Group #13101669 — subgroups

Subgroups — #13101669

Order 1600.

310 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 68 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	D₄×#1039	C₂	1	1	yes	maximal, characteristic
2	#11717037	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3964	C₂	1	2	yes	maximal
4	#11716616	C₂	1	2	yes	maximal, complemented
5	#11716613	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
7	C₂²×#1039	—	2	1	no	—
8	C₂×#6039	—	2	1	no	—
9	C₂²⋊#1039	—	2	1	no	—
10	C₄⋊#1039	C₂²	1	1	yes	characteristic
11	C₄×#1039	C₂²	1	1	yes	characteristic
12	C₅×C₄⋊Dic₅	C₂²	1	2	yes	—
13	C₂×C₅×#168	C₂²	1	2	yes	—
14	#87781	—	2	1	no	—
15	#65048	—	5	2	no	maximal
16	C₂³×C₅²	—	2	1	no	—
17	C₅²×D₄	—	2	1	no	—
18	C₅²×D₄	Q₈	1	2	yes	complemented
19	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
20	C₂×#1039	—	2	1	no	—
21	C₂×#1039	—	4	2	no	—
22	C₂×#1039	D₄	1	1	yes	characteristic
23	C₂×#1039	D₄	1	1	yes	characteristic
24	C₂×C₅×Dic₅	—	2	2	no	—
25	#6039	—	4	2	no	—
26	C₅×#168	—	2	2	no	—
27	D₄×Dic₅	—	5	2	no	—
28	D₄×Dic₅	—	10	2	no	—
29	#4034	—	5	2	no	—
30	C₅×#102	—	5	2	no	—
31	#3964	D₅	1	2	yes	—
32	#3950	—	5	2	no	—
33	#3946	—	5	2	no	—
34	#3939	—	5	2	no	—
35	C₂²×C₅²	—	2	2	no	—
36	C₂²×C₅²	—	4	1	no	—
37	C₂²×C₅²	—	4	1	no	—
38	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
39	C₄×C₅²	C₂×Q₈	1	1	yes	characteristic
40	C₄×C₅²	#41	1	1	yes	characteristic
41	#1039	—	2	1	no	—
42	#1039	—	4	1	no	—
43	#1039	D₈	1	2	yes	—
44	C₅×Dic₅	—	4	2	no	—
45	C₂×C₅×D₄	—	2	2	no	—
46	C₂×C₅×D₄	C₂×D₅	1	2	yes	—
47	C₂²×Dic₅	—	10	2	no	—
48	C₂²×Dic₅	—	10	4	no	—
49	C₂×Dic₁₀	—	10	2	no	—
50	C₂×C₅×C₈	—	5	2	no	—
51	C₅×C₄⋊C₄	—	5	2	no	—
52	C₂²⋊Dic₅	—	10	2	no	—
53	C₂²⋊Dic₅	—	10	4	no	—
54	C₄⋊Dic₅	—	5	2	no	—
55	C₄⋊Dic₅	—	10	2	no	—
56	C₄⋊Dic₅	C₂×D₅	1	2	yes	—
57	#683	—	10	2	no	—
58	C₄×Dic₅	—	5	2	no	—
59	C₄×Dic₅	—	10	2	no	—
60	C₂×#168	C₂×D₅	1	2	yes	—
61	#474	—	25	1	no	Sylow
62	C₂×C₅²	—	4	2	no	—
63	C₂×C₅²	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂×C₅²	C₂×D₈	1	1	yes	characteristic
65	C₂×C₅²	#137	1	1	yes	characteristic
66	C₂³×C₅	—	2	4	no	—
67	C₂³×C₅	—	2	2	no	—
68	C₅×D₄	—	2	4	no	—
69	C₅×D₄	—	2	2	no	—
70	C₅×D₄	—	2	4	no	—
71	C₅×D₄	Dic₁₀	1	4	yes	complemented
72	C₂×C₄×C₅	—	2	2	no	—
73	C₂×C₄×C₅	—	10	2	no	—
74	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
75	C₂×Dic₅	—	2	2	no	—
76	C₂×Dic₅	—	5	2	no	—
77	C₂×Dic₅	—	5	2	no	—
78	C₂×Dic₅	—	10	2	no	—
79	C₂×Dic₅	—	10	2	no	—
80	C₂×Dic₅	—	10	4	no	—
81	C₂×Dic₅	—	10	2	no	—
82	C₂×Dic₅	—	20	4	no	—
83	C₂×Dic₅	—	20	8	no	—
84	Dic₁₀	—	20	4	no	—
85	C₅×C₈	—	10	2	no	—
86	#168	—	2	2	no	—
87	C₄⋊Q₈	—	25	1	no	—
88	C₄×D₄	—	25	1	no	—
89	#107	—	25	2	no	—
90	C₄⋊C₈	—	25	1	no	—
91	#102	—	5	2	no	—
92	C₅²	#474	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₅	—	2	8	no	—
94	C₂²×C₅	—	2	4	no	—
95	C₂²×C₅	—	2	2	no	—
96	C₂²×C₅	—	4	4	no	—
97	C₂²×C₅	—	4	4	no	—
98	C₂²×C₅	—	4	2	no	—
99	C₂²×C₅	—	4	2	no	—
100	C₂²×C₅	D₄×D₅	1	2	yes	—
101	C₄×C₅	—	2	2	no	—
102	C₄×C₅	—	2	2	no	—
103	C₄×C₅	—	20	2	no	—
104	C₄×C₅	C₂×Dic₁₀	1	2	yes	—
105	C₄×C₅	#711	1	2	yes	—
106	Dic₅	—	4	2	no	—
107	Dic₅	—	5	4	no	—
108	Dic₅	—	10	4	no	—
109	Dic₅	—	10	2	no	—
110	Dic₅	—	20	2	no	—
111	Dic₅	—	20	4	no	—
112	Dic₅	—	20	2	no	—
113	C₂×Q₈	—	50	1	no	—
114	C₂×D₄	D₅²	1	1	yes	characteristic
115	C₂²×C₄	—	50	1	no	—
116	C₂×C₈	—	5	2	no	—
117	C₄⋊C₄	—	5	2	no	—
118	C₄⋊C₄	—	25	1	no	—
119	C₄⋊C₄	—	50	1	no	—
120	C₂²⋊C₄	—	50	1	no	—
121	C₄²	—	25	1	no	—
122	C₂×C₅	—	2	2	no	—
123	C₂×C₅	—	2	2	no	—
124	C₂×C₅	—	2	2	no	—
125	C₂×C₅	—	4	8	no	—
126	C₂×C₅	—	4	4	no	—
127	C₂×C₅	#4024	1	2	yes	—
128	C₂×C₅	D₅×D₈	1	2	yes	—
129	C₂×C₅	#4061	1	2	yes	—
130	C₂³	—	2	1	no	—
131	Q₈	—	100	2	no	—
132	D₄	—	2	1	no	—
133	D₄	#6039	1	2	yes	complemented
134	C₂×C₄	—	10	2	no	—
135	C₂×C₄	—	25	1	no	—
136	C₂×C₄	—	25	1	no	—
137	C₂×C₄	—	50	1	no	—
138	C₂×C₄	—	100	2	no	—
139	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
140	C₈	—	10	2	no	—
141	C₅	—	2	2	no	—
142	C₅	#65048	1	2	yes	complemented
143	C₂²	—	2	2	no	—
144	C₂²	—	4	1	no	—
145	C₂²	—	4	1	no	—
146	C₂²	#87886	1	1	yes	characteristic
147	C₄	—	20	2	no	—
148	C₄	—	25	2	no	—
149	C₄	—	50	1	no	—
150	C₄	—	100	1	no	—
151	C₄	#87880	1	1	yes	characteristic
152	C₄	C₂×#6039	1	1	yes	characteristic
153	C₂	—	4	2	no	—
154	C₂	#11717044	1	1	yes	characteristic
155	C₂	#11717049	1	1	yes	characteristic
156	C₂	#11717100	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₄×#1039

C₂

yes

maximal, characteristic

#11717037

C₂

yes

maximal, characteristic, complemented

C₅×#3964

C₂

yes

maximal

#11716616

C₂

yes

maximal, complemented

#11716613

C₂

yes

maximal, characteristic, complemented

C₂×C₅²×D₄

C₂²

yes

characteristic

C₂²×#1039

—

C₂×#6039

—

C₂²⋊#1039

—

C₄⋊#1039

C₂²

yes

characteristic

C₄×#1039

C₂²

yes

characteristic

C₅×C₄⋊Dic₅

C₂²

yes

—

C₂×C₅×#168

C₂²

yes

—

#87781

—

#65048

—

maximal

C₂³×C₅²

—

C₅²×D₄

—

C₅²×D₄

Q₈

yes

complemented

C₂×C₄×C₅²

C₂³

yes

—

—

D₄

yes

characteristic

C₂×#1039

D₄

yes

characteristic

C₂×C₅×Dic₅

—

#6039

—

C₅×#168

—

D₄×Dic₅

—

D₄×Dic₅

—

#4034

—

C₅×#102

—

#3964

D₅

yes

—

#3950

—

#3946

—

#3939

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

C₂×Q₈

yes

C₄×C₅²

yes

—

—

D₈

yes

—

C₅×Dic₅

—

C₂×C₅×D₄

—

C₂×C₅×D₄

C₂×D₅

yes

—

C₂²×Dic₅

—

C₂²×Dic₅

—

C₂×Dic₁₀

—

C₂×C₅×C₈

—

C₅×C₄⋊C₄

—

C₂²⋊Dic₅

—

C₂²⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

C₂×D₅

yes

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

—

C₂×#168

C₂×D₅

yes

—

#474

—

Sylow

C₂×C₅²

—

C₂×C₅²

C₂²⋊Q₈

yes

characteristic

C₂×C₅²

C₂×D₈

yes

characteristic

C₂×C₅²

#137

yes

characteristic

C₂³×C₅

—

C₂³×C₅

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

Dic₁₀

yes

complemented

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

C₅×C₈

—

#168

—

C₄⋊Q₈

—

C₄×D₄

—

#107

—

C₄⋊C₈

—

#102

—

C₅²

#474

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

100

C₂²×C₅

D₄×D₅

yes

—

101

C₄×C₅

—

102

C₄×C₅

—

103

C₄×C₅

—

104

C₄×C₅

C₂×Dic₁₀

yes

—

105

C₄×C₅

#711

yes

—

106

Dic₅

—

107

Dic₅

—

108

Dic₅

—

109

Dic₅

—

110

Dic₅

—

111

Dic₅

—

112

Dic₅

—

113

C₂×Q₈

—

114

C₂×D₄

D₅²

yes

characteristic

115

C₂²×C₄

—

116

C₂×C₈

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₂×C₅

—

123

C₂×C₅

—

124

C₂×C₅

—

125

C₂×C₅

—

126

C₂×C₅

—

127

C₂×C₅

#4024

yes

—

128

C₂×C₅

D₅×D₈

yes

—

129

C₂×C₅

#4061

yes

—

130

C₂³

—

131

Q₈

—

100

—

132

D₄

—

133

D₄

#6039

yes

complemented

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

100

—

139

C₂×C₄

C₂×D₅²

yes

characteristic

140

C₈

—

141

C₅

—

142

C₅

#65048

yes

complemented

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

#87886

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

100

—

151

C₄

#87880

yes

characteristic

152

C₄

C₂×#6039

yes

characteristic

153

C₂

—

154

C₂

#11717044

yes

characteristic

155

C₂

#11717049

yes

characteristic

156

C₂

#11717100

yes

characteristic

	subgroup	G/H
center	C₂²	#87886
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂×D₅²
Fitting, prime radical	C₂×C₅²×D₄	C₂²
socle	C₂²×C₅²	C₂×D₄
hypercenter	C₂×D₄	D₅²
hypocenter	C₅²	#474
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₄

subgroup

G/H

center

C₂²

#87886

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂×D₅²

Fitting, prime radical

C₂×C₅²×D₄

C₂²

socle

C₂²×C₅²

C₂×D₄

hypercenter

C₂×D₄

D₅²

hypocenter

C₅²

#474

LCS

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₄

N	H ≅ G/N	type
2. #11717037	C₂	semidirect
4. #11716616	C₂	semidirect
5. #11716613	C₂	semidirect
18. C₅²×D₄	Q₈	semidirect
71. C₅×D₄	Dic₁₀	semidirect
92. C₅²	#474	semidirect
133. D₄	#6039	semidirect
142. C₅	#65048	semidirect

H ≅ G/N

type

2. #11717037

C₂

semidirect

4. #11716616

C₂

semidirect

5. #11716613

C₂

semidirect

18. C₅²×D₄

Q₈

semidirect

71. C₅×D₄

Dic₁₀

semidirect

92. C₅²

#474

semidirect

133. D₄

#6039

semidirect

142. C₅

#65048

semidirect

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

subgroup	index
D₄×#1039	2
#11717037	2
C₅×#3964	2
#11716616	2
#11716613	2
#65048	5

subgroup

index

Subgroups — #13101669

Conjugacy classes of subgroups

Complements

Maximal subgroups