Group #13100901 — subgroups

Subgroups — #13100901

Order 1600.

288 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 74 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	C₄×#6037	C₂	1	1	yes	maximal, characteristic
2	C₂×#87768	C₂	1	2	yes	maximal, complemented
3	C₈×#1039	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#3952	C₂	1	2	yes	maximal, complemented
5	#11716613	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×C₅×D₅	C₂²	1	2	yes	—
7	C₂×#6037	C₄	1	1	yes	characteristic
8	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
9	C₄×#1039	C₂²	1	1	yes	characteristic
10	C₂×#6029	C₂²	1	1	yes	characteristic
11	C₂×C₅×#168	C₂²	1	2	yes	—
12	#87781	C₄	1	1	yes	characteristic
13	#87778	C₄	1	2	yes	—
14	#87768	—	2	4	no	—
15	#65018	—	5	2	no	maximal
16	C₂²×C₅×D₅	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
18	C₂×#1039	C₂×C₄	1	1	yes	characteristic
19	C₂×#1039	C₂×C₄	1	1	yes	characteristic
20	C₂×C₅×Dic₅	C₂×C₄	1	2	yes	—
21	C₄×C₅×D₅	—	2	2	no	—
22	C₄×C₅×D₅	—	2	2	no	—
23	#6037	—	2	2	no	—
24	C₅²×C₈	—	2	1	no	—
25	#6029	D₄	1	2	yes	complemented
26	C₅×#168	—	2	2	no	—
27	C₄×C₂²⋊D₅	—	5	2	no	—
28	C₂×#681	—	5	2	no	—
29	C₂×#676	—	5	2	no	—
30	C₅×C₂²⋊C₈	—	5	2	no	—
31	#3952	D₅	1	2	yes	complemented
32	#3946	—	5	2	no	—
33	C₈×Dic₅	—	5	2	no	—
34	C₈×Dic₅	—	10	2	no	—
35	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₅×D₅	—	2	2	no	—
37	C₂×C₅×D₅	—	2	2	no	—
38	C₂×C₅×D₅	—	2	2	no	—
39	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
40	C₄×C₅²	#41	1	1	yes	characteristic
41	#1039	—	2	1	no	—
42	#1039	M₁₆	1	2	yes	complemented
43	C₅×Dic₅	—	2	2	no	—
44	C₂²×C₄×C₅	—	5	2	no	—
45	C₂×C₂²⋊D₅	—	5	2	no	—
46	C₂×C₄×D₅	C₂×D₅	1	2	yes	—
47	C₂×C₅×C₈	—	2	2	no	—
48	C₂×C₅×C₈	—	5	2	no	—
49	C₂×C₅×C₈	C₂×D₅	1	2	yes	—
50	C₂²⋊Dic₅	—	5	2	no	—
51	#685	—	5	2	no	—
52	#683	—	5	2	no	—
53	C₄×Dic₅	—	5	2	no	—
54	C₄×Dic₅	—	10	2	no	—
55	#681	—	10	4	no	—
56	C₂×#168	—	5	2	no	—
57	C₂×#168	—	10	2	no	—
58	C₂×#168	C₂×D₅	1	2	yes	—
59	#676	—	10	4	no	—
60	#436	—	25	1	no	Sylow
61	C₂×C₅²	C₄×D₄	1	1	yes	characteristic
62	C₂×C₅²	C₂×M₁₆	1	1	yes	characteristic
63	C₂×C₅²	#131	1	1	yes	characteristic
64	C₅×D₅	—	4	2	no	—
65	C₂³×C₅	—	5	2	no	—
66	C₂²×D₅	C₄×D₅	1	2	yes	—
67	C₂×C₄×C₅	—	2	2	no	—
68	C₂×C₄×C₅	—	5	2	no	—
69	C₂×C₄×C₅	—	10	2	no	—
70	C₂×C₄×C₅	—	10	2	no	—
71	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
72	C₂²⋊D₅	—	10	4	no	—
73	C₂×Dic₅	—	5	2	no	—
74	C₂×Dic₅	—	5	2	no	—
75	C₂×Dic₅	—	10	2	no	—
76	C₂×Dic₅	—	10	2	no	—
77	C₂×Dic₅	C₄×D₅	1	2	yes	—
78	C₄×D₅	—	2	2	no	—
79	C₄×D₅	—	2	2	no	—
80	C₅×C₈	—	2	4	no	—
81	C₅×C₈	—	2	2	no	—
82	C₅×C₈	—	10	2	no	—
83	#168	—	2	2	no	—
84	#168	—	5	4	no	—
85	#168	—	10	4	no	—
86	C₂×M₁₆	—	25	2	no	—
87	C₄×D₄	—	25	1	no	—
88	C₄⋊C₈	—	25	1	no	—
89	C₂²⋊C₈	—	5	2	no	—
90	C₄×C₈	—	25	1	no	—
91	C₅²	#436	1	1	yes	characteristic, complemented, Sylow
92	C₂²×C₅	—	2	2	no	—
93	C₂²×C₅	—	10	2	no	—
94	C₂²×C₅	—	10	2	no	—
95	C₂²×C₅	—	10	2	no	—
96	C₂²×C₅	C₂×C₄×D₅	1	2	yes	—
97	C₂×D₅	—	2	2	no	—
98	C₂×D₅	—	2	2	no	—
99	C₂×D₅	—	2	2	no	—
100	C₄×C₅	—	2	2	no	—
101	C₄×C₅	—	2	2	no	—
102	C₄×C₅	—	10	2	no	—
103	C₄×C₅	D₄×D₅	1	2	yes	—
104	C₄×C₅	#711	1	2	yes	—
105	Dic₅	—	2	2	no	—
106	Dic₅	—	5	4	no	—
107	Dic₅	—	10	4	no	—
108	Dic₅	—	10	4	no	—
109	Dic₅	—	10	2	no	—
110	C₂×D₄	—	25	1	no	—
111	C₂²×C₄	—	5	2	no	—
112	M₁₆	—	50	4	no	—
113	C₂×C₈	—	5	2	no	—
114	C₂×C₈	—	25	1	no	—
115	C₂×C₈	D₅²	1	1	yes	characteristic
116	C₄⋊C₄	—	25	1	no	—
117	C₂²⋊C₄	—	25	2	no	—
118	C₄²	—	25	1	no	—
119	C₂×C₅	—	2	2	no	—
120	C₂×C₅	—	2	2	no	—
121	C₂×C₅	—	2	2	no	—
122	C₂×C₅	—	20	2	no	—
123	C₂×C₅	#4027	1	2	yes	—
124	C₂×C₅	#4047	1	2	yes	—
125	C₂×C₅	D₅×M₁₆	1	2	yes	—
126	D₅	—	4	2	no	—
127	C₂³	—	5	2	no	—
128	D₄	—	50	2	no	—
129	C₂×C₄	—	5	2	no	—
130	C₂×C₄	—	10	2	no	—
131	C₂×C₄	—	10	2	no	—
132	C₂×C₄	—	25	1	no	—
133	C₂×C₄	—	25	1	no	—
134	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
135	C₈	—	2	1	no	—
136	C₈	—	10	2	no	—
137	C₈	—	25	2	no	—
138	C₅	—	2	2	no	—
139	C₅	#65018	1	2	yes	complemented
140	C₂²	—	10	2	no	—
141	C₂²	—	10	2	no	—
142	C₂²	—	10	2	no	—
143	C₂²	C₄×D₅²	1	1	yes	characteristic
144	C₄	—	10	2	no	—
145	C₄	—	25	2	no	—
146	C₄	—	50	1	no	—
147	C₄	#87877	1	1	yes	characteristic
148	C₄	#87880	1	1	yes	characteristic
149	C₂	—	20	2	no	—
150	C₂	#11716909	1	1	yes	characteristic
151	C₂	#11716921	1	1	yes	characteristic
152	C₂	#11717020	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#6037

C₂

yes

maximal, characteristic

C₂×#87768

C₂

yes

maximal, complemented

C₈×#1039

C₂

yes

maximal, characteristic, complemented

C₅×#3952

C₂

yes

maximal, complemented

#11716613

C₂

yes

maximal, characteristic, complemented

C₂×C₄×C₅×D₅

C₂²

yes

—

C₂×#6037

C₄

yes

characteristic

C₂×C₅²×C₈

C₂²

yes

characteristic

C₄×#1039

C₂²

yes

characteristic

C₂×#6029

C₂²

yes

characteristic

C₂×C₅×#168

C₂²

yes

—

#87781

C₄

yes

characteristic

#87778

C₄

yes

—

#87768

—

#65018

—

maximal

C₂²×C₅×D₅

C₂×C₄

yes

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

C₂×C₄

yes

characteristic

C₂×#1039

C₂×C₄

yes

characteristic

C₂×C₅×Dic₅

C₂×C₄

yes

—

C₄×C₅×D₅

—

C₄×C₅×D₅

—

#6037

—

C₅²×C₈

—

#6029

D₄

yes

complemented

C₅×#168

—

C₄×C₂²⋊D₅

—

C₂×#681

—

C₂×#676

—

C₅×C₂²⋊C₈

—

#3952

D₅

yes

complemented

#3946

—

C₈×Dic₅

—

C₈×Dic₅

—

C₂²×C₅²

C₂²×C₄

yes

characteristic

C₂×C₅×D₅

—

C₂×C₅×D₅

—

C₂×C₅×D₅

—

C₄×C₅²

C₂×D₄

yes

C₄×C₅²

yes

—

M₁₆

yes

complemented

C₅×Dic₅

—

C₂²×C₄×C₅

—

C₂×C₂²⋊D₅

—

C₂×C₄×D₅

C₂×D₅

yes

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂×D₅

yes

—

C₂²⋊Dic₅

—

#685

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

—

—

—

—

C₂×D₅

yes

—

#676

—

#436

—

Sylow

C₂×C₅²

C₄×D₄

yes

characteristic

C₂×C₅²

C₂×M₁₆

yes

characteristic

C₂×C₅²

#131

yes

characteristic

C₅×D₅

—

C₂³×C₅

—

C₂²×D₅

C₄×D₅

yes

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂²⋊D₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

C₄×D₅

yes

—

C₄×D₅

—

C₄×D₅

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

#168

—

#168

—

#168

—

C₂×M₁₆

—

C₄×D₄

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₅²

#436

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂×C₄×D₅

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

100

C₄×C₅

—

101

C₄×C₅

—

102

C₄×C₅

—

103

C₄×C₅

D₄×D₅

yes

—

104

C₄×C₅

#711

yes

—

105

Dic₅

—

106

Dic₅

—

107

Dic₅

—

108

Dic₅

—

109

Dic₅

—

110

C₂×D₄

—

111

C₂²×C₄

—

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

D₅²

yes

characteristic

116

C₄⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₂×C₅

—

120

C₂×C₅

—

121

C₂×C₅

—

122

C₂×C₅

—

123

C₂×C₅

#4027

yes

—

124

C₂×C₅

#4047

yes

—

125

C₂×C₅

D₅×M₁₆

yes

—

126

D₅

—

127

C₂³

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

C₂×D₅²

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

—

138

C₅

—

139

C₅

#65018

yes

complemented

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₄×D₅²

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

#87877

yes

characteristic

148

C₄

#87880

yes

characteristic

149

C₂

—

150

C₂

#11716909

yes

characteristic

151

C₂

#11716921

yes

characteristic

152

C₂

#11717020

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×D₅²
commutator, socle	C₂²×C₅²	C₂²×C₄
Fitting	C₂×C₅²×C₈	C₂²
prime radical	C₂×#6037	C₄
cosocle	C₂×C₄×C₅²	C₂³
hypercenter	C₂×C₈	D₅²
hypocenter	C₅²	#436
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂×D₅²

commutator, socle

C₂²×C₅²

C₂²×C₄

Fitting

C₂×C₅²×C₈

C₂²

prime radical

C₂×#6037

C₄

cosocle

C₂×C₄×C₅²

C₂³

hypercenter

C₂×C₈

D₅²

C₅²

G ▷ C₂²×C₅² ▷ C₅²

C₁ ◁ C₂×C₄ ◁ C₂×C₈

N	H ≅ G/N	type
2. C₂×#87768	C₂	semidirect
3. C₈×#1039	C₂	semidirect
4. C₅×#3952	C₂	semidirect
5. #11716613	C₂	semidirect
25. #6029	D₄	semidirect
31. #3952	D₅	semidirect
42. #1039	M₁₆	semidirect
91. C₅²	#436	semidirect
139. C₅	#65018	semidirect

H ≅ G/N

type

2. C₂×#87768

C₂

semidirect

3. C₈×#1039

C₂

semidirect

4. C₅×#3952

C₂

semidirect

5. #11716613

C₂

semidirect

25. #6029

D₄

semidirect

31. #3952

D₅

semidirect

42. #1039

M₁₆

semidirect

91. C₅²

#436

semidirect

139. C₅

#65018

semidirect

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

subgroup	index
C₄×#6037	2
C₂×#87768	2
C₈×#1039	2
C₅×#3952	2
#11716613	2
#65018	5

subgroup

index

Subgroups — #13100901

Conjugacy classes of subgroups

Complements

Maximal subgroups