Group #46835693 — subgroups

Subgroups — #46835693

Order 34688.

172 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 45 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	C₂₇₁⋊#564	C₂	1	1	yes	maximal, characteristic
2	#31774306	C₂	1	2	yes	maximal
3	#31774267	C₂	1	1	yes	maximal, characteristic
4	#31773638	C₂	1	2	yes	maximal
5	#330×C₂₇₁	C₂	1	1	yes	maximal, characteristic
6	#22576160	C₂²	1	1	yes	characteristic
7	#22576152	C₂²	1	2	yes	—
8	#22576151	—	2	2	no	—
9	#22576150	—	2	2	no	—
10	#22576150	—	2	2	no	—
11	#22576094	—	2	2	no	—
12	#22576092	—	2	2	no	—
13	#22576083	—	2	2	no	—
14	#22576071	C₂²	1	1	yes	characteristic
15	#125×C₂₇₁	C₂²	1	1	yes	characteristic
16	#97×C₂₇₁	C₂²	1	2	yes	—
17	C₂×Dic₅₄₂	—	2	1	no	—
18	C₂×C₂₇₁⋊C₈	—	2	1	no	—
19	C₄⋊Dic₂₇₁	—	2	2	no	—
20	C₄⋊Dic₂₇₁	—	2	2	no	—
21	C₄⋊Dic₂₇₁	D₄	1	2	yes	—
22	C₄⋊Dic₂₇₁	D₄	1	2	yes	—
23	Dic₂₇₁⋊C₄	—	2	2	no	—
24	Dic₂₇₁⋊C₄	—	2	2	no	—
25	Dic₂₇₁⋊C₄	—	2	2	no	—
26	Dic₂₇₁⋊C₄	—	2	2	no	—
27	C₄×Dic₂₇₁	—	2	2	no	—
28	C₂×C₈×C₂₇₁	—	2	2	no	—
29	C₄⋊C₄×C₂₇₁	—	2	2	no	—
30	C₄⋊C₄×C₂₇₁	D₄	1	2	yes	—
31	C₄²×C₂₇₁	C₂³	1	1	yes	characteristic
32	Dic₅₄₂	—	4	2	no	—
33	C₂×Dic₂₇₁	—	2	2	no	—
34	C₂×Dic₂₇₁	—	2	2	no	—
35	C₂×Dic₂₇₁	—	4	1	no	—
36	C₂₇₁⋊C₈	—	4	1	no	—
37	C₂×C₄×C₂₇₁	—	2	2	no	—
38	C₂×C₄×C₂₇₁	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
40	C₈×C₂₇₁	—	2	4	no	—
41	C₂²×C₂₇₁	#120	1	1	yes	characteristic
42	C₄×C₂₇₁	—	2	1	no	—
43	C₄×C₂₇₁	—	2	2	no	—
44	C₄×C₂₇₁	—	4	2	no	—
45	Dic₂₇₁	—	4	2	no	—
46	Dic₂₇₁	—	4	2	no	—
47	Dic₂₇₁	—	8	1	no	—
48	C₂×C₂₇₁	#454	1	1	yes	characteristic
49	C₂×C₂₇₁	#456	1	2	yes	—
50	C₂₇₁	#1663	1	1	yes	characteristic, complemented, Sylow
51	#1663	—	271	1	no	maximal, Sylow
52	#564	—	271	1	no	—
53	#501	—	271	2	no	—
54	#491	—	271	1	no	—
55	#330	—	271	2	no	—
56	#330	D₂₇₁	1	1	yes	characteristic
57	C₄⋊Q₈	—	271	1	no	—
58	#125	—	271	2	no	—
59	#125	—	542	2	no	—
60	#125	—	542	2	no	—
61	#125	—	542	2	no	—
62	#125	C₂×D₂₇₁	1	1	yes	characteristic
63	#107	—	542	2	no	—
64	#106	—	542	2	no	—
65	#103	—	542	2	no	—
66	#97	—	271	1	no	—
67	#97	C₂×D₂₇₁	1	2	yes	—
68	C₂×Q₈	—	542	1	no	—
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	—	542	1	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	271	2	no	—
73	C₄⋊C₄	—	271	2	no	—
74	C₄⋊C₄	—	542	2	no	—
75	C₄⋊C₄	—	542	2	no	—
76	C₄⋊C₄	—	542	2	no	—
77	C₄⋊C₄	—	542	2	no	—
78	C₄⋊C₄	—	542	2	no	—
79	C₄⋊C₄	—	542	2	no	—
80	C₄⋊C₄	D₁₀₈₄	1	2	yes	—
81	C₄²	—	542	2	no	—
82	C₄²	C₂²×D₂₇₁	1	1	yes	characteristic
83	Q₈	—	1084	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	542	2	no	—
86	C₂×C₄	—	542	2	no	—
87	C₂×C₄	—	1084	1	no	—
88	C₂×C₄	D₄×D₂₇₁	1	2	yes	—
89	C₂×C₄	C₂×D₁₀₈₄	1	1	yes	characteristic
90	C₈	—	2	4	no	—
91	C₈	—	1084	1	no	—
92	C₂²	#22576130	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	2	no	—
95	C₄	—	4	2	no	—
96	C₄	—	1084	2	no	—
97	C₄	—	1084	2	no	—
98	C₄	—	2168	1	no	—
99	C₂	#31774072	1	1	yes	characteristic
100	C₂	#31774081	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₇₁⋊#564

C₂

yes

maximal, characteristic

#31774306

C₂

yes

maximal

#31774267

C₂

yes

maximal, characteristic

#31773638

C₂

yes

maximal

#330×C₂₇₁

C₂

yes

maximal, characteristic

#22576160

C₂²

yes

characteristic

#22576152

C₂²

yes

—

—

—

—

—

—

—

C₂²

yes

characteristic

#125×C₂₇₁

C₂²

yes

characteristic

#97×C₂₇₁

C₂²

yes

—

C₂×Dic₅₄₂

—

C₂×C₂₇₁⋊C₈

—

C₄⋊Dic₂₇₁

—

C₄⋊Dic₂₇₁

—

C₄⋊Dic₂₇₁

D₄

yes

—

C₄⋊Dic₂₇₁

D₄

yes

—

Dic₂₇₁⋊C₄

—

Dic₂₇₁⋊C₄

—

Dic₂₇₁⋊C₄

—

Dic₂₇₁⋊C₄

—

C₄×Dic₂₇₁

—

C₂×C₈×C₂₇₁

—

C₄⋊C₄×C₂₇₁

—

C₄⋊C₄×C₂₇₁

D₄

yes

—

C₄²×C₂₇₁

C₂³

yes

characteristic

Dic₅₄₂

—

C₂×Dic₂₇₁

—

C₂×Dic₂₇₁

—

C₂×Dic₂₇₁

—

C₂₇₁⋊C₈

—

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

C₂×D₄

yes

—

C₂×C₄×C₂₇₁

C₂×D₄

yes

characteristic

C₈×C₂₇₁

—

C₂²×C₂₇₁

#120

yes

characteristic

C₄×C₂₇₁

—

C₄×C₂₇₁

—

C₄×C₂₇₁

—

Dic₂₇₁

—

Dic₂₇₁

—

Dic₂₇₁

—

C₂×C₂₇₁

#454

yes

characteristic

C₂×C₂₇₁

#456

yes

—

C₂₇₁

#1663

yes

characteristic, complemented, Sylow

#1663

—

271

maximal, Sylow

#564

—

271

—

#501

—

271

—

#491

—

271

—

#330

—

271

—

#330

D₂₇₁

yes

characteristic

C₄⋊Q₈

—

271

—

#125

—

271

—

#125

—

542

—

#125

—

542

—

#125

—

542

—

#125

C₂×D₂₇₁

yes

characteristic

#107

—

542

—

#106

—

542

—

#103

—

542

—

#97

—

271

—

#97

C₂×D₂₇₁

yes

—

C₂×Q₈

—

542

—

C₂×C₈

—

C₂×C₈

—

542

—

C₄⋊C₄

—

C₄⋊C₄

—

271

—

C₄⋊C₄

—

271

—

C₄⋊C₄

—

542

—

C₄⋊C₄

—

542

—

C₄⋊C₄

—

542

—

C₄⋊C₄

—

542

—

C₄⋊C₄

—

542

—

C₄⋊C₄

—

542

—

C₄⋊C₄

D₁₀₈₄

yes

—

C₄²

—

542

—

C₄²

C₂²×D₂₇₁

yes

characteristic

Q₈

—

1084

—

C₂×C₄

—

C₂×C₄

—

542

—

C₂×C₄

—

542

—

C₂×C₄

—

1084

—

C₂×C₄

D₄×D₂₇₁

yes

—

C₂×C₄

C₂×D₁₀₈₄

yes

characteristic

C₈

—

C₈

—

1084

—

C₂²

#22576130

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

1084

—

C₄

—

1084

—

C₄

—

2168

—

C₂

#31774072

yes

characteristic

100

C₂

#31774081

yes

—

	subgroup	G/H
center	C₂²	#22576130
commutator, cosocle	C₄²×C₂₇₁	C₂³
Frattini	C₄²	C₂²×D₂₇₁
Fitting	#330×C₂₇₁	C₂
socle, prime radical	C₂²×C₂₇₁	#120
hypercenter	#330	D₂₇₁
hypocenter	C₂₇₁	#1663
LCS	G ▷ C₄²×C₂₇₁ ▷ C₂²×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #330

subgroup

G/H

center

C₂²

#22576130

commutator, cosocle

C₄²×C₂₇₁

C₂³

Frattini

C₄²

C₂²×D₂₇₁

Fitting

#330×C₂₇₁

C₂

C₂²×C₂₇₁

D₂₇₁

C₂₇₁

G ▷ C₄²×C₂₇₁ ▷ C₂²×C₂₇₁ ▷ C₂₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #330

N	H ≅ G/N	type
50. C₂₇₁	#1663	semidirect

H ≅ G/N

type

50. C₂₇₁

#1663

semidirect

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

subgroup	index
C₂₇₁⋊#564	2
#31774306	2
#31774267	2
#31773638	2
#330×C₂₇₁	2
#1663	271

subgroup

index

C₂₇₁⋊#564

#330×C₂₇₁

271

Subgroups — #46835693

Conjugacy classes of subgroups

Complements

Maximal subgroups