Subgroups — #20289658

Order 6784.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811468	C₂	1	2	yes	maximal
2	#16810492	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#88324	—	2	2	no	—
4	C₂×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
5	C₂×#11719919	—	2	2	no	—
6	C₂×C₄×Dic₅₃	C₄	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
8	C₂²×C₄×C₅₃	—	2	1	no	—
9	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
10	C₂²×Dic₅₃	—	2	1	no	—
11	C₂²×Dic₅₃	D₄	1	2	yes	—
12	C₂×#88324	—	4	2	no	—
13	C₄⋊C₄×C₅₃	—	4	1	no	—
14	C₄⋊Dic₅₃	—	4	1	no	—
15	#11719919	—	4	4	no	—
16	C₄×Dic₅₃	—	2	1	no	—
17	C₄×Dic₅₃	C₈	1	2	yes	complemented
18	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
19	C₂×C₄×C₅₃	—	2	1	no	—
20	C₂×C₄×C₅₃	—	2	1	no	—
21	C₂×C₄×C₅₃	—	4	2	no	—
22	C₂×C₄×C₅₃	—	4	1	no	—
23	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
24	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
25	C₂×Dic₅₃	—	2	2	no	—
26	C₂×Dic₅₃	—	2	2	no	—
27	C₂×Dic₅₃	—	2	2	no	—
28	C₂×Dic₅₃	—	2	2	no	—
29	C₂×Dic₅₃	—	4	2	no	—
30	#88324	—	8	2	no	—
31	C₂²×C₅₃	—	2	1	no	—
32	C₂²×C₅₃	—	2	1	no	—
33	C₂²×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
34	C₂²×C₅₃	#99	1	1	yes	characteristic
35	C₂²×C₅₃	#101	1	1	yes	characteristic
36	C₄×C₅₃	—	2	1	no	—
37	C₄×C₅₃	—	2	1	no	—
38	C₄×C₅₃	—	8	1	no	—
39	Dic₅₃	—	2	4	no	—
40	Dic₅₃	—	4	2	no	—
41	#1325	—	53	1	no	maximal, Sylow
42	C₂×C₅₃	—	2	1	no	—
43	C₂×C₅₃	—	2	1	no	—
44	C₂×C₅₃	#324	1	1	yes	characteristic
45	C₂×C₅₃	#353	1	1	yes	characteristic
46	C₂×C₅₃	#356	1	1	yes	characteristic
47	#384	—	53	1	no	—
48	#324	—	53	2	no	—
49	C₅₃	#1325	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₄⋊C₄	—	53	1	no	—
51	C₂×C₄⋊C₄	—	106	2	no	—
52	C₂×C₄⋊C₄	#6157	1	1	yes	characteristic
53	C₂×C₄²	—	53	1	no	—
54	C₂²⋊C₈	—	106	2	no	—
55	C₂²×C₄	—	2	1	no	—
56	C₂²×C₄	—	53	2	no	—
57	C₂²×C₄	—	106	1	no	—
58	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
59	C₂×C₈	—	212	2	no	—
60	C₄⋊C₄	—	4	1	no	—
61	C₄⋊C₄	—	212	1	no	—
62	C₄⋊C₄	—	212	4	no	—
63	C₄²	—	53	2	no	—
64	C₄²	—	106	1	no	—
65	C₂³	C₂²⋊#6157	1	1	yes	characteristic
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	—	4	1	no	—
70	C₂×C₄	—	106	2	no	—
71	C₂×C₄	—	106	2	no	—
72	C₂×C₄	—	106	2	no	—
73	C₂×C₄	—	106	2	no	—
74	C₂×C₄	—	212	2	no	—
75	C₂×C₄	D₅₃⋊C₈	1	1	yes	characteristic
76	C₂×C₄	#11719936	1	1	yes	characteristic
77	C₈	—	424	2	no	—
78	C₂²	—	2	1	no	—
79	C₂²	—	2	1	no	—
80	C₂²	#13133274	1	1	yes	characteristic
81	C₂²	#13133275	1	1	yes	characteristic
82	C₂²	#13133278	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	2	1	no	—
85	C₄	—	8	1	no	—
86	C₄	—	106	4	no	—
87	C₄	—	212	2	no	—
88	C₂	—	2	1	no	—
89	C₂	—	2	1	no	—
90	C₂	#16811469	1	1	yes	characteristic
91	C₂	#16811527	1	1	yes	characteristic
92	C₂	#16811533	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13133275
commutator	C₂×C₄×C₅₃	C₂×C₈
Frattini	C₂²×C₄	C₂×#6157
Fitting	C₂×C₄⋊C₄×C₅₃	C₄
socle	C₂²×C₅₃	#99
hypercenter	C₂×C₄⋊C₄	#6157
hypocenter	C₅₃	#1325
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
17. C₄×Dic₅₃	C₈	semidirect
49. C₅₃	#1325	semidirect