Subgroups — #20272887

Order 6784.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810991	C₂	1	1	yes	maximal, characteristic
2	#16810422	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#356	C₂	1	1	yes	maximal, characteristic
4	#13133209	C₄	1	1	yes	characteristic
5	#13133141	—	2	1	no	—
6	#13133141	—	2	1	no	—
7	#13133136	—	2	1	no	—
8	#13133128	C₄	1	1	yes	characteristic
9	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
10	#101×C₅₃	—	2	1	no	—
11	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
12	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
13	M₁₆×C₅₃	—	4	1	no	—
14	C₄⋊C₄×C₅₃	—	4	1	no	—
15	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₄⋊Dic₅₃	—	2	1	no	—
17	#11719919	—	4	1	no	—
18	#11719917	—	4	1	no	—
19	C₂×#88322	—	2	1	no	—
20	Q₈×C₅₃	—	4	1	no	—
21	Q₈×C₅₃	—	4	1	no	—
22	C₂×C₄×C₅₃	—	2	1	no	—
23	C₂×C₄×C₅₃	—	2	1	no	—
24	C₂×C₄×C₅₃	—	2	1	no	—
25	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×Dic₅₃	—	4	1	no	—
27	C₈×C₅₃	—	8	1	no	—
28	#88322	—	4	1	no	—
29	#88322	—	8	1	no	—
30	C₂²×C₅₃	#99	1	1	yes	characteristic
31	C₄×C₅₃	—	2	1	no	—
32	C₄×C₅₃	—	4	1	no	—
33	C₄×C₅₃	—	4	1	no	—
34	C₄×C₅₃	—	4	1	no	—
35	Dic₅₃	—	8	1	no	—
36	#1413	—	53	1	no	maximal, Sylow
37	C₂×C₅₃	—	2	1	no	—
38	C₂×C₅₃	#354	1	1	yes	characteristic
39	#491	—	53	1	no	—
40	#356	—	53	1	no	—
41	#356	D₅₃	1	1	yes	characteristic
42	C₅₃	#1413	1	1	yes	characteristic, complemented, Sylow
43	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
44	#125	—	53	1	no	—
45	#103	—	106	1	no	—
46	#103	—	106	1	no	—
47	#101	—	2	1	no	—
48	#101	—	106	1	no	—
49	#97	—	53	1	no	—
50	C₂×Q₈	D₂₁₂	1	1	yes	characteristic
51	C₂×Q₈	C₂²⋊D₅₃	1	1	yes	characteristic
52	M₁₆	—	4	1	no	—
53	M₁₆	—	212	1	no	—
54	C₂×C₈	—	106	1	no	—
55	C₄⋊C₄	—	4	1	no	—
56	C₄⋊C₄	—	106	1	no	—
57	C₄⋊C₄	—	212	1	no	—
58	C₄²	C₄×D₅₃	1	1	yes	characteristic
59	Q₈	—	4	1	no	—
60	Q₈	—	4	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	212	1	no	—
65	C₂×C₄	#11719921	1	1	yes	characteristic
66	C₈	—	8	1	no	—
67	C₈	—	424	1	no	—
68	C₈	—	212	1	no	—
69	C₂²	#13133131	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	4	1	no	—
72	C₄	—	4	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	424	1	no	—
75	C₂	—	2	1	no	—
76	C₂	#16810417	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810417
commutator	C₄²×C₅₃	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₅₃
Fitting	C₅₃×#356	C₂
socle	C₂×C₅₃	#354
hypercenter	#356	D₅₃
hypocenter	C₅₃	#1413
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #356

Complements

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

N	H ≅ G/N	type
42. C₅₃	#1413	semidirect