Group #13101155 — subgroups

Subgroups — #13101155

Order 1600.

272 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 70 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	M₁₆×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×#87769	C₂	1	1	yes	maximal, characteristic
3	#11716936	C₂	1	1	yes	maximal, characteristic
4	C₅×#3954	C₂	1	2	yes	maximal, complemented
5	#11716618	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
7	C₅²×M₁₆	C₂²	1	1	yes	characteristic
8	#87804	C₂²	1	1	yes	characteristic
9	#87799	—	2	1	no	—
10	C₈×C₅⋊D₅	—	2	1	no	—
11	C₅×#681	C₂²	1	2	yes	—
12	C₂×C₅×#168	C₂²	1	2	yes	—
13	#87769	—	2	1	no	—
14	#87769	C₄	1	2	yes	—
15	#87769	C₄	1	2	yes	—
16	#87767	—	2	1	no	—
17	#65182	—	5	2	no	maximal
18	C₂²×C₅⋊D₅	Q₈	1	1	yes	characteristic
19	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
20	C₂×#1039	Q₈	1	1	yes	characteristic
21	C₄×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
22	C₄×C₅⋊D₅	C₂×C₄	1	1	yes	characteristic
23	C₄×C₅⋊D₅	D₄	1	1	yes	characteristic
24	C₄×C₅⋊D₅	D₄	1	1	yes	characteristic
25	C₅²×C₈	—	2	1	no	—
26	#6029	—	2	1	no	—
27	C₅×#168	—	2	2	no	—
28	C₅×#168	C₂×C₄	1	4	yes	—
29	D₅×M₁₆	—	5	2	no	—
30	D₅×M₁₆	—	5	2	no	—
31	D₅×M₁₆	—	10	2	no	—
32	C₂×#676	—	5	2	no	—
33	C₅×#108	—	5	2	no	—
34	#3954	—	5	2	no	—
35	#3954	D₅	1	2	yes	complemented
36	#3951	—	5	2	no	—
37	C₂²×C₅²	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₅⋊D₅	—	4	1	no	—
39	C₂×C₅⋊D₅	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₅⋊D₅	C₄⋊C₄	1	1	yes	characteristic
41	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
42	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
43	#1039	C₄⋊C₄	1	1	yes	characteristic
44	#1039	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄×D₅	—	5	2	no	—
46	C₂×C₄×D₅	—	10	2	no	—
47	C₅×M₁₆	—	2	2	no	—
48	C₅×M₁₆	—	5	2	no	—
49	C₅×M₁₆	C₂×D₅	1	2	yes	—
50	C₂×C₅×C₈	—	5	2	no	—
51	#681	—	5	2	no	—
52	#681	—	10	2	no	—
53	#681	C₂×D₅	1	2	yes	—
54	C₂×#168	C₂×D₅	1	2	yes	—
55	#676	—	5	4	no	—
56	#676	—	5	4	no	—
57	#676	—	10	2	no	—
58	#676	—	10	2	no	—
59	#676	—	10	4	no	—
60	C₈×D₅	—	10	2	no	—
61	C₈×D₅	—	10	2	no	—
62	C₈×D₅	—	10	4	no	—
63	#430	—	25	1	no	Sylow
64	C₂×C₅²	—	2	1	no	—
65	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₅⋊D₅	—	2	1	no	—
67	C₅⋊D₅	—	2	1	no	—
68	C₂²×D₅	—	5	2	no	—
69	C₂²×D₅	—	10	2	no	—
70	C₂×C₄×C₅	—	2	2	no	—
71	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
72	C₂×Dic₅	—	5	2	no	—
73	C₂×Dic₅	—	10	2	no	—
74	C₄×D₅	—	5	2	no	—
75	C₄×D₅	—	5	2	no	—
76	C₄×D₅	—	5	2	no	—
77	C₄×D₅	—	5	2	no	—
78	C₄×D₅	—	10	2	no	—
79	C₄×D₅	—	10	2	no	—
80	C₄×D₅	—	10	2	no	—
81	C₄×D₅	—	10	2	no	—
82	C₅×C₈	—	2	4	no	—
83	C₅×C₈	—	2	2	no	—
84	C₅×C₈	—	5	4	no	—
85	C₅×C₈	—	10	2	no	—
86	#168	—	2	2	no	—
87	#168	—	10	4	no	—
88	#168	—	10	2	no	—
89	#168	C₄×D₅	1	4	yes	—
90	C₂×M₁₆	—	25	1	no	—
91	C₂×M₁₆	—	25	1	no	—
92	C₂×M₁₆	—	25	1	no	—
93	#108	—	5	2	no	—
94	#108	—	25	2	no	—
95	C₅²	#430	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₅	—	2	2	no	—
97	C₂²×C₅	Q₈×D₅	1	2	yes	—
98	C₂×D₅	—	5	2	no	—
99	C₂×D₅	—	5	2	no	—
100	C₂×D₅	—	10	2	no	—
101	C₂×D₅	—	10	2	no	—
102	C₂×D₅	—	20	2	no	—
103	C₂×D₅	—	20	4	no	—
104	C₄×C₅	—	2	2	no	—
105	C₄×C₅	—	2	2	no	—
106	C₄×C₅	C₂×C₄×D₅	1	2	yes	—
107	C₄×C₅	D₄×D₅	1	2	yes	—
108	Dic₅	—	5	2	no	—
109	Dic₅	—	5	2	no	—
110	Dic₅	—	10	2	no	—
111	Dic₅	—	10	2	no	—
112	C₂²×C₄	—	25	1	no	—
113	M₁₆	—	5	2	no	—
114	M₁₆	—	25	2	no	—
115	M₁₆	—	25	2	no	—
116	M₁₆	—	25	1	no	—
117	M₁₆	—	50	1	no	—
118	M₁₆	—	50	1	no	—
119	M₁₆	D₅²	1	1	yes	characteristic
120	C₂×C₈	—	5	2	no	—
121	C₂×C₈	—	50	1	no	—
122	C₂×C₈	—	50	1	no	—
123	C₂×C₅	—	2	2	no	—
124	C₂×C₅	—	2	2	no	—
125	C₂×C₅	—	4	2	no	—
126	C₂×C₅	D₅×C₄⋊C₄	1	2	yes	—
127	D₅	—	10	4	no	—
128	D₅	—	10	2	no	—
129	D₅	—	10	2	no	—
130	D₅	—	20	2	no	—
131	C₂³	—	25	1	no	—
132	C₂×C₄	—	25	1	no	—
133	C₂×C₄	—	25	1	no	—
134	C₂×C₄	—	25	1	no	—
135	C₂×C₄	—	25	1	no	—
136	C₂×C₄	—	25	1	no	—
137	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
138	C₈	—	2	1	no	—
139	C₈	—	5	4	no	—
140	C₈	—	10	2	no	—
141	C₈	—	50	1	no	—
142	C₅	—	2	2	no	—
143	C₅	#65182	1	2	yes	complemented
144	C₂²	—	25	1	no	—
145	C₂²	—	25	1	no	—
146	C₂²	—	100	1	no	—
147	C₂²	#87872	1	1	yes	characteristic
148	C₄	—	25	1	no	—
149	C₄	—	25	1	no	—
150	C₄	C₄×D₅²	1	1	yes	characteristic
151	C₄	#87886	1	1	yes	characteristic
152	C₂	—	2	1	no	—
153	C₂	—	50	1	no	—
154	C₂	—	50	1	no	—
155	C₂	#11717001	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

M₁₆×C₅⋊D₅

C₂

yes

maximal, characteristic

C₂×#87769

C₂

yes

maximal, characteristic

#11716936

C₂

yes

maximal, characteristic

C₅×#3954

C₂

yes

maximal, complemented

#11716618

C₂

yes

maximal, complemented

C₂×C₄×C₅⋊D₅

C₂²

yes

characteristic

C₅²×M₁₆

C₂²

yes

characteristic

#87804

C₂²

yes

characteristic

#87799

—

C₈×C₅⋊D₅

—

C₅×#681

C₂²

yes

—

C₂×C₅×#168

C₂²

yes

—

#87769

—

#87769

C₄

yes

—

#87769

C₄

yes

—

#87767

—

#65182

—

maximal

C₂²×C₅⋊D₅

Q₈

yes

characteristic

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

Q₈

yes

characteristic

C₄×C₅⋊D₅

C₂×C₄

yes

characteristic

C₄×C₅⋊D₅

C₂×C₄

yes

characteristic

C₄×C₅⋊D₅

D₄

yes

characteristic

C₄×C₅⋊D₅

D₄

yes

characteristic

C₅²×C₈

—

#6029

—

C₅×#168

—

C₅×#168

C₂×C₄

yes

—

D₅×M₁₆

—

D₅×M₁₆

—

D₅×M₁₆

—

—

—

—

D₅

yes

complemented

#3951

—

C₂²×C₅²

C₂×Q₈

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

C₄⋊C₄

yes

characteristic

C₂×C₅⋊D₅

C₄⋊C₄

yes

characteristic

C₄×C₅²

C₂²×C₄

yes

characteristic

C₄×C₅²

C₂×D₄

yes

characteristic

#1039

C₄⋊C₄

yes

characteristic

#1039

C₄⋊C₄

yes

characteristic

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₅×M₁₆

—

C₅×M₁₆

—

C₅×M₁₆

C₂×D₅

yes

—

C₂×C₅×C₈

—

#681

—

#681

—

#681

C₂×D₅

yes

—

C₂×#168

C₂×D₅

yes

—

—

—

—

—

—

C₈×D₅

—

C₈×D₅

—

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

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

—

—

—

C₄×D₅

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

#108

—

#108

—

C₅²

#430

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

Q₈×D₅

yes

—

C₂×D₅

—

C₂×D₅

—

100

C₂×D₅

—

101

C₂×D₅

—

102

C₂×D₅

—

103

C₂×D₅

—

104

C₄×C₅

—

105

C₄×C₅

—

106

C₄×C₅

C₂×C₄×D₅

yes

—

107

C₄×C₅

D₄×D₅

yes

—

108

Dic₅

—

109

Dic₅

—

110

Dic₅

—

111

Dic₅

—

112

C₂²×C₄

—

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

M₁₆

—

117

M₁₆

—

118

M₁₆

—

119

M₁₆

D₅²

yes

characteristic

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₅

—

124

C₂×C₅

—

125

C₂×C₅

—

126

C₂×C₅

D₅×C₄⋊C₄

yes

—

127

D₅

—

128

D₅

—

129

D₅

—

130

D₅

—

131

C₂³

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₂×D₅²

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₅

—

143

C₅

#65182

yes

complemented

144

C₂²

—

145

C₂²

—

146

C₂²

—

100

—

147

C₂²

#87872

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

C₄×D₅²

yes

characteristic

151

C₄

#87886

yes

characteristic

152

C₂

—

153

C₂

—

154

C₂

—

155

C₂

#11717001

yes

characteristic

	subgroup	G/H
center	C₄	#87886
commutator	C₄×C₅²	C₂²×C₄
Frattini	C₂×C₄	C₂×D₅²
Fitting	C₅²×M₁₆	C₂²
socle	C₂×C₅²	C₂×C₄⋊C₄
prime radical	C₂²×C₅⋊D₅	Q₈
cosocle	C₂×C₄×C₅²	C₂³
hypercenter	M₁₆	D₅²
hypocenter	C₅²	#430
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ M₁₆

subgroup

G/H

center

C₄

#87886

commutator

C₄×C₅²

C₂²×C₄

Frattini

C₂×C₄

C₂×D₅²

Fitting

C₅²×M₁₆

C₂²

socle

C₂×C₅²

C₂×C₄⋊C₄

prime radical

C₂²×C₅⋊D₅

Q₈

cosocle

C₂×C₄×C₅²

C₂³

hypercenter

M₁₆

D₅²

hypocenter

C₅²

#430

LCS

G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ M₁₆

N	H ≅ G/N	type
4. C₅×#3954	C₂	semidirect
5. #11716618	C₂	semidirect
35. #3954	D₅	semidirect
95. C₅²	#430	semidirect
143. C₅	#65182	semidirect

H ≅ G/N

type

4. C₅×#3954

C₂

semidirect

5. #11716618

C₂

semidirect

35. #3954

D₅

semidirect

95. C₅²

#430

semidirect

143. C₅

#65182

semidirect

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

subgroup	index
M₁₆×C₅⋊D₅	2
C₂×#87769	2
#11716936	2
C₅×#3954	2
#11716618	2
#65182	5

subgroup

index

M₁₆×C₅⋊D₅

Subgroups — #13101155

Conjugacy classes of subgroups

Complements

Maximal subgroups