Group #22550450 — subgroups

Subgroups — #22550450

Order 8128.

270 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 111 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×#14523248	C₂	1	1	yes	maximal, characteristic
2	#16861207	C₂	1	2	yes	maximal, complemented
3	#16861198	C₂	1	2	yes	maximal, complemented
4	#16861192	C₂	1	4	yes	maximal, complemented
5	#16861185	C₂	1	4	yes	maximal, complemented
6	#16861168	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14523248	—	2	4	no	—
9	C₂×Dic₂₅₄	C₂²	1	1	yes	characteristic
10	C₂×D₅₀₈	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊D₁₂₇	C₂²	1	2	yes	—
12	C₂×C₄×D₁₂₇	C₂²	1	2	yes	—
13	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
14	Dic₁₂₇⋊C₄	C₂²	1	8	yes	—
15	C₂²⋊Dic₁₂₇	C₂²	1	2	yes	—
16	(C₂×D₁₂₇)⋊C₄	C₂²	1	8	yes	—
17	C₄×Dic₁₂₇	C₂²	1	2	yes	—
18	C₂×Q₈×C₁₂₇	—	2	4	no	—
19	C₂×Q₈×C₁₂₇	C₂²	1	2	yes	—
20	C₂×Q₈×C₁₂₇	C₂²	1	2	yes	—
21	C₂²×C₄×C₁₂₇	C₂²	1	2	yes	—
22	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₂²×D₁₂₇	C₂³	1	2	yes	—
24	Dic₂₅₄	—	2	2	no	—
25	D₅₀₈	—	2	2	no	—
26	C₂²⋊D₁₂₇	—	2	4	no	—
27	C₂×Dic₁₂₇	C₂³	1	4	yes	—
28	C₂×Dic₁₂₇	C₂³	1	2	yes	—
29	C₄×D₁₂₇	—	2	4	no	—
30	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
31	Q₈×C₁₂₇	—	2	4	no	—
32	Q₈×C₁₂₇	—	2	4	no	—
33	C₂×C₄×C₁₂₇	—	2	4	no	—
34	C₂×C₄×C₁₂₇	D₄	1	4	yes	—
35	C₂×C₄×C₁₂₇	C₂³	1	4	yes	—
36	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
38	C₂²×C₁₂₇	—	2	2	no	—
39	C₂²×C₁₂₇	C₂×D₄	1	2	yes	—
40	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
41	C₂×D₁₂₇	—	2	2	no	—
42	C₂×D₁₂₇	—	2	4	no	—
43	C₄×C₁₂₇	—	2	4	no	—
44	C₄×C₁₂₇	C₂×D₄	1	2	yes	—
45	C₄×C₁₂₇	C₂×D₄	1	2	yes	—
46	Dic₁₂₇	—	2	2	no	—
47	Dic₁₂₇	—	2	4	no	—
48	C₂×C₁₂₇	—	2	2	no	—
49	C₂×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
50	C₂×C₁₂₇	2₋¹⁺⁴	1	2	yes	—
51	D₁₂₇	—	4	2	no	—
52	C₁₂₇	#536	1	1	yes	characteristic, complemented, Sylow
53	#536	—	127	1	no	maximal, Sylow
54	C₂×#41	—	127	1	no	—
55	C₂²×Q₈	D₁₂₇	1	1	yes	characteristic, complemented
56	C₄⋊Q₈	—	127	2	no	—
57	#124	—	127	2	no	—
58	#123	—	127	4	no	—
59	C₂²⋊Q₈	—	127	4	no	—
60	#117	—	127	1	no	—
61	#41	—	254	4	no	—
62	C₂×Q₈	—	2	4	no	—
63	C₂×Q₈	—	127	1	no	—
64	C₂×Q₈	C₂×D₁₂₇	1	2	yes	—
65	C₂×Q₈	C₂×D₁₂₇	1	2	yes	—
66	C₂×D₄	—	127	2	no	—
67	C₂×D₄	—	127	1	no	—
68	C₂²×C₄	—	127	2	no	—
69	C₂²×C₄	C₂×D₁₂₇	1	2	yes	—
70	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
71	C₄⋊C₄	—	127	8	no	—
72	C₄⋊C₄	—	127	2	no	—
73	C₂²⋊C₄	—	127	2	no	—
74	C₂²⋊C₄	—	127	8	no	—
75	C₄²	—	127	2	no	—
76	C₂³	—	127	2	no	—
77	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
78	Q₈	—	2	4	no	—
79	Q₈	—	2	4	no	—
80	Q₈	—	254	2	no	—
81	D₄	—	254	4	no	—
82	D₄	—	254	2	no	—
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	—	127	4	no	—
85	C₂×C₄	—	127	2	no	—
86	C₂×C₄	—	254	4	no	—
87	C₂×C₄	C₂²⋊D₁₂₇	1	4	yes	—
88	C₂×C₄	C₂²×D₁₂₇	1	4	yes	—
89	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
90	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
91	C₂²	—	2	2	no	—
92	C₂²	—	254	2	no	—
93	C₂²	—	254	4	no	—
94	C₂²	C₂×C₂²⋊D₁₂₇	1	2	yes	—
95	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
96	C₄	—	2	4	no	—
97	C₄	—	254	2	no	—
98	C₄	—	254	4	no	—
99	C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
100	C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
101	C₂	—	2	2	no	—
102	C₂	—	508	2	no	—
103	C₂	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
104	C₂	#16861257	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523248

C₂

yes

maximal, characteristic

#16861207

C₂

yes

maximal, complemented

#16861198

C₂

yes

maximal, complemented

#16861192

C₂

yes

maximal, complemented

#16861185

C₂

yes

maximal, complemented

#16861168

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₁₂₇

C₂

yes

maximal, characteristic, complemented

#14523248

—

C₂×Dic₂₅₄

C₂²

yes

characteristic

C₂×D₅₀₈

C₂²

yes

characteristic

C₂×C₂²⋊D₁₂₇

C₂²

yes

—

C₂×C₄×D₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

C₂²

yes

—

Dic₁₂₇⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₂₇

C₂²

yes

—

(C₂×D₁₂₇)⋊C₄

C₂²

yes

—

C₄×Dic₁₂₇

C₂²

yes

—

C₂×Q₈×C₁₂₇

—

C₂×Q₈×C₁₂₇

C₂²

yes

—

C₂×Q₈×C₁₂₇

C₂²

yes

—

C₂²×C₄×C₁₂₇

C₂²

yes

—

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₂²×D₁₂₇

C₂³

yes

—

Dic₂₅₄

—

D₅₀₈

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

C₂³

yes

—

C₂×Dic₁₂₇

C₂³

yes

—

C₄×D₁₂₇

—

C₂³×C₁₂₇

C₂³

yes

characteristic

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

D₄

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×D₄

yes

—

C₂²×C₁₂₇

C₂⁴

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

—

C₄×C₁₂₇

C₂×D₄

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂²×D₄

yes

characteristic

C₂×C₁₂₇

2₋¹⁺⁴

yes

—

D₁₂₇

—

C₁₂₇

#536

yes

characteristic, complemented, Sylow

#536

—

127

maximal, Sylow

C₂×#41

—

127

—

C₂²×Q₈

D₁₂₇

yes

characteristic, complemented

C₄⋊Q₈

—

127

—

#124

—

127

—

#123

—

127

—

C₂²⋊Q₈

—

127

—

#117

—

127

—

#41

—

254

—

C₂×Q₈

—

C₂×Q₈

—

127

—

C₂×Q₈

C₂×D₁₂₇

yes

—

C₂×Q₈

C₂×D₁₂₇

yes

—

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₂×D₁₂₇

yes

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

—

127

—

C₄⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₂²⋊C₄

—

127

—

C₄²

—

127

—

C₂³

—

127

—

C₂³

C₂²×D₁₂₇

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

254

—

D₄

—

254

—

D₄

—

254

—

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

C₂²⋊D₁₂₇

yes

—

C₂×C₄

C₂²×D₁₂₇

yes

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₂²

—

C₂²

—

254

—

C₂²

—

254

—

C₂²

C₂×C₂²⋊D₁₂₇

yes

—

C₂²

C₂³×D₁₂₇

yes

characteristic

C₄

—

C₄

—

254

—

C₄

—

254

—

C₄

C₂×C₂²⋊D₁₂₇

yes

—

100

C₄

C₂×C₂²⋊D₁₂₇

yes

—

101

C₂

—

102

C₂

—

508

—

103

C₂

C₂²×C₂²⋊D₁₂₇

yes

characteristic

104

C₂

#16861257

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂⁴
Fitting	C₂²×Q₈×C₁₂₇	C₂
hypercenter	C₂²×Q₈	D₁₂₇
hypocenter	C₁₂₇	#536
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂⁴

Fitting

C₂²×Q₈×C₁₂₇

C₂

hypercenter

C₂²×Q₈

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂² ◁ C₂²×Q₈

N	H ≅ G/N	type
2. #16861207	C₂	semidirect
3. #16861198	C₂	semidirect
4. #16861192	C₂	semidirect
5. #16861185	C₂	semidirect
6. #16861168	C₂	semidirect
7. C₂²×Q₈×C₁₂₇	C₂	semidirect
52. C₁₂₇	#536	semidirect
55. C₂²×Q₈	D₁₂₇	semidirect

H ≅ G/N

type

2. #16861207

C₂

semidirect

3. #16861198

C₂

semidirect

4. #16861192

C₂

semidirect

5. #16861185

C₂

semidirect

6. #16861168

C₂

semidirect

7. C₂²×Q₈×C₁₂₇

C₂

semidirect

52. C₁₂₇

#536

semidirect

55. C₂²×Q₈

D₁₂₇

semidirect

Subgroups — #22550450

Conjugacy classes of subgroups

Complements