Subgroups — #27270709

Order 13568.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20272633	C₂	1	1	yes	maximal, characteristic
2	#20272632	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1312	C₂	1	1	yes	maximal, characteristic
4	C₄×#11719908	C₄	1	2	yes	—
5	C₄×#11719908	C₄	1	1	yes	characteristic
6	C₄×#11719908	C₄	1	1	yes	characteristic
7	C₄×C₁₆×C₅₃	C₄	1	2	yes	—
8	C₈²×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×#11719908	—	2	2	no	—
10	C₂×#11719908	C₈	1	2	yes	—
11	C₂×#11719908	C₈	1	2	yes	—
12	C₂×C₁₆×C₅₃	—	2	2	no	—
13	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
17	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
18	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
19	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
20	C₄²×C₅₃	C₄²	1	1	yes	characteristic
21	C₁₆×C₅₃	—	4	2	no	—
22	#11719908	—	2	2	no	—
23	#11719908	—	4	2	no	—
24	#11719908	C₁₆	1	4	yes	complemented
25	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
26	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
27	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
28	C₈×C₅₃	—	2	2	no	—
29	C₈×C₅₃	—	2	2	no	—
30	C₈×C₅₃	C₂×C₁₆	1	2	yes	—
31	C₈×C₅₃	M₃₂	1	2	yes	—
32	#7053	—	53	1	no	maximal, Sylow
33	C₂²×C₅₃	#322	1	1	yes	characteristic
34	C₄×C₅₃	C₄×C₁₆	1	1	yes	characteristic
35	C₄×C₅₃	#346	1	1	yes	characteristic
36	C₄×C₅₃	#347	1	2	yes	—
37	C₄×C₅₃	#347	1	2	yes	—
38	#1312	—	53	1	no	—
39	#1312	D₅₃	1	1	yes	characteristic
40	#1311	—	53	1	no	—
41	C₂×C₅₃	#1312	1	1	yes	characteristic
42	C₂×C₅₃	#1313	1	1	yes	characteristic
43	C₂×C₅₃	#1313	1	1	yes	characteristic
44	C₄×C₁₆	—	53	1	no	—
45	C₄×C₁₆	—	53	1	no	—
46	C₄×C₁₆	—	53	2	no	—
47	C₄×C₁₆	Dic₅₃	1	2	yes	—
48	C₈²	C₂×D₅₃	1	1	yes	characteristic
49	C₅₃	#7053	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₁₆	—	2	2	no	—
51	C₂×C₁₆	—	53	2	no	—
52	C₂×C₁₆	—	53	2	no	—
53	C₂×C₁₆	—	106	2	no	—
54	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
55	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
56	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
57	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
58	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
59	C₂×C₈	#11719912	1	2	yes	—
60	C₂×C₈	#11719917	1	2	yes	—
61	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
62	C₁₆	—	4	2	no	—
63	C₁₆	—	53	4	no	—
64	C₁₆	—	106	2	no	—
65	C₁₆	—	212	2	no	—
66	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
67	C₂×C₄	#13133127	1	1	yes	characteristic
68	C₂×C₄	#13133128	1	1	yes	characteristic
69	C₈	—	2	2	no	—
70	C₈	—	2	2	no	—
71	C₈	C₁₆×D₅₃	1	2	yes	—
72	C₈	#13133156	1	2	yes	—
73	C₂²	#16810345	1	1	yes	characteristic
74	C₄	C₁₆×Dic₅₃	1	1	yes	characteristic
75	C₄	#16810398	1	1	yes	characteristic
76	C₄	#16810400	1	2	yes	—
77	C₄	#16810401	1	2	yes	—
78	C₂	#20272634	1	1	yes	characteristic
79	C₂	#20272636	1	1	yes	characteristic
80	C₂	#20272637	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₄×Dic₅₃
commutator	C₄×C₅₃	C₄×C₁₆
Frattini	C₈²	C₂×D₅₃
Fitting	C₅₃×#1312	C₂
socle, prime radical	C₂²×C₅₃	#322
cosocle	C₈²×C₅₃	C₂²
hypercenter	#1312	D₅₃
hypocenter	C₅₃	#7053
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ #1312

Complements

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

N	H ≅ G/N	type
24. #11719908	C₁₆	semidirect
49. C₅₃	#7053	semidirect

Maximal subgroups

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

subgroup	index
#20272633	2
#20272632	2
C₅₃×#1312	2
#7053	53