Group #20268603 — subgroups

Subgroups — #20268603

Order 6592.

258 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 107 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	#15693068	C₂	1	1	yes	maximal, characteristic, complemented
2	#15693067	C₂	1	1	yes	maximal, characteristic, complemented
3	#15693054	C₂	1	3	yes	maximal, complemented
4	#15693049	C₂	1	3	yes	maximal, complemented
5	#15693043	C₂	1	3	yes	maximal, complemented
6	C₂×#13110094	C₂	1	3	yes	maximal, complemented
7	#131×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#13110106	—	2	1	no	—
9	#13110105	C₂²	1	3	yes	—
10	#13110105	C₂²	1	3	yes	complemented
11	#13110104	—	2	3	no	—
12	C₂×Dic₂₀₆	C₂²	1	3	yes	complemented
13	Q₈×D₁₀₃	—	2	1	no	—
14	C₂×D₄₁₂	C₂²	1	3	yes	—
15	D₄×D₁₀₃	—	2	3	no	—
16	Dic₄₁₂	C₂²	1	3	yes	complemented
17	#13110094	C₂²	1	3	yes	complemented
18	#13110094	C₂²	1	6	yes	complemented
19	#13110094	C₂²	1	1	yes	characteristic, complemented
20	D₈₂₄	C₂²	1	3	yes	complemented
21	#41×C₁₀₃	C₂²	1	1	yes	characteristic
22	M₁₆×C₁₀₃	C₂²	1	3	yes	complemented
23	C₂×C₈×C₁₀₃	C₂²	1	3	yes	complemented
24	C₂²×D₁₀₃	—	2	3	no	—
25	Q₈×C₁₀₃	D₄	1	1	yes	characteristic
26	D₄×C₁₀₃	D₄	1	3	yes	—
27	C₂×C₄×C₁₀₃	C₂³	1	3	yes	—
28	C₂²⋊D₁₀₃	—	2	3	no	—
29	C₂²⋊D₁₀₃	—	2	3	no	—
30	C₂×Dic₁₀₃	—	2	3	no	—
31	D₄₁₂	—	2	3	no	—
32	D₄₁₂	C₂³	1	1	yes	characteristic
33	D₄₁₂	C₂³	1	3	yes	—
34	C₄×D₁₀₃	—	2	3	no	—
35	C₄×D₁₀₃	—	2	3	no	—
36	Dic₂₀₆	—	2	3	no	—
37	Dic₂₀₆	C₂³	1	3	yes	—
38	Dic₂₀₆	C₂³	1	1	yes	characteristic
39	C₈×C₁₀₃	C₂³	1	1	yes	characteristic
40	C₈×C₁₀₃	C₂³	1	3	yes	—
41	C₂²×C₁₀₃	C₂×D₄	1	3	yes	—
42	C₂×D₁₀₃	—	2	1	no	—
43	C₂×D₁₀₃	—	2	3	no	—
44	C₂×D₁₀₃	—	8	3	no	—
45	C₄×C₁₀₃	C₂×D₄	1	3	yes	—
46	C₄×C₁₀₃	C₂⁴	1	1	yes	characteristic
47	Dic₁₀₃	—	2	3	no	—
48	Dic₁₀₃	—	2	1	no	—
49	C₂×C₁₀₃	—	2	3	no	—
50	C₂×C₁₀₃	C₂²×D₄	1	1	yes	characteristic
51	D₁₀₃	—	4	1	no	—
52	D₁₀₃	—	4	3	no	—
53	C₁₀₃	#577	1	1	yes	characteristic, complemented, Sylow
54	#577	—	103	1	no	maximal, Sylow
55	2₋¹⁺⁴	—	103	1	no	—
56	H(5,𝔽₂)	—	103	1	no	—
57	#137	—	103	3	no	—
58	AGL(1,ℤ/8)	—	103	3	no	—
59	#135	—	103	3	no	—
60	C₂×SD₁₆	—	103	3	no	—
61	#131	D₁₀₃	1	1	yes	characteristic, complemented
62	#41	—	103	3	no	—
63	#41	—	103	3	no	—
64	#41	—	206	1	no	—
65	#41	—	206	3	no	—
66	#41	C₂×D₁₀₃	1	1	yes	characteristic
67	C₂×Q₈	—	103	3	no	—
68	C₂×Q₈	—	206	1	no	—
69	C₂×D₄	—	103	3	no	—
70	C₂×D₄	—	206	3	no	—
71	Q₁₆	—	103	3	no	—
72	SD₁₆	—	103	3	no	—
73	SD₁₆	—	103	6	no	—
74	SD₁₆	—	103	1	no	—
75	D₈	—	103	3	no	—
76	M₁₆	C₂×D₁₀₃	1	3	yes	complemented
77	C₂×C₈	C₂×D₁₀₃	1	3	yes	complemented
78	C₂³	—	206	3	no	—
79	Q₈	—	103	3	no	—
80	Q₈	—	103	1	no	—
81	Q₈	—	206	3	no	—
82	Q₈	D₄₁₂	1	1	yes	characteristic
83	D₄	—	103	1	no	—
84	D₄	—	103	3	no	—
85	D₄	—	206	3	no	—
86	D₄	—	206	3	no	—
87	D₄	—	206	3	no	—
88	D₄	D₄₁₂	1	3	yes	—
89	C₂×C₄	—	206	3	no	—
90	C₂×C₄	—	206	3	no	—
91	C₂×C₄	—	206	3	no	—
92	C₂×C₄	C₂²×D₁₀₃	1	3	yes	—
93	C₈	C₂²×D₁₀₃	1	1	yes	characteristic
94	C₈	C₂²×D₁₀₃	1	3	yes	—
95	C₂²	—	824	3	no	—
96	C₂²	—	206	1	no	—
97	C₂²	—	206	3	no	—
98	C₂²	C₂×D₄₁₂	1	3	yes	—
99	C₄	—	206	1	no	—
100	C₄	—	206	3	no	—
101	C₄	C₂×D₄₁₂	1	3	yes	—
102	C₄	C₂³×D₁₀₃	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	412	3	no	—
105	C₂	—	412	1	no	—
106	C₂	C₂²×D₄₁₂	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15693068

C₂

yes

maximal, characteristic, complemented

#15693067

C₂

yes

maximal, characteristic, complemented

#15693054

C₂

yes

maximal, complemented

#15693049

C₂

yes

maximal, complemented

#15693043

C₂

yes

maximal, complemented

C₂×#13110094

C₂

yes

maximal, complemented

#131×C₁₀₃

C₂

yes

maximal, characteristic, complemented

#13110106

—

#13110105

C₂²

yes

—

#13110105

C₂²

yes

complemented

#13110104

—

C₂×Dic₂₀₆

C₂²

yes

complemented

Q₈×D₁₀₃

—

C₂×D₄₁₂

C₂²

yes

—

D₄×D₁₀₃

—

Dic₄₁₂

C₂²

yes

complemented

#13110094

C₂²

yes

complemented

#13110094

C₂²

yes

complemented

#13110094

C₂²

yes

characteristic, complemented

D₈₂₄

C₂²

yes

complemented

#41×C₁₀₃

C₂²

yes

characteristic

M₁₆×C₁₀₃

C₂²

yes

complemented

C₂×C₈×C₁₀₃

C₂²

yes

complemented

C₂²×D₁₀₃

—

Q₈×C₁₀₃

D₄

yes

characteristic

D₄×C₁₀₃

D₄

yes

—

C₂×C₄×C₁₀₃

C₂³

yes

—

C₂²⋊D₁₀₃

—

C₂²⋊D₁₀₃

—

C₂×Dic₁₀₃

—

D₄₁₂

—

D₄₁₂

C₂³

yes

characteristic

D₄₁₂

C₂³

yes

—

C₄×D₁₀₃

—

C₄×D₁₀₃

—

Dic₂₀₆

—

Dic₂₀₆

C₂³

yes

—

Dic₂₀₆

C₂³

yes

characteristic

C₈×C₁₀₃

C₂³

yes

characteristic

C₈×C₁₀₃

C₂³

yes

—

C₂²×C₁₀₃

C₂×D₄

yes

—

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₄×C₁₀₃

C₂×D₄

yes

—

C₄×C₁₀₃

C₂⁴

yes

characteristic

Dic₁₀₃

—

Dic₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

C₂²×D₄

yes

characteristic

D₁₀₃

—

D₁₀₃

—

C₁₀₃

#577

yes

characteristic, complemented, Sylow

#577

—

103

maximal, Sylow

2₋¹⁺⁴

—

103

—

H(5,𝔽₂)

—

103

—

#137

—

103

—

AGL(1,ℤ/8)

—

103

—

#135

—

103

—

C₂×SD₁₆

—

103

—

#131

D₁₀₃

yes

characteristic, complemented

#41

—

103

—

#41

—

103

—

#41

—

206

—

#41

—

206

—

#41

C₂×D₁₀₃

yes

characteristic

C₂×Q₈

—

103

—

C₂×Q₈

—

206

—

C₂×D₄

—

103

—

C₂×D₄

—

206

—

Q₁₆

—

103

—

SD₁₆

—

103

—

SD₁₆

—

103

—

SD₁₆

—

103

—

D₈

—

103

—

M₁₆

C₂×D₁₀₃

yes

complemented

C₂×C₈

C₂×D₁₀₃

yes

complemented

C₂³

—

206

—

Q₈

—

103

—

Q₈

—

103

—

Q₈

—

206

—

Q₈

D₄₁₂

yes

characteristic

D₄

—

103

—

D₄

—

103

—

D₄

—

206

—

D₄

—

206

—

D₄

—

206

—

D₄

D₄₁₂

yes

—

C₂×C₄

—

206

—

C₂×C₄

—

206

—

C₂×C₄

—

206

—

C₂×C₄

C₂²×D₁₀₃

yes

—

C₈

C₂²×D₁₀₃

yes

characteristic

C₈

C₂²×D₁₀₃

yes

—

C₂²

—

824

—

C₂²

—

206

—

C₂²

—

206

—

C₂²

C₂×D₄₁₂

yes

—

C₄

—

206

—

100

C₄

—

206

—

101

C₄

C₂×D₄₁₂

yes

—

102

C₄

C₂³×D₁₀₃

yes

characteristic

103

C₂

—

104

C₂

—

412

—

105

C₂

—

412

—

106

C₂

C₂²×D₄₁₂

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×D₄₁₂
commutator	C₄×C₁₀₃	C₂⁴
Frattini	C₄	C₂³×D₁₀₃
Fitting	#131×C₁₀₃	C₂
socle	C₂×C₁₀₃	C₂²×D₄
hypercenter	#131	D₁₀₃
hypocenter	C₁₀₃	#577
LCS	G ▷ C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

subgroup

G/H

center

C₂

C₂²×D₄₁₂

commutator

C₄×C₁₀₃

C₂⁴

Frattini

C₄

C₂³×D₁₀₃

Fitting

#131×C₁₀₃

C₂

socle

C₂×C₁₀₃

C₂²×D₄

D₁₀₃

C₁₀₃

G ▷ C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131

N	H ≅ G/N	type
1. #15693068	C₂	semidirect
2. #15693067	C₂	semidirect
3. #15693054	C₂	semidirect
4. #15693049	C₂	semidirect
5. #15693043	C₂	semidirect
6. C₂×#13110094	C₂	semidirect
7. #131×C₁₀₃	C₂	semidirect
10. #13110105	C₂²	semidirect
12. C₂×Dic₂₀₆	C₂²	semidirect
16. Dic₄₁₂	C₂²	semidirect
17. #13110094	C₂²	semidirect
18. #13110094	C₂²	semidirect
19. #13110094	C₂²	semidirect
20. D₈₂₄	C₂²	semidirect
22. M₁₆×C₁₀₃	C₂²	semidirect
23. C₂×C₈×C₁₀₃	C₂²	semidirect
53. C₁₀₃	#577	semidirect
61. #131	D₁₀₃	semidirect
76. M₁₆	C₂×D₁₀₃	semidirect
77. C₂×C₈	C₂×D₁₀₃	semidirect

H ≅ G/N

type

1. #15693068

C₂

semidirect

2. #15693067

C₂

semidirect

3. #15693054

C₂

semidirect

4. #15693049

C₂

semidirect

5. #15693043

C₂

semidirect

6. C₂×#13110094

C₂

semidirect

7. #131×C₁₀₃

C₂

semidirect

10. #13110105

C₂²

semidirect

12. C₂×Dic₂₀₆

C₂²

semidirect

16. Dic₄₁₂

C₂²

semidirect

17. #13110094

C₂²

semidirect

18. #13110094

C₂²

semidirect

19. #13110094

C₂²

semidirect

20. D₈₂₄

C₂²

semidirect

22. M₁₆×C₁₀₃

C₂²

semidirect

23. C₂×C₈×C₁₀₃

C₂²

semidirect

53. C₁₀₃

#577

semidirect

61. #131

D₁₀₃

semidirect

76. M₁₆

C₂×D₁₀₃

semidirect

77. C₂×C₈

C₂×D₁₀₃

semidirect

Subgroups — #20268603

Conjugacy classes of subgroups

Complements