Subgroups — #28298867

Order 13568.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289823	C₂	1	1	yes	maximal, characteristic
2	#20289626	C₂	1	1	yes	maximal, characteristic
3	#20272848	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133267	C₄	1	2	yes	—
5	#16811509	C₄	1	1	yes	characteristic
6	C₈×#88324	C₄	1	1	yes	characteristic
7	C₂×#13133123	C₄	1	1	yes	characteristic
8	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₃₂×C₅₃	C₄	1	1	yes	characteristic
10	C₂×#11719910	C₈	1	2	yes	—
11	C₄×#88324	C₈	1	1	yes	characteristic
12	C₄×#88324	C₈	1	1	yes	characteristic
13	#13133267	—	2	2	no	—
14	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
15	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
16	#13133123	—	2	1	no	—
17	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₃₂×C₅₃	—	2	1	no	—
19	C₂×#88324	—	2	1	no	—
20	C₂×#88324	—	2	1	no	—
21	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
22	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
23	C₂×#88322	C₂×C₈	1	1	yes	characteristic
24	#11719910	C₁₆	1	4	yes	—
25	C₁₆×C₅₃	M₁₆	1	1	yes	characteristic
26	C₁₆×C₅₃	M₁₆	1	1	yes	characteristic
27	#11719908	M₁₆	1	2	yes	—
28	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
29	C₂×Dic₅₃	M₃₂	1	1	yes	characteristic
30	C₂×Dic₅₃	M₃₂	1	1	yes	characteristic
31	#88324	—	4	1	no	—
32	#88324	—	4	1	no	—
33	C₈×C₅₃	#97	1	1	yes	characteristic
34	C₈×C₅₃	#97	1	1	yes	characteristic
35	#88322	C₂×C₁₆	1	2	yes	—
36	#7332	—	53	1	no	maximal, Sylow
37	C₂²×C₅₃	#346	1	1	yes	characteristic
38	C₄×C₅₃	#322	1	1	yes	characteristic
39	C₄×C₅₃	C₄×C₁₆	1	1	yes	characteristic
40	Dic₅₃	—	2	1	no	—
41	Dic₅₃	—	2	1	no	—
42	#1397	—	53	1	no	—
43	#1397	—	53	1	no	—
44	#1311	—	53	1	no	—
45	C₂×C₅₃	#1312	1	1	yes	characteristic
46	C₂×C₅₃	#1398	1	1	yes	characteristic
47	C₂×C₅₃	#1398	1	1	yes	characteristic
48	C₂×C₃₂	—	53	2	no	—
49	C₂×C₃₂	—	53	1	no	—
50	C₂×C₃₂	#6157	1	1	yes	characteristic
51	C₄×C₁₆	—	53	1	no	—
52	C₄×C₁₆	—	53	1	no	—
53	C₈²	—	53	1	no	—
54	C₅₃	#7332	1	1	yes	characteristic, complemented, Sylow
55	C₂×C₁₆	—	53	2	no	—
56	C₂×C₁₆	—	53	1	no	—
57	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
58	C₄×C₈	—	53	1	no	—
59	C₄×C₈	—	53	1	no	—
60	C₄×C₈	—	53	1	no	—
61	C₃₂	—	2	1	no	—
62	C₃₂	—	106	2	no	—
63	C₃₂	—	106	1	no	—
64	C₂×C₈	—	53	1	no	—
65	C₂×C₈	—	106	1	no	—
66	C₂×C₈	—	106	1	no	—
67	C₂×C₈	C₄×#6157	1	1	yes	characteristic
68	C₄²	—	53	1	no	—
69	C₁₆	—	53	4	no	—
70	C₁₆	—	53	2	no	—
71	C₁₆	#11719936	1	1	yes	characteristic
72	C₁₆	#11719940	1	1	yes	characteristic
73	C₂×C₄	—	53	1	no	—
74	C₂×C₄	—	53	1	no	—
75	C₂×C₄	C₈×#6157	1	1	yes	characteristic
76	C₈	—	53	2	no	—
77	C₈	—	212	1	no	—
78	C₈	—	212	1	no	—
79	C₈	#13133271	1	1	yes	characteristic
80	C₈	#13133272	1	1	yes	characteristic
81	C₂²	#16811510	1	1	yes	characteristic
82	C₄	—	106	1	no	—
83	C₄	—	106	1	no	—
84	C₄	#88324⋊C₈	1	1	yes	characteristic
85	C₄	C₁₆×#6157	1	1	yes	characteristic
86	C₂	#88324⋊C₁₆	1	1	yes	characteristic
87	C₂	#20289824	1	1	yes	characteristic
88	C₂	#20289825	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄×#6157
commutator	C₄×C₅₃	C₄×C₁₆
Frattini	C₂×C₁₆	C₂×#6157
Fitting	C₂×C₃₂×C₅₃	C₄
socle, prime radical	C₂²×C₅₃	#346
cosocle	C₁₆×Dic₅₃	C₂²
hypercenter	C₂×C₃₂	#6157
hypocenter	C₅₃	#7332
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
54. C₅₃	#7332	semidirect

Maximal subgroups

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

subgroup	index
#20289823	2
#20289626	2
#20272848	2
#7332	53