Subgroups — #20272548

Order 6784.

208 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810676	C₂	1	2	yes	maximal
2	C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#88322	C₄	1	4	yes	complemented
4	C₂²⋊#88322	—	2	2	no	—
5	C₄×#88322	—	2	2	no	—
6	C₂×C₄²×C₅₃	—	2	2	no	—
7	C₂×C₄²×C₅₃	C₄	1	2	yes	—
8	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₅₃	—	2	2	no	—
10	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
11	C₂²×C₄×C₅₃	D₄	1	2	yes	—
12	C₄²×C₅₃	—	2	8	no	—
13	C₄²×C₅₃	—	2	2	no	—
14	C₄²×C₅₃	C₈	1	4	yes	complemented
15	C₄²×C₅₃	C₂×C₄	1	2	yes	—
16	C₄²×C₅₃	D₄	1	1	yes	characteristic
17	C₄²×C₅₃	Q₈	1	1	yes	characteristic
18	C₂×#88322	—	2	4	no	—
19	C₂³×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
20	C₂×C₄×C₅₃	—	2	8	no	—
21	C₂×C₄×C₅₃	—	2	4	no	—
22	C₂×C₄×C₅₃	—	2	4	no	—
23	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
24	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
25	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
26	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
28	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
29	#88322	—	4	4	no	—
30	C₂²×C₅₃	—	2	2	no	—
31	C₂²×C₅₃	#95	1	1	yes	characteristic
32	C₂²×C₅₃	C₂²⋊C₈	1	2	yes	—
33	C₄×C₅₃	—	2	8	no	—
34	C₄×C₅₃	—	2	2	no	—
35	C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
36	C₄×C₅₃	#97	1	1	yes	characteristic
37	C₄×C₅₃	#104	1	4	yes	—
38	C₄×C₅₃	C₄⋊C₈	1	2	yes	—
39	#1276	—	53	1	no	maximal, Sylow
40	C₂×C₅₃	—	2	2	no	—
41	C₂×C₅₃	#336	1	1	yes	characteristic
42	C₂×C₅₃	#339	1	2	yes	—
43	#431	—	53	2	no	—
44	C₄³	D₅₃	1	1	yes	characteristic
45	C₅₃	#1276	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₄²	—	2	2	no	—
47	C₂×C₄²	Dic₅₃	1	2	yes	—
48	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
49	C₄⋊C₈	—	53	4	no	—
50	C₂²⋊C₈	—	106	2	no	—
51	C₄×C₈	—	106	2	no	—
52	C₂²×C₄	—	2	2	no	—
53	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
54	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
55	C₂×C₈	—	106	4	no	—
56	C₄²	—	2	8	no	—
57	C₄²	—	2	2	no	—
58	C₄²	#88322	1	4	yes	complemented
59	C₄²	Dic₁₀₆	1	1	yes	characteristic
60	C₄²	C₄×D₅₃	1	2	yes	—
61	C₄²	D₂₁₂	1	1	yes	characteristic
62	C₂³	C₂²⋊Dic₅₃	1	1	yes	characteristic
63	C₂×C₄	—	2	8	no	—
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	C₂×#88322	1	2	yes	—
67	C₂×C₄	#11719917	1	2	yes	—
68	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
69	C₂×C₄	#11719919	1	2	yes	—
70	C₂×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
71	C₂×C₄	#11719921	1	2	yes	—
72	C₈	—	212	4	no	—
73	C₂²	—	2	2	no	—
74	C₂²	#13133124	1	1	yes	characteristic
75	C₂²	C₂²⋊#88322	1	2	yes	—
76	C₄	—	2	8	no	—
77	C₄	—	2	2	no	—
78	C₄	C₄×#88322	1	1	yes	characteristic
79	C₄	#13133128	1	1	yes	characteristic
80	C₄	#13133145	1	4	yes	—
81	C₄	C₄⋊#88322	1	2	yes	—
82	C₂	—	2	2	no	—
83	C₂	#16810378	1	1	yes	characteristic
84	C₂	#16810383	1	2	yes	—

	subgroup	G/H
center	C₄²	D₂₁₂
commutator	C₄×C₅₃	C₄×C₈
Frattini	C₂×C₄²	C₂×D₅₃
Fitting	C₄³×C₅₃	C₂
socle	C₂²×C₅₃	#95
hypercenter	C₄³	D₅₃
hypocenter	C₅₃	#1276
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄³

Complements

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

N	H ≅ G/N	type
3. C₄⋊#88322	C₄	semidirect
14. C₄²×C₅₃	C₈	semidirect
45. C₅₃	#1276	semidirect
58. C₄²	#88322	semidirect