Group #13102860 — subgroups

Subgroups — #13102860

Order 1600.

456 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 195 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×#117	C₂	1	1	yes	maximal, characteristic
2	#11717246	C₂	1	1	yes	maximal, characteristic
3	C₂²×#6029	C₂	1	1	yes	maximal, characteristic
4	#11716675	C₂	1	2	yes	maximal, complemented
5	#11716674	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
7	C₅²×C₄⋊C₄	C₂²	1	2	yes	—
8	C₅²×C₂²⋊C₄	—	2	1	no	—
9	C₄²×C₅²	—	2	1	no	—
10	C₂²⋊#1039	—	2	1	no	—
11	C₄⋊#1039	C₂²	1	2	yes	—
12	C₄×#1039	—	2	1	no	—
13	C₂×#6029	C₄	1	4	yes	complemented
14	C₂×#6029	C₂²	1	1	yes	characteristic
15	C₂×#6029	C₂²	1	1	yes	characteristic
16	#65287	—	5	6	no	maximal
17	C₂³×C₅²	D₄	1	1	yes	characteristic
18	C₂×C₄×C₅²	—	2	2	no	—
19	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
21	C₂×C₄×C₅²	Q₈	1	2	yes	—
22	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
23	C₂×#1039	—	2	2	no	—
24	#6029	C₂×C₄	1	2	yes	—
25	#6029	C₂×C₄	1	2	yes	—
26	C₅×#117	D₅	1	6	yes	—
27	#4072	—	5	6	no	—
28	C₂²×#168	—	5	6	no	—
29	#3940	—	5	12	no	—
30	#3939	—	5	12	no	—
31	C₂²×C₅²	—	2	2	no	—
32	C₂²×C₅²	C₄⋊C₄	1	2	yes	—
33	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
34	C₄×C₅²	—	4	2	no	—
35	C₄×C₅²	C₄⋊C₄	1	2	yes	—
36	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
37	C₄×C₅²	C₂×Q₈	1	1	yes	characteristic
38	#1039	—	4	2	no	—
39	C₂²×C₄×C₅	C₂×D₅	1	6	yes	—
40	C₅×C₄⋊C₄	C₂×D₅	1	12	yes	—
41	C₅×C₂²⋊C₄	—	2	6	no	—
42	C₄²×C₅	—	2	6	no	—
43	C₂²⋊Dic₅	—	10	6	no	—
44	C₄⋊Dic₅	—	5	12	no	—
45	C₄×Dic₅	—	10	6	no	—
46	C₂×#168	—	5	24	no	—
47	C₂×#168	—	5	6	no	—
48	C₂×#168	—	5	6	no	—
49	#427	—	25	1	no	Sylow
50	C₂×C₅²	—	2	2	no	—
51	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₅²	#135	1	2	yes	—
53	C₂³×C₅	C₂²⋊D₅	1	6	yes	—
54	C₂×C₄×C₅	—	2	12	no	—
55	C₂×C₄×C₅	Dic₁₀	1	12	yes	—
56	C₂×C₄×C₅	C₄×D₅	1	12	yes	—
57	C₂×C₄×C₅	C₂²⋊D₅	1	6	yes	—
58	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
59	C₂×Dic₅	—	10	12	no	—
60	#168	—	5	12	no	—
61	#168	—	5	12	no	—
62	C₂²×C₈	—	25	1	no	—
63	#117	—	25	1	no	—
64	#117	C₅⋊D₅	1	1	yes	characteristic
65	#107	—	25	2	no	—
66	#106	—	25	2	no	—
67	C₅²	#427	1	1	yes	characteristic, complemented, Sylow
68	C₂²×C₅	—	2	12	no	—
69	C₂²×C₅	#683	1	12	yes	—
70	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
71	C₄×C₅	—	4	12	no	—
72	C₄×C₅	#683	1	12	yes	—
73	C₄×C₅	C₂×Dic₁₀	1	6	yes	—
74	C₄×C₅	C₂×C₄×D₅	1	6	yes	—
75	Dic₅	—	20	12	no	—
76	C₂²×C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
77	C₂×C₈	—	25	4	no	—
78	C₂×C₈	—	25	1	no	—
79	C₂×C₈	—	25	1	no	—
80	C₄⋊C₄	—	25	2	no	—
81	C₄⋊C₄	C₂×C₅⋊D₅	1	2	yes	—
82	C₂²⋊C₄	—	2	1	no	—
83	C₂²⋊C₄	—	50	1	no	—
84	C₄²	—	2	1	no	—
85	C₄²	—	50	1	no	—
86	C₂×C₅	—	2	12	no	—
87	C₂×C₅	C₂×#683	1	6	yes	—
88	C₂×C₅	#4096	1	12	yes	—
89	C₂³	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	50	2	no	—
92	C₂×C₄	#6045	1	2	yes	—
93	C₂×C₄	C₄×C₅⋊D₅	1	2	yes	—
94	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
95	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
96	C₈	—	25	2	no	—
97	C₈	—	25	2	no	—
98	C₅	#65287	1	6	yes	complemented
99	C₂²	—	2	2	no	—
100	C₂²	#87806	1	2	yes	—
101	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
102	C₄	—	4	2	no	—
103	C₄	—	100	2	no	—
104	C₄	#87806	1	2	yes	—
105	C₄	C₂×#6045	1	1	yes	characteristic
106	C₄	C₂×C₄×C₅⋊D₅	1	1	yes	characteristic
107	C₂	—	2	2	no	—
108	C₂	C₂×#87806	1	1	yes	characteristic
109	C₂	#11717270	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²×#117

C₂

yes

maximal, characteristic

#11717246

C₂

yes

maximal, characteristic

C₂²×#6029

C₂

yes

maximal, characteristic

#11716675

C₂

yes

maximal, complemented

#11716674

C₂

yes

maximal, complemented

C₂²×C₄×C₅²

C₂²

yes

characteristic

C₅²×C₄⋊C₄

C₂²

yes

—

C₅²×C₂²⋊C₄

—

C₄²×C₅²

—

C₂²⋊#1039

—

C₄⋊#1039

C₂²

yes

—

C₄×#1039

—

C₂×#6029

C₄

yes

complemented

C₂×#6029

C₂²

yes

characteristic

C₂×#6029

C₂²

yes

characteristic

#65287

—

maximal

C₂³×C₅²

D₄

yes

characteristic

C₂×C₄×C₅²

—

C₂×C₄×C₅²

C₂×C₄

yes

—

C₂×C₄×C₅²

D₄

yes

characteristic

C₂×C₄×C₅²

Q₈

yes

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

—

#6029

C₂×C₄

yes

—

#6029

C₂×C₄

yes

—

C₅×#117

D₅

yes

—

#4072

—

C₂²×#168

—

#3940

—

#3939

—

C₂²×C₅²

—

C₂²×C₅²

C₄⋊C₄

yes

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

—

C₄×C₅²

C₄⋊C₄

yes

—

C₄×C₅²

C₂²×C₄

yes

characteristic

C₄×C₅²

C₂×Q₈

yes

characteristic

#1039

—

C₂²×C₄×C₅

C₂×D₅

yes

—

C₅×C₄⋊C₄

C₂×D₅

yes

—

C₅×C₂²⋊C₄

—

C₄²×C₅

—

C₂²⋊Dic₅

—

C₄⋊Dic₅

—

C₄×Dic₅

—

—

—

—

—

C₂×C₅²

—

C₂×C₅²

C₂×C₄⋊C₄

yes

characteristic

C₂×C₅²

#135

yes

—

C₂³×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

—

C₂×C₄×C₅

Dic₁₀

yes

—

C₂×C₄×C₅

C₄×D₅

yes

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

#168

—

#168

—

C₂²×C₈

—

#117

—

#117

C₅⋊D₅

yes

characteristic

#107

—

#106

—

C₅²

#427

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

#683

yes

—

C₂²×C₅

C₂×C₂²⋊D₅

yes

—

C₄×C₅

—

C₄×C₅

#683

yes

—

C₄×C₅

C₂×Dic₁₀

yes

—

C₄×C₅

C₂×C₄×D₅

yes

—

Dic₅

—

C₂²×C₄

C₂×C₅⋊D₅

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₅⋊D₅

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₂×C₅

—

C₂×C₅

C₂×#683

yes

—

C₂×C₅

#4096

yes

—

C₂³

C₂²⋊(C₅⋊D₅)

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#6045

yes

—

C₂×C₄

C₄×C₅⋊D₅

yes

—

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

characteristic

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

C₈

—

C₈

—

C₅

#65287

yes

complemented

C₂²

—

100

C₂²

#87806

yes

—

101

C₂²

C₂×C₂²⋊(C₅⋊D₅)

yes

characteristic

102

C₄

—

103

C₄

—

100

—

104

C₄

#87806

yes

—

105

C₄

C₂×#6045

yes

characteristic

106

C₄

C₂×C₄×C₅⋊D₅

yes

characteristic

107

C₂

—

108

C₂

C₂×#87806

yes

characteristic

109

C₂

#11717270

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²⋊(C₅⋊D₅)
commutator	C₄×C₅²	C₂²×C₄
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting	C₅²×#117	C₂
socle	C₂²×C₅²	C₂×D₄
prime radical	C₂³×C₅²	D₄
cosocle	C₂×C₄×C₅²	C₂³
hypercenter	#117	C₅⋊D₅
hypocenter	C₅²	#427
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #117

subgroup

G/H

center

C₂×C₄

C₂²⋊(C₅⋊D₅)

commutator

C₄×C₅²

C₂²×C₄

Frattini

C₂×C₄

C₂²×C₅⋊D₅

Fitting

C₅²×#117

C₂

socle

C₂²×C₅²

C₂×D₄

prime radical

C₂³×C₅²

D₄

cosocle

C₂×C₄×C₅²

C₂³

C₅⋊D₅

C₅²

G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #117

N	H ≅ G/N	type
4. #11716675	C₂	semidirect
5. #11716674	C₂	semidirect
13. C₂×#6029	C₄	semidirect
67. C₅²	#427	semidirect
98. C₅	#65287	semidirect

H ≅ G/N

type

4. #11716675

C₂

semidirect

5. #11716674

C₂

semidirect

13. C₂×#6029

C₄

semidirect

67. C₅²

#427

semidirect

98. C₅

#65287

semidirect

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

subgroup	index
C₅²×#117	2
#11717246	2
C₂²×#6029	2
#11716675	2
#11716674	2
#65287	5

subgroup

index

C₅²×#117

C₂²×#6029

Subgroups — #13102860

Conjugacy classes of subgroups

Complements

Maximal subgroups