Group #49120890 — subgroups

Subgroups — #49120890

Order 36224.

220 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 65 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	#31826842	C₂	1	1	yes	maximal, characteristic
2	#31826839	C₂	1	2	yes	maximal, complemented
3	#31826813	C₂	1	2	yes	maximal, complemented
4	#31826186	C₂	1	1	yes	maximal, characteristic, complemented
5	#498×C₂₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17985379	—	2	4	no	—
7	C₂×D₂₂₆₄	—	2	2	no	—
8	#22583607	C₂²	1	1	yes	characteristic
9	#22583607	C₂²	1	1	yes	characteristic
10	#22583530	—	2	4	no	—
11	C₄×C₂₈₃⋊C₈	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₂₈₃	C₂²	1	2	yes	—
13	#106×C₂₈₃	—	2	2	no	—
14	C₄×C₈×C₂₈₃	C₂²	1	1	yes	characteristic
15	C₂×D₁₁₃₂	—	2	1	no	—
16	C₂×D₁₁₃₂	—	2	1	no	—
17	C₂×D₁₁₃₂	—	2	2	no	—
18	C₂×D₁₁₃₂	—	2	2	no	—
19	#17985379	—	4	8	no	—
20	D₂₂₆₄	—	4	4	no	—
21	C₂×C₂₈₃⋊C₈	D₄	1	4	yes	—
22	C₂×Q₈×C₂₈₃	—	2	2	no	—
23	C₂×C₈×C₂₈₃	D₄	1	2	yes	—
24	C₄⋊C₄×C₂₈₃	—	2	2	no	—
25	C₄⋊C₄×C₂₈₃	—	2	2	no	—
26	C₄²×C₂₈₃	C₂³	1	1	yes	characteristic
27	C₂²×D₂₈₃	—	4	1	no	—
28	C₂²×D₂₈₃	—	4	1	no	—
29	D₁₁₃₂	—	4	2	no	—
30	D₁₁₃₂	—	4	2	no	—
31	D₁₁₃₂	—	4	4	no	—
32	D₁₁₃₂	—	4	4	no	—
33	C₂₈₃⋊C₈	—	2	4	no	—
34	Q₈×C₂₈₃	—	4	4	no	—
35	C₂×C₄×C₂₈₃	—	4	2	no	—
36	C₂×C₄×C₂₈₃	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₂₈₃	C₂×D₄	1	1	yes	characteristic
38	C₈×C₂₈₃	SD₁₆	1	4	yes	complemented
39	C₂²×C₂₈₃	#127	1	1	yes	characteristic
40	C₂×D₂₈₃	—	8	1	no	—
41	C₂×D₂₈₃	—	8	1	no	—
42	C₂×D₂₈₃	—	8	2	no	—
43	C₂×D₂₈₃	—	8	2	no	—
44	C₄×C₂₈₃	—	8	2	no	—
45	C₄×C₂₈₃	C₂×SD₁₆	1	2	yes	—
46	C₄×C₂₈₃	AGL(1,ℤ/8)	1	4	yes	—
47	C₂×C₂₈₃	#492	1	1	yes	characteristic
48	C₂×C₂₈₃	#496	1	2	yes	—
49	D₂₈₃	—	16	1	no	—
50	D₂₈₃	—	16	1	no	—
51	C₂₈₃	#1715	1	1	yes	characteristic, complemented, Sylow
52	#1715	—	283	1	no	maximal, Sylow
53	#498	D₂₈₃	1	1	yes	characteristic, complemented
54	#493	—	283	1	no	—
55	#492	—	283	2	no	—
56	#486	—	283	2	no	—
57	#322	—	283	1	no	—
58	C₂×SD₁₆	—	566	4	no	—
59	C₂×D₈	—	566	2	no	—
60	C₄⋊Q₈	C₂×D₂₈₃	1	2	yes	—
61	#127	—	283	1	no	—
62	#127	—	283	1	no	—
63	#106	—	2	2	no	—
64	#102	—	566	4	no	—
65	C₄×C₈	—	283	2	no	—
66	C₄×C₈	C₂×D₂₈₃	1	1	yes	characteristic
67	C₂×Q₈	—	2	2	no	—
68	C₂×D₄	—	566	2	no	—
69	C₂×D₄	—	566	2	no	—
70	C₂×D₄	—	566	1	no	—
71	C₂×D₄	—	566	1	no	—
72	SD₁₆	—	1132	8	no	—
73	D₈	—	1132	4	no	—
74	C₂×C₈	—	283	4	no	—
75	C₂×C₈	C₂²⋊D₂₈₃	1	2	yes	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄²	C₂²×D₂₈₃	1	1	yes	characteristic
79	C₂³	—	1132	1	no	—
80	C₂³	—	1132	1	no	—
81	Q₈	—	4	4	no	—
82	D₄	—	1132	2	no	—
83	D₄	—	1132	2	no	—
84	D₄	—	1132	4	no	—
85	D₄	—	1132	4	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	D₄×D₂₈₃	1	2	yes	—
88	C₂×C₄	C₂×C₂²⋊D₂₈₃	1	1	yes	characteristic
89	C₈	—	566	4	no	—
90	C₈	#17985379	1	4	yes	complemented
91	C₂²	—	2264	1	no	—
92	C₂²	—	2264	1	no	—
93	C₂²	—	2264	2	no	—
94	C₂²	—	2264	2	no	—
95	C₂²	#22583606	1	1	yes	characteristic
96	C₄	—	8	2	no	—
97	C₄	C₂×#17985379	1	2	yes	—
98	C₄	#22583637	1	4	yes	—
99	C₂	—	4528	1	no	—
100	C₂	—	4528	1	no	—
101	C₂	#31826839	1	1	yes	characteristic
102	C₂	#31826851	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#31826842

C₂

yes

maximal, characteristic

#31826839

C₂

yes

maximal, complemented

#31826813

C₂

yes

maximal, complemented

#31826186

C₂

yes

maximal, characteristic, complemented

#498×C₂₈₃

C₂

yes

maximal, characteristic, complemented

C₂×#17985379

—

C₂×D₂₂₆₄

—

#22583607

C₂²

yes

characteristic

#22583607

C₂²

yes

characteristic

#22583530

—

C₄×C₂₈₃⋊C₈

C₂²

yes

—

C₄⋊Q₈×C₂₈₃

C₂²

yes

—

#106×C₂₈₃

—

C₄×C₈×C₂₈₃

C₂²

yes

characteristic

C₂×D₁₁₃₂

—

C₂×D₁₁₃₂

—

C₂×D₁₁₃₂

—

C₂×D₁₁₃₂

—

#17985379

—

D₂₂₆₄

—

C₂×C₂₈₃⋊C₈

D₄

yes

—

C₂×Q₈×C₂₈₃

—

C₂×C₈×C₂₈₃

D₄

yes

—

C₄⋊C₄×C₂₈₃

—

C₄⋊C₄×C₂₈₃

—

C₄²×C₂₈₃

C₂³

yes

characteristic

C₂²×D₂₈₃

—

C₂²×D₂₈₃

—

D₁₁₃₂

—

D₁₁₃₂

—

D₁₁₃₂

—

D₁₁₃₂

—

C₂₈₃⋊C₈

—

Q₈×C₂₈₃

—

C₂×C₄×C₂₈₃

—

C₂×C₄×C₂₈₃

C₂×D₄

yes

—

C₂×C₄×C₂₈₃

C₂×D₄

yes

characteristic

C₈×C₂₈₃

SD₁₆

yes

complemented

C₂²×C₂₈₃

#127

yes

characteristic

C₂×D₂₈₃

—

C₂×D₂₈₃

—

C₂×D₂₈₃

—

C₂×D₂₈₃

—

C₄×C₂₈₃

—

C₄×C₂₈₃

C₂×SD₁₆

yes

—

C₄×C₂₈₃

AGL(1,ℤ/8)

yes

—

C₂×C₂₈₃

#492

yes

characteristic

C₂×C₂₈₃

#496

yes

—

D₂₈₃

—

D₂₈₃

—

C₂₈₃

#1715

yes

characteristic, complemented, Sylow

#1715

—

283

maximal, Sylow

#498

D₂₈₃

yes

characteristic, complemented

#493

—

283

—

#492

—

283

—

#486

—

283

—

#322

—

283

—

C₂×SD₁₆

—

566

—

C₂×D₈

—

566

—

C₄⋊Q₈

C₂×D₂₈₃

yes

—

#127

—

283

—

#127

—

283

—

#106

—

#102

—

566

—

C₄×C₈

—

283

—

C₄×C₈

C₂×D₂₈₃

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

566

—

C₂×D₄

—

566

—

C₂×D₄

—

566

—

C₂×D₄

—

566

—

SD₁₆

—

1132

—

D₈

—

1132

—

C₂×C₈

—

283

—

C₂×C₈

C₂²⋊D₂₈₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₂₈₃

yes

characteristic

C₂³

—

1132

—

C₂³

—

1132

—

Q₈

—

D₄

—

1132

—

D₄

—

1132

—

D₄

—

1132

—

D₄

—

1132

—

C₂×C₄

—

C₂×C₄

D₄×D₂₈₃

yes

—

C₂×C₄

C₂×C₂²⋊D₂₈₃

yes

characteristic

C₈

—

566

—

C₈

#17985379

yes

complemented

C₂²

—

2264

—

C₂²

—

2264

—

C₂²

—

2264

—

C₂²

—

2264

—

C₂²

#22583606

yes

characteristic

C₄

—

C₄

C₂×#17985379

yes

—

C₄

#22583637

yes

—

C₂

—

4528

—

100

C₂

—

4528

—

101

C₂

#31826839

yes

characteristic

102

C₂

#31826851

yes

—

	subgroup	G/H
center	C₂²	#22583606
commutator, cosocle	C₄²×C₂₈₃	C₂³
Frattini	C₄²	C₂²×D₂₈₃
Fitting	#498×C₂₈₃	C₂
socle	C₂²×C₂₈₃	#127
prime radical	#31826842	C₂
hypercenter	#498	D₂₈₃
hypocenter	C₂₈₃	#1715
LCS	G ▷ C₄²×C₂₈₃ ▷ C₂²×C₂₈₃ ▷ C₂₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#22583606

commutator, cosocle

C₄²×C₂₈₃

C₂³

Frattini

C₄²

C₂²×D₂₈₃

Fitting

#498×C₂₈₃

C₂

C₂²×C₂₈₃

C₂

D₂₈₃

C₂₈₃

G ▷ C₄²×C₂₈₃ ▷ C₂²×C₂₈₃ ▷ C₂₈₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
2. #31826839	C₂	semidirect
3. #31826813	C₂	semidirect
4. #31826186	C₂	semidirect
5. #498×C₂₈₃	C₂	semidirect
38. C₈×C₂₈₃	SD₁₆	semidirect
51. C₂₈₃	#1715	semidirect
53. #498	D₂₈₃	semidirect
90. C₈	#17985379	semidirect

H ≅ G/N

type

2. #31826839

C₂

semidirect

3. #31826813

C₂

semidirect

4. #31826186

C₂

semidirect

5. #498×C₂₈₃

C₂

semidirect

38. C₈×C₂₈₃

SD₁₆

semidirect

51. C₂₈₃

#1715

semidirect

53. #498

D₂₈₃

semidirect

90. C₈

#17985379

semidirect

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

subgroup	index
#31826842	2
#31826839	2
#31826813	2
#31826186	2
#498×C₂₈₃	2
#1715	283

subgroup

index

#498×C₂₈₃

283

Subgroups — #49120890

Conjugacy classes of subgroups

Complements

Maximal subgroups