	2	no	—
71	C₂×Dic₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
72	C₂×Dic₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
73	#170	—	10	4	no	—
74	#170	—	10	4	no	—
75	#170	—	10	8	no	—
76	C₂×M₁₆	—	25	1	no	—
77	C₂×M₁₆	—	25	1	no	—
78	C₂×C₄²	—	25	1	no	—
79	C₄⋊C₈	—	25	4	no	—
80	C₅²	#423	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₅	—	2	2	no	—
82	C₂²×C₅	—	2	2	no	—
83	C₂²×C₅	—	2	4	no	—
84	C₂²×C₅	—	2	2	no	—
85	C₂²×C₅	—	4	4	no	—
86	C₂²×C₅	C₄⋊AGL(1,𝔽₅)	1	4	yes	—
87	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	2	yes	—
88	C₄×C₅	—	5	4	no	—
89	C₄×C₅	—	10	2	no	—
90	Dic₅	—	2	2	no	—
91	Dic₅	—	5	4	no	—
92	Dic₅	—	5	2	no	—
93	Dic₅	—	5	2	no	—
94	Dic₅	—	10	4	no	—
95	Dic₅	—	10	2	no	—
96	Dic₅	—	10	2	no	—
97	Dic₅	#704	1	4	yes	complemented
98	C₂²×C₄	—	5	2	no	—
99	C₂²×C₄	—	25	1	no	—
100	M₁₆	—	50	2	no	—
101	M₁₆	—	50	2	no	—
102	C₂×C₈	—	25	2	no	—
103	C₂×C₈	—	25	2	no	—
104	C₄²	—	25	2	no	—
105	C₄²	—	25	1	no	—
106	C₄²	—	25	1	no	—
107	C₂×C₅	—	2	4	no	—
108	C₂×C₅	—	2	2	no	—
109	C₂×C₅	—	2	4	no	—
110	C₂×C₅	—	4	4	no	—
111	C₂×C₅	D₅⋊M₁₆	1	2	yes	—
112	C₂×C₅	C₂×C₄⋊AGL(1,𝔽₅)	1	2	yes	—
113	C₂×C₅	C₂×#704	1	2	yes	—
114	C₂³	D₅⋊AGL(1,𝔽₅)	1	1	yes	characteristic
115	C₂×C₄	—	5	2	no	—
116	C₂×C₄	—	5	2	no	—
117	C₂×C₄	—	10	4	no	—
118	C₂×C₄	—	25	2	no	—
119	C₂×C₄	—	25	2	no	—
120	C₂×C₄	—	25	1	no	—
121	C₂×C₄	—	25	1	no	—
122	C₈	—	50	2	no	—
123	C₈	—	50	2	no	—
124	C₅	—	2	2	no	—
125	C₅	#65758	1	2	yes	complemented
126	C₂²	—	2	2	no	—
127	C₂²	#87832	1	2	yes	—
128	C₂²	C₂×D₅⋊AGL(1,𝔽₅)	1	1	yes	characteristic
129	C₄	—	5	4	no	—
130	C₄	—	10	2	no	—
131	C₄	—	25	1	no	—
132	C₄	—	25	2	no	—
133	C₄	—	25	1	no	—
134	C₂	—	2	2	no	—
135	C₂	C₂×#87832	1	1	yes	characteristic
136	C₂	D₅⋊#704	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#87867

C₂

yes

maximal, characteristic

C₂×#87860

C₂

yes

maximal, characteristic

C₂×Dic₅²

C₂

yes

maximal, characteristic

#11716783

C₂

yes

maximal, complemented

C₂²×#1039

C₂²

yes

characteristic

C₂²×C₅×Dic₅

C₄

yes

—

#87867

—

C₂×#6034

C₂²

yes

—

#87860

—

C₂×#6033

C₂²

yes

—

Dic₅²

C₄

yes

characteristic

Dic₅²

C₄

yes

characteristic

Dic₅²

C₂²

yes

—

#65758

—

maximal

C₂³×C₅²

C₂×C₄

yes

characteristic

C₂×#1039

C₂×C₄

yes

characteristic

C₂×#1039

D₄

yes

—

C₂×#1039

Q₈

yes

—

C₂×#1039

C₂³

yes

characteristic

C₂×C₅×Dic₅

—

C₂×C₅×Dic₅

C₂×C₄

yes

—

C₂×C₅×Dic₅

C₂×C₄

yes

—

—

—

—

—

—

C₂×C₄×Dic₅

—

#4004

—

C₄⋊#170

—

C₂²×C₅²

—

C₂²×C₅²

C₄⋊C₄

yes

—

C₂²×C₅²

C₂²×C₄

yes

characteristic

#1039

C₄⋊C₄

yes

—

#1039

C₂×D₄

yes

characteristic

#1039

C₂×Q₈

yes

characteristic

C₅×Dic₅

—

C₅×Dic₅

M₁₆

yes

complemented

C₂²×C₄×C₅

—

C₂²×Dic₅

—

C₂²×Dic₅

—

C₂²×Dic₅

AGL(1,𝔽₅)

yes

—

—

—

—

—

—

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

#423

—

Sylow

C₂×C₅²

—

C₂×C₅²

C₂×C₄⋊C₄

yes

characteristic

C₂×C₅²

C₂×M₁₆

yes

—

C₂³×C₅

—

C₂³×C₅

C₂×AGL(1,𝔽₅)

yes

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

C₂×AGL(1,𝔽₅)

yes

—

C₂×Dic₅

C₂×AGL(1,𝔽₅)

yes

—

#170

—

#170

—

#170

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×C₄²

—

C₄⋊C₈

—

C₅²

#423

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₄⋊AGL(1,𝔽₅)

yes

—

C₂²×C₅

C₂²×AGL(1,𝔽₅)

yes

—

C₄×C₅

—

C₄×C₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

#704

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

100

M₁₆

—

101

M₁₆

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₄²

—

105

C₄²

—

106

C₄²

—

107

C₂×C₅

—

108

C₂×C₅

—

109

C₂×C₅

—

110

C₂×C₅

—

111

C₂×C₅

D₅⋊M₁₆

yes

—

112

C₂×C₅

C₂×C₄⋊AGL(1,𝔽₅)

yes

—

113

C₂×C₅

C₂×#704

yes

—

114

C₂³

D₅⋊AGL(1,𝔽₅)

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₈

—

123

C₈

—

124

C₅

—

125

C₅

#65758

yes

complemented

126

C₂²

—

127

C₂²

#87832

yes

—

128

C₂²

C₂×D₅⋊AGL(1,𝔽₅)

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₂

—

135

C₂

C₂×#87832

yes

characteristic

136

C₂

D₅⋊#704

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂×D₅⋊AGL(1,𝔽₅)
commutator, socle	C₂²×C₅²	C₂²×C₄
Fitting, prime radical	C₂³×C₅²	C₂×C₄
cosocle	C₂×#1039	C₂³
hypercenter	C₂³	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	#423
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂³

subgroup

G/H

center, Frattini

C₂²

C₂×D₅⋊AGL(1,𝔽₅)

commutator, socle

C₂²×C₅²

C₂²×C₄

Fitting, prime radical

C₂³×C₅²

C₂×C₄

C₂³

C₂³

C₅²

G ▷ C₂²×C₅² ▷ C₅²

C₁ ◁ C₂² ◁ C₂³

N	H ≅ G/N	type
4. #11716783	C₂	semidirect
38. C₅×Dic₅	M₁₆	semidirect
80. C₅²	#423	semidirect
97. Dic₅	#704	semidirect
125. C₅	#65758	semidirect

H ≅ G/N

type

4. #11716783

C₂

semidirect

38. C₅×Dic₅

M₁₆

semidirect

80. C₅²

#423

semidirect

97. Dic₅

#704

semidirect

125. C₅

#65758

semidirect

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

subgroup	index
C₂×#87867	2
C₂×#87860	2
C₂×Dic₅²	2
#11716783	2
#65758	5

subgroup

index

C₂×Dic₅²

Subgroups — #13104108

Conjugacy classes of subgroups

Complements

Maximal subgroups