Group #13103967 — subgroups

Subgroups — #13103967

Order 1600.

272 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 63 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×#87863	C₂	1	1	yes	maximal, characteristic
2	#11717435	C₂	1	1	yes	maximal, characteristic
3	#11716936	C₂	1	1	yes	maximal, characteristic
4	#11716773	C₂	1	4	yes	maximal, complemented
5	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
6	C₂×#6034	—	2	1	no	—
7	#87863	—	2	1	no	—
8	#87863	C₂²	1	2	yes	—
9	#87860	—	2	1	no	—
10	#87856	C₂²	1	2	yes	—
11	#87855	—	2	1	no	—
12	C₅×#681	C₄	1	2	yes	—
13	#87769	C₄	1	2	yes	—
14	#87767	C₂²	1	2	yes	—
15	#65729	—	5	2	no	maximal
16	C₂²×C₅⋊D₅	Q₈	1	1	yes	characteristic
17	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
18	C₂×#1039	D₄	1	1	yes	characteristic
19	C₄×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
20	C₄×C₅⋊D₅	D₄	1	1	yes	characteristic
21	C₄×C₅⋊D₅	Q₈	1	1	yes	characteristic
22	C₄×C₅⋊D₅	C₂³	1	1	yes	characteristic
23	#6034	—	2	2	no	—
24	#6033	—	2	2	no	—
25	C₅×#168	C₂×C₄	1	4	yes	—
26	D₅⋊M₁₆	—	5	2	no	—
27	D₅⋊M₁₆	—	10	2	no	—
28	C₂×#700	—	5	2	no	—
29	D₅×M₁₆	—	5	2	no	—
30	#3996	—	5	4	no	—
31	#3995	—	5	4	no	—
32	C₂²×C₅²	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₅⋊D₅	—	4	1	no	—
34	C₂×C₅⋊D₅	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₅⋊D₅	C₂×Q₈	1	1	yes	characteristic
36	C₄×C₅²	C₄⋊C₄	1	1	yes	characteristic
37	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
38	#1039	C₄⋊C₄	1	1	yes	characteristic
39	#1039	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×D₅	—	5	2	no	—
41	C₂×C₄×D₅	—	10	2	no	—
42	#704	—	10	2	no	—
43	#704	—	10	4	no	—
44	C₂×#170	—	10	2	no	—
45	#700	—	5	4	no	—
46	#700	—	5	2	no	—
47	#700	—	5	2	no	—
48	#700	—	10	4	no	—
49	#700	—	10	2	no	—
50	D₅⋊C₈	—	10	2	no	—
51	D₅⋊C₈	—	10	4	no	—
52	C₅×M₁₆	—	5	2	no	—
53	#681	AGL(1,𝔽₅)	1	2	yes	—
54	#676	—	5	4	no	—
55	C₈×D₅	—	5	4	no	—
56	#430	—	25	1	no	Sylow
57	C₂×C₅²	—	2	1	no	—
58	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₅⋊D₅	—	2	1	no	—
60	C₅⋊D₅	—	2	1	no	—
61	C₂²×D₅	—	5	2	no	—
62	C₂²×D₅	—	10	2	no	—
63	C₂×C₄×C₅	—	2	2	no	—
64	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
65	C₂×Dic₅	—	5	2	no	—
66	C₂×Dic₅	—	10	2	no	—
67	C₄×D₅	—	5	2	no	—
68	C₄×D₅	—	5	2	no	—
69	C₄×D₅	—	5	2	no	—
70	C₄×D₅	—	5	2	no	—
71	C₄×D₅	—	10	2	no	—
72	C₄×D₅	—	10	2	no	—
73	C₄×D₅	—	10	2	no	—
74	C₄×D₅	—	10	2	no	—
75	#170	—	10	2	no	—
76	#170	—	10	2	no	—
77	#170	—	10	4	no	—
78	#170	—	10	8	no	—
79	C₅×C₈	—	5	4	no	—
80	#168	C₂×AGL(1,𝔽₅)	1	4	yes	—
81	C₂×M₁₆	—	25	1	no	—
82	C₂×M₁₆	—	25	1	no	—
83	C₂×M₁₆	—	25	1	no	—
84	#108	—	25	4	no	—
85	C₅²	#430	1	1	yes	characteristic, complemented, Sylow
86	C₂²×C₅	—	2	2	no	—
87	C₂²×C₅	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
88	C₂×D₅	—	5	2	no	—
89	C₂×D₅	—	5	2	no	—
90	C₂×D₅	—	10	2	no	—
91	C₂×D₅	—	10	2	no	—
92	C₂×D₅	—	20	2	no	—
93	C₂×D₅	—	20	4	no	—
94	C₄×C₅	—	2	2	no	—
95	C₄×C₅	—	2	2	no	—
96	C₄×C₅	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
97	C₄×C₅	C₂²×AGL(1,𝔽₅)	1	2	yes	—
98	Dic₅	—	5	2	no	—
99	Dic₅	—	5	2	no	—
100	Dic₅	—	10	2	no	—
101	Dic₅	—	10	2	no	—
102	C₂²×C₄	—	25	1	no	—
103	M₁₆	—	5	2	no	—
104	M₁₆	—	25	2	no	—
105	M₁₆	—	25	2	no	—
106	M₁₆	—	25	2	no	—
107	M₁₆	—	50	1	no	—
108	M₁₆	—	50	1	no	—
109	C₂×C₈	—	25	2	no	—
110	C₂×C₈	—	50	1	no	—
111	C₂×C₈	—	50	1	no	—
112	C₂×C₅	—	2	2	no	—
113	C₂×C₅	—	2	2	no	—
114	C₂×C₅	—	4	2	no	—
115	C₂×C₅	C₂×C₄⋊AGL(1,𝔽₅)	1	2	yes	—
116	D₅	—	10	4	no	—
117	D₅	—	10	2	no	—
118	D₅	—	10	2	no	—
119	D₅	—	20	2	no	—
120	C₂³	—	25	1	no	—
121	C₂×C₄	—	25	1	no	—
122	C₂×C₄	—	25	1	no	—
123	C₂×C₄	—	25	1	no	—
124	C₂×C₄	—	25	1	no	—
125	C₂×C₄	—	25	1	no	—
126	C₂×C₄	D₅⋊AGL(1,𝔽₅)	1	1	yes	characteristic
127	C₈	—	5	4	no	—
128	C₈	—	50	2	no	—
129	C₈	—	50	2	no	—
130	C₅	—	2	2	no	—
131	C₅	#65729	1	2	yes	complemented
132	C₂²	—	25	1	no	—
133	C₂²	—	25	1	no	—
134	C₂²	—	100	1	no	—
135	C₂²	#87832	1	1	yes	characteristic
136	C₄	—	25	1	no	—
137	C₄	—	25	1	no	—
138	C₄	#87832	1	1	yes	characteristic
139	C₄	C₂×D₅⋊AGL(1,𝔽₅)	1	1	yes	characteristic
140	C₂	—	2	1	no	—
141	C₂	—	50	1	no	—
142	C₂	—	50	1	no	—
143	C₂	C₂×#87832	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#87863

C₂

yes

maximal, characteristic

#11717435

C₂

yes

maximal, characteristic

#11716936

C₂

yes

maximal, characteristic

#11716773

C₂

yes

maximal, complemented

C₂×C₄×C₅⋊D₅

C₂²

yes

—

—

C₂²

yes

—

#87860

—

#87856

C₂²

yes

—

#87855

—

C₅×#681

C₄

yes

—

#87769

C₄

yes

—

#87767

C₂²

yes

—

#65729

—

maximal

C₂²×C₅⋊D₅

Q₈

yes

characteristic

C₂×C₄×C₅²

C₂×C₄

yes

characteristic

C₂×#1039

D₄

yes

characteristic

C₄×C₅⋊D₅

C₂×C₄

yes

characteristic

C₄×C₅⋊D₅

D₄

yes

characteristic

C₄×C₅⋊D₅

Q₈

yes

characteristic

C₄×C₅⋊D₅

C₂³

yes

—

—

C₂×C₄

yes

—

D₅⋊M₁₆

—

D₅⋊M₁₆

—

C₂×#700

—

D₅×M₁₆

—

#3996

—

#3995

—

C₂²×C₅²

C₄⋊C₄

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

C₄⋊C₄

yes

characteristic

C₂×C₅⋊D₅

C₂×Q₈

yes

characteristic

C₄×C₅²

C₄⋊C₄

yes

characteristic

C₄×C₅²

C₂²×C₄

yes

characteristic

#1039

C₄⋊C₄

yes

characteristic

#1039

C₂×D₄

yes

characteristic

C₂×C₄×D₅

—

C₂×C₄×D₅

—

—

—

—

—

—

—

—

—

D₅⋊C₈

—

D₅⋊C₈

—

C₅×M₁₆

—

#681

AGL(1,𝔽₅)

yes

—

#676

—

C₈×D₅

—

#430

—

Sylow

C₂×C₅²

—

C₂×C₅²

C₂×C₄⋊C₄

yes

characteristic

C₅⋊D₅

—

C₅⋊D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×AGL(1,𝔽₅)

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

—

—

—

—

C₅×C₈

—

#168

C₂×AGL(1,𝔽₅)

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

#108

—

C₅²

#430

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

C₄⋊AGL(1,𝔽₅)

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₄⋊AGL(1,𝔽₅)

yes

—

C₄×C₅

C₂²×AGL(1,𝔽₅)

yes

—

Dic₅

—

Dic₅

—

100

Dic₅

—

101

Dic₅

—

102

C₂²×C₄

—

103

M₁₆

—

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₅

—

113

C₂×C₅

—

114

C₂×C₅

—

115

C₂×C₅

C₂×C₄⋊AGL(1,𝔽₅)

yes

—

116

D₅

—

117

D₅

—

118

D₅

—

119

D₅

—

120

C₂³

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

D₅⋊AGL(1,𝔽₅)

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

130

C₅

—

131

C₅

#65729

yes

complemented

132

C₂²

—

133

C₂²

—

134

C₂²

—

100

—

135

C₂²

#87832

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

#87832

yes

characteristic

139

C₄

C₂×D₅⋊AGL(1,𝔽₅)

yes

characteristic

140

C₂

—

141

C₂

—

142

C₂

—

143

C₂

C₂×#87832

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#87832
commutator	C₄×C₅²	C₂²×C₄
Frattini	C₄	C₂×D₅⋊AGL(1,𝔽₅)
Fitting	C₂×C₄×C₅²	C₂×C₄
socle	C₂×C₅²	C₂×C₄⋊C₄
prime radical	C₂²×C₅⋊D₅	Q₈
cosocle	C₄×C₅⋊D₅	C₂³
hypercenter	C₂×C₄	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	#430
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄

subgroup

G/H

center

C₂

C₂×#87832

commutator

C₄×C₅²

C₂²×C₄

Frattini

C₄

C₂×D₅⋊AGL(1,𝔽₅)

Fitting

C₂×C₄×C₅²

C₂×C₄

socle

C₂×C₅²

C₂×C₄⋊C₄

prime radical

C₂²×C₅⋊D₅

Q₈

cosocle

C₄×C₅⋊D₅

C₂³

C₂×C₄

C₅²

G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂ ◁ C₂×C₄

N	H ≅ G/N	type
4. #11716773	C₂	semidirect
85. C₅²	#430	semidirect
131. C₅	#65729	semidirect

H ≅ G/N

type

4. #11716773

C₂

semidirect

85. C₅²

#430

semidirect

131. C₅

#65729

semidirect

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

subgroup	index
C₂×#87863	2
#11717435	2
#11716936	2
#11716773	2
#65729	5

subgroup

index

Subgroups — #13103967

Conjugacy classes of subgroups

Complements

Maximal subgroups