Subgroups — #20283093

Order 6784.

504 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133217	C₂	1	2	yes	maximal
2	#16810529	C₂	1	8	yes	maximal
3	#16810489	C₂	1	4	yes	maximal
4	C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₀₆	C₂²	1	4	yes	—
6	#13133217	—	2	8	no	—
7	C₂×C₄⋊Dic₅₃	C₂²	1	4	yes	—
8	C₂×C₄⋊Dic₅₃	C₂²	1	4	yes	—
9	#13133124	C₂²	1	16	yes	—
10	C₂×C₄²×C₅₃	C₂²	1	4	yes	—
11	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
13	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
14	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
16	C₂²×Dic₅₃	C₂³	1	8	yes	—
17	C₂×Dic₁₀₆	—	2	8	no	—
18	C₂×Dic₁₀₆	—	2	16	no	—
19	C₄²×C₅₃	—	2	8	no	—
20	C₄²×C₅₃	D₄	1	4	yes	—
21	C₄²×C₅₃	Q₈	1	8	yes	complemented
22	C₄⋊Dic₅₃	—	2	8	no	—
23	C₄⋊Dic₅₃	—	2	8	no	—
24	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
25	C₂×C₄×C₅₃	—	2	8	no	—
26	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
27	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
28	C₂×C₄×C₅₃	C₂×Q₈	1	8	yes	—
29	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
30	C₂×C₄×C₅₃	#41	1	8	yes	—
31	C₂×Dic₅₃	—	2	8	no	—
32	C₂×Dic₅₃	—	2	16	no	—
33	Dic₁₀₆	—	4	16	no	—
34	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
35	C₂²×C₅₃	C₂²×Q₈	1	2	yes	—
36	C₂²×C₅₃	C₂×#41	1	4	yes	—
37	C₄×C₅₃	—	2	8	no	—
38	C₄×C₅₃	#124	1	8	yes	—
39	C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
40	Dic₅₃	—	4	8	no	—
41	#2862	—	53	1	no	maximal, Sylow
42	C₂×C₅₃	C₂×#124	1	2	yes	—
43	C₂×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
44	C₂×C₅₃	#533	1	4	yes	—
45	C₂×C₄⋊Q₈	—	53	2	no	—
46	#391	—	53	8	no	—
47	#382	—	53	4	no	—
48	C₄³	D₅₃	1	1	yes	characteristic
49	C₅₃	#2862	1	1	yes	characteristic, complemented, Sylow
50	C₂²×Q₈	—	53	4	no	—
51	C₄⋊Q₈	—	106	8	no	—
52	C₂×C₄⋊C₄	—	53	4	no	—
53	C₂×C₄⋊C₄	—	53	4	no	—
54	C₂×C₄²	C₂×D₅₃	1	4	yes	—
55	C₂×C₄²	C₂×D₅₃	1	2	yes	—
56	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
57	#95	—	53	16	no	—
58	C₂×Q₈	—	106	8	no	—
59	C₂×Q₈	—	106	16	no	—
60	C₂²×C₄	—	53	8	no	—
61	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
62	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
63	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
64	C₄⋊C₄	—	106	8	no	—
65	C₄⋊C₄	—	106	8	no	—
66	C₄²	—	2	8	no	—
67	C₄²	Dic₁₀₆	1	8	yes	complemented
68	C₄²	D₂₁₂	1	4	yes	—
69	C₂³	C₂³×D₅₃	1	1	yes	characteristic
70	Q₈	—	212	16	no	—
71	C₂×C₄	—	2	8	no	—
72	C₂×C₄	—	106	8	no	—
73	C₂×C₄	—	106	16	no	—
74	C₂×C₄	C₂×Dic₁₀₆	1	8	yes	—
75	C₂×C₄	C₂×Dic₁₀₆	1	4	yes	—
76	C₂×C₄	C₂×D₂₁₂	1	4	yes	—
77	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
78	C₂×C₄	#11719945	1	8	yes	—
79	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
80	C₂²	C₂²×Dic₁₀₆	1	2	yes	—
81	C₂²	C₂×#11719945	1	4	yes	—
82	C₄	—	2	8	no	—
83	C₄	—	212	8	no	—
84	C₄	#13133205	1	8	yes	—
85	C₄	#13133217	1	4	yes	—
86	C₂	C₂×#13133205	1	2	yes	—
87	C₂	C₂×#13133217	1	1	yes	characteristic
88	C₂	#16811180	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂⁴
Fitting	C₄³×C₅₃	C₂
hypercenter	C₄³	D₅₃
hypocenter	C₅₃	#2862
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³

Complements

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

N	H ≅ G/N	type
21. C₄²×C₅₃	Q₈	semidirect
49. C₅₃	#2862	semidirect
67. C₄²	Dic₁₀₆	semidirect