Group #49096273 — subgroups

Subgroups — #49096273

Order 35968.

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×#22582790	C₂	1	1	yes	maximal, characteristic
2	#31823333	C₂	1	1	yes	maximal, characteristic
3	#31823225	C₂	1	4	yes	maximal
4	C₂×#125×C₂₈₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×#14555886	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₂₈₁	—	2	2	no	—
7	C₂×C₄⋊Dic₂₈₁	C₂²	1	2	yes	—
8	#22582790	—	2	2	no	—
9	#125×C₂₈₁	—	2	2	no	—
10	#125×C₂₈₁	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₄×C₂₈₁	—	2	2	no	—
12	C₂×C₄⋊C₄×C₂₈₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₈₁	C₂²	1	1	yes	characteristic
14	C₂²×Dic₂₈₁	—	2	2	no	—
15	C₂×#14555886	—	2	2	no	—
16	C₂×#14555886	—	2	2	no	—
17	C₂×#14555886	Q₈	1	4	yes	—
18	C₄⋊Dic₂₈₁	—	2	4	no	—
19	C₄⋊Dic₂₈₁	—	4	4	no	—
20	C₂²×C₄×C₂₈₁	—	2	2	no	—
21	C₂²×C₄×C₂₈₁	D₄	1	2	yes	—
22	C₂²×C₄×C₂₈₁	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₂₈₁	—	2	8	no	—
24	C₄⋊C₄×C₂₈₁	—	2	2	no	—
25	C₄⋊C₄×C₂₈₁	C₂×C₄	1	4	yes	—
26	C₄²×C₂₈₁	C₂×C₄	1	2	yes	—
27	C₄²×C₂₈₁	D₄	1	2	yes	—
28	C₂×Dic₂₈₁	—	4	2	no	—
29	C₂×Dic₂₈₁	—	4	4	no	—
30	#14555886	—	2	4	no	—
31	C₂³×C₂₈₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₂₈₁	—	2	2	no	—
33	C₂×C₄×C₂₈₁	—	2	2	no	—
34	C₂×C₄×C₂₈₁	—	2	4	no	—
35	C₂×C₄×C₂₈₁	—	4	2	no	—
36	C₂×C₄×C₂₈₁	—	4	2	no	—
37	C₂×C₄×C₂₈₁	C₂²⋊C₄	1	4	yes	—
38	C₂×C₄×C₂₈₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₂₈₁	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₂₈₁	#41	1	2	yes	—
42	C₂²×C₂₈₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₂₈₁	#124	1	1	yes	characteristic
44	C₂²×C₂₈₁	C₄⋊Q₈	1	1	yes	characteristic
45	C₂²×C₂₈₁	AGL(1,ℤ/8)	1	2	yes	—
46	C₂²×C₂₈₁	#137	1	2	yes	—
47	C₄×C₂₈₁	—	2	4	no	—
48	C₄×C₂₈₁	—	4	4	no	—
49	C₄×C₂₈₁	C₄×Q₈	1	2	yes	—
50	C₄×C₂₈₁	C₂²⋊Q₈	1	2	yes	—
51	Dic₂₈₁	—	8	2	no	—
52	C₂×C₂₈₁	#391	1	1	yes	characteristic
53	C₂×C₂₈₁	#417	1	2	yes	—
54	C₂×C₂₈₁	#490	1	1	yes	characteristic
55	C₂×C₂₈₁	#491	1	1	yes	characteristic
56	C₂×C₂₈₁	#501	1	2	yes	—
57	C₂₈₁	#1992	1	1	yes	characteristic, complemented, Sylow
58	#1992	—	281	1	no	maximal, Sylow
59	C₂×#125	D₂₈₁	1	1	yes	characteristic
60	C₂×#97	—	281	1	no	—
61	#384	—	281	1	no	—
62	#340	—	281	4	no	—
63	C₂²×C₈	—	281	2	no	—
64	#125	—	2	2	no	—
65	#125	Dic₂₈₁	1	4	yes	complemented
66	C₂×C₄⋊C₄	—	2	2	no	—
67	C₂×C₄⋊C₄	—	281	2	no	—
68	C₂×C₄⋊C₄	—	562	2	no	—
69	C₂×C₄⋊C₄	C₂×D₂₈₁	1	2	yes	—
70	C₂×C₄²	C₂×D₂₈₁	1	1	yes	characteristic
71	#97	—	562	2	no	—
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	—	562	2	no	—
74	C₂²×C₄	C₂²⋊D₂₈₁	1	2	yes	—
75	C₂²×C₄	C₂²×D₂₈₁	1	1	yes	characteristic
76	C₂×C₈	—	281	4	no	—
77	C₂×C₈	—	562	2	no	—
78	C₂×C₈	—	562	2	no	—
79	C₄⋊C₄	—	2	8	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	562	4	no	—
82	C₄⋊C₄	—	1124	4	no	—
83	C₄⋊C₄	C₂×Dic₂₈₁	1	4	yes	—
84	C₄²	C₂×Dic₂₈₁	1	2	yes	—
85	C₄²	C₂²⋊D₂₈₁	1	2	yes	—
86	C₂³	C₂×C₂²⋊D₂₈₁	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	—	1124	2	no	—
93	C₂×C₄	—	1124	4	no	—
94	C₂×C₄	C₂²⋊Dic₂₈₁	1	4	yes	—
95	C₂×C₄	C₂²×Dic₂₈₁	1	1	yes	characteristic
96	C₂×C₄	C₂×C₂²⋊D₂₈₁	1	1	yes	characteristic
97	C₂×C₄	Q₈×D₂₈₁	1	2	yes	—
98	C₂×C₄	#17985139	1	2	yes	—
99	C₈	—	562	4	no	—
100	C₂²	C₂×C₂²⋊Dic₂₈₁	1	1	yes	characteristic
101	C₂²	#22582868	1	1	yes	characteristic
102	C₂²	#22582877	1	1	yes	characteristic
103	C₂²	#22582904	1	2	yes	—
104	C₂²	#22582909	1	2	yes	—
105	C₄	—	2	4	no	—
106	C₄	—	4	4	no	—
107	C₄	—	2248	2	no	—
108	C₄	Q₈×Dic₂₈₁	1	2	yes	—
109	C₄	#22582855	1	2	yes	—
110	C₂	#31823370	1	1	yes	characteristic
111	C₂	#31823464	1	2	yes	—
112	C₂	#31823829	1	1	yes	characteristic
113	C₂	#31823833	1	1	yes	characteristic
114	C₂	#31823873	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22582790

C₂

yes

maximal, characteristic

#31823333

C₂

yes

maximal, characteristic

#31823225

C₂

yes

maximal

C₂×#125×C₂₈₁

C₂

yes

maximal, characteristic

C₂²×#14555886

C₂²

yes

—

C₂×C₄⋊Dic₂₈₁

—

C₂×C₄⋊Dic₂₈₁

C₂²

yes

—

#22582790

—

#125×C₂₈₁

—

#125×C₂₈₁

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₈₁

—

C₂×C₄⋊C₄×C₂₈₁

C₂²

yes

—

C₂×C₄²×C₂₈₁

C₂²

yes

characteristic

C₂²×Dic₂₈₁

—

C₂×#14555886

—

C₂×#14555886

—

C₂×#14555886

Q₈

yes

—

C₄⋊Dic₂₈₁

—

C₄⋊Dic₂₈₁

—

C₂²×C₄×C₂₈₁

—

C₂²×C₄×C₂₈₁

D₄

yes

—

C₂²×C₄×C₂₈₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₈₁

—

C₄⋊C₄×C₂₈₁

—

C₄⋊C₄×C₂₈₁

C₂×C₄

yes

—

C₄²×C₂₈₁

C₂×C₄

yes

—

C₄²×C₂₈₁

D₄

yes

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

#14555886

—

C₂³×C₂₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₈₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×Q₈

yes

—

C₂×C₄×C₂₈₁

#41

yes

—

C₂²×C₂₈₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₈₁

#124

yes

characteristic

C₂²×C₂₈₁

C₄⋊Q₈

yes

characteristic

C₂²×C₂₈₁

AGL(1,ℤ/8)

yes

—

C₂²×C₂₈₁

#137

yes

—

C₄×C₂₈₁

—

C₄×C₂₈₁

—

C₄×C₂₈₁

C₄×Q₈

yes

—

C₄×C₂₈₁

C₂²⋊Q₈

yes

—

Dic₂₈₁

—

C₂×C₂₈₁

#391

yes

characteristic

C₂×C₂₈₁

#417

yes

—

C₂×C₂₈₁

yes

C₂×C₂₈₁

yes

C₂×C₂₈₁

yes

—

C₂₈₁

#1992

yes

characteristic, complemented, Sylow

#1992

—

281

maximal, Sylow

C₂×#125

D₂₈₁

yes

characteristic

C₂×#97

—

281

—

#384

—

281

—

#340

—

281

—

C₂²×C₈

—

281

—

#125

—

#125

Dic₂₈₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

281

—

C₂×C₄⋊C₄

—

562

—

C₂×C₄⋊C₄

C₂×D₂₈₁

yes

—

C₂×C₄²

C₂×D₂₈₁

yes

characteristic

#97

—

562

—

C₂²×C₄

—

C₂²×C₄

—

562

—

C₂²×C₄

C₂²⋊D₂₈₁

yes

—

C₂²×C₄

C₂²×D₂₈₁

yes

characteristic

C₂×C₈

—

281

—

C₂×C₈

—

562

—

C₂×C₈

—

562

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

562

—

C₄⋊C₄

—

1124

—

C₄⋊C₄

C₂×Dic₂₈₁

yes

—

C₄²

C₂×Dic₂₈₁

yes

—

C₄²

C₂²⋊D₂₈₁

yes

—

C₂³

C₂×C₂²⋊D₂₈₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

1124

—

C₂×C₄

—

1124

—

C₂×C₄

C₂²⋊Dic₂₈₁

yes

—

C₂×C₄

C₂²×Dic₂₈₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₈₁

yes

characteristic

C₂×C₄

Q₈×D₂₈₁

yes

—

C₂×C₄

#17985139

yes

—

C₈

—

562

—

100

C₂²

C₂×C₂²⋊Dic₂₈₁

yes

characteristic

101

C₂²

#22582868

yes

characteristic

102

C₂²

#22582877

yes

characteristic

103

C₂²

#22582904

yes

—

104

C₂²

#22582909

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

2248

—

108

C₄

Q₈×Dic₂₈₁

yes

—

109

C₄

#22582855

yes

—

110

C₂

#31823370

yes

characteristic

111

C₂

#31823464

yes

—

112

C₂

#31823829

yes

characteristic

113

C₂

#31823833

yes

characteristic

114

C₂

#31823873

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₈₁
commutator	C₂×C₄×C₂₈₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₈₁
Fitting	C₂×#125×C₂₈₁	C₂
socle, prime radical	C₂³×C₂₈₁	C₂×D₄
cosocle	C₂²×C₄×C₂₈₁	C₂³
hypercenter	C₂×#125	D₂₈₁
hypocenter	C₂₈₁	#1992
LCS	G ▷ C₂×C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₈₁

commutator

C₂×C₄×C₂₈₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₈₁

Fitting

C₂×#125×C₂₈₁

C₂

socle, prime radical

C₂³×C₂₈₁

C₂×D₄

cosocle

C₂²×C₄×C₂₈₁

C₂³

D₂₈₁

C₂₈₁

G ▷ C₂×C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
10. #125×C₂₈₁	C₄	semidirect
57. C₂₈₁	#1992	semidirect
65. #125	Dic₂₈₁	semidirect

H ≅ G/N

type

10. #125×C₂₈₁

C₄

semidirect

57. C₂₈₁

#1992

semidirect

65. #125

Dic₂₈₁

semidirect

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

subgroup	index
C₂×#22582790	2
#31823333	2
#31823225	2
C₂×#125×C₂₈₁	2
#1992	281

subgroup

index

C₂×#125×C₂₈₁

281

Subgroups — #49096273

Conjugacy classes of subgroups

Complements

Maximal subgroups