Group #49097472 — subgroups

Subgroups — #49097472

Order 35968.

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

normal subgroups only

This group has 126 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	#31823860	C₂	1	1	yes	maximal, characteristic, complemented
2	#31823580	C₂	1	1	yes	maximal, characteristic
3	#31823286	C₂	1	1	yes	maximal, characteristic, complemented
4	#31823285	C₂	1	1	yes	maximal, characteristic, complemented
5	#31823283	C₂	1	1	yes	maximal, characteristic, complemented
6	#31823273	C₂	1	1	yes	maximal, characteristic
7	#360×C₂₈₁	C₂	1	1	yes	maximal, characteristic
8	#22582886	—	2	1	no	—
9	#22582878	—	2	1	no	—
10	#22582819	C₂²	1	1	yes	characteristic
11	#22582815	C₂²	1	1	yes	characteristic
12	#22582811	C₂²	1	1	yes	characteristic
13	#22582810	—	2	1	no	—
14	#22582806	—	2	1	no	—
15	C₈×Dic₂₈₁	C₂²	1	1	yes	characteristic
16	#135×C₂₈₁	C₂²	1	1	yes	characteristic
17	M₃₂×C₂₈₁	C₂²	1	1	yes	characteristic
18	#106×C₂₈₁	C₂²	1	1	yes	characteristic
19	C₂×Dic₅₆₂	—	4	1	no	—
20	#17985122	—	2	1	no	—
21	#17985122	—	4	1	no	—
22	C₂×#14555886	—	4	1	no	—
23	C₂×#14555886	D₄	1	1	yes	characteristic
24	C₄⋊Dic₂₈₁	—	2	1	no	—
25	#17985117	—	4	1	no	—
26	C₄×Dic₂₈₁	D₄	1	1	yes	characteristic
27	#17985113	—	2	1	no	—
28	#41×C₂₈₁	—	2	1	no	—
29	Q₁₆×C₂₈₁	Q₈	1	1	yes	characteristic, complemented
30	SD₁₆×C₂₈₁	—	2	1	no	—
31	D₈×C₂₈₁	Q₈	1	1	yes	characteristic, complemented
32	C₂×C₈×C₂₈₁	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₂₈₁	—	2	1	no	—
34	C₁₆×C₂₈₁	—	2	1	no	—
35	Dic₅₆₂	—	8	2	no	—
36	C₂×Dic₂₈₁	—	2	1	no	—
37	C₂×Dic₂₈₁	—	4	1	no	—
38	#14555886	—	4	1	no	—
39	#14555886	SD₁₆	1	1	yes	characteristic
40	#14555886	SD₁₆	1	1	yes	characteristic
41	Q₈×C₂₈₁	—	2	1	no	—
42	D₄×C₂₈₁	—	2	1	no	—
43	D₄×C₂₈₁	—	4	1	no	—
44	C₂×C₄×C₂₈₁	—	4	1	no	—
45	C₂×C₄×C₂₈₁	—	4	1	no	—
46	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
47	C₈×C₂₈₁	C₂×Q₈	1	1	yes	characteristic
48	C₈×C₂₈₁	#41	1	1	yes	characteristic
49	C₂²×C₂₈₁	—	4	1	no	—
50	C₂²×C₂₈₁	AGL(1,ℤ/8)	1	1	yes	characteristic
51	C₄×C₂₈₁	—	4	1	no	—
52	C₄×C₂₈₁	—	8	1	no	—
53	C₄×C₂₈₁	C₂²⋊Q₈	1	1	yes	characteristic
54	C₄×C₂₈₁	C₂×SD₁₆	1	1	yes	characteristic
55	Dic₂₈₁	—	2	2	no	—
56	Dic₂₈₁	—	8	1	no	—
57	C₂×C₂₈₁	—	2	1	no	—
58	C₂×C₂₈₁	—	8	1	no	—
59	C₂×C₂₈₁	#476	1	1	yes	characteristic
60	C₂₈₁	#2230	1	1	yes	characteristic, complemented, Sylow
61	#2230	—	281	1	no	maximal, Sylow
62	#498	—	281	1	no	—
63	#443	—	281	1	no	—
64	#365	—	281	1	no	—
65	#365	—	281	1	no	—
66	#364	—	281	1	no	—
67	#360	—	281	1	no	—
68	#360	D₂₈₁	1	1	yes	characteristic
69	#135	C₂×D₂₈₁	1	1	yes	characteristic
70	#131	—	562	1	no	—
71	C₄⋊Q₈	—	562	1	no	—
72	M₃₂	—	281	1	no	—
73	M₃₂	C₂×D₂₈₁	1	1	yes	characteristic
74	#108	—	281	1	no	—
75	#106	—	281	1	no	—
76	#106	—	562	1	no	—
77	#106	C₂×D₂₈₁	1	1	yes	characteristic
78	#104	—	562	1	no	—
79	C₄×C₈	—	281	1	no	—
80	#41	—	2	1	no	—
81	C₂×Q₈	—	1124	1	no	—
82	Q₁₆	Dic₅₆₂	1	1	yes	characteristic, complemented
83	SD₁₆	—	2	1	no	—
84	D₈	Dic₅₆₂	1	1	yes	characteristic, complemented
85	M₁₆	—	562	1	no	—
86	M₁₆	—	1124	1	no	—
87	C₂×C₈	—	281	1	no	—
88	C₂×C₈	—	1124	1	no	—
89	C₂×C₈	C₂²×D₂₈₁	1	1	yes	characteristic
90	C₄⋊C₄	—	2	1	no	—
91	C₄⋊C₄	—	562	1	no	—
92	C₄⋊C₄	—	1124	1	no	—
93	C₄²	—	281	1	no	—
94	C₁₆	—	2	1	no	—
95	C₁₆	—	562	1	no	—
96	Q₈	—	2	1	no	—
97	Q₈	—	2248	2	no	—
98	D₄	—	2	1	no	—
99	D₄	—	4	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	—	562	1	no	—
103	C₂×C₄	—	1124	1	no	—
104	C₂×C₄	D₄×D₂₈₁	1	1	yes	characteristic
105	C₈	—	281	1	no	—
106	C₈	—	281	1	no	—
107	C₈	—	1124	1	no	—
108	C₈	C₂×Dic₅₆₂	1	1	yes	characteristic
109	C₈	#17985137	1	1	yes	characteristic
110	C₂²	—	4	1	no	—
111	C₂²	#22582903	1	1	yes	characteristic
112	C₄	—	4	1	no	—
113	C₄	—	8	1	no	—
114	C₄	—	562	2	no	—
115	C₄	—	2248	1	no	—
116	C₄	#22582856	1	1	yes	characteristic
117	C₄	SD₁₆×D₂₈₁	1	1	yes	characteristic
118	C₂	—	2	1	no	—
119	C₂	—	8	1	no	—
120	C₂	#31823751	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31823860

C₂

yes

maximal, characteristic, complemented

#31823580

C₂

yes

maximal, characteristic

#31823286

C₂

yes

maximal, characteristic, complemented

#31823285

C₂

yes

maximal, characteristic, complemented

#31823283

C₂

yes

maximal, characteristic, complemented

#31823273

C₂

yes

maximal, characteristic

#360×C₂₈₁

C₂

yes

maximal, characteristic

#22582886

—

#22582878

—

#22582819

C₂²

yes

characteristic

#22582815

C₂²

yes

characteristic

#22582811

C₂²

yes

characteristic

#22582810

—

#22582806

—

C₈×Dic₂₈₁

C₂²

yes

characteristic

#135×C₂₈₁

C₂²

yes

characteristic

M₃₂×C₂₈₁

C₂²

yes

characteristic

#106×C₂₈₁

C₂²

yes

characteristic

C₂×Dic₅₆₂

—

—

—

—

D₄

yes

characteristic

C₄⋊Dic₂₈₁

—

#17985117

—

C₄×Dic₂₈₁

D₄

yes

characteristic

#17985113

—

#41×C₂₈₁

—

Q₁₆×C₂₈₁

Q₈

yes

characteristic, complemented

SD₁₆×C₂₈₁

—

D₈×C₂₈₁

Q₈

yes

characteristic, complemented

C₂×C₈×C₂₈₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₈₁

—

C₁₆×C₂₈₁

—

Dic₅₆₂

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

#14555886

—

#14555886

SD₁₆

yes

characteristic

#14555886

SD₁₆

yes

characteristic

Q₈×C₂₈₁

—

D₄×C₂₈₁

—

D₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₈×C₂₈₁

C₂×Q₈

yes

characteristic

C₈×C₂₈₁

#41

yes

characteristic

C₂²×C₂₈₁

—

C₂²×C₂₈₁

AGL(1,ℤ/8)

yes

characteristic

C₄×C₂₈₁

—

C₄×C₂₈₁

—

C₄×C₂₈₁

C₂²⋊Q₈

yes

characteristic

C₄×C₂₈₁

C₂×SD₁₆

yes

characteristic

Dic₂₈₁

—

Dic₂₈₁

—

C₂×C₂₈₁

—

C₂×C₂₈₁

—

C₂×C₂₈₁

#476

yes

characteristic

C₂₈₁

#2230

yes

characteristic, complemented, Sylow

#2230

—

281

maximal, Sylow

#498

—

281

—

#443

—

281

—

#365

—

281

—

#365

—

281

—

#364

—

281

—

#360

—

281

—

#360

D₂₈₁

yes

characteristic

#135

C₂×D₂₈₁

yes

characteristic

#131

—

562

—

C₄⋊Q₈

—

562

—

M₃₂

—

281

—

M₃₂

C₂×D₂₈₁

yes

characteristic

#108

—

281

—

#106

—

281

—

#106

—

562

—

#106

C₂×D₂₈₁

yes

characteristic

#104

—

562

—

C₄×C₈

—

281

—

#41

—

C₂×Q₈

—

1124

—

Q₁₆

Dic₅₆₂

yes

characteristic, complemented

SD₁₆

—

D₈

Dic₅₆₂

yes

characteristic, complemented

M₁₆

—

562

—

M₁₆

—

1124

—

C₂×C₈

—

281

—

C₂×C₈

—

1124

—

C₂×C₈

C₂²×D₂₈₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

562

—

C₄⋊C₄

—

1124

—

C₄²

—

281

—

C₁₆

—

C₁₆

—

562

—

Q₈

—

Q₈

—

2248

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

562

—

103

C₂×C₄

—

1124

—

104

C₂×C₄

D₄×D₂₈₁

yes

characteristic

105

C₈

—

281

—

106

C₈

—

281

—

107

C₈

—

1124

—

108

C₈

C₂×Dic₅₆₂

yes

characteristic

109

C₈

#17985137

yes

characteristic

110

C₂²

—

111

C₂²

#22582903

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

562

—

115

C₄

—

2248

—

116

C₄

#22582856

yes

characteristic

117

C₄

SD₁₆×D₂₈₁

yes

characteristic

118

C₂

—

119

C₂

—

120

C₂

#31823751

yes

characteristic

	subgroup	G/H
center	C₂	#31823751
commutator, cosocle	C₂×C₈×C₂₈₁	C₂³
Frattini	C₂×C₈	C₂²×D₂₈₁
Fitting	#360×C₂₈₁	C₂
socle	C₂×C₂₈₁	#476
prime radical	#135×C₂₈₁	C₂²
hypercenter	#360	D₂₈₁
hypocenter	C₂₈₁	#2230
LCS	G ▷ C₂×C₈×C₂₈₁ ▷ C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #360

subgroup

G/H

center

C₂

#31823751

commutator, cosocle

C₂×C₈×C₂₈₁

C₂³

Frattini

C₂×C₈

C₂²×D₂₈₁

Fitting

#360×C₂₈₁

C₂

socle

C₂×C₂₈₁

#476

prime radical

#135×C₂₈₁

C₂²

D₂₈₁

C₂₈₁

G ▷ C₂×C₈×C₂₈₁ ▷ C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #360

N	H ≅ G/N	type
1. #31823860	C₂	semidirect
3. #31823286	C₂	semidirect
4. #31823285	C₂	semidirect
5. #31823283	C₂	semidirect
29. Q₁₆×C₂₈₁	Q₈	semidirect
31. D₈×C₂₈₁	Q₈	semidirect
60. C₂₈₁	#2230	semidirect
82. Q₁₆	Dic₅₆₂	semidirect
84. D₈	Dic₅₆₂	semidirect

H ≅ G/N

type

1. #31823860

C₂

semidirect

3. #31823286

C₂

semidirect

4. #31823285

C₂

semidirect

5. #31823283

C₂

semidirect

29. Q₁₆×C₂₈₁

Q₈

semidirect

31. D₈×C₂₈₁

Q₈

semidirect

60. C₂₈₁

#2230

semidirect

82. Q₁₆

Dic₅₆₂

semidirect

84. D₈

Dic₅₆₂

semidirect

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

subgroup	index
#31823860	2
#31823580	2
#31823286	2
#31823285	2
#31823283	2
#31823273	2
#360×C₂₈₁	2
#2230	281

subgroup

index

#360×C₂₈₁

281

Subgroups — #49097472

Conjugacy classes of subgroups

Complements

Maximal subgroups