Subgroups — #28311433

Order 13568.

278 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291056	C₂	1	3	yes	maximal
2	#20291052	C₂	1	3	yes	maximal
3	#20274471	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133291	C₂²	1	3	yes	—
5	C₂²×#11719910	C₂²	1	1	yes	characteristic
6	#16811545	—	2	6	no	—
7	#16811544	—	2	6	no	—
8	#16810581	C₂²	1	3	yes	—
9	#16810380	C₄	1	3	yes	—
10	C₅₃×#338	C₄	1	1	yes	characteristic
11	#13133291	—	2	6	no	—
12	C₂×#11719910	—	2	3	no	—
13	C₂×#11719910	—	2	3	no	—
14	C₂²×#88322	C₂³	1	1	yes	characteristic
15	#13133176	C₂×C₄	1	3	yes	—
16	#13133127	Q₈	1	6	yes	—
17	C₈×Dic₅₃	D₄	1	6	yes	—
18	C₂×M₁₆×C₅₃	C₂×C₄	1	3	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	C₂²⋊Dic₅₃	—	2	3	no	—
23	C₄⋊Dic₅₃	—	2	3	no	—
24	C₄×Dic₅₃	C₄⋊C₄	1	6	yes	—
25	C₂×#88322	C₂×D₄	1	3	yes	—
26	C₂×#88322	C₂×Q₈	1	3	yes	—
27	#11719910	—	4	1	no	—
28	#11719910	—	4	3	no	—
29	C₂³×C₅₃	#131	1	1	yes	characteristic
30	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
31	C₂×C₄×C₅₃	#131	1	3	yes	—
32	C₂×Dic₅₃	—	2	6	no	—
33	C₈×C₅₃	—	2	6	no	—
34	#88322	#127	1	1	yes	characteristic
35	#88322	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	#384	1	1	yes	characteristic
40	C₄×C₅₃	#424	1	3	yes	—
41	Dic₅₃	—	4	6	no	—
42	#2153	—	53	3	no	—
43	#2151	—	53	3	no	—
44	#1747	—	53	1	no	—
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	—	53	3	no	—
50	#364	—	106	6	no	—
51	C₄⋊C₁₆	—	106	6	no	—
52	#338	#6157	1	1	yes	characteristic
53	H(3,ℤ/4)	—	53	3	no	—
54	C₅₃	#12902	1	1	yes	characteristic, complemented, Sylow
55	C₂×M₁₆	C₂×#6157	1	3	yes	—
56	C₂²×C₈	—	53	1	no	—
57	#117	—	53	3	no	—
58	M₃₂	—	106	6	no	—
59	C₂×C₁₆	—	106	3	no	—
60	C₂×C₁₆	—	106	3	no	—
61	#97	—	53	6	no	—
62	C₄×C₈	—	53	6	no	—
63	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
64	M₁₆	—	2	6	no	—
65	C₂×C₈	—	53	3	no	—
66	C₂×C₈	—	53	3	no	—
67	C₂×C₈	C₄⋊#6157	1	6	yes	—
68	C₄⋊C₄	—	106	3	no	—
69	C₂²⋊C₄	—	106	3	no	—
70	C₄²	—	53	6	no	—
71	C₁₆	—	212	1	no	—
72	C₁₆	—	212	3	no	—
73	C₂³	#13133305	1	1	yes	characteristic
74	C₂×C₄	—	106	6	no	—
75	C₂×C₄	C₂×C₄⋊#6157	1	3	yes	—
76	C₂×C₄	#13133304	1	3	yes	—
77	C₈	—	2	6	no	—
78	C₈	—	53	1	no	—
79	C₈	—	53	3	no	—
80	C₂²	—	4	1	no	—
81	C₂²	#16811630	1	3	yes	—
82	C₄	—	212	6	no	—
83	C₄	#16811569	1	1	yes	characteristic
84	C₄	#16811629	1	3	yes	—
85	C₂	—	2	3	no	—
86	C₂	#20290649	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16811569
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₅₃×#338	C₄
socle	C₂×C₅₃	#1843
prime radical	C₂³×C₅₃	#131
cosocle	C₂²×#88322	C₂³
hypercenter	#338	#6157
hypocenter	C₅₃	#12902
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #338

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
#20291056	2
#20291052	2
#20274471	2
#12902	53