Subgroups — #20289652

Order 6784.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊#11719910	C₂	1	2	yes	maximal
2	C₂×#13133146	C₂	1	1	yes	maximal, characteristic
3	C₂×#11719910	—	2	2	no	—
4	C₂²×#88322	C₂²	1	1	yes	characteristic
5	C₂×C₄×Dic₅₃	C₄	1	1	yes	characteristic
6	#13133146	—	2	2	no	—
7	C₂²×C₈×C₅₃	C₄	1	1	yes	characteristic
8	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
9	C₂²×Dic₅₃	C₈	1	1	yes	characteristic
10	C₂²×Dic₅₃	C₈	1	1	yes	characteristic
11	C₂×C₈×C₅₃	—	2	1	no	—
12	C₂×C₈×C₅₃	—	4	1	no	—
13	C₄×Dic₅₃	—	2	1	no	—
14	C₄×Dic₅₃	—	2	1	no	—
15	C₂×#88322	—	4	1	no	—
16	C₂×#88322	D₄	1	2	yes	—
17	#11719910	—	4	2	no	—
18	C₂³×C₅₃	C₂×C₈	1	1	yes	characteristic
19	C₂×C₄×C₅₃	—	2	1	no	—
20	C₂×C₄×C₅₃	—	2	1	no	—
21	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
23	C₂×Dic₅₃	—	2	1	no	—
24	C₂×Dic₅₃	—	2	1	no	—
25	C₂×Dic₅₃	—	2	1	no	—
26	C₂×Dic₅₃	—	4	1	no	—
27	C₂×Dic₅₃	C₁₆	1	2	yes	complemented
28	C₈×C₅₃	—	4	1	no	—
29	#88322	—	2	2	no	—
30	C₂²×C₅₃	—	2	1	no	—
31	C₂²×C₅₃	—	2	1	no	—
32	C₂²×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
33	C₂²×C₅₃	C₂×C₁₆	1	1	yes	characteristic
34	C₂²×C₅₃	M₃₂	1	1	yes	characteristic
35	C₄×C₅₃	—	2	1	no	—
36	C₄×C₅₃	#99	1	1	yes	characteristic
37	C₄×C₅₃	#101	1	1	yes	characteristic
38	Dic₅₃	—	2	2	no	—
39	Dic₅₃	—	4	1	no	—
40	#1322	—	53	1	no	maximal, Sylow
41	C₂×C₅₃	—	2	1	no	—
42	C₂×C₅₃	—	2	1	no	—
43	C₂×C₅₃	#324	1	1	yes	characteristic
44	C₂×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
45	C₂×C₅₃	#349	1	1	yes	characteristic
46	C₂×C₄⋊C₈	—	53	1	no	—
47	C₂²⋊C₁₆	—	53	2	no	—
48	C₅₃	#1322	1	1	yes	characteristic, complemented, Sylow
49	C₂²×C₈	—	53	1	no	—
50	C₂²×C₈	#6157	1	1	yes	characteristic
51	C₂×C₄²	—	53	1	no	—
52	C₂×C₁₆	—	106	2	no	—
53	C₄⋊C₈	—	106	2	no	—
54	C₂²×C₄	—	53	1	no	—
55	C₂²×C₄	—	53	1	no	—
56	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	4	1	no	—
59	C₂×C₈	—	53	2	no	—
60	C₂×C₈	—	212	1	no	—
61	C₄²	—	106	1	no	—
62	C₄²	—	106	1	no	—
63	C₁₆	—	212	2	no	—
64	C₂³	D₅₃⋊C₈	1	1	yes	characteristic
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	53	2	no	—
68	C₂×C₄	—	106	1	no	—
69	C₂×C₄	—	106	1	no	—
70	C₂×C₄	—	106	1	no	—
71	C₂×C₄	—	212	1	no	—
72	C₂×C₄	#11719936	1	1	yes	characteristic
73	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
74	C₈	—	4	1	no	—
75	C₈	—	106	2	no	—
76	C₂²	—	2	1	no	—
77	C₂²	—	2	1	no	—
78	C₂²	#13133274	1	1	yes	characteristic
79	C₂²	D₅₃⋊C₁₆	1	1	yes	characteristic
80	C₂²	#13133292	1	1	yes	characteristic
81	C₄	—	2	1	no	—
82	C₄	—	106	2	no	—
83	C₄	—	212	1	no	—
84	C₄	#13133275	1	1	yes	characteristic
85	C₄	#13133278	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	2	1	no	—
88	C₂	#16811469	1	1	yes	characteristic
89	C₂	#16811518	1	1	yes	characteristic
90	C₂	#16811520	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#6157
commutator	C₂²×C₅₃	C₂×C₁₆
Frattini	C₂²×C₄	C₂×#6157
Fitting	C₂²×C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂²⋊C₈
hypercenter	C₂²×C₈	#6157
hypocenter	C₅₃	#1322
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
27. C₂×Dic₅₃	C₁₆	semidirect
48. C₅₃	#1322	semidirect