Group #13102933 — subgroups

Subgroups — #13102933

Order 1600.

472 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 155 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×#127	C₂	1	1	yes	maximal, characteristic
2	#11717189	C₂	1	1	yes	maximal, characteristic, complemented
3	#11716699	C₂	1	4	yes	maximal, complemented
4	C₄×#6029	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅²×D₄	—	2	1	no	—
6	C₂×C₅²×D₄	—	2	1	no	—
7	C₂×C₅²×D₄	C₂²	1	2	yes	—
8	C₂×#6045	—	2	1	no	—
9	C₄²×C₅²	C₂²	1	1	yes	characteristic
10	C₄⋊#1039	—	2	1	no	—
11	C₄⋊#1039	C₂²	1	2	yes	—
12	C₂×#6029	C₂²	1	2	yes	—
13	#65359	—	5	6	no	maximal
14	C₂³×C₅²	—	2	2	no	—
15	C₅²×D₄	—	2	4	no	—
16	C₅²×D₄	—	4	2	no	—
17	C₅²×D₄	—	4	2	no	—
18	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
19	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
20	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
21	C₂×#1039	—	2	2	no	—
22	#6045	—	4	2	no	—
23	#6029	—	2	2	no	—
24	C₅×#127	D₅	1	6	yes	—
25	#4015	—	5	6	no	—
26	#3964	—	5	24	no	—
27	C₄×#168	—	5	6	no	—
28	C₂²×C₅²	—	4	2	no	—
29	C₂²×C₅²	—	4	2	no	—
30	C₂²×C₅²	—	4	2	no	—
31	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
32	C₄×C₅²	D₈	1	2	yes	—
33	C₄×C₅²	SD₁₆	1	2	yes	—
34	C₄×C₅²	#41	1	2	yes	—
35	#1039	—	4	2	no	—
36	C₂×C₅×D₄	—	2	6	no	—
37	C₂×C₅×D₄	—	2	6	no	—
38	C₂×C₅×D₄	C₂×D₅	1	12	yes	—
39	C₂×Dic₁₀	—	10	6	no	—
40	C₄²×C₅	C₂×D₅	1	6	yes	—
41	C₄⋊Dic₅	—	5	12	no	—
42	C₄⋊Dic₅	—	10	6	no	—
43	C₂×#168	—	5	12	no	—
44	#486	—	25	1	no	Sylow
45	C₂×C₅²	—	8	2	no	—
46	C₂×C₅²	#124	1	1	yes	characteristic
47	C₂×C₅²	C₂×D₈	1	1	yes	characteristic
48	C₂×C₅²	C₂×SD₁₆	1	1	yes	characteristic
49	C₂³×C₅	—	2	12	no	—
50	C₅×D₄	—	2	24	no	—
51	C₅×D₄	—	4	12	no	—
52	C₅×D₄	—	4	12	no	—
53	C₂×C₄×C₅	C₂²⋊D₅	1	6	yes	—
54	C₂×C₄×C₅	C₂²⋊D₅	1	6	yes	—
55	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
56	C₂×Dic₅	—	10	12	no	—
57	Dic₁₀	—	20	12	no	—
58	#168	—	10	12	no	—
59	C₄⋊Q₈	—	25	1	no	—
60	#127	C₅⋊D₅	1	1	yes	characteristic
61	#102	—	25	4	no	—
62	C₄×C₈	—	25	1	no	—
63	C₅²	#486	1	1	yes	characteristic, complemented, Sylow
64	C₂²×C₅	—	4	12	no	—
65	C₂²×C₅	—	4	12	no	—
66	C₂²×C₅	—	4	12	no	—
67	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
68	C₄×C₅	#686	1	12	yes	—
69	C₄×C₅	#687	1	12	yes	—
70	C₄×C₅	#711	1	12	yes	—
71	Dic₅	—	20	12	no	—
72	C₂×Q₈	—	50	1	no	—
73	C₂×D₄	—	2	1	no	—
74	C₂×D₄	—	2	1	no	—
75	C₂×D₄	C₂×C₅⋊D₅	1	2	yes	—
76	C₂×C₈	—	25	2	no	—
77	C₄⋊C₄	—	25	2	no	—
78	C₄⋊C₄	—	50	1	no	—
79	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
80	C₂×C₅	—	8	12	no	—
81	C₂×C₅	C₂×#686	1	6	yes	—
82	C₂×C₅	C₂×#687	1	6	yes	—
83	C₂×C₅	#4082	1	6	yes	—
84	C₂³	—	2	2	no	—
85	Q₈	—	100	2	no	—
86	D₄	—	2	4	no	—
87	D₄	—	4	2	no	—
88	D₄	—	4	2	no	—
89	C₂×C₄	—	50	2	no	—
90	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
91	C₂×C₄	C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
92	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
93	C₈	—	50	2	no	—
94	C₅	#65359	1	6	yes	complemented
95	C₂²	—	4	2	no	—
96	C₂²	—	4	2	no	—
97	C₂²	—	4	2	no	—
98	C₂²	C₂×C₂²⋊(C₅⋊D₅)	1	1	yes	characteristic
99	C₄	—	100	2	no	—
100	C₄	#87809	1	2	yes	—
101	C₄	#87810	1	2	yes	—
102	C₄	#87902	1	2	yes	—
103	C₂	—	8	2	no	—
104	C₂	C₂×#87809	1	1	yes	characteristic
105	C₂	C₂×#87810	1	1	yes	characteristic
106	C₂	#11717256	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²×#127

C₂

yes

maximal, characteristic

#11717189

C₂

yes

maximal, characteristic, complemented

#11716699

C₂

yes

maximal, complemented

C₄×#6029

C₂

yes

maximal, characteristic, complemented

C₂×C₅²×D₄

—

C₂×C₅²×D₄

—

C₂×C₅²×D₄

C₂²

yes

—

C₂×#6045

—

C₄²×C₅²

C₂²

yes

characteristic

C₄⋊#1039

—

C₄⋊#1039

C₂²

yes

—

C₂×#6029

C₂²

yes

—

#65359

—

maximal

C₂³×C₅²

—

C₅²×D₄

—

C₅²×D₄

—

C₅²×D₄

—

C₂×C₄×C₅²

D₄

yes

characteristic

C₂×C₄×C₅²

D₄

yes

characteristic

C₂×C₄×C₅²

C₂³

yes

—

—

—

D₅

yes

—

#4015

—

#3964

—

C₄×#168

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

D₈

yes

—

C₄×C₅²

SD₁₆

yes

—

C₄×C₅²

#41

yes

—

#1039

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

C₂×D₅

yes

—

C₂×Dic₁₀

—

C₄²×C₅

C₂×D₅

yes

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₂×#168

—

#486

—

Sylow

C₂×C₅²

—

C₂×C₅²

#124

yes

characteristic

C₂×C₅²

C₂×D₈

yes

characteristic

C₂×C₅²

C₂×SD₁₆

yes

characteristic

C₂³×C₅

—

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

Dic₁₀

—

#168

—

C₄⋊Q₈

—

#127

C₅⋊D₅

yes

characteristic

#102

—

C₄×C₈

—

C₅²

#486

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂×C₂²⋊D₅

yes

—

C₄×C₅

#686

yes

—

C₄×C₅

#687

yes

—

C₄×C₅

#711

yes

—

Dic₅

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×C₅⋊D₅

yes

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₅⋊D₅

yes

characteristic

C₂×C₅

—

C₂×C₅

C₂×#686

yes

—

C₂×C₅

C₂×#687

yes

—

C₂×C₅

#4082

yes

—

C₂³

—

Q₈

—

100

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

characteristic

C₂×C₄

C₂²⋊(C₅⋊D₅)

yes

characteristic

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

C₈

—

C₅

#65359

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊(C₅⋊D₅)

yes

characteristic

C₄

—

100

—

100

C₄

#87809

yes

—

101

C₄

#87810

yes

—

102

C₄

#87902

yes

—

103

C₂

—

104

C₂

C₂×#87809

yes

characteristic

105

C₂

C₂×#87810

yes

characteristic

106

C₂

#11717256

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊(C₅⋊D₅)
commutator, cosocle	C₂×C₄×C₅²	C₂³
Frattini	C₂×C₄	C₂²×C₅⋊D₅
Fitting, prime radical	C₅²×#127	C₂
socle	C₂²×C₅²	C₂×D₄
hypercenter	#127	C₅⋊D₅
hypocenter	C₅²	#486
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

subgroup

G/H

center

C₂²

C₂×C₂²⋊(C₅⋊D₅)

commutator, cosocle

C₂×C₄×C₅²

C₂³

Frattini

C₂×C₄

C₂²×C₅⋊D₅

Fitting, prime radical

C₅²×#127

C₂

socle

C₂²×C₅²

C₂×D₄

C₅⋊D₅

C₅²

G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #127

N	H ≅ G/N	type
2. #11717189	C₂	semidirect
3. #11716699	C₂	semidirect
4. C₄×#6029	C₂	semidirect
63. C₅²	#486	semidirect
94. C₅	#65359	semidirect

H ≅ G/N

type

2. #11717189

C₂

semidirect

3. #11716699

C₂

semidirect

4. C₄×#6029

C₂

semidirect

63. C₅²

#486

semidirect

94. C₅

#65359

semidirect

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

subgroup	index
C₅²×#127	2
#11717189	2
#11716699	2
C₄×#6029	2
#65359	5

subgroup

index

C₅²×#127

Subgroups — #13102933

Conjugacy classes of subgroups

Complements

Maximal subgroups