Subgroups — #28298856

Order 13568.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289820	C₂	1	1	yes	maximal, characteristic
2	C₁₆×#88324	C₂	1	1	yes	maximal, characteristic
3	C₃₂×Dic₅₃	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	2	yes	—
12	#13133267	C₈	1	4	yes	complemented
13	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
14	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
15	#13133123	C₈	1	2	yes	complemented
16	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₃₂×C₅₃	C₈	1	2	yes	complemented
18	C₂×#88324	C₁₆	1	4	yes	—
19	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
20	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
21	C₂×#88322	C₂×C₈	1	1	yes	characteristic
22	#11719910	C₁₆	1	4	yes	—
23	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
24	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
25	#11719908	C₂×C₈	1	2	yes	—
26	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
27	C₂×Dic₅₃	C₂×C₁₆	1	2	yes	—
28	#88324	—	2	4	no	—
29	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
30	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
31	#88322	C₂×C₁₆	1	2	yes	—
32	#7329	—	53	1	no	maximal, Sylow
33	C₂²×C₅₃	C₄×C₁₆	1	1	yes	characteristic
34	C₄×C₅₃	C₈²	1	1	yes	characteristic
35	C₄×C₅₃	C₄×C₁₆	1	1	yes	characteristic
36	Dic₅₃	M₆₄	1	4	yes	—
37	C₄×C₃₂	—	53	1	no	—
38	C₄×C₃₂	—	53	1	no	—
39	C₈×C₁₆	—	53	1	no	—
40	C₂×C₅₃	C₈×C₁₆	1	1	yes	characteristic
41	C₂×C₅₃	#1397	1	2	yes	—
42	C₂×C₃₂	—	53	2	no	—
43	C₂×C₃₂	—	53	1	no	—
44	C₂×C₃₂	#6157	1	1	yes	characteristic
45	C₄×C₁₆	—	53	1	no	—
46	C₄×C₁₆	—	53	1	no	—
47	C₈²	—	53	1	no	—
48	C₅₃	#7329	1	1	yes	characteristic, complemented, Sylow
49	C₂×C₁₆	—	53	2	no	—
50	C₂×C₁₆	—	53	1	no	—
51	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
52	C₄×C₈	—	53	2	no	—
53	C₄×C₈	—	53	1	no	—
54	C₃₂	—	53	4	no	—
55	C₃₂	—	53	2	no	—
56	C₃₂	#88324	1	2	yes	complemented
57	C₂×C₈	—	53	1	no	—
58	C₂×C₈	—	53	4	no	—
59	C₂×C₈	C₄×#6157	1	1	yes	characteristic
60	C₄²	—	53	1	no	—
61	C₁₆	—	53	4	no	—
62	C₁₆	—	53	2	no	—
63	C₁₆	D₅₃⋊C₈	1	1	yes	characteristic
64	C₁₆	C₂×#88324	1	1	yes	characteristic
65	C₂×C₄	—	53	2	no	—
66	C₂×C₄	C₈×#6157	1	1	yes	characteristic
67	C₈	—	53	2	no	—
68	C₈	—	106	4	no	—
69	C₈	C₈×#6157	1	1	yes	characteristic
70	C₈	C₄×#88324	1	1	yes	characteristic
71	C₂²	C₁₆×#6157	1	1	yes	characteristic
72	C₄	—	53	4	no	—
73	C₄	C₈×#88324	1	1	yes	characteristic
74	C₄	C₁₆×#6157	1	1	yes	characteristic
75	C₂	C₁₆×#88324	1	1	yes	characteristic
76	C₂	#20289821	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂×#6157
commutator	C₂×C₅₃	C₈×C₁₆
Fitting	C₂×C₃₂×C₅₃	C₄
socle, prime radical	C₂²×C₅₃	C₄×C₁₆
cosocle	C₁₆×Dic₅₃	C₂²
hypercenter	C₂×C₃₂	#6157
hypocenter	C₅₃	#7329
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
12. #13133267	C₈	semidirect
15. #13133123	C₈	semidirect
17. C₃₂×C₅₃	C₈	semidirect
48. C₅₃	#7329	semidirect
56. C₃₂	#88324	semidirect

Maximal subgroups

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

subgroup	index
#20289820	2
C₁₆×#88324	2
C₃₂×Dic₅₃	2
#7329	53