Group #20265918 — subgroups

Subgroups — #20265918

Order 6464.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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	#15692252	C₂	1	1	yes	maximal, characteristic, complemented
2	#15692249	C₂	1	1	yes	maximal, characteristic, complemented
3	#15692240	C₂	1	3	yes	maximal, complemented
4	#15692230	C₂	1	3	yes	maximal, complemented
5	#15692228	C₂	1	3	yes	maximal, complemented
6	#15692225	C₂	1	3	yes	maximal, complemented
7	#126×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₁₀₁	C₂²	1	1	yes	characteristic
9	C₂×D₄₀₄	C₂²	1	3	yes	—
10	C₂×D₄₀₄	C₂²	1	3	yes	—
11	C₂×D₄₀₄	C₂²	1	3	yes	complemented
12	C₂×C₂²⋊D₁₀₁	C₂²	1	3	yes	—
13	C₂×C₄×D₁₀₁	C₂²	1	3	yes	complemented
14	#13108053	C₂²	1	3	yes	complemented
15	#13108051	C₂²	1	3	yes	—
16	#13108051	C₂²	1	3	yes	—
17	#13108051	C₂²	1	3	yes	complemented
18	C₄⋊C₄×C₁₀₁	C₂²	1	3	yes	complemented
19	C₂²⋊C₄×C₁₀₁	C₂²	1	3	yes	—
20	C₄²×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
21	C₂³×C₁₀₁	C₂³	1	1	yes	characteristic
22	C₂²×D₁₀₁	—	4	3	no	—
23	C₂²×D₁₀₁	C₂³	1	1	yes	characteristic
24	C₂²×D₁₀₁	C₂³	1	1	yes	characteristic
25	C₂²×D₁₀₁	C₂³	1	3	yes	—
26	C₂×C₄×C₁₀₁	C₂³	1	3	yes	—
27	C₂×C₄×C₁₀₁	C₂³	1	3	yes	—
28	C₂²⋊D₁₀₁	—	4	3	no	—
29	C₂×Dic₁₀₁	C₂³	1	3	yes	—
30	D₄₀₄	—	4	3	no	—
31	D₄₀₄	—	4	3	no	—
32	D₄₀₄	—	4	3	no	—
33	C₄×D₁₀₁	—	4	3	no	—
34	C₂²×C₁₀₁	—	2	3	no	—
35	C₂²×C₁₀₁	C₂⁴	1	1	yes	characteristic
36	C₂×D₁₀₁	—	2	3	no	—
37	C₂×D₁₀₁	—	2	3	no	—
38	C₂×D₁₀₁	—	2	3	no	—
39	C₂×D₁₀₁	—	2	3	no	—
40	C₂×D₁₀₁	—	2	3	no	—
41	C₂×D₁₀₁	—	4	4	no	—
42	C₄×C₁₀₁	—	2	3	no	—
43	C₄×C₁₀₁	—	2	3	no	—
44	Dic₁₀₁	—	2	3	no	—
45	C₂×C₁₀₁	—	4	1	no	—
46	C₂×C₁₀₁	H(5,𝔽₂)	1	3	yes	—
47	D₁₀₁	—	4	3	no	—
48	D₁₀₁	—	4	1	no	—
49	D₁₀₁	—	4	1	no	—
50	C₁₀₁	#560	1	1	yes	characteristic, complemented, Sylow
51	#560	—	101	1	no	maximal, Sylow
52	#127	—	101	1	no	—
53	#126	—	101	1	no	—
54	#126	D₁₀₁	1	1	yes	characteristic, complemented
55	#123	—	101	3	no	—
56	#121	—	101	3	no	—
57	#121	—	101	3	no	—
58	#120	—	101	3	no	—
59	C₂⁴	—	101	1	no	—
60	C₂×D₄	—	101	3	no	—
61	C₂×D₄	—	101	3	no	—
62	C₂×D₄	—	101	3	no	—
63	C₂×D₄	—	101	3	no	—
64	C₂²×C₄	—	101	3	no	—
65	C₄⋊C₄	—	101	3	no	—
66	C₄⋊C₄	C₂×D₁₀₁	1	3	yes	complemented
67	C₂²⋊C₄	—	101	3	no	—
68	C₂²⋊C₄	—	101	3	no	—
69	C₂²⋊C₄	—	101	3	no	—
70	C₂²⋊C₄	C₂×D₁₀₁	1	3	yes	—
71	C₄²	C₂×D₁₀₁	1	1	yes	characteristic, complemented
72	C₂³	—	101	1	no	—
73	C₂³	—	101	1	no	—
74	C₂³	—	101	3	no	—
75	C₂³	—	404	3	no	—
76	C₂³	C₂²×D₁₀₁	1	1	yes	characteristic
77	D₄	—	404	3	no	—
78	D₄	—	404	3	no	—
79	D₄	—	404	3	no	—
80	D₄	—	404	3	no	—
81	C₂×C₄	—	101	3	no	—
82	C₂×C₄	—	404	3	no	—
83	C₂×C₄	C₂²×D₁₀₁	1	3	yes	—
84	C₂×C₄	C₂²×D₁₀₁	1	3	yes	—
85	C₂²	—	2	3	no	—
86	C₂²	—	404	4	no	—
87	C₂²	—	202	3	no	—
88	C₂²	—	202	3	no	—
89	C₂²	—	202	3	no	—
90	C₂²	—	202	3	no	—
91	C₂²	—	202	3	no	—
92	C₂²	C₂³×D₁₀₁	1	1	yes	characteristic
93	C₄	—	2	3	no	—
94	C₄	—	2	3	no	—
95	C₄	—	202	3	no	—
96	C₂	—	4	1	no	—
97	C₂	—	404	1	no	—
98	C₂	—	404	1	no	—
99	C₂	—	404	3	no	—
100	C₂	#15692301	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#15692252

C₂

yes

maximal, characteristic, complemented

#15692249

C₂

yes

maximal, characteristic, complemented

#15692240

C₂

yes

maximal, complemented

#15692230

C₂

yes

maximal, complemented

#15692228

C₂

yes

maximal, complemented

#15692225

C₂

yes

maximal, complemented

#126×C₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₀₁

C₂²

yes

characteristic

C₂×D₄₀₄

C₂²

yes

—

C₂×D₄₀₄

C₂²

yes

—

C₂×D₄₀₄

C₂²

yes

complemented

C₂×C₂²⋊D₁₀₁

C₂²

yes

—

C₂×C₄×D₁₀₁

C₂²

yes

complemented

#13108053

C₂²

yes

complemented

#13108051

C₂²

yes

—

#13108051

C₂²

yes

—

#13108051

C₂²

yes

complemented

C₄⋊C₄×C₁₀₁

C₂²

yes

complemented

C₂²⋊C₄×C₁₀₁

C₂²

yes

—

C₄²×C₁₀₁

C₂²

yes

characteristic, complemented

C₂³×C₁₀₁

C₂³

yes

characteristic

C₂²×D₁₀₁

—

C₂²×D₁₀₁

C₂³

yes

characteristic

C₂²×D₁₀₁

C₂³

yes

characteristic

C₂²×D₁₀₁

C₂³

yes

—

C₂×C₄×C₁₀₁

C₂³

yes

—

C₂×C₄×C₁₀₁

C₂³

yes

—

C₂²⋊D₁₀₁

—

C₂×Dic₁₀₁

C₂³

yes

—

D₄₀₄

—

D₄₀₄

—

D₄₀₄

—

C₄×D₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂⁴

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

H(5,𝔽₂)

yes

—

D₁₀₁

—

D₁₀₁

—

D₁₀₁

—

C₁₀₁

#560

yes

characteristic, complemented, Sylow

#560

—

101

maximal, Sylow

#127

—

101

—

#126

—

101

—

#126

D₁₀₁

yes

characteristic, complemented

#123

—

101

—

#121

—

101

—

#121

—

101

—

#120

—

101

—

C₂⁴

—

101

—

C₂×D₄

—

101

—

C₂×D₄

—

101

—

C₂×D₄

—

101

—

C₂×D₄

—

101

—

C₂²×C₄

—

101

—

C₄⋊C₄

—

101

—

C₄⋊C₄

C₂×D₁₀₁

yes

complemented

C₂²⋊C₄

—

101

—

C₂²⋊C₄

—

101

—

C₂²⋊C₄

—

101

—

C₂²⋊C₄

C₂×D₁₀₁

yes

—

C₄²

C₂×D₁₀₁

yes

characteristic, complemented

C₂³

—

101

—

C₂³

—

101

—

C₂³

—

101

—

C₂³

—

404

—

C₂³

C₂²×D₁₀₁

yes

characteristic

D₄

—

404

—

D₄

—

404

—

D₄

—

404

—

D₄

—

404

—

C₂×C₄

—

101

—

C₂×C₄

—

404

—

C₂×C₄

C₂²×D₁₀₁

yes

—

C₂×C₄

C₂²×D₁₀₁

yes

—

C₂²

—

C₂²

—

404

—

C₂²

—

202

—

C₂²

—

202

—

C₂²

—

202

—

C₂²

—

202

—

C₂²

—

202

—

C₂²

C₂³×D₁₀₁

yes

characteristic

C₄

—

C₄

—

C₄

—

202

—

C₂

—

C₂

—

404

—

C₂

—

404

—

C₂

—

404

—

100

C₂

#15692301

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₁
commutator, socle	C₂²×C₁₀₁	C₂⁴
Fitting	#126×C₁₀₁	C₂
hypercenter	#126	D₁₀₁
hypocenter	C₁₀₁	#560
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ #126

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₀₁

commutator, socle

C₂²×C₁₀₁

C₂⁴

Fitting

#126×C₁₀₁

C₂

D₁₀₁

C₁₀₁

G ▷ C₂²×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂² ◁ #126

N	H ≅ G/N	type
1. #15692252	C₂	semidirect
2. #15692249	C₂	semidirect
3. #15692240	C₂	semidirect
4. #15692230	C₂	semidirect
5. #15692228	C₂	semidirect
6. #15692225	C₂	semidirect
7. #126×C₁₀₁	C₂	semidirect
11. C₂×D₄₀₄	C₂²	semidirect
13. C₂×C₄×D₁₀₁	C₂²	semidirect
14. #13108053	C₂²	semidirect
17. #13108051	C₂²	semidirect
18. C₄⋊C₄×C₁₀₁	C₂²	semidirect
20. C₄²×C₁₀₁	C₂²	semidirect
50. C₁₀₁	#560	semidirect
54. #126	D₁₀₁	semidirect
66. C₄⋊C₄	C₂×D₁₀₁	semidirect
71. C₄²	C₂×D₁₀₁	semidirect

H ≅ G/N

type

1. #15692252

C₂

semidirect

2. #15692249

C₂

semidirect

3. #15692240

C₂

semidirect

4. #15692230

C₂

semidirect

5. #15692228

C₂

semidirect

6. #15692225

C₂

semidirect

7. #126×C₁₀₁

C₂

semidirect

11. C₂×D₄₀₄

C₂²

semidirect

13. C₂×C₄×D₁₀₁

C₂²

semidirect

14. #13108053

C₂²

semidirect

17. #13108051

C₂²

semidirect

18. C₄⋊C₄×C₁₀₁

C₂²

semidirect

20. C₄²×C₁₀₁

C₂²

semidirect

50. C₁₀₁

#560

semidirect

54. #126

D₁₀₁

semidirect

66. C₄⋊C₄

C₂×D₁₀₁

semidirect

71. C₄²

C₂×D₁₀₁

semidirect

Subgroups — #20265918

Conjugacy classes of subgroups

Complements