Group #13103042 — subgroups

Subgroups — #13103042

Order 1600.

432 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 127 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	C₂×C₅²×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	Q₈×#1039	C₂	1	1	yes	maximal, characteristic
3	#11717264	C₂	1	1	yes	maximal, characteristic
4	C₂×#87812	C₂	1	1	yes	maximal, characteristic
5	#11716702	C₂	1	1	yes	maximal, characteristic
6	#11716683	C₂	1	1	yes	maximal, characteristic
7	#11716681	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
10	C₂×#6045	C₂²	1	1	yes	characteristic
11	C₅²×Q₁₆	—	2	2	no	—
12	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic
13	#87812	—	2	2	no	—
14	C₄⋊#1039	—	2	1	no	—
15	C₄⋊#1039	C₂²	1	1	yes	characteristic
16	#87806	—	2	1	no	—
17	#87806	—	2	1	no	—
18	C₄×#1039	—	2	1	no	—
19	C₄×#1039	C₂²	1	1	yes	characteristic
20	C₂×#6029	C₂²	1	1	yes	characteristic
21	#65471	—	5	6	no	maximal
22	C₅²×Q₈	—	2	1	no	—
23	C₅²×Q₈	—	2	1	no	—
24	C₅²×Q₈	—	2	1	no	—
25	C₅²×Q₈	D₄	1	2	yes	—
26	C₂×C₄×C₅²	—	2	1	no	—
27	C₂×C₄×C₅²	—	2	1	no	—
28	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
29	C₂×#1039	—	2	1	no	—
30	C₂×#1039	—	2	1	no	—
31	C₂×#1039	D₄	1	1	yes	characteristic
32	C₂×#1039	D₄	1	1	yes	characteristic
33	#6045	—	2	1	no	—
34	#6045	—	2	1	no	—
35	C₅²×C₈	—	2	1	no	—
36	#6029	—	2	1	no	—
37	C₂×C₅×Q₁₆	D₅	1	6	yes	—
38	Q₈×Dic₅	—	5	6	no	—
39	#4090	—	5	6	no	—
40	C₂×#689	—	5	6	no	—
41	#3967	—	5	6	no	—
42	#3948	—	5	6	no	—
43	#3946	—	5	6	no	—
44	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
45	C₄×C₅²	—	2	2	no	—
46	C₄×C₅²	—	4	1	no	—
47	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
48	C₄×C₅²	#41	1	1	yes	characteristic
49	#1039	—	2	1	no	—
50	#1039	—	4	1	no	—
51	#1039	—	4	1	no	—
52	#1039	Q₁₆	1	2	yes	complemented
53	C₂×C₅×Q₈	C₂×D₅	1	6	yes	—
54	C₂×C₅×Q₈	C₂×D₅	1	6	yes	—
55	C₂×Dic₁₀	—	5	6	no	—
56	C₅×Q₁₆	—	2	12	no	—
57	C₂×C₅×C₈	C₂×D₅	1	6	yes	—
58	#689	—	10	12	no	—
59	C₄⋊Dic₅	—	5	6	no	—
60	C₄⋊Dic₅	—	10	6	no	—
61	#683	—	10	6	no	—
62	#683	—	10	6	no	—
63	C₄×Dic₅	—	5	6	no	—
64	C₄×Dic₅	—	10	6	no	—
65	C₂×#168	—	5	6	no	—
66	#462	—	25	1	no	Sylow
67	C₂×C₅²	#121	1	1	yes	characteristic
68	C₂×C₅²	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₅²	#137	1	1	yes	characteristic
70	C₅×Q₈	—	2	6	no	—
71	C₅×Q₈	—	2	6	no	—
72	C₅×Q₈	—	2	6	no	—
73	C₅×Q₈	C₂²⋊D₅	1	12	yes	—
74	C₂×C₄×C₅	—	2	6	no	—
75	C₂×C₄×C₅	—	2	6	no	—
76	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
77	C₂×Dic₅	—	5	6	no	—
78	C₂×Dic₅	—	5	6	no	—
79	C₂×Dic₅	—	10	6	no	—
80	C₂×Dic₅	—	10	6	no	—
81	Dic₁₀	—	10	6	no	—
82	Dic₁₀	—	10	6	no	—
83	C₅×C₈	—	2	6	no	—
84	#168	—	10	6	no	—
85	C₂×Q₁₆	—	25	1	no	—
86	C₂×Q₁₆	C₅⋊D₅	1	1	yes	characteristic
87	C₄⋊Q₈	—	25	1	no	—
88	C₄×Q₈	—	25	1	no	—
89	C₄⋊C₈	—	25	1	no	—
90	#103	—	25	1	no	—
91	#103	—	25	1	no	—
92	C₅²	#462	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₅	D₄×D₅	1	6	yes	—
94	C₄×C₅	—	2	12	no	—
95	C₄×C₅	—	4	6	no	—
96	C₄×C₅	#711	1	6	yes	—
97	C₄×C₅	C₂×C₂²⋊D₅	1	6	yes	—
98	Dic₅	—	5	12	no	—
99	Dic₅	—	10	6	no	—
100	Dic₅	—	20	6	no	—
101	Dic₅	—	20	6	no	—
102	C₂×Q₈	—	25	1	no	—
103	C₂×Q₈	C₂×C₅⋊D₅	1	1	yes	characteristic
104	C₂×Q₈	C₂×C₅⋊D₅	1	1	yes	characteristic
105	Q₁₆	—	2	2	no	—
106	Q₁₆	—	50	2	no	—
107	C₂×C₈	—	25	1	no	—
108	C₂×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
109	C₄⋊C₄	—	25	1	no	—
110	C₄⋊C₄	—	50	1	no	—
111	C₄⋊C₄	—	50	1	no	—
112	C₄⋊C₄	—	50	1	no	—
113	C₄²	—	25	1	no	—
114	C₄²	—	50	1	no	—
115	C₂×C₅	D₅×Q₁₆	1	6	yes	—
116	C₂×C₅	#4064	1	6	yes	—
117	C₂×C₅	#4085	1	6	yes	—
118	Q₈	—	2	1	no	—
119	Q₈	—	2	1	no	—
120	Q₈	—	2	1	no	—
121	Q₈	—	50	1	no	—
122	Q₈	—	50	1	no	—
123	Q₈	C₂²⋊(C₅⋊D₅)	1	2	yes	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	25	1	no	—
127	C₂×C₄	—	25	1	no	—
128	C₂×C₄	—	50	1	no	—
129	C₂×C₄	—	50	1	no	—
130	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
131	C₈	—	2	1	no	—
132	C₈	—	50	1	no	—
133	C₅	#65471	1	6	yes	complemented
134	C₂²	D₄×C₅⋊D₅	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	4	1	no	—
137	C₄	—	25	2	no	—
138	C₄	—	50	1	no	—
139	C₄	—	100	1	no	—
140	C₄	—	100	1	no	—
141	C₄	#87902	1	1	yes	characteristic
142	C₄	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
143	C₂	Q₁₆×C₅⋊D₅	1	1	yes	characteristic
144	C₂	#11717238	1	1	yes	characteristic
145	C₂	#11717259	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅²×Q₁₆

C₂

yes

maximal, characteristic

Q₈×#1039

C₂

yes

maximal, characteristic

#11717264

C₂

yes

maximal, characteristic

C₂×#87812

C₂

yes

maximal, characteristic

#11716702

C₂

yes

maximal, characteristic

#11716683

C₂

yes

maximal, characteristic

#11716681

C₂

yes

maximal, characteristic

C₂×C₅²×Q₈

C₂²

yes

characteristic

C₂×C₅²×Q₈

C₂²

yes

characteristic

C₂×#6045

C₂²

yes

characteristic

C₅²×Q₁₆

—

C₂×C₅²×C₈

C₂²

yes

—

—

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₂×#6029

C₂²

yes

characteristic

#65471

—

maximal

C₅²×Q₈

—

C₅²×Q₈

—

C₅²×Q₈

—

C₅²×Q₈

D₄

yes

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

C₂³

yes

—

—

D₄

yes

characteristic

C₂×#1039

D₄

yes

characteristic

#6045

—

#6045

—

C₅²×C₈

—

#6029

—

C₂×C₅×Q₁₆

D₅

yes

—

Q₈×Dic₅

—

—

—

—

—

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

—

C₄×C₅²

—

C₄×C₅²

C₂×D₄

yes

C₄×C₅²

yes

—

—

—

Q₁₆

yes

complemented

C₂×C₅×Q₈

C₂×D₅

yes

—

C₂×C₅×Q₈

C₂×D₅

yes

—

C₂×Dic₁₀

—

C₅×Q₁₆

—

C₂×C₅×C₈

C₂×D₅

yes

—

#689

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

#683

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

—

—

—

C₂×C₅²

yes

C₂×C₅²

C₂×Q₁₆

yes

characteristic

C₂×C₅²

#137

yes

characteristic

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

C₂²⋊D₅

yes

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

#168

—

C₂×Q₁₆

—

C₂×Q₁₆

C₅⋊D₅

yes

characteristic

C₄⋊Q₈

—

C₄×Q₈

—

C₄⋊C₈

—

#103

—

#103

—

C₅²

#462

yes

characteristic, complemented, Sylow

C₂²×C₅

D₄×D₅

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#711

yes

—

C₄×C₅

C₂×C₂²⋊D₅

yes

—

Dic₅

—

Dic₅

—

100

Dic₅

—

101

Dic₅

—

102

C₂×Q₈

—

103

C₂×Q₈

C₂×C₅⋊D₅

yes

characteristic

104

C₂×Q₈

C₂×C₅⋊D₅

yes

characteristic

105

Q₁₆

—

106

Q₁₆

—

107

C₂×C₈

—

108

C₂×C₈

C₂×C₅⋊D₅

yes

characteristic

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄²

—

114

C₄²

—

115

C₂×C₅

D₅×Q₁₆

yes

—

116

C₂×C₅

#4064

yes

—

117

C₂×C₅

#4085

yes

—

118

Q₈

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

Q₈

—

123

Q₈

C₂²⋊(C₅⋊D₅)

yes

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

131

C₈

—

132

C₈

—

133

C₅

#65471

yes

complemented

134

C₂²

D₄×C₅⋊D₅

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

100

—

140

C₄

—

100

—

141

C₄

#87902

yes

characteristic

142

C₄

C₂×C₂²⋊(C₅⋊D₅)

yes

characteristic

143

C₂

Q₁₆×C₅⋊D₅

yes

characteristic

144

C₂

#11717238

yes

characteristic

145

C₂

#11717259

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×C₅⋊D₅
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₂×C₅²×Q₁₆	C₂
socle, prime radical	C₂²×C₅²	C₂×D₄
hypercenter	C₂×Q₁₆	C₅⋊D₅
hypocenter	C₅²	#462
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

subgroup

G/H

center

C₂²

D₄×C₅⋊D₅

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂²×C₅⋊D₅

Fitting

C₂×C₅²×Q₁₆

C₂

socle, prime radical

C₂²×C₅²

C₂×D₄

hypercenter

C₂×Q₁₆

C₅⋊D₅

hypocenter

C₅²

#462

LCS

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

N	H ≅ G/N	type
52. #1039	Q₁₆	semidirect
92. C₅²	#462	semidirect
133. C₅	#65471	semidirect

H ≅ G/N

type

52. #1039

Q₁₆

semidirect

92. C₅²

#462

semidirect

133. C₅

#65471

semidirect

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

subgroup	index
C₂×C₅²×Q₁₆	2
Q₈×#1039	2
#11717264	2
C₂×#87812	2
#11716702	2
#11716683	2
#11716681	2
#65471	5

subgroup

index

C₂×C₅²×Q₁₆

Subgroups — #13103042

Conjugacy classes of subgroups

Complements

Maximal subgroups