Subgroups — C₂³×#11719940 #20291700

Order 6784.

1720 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#11719940	C₂	1	28	yes	maximal, direct product factor
2	C₂³×#88324	C₂	1	2	yes	maximal, complemented
3	C₂⁴×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719940	C₂²	1	112	yes	direct product factor
5	C₂²×#88324	C₂²	1	28	yes	complemented
6	C₂³×Dic₅₃	C₄	1	14	yes	—
7	C₂³×Dic₅₃	C₄	1	1	yes	characteristic
8	C₂³×Dic₅₃	C₂²	1	1	yes	characteristic
9	C₂³×Dic₅₃	C₂²	1	14	yes	—
10	C₂⁵×C₅₃	C₄	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×Dic₅₃	C₂×C₄	1	84	yes	—
15	C₂²×Dic₅₃	C₂×C₄	1	7	yes	—
16	C₂²×Dic₅₃	C₂×C₄	1	14	yes	—
17	C₂²×Dic₅₃	C₂³	1	7	yes	—
18	C₂²×Dic₅₃	C₂³	1	28	yes	—
19	#11719940	C₂³	1	64	yes	direct product factor
20	C₂×#88324	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×Dic₅₃	C₂²×C₄	1	56	yes	—
26	C₂×Dic₅₃	C₂²×C₄	1	21	yes	—
27	C₂×Dic₅₃	C₂²×C₄	1	28	yes	—
28	C₂×Dic₅₃	C₂⁴	1	7	yes	—
29	C₂×Dic₅₃	C₂⁴	1	8	yes	—
30	#88324	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	Dic₅₃	C₂³×C₄	1	7	yes	—
36	Dic₅₃	C₂³×C₄	1	8	yes	—
37	Dic₅₃	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	—	53	1	no	—
43	C₂²×M₁₆	—	53	28	no	—
44	C₂³×C₈	—	53	2	no	—
45	C₅₃	C₂³×M₁₆	1	1	yes	characteristic, complemented, Sylow
46	C₂⁵	#6157	1	1	yes	characteristic
47	C₂³×C₄	—	53	14	no	—
48	C₂³×C₄	—	53	14	no	—
49	C₂³×C₄	—	53	1	no	—
50	C₂³×C₄	—	53	1	no	—
51	C₂×M₁₆	—	53	112	no	—
52	C₂²×C₈	—	53	28	no	—
53	C₂⁴	—	2	8	no	—
54	C₂⁴	C₂×#6157	1	14	yes	—
55	C₂⁴	C₂×#6157	1	1	yes	characteristic
56	C₂²×C₄	—	53	84	no	—
57	C₂²×C₄	—	53	14	no	—
58	C₂²×C₄	—	53	7	no	—
59	C₂²×C₄	—	53	7	no	—
60	C₂²×C₄	—	53	28	no	—
61	M₁₆	—	53	64	no	—
62	C₂×C₈	—	53	56	no	—
63	C₂³	—	2	56	no	—
64	C₂³	#11719940	1	8	yes	direct product factor
65	C₂³	C₂²×#6157	1	28	yes	—
66	C₂³	C₂²×#6157	1	7	yes	—
67	C₂×C₄	—	53	56	no	—
68	C₂×C₄	—	53	21	no	—
69	C₂×C₄	—	53	28	no	—
70	C₂×C₄	—	53	7	no	—
71	C₂×C₄	—	53	8	no	—
72	C₈	—	53	16	no	—
73	C₂²	—	2	56	no	—
74	C₂²	C₂×#11719940	1	28	yes	direct product factor
75	C₂²	C₂³×#6157	1	8	yes	—
76	C₂²	C₂³×#6157	1	7	yes	—
77	C₄	—	53	1	no	—
78	C₄	—	53	7	no	—
79	C₄	—	53	8	no	—
80	C₂	—	2	8	no	—
81	C₂	C₂²×#11719940	1	14	yes	direct product factor
82	C₂	C₂⁴×#6157	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×#6157
commutator	C₂×C₅₃	C₂⁴×C₄
Frattini	C₂	C₂⁴×#6157
Fitting	C₂⁵×C₅₃	C₄
socle	C₂⁴×C₅₃	C₂×C₄
hypercenter	C₂⁵	#6157
hypocenter	C₅₃	C₂³×M₁₆
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵

Complements

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

N	H ≅ G/N	type
1. C₂²×#11719940	C₂	direct (N × H)
2. C₂³×#88324	C₂	semidirect
4. C₂×#11719940	C₂²	direct (N × H)
5. C₂²×#88324	C₂²	semidirect
19. #11719940	C₂³	direct (N × H)
20. C₂×#88324	C₂³	semidirect
22. C₂³×C₅₃	M₁₆	semidirect
30. #88324	C₂⁴	semidirect
32. C₂²×C₅₃	C₂×M₁₆	semidirect
40. C₂×C₅₃	C₂²×M₁₆	semidirect
45. C₅₃	C₂³×M₁₆	semidirect
64. C₂³	#11719940	direct (N × H)
74. C₂²	C₂×#11719940	direct (N × H)
81. C₂	C₂²×#11719940	direct (N × H)

Subgroups — C23×#11719940 #20291700

Conjugacy classes of subgroups

Complements

Subgroups — C₂³×#11719940 #20291700