Subgroups — #27300514

Order 13568.

278 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276481	C₂	1	3	yes	maximal
2	#20276474	C₂	1	3	yes	maximal
3	C₅₃×#1747	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133157	C₂²	1	3	yes	—
5	C₂²×#11719908	C₂²	1	1	yes	characteristic
6	#16810444	—	2	6	no	—
7	C₄⋊#11719908	—	2	6	no	—
8	C₅₃×#405	C₂²	1	3	yes	—
9	C₅₃×#338	C₄	1	1	yes	characteristic
10	C₅₃×H(3,ℤ/4)	C₄	1	3	yes	—
11	#13133157	—	2	6	no	—
12	C₂×#11719908	—	2	3	no	—
13	C₂×#11719908	—	2	3	no	—
14	C₂×M₁₆×C₅₃	C₂×C₄	1	3	yes	—
15	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
16	#117×C₅₃	C₂×C₄	1	3	yes	—
17	#97×C₅₃	Q₈	1	6	yes	—
18	C₄×C₈×C₅₃	D₄	1	6	yes	—
19	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
20	M₁₆×C₅₃	—	2	6	no	—
21	C₂×C₈×C₅₃	C₄⋊C₄	1	6	yes	—
22	C₂×C₈×C₅₃	C₂×D₄	1	3	yes	—
23	C₂×C₈×C₅₃	C₂×Q₈	1	3	yes	—
24	C₄⋊C₄×C₅₃	—	2	3	no	—
25	C₂²⋊C₄×C₅₃	—	2	3	no	—
26	C₄²×C₅₃	C₄⋊C₄	1	6	yes	—
27	#11719908	—	4	1	no	—
28	#11719908	—	4	3	no	—
29	C₂³×C₅₃	#131	1	1	yes	characteristic
30	C₂×C₄×C₅₃	—	2	6	no	—
31	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
32	C₂×C₄×C₅₃	#131	1	3	yes	—
33	C₈×C₅₃	—	2	6	no	—
34	C₈×C₅₃	#127	1	1	yes	characteristic
35	C₈×C₅₃	C₄⋊Q₈	1	3	yes	—
36	#12902	—	53	1	no	maximal, Sylow
37	C₂²×C₅₃	—	4	1	no	—
38	C₂²×C₅₃	#424	1	3	yes	—
39	C₄×C₅₃	—	4	6	no	—
40	C₄×C₅₃	#384	1	1	yes	characteristic
41	C₄×C₅₃	#424	1	3	yes	—
42	#2153	—	53	3	no	—
43	#2151	—	53	3	no	—
44	#1747	D₅₃	1	1	yes	characteristic
45	C₂×C₅₃	—	2	3	no	—
46	C₂×C₅₃	#1843	1	1	yes	characteristic
47	C₂×M₃₂	—	53	3	no	—
48	C₂²×C₁₆	—	53	1	no	—
49	#405	C₂×D₅₃	1	3	yes	—
50	#364	—	106	6	no	—
51	C₄⋊C₁₆	—	106	6	no	—
52	#338	Dic₅₃	1	1	yes	characteristic
53	H(3,ℤ/4)	Dic₅₃	1	3	yes	—
54	C₅₃	#12902	1	1	yes	characteristic, complemented, Sylow
55	C₂×M₁₆	C₂×Dic₅₃	1	3	yes	—
56	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
57	#117	C₂×Dic₅₃	1	3	yes	—
58	M₃₂	—	106	6	no	—
59	C₂×C₁₆	—	106	3	no	—
60	C₂×C₁₆	—	106	3	no	—
61	#97	Dic₁₀₆	1	6	yes	—
62	C₄×C₈	D₂₁₂	1	6	yes	—
63	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
64	M₁₆	—	2	6	no	—
65	C₂×C₈	C₄⋊Dic₅₃	1	6	yes	—
66	C₂×C₈	C₂×Dic₁₀₆	1	3	yes	—
67	C₂×C₈	C₂×D₂₁₂	1	3	yes	—
68	C₄⋊C₄	—	2	3	no	—
69	C₂²⋊C₄	—	2	3	no	—
70	C₄²	C₄⋊Dic₅₃	1	6	yes	—
71	C₁₆	—	212	1	no	—
72	C₁₆	—	212	3	no	—
73	C₂³	#13133224	1	1	yes	characteristic
74	C₂×C₄	—	2	6	no	—
75	C₂×C₄	C₂×C₄⋊Dic₅₃	1	3	yes	—
76	C₂×C₄	#13133224	1	3	yes	—
77	C₈	—	2	6	no	—
78	C₈	#13133214	1	1	yes	characteristic
79	C₈	#13133217	1	3	yes	—
80	C₂²	—	4	1	no	—
81	C₂²	#16810650	1	3	yes	—
82	C₄	—	4	6	no	—
83	C₄	#16810493	1	1	yes	characteristic
84	C₄	#16810650	1	3	yes	—
85	C₂	—	2	3	no	—
86	C₂	#20274869	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13133214
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1747	C₂
socle	C₂×C₅₃	#1843
prime radical	C₂³×C₅₃	#131
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1747	D₅₃
hypocenter	C₅₃	#12902
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1747

Complements

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

N	H ≅ G/N	type
54. C₅₃	#12902	semidirect

Maximal subgroups

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

subgroup	index
#20276481	2
#20276474	2
C₅₃×#1747	2
#12902	53