Group #22549877 — subgroups

Subgroups — #22549877

Order 8128.

162 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 71 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×#14523232	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄⋊Dic₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#16861133	C₂	1	2	yes	maximal, complemented
4	#16861130	C₂	1	2	yes	maximal, complemented
5	C₂×#41×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₂₇	—	2	1	no	—
7	#14523232	—	2	2	no	—
8	C₂×C₁₂₇⋊C₈	C₂²	1	2	yes	—
9	C₄⋊Dic₁₂₇	—	2	1	no	—
10	C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
11	#41×C₁₂₇	—	2	2	no	—
12	#41×C₁₂₇	C₄	1	4	yes	complemented
13	C₂×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×D₄×C₁₂₇	—	2	1	no	—
15	C₂×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₁₂₇	—	2	1	no	—
17	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂×Dic₁₂₇	—	2	2	no	—
19	C₂×Dic₁₂₇	—	4	2	no	—
20	C₁₂₇⋊C₈	—	2	2	no	—
21	C₂³×C₁₂₇	—	2	1	no	—
22	C₂³×C₁₂₇	D₄	1	1	yes	characteristic
23	Q₈×C₁₂₇	—	2	1	no	—
24	Q₈×C₁₂₇	C₂×C₄	1	2	yes	complemented
25	D₄×C₁₂₇	—	2	4	no	—
26	D₄×C₁₂₇	—	2	1	no	—
27	D₄×C₁₂₇	C₂×C₄	1	2	yes	complemented
28	C₂×C₄×C₁₂₇	—	2	1	no	—
29	C₂×C₄×C₁₂₇	—	2	4	no	—
30	C₂×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
31	C₂×C₄×C₁₂₇	D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₂₇	D₄	1	2	yes	—
33	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
34	C₂²×C₁₂₇	—	2	1	no	—
35	C₂²×C₁₂₇	—	2	1	no	—
36	C₂²×C₁₂₇	—	2	2	no	—
37	C₂²×C₁₂₇	—	4	1	no	—
38	C₂²×C₁₂₇	—	4	1	no	—
39	C₂²×C₁₂₇	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
41	C₄×C₁₂₇	—	2	2	no	—
42	C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
43	C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
44	C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
45	Dic₁₂₇	—	4	2	no	—
46	C₂×C₁₂₇	—	2	2	no	—
47	C₂×C₁₂₇	—	4	2	no	—
48	C₂×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₂×C₁₂₇	#137	1	1	yes	characteristic
51	C₁₂₇	#417	1	1	yes	characteristic, complemented, Sylow
52	#417	—	127	1	no	maximal, Sylow
53	C₂×#41	D₁₂₇	1	1	yes	characteristic
54	C₂×M₁₆	—	127	1	no	—
55	C₂×C₄⋊C₄	—	127	1	no	—
56	#103	—	127	2	no	—
57	#102	—	127	2	no	—
58	#41	—	2	2	no	—
59	#41	Dic₁₂₇	1	4	yes	complemented
60	C₂×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
61	C₂×D₄	—	2	1	no	—
62	C₂×D₄	C₂×D₁₂₇	1	1	yes	characteristic
63	C₂²×C₄	—	2	1	no	—
64	C₂²×C₄	—	254	1	no	—
65	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
66	M₁₆	—	254	2	no	—
67	C₂×C₈	—	127	2	no	—
68	C₄⋊C₄	—	127	2	no	—
69	C₄⋊C₄	—	254	1	no	—
70	C₂³	—	2	1	no	—
71	C₂³	C₂²⋊D₁₂₇	1	1	yes	characteristic
72	Q₈	—	2	1	no	—
73	Q₈	C₂×Dic₁₂₇	1	2	yes	complemented
74	D₄	—	2	4	no	—
75	D₄	—	2	1	no	—
76	D₄	C₂×Dic₁₂₇	1	2	yes	complemented
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	2	4	no	—
79	C₂×C₄	—	508	2	no	—
80	C₂×C₄	—	254	2	no	—
81	C₂×C₄	C₂×Dic₁₂₇	1	2	yes	—
82	C₂×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
83	C₂×C₄	C₂²⋊D₁₂₇	1	2	yes	—
84	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
85	C₈	—	254	2	no	—
86	C₂²	—	2	1	no	—
87	C₂²	—	2	1	no	—
88	C₂²	—	2	2	no	—
89	C₂²	—	4	1	no	—
90	C₂²	—	4	1	no	—
91	C₂²	C₂²⋊Dic₁₂₇	1	2	yes	—
92	C₂²	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
93	C₄	—	2	2	no	—
94	C₄	—	508	2	no	—
95	C₄	C₂²⋊Dic₁₂₇	1	2	yes	—
96	C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
97	C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
98	C₂	—	2	2	no	—
99	C₂	—	4	2	no	—
100	C₂	C₂×C₂²⋊Dic₁₂₇	1	1	yes	characteristic
101	C₂	#16861234	1	1	yes	characteristic
102	C₂	#16861239	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523232

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊Dic₁₂₇

C₂

yes

maximal, characteristic, complemented

#16861133

C₂

yes

maximal, complemented

#16861130

C₂

yes

maximal, complemented

C₂×#41×C₁₂₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₇

—

#14523232

—

C₂×C₁₂₇⋊C₈

C₂²

yes

—

C₄⋊Dic₁₂₇

—

C₄⋊Dic₁₂₇

C₂²

yes

—

#41×C₁₂₇

—

#41×C₁₂₇

C₄

yes

complemented

C₂×Q₈×C₁₂₇

C₂²

yes

characteristic

C₂×D₄×C₁₂₇

—

C₂×D₄×C₁₂₇

C₂²

yes

characteristic

C₂²×C₄×C₁₂₇

—

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

—

C₂³×C₁₂₇

D₄

yes

characteristic

Q₈×C₁₂₇

—

Q₈×C₁₂₇

C₂×C₄

yes

complemented

D₄×C₁₂₇

—

D₄×C₁₂₇

—

D₄×C₁₂₇

C₂×C₄

yes

complemented

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

—

C₂×C₄×C₁₂₇

C₂×C₄

yes

—

C₂×C₄×C₁₂₇

D₄

yes

characteristic

C₂×C₄×C₁₂₇

D₄

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂²⋊C₄

yes

—

C₂²×C₁₂₇

C₂×D₄

yes

characteristic

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂²⋊C₄

yes

—

C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₄×C₁₂₇

C₂×D₄

yes

characteristic

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂×C₂²⋊C₄

yes

C₂×C₁₂₇

yes

C₂×C₁₂₇

yes

C₁₂₇

yes

characteristic, complemented, Sylow

#417

—

127

maximal, Sylow

C₂×#41

D₁₂₇

yes

characteristic

C₂×M₁₆

—

127

—

C₂×C₄⋊C₄

—

127

—

#103

—

127

—

#102

—

127

—

#41

—

#41

Dic₁₂₇

yes

complemented

C₂×Q₈

C₂×D₁₂₇

yes

characteristic

C₂×D₄

—

C₂×D₄

C₂×D₁₂₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

M₁₆

—

254

—

C₂×C₈

—

127

—

C₄⋊C₄

—

127

—

C₄⋊C₄

—

254

—

C₂³

—

C₂³

C₂²⋊D₁₂₇

yes

characteristic

Q₈

—

Q₈

C₂×Dic₁₂₇

yes

complemented

D₄

—

D₄

—

D₄

C₂×Dic₁₂₇

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

508

—

C₂×C₄

—

254

—

C₂×C₄

C₂×Dic₁₂₇

yes

—

C₂×C₄

C₂²⋊D₁₂₇

yes

characteristic

C₂×C₄

C₂²⋊D₁₂₇

yes

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₈

—

254

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²⋊Dic₁₂₇

yes

—

C₂²

C₂×C₂²⋊D₁₂₇

yes

characteristic

C₄

—

C₄

—

508

—

C₄

C₂²⋊Dic₁₂₇

yes

—

C₄

C₂²×Dic₁₂₇

yes

characteristic

C₄

C₂×C₂²⋊D₁₂₇

yes

characteristic

C₂

—

C₂

—

100

C₂

C₂×C₂²⋊Dic₁₂₇

yes

characteristic

101

C₂

#16861234

yes

characteristic

102

C₂

#16861239

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₂₇
commutator	C₄×C₁₂₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₂₇
Fitting	C₂×#41×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₄
hypercenter	C₂×#41	D₁₂₇
hypocenter	C₁₂₇	#417
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#41

subgroup

G/H

center

C₂²

C₂×C₂²⋊D₁₂₇

commutator

C₄×C₁₂₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×D₁₂₇

Fitting

C₂×#41×C₁₂₇

C₂

socle

C₂²×C₁₂₇

C₂×D₄

hypercenter

C₂×#41

D₁₂₇

hypocenter

C₁₂₇

#417

LCS

G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#41

N	H ≅ G/N	type
1. C₂×#14523232	C₂	semidirect
2. C₂×C₄⋊Dic₁₂₇	C₂	semidirect
3. #16861133	C₂	semidirect
4. #16861130	C₂	semidirect
12. #41×C₁₂₇	C₄	semidirect
24. Q₈×C₁₂₇	C₂×C₄	semidirect
27. D₄×C₁₂₇	C₂×C₄	semidirect
51. C₁₂₇	#417	semidirect
59. #41	Dic₁₂₇	semidirect
73. Q₈	C₂×Dic₁₂₇	semidirect
76. D₄	C₂×Dic₁₂₇	semidirect

H ≅ G/N

type

1. C₂×#14523232

C₂

semidirect

2. C₂×C₄⋊Dic₁₂₇

C₂

semidirect

3. #16861133

C₂

semidirect

4. #16861130

C₂

semidirect

12. #41×C₁₂₇

C₄

semidirect

24. Q₈×C₁₂₇

C₂×C₄

semidirect

27. D₄×C₁₂₇

C₂×C₄

semidirect

51. C₁₂₇

#417

semidirect

59. #41

Dic₁₂₇

semidirect

73. Q₈

C₂×Dic₁₂₇

semidirect

76. D₄

C₂×Dic₁₂₇

semidirect

Subgroups — #22549877

Conjugacy classes of subgroups

Complements