Group #22549735 — subgroups

Subgroups — #22549735

Order 8128.

234 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 79 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×C₄×D₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₂₇⋊C₄	C₂	1	2	yes	maximal, complemented
3	C₂×(C₂×D₁₂₇)⋊C₄	C₂	1	2	yes	maximal
4	C₁₂₇⋊#95	C₂	1	1	yes	maximal, characteristic, complemented
5	#95×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₁₂₇	C₂²	1	1	yes	characteristic
7	C₂²×Dic₁₂₇	C₂²	1	2	yes	—
8	C₂²×Dic₁₂₇	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₁₂₇	—	2	4	no	—
10	C₂×C₄×D₁₂₇	C₄	1	4	yes	complemented
11	Dic₁₂₇⋊C₄	—	2	4	no	—
12	(C₂×D₁₂₇)⋊C₄	—	2	4	no	—
13	C₂²×C₄×C₁₂₇	C₂²	1	2	yes	—
14	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂²×D₁₂₇	—	2	2	no	—
16	C₂²×D₁₂₇	—	2	2	no	—
17	C₂²×D₁₂₇	C₂×C₄	1	2	yes	—
18	C₂²×D₁₂₇	D₄	1	2	yes	—
19	C₂²×D₁₂₇	Q₈	1	2	yes	—
20	C₂×Dic₁₂₇	—	2	2	no	—
21	C₂×Dic₁₂₇	—	2	2	no	—
22	C₂×Dic₁₂₇	—	2	2	no	—
23	C₂×Dic₁₂₇	C₂×C₄	1	2	yes	—
24	C₂×Dic₁₂₇	D₄	1	4	yes	—
25	C₄×D₁₂₇	—	2	8	no	—
26	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
27	C₂×C₄×C₁₂₇	—	2	2	no	—
28	C₂×C₄×C₁₂₇	—	2	2	no	—
29	C₂×C₄×C₁₂₇	—	2	2	no	—
30	C₂×C₄×C₁₂₇	—	2	2	no	—
31	C₂×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
32	C₂²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₁₂₇	C₂×D₄	1	2	yes	—
34	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₁₂₇	#41	1	2	yes	—
37	C₂×D₁₂₇	—	2	2	no	—
38	C₂×D₁₂₇	—	2	2	no	—
39	C₂×D₁₂₇	—	2	4	no	—
40	C₂×D₁₂₇	—	2	4	no	—
41	C₂×D₁₂₇	C₄⋊C₄	1	4	yes	complemented
42	C₄×C₁₂₇	—	2	2	no	—
43	C₄×C₁₂₇	—	4	2	no	—
44	Dic₁₂₇	—	4	2	no	—
45	Dic₁₂₇	C₂²⋊C₄	1	4	yes	complemented
46	C₂×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₁₂₇	#117	1	1	yes	characteristic
49	C₂×C₁₂₇	#121	1	2	yes	—
50	C₂×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
51	D₁₂₇	—	2	4	no	—
52	C₁₂₇	#380	1	1	yes	characteristic, complemented, Sylow
53	#380	—	127	1	no	maximal, Sylow
54	C₂³×C₄	—	127	1	no	—
55	C₂×C₄⋊C₄	—	127	2	no	—
56	C₂×C₂²⋊C₄	—	127	2	no	—
57	#95	—	127	1	no	—
58	#95	D₁₂₇	1	1	yes	characteristic, complemented
59	C₂⁴	—	127	1	no	—
60	C₂²×C₄	—	127	1	no	—
61	C₂²×C₄	—	127	4	no	—
62	C₂²×C₄	—	127	2	no	—
63	C₂²×C₄	—	254	4	no	—
64	C₂²×C₄	C₂×D₁₂₇	1	2	yes	—
65	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
66	C₄⋊C₄	—	254	4	no	—
67	C₂²⋊C₄	—	254	4	no	—
68	C₂³	—	127	2	no	—
69	C₂³	—	127	2	no	—
70	C₂³	—	127	2	no	—
71	C₂³	—	254	2	no	—
72	C₂³	—	254	2	no	—
73	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	127	2	no	—
79	C₂×C₄	—	127	4	no	—
80	C₂×C₄	—	254	8	no	—
81	C₂×C₄	—	254	2	no	—
82	C₂×C₄	—	254	2	no	—
83	C₂×C₄	—	254	2	no	—
84	C₂×C₄	C₄×D₁₂₇	1	2	yes	—
85	C₂²	—	127	4	no	—
86	C₂²	—	254	4	no	—
87	C₂²	—	254	2	no	—
88	C₂²	—	254	2	no	—
89	C₂²	—	254	4	no	—
90	C₂²	C₂×C₄×D₁₂₇	1	1	yes	characteristic
91	C₂²	D₄×D₁₂₇	1	2	yes	—
92	C₂²	D₄×D₁₂₇	1	1	yes	characteristic
93	C₂²	Q₈×D₁₂₇	1	1	yes	characteristic
94	C₂²	#14523248	1	2	yes	—
95	C₄	—	2	2	no	—
96	C₄	—	4	2	no	—
97	C₄	—	127	4	no	—
98	C₄	—	508	2	no	—
99	C₂	—	254	4	no	—
100	C₂	C₂²⋊C₄×D₁₂₇	1	1	yes	characteristic
101	C₂	C₄⋊C₄×D₁₂₇	1	1	yes	characteristic
102	C₂	#16861166	1	1	yes	characteristic
103	C₂	#16861182	1	2	yes	—
104	C₂	#16861187	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄×D₁₂₇

C₂

yes

maximal, characteristic

C₂×Dic₁₂₇⋊C₄

C₂

yes

maximal, complemented

C₂×(C₂×D₁₂₇)⋊C₄

C₂

yes

maximal

C₁₂₇⋊#95

C₂

yes

maximal, characteristic, complemented

#95×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₂₇

C₂²

yes

characteristic

C₂²×Dic₁₂₇

C₂²

yes

—

C₂²×Dic₁₂₇

C₂²

yes

characteristic

C₂×C₄×D₁₂₇

—

C₂×C₄×D₁₂₇

C₄

yes

complemented

Dic₁₂₇⋊C₄

—

(C₂×D₁₂₇)⋊C₄

—

C₂²×C₄×C₁₂₇

C₂²

yes

—

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

C₂×C₄

yes

—

C₂²×D₁₂₇

D₄

yes

—

C₂²×D₁₂₇

Q₈

yes

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

C₂×C₄

yes

—

C₂×Dic₁₂₇

D₄

yes

—

C₄×D₁₂₇

—

C₂³×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×C₄

yes

—

C₂²×C₁₂₇

C₂²×C₄

yes

characteristic

C₂²×C₁₂₇

C₂×D₄

yes

—

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₂²×C₁₂₇

C₂×Q₈

yes

characteristic

C₂²×C₁₂₇

#41

yes

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

C₄⋊C₄

yes

complemented

C₄×C₁₂₇

—

C₄×C₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

C₂²⋊C₄

yes

complemented

C₂×C₁₂₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₁₂₇

C₂×C₄⋊C₄

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

C₂×C₁₂₇

C₂²⋊Q₈

yes

—

D₁₂₇

—

C₁₂₇

#380

yes

characteristic, complemented, Sylow

#380

—

127

maximal, Sylow

C₂³×C₄

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₂²⋊C₄

—

127

—

#95

—

127

—

#95

D₁₂₇

yes

characteristic, complemented

C₂⁴

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂×D₁₂₇

yes

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

—

254

—

C₂²⋊C₄

—

254

—

C₂³

—

127

—

C₂³

—

127

—

C₂³

—

127

—

C₂³

—

254

—

C₂³

—

254

—

C₂³

C₂²×D₁₂₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₄×D₁₂₇

yes

—

C₂²

—

127

—

C₂²

—

254

—

C₂²

—

254

—

C₂²

—

254

—

C₂²

—

254

—

C₂²

C₂×C₄×D₁₂₇

yes

characteristic

C₂²

D₄×D₁₂₇

yes

—

C₂²

D₄×D₁₂₇

yes

characteristic

C₂²

Q₈×D₁₂₇

yes

characteristic

C₂²

#14523248

yes

—

C₄

—

C₄

—

C₄

—

127

—

C₄

—

508

—

C₂

—

254

—

100

C₂

C₂²⋊C₄×D₁₂₇

yes

characteristic

101

C₂

C₄⋊C₄×D₁₂₇

yes

characteristic

102

C₂

#16861166

yes

characteristic

103

C₂

#16861182

yes

—

104

C₂

#16861187

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₂₇
commutator	C₂²×C₁₂₇	C₂²×C₄
Fitting	#95×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂³
hypercenter	#95	D₁₂₇
hypocenter	C₁₂₇	#380
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ #95

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₂₇

commutator

C₂²×C₁₂₇

C₂²×C₄

Fitting

#95×C₁₂₇

C₂

socle

C₂³×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂³ ◁ #95

N	H ≅ G/N	type
2. C₂×Dic₁₂₇⋊C₄	C₂	semidirect
4. C₁₂₇⋊#95	C₂	semidirect
5. #95×C₁₂₇	C₂	semidirect
10. C₂×C₄×D₁₂₇	C₄	semidirect
41. C₂×D₁₂₇	C₄⋊C₄	semidirect
45. Dic₁₂₇	C₂²⋊C₄	semidirect
52. C₁₂₇	#380	semidirect
58. #95	D₁₂₇	semidirect

H ≅ G/N

type

2. C₂×Dic₁₂₇⋊C₄

C₂

semidirect

4. C₁₂₇⋊#95

C₂

semidirect

5. #95×C₁₂₇

C₂

semidirect

10. C₂×C₄×D₁₂₇

C₄

semidirect

41. C₂×D₁₂₇

C₄⋊C₄

semidirect

45. Dic₁₂₇

C₂²⋊C₄

semidirect

52. C₁₂₇

#380

semidirect

58. #95

D₁₂₇

semidirect

Subgroups — #22549735

Conjugacy classes of subgroups

Complements