Subgroups — #20289677

Order 6784.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811545	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈×#88324	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810740	C₂	1	1	yes	maximal, characteristic
4	#13133291	—	2	1	no	—
5	C₄×#88324	—	2	1	no	—
6	#13133224	—	2	1	no	—
7	#13133153	C₄	1	1	yes	characteristic
8	#13133144	—	2	1	no	—
9	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
10	#135×C₅₃	C₄	1	1	yes	characteristic
11	#41×C₅₃	—	2	1	no	—
12	C₂×#88324	—	2	2	no	—
13	Q₁₆×C₅₃	C₈	1	1	yes	characteristic, complemented
14	SD₁₆×C₅₃	—	2	1	no	—
15	D₈×C₅₃	C₈	1	1	yes	characteristic, complemented
16	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₄×Dic₅₃	D₄	1	1	yes	characteristic
18	#11719917	—	2	1	no	—
19	#11719917	—	4	1	no	—
20	C₂×#88322	—	4	1	no	—
21	C₂×#88322	D₄	1	1	yes	characteristic
22	#11719910	—	4	1	no	—
23	Q₈×C₅₃	—	2	1	no	—
24	D₄×C₅₃	—	2	1	no	—
25	D₄×C₅₃	—	4	1	no	—
26	C₂×C₄×C₅₃	—	4	1	no	—
27	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×Dic₅₃	—	2	1	no	—
29	#88324	—	2	4	no	—
30	C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
31	C₈×C₅₃	M₁₆	1	1	yes	characteristic
32	#88322	—	4	1	no	—
33	#88322	D₈	1	1	yes	characteristic
34	#88322	SD₁₆	1	1	yes	characteristic
35	C₂²×C₅₃	—	4	1	no	—
36	C₂²×C₅₃	#104	1	1	yes	characteristic
37	C₄×C₅₃	—	4	1	no	—
38	C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₅₃	#102	1	1	yes	characteristic
40	Dic₅₃	—	2	2	no	—
41	#1335	—	53	1	no	maximal, Sylow
42	C₂×C₅₃	—	2	1	no	—
43	C₂×C₅₃	—	8	1	no	—
44	C₂×C₅₃	D₄⋊C₈	1	1	yes	characteristic
45	#443	—	53	1	no	—
46	#364	—	53	1	no	—
47	C₈²	—	53	1	no	—
48	C₅₃	#1335	1	1	yes	characteristic, complemented, Sylow
49	#135	#6157	1	1	yes	characteristic
50	#131	—	106	1	no	—
51	M₃₂	—	106	1	no	—
52	#108	—	53	1	no	—
53	#104	—	106	1	no	—
54	C₄×C₈	—	53	1	no	—
55	C₄×C₈	—	106	1	no	—
56	#41	—	2	1	no	—
57	Q₁₆	#88324	1	1	yes	characteristic, complemented
58	SD₁₆	—	2	1	no	—
59	D₈	#88324	1	1	yes	characteristic, complemented
60	M₁₆	—	106	1	no	—
61	M₁₆	—	212	1	no	—
62	C₂×C₈	—	53	1	no	—
63	C₂×C₈	—	106	2	no	—
64	C₂×C₈	—	212	1	no	—
65	C₂×C₈	C₂×#6157	1	1	yes	characteristic
66	C₄²	—	53	1	no	—
67	C₁₆	—	212	1	no	—
68	Q₈	—	2	1	no	—
69	D₄	—	2	1	no	—
70	D₄	—	4	1	no	—
71	C₂×C₄	—	4	1	no	—
72	C₂×C₄	—	106	1	no	—
73	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
74	C₈	—	53	1	no	—
75	C₈	—	53	1	no	—
76	C₈	—	106	4	no	—
77	C₈	—	212	1	no	—
78	C₈	C₂×#88324	1	1	yes	characteristic
79	C₈	#11719940	1	1	yes	characteristic
80	C₂²	—	4	1	no	—
81	C₂²	#13133281	1	1	yes	characteristic
82	C₄	—	4	1	no	—
83	C₄	—	106	2	no	—
84	C₄	C₂²⋊#88324	1	1	yes	characteristic
85	C₄	#13133279	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	8	1	no	—
88	C₂	D₄⋊#88324	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#13133279
commutator	C₈×C₅₃	C₂×C₈
Frattini	C₂×C₈	C₂×#6157
Fitting	#135×C₅₃	C₄
socle	C₂×C₅₃	D₄⋊C₈
hypercenter	#135	#6157
hypocenter	C₅₃	#1335
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #16811545	C₂	semidirect
2. C₈×#88324	C₂	semidirect
13. Q₁₆×C₅₃	C₈	semidirect
15. D₈×C₅₃	C₈	semidirect
48. C₅₃	#1335	semidirect
57. Q₁₆	#88324	semidirect
59. D₈	#88324	semidirect