Group #13104228 — subgroups

Subgroups — #13104228

Order 1600.

228 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 49 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×#87862	C₂	1	1	yes	maximal, characteristic
2	#11716992	C₂	1	2	yes	maximal
3	#11716801	C₂	1	2	yes	maximal, complemented
4	#11716800	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
6	C₂×#6034	C₂²	1	1	yes	characteristic
7	C₂×#6034	C₂²	1	1	yes	characteristic
8	#87862	C₄	1	4	yes	complemented
9	C₄×C₅×Dic₅	—	2	2	no	—
10	#87781	C₂²	1	4	yes	—
11	#87779	—	2	2	no	—
12	C₂²×C₅⋊D₅	D₄	1	1	yes	characteristic
13	C₂×C₄×C₅²	D₄	1	1	yes	characteristic
14	C₂×#1039	C₂³	1	1	yes	characteristic
15	C₄×C₅⋊D₅	C₂×C₄	1	2	yes	—
16	C₄×C₅⋊D₅	Q₈	1	2	yes	—
17	C₂×C₅×Dic₅	—	2	4	no	—
18	#6034	C₂×C₄	1	2	yes	—
19	#6034	C₂×C₄	1	2	yes	—
20	C₂×D₅⋊C₈	—	10	1	no	—
21	#4018	—	10	2	no	—
22	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
23	C₂×C₅⋊D₅	—	2	2	no	—
24	C₂×C₅⋊D₅	C₄⋊C₄	1	2	yes	—
25	C₄×C₅²	C₄⋊C₄	1	2	yes	—
26	#1039	C₂²×C₄	1	1	yes	characteristic
27	#1039	C₂×Q₈	1	1	yes	characteristic
28	C₅×Dic₅	—	4	4	no	—
29	C₂×C₄×D₅	—	10	1	no	—
30	C₂×C₄×D₅	—	10	2	no	—
31	C₂×#170	—	10	1	no	—
32	C₂×#170	—	10	1	no	—
33	D₅⋊C₈	—	10	4	no	—
34	C₄²×C₅	—	10	2	no	—
35	#685	—	10	4	no	—
36	#683	—	10	4	no	—
37	C₄×Dic₅	—	2	2	no	—
38	#427	—	25	1	no	maximal, Sylow
39	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₅²	#135	1	2	yes	—
41	C₅⋊D₅	—	2	2	no	—
42	C₂²×D₅	—	10	1	no	—
43	C₂²×D₅	—	10	2	no	—
44	C₂×C₄×C₅	—	2	2	no	—
45	C₂×C₄×C₅	—	2	1	no	—
46	C₂×C₄×C₅	—	10	4	no	—
47	C₂×Dic₅	—	2	4	no	—
48	C₂×Dic₅	—	10	2	no	—
49	C₂×Dic₅	—	10	1	no	—
50	C₄×D₅	—	10	2	no	—
51	C₄×D₅	—	10	4	no	—
52	C₄×D₅	—	10	2	no	—
53	C₄×D₅	—	10	4	no	—
54	#170	—	10	2	no	—
55	#170	—	10	2	no	—
56	C₂²×C₈	—	25	1	no	—
57	#117	—	25	2	no	—
58	#107	—	25	2	no	—
59	#106	—	25	2	no	—
60	C₅²	#427	1	1	yes	characteristic, complemented, Sylow
61	C₂²×C₅	—	2	2	no	—
62	C₂²×C₅	—	2	1	no	—
63	C₂×D₅	—	10	2	no	—
64	C₂×D₅	—	10	4	no	—
65	C₂×D₅	—	10	4	no	—
66	C₂×D₅	—	20	4	no	—
67	C₄×C₅	—	2	2	no	—
68	C₄×C₅	—	2	4	no	—
69	C₄×C₅	—	20	4	no	—
70	Dic₅	—	4	4	no	—
71	Dic₅	—	10	1	no	—
72	Dic₅	—	10	2	no	—
73	Dic₅	—	10	2	no	—
74	Dic₅	—	10	1	no	—
75	C₂²×C₄	—	25	1	no	—
76	C₂×C₈	—	25	1	no	—
77	C₂×C₈	—	25	1	no	—
78	C₂×C₈	—	25	4	no	—
79	C₄⋊C₄	—	25	4	no	—
80	C₂²⋊C₄	—	50	2	no	—
81	C₄²	—	10	2	no	—
82	C₂×C₅	—	2	4	no	—
83	C₂×C₅	—	2	2	no	—
84	C₂×C₅	—	2	1	no	—
85	C₂×C₅	—	2	2	no	—
86	D₅	—	10	4	no	—
87	D₅	—	20	4	no	—
88	C₂³	—	25	1	no	—
89	C₂×C₄	—	10	4	no	—
90	C₂×C₄	—	25	1	no	—
91	C₂×C₄	—	25	2	no	—
92	C₂×C₄	—	25	2	no	—
93	C₂×C₄	D₅²⋊C₂	1	1	yes	characteristic
94	C₈	—	25	2	no	—
95	C₈	—	25	2	no	—
96	C₅	—	2	1	no	—
97	C₅	—	2	2	no	—
98	C₂²	—	25	2	no	—
99	C₂²	—	50	2	no	—
100	C₂²	C₂×D₅²⋊C₂	1	1	yes	characteristic
101	C₄	—	20	4	no	—
102	C₄	—	25	1	no	—
103	C₄	—	25	1	no	—
104	C₄	#87836	1	2	yes	—
105	C₂	—	50	2	no	—
106	C₂	#11717381	1	2	yes	—
107	C₂	C₂×#87836	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#87862

C₂

yes

maximal, characteristic

#11716992

C₂

yes

maximal

#11716801

C₂

yes

maximal, complemented

#11716800

C₂

yes

maximal, complemented

C₂×C₄×C₅⋊D₅

C₂²

yes

characteristic

C₂×#6034

C₂²

yes

characteristic

C₂×#6034

C₂²

yes

characteristic

#87862

C₄

yes

complemented

C₄×C₅×Dic₅

—

#87781

C₂²

yes

—

#87779

—

C₂²×C₅⋊D₅

D₄

yes

characteristic

C₂×C₄×C₅²

D₄

yes

characteristic

C₂×#1039

C₂³

yes

characteristic

C₄×C₅⋊D₅

C₂×C₄

yes

—

C₄×C₅⋊D₅

Q₈

yes

—

C₂×C₅×Dic₅

—

#6034

C₂×C₄

yes

—

#6034

C₂×C₄

yes

—

C₂×D₅⋊C₈

—

#4018

—

C₂²×C₅²

C₂×D₄

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

C₄⋊C₄

yes

—

C₄×C₅²

C₄⋊C₄

yes

—

#1039

C₂²×C₄

yes

characteristic

#1039

C₂×Q₈

yes

characteristic

C₅×Dic₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×#170

—

C₂×#170

—

D₅⋊C₈

—

C₄²×C₅

—

#685

—

#683

—

C₄×Dic₅

—

#427

—

maximal, Sylow

C₂×C₅²

C₂×C₄⋊C₄

yes

characteristic

C₂×C₅²

#135

yes

—

C₅⋊D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

#170

—

#170

—

C₂²×C₈

—

#117

—

#107

—

#106

—

C₅²

#427

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

—

D₅

—

D₅

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₅²⋊C₂

yes

characteristic

C₈

—

C₈

—

C₅

—

C₅

—

C₂²

—

C₂²

—

100

C₂²

C₂×D₅²⋊C₂

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

#87836

yes

—

105

C₂

—

106

C₂

#11717381

yes

—

107

C₂

C₂×#87836

yes

characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄	D₅²⋊C₂
commutator	#1039	C₂²×C₄
Frattini	C₂²	C₂×D₅²⋊C₂
Fitting	C₂×C₄×C₅²	D₄
socle	C₂²×C₅²	C₂×D₄
prime radical	C₂²×C₅⋊D₅	D₄
cosocle	C₂×#1039	C₂³
hypocenter	C₅²	#427
LCS	G ▷ #1039 ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄

subgroup

G/H

center, hypercenter

C₂×C₄

D₅²⋊C₂

commutator

#1039

C₂²×C₄

Frattini

C₂²

C₂×D₅²⋊C₂

Fitting

C₂×C₄×C₅²

D₄

socle

C₂²×C₅²

C₂×D₄

prime radical

C₂²×C₅⋊D₅

D₄

C₂³

C₅²

G ▷ #1039 ▷ C₂×C₅² ▷ C₅²

UCS

C₁ ◁ C₂×C₄

N	H ≅ G/N	type
3. #11716801	C₂	semidirect
4. #11716800	C₂	semidirect
8. #87862	C₄	semidirect
60. C₅²	#427	semidirect

H ≅ G/N

type

3. #11716801

C₂

semidirect

4. #11716800

C₂

semidirect

8. #87862

C₄

semidirect

60. C₅²

#427

semidirect

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

subgroup	index
C₂×#87862	2
#11716992	2
#11716801	2
#11716800	2
#427	25

subgroup

index

Subgroups — #13104228

Conjugacy classes of subgroups

Complements

Maximal subgroups