Subgroups — #20272661

Order 6784.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810349	C₂	1	2	yes	maximal
2	C₅₃×#384	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#88322	—	2	2	no	—
4	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
5	C₂×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
6	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×C₄²×C₅₃	C₄	1	1	yes	characteristic
8	C₂²×C₄×C₅₃	—	2	1	no	—
9	C₂²×C₄×C₅₃	—	2	1	no	—
10	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
11	C₂²×C₄×C₅₃	D₄	1	2	yes	—
12	C₄⋊C₄×C₅₃	—	4	1	no	—
13	C₄⋊C₄×C₅₃	—	4	4	no	—
14	C₄⋊C₄×C₅₃	—	4	1	no	—
15	C₄²×C₅₃	—	2	1	no	—
16	C₄²×C₅₃	C₈	1	2	yes	complemented
17	C₂×#88322	—	4	2	no	—
18	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
19	C₂×C₄×C₅₃	—	2	2	no	—
20	C₂×C₄×C₅₃	—	2	2	no	—
21	C₂×C₄×C₅₃	—	2	2	no	—
22	C₂×C₄×C₅₃	—	2	1	no	—
23	C₂×C₄×C₅₃	—	2	1	no	—
24	C₂×C₄×C₅₃	—	2	2	no	—
25	C₂×C₄×C₅₃	—	4	2	no	—
26	C₂×C₄×C₅₃	—	4	2	no	—
27	C₂×C₄×C₅₃	—	4	1	no	—
28	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
30	#88322	—	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	4	no	—
37	C₄×C₅₃	—	2	1	no	—
38	C₄×C₅₃	—	2	1	no	—
39	C₄×C₅₃	—	4	2	no	—
40	C₄×C₅₃	—	8	1	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	D₅₃	1	1	yes	characteristic
48	#324	—	53	2	no	—
49	C₅₃	#1325	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₄⋊C₄	—	2	2	no	—
51	C₂×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
52	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
53	C₂×C₄²	Dic₅₃	1	1	yes	characteristic
54	C₂²⋊C₈	—	106	2	no	—
55	C₂²×C₄	—	2	1	no	—
56	C₂²×C₄	—	2	1	no	—
57	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
58	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
59	C₂×C₈	—	212	2	no	—
60	C₄⋊C₄	—	4	1	no	—
61	C₄⋊C₄	—	4	1	no	—
62	C₄⋊C₄	—	4	4	no	—
63	C₄²	—	2	1	no	—
64	C₄²	#88322	1	2	yes	complemented
65	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	—	4	1	no	—
75	C₂×C₄	C₂×#88322	1	1	yes	characteristic
76	C₂×C₄	#11719917	1	1	yes	characteristic
77	C₈	—	424	2	no	—
78	C₂²	—	2	1	no	—
79	C₂²	—	2	1	no	—
80	C₂²	C₂²⋊#88322	1	1	yes	characteristic
81	C₂²	#13133132	1	1	yes	characteristic
82	C₂²	#13133136	1	1	yes	characteristic
83	C₄	—	2	4	no	—
84	C₄	—	2	1	no	—
85	C₄	—	2	1	no	—
86	C₄	—	4	2	no	—
87	C₄	—	8	1	no	—
88	C₂	—	2	1	no	—
89	C₂	—	2	1	no	—
90	C₂	#16810349	1	1	yes	characteristic
91	C₂	#16810416	1	1	yes	characteristic
92	C₂	#16810422	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13133132
commutator	C₂×C₄×C₅₃	C₂×C₈
Frattini	C₂×C₄⋊C₄	C₂×D₅₃
Fitting	C₅₃×#384	C₂
socle	C₂²×C₅₃	#99
hypercenter	#384	D₅₃
hypocenter	C₅₃	#1325
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #384

Complements

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

N	H ≅ G/N	type
16. C₄²×C₅₃	C₈	semidirect
49. C₅₃	#1325	semidirect
64. C₄²	#88322	semidirect