Subgroups — C₂³×#11719924 #20289433

Order 6784.

1720 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₂₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×#11719924	C₂	1	28	yes	maximal, direct product factor
3	C₂³×#88322	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×D₅₃	—	2	1	no	—
6	C₂²×D₂₁₂	—	2	7	no	—
7	C₂²×D₂₁₂	C₂²	1	14	yes	—
8	C₂×#11719924	C₂²	1	112	yes	direct product factor
9	C₂²×#88322	C₂²	1	14	yes	complemented
10	C₂²×Q₈×C₅₃	—	2	7	no	—
11	C₂²×Q₈×C₅₃	C₂²	1	14	yes	complemented
12	C₂³×C₄×C₅₃	—	2	1	no	—
13	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂⁴×C₅₃	D₄	1	1	yes	characteristic
15	C₂³×D₅₃	—	2	14	no	—
16	C₂³×D₅₃	—	4	8	no	—
17	C₂×Q₈×C₅₃	—	2	42	no	—
18	C₂×Q₈×C₅₃	C₂³	1	28	yes	complemented
19	C₂²×C₄×C₅₃	—	2	14	no	—
20	C₂²×C₄×C₅₃	D₄	1	7	yes	—
21	C₂²×C₄×C₅₃	C₂³	1	7	yes	—
22	C₂×D₂₁₂	—	2	42	no	—
23	C₂×D₂₁₂	C₂³	1	28	yes	—
24	#11719924	C₂³	1	64	yes	direct product factor
25	C₂×#88322	C₂³	1	28	yes	complemented
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	Q₈×C₅₃	—	2	28	no	—
31	Q₈×C₅₃	C₂⁴	1	8	yes	complemented
32	C₂×C₄×C₅₃	—	2	28	no	—
33	C₂×C₄×C₅₃	C₂×D₄	1	21	yes	—
34	C₂×C₄×C₅₃	C₂⁴	1	7	yes	—
35	D₂₁₂	—	2	28	no	—
36	D₂₁₂	C₂⁴	1	8	yes	—
37	#88322	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	—	2	8	no	—
43	C₄×C₅₃	C₂²×D₄	1	7	yes	—
44	C₄×C₅₃	C₂⁵	1	1	yes	characteristic
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	D₅₃	1	1	yes	characteristic, complemented
50	C₂³×D₄	—	53	1	no	—
51	C₂²×SD₁₆	—	53	28	no	—
52	C₂³×C₈	—	53	1	no	—
53	C₅₃	C₂³×SD₁₆	1	1	yes	characteristic, complemented, Sylow
54	C₂⁵	—	106	1	no	—
55	C₂²×Q₈	—	2	7	no	—
56	C₂²×Q₈	C₂×D₅₃	1	14	yes	complemented
57	C₂²×D₄	—	53	14	no	—
58	C₂²×D₄	—	106	7	no	—
59	C₂³×C₄	—	2	1	no	—
60	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
61	C₂×SD₁₆	—	53	112	no	—
62	C₂²×C₈	—	53	14	no	—
63	C₂⁴	—	106	14	no	—
64	C₂⁴	—	212	8	no	—
65	C₂⁴	C₂²⋊D₅₃	1	1	yes	characteristic
66	C₂×Q₈	—	2	42	no	—
67	C₂×Q₈	C₂²×D₅₃	1	28	yes	complemented
68	C₂×D₄	—	53	28	no	—
69	C₂×D₄	—	106	42	no	—
70	C₂²×C₄	—	2	14	no	—
71	C₂²×C₄	C₂²⋊D₅₃	1	7	yes	—
72	C₂²×C₄	C₂²×D₅₃	1	7	yes	—
73	SD₁₆	—	53	64	no	—
74	C₂×C₈	—	53	28	no	—
75	C₂³	—	106	28	no	—
76	C₂³	—	212	56	no	—
77	C₂³	#11719924	1	8	yes	direct product factor
78	C₂³	C₂×C₂²⋊D₅₃	1	7	yes	—
79	Q₈	—	2	28	no	—
80	Q₈	C₂³×D₅₃	1	8	yes	complemented
81	D₄	—	53	8	no	—
82	D₄	—	106	28	no	—
83	C₂×C₄	—	2	28	no	—
84	C₂×C₄	C₂×C₂²⋊D₅₃	1	21	yes	—
85	C₂×C₄	C₂³×D₅₃	1	7	yes	—
86	C₈	—	53	8	no	—
87	C₂²	—	106	8	no	—
88	C₂²	—	212	56	no	—
89	C₂²	C₂×#11719924	1	28	yes	direct product factor
90	C₂²	C₂²×C₂²⋊D₅₃	1	7	yes	—
91	C₄	—	2	8	no	—
92	C₄	C₂²×C₂²⋊D₅₃	1	7	yes	—
93	C₄	C₂⁴×D₅₃	1	1	yes	characteristic
94	C₂	—	212	8	no	—
95	C₂	C₂²×#11719924	1	14	yes	direct product factor
96	C₂	C₂³×C₂²⋊D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂⁵
Frattini	C₄	C₂⁴×D₅₃
Fitting	C₂³×Q₈×C₅₃	C₂
socle	C₂⁴×C₅₃	D₄
hypercenter	C₂³×Q₈	D₅₃
hypocenter	C₅₃	C₂³×SD₁₆
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂²×#11719924	C₂	direct (N × H)
3. C₂³×#88322	C₂	semidirect
4. C₂³×Q₈×C₅₃	C₂	semidirect
8. C₂×#11719924	C₂²	direct (N × H)
9. C₂²×#88322	C₂²	semidirect
11. C₂²×Q₈×C₅₃	C₂²	semidirect
18. C₂×Q₈×C₅₃	C₂³	semidirect
24. #11719924	C₂³	direct (N × H)
25. C₂×#88322	C₂³	semidirect
26. C₂³×C₅₃	SD₁₆	semidirect
31. Q₈×C₅₃	C₂⁴	semidirect
37. #88322	C₂⁴	semidirect
38. C₂²×C₅₃	C₂×SD₁₆	semidirect
46. C₂×C₅₃	C₂²×SD₁₆	semidirect
49. C₂³×Q₈	D₅₃	semidirect
53. C₅₃	C₂³×SD₁₆	semidirect
56. C₂²×Q₈	C₂×D₅₃	semidirect
67. C₂×Q₈	C₂²×D₅₃	semidirect
77. C₂³	#11719924	direct (N × H)
80. Q₈	C₂³×D₅₃	semidirect
89. C₂²	C₂×#11719924	direct (N × H)
95. C₂	C₂²×#11719924	direct (N × H)

Subgroups — C23×#11719924 #20289433

Conjugacy classes of subgroups

Complements

Subgroups — C₂³×#11719924 #20289433