Group #49097473 — subgroups

Subgroups — #49097473

Order 35968.

126 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 43 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	#31823579	C₂	1	1	yes	maximal, characteristic
3	#31823287	C₂	1	1	yes	maximal, characteristic, complemented
4	#31823283	C₂	1	1	yes	maximal, characteristic, complemented
5	#31823274	C₂	1	1	yes	maximal, characteristic
6	#31823272	C₂	1	1	yes	maximal, characteristic
7	#365×C₂₈₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#22582901	C₂²	1	1	yes	characteristic
9	#22582887	—	2	1	no	—
10	#22582878	—	2	1	no	—
11	#22582819	C₂²	1	1	yes	characteristic
12	#22582811	C₂²	1	1	yes	characteristic
13	#22582810	—	2	1	no	—
14	#22582805	—	2	1	no	—
15	C₈×Dic₂₈₁	C₂²	1	1	yes	characteristic
16	M₃₂×C₂₈₁	C₂²	1	1	yes	characteristic
17	#108×C₂₈₁	C₂²	1	1	yes	characteristic
18	#106×C₂₈₁	C₂²	1	1	yes	characteristic
19	#17985138	—	2	1	no	—
20	C₂×Dic₅₆₂	—	4	1	no	—
21	Dic₁₁₂₄	Q₈	1	1	yes	characteristic, complemented
22	#17985127	—	2	1	no	—
23	D₂₂₄₈	Q₈	1	1	yes	characteristic, complemented
24	#17985121	—	4	1	no	—
25	C₂×#14555886	D₄	1	1	yes	characteristic
26	C₈×D₂₈₁	—	4	1	no	—
27	C₄⋊Dic₂₈₁	—	2	1	no	—
28	#17985117	—	4	1	no	—
29	C₄×Dic₂₈₁	D₄	1	1	yes	characteristic
30	#17985113	—	2	1	no	—
31	M₁₆×C₂₈₁	—	2	1	no	—
32	C₂×C₈×C₂₈₁	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₂₈₁	—	2	1	no	—
34	C₁₆×C₂₈₁	—	2	1	no	—
35	Dic₅₆₂	—	2	1	no	—
36	Dic₅₆₂	—	8	2	no	—
37	D₁₁₂₄	—	2	1	no	—
38	C₂²⋊D₂₈₁	—	4	1	no	—
39	C₂×Dic₂₈₁	—	2	1	no	—
40	C₂×Dic₂₈₁	—	4	1	no	—
41	C₄×D₂₈₁	—	4	1	no	—
42	#14555886	SD₁₆	1	1	yes	characteristic
43	#14555886	SD₁₆	1	1	yes	characteristic
44	C₂×C₄×C₂₈₁	—	4	1	no	—
45	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
46	C₈×C₂₈₁	—	4	1	no	—
47	C₈×C₂₈₁	C₂×Q₈	1	1	yes	characteristic
48	C₈×C₂₈₁	#41	1	1	yes	characteristic
49	C₂²×C₂₈₁	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₂×D₂₈₁	—	4	1	no	—
51	C₄×C₂₈₁	—	8	1	no	—
52	C₄×C₂₈₁	C₂²⋊Q₈	1	1	yes	characteristic
53	C₄×C₂₈₁	C₂×SD₁₆	1	1	yes	characteristic
54	Dic₂₈₁	—	2	2	no	—
55	Dic₂₈₁	—	4	1	no	—
56	Dic₂₈₁	—	8	1	no	—
57	C₂×C₂₈₁	—	2	1	no	—
58	C₂×C₂₈₁	#476	1	1	yes	characteristic
59	D₂₈₁	—	8	1	no	—
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	D₂₈₁	1	1	yes	characteristic, complemented
66	#364	—	281	1	no	—
67	#360	—	281	1	no	—
68	#360	—	281	1	no	—
69	#135	—	281	1	no	—
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	C₂×D₂₈₁	1	1	yes	characteristic
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	—	562	1	no	—
81	C₂×Q₈	—	1124	1	no	—
82	Q₁₆	—	281	1	no	—
83	SD₁₆	—	562	1	no	—
84	D₈	—	281	1	no	—
85	M₁₆	—	2	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	—	562	1	no	—
97	Q₈	—	2248	2	no	—
98	D₄	—	562	1	no	—
99	D₄	—	1124	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	562	1	no	—
102	C₂×C₄	—	1124	1	no	—
103	C₂×C₄	—	1124	1	no	—
104	C₂×C₄	D₄×D₂₈₁	1	1	yes	characteristic
105	C₈	—	4	1	no	—
106	C₈	—	281	1	no	—
107	C₈	—	281	1	no	—
108	C₈	Q₈×D₂₈₁	1	1	yes	characteristic
109	C₈	#17985138	1	1	yes	characteristic
110	C₂²	—	1124	1	no	—
111	C₂²	#22582906	1	1	yes	characteristic
112	C₄	—	8	1	no	—
113	C₄	—	562	2	no	—
114	C₄	—	1124	1	no	—
115	C₄	—	2248	1	no	—
116	C₄	#22582857	1	1	yes	characteristic
117	C₄	SD₁₆×D₂₈₁	1	1	yes	characteristic
118	C₂	—	2	1	no	—
119	C₂	—	2248	1	no	—
120	C₂	#31823752	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31823860

C₂

yes

maximal, characteristic, complemented

#31823579

C₂

yes

maximal, characteristic

#31823287

C₂

yes

maximal, characteristic, complemented

#31823283

C₂

yes

maximal, characteristic, complemented

#31823274

C₂

yes

maximal, characteristic

#31823272

C₂

yes

maximal, characteristic

#365×C₂₈₁

C₂

yes

maximal, characteristic, complemented

#22582901

C₂²

yes

—

—

C₂²

yes

characteristic

#22582811

C₂²

yes

characteristic

#22582810

—

#22582805

—

C₈×Dic₂₈₁

C₂²

yes

characteristic

M₃₂×C₂₈₁

C₂²

yes

characteristic

#108×C₂₈₁

C₂²

yes

characteristic

#106×C₂₈₁

C₂²

yes

characteristic

#17985138

—

C₂×Dic₅₆₂

—

Dic₁₁₂₄

Q₈

yes

characteristic, complemented

#17985127

—

D₂₂₄₈

Q₈

yes

characteristic, complemented

#17985121

—

C₂×#14555886

D₄

yes

characteristic

C₈×D₂₈₁

—

C₄⋊Dic₂₈₁

—

#17985117

—

C₄×Dic₂₈₁

D₄

yes

characteristic

#17985113

—

M₁₆×C₂₈₁

—

C₂×C₈×C₂₈₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₈₁

—

C₁₆×C₂₈₁

—

Dic₅₆₂

—

Dic₅₆₂

—

D₁₁₂₄

—

C₂²⋊D₂₈₁

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

C₄×D₂₈₁

—

#14555886

SD₁₆

yes

characteristic

#14555886

SD₁₆

yes

characteristic

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₈×C₂₈₁

—

C₈×C₂₈₁

C₂×Q₈

yes

C₈×C₂₈₁

yes

C₂²×C₂₈₁

yes

C₂×D₂₈₁

—

C₄×C₂₈₁

—

C₄×C₂₈₁

C₂²⋊Q₈

yes

characteristic

C₄×C₂₈₁

C₂×SD₁₆

yes

characteristic

Dic₂₈₁

—

Dic₂₈₁

—

Dic₂₈₁

—

C₂×C₂₈₁

—

C₂×C₂₈₁

#476

yes

characteristic

D₂₈₁

—

C₂₈₁

#2230

yes

characteristic, complemented, Sylow

#2230

—

281

maximal, Sylow

#498

—

281

—

#443

—

281

—

#365

—

281

—

#365

D₂₈₁

yes

characteristic, complemented

#364

—

281

—

#360

—

281

—

#360

—

281

—

#135

—

281

—

#131

—

562

—

C₄⋊Q₈

—

562

—

M₃₂

—

281

—

M₃₂

C₂×D₂₈₁

yes

characteristic

#108

C₂×D₂₈₁

yes

characteristic

#106

—

281

—

#106

—

562

—

#106

C₂×D₂₈₁

yes

characteristic

#104

—

562

—

C₄×C₈

—

281

—

#41

—

562

—

C₂×Q₈

—

1124

—

Q₁₆

—

281

—

SD₁₆

—

562

—

D₈

—

281

—

M₁₆

—

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

—

562

—

Q₈

—

2248

—

D₄

—

562

—

D₄

—

1124

—

100

C₂×C₄

—

101

C₂×C₄

—

562

—

102

C₂×C₄

—

1124

—

103

C₂×C₄

—

1124

—

104

C₂×C₄

D₄×D₂₈₁

yes

characteristic

105

C₈

—

106

C₈

—

281

—

107

C₈

—

281

—

108

C₈

Q₈×D₂₈₁

yes

characteristic

109

C₈

#17985138

yes

characteristic

110

C₂²

—

1124

—

111

C₂²

#22582906

yes

characteristic

112

C₄

—

113

C₄

—

562

—

114

C₄

—

1124

—

115

C₄

—

2248

—

116

C₄

#22582857

yes

characteristic

117

C₄

SD₁₆×D₂₈₁

yes

characteristic

118

C₂

—

119

C₂

—

2248

—

120

C₂

#31823752

yes

characteristic

	subgroup	G/H
center	C₂	#31823752
commutator, cosocle	C₂×C₈×C₂₈₁	C₂³
Frattini	C₂×C₈	C₂²×D₂₈₁
Fitting	#365×C₂₈₁	C₂
socle	C₂×C₂₈₁	#476
prime radical	#22582901	C₂²
hypercenter	#365	D₂₈₁
hypocenter	C₂₈₁	#2230
LCS	G ▷ C₂×C₈×C₂₈₁ ▷ C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

subgroup

G/H

center

C₂

#31823752

commutator, cosocle

C₂×C₈×C₂₈₁

C₂³

Frattini

C₂×C₈

C₂²×D₂₈₁

Fitting

#365×C₂₈₁

C₂

C₂×C₂₈₁

C₂²

D₂₈₁

C₂₈₁

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

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

N	H ≅ G/N	type
1. #31823860	C₂	semidirect
3. #31823287	C₂	semidirect
4. #31823283	C₂	semidirect
7. #365×C₂₈₁	C₂	semidirect
21. Dic₁₁₂₄	Q₈	semidirect
23. D₂₂₄₈	Q₈	semidirect
60. C₂₈₁	#2230	semidirect
65. #365	D₂₈₁	semidirect

H ≅ G/N

type

1. #31823860

C₂

semidirect

3. #31823287

C₂

semidirect

4. #31823283

C₂

semidirect

7. #365×C₂₈₁

C₂

semidirect

21. Dic₁₁₂₄

Q₈

semidirect

23. D₂₂₄₈

Q₈

semidirect

60. C₂₈₁

#2230

semidirect

65. #365

D₂₈₁

semidirect

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

subgroup	index
#31823860	2
#31823579	2
#31823287	2
#31823283	2
#31823274	2
#31823272	2
#365×C₂₈₁	2
#2230	281

subgroup

index

#365×C₂₈₁

281

Subgroups — #49097473

Conjugacy classes of subgroups

Complements

Maximal subgroups