Subgroups — #44567010

Order 32192.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632864	C₂	1	3	yes	maximal
2	#30632855	C₂	1	6	yes	maximal
3	Q₈×Dic₅₀₃	C₂	1	2	yes	maximal
4	C₄×Dic₁₀₀₆	C₂	1	3	yes	maximal
5	C₄×Q₈×C₅₀₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₀₀₆	C₂²	1	3	yes	—
7	C₄⋊Dic₅₀₃	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₅₀₃	C₂²	1	6	yes	—
9	C₄⋊Dic₅₀₃	C₂²	1	6	yes	—
10	Dic₅₀₃⋊C₄	C₂²	1	6	yes	—
11	C₄×Dic₅₀₃	C₂²	1	6	yes	—
12	C₂×Q₈×C₅₀₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₅₀₃	C₂²	1	3	yes	—
14	C₄²×C₅₀₃	C₂²	1	3	yes	—
15	Dic₁₀₀₆	—	2	6	no	—
16	C₂×Dic₅₀₃	C₂³	1	2	yes	—
17	C₂×Dic₅₀₃	C₂³	1	6	yes	—
18	Q₈×C₅₀₃	Q₈	1	4	yes	complemented
19	C₂×C₄×C₅₀₃	C₂³	1	3	yes	—
20	C₂×C₄×C₅₀₃	C₂³	1	3	yes	—
21	C₂×C₄×C₅₀₃	C₂³	1	1	yes	characteristic
22	C₂²×C₅₀₃	C₂⁴	1	1	yes	characteristic
23	C₄×C₅₀₃	—	2	3	no	—
24	C₄×C₅₀₃	C₂×Q₈	1	6	yes	—
25	C₄×C₅₀₃	#41	1	2	yes	—
26	Dic₅₀₃	—	2	6	no	—
27	Dic₅₀₃	—	2	2	no	—
28	C₂×C₅₀₃	C₂²×Q₈	1	1	yes	characteristic
29	C₂×C₅₀₃	C₂×#41	1	1	yes	characteristic
30	C₂×C₅₀₃	2₋¹⁺⁴	1	1	yes	characteristic
31	C₅₀₃	#557	1	1	yes	characteristic, complemented, Sylow
32	#557	—	503	1	no	maximal, Sylow
33	C₄⋊Q₈	—	503	3	no	—
34	#125	—	503	6	no	—
35	C₄×Q₈	—	503	2	no	—
36	C₄×Q₈	—	503	3	no	—
37	C₄×Q₈	D₅₀₃	1	1	yes	characteristic
38	C₂×Q₈	—	503	3	no	—
39	C₂×Q₈	C₂×D₅₀₃	1	1	yes	characteristic
40	C₄⋊C₄	—	503	1	no	—
41	C₄⋊C₄	—	503	6	no	—
42	C₄⋊C₄	—	503	6	no	—
43	C₄⋊C₄	—	503	6	no	—
44	C₄⋊C₄	C₂×D₅₀₃	1	3	yes	—
45	C₄²	—	503	6	no	—
46	C₄²	C₂×D₅₀₃	1	3	yes	—
47	Q₈	—	1006	6	no	—
48	Q₈	Dic₁₀₀₆	1	4	yes	complemented
49	C₂×C₄	—	503	2	no	—
50	C₂×C₄	—	503	6	no	—
51	C₂×C₄	C₂²×D₅₀₃	1	3	yes	—
52	C₂×C₄	C₂²×D₅₀₃	1	3	yes	—
53	C₂×C₄	C₂²×D₅₀₃	1	1	yes	characteristic
54	C₂²	C₂³×D₅₀₃	1	1	yes	characteristic
55	C₄	—	2	3	no	—
56	C₄	—	1006	6	no	—
57	C₄	—	1006	2	no	—
58	C₄	C₂×Dic₁₀₀₆	1	6	yes	—
59	C₄	#22548884	1	2	yes	—
60	C₂	C₂²×Dic₁₀₀₆	1	1	yes	characteristic
61	C₂	C₂×#22548884	1	1	yes	characteristic
62	C₂	#30632911	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₀₃
commutator, socle, prime radical, cosocle	C₂²×C₅₀₃	C₂⁴
Fitting	C₄×Q₈×C₅₀₃	C₂
hypercenter	C₄×Q₈	D₅₀₃
hypocenter	C₅₀₃	#557
LCS	G ▷ C₂²×C₅₀₃ ▷ C₅₀₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
18. Q₈×C₅₀₃	Q₈	semidirect
31. C₅₀₃	#557	semidirect
48. Q₈	Dic₁₀₀₆	semidirect

Maximal subgroups

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

subgroup	index
#30632864	2
#30632855	2
Q₈×Dic₅₀₃	2
C₄×Dic₁₀₀₆	2
C₄×Q₈×C₅₀₃	2
#557	503