Subgroups — #20276510

Order 6784.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133153	C₂	1	2	yes	maximal
2	#16810453	C₂	1	4	yes	maximal, complemented
3	C₂×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719917	—	2	2	no	—
5	#13133153	—	2	2	no	—
6	#13133153	C₂²	1	4	yes	—
7	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	M₃₂×C₅₃	C₄	1	4	yes	—
9	C₂×C₁₆×C₅₃	C₂²	1	2	yes	—
10	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
11	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
12	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
13	C₂×C₈×C₅₃	Q₈	1	2	yes	—
14	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
15	#11719917	—	2	4	no	—
16	#11719917	—	4	2	no	—
17	C₂×#88322	—	4	2	no	—
18	C₁₆×C₅₃	C₂×C₄	1	4	yes	—
19	C₂³×C₅₃	Q₁₆	1	1	yes	characteristic
20	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
21	C₂×C₄×C₅₃	D₈	1	2	yes	—
22	C₂×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
23	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
24	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
25	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
26	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
27	#88322	—	4	4	no	—
28	C₂²×C₅₃	—	4	1	no	—
29	C₂²×C₅₃	#107	1	2	yes	—
30	C₂²×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
31	C₄×C₅₃	#107	1	2	yes	—
32	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
33	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
34	#2161	—	53	1	no	maximal, Sylow
35	C₂×C₅₃	—	2	2	no	—
36	C₂×C₅₃	—	2	1	no	—
37	C₂×C₅₃	C₂×#107	1	1	yes	characteristic
38	C₂×M₃₂	D₅₃	1	1	yes	characteristic
39	C₂×#108	—	53	2	no	—
40	#368	—	53	4	no	—
41	C₅₃	#2161	1	1	yes	characteristic, complemented, Sylow
42	C₂×M₁₆	—	106	2	no	—
43	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
44	M₃₂	Dic₅₃	1	4	yes	—
45	C₂×C₁₆	C₂×D₅₃	1	2	yes	—
46	#108	—	53	4	no	—
47	#108	—	106	2	no	—
48	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
49	M₁₆	—	106	4	no	—
50	M₁₆	—	212	2	no	—
51	C₂×C₈	—	212	2	no	—
52	C₂×C₈	Dic₁₀₆	1	2	yes	—
53	C₂×C₈	D₂₁₂	1	1	yes	characteristic
54	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
55	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
56	C₁₆	C₂×Dic₅₃	1	4	yes	—
57	C₂³	Dic₂₁₂	1	1	yes	characteristic
58	C₂×C₄	D₄₂₄	1	2	yes	—
59	C₂×C₄	Dic₂₁₂	1	1	yes	characteristic
60	C₂×C₄	C₄⋊Dic₅₃	1	2	yes	—
61	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
62	C₈	—	212	4	no	—
63	C₈	C₄⋊Dic₅₃	1	2	yes	—
64	C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
65	C₈	C₂²×Dic₅₃	1	1	yes	characteristic
66	C₂²	—	4	1	no	—
67	C₂²	#13133151	1	2	yes	—
68	C₂²	C₂×Dic₂₁₂	1	1	yes	characteristic
69	C₄	#13133151	1	2	yes	—
70	C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
71	C₄	C₂×D₄₂₄	1	1	yes	characteristic
72	C₂	—	2	2	no	—
73	C₂	—	2	1	no	—
74	C₂	C₂×#13133151	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄₂₄
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₂×M₃₂×C₅₃	C₂
socle	C₂×C₅₃	C₂×#107
hypercenter	C₂×M₃₂	D₅₃
hypocenter	C₅₃	#2161
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
2. #16810453	C₂	semidirect
41. C₅₃	#2161	semidirect