Subgroups — #20289672

Order 6784.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811509	C₂	1	1	yes	maximal, characteristic
2	#16811486	C₂	1	1	yes	maximal, characteristic
3	#16810717	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719910	—	2	1	no	—
5	C₄⋊#88324	—	2	1	no	—
6	C₂×Dic₂₁₂	C₄	1	1	yes	characteristic
7	#13133182	—	2	1	no	—
8	#13133140	—	2	1	no	—
9	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
10	#107×C₅₃	C₄	1	1	yes	characteristic
11	C₂×Dic₁₀₆	—	2	1	no	—
12	C₂×#88324	—	4	1	no	—
13	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₄⋊C₄×C₅₃	—	2	1	no	—
15	#11719919	—	4	1	no	—
16	C₄×Dic₅₃	—	4	1	no	—
17	C₄×Dic₅₃	D₄	1	1	yes	characteristic
18	C₂×#88322	D₄	1	1	yes	characteristic
19	Dic₂₁₂	—	2	1	no	—
20	Dic₂₁₂	C₈	1	2	yes	complemented
21	#11719910	—	2	2	no	—
22	C₂×C₄×C₅₃	—	4	1	no	—
23	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×Dic₅₃	—	4	1	no	—
25	C₂×Dic₅₃	D₈	1	1	yes	characteristic
26	C₂×Dic₅₃	SD₁₆	1	1	yes	characteristic
27	Dic₁₀₆	—	2	2	no	—
28	Dic₁₀₆	—	4	1	no	—
29	#88324	—	8	1	no	—
30	C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
31	C₈×C₅₃	M₁₆	1	1	yes	characteristic
32	#88322	—	2	1	no	—
33	C₂²×C₅₃	#102	1	1	yes	characteristic
34	C₄×C₅₃	—	8	1	no	—
35	C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
36	C₄×C₅₃	#104	1	1	yes	characteristic
37	Dic₅₃	—	2	1	no	—
38	Dic₅₃	—	4	2	no	—
39	Dic₅₃	SD₃₂	1	1	yes	characteristic
40	Dic₅₃	Q₃₂	1	1	yes	characteristic
41	#1332	—	53	1	no	maximal, Sylow
42	C₂×C₅₃	D₄⋊C₈	1	1	yes	characteristic
43	C₂×C₅₃	#358	1	1	yes	characteristic
44	C₂×C₅₃	#359	1	1	yes	characteristic
45	C₄×Q₁₆	—	53	1	no	—
46	C₄×C₁₆	—	53	1	no	—
47	#335	—	53	1	no	—
48	C₅₃	#1332	1	1	yes	characteristic, complemented, Sylow
49	C₂×Q₁₆	—	53	1	no	—
50	C₄×Q₈	—	106	1	no	—
51	C₂×C₁₆	—	106	1	no	—
52	#107	#6157	1	1	yes	characteristic
53	C₄⋊C₈	—	106	1	no	—
54	#103	—	106	1	no	—
55	C₄×C₈	—	53	1	no	—
56	C₂×Q₈	—	106	1	no	—
57	Q₁₆	—	53	2	no	—
58	Q₁₆	—	106	1	no	—
59	C₂×C₈	—	53	1	no	—
60	C₂×C₈	—	212	1	no	—
61	C₂×C₈	C₂×#6157	1	1	yes	characteristic
62	C₄⋊C₄	—	2	1	no	—
63	C₄⋊C₄	—	212	1	no	—
64	C₄²	—	53	1	no	—
65	C₄²	—	212	1	no	—
66	C₁₆	—	106	2	no	—
67	Q₈	—	106	2	no	—
68	Q₈	—	212	1	no	—
69	C₂×C₄	—	4	1	no	—
70	C₂×C₄	—	53	1	no	—
71	C₂×C₄	—	53	1	no	—
72	C₂×C₄	—	212	1	no	—
73	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
74	C₈	—	106	1	no	—
75	C₈	—	424	1	no	—
76	C₈	D₅₃⋊C₈	1	1	yes	characteristic
77	C₈	#11719936	1	1	yes	characteristic
78	C₂²	#13133279	1	1	yes	characteristic
79	C₄	—	8	1	no	—
80	C₄	—	53	1	no	—
81	C₄	—	53	1	no	—
82	C₄	—	106	1	no	—
83	C₄	—	212	2	no	—
84	C₄	#13133274	1	1	yes	characteristic
85	C₄	#13133282	1	1	yes	characteristic
86	C₂	#16811471	1	1	yes	characteristic
87	C₂	#16811536	1	1	yes	characteristic
88	C₂	#16811537	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13133279
commutator	C₈×C₅₃	C₂×C₈
Frattini	C₂×C₈	C₂×#6157
Fitting	#107×C₅₃	C₄
socle	C₂²×C₅₃	#102
hypercenter	#107	#6157
hypocenter	C₅₃	#1332
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #107

Complements

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

N	H ≅ G/N	type
20. Dic₂₁₂	C₈	semidirect
48. C₅₃	#1332	semidirect