Subgroups — #20276473

Order 6784.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133157	C₂	1	1	yes	maximal, characteristic
2	#16810578	C₂	1	1	yes	maximal, characteristic
3	#16810441	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719917	C₄	1	1	yes	characteristic
6	#13133176	C₄	1	1	yes	characteristic
7	#13133157	—	2	2	no	—
8	C₂×#11719908	C₂²	1	2	yes	—
9	#13133127	C₄	1	2	yes	—
10	C₈×Dic₅₃	C₂²	1	2	yes	—
11	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₅₃	—	2	2	no	—
13	C₂×C₁₆×C₅₃	C₂²	1	2	yes	—
14	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₅₃	D₄	1	2	yes	—
17	C₂×C₈×C₅₃	Q₈	1	2	yes	—
18	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
19	C₂²⋊Dic₅₃	C₈	1	2	yes	—
20	C₄⋊Dic₅₃	C₈	1	2	yes	—
21	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
22	#11719917	—	2	2	no	—
23	C₂×#88322	C₂×C₄	1	2	yes	—
24	C₁₆×C₅₃	—	2	2	no	—
25	#11719908	—	2	2	no	—
26	C₂³×C₅₃	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
28	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
30	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×Dic₅₃	C₂×C₈	1	4	yes	—
32	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
33	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
35	#88322	—	2	2	no	—
36	C₂²×C₅₃	—	2	2	no	—
37	C₂²×C₅₃	C₄⋊C₈	1	2	yes	—
38	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
39	C₄×C₅₃	C₄⋊C₈	1	2	yes	—
40	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
42	Dic₅₃	—	2	4	no	—
43	#2151	—	53	1	no	maximal, Sylow
44	C₂×C₅₃	—	2	2	no	—
45	C₂×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
46	C₂×C₅₃	#504	1	2	yes	—
47	C₂×M₃₂	—	53	1	no	—
48	C₂²×C₁₆	D₅₃	1	1	yes	characteristic
49	#405	—	53	1	no	—
50	C₄⋊C₁₆	—	53	4	no	—
51	C₅₃	#2151	1	1	yes	characteristic, complemented, Sylow
52	C₂×M₁₆	—	53	1	no	—
53	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
54	#117	—	53	1	no	—
55	M₃₂	—	106	2	no	—
56	C₂×C₁₆	—	2	2	no	—
57	C₂×C₁₆	—	53	2	no	—
58	C₂×C₁₆	C₂×D₅₃	1	2	yes	—
59	#97	—	53	2	no	—
60	C₄×C₈	—	53	2	no	—
61	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
62	M₁₆	—	106	2	no	—
63	C₂×C₈	—	53	2	no	—
64	C₂×C₈	Dic₁₀₆	1	2	yes	—
65	C₂×C₈	C₄×D₅₃	1	1	yes	characteristic
66	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
67	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
68	C₄⋊C₄	—	53	2	no	—
69	C₂²⋊C₄	—	53	2	no	—
70	C₄²	—	53	2	no	—
71	C₁₆	—	2	2	no	—
72	C₁₆	—	106	2	no	—
73	C₂³	C₈×D₅₃	1	1	yes	characteristic
74	C₂×C₄	—	53	4	no	—
75	C₂×C₄	C₈×D₅₃	1	1	yes	characteristic
76	C₂×C₄	#11719912	1	2	yes	—
77	C₂×C₄	#11719919	1	2	yes	—
78	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
79	C₈	—	106	2	no	—
80	C₈	#11719919	1	2	yes	—
81	C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
82	C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
83	C₂²	—	2	2	no	—
84	C₂²	#13133146	1	2	yes	—
85	C₂²	C₂×C₈×D₅₃	1	1	yes	characteristic
86	C₄	—	106	4	no	—
87	C₄	#13133146	1	2	yes	—
88	C₄	C₂×#11719919	1	1	yes	characteristic
89	C₄	C₂×#11719912	1	1	yes	characteristic
90	C₂	—	2	2	no	—
91	C₂	C₂×#13133146	1	1	yes	characteristic
92	C₂	#16811039	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₈
Fitting	C₂²×C₁₆×C₅₃	C₂
hypercenter	C₂²×C₁₆	D₅₃
hypocenter	C₅₃	#2151
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
3. #16810441	C₂	semidirect
51. C₅₃	#2151	semidirect