Group #49122226 — subgroups

Subgroups — #49122226

Order 36224.

252 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂×#22583602	C₂	1	1	yes	maximal, characteristic
2	#31826331	C₂	1	1	yes	maximal, characteristic
3	#31826228	C₂	1	4	yes	maximal
4	C₂×C₄×C₈×C₂₈₃	C₂	1	1	yes	maximal, characteristic
5	#22583602	—	2	2	no	—
6	#22583602	C₄	1	4	yes	complemented
7	C₂×C₄⋊Dic₂₈₃	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₂₈₃	C₂²	1	2	yes	—
9	C₂×Dic₂₈₃⋊C₄	—	2	2	no	—
10	C₂₈₃⋊#95	—	2	2	no	—
11	C₂²×C₈×C₂₈₃	C₂²	1	2	yes	—
12	C₂×C₄²×C₂₈₃	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₂₈₃	—	2	2	no	—
14	C₂²×Dic₂₈₃	—	2	2	no	—
15	C₂²×Dic₂₈₃	—	2	2	no	—
16	C₄⋊Dic₂₈₃	—	2	2	no	—
17	C₄⋊Dic₂₈₃	—	2	4	no	—
18	C₄⋊Dic₂₈₃	C₂×C₄	1	4	yes	—
19	Dic₂₈₃⋊C₄	—	2	8	no	—
20	C₂²×C₄×C₂₈₃	D₄	1	2	yes	—
21	C₂²×C₄×C₂₈₃	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₂₈₃	—	2	2	no	—
23	C₂×C₈×C₂₈₃	—	2	2	no	—
24	C₂×C₈×C₂₈₃	Q₈	1	4	yes	—
25	C₄²×C₂₈₃	C₂×C₄	1	2	yes	—
26	C₄²×C₂₈₃	D₄	1	2	yes	—
27	C₂×Dic₂₈₃	—	2	4	no	—
28	C₂×Dic₂₈₃	—	4	4	no	—
29	C₂×Dic₂₈₃	—	4	2	no	—
30	C₂×Dic₂₈₃	—	4	4	no	—
31	C₂³×C₂₈₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₂₈₃	—	2	4	no	—
33	C₂×C₄×C₂₈₃	C₂²⋊C₄	1	4	yes	—
34	C₂×C₄×C₂₈₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₈₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₂₈₃	C₂×Q₈	1	2	yes	—
37	C₂×C₄×C₂₈₃	#41	1	2	yes	—
38	C₈×C₂₈₃	—	2	4	no	—
39	C₂²×C₂₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₂₈₃	#124	1	1	yes	characteristic
41	C₂²×C₂₈₃	C₄⋊Q₈	1	1	yes	characteristic
42	C₂²×C₂₈₃	#135	1	4	yes	—
43	C₄×C₂₈₃	—	2	4	no	—
44	C₄×C₂₈₃	C₄×Q₈	1	2	yes	—
45	C₄×C₂₈₃	C₂²⋊Q₈	1	2	yes	—
46	Dic₂₈₃	—	4	4	no	—
47	Dic₂₈₃	—	8	2	no	—
48	C₂×C₂₈₃	#391	1	1	yes	characteristic
49	C₂×C₂₈₃	#416	1	2	yes	—
50	C₂×C₂₈₃	#488	1	2	yes	—
51	C₂×C₂₈₃	#499	1	2	yes	—
52	C₂₈₃	#1991	1	1	yes	characteristic, complemented, Sylow
53	#1991	—	283	1	no	maximal, Sylow
54	C₂×#125	—	283	1	no	—
55	C₂×C₄×C₈	D₂₈₃	1	1	yes	characteristic
56	#382	—	283	1	no	—
57	#340	—	283	4	no	—
58	C₂²×C₈	C₂×D₂₈₃	1	2	yes	—
59	#125	—	283	4	no	—
60	#125	—	566	2	no	—
61	C₂×C₄⋊C₄	—	283	2	no	—
62	C₂×C₄⋊C₄	—	283	2	no	—
63	C₂×C₄⋊C₄	—	566	2	no	—
64	C₂×C₄²	C₂×D₂₈₃	1	1	yes	characteristic
65	C₄×C₈	—	2	2	no	—
66	#95	—	566	2	no	—
67	C₂²×C₄	—	566	2	no	—
68	C₂²×C₄	—	566	2	no	—
69	C₂²×C₄	D₁₁₃₂	1	2	yes	—
70	C₂²×C₄	C₂²×D₂₈₃	1	1	yes	characteristic
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	Dic₅₆₆	1	4	yes	—
74	C₄⋊C₄	—	283	4	no	—
75	C₄⋊C₄	—	566	8	no	—
76	C₄⋊C₄	—	566	2	no	—
77	C₄⋊C₄	—	566	4	no	—
78	C₄²	C₄×D₂₈₃	1	2	yes	—
79	C₄²	C₂²⋊D₂₈₃	1	2	yes	—
80	C₂³	C₂×D₁₁₃₂	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	566	4	no	—
83	C₂×C₄	—	1132	4	no	—
84	C₂×C₄	—	1132	2	no	—
85	C₂×C₄	—	1132	4	no	—
86	C₂×C₄	(C₂×D₂₈₃)⋊C₄	1	4	yes	—
87	C₂×C₄	C₂×C₄×D₂₈₃	1	1	yes	characteristic
88	C₂×C₄	C₂×C₂²⋊D₂₈₃	1	1	yes	characteristic
89	C₂×C₄	C₂×Dic₅₆₆	1	2	yes	—
90	C₂×C₄	#17985392	1	2	yes	—
91	C₈	—	2	4	no	—
92	C₂²	C₂×(C₂×D₂₈₃)⋊C₄	1	1	yes	characteristic
93	C₂²	#22583598	1	1	yes	characteristic
94	C₂²	#22583610	1	1	yes	characteristic
95	C₂²	#22583632	1	4	yes	—
96	C₄	—	2	4	no	—
97	C₄	—	1132	4	no	—
98	C₄	—	2264	2	no	—
99	C₄	C₄×Dic₅₆₆	1	2	yes	—
100	C₄	#22583589	1	2	yes	—
101	C₂	#31826371	1	1	yes	characteristic
102	C₂	#31826462	1	2	yes	—
103	C₂	#31826820	1	2	yes	—
104	C₂	#31826867	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22583602

C₂

yes

maximal, characteristic

#31826331

C₂

yes

maximal, characteristic

#31826228

C₂

yes

maximal

C₂×C₄×C₈×C₂₈₃

C₂

yes

maximal, characteristic

#22583602

—

#22583602

C₄

yes

complemented

C₂×C₄⋊Dic₂₈₃

C₂²

yes

—

C₂×C₄⋊Dic₂₈₃

C₂²

yes

—

C₂×Dic₂₈₃⋊C₄

—

C₂₈₃⋊#95

—

C₂²×C₈×C₂₈₃

C₂²

yes

—

C₂×C₄²×C₂₈₃

C₂²

yes

characteristic

C₄×C₈×C₂₈₃

—

C₂²×Dic₂₈₃

—

C₂²×Dic₂₈₃

—

C₄⋊Dic₂₈₃

—

C₄⋊Dic₂₈₃

—

C₄⋊Dic₂₈₃

C₂×C₄

yes

—

Dic₂₈₃⋊C₄

—

C₂²×C₄×C₂₈₃

D₄

yes

—

C₂²×C₄×C₂₈₃

C₂³

yes

characteristic

C₂×C₈×C₂₈₃

—

C₂×C₈×C₂₈₃

—

C₂×C₈×C₂₈₃

Q₈

yes

—

C₄²×C₂₈₃

C₂×C₄

yes

—

C₄²×C₂₈₃

D₄

yes

—

C₂×Dic₂₈₃

—

C₂×Dic₂₈₃

—

C₂×Dic₂₈₃

—

C₂×Dic₂₈₃

—

C₂³×C₂₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₃

—

C₂×C₄×C₂₈₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₃

C₂×Q₈

yes

—

C₂×C₄×C₂₈₃

#41

yes

—

C₈×C₂₈₃

—

C₂²×C₂₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₈₃

#124

yes

characteristic

C₂²×C₂₈₃

C₄⋊Q₈

yes

characteristic

C₂²×C₂₈₃

#135

yes

—

C₄×C₂₈₃

—

C₄×C₂₈₃

C₄×Q₈

yes

—

C₄×C₂₈₃

C₂²⋊Q₈

yes

—

Dic₂₈₃

—

Dic₂₈₃

—

C₂×C₂₈₃

#391

yes

characteristic

C₂×C₂₈₃

#416

yes

—

C₂×C₂₈₃

#488

yes

—

C₂×C₂₈₃

#499

yes

—

C₂₈₃

#1991

yes

characteristic, complemented, Sylow

#1991

—

283

maximal, Sylow

C₂×#125

—

283

—

C₂×C₄×C₈

D₂₈₃

yes

characteristic

#382

—

283

—

#340

—

283

—

C₂²×C₈

C₂×D₂₈₃

yes

—

#125

—

283

—

#125

—

566

—

C₂×C₄⋊C₄

—

283

—

C₂×C₄⋊C₄

—

283

—

C₂×C₄⋊C₄

—

566

—

C₂×C₄²

C₂×D₂₈₃

yes

characteristic

C₄×C₈

—

#95

—

566

—

C₂²×C₄

—

566

—

C₂²×C₄

—

566

—

C₂²×C₄

D₁₁₃₂

yes

—

C₂²×C₄

C₂²×D₂₈₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₅₆₆

yes

—

C₄⋊C₄

—

283

—

C₄⋊C₄

—

566

—

C₄⋊C₄

—

566

—

C₄⋊C₄

—

566

—

C₄²

C₄×D₂₈₃

yes

—

C₄²

C₂²⋊D₂₈₃

yes

—

C₂³

C₂×D₁₁₃₂

yes

characteristic

C₂×C₄

—

C₂×C₄

—

566

—

C₂×C₄

—

1132

—

C₂×C₄

—

1132

—

C₂×C₄

—

1132

—

C₂×C₄

(C₂×D₂₈₃)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₂₈₃

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₈₃

yes

characteristic

C₂×C₄

C₂×Dic₅₆₆

yes

—

C₂×C₄

#17985392

yes

—

C₈

—

C₂²

C₂×(C₂×D₂₈₃)⋊C₄

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₄

—

C₄

—

1132

—

C₄

—

2264

—

C₄

C₄×Dic₅₆₆

yes

—

100

C₄

#22583589

yes

—

101

C₂

#31826371

yes

characteristic

102

C₂

#31826462

yes

—

103

C₂

#31826820

yes

—

104

C₂

#31826867

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₁₃₂
commutator	C₂×C₄×C₂₈₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₈₃
Fitting	C₂×C₄×C₈×C₂₈₃	C₂
socle, prime radical	C₂³×C₂₈₃	C₂×D₄
cosocle	C₂²×C₄×C₂₈₃	C₂³
hypercenter	C₂×C₄×C₈	D₂₈₃
hypocenter	C₂₈₃	#1991
LCS	G ▷ C₂×C₄×C₂₈₃ ▷ C₂×C₂₈₃ ▷ C₂₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₁₁₃₂

commutator

C₂×C₄×C₂₈₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₈₃

Fitting

C₂×C₄×C₈×C₂₈₃

C₂

socle, prime radical

C₂³×C₂₈₃

C₂×D₄

cosocle

C₂²×C₄×C₂₈₃

C₂³

hypercenter

C₂×C₄×C₈

D₂₈₃

hypocenter

C₂₈₃

#1991

LCS

G ▷ C₂×C₄×C₂₈₃ ▷ C₂×C₂₈₃ ▷ C₂₈₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
6. #22583602	C₄	semidirect
52. C₂₈₃	#1991	semidirect

H ≅ G/N

type

6. #22583602

C₄

semidirect

52. C₂₈₃

#1991

semidirect

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

subgroup	index
C₂×#22583602	2
#31826331	2
#31826228	2
C₂×C₄×C₈×C₂₈₃	2
#1991	283

subgroup

index

C₂×C₄×C₈×C₂₈₃

283

Subgroups — #49122226

Conjugacy classes of subgroups

Complements

Maximal subgroups