Group #13097989 — subgroups

Subgroups — #13097989

Order 1600.

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

normal subgroups only

This group has 216 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×D₂₀₀	C₂	1	2	yes	maximal, complemented
2	C₂×#87713	C₂	1	2	yes	maximal, complemented
3	#11716437	C₂	1	1	yes	maximal, characteristic, complemented
4	#11716436	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₅×#97	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₀₀	—	2	2	no	—
7	C₂×D₁₀₀	C₂²	1	2	yes	—
8	C₂×D₁₀₀	C₂²	1	1	yes	characteristic
9	C₂×Dic₅₀	C₂²	1	1	yes	characteristic
10	C₂×C₈×C₂₅	C₂²	1	2	yes	—
11	C₄²×C₂₅	C₂²	1	1	yes	characteristic
12	#87720	—	2	2	no	—
13	D₂₀₀	—	2	4	no	—
14	#87713	—	2	4	no	—
15	#65001	—	5	1	no	maximal
16	C₂²×D₂₅	—	2	1	no	—
17	C₂²×D₂₅	—	2	2	no	—
18	C₂×C₄×C₂₅	D₄	1	2	yes	—
19	C₂×C₄×C₂₅	C₂³	1	1	yes	characteristic
20	C₂×Dic₂₅	—	2	1	no	—
21	D₁₀₀	—	2	2	no	—
22	D₁₀₀	—	2	2	no	—
23	D₁₀₀	—	2	2	no	—
24	D₁₀₀	—	4	4	no	—
25	Dic₅₀	—	2	2	no	—
26	C₈×C₂₅	D₄	1	4	yes	complemented
27	C₂×D₄₀	—	5	2	no	—
28	C₂×#677	—	5	2	no	—
29	#4021	—	5	1	no	—
30	#4020	—	5	1	no	—
31	C₅×#97	D₅	1	1	yes	characteristic
32	C₂²×C₂₅	C₂×D₄	1	1	yes	characteristic
33	C₂×D₂₅	—	4	2	no	—
34	C₂×D₂₅	—	4	1	no	—
35	C₂×D₂₅	—	4	4	no	—
36	C₂×D₂₅	—	4	2	no	—
37	C₄×C₂₅	—	2	2	no	—
38	C₄×C₂₅	C₂×D₄	1	2	yes	—
39	Dic₂₅	—	4	1	no	—
40	C₂×D₂₀	—	5	2	no	—
41	C₂×D₂₀	—	5	1	no	—
42	C₂×D₂₀	—	10	2	no	—
43	C₂×Dic₁₀	—	5	1	no	—
44	C₂×C₅×C₈	C₂×D₅	1	2	yes	—
45	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
46	#685	—	10	2	no	—
47	D₄₀	—	10	4	no	—
48	#677	—	10	4	no	—
49	#496	—	25	1	no	Sylow
50	C₂×C₂₅	#127	1	1	yes	characteristic
51	C₂×C₂₅	AGL(1,ℤ/8)	1	2	yes	—
52	D₂₅	—	8	2	no	—
53	D₂₅	—	8	1	no	—
54	C₂²×D₅	—	10	1	no	—
55	C₂²×D₅	—	10	2	no	—
56	C₂×C₄×C₅	D₂₀	1	2	yes	—
57	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
58	C₂×Dic₅	—	10	1	no	—
59	D₂₀	—	10	2	no	—
60	D₂₀	—	10	2	no	—
61	D₂₀	—	10	2	no	—
62	D₂₀	—	20	4	no	—
63	Dic₁₀	—	10	2	no	—
64	C₅×C₈	D₂₀	1	4	yes	—
65	C₂×SD₁₆	—	25	2	no	—
66	C₂×D₈	—	25	2	no	—
67	#127	—	25	1	no	—
68	#124	—	25	1	no	—
69	#97	D₂₅	1	1	yes	characteristic, complemented
70	C₂₅	#496	1	1	yes	characteristic, complemented, Sylow
71	C₂²×C₅	C₂×D₂₀	1	1	yes	characteristic
72	C₂×D₅	—	20	2	no	—
73	C₂×D₅	—	20	1	no	—
74	C₂×D₅	—	20	4	no	—
75	C₂×D₅	—	20	2	no	—
76	C₄×C₅	—	2	2	no	—
77	C₄×C₅	C₂×D₂₀	1	2	yes	—
78	Dic₅	—	20	1	no	—
79	C₂×Q₈	—	25	1	no	—
80	C₂×D₄	—	25	1	no	—
81	C₂×D₄	—	25	2	no	—
82	C₂×D₄	—	50	2	no	—
83	SD₁₆	—	50	4	no	—
84	D₈	—	50	4	no	—
85	C₂×C₈	C₂×D₂₅	1	2	yes	—
86	C₂²⋊C₄	—	50	2	no	—
87	C₄²	C₂×D₂₅	1	1	yes	characteristic
88	C₂×C₅	#4020	1	1	yes	characteristic
89	C₂×C₅	#4054	1	2	yes	—
90	D₅	—	40	2	no	—
91	D₅	—	40	1	no	—
92	C₂³	—	50	1	no	—
93	C₂³	—	50	2	no	—
94	Q₈	—	50	2	no	—
95	D₄	—	50	2	no	—
96	D₄	—	50	2	no	—
97	D₄	—	50	2	no	—
98	D₄	—	100	4	no	—
99	C₂×C₄	—	50	1	no	—
100	C₂×C₄	D₁₀₀	1	2	yes	—
101	C₂×C₄	C₂²×D₂₅	1	1	yes	characteristic
102	C₈	D₁₀₀	1	4	yes	complemented
103	C₅	#65001	1	1	yes	characteristic
104	C₂²	—	100	1	no	—
105	C₂²	—	100	2	no	—
106	C₂²	—	100	4	no	—
107	C₂²	—	100	2	no	—
108	C₂²	C₂×D₁₀₀	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	—	100	1	no	—
111	C₄	C₂×D₁₀₀	1	2	yes	—
112	C₂	—	200	2	no	—
113	C₂	—	200	1	no	—
114	C₂	#11716436	1	1	yes	characteristic
115	C₂	#11716470	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₂₀₀

C₂

yes

maximal, complemented

C₂×#87713

C₂

yes

maximal, complemented

#11716437

C₂

yes

maximal, characteristic, complemented

#11716436

C₂

yes

maximal, characteristic, complemented

C₂₅×#97

C₂

yes

maximal, characteristic, complemented

C₂×D₁₀₀

—

C₂×D₁₀₀

C₂²

yes

—

C₂×D₁₀₀

C₂²

yes

characteristic

C₂×Dic₅₀

C₂²

yes

characteristic

C₂×C₈×C₂₅

C₂²

yes

—

C₄²×C₂₅

C₂²

yes

characteristic

#87720

—

D₂₀₀

—

#87713

—

#65001

—

maximal

C₂²×D₂₅

—

C₂²×D₂₅

—

C₂×C₄×C₂₅

D₄

yes

—

C₂×C₄×C₂₅

C₂³

yes

characteristic

C₂×Dic₂₅

—

D₁₀₀

—

D₁₀₀

—

D₁₀₀

—

D₁₀₀

—

Dic₅₀

—

C₈×C₂₅

D₄

yes

complemented

C₂×D₄₀

—

C₂×#677

—

#4021

—

#4020

—

C₅×#97

D₅

yes

characteristic

C₂²×C₂₅

C₂×D₄

yes

characteristic

C₂×D₂₅

—

C₂×D₂₅

—

C₂×D₂₅

—

C₂×D₂₅

—

C₄×C₂₅

—

C₄×C₂₅

C₂×D₄

yes

—

Dic₂₅

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×D₂₀

—

C₂×Dic₁₀

—

C₂×C₅×C₈

C₂×D₅

yes

—

C₄²×C₅

C₂×D₅

yes

characteristic

#685

—

D₄₀

—

—

—

C₂×C₂₅

yes

C₂×C₂₅

yes

—

D₂₅

—

D₂₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂×C₄×C₅

D₂₀

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

characteristic

C₂×Dic₅

—

D₂₀

—

D₂₀

—

D₂₀

—

D₂₀

—

Dic₁₀

—

C₅×C₈

D₂₀

yes

—

C₂×SD₁₆

—

C₂×D₈

—

#127

—

#124

—

#97

D₂₅

yes

characteristic, complemented

C₂₅

#496

yes

characteristic, complemented, Sylow

C₂²×C₅

C₂×D₂₀

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

C₂×D₂₀

yes

—

Dic₅

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

D₈

—

C₂×C₈

C₂×D₂₅

yes

—

C₂²⋊C₄

—

C₄²

C₂×D₂₅

yes

C₂×C₅

yes

C₂×C₅

yes

—

D₅

—

D₅

—

C₂³

—

C₂³

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

100

—

C₂×C₄

—

100

C₂×C₄

D₁₀₀

yes

—

101

C₂×C₄

C₂²×D₂₅

yes

characteristic

102

C₈

D₁₀₀

yes

complemented

103

C₅

#65001

yes

characteristic

104

C₂²

—

100

—

105

C₂²

—

100

—

106

C₂²

—

100

—

107

C₂²

—

100

—

108

C₂²

C₂×D₁₀₀

yes

characteristic

109

C₄

—

110

C₄

—

100

—

111

C₄

C₂×D₁₀₀

yes

—

112

C₂

—

200

—

113

C₂

—

200

—

114

C₂

#11716436

yes

characteristic

115

C₂

#11716470

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
hypercenter	#97	D₂₅
hypocenter	C₂₅	#496
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

D₂₅

C₂₅

G ▷ C₂×C₄×C₂₅ ▷ C₂×C₂₅ ▷ C₂₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
1. C₂×D₂₀₀	C₂	semidirect
2. C₂×#87713	C₂	semidirect
3. #11716437	C₂	semidirect
4. #11716436	C₂	semidirect
5. C₂₅×#97	C₂	semidirect
26. C₈×C₂₅	D₄	semidirect
69. #97	D₂₅	semidirect
70. C₂₅	#496	semidirect
102. C₈	D₁₀₀	semidirect

H ≅ G/N

type

1. C₂×D₂₀₀

C₂

semidirect

2. C₂×#87713

C₂

semidirect

3. #11716437

C₂

semidirect

4. #11716436

C₂

semidirect

5. C₂₅×#97

C₂

semidirect

26. C₈×C₂₅

D₄

semidirect

69. #97

D₂₅

semidirect

70. C₂₅

#496

semidirect

102. C₈

D₁₀₀

semidirect

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

subgroup	index
C₂×D₂₀₀	2
C₂×#87713	2
#11716437	2
#11716436	2
C₂₅×#97	2
#65001	5

subgroup

index

C₂×D₂₀₀

C₂₅×#97

Subgroups — #13097989

Conjugacy classes of subgroups

Complements

Maximal subgroups