Subgroups — #20274829

Order 6784.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133136	C₂	1	2	yes	maximal
2	#16810393	C₂	1	4	yes	maximal
3	C₅₃×#519	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719917	C₂²	1	4	yes	—
5	#13133136	—	2	4	no	—
6	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₄×Q₈×C₅₃	—	2	4	no	—
8	#117×C₅₃	C₄	1	4	yes	—
9	#117×C₅₃	C₂²	1	2	yes	—
10	C₂×Q₈×C₅₃	—	2	4	no	—
11	C₂×Q₈×C₅₃	D₄	1	4	yes	—
12	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
13	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
14	C₄⋊C₄×C₅₃	—	2	4	no	—
15	C₄⋊C₄×C₅₃	—	2	2	no	—
16	C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
17	C₄²×C₅₃	—	2	4	no	—
18	C₄²×C₅₃	—	2	2	no	—
19	#11719917	—	2	4	no	—
20	#11719917	—	2	4	no	—
21	C₂×#88322	—	2	4	no	—
22	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
23	Q₈×C₅₃	—	2	8	no	—
24	C₂×C₄×C₅₃	—	2	4	no	—
25	C₂×C₄×C₅₃	—	2	4	no	—
26	C₂×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
27	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
28	C₂×C₄×C₅₃	#41	1	4	yes	—
29	#88322	—	4	4	no	—
30	C₂²×C₅₃	—	4	1	no	—
31	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₅₃	#117	1	2	yes	—
33	C₄×C₅₃	—	2	4	no	—
34	C₄×C₅₃	—	4	4	no	—
35	C₄×C₅₃	#123	1	4	yes	—
36	#1830	—	53	1	no	maximal, Sylow
37	C₂×C₅₃	—	2	1	no	—
38	C₂×C₅₃	—	2	2	no	—
39	C₂×C₅₃	#381	1	1	yes	characteristic
40	#519	D₅₃	1	1	yes	characteristic
41	C₂×#101	—	53	2	no	—
42	#344	—	53	4	no	—
43	C₅₃	#1830	1	1	yes	characteristic, complemented, Sylow
44	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
45	C₂×M₁₆	—	53	4	no	—
46	C₄×Q₈	—	2	4	no	—
47	#117	Dic₅₃	1	4	yes	—
48	#117	C₂×D₅₃	1	2	yes	—
49	#101	—	106	4	no	—
50	C₂×Q₈	—	2	4	no	—
51	C₂×Q₈	C₂²⋊D₅₃	1	4	yes	—
52	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
53	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
54	M₁₆	—	106	4	no	—
55	M₁₆	—	106	4	no	—
56	C₂×C₈	—	106	4	no	—
57	C₄⋊C₄	—	2	2	no	—
58	C₄⋊C₄	—	2	4	no	—
59	C₂²⋊C₄	C₂×Dic₅₃	1	4	yes	—
60	C₄²	—	2	2	no	—
61	C₄²	—	2	4	no	—
62	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
63	Q₈	—	2	8	no	—
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	C₂²⋊Dic₅₃	1	4	yes	—
67	C₂×C₄	#11719945	1	4	yes	—
68	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
69	C₈	—	212	4	no	—
70	C₂²	—	4	1	no	—
71	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
72	C₂²	#13133176	1	2	yes	—
73	C₄	—	2	4	no	—
74	C₄	—	4	4	no	—
75	C₄	#13133200	1	4	yes	—
76	C₂	—	2	1	no	—
77	C₂	—	2	2	no	—
78	C₂	#16810486	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810486
commutator	C₂³×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#519	C₂
socle	C₂×C₅₃	#381
hypercenter	#519	D₅₃
hypocenter	C₅₃	#1830
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #519

Complements

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

N	H ≅ G/N	type
43. C₅₃	#1830	semidirect