Subgroups — C₂²×#13133157 #20287811

Order 6784.

400 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133157	C₂	1	12	yes	maximal, direct product factor
2	C₂²×#11719908	C₂	1	2	yes	maximal, complemented
3	C₂³×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	#13133157	C₂²	1	16	yes	direct product factor
5	C₂×#11719908	C₂²	1	12	yes	complemented
6	C₂³×C₄×C₅₃	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₅₃	C₄	1	6	yes	—
8	C₂²×C₈×C₅₃	C₄	1	1	yes	characteristic
9	C₂²×C₈×C₅₃	C₂²	1	6	yes	—
10	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂⁴×C₅₃	C₈	1	1	yes	characteristic
12	C₂²×C₄×C₅₃	C₈	1	6	yes	—
13	C₂²×C₄×C₅₃	C₈	1	1	yes	characteristic
14	C₂²×C₄×C₅₃	C₂×C₄	1	6	yes	—
15	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₅₃	C₂×C₄	1	12	yes	—
17	C₂×C₈×C₅₃	C₂×C₄	1	6	yes	—
18	C₂×C₈×C₅₃	C₂×C₄	1	3	yes	—
19	C₂×C₈×C₅₃	C₂³	1	3	yes	—
20	C₂×C₈×C₅₃	C₂³	1	4	yes	—
21	#11719908	C₂³	1	8	yes	complemented
22	C₂³×C₅₃	—	2	4	no	—
23	C₂³×C₅₃	C₂×C₈	1	6	yes	—
24	C₂³×C₅₃	C₂×C₈	1	1	yes	characteristic
25	C₂×C₄×C₅₃	C₂×C₈	1	12	yes	—
26	C₂×C₄×C₅₃	C₂×C₈	1	6	yes	—
27	C₂×C₄×C₅₃	C₂×C₈	1	3	yes	—
28	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
29	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
30	C₈×C₅₃	C₂²×C₄	1	3	yes	—
31	C₈×C₅₃	C₂²×C₄	1	4	yes	—
32	C₈×C₅₃	C₂⁴	1	1	yes	characteristic
33	C₂²×C₅₃	—	2	12	no	—
34	C₂²×C₅₃	M₃₂	1	4	yes	complemented
35	C₂²×C₅₃	C₂²×C₈	1	4	yes	—
36	C₂²×C₅₃	C₂²×C₈	1	3	yes	—
37	C₄×C₅₃	C₂²×C₈	1	3	yes	—
38	C₄×C₅₃	C₂²×C₈	1	4	yes	—
39	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
40	C₂²×M₃₂	—	53	1	no	maximal, Sylow
41	C₂×C₅₃	—	2	4	no	—
42	C₂×C₅₃	C₂×M₃₂	1	6	yes	complemented
43	C₂×C₅₃	C₂³×C₈	1	1	yes	characteristic
44	C₂³×C₈	D₅₃	1	1	yes	characteristic
45	C₂×M₃₂	—	53	12	no	—
46	C₂²×C₁₆	—	53	2	no	—
47	C₅₃	C₂²×M₃₂	1	1	yes	characteristic, complemented, Sylow
48	C₂³×C₄	Dic₅₃	1	1	yes	characteristic
49	C₂²×C₈	Dic₅₃	1	1	yes	characteristic
50	C₂²×C₈	Dic₅₃	1	6	yes	—
51	C₂²×C₈	C₂×D₅₃	1	6	yes	—
52	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
53	M₃₂	—	53	16	no	—
54	C₂×C₁₆	—	53	12	no	—
55	C₂⁴	#88322	1	1	yes	characteristic
56	C₂²×C₄	#88322	1	6	yes	—
57	C₂²×C₄	#88322	1	1	yes	characteristic
58	C₂²×C₄	C₂×Dic₅₃	1	6	yes	—
59	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
60	C₂×C₈	C₂×Dic₅₃	1	6	yes	—
61	C₂×C₈	C₂×Dic₅₃	1	12	yes	—
62	C₂×C₈	C₂×Dic₅₃	1	3	yes	—
63	C₂×C₈	C₂²×D₅₃	1	3	yes	—
64	C₂×C₈	C₂²×D₅₃	1	4	yes	—
65	C₁₆	—	53	8	no	—
66	C₂³	—	2	4	no	—
67	C₂³	C₂×#88322	1	6	yes	—
68	C₂³	C₂×#88322	1	1	yes	characteristic
69	C₂×C₄	C₂×#88322	1	12	yes	—
70	C₂×C₄	C₂×#88322	1	6	yes	—
71	C₂×C₄	C₂×#88322	1	3	yes	—
72	C₂×C₄	C₂²×Dic₅₃	1	4	yes	—
73	C₂×C₄	C₂²×Dic₅₃	1	3	yes	—
74	C₈	C₂²×Dic₅₃	1	4	yes	—
75	C₈	C₂²×Dic₅₃	1	3	yes	—
76	C₈	C₂³×D₅₃	1	1	yes	characteristic
77	C₂²	—	2	12	no	—
78	C₂²	#13133157	1	4	yes	direct product factor
79	C₂²	C₂²×#88322	1	4	yes	—
80	C₂²	C₂²×#88322	1	3	yes	—
81	C₄	C₂²×#88322	1	4	yes	—
82	C₄	C₂²×#88322	1	3	yes	—
83	C₄	C₂³×Dic₅₃	1	1	yes	characteristic
84	C₂	—	2	4	no	—
85	C₂	C₂×#13133157	1	6	yes	direct product factor
86	C₂	C₂³×#88322	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₅₃
commutator	C₂×C₅₃	C₂³×C₈
Frattini	C₈	C₂³×D₅₃
Fitting	C₂³×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₈
hypercenter	C₂³×C₈	D₅₃
hypocenter	C₅₃	C₂²×M₃₂
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#13133157	C₂	direct (N × H)
2. C₂²×#11719908	C₂	semidirect
4. #13133157	C₂²	direct (N × H)
5. C₂×#11719908	C₂²	semidirect
21. #11719908	C₂³	semidirect
34. C₂²×C₅₃	M₃₂	semidirect
42. C₂×C₅₃	C₂×M₃₂	semidirect
47. C₅₃	C₂²×M₃₂	semidirect
78. C₂²	#13133157	direct (N × H)
85. C₂	C₂×#13133157	direct (N × H)

Subgroups — C22×#13133157 #20287811

Conjugacy classes of subgroups

Complements

Subgroups — C₂²×#13133157 #20287811