Group #22550538 — subgroups

Subgroups — #22550538

Order 8128.

290 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 113 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×#14523249	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×Q₈×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#16861197	C₂	1	3	yes	maximal, complemented
4	#16861189	C₂	1	3	yes	maximal, complemented
5	#16861184	C₂	1	3	yes	maximal, complemented
6	C₄×D₅₀₈	C₂	1	3	yes	maximal, complemented
7	C₄×Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14523249	—	2	4	no	—
9	C₂×Dic₂₅₄	C₂²	1	3	yes	—
10	Q₈×D₁₂₇	—	2	4	no	—
11	C₂×D₅₀₈	C₂²	1	3	yes	—
12	C₂×D₅₀₈	C₂²	1	3	yes	—
13	C₂×C₄×D₁₂₇	C₂²	1	3	yes	—
14	C₂×C₄×D₁₂₇	C₂²	1	3	yes	—
15	C₄⋊Dic₁₂₇	C₂²	1	3	yes	—
16	(C₂×D₁₂₇)⋊C₄	C₂²	1	6	yes	—
17	(C₂×D₁₂₇)⋊C₄	C₂²	1	3	yes	—
18	(C₂×D₁₂₇)⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₁₂₇	C₂²	1	3	yes	—
21	C₄²×C₁₂₇	C₂²	1	3	yes	—
22	C₂²×D₁₂₇	C₂³	1	3	yes	—
23	C₂²×D₁₂₇	C₂³	1	1	yes	characteristic
24	Dic₂₅₄	—	2	6	no	—
25	D₅₀₈	—	2	6	no	—
26	D₅₀₈	—	2	6	no	—
27	C₂×Dic₁₂₇	C₂³	1	3	yes	—
28	C₂×Dic₁₂₇	C₂³	1	1	yes	characteristic
29	C₄×D₁₂₇	—	2	6	no	—
30	C₄×D₁₂₇	—	2	6	no	—
31	Q₈×C₁₂₇	D₄	1	4	yes	complemented
32	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₁₂₇	C₂³	1	3	yes	—
34	C₂×C₄×C₁₂₇	C₂³	1	3	yes	—
35	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
36	C₂×D₁₂₇	—	2	1	no	—
37	C₂×D₁₂₇	—	2	1	no	—
38	C₂×D₁₂₇	—	2	3	no	—
39	C₂×D₁₂₇	—	2	3	no	—
40	C₂×D₁₂₇	—	2	3	no	—
41	C₂×D₁₂₇	#41	1	2	yes	complemented
42	C₄×C₁₂₇	—	2	3	no	—
43	C₄×C₁₂₇	—	2	1	no	—
44	C₄×C₁₂₇	C₂×D₄	1	6	yes	—
45	Dic₁₂₇	—	2	1	no	—
46	Dic₁₂₇	—	2	3	no	—
47	C₂×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
48	C₂×C₁₂₇	C₂×#41	1	1	yes	characteristic
49	C₂×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
50	D₁₂₇	—	2	2	no	—
51	D₁₂₇	—	4	3	no	—
52	C₁₂₇	#548	1	1	yes	characteristic, complemented, Sylow
53	#548	—	127	1	no	maximal, Sylow
54	C₂×#41	—	127	1	no	—
55	C₂²×Q₈	—	127	1	no	—
56	#124	—	127	3	no	—
57	C₂²⋊Q₈	—	127	3	no	—
58	#121	—	127	3	no	—
59	C₄×Q₈	D₁₂₇	1	1	yes	characteristic, complemented
60	C₄×D₄	—	127	3	no	—
61	#41	—	254	4	no	—
62	C₂×Q₈	—	127	3	no	—
63	C₂×Q₈	—	254	4	no	—
64	C₂×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
65	C₂×D₄	—	127	3	no	—
66	C₂×D₄	—	127	3	no	—
67	C₂²×C₄	—	127	3	no	—
68	C₂²×C₄	—	127	3	no	—
69	C₄⋊C₄	—	127	3	no	—
70	C₄⋊C₄	C₂×D₁₂₇	1	3	yes	—
71	C₂²⋊C₄	—	127	1	no	—
72	C₂²⋊C₄	—	127	3	no	—
73	C₂²⋊C₄	—	127	6	no	—
74	C₄²	C₂×D₁₂₇	1	3	yes	—
75	C₂³	—	127	1	no	—
76	C₂³	—	127	3	no	—
77	Q₈	—	254	6	no	—
78	Q₈	D₅₀₈	1	4	yes	complemented
79	D₄	—	254	6	no	—
80	D₄	—	254	6	no	—
81	C₂×C₄	—	127	3	no	—
82	C₂×C₄	—	127	1	no	—
83	C₂×C₄	—	254	6	no	—
84	C₂×C₄	—	254	6	no	—
85	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
86	C₂×C₄	C₂²×D₁₂₇	1	3	yes	—
87	C₂×C₄	C₂²×D₁₂₇	1	3	yes	—
88	C₂²	—	127	2	no	—
89	C₂²	—	254	3	no	—
90	C₂²	—	254	3	no	—
91	C₂²	—	254	3	no	—
92	C₂²	—	254	1	no	—
93	C₂²	—	254	1	no	—
94	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
95	C₄	—	2	3	no	—
96	C₄	—	2	1	no	—
97	C₄	—	254	1	no	—
98	C₄	—	254	3	no	—
99	C₄	C₂×D₅₀₈	1	6	yes	—
100	C₂	—	508	3	no	—
101	C₂	—	254	2	no	—
102	C₂	C₂²×D₅₀₈	1	1	yes	characteristic
103	C₂	#41×D₁₂₇	1	1	yes	characteristic
104	C₂	#16861257	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523249

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₁₂₇

C₂

yes

maximal, characteristic, complemented

#16861197

C₂

yes

maximal, complemented

#16861189

C₂

yes

maximal, complemented

#16861184

C₂

yes

maximal, complemented

C₄×D₅₀₈

C₂

yes

maximal, complemented

C₄×Q₈×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#14523249

—

C₂×Dic₂₅₄

C₂²

yes

—

Q₈×D₁₂₇

—

C₂×D₅₀₈

C₂²

yes

—

C₂×D₅₀₈

C₂²

yes

—

C₂×C₄×D₁₂₇

C₂²

yes

—

C₂×C₄×D₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

characteristic

C₂×Q₈×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂²

yes

—

C₄²×C₁₂₇

C₂²

yes

—

C₂²×D₁₂₇

C₂³

yes

—

C₂²×D₁₂₇

C₂³

yes

characteristic

Dic₂₅₄

—

D₅₀₈

—

D₅₀₈

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

characteristic

C₄×D₁₂₇

—

C₄×D₁₂₇

—

Q₈×C₁₂₇

D₄

yes

complemented

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂²×C₁₂₇

C₂⁴

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

#41

yes

complemented

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

C₂²×D₄

yes

characteristic

C₂×C₁₂₇

C₂×#41

yes

characteristic

C₂×C₁₂₇

2₋¹⁺⁴

yes

characteristic

D₁₂₇

—

D₁₂₇

—

C₁₂₇

#548

yes

characteristic, complemented, Sylow

#548

—

127

maximal, Sylow

C₂×#41

—

127

—

C₂²×Q₈

—

127

—

#124

—

127

—

C₂²⋊Q₈

—

127

—

#121

—

127

—

C₄×Q₈

D₁₂₇

yes

characteristic, complemented

C₄×D₄

—

127

—

#41

—

254

—

C₂×Q₈

—

127

—

C₂×Q₈

—

254

—

C₂×Q₈

C₂×D₁₂₇

yes

characteristic

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

C₂×D₁₂₇

yes

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₄²

C₂×D₁₂₇

yes

—

C₂³

—

127

—

C₂³

—

127

—

Q₈

—

254

—

Q₈

D₅₀₈

yes

complemented

D₄

—

254

—

D₄

—

254

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₄

C₂²×D₁₂₇

yes

—

C₂×C₄

C₂²×D₁₂₇

yes

—

C₂²

—

127

—

C₂²

—

254

—

C₂²

—

254

—

C₂²

—

254

—

C₂²

—

254

—

C₂²

—

254

—

C₂²

C₂³×D₁₂₇

yes

characteristic

C₄

—

C₄

—

C₄

—

254

—

C₄

—

254

—

C₄

C₂×D₅₀₈

yes

—

100

C₂

—

508

—

101

C₂

—

254

—

102

C₂

C₂²×D₅₀₈

yes

characteristic

103

C₂

#41×D₁₂₇

yes

characteristic

104

C₂

#16861257

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂⁴
Fitting	C₄×Q₈×C₁₂₇	C₂
hypercenter	C₄×Q₈	D₁₂₇
hypocenter	C₁₂₇	#548
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂⁴

Fitting

C₄×Q₈×C₁₂₇

C₂

hypercenter

C₄×Q₈

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
1. C₂×#14523249	C₂	semidirect
2. C₂×Q₈×D₁₂₇	C₂	semidirect
3. #16861197	C₂	semidirect
4. #16861189	C₂	semidirect
5. #16861184	C₂	semidirect
6. C₄×D₅₀₈	C₂	semidirect
7. C₄×Q₈×C₁₂₇	C₂	semidirect
31. Q₈×C₁₂₇	D₄	semidirect
41. C₂×D₁₂₇	#41	semidirect
52. C₁₂₇	#548	semidirect
59. C₄×Q₈	D₁₂₇	semidirect
78. Q₈	D₅₀₈	semidirect

H ≅ G/N

type

1. C₂×#14523249

C₂

semidirect

2. C₂×Q₈×D₁₂₇

C₂

semidirect

3. #16861197

C₂

semidirect

4. #16861189

C₂

semidirect

5. #16861184

C₂

semidirect

6. C₄×D₅₀₈

C₂

semidirect

7. C₄×Q₈×C₁₂₇

C₂

semidirect

31. Q₈×C₁₂₇

D₄

semidirect

41. C₂×D₁₂₇

#41

semidirect

52. C₁₂₇

#548

semidirect

59. C₄×Q₈

D₁₂₇

semidirect

78. Q₈

D₅₀₈

semidirect

Subgroups — #22550538

Conjugacy classes of subgroups

Complements