Group #49120927 — subgroups

Subgroups — #49120927

Order 36224.

242 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 69 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, complemented
2	#31826842	C₂	1	2	yes	maximal, complemented
3	#31826838	C₂	1	1	yes	maximal, characteristic, complemented
4	#31826812	C₂	1	2	yes	maximal, complemented
5	#322×C₂₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17985381	—	2	2	no	—
7	C₂×D₂₂₆₄	—	2	4	no	—
8	C₂×D₂₂₆₄	—	2	2	no	—
9	#22583610	C₂²	1	1	yes	characteristic
10	#22583607	C₂²	1	2	yes	—
11	#22583607	C₂²	1	1	yes	characteristic
12	#22583532	—	2	4	no	—
13	C₄×C₈×C₂₈₃	C₂²	1	2	yes	—
14	C₄×C₈×C₂₈₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₅₆₆	—	2	1	no	—
16	C₂×D₁₁₃₂	—	2	2	no	—
17	C₂×D₁₁₃₂	—	2	2	no	—
18	C₂×D₁₁₃₂	—	2	2	no	—
19	C₂×D₁₁₃₂	—	2	1	no	—
20	C₂×D₁₁₃₂	—	2	2	no	—
21	#17985381	—	4	4	no	—
22	D₂₂₆₄	—	4	8	no	—
23	D₂₂₆₄	—	4	4	no	—
24	C₄⋊Dic₂₈₃	—	2	2	no	—
25	C₂×C₈×C₂₈₃	D₄	1	4	yes	—
26	C₂×C₈×C₂₈₃	D₄	1	2	yes	—
27	C₄²×C₂₈₃	C₂³	1	1	yes	characteristic
28	C₂²×D₂₈₃	—	4	1	no	—
29	C₂²×D₂₈₃	—	4	2	no	—
30	Dic₅₆₆	—	4	2	no	—
31	D₁₁₃₂	—	4	4	no	—
32	D₁₁₃₂	—	4	2	no	—
33	D₁₁₃₂	—	4	4	no	—
34	D₁₁₃₂	—	4	4	no	—
35	D₁₁₃₂	—	4	4	no	—
36	C₂×Dic₂₈₃	—	4	1	no	—
37	C₂×C₄×C₂₈₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₂₈₃	C₂×D₄	1	2	yes	—
39	C₈×C₂₈₃	—	2	4	no	—
40	C₈×C₂₈₃	D₈	1	4	yes	complemented
41	C₂²×C₂₈₃	#127	1	1	yes	characteristic
42	C₂×D₂₈₃	—	8	2	no	—
43	C₂×D₂₈₃	—	8	1	no	—
44	C₂×D₂₈₃	—	8	2	no	—
45	C₂×D₂₈₃	—	8	2	no	—
46	C₂×D₂₈₃	—	8	2	no	—
47	C₄×C₂₈₃	C₂×D₈	1	2	yes	—
48	C₄×C₂₈₃	AGL(1,ℤ/8)	1	4	yes	—
49	Dic₂₈₃	—	8	1	no	—
50	C₂×C₂₈₃	#493	1	1	yes	characteristic
51	C₂×C₂₈₃	#496	1	2	yes	—
52	D₂₈₃	—	16	2	no	—
53	D₂₈₃	—	16	1	no	—
54	C₂₈₃	#1721	1	1	yes	characteristic, complemented, Sylow
55	#1721	—	283	1	no	maximal, Sylow
56	#493	—	283	2	no	—
57	#493	—	283	1	no	—
58	#492	—	283	1	no	—
59	#486	—	283	2	no	—
60	#322	D₂₈₃	1	1	yes	characteristic, complemented
61	C₂×SD₁₆	—	566	2	no	—
62	C₂×D₈	—	566	4	no	—
63	C₂×D₈	—	566	2	no	—
64	C₄⋊Q₈	—	283	1	no	—
65	#127	—	283	2	no	—
66	#127	—	283	1	no	—
67	#102	—	566	4	no	—
68	C₄×C₈	C₂×D₂₈₃	1	2	yes	—
69	C₄×C₈	C₂×D₂₈₃	1	1	yes	characteristic
70	C₂×Q₈	—	566	1	no	—
71	C₂×D₄	—	566	2	no	—
72	C₂×D₄	—	566	2	no	—
73	C₂×D₄	—	566	2	no	—
74	C₂×D₄	—	566	2	no	—
75	C₂×D₄	—	566	1	no	—
76	SD₁₆	—	1132	4	no	—
77	D₈	—	1132	8	no	—
78	D₈	—	1132	4	no	—
79	C₂×C₈	D₁₁₃₂	1	4	yes	—
80	C₂×C₈	D₁₁₃₂	1	2	yes	—
81	C₄⋊C₄	—	566	2	no	—
82	C₄²	C₂²×D₂₈₃	1	1	yes	characteristic
83	C₂³	—	1132	2	no	—
84	C₂³	—	1132	1	no	—
85	Q₈	—	1132	2	no	—
86	D₄	—	1132	4	no	—
87	D₄	—	1132	2	no	—
88	D₄	—	1132	4	no	—
89	D₄	—	1132	4	no	—
90	D₄	—	1132	4	no	—
91	C₂×C₄	—	1132	1	no	—
92	C₂×C₄	C₂×D₁₁₃₂	1	1	yes	characteristic
93	C₂×C₄	C₂×D₁₁₃₂	1	2	yes	—
94	C₈	—	2	4	no	—
95	C₈	D₂₂₆₄	1	4	yes	complemented
96	C₂²	—	2264	2	no	—
97	C₂²	—	2264	1	no	—
98	C₂²	—	2264	2	no	—
99	C₂²	—	2264	2	no	—
100	C₂²	—	2264	2	no	—
101	C₂²	#22583607	1	1	yes	characteristic
102	C₄	—	2264	1	no	—
103	C₄	C₂×D₂₂₆₄	1	2	yes	—
104	C₄	#22583638	1	4	yes	—
105	C₂	—	4528	2	no	—
106	C₂	—	4528	1	no	—
107	C₂	#31826842	1	1	yes	characteristic
108	C₂	#31826854	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#31826842

C₂

yes

maximal, characteristic, complemented

#31826842

C₂

yes

maximal, complemented

#31826838

C₂

yes

maximal, characteristic, complemented

#31826812

C₂

yes

maximal, complemented

#322×C₂₈₃

C₂

yes

maximal, characteristic, complemented

C₂×#17985381

—

C₂×D₂₂₆₄

—

C₂×D₂₂₆₄

—

#22583610

C₂²

yes

characteristic

#22583607

C₂²

yes

—

#22583607

C₂²

yes

characteristic

#22583532

—

C₄×C₈×C₂₈₃

C₂²

yes

—

C₄×C₈×C₂₈₃

C₂²

yes

characteristic

C₂×Dic₅₆₆

—

C₂×D₁₁₃₂

—

C₂×D₁₁₃₂

—

C₂×D₁₁₃₂

—

C₂×D₁₁₃₂

—

C₂×D₁₁₃₂

—

#17985381

—

D₂₂₆₄

—

D₂₂₆₄

—

C₄⋊Dic₂₈₃

—

C₂×C₈×C₂₈₃

D₄

yes

—

C₂×C₈×C₂₈₃

D₄

yes

—

C₄²×C₂₈₃

C₂³

yes

characteristic

C₂²×D₂₈₃

—

C₂²×D₂₈₃

—

Dic₅₆₆

—

D₁₁₃₂

—

D₁₁₃₂

—

D₁₁₃₂

—

D₁₁₃₂

—

D₁₁₃₂

—

C₂×Dic₂₈₃

—

C₂×C₄×C₂₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₃

C₂×D₄

yes

—

C₈×C₂₈₃

—

C₈×C₂₈₃

D₈

yes

complemented

C₂²×C₂₈₃

#127

yes

characteristic

C₂×D₂₈₃

—

C₂×D₂₈₃

—

C₂×D₂₈₃

—

C₂×D₂₈₃

—

C₂×D₂₈₃

—

C₄×C₂₈₃

C₂×D₈

yes

—

C₄×C₂₈₃

AGL(1,ℤ/8)

yes

—

Dic₂₈₃

—

C₂×C₂₈₃

#493

yes

characteristic

C₂×C₂₈₃

#496

yes

—

D₂₈₃

—

D₂₈₃

—

C₂₈₃

#1721

yes

characteristic, complemented, Sylow

#1721

—

283

maximal, Sylow

#493

—

283

—

#493

—

283

—

#492

—

283

—

#486

—

283

—

#322

D₂₈₃

yes

characteristic, complemented

C₂×SD₁₆

—

566

—

C₂×D₈

—

566

—

C₂×D₈

—

566

—

C₄⋊Q₈

—

283

—

#127

—

283

—

#127

—

283

—

#102

—

566

—

C₄×C₈

C₂×D₂₈₃

yes

—

C₄×C₈

C₂×D₂₈₃

yes

characteristic

C₂×Q₈

—

566

—

C₂×D₄

—

566

—

C₂×D₄

—

566

—

C₂×D₄

—

566

—

C₂×D₄

—

566

—

C₂×D₄

—

566

—

SD₁₆

—

1132

—

D₈

—

1132

—

D₈

—

1132

—

C₂×C₈

D₁₁₃₂

yes

—

C₂×C₈

D₁₁₃₂

yes

—

C₄⋊C₄

—

566

—

C₄²

C₂²×D₂₈₃

yes

characteristic

C₂³

—

1132

—

C₂³

—

1132

—

Q₈

—

1132

—

D₄

—

1132

—

D₄

—

1132

—

D₄

—

1132

—

D₄

—

1132

—

D₄

—

1132

—

C₂×C₄

—

1132

—

C₂×C₄

C₂×D₁₁₃₂

yes

characteristic

C₂×C₄

C₂×D₁₁₃₂

yes

—

C₈

—

C₈

D₂₂₆₄

yes

complemented

C₂²

—

2264

—

C₂²

—

2264

—

C₂²

—

2264

—

C₂²

—

2264

—

100

C₂²

—

2264

—

101

C₂²

#22583607

yes

characteristic

102

C₄

—

2264

—

103

C₄

C₂×D₂₂₆₄

yes

—

104

C₄

#22583638

yes

—

105

C₂

—

4528

—

106

C₂

—

4528

—

107

C₂

#31826842

yes

characteristic

108

C₂

#31826854

yes

—

	subgroup	G/H
center	C₂²	#22583607
commutator, cosocle	C₄²×C₂₈₃	C₂³
Frattini	C₄²	C₂²×D₂₈₃
Fitting	#322×C₂₈₃	C₂
socle	C₂²×C₂₈₃	#127
hypercenter	#322	D₂₈₃
hypocenter	C₂₈₃	#1721
LCS	G ▷ C₄²×C₂₈₃ ▷ C₂²×C₂₈₃ ▷ C₂₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#22583607

commutator, cosocle

C₄²×C₂₈₃

C₂³

Frattini

C₄²

C₂²×D₂₈₃

Fitting

#322×C₂₈₃

C₂

C₂²×C₂₈₃

D₂₈₃

C₂₈₃

G ▷ C₄²×C₂₈₃ ▷ C₂²×C₂₈₃ ▷ C₂₈₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
1. #31826842	C₂	semidirect
2. #31826842	C₂	semidirect
3. #31826838	C₂	semidirect
4. #31826812	C₂	semidirect
5. #322×C₂₈₃	C₂	semidirect
40. C₈×C₂₈₃	D₈	semidirect
54. C₂₈₃	#1721	semidirect
60. #322	D₂₈₃	semidirect
95. C₈	D₂₂₆₄	semidirect

H ≅ G/N

type

1. #31826842

C₂

semidirect

2. #31826842

C₂

semidirect

3. #31826838

C₂

semidirect

4. #31826812

C₂

semidirect

5. #322×C₂₈₃

C₂

semidirect

40. C₈×C₂₈₃

D₈

semidirect

54. C₂₈₃

#1721

semidirect

60. #322

D₂₈₃

semidirect

95. C₈

D₂₂₆₄

semidirect

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

subgroup	index
#31826842	2
#31826838	2
#31826812	2
#322×C₂₈₃	2
#1721	283

subgroup

index

#322×C₂₈₃

283

Subgroups — #49120927

Conjugacy classes of subgroups

Complements

Maximal subgroups