Subgroups — #20274856

Order 6784.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810647	C₂	1	2	yes	maximal
2	#16810388	C₂	1	4	yes	maximal
3	C₂×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719917	C₂²	1	4	yes	—
5	C₄⋊#88322	—	2	4	no	—
6	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
7	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
8	#97×C₅₃	C₄	1	4	yes	—
9	C₂²×C₄×C₅₃	Q₈	1	2	yes	—
10	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
11	C₂×C₈×C₅₃	—	2	4	no	—
12	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
13	C₄²×C₅₃	C₂×C₄	1	2	yes	—
14	C₄²×C₅₃	D₄	1	2	yes	—
15	#11719917	—	2	8	no	—
16	C₂×#88322	—	2	4	no	—
17	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
18	C₂×C₄×C₅₃	—	4	2	no	—
19	C₂×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
20	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
21	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
22	C₂×C₄×C₅₃	#41	1	4	yes	—
23	C₈×C₅₃	—	2	4	no	—
24	#88322	—	4	4	no	—
25	C₂²×C₅₃	—	2	2	no	—
26	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₅₃	#125	1	2	yes	—
28	C₄×C₅₃	—	2	4	no	—
29	C₄×C₅₃	#117	1	2	yes	—
30	C₄×C₅₃	#124	1	2	yes	—
31	#1838	—	53	1	no	maximal, Sylow
32	C₂×C₅₃	—	2	2	no	—
33	C₂×C₅₃	#382	1	1	yes	characteristic
34	C₂×C₅₃	#430	1	2	yes	—
35	#423	—	53	2	no	—
36	C₂×#97	D₅₃	1	1	yes	characteristic
37	#341	—	53	4	no	—
38	C₅₃	#1838	1	1	yes	characteristic, complemented, Sylow
39	C₂×M₁₆	—	53	4	no	—
40	C₂²×C₈	C₂×D₅₃	1	2	yes	—
41	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
42	C₄⋊C₈	—	106	4	no	—
43	#97	Dic₅₃	1	4	yes	—
44	C₂²×C₄	Dic₁₀₆	1	2	yes	—
45	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
46	M₁₆	—	106	8	no	—
47	C₂×C₈	—	2	4	no	—
48	C₂×C₈	—	106	4	no	—
49	C₂×C₈	C₂×Dic₅₃	1	4	yes	—
50	C₄²	D₂₁₂	1	2	yes	—
51	C₄²	C₂×Dic₅₃	1	2	yes	—
52	C₂³	C₂×Dic₁₀₆	1	1	yes	characteristic
53	C₂×C₄	—	4	2	no	—
54	C₂×C₄	C₄⋊Dic₅₃	1	4	yes	—
55	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
56	C₂×C₄	#11719945	1	4	yes	—
57	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
58	C₈	—	2	4	no	—
59	C₈	—	212	4	no	—
60	C₂²	—	2	2	no	—
61	C₂²	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
62	C₂²	#13133209	1	2	yes	—
63	C₄	—	2	4	no	—
64	C₄	#13133176	1	2	yes	—
65	C₄	#13133205	1	2	yes	—
66	C₂	—	2	2	no	—
67	C₂	#16810489	1	1	yes	characteristic
68	C₂	#16810672	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#97×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	C₂×#97	D₅₃
hypocenter	C₅₃	#1838
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
38. C₅₃	#1838	semidirect