Subgroups — C₂³×#11719913 #20289435

Order 6784.

1720 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₂₁₂	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11719913	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×D₅₃	—	2	1	no	—
6	C₂²×Dic₁₀₆	—	2	7	no	—
7	C₂²×Dic₁₀₆	C₂²	1	14	yes	complemented
8	C₂²×D₂₁₂	—	2	7	no	—
9	C₂²×D₂₁₂	C₂²	1	14	yes	—
10	C₂³×Dic₅₃	—	2	1	no	—
11	C₂×#11719913	C₂²	1	112	yes	direct product factor
12	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₅₃	C₂²	1	14	yes	complemented
14	C₂⁴×C₅₃	D₄	1	1	yes	characteristic
15	C₂³×D₅₃	—	2	14	no	—
16	C₂³×D₅₃	—	4	8	no	—
17	C₂²×C₄×C₅₃	D₄	1	7	yes	—
18	C₂²×C₄×C₅₃	C₂³	1	7	yes	—
19	C₂²×Dic₅₃	—	2	14	no	—
20	C₂×D₂₁₂	—	2	42	no	—
21	C₂×D₂₁₂	C₂³	1	28	yes	—
22	C₂×Dic₁₀₆	—	2	42	no	—
23	C₂×Dic₁₀₆	C₂³	1	28	yes	complemented
24	C₂×C₈×C₅₃	C₂³	1	28	yes	complemented
25	#11719913	C₂³	1	64	yes	direct product factor
26	C₂³×C₅₃	SD₁₆	1	8	yes	complemented
27	C₂³×C₅₃	C₂×D₄	1	7	yes	—
28	C₂²×D₅₃	—	2	28	no	—
29	C₂²×D₅₃	—	4	56	no	—
30	C₂×C₄×C₅₃	C₂×D₄	1	21	yes	—
31	C₂×C₄×C₅₃	C₂⁴	1	7	yes	—
32	C₂×Dic₅₃	—	2	28	no	—
33	D₂₁₂	—	2	28	no	—
34	D₂₁₂	C₂⁴	1	8	yes	—
35	Dic₁₀₆	—	2	28	no	—
36	Dic₁₀₆	C₂⁴	1	8	yes	complemented
37	C₈×C₅₃	C₂⁴	1	8	yes	complemented
38	C₂²×C₅₃	C₂×SD₁₆	1	28	yes	complemented
39	C₂²×C₅₃	C₂²×D₄	1	7	yes	—
40	C₂×D₅₃	—	2	8	no	—
41	C₂×D₅₃	—	4	56	no	—
42	C₄×C₅₃	C₂²×D₄	1	7	yes	—
43	C₄×C₅₃	C₂⁵	1	1	yes	characteristic
44	Dic₅₃	—	2	8	no	—
45	C₂³×SD₁₆	—	53	1	no	maximal, Sylow
46	C₂×C₅₃	C₂²×SD₁₆	1	14	yes	complemented
47	C₂×C₅₃	C₂³×D₄	1	1	yes	characteristic
48	D₅₃	—	4	8	no	—
49	C₂³×Q₈	—	53	1	no	—
50	C₂³×D₄	—	53	1	no	—
51	C₂²×SD₁₆	—	53	28	no	—
52	C₂³×C₈	D₅₃	1	1	yes	characteristic, complemented
53	C₅₃	C₂³×SD₁₆	1	1	yes	characteristic, complemented, Sylow
54	C₂⁵	—	106	1	no	—
55	C₂²×Q₈	—	53	14	no	—
56	C₂²×Q₈	—	106	7	no	—
57	C₂²×D₄	—	53	14	no	—
58	C₂²×D₄	—	106	7	no	—
59	C₂³×C₄	—	106	1	no	—
60	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
61	C₂×SD₁₆	—	53	112	no	—
62	C₂²×C₈	C₂×D₅₃	1	14	yes	complemented
63	C₂⁴	—	106	14	no	—
64	C₂⁴	—	212	8	no	—
65	C₂⁴	D₂₁₂	1	1	yes	characteristic
66	C₂×Q₈	—	53	28	no	—
67	C₂×Q₈	—	106	42	no	—
68	C₂×D₄	—	53	28	no	—
69	C₂×D₄	—	106	42	no	—
70	C₂²×C₄	—	106	14	no	—
71	C₂²×C₄	D₂₁₂	1	7	yes	—
72	C₂²×C₄	C₂²×D₅₃	1	7	yes	—
73	SD₁₆	—	53	64	no	—
74	C₂×C₈	C₂²×D₅₃	1	28	yes	complemented
75	C₂³	—	106	28	no	—
76	C₂³	—	212	56	no	—
77	C₂³	#11719913	1	8	yes	direct product factor
78	C₂³	C₂×D₂₁₂	1	7	yes	—
79	Q₈	—	53	8	no	—
80	Q₈	—	106	28	no	—
81	D₄	—	53	8	no	—
82	D₄	—	106	28	no	—
83	C₂×C₄	—	106	28	no	—
84	C₂×C₄	C₂×D₂₁₂	1	21	yes	—
85	C₂×C₄	C₂³×D₅₃	1	7	yes	—
86	C₈	C₂³×D₅₃	1	8	yes	complemented
87	C₂²	—	106	8	no	—
88	C₂²	—	212	56	no	—
89	C₂²	C₂×#11719913	1	28	yes	direct product factor
90	C₂²	C₂²×D₂₁₂	1	7	yes	—
91	C₄	—	106	8	no	—
92	C₄	C₂²×D₂₁₂	1	7	yes	—
93	C₄	C₂⁴×D₅₃	1	1	yes	characteristic
94	C₂	—	212	8	no	—
95	C₂	C₂²×#11719913	1	14	yes	direct product factor
96	C₂	C₂³×D₂₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₂₁₂
commutator	C₄×C₅₃	C₂⁵
Frattini	C₄	C₂⁴×D₅₃
Fitting	C₂³×C₈×C₅₃	C₂
socle	C₂⁴×C₅₃	D₄
hypercenter	C₂³×C₈	D₅₃
hypocenter	C₅₃	C₂³×SD₁₆
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
1. C₂³×Dic₁₀₆	C₂	semidirect
3. C₂²×#11719913	C₂	direct (N × H)
4. C₂³×C₈×C₅₃	C₂	semidirect
7. C₂²×Dic₁₀₆	C₂²	semidirect
11. C₂×#11719913	C₂²	direct (N × H)
13. C₂²×C₈×C₅₃	C₂²	semidirect
23. C₂×Dic₁₀₆	C₂³	semidirect
24. C₂×C₈×C₅₃	C₂³	semidirect
25. #11719913	C₂³	direct (N × H)
26. C₂³×C₅₃	SD₁₆	semidirect
36. Dic₁₀₆	C₂⁴	semidirect
37. C₈×C₅₃	C₂⁴	semidirect
38. C₂²×C₅₃	C₂×SD₁₆	semidirect
46. C₂×C₅₃	C₂²×SD₁₆	semidirect
52. C₂³×C₈	D₅₃	semidirect
53. C₅₃	C₂³×SD₁₆	semidirect
62. C₂²×C₈	C₂×D₅₃	semidirect
74. C₂×C₈	C₂²×D₅₃	semidirect
77. C₂³	#11719913	direct (N × H)
86. C₈	C₂³×D₅₃	semidirect
89. C₂²	C₂×#11719913	direct (N × H)
95. C₂	C₂²×#11719913	direct (N × H)

Subgroups — C23×#11719913 #20289435

Conjugacy classes of subgroups

Complements

Subgroups — C₂³×#11719913 #20289435