Group #22550688 — subgroups

Subgroups — #22550688

Order 8128.

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	#16861256	C₂	1	1	yes	maximal, characteristic, complemented
2	#16861255	C₂	1	1	yes	maximal, characteristic, complemented
3	#16861242	C₂	1	3	yes	maximal, complemented
4	#16861237	C₂	1	3	yes	maximal, complemented
5	#16861231	C₂	1	3	yes	maximal, complemented
6	C₂×#14523237	C₂	1	3	yes	maximal, complemented
7	#131×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14523249	—	2	1	no	—
9	#14523248	C₂²	1	3	yes	—
10	#14523248	C₂²	1	3	yes	complemented
11	#14523247	—	2	3	no	—
12	C₂×Dic₂₅₄	C₂²	1	3	yes	complemented
13	Q₈×D₁₂₇	—	2	1	no	—
14	C₂×D₅₀₈	C₂²	1	3	yes	—
15	D₄×D₁₂₇	—	2	3	no	—
16	Dic₅₀₈	C₂²	1	3	yes	complemented
17	#14523237	C₂²	1	3	yes	complemented
18	#14523237	C₂²	1	6	yes	complemented
19	#14523237	C₂²	1	1	yes	characteristic, complemented
20	D₁₀₁₆	C₂²	1	3	yes	complemented
21	#41×C₁₂₇	C₂²	1	1	yes	characteristic
22	M₁₆×C₁₂₇	C₂²	1	3	yes	complemented
23	C₂×C₈×C₁₂₇	C₂²	1	3	yes	complemented
24	C₂²×D₁₂₇	—	2	3	no	—
25	Dic₂₅₄	—	2	3	no	—
26	Dic₂₅₄	C₂³	1	3	yes	—
27	Dic₂₅₄	C₂³	1	1	yes	characteristic
28	D₅₀₈	—	2	3	no	—
29	D₅₀₈	C₂³	1	1	yes	characteristic
30	D₅₀₈	C₂³	1	3	yes	—
31	C₂²⋊D₁₂₇	—	2	3	no	—
32	C₂²⋊D₁₂₇	—	2	3	no	—
33	C₂×Dic₁₂₇	—	2	3	no	—
34	C₄×D₁₂₇	—	2	3	no	—
35	C₄×D₁₂₇	—	2	3	no	—
36	Q₈×C₁₂₇	D₄	1	1	yes	characteristic
37	D₄×C₁₂₇	D₄	1	3	yes	—
38	C₂×C₄×C₁₂₇	C₂³	1	3	yes	—
39	C₈×C₁₂₇	C₂³	1	1	yes	characteristic
40	C₈×C₁₂₇	C₂³	1	3	yes	—
41	C₂²×C₁₂₇	C₂×D₄	1	3	yes	—
42	C₂×D₁₂₇	—	2	1	no	—
43	C₂×D₁₂₇	—	2	3	no	—
44	C₂×D₁₂₇	—	8	3	no	—
45	C₄×C₁₂₇	C₂×D₄	1	3	yes	—
46	C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
47	Dic₁₂₇	—	2	1	no	—
48	Dic₁₂₇	—	2	3	no	—
49	C₂×C₁₂₇	—	2	3	no	—
50	C₂×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
51	D₁₂₇	—	4	1	no	—
52	D₁₂₇	—	4	3	no	—
53	C₁₂₇	#577	1	1	yes	characteristic, complemented, Sylow
54	#577	—	127	1	no	maximal, Sylow
55	2₋¹⁺⁴	—	127	1	no	—
56	H(5,𝔽₂)	—	127	1	no	—
57	#137	—	127	3	no	—
58	AGL(1,ℤ/8)	—	127	3	no	—
59	#135	—	127	3	no	—
60	C₂×SD₁₆	—	127	3	no	—
61	#131	D₁₂₇	1	1	yes	characteristic, complemented
62	#41	—	127	3	no	—
63	#41	—	127	3	no	—
64	#41	—	254	1	no	—
65	#41	—	254	3	no	—
66	#41	C₂×D₁₂₇	1	1	yes	characteristic
67	C₂×Q₈	—	127	3	no	—
68	C₂×Q₈	—	254	1	no	—
69	C₂×D₄	—	127	3	no	—
70	C₂×D₄	—	254	3	no	—
71	Q₁₆	—	127	3	no	—
72	SD₁₆	—	127	3	no	—
73	SD₁₆	—	127	6	no	—
74	SD₁₆	—	127	1	no	—
75	D₈	—	127	3	no	—
76	M₁₆	C₂×D₁₂₇	1	3	yes	complemented
77	C₂×C₈	C₂×D₁₂₇	1	3	yes	complemented
78	C₂³	—	254	3	no	—
79	Q₈	—	127	3	no	—
80	Q₈	—	127	1	no	—
81	Q₈	—	254	3	no	—
82	Q₈	D₅₀₈	1	1	yes	characteristic
83	D₄	—	127	1	no	—
84	D₄	—	127	3	no	—
85	D₄	—	254	3	no	—
86	D₄	—	254	3	no	—
87	D₄	—	254	3	no	—
88	D₄	D₅₀₈	1	3	yes	—
89	C₂×C₄	—	254	3	no	—
90	C₂×C₄	—	254	3	no	—
91	C₂×C₄	—	254	3	no	—
92	C₂×C₄	C₂²×D₁₂₇	1	3	yes	—
93	C₈	C₂²×D₁₂₇	1	1	yes	characteristic
94	C₈	C₂²×D₁₂₇	1	3	yes	—
95	C₂²	—	1016	3	no	—
96	C₂²	—	254	1	no	—
97	C₂²	—	254	3	no	—
98	C₂²	C₂×D₅₀₈	1	3	yes	—
99	C₄	—	254	1	no	—
100	C₄	—	254	3	no	—
101	C₄	C₂×D₅₀₈	1	3	yes	—
102	C₄	C₂³×D₁₂₇	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	508	3	no	—
105	C₂	—	508	1	no	—
106	C₂	C₂²×D₅₀₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16861256

C₂

yes

maximal, characteristic, complemented

#16861255

C₂

yes

maximal, characteristic, complemented

#16861242

C₂

yes

maximal, complemented

#16861237

C₂

yes

maximal, complemented

#16861231

C₂

yes

maximal, complemented

C₂×#14523237

C₂

yes

maximal, complemented

#131×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#14523249

—

#14523248

C₂²

yes

—

#14523248

C₂²

yes

complemented

#14523247

—

C₂×Dic₂₅₄

C₂²

yes

complemented

Q₈×D₁₂₇

—

C₂×D₅₀₈

C₂²

yes

—

D₄×D₁₂₇

—

Dic₅₀₈

C₂²

yes

complemented

#14523237

C₂²

yes

complemented

#14523237

C₂²

yes

complemented

#14523237

C₂²

yes

characteristic, complemented

D₁₀₁₆

C₂²

yes

complemented

#41×C₁₂₇

C₂²

yes

characteristic

M₁₆×C₁₂₇

C₂²

yes

complemented

C₂×C₈×C₁₂₇

C₂²

yes

complemented

C₂²×D₁₂₇

—

Dic₂₅₄

—

Dic₂₅₄

C₂³

yes

—

Dic₂₅₄

C₂³

yes

characteristic

D₅₀₈

—

D₅₀₈

C₂³

yes

characteristic

D₅₀₈

C₂³

yes

—

C₂²⋊D₁₂₇

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₄×D₁₂₇

—

C₄×D₁₂₇

—

Q₈×C₁₂₇

D₄

yes

characteristic

D₄×C₁₂₇

D₄

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

—

C₈×C₁₂₇

C₂³

yes

characteristic

C₈×C₁₂₇

C₂³

yes

—

C₂²×C₁₂₇

C₂×D₄

yes

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

—

C₄×C₁₂₇

C₂⁴

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂²×D₄

yes

characteristic

D₁₂₇

—

D₁₂₇

—

C₁₂₇

#577

yes

characteristic, complemented, Sylow

#577

—

127

maximal, Sylow

2₋¹⁺⁴

—

127

—

H(5,𝔽₂)

—

127

—

#137

—

127

—

AGL(1,ℤ/8)

—

127

—

#135

—

127

—

C₂×SD₁₆

—

127

—

#131

D₁₂₇

yes

characteristic, complemented

#41

—

127

—

#41

—

127

—

#41

—

254

—

#41

—

254

—

#41

C₂×D₁₂₇

yes

characteristic

C₂×Q₈

—

127

—

C₂×Q₈

—

254

—

C₂×D₄

—

127

—

C₂×D₄

—

254

—

Q₁₆

—

127

—

SD₁₆

—

127

—

SD₁₆

—

127

—

SD₁₆

—

127

—

D₈

—

127

—

M₁₆

C₂×D₁₂₇

yes

complemented

C₂×C₈

C₂×D₁₂₇

yes

complemented

C₂³

—

254

—

Q₈

—

127

—

Q₈

—

127

—

Q₈

—

254

—

Q₈

D₅₀₈

yes

characteristic

D₄

—

127

—

D₄

—

127

—

D₄

—

254

—

D₄

—

254

—

D₄

—

254

—

D₄

D₅₀₈

yes

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂²×D₁₂₇

yes

—

C₈

C₂²×D₁₂₇

yes

characteristic

C₈

C₂²×D₁₂₇

yes

—

C₂²

—

1016

—

C₂²

—

254

—

C₂²

—

254

—

C₂²

C₂×D₅₀₈

yes

—

C₄

—

254

—

100

C₄

—

254

—

101

C₄

C₂×D₅₀₈

yes

—

102

C₄

C₂³×D₁₂₇

yes

characteristic

103

C₂

—

104

C₂

—

508

—

105

C₂

—

508

—

106

C₂

C₂²×D₅₀₈

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×D₅₀₈
commutator	C₄×C₁₂₇	C₂⁴
Frattini	C₄	C₂³×D₁₂₇
Fitting	#131×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂²×D₄
hypercenter	#131	D₁₂₇
hypocenter	C₁₂₇	#577
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

subgroup

G/H

center

C₂

C₂²×D₅₀₈

commutator

C₄×C₁₂₇

C₂⁴

Frattini

C₄

C₂³×D₁₂₇

Fitting

#131×C₁₂₇

C₂

socle

C₂×C₁₂₇

C₂²×D₄

D₁₂₇

C₁₂₇

G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131

N	H ≅ G/N	type
1. #16861256	C₂	semidirect
2. #16861255	C₂	semidirect
3. #16861242	C₂	semidirect
4. #16861237	C₂	semidirect
5. #16861231	C₂	semidirect
6. C₂×#14523237	C₂	semidirect
7. #131×C₁₂₇	C₂	semidirect
10. #14523248	C₂²	semidirect
12. C₂×Dic₂₅₄	C₂²	semidirect
16. Dic₅₀₈	C₂²	semidirect
17. #14523237	C₂²	semidirect
18. #14523237	C₂²	semidirect
19. #14523237	C₂²	semidirect
20. D₁₀₁₆	C₂²	semidirect
22. M₁₆×C₁₂₇	C₂²	semidirect
23. C₂×C₈×C₁₂₇	C₂²	semidirect
53. C₁₂₇	#577	semidirect
61. #131	D₁₂₇	semidirect
76. M₁₆	C₂×D₁₂₇	semidirect
77. C₂×C₈	C₂×D₁₂₇	semidirect

H ≅ G/N

type

1. #16861256

C₂

semidirect

2. #16861255

C₂

semidirect

3. #16861242

C₂

semidirect

4. #16861237

C₂

semidirect

5. #16861231

C₂

semidirect

6. C₂×#14523237

C₂

semidirect

7. #131×C₁₂₇

C₂

semidirect

10. #14523248

C₂²

semidirect

12. C₂×Dic₂₅₄

C₂²

semidirect

16. Dic₅₀₈

C₂²

semidirect

17. #14523237

C₂²

semidirect

18. #14523237

C₂²

semidirect

19. #14523237

C₂²

semidirect

20. D₁₀₁₆

C₂²

semidirect

22. M₁₆×C₁₂₇

C₂²

semidirect

23. C₂×C₈×C₁₂₇

C₂²

semidirect

53. C₁₂₇

#577

semidirect

61. #131

D₁₂₇

semidirect

76. M₁₆

C₂×D₁₂₇

semidirect

77. C₂×C₈

C₂×D₁₂₇

semidirect

Subgroups — #22550688

Conjugacy classes of subgroups

Complements