Group #22550026 — subgroups

Subgroups — #22550026

Order 8128.

188 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 47 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	C₂²×D₅₀₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523235	C₂	1	2	yes	maximal, complemented
3	#16861129	C₂	1	2	yes	maximal, complemented
4	C₂²⋊(C₁₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₁₂₇	—	2	1	no	—
7	C₂×D₅₀₈	—	2	4	no	—
8	C₂×D₅₀₈	—	2	1	no	—
9	C₂×D₅₀₈	C₂²	1	2	yes	—
10	#14523235	—	2	4	no	—
11	C₂×C₁₂₇⋊C₈	C₂²	1	2	yes	complemented
12	C₂×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₁₂₇	—	2	1	no	—
15	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
16	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
17	C₂²×D₁₂₇	—	2	4	no	—
18	C₂²×D₁₂₇	—	2	2	no	—
19	C₂²×D₁₂₇	—	4	2	no	—
20	C₂²×D₁₂₇	—	4	2	no	—
21	D₅₀₈	—	2	4	no	—
22	D₅₀₈	—	2	2	no	—
23	D₅₀₈	D₄	1	4	yes	—
24	C₁₂₇⋊C₈	—	2	2	no	—
25	C₂³×C₁₂₇	D₄	1	1	yes	characteristic
26	Q₈×C₁₂₇	—	2	2	no	—
27	C₂×C₄×C₁₂₇	—	2	2	no	—
28	C₂×C₄×C₁₂₇	—	2	1	no	—
29	C₂×C₄×C₁₂₇	—	2	1	no	—
30	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
32	C₂²×C₁₂₇	—	2	1	no	—
33	C₂²×C₁₂₇	—	2	1	no	—
34	C₂²×C₁₂₇	SD₁₆	1	2	yes	complemented
35	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
36	C₂×D₁₂₇	—	2	4	no	—
37	C₂×D₁₂₇	—	4	8	no	—
38	C₂×D₁₂₇	—	4	2	no	—
39	C₂×D₁₂₇	—	4	2	no	—
40	C₄×C₁₂₇	—	2	1	no	—
41	C₄×C₁₂₇	—	4	1	no	—
42	C₄×C₁₂₇	—	4	1	no	—
43	C₄×C₁₂₇	C₂×D₄	1	2	yes	—
44	C₂×C₁₂₇	—	2	2	no	—
45	C₂×C₁₂₇	#120	1	1	yes	characteristic
46	C₂×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
47	C₂×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
48	D₁₂₇	—	4	4	no	—
49	C₁₂₇	#450	1	1	yes	characteristic, complemented, Sylow
50	#450	—	127	1	no	maximal, Sylow
51	C₂²×D₄	—	127	1	no	—
52	C₂×SD₁₆	—	127	2	no	—
53	C₂²⋊Q₈	D₁₂₇	1	1	yes	characteristic, complemented
54	#102	—	127	2	no	—
55	C₂²⋊C₈	—	127	1	no	—
56	C₂⁴	—	254	1	no	—
57	C₂×Q₈	C₂×D₁₂₇	1	1	yes	characteristic, complemented
58	C₂×D₄	—	127	2	no	—
59	C₂×D₄	—	254	4	no	—
60	C₂×D₄	—	254	1	no	—
61	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
62	SD₁₆	—	254	4	no	—
63	C₂×C₈	—	127	2	no	—
64	C₄⋊C₄	—	2	1	no	—
65	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic, complemented
66	C₂²⋊C₄	—	2	1	no	—
67	C₂³	—	508	2	no	—
68	C₂³	—	508	2	no	—
69	C₂³	—	254	4	no	—
70	C₂³	—	254	2	no	—
71	C₂³	C₂²⋊D₁₂₇	1	1	yes	characteristic
72	Q₈	—	2	2	no	—
73	D₄	—	127	4	no	—
74	D₄	—	254	4	no	—
75	D₄	—	254	2	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
80	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
81	C₈	—	254	2	no	—
82	C₂²	—	2	1	no	—
83	C₂²	—	2	1	no	—
84	C₂²	—	508	8	no	—
85	C₂²	—	508	2	no	—
86	C₂²	—	508	2	no	—
87	C₂²	—	254	4	no	—
88	C₂²	#14523235	1	2	yes	complemented
89	C₂²	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
90	C₄	—	2	1	no	—
91	C₄	—	4	1	no	—
92	C₄	—	4	1	no	—
93	C₄	D₄×D₁₂₇	1	2	yes	—
94	C₂	—	2	2	no	—
95	C₂	—	508	4	no	—
96	C₂	#16861177	1	1	yes	characteristic
97	C₂	C₂×#14523235	1	1	yes	characteristic
98	C₂	#16861234	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₅₀₈

C₂

yes

maximal, characteristic

C₂×#14523235

C₂

yes

maximal, complemented

#16861129

C₂

yes

maximal, complemented

C₂²⋊(C₁₂₇⋊C₈)

C₂

yes

maximal, characteristic, complemented

C₂²⋊Q₈×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₂₇

—

C₂×D₅₀₈

—

C₂×D₅₀₈

—

C₂×D₅₀₈

C₂²

yes

—

#14523235

—

C₂×C₁₂₇⋊C₈

C₂²

yes

complemented

C₂×Q₈×C₁₂₇

C₂²

yes

characteristic, complemented

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic, complemented

C₂²⋊C₄×C₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

D₅₀₈

—

D₅₀₈

—

D₅₀₈

D₄

yes

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

D₄

yes

characteristic

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

SD₁₆

yes

complemented

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#120

yes

characteristic

C₂×C₁₂₇

C₂×SD₁₆

yes

characteristic

C₂×C₁₂₇

AGL(1,ℤ/8)

yes

characteristic

D₁₂₇

—

C₁₂₇

#450

yes

characteristic, complemented, Sylow

#450

—

127

maximal, Sylow

C₂²×D₄

—

127

—

C₂×SD₁₆

—

127

—

C₂²⋊Q₈

D₁₂₇

yes

characteristic, complemented

#102

—

127

—

C₂²⋊C₈

—

127

—

C₂⁴

—

254

—

C₂×Q₈

C₂×D₁₂₇

yes

characteristic, complemented

C₂×D₄

—

127

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

SD₁₆

—

254

—

C₂×C₈

—

127

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic, complemented

C₂²⋊C₄

—

C₂³

—

508

—

C₂³

—

508

—

C₂³

—

254

—

C₂³

—

254

—

C₂³

C₂²⋊D₁₂₇

yes

characteristic

Q₈

—

D₄

—

127

—

D₄

—

254

—

D₄

—

254

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊D₁₂₇

yes

characteristic

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₈

—

254

—

C₂²

—

C₂²

—

C₂²

—

508

—

C₂²

—

508

—

C₂²

—

508

—

C₂²

—

254

—

C₂²

#14523235

yes

complemented

C₂²

C₂×C₂²⋊D₁₂₇

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

D₄×D₁₂₇

yes

—

C₂

—

C₂

—

508

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	C₂²⋊Q₈×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	C₂²⋊Q₈	D₁₂₇
hypocenter	C₁₂₇	#450
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊D₁₂₇

commutator

C₂×C₄×C₁₂₇

C₂³

Frattini

C₂×C₄

C₂²×D₁₂₇

Fitting

C₂²⋊Q₈×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×D₄

hypercenter

C₂²⋊Q₈

D₁₂₇

hypocenter

C₁₂₇

#450

LCS

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊Q₈

N	H ≅ G/N	type
2. C₂×#14523235	C₂	semidirect
3. #16861129	C₂	semidirect
4. C₂²⋊(C₁₂₇⋊C₈)	C₂	semidirect
5. C₂²⋊Q₈×C₁₂₇	C₂	semidirect
11. C₂×C₁₂₇⋊C₈	C₂²	semidirect
12. C₂×Q₈×C₁₂₇	C₂²	semidirect
15. C₄⋊C₄×C₁₂₇	C₂²	semidirect
34. C₂²×C₁₂₇	SD₁₆	semidirect
49. C₁₂₇	#450	semidirect
53. C₂²⋊Q₈	D₁₂₇	semidirect
57. C₂×Q₈	C₂×D₁₂₇	semidirect
65. C₄⋊C₄	C₂×D₁₂₇	semidirect
88. C₂²	#14523235	semidirect

H ≅ G/N

type

2. C₂×#14523235

C₂

semidirect

3. #16861129

C₂

semidirect

4. C₂²⋊(C₁₂₇⋊C₈)

C₂

semidirect

5. C₂²⋊Q₈×C₁₂₇

C₂

semidirect

11. C₂×C₁₂₇⋊C₈

C₂²

semidirect

12. C₂×Q₈×C₁₂₇

C₂²

semidirect

15. C₄⋊C₄×C₁₂₇

C₂²

semidirect

34. C₂²×C₁₂₇

SD₁₆

semidirect

49. C₁₂₇

#450

semidirect

53. C₂²⋊Q₈

D₁₂₇

semidirect

57. C₂×Q₈

C₂×D₁₂₇

semidirect

65. C₄⋊C₄

C₂×D₁₂₇

semidirect

88. C₂²

#14523235

semidirect

Subgroups — #22550026

Conjugacy classes of subgroups

Complements