Group #22549967 — subgroups

Subgroups — #22549967

Order 8128.

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

normal subgroups only

This group has 122 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×#14523236	C₂	1	1	yes	maximal, characteristic
2	#16861174	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#16861140	C₂	1	1	yes	maximal, characteristic, complemented
5	#16861134	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#102×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₅₄	C₂²	1	1	yes	characteristic
9	C₂²×Dic₁₂₇	—	2	1	no	—
10	#14523236	C₄	1	4	yes	complemented
11	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
13	Dic₁₂₇⋊C₄	—	2	1	no	—
14	C₂²⋊Dic₁₂₇	—	2	1	no	—
15	C₄×Dic₁₂₇	—	2	1	no	—
16	C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
17	C₂×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
20	Dic₂₅₄	—	2	1	no	—
21	Dic₂₅₄	C₂×C₄	1	2	yes	complemented
22	C₂×Dic₁₂₇	—	2	1	no	—
23	C₂×Dic₁₂₇	—	2	1	no	—
24	C₂×Dic₁₂₇	—	4	2	no	—
25	C₂×Dic₁₂₇	D₄	1	1	yes	characteristic
26	C₂×Dic₁₂₇	D₄	1	1	yes	characteristic
27	C₁₂₇⋊C₈	C₂×C₄	1	2	yes	complemented
28	C₂³×C₁₂₇	—	2	1	no	—
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	—	2	1	no	—
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	—	4	1	no	—
39	C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
40	C₄×C₁₂₇	#41	1	1	yes	characteristic
41	Dic₁₂₇	—	2	2	no	—
42	Dic₁₂₇	—	2	1	no	—
43	Dic₁₂₇	—	4	1	no	—
44	Dic₁₂₇	SD₁₆	1	2	yes	complemented
45	C₂×C₁₂₇	—	4	2	no	—
46	C₂×C₁₂₇	C₄×D₄	1	1	yes	characteristic
47	C₂×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
48	C₂×C₁₂₇	#135	1	1	yes	characteristic
49	C₁₂₇	C₄×SD₁₆	1	1	yes	characteristic, complemented, Sylow
50	C₄×SD₁₆	—	127	1	no	maximal, Sylow
51	C₂×SD₁₆	—	127	1	no	—
52	C₄×Q₈	—	127	1	no	—
53	C₄×D₄	—	127	1	no	—
54	#106	—	127	1	no	—
55	#103	—	127	1	no	—
56	#102	D₁₂₇	1	1	yes	characteristic
57	C₄×C₈	—	127	1	no	—
58	C₂×Q₈	—	127	1	no	—
59	C₂×D₄	C₂×D₁₂₇	1	1	yes	characteristic
60	C₂²×C₄	—	254	1	no	—
61	SD₁₆	—	127	4	no	—
62	C₂×C₈	—	127	1	no	—
63	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
64	C₄⋊C₄	—	127	1	no	—
65	C₄⋊C₄	—	254	1	no	—
66	C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
67	C₂²⋊C₄	—	254	1	no	—
68	C₄²	—	127	1	no	—
69	C₄²	—	254	1	no	—
70	C₂³	—	2	1	no	—
71	Q₈	—	127	2	no	—
72	Q₈	—	254	1	no	—
73	D₄	—	2	1	no	—
74	D₄	C₄×D₁₂₇	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	—	2	1	no	—
83	C₈	—	127	2	no	—
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	—	4	1	no	—
89	C₄	—	127	2	no	—
90	C₄	—	508	1	no	—
91	C₄	—	254	1	no	—
92	C₄	—	254	2	no	—
93	C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
94	C₄	#14523247	1	1	yes	characteristic
95	C₂	—	4	2	no	—
96	C₂	(C₂²⋊D₁₂₇)⋊C₄	1	1	yes	characteristic
97	C₂	SD₁₆×D₁₂₇	1	1	yes	characteristic
98	C₂	#16861228	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523236

C₂

yes

maximal, characteristic

#16861174

C₂

yes

maximal, characteristic, complemented

D₄×Dic₁₂₇

C₂

yes

maximal, characteristic

#16861140

C₂

yes

maximal, characteristic, complemented

#16861134

C₂

yes

maximal, characteristic, complemented

C₈×Dic₁₂₇

C₂

yes

maximal, characteristic, complemented

#102×C₁₂₇

C₂

yes

maximal, characteristic

C₂×Dic₂₅₄

C₂²

yes

characteristic

C₂²×Dic₁₂₇

—

#14523236

C₄

yes

complemented

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

C₄⋊Dic₁₂₇

C₂²

yes

characteristic

Dic₁₂₇⋊C₄

—

C₂²⋊Dic₁₂₇

—

C₄×Dic₁₂₇

—

C₄×Dic₁₂₇

C₂²

yes

characteristic

C₂×D₄×C₁₂₇

C₂²

yes

characteristic

C₂×C₈×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂²

yes

characteristic

Dic₂₅₄

—

Dic₂₅₄

C₂×C₄

yes

complemented

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

D₄

yes

characteristic

C₂×Dic₁₂₇

D₄

yes

characteristic

C₁₂₇⋊C₈

C₂×C₄

yes

complemented

C₂³×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

C₂×C₄

yes

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₈×C₁₂₇

—

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

SD₁₆

yes

complemented

C₂×C₁₂₇

—

C₂×C₁₂₇

C₄×D₄

yes

characteristic

C₂×C₁₂₇

C₂×SD₁₆

yes

characteristic

C₂×C₁₂₇

#135

yes

characteristic

C₁₂₇

C₄×SD₁₆

yes

characteristic, complemented, Sylow

C₄×SD₁₆

—

127

maximal, Sylow

C₂×SD₁₆

—

127

—

C₄×Q₈

—

127

—

C₄×D₄

—

127

—

#106

—

127

—

#103

—

127

—

#102

D₁₂₇

yes

characteristic

C₄×C₈

—

127

—

C₂×Q₈

—

127

—

C₂×D₄

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

—

254

—

SD₁₆

—

127

—

C₂×C₈

—

127

—

C₂×C₈

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

C₄⋊C₄

C₂×D₁₂₇

yes

characteristic

C₂²⋊C₄

—

254

—

C₄²

—

127

—

C₄²

—

254

—

C₂³

—

Q₈

—

127

—

Q₈

—

254

—

D₄

—

D₄

C₄×D₁₂₇

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

—

C₈

—

127

—

C₂²

—

C₂²

—

C₂²

—

C₂²

D₄×D₁₂₇

yes

characteristic

C₄

—

C₄

—

127

—

C₄

—

508

—

C₄

—

254

—

C₄

—

254

—

C₄

C₂×C₄×D₁₂₇

yes

characteristic

C₄

#14523247

yes

characteristic

C₂

—

C₂

(C₂²⋊D₁₂₇)⋊C₄

yes

characteristic

C₂

SD₁₆×D₁₂₇

yes

characteristic

C₂

#16861228

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₂₇
commutator	C₄×C₁₂₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	#102×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	#102	D₁₂₇
hypocenter	C₁₂₇	C₄×SD₁₆
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

subgroup

G/H

center

C₂²

D₄×D₁₂₇

commutator

C₄×C₁₂₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×D₁₂₇

Fitting

#102×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×D₄

hypercenter

#102

D₁₂₇

hypocenter

C₁₂₇

C₄×SD₁₆

LCS

G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

N	H ≅ G/N	type
2. #16861174	C₂	semidirect
4. #16861140	C₂	semidirect
5. #16861134	C₂	semidirect
6. C₈×Dic₁₂₇	C₂	semidirect
10. #14523236	C₄	semidirect
21. Dic₂₅₄	C₂×C₄	semidirect
27. C₁₂₇⋊C₈	C₂×C₄	semidirect
44. Dic₁₂₇	SD₁₆	semidirect
49. C₁₂₇	C₄×SD₁₆	semidirect

H ≅ G/N

type

2. #16861174

C₂

semidirect

4. #16861140

C₂

semidirect

5. #16861134

C₂

semidirect

6. C₈×Dic₁₂₇

C₂

semidirect

10. #14523236

C₄

semidirect

21. Dic₂₅₄

C₂×C₄

semidirect

27. C₁₂₇⋊C₈

C₂×C₄

semidirect

44. Dic₁₂₇

SD₁₆

semidirect

49. C₁₂₇

C₄×SD₁₆

semidirect

Subgroups — #22549967

Conjugacy classes of subgroups

Complements