Subgroups — #20272761

Order 6784.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810376	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#493	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719908	—	2	1	no	—
5	C₄⋊#88322	—	2	1	no	—
6	C₂×D₈×C₅₃	—	2	1	no	—
7	C₂×D₈×C₅₃	C₄	1	2	yes	—
8	#127×C₅₃	—	2	1	no	—
9	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×D₄×C₅₃	—	2	2	no	—
11	C₂×D₄×C₅₃	—	4	1	no	—
12	C₂×D₄×C₅₃	—	4	1	no	—
13	D₈×C₅₃	—	4	2	no	—
14	D₈×C₅₃	—	4	1	no	—
15	D₈×C₅₃	—	4	1	no	—
16	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
18	C₄²×C₅₃	D₄	1	1	yes	characteristic
19	C₂×#88322	—	4	1	no	—
20	#11719908	—	4	1	no	—
21	C₂³×C₅₃	—	4	2	no	—
22	D₄×C₅₃	—	4	2	no	—
23	D₄×C₅₃	—	4	2	no	—
24	D₄×C₅₃	—	8	2	no	—
25	D₄×C₅₃	—	8	1	no	—
26	D₄×C₅₃	—	8	1	no	—
27	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
29	C₂×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
30	C₈×C₅₃	—	2	1	no	—
31	C₈×C₅₃	D₈	1	1	yes	characteristic
32	C₈×C₅₃	SD₁₆	1	1	yes	characteristic
33	#88322	—	8	1	no	—
34	C₂²×C₅₃	—	8	2	no	—
35	C₂²×C₅₃	—	8	2	no	—
36	C₂²×C₅₃	—	8	2	no	—
37	C₂²×C₅₃	#102	1	1	yes	characteristic
38	C₄×C₅₃	—	2	1	no	—
39	C₄×C₅₃	#100	1	1	yes	characteristic
40	C₄×C₅₃	#102	1	1	yes	characteristic
41	C₄×C₅₃	D₁₆	1	1	yes	characteristic
42	C₄×C₅₃	SD₃₂	1	1	yes	characteristic
43	#1361	—	53	1	no	maximal, Sylow
44	C₂×C₅₃	—	16	2	no	—
45	C₂×C₅₃	#331	1	1	yes	characteristic
46	C₂×C₅₃	#357	1	1	yes	characteristic
47	C₂×C₅₃	#361	1	1	yes	characteristic
48	#493	D₅₃	1	1	yes	characteristic
49	C₄⋊C₁₆	—	53	1	no	—
50	#335	—	53	1	no	—
51	C₅₃	#1361	1	1	yes	characteristic, complemented, Sylow
52	C₂×D₈	—	2	1	no	—
53	C₂×D₈	Dic₅₃	1	2	yes	—
54	#127	—	2	1	no	—
55	C₂×C₁₆	—	106	1	no	—
56	C₄⋊C₈	—	106	1	no	—
57	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
58	C₂×D₄	—	2	2	no	—
59	C₂×D₄	—	4	1	no	—
60	C₂×D₄	—	4	1	no	—
61	D₈	—	4	2	no	—
62	D₈	—	4	1	no	—
63	D₈	—	4	1	no	—
64	C₂×C₈	—	212	1	no	—
65	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
66	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
67	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
68	C₁₆	—	212	1	no	—
69	C₂³	—	4	2	no	—
70	D₄	—	4	2	no	—
71	D₄	—	4	2	no	—
72	D₄	—	8	2	no	—
73	D₄	—	8	1	no	—
74	D₄	—	8	1	no	—
75	C₂×C₄	#11719922	1	1	yes	characteristic
76	C₂×C₄	#11719923	1	1	yes	characteristic
77	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	424	1	no	—
80	C₈	#11719922	1	1	yes	characteristic
81	C₈	#11719923	1	1	yes	characteristic
82	C₂²	—	8	2	no	—
83	C₂²	—	8	2	no	—
84	C₂²	—	8	2	no	—
85	C₂²	#13133138	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	#13133134	1	1	yes	characteristic
88	C₄	#13133138	1	1	yes	characteristic
89	C₄	#13133158	1	1	yes	characteristic
90	C₄	#13133161	1	1	yes	characteristic
91	C₂	—	16	2	no	—
92	C₂	#16810367	1	1	yes	characteristic
93	C₂	#16810424	1	1	yes	characteristic
94	C₂	#16810436	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13133138
commutator	C₂×C₈×C₅₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₅₃
Fitting	C₅₃×#493	C₂
socle	C₂²×C₅₃	#102
hypercenter	#493	D₅₃
hypocenter	C₅₃	#1361
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #493

Complements

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

N	H ≅ G/N	type
1. C₄⋊#11719908	C₂	semidirect
2. #16810376	C₂	semidirect
51. C₅₃	#1361	semidirect