Subgroups — #20273012

Order 6784.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810676	C₂	1	2	yes	maximal
2	#16810345	C₂	1	2	yes	maximal, complemented
3	#16810344	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#88322	C₄	1	4	yes	—
7	C₂²⋊#88322	C₄	1	4	yes	—
8	C₄×#88322	C₂²	1	2	yes	—
9	C₄×#88322	C₂²	1	2	yes	—
10	C₂×M₁₆×C₅₃	C₄	1	2	yes	—
11	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	#97×C₅₃	C₄	1	2	yes	—
13	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
16	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
17	M₁₆×C₅₃	—	2	4	no	—
18	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
20	C₄²×C₅₃	C₂×C₄	1	2	yes	—
21	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₅₃	C₂³	1	1	yes	characteristic
23	C₂×#88322	C₂×C₄	1	4	yes	—
24	C₂×#88322	C₂×C₄	1	4	yes	—
25	C₂³×C₅₃	C₄²	1	1	yes	characteristic
26	C₂×C₄×C₅₃	—	2	2	no	—
27	C₂×C₄×C₅₃	—	2	4	no	—
28	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
29	C₂×C₄×C₅₃	C₄²	1	2	yes	—
30	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
31	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₈×C₅₃	—	2	2	no	—
33	C₈×C₅₃	M₁₆	1	4	yes	—
34	#88322	—	2	4	no	—
35	#88322	—	2	4	no	—
36	C₂²×C₅₃	—	2	1	no	—
37	C₂²×C₅₃	—	2	2	no	—
38	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
39	C₄×C₅₃	—	2	1	no	—
40	C₄×C₅₃	—	2	2	no	—
41	C₄×C₅₃	C₂×M₁₆	1	2	yes	—
42	C₄×C₅₃	#131	1	4	yes	—
43	#1453	—	53	1	no	maximal, Sylow
44	C₂×C₅₃	—	4	1	no	—
45	C₂×C₅₃	C₄×M₁₆	1	1	yes	characteristic
46	C₂×C₅₃	#405	1	2	yes	—
47	#431	—	53	2	no	—
48	C₄×M₁₆	D₅₃	1	1	yes	characteristic
49	#322	—	53	1	no	—
50	#322	—	53	2	no	—
51	C₈²	—	53	1	no	—
52	C₅₃	#1453	1	1	yes	characteristic, complemented, Sylow
53	C₂×M₁₆	Dic₅₃	1	2	yes	—
54	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
55	C₄⋊C₈	—	53	4	no	—
56	C₂²⋊C₈	—	53	4	no	—
57	#97	Dic₅₃	1	2	yes	—
58	C₄×C₈	—	53	2	no	—
59	C₄×C₈	—	53	2	no	—
60	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
61	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
62	C₂²×C₄	C₄×D₅₃	1	2	yes	—
63	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
64	M₁₆	—	2	4	no	—
65	C₂×C₈	—	53	4	no	—
66	C₂×C₈	—	53	4	no	—
67	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
68	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
69	C₄²	C₄×D₅₃	1	2	yes	—
70	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
71	C₄²	C₂²×D₅₃	1	1	yes	characteristic
72	C₂³	C₄×Dic₅₃	1	1	yes	characteristic
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
76	C₂×C₄	C₄×Dic₅₃	1	2	yes	—
77	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
78	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₈	—	106	4	no	—
81	C₈	—	106	4	no	—
82	C₈	#11719917	1	4	yes	—
83	C₂²	—	2	1	no	—
84	C₂²	—	2	2	no	—
85	C₂²	C₂×C₄×Dic₅₃	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	2	2	no	—
88	C₄	C₂×#11719917	1	2	yes	—
89	C₄	#13133225	1	4	yes	—
90	C₂	—	4	1	no	—
91	C₂	C₄×#11719917	1	1	yes	characteristic
92	C₂	#16810581	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂×C₄²
Fitting	C₄×M₁₆×C₅₃	C₂
hypercenter	C₄×M₁₆	D₅₃
hypocenter	C₅₃	#1453
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #16810345	C₂	semidirect
3. #16810344	C₂	semidirect
4. C₈×#88322	C₂	semidirect
52. C₅₃	#1453	semidirect