Group #46837334 — subgroups

Subgroups — #46837334

Order 34688.

260 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 113 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×#22576071	C₂	1	1	yes	maximal, characteristic
2	#31773767	C₂	1	1	yes	maximal, characteristic
3	#31773659	C₂	1	4	yes	maximal
4	C₂×#125×C₂₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₂₇₁⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₂₇₁	—	2	2	no	—
7	C₂×C₄⋊Dic₂₇₁	C₂²	1	2	yes	—
8	#22576071	—	2	2	no	—
9	#125×C₂₇₁	—	2	2	no	—
10	#125×C₂₇₁	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₄×C₂₇₁	—	2	2	no	—
12	C₂×C₄⋊C₄×C₂₇₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₇₁	C₂²	1	1	yes	characteristic
14	C₂²×Dic₂₇₁	—	2	2	no	—
15	C₂×C₂₇₁⋊C₈	—	2	2	no	—
16	C₂×C₂₇₁⋊C₈	—	2	2	no	—
17	C₂×C₂₇₁⋊C₈	Q₈	1	4	yes	—
18	C₄⋊Dic₂₇₁	—	2	4	no	—
19	C₄⋊Dic₂₇₁	—	4	4	no	—
20	C₂²×C₄×C₂₇₁	—	2	2	no	—
21	C₂²×C₄×C₂₇₁	D₄	1	2	yes	—
22	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₂₇₁	—	2	8	no	—
24	C₄⋊C₄×C₂₇₁	—	2	2	no	—
25	C₄⋊C₄×C₂₇₁	C₂×C₄	1	4	yes	—
26	C₄²×C₂₇₁	C₂×C₄	1	2	yes	—
27	C₄²×C₂₇₁	D₄	1	2	yes	—
28	C₂×Dic₂₇₁	—	4	2	no	—
29	C₂×Dic₂₇₁	—	4	4	no	—
30	C₂₇₁⋊C₈	—	2	4	no	—
31	C₂³×C₂₇₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₂₇₁	—	2	2	no	—
33	C₂×C₄×C₂₇₁	—	2	2	no	—
34	C₂×C₄×C₂₇₁	—	2	4	no	—
35	C₂×C₄×C₂₇₁	—	4	2	no	—
36	C₂×C₄×C₂₇₁	—	4	2	no	—
37	C₂×C₄×C₂₇₁	C₂²⋊C₄	1	4	yes	—
38	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₂₇₁	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₂₇₁	#41	1	2	yes	—
42	C₂²×C₂₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₂₇₁	#124	1	1	yes	characteristic
44	C₂²×C₂₇₁	C₄⋊Q₈	1	1	yes	characteristic
45	C₂²×C₂₇₁	AGL(1,ℤ/8)	1	2	yes	—
46	C₂²×C₂₇₁	#137	1	2	yes	—
47	C₄×C₂₇₁	—	2	4	no	—
48	C₄×C₂₇₁	—	4	4	no	—
49	C₄×C₂₇₁	C₄×Q₈	1	2	yes	—
50	C₄×C₂₇₁	C₂²⋊Q₈	1	2	yes	—
51	Dic₂₇₁	—	8	2	no	—
52	C₂×C₂₇₁	#391	1	1	yes	characteristic
53	C₂×C₂₇₁	#417	1	2	yes	—
54	C₂×C₂₇₁	#490	1	1	yes	characteristic
55	C₂×C₂₇₁	#491	1	1	yes	characteristic
56	C₂×C₂₇₁	#501	1	2	yes	—
57	C₂₇₁	#1992	1	1	yes	characteristic, complemented, Sylow
58	#1992	—	271	1	no	maximal, Sylow
59	C₂×#125	D₂₇₁	1	1	yes	characteristic
60	C₂×#97	—	271	1	no	—
61	#384	—	271	1	no	—
62	#340	—	271	4	no	—
63	C₂²×C₈	—	271	2	no	—
64	#125	—	2	2	no	—
65	#125	Dic₂₇₁	1	4	yes	complemented
66	C₂×C₄⋊C₄	—	2	2	no	—
67	C₂×C₄⋊C₄	—	271	2	no	—
68	C₂×C₄⋊C₄	—	542	2	no	—
69	C₂×C₄⋊C₄	C₂×D₂₇₁	1	2	yes	—
70	C₂×C₄²	C₂×D₂₇₁	1	1	yes	characteristic
71	#97	—	542	2	no	—
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	—	542	2	no	—
74	C₂²×C₄	C₂²⋊D₂₇₁	1	2	yes	—
75	C₂²×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
76	C₂×C₈	—	271	4	no	—
77	C₂×C₈	—	542	2	no	—
78	C₂×C₈	—	542	2	no	—
79	C₄⋊C₄	—	2	8	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	542	4	no	—
82	C₄⋊C₄	—	1084	4	no	—
83	C₄⋊C₄	C₂×Dic₂₇₁	1	4	yes	—
84	C₄²	C₂×Dic₂₇₁	1	2	yes	—
85	C₄²	C₂²⋊D₂₇₁	1	2	yes	—
86	C₂³	C₂×C₂²⋊D₂₇₁	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	1084	2	no	—
93	C₂×C₄	—	1084	4	no	—
94	C₂×C₄	C₂²⋊Dic₂₇₁	1	4	yes	—
95	C₂×C₄	C₂²×Dic₂₇₁	1	1	yes	characteristic
96	C₂×C₄	C₂×C₂²⋊D₂₇₁	1	1	yes	characteristic
97	C₂×C₄	Q₈×D₂₇₁	1	2	yes	—
98	C₂×C₄	#16864655	1	2	yes	—
99	C₈	—	542	4	no	—
100	C₂²	C₂×C₂²⋊Dic₂₇₁	1	1	yes	characteristic
101	C₂²	#22576149	1	1	yes	characteristic
102	C₂²	#22576158	1	1	yes	characteristic
103	C₂²	#22576185	1	2	yes	—
104	C₂²	#22576190	1	2	yes	—
105	C₄	—	2	4	no	—
106	C₄	—	4	4	no	—
107	C₄	—	2168	2	no	—
108	C₄	Q₈×Dic₂₇₁	1	2	yes	—
109	C₄	#22576136	1	2	yes	—
110	C₂	#31773804	1	1	yes	characteristic
111	C₂	#31773898	1	2	yes	—
112	C₂	#31774263	1	1	yes	characteristic
113	C₂	#31774267	1	1	yes	characteristic
114	C₂	#31774307	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22576071

C₂

yes

maximal, characteristic

#31773767

C₂

yes

maximal, characteristic

#31773659

C₂

yes

maximal

C₂×#125×C₂₇₁

C₂

yes

maximal, characteristic

C₂²×C₂₇₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₂₇₁

—

C₂×C₄⋊Dic₂₇₁

C₂²

yes

—

#22576071

—

#125×C₂₇₁

—

#125×C₂₇₁

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₇₁

—

C₂×C₄⋊C₄×C₂₇₁

C₂²

yes

—

C₂×C₄²×C₂₇₁

C₂²

yes

characteristic

C₂²×Dic₂₇₁

—

C₂×C₂₇₁⋊C₈

—

C₂×C₂₇₁⋊C₈

—

C₂×C₂₇₁⋊C₈

Q₈

yes

—

C₄⋊Dic₂₇₁

—

C₄⋊Dic₂₇₁

—

C₂²×C₄×C₂₇₁

—

C₂²×C₄×C₂₇₁

D₄

yes

—

C₂²×C₄×C₂₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₇₁

—

C₄⋊C₄×C₂₇₁

—

C₄⋊C₄×C₂₇₁

C₂×C₄

yes

—

C₄²×C₂₇₁

C₂×C₄

yes

—

C₄²×C₂₇₁

D₄

yes

—

C₂×Dic₂₇₁

—

C₂×Dic₂₇₁

—

C₂₇₁⋊C₈

—

C₂³×C₂₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₁

C₂×Q₈

yes

—

C₂×C₄×C₂₇₁

#41

yes

—

C₂²×C₂₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₇₁

#124

yes

characteristic

C₂²×C₂₇₁

C₄⋊Q₈

yes

characteristic

C₂²×C₂₇₁

AGL(1,ℤ/8)

yes

—

C₂²×C₂₇₁

#137

yes

—

C₄×C₂₇₁

—

C₄×C₂₇₁

—

C₄×C₂₇₁

C₄×Q₈

yes

—

C₄×C₂₇₁

C₂²⋊Q₈

yes

—

Dic₂₇₁

—

C₂×C₂₇₁

#391

yes

characteristic

C₂×C₂₇₁

#417

yes

—

C₂×C₂₇₁

yes

C₂×C₂₇₁

yes

C₂×C₂₇₁

yes

—

C₂₇₁

#1992

yes

characteristic, complemented, Sylow

#1992

—

271

maximal, Sylow

C₂×#125

D₂₇₁

yes

characteristic

C₂×#97

—

271

—

#384

—

271

—

#340

—

271

—

C₂²×C₈

—

271

—

#125

—

#125

Dic₂₇₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

271

—

C₂×C₄⋊C₄

—

542

—

C₂×C₄⋊C₄

C₂×D₂₇₁

yes

—

C₂×C₄²

C₂×D₂₇₁

yes

characteristic

#97

—

542

—

C₂²×C₄

—

C₂²×C₄

—

542

—

C₂²×C₄

C₂²⋊D₂₇₁

yes

—

C₂²×C₄

C₂²×D₂₇₁

yes

characteristic

C₂×C₈

—

271

—

C₂×C₈

—

542

—

C₂×C₈

—

542

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

542

—

C₄⋊C₄

—

1084

—

C₄⋊C₄

C₂×Dic₂₇₁

yes

—

C₄²

C₂×Dic₂₇₁

yes

—

C₄²

C₂²⋊D₂₇₁

yes

—

C₂³

C₂×C₂²⋊D₂₇₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

1084

—

C₂×C₄

—

1084

—

C₂×C₄

C₂²⋊Dic₂₇₁

yes

—

C₂×C₄

C₂²×Dic₂₇₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₇₁

yes

characteristic

C₂×C₄

Q₈×D₂₇₁

yes

—

C₂×C₄

#16864655

yes

—

C₈

—

542

—

100

C₂²

C₂×C₂²⋊Dic₂₇₁

yes

characteristic

101

C₂²

#22576149

yes

characteristic

102

C₂²

#22576158

yes

characteristic

103

C₂²

#22576185

yes

—

104

C₂²

#22576190

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

2168

—

108

C₄

Q₈×Dic₂₇₁

yes

—

109

C₄

#22576136

yes

—

110

C₂

#31773804

yes

characteristic

111

C₂

#31773898

yes

—

112

C₂

#31774263

yes

characteristic

113

C₂

#31774267

yes

characteristic

114

C₂

#31774307

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₇₁
commutator	C₂×C₄×C₂₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₇₁
Fitting	C₂×#125×C₂₇₁	C₂
socle, prime radical	C₂³×C₂₇₁	C₂×D₄
cosocle	C₂²×C₄×C₂₇₁	C₂³
hypercenter	C₂×#125	D₂₇₁
hypocenter	C₂₇₁	#1992
LCS	G ▷ C₂×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₇₁

commutator

C₂×C₄×C₂₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₇₁

Fitting

C₂×#125×C₂₇₁

C₂

socle, prime radical

C₂³×C₂₇₁

C₂×D₄

cosocle

C₂²×C₄×C₂₇₁

C₂³

D₂₇₁

C₂₇₁

G ▷ C₂×C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
10. #125×C₂₇₁	C₄	semidirect
57. C₂₇₁	#1992	semidirect
65. #125	Dic₂₇₁	semidirect

H ≅ G/N

type

10. #125×C₂₇₁

C₄

semidirect

57. C₂₇₁

#1992

semidirect

65. #125

Dic₂₇₁

semidirect

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

subgroup	index
C₂×#22576071	2
#31773767	2
#31773659	2
C₂×#125×C₂₇₁	2
#1992	271

subgroup

index

C₂×#125×C₂₇₁

271

Subgroups — #46837334

Conjugacy classes of subgroups

Complements

Maximal subgroups