Subgroups — #20289730

Order 6784.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811544	C₂	1	1	yes	maximal, characteristic
2	#16811487	C₂	1	1	yes	maximal, characteristic
3	#16811001	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	C₂×#11719925	—	2	1	no	—
8	#13133215	—	2	1	no	—
9	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
10	C₂×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
11	C₂×Q₈×C₅₃	—	2	1	no	—
12	C₂×Dic₁₀₆	—	2	1	no	—
13	C₂×#88324	—	4	1	no	—
14	Q₁₆×C₅₃	—	4	1	no	—
15	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	#11719925	—	4	1	no	—
17	#11719925	—	4	1	no	—
18	#11719919	—	4	1	no	—
19	#11719919	—	4	1	no	—
20	C₄×Dic₅₃	D₄	1	1	yes	characteristic
21	C₂×#88322	D₄	1	1	yes	characteristic
22	Dic₂₁₂	—	4	1	no	—
23	#11719910	—	4	1	no	—
24	Q₈×C₅₃	—	4	1	no	—
25	Q₈×C₅₃	—	4	1	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	D₈	1	1	yes	characteristic
30	C₂×Dic₅₃	SD₁₆	1	1	yes	characteristic
31	Dic₁₀₆	—	4	1	no	—
32	Dic₁₀₆	—	4	1	no	—
33	#88324	—	8	1	no	—
34	C₈×C₅₃	—	2	1	no	—
35	#88322	D₈	1	1	yes	characteristic
36	#88322	SD₁₆	1	1	yes	characteristic
37	C₂²×C₅₃	#102	1	1	yes	characteristic
38	C₄×C₅₃	—	8	1	no	—
39	C₄×C₅₃	#100	1	1	yes	characteristic
40	C₄×C₅₃	#102	1	1	yes	characteristic
41	Dic₅₃	—	2	1	no	—
42	Dic₅₃	—	8	1	no	—
43	Dic₅₃	SD₃₂	1	1	yes	characteristic
44	Dic₅₃	Q₃₂	1	1	yes	characteristic
45	#1362	—	53	1	no	maximal, Sylow
46	C₂×C₅₃	#331	1	1	yes	characteristic
47	C₂×C₅₃	#358	1	1	yes	characteristic
48	C₂×C₅₃	#362	1	1	yes	characteristic
49	#494	—	53	1	no	—
50	C₄⋊C₁₆	—	53	1	no	—
51	#335	—	53	1	no	—
52	C₅₃	#1362	1	1	yes	characteristic, complemented, Sylow
53	C₂×Q₁₆	—	53	1	no	—
54	C₂×Q₁₆	—	106	1	no	—
55	C₂×Q₁₆	#6157	1	1	yes	characteristic
56	C₄⋊Q₈	—	106	1	no	—
57	C₂×C₁₆	—	106	1	no	—
58	C₄⋊C₈	—	106	1	no	—
59	C₄×C₈	—	53	1	no	—
60	C₂×Q₈	—	2	1	no	—
61	C₂×Q₈	—	106	1	no	—
62	Q₁₆	—	4	1	no	—
63	Q₁₆	—	212	1	no	—
64	Q₁₆	—	212	1	no	—
65	Q₁₆	—	212	1	no	—
66	C₂×C₈	—	53	1	no	—
67	C₂×C₈	—	212	1	no	—
68	C₂×C₈	C₂×#6157	1	1	yes	characteristic
69	C₄⋊C₄	—	212	1	no	—
70	C₄⋊C₄	—	212	1	no	—
71	C₄²	—	53	1	no	—
72	C₁₆	—	212	1	no	—
73	Q₈	—	4	1	no	—
74	Q₈	—	4	1	no	—
75	Q₈	—	212	1	no	—
76	Q₈	—	212	1	no	—
77	C₂×C₄	—	4	1	no	—
78	C₂×C₄	—	53	1	no	—
79	C₂×C₄	—	53	1	no	—
80	C₂×C₄	—	212	1	no	—
81	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
82	C₈	—	2	1	no	—
83	C₈	—	53	1	no	—
84	C₈	—	53	1	no	—
85	C₈	—	424	1	no	—
86	C₂²	#13133279	1	1	yes	characteristic
87	C₄	—	8	1	no	—
88	C₄	—	53	1	no	—
89	C₄	—	53	1	no	—
90	C₄	—	106	1	no	—
91	C₄	—	424	1	no	—
92	C₄	#13133277	1	1	yes	characteristic
93	C₄	#13133279	1	1	yes	characteristic
94	C₂	#16811481	1	1	yes	characteristic
95	C₂	#16811536	1	1	yes	characteristic
96	C₂	#16811542	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13133279
commutator	C₂×C₈×C₅₃	C₂×C₄
Frattini	C₂×C₈	C₂×#6157
Fitting	C₂×Q₁₆×C₅₃	C₄
socle	C₂²×C₅₃	#102
hypercenter	C₂×Q₁₆	#6157
hypocenter	C₅₃	#1362
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
52. C₅₃	#1362	semidirect