Group #13103965 — subgroups

Subgroups — #13103965

Order 1600.

280 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 65 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	#11717448	C₂	1	1	yes	maximal, characteristic
2	#11717435	C₂	1	1	yes	maximal, characteristic
3	C₂×#87767	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	#87867	—	2	1	no	—
7	#87863	C₂²	1	2	yes	—
8	#87862	—	2	1	no	—
9	#87860	—	2	1	no	—
10	#87856	C₂²	1	2	yes	—
11	#87855	—	2	1	no	—
12	C₂×C₅×#168	C₄	1	2	yes	—
13	#87767	C₄	1	1	yes	characteristic
14	#87767	C₄	1	1	yes	characteristic
15	#87767	C₂²	1	2	yes	—
16	D₅⋊#108	—	5	2	no	maximal
17	C₂²×C₅⋊D₅	Q₈	1	1	yes	characteristic
18	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
19	C₂×#1039	D₄	1	1	yes	characteristic
20	C₄×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
21	C₄×C₅⋊D₅	D₄	1	1	yes	characteristic
22	C₄×C₅⋊D₅	Q₈	1	1	yes	characteristic
23	C₄×C₅⋊D₅	C₂³	1	1	yes	characteristic
24	#6034	—	2	2	no	—
25	#6033	—	2	2	no	—
26	C₅×#168	C₂×C₄	1	2	yes	—
27	C₅×#168	C₂×C₄	1	2	yes	—
28	D₅⋊M₁₆	—	5	2	no	—
29	D₅⋊M₁₆	—	5	2	no	—
30	D₅⋊M₁₆	—	10	2	no	—
31	C₂×C₈×D₅	—	5	2	no	—
32	#3996	—	5	4	no	—
33	#3995	—	5	4	no	—
34	C₂²×C₅²	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₅⋊D₅	—	2	2	no	—
36	C₂×C₅⋊D₅	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₅⋊D₅	C₂×Q₈	1	1	yes	characteristic
38	C₄×C₅²	C₄⋊C₄	1	1	yes	characteristic
39	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
40	#1039	C₄⋊C₄	1	1	yes	characteristic
41	#1039	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×D₅	—	5	2	no	—
43	C₂×C₄×D₅	—	10	2	no	—
44	#704	—	10	2	no	—
45	#704	—	10	2	no	—
46	#704	—	10	4	no	—
47	#700	—	5	4	no	—
48	#700	—	5	4	no	—
49	#700	—	10	4	no	—
50	D₅⋊C₈	—	10	2	no	—
51	D₅⋊C₈	—	10	2	no	—
52	D₅⋊C₈	—	10	4	no	—
53	C₂×C₅×C₈	—	5	2	no	—
54	C₂×#168	AGL(1,𝔽₅)	1	2	yes	—
55	C₈×D₅	—	5	2	no	—
56	C₈×D₅	—	5	2	no	—
57	C₈×D₅	—	5	2	no	—
58	C₈×D₅	—	5	2	no	—
59	C₂×#108	—	25	1	no	Sylow
60	C₂×C₅²	—	2	1	no	—
61	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₅⋊D₅	—	2	1	no	—
63	C₅⋊D₅	#108	1	2	yes	complemented
64	C₂²×D₅	—	5	2	no	—
65	C₂²×D₅	—	10	2	no	—
66	C₂×C₄×C₅	—	2	2	no	—
67	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	2	yes	—
68	C₂×Dic₅	—	5	2	no	—
69	C₂×Dic₅	—	10	2	no	—
70	C₄×D₅	—	5	2	no	—
71	C₄×D₅	—	5	2	no	—
72	C₄×D₅	—	5	2	no	—
73	C₄×D₅	—	5	2	no	—
74	C₄×D₅	—	10	2	no	—
75	C₄×D₅	—	10	2	no	—
76	C₄×D₅	—	10	2	no	—
77	C₄×D₅	—	10	2	no	—
78	#170	—	10	4	no	—
79	#170	—	10	4	no	—
80	#170	—	10	8	no	—
81	C₅×C₈	—	5	2	no	—
82	C₅×C₈	—	5	2	no	—
83	#168	C₂×AGL(1,𝔽₅)	1	2	yes	—
84	#168	C₂×AGL(1,𝔽₅)	1	2	yes	—
85	C₂×M₁₆	—	25	1	no	—
86	C₂×M₁₆	—	25	1	no	—
87	C₂²×C₈	—	25	1	no	—
88	#108	—	25	4	no	—
89	C₅²	C₂×#108	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₅	—	2	2	no	—
91	C₂²×C₅	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
92	C₂×D₅	—	5	2	no	—
93	C₂×D₅	—	5	2	no	—
94	C₂×D₅	—	10	2	no	—
95	C₂×D₅	—	10	2	no	—
96	C₂×D₅	—	10	4	no	—
97	C₂×D₅	—	20	4	no	—
98	C₄×C₅	—	2	2	no	—
99	C₄×C₅	—	2	2	no	—
100	C₄×C₅	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
101	C₄×C₅	C₂²×AGL(1,𝔽₅)	1	2	yes	—
102	Dic₅	—	5	2	no	—
103	Dic₅	—	5	2	no	—
104	Dic₅	—	10	2	no	—
105	Dic₅	—	10	2	no	—
106	C₂²×C₄	—	25	1	no	—
107	M₁₆	—	25	2	no	—
108	M₁₆	—	25	2	no	—
109	M₁₆	—	50	1	no	—
110	M₁₆	—	50	1	no	—
111	C₂×C₈	—	5	2	no	—
112	C₂×C₈	—	25	2	no	—
113	C₂×C₈	—	25	1	no	—
114	C₂×C₈	—	25	1	no	—
115	C₂×C₈	—	50	1	no	—
116	C₂×C₈	—	50	1	no	—
117	C₂×C₅	—	2	2	no	—
118	C₂×C₅	—	2	2	no	—
119	C₂×C₅	—	4	2	no	—
120	C₂×C₅	C₂×C₄⋊AGL(1,𝔽₅)	1	2	yes	—
121	D₅	—	5	4	no	—
122	D₅	—	10	4	no	—
123	D₅	—	10	2	no	—
124	D₅	—	20	2	no	—
125	C₂³	—	25	1	no	—
126	C₂×C₄	—	25	1	no	—
127	C₂×C₄	—	25	1	no	—
128	C₂×C₄	—	25	1	no	—
129	C₂×C₄	—	25	1	no	—
130	C₂×C₄	—	25	1	no	—
131	C₂×C₄	D₅⋊AGL(1,𝔽₅)	1	1	yes	characteristic
132	C₈	—	5	2	no	—
133	C₈	—	5	2	no	—
134	C₈	—	50	2	no	—
135	C₈	—	50	2	no	—
136	C₅	—	2	2	no	—
137	C₅	D₅⋊#108	1	2	yes	complemented
138	C₂²	—	25	1	no	—
139	C₂²	—	25	1	no	—
140	C₂²	—	50	2	no	—
141	C₂²	#87832	1	1	yes	characteristic
142	C₄	—	25	1	no	—
143	C₄	—	25	1	no	—
144	C₄	#87832	1	1	yes	characteristic
145	C₄	C₂×D₅⋊AGL(1,𝔽₅)	1	1	yes	characteristic
146	C₂	—	2	1	no	—
147	C₂	—	25	2	no	—
148	C₂	—	50	1	no	—
149	C₂	C₂×#87832	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11717448

C₂

yes

maximal, characteristic

#11717435

C₂

yes

maximal, characteristic

C₂×#87767

C₂

yes

maximal, characteristic

#11716773

C₂

yes

maximal, complemented

C₂×C₄×C₅⋊D₅

C₂²

yes

characteristic

#87867

—

#87863

C₂²

yes

—

#87862

—

#87860

—

#87856

C₂²

yes

—

#87855

—

C₂×C₅×#168

C₄

yes

—

#87767

C₄

yes

characteristic

#87767

C₄

yes

characteristic

#87767

C₂²

yes

—

D₅⋊#108

—

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

—

C₅×#168

C₂×C₄

yes

—

D₅⋊M₁₆

—

D₅⋊M₁₆

—

D₅⋊M₁₆

—

C₂×C₈×D₅

—

#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

—

D₅⋊C₈

—

C₂×C₅×C₈

—

C₂×#168

AGL(1,𝔽₅)

yes

—

C₈×D₅

—

C₈×D₅

—

C₈×D₅

—

C₈×D₅

—

C₂×#108

—

Sylow

C₂×C₅²

—

C₂×C₅²

C₂×C₄⋊C₄

yes

characteristic

C₅⋊D₅

—

C₅⋊D₅

#108

yes

complemented

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

—

#170

—

#170

—

#170

—

C₅×C₈

—

C₅×C₈

—

#168

C₂×AGL(1,𝔽₅)

yes

—

#168

C₂×AGL(1,𝔽₅)

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

#108

—

C₅²

C₂×#108

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

—

100

C₄×C₅

C₄⋊AGL(1,𝔽₅)

yes

—

101

C₄×C₅

C₂²×AGL(1,𝔽₅)

yes

—

102

Dic₅

—

103

Dic₅

—

104

Dic₅

—

105

Dic₅

—

106

C₂²×C₄

—

107

M₁₆

—

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₅

—

118

C₂×C₅

—

119

C₂×C₅

—

120

C₂×C₅

C₂×C₄⋊AGL(1,𝔽₅)

yes

—

121

D₅

—

122

D₅

—

123

D₅

—

124

D₅

—

125

C₂³

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

D₅⋊AGL(1,𝔽₅)

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₅

—

137

C₅

D₅⋊#108

yes

complemented

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

#87832

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

#87832

yes

characteristic

145

C₄

C₂×D₅⋊AGL(1,𝔽₅)

yes

characteristic

146

C₂

—

147

C₂

—

148

C₂

—

149

C₂

C₂×#87832

yes

characteristic

	subgroup	G/H
center	C₄	#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	C₂×#108
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂×C₄

subgroup

G/H

center

C₄

#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 ◁ C₄ ◁ C₂×C₄

N	H ≅ G/N	type
4. #11716773	C₂	semidirect
63. C₅⋊D₅	#108	semidirect
89. C₅²	C₂×#108	semidirect
137. C₅	D₅⋊#108	semidirect

H ≅ G/N

type

4. #11716773

C₂

semidirect

63. C₅⋊D₅

#108

semidirect

89. C₅²

C₂×#108

semidirect

137. C₅

D₅⋊#108

semidirect

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

subgroup	index
#11717448	2
#11717435	2
C₂×#87767	2
#11716773	2
D₅⋊#108	5

subgroup

index

Subgroups — #13103965

Conjugacy classes of subgroups

Complements

Maximal subgroups