Group #13103089 — subgroups

Subgroups — #13103089

Order 1600.

592 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 171 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	C₂²×C₅²×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#87812	C₂	1	2	yes	maximal, complemented
3	#11717244	C₂	1	1	yes	maximal, characteristic
4	#11716702	C₂	1	2	yes	maximal, complemented
5	C₂²⋊#6029	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅²×Q₈	—	2	4	no	—
7	C₂×C₅²×Q₈	—	2	1	no	—
8	C₂×C₅²×Q₈	C₂²	1	2	yes	—
9	C₂²×C₄×C₅²	—	2	1	no	—
10	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
11	C₂×#6045	C₂²	1	1	yes	characteristic
12	C₂²⋊#1039	—	2	1	no	—
13	#87812	—	2	4	no	—
14	C₄⋊#1039	C₂²	1	1	yes	characteristic
15	#87806	—	2	1	no	—
16	C₂×#6029	C₂²	1	2	yes	—
17	#65517	—	5	6	no	maximal
18	C₂³×C₅²	D₄	1	1	yes	characteristic
19	C₅²×Q₈	—	2	4	no	—
20	C₅²×Q₈	—	2	2	no	—
21	C₅²×Q₈	D₄	1	4	yes	—
22	C₂×C₄×C₅²	—	2	4	no	—
23	C₂×C₄×C₅²	—	2	2	no	—
24	C₂×C₄×C₅²	—	2	2	no	—
25	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
26	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
27	C₂×#1039	—	2	1	no	—
28	C₂×#1039	—	2	1	no	—
29	#6045	—	2	2	no	—
30	#6029	—	2	2	no	—
31	C₂²×C₅×Q₈	D₅	1	6	yes	—
32	C₂×#689	—	5	12	no	—
33	#4070	—	5	6	no	—
34	#3967	—	5	12	no	—
35	C₂²⋊#168	—	5	6	no	—
36	C₂²×C₅²	—	2	1	no	—
37	C₂²×C₅²	—	2	1	no	—
38	C₂²×C₅²	Q₁₆	1	2	yes	complemented
39	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
40	C₄×C₅²	—	2	4	no	—
41	C₄×C₅²	—	2	1	no	—
42	C₄×C₅²	C₂×D₄	1	2	yes	—
43	#1039	—	4	1	no	—
44	#1039	—	4	1	no	—
45	C₂×C₅×Q₈	—	2	24	no	—
46	C₂×C₅×Q₈	—	2	6	no	—
47	C₂×C₅×Q₈	C₂×D₅	1	12	yes	—
48	C₂²×C₄×C₅	—	2	6	no	—
49	C₂²×C₄×C₅	C₂×D₅	1	6	yes	—
50	C₂×Dic₁₀	—	5	6	no	—
51	C₂²⋊Dic₅	—	10	6	no	—
52	#689	—	10	24	no	—
53	C₄⋊Dic₅	—	5	6	no	—
54	#683	—	10	6	no	—
55	C₂×#168	—	5	12	no	—
56	#451	—	25	1	no	Sylow
57	C₂×C₅²	—	2	2	no	—
58	C₂×C₅²	#120	1	1	yes	characteristic
59	C₂×C₅²	C₂×Q₁₆	1	1	yes	characteristic
60	C₂×C₅²	#137	1	1	yes	characteristic
61	C₂³×C₅	C₂²⋊D₅	1	6	yes	—
62	C₅×Q₈	—	2	24	no	—
63	C₅×Q₈	—	2	12	no	—
64	C₅×Q₈	C₂²⋊D₅	1	24	yes	—
65	C₂×C₄×C₅	—	2	24	no	—
66	C₂×C₄×C₅	—	2	12	no	—
67	C₂×C₄×C₅	—	2	12	no	—
68	C₂×C₄×C₅	C₂²⋊D₅	1	6	yes	—
69	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
70	C₂×Dic₅	—	10	6	no	—
71	C₂×Dic₅	—	10	6	no	—
72	Dic₁₀	—	10	12	no	—
73	#168	—	10	12	no	—
74	C₂²×Q₈	C₅⋊D₅	1	1	yes	characteristic
75	C₂×Q₁₆	—	25	2	no	—
76	C₂²⋊Q₈	—	25	1	no	—
77	#103	—	25	2	no	—
78	C₂²⋊C₈	—	25	1	no	—
79	C₅²	#451	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₅	—	2	6	no	—
81	C₂²×C₅	—	2	6	no	—
82	C₂²×C₅	#689	1	12	yes	complemented
83	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
84	C₄×C₅	—	2	24	no	—
85	C₄×C₅	—	2	6	no	—
86	C₄×C₅	C₂×C₂²⋊D₅	1	12	yes	—
87	Dic₅	—	20	6	no	—
88	Dic₅	—	20	6	no	—
89	C₂×Q₈	—	2	4	no	—
90	C₂×Q₈	—	2	1	no	—
91	C₂×Q₈	—	25	1	no	—
92	C₂×Q₈	C₂×C₅⋊D₅	1	2	yes	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
95	Q₁₆	—	50	4	no	—
96	C₂×C₈	—	25	2	no	—
97	C₄⋊C₄	—	25	1	no	—
98	C₄⋊C₄	—	50	1	no	—
99	C₂²⋊C₄	—	50	1	no	—
100	C₂×C₅	—	2	12	no	—
101	C₂×C₅	#4088	1	6	yes	—
102	C₂×C₅	C₂×#689	1	6	yes	—
103	C₂×C₅	#4099	1	6	yes	—
104	C₂³	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
105	Q₈	—	2	4	no	—
106	Q₈	—	2	2	no	—
107	Q₈	—	50	2	no	—
108	Q₈	C₂²⋊(C₅⋊D₅)	1	4	yes	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	50	1	no	—
113	C₂×C₄	—	50	1	no	—
114	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
115	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
116	C₈	—	50	2	no	—
117	C₅	#65517	1	6	yes	complemented
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	#87812	1	2	yes	complemented
121	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
122	C₄	—	2	4	no	—
123	C₄	—	2	1	no	—
124	C₄	—	100	1	no	—
125	C₄	—	100	1	no	—
126	C₄	C₂×C₂²⋊(C₅⋊D₅)	1	2	yes	—
127	C₂	—	2	2	no	—
128	C₂	#11717262	1	1	yes	characteristic
129	C₂	C₂×#87812	1	1	yes	characteristic
130	C₂	#11717273	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₅²×Q₈

C₂

yes

maximal, characteristic

C₂×#87812

C₂

yes

maximal, complemented

#11717244

C₂

yes

maximal, characteristic

#11716702

C₂

yes

maximal, complemented

C₂²⋊#6029

C₂

yes

maximal, characteristic

C₂×C₅²×Q₈

—

C₂×C₅²×Q₈

—

C₂×C₅²×Q₈

C₂²

yes

—

C₂²×C₄×C₅²

—

C₂²×C₄×C₅²

C₂²

yes

characteristic

C₂×#6045

C₂²

yes

characteristic

C₂²⋊#1039

—

#87812

—

C₄⋊#1039

C₂²

yes

characteristic

#87806

—

C₂×#6029

C₂²

yes

—

#65517

—

maximal

C₂³×C₅²

D₄

yes

characteristic

C₅²×Q₈

—

C₅²×Q₈

—

C₅²×Q₈

D₄

yes

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

D₄

yes

characteristic

C₂×C₄×C₅²

C₂³

yes

—

—

—

—

C₂²×C₅×Q₈

D₅

yes

—

C₂×#689

—

#4070

—

#3967

—

C₂²⋊#168

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

Q₁₆

yes

complemented

C₂²×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

—

C₄×C₅²

—

C₄×C₅²

C₂×D₄

yes

—

#1039

—

#1039

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

C₂×D₅

yes

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₅

yes

—

C₂×Dic₁₀

—

C₂²⋊Dic₅

—

#689

—

C₄⋊Dic₅

—

—

—

—

C₂×C₅²

—

C₂×C₅²

#120

yes

characteristic

C₂×C₅²

C₂×Q₁₆

yes

characteristic

C₂×C₅²

#137

yes

characteristic

C₂³×C₅

C₂²⋊D₅

yes

—

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

C₂²⋊D₅

yes

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

#168

—

C₂²×Q₈

C₅⋊D₅

yes

characteristic

C₂×Q₁₆

—

C₂²⋊Q₈

—

#103

—

C₂²⋊C₈

—

C₅²

#451

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

#689

yes

complemented

C₂²×C₅

C₂×C₂²⋊D₅

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₂×C₂²⋊D₅

yes

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×C₅⋊D₅

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×C₅⋊D₅

yes

characteristic

Q₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

100

C₂×C₅

—

101

C₂×C₅

#4088

yes

—

102

C₂×C₅

C₂×#689

yes

—

103

C₂×C₅

#4099

yes

—

104

C₂³

C₂²⋊(C₅⋊D₅)

yes

characteristic

105

Q₈

—

106

Q₈

—

107

Q₈

—

108

Q₈

C₂²⋊(C₅⋊D₅)

yes

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

characteristic

115

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

116

C₈

—

117

C₅

#65517

yes

complemented

118

C₂²

—

119

C₂²

—

120

C₂²

#87812

yes

complemented

121

C₂²

C₂×C₂²⋊(C₅⋊D₅)

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

100

—

125

C₄

—

100

—

126

C₄

C₂×C₂²⋊(C₅⋊D₅)

yes

—

127

C₂

—

128

C₂

#11717262

yes

characteristic

129

C₂

C₂×#87812

yes

characteristic

130

C₂

#11717273

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊(C₅⋊D₅)
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₂²×C₅²×Q₈	C₂
socle	C₂²×C₅²	C₂×D₄
prime radical	C₂³×C₅²	D₄
hypercenter	C₂²×Q₈	C₅⋊D₅
hypocenter	C₅²	#451
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊(C₅⋊D₅)

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂²×C₅⋊D₅

Fitting

C₂²×C₅²×Q₈

C₂

socle

C₂²×C₅²

C₂×D₄

prime radical

C₂³×C₅²

D₄

hypercenter

C₂²×Q₈

C₅⋊D₅

hypocenter

C₅²

#451

LCS

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₈

N	H ≅ G/N	type
2. C₂×#87812	C₂	semidirect
4. #11716702	C₂	semidirect
38. C₂²×C₅²	Q₁₆	semidirect
79. C₅²	#451	semidirect
82. C₂²×C₅	#689	semidirect
117. C₅	#65517	semidirect
120. C₂²	#87812	semidirect

H ≅ G/N

type

2. C₂×#87812

C₂

semidirect

4. #11716702

C₂

semidirect

38. C₂²×C₅²

Q₁₆

semidirect

79. C₅²

#451

semidirect

82. C₂²×C₅

#689

semidirect

117. C₅

#65517

semidirect

120. C₂²

#87812

semidirect

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

subgroup	index
C₂²×C₅²×Q₈	2
C₂×#87812	2
#11717244	2
#11716702	2
C₂²⋊#6029	2
#65517	5

subgroup

index

C₂²×C₅²×Q₈

C₂²⋊#6029

Subgroups — #13103089

Conjugacy classes of subgroups

Complements

Maximal subgroups