Subgroups — #20276966

Order 6784.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₄₂₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133163	C₂	1	2	yes	maximal
3	#16811001	C₂	1	2	yes	maximal
4	C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₁₂	C₂²	1	2	yes	—
7	C₂×#11719925	—	2	2	no	—
8	#13133215	—	2	2	no	—
9	Dic₄₂₄	—	2	2	no	—
10	#13133163	—	2	4	no	—
11	C₂×#11719908	C₂²	1	1	yes	characteristic
12	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
13	C₂×Q₁₆×C₅₃	C₂²	1	2	yes	—
14	Q₃₂×C₅₃	—	2	2	no	—
15	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₅₃	—	2	2	no	—
17	C₂×Dic₁₀₆	—	2	2	no	—
18	Q₁₆×C₅₃	—	2	2	no	—
19	Q₁₆×C₅₃	—	2	2	no	—
20	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
21	#11719925	—	4	4	no	—
22	#11719919	—	4	4	no	—
23	C₄×Dic₅₃	D₄	1	1	yes	characteristic
24	C₂×#88322	D₄	1	1	yes	characteristic
25	Dic₂₁₂	—	2	2	no	—
26	Dic₂₁₂	—	2	2	no	—
27	C₁₆×C₅₃	D₄	1	2	yes	—
28	#11719908	D₄	1	2	yes	—
29	Q₈×C₅₃	—	4	2	no	—
30	Q₈×C₅₃	—	4	2	no	—
31	C₂×C₄×C₅₃	—	4	2	no	—
32	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
33	C₂×Dic₅₃	—	4	2	no	—
34	C₂×Dic₅₃	D₈	1	2	yes	—
35	Dic₁₀₆	—	4	2	no	—
36	Dic₁₀₆	—	4	2	no	—
37	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
39	#88322	D₈	1	2	yes	—
40	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
41	C₄×C₅₃	—	8	2	no	—
42	C₄×C₅₃	#127	1	1	yes	characteristic
43	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
44	Dic₅₃	—	8	2	no	—
45	Dic₅₃	Q₃₂	1	4	yes	complemented
46	#2247	—	53	1	no	maximal, Sylow
47	C₂×C₅₃	#493	1	1	yes	characteristic
48	C₂×C₅₃	C₂×Q₃₂	1	2	yes	—
49	C₂×Q₃₂	—	53	2	no	—
50	C₂×Q₃₂	—	53	1	no	—
51	C₂×Q₃₂	D₅₃	1	1	yes	characteristic
52	#494	—	53	2	no	—
53	C₄×C₁₆	—	53	1	no	—
54	C₅₃	#2247	1	1	yes	characteristic, complemented, Sylow
55	C₂×Q₁₆	—	53	2	no	—
56	C₂×Q₁₆	—	106	2	no	—
57	C₂×Q₁₆	C₂×D₅₃	1	2	yes	—
58	C₄⋊Q₈	—	106	2	no	—
59	Q₃₂	—	2	2	no	—
60	Q₃₂	—	106	4	no	—
61	Q₃₂	—	106	2	no	—
62	C₂×C₁₆	—	53	1	no	—
63	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
64	C₄×C₈	—	53	1	no	—
65	C₂×Q₈	—	2	2	no	—
66	C₂×Q₈	—	106	2	no	—
67	Q₁₆	—	2	2	no	—
68	Q₁₆	—	2	2	no	—
69	Q₁₆	—	106	2	no	—
70	Q₁₆	—	106	2	no	—
71	Q₁₆	—	212	4	no	—
72	C₂×C₈	—	53	1	no	—
73	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
74	C₄⋊C₄	—	212	4	no	—
75	C₄²	—	53	1	no	—
76	C₁₆	—	53	2	no	—
77	C₁₆	C₂²⋊D₅₃	1	2	yes	—
78	Q₈	—	4	2	no	—
79	Q₈	—	4	2	no	—
80	Q₈	—	212	2	no	—
81	Q₈	—	212	2	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	53	2	no	—
84	C₂×C₄	—	212	2	no	—
85	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
86	C₈	—	53	2	no	—
87	C₈	D₄×D₅₃	1	1	yes	characteristic
88	C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
89	C₂²	D₈×D₅₃	1	1	yes	characteristic
90	C₄	—	8	2	no	—
91	C₄	—	53	4	no	—
92	C₄	—	424	2	no	—
93	C₄	#13133213	1	1	yes	characteristic
94	C₄	D₈×D₅₃	1	1	yes	characteristic
95	C₂	#16810998	1	1	yes	characteristic
96	C₂	Q₃₂×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₅₃
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₂×Q₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	C₂×Q₃₂	D₅₃
hypocenter	C₅₃	#2247
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
45. Dic₅₃	Q₃₂	semidirect
54. C₅₃	#2247	semidirect