Group #22549986 — subgroups

Subgroups — #22549986

Order 8128.

132 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 51 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×#14523233	C₂	1	1	yes	maximal, characteristic
2	C₄×D₅₀₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#16861140	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861130	C₂	1	1	yes	maximal, characteristic, complemented
5	#16861129	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861120	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×D₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₅₀₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₁₂₇	—	2	1	no	—
10	#14523233	C₄	1	4	yes	complemented
11	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
14	(C₂×D₁₂₇)⋊C₄	—	2	1	no	—
15	C₂×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₁₂₇	—	2	1	no	—
17	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
19	C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₂²×D₁₂₇	—	2	1	no	—
21	D₅₀₈	—	2	1	no	—
22	D₅₀₈	C₂×C₄	1	2	yes	complemented
23	C₂×Dic₁₂₇	—	2	1	no	—
24	C₄×D₁₂₇	—	4	2	no	—
25	C₁₂₇⋊C₈	—	2	1	no	—
26	C₁₂₇⋊C₈	C₂×C₄	1	2	yes	complemented
27	C₂³×C₁₂₇	—	2	1	no	—
28	D₄×C₁₂₇	—	2	1	no	—
29	D₄×C₁₂₇	C₂×C₄	1	2	yes	complemented
30	C₂×C₄×C₁₂₇	—	2	1	no	—
31	C₂×C₄×C₁₂₇	—	4	2	no	—
32	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
35	C₂²×C₁₂₇	—	2	2	no	—
36	C₂²×C₁₂₇	—	4	1	no	—
37	C₂²×C₁₂₇	—	4	1	no	—
38	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
39	C₂×D₁₂₇	—	2	2	no	—
40	C₂×D₁₂₇	—	4	1	no	—
41	C₂×D₁₂₇	—	4	1	no	—
42	C₄×C₁₂₇	—	2	1	no	—
43	C₄×C₁₂₇	—	2	1	no	—
44	C₄×C₁₂₇	—	4	1	no	—
45	C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
46	C₄×C₁₂₇	#41	1	1	yes	characteristic
47	Dic₁₂₇	—	4	1	no	—
48	C₂×C₁₂₇	—	4	2	no	—
49	C₂×C₁₂₇	C₄×D₄	1	1	yes	characteristic
50	C₂×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
51	C₂×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
52	D₁₂₇	—	4	2	no	—
53	C₁₂₇	#442	1	1	yes	characteristic, complemented, Sylow
54	#442	—	127	1	no	maximal, Sylow
55	C₂×D₈	—	127	1	no	—
56	C₄×D₄	—	127	1	no	—
57	C₄×D₄	D₁₂₇	1	1	yes	characteristic, complemented
58	#106	—	127	1	no	—
59	#102	—	127	1	no	—
60	#102	—	127	1	no	—
61	#97	—	127	1	no	—
62	C₂×D₄	—	127	1	no	—
63	C₂×D₄	C₂×D₁₂₇	1	1	yes	characteristic
64	C₂²×C₄	—	2	1	no	—
65	C₂²×C₄	—	254	1	no	—
66	D₈	—	127	4	no	—
67	C₂×C₈	—	127	1	no	—
68	C₂×C₈	—	127	1	no	—
69	C₄⋊C₄	—	127	1	no	—
70	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
71	C₂²⋊C₄	—	2	1	no	—
72	C₂²⋊C₄	—	254	1	no	—
73	C₄²	C₂×D₁₂₇	1	1	yes	characteristic
74	C₂³	—	2	1	no	—
75	C₂³	—	254	1	no	—
76	D₄	—	2	1	no	—
77	D₄	—	127	2	no	—
78	D₄	—	254	1	no	—
79	D₄	C₄×D₁₂₇	1	2	yes	complemented
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	508	2	no	—
83	C₂×C₄	—	254	1	no	—
84	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
85	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
86	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
87	C₈	—	127	2	no	—
88	C₈	—	254	1	no	—
89	C₂²	—	2	2	no	—
90	C₂²	—	4	1	no	—
91	C₂²	—	4	1	no	—
92	C₂²	—	508	1	no	—
93	C₂²	—	508	1	no	—
94	C₂²	—	254	2	no	—
95	C₂²	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
96	C₄	—	2	1	no	—
97	C₄	—	2	1	no	—
98	C₄	—	4	1	no	—
99	C₄	—	508	1	no	—
100	C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
101	C₄	#14523248	1	1	yes	characteristic
102	C₂	—	4	2	no	—
103	C₂	—	508	2	no	—
104	C₂	C₄×C₂²⋊D₁₂₇	1	1	yes	characteristic
105	C₂	#16861234	1	1	yes	characteristic
106	C₂	#16861235	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523233

C₂

yes

maximal, characteristic

C₄×D₅₀₈

C₂

yes

maximal, characteristic, complemented

#16861140

C₂

yes

maximal, characteristic, complemented

#16861130

C₂

yes

maximal, characteristic, complemented

#16861129

C₂

yes

maximal, characteristic, complemented

#16861120

C₂

yes

maximal, characteristic, complemented

C₄×D₄×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×D₅₀₈

C₂²

yes

characteristic

C₂×C₄×D₁₂₇

—

#14523233

C₄

yes

complemented

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₄⋊Dic₁₂₇

C₂²

yes

characteristic

(C₂×D₁₂₇)⋊C₄

—

C₂×D₄×C₁₂₇

C₂²

yes

characteristic

C₂²×C₄×C₁₂₇

—

C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

C₂²⋊C₄×C₁₂₇

—

C₄²×C₁₂₇

C₂²

yes

characteristic

C₂²×D₁₂₇

—

D₅₀₈

—

D₅₀₈

C₂×C₄

yes

complemented

C₂×Dic₁₂₇

—

C₄×D₁₂₇

—

C₁₂₇⋊C₈

—

C₁₂₇⋊C₈

C₂×C₄

yes

complemented

C₂³×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

C₂×C₄

yes

complemented

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

D₄

yes

characteristic

C₂×C₄×C₁₂₇

D₄

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₄×C₁₂₇

#41

yes

characteristic

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₄×D₄

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

D₁₂₇

—

C₁₂₇

#442

yes

characteristic, complemented, Sylow

#442

—

127

maximal, Sylow

C₂×D₈

—

127

—

C₄×D₄

—

127

—

C₄×D₄

D₁₂₇

yes

characteristic, complemented

#106

—

127

—

#102

—

127

—

#102

—

127

—

#97

—

127

—

C₂×D₄

—

127

—

C₂×D₄

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

254

—

D₈

—

127

—

C₂×C₈

—

127

—

C₂×C₈

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

254

—

C₄²

C₂×D₁₂₇

yes

characteristic

C₂³

—

C₂³

—

254

—

D₄

—

D₄

—

127

—

D₄

—

254

—

D₄

C₄×D₁₂₇

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

508

—

C₂×C₄

—

254

—

C₂×C₄

C₂²⋊D₁₂₇

yes

characteristic

C₂×C₄

C₂²⋊D₁₂₇

yes

characteristic

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₈

—

127

—

C₈

—

254

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

508

—

C₂²

—

508

—

C₂²

—

254

—

C₂²

C₂×C₂²⋊D₁₂₇

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

508

—

100

C₄

C₂×C₄×D₁₂₇

yes

characteristic

101

C₄

#14523248

yes

characteristic

102

C₂

—

103

C₂

—

508

—

104

C₂

C₄×C₂²⋊D₁₂₇

yes

characteristic

105

C₂

#16861234

yes

characteristic

106

C₂

#16861235

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₂₇
commutator	C₄×C₁₂₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	C₄×D₄×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	C₄×D₄	D₁₂₇
hypocenter	C₁₂₇	#442
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₄

subgroup

G/H

center

C₂²

C₂×C₂²⋊D₁₂₇

commutator

C₄×C₁₂₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×D₁₂₇

Fitting

C₄×D₄×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×D₄

hypercenter

C₄×D₄

D₁₂₇

hypocenter

C₁₂₇

#442

LCS

G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₄

N	H ≅ G/N	type
2. C₄×D₅₀₈	C₂	semidirect
3. #16861140	C₂	semidirect
4. #16861130	C₂	semidirect
5. #16861129	C₂	semidirect
6. #16861120	C₂	semidirect
7. C₄×D₄×C₁₂₇	C₂	semidirect
10. #14523233	C₄	semidirect
22. D₅₀₈	C₂×C₄	semidirect
26. C₁₂₇⋊C₈	C₂×C₄	semidirect
29. D₄×C₁₂₇	C₂×C₄	semidirect
53. C₁₂₇	#442	semidirect
57. C₄×D₄	D₁₂₇	semidirect
79. D₄	C₄×D₁₂₇	semidirect

H ≅ G/N

type

2. C₄×D₅₀₈

C₂

semidirect

3. #16861140

C₂

semidirect

4. #16861130

C₂

semidirect

5. #16861129

C₂

semidirect

6. #16861120

C₂

semidirect

7. C₄×D₄×C₁₂₇

C₂

semidirect

10. #14523233

C₄

semidirect

22. D₅₀₈

C₂×C₄

semidirect

26. C₁₂₇⋊C₈

C₂×C₄

semidirect

29. D₄×C₁₂₇

C₂×C₄

semidirect

53. C₁₂₇

#442

semidirect

57. C₄×D₄

D₁₂₇

semidirect

79. D₄

C₄×D₁₂₇

semidirect

Subgroups — #22549986

Conjugacy classes of subgroups

Complements