Group #22550269 — subgroups

Subgroups — #22550269

Order 8128.

124 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 43 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×#14523238	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14523237	C₂	1	1	yes	maximal, characteristic
3	C₂×#14523235	C₂	1	1	yes	maximal, characteristic
4	#16861207	C₂	1	1	yes	maximal, characteristic, complemented
5	#16861198	C₂	1	1	yes	maximal, characteristic
6	#16861119	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×Q₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₅₄	C₂²	1	1	yes	characteristic
9	C₂×D₅₀₈	C₂²	1	1	yes	characteristic
10	#14523238	—	2	2	no	—
11	#14523237	—	2	2	no	—
12	#14523235	—	2	2	no	—
13	C₂×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
14	Dic₁₂₇⋊C₄	—	2	2	no	—
15	(C₂×D₁₂₇)⋊C₄	—	2	2	no	—
16	C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
19	Q₁₆×C₁₂₇	—	2	2	no	—
20	C₂×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₂²×D₁₂₇	—	2	1	no	—
22	Dic₂₅₄	—	2	1	no	—
23	Dic₂₅₄	—	2	1	no	—
24	D₅₀₈	—	2	1	no	—
25	D₅₀₈	—	2	1	no	—
26	C₂×Dic₁₂₇	—	2	1	no	—
27	C₂×Dic₁₂₇	D₄	1	2	yes	—
28	C₁₂₇⋊C₈	D₄	1	2	yes	—
29	Q₈×C₁₂₇	—	2	1	no	—
30	Q₈×C₁₂₇	—	2	1	no	—
31	Q₈×C₁₂₇	—	2	1	no	—
32	Q₈×C₁₂₇	—	2	1	no	—
33	C₂×C₄×C₁₂₇	—	2	1	no	—
34	C₂×C₄×C₁₂₇	—	2	1	no	—
35	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
36	C₈×C₁₂₇	D₄	1	2	yes	—
37	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
38	C₂×D₁₂₇	—	4	1	no	—
39	C₂×D₁₂₇	—	4	2	no	—
40	C₄×C₁₂₇	—	4	1	no	—
41	C₄×C₁₂₇	—	4	1	no	—
42	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
43	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
44	Dic₁₂₇	—	2	2	no	—
45	Dic₁₂₇	—	4	1	no	—
46	C₂×C₁₂₇	#127	1	1	yes	characteristic
47	C₂×C₁₂₇	#137	1	2	yes	—
48	D₁₂₇	—	8	1	no	—
49	C₁₂₇	#497	1	1	yes	characteristic, complemented, Sylow
50	#497	—	127	1	no	maximal, Sylow
51	C₂×Q₁₆	—	127	1	no	—
52	C₂×Q₁₆	D₁₂₇	1	1	yes	characteristic, complemented
53	C₂×SD₁₆	—	127	1	no	—
54	C₂×SD₁₆	—	127	1	no	—
55	C₄⋊Q₈	—	127	1	no	—
56	#124	—	127	1	no	—
57	#97	—	127	1	no	—
58	C₂×Q₈	—	127	1	no	—
59	C₂×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
60	C₂×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
61	C₂×D₄	—	127	1	no	—
62	Q₁₆	—	2	2	no	—
63	Q₁₆	—	254	2	no	—
64	SD₁₆	—	254	2	no	—
65	SD₁₆	—	254	2	no	—
66	C₂×C₈	—	127	1	no	—
67	C₂×C₈	C₂×D₁₂₇	1	1	yes	characteristic
68	C₄⋊C₄	—	254	2	no	—
69	C₂²⋊C₄	—	254	2	no	—
70	C₄²	—	127	1	no	—
71	C₂³	—	254	1	no	—
72	Q₈	—	2	1	no	—
73	Q₈	—	2	1	no	—
74	Q₈	—	2	1	no	—
75	Q₈	—	2	1	no	—
76	Q₈	—	254	1	no	—
77	Q₈	—	254	1	no	—
78	D₄	—	254	1	no	—
79	D₄	—	254	1	no	—
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	127	2	no	—
83	C₂×C₄	—	254	1	no	—
84	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
85	C₈	—	127	2	no	—
86	C₈	C₂²⋊D₁₂₇	1	2	yes	—
87	C₂²	—	508	1	no	—
88	C₂²	—	508	2	no	—
89	C₂²	D₄×D₁₂₇	1	1	yes	characteristic
90	C₄	—	4	1	no	—
91	C₄	—	4	1	no	—
92	C₄	—	508	1	no	—
93	C₄	—	254	2	no	—
94	C₄	D₄×D₁₂₇	1	1	yes	characteristic
95	C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
96	C₂	—	1016	1	no	—
97	C₂	#16861205	1	1	yes	characteristic
98	C₂	#16861241	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523238

C₂

yes

maximal, characteristic, complemented

C₂×#14523237

C₂

yes

maximal, characteristic

C₂×#14523235

C₂

yes

maximal, characteristic

#16861207

C₂

yes

maximal, characteristic, complemented

#16861198

C₂

yes

maximal, characteristic

#16861119

C₂

yes

maximal, characteristic, complemented

C₂×Q₁₆×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₅₄

C₂²

yes

characteristic

C₂×D₅₀₈

C₂²

yes

—

—

—

C₂×C₁₂₇⋊C₈

C₂²

yes

characteristic

Dic₁₂₇⋊C₄

—

(C₂×D₁₂₇)⋊C₄

—

C₄×Dic₁₂₇

C₂²

yes

characteristic

C₂×Q₈×C₁₂₇

C₂²

yes

characteristic

C₂×Q₈×C₁₂₇

C₂²

yes

characteristic

Q₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂²

yes

characteristic

C₂²×D₁₂₇

—

Dic₂₅₄

—

Dic₂₅₄

—

D₅₀₈

—

D₅₀₈

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

D₄

yes

—

C₁₂₇⋊C₈

D₄

yes

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

Q₈×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₈×C₁₂₇

D₄

yes

—

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

characteristic

C₄×C₁₂₇

C₂×D₄

yes

characteristic

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

#127

yes

characteristic

C₂×C₁₂₇

#137

yes

—

D₁₂₇

—

C₁₂₇

#497

yes

characteristic, complemented, Sylow

#497

—

127

maximal, Sylow

C₂×Q₁₆

—

127

—

C₂×Q₁₆

D₁₂₇

yes

characteristic, complemented

C₂×SD₁₆

—

127

—

C₂×SD₁₆

—

127

—

C₄⋊Q₈

—

127

—

#124

—

127

—

#97

—

127

—

C₂×Q₈

—

127

—

C₂×Q₈

C₂×D₁₂₇

yes

characteristic

C₂×Q₈

C₂×D₁₂₇

yes

characteristic

C₂×D₄

—

127

—

Q₁₆

—

Q₁₆

—

254

—

SD₁₆

—

254

—

SD₁₆

—

254

—

C₂×C₈

—

127

—

C₂×C₈

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

—

254

—

C₂²⋊C₄

—

254

—

C₄²

—

127

—

C₂³

—

254

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

254

—

Q₈

—

254

—

D₄

—

254

—

D₄

—

254

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₈

—

127

—

C₈

C₂²⋊D₁₂₇

yes

—

C₂²

—

508

—

C₂²

—

508

—

C₂²

D₄×D₁₂₇

yes

characteristic

C₄

—

C₄

—

C₄

—

508

—

C₄

—

254

—

C₄

D₄×D₁₂₇

yes

characteristic

C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

C₂

—

1016

—

C₂

#16861205

yes

characteristic

C₂

#16861241

yes

—

	subgroup	G/H
center	C₂²	D₄×D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	C₂×Q₁₆×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	C₂×Q₁₆	D₁₂₇
hypocenter	C₁₂₇	#497
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

subgroup

G/H

center

C₂²

D₄×D₁₂₇

commutator

C₂×C₄×C₁₂₇

C₂³

Frattini

C₂×C₄

C₂²×D₁₂₇

Fitting

C₂×Q₁₆×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×D₄

hypercenter

C₂×Q₁₆

D₁₂₇

hypocenter

C₁₂₇

#497

LCS

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

N	H ≅ G/N	type
1. C₂×#14523238	C₂	semidirect
4. #16861207	C₂	semidirect
6. #16861119	C₂	semidirect
7. C₂×Q₁₆×C₁₂₇	C₂	semidirect
49. C₁₂₇	#497	semidirect
52. C₂×Q₁₆	D₁₂₇	semidirect

H ≅ G/N

type

1. C₂×#14523238

C₂

semidirect

4. #16861207

C₂

semidirect

6. #16861119

C₂

semidirect

7. C₂×Q₁₆×C₁₂₇

C₂

semidirect

49. C₁₂₇

#497

semidirect

52. C₂×Q₁₆

D₁₂₇

semidirect

Subgroups — #22550269

Conjugacy classes of subgroups

Complements