Subgroups — #45688351

Order 33344.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30659783	C₂	1	1	yes	maximal, characteristic
2	#30659778	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×D₅₂₁	C₂	1	1	yes	maximal, characteristic
4	#22552374	—	2	1	no	—
5	#22552367	—	2	1	no	—
6	Q₈×D₅₂₁	—	2	1	no	—
7	Dic₂₀₈₄	C₄	1	1	yes	characteristic
8	#22552354	—	2	1	no	—
9	C₈×D₅₂₁	C₂²	1	1	yes	characteristic
10	Q₁₆×C₅₂₁	C₄	1	1	yes	characteristic
11	#16861967	—	4	1	no	—
12	Dic₁₀₄₂	—	2	1	no	—
13	Dic₁₀₄₂	—	4	1	no	—
14	C₄×D₅₂₁	—	4	1	no	—
15	C₄×D₅₂₁	D₄	1	1	yes	characteristic
16	#16861960	D₄	1	1	yes	characteristic
17	Q₈×C₅₂₁	—	2	1	no	—
18	C₈×C₅₂₁	C₂×C₄	1	1	yes	characteristic
19	C₂×D₅₂₁	SD₁₆	1	1	yes	characteristic
20	C₄×C₅₂₁	—	4	1	no	—
21	C₄×C₅₂₁	C₂²⋊C₄	1	1	yes	characteristic
22	Dic₅₂₁	—	4	1	no	—
23	Dic₅₂₁	D₈	1	1	yes	characteristic
24	C₂×C₅₂₁	#102	1	1	yes	characteristic
25	D₅₂₁	—	2	1	no	—
26	C₅₂₁	#362	1	1	yes	characteristic, complemented, Sylow
27	#362	—	521	1	no	maximal, Sylow
28	C₂×Q₁₆	—	521	1	no	—
29	M₃₂	—	521	1	no	—
30	#108	—	521	1	no	—
31	C₂×Q₈	—	1042	1	no	—
32	Q₁₆	—	521	1	no	—
33	Q₁₆	—	1042	1	no	—
34	Q₁₆	#14523438	1	1	yes	characteristic
35	M₁₆	—	1042	1	no	—
36	C₂×C₈	—	521	1	no	—
37	C₁₆	—	1042	1	no	—
38	Q₈	—	2	1	no	—
39	Q₈	—	1042	1	no	—
40	Q₈	—	2084	1	no	—
41	C₂×C₄	—	521	1	no	—
42	C₂×C₄	—	2084	1	no	—
43	C₈	—	521	1	no	—
44	C₈	—	2084	1	no	—
45	C₈	C₂×#14523438	1	1	yes	characteristic
46	C₂²	—	521	1	no	—
47	C₄	—	4	1	no	—
48	C₄	—	521	1	no	—
49	C₄	—	2084	1	no	—
50	C₄	C₂²⋊#14523438	1	1	yes	characteristic
51	C₂	—	1042	1	no	—
52	C₂	#30659770	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#30659770
commutator	C₈×C₅₂₁	C₂×C₄
Frattini	C₈	C₂×#14523438
Fitting	Q₁₆×C₅₂₁	C₄
socle	C₂×C₅₂₁	#102
prime radical	C₂×D₅₂₁	SD₁₆
cosocle	C₈×D₅₂₁	C₂²
hypercenter	Q₁₆	#14523438
hypocenter	C₅₂₁	#362
LCS	G ▷ C₈×C₅₂₁ ▷ C₄×C₅₂₁ ▷ C₂×C₅₂₁ ▷ C₅₂₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
26. C₅₂₁	#362	semidirect

Maximal subgroups

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

subgroup	index
#30659783	2
#30659778	2
Q₁₆×D₅₂₁	2
#362	521