Subgroups — #20274461

Order 6784.

296 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	#16810378	C₂	1	4	yes	maximal
3	C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#88322	C₂²	1	4	yes	—
5	C₄×#88322	C₄	1	8	yes	complemented
6	C₂×C₄²×C₅₃	C₄	1	2	yes	—
7	C₂×C₄²×C₅₃	C₄	1	2	yes	—
8	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
9	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₅₃	C₂×C₄	1	4	yes	—
11	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
14	C₄²×C₅₃	—	2	4	no	—
15	C₄²×C₅₃	—	2	4	no	—
16	C₄²×C₅₃	C₈	1	8	yes	complemented
17	C₄²×C₅₃	C₂×C₄	1	4	yes	—
18	C₂×#88322	—	2	8	no	—
19	C₂×#88322	C₂×C₄	1	8	yes	—
20	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
21	C₂×C₄×C₅₃	—	2	4	no	—
22	C₂×C₄×C₅₃	—	2	4	no	—
23	C₂×C₄×C₅₃	C₄²	1	2	yes	—
24	C₂×C₄×C₅₃	C₄²	1	2	yes	—
25	C₂×C₄×C₅₃	C₂×C₈	1	8	yes	—
26	C₂×C₄×C₅₃	C₂×C₈	1	4	yes	—
27	C₂×C₄×C₅₃	M₁₆	1	4	yes	—
28	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
29	C₂×C₄×C₅₃	#41	1	4	yes	—
30	#88322	—	2	8	no	—
31	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
32	C₂²×C₅₃	#117	1	2	yes	—
33	C₂²×C₅₃	C₂²×C₈	1	2	yes	—
34	C₂²×C₅₃	C₂×M₁₆	1	2	yes	—
35	C₄×C₅₃	—	2	8	no	—
36	C₄×C₅₃	C₄×C₈	1	4	yes	—
37	C₄×C₅₃	#97	1	4	yes	—
38	C₄×C₅₃	#117	1	4	yes	—
39	#1744	—	53	1	no	maximal, Sylow
40	C₂×C₅₃	#376	1	1	yes	characteristic
41	C₂×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
42	C₂×C₅₃	C₂×#97	1	1	yes	characteristic
43	C₂×C₅₃	#431	1	4	yes	—
44	C₂×C₄×C₈	—	53	2	no	—
45	C₄³	D₅₃	1	1	yes	characteristic
46	#336	—	53	4	no	—
47	C₅₃	#1744	1	1	yes	characteristic, complemented, Sylow
48	C₂²×C₈	—	53	4	no	—
49	C₂×C₄²	Dic₅₃	1	2	yes	—
50	C₂×C₄²	Dic₅₃	1	2	yes	—
51	C₂×C₄²	C₂×D₅₃	1	2	yes	—
52	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
53	C₄×C₈	—	53	8	no	—
54	C₂²×C₄	C₂×Dic₅₃	1	4	yes	—
55	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
56	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
57	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
58	C₂×C₈	—	53	8	no	—
59	C₂×C₈	—	106	8	no	—
60	C₄²	—	2	4	no	—
61	C₄²	—	2	4	no	—
62	C₄²	#88322	1	8	yes	complemented
63	C₄²	C₄×D₅₃	1	4	yes	—
64	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	C₂×#88322	1	8	yes	—
68	C₂×C₄	C₂×#88322	1	4	yes	—
69	C₂×C₄	#11719917	1	4	yes	—
70	C₂×C₄	C₄×Dic₅₃	1	2	yes	—
71	C₂×C₄	C₄×Dic₅₃	1	2	yes	—
72	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
73	C₂×C₄	#11719945	1	4	yes	—
74	C₈	—	106	8	no	—
75	C₂²	C₂×C₄×Dic₅₃	1	1	yes	characteristic
76	C₂²	#13133176	1	2	yes	—
77	C₂²	C₂²×#88322	1	2	yes	—
78	C₂²	C₂×#11719917	1	2	yes	—
79	C₄	—	2	8	no	—
80	C₄	C₄×#88322	1	4	yes	—
81	C₄	#13133128	1	4	yes	—
82	C₄	#13133174	1	4	yes	—
83	C₂	#16810469	1	1	yes	characteristic
84	C₂	C₂×C₄×#88322	1	1	yes	characteristic
85	C₂	C₂×#13133128	1	1	yes	characteristic
86	C₂	#16810676	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂×D₅₃
commutator	C₂×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₄³×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	C₄³	D₅₃
hypocenter	C₅₃	#1744
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄³

Complements

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

N	H ≅ G/N	type
5. C₄×#88322	C₄	semidirect
16. C₄²×C₅₃	C₈	semidirect
47. C₅₃	#1744	semidirect
62. C₄²	#88322	semidirect