Subgroups — #20274855

Order 6784.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810647	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133128	C₂	1	1	yes	maximal, characteristic
3	#16810387	C₂	1	4	yes	maximal
4	C₅₃×#423	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719917	C₂²	1	2	yes	—
6	C₂²×#88322	C₂²	1	2	yes	—
7	C₄⋊#88322	—	2	2	no	—
8	#13133128	C₄	1	4	yes	—
9	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
10	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₅₃	—	2	2	no	—
12	C₂²×C₄×C₅₃	Q₈	1	2	yes	—
13	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
14	M₁₆×C₅₃	—	2	4	no	—
15	C₂×C₈×C₅₃	—	2	2	no	—
16	C₄²×C₅₃	C₂×C₄	1	2	yes	—
17	C₄²×C₅₃	D₄	1	2	yes	—
18	#11719917	—	2	4	no	—
19	C₂×#88322	—	2	2	no	—
20	C₂×#88322	—	2	2	no	—
21	C₂×#88322	—	2	2	no	—
22	C₂×#88322	C₂×C₄	1	4	yes	—
23	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
24	C₂×C₄×C₅₃	—	4	2	no	—
25	C₂×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
26	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₅₃	#41	1	2	yes	—
29	C₂×C₄×C₅₃	#41	1	2	yes	—
30	C₈×C₅₃	—	4	2	no	—
31	#88322	—	2	4	no	—
32	#88322	—	4	2	no	—
33	C₂²×C₅₃	—	2	2	no	—
34	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₅₃	#125	1	2	yes	—
36	C₄×C₅₃	—	2	4	no	—
37	C₄×C₅₃	#117	1	2	yes	—
38	C₄×C₅₃	#124	1	1	yes	characteristic
39	C₄×C₅₃	#124	1	1	yes	characteristic
40	#1838	—	53	1	no	maximal, Sylow
41	C₂×C₅₃	—	2	2	no	—
42	C₂×C₅₃	#382	1	1	yes	characteristic
43	C₂×C₅₃	#430	1	2	yes	—
44	#423	—	53	1	no	—
45	#423	D₅₃	1	1	yes	characteristic
46	C₂×#97	—	53	1	no	—
47	#341	—	53	4	no	—
48	C₅₃	#1838	1	1	yes	characteristic, complemented, Sylow
49	C₂×M₁₆	—	53	2	no	—
50	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
51	C₂²×C₈	—	53	2	no	—
52	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
53	C₄⋊C₈	—	2	2	no	—
54	C₄⋊C₈	—	106	2	no	—
55	#97	—	53	4	no	—
56	C₂²×C₄	Dic₁₀₆	1	2	yes	—
57	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
58	M₁₆	—	2	4	no	—
59	M₁₆	—	106	4	no	—
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	53	4	no	—
62	C₂×C₈	—	106	2	no	—
63	C₂×C₈	—	106	2	no	—
64	C₂×C₈	—	106	2	no	—
65	C₄²	C₄×D₅₃	1	2	yes	—
66	C₄²	C₂²⋊D₅₃	1	2	yes	—
67	C₂³	C₂×Dic₁₀₆	1	1	yes	characteristic
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	#11719919	1	4	yes	—
70	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
71	C₂×C₄	#11719947	1	2	yes	—
72	C₂×C₄	#11719949	1	2	yes	—
73	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
74	C₈	—	4	2	no	—
75	C₈	—	106	4	no	—
76	C₈	—	212	2	no	—
77	C₂²	—	2	2	no	—
78	C₂²	C₂×#11719919	1	1	yes	characteristic
79	C₂²	#13133208	1	2	yes	—
80	C₄	—	2	4	no	—
81	C₄	#13133175	1	2	yes	—
82	C₄	#13133204	1	1	yes	characteristic
83	C₄	#13133206	1	1	yes	characteristic
84	C₂	—	2	2	no	—
85	C₂	#16810488	1	1	yes	characteristic
86	C₂	#16810671	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#423	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	#423	D₅₃
hypocenter	C₅₃	#1838
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

Complements

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

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