Subgroups — #20272542

Order 6784.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#88322	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#88322	C₄	1	4	yes	—
4	C₄×#88322	—	2	2	no	—
5	C₄×#88322	—	2	2	no	—
6	C₂²×C₈×C₅₃	C₄	1	2	yes	—
7	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
8	C₄×C₈×C₅₃	—	2	2	no	—
9	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
10	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
11	C₂×C₈×C₅₃	—	2	2	no	—
12	C₂×C₈×C₅₃	—	2	2	no	—
13	C₂×C₈×C₅₃	C₈	1	4	yes	complemented
14	C₄²×C₅₃	D₄	1	1	yes	characteristic
15	C₄²×C₅₃	D₄	1	2	yes	—
16	C₄²×C₅₃	Q₈	1	1	yes	characteristic
17	C₂×#88322	—	2	4	no	—
18	C₂×#88322	—	2	4	no	—
19	C₂×#88322	C₈	1	8	yes	complemented
20	C₂³×C₅₃	C₄²	1	1	yes	characteristic
21	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
22	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
24	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₅₃	C₂×C₈	1	4	yes	—
26	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₅₃	M₁₆	1	4	yes	—
28	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
29	C₈×C₅₃	—	2	4	no	—
30	#88322	—	2	8	no	—
31	C₂²×C₅₃	#95	1	1	yes	characteristic
32	C₂²×C₅₃	C₄×C₈	1	2	yes	—
33	C₂²×C₅₃	C₄×C₈	1	1	yes	characteristic
34	C₂²×C₅₃	#97	1	2	yes	—
35	C₂²×C₅₃	#97	1	1	yes	characteristic
36	C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
37	C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
38	C₄×C₅₃	C₄⋊C₈	1	4	yes	—
39	C₄×C₅₃	C₄⋊C₈	1	2	yes	—
40	#1273	—	53	1	no	maximal, Sylow
41	C₂×C₅₃	C₈²	1	1	yes	characteristic
42	C₂×C₅₃	#322	1	1	yes	characteristic
43	C₂×C₅₃	#322	1	2	yes	—
44	C₂×C₅₃	#336	1	2	yes	—
45	C₂×C₅₃	#336	1	1	yes	characteristic
46	C₂×C₄×C₈	—	53	2	no	—
47	C₂×C₄×C₈	D₅₃	1	1	yes	characteristic
48	C₅₃	#1273	1	1	yes	characteristic, complemented, Sylow
49	C₂²×C₈	—	53	4	no	—
50	C₂²×C₈	Dic₅₃	1	2	yes	—
51	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
52	C₄×C₈	—	2	2	no	—
53	C₄×C₈	—	106	2	no	—
54	C₄×C₈	—	106	2	no	—
55	C₂²×C₄	C₄×D₅₃	1	2	yes	—
56	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
57	C₂×C₈	—	2	2	no	—
58	C₂×C₈	—	2	2	no	—
59	C₂×C₈	—	53	8	no	—
60	C₂×C₈	—	106	4	no	—
61	C₂×C₈	—	106	4	no	—
62	C₂×C₈	#88322	1	4	yes	complemented
63	C₄²	Dic₁₀₆	1	1	yes	characteristic
64	C₄²	D₂₁₂	1	1	yes	characteristic
65	C₄²	C₂²⋊D₅₃	1	2	yes	—
66	C₂³	C₄×Dic₅₃	1	1	yes	characteristic
67	C₂×C₄	C₈×D₅₃	1	4	yes	—
68	C₂×C₄	#11719912	1	4	yes	—
69	C₂×C₄	C₂×#88322	1	2	yes	—
70	C₂×C₄	#11719917	1	2	yes	—
71	C₂×C₄	#11719919	1	2	yes	—
72	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
73	C₂×C₄	#11719921	1	2	yes	—
74	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
75	C₈	—	2	4	no	—
76	C₈	—	106	8	no	—
77	C₂²	#13133124	1	1	yes	characteristic
78	C₂²	C₈×Dic₅₃	1	2	yes	—
79	C₂²	C₄×#88322	1	1	yes	characteristic
80	C₂²	#13133127	1	2	yes	—
81	C₂²	#13133128	1	1	yes	characteristic
82	C₄	#13133129	1	4	yes	—
83	C₄	C₂²⋊#88322	1	2	yes	—
84	C₄	#13133146	1	4	yes	—
85	C₄	C₄⋊#88322	1	2	yes	—
86	C₂	C₈×#88322	1	1	yes	characteristic
87	C₂	#16810344	1	1	yes	characteristic
88	C₂	#16810345	1	2	yes	—
89	C₂	#16810377	1	2	yes	—
90	C₂	#16810378	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂×D₅₃
commutator	C₂×C₅₃	C₈²
Fitting	C₂×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₄²
hypercenter	C₂×C₄×C₈	D₅₃
hypocenter	C₅₃	#1273
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
13. C₂×C₈×C₅₃	C₈	semidirect
19. C₂×#88322	C₈	semidirect
48. C₅₃	#1273	semidirect
62. C₂×C₈	#88322	semidirect