Subgroups — #20272986

Order 6784.

736 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133124	C₂	1	6	yes	maximal
2	C₂³×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂³×Dic₅₃	C₂²	1	4	yes	—
4	#13133124	—	2	12	no	—
5	C₂²×Q₈×C₅₃	—	2	12	no	—
6	C₂²×Q₈×C₅₃	C₄	1	4	yes	complemented
7	C₂³×C₄×C₅₃	C₂²	1	3	yes	—
8	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
9	C₂×Q₈×C₅₃	—	2	32	no	—
10	C₂×Q₈×C₅₃	—	2	24	no	—
11	C₂²×C₄×C₅₃	—	2	12	no	—
12	C₂²×C₄×C₅₃	C₂×C₄	1	6	yes	—
13	C₂²×C₄×C₅₃	D₄	1	12	yes	—
14	C₂²×Dic₅₃	—	2	12	no	—
15	C₂²×Dic₅₃	—	2	12	no	—
16	C₂²×Dic₅₃	Q₈	1	8	yes	complemented
17	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
18	C₂³×C₅₃	C₂×D₄	1	6	yes	—
19	C₂³×C₅₃	C₂×Q₈	1	4	yes	—
20	C₂³×C₅₃	#41	1	4	yes	—
21	Q₈×C₅₃	—	2	32	no	—
22	C₂×C₄×C₅₃	—	2	12	no	—
23	C₂×C₄×C₅₃	—	2	24	no	—
24	C₂×C₄×C₅₃	C₂²⋊C₄	1	12	yes	—
25	C₂×Dic₅₃	—	2	24	no	—
26	C₂×Dic₅₃	—	4	4	no	—
27	C₂×Dic₅₃	—	4	12	no	—
28	C₂²×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
29	C₂²×C₅₃	C₄×Q₈	1	4	yes	—
30	C₂²×C₅₃	#120	1	4	yes	—
31	C₂²×C₅₃	C₂²⋊Q₈	1	12	yes	—
32	C₂²×C₅₃	#124	1	6	yes	—
33	C₂²×C₅₃	C₄⋊Q₈	1	6	yes	—
34	C₄×C₅₃	—	2	12	no	—
35	Dic₅₃	—	4	8	no	—
36	#1446	—	53	1	no	maximal, Sylow
37	C₂×C₅₃	#379	1	1	yes	characteristic
38	C₂×C₅₃	#391	1	6	yes	—
39	C₂×C₅₃	#393	1	4	yes	—
40	C₂×C₅₃	#395	1	4	yes	—
41	C₂³×Q₈	D₅₃	1	1	yes	characteristic
42	C₂×#95	—	53	6	no	—
43	C₅₃	#1446	1	1	yes	characteristic, complemented, Sylow
44	C₂²×Q₈	—	2	12	no	—
45	C₂²×Q₈	Dic₅₃	1	4	yes	complemented
46	C₂³×C₄	—	53	4	no	—
47	C₂³×C₄	C₂×D₅₃	1	3	yes	—
48	#95	—	106	12	no	—
49	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
50	C₂×Q₈	—	2	32	no	—
51	C₂×Q₈	—	2	24	no	—
52	C₂²×C₄	—	2	12	no	—
53	C₂²×C₄	—	53	8	no	—
54	C₂²×C₄	—	106	12	no	—
55	C₂²×C₄	—	106	12	no	—
56	C₂²×C₄	C₂×Dic₅₃	1	6	yes	—
57	C₂²×C₄	C₂²⋊D₅₃	1	12	yes	—
58	C₂³	Q₈×D₅₃	1	4	yes	—
59	C₂³	#11719949	1	4	yes	—
60	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
61	C₂³	C₂×C₂²⋊D₅₃	1	6	yes	—
62	Q₈	—	2	32	no	—
63	C₂×C₄	—	2	12	no	—
64	C₂×C₄	—	2	24	no	—
65	C₂×C₄	—	106	24	no	—
66	C₂×C₄	—	212	4	no	—
67	C₂×C₄	—	212	12	no	—
68	C₂×C₄	C₂²⋊Dic₅₃	1	12	yes	—
69	C₂²	C₂×C₂²⋊Dic₅₃	1	3	yes	—
70	C₂²	Q₈×Dic₅₃	1	4	yes	—
71	C₂²	#13133186	1	4	yes	—
72	C₂²	#13133193	1	12	yes	—
73	C₂²	#13133206	1	6	yes	—
74	C₂²	#13133215	1	6	yes	—
75	C₄	—	2	12	no	—
76	C₄	—	212	8	no	—
77	C₂	#16810479	1	1	yes	characteristic
78	C₂	#16810530	1	6	yes	—
79	C₂	#16810534	1	4	yes	—
80	C₂	#16810542	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂²×D₅₃
commutator	C₂³×C₅₃	C₂²×C₄
Fitting	C₂³×Q₈×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂³
hypercenter	C₂³×Q₈	D₅₃
hypocenter	C₅₃	#1446
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
6. C₂²×Q₈×C₅₃	C₄	semidirect
16. C₂²×Dic₅₃	Q₈	semidirect
43. C₅₃	#1446	semidirect
45. C₂²×Q₈	Dic₅₃	semidirect