Subgroups — #44566308

Order 32192.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22548867	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅₀₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₅₀₃⋊C₈)	C₂	1	4	yes	maximal, complemented
4	#117×C₅₀₃	C₂	1	1	yes	maximal, characteristic
5	#22548867	—	2	2	no	—
6	C₂×C₅₀₃⋊C₈	—	2	2	no	—
7	C₂×C₅₀₃⋊C₈	C₂²	1	2	yes	—
8	C₂×C₅₀₃⋊C₈	C₂²	1	2	yes	—
9	C₂²×C₄×C₅₀₃	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₅₀₃	C₄	1	2	yes	—
11	C₂²⋊C₄×C₅₀₃	C₄	1	2	yes	—
12	C₄²×C₅₀₃	C₂²	1	2	yes	—
13	C₅₀₃⋊C₈	—	2	2	no	—
14	C₅₀₃⋊C₈	—	2	2	no	—
15	C₂³×C₅₀₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₅₀₃	C₂×C₄	1	4	yes	—
17	C₂×C₄×C₅₀₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₅₀₃	D₄	1	2	yes	—
19	C₂×C₄×C₅₀₃	Q₈	1	2	yes	—
20	C₂×C₄×C₅₀₃	C₂³	1	1	yes	characteristic
21	C₂²×C₅₀₃	—	2	2	no	—
22	C₂²×C₅₀₃	C₄⋊C₄	1	2	yes	—
23	C₂²×C₅₀₃	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₅₀₃	—	2	4	no	—
25	C₄×C₅₀₃	C₄⋊C₄	1	2	yes	—
26	C₄×C₅₀₃	C₂×D₄	1	1	yes	characteristic
27	C₄×C₅₀₃	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₅₀₃	—	2	2	no	—
29	C₂×C₅₀₃	C₂×C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₅₀₃	#131	1	2	yes	—
31	C₅₀₃	#424	1	1	yes	characteristic, complemented, Sylow
32	#424	—	503	1	no	maximal, Sylow
33	C₂×M₁₆	—	503	1	no	—
34	C₂²×C₈	—	503	1	no	—
35	#117	D₅₀₃	1	1	yes	characteristic
36	C₄⋊C₈	—	503	4	no	—
37	C₂²×C₄	C₂×D₅₀₃	1	1	yes	characteristic
38	M₁₆	—	1006	2	no	—
39	C₂×C₈	—	1006	2	no	—
40	C₂×C₈	—	503	2	no	—
41	C₂×C₈	—	503	2	no	—
42	C₄⋊C₄	Dic₅₀₃	1	2	yes	—
43	C₂²⋊C₄	Dic₅₀₃	1	2	yes	—
44	C₄²	C₂×D₅₀₃	1	2	yes	—
45	C₂³	C₂×Dic₅₀₃	1	1	yes	characteristic
46	C₂×C₄	C₂×Dic₅₀₃	1	4	yes	—
47	C₂×C₄	C₂×Dic₅₀₃	1	1	yes	characteristic
48	C₂×C₄	D₂₀₁₂	1	2	yes	—
49	C₂×C₄	Dic₁₀₀₆	1	2	yes	—
50	C₂×C₄	C₂²×D₅₀₃	1	1	yes	characteristic
51	C₈	—	1006	2	no	—
52	C₈	—	1006	2	no	—
53	C₂²	—	2	2	no	—
54	C₂²	C₄⋊Dic₅₀₃	1	2	yes	—
55	C₂²	C₂²×Dic₅₀₃	1	1	yes	characteristic
56	C₄	—	2	4	no	—
57	C₄	C₄⋊Dic₅₀₃	1	2	yes	—
58	C₄	C₂×D₂₀₁₂	1	1	yes	characteristic
59	C₄	C₂×Dic₁₀₀₆	1	1	yes	characteristic
60	C₂	—	2	2	no	—
61	C₂	C₂×C₄⋊Dic₅₀₃	1	1	yes	characteristic
62	C₂	#30632871	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₅₀₃
commutator, socle	C₂²×C₅₀₃	C₂²×C₄
Fitting	#117×C₅₀₃	C₂
prime radical	C₂³×C₅₀₃	C₂×C₄
cosocle	C₂×C₄×C₅₀₃	C₂³
hypercenter	#117	D₅₀₃
hypocenter	C₅₀₃	#424
LCS	G ▷ C₂²×C₅₀₃ ▷ C₅₀₃
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₄⋊(C₅₀₃⋊C₈)	C₂	semidirect
31. C₅₀₃	#424	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22548867	2
C₂²×C₅₀₃⋊C₈	2
C₄⋊(C₅₀₃⋊C₈)	2
#117×C₅₀₃	2
#424	503