Group #13104093 — subgroups

Subgroups — #13104093

Order 1600.

352 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 95 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	#11716778	C₂	1	2	yes	maximal, complemented
5	#11716777	C₂	1	2	yes	maximal, complemented
6	C₂²×#1039	C₂²	1	1	yes	characteristic
7	C₂²×C₅×Dic₅	C₄	1	2	yes	—
8	#87867	C₄	1	4	yes	complemented
9	C₂×#6034	C₂²	1	2	yes	—
10	#87860	C₄	1	4	yes	complemented
11	C₂×#6033	C₂²	1	2	yes	—
12	Dic₅²	C₄	1	2	yes	—
13	Dic₅²	C₂²	1	1	yes	characteristic
14	Dic₅²	C₂²	1	1	yes	characteristic
15	C₄×#704	—	5	2	no	maximal
16	C₂³×C₅²	C₂×C₄	1	1	yes	characteristic
17	C₂×#1039	C₂×C₄	1	1	yes	characteristic
18	C₂×#1039	C₂×C₄	1	2	yes	—
19	C₂×#1039	C₂×C₄	1	2	yes	—
20	C₂×#1039	C₂³	1	1	yes	characteristic
21	C₂×C₅×Dic₅	—	2	4	no	—
22	C₂×C₅×Dic₅	C₂×C₄	1	2	yes	—
23	C₂×C₅×Dic₅	C₂×C₄	1	2	yes	—
24	#6034	C₂×C₄	1	4	yes	complemented
25	#6033	C₂×C₄	1	4	yes	complemented
26	C₂×#704	—	5	2	no	—
27	C₂×#704	—	5	2	no	—
28	C₂×#704	—	10	2	no	—
29	C₂×C₄×Dic₅	—	5	2	no	—
30	#4002	—	5	4	no	—
31	C₄×#170	—	5	4	no	—
32	C₂²×C₅²	—	2	2	no	—
33	C₂²×C₅²	C₄²	1	2	yes	—
34	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
35	#1039	C₄²	1	2	yes	—
36	#1039	C₂²×C₄	1	1	yes	characteristic
37	#1039	C₂²×C₄	1	1	yes	characteristic
38	C₅×Dic₅	—	2	2	no	—
39	C₅×Dic₅	M₁₆	1	4	yes	complemented
40	C₂²×C₄×C₅	—	5	2	no	—
41	C₂²×Dic₅	—	5	2	no	—
42	C₂²×Dic₅	—	10	2	no	—
43	C₂²×Dic₅	AGL(1,𝔽₅)	1	2	yes	—
44	#704	—	5	8	no	—
45	#704	—	5	8	no	—
46	#704	—	10	8	no	—
47	C₂×#170	—	5	4	no	—
48	C₂×#170	—	5	4	no	—
49	C₂×#170	—	10	4	no	—
50	C₄×Dic₅	—	5	2	no	—
51	C₄×Dic₅	—	5	2	no	—
52	C₄×Dic₅	—	5	2	no	—
53	C₄×Dic₅	—	5	2	no	—
54	C₄×M₁₆	—	25	1	no	Sylow
55	C₂×C₅²	—	2	2	no	—
56	C₂×C₅²	C₂×C₄²	1	1	yes	characteristic
57	C₂×C₅²	C₂×M₁₆	1	2	yes	—
58	C₂³×C₅	—	2	2	no	—
59	C₂³×C₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
60	C₂×C₄×C₅	—	5	2	no	—
61	C₂×C₄×C₅	—	5	2	no	—
62	C₂×C₄×C₅	—	10	4	no	—
63	C₂×Dic₅	—	2	4	no	—
64	C₂×Dic₅	—	5	4	no	—
65	C₂×Dic₅	—	5	4	no	—
66	C₂×Dic₅	—	5	2	no	—
67	C₂×Dic₅	—	5	2	no	—
68	C₂×Dic₅	—	10	4	no	—
69	C₂×Dic₅	—	10	4	no	—
70	C₂×Dic₅	—	10	2	no	—
71	C₂×Dic₅	—	10	2	no	—
72	C₂×Dic₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
73	C₂×Dic₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
74	#170	—	5	8	no	—
75	#170	—	5	8	no	—
76	#170	—	10	8	no	—
77	C₂×M₁₆	—	25	1	no	—
78	C₂×M₁₆	—	25	1	no	—
79	C₂×C₄²	—	25	1	no	—
80	#97	—	25	2	no	—
81	C₄×C₈	—	25	2	no	—
82	C₅²	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₅	—	2	2	no	—
84	C₂²×C₅	—	2	2	no	—
85	C₂²×C₅	—	2	4	no	—
86	C₂²×C₅	—	2	2	no	—
87	C₂²×C₅	—	4	4	no	—
88	C₂²×C₅	C₄×AGL(1,𝔽₅)	1	4	yes	—
89	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	2	yes	—
90	C₄×C₅	—	5	4	no	—
91	C₄×C₅	—	10	2	no	—
92	Dic₅	—	2	2	no	—
93	Dic₅	—	5	4	no	—
94	Dic₅	—	5	2	no	—
95	Dic₅	—	5	2	no	—
96	Dic₅	—	10	4	no	—
97	Dic₅	—	10	2	no	—
98	Dic₅	—	10	2	no	—
99	Dic₅	#704	1	4	yes	complemented
100	C₂²×C₄	—	5	2	no	—
101	C₂²×C₄	—	25	1	no	—
102	M₁₆	—	25	4	no	—
103	M₁₆	—	25	4	no	—
104	C₂×C₈	—	25	2	no	—
105	C₂×C₈	—	25	2	no	—
106	C₄²	—	25	1	no	—
107	C₄²	—	25	1	no	—
108	C₄²	—	25	2	no	—
109	C₂×C₅	—	2	4	no	—
110	C₂×C₅	—	2	2	no	—
111	C₂×C₅	—	2	4	no	—
112	C₂×C₅	—	4	4	no	—
113	C₂×C₅	D₅⋊M₁₆	1	2	yes	—
114	C₂×C₅	C₂×C₄×AGL(1,𝔽₅)	1	2	yes	—
115	C₂×C₅	C₂×#704	1	2	yes	—
116	C₂³	D₅⋊AGL(1,𝔽₅)	1	1	yes	characteristic
117	C₂×C₄	—	5	2	no	—
118	C₂×C₄	—	5	2	no	—
119	C₂×C₄	—	10	4	no	—
120	C₂×C₄	—	25	2	no	—
121	C₂×C₄	—	25	2	no	—
122	C₂×C₄	—	25	1	no	—
123	C₂×C₄	—	25	1	no	—
124	C₈	—	25	4	no	—
125	C₈	—	25	4	no	—
126	C₅	—	2	2	no	—
127	C₅	C₄×#704	1	2	yes	complemented
128	C₂²	—	2	2	no	—
129	C₂²	#87830	1	2	yes	—
130	C₂²	C₂×D₅⋊AGL(1,𝔽₅)	1	1	yes	characteristic
131	C₄	—	5	4	no	—
132	C₄	—	10	2	no	—
133	C₄	—	25	1	no	—
134	C₄	—	25	1	no	—
135	C₄	—	25	2	no	—
136	C₂	—	2	2	no	—
137	C₂	C₂×#87830	1	1	yes	characteristic
138	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

#11716778

C₂

yes

maximal, complemented

#11716777

C₂

yes

maximal, complemented

C₂²×#1039

C₂²

yes

characteristic

C₂²×C₅×Dic₅

C₄

yes

—

#87867

C₄

yes

complemented

C₂×#6034

C₂²

yes

—

#87860

C₄

yes

complemented

C₂×#6033

C₂²

yes

—

Dic₅²

C₄

yes

—

Dic₅²

C₂²

yes

characteristic

Dic₅²

C₂²

yes

characteristic

C₄×#704

—

maximal

C₂³×C₅²

C₂×C₄

yes

characteristic

C₂×#1039

C₂×C₄

yes

characteristic

C₂×#1039

C₂×C₄

yes

—

C₂×#1039

C₂×C₄

yes

—

C₂×#1039

C₂³

yes

characteristic

C₂×C₅×Dic₅

—

C₂×C₅×Dic₅

C₂×C₄

yes

—

C₂×C₅×Dic₅

C₂×C₄

yes

—

#6034

C₂×C₄

yes

complemented

#6033

C₂×C₄

yes

—

—

—

C₂×C₄×Dic₅

—

#4002

—

C₄×#170

—

C₂²×C₅²

—

C₂²×C₅²

C₄²

yes

—

C₂²×C₅²

C₂²×C₄

yes

characteristic

#1039

C₄²

yes

—

#1039

C₂²×C₄

yes

characteristic

#1039

C₂²×C₄

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

—

C₄×M₁₆

—

Sylow

C₂×C₅²

—

C₂×C₅²

C₂×C₄²

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

—

#97

—

C₄×C₈

—

C₅²

C₄×M₁₆

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

100

C₂²×C₄

—

101

C₂²×C₄

—

102

M₁₆

—

103

M₁₆

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₂×C₅

—

110

C₂×C₅

—

111

C₂×C₅

—

112

C₂×C₅

—

113

C₂×C₅

D₅⋊M₁₆

yes

—

114

C₂×C₅

C₂×C₄×AGL(1,𝔽₅)

yes

—

115

C₂×C₅

C₂×#704

yes

—

116

C₂³

D₅⋊AGL(1,𝔽₅)

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₈

—

125

C₈

—

126

C₅

—

127

C₅

C₄×#704

yes

complemented

128

C₂²

—

129

C₂²

#87830

yes

—

130

C₂²

C₂×D₅⋊AGL(1,𝔽₅)

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₂

—

137

C₂

C₂×#87830

yes

characteristic

138

C₂

D₅⋊#704

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂×D₅⋊AGL(1,𝔽₅)
commutator	C₂×C₅²	C₂×C₄²
Fitting, prime radical	C₂³×C₅²	C₂×C₄
socle	C₂²×C₅²	C₂²×C₄
cosocle	C₂×#1039	C₂³
hypercenter	C₂³	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	C₄×M₁₆
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂³

subgroup

G/H

center, Frattini

C₂²

C₂×D₅⋊AGL(1,𝔽₅)

commutator

C₂×C₅²

C₂×C₄²

Fitting, prime radical

C₂³×C₅²

C₂×C₄

socle

C₂²×C₅²

C₂²×C₄

C₂³

C₂³

C₅²

C₄×M₁₆

LCS

G ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₂³

N	H ≅ G/N	type
4. #11716778	C₂	semidirect
5. #11716777	C₂	semidirect
8. #87867	C₄	semidirect
10. #87860	C₄	semidirect
24. #6034	C₂×C₄	semidirect
25. #6033	C₂×C₄	semidirect
39. C₅×Dic₅	M₁₆	semidirect
82. C₅²	C₄×M₁₆	semidirect
99. Dic₅	#704	semidirect
127. C₅	C₄×#704	semidirect

H ≅ G/N

type

4. #11716778

C₂

semidirect

5. #11716777

C₂

semidirect

8. #87867

C₄

semidirect

10. #87860

C₄

semidirect

24. #6034

C₂×C₄

semidirect

25. #6033

C₂×C₄

semidirect

39. C₅×Dic₅

M₁₆

semidirect

82. C₅²

C₄×M₁₆

semidirect

99. Dic₅

#704

semidirect

127. C₅

C₄×#704

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
#11716778	2
#11716777	2
C₄×#704	5

subgroup

index

C₂×Dic₅²

Subgroups — #13104093

Conjugacy classes of subgroups

Complements

Maximal subgroups