Group #13097991 — subgroups

Subgroups — #13097991

Order 1600.

186 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 65 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×#87713	C₂	1	2	yes	maximal
2	C₂×Dic₁₀₀	C₂	1	2	yes	maximal, complemented
3	#11716437	C₂	1	1	yes	maximal, characteristic
4	#11716431	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₅×#97	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₀₀	C₂²	1	1	yes	characteristic
7	C₂×Dic₅₀	C₂²	1	2	yes	—
8	C₂×Dic₅₀	C₂²	1	1	yes	characteristic
9	C₂×C₈×C₂₅	C₂²	1	2	yes	—
10	C₄²×C₂₅	C₂²	1	1	yes	characteristic
11	#87720	—	2	2	no	—
12	C₄⋊Dic₂₅	—	2	2	no	—
13	#87713	—	2	4	no	—
14	Dic₁₀₀	—	2	4	no	—
15	#65004	—	5	1	no	maximal
16	C₂²×D₂₅	—	2	1	no	—
17	C₂×C₄×C₂₅	D₄	1	2	yes	—
18	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
19	C₂×Dic₂₅	—	2	2	no	—
20	C₂×Dic₂₅	—	2	1	no	—
21	D₁₀₀	—	2	2	no	—
22	Dic₅₀	—	2	2	no	—
23	Dic₅₀	—	2	2	no	—
24	Dic₅₀	—	2	2	no	—
25	C₈×C₂₅	D₄	1	4	yes	—
26	C₂×Dic₂₀	—	5	2	no	—
27	C₂×#677	—	5	2	no	—
28	#4021	—	5	1	no	—
29	#4015	—	5	1	no	—
30	C₅×#97	D₅	1	1	yes	characteristic
31	C₂²×C₂₅	C₂×D₄	1	1	yes	characteristic
32	C₂×D₂₅	—	4	1	no	—
33	C₂×D₂₅	—	4	2	no	—
34	C₄×C₂₅	—	2	2	no	—
35	C₄×C₂₅	C₂×D₄	1	2	yes	—
36	Dic₂₅	—	4	1	no	—
37	Dic₂₅	—	4	2	no	—
38	C₂×D₂₀	—	5	1	no	—
39	C₂×Dic₁₀	—	5	2	no	—
40	C₂×Dic₁₀	—	5	1	no	—
41	C₂×C₅×C₈	C₂×D₅	1	2	yes	—
42	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
43	#685	—	10	2	no	—
44	C₄⋊Dic₅	—	10	2	no	—
45	Dic₂₀	—	10	4	no	—
46	#677	—	10	4	no	—
47	#497	—	25	1	no	Sylow
48	C₂×C₂₅	#127	1	1	yes	characteristic
49	C₂×C₂₅	#137	1	2	yes	—
50	D₂₅	—	8	1	no	—
51	C₂²×D₅	—	10	1	no	—
52	C₂×C₄×C₅	D₂₀	1	2	yes	—
53	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
54	C₂×Dic₅	—	10	2	no	—
55	C₂×Dic₅	—	10	1	no	—
56	D₂₀	—	10	2	no	—
57	Dic₁₀	—	10	2	no	—
58	Dic₁₀	—	10	2	no	—
59	Dic₁₀	—	10	2	no	—
60	C₅×C₈	D₂₀	1	4	yes	—
61	C₂×Q₁₆	—	25	2	no	—
62	C₂×SD₁₆	—	25	2	no	—
63	C₄⋊Q₈	—	25	1	no	—
64	#124	—	25	1	no	—
65	#97	D₂₅	1	1	yes	characteristic, complemented
66	C₂₅	#497	1	1	yes	characteristic, complemented, Sylow
67	C₂²×C₅	C₂×D₂₀	1	1	yes	characteristic
68	C₂×D₅	—	20	1	no	—
69	C₂×D₅	—	20	2	no	—
70	C₄×C₅	—	2	2	no	—
71	C₄×C₅	C₂×D₂₀	1	2	yes	—
72	Dic₅	—	20	2	no	—
73	Dic₅	—	20	1	no	—
74	C₂×Q₈	—	25	1	no	—
75	C₂×Q₈	—	25	2	no	—
76	C₂×D₄	—	25	1	no	—
77	Q₁₆	—	50	4	no	—
78	SD₁₆	—	50	4	no	—
79	C₂×C₈	C₂×D₂₅	1	2	yes	—
80	C₄⋊C₄	—	50	2	no	—
81	C₂²⋊C₄	—	50	2	no	—
82	C₄²	C₂×D₂₅	1	1	yes	characteristic
83	C₂×C₅	#4020	1	1	yes	characteristic
84	C₂×C₅	#4055	1	2	yes	—
85	D₅	—	40	1	no	—
86	C₂³	—	50	1	no	—
87	Q₈	—	50	2	no	—
88	Q₈	—	50	2	no	—
89	Q₈	—	50	2	no	—
90	D₄	—	50	2	no	—
91	C₂×C₄	—	50	1	no	—
92	C₂×C₄	—	50	2	no	—
93	C₂×C₄	D₁₀₀	1	2	yes	—
94	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
95	C₈	D₁₀₀	1	4	yes	—
96	C₅	#65004	1	1	yes	characteristic
97	C₂²	—	100	1	no	—
98	C₂²	—	100	2	no	—
99	C₂²	C₂×D₁₀₀	1	1	yes	characteristic
100	C₄	—	2	2	no	—
101	C₄	—	100	1	no	—
102	C₄	—	100	2	no	—
103	C₄	C₂×D₁₀₀	1	2	yes	—
104	C₂	—	200	1	no	—
105	C₂	#11716436	1	1	yes	characteristic
106	C₂	#11716471	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#87713

C₂

yes

maximal

C₂×Dic₁₀₀

C₂

yes

maximal, complemented

#11716437

C₂

yes

maximal, characteristic

#11716431

C₂

yes

maximal, characteristic, complemented

C₂₅×#97

C₂

yes

maximal, characteristic, complemented

C₂×D₁₀₀

C₂²

yes

characteristic

C₂×Dic₅₀

C₂²

yes

—

C₂×Dic₅₀

C₂²

yes

characteristic

C₂×C₈×C₂₅

C₂²

yes

—

C₄²×C₂₅

C₂²

yes

characteristic

#87720

—

C₄⋊Dic₂₅

—

#87713

—

Dic₁₀₀

—

#65004

—

maximal

C₂²×D₂₅

—

C₂×C₄×C₂₅

D₄

yes

—

C₂×C₄×C₂₅

C₂³

yes

characteristic

C₂×Dic₂₅

—

C₂×Dic₂₅

—

D₁₀₀

—

Dic₅₀

—

Dic₅₀

—

Dic₅₀

—

C₈×C₂₅

D₄

yes

—

C₂×Dic₂₀

—

C₂×#677

—

#4021

—

#4015

—

C₅×#97

D₅

yes

characteristic

C₂²×C₂₅

C₂×D₄

yes

characteristic

C₂×D₂₅

—

C₂×D₂₅

—

C₄×C₂₅

—

C₄×C₂₅

C₂×D₄

yes

—

Dic₂₅

—

Dic₂₅

—

C₂×D₂₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

C₂×D₅

yes

—

C₄²×C₅

C₂×D₅

yes

characteristic

#685

—

C₄⋊Dic₅

—

Dic₂₀

—

—

—

C₂×C₂₅

yes

C₂×C₂₅

yes

—

D₂₅

—

C₂²×D₅

—

C₂×C₄×C₅

D₂₀

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

—

D₂₀

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

D₂₀

yes

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

—

#124

—

#97

D₂₅

yes

characteristic, complemented

C₂₅

#497

yes

characteristic, complemented, Sylow

C₂²×C₅

C₂×D₂₀

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

C₂×D₂₀

yes

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

Q₁₆

—

SD₁₆

—

C₂×C₈

C₂×D₂₅

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×D₂₅

yes

C₂×C₅

yes

C₂×C₅

yes

—

D₅

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₁₀₀

yes

—

C₂×C₄

C₂²×D₂₅

yes

characteristic

C₈

D₁₀₀

yes

—

C₅

#65004

yes

characteristic

C₂²

—

100

—

C₂²

—

100

—

C₂²

C₂×D₁₀₀

yes

characteristic

100

C₄

—

101

C₄

—

100

—

102

C₄

—

100

—

103

C₄

C₂×D₁₀₀

yes

—

104

C₂

—

200

—

105

C₂

#11716436

yes

characteristic

106

C₂

#11716471

yes

—

	subgroup	G/H
center	C₂²	C₂×D₁₀₀
commutator, cosocle	C₂×C₄×C₂₅	C₂³
Frattini	C₂×C₄×C₅	C₂²×D₅
Fitting	C₂₅×#97	C₂
socle	C₂²×C₅	C₂×D₂₀
prime radical	C₂×D₁₀₀	C₂²
hypercenter	#97	D₂₅
hypocenter	C₂₅	#497
LCS	G ▷ C₂×C₄×C₂₅ ▷ C₂×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

C₂×D₁₀₀

commutator, cosocle

C₂×C₄×C₂₅

C₂³

Frattini

C₂×C₄×C₅

C₂²×D₅

Fitting

C₂₅×#97

C₂

socle

C₂²×C₅

C₂×D₂₀

prime radical

C₂×D₁₀₀

C₂²

D₂₅

C₂₅

G ▷ C₂×C₄×C₂₅ ▷ C₂×C₂₅ ▷ C₂₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
2. C₂×Dic₁₀₀	C₂	semidirect
4. #11716431	C₂	semidirect
5. C₂₅×#97	C₂	semidirect
65. #97	D₂₅	semidirect
66. C₂₅	#497	semidirect

H ≅ G/N

type

2. C₂×Dic₁₀₀

C₂

semidirect

4. #11716431

C₂

semidirect

5. C₂₅×#97

C₂

semidirect

65. #97

D₂₅

semidirect

66. C₂₅

#497

semidirect

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

subgroup	index
C₂×#87713	2
C₂×Dic₁₀₀	2
#11716437	2
#11716431	2
C₂₅×#97	2
#65004	5

subgroup

index

C₂×Dic₁₀₀

C₂₅×#97

Subgroups — #13097991

Conjugacy classes of subgroups

Complements

Maximal subgroups