Group #13100716 — subgroups

Subgroups — #13100716

Order 1600.

262 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 62 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	#11717207	C₂	1	1	yes	maximal, characteristic
2	#11717037	C₂	1	1	yes	maximal, characteristic
3	C₅×#3940	C₂	1	2	yes	maximal
4	#11716613	C₂	1	1	yes	maximal, characteristic
5	#11716608	C₂	1	2	yes	maximal
6	C₂×#6045	C₂²	1	1	yes	characteristic
7	C₂×#6039	—	2	1	no	—
8	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
9	#87806	—	2	1	no	—
10	C₄×#1039	—	2	1	no	—
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	#65143	—	5	2	no	maximal
16	C₂×C₄×C₅²	—	2	1	no	—
17	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
18	C₂×#1039	—	2	1	no	—
19	C₂×#1039	D₄	1	1	yes	characteristic
20	C₂×#1039	D₄	1	1	yes	characteristic
21	#6045	—	2	1	no	—
22	#6045	Q₈	1	2	yes	complemented
23	C₂×C₅×Dic₅	—	2	2	no	—
24	#6039	—	4	2	no	—
25	C₅×#168	—	2	2	no	—
26	#4034	—	5	2	no	—
27	#4033	—	5	2	no	—
28	#4033	—	10	2	no	—
29	C₅×#106	—	5	2	no	—
30	#3948	—	5	2	no	—
31	#3946	—	5	2	no	—
32	#3942	—	5	2	no	—
33	#3940	D₅	1	2	yes	—
34	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
35	C₄×C₅²	—	4	1	no	—
36	C₄×C₅²	C₂×Q₈	1	1	yes	characteristic
37	C₄×C₅²	#41	1	1	yes	characteristic
38	#1039	—	2	2	no	—
39	#1039	—	2	1	no	—
40	#1039	SD₁₆	1	2	yes	—
41	C₅×Dic₅	—	4	2	no	—
42	C₂×Dic₁₀	—	5	2	no	—
43	C₂×Dic₁₀	—	10	2	no	—
44	C₂×Dic₁₀	—	10	2	no	—
45	C₂×C₅×C₈	—	5	2	no	—
46	C₅×C₄⋊C₄	—	2	2	no	—
47	C₅×C₄⋊C₄	—	5	2	no	—
48	C₅×C₄⋊C₄	C₂×D₅	1	2	yes	—
49	C₄⋊Dic₅	C₂×D₅	1	2	yes	—
50	#683	—	10	2	no	—
51	#683	—	10	2	no	—
52	#683	—	10	4	no	—
53	C₄×Dic₅	—	5	2	no	—
54	C₄×Dic₅	—	10	2	no	—
55	C₄×Dic₅	—	10	2	no	—
56	C₄×Dic₅	—	10	4	no	—
57	C₂×#168	C₂×D₅	1	2	yes	—
58	#475	—	25	1	no	Sylow
59	C₂×C₅²	C₂²⋊Q₈	1	1	yes	characteristic
60	C₂×C₅²	C₂×SD₁₆	1	1	yes	characteristic
61	C₂×C₅²	#137	1	1	yes	characteristic
62	C₂×C₄×C₅	—	2	4	no	—
63	C₂×C₄×C₅	—	2	2	no	—
64	C₂×C₄×C₅	—	2	2	no	—
65	C₂×C₄×C₅	—	10	2	no	—
66	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
67	C₂×Dic₅	—	2	2	no	—
68	C₂×Dic₅	—	5	2	no	—
69	C₂×Dic₅	—	5	2	no	—
70	C₂×Dic₅	—	10	2	no	—
71	C₂×Dic₅	—	10	2	no	—
72	C₂×Dic₅	—	10	4	no	—
73	C₂×Dic₅	—	10	2	no	—
74	Dic₁₀	—	5	4	no	—
75	Dic₁₀	—	10	4	no	—
76	Dic₁₀	—	10	4	no	—
77	Dic₁₀	—	10	2	no	—
78	Dic₁₀	—	20	4	no	—
79	C₅×C₈	—	10	2	no	—
80	#168	—	2	2	no	—
81	C₄⋊Q₈	—	25	1	no	—
82	C₄×Q₈	—	25	1	no	—
83	#106	—	5	2	no	—
84	C₄⋊C₈	—	25	1	no	—
85	#103	—	25	2	no	—
86	C₅²	#475	1	1	yes	characteristic, complemented, Sylow
87	C₂²×C₅	—	2	2	no	—
88	C₂²×C₅	D₄×D₅	1	2	yes	—
89	C₄×C₅	—	2	2	no	—
90	C₄×C₅	—	2	2	no	—
91	C₄×C₅	—	4	4	no	—
92	C₄×C₅	—	4	2	no	—
93	C₄×C₅	—	20	2	no	—
94	C₄×C₅	#709	1	2	yes	—
95	C₄×C₅	Q₈×D₅	1	2	yes	—
96	Dic₅	—	4	2	no	—
97	Dic₅	—	5	4	no	—
98	Dic₅	—	10	4	no	—
99	Dic₅	—	10	8	no	—
100	Dic₅	—	10	4	no	—
101	Dic₅	—	10	2	no	—
102	Dic₅	—	20	2	no	—
103	C₂×Q₈	—	25	1	no	—
104	C₂×Q₈	—	50	1	no	—
105	C₂×C₈	—	5	2	no	—
106	C₄⋊C₄	—	5	2	no	—
107	C₄⋊C₄	—	50	1	no	—
108	C₄⋊C₄	—	50	1	no	—
109	C₄⋊C₄	D₅²	1	1	yes	characteristic
110	C₄²	—	25	1	no	—
111	C₄²	—	50	1	no	—
112	C₂×C₅	—	2	2	no	—
113	C₂×C₅	—	2	2	no	—
114	C₂×C₅	—	2	2	no	—
115	C₂×C₅	#4042	1	2	yes	—
116	C₂×C₅	D₅×SD₁₆	1	2	yes	—
117	C₂×C₅	#4061	1	2	yes	—
118	Q₈	—	25	2	no	—
119	Q₈	—	50	1	no	—
120	Q₈	—	100	2	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	10	2	no	—
123	C₂×C₄	—	25	1	no	—
124	C₂×C₄	—	25	1	no	—
125	C₂×C₄	—	50	1	no	—
126	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
127	C₈	—	10	2	no	—
128	C₅	—	2	2	no	—
129	C₅	#65143	1	2	yes	complemented
130	C₂²	#87886	1	1	yes	characteristic
131	C₄	—	4	1	no	—
132	C₄	—	20	2	no	—
133	C₄	—	25	2	no	—
134	C₄	—	50	2	no	—
135	C₄	—	50	1	no	—
136	C₄	#87872	1	1	yes	characteristic
137	C₄	#87873	1	1	yes	characteristic
138	C₂	#11716984	1	1	yes	characteristic
139	C₂	#11717049	1	1	yes	characteristic
140	C₂	#11717058	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11717207

C₂

yes

maximal, characteristic

#11717037

C₂

yes

maximal, characteristic

C₅×#3940

C₂

yes

maximal

#11716613

C₂

yes

maximal, characteristic

#11716608

C₂

yes

maximal

C₂×#6045

C₂²

yes

characteristic

C₂×#6039

—

C₅²×C₄⋊C₄

C₂²

yes

—

—

C₂²

yes

characteristic

C₅×C₄⋊Dic₅

C₂²

yes

—

C₂×C₅×#168

C₂²

yes

—

#87781

—

#65143

—

maximal

C₂×C₄×C₅²

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

—

C₂×#1039

D₄

yes

characteristic

C₂×#1039

D₄

yes

characteristic

#6045

—

#6045

Q₈

yes

complemented

C₂×C₅×Dic₅

—

—

—

—

—

—

—

—

—

—

D₅

yes

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

—

C₄×C₅²

C₂×Q₈

yes

C₄×C₅²

yes

—

—

SD₁₆

yes

—

C₅×Dic₅

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

C₂×D₅

yes

—

C₄⋊Dic₅

C₂×D₅

yes

—

#683

—

#683

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₂×#168

C₂×D₅

yes

—

#475

—

Sylow

C₂×C₅²

C₂²⋊Q₈

yes

characteristic

C₂×C₅²

C₂×SD₁₆

yes

characteristic

C₂×C₅²

#137

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

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

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

#168

—

C₄⋊Q₈

—

C₄×Q₈

—

#106

—

C₄⋊C₈

—

#103

—

C₅²

#475

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

D₄×D₅

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#709

yes

—

C₄×C₅

Q₈×D₅

yes

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

100

Dic₅

—

101

Dic₅

—

102

Dic₅

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×C₈

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

D₅²

yes

characteristic

110

C₄²

—

111

C₄²

—

112

C₂×C₅

—

113

C₂×C₅

—

114

C₂×C₅

—

115

C₂×C₅

#4042

yes

—

116

C₂×C₅

D₅×SD₁₆

yes

—

117

C₂×C₅

#4061

yes

—

118

Q₈

—

119

Q₈

—

120

Q₈

—

100

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

C₂×D₅²

yes

characteristic

127

C₈

—

128

C₅

—

129

C₅

#65143

yes

complemented

130

C₂²

#87886

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#87872

yes

characteristic

137

C₄

#87873

yes

characteristic

138

C₂

#11716984

yes

characteristic

139

C₂

#11717049

yes

characteristic

140

C₂

#11717058

yes

characteristic

	subgroup	G/H
center	C₂²	#87886
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂×D₅²
Fitting	C₅²×C₄⋊C₄	C₂²
socle, prime radical	C₂²×C₅²	C₂×D₄
hypercenter	C₄⋊C₄	D₅²
hypocenter	C₅²	#475
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

subgroup

G/H

center

C₂²

#87886

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂×D₅²

Fitting

C₅²×C₄⋊C₄

C₂²

socle, prime radical

C₂²×C₅²

C₂×D₄

hypercenter

C₄⋊C₄

D₅²

hypocenter

C₅²

#475

LCS

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

N	H ≅ G/N	type
22. #6045	Q₈	semidirect
86. C₅²	#475	semidirect
129. C₅	#65143	semidirect

H ≅ G/N

type

22. #6045

Q₈

semidirect

86. C₅²

#475

semidirect

129. C₅

#65143

semidirect

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

subgroup	index
#11717207	2
#11717037	2
C₅×#3940	2
#11716613	2
#11716608	2
#65143	5

subgroup

index

Subgroups — #13100716

Conjugacy classes of subgroups

Complements

Maximal subgroups