Subgroups — #45687816

Order 33344.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₀₈₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#22552353	C₂	1	2	yes	maximal, complemented
3	C₂×D₄₁₆₈	C₂	1	1	yes	maximal, characteristic
4	#30659696	C₂	1	2	yes	maximal, complemented
5	C₄×C₈×C₅₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₁₀₄₂	C₂²	1	2	yes	—
7	C₂×D₂₀₈₄	C₂²	1	2	yes	—
8	Dic₂₀₈₄	—	2	2	no	—
9	#22552353	—	2	4	no	—
10	D₄₁₆₈	—	2	2	no	—
11	#22552341	—	2	4	no	—
12	C₂×C₈×C₅₂₁	C₂²	1	1	yes	characteristic
13	C₂×C₈×C₅₂₁	C₂²	1	1	yes	characteristic
14	C₄²×C₅₂₁	C₂²	1	1	yes	characteristic
15	C₂²×D₅₂₁	—	2	2	no	—
16	Dic₁₀₄₂	—	2	2	no	—
17	Dic₁₀₄₂	—	2	2	no	—
18	D₂₀₈₄	—	2	2	no	—
19	D₂₀₈₄	—	2	2	no	—
20	C₂×Dic₅₂₁	—	2	2	no	—
21	C₂×C₄×C₅₂₁	D₄	1	2	yes	—
22	C₂×C₄×C₅₂₁	C₂³	1	1	yes	characteristic
23	C₈×C₅₂₁	D₄	1	2	yes	—
24	C₈×C₅₂₁	D₄	1	2	yes	—
25	C₂²×C₅₂₁	C₂×D₄	1	1	yes	characteristic
26	C₂×D₅₂₁	—	4	2	no	—
27	C₂×D₅₂₁	—	4	4	no	—
28	C₄×C₅₂₁	—	2	2	no	—
29	C₄×C₅₂₁	C₂×D₄	1	1	yes	characteristic
30	C₄×C₅₂₁	C₂×D₄	1	1	yes	characteristic
31	Dic₅₂₁	—	4	2	no	—
32	C₂×C₅₂₁	#127	1	1	yes	characteristic
33	C₂×C₅₂₁	#135	1	2	yes	—
34	D₅₂₁	—	8	2	no	—
35	C₅₂₁	#495	1	1	yes	characteristic, complemented, Sylow
36	#495	—	521	1	no	maximal, Sylow
37	C₂×Q₁₆	—	521	1	no	—
38	C₂×SD₁₆	—	521	2	no	—
39	C₂×D₈	—	521	1	no	—
40	#124	—	521	2	no	—
41	C₄×C₈	D₅₂₁	1	1	yes	characteristic, complemented
42	C₂×Q₈	—	521	2	no	—
43	C₂×D₄	—	521	2	no	—
44	Q₁₆	—	1042	2	no	—
45	SD₁₆	—	1042	4	no	—
46	D₈	—	1042	2	no	—
47	C₂×C₈	C₂×D₅₂₁	1	1	yes	characteristic
48	C₂×C₈	C₂×D₅₂₁	1	1	yes	characteristic
49	C₂²⋊C₄	—	1042	4	no	—
50	C₄²	C₂×D₅₂₁	1	1	yes	characteristic
51	C₂³	—	1042	2	no	—
52	Q₈	—	1042	2	no	—
53	Q₈	—	1042	2	no	—
54	D₄	—	1042	2	no	—
55	D₄	—	1042	2	no	—
56	C₂×C₄	—	1042	2	no	—
57	C₂×C₄	D₂₀₈₄	1	2	yes	—
58	C₂×C₄	C₂²×D₅₂₁	1	1	yes	characteristic
59	C₈	D₂₀₈₄	1	2	yes	—
60	C₈	D₂₀₈₄	1	2	yes	—
61	C₂²	—	2084	2	no	—
62	C₂²	—	2084	4	no	—
63	C₂²	C₂×D₂₀₈₄	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	2084	2	no	—
66	C₄	C₂×D₂₀₈₄	1	1	yes	characteristic
67	C₄	C₂×D₂₀₈₄	1	1	yes	characteristic
68	C₂	—	4168	2	no	—
69	C₂	#30659705	1	1	yes	characteristic
70	C₂	#30659730	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₂₀₈₄
commutator, cosocle	C₂×C₄×C₅₂₁	C₂³
Frattini	C₂×C₄	C₂²×D₅₂₁
Fitting	C₄×C₈×C₅₂₁	C₂
socle	C₂²×C₅₂₁	C₂×D₄
prime radical	C₂×D₄₁₆₈	C₂
hypercenter	C₄×C₈	D₅₂₁
hypocenter	C₅₂₁	#495
LCS	G ▷ C₂×C₄×C₅₂₁ ▷ C₂×C₅₂₁ ▷ C₅₂₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₂₀₈₄	C₂	semidirect
2. C₂×#22552353	C₂	semidirect
4. #30659696	C₂	semidirect
5. C₄×C₈×C₅₂₁	C₂	semidirect
35. C₅₂₁	#495	semidirect
41. C₄×C₈	D₅₂₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₀₈₄	2
C₂×#22552353	2
C₂×D₄₁₆₈	2
#30659696	2
C₄×C₈×C₅₂₁	2
#495	521