Subgroups — #20274886

Order 6784.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133152	C₂	1	4	yes	maximal
2	#16810647	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#88322	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#423	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719917	C₂²	1	2	yes	—
6	C₂²×#88322	C₂²	1	2	yes	—
7	#13133152	—	2	8	no	—
8	C₄⋊#88322	—	2	2	no	—
9	C₄×#88322	C₄	1	2	yes	—
10	C₄×#88322	C₄	1	2	yes	—
11	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₅₃	—	2	2	no	—
14	C₂²×C₄×C₅₃	Q₈	1	2	yes	—
15	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
16	M₁₆×C₅₃	—	2	4	no	—
17	C₂×C₈×C₅₃	—	2	2	no	—
18	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₅₃	D₄	1	2	yes	—
21	#11719917	—	2	4	no	—
22	C₂×#88322	—	2	2	no	—
23	C₂×#88322	C₂×C₄	1	4	yes	—
24	C₂×#88322	D₄	1	4	yes	—
25	C₂×#88322	Q₈	1	4	yes	—
26	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
27	C₂×C₄×C₅₃	—	2	4	no	—
28	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
32	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
34	C₈×C₅₃	—	4	2	no	—
35	#88322	—	4	2	no	—
36	#88322	C₄⋊C₄	1	8	yes	—
37	C₂²×C₅₃	—	2	2	no	—
38	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₅₃	C₄⋊Q₈	1	2	yes	—
40	C₄×C₅₃	—	2	2	no	—
41	C₄×C₅₃	#108	1	4	yes	complemented
42	C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
43	C₄×C₅₃	#127	1	1	yes	characteristic
44	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
45	#1848	—	53	1	no	maximal, Sylow
46	C₂×C₅₃	—	2	2	no	—
47	C₂×C₅₃	#384	1	1	yes	characteristic
48	C₂×C₅₃	C₂×#108	1	2	yes	—
49	C₂×#108	—	53	4	no	—
50	#423	—	53	1	no	—
51	#423	D₅₃	1	1	yes	characteristic
52	C₂×C₄×C₈	—	53	1	no	—
53	C₅₃	#1848	1	1	yes	characteristic, complemented, Sylow
54	C₂×M₁₆	—	53	2	no	—
55	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
56	C₂²×C₈	—	53	2	no	—
57	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
58	#108	—	106	8	no	—
59	C₄⋊C₈	—	2	2	no	—
60	C₄⋊C₈	—	106	2	no	—
61	C₄×C₈	—	53	2	no	—
62	C₄×C₈	—	53	2	no	—
63	C₂²×C₄	Dic₁₀₆	1	2	yes	—
64	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
65	M₁₆	—	2	4	no	—
66	M₁₆	—	106	4	no	—
67	C₂×C₈	—	2	2	no	—
68	C₂×C₈	—	53	4	no	—
69	C₂×C₈	—	53	4	no	—
70	C₂×C₈	—	53	4	no	—
71	C₂×C₈	—	106	2	no	—
72	C₄²	C₄×D₅₃	1	1	yes	characteristic
73	C₄²	C₄×D₅₃	1	1	yes	characteristic
74	C₄²	C₂²⋊D₅₃	1	2	yes	—
75	C₂³	C₂×Dic₁₀₆	1	1	yes	characteristic
76	C₂×C₄	—	2	4	no	—
77	C₂×C₄	#11719919	1	2	yes	—
78	C₂×C₄	#11719919	1	2	yes	—
79	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
80	C₂×C₄	D₄×D₅₃	1	2	yes	—
81	C₂×C₄	Q₈×D₅₃	1	2	yes	—
82	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
83	C₈	—	4	2	no	—
84	C₈	—	53	8	no	—
85	C₈	—	212	2	no	—
86	C₂²	—	2	2	no	—
87	C₂²	C₂×#11719919	1	1	yes	characteristic
88	C₂²	#13133216	1	2	yes	—
89	C₄	—	2	2	no	—
90	C₄	#13133152	1	4	yes	complemented
91	C₄	C₄⋊C₄×D₅₃	1	2	yes	—
92	C₄	#13133213	1	1	yes	characteristic
93	C₄	#13133215	1	1	yes	characteristic
94	C₂	—	2	2	no	—
95	C₂	#16810492	1	1	yes	characteristic
96	C₂	C₂×#13133152	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#423	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	#423	D₅₃
hypocenter	C₅₃	#1848
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
41. C₄×C₅₃	#108	semidirect
53. C₅₃	#1848	semidirect
90. C₄	#13133152	semidirect