Subgroups — #20274468

Order 6784.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#88322	C₂	1	1	yes	maximal, characteristic
2	#16810469	C₂	1	1	yes	maximal, characteristic
3	#16810377	C₂	1	4	yes	maximal
4	C₂×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#88322	C₂²	1	2	yes	—
6	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
7	C₄×#88322	C₄	1	4	yes	complemented
8	#13133124	C₄	1	4	yes	—
9	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
10	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	#97×C₅₃	C₄	1	4	yes	complemented
12	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
13	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
14	C₂²×Dic₅₃	C₂×C₄	1	4	yes	—
15	C₂×C₈×C₅₃	—	2	4	no	—
16	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
17	C₄²×C₅₃	C₂×C₄	1	2	yes	—
18	C₄²×C₅₃	C₂×C₄	1	2	yes	—
19	C₄×Dic₅₃	—	2	4	no	—
20	C₂×#88322	—	2	4	no	—
21	C₂×#88322	C₂×C₄	1	4	yes	—
22	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
23	C₂×C₄×C₅₃	—	2	4	no	—
24	C₂×C₄×C₅₃	C₄²	1	4	yes	—
25	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₅₃	#41	1	4	yes	—
28	C₂×Dic₅₃	—	2	4	no	—
29	C₂×Dic₅₃	—	2	8	no	—
30	C₈×C₅₃	—	2	4	no	—
31	#88322	—	2	4	no	—
32	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
33	C₂²×C₅₃	#117	1	2	yes	—
34	C₂²×C₅₃	#131	1	4	yes	—
35	C₄×C₅₃	—	2	4	no	—
36	C₄×C₅₃	#117	1	2	yes	—
37	C₄×C₅₃	#117	1	2	yes	—
38	Dic₅₃	—	4	4	no	—
39	#1746	—	53	1	no	maximal, Sylow
40	C₂×C₅₃	#376	1	1	yes	characteristic
41	C₂×C₅₃	#405	1	2	yes	—
42	C₂×C₅₃	#433	1	4	yes	—
43	C₂×#97	D₅₃	1	1	yes	characteristic
44	C₂×C₄×C₈	—	53	1	no	—
45	#376	—	53	1	no	—
46	#336	—	53	4	no	—
47	C₅₃	#1746	1	1	yes	characteristic, complemented, Sylow
48	C₂²×C₈	—	53	2	no	—
49	C₂²×C₈	C₂×D₅₃	1	2	yes	—
50	C₂×C₄²	—	53	2	no	—
51	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
52	#97	Dic₅₃	1	4	yes	complemented
53	C₄×C₈	—	53	4	no	—
54	#95	—	53	4	no	—
55	C₂²×C₄	—	53	4	no	—
56	C₂²×C₄	C₄×D₅₃	1	2	yes	—
57	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
58	C₂×C₈	—	2	4	no	—
59	C₂×C₈	—	53	4	no	—
60	C₂×C₈	—	106	4	no	—
61	C₂×C₈	C₂×Dic₅₃	1	4	yes	—
62	C₄²	—	106	4	no	—
63	C₄²	C₄×D₅₃	1	2	yes	—
64	C₄²	C₂×Dic₅₃	1	2	yes	—
65	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	—	106	4	no	—
68	C₂×C₄	—	106	8	no	—
69	C₂×C₄	C₄×Dic₅₃	1	4	yes	—
70	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
71	C₂×C₄	#11719945	1	4	yes	—
72	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
73	C₈	—	2	4	no	—
74	C₈	—	106	4	no	—
75	C₂²	C₂×C₄×Dic₅₃	1	1	yes	characteristic
76	C₂²	#13133174	1	2	yes	—
77	C₂²	#13133225	1	4	yes	—
78	C₄	—	2	4	no	—
79	C₄	—	212	4	no	—
80	C₄	#13133174	1	2	yes	—
81	C₄	#13133176	1	2	yes	—
82	C₂	#16810469	1	1	yes	characteristic
83	C₂	#16810581	1	2	yes	—
84	C₂	#16810685	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄²
Fitting	C₂×#97×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	C₂×#97	D₅₃
hypocenter	C₅₃	#1746
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
7. C₄×#88322	C₄	semidirect
11. #97×C₅₃	C₄	semidirect
47. C₅₃	#1746	semidirect
52. #97	Dic₅₃	semidirect