Group #22554124 — subgroups

Subgroups — #22554124

Order 8384.

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	#16862293	C₂	1	1	yes	maximal, characteristic, complemented
2	#16862292	C₂	1	1	yes	maximal, characteristic, complemented
3	#16862279	C₂	1	3	yes	maximal, complemented
4	#16862274	C₂	1	3	yes	maximal, complemented
5	#16862268	C₂	1	3	yes	maximal, complemented
6	C₂×#14523502	C₂	1	3	yes	maximal, complemented
7	#131×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#14523514	—	2	1	no	—
9	#14523513	C₂²	1	3	yes	—
10	#14523513	C₂²	1	3	yes	complemented
11	#14523512	—	2	3	no	—
12	C₂×Dic₂₆₂	C₂²	1	3	yes	complemented
13	Q₈×D₁₃₁	—	2	1	no	—
14	C₂×D₅₂₄	C₂²	1	3	yes	—
15	D₄×D₁₃₁	—	2	3	no	—
16	Dic₅₂₄	C₂²	1	3	yes	complemented
17	#14523502	C₂²	1	3	yes	complemented
18	#14523502	C₂²	1	6	yes	complemented
19	#14523502	C₂²	1	1	yes	characteristic, complemented
20	D₁₀₄₈	C₂²	1	3	yes	complemented
21	#41×C₁₃₁	C₂²	1	1	yes	characteristic
22	M₁₆×C₁₃₁	C₂²	1	3	yes	complemented
23	C₂×C₈×C₁₃₁	C₂²	1	3	yes	complemented
24	C₂²×D₁₃₁	—	2	3	no	—
25	Dic₂₆₂	—	2	3	no	—
26	Dic₂₆₂	C₂³	1	3	yes	—
27	Dic₂₆₂	C₂³	1	1	yes	characteristic
28	D₅₂₄	—	2	3	no	—
29	D₅₂₄	C₂³	1	1	yes	characteristic
30	D₅₂₄	C₂³	1	3	yes	—
31	C₂²⋊D₁₃₁	—	2	3	no	—
32	C₂²⋊D₁₃₁	—	2	3	no	—
33	C₂×Dic₁₃₁	—	2	3	no	—
34	C₄×D₁₃₁	—	2	3	no	—
35	C₄×D₁₃₁	—	2	3	no	—
36	Q₈×C₁₃₁	D₄	1	1	yes	characteristic
37	D₄×C₁₃₁	D₄	1	3	yes	—
38	C₂×C₄×C₁₃₁	C₂³	1	3	yes	—
39	C₈×C₁₃₁	C₂³	1	1	yes	characteristic
40	C₈×C₁₃₁	C₂³	1	3	yes	—
41	C₂²×C₁₃₁	C₂×D₄	1	3	yes	—
42	C₂×D₁₃₁	—	2	1	no	—
43	C₂×D₁₃₁	—	2	3	no	—
44	C₂×D₁₃₁	—	8	3	no	—
45	C₄×C₁₃₁	C₂×D₄	1	3	yes	—
46	C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
47	Dic₁₃₁	—	2	1	no	—
48	Dic₁₃₁	—	2	3	no	—
49	C₂×C₁₃₁	—	2	3	no	—
50	C₂×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
51	D₁₃₁	—	4	1	no	—
52	D₁₃₁	—	4	3	no	—
53	C₁₃₁	#577	1	1	yes	characteristic, complemented, Sylow
54	#577	—	131	1	no	maximal, Sylow
55	2₋¹⁺⁴	—	131	1	no	—
56	H(5,𝔽₂)	—	131	1	no	—
57	#137	—	131	3	no	—
58	AGL(1,ℤ/8)	—	131	3	no	—
59	#135	—	131	3	no	—
60	C₂×SD₁₆	—	131	3	no	—
61	#131	D₁₃₁	1	1	yes	characteristic, complemented
62	#41	—	262	1	no	—
63	#41	—	262	3	no	—
64	#41	—	131	3	no	—
65	#41	—	131	3	no	—
66	#41	C₂×D₁₃₁	1	1	yes	characteristic
67	C₂×Q₈	—	262	1	no	—
68	C₂×Q₈	—	131	3	no	—
69	C₂×D₄	—	262	3	no	—
70	C₂×D₄	—	131	3	no	—
71	Q₁₆	—	131	3	no	—
72	SD₁₆	—	131	3	no	—
73	SD₁₆	—	131	6	no	—
74	SD₁₆	—	131	1	no	—
75	D₈	—	131	3	no	—
76	M₁₆	C₂×D₁₃₁	1	3	yes	complemented
77	C₂×C₈	C₂×D₁₃₁	1	3	yes	complemented
78	C₂³	—	262	3	no	—
79	Q₈	—	262	3	no	—
80	Q₈	—	131	3	no	—
81	Q₈	—	131	1	no	—
82	Q₈	D₅₂₄	1	1	yes	characteristic
83	D₄	—	262	3	no	—
84	D₄	—	262	3	no	—
85	D₄	—	262	3	no	—
86	D₄	—	131	1	no	—
87	D₄	—	131	3	no	—
88	D₄	D₅₂₄	1	3	yes	—
89	C₂×C₄	—	262	3	no	—
90	C₂×C₄	—	262	3	no	—
91	C₂×C₄	—	262	3	no	—
92	C₂×C₄	C₂²×D₁₃₁	1	3	yes	—
93	C₈	C₂²×D₁₃₁	1	1	yes	characteristic
94	C₈	C₂²×D₁₃₁	1	3	yes	—
95	C₂²	—	262	1	no	—
96	C₂²	—	262	3	no	—
97	C₂²	—	1048	3	no	—
98	C₂²	C₂×D₅₂₄	1	3	yes	—
99	C₄	—	262	1	no	—
100	C₄	—	262	3	no	—
101	C₄	C₂×D₅₂₄	1	3	yes	—
102	C₄	C₂³×D₁₃₁	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	524	3	no	—
105	C₂	—	524	1	no	—
106	C₂	C₂²×D₅₂₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16862293

C₂

yes

maximal, characteristic, complemented

#16862292

C₂

yes

maximal, characteristic, complemented

#16862279

C₂

yes

maximal, complemented

#16862274

C₂

yes

maximal, complemented

#16862268

C₂

yes

maximal, complemented

C₂×#14523502

C₂

yes

maximal, complemented

#131×C₁₃₁

C₂

yes

maximal, characteristic, complemented

#14523514

—

#14523513

C₂²

yes

—

#14523513

C₂²

yes

complemented

#14523512

—

C₂×Dic₂₆₂

C₂²

yes

complemented

Q₈×D₁₃₁

—

C₂×D₅₂₄

C₂²

yes

—

D₄×D₁₃₁

—

Dic₅₂₄

C₂²

yes

complemented

#14523502

C₂²

yes

complemented

#14523502

C₂²

yes

complemented

#14523502

C₂²

yes

characteristic, complemented

D₁₀₄₈

C₂²

yes

complemented

#41×C₁₃₁

C₂²

yes

characteristic

M₁₆×C₁₃₁

C₂²

yes

complemented

C₂×C₈×C₁₃₁

C₂²

yes

complemented

C₂²×D₁₃₁

—

Dic₂₆₂

—

Dic₂₆₂

C₂³

yes

—

Dic₂₆₂

C₂³

yes

characteristic

D₅₂₄

—

D₅₂₄

C₂³

yes

characteristic

D₅₂₄

C₂³

yes

—

C₂²⋊D₁₃₁

—

C₂²⋊D₁₃₁

—

C₂×Dic₁₃₁

—

C₄×D₁₃₁

—

C₄×D₁₃₁

—

Q₈×C₁₃₁

D₄

yes

characteristic

D₄×C₁₃₁

D₄

yes

—

C₂×C₄×C₁₃₁

C₂³

yes

—

C₈×C₁₃₁

C₂³

yes

characteristic

C₈×C₁₃₁

C₂³

yes

—

C₂²×C₁₃₁

C₂×D₄

yes

—

C₂×D₁₃₁

—

C₂×D₁₃₁

—

C₂×D₁₃₁

—

C₄×C₁₃₁

C₂×D₄

yes

—

C₄×C₁₃₁

C₂⁴

yes

characteristic

Dic₁₃₁

—

Dic₁₃₁

—

C₂×C₁₃₁

—

C₂×C₁₃₁

C₂²×D₄

yes

characteristic

D₁₃₁

—

D₁₃₁

—

C₁₃₁

#577

yes

characteristic, complemented, Sylow

#577

—

131

maximal, Sylow

2₋¹⁺⁴

—

131

—

H(5,𝔽₂)

—

131

—

#137

—

131

—

AGL(1,ℤ/8)

—

131

—

#135

—

131

—

C₂×SD₁₆

—

131

—

#131

D₁₃₁

yes

characteristic, complemented

#41

—

262

—

#41

—

262

—

#41

—

131

—

#41

—

131

—

#41

C₂×D₁₃₁

yes

characteristic

C₂×Q₈

—

262

—

C₂×Q₈

—

131

—

C₂×D₄

—

262

—

C₂×D₄

—

131

—

Q₁₆

—

131

—

SD₁₆

—

131

—

SD₁₆

—

131

—

SD₁₆

—

131

—

D₈

—

131

—

M₁₆

C₂×D₁₃₁

yes

complemented

C₂×C₈

C₂×D₁₃₁

yes

complemented

C₂³

—

262

—

Q₈

—

262

—

Q₈

—

131

—

Q₈

—

131

—

Q₈

D₅₂₄

yes

characteristic

D₄

—

262

—

D₄

—

262

—

D₄

—

262

—

D₄

—

131

—

D₄

—

131

—

D₄

D₅₂₄

yes

—

C₂×C₄

—

262

—

C₂×C₄

—

262

—

C₂×C₄

—

262

—

C₂×C₄

C₂²×D₁₃₁

yes

—

C₈

C₂²×D₁₃₁

yes

characteristic

C₈

C₂²×D₁₃₁

yes

—

C₂²

—

262

—

C₂²

—

262

—

C₂²

—

1048

—

C₂²

C₂×D₅₂₄

yes

—

C₄

—

262

—

100

C₄

—

262

—

101

C₄

C₂×D₅₂₄

yes

—

102

C₄

C₂³×D₁₃₁

yes

characteristic

103

C₂

—

104

C₂

—

524

—

105

C₂

—

524

—

106

C₂

C₂²×D₅₂₄

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×D₅₂₄
commutator	C₄×C₁₃₁	C₂⁴
Frattini	C₄	C₂³×D₁₃₁
Fitting	#131×C₁₃₁	C₂
socle	C₂×C₁₃₁	C₂²×D₄
hypercenter	#131	D₁₃₁
hypocenter	C₁₃₁	#577
LCS	G ▷ C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

subgroup

G/H

center

C₂

C₂²×D₅₂₄

commutator

C₄×C₁₃₁

C₂⁴

Frattini

C₄

C₂³×D₁₃₁

Fitting

#131×C₁₃₁

C₂

socle

C₂×C₁₃₁

C₂²×D₄

D₁₃₁

C₁₃₁

G ▷ C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131

N	H ≅ G/N	type
1. #16862293	C₂	semidirect
2. #16862292	C₂	semidirect
3. #16862279	C₂	semidirect
4. #16862274	C₂	semidirect
5. #16862268	C₂	semidirect
6. C₂×#14523502	C₂	semidirect
7. #131×C₁₃₁	C₂	semidirect
10. #14523513	C₂²	semidirect
12. C₂×Dic₂₆₂	C₂²	semidirect
16. Dic₅₂₄	C₂²	semidirect
17. #14523502	C₂²	semidirect
18. #14523502	C₂²	semidirect
19. #14523502	C₂²	semidirect
20. D₁₀₄₈	C₂²	semidirect
22. M₁₆×C₁₃₁	C₂²	semidirect
23. C₂×C₈×C₁₃₁	C₂²	semidirect
53. C₁₃₁	#577	semidirect
61. #131	D₁₃₁	semidirect
76. M₁₆	C₂×D₁₃₁	semidirect
77. C₂×C₈	C₂×D₁₃₁	semidirect

H ≅ G/N

type

1. #16862293

C₂

semidirect

2. #16862292

C₂

semidirect

3. #16862279

C₂

semidirect

4. #16862274

C₂

semidirect

5. #16862268

C₂

semidirect

6. C₂×#14523502

C₂

semidirect

7. #131×C₁₃₁

C₂

semidirect

10. #14523513

C₂²

semidirect

12. C₂×Dic₂₆₂

C₂²

semidirect

16. Dic₅₂₄

C₂²

semidirect

17. #14523502

C₂²

semidirect

18. #14523502

C₂²

semidirect

19. #14523502

C₂²

semidirect

20. D₁₀₄₈

C₂²

semidirect

22. M₁₆×C₁₃₁

C₂²

semidirect

23. C₂×C₈×C₁₃₁

C₂²

semidirect

53. C₁₃₁

#577

semidirect

61. #131

D₁₃₁

semidirect

76. M₁₆

C₂×D₁₃₁

semidirect

77. C₂×C₈

C₂×D₁₃₁

semidirect

Subgroups — #22554124

Conjugacy classes of subgroups

Complements