Subgroups — C₂³×#11719917 #20289417

Order 6784.

1720 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#11719917	C₂	1	28	yes	maximal, direct product factor
2	C₂³×#88322	C₂	1	2	yes	maximal, complemented
3	C₂⁴×C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719917	C₂²	1	112	yes	direct product factor
5	C₂²×#88322	C₂²	1	28	yes	complemented
6	C₂⁵×C₅₃	C₄	1	1	yes	characteristic
7	C₂³×C₄×C₅₃	C₄	1	14	yes	—
8	C₂³×C₄×C₅₃	C₄	1	1	yes	characteristic
9	C₂³×C₄×C₅₃	C₂²	1	14	yes	—
10	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
11	C₂⁴×C₅₃	—	2	8	no	—
12	C₂⁴×C₅₃	C₂×C₄	1	14	yes	—
13	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₅₃	C₂×C₄	1	84	yes	—
15	C₂²×C₄×C₅₃	C₂×C₄	1	14	yes	—
16	C₂²×C₄×C₅₃	C₂×C₄	1	7	yes	—
17	C₂²×C₄×C₅₃	C₂³	1	7	yes	—
18	C₂²×C₄×C₅₃	C₂³	1	28	yes	—
19	#11719917	C₂³	1	64	yes	direct product factor
20	C₂×#88322	C₂³	1	56	yes	complemented
21	C₂³×C₅₃	—	2	56	no	—
22	C₂³×C₅₃	M₁₆	1	8	yes	complemented
23	C₂³×C₅₃	C₂²×C₄	1	28	yes	—
24	C₂³×C₅₃	C₂²×C₄	1	7	yes	—
25	C₂×C₄×C₅₃	C₂²×C₄	1	56	yes	—
26	C₂×C₄×C₅₃	C₂²×C₄	1	21	yes	—
27	C₂×C₄×C₅₃	C₂²×C₄	1	28	yes	—
28	C₂×C₄×C₅₃	C₂⁴	1	7	yes	—
29	C₂×C₄×C₅₃	C₂⁴	1	8	yes	—
30	#88322	C₂⁴	1	16	yes	complemented
31	C₂²×C₅₃	—	2	56	no	—
32	C₂²×C₅₃	C₂×M₁₆	1	28	yes	complemented
33	C₂²×C₅₃	C₂³×C₄	1	8	yes	—
34	C₂²×C₅₃	C₂³×C₄	1	7	yes	—
35	C₄×C₅₃	C₂³×C₄	1	7	yes	—
36	C₄×C₅₃	C₂³×C₄	1	8	yes	—
37	C₄×C₅₃	C₂⁵	1	1	yes	characteristic
38	C₂³×M₁₆	—	53	1	no	maximal, Sylow
39	C₂×C₅₃	—	2	8	no	—
40	C₂×C₅₃	C₂²×M₁₆	1	14	yes	complemented
41	C₂×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
42	C₂⁴×C₄	D₅₃	1	1	yes	characteristic
43	C₂²×M₁₆	—	53	28	no	—
44	C₂³×C₈	—	53	2	no	—
45	C₅₃	C₂³×M₁₆	1	1	yes	characteristic, complemented, Sylow
46	C₂⁵	Dic₅₃	1	1	yes	characteristic
47	C₂³×C₄	Dic₅₃	1	1	yes	characteristic
48	C₂³×C₄	Dic₅₃	1	14	yes	—
49	C₂³×C₄	C₂×D₅₃	1	14	yes	—
50	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
51	C₂×M₁₆	—	53	112	no	—
52	C₂²×C₈	—	53	28	no	—
53	C₂⁴	—	2	8	no	—
54	C₂⁴	C₂×Dic₅₃	1	14	yes	—
55	C₂⁴	C₂×Dic₅₃	1	1	yes	characteristic
56	C₂²×C₄	C₂×Dic₅₃	1	84	yes	—
57	C₂²×C₄	C₂×Dic₅₃	1	14	yes	—
58	C₂²×C₄	C₂×Dic₅₃	1	7	yes	—
59	C₂²×C₄	C₂²×D₅₃	1	7	yes	—
60	C₂²×C₄	C₂²×D₅₃	1	28	yes	—
61	M₁₆	—	53	64	no	—
62	C₂×C₈	—	53	56	no	—
63	C₂³	—	2	56	no	—
64	C₂³	#11719917	1	8	yes	direct product factor
65	C₂³	C₂²×Dic₅₃	1	28	yes	—
66	C₂³	C₂²×Dic₅₃	1	7	yes	—
67	C₂×C₄	C₂²×Dic₅₃	1	56	yes	—
68	C₂×C₄	C₂²×Dic₅₃	1	28	yes	—
69	C₂×C₄	C₂²×Dic₅₃	1	21	yes	—
70	C₂×C₄	C₂³×D₅₃	1	7	yes	—
71	C₂×C₄	C₂³×D₅₃	1	8	yes	—
72	C₈	—	53	16	no	—
73	C₂²	—	2	56	no	—
74	C₂²	C₂×#11719917	1	28	yes	direct product factor
75	C₂²	C₂³×Dic₅₃	1	8	yes	—
76	C₂²	C₂³×Dic₅₃	1	7	yes	—
77	C₄	C₂³×Dic₅₃	1	7	yes	—
78	C₄	C₂³×Dic₅₃	1	8	yes	—
79	C₄	C₂⁴×D₅₃	1	1	yes	characteristic
80	C₂	—	2	8	no	—
81	C₂	C₂²×#11719917	1	14	yes	direct product factor
82	C₂	C₂⁴×Dic₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂×D₅₃
commutator	C₂×C₅₃	C₂⁴×C₄
Frattini	C₄	C₂⁴×D₅₃
Fitting	C₂⁴×C₄×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂×C₄
hypercenter	C₂⁴×C₄	D₅₃
hypocenter	C₅₃	C₂³×M₁₆
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#11719917	C₂	direct (N × H)
2. C₂³×#88322	C₂	semidirect
4. C₂×#11719917	C₂²	direct (N × H)
5. C₂²×#88322	C₂²	semidirect
19. #11719917	C₂³	direct (N × H)
20. C₂×#88322	C₂³	semidirect
22. C₂³×C₅₃	M₁₆	semidirect
30. #88322	C₂⁴	semidirect
32. C₂²×C₅₃	C₂×M₁₆	semidirect
40. C₂×C₅₃	C₂²×M₁₆	semidirect
45. C₅₃	C₂³×M₁₆	semidirect
64. C₂³	#11719917	direct (N × H)
74. C₂²	C₂×#11719917	direct (N × H)
81. C₂	C₂²×#11719917	direct (N × H)

Subgroups — C23×#11719917 #20289417

Conjugacy classes of subgroups

Complements

Subgroups — C₂³×#11719917 #20289417