Group #13103109 — subgroups

Subgroups — #13103109

Order 1600.

568 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 139 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	C₅²×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic
2	#11717271	C₂	1	2	yes	maximal, complemented
3	#11717264	C₂	1	1	yes	maximal, characteristic, complemented
4	#11716704	C₂	1	2	yes	maximal, complemented
5	#11716703	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅²×#41	—	2	2	no	—
7	C₅²×#41	—	2	2	no	—
8	C₅²×#41	C₂²	1	2	yes	—
9	C₂×C₅²×Q₈	—	2	2	no	—
10	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
11	C₂×#6045	C₂²	1	1	yes	characteristic
12	#87812	—	2	2	no	—
13	#87810	—	2	2	no	—
14	#87806	—	2	2	no	—
15	C₄×#1039	C₂²	1	1	yes	characteristic
16	#87804	C₂²	1	2	yes	—
17	#65535	—	5	6	no	maximal
18	C₅²×Q₈	—	2	2	no	—
19	C₅²×Q₈	—	2	2	no	—
20	C₅²×Q₈	D₄	1	2	yes	—
21	C₅²×D₄	—	2	2	no	—
22	C₅²×D₄	—	2	2	no	—
23	C₅²×D₄	D₄	1	2	yes	—
24	C₂×C₄×C₅²	—	2	4	no	—
25	C₂×C₄×C₅²	—	2	2	no	—
26	C₂×C₄×C₅²	D₄	1	2	yes	—
27	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
28	C₂×#1039	—	2	1	no	—
29	C₂×#1039	—	2	1	no	—
30	#6045	—	2	2	no	—
31	#6029	—	2	2	no	—
32	C₅×2₋¹⁺⁴	D₅	1	6	yes	—
33	#4097	—	5	12	no	—
34	#4090	—	5	6	no	—
35	#3969	—	5	12	no	—
36	#3968	—	5	6	no	—
37	C₂²×C₅²	—	2	2	no	—
38	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
39	C₄×C₅²	—	2	2	no	—
40	C₄×C₅²	—	2	2	no	—
41	C₄×C₅²	C₂×D₄	1	2	yes	—
42	#1039	—	2	2	no	—
43	#1039	—	4	1	no	—
44	C₅×#41	—	2	12	no	—
45	C₅×#41	—	2	12	no	—
46	C₅×#41	C₂×D₅	1	12	yes	—
47	C₂×C₅×Q₈	—	2	12	no	—
48	C₂×C₅×Q₈	C₂×D₅	1	6	yes	—
49	C₂×Dic₁₀	—	5	6	no	—
50	#689	—	10	12	no	—
51	#687	—	10	12	no	—
52	#683	—	10	12	no	—
53	C₄×Dic₅	—	5	6	no	—
54	#681	—	5	12	no	—
55	#456	—	25	1	no	Sylow
56	C₂×C₅²	—	2	1	no	—
57	C₂×C₅²	—	4	2	no	—
58	C₂×C₅²	#120	1	1	yes	characteristic
59	C₅×Q₈	—	2	12	no	—
60	C₅×Q₈	—	2	12	no	—
61	C₅×Q₈	C₂²⋊D₅	1	12	yes	—
62	C₅×D₄	—	2	12	no	—
63	C₅×D₄	—	2	12	no	—
64	C₅×D₄	C₂²⋊D₅	1	12	yes	—
65	C₂×C₄×C₅	—	2	24	no	—
66	C₂×C₄×C₅	—	2	12	no	—
67	C₂×C₄×C₅	C₂²⋊D₅	1	12	yes	—
68	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
69	C₂×Dic₅	—	10	6	no	—
70	C₂×Dic₅	—	10	6	no	—
71	Dic₁₀	—	10	12	no	—
72	#168	—	10	12	no	—
73	2₋¹⁺⁴	C₅⋊D₅	1	1	yes	characteristic
74	#137	—	25	2	no	—
75	C₄⋊Q₈	—	25	1	no	—
76	#104	—	25	2	no	—
77	#101	—	25	1	no	—
78	C₅²	#456	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₅	—	2	12	no	—
80	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
81	C₄×C₅	—	2	12	no	—
82	C₄×C₅	—	2	12	no	—
83	C₄×C₅	C₂×C₂²⋊D₅	1	12	yes	—
84	Dic₅	—	10	12	no	—
85	Dic₅	—	20	6	no	—
86	#41	—	2	2	no	—
87	#41	—	2	2	no	—
88	#41	C₂×C₅⋊D₅	1	2	yes	—
89	C₂×Q₈	—	2	2	no	—
90	C₂×Q₈	—	25	1	no	—
91	C₂×Q₈	C₂×C₅⋊D₅	1	1	yes	characteristic
92	Q₁₆	—	50	2	no	—
93	SD₁₆	—	50	2	no	—
94	M₁₆	—	25	2	no	—
95	C₄⋊C₄	—	50	2	no	—
96	C₄²	—	25	1	no	—
97	C₂×C₅	—	2	6	no	—
98	C₂×C₅	—	4	12	no	—
99	C₂×C₅	#4099	1	6	yes	—
100	Q₈	—	2	2	no	—
101	Q₈	—	2	2	no	—
102	Q₈	—	50	2	no	—
103	Q₈	C₂²⋊(C₅⋊D₅)	1	2	yes	—
104	D₄	—	2	2	no	—
105	D₄	—	2	2	no	—
106	D₄	C₂²⋊(C₅⋊D₅)	1	2	yes	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	50	1	no	—
110	C₂×C₄	—	50	1	no	—
111	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	2	yes	—
112	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
113	C₈	—	50	2	no	—
114	C₅	#65535	1	6	yes	complemented
115	C₂²	—	2	2	no	—
116	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	50	2	no	—
120	C₄	—	100	1	no	—
121	C₄	C₂×C₂²⋊(C₅⋊D₅)	1	2	yes	—
122	C₂	—	2	1	no	—
123	C₂	—	4	2	no	—
124	C₂	#11717273	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²×2₋¹⁺⁴

C₂

yes

maximal, characteristic

#11717271

C₂

yes

maximal, complemented

#11717264

C₂

yes

maximal, characteristic, complemented

#11716704

C₂

yes

maximal, complemented

#11716703

C₂

yes

maximal, characteristic, complemented

C₅²×#41

—

C₅²×#41

—

C₅²×#41

C₂²

yes

—

C₂×C₅²×Q₈

—

C₂×C₅²×Q₈

C₂²

yes

characteristic

C₂×#6045

C₂²

yes

—

—

—

C₂²

yes

characteristic

#87804

C₂²

yes

—

#65535

—

maximal

C₅²×Q₈

—

C₅²×Q₈

—

C₅²×Q₈

D₄

yes

—

C₅²×D₄

—

C₅²×D₄

—

C₅²×D₄

D₄

yes

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

—

C₂×C₄×C₅²

D₄

yes

—

C₂×C₄×C₅²

C₂³

yes

—

—

—

—

C₅×2₋¹⁺⁴

D₅

yes

—

—

—

—

—

C₂²×C₅²

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

—

C₄×C₅²

—

C₄×C₅²

C₂×D₄

yes

—

#1039

—

#1039

—

C₅×#41

—

C₅×#41

—

C₅×#41

C₂×D₅

yes

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

C₂×D₅

yes

—

C₂×Dic₁₀

—

#689

—

#687

—

#683

—

C₄×Dic₅

—

#681

—

#456

—

Sylow

C₂×C₅²

—

C₂×C₅²

—

C₂×C₅²

#120

yes

characteristic

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

C₂²⋊D₅

yes

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

C₂²⋊D₅

yes

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

#168

—

2₋¹⁺⁴

C₅⋊D₅

yes

characteristic

#137

—

C₄⋊Q₈

—

#104

—

#101

—

C₅²

#456

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

C₂×C₂²⋊D₅

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₂×C₂²⋊D₅

yes

—

Dic₅

—

Dic₅

—

#41

—

#41

—

#41

C₂×C₅⋊D₅

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×C₅⋊D₅

yes

characteristic

Q₁₆

—

SD₁₆

—

M₁₆

—

C₄⋊C₄

—

C₄²

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

#4099

yes

—

100

Q₈

—

101

Q₈

—

102

Q₈

—

103

Q₈

C₂²⋊(C₅⋊D₅)

yes

—

104

D₄

—

105

D₄

—

106

D₄

C₂²⋊(C₅⋊D₅)

yes

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

—

112

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

113

C₈

—

114

C₅

#65535

yes

complemented

115

C₂²

—

116

C₂²

C₂×C₂²⋊(C₅⋊D₅)

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

100

—

121

C₄

C₂×C₂²⋊(C₅⋊D₅)

yes

—

122

C₂

—

123

C₂

—

124

C₂

#11717273

yes

characteristic

	subgroup	G/H
center	C₂	#11717273
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting, prime radical	C₅²×2₋¹⁺⁴	C₂
socle	C₂×C₅²	#120
hypercenter	2₋¹⁺⁴	C₅⋊D₅
hypocenter	C₅²	#456
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ 2₋¹⁺⁴

subgroup

G/H

center

C₂

#11717273

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂²×C₅⋊D₅

Fitting, prime radical

C₅²×2₋¹⁺⁴

C₂

socle

C₂×C₅²

#120

hypercenter

2₋¹⁺⁴

C₅⋊D₅

hypocenter

C₅²

#456

LCS

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ 2₋¹⁺⁴

N	H ≅ G/N	type
2. #11717271	C₂	semidirect
3. #11717264	C₂	semidirect
4. #11716704	C₂	semidirect
5. #11716703	C₂	semidirect
78. C₅²	#456	semidirect
114. C₅	#65535	semidirect

H ≅ G/N

type

2. #11717271

C₂

semidirect

3. #11717264

C₂

semidirect

4. #11716704

C₂

semidirect

5. #11716703

C₂

semidirect

78. C₅²

#456

semidirect

114. C₅

#65535

semidirect

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

subgroup	index
C₅²×2₋¹⁺⁴	2
#11717271	2
#11717264	2
#11716704	2
#11716703	2
#65535	5

subgroup

index

C₅²×2₋¹⁺⁴

Subgroups — #13103109

Conjugacy classes of subgroups

Complements

Maximal subgroups