Group #30638880 — subgroups

Subgroups — #30638880

Order 16256.

162 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 65 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	M₁₆×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
2	#22549648	C₂	1	2	yes	maximal
3	#22549637	C₂	1	2	yes	maximal
4	#22549633	C₂	1	1	yes	maximal, characteristic
5	#338×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523232	C₂²	1	1	yes	characteristic
7	#16861168	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
9	#16861119	C₄	1	2	yes	complemented
10	C₈×Dic₁₂₇	C₄	1	2	yes	complemented
11	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
12	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂²×Dic₁₂₇	—	2	1	no	—
14	#14523232	—	2	2	no	—
15	C₂×C₁₂₇⋊C₈	C₂×C₄	1	2	yes	—
16	C₄⋊Dic₁₂₇	—	2	2	no	—
17	C₂²⋊Dic₁₂₇	—	2	2	no	—
18	C₄×Dic₁₂₇	—	2	1	no	—
19	C₄×Dic₁₂₇	—	2	2	no	—
20	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
23	M₁₆×C₁₂₇	—	2	2	no	—
24	M₁₆×C₁₂₇	—	2	2	no	—
25	M₁₆×C₁₂₇	—	2	2	no	—
26	C₂×C₈×C₁₂₇	—	2	2	no	—
27	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
28	C₂×Dic₁₂₇	—	2	1	no	—
29	C₂×Dic₁₂₇	—	2	1	no	—
30	C₂×Dic₁₂₇	—	4	2	no	—
31	C₂×Dic₁₂₇	—	4	2	no	—
32	C₁₂₇⋊C₈	—	2	2	no	—
33	C₂³×C₁₂₇	#41	1	1	yes	characteristic
34	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₂₇	#41	1	2	yes	—
36	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
37	C₂×C₄×C₁₂₇	#41	1	2	yes	—
38	C₈×C₁₂₇	—	2	2	no	—
39	C₈×C₁₂₇	—	4	2	no	—
40	C₂²×C₁₂₇	—	4	1	no	—
41	C₂²×C₁₂₇	#117	1	1	yes	characteristic
42	C₂²×C₁₂₇	#126	1	2	yes	—
43	C₄×C₁₂₇	#117	1	1	yes	characteristic
44	C₄×C₁₂₇	#117	1	1	yes	characteristic
45	C₄×C₁₂₇	#126	1	2	yes	—
46	Dic₁₂₇	—	2	2	no	—
47	Dic₁₂₇	—	4	1	no	—
48	Dic₁₂₇	—	8	2	no	—
49	C₂×C₁₂₇	—	2	1	no	—
50	C₂×C₁₂₇	—	2	2	no	—
51	C₂×C₁₂₇	#383	1	1	yes	characteristic
52	#1841	—	127	1	no	maximal, Sylow
53	C₁₂₇	#1841	1	1	yes	characteristic, complemented, Sylow
54	C₄×M₁₆	—	127	1	no	—
55	#344	—	127	2	no	—
56	#339	—	127	2	no	—
57	#338	D₁₂₇	1	1	yes	characteristic
58	H(3,ℤ/4)	—	127	1	no	—
59	C₂×M₁₆	—	127	1	no	—
60	C₂×M₁₆	C₂×D₁₂₇	1	2	yes	—
61	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
62	#117	—	127	2	no	—
63	C₂×C₄²	—	127	1	no	—
64	#97	—	127	2	no	—
65	C₄×C₈	—	127	2	no	—
66	C₂²×C₄	—	254	1	no	—
67	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	2	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	254	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	127	2	no	—
74	C₂×C₈	C₄×D₁₂₇	1	2	yes	—
75	C₄⋊C₄	—	254	2	no	—
76	C₂²⋊C₄	—	254	2	no	—
77	C₄²	—	127	1	no	—
78	C₄²	—	127	1	no	—
79	C₄²	—	254	2	no	—
80	C₄²	—	254	1	no	—
81	C₂³	#14523249	1	1	yes	characteristic
82	C₂×C₄	—	508	2	no	—
83	C₂×C₄	—	508	2	no	—
84	C₂×C₄	—	254	1	no	—
85	C₂×C₄	—	254	1	no	—
86	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
87	C₂×C₄	#14523247	1	1	yes	characteristic
88	C₂×C₄	#14523247	1	2	yes	—
89	C₂×C₄	#14523248	1	2	yes	—
90	C₈	—	2	2	no	—
91	C₈	—	4	2	no	—
92	C₈	—	254	2	no	—
93	C₂²	—	4	1	no	—
94	C₂²	#16861167	1	1	yes	characteristic
95	C₂²	#16861204	1	2	yes	—
96	C₄	—	1016	2	no	—
97	C₄	—	508	1	no	—
98	C₄	—	254	2	no	—
99	C₄	#16861165	1	1	yes	characteristic
100	C₄	#16861166	1	1	yes	characteristic
101	C₄	#16861202	1	2	yes	—
102	C₂	—	2	2	no	—
103	C₂	—	2	1	no	—
104	C₂	#22549743	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

M₁₆×Dic₁₂₇

C₂

yes

maximal, characteristic

#22549648

C₂

yes

maximal

#22549637

C₂

yes

maximal

#22549633

C₂

yes

maximal, characteristic

#338×C₁₂₇

C₂

yes

maximal, characteristic

C₂×#14523232

C₂²

yes

characteristic

#16861168

C₂²

yes

—

C₂×C₄×Dic₁₂₇

C₂²

yes

characteristic

#16861119

C₄

yes

complemented

C₈×Dic₁₂₇

C₄

yes

complemented

C₂×M₁₆×C₁₂₇

C₂²

yes

—

C₂×M₁₆×C₁₂₇

C₂²

yes

characteristic

C₂²×Dic₁₂₇

—

#14523232

—

C₂×C₁₂₇⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₁₂₇

—

C₂²⋊Dic₁₂₇

—

C₄×Dic₁₂₇

—

C₄×Dic₁₂₇

—

C₄×Dic₁₂₇

C₂×C₄

yes

characteristic

C₄×Dic₁₂₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₂₇

C₂³

yes

characteristic

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

—

M₁₆×C₁₂₇

—

C₂×C₈×C₁₂₇

—

C₂×C₈×C₁₂₇

C₂×C₄

yes

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₂×Dic₁₂₇

—

C₁₂₇⋊C₈

—

C₂³×C₁₂₇

#41

yes

characteristic

C₂×C₄×C₁₂₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₂×C₄×C₁₂₇

#41

yes

characteristic

C₂×C₄×C₁₂₇

#41

yes

—

C₈×C₁₂₇

—

C₈×C₁₂₇

—

C₂²×C₁₂₇

—

C₂²×C₁₂₇

#117

yes

characteristic

C₂²×C₁₂₇

#126

yes

—

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

C₄×C₁₂₇

yes

—

Dic₁₂₇

—

Dic₁₂₇

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

#383

yes

characteristic

#1841

—

127

maximal, Sylow

C₁₂₇

#1841

yes

characteristic, complemented, Sylow

C₄×M₁₆

—

127

—

#344

—

127

—

#339

—

127

—

#338

D₁₂₇

yes

characteristic

H(3,ℤ/4)

—

127

—

C₂×M₁₆

—

127

—

C₂×M₁₆

C₂×D₁₂₇

yes

—

C₂×M₁₆

C₂×D₁₂₇

yes

characteristic

#117

—

127

—

C₂×C₄²

—

127

—

#97

—

127

—

C₄×C₈

—

127

—

C₂²×C₄

—

254

—

C₂²×C₄

C₂²×D₁₂₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

254

—

C₂×C₈

—

C₂×C₈

—

127

—

C₂×C₈

C₄×D₁₂₇

yes

—

C₄⋊C₄

—

254

—

C₂²⋊C₄

—

254

—

C₄²

—

127

—

C₄²

—

127

—

C₄²

—

254

—

C₄²

—

254

—

C₂³

#14523249

yes

characteristic

C₂×C₄

—

508

—

C₂×C₄

—

508

—

C₂×C₄

—

254

—

C₂×C₄

—

254

—

C₂×C₄

C₂×C₄×D₁₂₇

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

#14523248

yes

—

C₈

—

C₈

—

C₈

—

254

—

C₂²

—

C₂²

#16861167

yes

characteristic

C₂²

#16861204

yes

—

C₄

—

1016

—

C₄

—

508

—

C₄

—

254

—

C₄

#16861165

yes

characteristic

100

C₄

#16861166

yes

characteristic

101

C₄

#16861202

yes

—

102

C₂

—

103

C₂

—

104

C₂

#22549743

yes

characteristic

	subgroup	G/H
center	C₄	#16861166
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#338×C₁₂₇	C₂
socle	C₂×C₁₂₇	#383
prime radical	C₂³×C₁₂₇	#41
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	#338	D₁₂₇
hypocenter	C₁₂₇	#1841
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#16861166

commutator

C₂×C₄×C₁₂₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₂₇

Fitting

#338×C₁₂₇

C₂

C₂×C₁₂₇

C₂³×C₁₂₇

C₂²×C₄×C₁₂₇

C₂³

D₁₂₇

C₁₂₇

G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
9. #16861119	C₄	semidirect
10. C₈×Dic₁₂₇	C₄	semidirect
53. C₁₂₇	#1841	semidirect

H ≅ G/N

type

9. #16861119

C₄

semidirect

10. C₈×Dic₁₂₇

C₄

semidirect

53. C₁₂₇

#1841

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
M₁₆×Dic₁₂₇	2
#22549648	2
#22549637	2
#22549633	2
#338×C₁₂₇	2
#1841	127

subgroup

index

M₁₆×Dic₁₂₇

#338×C₁₂₇

127

Subgroups — #30638880

Conjugacy classes of subgroups

Complements

Maximal subgroups