Subgroups — #45655628

Order 32576.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22550848	C₂	1	2	yes	maximal, complemented
2	C₂×#22550846	C₂	1	2	yes	maximal, complemented
3	#30654212	C₂	1	1	yes	maximal, characteristic, complemented
4	#30654126	C₂	1	1	yes	maximal, characteristic, complemented
5	#124×C₅₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₂₀₃₆	—	2	2	no	—
7	C₂×D₂₀₃₆	C₂²	1	2	yes	—
8	#22550848	—	2	4	no	—
9	#22550846	—	2	4	no	—
10	C₂×#16861325	C₂²	1	2	yes	—
11	C₂×Q₈×C₅₀₉	C₂²	1	1	yes	characteristic
12	C₂×D₄×C₅₀₉	C₂²	1	1	yes	characteristic
13	C₂²⋊C₄×C₅₀₉	—	2	2	no	—
14	C₄²×C₅₀₉	C₂²	1	1	yes	characteristic
15	C₂²×D₅₀₉	—	2	2	no	—
16	D₂₀₃₆	—	2	2	no	—
17	D₂₀₃₆	—	2	2	no	—
18	D₂₀₃₆	—	4	4	no	—
19	#16861325	D₄	1	4	yes	complemented
20	C₂³×C₅₀₉	—	2	1	no	—
21	Q₈×C₅₀₉	—	2	2	no	—
22	D₄×C₅₀₉	—	2	2	no	—
23	C₂×C₄×C₅₀₉	—	2	1	no	—
24	C₂×C₄×C₅₀₉	D₄	1	2	yes	—
25	C₂×C₄×C₅₀₉	C₂³	1	1	yes	characteristic
26	C₂²×C₅₀₉	—	4	1	no	—
27	C₂²×C₅₀₉	—	4	2	no	—
28	C₂²×C₅₀₉	C₂×D₄	1	1	yes	characteristic
29	C₂×D₅₀₉	—	4	2	no	—
30	C₂×D₅₀₉	—	4	4	no	—
31	C₄×C₅₀₉	—	2	2	no	—
32	C₄×C₅₀₉	—	4	1	no	—
33	C₄×C₅₀₉	C₂×D₄	1	2	yes	—
34	C₂×C₅₀₉	—	8	1	no	—
35	C₂×C₅₀₉	#127	1	1	yes	characteristic
36	C₂×C₅₀₉	AGL(1,ℤ/8)	1	2	yes	—
37	D₅₀₉	—	8	2	no	—
38	C₅₀₉	#496	1	1	yes	characteristic, complemented, Sylow
39	#496	—	509	1	no	maximal, Sylow
40	C₂×SD₁₆	—	509	2	no	—
41	C₂×D₈	—	509	2	no	—
42	#127	—	509	1	no	—
43	#124	D₅₀₉	1	1	yes	characteristic, complemented
44	#97	—	509	1	no	—
45	C₂×Q₈	C₂×D₅₀₉	1	1	yes	characteristic
46	C₂×D₄	—	1018	2	no	—
47	C₂×D₄	—	509	2	no	—
48	C₂×D₄	C₂×D₅₀₉	1	1	yes	characteristic
49	SD₁₆	—	1018	4	no	—
50	D₈	—	1018	4	no	—
51	C₂×C₈	—	509	2	no	—
52	C₂²⋊C₄	—	2	2	no	—
53	C₄²	C₂×D₅₀₉	1	1	yes	characteristic
54	C₂³	—	2	1	no	—
55	C₂³	—	1018	2	no	—
56	Q₈	—	2	2	no	—
57	D₄	—	2	2	no	—
58	D₄	—	2036	4	no	—
59	D₄	—	1018	2	no	—
60	D₄	—	1018	2	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	C₂²⋊D₅₀₉	1	2	yes	—
63	C₂×C₄	C₂²×D₅₀₉	1	1	yes	characteristic
64	C₈	—	509	4	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	4	2	no	—
67	C₂²	—	2036	2	no	—
68	C₂²	—	2036	4	no	—
69	C₂²	C₂×C₂²⋊D₅₀₉	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	4	1	no	—
72	C₄	D₄×D₅₀₉	1	2	yes	—
73	C₂	—	8	1	no	—
74	C₂	—	4072	2	no	—
75	C₂	#30654211	1	1	yes	characteristic
76	C₂	#30654240	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅₀₉
commutator, cosocle	C₂×C₄×C₅₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₅₀₉
Fitting	#124×C₅₀₉	C₂
socle	C₂²×C₅₀₉	C₂×D₄
hypercenter	#124	D₅₀₉
hypocenter	C₅₀₉	#496
LCS	G ▷ C₂×C₄×C₅₀₉ ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

Complements

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

N	H ≅ G/N	type
1. C₂×#22550848	C₂	semidirect
2. C₂×#22550846	C₂	semidirect
3. #30654212	C₂	semidirect
4. #30654126	C₂	semidirect
5. #124×C₅₀₉	C₂	semidirect
19. #16861325	D₄	semidirect
38. C₅₀₉	#496	semidirect
43. #124	D₅₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22550848	2
C₂×#22550846	2
#30654212	2
#30654126	2
#124×C₅₀₉	2
#496	509