Subgroups — #20277891

Order 6784.

648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133214	C₂	1	1	yes	maximal, characteristic
2	#16810501	C₂	1	12	yes	maximal, complemented
3	#16810491	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#383	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₂₁₂	C₂²	1	6	yes	—
6	C₂²×C₄×D₅₃	C₂²	1	4	yes	—
7	#13133214	C₄	1	8	yes	complemented
8	C₂×#11719921	C₂²	1	12	yes	—
9	#13133124	C₂²	1	6	yes	—
10	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	#95×C₅₃	C₂²	1	6	yes	—
12	C₂³×D₅₃	C₂³	1	4	yes	—
13	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
14	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
15	C₂²×Dic₅₃	C₂³	1	4	yes	—
16	C₂×D₂₁₂	—	4	6	no	—
17	C₂×D₂₁₂	—	4	6	no	—
18	C₂×D₂₁₂	C₂×C₄	1	24	yes	complemented
19	C₂×C₄×D₅₃	—	4	12	no	—
20	C₄²×C₅₃	C₂×C₄	1	4	yes	complemented
21	#11719921	—	4	12	no	—
22	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
23	C₂²×D₅₃	—	2	12	no	—
24	C₂²×D₅₃	—	2	12	no	—
25	C₂²×D₅₃	C₂²×C₄	1	8	yes	—
26	C₂×C₄×C₅₃	—	2	6	no	—
27	C₂×C₄×C₅₃	—	2	12	no	—
28	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
29	C₂×Dic₅₃	—	2	12	no	—
30	D₂₁₂	—	4	24	no	—
31	C₄×D₅₃	—	4	16	no	—
32	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
33	C₂²×C₅₃	H(5,𝔽₂)	1	6	yes	—
34	C₂×D₅₃	—	2	24	no	—
35	C₂×D₅₃	—	4	4	no	—
36	C₂×D₅₃	—	4	12	no	—
37	C₄×C₅₃	—	2	6	no	—
38	C₄×C₅₃	—	4	4	no	—
39	Dic₅₃	—	4	4	no	—
40	#2380	—	53	1	no	maximal, Sylow
41	C₂×C₅₃	#518	1	3	yes	—
42	C₂×C₅₃	#560	1	4	yes	—
43	D₅₃	—	4	8	no	—
44	C₂×#127	—	53	1	no	—
45	#386	—	53	12	no	—
46	#383	—	53	1	no	—
47	#383	D₅₃	1	1	yes	characteristic, complemented
48	C₅₃	#2380	1	1	yes	characteristic, complemented, Sylow
49	C₂²×D₄	—	53	6	no	—
50	C₂³×C₄	—	53	4	no	—
51	#127	—	53	8	no	—
52	C₂×C₂²⋊C₄	—	53	12	no	—
53	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
54	#95	—	53	6	no	—
55	#95	C₂×D₅₃	1	6	yes	—
56	C₂⁴	—	53	4	no	—
57	C₂×D₄	—	53	24	no	—
58	C₂×D₄	—	212	6	no	—
59	C₂×D₄	—	212	6	no	—
60	C₂²×C₄	—	53	4	no	—
61	C₂²×C₄	—	212	12	no	—
62	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
63	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
64	C₂²⋊C₄	—	212	12	no	—
65	C₄²	C₄×D₅₃	1	4	yes	complemented
66	C₂³	—	53	8	no	—
67	C₂³	—	106	12	no	—
68	C₂³	—	106	12	no	—
69	C₂³	C₂³×D₅₃	1	1	yes	characteristic
70	D₄	—	212	24	no	—
71	C₂×C₄	—	2	6	no	—
72	C₂×C₄	—	2	12	no	—
73	C₂×C₄	—	106	12	no	—
74	C₂×C₄	—	212	16	no	—
75	C₂×C₄	C₂×C₄×D₅₃	1	6	yes	—
76	C₂²	—	106	24	no	—
77	C₂²	—	212	4	no	—
78	C₂²	—	212	12	no	—
79	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
80	C₂²	#13133263	1	6	yes	—
81	C₄	—	2	6	no	—
82	C₄	—	4	4	no	—
83	C₄	—	212	4	no	—
84	C₂	—	212	8	no	—
85	C₂	#16811097	1	3	yes	—
86	C₂	#16811364	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#383	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	#383	D₅₃
hypocenter	C₅₃	#2380
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ #383

Complements

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

N	H ≅ G/N	type
2. #16810501	C₂	semidirect
3. #16810491	C₂	semidirect
4. C₅₃×#383	C₂	semidirect
7. #13133214	C₄	semidirect
18. C₂×D₂₁₂	C₂×C₄	semidirect
20. C₄²×C₅₃	C₂×C₄	semidirect
47. #383	D₅₃	semidirect
48. C₅₃	#2380	semidirect
65. C₄²	C₄×D₅₃	semidirect