Group #22549812 — subgroups

Subgroups — #22549812

Order 8128.

186 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 63 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×C₄⋊Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×Dic₁₂₇⋊C₄	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊Dic₁₂₇	C₂	1	2	yes	maximal
4	C₁₂₇⋊#95	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂²⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₂₇	C₂²	1	2	yes	—
7	C₂²×Dic₁₂₇	C₂²	1	2	yes	—
8	C₄⋊Dic₁₂₇	—	2	2	no	—
9	Dic₁₂₇⋊C₄	—	2	4	no	—
10	C₂²⋊Dic₁₂₇	—	2	4	no	—
11	C₂⁴×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
15	C₂×Dic₁₂₇	—	2	2	no	—
16	C₂×Dic₁₂₇	—	2	2	no	—
17	C₂×Dic₁₂₇	—	2	2	no	—
18	C₂×Dic₁₂₇	—	2	2	no	—
19	C₂×Dic₁₂₇	—	2	2	no	—
20	C₂×Dic₁₂₇	D₄	1	4	yes	—
21	C₂³×C₁₂₇	—	2	1	no	—
22	C₂³×C₁₂₇	—	2	2	no	—
23	C₂³×C₁₂₇	—	2	2	no	—
24	C₂³×C₁₂₇	—	2	1	no	—
25	C₂³×C₁₂₇	Q₈	1	2	yes	—
26	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
27	C₂×C₄×C₁₂₇	—	2	2	no	—
28	C₂×C₄×C₁₂₇	—	2	1	no	—
29	C₂×C₄×C₁₂₇	—	2	2	no	—
30	C₂×C₄×C₁₂₇	D₄	1	2	yes	—
31	C₂²×C₁₂₇	—	2	4	no	—
32	C₂²×C₁₂₇	—	2	2	no	—
33	C₂²×C₁₂₇	—	4	1	no	—
34	C₂²×C₁₂₇	—	4	2	no	—
35	C₂²×C₁₂₇	—	4	1	no	—
36	C₂²×C₁₂₇	C₂×D₄	1	2	yes	—
37	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₁₂₇	#41	1	2	yes	—
40	C₂²×C₁₂₇	#41	1	1	yes	characteristic
41	C₄×C₁₂₇	—	2	2	no	—
42	C₄×C₁₂₇	—	4	1	no	—
43	Dic₁₂₇	—	2	4	no	—
44	Dic₁₂₇	—	4	2	no	—
45	C₂×C₁₂₇	—	4	2	no	—
46	C₂×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
47	C₂×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
48	C₂×C₁₂₇	#123	1	1	yes	characteristic
49	C₂×C₁₂₇	#123	1	2	yes	—
50	C₂×C₁₂₇	#127	1	1	yes	characteristic
51	C₁₂₇	#399	1	1	yes	characteristic, complemented, Sylow
52	#399	—	127	1	no	maximal, Sylow
53	C₂×C₄⋊C₄	—	127	2	no	—
54	C₂×C₄⋊C₄	—	127	1	no	—
55	C₂×C₂²⋊C₄	—	127	2	no	—
56	C₂×C₂²⋊C₄	D₁₂₇	1	1	yes	characteristic
57	#95	—	127	1	no	—
58	C₂⁴	C₂×D₁₂₇	1	1	yes	characteristic
59	C₂²×C₄	—	127	2	no	—
60	C₂²×C₄	—	127	2	no	—
61	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
62	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
63	C₄⋊C₄	—	254	4	no	—
64	C₄⋊C₄	—	254	2	no	—
65	C₂²⋊C₄	—	2	2	no	—
66	C₂²⋊C₄	—	254	4	no	—
67	C₂³	—	2	2	no	—
68	C₂³	—	2	1	no	—
69	C₂³	—	2	2	no	—
70	C₂³	—	2	1	no	—
71	C₂³	Dic₂₅₄	1	2	yes	—
72	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	127	4	no	—
77	C₂×C₄	—	254	2	no	—
78	C₂×C₄	—	254	2	no	—
79	C₂×C₄	—	254	2	no	—
80	C₂×C₄	—	254	2	no	—
81	C₂×C₄	—	254	2	no	—
82	C₂×C₄	C₂²⋊D₁₂₇	1	2	yes	—
83	C₂²	—	2	4	no	—
84	C₂²	—	2	2	no	—
85	C₂²	—	4	1	no	—
86	C₂²	—	4	2	no	—
87	C₂²	—	4	1	no	—
88	C₂²	D₄×D₁₂₇	1	2	yes	—
89	C₂²	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
90	C₂²	C₂×Dic₂₅₄	1	1	yes	characteristic
91	C₂²	#14523247	1	2	yes	—
92	C₂²	#14523248	1	1	yes	characteristic
93	C₄	—	2	2	no	—
94	C₄	—	4	1	no	—
95	C₄	—	508	2	no	—
96	C₄	—	254	4	no	—
97	C₂	—	4	2	no	—
98	C₂	#16861186	1	2	yes	—
99	C₂	#16861188	1	1	yes	characteristic
100	C₂	#16861190	1	1	yes	characteristic
101	C₂	#16861191	1	2	yes	—
102	C₂	#16861205	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Dic₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₂₇⋊C₄

C₂

yes

maximal, complemented

C₂×C₂²⋊Dic₁₂₇

C₂

yes

maximal

C₁₂₇⋊#95

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊C₄×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₇

C₂²

yes

—

C₂²×Dic₁₂₇

C₂²

yes

—

C₄⋊Dic₁₂₇

—

Dic₁₂₇⋊C₄

—

C₂²⋊Dic₁₂₇

—

C₂⁴×C₁₂₇

C₂²

yes

characteristic

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₂²⋊C₄×C₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

D₄

yes

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

Q₈

yes

—

C₂³×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

D₄

yes

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×D₄

yes

—

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₂²×C₁₂₇

C₂×Q₈

yes

characteristic

C₂²×C₁₂₇

#41

yes

—

C₂²×C₁₂₇

#41

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂²⋊Q₈

yes

—

C₂×C₁₂₇

C₂²⋊Q₈

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

—

C₂×C₁₂₇

#127

yes

characteristic

C₁₂₇

#399

yes

characteristic, complemented, Sylow

#399

—

127

maximal, Sylow

C₂×C₄⋊C₄

—

127

—

C₂×C₄⋊C₄

—

127

—

C₂×C₂²⋊C₄

—

127

—

C₂×C₂²⋊C₄

D₁₂₇

yes

characteristic

#95

—

127

—

C₂⁴

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

—

127

—

C₂²×C₄

—

127

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

—

254

—

C₄⋊C₄

—

254

—

C₂²⋊C₄

—

C₂²⋊C₄

—

254

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

Dic₂₅₄

yes

—

C₂³

C₂²×D₁₂₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂²⋊D₁₂₇

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

D₄×D₁₂₇

yes

—

C₂²

C₂×C₂²⋊D₁₂₇

yes

characteristic

C₂²

C₂×Dic₂₅₄

yes

characteristic

C₂²

#14523247

yes

—

C₂²

#14523248

yes

characteristic

C₄

—

C₄

—

C₄

—

508

—

C₄

—

254

—

C₂

—

C₂

#16861186

yes

—

C₂

#16861188

yes

characteristic

100

C₂

#16861190

yes

characteristic

101

C₂

#16861191

yes

—

102

C₂

#16861205

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₂₇
commutator, socle	C₂³×C₁₂₇	C₂³
Fitting	C₂×C₂²⋊C₄×C₁₂₇	C₂
hypercenter	C₂×C₂²⋊C₄	D₁₂₇
hypocenter	C₁₂₇	#399
LCS	G ▷ C₂³×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₂₇

commutator, socle

C₂³×C₁₂₇

C₂³

Fitting

C₂×C₂²⋊C₄×C₁₂₇

C₂

hypercenter

C₂×C₂²⋊C₄

D₁₂₇

C₁₂₇

G ▷ C₂³×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂³ ◁ C₂×C₂²⋊C₄

N	H ≅ G/N	type
1. C₂×C₄⋊Dic₁₂₇	C₂	semidirect
2. C₂×Dic₁₂₇⋊C₄	C₂	semidirect
4. C₁₂₇⋊#95	C₂	semidirect
51. C₁₂₇	#399	semidirect

H ≅ G/N

type

1. C₂×C₄⋊Dic₁₂₇

C₂

semidirect

2. C₂×Dic₁₂₇⋊C₄

C₂

semidirect

4. C₁₂₇⋊#95

C₂

semidirect

51. C₁₂₇

#399

semidirect

Subgroups — #22549812

Conjugacy classes of subgroups

Complements