Subgroups — #20289688

Order 6784.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811520	C₂	1	1	yes	maximal, characteristic
2	#16811491	C₂	1	1	yes	maximal, characteristic
3	#16810882	C₂	1	1	yes	maximal, characteristic
4	#13133296	—	2	1	no	—
5	#13133291	—	2	1	no	—
6	C₂×Dic₂₁₂	C₄	1	1	yes	characteristic, complemented
7	C₂×#11719912	C₂²	1	1	yes	characteristic
8	#13133197	—	2	1	no	—
9	#13133140	—	2	1	no	—
10	#106×C₅₃	C₄	1	1	yes	characteristic, complemented
11	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
12	C₂×Dic₁₀₆	—	2	1	no	—
13	C₂²⋊#6157	—	4	1	no	—
14	C₄⋊#6157	—	4	1	no	—
15	C₄×#6157	—	2	2	no	—
16	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₄⋊C₄×C₅₃	—	2	1	no	—
18	#11719921	—	4	1	no	—
19	C₄⋊Dic₅₃	—	4	1	no	—
20	C₂×#88322	D₄	1	1	yes	characteristic
21	Dic₂₁₂	—	4	1	no	—
22	#11719912	—	4	1	no	—
23	#11719910	—	4	1	no	—
24	C₂²×D₅₃	D₈	1	1	yes	characteristic
25	C₂×#6157	—	4	2	no	—
26	C₂×C₄×C₅₃	—	4	1	no	—
27	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×Dic₅₃	—	4	1	no	—
29	C₂×Dic₅₃	SD₁₆	1	1	yes	characteristic
30	C₄×D₅₃	—	2	1	no	—
31	C₄×D₅₃	—	2	1	no	—
32	Dic₁₀₆	—	4	1	no	—
33	Dic₁₀₆	—	4	1	no	—
34	C₈×C₅₃	—	2	1	no	—
35	#88322	—	2	1	no	—
36	C₂²×C₅₃	#102	1	1	yes	characteristic
37	C₂×D₅₃	—	2	1	no	—
38	C₂×D₅₃	—	4	1	no	—
39	#6157	—	8	2	no	—
40	C₄×C₅₃	—	8	1	no	—
41	C₄×C₅₃	#99	1	1	yes	characteristic
42	C₄×C₅₃	#104	1	1	yes	characteristic
43	Dic₅₃	—	2	1	no	—
44	Dic₅₃	—	8	1	no	—
45	#1340	—	53	1	no	maximal, Sylow
46	C₂×C₅₃	—	2	1	no	—
47	C₂×C₅₃	#327	1	1	yes	characteristic
48	D₅₃	—	4	1	no	—
49	#470	—	53	1	no	—
50	#349	—	53	1	no	—
51	H(3,ℤ/4)	—	53	1	no	—
52	C₅₃	#1340	1	1	yes	characteristic, complemented, Sylow
53	C₂×Q₁₆	—	53	1	no	—
54	C₂×M₁₆	—	53	1	no	—
55	C₂²⋊Q₈	—	106	1	no	—
56	#117	—	106	1	no	—
57	M₃₂	—	106	1	no	—
58	#106	#6157	1	1	yes	characteristic, complemented
59	#103	—	106	1	no	—
60	C₂×Q₈	—	106	1	no	—
61	C₂²×C₄	—	53	1	no	—
62	Q₁₆	—	212	1	no	—
63	M₁₆	—	212	1	no	—
64	C₂×C₈	—	53	1	no	—
65	C₂×C₈	C₂×#6157	1	1	yes	characteristic
66	C₄⋊C₄	—	2	1	no	—
67	C₄⋊C₄	—	212	1	no	—
68	C₄⋊C₄	—	212	1	no	—
69	C₂²⋊C₄	—	212	1	no	—
70	C₂²⋊C₄	—	212	1	no	—
71	C₄²	—	106	2	no	—
72	C₁₆	—	212	1	no	—
73	C₂³	—	53	1	no	—
74	Q₈	—	212	1	no	—
75	Q₈	—	212	1	no	—
76	C₂×C₄	—	4	1	no	—
77	C₂×C₄	—	53	1	no	—
78	C₂×C₄	—	106	1	no	—
79	C₂×C₄	—	106	1	no	—
80	C₂×C₄	—	212	2	no	—
81	C₂×C₄	—	212	1	no	—
82	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
83	C₈	—	2	1	no	—
84	C₈	—	106	1	no	—
85	C₂²	—	106	1	no	—
86	C₂²	—	212	1	no	—
87	C₂²	#13133279	1	1	yes	characteristic
88	C₄	—	8	1	no	—
89	C₄	—	106	1	no	—
90	C₄	—	424	2	no	—
91	C₄	—	424	1	no	—
92	C₄	#13133276	1	1	yes	characteristic
93	C₄	#13133282	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	212	1	no	—
96	C₂	#16811475	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16811475
commutator	C₂×C₈×C₅₃	C₂×C₄
Frattini	C₂×C₈	C₂×#6157
Fitting	#106×C₅₃	C₄
socle	C₂×C₅₃	#327
hypercenter	#106	#6157
hypocenter	C₅₃	#1340
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #106

Complements

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

N	H ≅ G/N	type
6. C₂×Dic₂₁₂	C₄	semidirect
10. #106×C₅₃	C₄	semidirect
52. C₅₃	#1340	semidirect
58. #106	#6157	semidirect