Subgroups — #20274464

Order 6784.

272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133128	C₂	1	2	yes	maximal
2	#16810378	C₂	1	4	yes	maximal
3	C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#88322	C₂²	1	4	yes	—
5	#13133128	C₄	1	8	yes	complemented
6	C₂×C₄²×C₅₃	C₄	1	4	yes	—
7	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
8	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
10	C₂²×C₄×C₅₃	C₂×C₄	1	4	yes	—
11	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
12	C₄²×C₅₃	—	2	8	no	—
13	C₄²×C₅₃	—	2	4	no	—
14	C₄²×C₅₃	C₂×C₄	1	4	yes	—
15	C₂×#88322	—	2	8	no	—
16	C₂×#88322	C₂×C₄	1	8	yes	—
17	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
18	C₂×C₄×C₅₃	—	2	4	no	—
19	C₂×C₄×C₅₃	—	2	4	no	—
20	C₂×C₄×C₅₃	—	2	4	no	—
21	C₂×C₄×C₅₃	C₄²	1	4	yes	—
22	C₂×C₄×C₅₃	M₁₆	1	8	yes	—
23	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
24	C₂×C₄×C₅₃	#41	1	4	yes	—
25	#88322	—	2	8	no	—
26	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
27	C₂²×C₅₃	#117	1	2	yes	—
28	C₂²×C₅₃	C₂×M₁₆	1	4	yes	—
29	C₄×C₅₃	—	2	4	no	—
30	C₄×C₅₃	—	2	8	no	—
31	C₄×C₅₃	#117	1	4	yes	—
32	#1745	—	53	1	no	maximal, Sylow
33	C₂×C₅₃	#376	1	1	yes	characteristic
34	C₂×C₅₃	C₄×M₁₆	1	2	yes	—
35	C₂×C₅₃	#432	1	4	yes	—
36	C₂×#97	—	53	2	no	—
37	C₄³	D₅₃	1	1	yes	characteristic
38	#336	—	53	4	no	—
39	C₅₃	#1745	1	1	yes	characteristic, complemented, Sylow
40	C₂²×C₈	—	53	4	no	—
41	C₂×C₄²	Dic₅₃	1	4	yes	—
42	C₂×C₄²	C₂×D₅₃	1	2	yes	—
43	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
44	#97	—	53	8	no	—
45	C₂²×C₄	C₂×Dic₅₃	1	4	yes	—
46	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
47	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
48	C₂×C₈	—	53	8	no	—
49	C₂×C₈	—	106	8	no	—
50	C₄²	—	2	4	no	—
51	C₄²	—	2	8	no	—
52	C₄²	C₄×D₅₃	1	4	yes	—
53	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
54	C₂×C₄	—	2	4	no	—
55	C₂×C₄	—	2	4	no	—
56	C₂×C₄	—	2	4	no	—
57	C₂×C₄	#11719917	1	8	yes	—
58	C₂×C₄	C₄×Dic₅₃	1	4	yes	—
59	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
60	C₂×C₄	#11719945	1	4	yes	—
61	C₈	—	106	8	no	—
62	C₂²	C₂×C₄×Dic₅₃	1	1	yes	characteristic
63	C₂²	#13133176	1	2	yes	—
64	C₂²	C₂×#11719917	1	4	yes	—
65	C₄	—	2	4	no	—
66	C₄	—	2	8	no	—
67	C₄	#13133174	1	4	yes	—
68	C₂	#16810469	1	1	yes	characteristic
69	C₂	C₄×#11719917	1	2	yes	—
70	C₂	#16810680	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄²
Fitting	C₄³×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	C₄³	D₅₃
hypocenter	C₅₃	#1745
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
5. #13133128	C₄	semidirect
39. C₅₃	#1745	semidirect