Subgroups — #45687190

Order 33344.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22552348	C₂	1	2	yes	maximal
2	C₂×M₁₆×C₅₂₁	C₂	1	1	yes	maximal, characteristic
3	#22552348	—	2	2	no	—
4	#22552348	—	2	2	no	—
5	C₂×#16861960	C₄	1	4	yes	—
6	C₂²×C₄×C₅₂₁	C₂²	1	1	yes	characteristic
7	M₁₆×C₅₂₁	—	2	2	no	—
8	C₂×C₈×C₅₂₁	C₄	1	2	yes	—
9	#16861960	—	2	4	no	—
10	C₂³×C₅₂₁	Q₈	1	1	yes	characteristic
11	C₂×C₄×C₅₂₁	C₂×C₄	1	2	yes	—
12	C₂×C₄×C₅₂₁	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₅₂₁	D₄	1	2	yes	—
14	C₂×C₄×C₅₂₁	D₄	1	1	yes	characteristic
15	C₈×C₅₂₁	—	2	2	no	—
16	C₂²×C₅₂₁	—	4	1	no	—
17	C₂²×C₅₂₁	C₄⋊C₄	1	2	yes	—
18	C₂²×C₅₂₁	C₄⋊C₄	1	1	yes	characteristic
19	C₄×C₅₂₁	C₄²	1	1	yes	characteristic
20	C₄×C₅₂₁	C₂²⋊C₄	1	2	yes	—
21	C₄×C₅₂₁	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₅₂₁	—	2	2	no	—
23	C₂×C₅₂₁	—	2	1	no	—
24	C₂×C₅₂₁	#95	1	1	yes	characteristic
25	C₅₂₁	#338	1	1	yes	characteristic, complemented, Sylow
26	#338	—	521	1	no	maximal, Sylow
27	C₂×M₁₆	—	521	2	no	—
28	C₂×M₁₆	D₅₂₁	1	1	yes	characteristic
29	C₂²×C₄	C₂×D₅₂₁	1	1	yes	characteristic
30	M₁₆	—	2	2	no	—
31	M₁₆	—	1042	2	no	—
32	M₁₆	—	1042	2	no	—
33	C₂×C₈	—	521	4	no	—
34	C₂×C₈	Dic₅₂₁	1	2	yes	—
35	C₂³	Dic₁₀₄₂	1	1	yes	characteristic
36	C₂×C₄	C₄×D₅₂₁	1	2	yes	—
37	C₂×C₄	C₂×Dic₅₂₁	1	1	yes	characteristic
38	C₂×C₄	C₂²⋊D₅₂₁	1	2	yes	—
39	C₂×C₄	D₂₀₈₄	1	1	yes	characteristic
40	C₈	—	2	2	no	—
41	C₈	—	1042	4	no	—
42	C₂²	—	4	1	no	—
43	C₂²	#22552343	1	2	yes	—
44	C₂²	C₄⋊Dic₅₂₁	1	1	yes	characteristic
45	C₄	C₄×Dic₅₂₁	1	1	yes	characteristic
46	C₄	#22552341	1	2	yes	—
47	C₄	C₂²⋊Dic₅₂₁	1	1	yes	characteristic
48	C₂	—	2	2	no	—
49	C₂	—	2	1	no	—
50	C₂	#30659615	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×Dic₅₂₁
commutator	C₄×C₅₂₁	C₄²
Frattini	C₂²×C₄	C₂×D₅₂₁
Fitting	C₂×M₁₆×C₅₂₁	C₂
socle	C₂×C₅₂₁	#95
prime radical	C₂³×C₅₂₁	Q₈
cosocle	C₂²×C₄×C₅₂₁	C₂²
hypercenter	C₂×M₁₆	D₅₂₁
hypocenter	C₅₂₁	#338
LCS	G ▷ C₄×C₅₂₁ ▷ C₅₂₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
25. C₅₂₁	#338	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22552348	2
C₂×M₁₆×C₅₂₁	2
#338	521