Group #46835502 — subgroups

Subgroups — #46835502

Order 34688.

206 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 57 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	#31774627	C₂	1	1	yes	maximal, characteristic
2	#31774189	C₂	1	2	yes	maximal
3	#31773646	C₂	1	1	yes	maximal, characteristic
4	#31773629	C₂	1	2	yes	maximal
5	#494×C₂₇₁	C₂	1	1	yes	maximal, characteristic
6	#22576159	—	2	2	no	—
7	#22576158	—	2	2	no	—
8	C₄×Dic₅₄₂	C₂²	1	2	yes	—
9	#22576125	—	2	2	no	—
10	#22576093	—	2	2	no	—
11	C₄⋊(C₂₇₁⋊C₈)	C₂²	1	2	yes	—
12	#22576083	—	2	2	no	—
13	C₂×Q₁₆×C₂₇₁	—	2	2	no	—
14	C₄⋊Q₈×C₂₇₁	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₂₇₁	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₅₄₂	—	2	2	no	—
18	C₂×Dic₅₄₂	D₄	1	2	yes	—
19	C₂×C₂₇₁⋊C₈	—	2	2	no	—
20	C₄⋊Dic₂₇₁	D₄	1	2	yes	—
21	Dic₂₇₁⋊C₄	—	2	4	no	—
22	Dic₂₇₁⋊C₄	—	2	2	no	—
23	C₄×Dic₂₇₁	—	2	2	no	—
24	C₄×Dic₂₇₁	—	2	2	no	—
25	C₂×Q₈×C₂₇₁	—	2	1	no	—
26	C₂×Q₈×C₂₇₁	D₄	1	2	yes	—
27	Q₁₆×C₂₇₁	—	4	4	no	—
28	C₂×C₈×C₂₇₁	—	2	1	no	—
29	C₄⋊C₄×C₂₇₁	—	2	2	no	—
30	C₄⋊C₄×C₂₇₁	—	2	2	no	—
31	C₄²×C₂₇₁	C₂³	1	1	yes	characteristic
32	Dic₅₄₂	—	2	2	no	—
33	Dic₅₄₂	—	8	2	no	—
34	Dic₅₄₂	Q₁₆	1	4	yes	complemented
35	C₂×Dic₂₇₁	—	2	2	no	—
36	C₂×Dic₂₇₁	—	2	2	no	—
37	C₂₇₁⋊C₈	—	4	2	no	—
38	Q₈×C₂₇₁	—	4	2	no	—
39	Q₈×C₂₇₁	—	4	2	no	—
40	C₂×C₄×C₂₇₁	—	2	2	no	—
41	C₂×C₄×C₂₇₁	—	4	1	no	—
42	C₂×C₄×C₂₇₁	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
44	C₈×C₂₇₁	—	2	2	no	—
45	C₂²×C₂₇₁	#120	1	1	yes	characteristic
46	C₄×C₂₇₁	—	2	1	no	—
47	C₄×C₂₇₁	—	4	2	no	—
48	C₄×C₂₇₁	—	8	1	no	—
49	C₄×C₂₇₁	C₂×Q₁₆	1	2	yes	—
50	C₄×C₂₇₁	#137	1	2	yes	—
51	Dic₂₇₁	—	2	4	no	—
52	Dic₂₇₁	—	4	2	no	—
53	C₂×C₂₇₁	#451	1	2	yes	—
54	C₂×C₂₇₁	#453	1	1	yes	characteristic
55	C₂₇₁	#1633	1	1	yes	characteristic, complemented, Sylow
56	#1633	—	271	1	no	maximal, Sylow
57	Q₈²	—	271	1	no	—
58	#494	D₂₇₁	1	1	yes	characteristic
59	#477	—	271	2	no	—
60	#333	—	271	1	no	—
61	Q₈⋊C₈	—	271	2	no	—
62	C₂×Q₁₆	—	2	2	no	—
63	C₄⋊Q₈	—	542	2	no	—
64	C₄⋊Q₈	—	542	2	no	—
65	C₄⋊Q₈	C₂×D₂₇₁	1	1	yes	characteristic
66	C₄⋊Q₈	C₂×D₂₇₁	1	1	yes	characteristic
67	C₄×Q₈	—	271	2	no	—
68	C₄×Q₈	—	542	2	no	—
69	#107	—	542	2	no	—
70	C₄⋊C₈	—	271	2	no	—
71	#103	—	542	2	no	—
72	C₄×C₈	C₂×D₂₇₁	1	1	yes	characteristic
73	C₂×Q₈	—	2	1	no	—
74	C₂×Q₈	—	271	2	no	—
75	C₂×Q₈	—	542	2	no	—
76	C₂×Q₈	C₂²⋊D₂₇₁	1	2	yes	—
77	Q₁₆	—	4	4	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	542	2	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	271	2	no	—
83	C₄⋊C₄	—	542	4	no	—
84	C₄⋊C₄	—	542	2	no	—
85	C₄²	—	542	2	no	—
86	C₄²	—	542	2	no	—
87	C₄²	C₂²×D₂₇₁	1	1	yes	characteristic
88	Q₈	—	4	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	271	4	no	—
91	Q₈	—	542	2	no	—
92	Q₈	—	2168	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	4	1	no	—
95	C₂×C₄	—	542	2	no	—
96	C₂×C₄	—	542	2	no	—
97	C₂×C₄	D₄×D₂₇₁	1	2	yes	—
98	C₂×C₄	C₂×C₂²⋊D₂₇₁	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	—	1084	2	no	—
101	C₂²	#22576128	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	—	8	1	no	—
105	C₄	—	542	4	no	—
106	C₄	—	1084	2	no	—
107	C₄	Q₁₆×D₂₇₁	1	2	yes	—
108	C₄	#22576192	1	2	yes	—
109	C₂	#31774051	1	2	yes	—
110	C₂	#31774064	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31774627

C₂

yes

maximal, characteristic

#31774189

C₂

yes

maximal

#31773646

C₂

yes

maximal, characteristic

#31773629

C₂

yes

maximal

#494×C₂₇₁

C₂

yes

maximal, characteristic

#22576159

—

#22576158

—

C₄×Dic₅₄₂

C₂²

yes

—

#22576125

—

#22576093

—

C₄⋊(C₂₇₁⋊C₈)

C₂²

yes

—

#22576083

—

C₂×Q₁₆×C₂₇₁

—

C₄⋊Q₈×C₂₇₁

C₂²

yes

characteristic

C₄⋊Q₈×C₂₇₁

C₂²

yes

characteristic

C₄×C₈×C₂₇₁

C₂²

yes

characteristic

C₂×Dic₅₄₂

—

C₂×Dic₅₄₂

D₄

yes

—

C₂×C₂₇₁⋊C₈

—

C₄⋊Dic₂₇₁

D₄

yes

—

Dic₂₇₁⋊C₄

—

Dic₂₇₁⋊C₄

—

C₄×Dic₂₇₁

—

C₄×Dic₂₇₁

—

C₂×Q₈×C₂₇₁

—

C₂×Q₈×C₂₇₁

D₄

yes

—

Q₁₆×C₂₇₁

—

C₂×C₈×C₂₇₁

—

C₄⋊C₄×C₂₇₁

—

C₄⋊C₄×C₂₇₁

—

C₄²×C₂₇₁

C₂³

yes

characteristic

Dic₅₄₂

—

Dic₅₄₂

—

Dic₅₄₂

Q₁₆

yes

complemented

C₂×Dic₂₇₁

—

C₂×Dic₂₇₁

—

C₂₇₁⋊C₈

—

Q₈×C₂₇₁

—

Q₈×C₂₇₁

—

C₂×C₄×C₂₇₁

—

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

—

C₄×C₂₇₁

C₂×Q₁₆

yes

—

C₄×C₂₇₁

#137

yes

—

Dic₂₇₁

—

Dic₂₇₁

—

C₂×C₂₇₁

#451

yes

—

C₂×C₂₇₁

#453

yes

characteristic

C₂₇₁

#1633

yes

characteristic, complemented, Sylow

#1633

—

271

maximal, Sylow

Q₈²

—

271

—

#494

D₂₇₁

yes

characteristic

#477

—

271

—

#333

—

271

—

Q₈⋊C₈

—

271

—

C₂×Q₁₆

—

C₄⋊Q₈

—

542

—

C₄⋊Q₈

—

542

—

C₄⋊Q₈

C₂×D₂₇₁

yes

characteristic

C₄⋊Q₈

C₂×D₂₇₁

yes

characteristic

C₄×Q₈

—

271

—

C₄×Q₈

—

542

—

#107

—

542

—

C₄⋊C₈

—

271

—

#103

—

542

—

C₄×C₈

C₂×D₂₇₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

271

—

C₂×Q₈

—

542

—

C₂×Q₈

C₂²⋊D₂₇₁

yes

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

542

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

271

—

C₄⋊C₄

—

542

—

C₄⋊C₄

—

542

—

C₄²

—

542

—

C₄²

—

542

—

C₄²

C₂²×D₂₇₁

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

271

—

Q₈

—

542

—

Q₈

—

2168

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

542

—

C₂×C₄

—

542

—

C₂×C₄

D₄×D₂₇₁

yes

—

C₂×C₄

C₂×C₂²⋊D₂₇₁

yes

characteristic

C₈

—

100

C₈

—

1084

—

101

C₂²

#22576128

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

542

—

106

C₄

—

1084

—

107

C₄

Q₁₆×D₂₇₁

yes

—

108

C₄

#22576192

yes

—

109

C₂

#31774051

yes

—

110

C₂

#31774064

yes

characteristic

	subgroup	G/H
center	C₂²	#22576128
commutator, cosocle	C₄²×C₂₇₁	C₂³
Frattini	C₄²	C₂²×D₂₇₁
Fitting	#494×C₂₇₁	C₂
socle, prime radical	C₂²×C₂₇₁	#120
hypercenter	#494	D₂₇₁
hypocenter	C₂₇₁	#1633
LCS	G ▷ C₄²×C₂₇₁ ▷ C₂²×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#22576128

commutator, cosocle

C₄²×C₂₇₁

C₂³

Frattini

C₄²

C₂²×D₂₇₁

Fitting

#494×C₂₇₁

C₂

C₂²×C₂₇₁

D₂₇₁

C₂₇₁

G ▷ C₄²×C₂₇₁ ▷ C₂²×C₂₇₁ ▷ C₂₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
34. Dic₅₄₂	Q₁₆	semidirect
55. C₂₇₁	#1633	semidirect

H ≅ G/N

type

34. Dic₅₄₂

Q₁₆

semidirect

55. C₂₇₁

#1633

semidirect

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

subgroup	index
#31774627	2
#31774189	2
#31773646	2
#31773629	2
#494×C₂₇₁	2
#1633	271

subgroup

index

#494×C₂₇₁

271

Subgroups — #46835502

Conjugacy classes of subgroups

Complements

Maximal subgroups