Subgroups — #20282968

Order 6784.

580 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810535	C₂	1	12	yes	maximal, complemented
2	#16810479	C₂	1	2	yes	maximal
3	C₅₃×#379	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₁₀₆	C₂²	1	4	yes	complemented
5	C₂×C₂²⋊Dic₅₃	C₂²	1	12	yes	—
6	#13133124	C₂²	1	12	yes	complemented
7	C₂⁵×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊C₄×C₅₃	C₂²	1	6	yes	—
9	C₂⁴×C₅₃	—	4	6	no	—
10	C₂⁴×C₅₃	Q₈	1	4	yes	complemented
11	C₂⁴×C₅₃	C₂³	1	3	yes	—
12	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
13	C₂²×Dic₅₃	C₂³	1	8	yes	—
14	C₂×Dic₁₀₆	—	4	12	no	—
15	C₂²⋊C₄×C₅₃	—	4	6	no	—
16	C₂²⋊Dic₅₃	—	4	12	no	—
17	C₂³×C₅₃	—	2	12	no	—
18	C₂³×C₅₃	—	2	6	no	—
19	C₂³×C₅₃	—	4	12	no	—
20	C₂³×C₅₃	—	4	16	no	—
21	C₂³×C₅₃	C₂×Q₈	1	6	yes	—
22	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
23	C₂×C₄×C₅₃	—	2	12	no	—
24	C₂×Dic₅₃	—	2	24	no	—
25	Dic₁₀₆	—	4	16	no	—
26	C₂²×C₅₃	—	2	12	no	—
27	C₂²×C₅₃	—	2	6	no	—
28	C₂²×C₅₃	—	4	16	no	—
29	C₂²×C₅₃	—	4	12	no	—
30	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
31	C₂²×C₅₃	H(5,𝔽₂)	1	6	yes	—
32	C₄×C₅₃	—	4	4	no	—
33	Dic₅₃	—	4	8	no	—
34	#2840	—	53	1	no	maximal, Sylow
35	C₂×C₅₃	—	4	6	no	—
36	C₂×C₅₃	#543	1	3	yes	—
37	C₂×C₅₃	H(3,𝔽₄)	1	4	yes	—
38	#393	—	53	12	no	—
39	#379	—	53	2	no	—
40	#379	D₅₃	1	1	yes	characteristic
41	C₅₃	#2840	1	1	yes	characteristic, complemented, Sylow
42	C₂⁵	C₂×D₅₃	1	1	yes	characteristic
43	C₂²×Q₈	—	53	4	no	—
44	C₂×C₂²⋊C₄	—	53	12	no	—
45	C₂×C₂²⋊C₄	C₂×D₅₃	1	6	yes	—
46	#95	—	53	12	no	—
47	C₂⁴	—	4	6	no	—
48	C₂⁴	Dic₁₀₆	1	4	yes	complemented
49	C₂⁴	C₂²×D₅₃	1	3	yes	—
50	C₂×Q₈	—	212	12	no	—
51	C₂²×C₄	—	53	8	no	—
52	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
53	C₂²⋊C₄	—	4	6	no	—
54	C₂²⋊C₄	—	212	12	no	—
55	C₂³	—	2	6	no	—
56	C₂³	—	2	12	no	—
57	C₂³	—	4	12	no	—
58	C₂³	—	4	16	no	—
59	C₂³	C₂×Dic₁₀₆	1	6	yes	—
60	C₂³	C₂³×D₅₃	1	1	yes	characteristic
61	Q₈	—	212	16	no	—
62	C₂×C₄	—	2	12	no	—
63	C₂×C₄	—	106	24	no	—
64	C₂²	—	2	6	no	—
65	C₂²	—	2	12	no	—
66	C₂²	—	4	16	no	—
67	C₂²	—	4	12	no	—
68	C₂²	C₂²×Dic₁₀₆	1	1	yes	characteristic
69	C₂²	#13133262	1	6	yes	—
70	C₄	—	4	4	no	—
71	C₄	—	212	8	no	—
72	C₂	—	4	6	no	—
73	C₂	#16811249	1	3	yes	—
74	C₂	#16811366	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₅₃×#379	C₂
hypercenter	#379	D₅₃
hypocenter	C₅₃	#2840
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #379

Complements

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

N	H ≅ G/N	type
1. #16810535	C₂	semidirect
4. C₂²×Dic₁₀₆	C₂²	semidirect
6. #13133124	C₂²	semidirect
10. C₂⁴×C₅₃	Q₈	semidirect
41. C₅₃	#2840	semidirect
48. C₂⁴	Dic₁₀₆	semidirect