Group #22550641 — subgroups

Subgroups — #22550641

Order 8128.

258 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 147 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	#41×D₁₂₇	C₂	1	1	yes	maximal, characteristic
2	#16861217	C₂	1	3	yes	maximal, complemented
3	#16861216	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861215	C₂	1	3	yes	maximal, complemented
5	C₂×#14523231	C₂	1	3	yes	maximal, complemented
6	M₁₆×D₁₂₇	C₂	1	3	yes	maximal, complemented
7	#131×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14523249	C₄	1	1	yes	characteristic
9	#14523248	C₂²	1	3	yes	—
10	#14523247	C₄	1	3	yes	—
11	Q₈×D₁₂₇	C₄	1	1	yes	characteristic
12	D₄×D₁₂₇	C₄	1	3	yes	—
13	C₂×C₄×D₁₂₇	C₂²	1	3	yes	—
14	#14523232	C₂²	1	3	yes	complemented
15	#14523231	C₂²	1	3	yes	—
16	#14523231	C₂²	1	1	yes	characteristic
17	#14523231	C₂²	1	3	yes	complemented
18	#14523231	C₂²	1	3	yes	complemented
19	C₂×C₁₂₇⋊C₈	C₂²	1	3	yes	complemented
20	C₈×D₁₂₇	C₂²	1	6	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	C₂×C₄	1	3	yes	—
25	Dic₂₅₄	C₂×C₄	1	3	yes	—
26	D₅₀₈	C₂×C₄	1	3	yes	—
27	C₂²⋊D₁₂₇	C₂×C₄	1	6	yes	—
28	C₂×Dic₁₂₇	C₂×C₄	1	3	yes	—
29	C₄×D₁₂₇	C₂×C₄	1	3	yes	—
30	C₄×D₁₂₇	C₂×C₄	1	3	yes	—
31	C₄×D₁₂₇	C₂³	1	3	yes	—
32	C₄×D₁₂₇	C₂³	1	1	yes	characteristic
33	C₁₂₇⋊C₈	C₂³	1	3	yes	—
34	C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
35	Q₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
36	D₄×C₁₂₇	C₂×C₄	1	3	yes	—
37	C₂×C₄×C₁₂₇	C₂³	1	3	yes	—
38	C₈×C₁₂₇	C₂³	1	1	yes	characteristic
39	C₈×C₁₂₇	C₂³	1	3	yes	—
40	C₂²×C₁₂₇	C₂²×C₄	1	3	yes	—
41	C₂×D₁₂₇	—	4	3	no	—
42	C₂×D₁₂₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×D₁₂₇	C₂²×C₄	1	3	yes	—
44	C₄×C₁₂₇	C₂²×C₄	1	3	yes	—
45	C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
46	Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
47	Dic₁₂₇	C₂²×C₄	1	3	yes	—
48	C₂×C₁₂₇	—	2	3	no	—
49	C₂×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
50	D₁₂₇	—	2	1	no	—
51	D₁₂₇	—	2	3	no	—
52	C₁₂₇	#568	1	1	yes	characteristic, complemented, Sylow
53	#568	—	127	1	no	maximal, Sylow
54	C₂×#41	—	127	1	no	—
55	#131	—	127	1	no	—
56	#131	—	127	3	no	—
57	#131	—	127	3	no	—
58	#131	D₁₂₇	1	1	yes	characteristic, complemented
59	C₂×M₁₆	—	127	3	no	—
60	C₂×M₁₆	—	127	3	no	—
61	#41	—	127	1	no	—
62	#41	—	127	3	no	—
63	#41	—	127	3	no	—
64	#41	C₂×D₁₂₇	1	1	yes	characteristic
65	C₂×Q₈	—	127	1	no	—
66	C₂×D₄	—	127	3	no	—
67	C₂²×C₄	—	127	3	no	—
68	M₁₆	—	127	3	no	—
69	M₁₆	—	127	3	no	—
70	M₁₆	—	127	1	no	—
71	M₁₆	—	127	3	no	—
72	M₁₆	—	127	3	no	—
73	M₁₆	C₂×D₁₂₇	1	3	yes	complemented
74	C₂×C₈	—	127	3	no	—
75	C₂×C₈	—	127	6	no	—
76	C₂×C₈	C₂×D₁₂₇	1	3	yes	complemented
77	C₂³	—	127	3	no	—
78	Q₈	—	127	3	no	—
79	Q₈	C₄×D₁₂₇	1	1	yes	characteristic
80	D₄	—	127	6	no	—
81	D₄	—	127	3	no	—
82	D₄	C₄×D₁₂₇	1	3	yes	—
83	C₂×C₄	—	127	3	no	—
84	C₂×C₄	—	127	3	no	—
85	C₂×C₄	—	127	3	no	—
86	C₂×C₄	—	127	3	no	—
87	C₂×C₄	—	127	1	no	—
88	C₂×C₄	C₂²×D₁₂₇	1	3	yes	—
89	C₈	—	127	1	no	—
90	C₈	—	127	3	no	—
91	C₈	C₂²×D₁₂₇	1	3	yes	—
92	C₈	C₂²×D₁₂₇	1	1	yes	characteristic
93	C₂²	—	127	1	no	—
94	C₂²	—	127	3	no	—
95	C₂²	—	508	3	no	—
96	C₂²	C₂×C₄×D₁₂₇	1	3	yes	—
97	C₄	—	127	1	no	—
98	C₄	—	127	3	no	—
99	C₄	C₂×C₄×D₁₂₇	1	3	yes	—
100	C₄	C₂³×D₁₂₇	1	1	yes	characteristic
101	C₂	—	2	3	no	—
102	C₂	—	254	1	no	—
103	C₂	—	254	3	no	—
104	C₂	C₂²×C₄×D₁₂₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#41×D₁₂₇

C₂

yes

maximal, characteristic

#16861217

C₂

yes

maximal, complemented

#16861216

C₂

yes

maximal, characteristic, complemented

#16861215

C₂

yes

maximal, complemented

C₂×#14523231

C₂

yes

maximal, complemented

M₁₆×D₁₂₇

C₂

yes

maximal, complemented

#131×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#14523249

C₄

yes

characteristic

#14523248

C₂²

yes

—

#14523247

C₄

yes

—

Q₈×D₁₂₇

C₄

yes

characteristic

D₄×D₁₂₇

C₄

yes

—

C₂×C₄×D₁₂₇

C₂²

yes

—

#14523232

C₂²

yes

complemented

#14523231

C₂²

yes

—

#14523231

C₂²

yes

characteristic

#14523231

C₂²

yes

complemented

#14523231

C₂²

yes

complemented

C₂×C₁₂₇⋊C₈

C₂²

yes

complemented

C₈×D₁₂₇

C₂²

yes

complemented

#41×C₁₂₇

C₂²

yes

characteristic

M₁₆×C₁₂₇

C₂²

yes

complemented

C₂×C₈×C₁₂₇

C₂²

yes

complemented

C₂²×D₁₂₇

C₂×C₄

yes

—

Dic₂₅₄

C₂×C₄

yes

—

D₅₀₈

C₂×C₄

yes

—

C₂²⋊D₁₂₇

C₂×C₄

yes

—

C₂×Dic₁₂₇

C₂×C₄

yes

—

C₄×D₁₂₇

C₂×C₄

yes

—

C₄×D₁₂₇

C₂×C₄

yes

—

C₄×D₁₂₇

C₂³

yes

—

C₄×D₁₂₇

C₂³

yes

characteristic

C₁₂₇⋊C₈

C₂³

yes

—

C₁₂₇⋊C₈

C₂³

yes

characteristic

Q₈×C₁₂₇

C₂×C₄

yes

characteristic

D₄×C₁₂₇

C₂×C₄

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

—

C₈×C₁₂₇

C₂³

yes

characteristic

C₈×C₁₂₇

C₂³

yes

—

C₂²×C₁₂₇

C₂²×C₄

yes

—

C₂×D₁₂₇

—

C₂×D₁₂₇

C₂²×C₄

yes

characteristic

C₂×D₁₂₇

C₂²×C₄

yes

—

C₄×C₁₂₇

C₂²×C₄

yes

—

C₄×C₁₂₇

C₂⁴

yes

characteristic

Dic₁₂₇

C₂²×C₄

yes

characteristic

Dic₁₂₇

C₂²×C₄

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂³×C₄

yes

characteristic

D₁₂₇

—

D₁₂₇

—

C₁₂₇

#568

yes

characteristic, complemented, Sylow

#568

—

127

maximal, Sylow

C₂×#41

—

127

—

#131

—

127

—

#131

—

127

—

#131

—

127

—

#131

D₁₂₇

yes

characteristic, complemented

C₂×M₁₆

—

127

—

C₂×M₁₆

—

127

—

#41

—

127

—

#41

—

127

—

#41

—

127

—

#41

C₂×D₁₂₇

yes

characteristic

C₂×Q₈

—

127

—

C₂×D₄

—

127

—

C₂²×C₄

—

127

—

M₁₆

—

127

—

M₁₆

—

127

—

M₁₆

—

127

—

M₁₆

—

127

—

M₁₆

—

127

—

M₁₆

C₂×D₁₂₇

yes

complemented

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

C₂×D₁₂₇

yes

complemented

C₂³

—

127

—

Q₈

—

127

—

Q₈

C₄×D₁₂₇

yes

characteristic

D₄

—

127

—

D₄

—

127

—

D₄

C₄×D₁₂₇

yes

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

C₂²×D₁₂₇

yes

—

C₈

—

127

—

C₈

—

127

—

C₈

C₂²×D₁₂₇

yes

—

C₈

C₂²×D₁₂₇

yes

characteristic

C₂²

—

127

—

C₂²

—

127

—

C₂²

—

508

—

C₂²

C₂×C₄×D₁₂₇

yes

—

C₄

—

127

—

C₄

—

127

—

C₄

C₂×C₄×D₁₂₇

yes

—

100

C₄

C₂³×D₁₂₇

yes

characteristic

101

C₂

—

102

C₂

—

254

—

103

C₂

—

254

—

104

C₂

C₂²×C₄×D₁₂₇

yes

characteristic

	subgroup	G/H
center, Frattini	C₄	C₂³×D₁₂₇
commutator, socle	C₂×C₁₂₇	C₂³×C₄
Fitting	#131×C₁₂₇	C₂
hypercenter	#131	D₁₂₇
hypocenter	C₁₂₇	#568
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ #131

subgroup

G/H

center, Frattini

C₄

C₂³×D₁₂₇

commutator, socle

C₂×C₁₂₇

C₂³×C₄

Fitting

#131×C₁₂₇

C₂

D₁₂₇

C₁₂₇

G ▷ C₂×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₄ ◁ #131

N	H ≅ G/N	type
2. #16861217	C₂	semidirect
3. #16861216	C₂	semidirect
4. #16861215	C₂	semidirect
5. C₂×#14523231	C₂	semidirect
6. M₁₆×D₁₂₇	C₂	semidirect
7. #131×C₁₂₇	C₂	semidirect
14. #14523232	C₂²	semidirect
17. #14523231	C₂²	semidirect
18. #14523231	C₂²	semidirect
19. C₂×C₁₂₇⋊C₈	C₂²	semidirect
20. C₈×D₁₂₇	C₂²	semidirect
22. M₁₆×C₁₂₇	C₂²	semidirect
23. C₂×C₈×C₁₂₇	C₂²	semidirect
52. C₁₂₇	#568	semidirect
58. #131	D₁₂₇	semidirect
73. M₁₆	C₂×D₁₂₇	semidirect
76. C₂×C₈	C₂×D₁₂₇	semidirect

H ≅ G/N

type

2. #16861217

C₂

semidirect

3. #16861216

C₂

semidirect

4. #16861215

C₂

semidirect

5. C₂×#14523231

C₂

semidirect

6. M₁₆×D₁₂₇

C₂

semidirect

7. #131×C₁₂₇

C₂

semidirect

14. #14523232

C₂²

semidirect

17. #14523231

C₂²

semidirect

18. #14523231

C₂²

semidirect

19. C₂×C₁₂₇⋊C₈

C₂²

semidirect

20. C₈×D₁₂₇

C₂²

semidirect

22. M₁₆×C₁₂₇

C₂²

semidirect

23. C₂×C₈×C₁₂₇

C₂²

semidirect

52. C₁₂₇

#568

semidirect

58. #131

D₁₂₇

semidirect

73. M₁₆

C₂×D₁₂₇

semidirect

76. C₂×C₈

C₂×D₁₂₇

semidirect

Subgroups — #22550641

Conjugacy classes of subgroups

Complements