Subgroups — #44566486

Order 32192.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22548871	C₂	1	2	yes	maximal
2	#30632864	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊(C₅₀₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#30632787	C₂	1	2	yes	maximal, complemented
5	C₄×D₄×C₅₀₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₀₀₆	C₂²	1	2	yes	—
7	#22548871	—	2	4	no	—
8	C₂×C₅₀₃⋊C₈	C₂²	1	2	yes	—
9	C₄⋊Dic₅₀₃	—	2	1	no	—
10	C₄⋊Dic₅₀₃	—	2	1	no	—
11	C₂×D₄×C₅₀₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₅₀₃	—	2	1	no	—
13	C₄⋊C₄×C₅₀₃	C₂²	1	1	yes	characteristic
14	C₂²⋊C₄×C₅₀₃	—	2	1	no	—
15	C₄²×C₅₀₃	C₂²	1	1	yes	characteristic
16	Dic₁₀₀₆	—	2	2	no	—
17	Dic₁₀₀₆	—	2	2	no	—
18	C₂×Dic₅₀₃	—	2	2	no	—
19	C₅₀₃⋊C₈	—	2	2	no	—
20	C₂³×C₅₀₃	—	2	1	no	—
21	D₄×C₅₀₃	—	2	1	no	—
22	D₄×C₅₀₃	D₄	1	2	yes	—
23	C₂×C₄×C₅₀₃	—	2	1	no	—
24	C₂×C₄×C₅₀₃	—	4	2	no	—
25	C₂×C₄×C₅₀₃	D₄	1	1	yes	characteristic
26	C₂×C₄×C₅₀₃	D₄	1	1	yes	characteristic
27	C₂×C₄×C₅₀₃	C₂³	1	1	yes	characteristic
28	C₂²×C₅₀₃	—	2	2	no	—
29	C₂²×C₅₀₃	—	4	1	no	—
30	C₂²×C₅₀₃	—	4	1	no	—
31	C₂²×C₅₀₃	C₂×D₄	1	1	yes	characteristic
32	C₄×C₅₀₃	—	2	1	no	—
33	C₄×C₅₀₃	—	4	1	no	—
34	C₄×C₅₀₃	SD₁₆	1	2	yes	complemented
35	C₄×C₅₀₃	C₂×D₄	1	1	yes	characteristic
36	C₄×C₅₀₃	#41	1	1	yes	characteristic
37	Dic₅₀₃	—	4	2	no	—
38	C₂×C₅₀₃	—	4	2	no	—
39	C₂×C₅₀₃	#121	1	1	yes	characteristic
40	C₂×C₅₀₃	C₂×SD₁₆	1	1	yes	characteristic
41	C₂×C₅₀₃	#137	1	1	yes	characteristic
42	C₅₀₃	#461	1	1	yes	characteristic, complemented, Sylow
43	#461	—	503	1	no	maximal, Sylow
44	C₂×SD₁₆	—	503	2	no	—
45	C₄⋊Q₈	—	503	1	no	—
46	C₄×D₄	D₅₀₃	1	1	yes	characteristic
47	C₄⋊C₈	—	503	1	no	—
48	#103	—	503	2	no	—
49	C₂×Q₈	—	503	2	no	—
50	C₂×D₄	C₂×D₅₀₃	1	1	yes	characteristic
51	C₂²×C₄	—	2	1	no	—
52	SD₁₆	—	1006	4	no	—
53	C₂×C₈	—	503	2	no	—
54	C₄⋊C₄	—	1006	1	no	—
55	C₄⋊C₄	—	1006	1	no	—
56	C₄⋊C₄	C₂×D₅₀₃	1	1	yes	characteristic
57	C₂²⋊C₄	—	2	1	no	—
58	C₄²	C₂×D₅₀₃	1	1	yes	characteristic
59	C₂³	—	2	1	no	—
60	Q₈	—	1006	2	no	—
61	Q₈	—	1006	2	no	—
62	D₄	—	2	1	no	—
63	D₄	D₂₀₁₂	1	2	yes	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	1006	2	no	—
67	C₂×C₄	C₂²⋊D₅₀₃	1	1	yes	characteristic
68	C₂×C₄	C₂²⋊D₅₀₃	1	1	yes	characteristic
69	C₂×C₄	C₂²×D₅₀₃	1	1	yes	characteristic
70	C₈	—	1006	2	no	—
71	C₂²	—	2	2	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	4	1	no	—
74	C₂²	C₂×C₂²⋊D₅₀₃	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	2012	2	no	—
78	C₄	#22548871	1	2	yes	complemented
79	C₄	C₂×D₂₀₁₂	1	1	yes	characteristic
80	C₄	#22548883	1	1	yes	characteristic
81	C₂	—	4	2	no	—
82	C₂	#30632838	1	1	yes	characteristic
83	C₂	C₂×#22548871	1	1	yes	characteristic
84	C₂	#30632894	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅₀₃
commutator, cosocle	C₂×C₄×C₅₀₃	C₂³
Frattini	C₂×C₄	C₂²×D₅₀₃
Fitting	C₄×D₄×C₅₀₃	C₂
socle	C₂²×C₅₀₃	C₂×D₄
prime radical	C₂×D₄×C₅₀₃	C₂²
hypercenter	C₄×D₄	D₅₀₃
hypocenter	C₅₀₃	#461
LCS	G ▷ C₂×C₄×C₅₀₃ ▷ C₂×C₅₀₃ ▷ C₅₀₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. #30632864	C₂	semidirect
3. C₄⋊(C₅₀₃⋊C₈)	C₂	semidirect
4. #30632787	C₂	semidirect
34. C₄×C₅₀₃	SD₁₆	semidirect
42. C₅₀₃	#461	semidirect
78. C₄	#22548871	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22548871	2
#30632864	2
C₄⋊(C₅₀₃⋊C₈)	2
#30632787	2
C₄×D₄×C₅₀₃	2
#461	503