Group #22550009 — subgroups

Subgroups — #22550009

Order 8128.

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

normal subgroups only

This group has 198 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, complemented
2	C₂×#14523233	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	#121×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	#14523233	—	2	4	no	—
11	C₂×C₁₂₇⋊C₈	C₂²	1	2	yes	complemented
12	C₂×D₄×C₁₂₇	—	2	1	no	—
13	C₂×D₄×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
14	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
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	complemented
24	C₁₂₇⋊C₈	—	2	2	no	—
25	C₂³×C₁₂₇	—	2	1	no	—
26	C₂³×C₁₂₇	D₄	1	1	yes	characteristic
27	D₄×C₁₂₇	—	2	2	no	—
28	D₄×C₁₂₇	—	4	2	no	—
29	C₂×C₄×C₁₂₇	—	2	2	no	—
30	C₂×C₄×C₁₂₇	—	2	1	no	—
31	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
33	C₂²×C₁₂₇	—	2	1	no	—
34	C₂²×C₁₂₇	—	2	1	no	—
35	C₂²×C₁₂₇	—	4	1	no	—
36	C₂²×C₁₂₇	—	4	1	no	—
37	C₂²×C₁₂₇	—	4	1	no	—
38	C₂²×C₁₂₇	D₈	1	2	yes	complemented
39	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
40	C₂×D₁₂₇	—	2	4	no	—
41	C₂×D₁₂₇	—	4	8	no	—
42	C₂×D₁₂₇	—	4	2	no	—
43	C₂×D₁₂₇	—	4	2	no	—
44	C₄×C₁₂₇	—	2	1	no	—
45	C₄×C₁₂₇	—	4	1	no	—
46	C₄×C₁₂₇	C₂×D₄	1	2	yes	—
47	C₂×C₁₂₇	—	2	2	no	—
48	C₂×C₁₂₇	—	8	1	no	—
49	C₂×C₁₂₇	#120	1	1	yes	characteristic
50	C₂×C₁₂₇	C₂×D₈	1	1	yes	characteristic
51	C₂×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
52	D₁₂₇	—	4	4	no	—
53	C₁₂₇	#447	1	1	yes	characteristic, complemented, Sylow
54	#447	—	127	1	no	maximal, Sylow
55	C₂²×D₄	—	127	1	no	—
56	C₂×D₈	—	127	2	no	—
57	#121	D₁₂₇	1	1	yes	characteristic, complemented
58	#102	—	127	2	no	—
59	C₂²⋊C₈	—	127	1	no	—
60	C₂⁴	—	254	1	no	—
61	C₂×D₄	—	2	1	no	—
62	C₂×D₄	—	127	2	no	—
63	C₂×D₄	—	254	4	no	—
64	C₂×D₄	—	254	1	no	—
65	C₂×D₄	C₂×D₁₂₇	1	1	yes	characteristic, complemented
66	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
67	D₈	—	254	4	no	—
68	C₂×C₈	—	127	2	no	—
69	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic, complemented
70	C₂²⋊C₄	—	2	1	no	—
71	C₂³	—	2	1	no	—
72	C₂³	—	508	2	no	—
73	C₂³	—	508	2	no	—
74	C₂³	—	254	4	no	—
75	C₂³	—	254	2	no	—
76	C₂³	C₂²⋊D₁₂₇	1	1	yes	characteristic
77	D₄	—	2	2	no	—
78	D₄	—	4	2	no	—
79	D₄	—	127	4	no	—
80	D₄	—	254	4	no	—
81	D₄	—	254	2	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
85	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
86	C₈	—	254	2	no	—
87	C₂²	—	2	1	no	—
88	C₂²	—	2	1	no	—
89	C₂²	—	4	1	no	—
90	C₂²	—	4	1	no	—
91	C₂²	—	4	1	no	—
92	C₂²	—	508	8	no	—
93	C₂²	—	508	2	no	—
94	C₂²	—	508	2	no	—
95	C₂²	—	254	4	no	—
96	C₂²	#14523233	1	2	yes	complemented
97	C₂²	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	4	1	no	—
100	C₄	D₄×D₁₂₇	1	2	yes	—
101	C₂	—	2	2	no	—
102	C₂	—	8	1	no	—
103	C₂	—	508	4	no	—
104	C₂	#16861177	1	1	yes	characteristic
105	C₂	C₂×#14523233	1	1	yes	characteristic
106	C₂	#16861234	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₅₀₈

C₂

yes

maximal, characteristic, complemented

C₂×#14523233

C₂

yes

maximal, complemented

#16861129

C₂

yes

maximal, complemented

C₂²⋊(C₁₂₇⋊C₈)

C₂

yes

maximal, characteristic, complemented

#121×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₂₇

—

C₂×D₅₀₈

—

C₂×D₅₀₈

—

C₂×D₅₀₈

C₂²

yes

—

#14523233

—

C₂×C₁₂₇⋊C₈

C₂²

yes

complemented

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

C₂²

yes

characteristic, complemented

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic, complemented

C₂²⋊C₄×C₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

D₅₀₈

—

D₅₀₈

—

D₅₀₈

D₄

yes

complemented

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

D₄

yes

characteristic

D₄×C₁₂₇

—

D₄×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

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

D₈

yes

complemented

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#120

yes

characteristic

C₂×C₁₂₇

C₂×D₈

yes

characteristic

C₂×C₁₂₇

AGL(1,ℤ/8)

yes

characteristic

D₁₂₇

—

C₁₂₇

#447

yes

characteristic, complemented, Sylow

#447

—

127

maximal, Sylow

C₂²×D₄

—

127

—

C₂×D₈

—

127

—

#121

D₁₂₇

yes

characteristic, complemented

#102

—

127

—

C₂²⋊C₈

—

127

—

C₂⁴

—

254

—

C₂×D₄

—

C₂×D₄

—

127

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

C₂×D₄

C₂×D₁₂₇

yes

characteristic, complemented

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

D₈

—

254

—

C₂×C₈

—

127

—

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic, complemented

C₂²⋊C₄

—

C₂³

—

C₂³

—

508

—

C₂³

—

508

—

C₂³

—

254

—

C₂³

—

254

—

C₂³

C₂²⋊D₁₂₇

yes

characteristic

D₄

—

D₄

—

D₄

—

127

—

D₄

—

254

—

D₄

—

254

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊D₁₂₇

yes

characteristic

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₈

—

254

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

508

—

C₂²

—

508

—

C₂²

—

508

—

C₂²

—

254

—

C₂²

#14523233

yes

complemented

C₂²

C₂×C₂²⋊D₁₂₇

yes

characteristic

C₄

—

C₄

—

100

C₄

D₄×D₁₂₇

yes

—

101

C₂

—

102

C₂

—

103

C₂

—

508

—

104

C₂

#16861177

yes

characteristic

105

C₂

C₂×#14523233

yes

characteristic

106

C₂

#16861234

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	#121×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	#121	D₁₂₇
hypocenter	C₁₂₇	#447
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #121

subgroup

G/H

center

C₂²

C₂×C₂²⋊D₁₂₇

commutator

C₂×C₄×C₁₂₇

C₂³

Frattini

C₂×C₄

C₂²×D₁₂₇

Fitting

#121×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×D₄

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #121

N	H ≅ G/N	type
1. C₂²×D₅₀₈	C₂	semidirect
2. C₂×#14523233	C₂	semidirect
3. #16861129	C₂	semidirect
4. C₂²⋊(C₁₂₇⋊C₈)	C₂	semidirect
5. #121×C₁₂₇	C₂	semidirect
11. C₂×C₁₂₇⋊C₈	C₂²	semidirect
13. C₂×D₄×C₁₂₇	C₂²	semidirect
15. C₄⋊C₄×C₁₂₇	C₂²	semidirect
23. D₅₀₈	D₄	semidirect
38. C₂²×C₁₂₇	D₈	semidirect
53. C₁₂₇	#447	semidirect
57. #121	D₁₂₇	semidirect
65. C₂×D₄	C₂×D₁₂₇	semidirect
69. C₄⋊C₄	C₂×D₁₂₇	semidirect
96. C₂²	#14523233	semidirect

H ≅ G/N

type

1. C₂²×D₅₀₈

C₂

semidirect

2. C₂×#14523233

C₂

semidirect

3. #16861129

C₂

semidirect

4. C₂²⋊(C₁₂₇⋊C₈)

C₂

semidirect

5. #121×C₁₂₇

C₂

semidirect

11. C₂×C₁₂₇⋊C₈

C₂²

semidirect

13. C₂×D₄×C₁₂₇

C₂²

semidirect

15. C₄⋊C₄×C₁₂₇

C₂²

semidirect

23. D₅₀₈

D₄

semidirect

38. C₂²×C₁₂₇

D₈

semidirect

53. C₁₂₇

#447

semidirect

57. #121

D₁₂₇

semidirect

65. C₂×D₄

C₂×D₁₂₇

semidirect

69. C₄⋊C₄

C₂×D₁₂₇

semidirect

96. C₂²

#14523233

semidirect

Subgroups — #22550009

Conjugacy classes of subgroups

Complements