Group #13099329 — subgroups

Subgroups — #13099329

Order 1600.

240 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 84 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×#87804	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#681	C₂	1	2	yes	maximal
3	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
4	#87804	—	2	2	no	—
5	C₂×#6029	C₄	1	2	yes	—
6	C₅×#681	—	2	2	no	—
7	C₅×#681	—	2	2	no	—
8	C₂×C₅×#168	C₄	1	4	yes	—
9	#64780	—	5	2	no	maximal
10	C₂³×C₅²	Q₈	1	1	yes	characteristic
11	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
13	C₂×C₄×C₅²	D₄	1	2	yes	—
14	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
15	#6029	—	2	2	no	—
16	C₅×#168	—	2	4	no	—
17	C₂×C₅×M₁₆	—	5	2	no	—
18	C₂×#681	—	5	2	no	—
19	C₂×#681	—	10	2	no	—
20	C₂×#681	D₅	1	2	yes	—
21	C₂²×C₅²	—	4	1	no	—
22	C₂²×C₅²	C₄⋊C₄	1	1	yes	characteristic
23	C₂²×C₅²	C₄⋊C₄	1	2	yes	—
24	C₄×C₅²	C₄²	1	1	yes	characteristic
25	C₄×C₅²	C₂²⋊C₄	1	1	yes	characteristic
26	C₄×C₅²	C₂²⋊C₄	1	2	yes	—
27	C₂²×C₄×C₅	—	2	2	no	—
28	C₂²×C₄×C₅	C₂×D₅	1	2	yes	—
29	C₅×M₁₆	—	10	2	no	—
30	C₅×M₁₆	—	10	2	no	—
31	C₂×C₅×C₈	—	5	4	no	—
32	#681	—	2	2	no	—
33	#681	—	2	2	no	—
34	#681	—	10	8	no	—
35	#681	—	10	2	no	—
36	#681	—	10	2	no	—
37	C₂×#168	—	5	4	no	—
38	C₂×#168	—	10	4	no	—
39	C₂×#168	Dic₅	1	4	yes	—
40	#338	—	25	1	no	Sylow
41	C₂×C₅²	—	2	1	no	—
42	C₂×C₅²	—	2	2	no	—
43	C₂×C₅²	#95	1	1	yes	characteristic
44	C₂³×C₅	—	2	2	no	—
45	C₂³×C₅	Dic₁₀	1	2	yes	—
46	C₂×C₄×C₅	—	2	4	no	—
47	C₂×C₄×C₅	—	2	2	no	—
48	C₂×C₄×C₅	—	2	2	no	—
49	C₂×C₄×C₅	—	2	4	no	—
50	C₂×C₄×C₅	C₄×D₅	1	2	yes	—
51	C₂×C₄×C₅	C₄×D₅	1	2	yes	—
52	C₂×C₄×C₅	D₂₀	1	2	yes	—
53	C₂×C₄×C₅	C₂×Dic₅	1	2	yes	—
54	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
55	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
56	C₅×C₈	—	10	4	no	—
57	#168	—	2	4	no	—
58	#168	—	10	8	no	—
59	#168	—	10	4	no	—
60	C₂×M₁₆	—	5	2	no	—
61	C₂×M₁₆	—	25	1	no	—
62	C₅²	#338	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₅	—	2	4	no	—
64	C₂²×C₅	—	2	2	no	—
65	C₂²×C₅	—	4	2	no	—
66	C₂²×C₅	—	4	4	no	—
67	C₂²×C₅	#683	1	2	yes	—
68	C₂²×C₅	#683	1	2	yes	—
69	C₂²×C₅	C₄⋊Dic₅	1	2	yes	—
70	C₄×C₅	—	2	2	no	—
71	C₄×C₅	—	2	4	no	—
72	C₄×C₅	—	2	2	no	—
73	C₄×C₅	C₄×Dic₅	1	2	yes	—
74	C₄×C₅	#685	1	2	yes	—
75	C₄×C₅	#685	1	2	yes	—
76	C₄×C₅	C₂²⋊Dic₅	1	2	yes	—
77	C₂²×C₄	D₅²	1	1	yes	characteristic
78	M₁₆	—	10	2	no	—
79	M₁₆	—	10	2	no	—
80	M₁₆	—	50	2	no	—
81	C₂×C₈	—	5	4	no	—
82	C₂×C₈	—	25	2	no	—
83	C₂×C₅	—	2	4	no	—
84	C₂×C₅	—	2	2	no	—
85	C₂×C₅	—	2	2	no	—
86	C₂×C₅	—	2	2	no	—
87	C₂×C₅	—	2	2	no	—
88	C₂×C₅	—	4	4	no	—
89	C₂×C₅	#3963	1	2	yes	—
90	C₂³	#6039	1	1	yes	characteristic
91	C₂×C₄	D₅×Dic₅	1	2	yes	—
92	C₂×C₄	#6036	1	1	yes	characteristic
93	C₂×C₄	#6037	1	1	yes	characteristic
94	C₂×C₄	#6038	1	2	yes	—
95	C₈	—	10	4	no	—
96	C₈	—	50	2	no	—
97	C₅	—	2	2	no	—
98	C₅	#64780	1	2	yes	complemented
99	C₂²	—	4	1	no	—
100	C₂²	#87780	1	2	yes	—
101	C₂²	#87781	1	1	yes	characteristic
102	C₄	Dic₅²	1	1	yes	characteristic
103	C₄	#87778	1	2	yes	—
104	C₄	#87779	1	1	yes	characteristic
105	C₂	—	2	2	no	—
106	C₂	—	2	1	no	—
107	C₂	#11716620	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#87804

C₂

yes

maximal, characteristic

C₂×C₅×#681

C₂

yes

maximal

C₂²×C₄×C₅²

C₂²

yes

characteristic

#87804

—

C₂×#6029

C₄

yes

—

C₅×#681

—

C₅×#681

—

C₂×C₅×#168

C₄

yes

—

#64780

—

maximal

C₂³×C₅²

Q₈

yes

characteristic

C₂×C₄×C₅²

C₂×C₄

yes

characteristic

C₂×C₄×C₅²

C₂×C₄

yes

—

C₂×C₄×C₅²

D₄

yes

—

C₂×C₄×C₅²

D₄

yes

characteristic

#6029

—

C₅×#168

—

C₂×C₅×M₁₆

—

C₂×#681

—

C₂×#681

—

C₂×#681

D₅

yes

—

C₂²×C₅²

—

C₂²×C₅²

C₄⋊C₄

yes

characteristic

C₂²×C₅²

C₄⋊C₄

yes

—

C₄×C₅²

C₄²

yes

characteristic

C₄×C₅²

C₂²⋊C₄

yes

characteristic

C₄×C₅²

C₂²⋊C₄

yes

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₅

yes

—

C₅×M₁₆

—

C₅×M₁₆

—

C₂×C₅×C₈

—

—

—

—

—

—

—

—

Dic₅

yes

—

#338

—

Sylow

C₂×C₅²

—

C₂×C₅²

—

C₂×C₅²

#95

yes

characteristic

C₂³×C₅

—

C₂³×C₅

Dic₁₀

yes

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₄×D₅

yes

—

C₂×C₄×C₅

C₄×D₅

yes

—

C₂×C₄×C₅

D₂₀

yes

—

C₂×C₄×C₅

C₂×Dic₅

yes

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₅×C₈

—

#168

—

#168

—

#168

—

C₂×M₁₆

—

C₂×M₁₆

—

C₅²

#338

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

#683

yes

—

C₂²×C₅

#683

yes

—

C₂²×C₅

C₄⋊Dic₅

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₄×Dic₅

yes

—

C₄×C₅

#685

yes

—

C₄×C₅

#685

yes

—

C₄×C₅

C₂²⋊Dic₅

yes

—

C₂²×C₄

D₅²

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

#3963

yes

—

C₂³

#6039

yes

characteristic

C₂×C₄

D₅×Dic₅

yes

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₈

—

C₈

—

C₅

—

C₅

#64780

yes

complemented

C₂²

—

100

C₂²

#87780

yes

—

101

C₂²

#87781

yes

characteristic

102

C₄

Dic₅²

yes

characteristic

103

C₄

#87778

yes

—

104

C₄

#87779

yes

characteristic

105

C₂

—

106

C₂

—

107

C₂

#11716620

yes

characteristic

	subgroup	G/H
center	C₄	Dic₅²
commutator	C₄×C₅²	C₄²
Frattini, hypercenter	C₂²×C₄	D₅²
Fitting, cosocle	C₂²×C₄×C₅²	C₂²
socle	C₂×C₅²	#95
prime radical	C₂³×C₅²	Q₈
hypocenter	C₅²	#338
LCS	G ▷ C₄×C₅² ▷ C₅²
UCS	C₁ ◁ C₄ ◁ C₂²×C₄

subgroup

G/H

center

C₄

Dic₅²

commutator

C₄×C₅²

C₄²

Frattini, hypercenter

C₂²×C₄

D₅²

Fitting, cosocle

C₂²×C₄×C₅²

C₂²

socle

C₂×C₅²

#95

prime radical

C₂³×C₅²

Q₈

C₅²

G ▷ C₄×C₅² ▷ C₅²

C₁ ◁ C₄ ◁ C₂²×C₄

N	H ≅ G/N	type
62. C₅²	#338	semidirect
98. C₅	#64780	semidirect

H ≅ G/N

type

62. C₅²

#338

semidirect

98. C₅

#64780

semidirect

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

subgroup	index
C₂×#87804	2
C₂×C₅×#681	2
#64780	5

subgroup

index

C₂×#87804

C₂×C₅×#681

#64780

Subgroups — #13099329

Conjugacy classes of subgroups

Complements

Maximal subgroups