Subgroups — #20276296

Order 6784.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810672	C₂	1	1	yes	maximal, characteristic
2	#16810651	C₂	1	1	yes	maximal, characteristic
3	#16810395	C₂	1	2	yes	maximal
4	#16810387	C₂	1	2	yes	maximal
5	C₅₃×#338	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₂²	1	2	yes	—
7	C₂²×#88322	C₂²	1	1	yes	characteristic
8	#13133176	C₂²	1	1	yes	characteristic
9	#13133153	—	2	2	no	—
10	#13133146	—	2	2	no	—
11	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
12	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
14	M₁₆×C₅₃	—	2	2	no	—
15	M₁₆×C₅₃	—	2	2	no	—
16	M₁₆×C₅₃	—	2	2	no	—
17	C₂×C₈×C₅₃	—	2	2	no	—
18	C₂×C₈×C₅₃	D₄	1	2	yes	—
19	C₂²⋊Dic₅₃	Q₈	1	2	yes	—
20	C₄⋊Dic₅₃	—	2	1	no	—
21	C₄×Dic₅₃	—	2	1	no	—
22	#11719917	—	2	2	no	—
23	#11719917	—	2	2	no	—
24	C₂×#88322	—	2	1	no	—
25	C₂×#88322	—	2	2	no	—
26	C₂×#88322	—	2	2	no	—
27	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₅₃	#41	1	2	yes	—
30	C₂×C₄×C₅₃	#41	1	2	yes	—
31	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
32	C₂×Dic₅₃	—	2	2	no	—
33	C₈×C₅₃	—	2	2	no	—
34	C₈×C₅₃	—	4	2	no	—
35	#88322	—	4	1	no	—
36	#88322	—	4	2	no	—
37	C₂²×C₅₃	—	4	1	no	—
38	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
39	C₂²×C₅₃	#125	1	2	yes	—
40	C₄×C₅₃	#123	1	1	yes	characteristic
41	C₄×C₅₃	#124	1	1	yes	characteristic
42	C₄×C₅₃	#126	1	2	yes	—
43	Dic₅₃	—	4	2	no	—
44	#2103	—	53	1	no	maximal, Sylow
45	C₂×C₅₃	—	2	1	no	—
46	C₂×C₅₃	—	2	2	no	—
47	C₂×C₅₃	#400	1	1	yes	characteristic
48	#430	—	53	1	no	—
49	#424	—	53	1	no	—
50	#344	—	53	2	no	—
51	#341	—	53	2	no	—
52	#338	D₅₃	1	1	yes	characteristic
53	C₅₃	#2103	1	1	yes	characteristic, complemented, Sylow
54	C₂×M₁₆	—	53	2	no	—
55	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
56	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
57	C₂²×C₈	—	53	1	no	—
58	#117	—	53	1	no	—
59	#108	—	106	2	no	—
60	C₄⋊C₈	—	106	2	no	—
61	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
62	M₁₆	—	2	2	no	—
63	M₁₆	—	2	2	no	—
64	M₁₆	—	2	2	no	—
65	M₁₆	—	106	2	no	—
66	M₁₆	—	106	2	no	—
67	C₂×C₈	—	2	2	no	—
68	C₂×C₈	—	106	1	no	—
69	C₂×C₈	—	106	2	no	—
70	C₂×C₈	—	106	2	no	—
71	C₂×C₈	D₂₁₂	1	2	yes	—
72	C₄⋊C₄	—	106	1	no	—
73	C₂²⋊C₄	—	53	2	no	—
74	C₄²	—	106	1	no	—
75	C₂³	Q₈×D₅₃	1	1	yes	characteristic
76	C₂×C₄	—	106	2	no	—
77	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
78	C₂×C₄	#11719945	1	2	yes	—
79	C₂×C₄	#11719947	1	2	yes	—
80	C₂×C₄	#11719947	1	1	yes	characteristic
81	C₈	—	2	2	no	—
82	C₈	—	4	2	no	—
83	C₈	—	212	1	no	—
84	C₈	—	212	2	no	—
85	C₂²	—	4	1	no	—
86	C₂²	#13133197	1	1	yes	characteristic
87	C₂²	#13133207	1	2	yes	—
88	C₄	—	212	2	no	—
89	C₄	#13133201	1	1	yes	characteristic
90	C₄	#13133205	1	1	yes	characteristic
91	C₄	#13133210	1	2	yes	—
92	C₂	—	2	1	no	—
93	C₂	—	2	2	no	—
94	C₂	#16810563	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#13133205
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#338	C₂
socle	C₂×C₅₃	#400
hypercenter	#338	D₅₃
hypocenter	C₅₃	#2103
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
53. C₅₃	#2103	semidirect