Group #22549851 — subgroups

Subgroups — #100×D₁₂₇ #22549851

Order 8128.

186 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 53 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×D₄×D₁₂₇	C₂	1	1	yes	maximal, characteristic
2	M₁₆×D₁₂₇	C₂	1	2	yes	maximal, complemented
3	#16861126	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861125	C₂	1	2	yes	maximal, complemented
5	#100×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₁₂₇	C₄	1	2	yes	—
7	C₂×D₅₀₈	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊D₁₂₇	C₄	1	2	yes	—
9	D₄×D₁₂₇	—	2	2	no	—
10	D₄×D₁₂₇	—	2	2	no	—
11	C₂×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
12	#14523232	C₂²	1	2	yes	complemented
13	#14523231	—	2	2	no	—
14	C₈×D₁₂₇	—	2	2	no	—
15	C₂×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
16	M₁₆×C₁₂₇	C₂²	1	2	yes	complemented
17	C₂²×D₁₂₇	—	2	2	no	—
18	C₂²×D₁₂₇	—	2	2	no	—
19	C₂²×D₁₂₇	—	4	4	no	—
20	C₂²×D₁₂₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×D₁₂₇	C₂×C₄	1	2	yes	—
22	D₅₀₈	—	2	2	no	—
23	C₂²⋊D₁₂₇	—	2	4	no	—
24	C₂×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
25	C₄×D₁₂₇	D₄	1	2	yes	—
26	C₄×D₁₂₇	D₄	1	2	yes	—
27	C₁₂₇⋊C₈	—	2	2	no	—
28	C₂³×C₁₂₇	C₂×C₄	1	2	yes	—
29	D₄×C₁₂₇	—	2	2	no	—
30	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
31	C₈×C₁₂₇	—	2	2	no	—
32	C₂²×C₁₂₇	—	2	2	no	—
33	C₂²×C₁₂₇	—	4	2	no	—
34	C₂²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×D₁₂₇	—	2	2	no	—
36	C₂×D₁₂₇	—	2	2	no	—
37	C₂×D₁₂₇	—	4	4	no	—
38	C₂×D₁₂₇	—	4	4	no	—
39	C₂×D₁₂₇	—	4	2	no	—
40	C₂×D₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×D₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₄×C₁₂₇	C₂×D₄	1	2	yes	—
43	Dic₁₂₇	C₂²⋊C₄	1	2	yes	—
44	C₂×C₁₂₇	—	2	1	no	—
45	C₂×C₁₂₇	—	4	2	no	—
46	C₂×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	D₁₂₇	—	2	1	no	—
48	D₁₂₇	—	4	2	no	—
49	D₁₂₇	#100	1	2	yes	direct product factor
50	C₁₂₇	C₂×#100	1	1	yes	characteristic, complemented, Sylow
51	C₂×#100	—	127	1	no	maximal, Sylow
52	C₂²×D₄	—	127	1	no	—
53	C₂×M₁₆	—	127	2	no	—
54	#100	—	127	1	no	—
55	#100	—	127	2	no	—
56	#100	D₁₂₇	1	1	yes	characteristic, direct product factor
57	C₂⁴	—	127	2	no	—
58	C₂×D₄	—	127	2	no	—
59	C₂×D₄	—	127	1	no	—
60	C₂×D₄	—	254	2	no	—
61	C₂×D₄	—	254	2	no	—
62	C₂×D₄	C₂×D₁₂₇	1	1	yes	characteristic
63	C₂²×C₄	—	127	1	no	—
64	M₁₆	—	127	2	no	—
65	M₁₆	—	254	2	no	—
66	M₁₆	C₂×D₁₂₇	1	2	yes	complemented
67	C₂×C₈	—	254	2	no	—
68	C₂³	—	127	1	no	—
69	C₂³	—	127	2	no	—
70	C₂³	—	508	4	no	—
71	C₂³	—	254	2	no	—
72	C₂³	—	254	2	no	—
73	C₂³	C₄×D₁₂₇	1	2	yes	—
74	D₄	—	2	2	no	—
75	D₄	—	254	4	no	—
76	D₄	—	254	2	no	—
77	C₂×C₄	—	127	1	no	—
78	C₂×C₄	—	127	2	no	—
79	C₂×C₄	—	127	2	no	—
80	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
81	C₈	—	2	2	no	—
82	C₈	—	254	2	no	—
83	C₂²	—	2	2	no	—
84	C₂²	—	4	2	no	—
85	C₂²	—	127	1	no	—
86	C₂²	—	127	1	no	—
87	C₂²	—	508	2	no	—
88	C₂²	—	508	4	no	—
89	C₂²	—	508	4	no	—
90	C₂²	—	254	2	no	—
91	C₂²	—	254	2	no	—
92	C₂²	C₂×C₄×D₁₂₇	1	1	yes	characteristic
93	C₄	—	127	2	no	—
94	C₄	D₄×D₁₂₇	1	2	yes	—
95	C₂	—	2	1	no	—
96	C₂	—	4	2	no	—
97	C₂	—	127	2	no	—
98	C₂	—	508	2	no	—
99	C₂	—	254	1	no	—
100	C₂	C₂²⋊C₄×D₁₂₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₄×D₁₂₇

C₂

yes

maximal, characteristic

M₁₆×D₁₂₇

C₂

yes

maximal, complemented

#16861126

C₂

yes

maximal, characteristic, complemented

#16861125

C₂

yes

maximal, complemented

#100×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₂₇

C₄

yes

—

C₂×D₅₀₈

C₂²

yes

characteristic

C₂×C₂²⋊D₁₂₇

C₄

yes

—

D₄×D₁₂₇

—

D₄×D₁₂₇

—

C₂×C₄×D₁₂₇

C₂²

yes

characteristic

#14523232

C₂²

yes

complemented

#14523231

—

C₈×D₁₂₇

—

C₂×D₄×C₁₂₇

C₂²

yes

characteristic

M₁₆×C₁₂₇

C₂²

yes

complemented

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

C₂×C₄

yes

characteristic

C₂²×D₁₂₇

C₂×C₄

yes

—

D₅₀₈

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

C₂×C₄

yes

characteristic

C₄×D₁₂₇

D₄

yes

—

C₄×D₁₂₇

D₄

yes

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

C₂×C₄

yes

—

D₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

C₂²⋊C₄

yes

characteristic

C₂×D₁₂₇

C₂²⋊C₄

yes

characteristic

C₄×C₁₂₇

C₂×D₄

yes

—

Dic₁₂₇

C₂²⋊C₄

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

D₁₂₇

—

D₁₂₇

—

D₁₂₇

#100

yes

direct product factor

C₁₂₇

C₂×#100

yes

characteristic, complemented, Sylow

C₂×#100

—

127

maximal, Sylow

C₂²×D₄

—

127

—

C₂×M₁₆

—

127

—

#100

—

127

—

#100

—

127

—

#100

D₁₂₇

yes

characteristic, direct product factor

C₂⁴

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

C₂×D₄

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

—

127

—

M₁₆

—

127

—

M₁₆

—

254

—

M₁₆

C₂×D₁₂₇

yes

complemented

C₂×C₈

—

254

—

C₂³

—

127

—

C₂³

—

127

—

C₂³

—

508

—

C₂³

—

254

—

C₂³

—

254

—

C₂³

C₄×D₁₂₇

yes

—

D₄

—

D₄

—

254

—

D₄

—

254

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₈

—

C₈

—

254

—

C₂²

—

C₂²

—

C₂²

—

127

—

C₂²

—

127

—

C₂²

—

508

—

C₂²

—

508

—

C₂²

—

508

—

C₂²

—

254

—

C₂²

—

254

—

C₂²

C₂×C₄×D₁₂₇

yes

characteristic

C₄

—

127

—

C₄

D₄×D₁₂₇

yes

—

C₂

—

C₂

—

C₂

—

127

—

C₂

—

508

—

C₂

—

254

—

100

C₂

C₂²⋊C₄×D₁₂₇

yes

characteristic

	subgroup	G/H
center	C₂	C₂²⋊C₄×D₁₂₇
commutator	C₂²×C₁₂₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	#100×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂×C₂²⋊C₄
hypercenter	#100	D₁₂₇
hypocenter	C₁₂₇	C₂×#100
LCS	G ▷ C₂²×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #100

subgroup

G/H

center

C₂

C₂²⋊C₄×D₁₂₇

commutator

C₂²×C₁₂₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×D₁₂₇

Fitting

#100×C₁₂₇

C₂

socle

C₂×C₁₂₇

C₂×C₂²⋊C₄

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #100

N	H ≅ G/N	type
2. M₁₆×D₁₂₇	C₂	semidirect
3. #16861126	C₂	semidirect
4. #16861125	C₂	semidirect
5. #100×C₁₂₇	C₂	semidirect
12. #14523232	C₂²	semidirect
16. M₁₆×C₁₂₇	C₂²	semidirect
49. D₁₂₇	#100	direct (N × H)
50. C₁₂₇	C₂×#100	semidirect
56. #100	D₁₂₇	direct (N × H)
66. M₁₆	C₂×D₁₂₇	semidirect

H ≅ G/N

type

2. M₁₆×D₁₂₇

C₂

semidirect

3. #16861126

C₂

semidirect

4. #16861125

C₂

semidirect

5. #100×C₁₂₇

C₂

semidirect

12. #14523232

C₂²

semidirect

16. M₁₆×C₁₂₇

C₂²

semidirect

49. D₁₂₇

#100

direct (N × H)

50. C₁₂₇

C₂×#100

semidirect

56. #100

D₁₂₇

direct (N × H)

66. M₁₆

C₂×D₁₂₇

semidirect

Subgroups — #100×D₁₂₇ #22549851

Conjugacy classes of subgroups

Complements

Subgroups — #100×D127 #22549851

Conjugacy classes of subgroups

Complements

Subgroups — #100×D₁₂₇ #22549851