Subgroups — #45687556

Order 33344.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₁₀₄₂	C₂	1	1	yes	maximal, characteristic
2	#30659637	C₂	1	2	yes	maximal
3	C₄×#16861960	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₅₂₁	C₂	1	2	yes	maximal
5	C₄⋊C₈×C₅₂₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₀₄₂	C₄	1	1	yes	characteristic
7	C₂×#16861960	C₂²	1	1	yes	characteristic
8	C₂×#16861960	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₅₂₁	C₄	1	1	yes	characteristic
10	#22552343	C₄	1	2	yes	—
11	C₄×Dic₅₂₁	C₂²	1	2	yes	—
12	C₂×C₈×C₅₂₁	C₂²	1	2	yes	—
13	C₄²×C₅₂₁	C₂²	1	1	yes	characteristic
14	Dic₁₀₄₂	C₈	1	4	yes	complemented
15	C₂×Dic₅₂₁	C₂×C₄	1	2	yes	—
16	C₂×Dic₅₂₁	C₂×C₄	1	2	yes	—
17	#16861960	—	2	1	no	—
18	#16861960	Q₈	1	2	yes	complemented
19	C₂×C₄×C₅₂₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₅₂₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₅₂₁	C₂³	1	1	yes	characteristic
22	C₈×C₅₂₁	—	2	2	no	—
23	C₂²×C₅₂₁	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₅₂₁	—	2	1	no	—
25	C₄×C₅₂₁	C₂×C₈	1	2	yes	—
26	C₄×C₅₂₁	C₂×Q₈	1	1	yes	characteristic
27	C₄×C₅₂₁	#41	1	1	yes	characteristic
28	Dic₅₂₁	—	2	2	no	—
29	Dic₅₂₁	C₂×C₈	1	4	yes	—
30	C₂×C₅₂₁	C₄×Q₈	1	1	yes	characteristic
31	C₂×C₅₂₁	C₂²×C₈	1	1	yes	characteristic
32	C₂×C₅₂₁	#131	1	1	yes	characteristic
33	C₅₂₁	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
34	C₈×Q₈	—	521	1	no	maximal, Sylow
35	C₄×Q₈	—	521	1	no	—
36	C₄⋊C₈	—	521	2	no	—
37	C₄⋊C₈	D₅₂₁	1	1	yes	characteristic
38	C₄×C₈	—	521	2	no	—
39	C₄×C₈	—	521	1	no	—
40	C₂×Q₈	—	521	1	no	—
41	C₂×C₈	—	521	1	no	—
42	C₂×C₈	—	521	1	no	—
43	C₂×C₈	C₂×D₅₂₁	1	2	yes	—
44	C₄⋊C₄	—	521	2	no	—
45	C₄⋊C₄	—	521	1	no	—
46	C₄²	—	521	2	no	—
47	C₄²	C₂×D₅₂₁	1	1	yes	characteristic
48	Q₈	—	521	4	no	—
49	C₂×C₄	—	521	2	no	—
50	C₂×C₄	—	521	2	no	—
51	C₂×C₄	C₄×D₅₂₁	1	1	yes	characteristic
52	C₂×C₄	C₄×D₅₂₁	1	1	yes	characteristic
53	C₂×C₄	C₂²×D₅₂₁	1	1	yes	characteristic
54	C₈	—	2	2	no	—
55	C₈	—	521	2	no	—
56	C₈	—	1042	1	no	—
57	C₂²	C₂×C₄×D₅₂₁	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	521	4	no	—
60	C₄	—	1042	2	no	—
61	C₄	C₈×D₅₂₁	1	2	yes	—
62	C₄	Q₈×D₅₂₁	1	1	yes	characteristic
63	C₄	#22552363	1	1	yes	characteristic
64	C₂	#30659673	1	1	yes	characteristic
65	C₂	C₂×C₈×D₅₂₁	1	1	yes	characteristic
66	C₂	#30659716	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₅₂₁
commutator	C₂×C₅₂₁	C₂²×C₈
Fitting	C₄⋊C₈×C₅₂₁	C₂
socle, prime radical	C₂²×C₅₂₁	C₂²×C₄
cosocle	C₂×C₄×C₅₂₁	C₂³
hypercenter	C₄⋊C₈	D₅₂₁
hypocenter	C₅₂₁	C₈×Q₈
LCS	G ▷ C₂×C₅₂₁ ▷ C₅₂₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
14. Dic₁₀₄₂	C₈	semidirect
18. #16861960	Q₈	semidirect
33. C₅₂₁	C₈×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₁₀₄₂	2
#30659637	2
C₄×#16861960	2
C₈×Dic₅₂₁	2
C₄⋊C₈×C₅₂₁	2
C₈×Q₈	521