Group #22549977 — subgroups

Subgroups — #22549977

Order 8128.

120 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 57 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	Q₈×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#16861143	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861133	C₂	1	1	yes	maximal, characteristic, complemented
5	#16861130	C₂	1	1	yes	maximal, characteristic
6	#16861119	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×SD₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₂₇	—	2	1	no	—
9	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₁₂₇	—	2	1	no	—
11	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
13	C₂²⋊Dic₁₂₇	—	2	1	no	—
14	C₄×Dic₁₂₇	—	2	1	no	—
15	C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
18	SD₁₆×C₁₂₇	C₄	1	4	yes	complemented
19	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₂×Dic₁₂₇	—	2	1	no	—
21	C₂×Dic₁₂₇	—	2	1	no	—
22	C₂×Dic₁₂₇	—	4	2	no	—
23	C₂×Dic₁₂₇	D₄	1	1	yes	characteristic
24	C₂×Dic₁₂₇	D₄	1	1	yes	characteristic
25	C₁₂₇⋊C₈	—	2	1	no	—
26	C₂³×C₁₂₇	—	2	1	no	—
27	Q₈×C₁₂₇	—	2	1	no	—
28	Q₈×C₁₂₇	C₂×C₄	1	2	yes	complemented
29	D₄×C₁₂₇	—	2	1	no	—
30	D₄×C₁₂₇	C₂×C₄	1	2	yes	—
31	C₂×C₄×C₁₂₇	—	2	1	no	—
32	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
33	C₈×C₁₂₇	C₂×C₄	1	2	yes	complemented
34	C₂²×C₁₂₇	—	2	2	no	—
35	C₂²×C₁₂₇	—	4	1	no	—
36	C₂²×C₁₂₇	—	4	1	no	—
37	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
38	C₄×C₁₂₇	—	2	2	no	—
39	C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
40	C₄×C₁₂₇	#41	1	1	yes	characteristic
41	Dic₁₂₇	—	2	1	no	—
42	Dic₁₂₇	—	2	1	no	—
43	Dic₁₂₇	—	4	1	no	—
44	Dic₁₂₇	—	4	1	no	—
45	C₂×C₁₂₇	—	4	2	no	—
46	C₂×C₁₂₇	C₄×D₄	1	1	yes	characteristic
47	C₂×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
48	C₂×C₁₂₇	#137	1	1	yes	characteristic
49	C₁₂₇	#440	1	1	yes	characteristic, complemented, Sylow
50	#440	—	127	1	no	maximal, Sylow
51	C₂×SD₁₆	D₁₂₇	1	1	yes	characteristic
52	C₄×Q₈	—	127	1	no	—
53	C₄×D₄	—	127	1	no	—
54	#107	—	127	1	no	—
55	#103	—	127	1	no	—
56	#102	—	127	1	no	—
57	#97	—	127	1	no	—
58	C₂×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
59	C₂×D₄	C₂×D₁₂₇	1	1	yes	characteristic
60	C₂²×C₄	—	254	1	no	—
61	SD₁₆	Dic₁₂₇	1	4	yes	complemented
62	C₂×C₈	—	127	1	no	—
63	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
64	C₄⋊C₄	—	127	1	no	—
65	C₄⋊C₄	—	127	1	no	—
66	C₄⋊C₄	—	254	1	no	—
67	C₂²⋊C₄	—	254	1	no	—
68	C₄²	—	127	1	no	—
69	C₄²	—	254	1	no	—
70	C₂³	—	2	1	no	—
71	Q₈	—	2	1	no	—
72	Q₈	C₂×Dic₁₂₇	1	2	yes	complemented
73	D₄	—	2	1	no	—
74	D₄	C₂×Dic₁₂₇	1	2	yes	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	127	1	no	—
77	C₂×C₄	—	127	1	no	—
78	C₂×C₄	—	508	2	no	—
79	C₂×C₄	—	254	1	no	—
80	C₂×C₄	—	254	1	no	—
81	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
82	C₈	—	254	1	no	—
83	C₈	C₂×Dic₁₂₇	1	2	yes	complemented
84	C₂²	—	2	2	no	—
85	C₂²	—	4	1	no	—
86	C₂²	—	4	1	no	—
87	C₂²	D₄×D₁₂₇	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	508	1	no	—
90	C₄	—	508	1	no	—
91	C₄	—	254	1	no	—
92	C₄	—	254	1	no	—
93	C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
94	C₄	#14523247	1	1	yes	characteristic
95	C₂	—	4	2	no	—
96	C₂	D₄×Dic₁₂₇	1	1	yes	characteristic
97	C₂	#16861236	1	1	yes	characteristic
98	C₂	#16861238	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₂₇

C₂

yes

maximal, characteristic, complemented

D₄×Dic₁₂₇

C₂

yes

maximal, characteristic

#16861143

C₂

yes

maximal, characteristic, complemented

#16861133

C₂

yes

maximal, characteristic, complemented

#16861130

C₂

yes

maximal, characteristic

#16861119

C₂

yes

maximal, characteristic, complemented

C₂×SD₁₆×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₇

—

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

C₂²

yes

characteristic

C₄⋊Dic₁₂₇

C₂²

yes

characteristic

C₂²⋊Dic₁₂₇

—

C₄×Dic₁₂₇

—

C₄×Dic₁₂₇

C₂²

yes

characteristic

C₂×Q₈×C₁₂₇

C₂²

yes

characteristic

C₂×D₄×C₁₂₇

C₂²

yes

characteristic

SD₁₆×C₁₂₇

C₄

yes

complemented

C₂×C₈×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

D₄

yes

characteristic

C₂×Dic₁₂₇

D₄

yes

characteristic

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

C₂×C₄

yes

complemented

D₄×C₁₂₇

—

D₄×C₁₂₇

C₂×C₄

yes

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₈×C₁₂₇

C₂×C₄

yes

complemented

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₄×C₁₂₇

#41

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₄×D₄

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₁₂₇

yes

characteristic, complemented, Sylow

#440

—

127

maximal, Sylow

C₂×SD₁₆

D₁₂₇

yes

characteristic

C₄×Q₈

—

127

—

C₄×D₄

—

127

—

#107

—

127

—

#103

—

127

—

#102

—

127

—

#97

—

127

—

C₂×Q₈

C₂×D₁₂₇

yes

characteristic

C₂×D₄

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

—

254

—

SD₁₆

Dic₁₂₇

yes

complemented

C₂×C₈

—

127

—

C₂×C₈

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

C₂²⋊C₄

—

254

—

C₄²

—

127

—

C₄²

—

254

—

C₂³

—

Q₈

—

Q₈

C₂×Dic₁₂₇

yes

complemented

D₄

—

D₄

C₂×Dic₁₂₇

yes

—

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

508

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₈

—

254

—

C₈

C₂×Dic₁₂₇

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

D₄×D₁₂₇

yes

characteristic

C₄

—

C₄

—

508

—

C₄

—

508

—

C₄

—

254

—

C₄

—

254

—

C₄

C₂²×Dic₁₂₇

yes

characteristic

C₄

#14523247

yes

characteristic

C₂

—

C₂

D₄×Dic₁₂₇

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²	D₄×D₁₂₇
commutator	C₄×C₁₂₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	C₂×SD₁₆×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	C₂×SD₁₆	D₁₂₇
hypocenter	C₁₂₇	#440
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

subgroup

G/H

center

C₂²

D₄×D₁₂₇

commutator

C₄×C₁₂₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×D₁₂₇

Fitting

C₂×SD₁₆×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×D₄

hypercenter

C₂×SD₁₆

D₁₂₇

hypocenter

C₁₂₇

#440

LCS

G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

N	H ≅ G/N	type
1. Q₈×Dic₁₂₇	C₂	semidirect
3. #16861143	C₂	semidirect
4. #16861133	C₂	semidirect
6. #16861119	C₂	semidirect
18. SD₁₆×C₁₂₇	C₄	semidirect
28. Q₈×C₁₂₇	C₂×C₄	semidirect
33. C₈×C₁₂₇	C₂×C₄	semidirect
49. C₁₂₇	#440	semidirect
61. SD₁₆	Dic₁₂₇	semidirect
72. Q₈	C₂×Dic₁₂₇	semidirect
83. C₈	C₂×Dic₁₂₇	semidirect

H ≅ G/N

type

1. Q₈×Dic₁₂₇

C₂

semidirect

3. #16861143

C₂

semidirect

4. #16861133

C₂

semidirect

6. #16861119

C₂

semidirect

18. SD₁₆×C₁₂₇

C₄

semidirect

28. Q₈×C₁₂₇

C₂×C₄

semidirect

33. C₈×C₁₂₇

C₂×C₄

semidirect

49. C₁₂₇

#440

semidirect

61. SD₁₆

Dic₁₂₇

semidirect

72. Q₈

C₂×Dic₁₂₇

semidirect

83. C₈

C₂×Dic₁₂₇

semidirect

Subgroups — #22549977

Conjugacy classes of subgroups

Complements