Group #22549778 — subgroups

Subgroups — #22549778

Order 8128.

258 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 87 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
2	C₂×C₄⋊Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×(C₂×D₁₂₇)⋊C₄	C₂	1	4	yes	maximal, complemented
4	C₂×C₄²×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₁₂₇	C₂²	1	2	yes	—
6	C₂×D₅₀₈	—	2	2	no	—
7	C₂×D₅₀₈	—	2	2	no	—
8	C₂×D₅₀₈	C₄	1	4	yes	complemented
9	C₂²×Dic₁₂₇	C₂²	1	2	yes	—
10	C₄⋊Dic₁₂₇	—	2	2	no	—
11	(C₂×D₁₂₇)⋊C₄	—	2	8	no	—
12	C₂²×C₄×C₁₂₇	C₂²	1	2	yes	—
13	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₄²×C₁₂₇	—	2	2	no	—
15	C₂²×D₁₂₇	—	2	4	no	—
16	C₂²×D₁₂₇	—	2	2	no	—
17	C₂²×D₁₂₇	—	2	2	no	—
18	C₂²×D₁₂₇	—	2	4	no	—
19	C₂²×D₁₂₇	C₂×C₄	1	4	yes	—
20	D₅₀₈	—	2	8	no	—
21	C₂×Dic₁₂₇	—	2	4	no	—
22	C₂×Dic₁₂₇	—	2	2	no	—
23	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₁₂₇	—	2	2	no	—
25	C₂×C₄×C₁₂₇	—	2	2	no	—
26	C₂×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
27	C₂×C₄×C₁₂₇	D₄	1	2	yes	—
28	C₂×C₄×C₁₂₇	D₄	1	2	yes	—
29	C₂×C₄×C₁₂₇	D₄	1	4	yes	complemented
30	C₂²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
32	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₁₂₇	C₂×D₄	1	2	yes	—
34	C₂²×C₁₂₇	#41	1	2	yes	—
35	C₂×D₁₂₇	—	2	4	no	—
36	C₂×D₁₂₇	—	2	8	no	—
37	C₂×D₁₂₇	—	4	4	no	—
38	C₂×D₁₂₇	—	4	2	no	—
39	C₂×D₁₂₇	—	4	2	no	—
40	C₄×C₁₂₇	—	2	4	no	—
41	C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	complemented
42	Dic₁₂₇	—	4	2	no	—
43	C₂×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₁₂₇	C₄×D₄	1	2	yes	—
45	C₂×C₁₂₇	#121	1	2	yes	—
46	C₂×C₁₂₇	#124	1	1	yes	characteristic
47	C₂×C₁₂₇	#127	1	1	yes	characteristic
48	D₁₂₇	—	4	4	no	—
49	C₁₂₇	#390	1	1	yes	characteristic, complemented, Sylow
50	#390	—	127	1	no	maximal, Sylow
51	C₂²×D₄	—	127	1	no	—
52	C₂×C₄⋊C₄	—	127	1	no	—
53	C₂×C₂²⋊C₄	—	127	4	no	—
54	C₂×C₄²	D₁₂₇	1	1	yes	characteristic, complemented
55	C₂⁴	—	127	2	no	—
56	C₂×D₄	—	127	4	no	—
57	C₂×D₄	—	254	2	no	—
58	C₂×D₄	—	254	2	no	—
59	C₂²×C₄	—	127	2	no	—
60	C₂²×C₄	C₂×D₁₂₇	1	2	yes	—
61	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
62	C₄⋊C₄	—	254	2	no	—
63	C₂²⋊C₄	—	254	8	no	—
64	C₄²	—	2	2	no	—
65	C₂³	—	127	4	no	—
66	C₂³	—	254	2	no	—
67	C₂³	—	254	4	no	—
68	C₂³	—	254	2	no	—
69	C₂³	—	254	4	no	—
70	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
71	D₄	—	254	8	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	254	4	no	—
75	C₂×C₄	—	254	2	no	—
76	C₂×C₄	C₄×D₁₂₇	1	2	yes	—
77	C₂×C₄	C₂²⋊D₁₂₇	1	2	yes	—
78	C₂×C₄	D₅₀₈	1	2	yes	—
79	C₂×C₄	D₅₀₈	1	4	yes	complemented
80	C₂²	—	508	2	no	—
81	C₂²	—	508	2	no	—
82	C₂²	—	508	4	no	—
83	C₂²	—	254	4	no	—
84	C₂²	—	254	8	no	—
85	C₂²	C₂×C₄×D₁₂₇	1	1	yes	characteristic
86	C₂²	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
87	C₂²	C₂×D₅₀₈	1	1	yes	characteristic
88	C₂²	C₂×D₅₀₈	1	2	yes	—
89	C₂²	#14523248	1	2	yes	—
90	C₄	—	2	4	no	—
91	C₄	—	508	2	no	—
92	C₄	(C₂×D₁₂₇)⋊C₄	1	4	yes	complemented
93	C₂	—	508	4	no	—
94	C₂	C₂×(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
95	C₂	C₄×D₅₀₈	1	2	yes	—
96	C₂	#16861183	1	2	yes	—
97	C₂	#16861197	1	1	yes	characteristic
98	C₂	#16861206	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₅₀₈

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×(C₂×D₁₂₇)⋊C₄

C₂

yes

maximal, complemented

C₂×C₄²×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₂₇

C₂²

yes

—

C₂×D₅₀₈

—

C₂×D₅₀₈

—

C₂×D₅₀₈

C₄

yes

complemented

C₂²×Dic₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

—

(C₂×D₁₂₇)⋊C₄

—

C₂²×C₄×C₁₂₇

C₂²

yes

—

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₄²×C₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

C₂×C₄

yes

—

D₅₀₈

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂³×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×C₄

yes

—

C₂×C₄×C₁₂₇

D₄

yes

—

C₂×C₄×C₁₂₇

D₄

yes

—

C₂×C₄×C₁₂₇

D₄

yes

complemented

C₂²×C₁₂₇

C₂²×C₄

yes

characteristic

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₂²×C₁₂₇

C₂×D₄

yes

—

C₂²×C₁₂₇

#41

yes

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂²⋊C₄

yes

complemented

Dic₁₂₇

—

C₂×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₁₂₇

C₄×D₄

yes

—

C₂×C₁₂₇

#121

yes

—

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

D₁₂₇

—

C₁₂₇

#390

yes

characteristic, complemented, Sylow

#390

—

127

maximal, Sylow

C₂²×D₄

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₂²⋊C₄

—

127

—

C₂×C₄²

D₁₂₇

yes

characteristic, complemented

C₂⁴

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

254

—

C₂×D₄

—

254

—

C₂²×C₄

—

127

—

C₂²×C₄

C₂×D₁₂₇

yes

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

—

254

—

C₂²⋊C₄

—

254

—

C₄²

—

C₂³

—

127

—

C₂³

—

254

—

C₂³

—

254

—

C₂³

—

254

—

C₂³

—

254

—

C₂³

C₂²×D₁₂₇

yes

characteristic

D₄

—

254

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₄×D₁₂₇

yes

—

C₂×C₄

C₂²⋊D₁₂₇

yes

—

C₂×C₄

D₅₀₈

yes

—

C₂×C₄

D₅₀₈

yes

complemented

C₂²

—

508

—

C₂²

—

508

—

C₂²

—

508

—

C₂²

—

254

—

C₂²

—

254

—

C₂²

C₂×C₄×D₁₂₇

yes

characteristic

C₂²

C₂×C₂²⋊D₁₂₇

yes

characteristic

C₂²

C₂×D₅₀₈

yes

characteristic

C₂²

C₂×D₅₀₈

yes

—

C₂²

#14523248

yes

—

C₄

—

C₄

—

508

—

C₄

(C₂×D₁₂₇)⋊C₄

yes

complemented

C₂

—

508

—

C₂

C₂×(C₂×D₁₂₇)⋊C₄

yes

characteristic

C₂

C₄×D₅₀₈

yes

—

C₂

#16861183

yes

—

C₂

yes

C₂

yes

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₂₇
commutator	C₂²×C₁₂₇	C₂²×C₄
Fitting	C₂×C₄²×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂³
hypercenter	C₂×C₄²	D₁₂₇
hypocenter	C₁₂₇	#390
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₂₇

commutator

C₂²×C₁₂₇

C₂²×C₄

Fitting

C₂×C₄²×C₁₂₇

C₂

socle

C₂³×C₁₂₇

C₂³

hypercenter

C₂×C₄²

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂³ ◁ C₂×C₄²

N	H ≅ G/N	type
2. C₂×C₄⋊Dic₁₂₇	C₂	semidirect
3. C₂×(C₂×D₁₂₇)⋊C₄	C₂	semidirect
4. C₂×C₄²×C₁₂₇	C₂	semidirect
8. C₂×D₅₀₈	C₄	semidirect
29. C₂×C₄×C₁₂₇	D₄	semidirect
41. C₄×C₁₂₇	C₂²⋊C₄	semidirect
49. C₁₂₇	#390	semidirect
54. C₂×C₄²	D₁₂₇	semidirect
79. C₂×C₄	D₅₀₈	semidirect
92. C₄	(C₂×D₁₂₇)⋊C₄	semidirect

H ≅ G/N

type

2. C₂×C₄⋊Dic₁₂₇

C₂

semidirect

3. C₂×(C₂×D₁₂₇)⋊C₄

C₂

semidirect

4. C₂×C₄²×C₁₂₇

C₂

semidirect

8. C₂×D₅₀₈

C₄

semidirect

29. C₂×C₄×C₁₂₇

D₄

semidirect

41. C₄×C₁₂₇

C₂²⋊C₄

semidirect

49. C₁₂₇

#390

semidirect

54. C₂×C₄²

D₁₂₇

semidirect

79. C₂×C₄

D₅₀₈

semidirect

92. C₄

(C₂×D₁₂₇)⋊C₄

semidirect

Subgroups — #22549778

Conjugacy classes of subgroups

Complements