Subgroups — #45688301

Order 33344.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅₂₁⋊M₁₆	C₂	1	2	yes	maximal
2	C₂×#22552347	C₂	1	1	yes	maximal, characteristic
3	#22552374	—	2	2	no	—
4	#22552373	—	2	2	no	—
5	#22552372	C₄	1	4	yes	—
6	C₂×C₄×D₅₂₁	C₂²	1	1	yes	characteristic
7	#22552347	—	2	2	no	—
8	C₂×#16861960	C₄	1	1	yes	characteristic
9	C₂×C₈×C₅₂₁	C₄	1	1	yes	characteristic
10	#16861967	—	2	4	no	—
11	C₂²×D₅₂₁	Q₈	1	1	yes	characteristic
12	C₂×Dic₅₂₁	D₄	1	1	yes	characteristic
13	C₄×D₅₂₁	C₂×C₄	1	2	yes	—
14	C₄×D₅₂₁	D₄	1	2	yes	—
15	#16861960	—	2	1	no	—
16	C₂×C₄×C₅₂₁	C₂×C₄	1	1	yes	characteristic
17	C₈×C₅₂₁	—	2	1	no	—
18	C₂²×C₅₂₁	C₄⋊C₄	1	1	yes	characteristic
19	C₂×D₅₂₁	—	4	1	no	—
20	C₂×D₅₂₁	C₄⋊C₄	1	2	yes	—
21	C₄×C₅₂₁	C₄²	1	1	yes	characteristic
22	C₄×C₅₂₁	C₂²⋊C₄	1	1	yes	characteristic
23	Dic₅₂₁	C₂²⋊C₄	1	2	yes	—
24	C₂×C₅₂₁	—	2	1	no	—
25	C₂×C₅₂₁	#95	1	1	yes	characteristic
26	D₅₂₁	—	2	2	no	—
27	C₅₂₁	#338	1	1	yes	characteristic, complemented, Sylow
28	#338	—	521	1	no	maximal, Sylow
29	C₂×M₁₆	—	521	2	no	—
30	C₂×M₁₆	—	521	1	no	—
31	C₂²×C₄	—	521	1	no	—
32	M₁₆	—	1042	2	no	—
33	M₁₆	—	1042	2	no	—
34	M₁₆	—	1042	2	no	—
35	C₂×C₈	—	521	4	no	—
36	C₂×C₈	—	521	1	no	—
37	C₂×C₈	#14523438	1	1	yes	characteristic
38	C₂³	—	521	1	no	—
39	C₂×C₄	—	521	1	no	—
40	C₂×C₄	—	521	2	no	—
41	C₂×C₄	—	521	2	no	—
42	C₂×C₄	C₂×#14523438	1	1	yes	characteristic
43	C₈	—	2	1	no	—
44	C₈	—	1042	4	no	—
45	C₈	—	1042	1	no	—
46	C₂²	—	521	2	no	—
47	C₂²	—	2084	1	no	—
48	C₂²	C₄⋊#14523438	1	1	yes	characteristic
49	C₄	—	521	2	no	—
50	C₄	C₄×#14523438	1	1	yes	characteristic
51	C₄	C₂²⋊#14523438	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	1042	2	no	—
54	C₂	(C₂×#14523438)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×#14523438
commutator	C₄×C₅₂₁	C₄²
Frattini	C₂×C₄	C₂×#14523438
Fitting	C₂×C₈×C₅₂₁	C₄
socle	C₂×C₅₂₁	#95
prime radical	C₂²×D₅₂₁	Q₈
cosocle	C₂×C₄×D₅₂₁	C₂²
hypercenter	C₂×C₈	#14523438
hypocenter	C₅₂₁	#338
LCS	G ▷ C₄×C₅₂₁ ▷ C₅₂₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈

Complements

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

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

Maximal subgroups

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

subgroup	index
D₅₂₁⋊M₁₆	2
C₂×#22552347	2
#338	521