Subgroups — #27298807

Order 13568.

310 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276968	C₂	1	4	yes	maximal
2	#20274333	C₂	1	2	yes	maximal
3	C₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₄₂₄	—	2	8	no	—
5	#16811028	—	2	4	no	—
6	#16811003	C₂²	1	4	yes	—
7	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
8	C₈²×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×Dic₂₁₂	—	2	8	no	—
10	#13133217	—	2	4	no	—
11	Dic₄₂₄	—	4	16	no	—
12	#13133151	—	4	8	no	—
13	C₂×C₁₆×C₅₃	D₄	1	4	yes	—
14	C₄×C₈×C₅₃	D₄	1	2	yes	—
15	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
16	C₂×Dic₁₀₆	—	4	4	no	—
17	C₂×C₈×C₅₃	D₈	1	4	yes	—
18	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
19	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
20	C₄⋊Dic₅₃	—	4	8	no	—
21	Dic₂₁₂	—	4	16	no	—
22	C₁₆×C₅₃	D₈	1	8	yes	—
23	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
24	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
25	C₂×Dic₅₃	—	8	4	no	—
26	Dic₁₀₆	—	8	8	no	—
27	C₈×C₅₃	Q₃₂	1	8	yes	complemented
28	C₈×C₅₃	C₂×D₈	1	4	yes	—
29	#12516	—	53	1	no	maximal, Sylow
30	C₂²×C₅₃	#493	1	1	yes	characteristic
31	C₄×C₅₃	#493	1	2	yes	—
32	C₄×C₅₃	C₂×Q₃₂	1	4	yes	—
33	Dic₅₃	—	16	4	no	—
34	#2247	—	53	4	no	—
35	#1713	—	53	2	no	—
36	C₈×C₁₆	D₅₃	1	1	yes	characteristic
37	C₂×C₅₃	#1712	1	1	yes	characteristic
38	C₂×C₅₃	#2247	1	2	yes	—
39	C₂×Q₃₂	—	106	8	no	—
40	#500	—	106	4	no	—
41	#494	—	53	4	no	—
42	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
43	C₈²	C₂×D₅₃	1	1	yes	characteristic
44	C₅₃	#12516	1	1	yes	characteristic, complemented, Sylow
45	C₂×Q₁₆	—	106	8	no	—
46	C₄⋊Q₈	—	106	4	no	—
47	Q₃₂	—	212	16	no	—
48	C₂×C₁₆	D₂₁₂	1	4	yes	—
49	#107	—	212	8	no	—
50	C₄×C₈	D₂₁₂	1	2	yes	—
51	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
52	C₂×Q₈	—	212	4	no	—
53	Q₁₆	—	212	16	no	—
54	C₂×C₈	D₄₂₄	1	4	yes	—
55	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
56	C₄⋊C₄	—	212	8	no	—
57	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
58	C₁₆	D₄₂₄	1	8	yes	—
59	Q₈	—	424	8	no	—
60	C₂×C₄	—	424	4	no	—
61	C₂×C₄	#13133214	1	1	yes	characteristic
62	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
63	C₈	Dic₄₂₄	1	8	yes	complemented
64	C₈	C₂×D₄₂₄	1	4	yes	—
65	C₂²	#16811000	1	1	yes	characteristic
66	C₄	—	848	4	no	—
67	C₄	#16811000	1	2	yes	—
68	C₄	C₂×Dic₄₂₄	1	4	yes	—
69	C₂	#20274330	1	1	yes	characteristic
70	C₂	#20276968	1	2	yes	—

	subgroup	G/H
center	C₂²	#16811000
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₈×C₁₆×C₅₃	C₂
socle, prime radical	C₂²×C₅₃	#493
hypercenter	C₈×C₁₆	D₅₃
hypocenter	C₅₃	#12516
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
27. C₈×C₅₃	Q₃₂	semidirect
44. C₅₃	#12516	semidirect
63. C₈	Dic₄₂₄	semidirect

Maximal subgroups

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

subgroup	index
#20276968	2
#20274333	2
C₈×C₁₆×C₅₃	2
#12516	53