Subgroups — #27298801

Order 13568.

246 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276944	C₂	1	4	yes	maximal
2	#20274333	C₂	1	1	yes	maximal, characteristic
3	#20272632	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1710	C₂	1	1	yes	maximal, characteristic
5	#16811028	—	2	2	no	—
6	#16811003	C₂²	1	2	yes	—
7	#16810426	—	2	8	no	—
8	C₄×#11719908	C₂²	1	2	yes	—
9	C₅₃×#500	C₂²	1	2	yes	—
10	C₅₃×#487	—	2	2	no	—
11	C₈²×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×Dic₂₁₂	—	2	4	no	—
13	#13133217	—	2	2	no	—
14	C₂×#11719908	D₄	1	4	yes	—
15	#13133151	—	4	4	no	—
16	C₄⋊Q₈×C₅₃	—	2	2	no	—
17	#107×C₅₃	—	2	4	no	—
18	#103×C₅₃	—	4	4	no	—
19	C₄×C₈×C₅₃	D₄	1	2	yes	—
20	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
21	C₂×Q₈×C₅₃	—	4	2	no	—
22	C₂×Dic₁₀₆	—	4	2	no	—
23	C₂×C₈×C₅₃	D₈	1	4	yes	—
24	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
25	C₄⋊C₄×C₅₃	—	4	2	no	—
26	C₄⋊C₄×C₅₃	—	4	2	no	—
27	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
28	C₄⋊Dic₅₃	—	4	4	no	—
29	Dic₂₁₂	—	4	8	no	—
30	#11719908	—	2	4	no	—
31	Q₈×C₅₃	—	8	4	no	—
32	C₂×C₄×C₅₃	—	8	2	no	—
33	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
35	C₂×Dic₅₃	—	8	2	no	—
36	Dic₁₀₆	—	8	4	no	—
37	C₈×C₅₃	—	2	4	no	—
38	C₈×C₅₃	#135	1	4	yes	—
39	#12514	—	53	1	no	maximal, Sylow
40	C₂²×C₅₃	#493	1	1	yes	characteristic
41	C₄×C₅₃	—	16	2	no	—
42	C₄×C₅₃	#495	1	2	yes	—
43	C₄×C₅₃	#510	1	4	yes	—
44	Dic₅₃	—	16	2	no	—
45	#2241	—	53	4	no	—
46	#1713	—	53	1	no	—
47	#1710	D₅₃	1	1	yes	characteristic
48	#1311	—	53	1	no	—
49	C₂×C₅₃	#1709	1	1	yes	characteristic
50	C₂×C₅₃	#2251	1	2	yes	—
51	#500	—	106	2	no	—
52	#500	C₂×D₅₃	1	2	yes	—
53	#494	—	53	2	no	—
54	#487	—	2	2	no	—
55	#358	—	106	8	no	—
56	C₄×C₁₆	—	53	2	no	—
57	C₈²	C₂×D₅₃	1	1	yes	characteristic
58	C₅₃	#12514	1	1	yes	characteristic, complemented, Sylow
59	C₂×Q₁₆	—	106	4	no	—
60	C₄⋊Q₈	—	2	2	no	—
61	C₄⋊Q₈	—	106	2	no	—
62	C₂×C₁₆	—	53	4	no	—
63	#107	—	2	4	no	—
64	#107	—	212	4	no	—
65	#103	—	4	4	no	—
66	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
67	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
68	C₂×Q₈	—	4	2	no	—
69	C₂×Q₈	—	212	2	no	—
70	Q₁₆	—	212	8	no	—
71	C₂×C₈	#11719922	1	4	yes	—
72	C₂×C₈	D₄×D₅₃	1	2	yes	—
73	C₄⋊C₄	—	4	2	no	—
74	C₄⋊C₄	—	4	2	no	—
75	C₄⋊C₄	—	212	4	no	—
76	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
77	C₁₆	—	106	4	no	—
78	Q₈	—	8	4	no	—
79	Q₈	—	424	4	no	—
80	C₂×C₄	—	8	2	no	—
81	C₂×C₄	—	424	2	no	—
82	C₂×C₄	#13133213	1	1	yes	characteristic
83	C₂×C₄	C₂×#11719922	1	2	yes	—
84	C₈	—	2	4	no	—
85	C₈	#13133240	1	4	yes	—
86	C₂²	#16810999	1	1	yes	characteristic
87	C₄	—	16	2	no	—
88	C₄	—	848	2	no	—
89	C₄	#16811008	1	2	yes	—
90	C₄	#16811063	1	4	yes	—
91	C₂	#20274322	1	1	yes	characteristic
92	C₂	#20276987	1	2	yes	—

	subgroup	G/H
center	C₂²	#16810999
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1710	C₂
socle, prime radical	C₂²×C₅₃	#493
hypercenter	#1710	D₅₃
hypocenter	C₅₃	#12514
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

Complements

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

N	H ≅ G/N	type
58. C₅₃	#12514	semidirect

Maximal subgroups

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

subgroup	index
#20276944	2
#20274333	2
#20272632	2
C₅₃×#1710	2
#12514	53