Subgroups — #20272797

Order 6784.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133157	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	#13133157	—	2	2	no	—
4	#13133157	C₄	1	4	yes	complemented
5	C₂×#11719908	—	2	2	no	—
6	C₂²×C₈×C₅₃	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₅₃	C₄	1	1	yes	characteristic
9	C₄×C₈×C₅₃	—	2	2	no	—
10	C₂²×C₄×C₅₃	—	2	1	no	—
11	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₈×C₅₃	—	2	1	no	—
13	C₂×C₈×C₅₃	—	2	1	no	—
14	C₂×C₈×C₅₃	C₈	1	2	yes	—
15	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
16	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
17	C₂×C₈×C₅₃	D₄	1	2	yes	—
18	C₂×C₈×C₅₃	Q₈	1	1	yes	characteristic
19	C₄²×C₅₃	—	2	1	no	—
20	C₄²×C₅₃	C₈	1	2	yes	—
21	#11719908	—	2	4	no	—
22	C₂³×C₅₃	M₁₆	1	1	yes	characteristic
23	C₂×C₄×C₅₃	—	2	4	no	—
24	C₂×C₄×C₅₃	—	2	2	no	—
25	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
26	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
31	C₈×C₅₃	—	2	2	no	—
32	C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
33	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
34	C₂²×C₅₃	—	2	2	no	—
35	C₂²×C₅₃	#97	1	1	yes	characteristic
36	C₂²×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
37	C₂²×C₅₃	C₄⋊C₈	1	1	yes	characteristic
38	C₄×C₅₃	—	2	4	no	—
39	C₄×C₅₃	#95	1	1	yes	characteristic
40	C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
41	C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
42	C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
43	#1376	—	53	1	no	maximal, Sylow
44	C₂×C₅₃	—	2	1	no	—
45	C₂×C₅₃	—	2	1	no	—
46	C₂×C₅₃	#336	1	1	yes	characteristic
47	C₂×C₅₃	#364	1	2	yes	—
48	C₂×M₃₂	—	53	2	no	—
49	C₂×C₄×C₈	D₅₃	1	1	yes	characteristic
50	C₅₃	#1376	1	1	yes	characteristic, complemented, Sylow
51	C₂²×C₈	Dic₅₃	1	1	yes	characteristic
52	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
53	C₂×C₄²	Dic₅₃	1	1	yes	characteristic
54	M₃₂	—	53	4	no	—
55	M₃₂	—	106	2	no	—
56	C₂×C₁₆	—	106	2	no	—
57	C₄×C₈	—	2	2	no	—
58	C₂²×C₄	—	2	1	no	—
59	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
60	C₂×C₈	—	2	1	no	—
61	C₂×C₈	—	2	1	no	—
62	C₂×C₈	#88322	1	2	yes	—
63	C₂×C₈	Dic₁₀₆	1	1	yes	characteristic
64	C₂×C₈	C₄×D₅₃	1	2	yes	—
65	C₂×C₈	D₂₁₂	1	1	yes	characteristic
66	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
67	C₄²	—	2	1	no	—
68	C₄²	#88322	1	2	yes	—
69	C₁₆	—	106	4	no	—
70	C₂³	#11719917	1	1	yes	characteristic
71	C₂×C₄	—	2	4	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	C₂×#88322	1	1	yes	characteristic
74	C₂×C₄	C₂×#88322	1	1	yes	characteristic
75	C₂×C₄	#11719917	1	1	yes	characteristic
76	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
77	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
78	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₈	#11719919	1	2	yes	—
81	C₈	#11719921	1	2	yes	—
82	C₂²	—	2	2	no	—
83	C₂²	#13133128	1	1	yes	characteristic
84	C₂²	C₂²⋊#88322	1	1	yes	characteristic
85	C₂²	C₄⋊#88322	1	1	yes	characteristic
86	C₄	—	2	4	no	—
87	C₄	#13133124	1	1	yes	characteristic
88	C₄	C₄×#88322	1	1	yes	characteristic
89	C₄	C₂²⋊#88322	1	1	yes	characteristic
90	C₄	C₄⋊#88322	1	1	yes	characteristic
91	C₂	—	2	1	no	—
92	C₂	—	2	1	no	—
93	C₂	#16810378	1	1	yes	characteristic
94	C₂	#16810444	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₂₁₂
commutator	C₄×C₅₃	C₄×C₈
Frattini	C₂²×C₈	C₂×D₅₃
Fitting	C₂×C₄×C₈×C₅₃	C₂
socle	C₂²×C₅₃	C₄⋊C₈
hypercenter	C₂×C₄×C₈	D₅₃
hypocenter	C₅₃	#1376
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #13133157	C₄	semidirect
50. C₅₃	#1376	semidirect