Subgroups — C₂²×C₄×#88322 #20283116

Order 6784.

760 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#88322	C₂	1	2	yes	maximal
2	C₂×C₄×#88322	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#88322	C₄	1	16	yes	direct product factor
5	C₂²×#88322	C₂²	1	12	yes	—
6	C₄×#88322	C₂²	1	16	yes	direct product factor
7	C₂³×C₄×C₅₃	C₄	1	2	yes	—
8	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₄²×C₅₃	C₄	1	6	yes	—
10	C₂×C₄²×C₅₃	C₂²	1	6	yes	—
11	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₅₃	C₈	1	16	yes	complemented
13	C₂²×C₄×C₅₃	C₂×C₄	1	3	yes	—
14	C₂²×C₄×C₅₃	C₂×C₄	1	8	yes	—
15	C₂²×C₄×C₅₃	C₂×C₄	1	12	yes	—
16	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
17	C₄²×C₅₃	C₂×C₄	1	12	yes	—
18	C₄²×C₅₃	C₂³	1	4	yes	—
19	C₂×#88322	C₂×C₄	1	48	yes	direct product factor
20	C₂×#88322	C₂³	1	8	yes	—
21	C₂³×C₅₃	C₄²	1	4	yes	—
22	C₂³×C₅₃	C₂×C₈	1	8	yes	—
23	C₂³×C₅₃	C₂²×C₄	1	3	yes	—
24	C₂×C₄×C₅₃	C₄²	1	12	yes	—
25	C₂×C₄×C₅₃	C₂×C₈	1	48	yes	complemented
26	C₂×C₄×C₅₃	C₂²×C₄	1	12	yes	—
27	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
28	C₂×C₄×C₅₃	C₂²×C₄	1	8	yes	—
29	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
30	#88322	C₂²×C₄	1	16	yes	direct product factor
31	C₂²×C₅₃	C₄×C₈	1	16	yes	complemented
32	C₂²×C₅₃	C₂×C₄²	1	6	yes	—
33	C₂²×C₅₃	C₂²×C₈	1	12	yes	—
34	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
35	C₄×C₅₃	C₂×C₄²	1	6	yes	—
36	C₄×C₅₃	C₂²×C₈	1	16	yes	complemented
37	C₄×C₅₃	C₂³×C₄	1	2	yes	—
38	C₂²×C₄×C₈	—	53	1	no	maximal, Sylow
39	C₂×C₅₃	C₂×C₄×C₈	1	12	yes	complemented
40	C₂×C₅₃	C₂²×C₄²	1	1	yes	characteristic
41	C₂×C₅₃	C₂³×C₈	1	2	yes	—
42	C₂³×C₈	—	53	2	no	—
43	C₂²×C₄²	D₅₃	1	1	yes	characteristic
44	C₂×C₄×C₈	—	53	12	no	—
45	C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
46	C₂³×C₄	Dic₅₃	1	2	yes	—
47	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
48	C₂²×C₈	—	53	16	no	—
49	C₂²×C₈	—	53	12	no	—
50	C₂×C₄²	Dic₅₃	1	6	yes	—
51	C₂×C₄²	C₂×D₅₃	1	6	yes	—
52	C₄×C₈	—	53	16	no	—
53	C₂⁴	C₂×Dic₅₃	1	1	yes	characteristic
54	C₂²×C₄	#88322	1	16	yes	direct product factor
55	C₂²×C₄	C₄×D₅₃	1	8	yes	—
56	C₂²×C₄	C₂×Dic₅₃	1	12	yes	—
57	C₂²×C₄	C₂×Dic₅₃	1	3	yes	—
58	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
59	C₂×C₈	—	53	48	no	—
60	C₂×C₈	—	53	8	no	—
61	C₄²	C₂×Dic₅₃	1	12	yes	—
62	C₄²	C₂²×D₅₃	1	4	yes	—
63	C₂³	C₂×#88322	1	8	yes	—
64	C₂³	C₄×Dic₅₃	1	4	yes	—
65	C₂³	C₂²×Dic₅₃	1	3	yes	—
66	C₂×C₄	C₂×#88322	1	48	yes	direct product factor
67	C₂×C₄	C₄×Dic₅₃	1	12	yes	—
68	C₂×C₄	C₂×C₄×D₅₃	1	12	yes	—
69	C₂×C₄	C₂²×Dic₅₃	1	3	yes	—
70	C₂×C₄	C₂²×Dic₅₃	1	8	yes	—
71	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
72	C₈	—	53	16	no	—
73	C₂²	C₄×#88322	1	16	yes	direct product factor
74	C₂²	C₂×C₄×Dic₅₃	1	6	yes	—
75	C₂²	C₂²×#88322	1	12	yes	—
76	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
77	C₄	C₂×C₄×Dic₅₃	1	6	yes	—
78	C₄	C₂²×#88322	1	16	yes	direct product factor
79	C₄	C₂²×C₄×D₅₃	1	2	yes	—
80	C₂	C₂×C₄×#88322	1	12	yes	direct product factor
81	C₂	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
82	C₂	C₂³×#88322	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂²×C₄²	D₅₃
commutator, hypocenter	C₅₃	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂²×C₄²×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂×C₄
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#88322	C₂	direct (N × H)
4. C₂²×#88322	C₄	direct (N × H)
6. C₄×#88322	C₂²	direct (N × H)
12. C₂²×C₄×C₅₃	C₈	semidirect
19. C₂×#88322	C₂×C₄	direct (N × H)
25. C₂×C₄×C₅₃	C₂×C₈	semidirect
30. #88322	C₂²×C₄	direct (N × H)
31. C₂²×C₅₃	C₄×C₈	semidirect
36. C₄×C₅₃	C₂²×C₈	semidirect
39. C₂×C₅₃	C₂×C₄×C₈	semidirect
45. C₅₃	C₂²×C₄×C₈	semidirect
54. C₂²×C₄	#88322	direct (N × H)
66. C₂×C₄	C₂×#88322	direct (N × H)
73. C₂²	C₄×#88322	direct (N × H)
78. C₄	C₂²×#88322	direct (N × H)
80. C₂	C₂×C₄×#88322	direct (N × H)

Subgroups — C22×C4×#88322 #20283116

Conjugacy classes of subgroups

Complements

Subgroups — C₂²×C₄×#88322 #20283116