Subgroups — #20272871

Order 6784.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811013	C₂	1	1	yes	maximal, characteristic
2	#16810420	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#354	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11719924	—	2	1	no	—
5	C₂×#11719922	—	2	1	no	—
6	#13133214	C₄	1	1	yes	characteristic, complemented
7	#13133136	—	2	1	no	—
8	#13133128	C₄	1	1	yes	characteristic, complemented
9	#124×C₅₃	C₂²	1	1	yes	characteristic
10	#99×C₅₃	—	2	1	no	—
11	C₂×Q₈×C₅₃	D₄	1	1	yes	characteristic
12	C₂×D₄×C₅₃	D₄	1	1	yes	characteristic
13	C₂×D₂₁₂	—	2	1	no	—
14	C₂×D₂₁₂	—	4	1	no	—
15	C₂²⋊C₄×C₅₃	—	4	1	no	—
16	C₂²⋊C₄×C₅₃	—	4	1	no	—
17	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
18	#11719924	—	4	2	no	—
19	#11719922	—	4	2	no	—
20	#11719917	—	4	1	no	—
21	C₂×#88322	—	2	1	no	—
22	C₂³×C₅₃	—	2	1	no	—
23	C₂²×D₅₃	—	4	1	no	—
24	Q₈×C₅₃	—	4	1	no	—
25	D₄×C₅₃	—	4	1	no	—
26	C₂×C₄×C₅₃	—	2	1	no	—
27	C₂×C₄×C₅₃	—	2	1	no	—
28	C₂×C₄×C₅₃	—	8	1	no	—
29	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
30	D₂₁₂	—	4	1	no	—
31	D₂₁₂	—	4	1	no	—
32	D₂₁₂	—	8	2	no	—
33	#88322	—	2	2	no	—
34	#88322	—	8	1	no	—
35	C₂²×C₅₃	—	4	1	no	—
36	C₂²×C₅₃	—	8	1	no	—
37	C₂²×C₅₃	#99	1	1	yes	characteristic
38	C₂×D₅₃	—	8	2	no	—
39	C₂×D₅₃	—	8	1	no	—
40	C₄×C₅₃	—	2	1	no	—
41	C₄×C₅₃	—	4	1	no	—
42	C₄×C₅₃	—	4	1	no	—
43	C₄×C₅₃	—	16	1	no	—
44	#1406	—	53	1	no	maximal, Sylow
45	C₂×C₅₃	—	2	1	no	—
46	C₂×C₅₃	—	8	1	no	—
47	C₂×C₅₃	#353	1	1	yes	characteristic
48	D₅₃	—	16	1	no	—
49	#496	—	53	1	no	—
50	#355	—	53	1	no	—
51	#354	D₅₃	1	1	yes	characteristic, complemented
52	C₅₃	#1406	1	1	yes	characteristic, complemented, Sylow
53	C₂×SD₁₆	—	106	1	no	—
54	C₂×D₈	—	106	1	no	—
55	#127	—	53	1	no	—
56	#124	C₂×D₅₃	1	1	yes	characteristic
57	#101	—	106	1	no	—
58	#99	—	2	1	no	—
59	#97	—	53	1	no	—
60	C₂×Q₈	D₂₁₂	1	1	yes	characteristic
61	C₂×D₄	—	106	1	no	—
62	C₂×D₄	—	212	1	no	—
63	C₂×D₄	C₂²⋊D₅₃	1	1	yes	characteristic
64	SD₁₆	—	212	2	no	—
65	D₈	—	212	2	no	—
66	M₁₆	—	212	1	no	—
67	C₂×C₈	—	106	1	no	—
68	C₂²⋊C₄	—	4	1	no	—
69	C₂²⋊C₄	—	4	1	no	—
70	C₄²	C₄×D₅₃	1	1	yes	characteristic
71	C₂³	—	2	1	no	—
72	C₂³	—	212	1	no	—
73	Q₈	—	4	1	no	—
74	D₄	—	4	1	no	—
75	D₄	—	424	2	no	—
76	D₄	—	212	1	no	—
77	D₄	—	212	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	2	1	no	—
80	C₂×C₄	—	8	1	no	—
81	C₂×C₄	#11719921	1	1	yes	characteristic
82	C₈	—	106	2	no	—
83	C₈	—	424	1	no	—
84	C₂²	—	4	1	no	—
85	C₂²	—	8	1	no	—
86	C₂²	—	424	2	no	—
87	C₂²	—	424	1	no	—
88	C₂²	#13133131	1	1	yes	characteristic
89	C₄	—	2	1	no	—
90	C₄	—	4	1	no	—
91	C₄	—	4	1	no	—
92	C₄	—	16	1	no	—
93	C₂	—	2	1	no	—
94	C₂	—	8	1	no	—
95	C₂	—	848	1	no	—
96	C₂	#16810415	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16810415
commutator	C₄²×C₅₃	C₂×C₄
Frattini	#124	C₂×D₅₃
Fitting	C₅₃×#354	C₂
socle	C₂×C₅₃	#353
hypercenter	#354	D₅₃
hypocenter	C₅₃	#1406
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #354

Complements

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

N	H ≅ G/N	type
2. #16810420	C₂	semidirect
3. C₅₃×#354	C₂	semidirect
6. #13133214	C₄	semidirect
8. #13133128	C₄	semidirect
51. #354	D₅₃	semidirect
52. C₅₃	#1406	semidirect