Group #22550676 — subgroups

Subgroups — #22550676

Order 8128.

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

normal subgroups only

This group has 262 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	C₂×#14523248	C₂	1	2	yes	maximal, complemented
2	#16861242	C₂	1	4	yes	maximal, complemented
3	#16861237	C₂	1	4	yes	maximal, complemented
4	#16861231	C₂	1	4	yes	maximal, complemented
5	C₂×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#14523248	—	2	2	no	—
7	#14523248	—	2	2	no	—
8	#14523248	C₂²	1	4	yes	—
9	#14523248	C₂²	1	4	yes	complemented
10	C₂×Dic₂₅₄	C₂²	1	2	yes	complemented
11	C₂×D₅₀₈	C₂²	1	2	yes	complemented
12	C₂×C₂²⋊D₁₂₇	—	2	2	no	—
13	C₂×C₄×D₁₂₇	—	2	2	no	—
14	Dic₅₀₈	C₂²	1	4	yes	complemented
15	#14523237	C₂²	1	8	yes	complemented
16	D₁₀₁₆	C₂²	1	4	yes	complemented
17	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
18	M₁₆×C₁₂₇	C₂²	1	4	yes	complemented
19	C₂×C₈×C₁₂₇	C₂²	1	2	yes	complemented
20	C₂²×D₁₂₇	—	2	2	no	—
21	Dic₂₅₄	—	2	2	no	—
22	Dic₂₅₄	C₂³	1	4	yes	—
23	D₅₀₈	—	2	2	no	—
24	D₅₀₈	C₂³	1	4	yes	—
25	C₂²⋊D₁₂₇	—	2	4	no	—
26	C₂²⋊D₁₂₇	—	2	4	no	—
27	C₂×Dic₁₂₇	—	2	2	no	—
28	C₄×D₁₂₇	—	2	4	no	—
29	C₄×D₁₂₇	—	2	4	no	—
30	C₂³×C₁₂₇	D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₂₇	D₄	1	2	yes	—
32	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
34	C₂×C₄×C₁₂₇	C₂³	1	2	yes	—
35	C₈×C₁₂₇	C₂³	1	4	yes	—
36	C₂²×C₁₂₇	—	4	1	no	—
37	C₂²×C₁₂₇	C₂×D₄	1	2	yes	—
38	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
39	C₂×D₁₂₇	—	2	4	no	—
40	C₂×D₁₂₇	—	4	4	no	—
41	C₄×C₁₂₇	C₂×D₄	1	2	yes	—
42	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
43	C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
44	Dic₁₂₇	—	2	4	no	—
45	C₂×C₁₂₇	—	2	2	no	—
46	C₂×C₁₂₇	—	2	1	no	—
47	C₂×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
48	D₁₂₇	—	4	4	no	—
49	C₁₂₇	#575	1	1	yes	characteristic, complemented, Sylow
50	#575	—	127	1	no	maximal, Sylow
51	C₂×#41	—	127	2	no	—
52	#137	—	127	4	no	—
53	AGL(1,ℤ/8)	—	127	4	no	—
54	#135	—	127	4	no	—
55	C₂×M₁₆	D₁₂₇	1	1	yes	characteristic, complemented
56	#41	—	127	4	no	—
57	#41	—	127	4	no	—
58	#41	—	254	2	no	—
59	#41	—	254	2	no	—
60	C₂×Q₈	—	127	2	no	—
61	C₂×D₄	—	127	2	no	—
62	C₂×D₄	—	254	2	no	—
63	C₂²×C₄	—	254	2	no	—
64	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
65	Q₁₆	—	127	4	no	—
66	SD₁₆	—	127	8	no	—
67	D₈	—	127	4	no	—
68	M₁₆	C₂×D₁₂₇	1	4	yes	complemented
69	C₂×C₈	C₂×D₁₂₇	1	2	yes	complemented
70	C₂³	—	254	2	no	—
71	C₂³	D₅₀₈	1	1	yes	characteristic
72	Q₈	—	127	4	no	—
73	Q₈	—	254	2	no	—
74	D₄	—	127	4	no	—
75	D₄	—	254	4	no	—
76	D₄	—	254	4	no	—
77	D₄	—	254	2	no	—
78	C₂×C₄	—	254	2	no	—
79	C₂×C₄	—	254	4	no	—
80	C₂×C₄	—	254	4	no	—
81	C₂×C₄	D₅₀₈	1	1	yes	characteristic
82	C₂×C₄	D₅₀₈	1	2	yes	—
83	C₂×C₄	C₂²×D₁₂₇	1	2	yes	—
84	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
85	C₈	C₂²×D₁₂₇	1	4	yes	—
86	C₂²	—	4	1	no	—
87	C₂²	—	508	4	no	—
88	C₂²	—	254	4	no	—
89	C₂²	C₂×D₅₀₈	1	2	yes	—
90	C₂²	C₂×D₅₀₈	1	1	yes	characteristic
91	C₄	—	254	4	no	—
92	C₄	C₂×D₅₀₈	1	2	yes	—
93	C₄	C₂×D₅₀₈	1	1	yes	characteristic
94	C₄	C₂³×D₁₂₇	1	1	yes	characteristic
95	C₂	—	2	2	no	—
96	C₂	—	2	1	no	—
97	C₂	—	508	4	no	—
98	C₂	C₂²×D₅₀₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523248

C₂

yes

maximal, complemented

#16861242

C₂

yes

maximal, complemented

#16861237

C₂

yes

maximal, complemented

#16861231

C₂

yes

maximal, complemented

C₂×M₁₆×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#14523248

—

#14523248

—

#14523248

C₂²

yes

—

#14523248

C₂²

yes

complemented

C₂×Dic₂₅₄

C₂²

yes

complemented

C₂×D₅₀₈

C₂²

yes

complemented

C₂×C₂²⋊D₁₂₇

—

C₂×C₄×D₁₂₇

—

Dic₅₀₈

C₂²

yes

complemented

#14523237

C₂²

yes

complemented

D₁₀₁₆

C₂²

yes

complemented

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

M₁₆×C₁₂₇

C₂²

yes

complemented

C₂×C₈×C₁₂₇

C₂²

yes

complemented

C₂²×D₁₂₇

—

Dic₂₅₄

—

Dic₂₅₄

C₂³

yes

—

D₅₀₈

—

D₅₀₈

C₂³

yes

—

C₂²⋊D₁₂₇

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

—

C₄×D₁₂₇

—

C₄×D₁₂₇

—

C₂³×C₁₂₇

D₄

yes

characteristic

C₂×C₄×C₁₂₇

D₄

yes

—

C₂×C₄×C₁₂₇

D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

—

C₈×C₁₂₇

C₂³

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×D₄

yes

—

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

—

C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₄×C₁₂₇

C₂⁴

yes

characteristic

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂²×D₄

yes

characteristic

D₁₂₇

—

C₁₂₇

#575

yes

characteristic, complemented, Sylow

#575

—

127

maximal, Sylow

C₂×#41

—

127

—

#137

—

127

—

AGL(1,ℤ/8)

—

127

—

#135

—

127

—

C₂×M₁₆

D₁₂₇

yes

characteristic, complemented

#41

—

127

—

#41

—

127

—

#41

—

254

—

#41

—

254

—

C₂×Q₈

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

254

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

Q₁₆

—

127

—

SD₁₆

—

127

—

D₈

—

127

—

M₁₆

C₂×D₁₂₇

yes

complemented

C₂×C₈

C₂×D₁₂₇

yes

complemented

C₂³

—

254

—

C₂³

D₅₀₈

yes

characteristic

Q₈

—

127

—

Q₈

—

254

—

D₄

—

127

—

D₄

—

254

—

D₄

—

254

—

D₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

D₅₀₈

yes

characteristic

C₂×C₄

D₅₀₈

yes

—

C₂×C₄

C₂²×D₁₂₇

yes

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₈

C₂²×D₁₂₇

yes

—

C₂²

—

C₂²

—

508

—

C₂²

—

254

—

C₂²

C₂×D₅₀₈

yes

—

C₂²

C₂×D₅₀₈

yes

characteristic

C₄

—

254

—

C₄

C₂×D₅₀₈

yes

—

C₄

C₂×D₅₀₈

yes

characteristic

C₄

C₂³×D₁₂₇

yes

characteristic

C₂

—

C₂

—

C₂

—

508

—

C₂

C₂²×D₅₀₈

yes

characteristic

	subgroup	G/H
center	C₄	C₂×D₅₀₈
commutator	C₄×C₁₂₇	C₂⁴
Frattini	C₄	C₂³×D₁₂₇
Fitting	C₂×M₁₆×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂²×D₄
hypercenter	C₂×M₁₆	D₁₂₇
hypocenter	C₁₂₇	#575
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

subgroup

G/H

center

C₄

C₂×D₅₀₈

commutator

C₄×C₁₂₇

C₂⁴

Frattini

C₄

C₂³×D₁₂₇

Fitting

C₂×M₁₆×C₁₂₇

C₂

socle

C₂×C₁₂₇

C₂²×D₄

hypercenter

C₂×M₁₆

D₁₂₇

hypocenter

C₁₂₇

#575

LCS

G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
1. C₂×#14523248	C₂	semidirect
2. #16861242	C₂	semidirect
3. #16861237	C₂	semidirect
4. #16861231	C₂	semidirect
5. C₂×M₁₆×C₁₂₇	C₂	semidirect
9. #14523248	C₂²	semidirect
10. C₂×Dic₂₅₄	C₂²	semidirect
11. C₂×D₅₀₈	C₂²	semidirect
14. Dic₅₀₈	C₂²	semidirect
15. #14523237	C₂²	semidirect
16. D₁₀₁₆	C₂²	semidirect
18. M₁₆×C₁₂₇	C₂²	semidirect
19. C₂×C₈×C₁₂₇	C₂²	semidirect
49. C₁₂₇	#575	semidirect
55. C₂×M₁₆	D₁₂₇	semidirect
68. M₁₆	C₂×D₁₂₇	semidirect
69. C₂×C₈	C₂×D₁₂₇	semidirect

H ≅ G/N

type

1. C₂×#14523248

C₂

semidirect

2. #16861242

C₂

semidirect

3. #16861237

C₂

semidirect

4. #16861231

C₂

semidirect

5. C₂×M₁₆×C₁₂₇

C₂

semidirect

9. #14523248

C₂²

semidirect

10. C₂×Dic₂₅₄

C₂²

semidirect

11. C₂×D₅₀₈

C₂²

semidirect

14. Dic₅₀₈

C₂²

semidirect

15. #14523237

C₂²

semidirect

16. D₁₀₁₆

C₂²

semidirect

18. M₁₆×C₁₂₇

C₂²

semidirect

19. C₂×C₈×C₁₂₇

C₂²

semidirect

49. C₁₂₇

#575

semidirect

55. C₂×M₁₆

D₁₂₇

semidirect

68. M₁₆

C₂×D₁₂₇

semidirect

69. C₂×C₈

C₂×D₁₂₇

semidirect

Subgroups — #22550676

Conjugacy classes of subgroups

Complements