Subgroups — #27271424

Order 13568.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133123	C₂	1	1	yes	maximal, characteristic
2	C₁₆×#88322	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133123	C₄	1	2	yes	—
5	C₄×#11719908	C₄	1	1	yes	characteristic
6	C₈×#88322	C₄	1	1	yes	characteristic
7	C₂×C₃₂×C₅₃	C₄	1	2	yes	—
8	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×#11719908	C₈	1	2	yes	—
10	C₄×#88322	C₈	1	1	yes	characteristic
11	C₄×#88322	C₈	1	1	yes	characteristic
12	#13133123	—	2	2	no	—
13	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₅₃	—	2	2	no	—
17	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
18	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
19	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
20	C₂×#88322	C₁₆	1	2	yes	—
21	C₂×#88322	C₁₆	1	2	yes	—
22	C₁₆×C₅₃	M₁₆	1	2	yes	—
23	C₁₆×C₅₃	M₁₆	1	2	yes	—
24	#11719908	—	2	2	no	—
25	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
26	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
27	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
28	C₈×C₅₃	#97	1	1	yes	characteristic
29	C₈×C₅₃	#97	1	1	yes	characteristic
30	C₈×C₅₃	M₃₂	1	2	yes	—
31	#88322	—	2	2	no	—
32	#88322	C₃₂	1	4	yes	complemented
33	#7331	—	53	1	no	maximal, Sylow
34	C₂²×C₅₃	C₄×C₁₆	1	1	yes	characteristic
35	C₄×C₅₃	#322	1	1	yes	characteristic
36	C₄×C₅₃	#346	1	1	yes	characteristic
37	C₄×C₅₃	C₂×C₃₂	1	2	yes	—
38	C₄×C₅₃	M₆₄	1	2	yes	—
39	C₄×C₃₂	—	53	1	no	—
40	C₄×C₃₂	D₅₃	1	1	yes	characteristic
41	C₈×C₁₆	—	53	1	no	—
42	C₂×C₅₃	#1312	1	1	yes	characteristic
43	C₂×C₅₃	C₄×C₃₂	1	1	yes	characteristic
44	C₂×C₅₃	#1397	1	1	yes	characteristic
45	C₂×C₃₂	—	53	2	no	—
46	C₂×C₃₂	Dic₅₃	1	2	yes	—
47	C₄×C₁₆	—	53	1	no	—
48	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
49	C₈²	—	53	1	no	—
50	C₅₃	#7331	1	1	yes	characteristic, complemented, Sylow
51	C₂×C₁₆	—	53	2	no	—
52	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
53	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
54	C₄×C₈	—	53	1	no	—
55	C₄×C₈	—	53	1	no	—
56	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
57	C₃₂	—	2	2	no	—
58	C₃₂	—	106	2	no	—
59	C₂×C₈	—	53	2	no	—
60	C₂×C₈	—	53	2	no	—
61	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
62	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
63	C₄²	C₈×D₅₃	1	1	yes	characteristic
64	C₁₆	—	106	2	no	—
65	C₁₆	#11719912	1	2	yes	—
66	C₁₆	#11719917	1	2	yes	—
67	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
68	C₂×C₄	C₁₆×D₅₃	1	1	yes	characteristic
69	C₂×C₄	C₁₆×D₅₃	1	1	yes	characteristic
70	C₈	—	53	4	no	—
71	C₈	—	106	2	no	—
72	C₈	#13133127	1	1	yes	characteristic
73	C₈	#13133128	1	1	yes	characteristic
74	C₈	#13133156	1	2	yes	—
75	C₂²	C₁₆×Dic₅₃	1	1	yes	characteristic
76	C₄	#16810345	1	1	yes	characteristic
77	C₄	#16810398	1	1	yes	characteristic
78	C₄	C₃₂×D₅₃	1	2	yes	—
79	C₄	#16810456	1	2	yes	—
80	C₂	#20272634	1	1	yes	characteristic
81	C₂	C₃₂×Dic₅₃	1	1	yes	characteristic
82	C₂	#20272848	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄×C₁₆	C₂×D₅₃
commutator	C₂×C₅₃	C₄×C₃₂
Fitting	C₄×C₃₂×C₅₃	C₂
socle, prime radical	C₂²×C₅₃	C₄×C₁₆
cosocle	C₄×C₁₆×C₅₃	C₂²
hypercenter	C₄×C₃₂	D₅₃
hypocenter	C₅₃	#7331
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
32. #88322	C₃₂	semidirect
50. C₅₃	#7331	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133123	2
C₁₆×#88322	2
C₄×C₃₂×C₅₃	2
#7331	53