Group #22549779 — subgroups

Subgroups — #22549779

Order 8128.

258 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 91 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×C₄⋊Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊Dic₁₂₇	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₂₇	C₂²	1	2	yes	—
6	C₂²×Dic₁₂₇	C₂²	1	2	yes	—
7	C₄⋊Dic₁₂₇	—	2	2	no	—
8	C₂²⋊Dic₁₂₇	—	2	8	no	—
9	C₄×Dic₁₂₇	—	2	2	no	—
10	C₂⁴×C₁₂₇	C₂²	1	2	yes	—
11	C₂×D₄×C₁₂₇	—	2	2	no	—
12	C₂×D₄×C₁₂₇	—	2	2	no	—
13	C₂×D₄×C₁₂₇	C₄	1	4	yes	complemented
14	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₂₇	—	2	2	no	—
16	C₂×Dic₁₂₇	—	2	2	no	—
17	C₂×Dic₁₂₇	—	2	4	no	—
18	C₂×Dic₁₂₇	—	2	2	no	—
19	C₂×Dic₁₂₇	D₄	1	4	yes	complemented
20	C₂³×C₁₂₇	—	2	2	no	—
21	C₂³×C₁₂₇	—	2	2	no	—
22	C₂³×C₁₂₇	—	2	4	no	—
23	C₂³×C₁₂₇	—	2	4	no	—
24	C₂³×C₁₂₇	C₂×C₄	1	4	yes	—
25	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
26	D₄×C₁₂₇	—	2	8	no	—
27	C₂×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
28	C₂×C₄×C₁₂₇	D₄	1	2	yes	—
29	C₂×C₄×C₁₂₇	D₄	1	2	yes	—
30	C₂²×C₁₂₇	—	2	4	no	—
31	C₂²×C₁₂₇	—	2	8	no	—
32	C₂²×C₁₂₇	—	4	4	no	—
33	C₂²×C₁₂₇	—	4	2	no	—
34	C₂²×C₁₂₇	—	4	2	no	—
35	C₂²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₁₂₇	C₂×D₄	1	2	yes	—
37	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₁₂₇	#41	1	2	yes	—
40	C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	complemented
41	Dic₁₂₇	—	2	4	no	—
42	Dic₁₂₇	—	4	2	no	—
43	C₂×C₁₂₇	—	4	4	no	—
44	C₂×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₁₂₇	C₄×D₄	1	2	yes	—
46	C₂×C₁₂₇	#121	1	2	yes	—
47	C₂×C₁₂₇	#124	1	1	yes	characteristic
48	C₂×C₁₂₇	#127	1	1	yes	characteristic
49	C₁₂₇	#390	1	1	yes	characteristic, complemented, Sylow
50	#390	—	127	1	no	maximal, Sylow
51	C₂²×D₄	D₁₂₇	1	1	yes	characteristic
52	C₂×C₄⋊C₄	—	127	1	no	—
53	C₂×C₂²⋊C₄	—	127	4	no	—
54	C₂×C₄²	—	127	1	no	—
55	C₂⁴	C₂×D₁₂₇	1	2	yes	—
56	C₂×D₄	—	2	2	no	—
57	C₂×D₄	—	2	2	no	—
58	C₂×D₄	Dic₁₂₇	1	4	yes	complemented
59	C₂²×C₄	—	127	2	no	—
60	C₂²×C₄	—	127	2	no	—
61	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
62	C₄⋊C₄	—	254	2	no	—
63	C₂²⋊C₄	—	254	8	no	—
64	C₄²	—	254	2	no	—
65	C₂³	—	2	4	no	—
66	C₂³	—	2	2	no	—
67	C₂³	—	2	2	no	—
68	C₂³	—	2	4	no	—
69	C₂³	C₂×Dic₁₂₇	1	4	yes	—
70	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
71	D₄	—	2	8	no	—
72	C₂×C₄	—	127	4	no	—
73	C₂×C₄	—	254	4	no	—
74	C₂×C₄	—	254	2	no	—
75	C₂×C₄	—	254	2	no	—
76	C₂×C₄	—	254	2	no	—
77	C₂×C₄	C₂×Dic₁₂₇	1	2	yes	—
78	C₂×C₄	C₂²⋊D₁₂₇	1	2	yes	—
79	C₂×C₄	C₂²⋊D₁₂₇	1	2	yes	—
80	C₂²	—	2	4	no	—
81	C₂²	—	2	8	no	—
82	C₂²	—	4	2	no	—
83	C₂²	—	4	2	no	—
84	C₂²	—	4	4	no	—
85	C₂²	C₂²×Dic₁₂₇	1	1	yes	characteristic
86	C₂²	D₄×D₁₂₇	1	2	yes	—
87	C₂²	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
88	C₂²	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
89	C₂²	#14523247	1	2	yes	—
90	C₄	—	508	2	no	—
91	C₄	—	254	4	no	—
92	C₄	C₂²⋊Dic₁₂₇	1	4	yes	complemented
93	C₂	—	4	4	no	—
94	C₂	C₂×C₂²⋊Dic₁₂₇	1	1	yes	characteristic
95	C₂	D₄×Dic₁₂₇	1	2	yes	—
96	C₂	#16861180	1	2	yes	—
97	C₂	#16861196	1	1	yes	characteristic
98	C₂	#16861205	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Dic₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊Dic₁₂₇

C₂

yes

maximal, complemented

C₂×C₄×Dic₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₇

C₂²

yes

—

C₂²×Dic₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

—

C₂²⋊Dic₁₂₇

—

C₄×Dic₁₂₇

—

C₂⁴×C₁₂₇

C₂²

yes

—

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

C₄

yes

complemented

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

D₄

yes

complemented

C₂³×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

C₂×C₄

yes

—

C₂³×C₁₂₇

C₂³

yes

characteristic

D₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×C₄

yes

—

C₂×C₄×C₁₂₇

D₄

yes

—

C₂×C₄×C₁₂₇

D₄

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂²×C₄

yes

characteristic

C₂²×C₁₂₇

C₂×D₄

yes

—

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₂²×C₁₂₇

#41

yes

—

C₄×C₁₂₇

C₂²⋊C₄

yes

complemented

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₁₂₇

C₄×D₄

yes

—

C₂×C₁₂₇

#121

yes

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₁₂₇

yes

characteristic, complemented, Sylow

#390

—

127

maximal, Sylow

C₂²×D₄

D₁₂₇

yes

characteristic

C₂×C₄⋊C₄

—

127

—

C₂×C₂²⋊C₄

—

127

—

C₂×C₄²

—

127

—

C₂⁴

C₂×D₁₂₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

Dic₁₂₇

yes

complemented

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

—

254

—

C₂²⋊C₄

—

254

—

C₄²

—

254

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂×Dic₁₂₇

yes

—

C₂³

C₂²×D₁₂₇

yes

characteristic

D₄

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂×Dic₁₂₇

yes

—

C₂×C₄

C₂²⋊D₁₂₇

yes

—

C₂×C₄

C₂²⋊D₁₂₇

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²×Dic₁₂₇

yes

characteristic

C₂²

D₄×D₁₂₇

yes

—

C₂²

C₂×C₂²⋊D₁₂₇

yes

characteristic

C₂²

C₂×C₂²⋊D₁₂₇

yes

characteristic

C₂²

#14523247

yes

—

C₄

—

508

—

C₄

—

254

—

C₄

C₂²⋊Dic₁₂₇

yes

complemented

C₂

—

C₂

C₂×C₂²⋊Dic₁₂₇

yes

characteristic

C₂

D₄×Dic₁₂₇

yes

—

C₂

#16861180

yes

—

C₂

yes

C₂

yes

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₂₇
commutator	C₂²×C₁₂₇	C₂²×C₄
Fitting	C₂²×D₄×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂³
hypercenter	C₂²×D₄	D₁₂₇
hypocenter	C₁₂₇	#390
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂²×D₄

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₂₇

commutator

C₂²×C₁₂₇

C₂²×C₄

Fitting

C₂²×D₄×C₁₂₇

C₂

socle

C₂³×C₁₂₇

C₂³

hypercenter

C₂²×D₄

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂³ ◁ C₂²×D₄

N	H ≅ G/N	type
1. C₂×C₄⋊Dic₁₂₇	C₂	semidirect
2. C₂×C₂²⋊Dic₁₂₇	C₂	semidirect
3. C₂×C₄×Dic₁₂₇	C₂	semidirect
13. C₂×D₄×C₁₂₇	C₄	semidirect
19. C₂×Dic₁₂₇	D₄	semidirect
40. C₄×C₁₂₇	C₂²⋊C₄	semidirect
49. C₁₂₇	#390	semidirect
58. C₂×D₄	Dic₁₂₇	semidirect
92. C₄	C₂²⋊Dic₁₂₇	semidirect

H ≅ G/N

type

1. C₂×C₄⋊Dic₁₂₇

C₂

semidirect

2. C₂×C₂²⋊Dic₁₂₇

C₂

semidirect

3. C₂×C₄×Dic₁₂₇

C₂

semidirect

13. C₂×D₄×C₁₂₇

C₄

semidirect

19. C₂×Dic₁₂₇

D₄

semidirect

40. C₄×C₁₂₇

C₂²⋊C₄

semidirect

49. C₁₂₇

#390

semidirect

58. C₂×D₄

Dic₁₂₇

semidirect

92. C₄

C₂²⋊Dic₁₂₇

semidirect

Subgroups — #22549779

Conjugacy classes of subgroups

Complements