Group #49123431 — subgroups

Subgroups — #49123431

Order 36224.

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	#31826862	C₂	1	1	yes	maximal, characteristic, complemented
2	#31826581	C₂	1	1	yes	maximal, characteristic
3	#31826289	C₂	1	1	yes	maximal, characteristic, complemented
4	#31826285	C₂	1	1	yes	maximal, characteristic, complemented
5	#31826276	C₂	1	1	yes	maximal, characteristic
6	#31826274	C₂	1	1	yes	maximal, characteristic
7	#365×C₂₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#22583632	C₂²	1	1	yes	characteristic
9	#22583618	—	2	1	no	—
10	#22583609	—	2	1	no	—
11	#22583550	C₂²	1	1	yes	characteristic
12	#22583542	C₂²	1	1	yes	characteristic
13	#22583541	—	2	1	no	—
14	#22583536	—	2	1	no	—
15	C₈×Dic₂₈₃	C₂²	1	1	yes	characteristic
16	M₃₂×C₂₈₃	C₂²	1	1	yes	characteristic
17	#108×C₂₈₃	C₂²	1	1	yes	characteristic
18	#106×C₂₈₃	C₂²	1	1	yes	characteristic
19	#17985392	—	2	1	no	—
20	C₂×Dic₅₆₆	—	4	1	no	—
21	Dic₁₁₃₂	Q₈	1	1	yes	characteristic, complemented
22	#17985381	—	2	1	no	—
23	D₂₂₆₄	Q₈	1	1	yes	characteristic, complemented
24	#17985375	—	4	1	no	—
25	C₂×C₂₈₃⋊C₈	D₄	1	1	yes	characteristic
26	C₈×D₂₈₃	—	4	1	no	—
27	C₄⋊Dic₂₈₃	—	2	1	no	—
28	Dic₂₈₃⋊C₄	—	4	1	no	—
29	C₄×Dic₂₈₃	D₄	1	1	yes	characteristic
30	C₂₈₃⋊C₁₆	—	2	1	no	—
31	M₁₆×C₂₈₃	—	2	1	no	—
32	C₂×C₈×C₂₈₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₂₈₃	—	2	1	no	—
34	C₁₆×C₂₈₃	—	2	1	no	—
35	Dic₅₆₆	—	2	1	no	—
36	Dic₅₆₆	—	8	2	no	—
37	D₁₁₃₂	—	2	1	no	—
38	C₂²⋊D₂₈₃	—	4	1	no	—
39	C₂×Dic₂₈₃	—	2	1	no	—
40	C₂×Dic₂₈₃	—	4	1	no	—
41	C₄×D₂₈₃	—	4	1	no	—
42	C₂₈₃⋊C₈	SD₁₆	1	1	yes	characteristic
43	C₂₈₃⋊C₈	SD₁₆	1	1	yes	characteristic
44	C₂×C₄×C₂₈₃	—	4	1	no	—
45	C₂×C₄×C₂₈₃	C₂×D₄	1	1	yes	characteristic
46	C₈×C₂₈₃	—	4	1	no	—
47	C₈×C₂₈₃	C₂×Q₈	1	1	yes	characteristic
48	C₈×C₂₈₃	#41	1	1	yes	characteristic
49	C₂²×C₂₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₂×D₂₈₃	—	4	1	no	—
51	C₄×C₂₈₃	—	8	1	no	—
52	C₄×C₂₈₃	C₂²⋊Q₈	1	1	yes	characteristic
53	C₄×C₂₈₃	C₂×SD₁₆	1	1	yes	characteristic
54	Dic₂₈₃	—	2	2	no	—
55	Dic₂₈₃	—	4	1	no	—
56	Dic₂₈₃	—	8	1	no	—
57	C₂×C₂₈₃	—	2	1	no	—
58	C₂×C₂₈₃	#476	1	1	yes	characteristic
59	D₂₈₃	—	8	1	no	—
60	C₂₈₃	#2230	1	1	yes	characteristic, complemented, Sylow
61	#2230	—	283	1	no	maximal, Sylow
62	#498	—	283	1	no	—
63	#443	—	283	1	no	—
64	#365	—	283	1	no	—
65	#365	D₂₈₃	1	1	yes	characteristic, complemented
66	#364	—	283	1	no	—
67	#360	—	283	1	no	—
68	#360	—	283	1	no	—
69	#135	—	283	1	no	—
70	#131	—	566	1	no	—
71	C₄⋊Q₈	—	566	1	no	—
72	M₃₂	—	283	1	no	—
73	M₃₂	C₂×D₂₈₃	1	1	yes	characteristic
74	#108	C₂×D₂₈₃	1	1	yes	characteristic
75	#106	—	283	1	no	—
76	#106	—	566	1	no	—
77	#106	C₂×D₂₈₃	1	1	yes	characteristic
78	#104	—	566	1	no	—
79	C₄×C₈	—	283	1	no	—
80	#41	—	566	1	no	—
81	C₂×Q₈	—	1132	1	no	—
82	Q₁₆	—	283	1	no	—
83	SD₁₆	—	566	1	no	—
84	D₈	—	283	1	no	—
85	M₁₆	—	2	1	no	—
86	M₁₆	—	1132	1	no	—
87	C₂×C₈	—	283	1	no	—
88	C₂×C₈	—	1132	1	no	—
89	C₂×C₈	C₂²×D₂₈₃	1	1	yes	characteristic
90	C₄⋊C₄	—	2	1	no	—
91	C₄⋊C₄	—	566	1	no	—
92	C₄⋊C₄	—	1132	1	no	—
93	C₄²	—	283	1	no	—
94	C₁₆	—	2	1	no	—
95	C₁₆	—	566	1	no	—
96	Q₈	—	566	1	no	—
97	Q₈	—	2264	2	no	—
98	D₄	—	566	1	no	—
99	D₄	—	1132	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	566	1	no	—
102	C₂×C₄	—	1132	1	no	—
103	C₂×C₄	—	1132	1	no	—
104	C₂×C₄	D₄×D₂₈₃	1	1	yes	characteristic
105	C₈	—	4	1	no	—
106	C₈	—	283	1	no	—
107	C₈	—	283	1	no	—
108	C₈	Q₈×D₂₈₃	1	1	yes	characteristic
109	C₈	#17985392	1	1	yes	characteristic
110	C₂²	—	1132	1	no	—
111	C₂²	#22583637	1	1	yes	characteristic
112	C₄	—	8	1	no	—
113	C₄	—	566	2	no	—
114	C₄	—	1132	1	no	—
115	C₄	—	2264	1	no	—
116	C₄	#22583588	1	1	yes	characteristic
117	C₄	SD₁₆×D₂₈₃	1	1	yes	characteristic
118	C₂	—	2	1	no	—
119	C₂	—	2264	1	no	—
120	C₂	#31826754	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31826862

C₂

yes

maximal, characteristic, complemented

#31826581

C₂

yes

maximal, characteristic

#31826289

C₂

yes

maximal, characteristic, complemented

#31826285

C₂

yes

maximal, characteristic, complemented

#31826276

C₂

yes

maximal, characteristic

#31826274

C₂

yes

maximal, characteristic

#365×C₂₈₃

C₂

yes

maximal, characteristic, complemented

#22583632

C₂²

yes

—

—

C₂²

yes

characteristic

#22583542

C₂²

yes

characteristic

#22583541

—

#22583536

—

C₈×Dic₂₈₃

C₂²

yes

characteristic

M₃₂×C₂₈₃

C₂²

yes

characteristic

#108×C₂₈₃

C₂²

yes

characteristic

#106×C₂₈₃

C₂²

yes

characteristic

#17985392

—

C₂×Dic₅₆₆

—

Dic₁₁₃₂

Q₈

yes

characteristic, complemented

#17985381

—

D₂₂₆₄

Q₈

yes

characteristic, complemented

#17985375

—

C₂×C₂₈₃⋊C₈

D₄

yes

characteristic

C₈×D₂₈₃

—

C₄⋊Dic₂₈₃

—

Dic₂₈₃⋊C₄

—

C₄×Dic₂₈₃

D₄

yes

characteristic

C₂₈₃⋊C₁₆

—

M₁₆×C₂₈₃

—

C₂×C₈×C₂₈₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₈₃

—

C₁₆×C₂₈₃

—

Dic₅₆₆

—

Dic₅₆₆

—

D₁₁₃₂

—

C₂²⋊D₂₈₃

—

C₂×Dic₂₈₃

—

C₂×Dic₂₈₃

—

C₄×D₂₈₃

—

C₂₈₃⋊C₈

SD₁₆

yes

characteristic

C₂₈₃⋊C₈

SD₁₆

yes

characteristic

C₂×C₄×C₂₈₃

—

C₂×C₄×C₂₈₃

C₂×D₄

yes

characteristic

C₈×C₂₈₃

—

C₈×C₂₈₃

C₂×Q₈

yes

C₈×C₂₈₃

yes

C₂²×C₂₈₃

yes

C₂×D₂₈₃

—

C₄×C₂₈₃

—

C₄×C₂₈₃

C₂²⋊Q₈

yes

characteristic

C₄×C₂₈₃

C₂×SD₁₆

yes

characteristic

Dic₂₈₃

—

Dic₂₈₃

—

Dic₂₈₃

—

C₂×C₂₈₃

—

C₂×C₂₈₃

#476

yes

characteristic

D₂₈₃

—

C₂₈₃

#2230

yes

characteristic, complemented, Sylow

#2230

—

283

maximal, Sylow

#498

—

283

—

#443

—

283

—

#365

—

283

—

#365

D₂₈₃

yes

characteristic, complemented

#364

—

283

—

#360

—

283

—

#360

—

283

—

#135

—

283

—

#131

—

566

—

C₄⋊Q₈

—

566

—

M₃₂

—

283

—

M₃₂

C₂×D₂₈₃

yes

characteristic

#108

C₂×D₂₈₃

yes

characteristic

#106

—

283

—

#106

—

566

—

#106

C₂×D₂₈₃

yes

characteristic

#104

—

566

—

C₄×C₈

—

283

—

#41

—

566

—

C₂×Q₈

—

1132

—

Q₁₆

—

283

—

SD₁₆

—

566

—

D₈

—

283

—

M₁₆

—

M₁₆

—

1132

—

C₂×C₈

—

283

—

C₂×C₈

—

1132

—

C₂×C₈

C₂²×D₂₈₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

566

—

C₄⋊C₄

—

1132

—

C₄²

—

283

—

C₁₆

—

C₁₆

—

566

—

Q₈

—

566

—

Q₈

—

2264

—

D₄

—

566

—

D₄

—

1132

—

100

C₂×C₄

—

101

C₂×C₄

—

566

—

102

C₂×C₄

—

1132

—

103

C₂×C₄

—

1132

—

104

C₂×C₄

D₄×D₂₈₃

yes

characteristic

105

C₈

—

106

C₈

—

283

—

107

C₈

—

283

—

108

C₈

Q₈×D₂₈₃

yes

characteristic

109

C₈

#17985392

yes

characteristic

110

C₂²

—

1132

—

111

C₂²

#22583637

yes

characteristic

112

C₄

—

113

C₄

—

566

—

114

C₄

—

1132

—

115

C₄

—

2264

—

116

C₄

#22583588

yes

characteristic

117

C₄

SD₁₆×D₂₈₃

yes

characteristic

118

C₂

—

119

C₂

—

2264

—

120

C₂

#31826754

yes

characteristic

	subgroup	G/H
center	C₂	#31826754
commutator, cosocle	C₂×C₈×C₂₈₃	C₂³
Frattini	C₂×C₈	C₂²×D₂₈₃
Fitting	#365×C₂₈₃	C₂
socle	C₂×C₂₈₃	#476
prime radical	#22583632	C₂²
hypercenter	#365	D₂₈₃
hypocenter	C₂₈₃	#2230
LCS	G ▷ C₂×C₈×C₂₈₃ ▷ C₄×C₂₈₃ ▷ C₂×C₂₈₃ ▷ C₂₈₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

subgroup

G/H

center

C₂

#31826754

commutator, cosocle

C₂×C₈×C₂₈₃

C₂³

Frattini

C₂×C₈

C₂²×D₂₈₃

Fitting

#365×C₂₈₃

C₂

C₂×C₂₈₃

C₂²

D₂₈₃

C₂₈₃

G ▷ C₂×C₈×C₂₈₃ ▷ C₄×C₂₈₃ ▷ C₂×C₂₈₃ ▷ C₂₈₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

N	H ≅ G/N	type
1. #31826862	C₂	semidirect
3. #31826289	C₂	semidirect
4. #31826285	C₂	semidirect
7. #365×C₂₈₃	C₂	semidirect
21. Dic₁₁₃₂	Q₈	semidirect
23. D₂₂₆₄	Q₈	semidirect
60. C₂₈₃	#2230	semidirect
65. #365	D₂₈₃	semidirect

H ≅ G/N

type

1. #31826862

C₂

semidirect

3. #31826289

C₂

semidirect

4. #31826285

C₂

semidirect

7. #365×C₂₈₃

C₂

semidirect

21. Dic₁₁₃₂

Q₈

semidirect

23. D₂₂₆₄

Q₈

semidirect

60. C₂₈₃

#2230

semidirect

65. #365

D₂₈₃

semidirect

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

subgroup	index
#31826862	2
#31826581	2
#31826289	2
#31826285	2
#31826276	2
#31826274	2
#365×C₂₈₃	2
#2230	283

subgroup

index

#365×C₂₈₃

283

Subgroups — #49123431

Conjugacy classes of subgroups

Complements

Maximal subgroups