Subgroups — #27270828

Order 13568.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20272785	C₂	1	2	yes	maximal
2	C₅₃×#1713	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11719908	—	2	2	no	—
4	C₅₃×#500	—	2	2	no	—
5	C₅₃×#494	C₄	1	2	yes	—
6	C₈²×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×#11719908	—	4	2	no	—
8	C₂×Q₁₆×C₅₃	—	4	2	no	—
9	C₄⋊Q₈×C₅₃	—	2	2	no	—
10	#107×C₅₃	—	4	2	no	—
11	#107×C₅₃	—	4	2	no	—
12	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₄×C₈×C₅₃	D₄	1	2	yes	—
14	C₂×Q₈×C₅₃	—	4	2	no	—
15	Q₁₆×C₅₃	—	8	4	no	—
16	C₂×C₈×C₅₃	—	2	1	no	—
17	C₂×C₈×C₅₃	D₈	1	2	yes	—
18	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
19	C₄⋊C₄×C₅₃	—	4	4	no	—
20	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
21	#11719908	—	8	2	no	—
22	Q₈×C₅₃	—	8	4	no	—
23	C₂×C₄×C₅₃	—	8	2	no	—
24	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
25	C₂×C₄×C₅₃	#102	1	2	yes	—
26	C₈×C₅₃	—	2	2	no	—
27	C₈×C₅₃	—	2	2	no	—
28	C₈×C₅₃	SD₃₂	1	2	yes	—
29	C₈×C₅₃	Q₃₂	1	2	yes	—
30	#7098	—	53	1	no	maximal, Sylow
31	C₂²×C₅₃	#331	1	1	yes	characteristic
32	C₄×C₅₃	—	2	1	no	—
33	C₄×C₅₃	—	16	2	no	—
34	C₄×C₅₃	#358	1	2	yes	—
35	C₄×C₅₃	#362	1	2	yes	—
36	#1713	D₅₃	1	1	yes	characteristic
37	#1370	—	53	2	no	—
38	C₂×C₅₃	#1357	1	1	yes	characteristic
39	C₂×C₅₃	#1362	1	2	yes	—
40	#500	—	2	2	no	—
41	#494	Dic₅₃	1	2	yes	—
42	C₄⋊C₁₆	—	106	2	no	—
43	C₈²	C₂×D₅₃	1	1	yes	characteristic
44	C₅₃	#7098	1	1	yes	characteristic, complemented, Sylow
45	C₂×Q₁₆	—	4	2	no	—
46	C₄⋊Q₈	—	2	2	no	—
47	C₂×C₁₆	—	212	2	no	—
48	#107	—	4	2	no	—
49	#107	—	4	2	no	—
50	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
51	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
52	C₂×Q₈	—	4	2	no	—
53	Q₁₆	—	8	4	no	—
54	C₂×C₈	—	2	1	no	—
55	C₂×C₈	#11719922	1	2	yes	—
56	C₂×C₈	#11719923	1	2	yes	—
57	C₄⋊C₄	—	4	4	no	—
58	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
59	C₁₆	—	424	2	no	—
60	Q₈	—	8	4	no	—
61	C₂×C₄	—	8	2	no	—
62	C₂×C₄	#13133134	1	1	yes	characteristic
63	C₂×C₄	#13133138	1	2	yes	—
64	C₈	—	2	2	no	—
65	C₈	—	2	2	no	—
66	C₈	#13133160	1	2	yes	—
67	C₈	#13133163	1	2	yes	—
68	C₂²	#16810367	1	1	yes	characteristic
69	C₄	—	2	1	no	—
70	C₄	—	16	2	no	—
71	C₄	#16810427	1	2	yes	—
72	C₄	#16810439	1	2	yes	—
73	C₂	#20272752	1	1	yes	characteristic
74	C₂	#20272764	1	2	yes	—

	subgroup	G/H
center	C₂²	#16810367
commutator	C₄×C₈×C₅₃	C₂×C₄
Frattini	C₈²	C₂×D₅₃
Fitting	C₅₃×#1713	C₂
socle, prime radical	C₂²×C₅₃	#331
cosocle	C₈²×C₅₃	C₂²
hypercenter	#1713	D₅₃
hypocenter	C₅₃	#7098
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

Complements

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

N	H ≅ G/N	type
44. C₅₃	#7098	semidirect

Maximal subgroups

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

subgroup	index
#20272785	2
C₅₃×#1713	2
#7098	53