Subgroups — C₂×#30654126 #45655187

Order 32576.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#16861325	C₂	1	2	yes	maximal
2	#30654126	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄²×C₅₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#16861325	C₄	1	8	yes	complemented
5	C₂×#16861325	C₂²	1	4	yes	—
6	C₂²×C₄×C₅₀₉	C₄	1	2	yes	—
7	C₂²×C₄×C₅₀₉	C₂²	1	1	yes	characteristic
8	C₄²×C₅₀₉	C₄	1	2	yes	—
9	C₄²×C₅₀₉	C₂²	1	2	yes	—
10	#16861325	C₂×C₄	1	8	yes	complemented
11	C₂³×C₅₀₉	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×C₅₀₉	—	2	4	no	—
13	C₂×C₄×C₅₀₉	C₂×C₄	1	4	yes	—
14	C₂×C₄×C₅₀₉	C₂×C₄	1	4	yes	—
15	C₂×C₄×C₅₀₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₅₀₉	C₂³	1	1	yes	characteristic
17	C₂²×C₅₀₉	C₄²	1	2	yes	—
18	C₂²×C₅₀₉	M₁₆	1	4	yes	—
19	C₂²×C₅₀₉	C₂²×C₄	1	1	yes	characteristic
20	C₄×C₅₀₉	—	2	4	no	—
21	C₄×C₅₀₉	C₄²	1	2	yes	—
22	C₄×C₅₀₉	C₂²×C₄	1	2	yes	—
23	C₂×C₅₀₉	#97	1	4	yes	complemented
24	C₂×C₅₀₉	C₂×C₄²	1	1	yes	characteristic
25	C₂×C₅₀₉	C₂×M₁₆	1	2	yes	—
26	C₅₀₉	C₂×#97	1	1	yes	characteristic, complemented, Sylow
27	C₂×#97	—	509	1	no	maximal, Sylow
28	C₂²×C₈	—	509	2	no	—
29	C₂×C₄²	D₅₀₉	1	1	yes	characteristic
30	#97	—	509	4	no	—
31	C₂²×C₄	Dic₅₀₉	1	2	yes	—
32	C₂²×C₄	C₂×D₅₀₉	1	1	yes	characteristic
33	C₂×C₈	—	509	8	no	—
34	C₂×C₈	—	509	4	no	—
35	C₄²	Dic₅₀₉	1	2	yes	—
36	C₄²	C₂×D₅₀₉	1	2	yes	—
37	C₂³	C₂×Dic₅₀₉	1	1	yes	characteristic
38	C₂×C₄	—	2	4	no	—
39	C₂×C₄	C₄×D₅₀₉	1	4	yes	—
40	C₂×C₄	C₂×Dic₅₀₉	1	4	yes	—
41	C₂×C₄	C₂×Dic₅₀₉	1	1	yes	characteristic
42	C₂×C₄	C₂²×D₅₀₉	1	1	yes	characteristic
43	C₈	—	509	8	no	—
44	C₂²	C₄×Dic₅₀₉	1	2	yes	—
45	C₂²	#22550845	1	4	yes	—
46	C₂²	C₂²×Dic₅₀₉	1	1	yes	characteristic
47	C₄	—	2	4	no	—
48	C₄	C₄×Dic₅₀₉	1	2	yes	—
49	C₄	C₂×C₄×D₅₀₉	1	2	yes	—
50	C₂	#30654126	1	4	yes	direct product factor
51	C₂	C₂×C₄×Dic₅₀₉	1	1	yes	characteristic
52	C₂	C₂×#22550845	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₀₉
commutator	C₂×C₅₀₉	C₂×C₄²
Frattini	C₂×C₄	C₂²×D₅₀₉
Fitting	C₂×C₄²×C₅₀₉	C₂
socle, prime radical	C₂³×C₅₀₉	C₂×C₄
cosocle	C₂×C₄×C₅₀₉	C₂³
hypercenter	C₂×C₄²	D₅₀₉
hypocenter	C₅₀₉	C₂×#97
LCS	G ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. #30654126	C₂	direct (N × H)
4. C₂×#16861325	C₄	semidirect
10. #16861325	C₂×C₄	semidirect
23. C₂×C₅₀₉	#97	semidirect
26. C₅₀₉	C₂×#97	semidirect
50. C₂	#30654126	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#16861325	2
#30654126	2
C₂×C₄²×C₅₀₉	2
C₂×#97	509

Subgroups — C2×#30654126 #45655187

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#30654126 #45655187