Group #49096124 — subgroups

Subgroups — #49096124

Order 35968.

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

normal subgroups only

This group has 308 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×#22582803	C₂	1	4	yes	maximal
2	C₂×#22582790	C₂	1	1	yes	maximal, characteristic
3	#31823329	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Q₈×C₂₈₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×#14555886	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₂₈₁	C₂²	1	2	yes	—
7	#22582803	—	2	8	no	—
8	#22582790	—	2	2	no	—
9	#22582786	—	2	2	no	—
10	C₂²×Q₈×C₂₈₁	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₂₈₁	—	2	2	no	—
12	C₄⋊Q₈×C₂₈₁	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₂₈₁	—	2	2	no	—
14	C₂×C₄²×C₂₈₁	C₂²	1	1	yes	characteristic
15	C₂²×Dic₂₈₁	—	2	2	no	—
16	C₂×#14555886	—	2	2	no	—
17	C₂×#14555886	—	2	2	no	—
18	C₂×#14555886	D₄	1	4	yes	—
19	C₄⋊Dic₂₈₁	—	2	4	no	—
20	C₂×Q₈×C₂₈₁	—	2	2	no	—
21	C₂×Q₈×C₂₈₁	—	2	4	no	—
22	C₂×Q₈×C₂₈₁	—	2	4	no	—
23	C₂×Q₈×C₂₈₁	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₂₈₁	—	2	2	no	—
25	C₂²×C₄×C₂₈₁	D₄	1	2	yes	—
26	C₂²×C₄×C₂₈₁	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₂₈₁	—	2	8	no	—
28	C₄²×C₂₈₁	C₂×C₄	1	2	yes	—
29	C₄²×C₂₈₁	D₄	1	2	yes	—
30	C₂×Dic₂₈₁	—	4	2	no	—
31	C₂×Dic₂₈₁	—	4	4	no	—
32	#14555886	—	2	4	no	—
33	C₂³×C₂₈₁	C₂×D₄	1	1	yes	characteristic
34	Q₈×C₂₈₁	—	2	8	no	—
35	Q₈×C₂₈₁	—	4	4	no	—
36	C₂×C₄×C₂₈₁	—	2	4	no	—
37	C₂×C₄×C₂₈₁	—	2	4	no	—
38	C₂×C₄×C₂₈₁	—	4	2	no	—
39	C₂×C₄×C₂₈₁	—	4	2	no	—
40	C₂×C₄×C₂₈₁	C₂²⋊C₄	1	4	yes	—
41	C₂×C₄×C₂₈₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₂₈₁	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₈₁	#41	1	2	yes	—
45	C₂²×C₂₈₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₂₈₁	#124	1	1	yes	characteristic
47	C₂²×C₂₈₁	#127	1	1	yes	characteristic
48	C₂²×C₂₈₁	#137	1	4	yes	—
49	C₄×C₂₈₁	—	2	4	no	—
50	C₄×C₂₈₁	—	4	4	no	—
51	C₄×C₂₈₁	C₄×D₄	1	2	yes	—
52	C₄×C₂₈₁	#121	1	2	yes	—
53	Dic₂₈₁	—	8	2	no	—
54	C₂×C₂₈₁	#390	1	1	yes	characteristic
55	C₂×C₂₈₁	#419	1	2	yes	—
56	C₂×C₂₈₁	#491	1	2	yes	—
57	C₂×C₂₈₁	#497	1	2	yes	—
58	C₂₈₁	#1960	1	1	yes	characteristic, complemented, Sylow
59	#1960	—	281	1	no	maximal, Sylow
60	C₂×C₄⋊Q₈	D₂₈₁	1	1	yes	characteristic
61	C₂×#103	—	281	4	no	—
62	C₂×#97	—	281	1	no	—
63	#382	—	281	1	no	—
64	C₂²×Q₈	C₂×D₂₈₁	1	2	yes	—
65	C₂²×C₈	—	281	2	no	—
66	C₄⋊Q₈	—	2	2	no	—
67	C₄⋊Q₈	Dic₂₈₁	1	4	yes	complemented
68	C₂×C₄⋊C₄	—	2	2	no	—
69	C₂×C₄⋊C₄	—	281	2	no	—
70	C₂×C₄²	C₂×D₂₈₁	1	1	yes	characteristic
71	#103	—	562	8	no	—
72	#97	—	562	2	no	—
73	#95	—	562	2	no	—
74	C₂×Q₈	—	2	2	no	—
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	2	4	no	—
77	C₂×Q₈	C₂×Dic₂₈₁	1	4	yes	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	—	562	2	no	—
80	C₂²×C₄	C₂²⋊D₂₈₁	1	2	yes	—
81	C₂²×C₄	C₂²×D₂₈₁	1	1	yes	characteristic
82	C₂×C₈	—	281	4	no	—
83	C₂×C₈	—	562	2	no	—
84	C₂×C₈	—	562	2	no	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	562	4	no	—
87	C₄²	C₂×Dic₂₈₁	1	2	yes	—
88	C₄²	C₂²⋊D₂₈₁	1	2	yes	—
89	C₂³	C₂×C₂²⋊D₂₈₁	1	1	yes	characteristic
90	Q₈	—	2	8	no	—
91	Q₈	—	4	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	1124	2	no	—
97	C₂×C₄	—	1124	4	no	—
98	C₂×C₄	C₂²⋊Dic₂₈₁	1	4	yes	—
99	C₂×C₄	C₂²×Dic₂₈₁	1	1	yes	characteristic
100	C₂×C₄	D₄×D₂₈₁	1	2	yes	—
101	C₂×C₄	C₂×C₂²⋊D₂₈₁	1	1	yes	characteristic
102	C₂×C₄	#17985137	1	2	yes	—
103	C₈	—	562	4	no	—
104	C₂²	C₂×C₂²⋊Dic₂₈₁	1	1	yes	characteristic
105	C₂²	#22582866	1	1	yes	characteristic
106	C₂²	#22582875	1	1	yes	characteristic
107	C₂²	#22582910	1	4	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	4	4	no	—
110	C₄	—	2248	2	no	—
111	C₄	D₄×Dic₂₈₁	1	2	yes	—
112	C₄	#22582850	1	2	yes	—
113	C₂	#31823366	1	1	yes	characteristic
114	C₂	#31823472	1	2	yes	—
115	C₂	#31823831	1	2	yes	—
116	C₂	#31823855	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22582803

C₂

yes

maximal

C₂×#22582790

C₂

yes

maximal, characteristic

#31823329

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₂₈₁

C₂

yes

maximal, characteristic

C₂²×#14555886

C₂²

yes

—

C₂×C₄⋊Dic₂₈₁

C₂²

yes

—

#22582803

—

#22582790

—

#22582786

—

C₂²×Q₈×C₂₈₁

C₂²

yes

—

C₄⋊Q₈×C₂₈₁

—

C₄⋊Q₈×C₂₈₁

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₈₁

—

C₂×C₄²×C₂₈₁

C₂²

yes

characteristic

C₂²×Dic₂₈₁

—

C₂×#14555886

—

C₂×#14555886

—

C₂×#14555886

D₄

yes

—

C₄⋊Dic₂₈₁

—

C₂×Q₈×C₂₈₁

—

C₂×Q₈×C₂₈₁

—

C₂×Q₈×C₂₈₁

—

C₂×Q₈×C₂₈₁

C₂×C₄

yes

—

C₂²×C₄×C₂₈₁

—

C₂²×C₄×C₂₈₁

D₄

yes

—

C₂²×C₄×C₂₈₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₈₁

—

C₄²×C₂₈₁

C₂×C₄

yes

—

C₄²×C₂₈₁

D₄

yes

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

#14555886

—

C₂³×C₂₈₁

C₂×D₄

yes

characteristic

Q₈×C₂₈₁

—

Q₈×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

—

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₁

#41

yes

—

C₂²×C₂₈₁

C₂×C₂²⋊C₄

yes

C₂²×C₂₈₁

yes

C₂²×C₂₈₁

yes

C₂²×C₂₈₁

yes

—

C₄×C₂₈₁

—

C₄×C₂₈₁

—

C₄×C₂₈₁

C₄×D₄

yes

—

C₄×C₂₈₁

#121

yes

—

Dic₂₈₁

—

C₂×C₂₈₁

#390

yes

characteristic

C₂×C₂₈₁

#419

yes

—

C₂×C₂₈₁

#491

yes

—

C₂×C₂₈₁

#497

yes

—

C₂₈₁

#1960

yes

characteristic, complemented, Sylow

#1960

—

281

maximal, Sylow

C₂×C₄⋊Q₈

D₂₈₁

yes

characteristic

C₂×#103

—

281

—

C₂×#97

—

281

—

#382

—

281

—

C₂²×Q₈

C₂×D₂₈₁

yes

—

C₂²×C₈

—

281

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₂₈₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

281

—

C₂×C₄²

C₂×D₂₈₁

yes

characteristic

#103

—

562

—

#97

—

562

—

#95

—

562

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₂₈₁

yes

—

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

—

562

—

C₄²

C₂×Dic₂₈₁

yes

—

C₄²

C₂²⋊D₂₈₁

yes

—

C₂³

C₂×C₂²⋊D₂₈₁

yes

characteristic

Q₈

—

Q₈

—

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

100

C₂×C₄

D₄×D₂₈₁

yes

—

101

C₂×C₄

C₂×C₂²⋊D₂₈₁

yes

characteristic

102

C₂×C₄

#17985137

yes

—

103

C₈

—

562

—

104

C₂²

C₂×C₂²⋊Dic₂₈₁

yes

characteristic

105

C₂²

#22582866

yes

characteristic

106

C₂²

#22582875

yes

characteristic

107

C₂²

#22582910

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

2248

—

111

C₄

D₄×Dic₂₈₁

yes

—

112

C₄

#22582850

yes

—

113

C₂

#31823366

yes

characteristic

114

C₂

#31823472

yes

—

115

C₂

#31823831

yes

—

116

C₂

#31823855

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₈₁
commutator	C₂×C₄×C₂₈₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₈₁
Fitting	C₂×C₄⋊Q₈×C₂₈₁	C₂
socle, prime radical	C₂³×C₂₈₁	C₂×D₄
cosocle	C₂²×C₄×C₂₈₁	C₂³
hypercenter	C₂×C₄⋊Q₈	D₂₈₁
hypocenter	C₂₈₁	#1960
LCS	G ▷ C₂×C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₈₁

commutator

C₂×C₄×C₂₈₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₈₁

Fitting

C₂×C₄⋊Q₈×C₂₈₁

C₂

socle, prime radical

C₂³×C₂₈₁

C₂×D₄

cosocle

C₂²×C₄×C₂₈₁

C₂³

hypercenter

C₂×C₄⋊Q₈

D₂₈₁

hypocenter

C₂₈₁

#1960

LCS

G ▷ C₂×C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₂₈₁	C₄	semidirect
58. C₂₈₁	#1960	semidirect
67. C₄⋊Q₈	Dic₂₈₁	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₂₈₁

C₄

semidirect

58. C₂₈₁

#1960

semidirect

67. C₄⋊Q₈

Dic₂₈₁

semidirect

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

subgroup	index
C₂×#22582803	2
C₂×#22582790	2
#31823329	2
C₂×C₄⋊Q₈×C₂₈₁	2
#1960	281

subgroup

index

C₂×C₄⋊Q₈×C₂₈₁

281

Subgroups — #49096124

Conjugacy classes of subgroups

Complements

Maximal subgroups