Subgroups — #20273018

Order 6784.

214 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#11719917	C₂	1	2	yes	maximal
2	#16810345	C₂	1	2	yes	maximal, complemented
3	#16810344	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₄	1	4	yes	—
7	#13133128	C₄	1	4	yes	—
8	C₄×#88322	C₂²	1	2	yes	—
9	C₄×#88322	C₂²	1	2	yes	—
10	C₂×M₁₆×C₅₃	C₄	1	2	yes	—
11	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	#97×C₅₃	C₄	1	2	yes	—
13	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
16	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
17	M₁₆×C₅₃	—	2	4	no	—
18	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
20	C₄²×C₅₃	C₂×C₄	1	2	yes	—
21	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₅₃	C₂³	1	1	yes	characteristic
23	#11719917	—	2	8	no	—
24	C₂×#88322	C₂×C₄	1	4	yes	—
25	C₂×#88322	C₂×C₄	1	4	yes	—
26	C₂³×C₅₃	C₄²	1	1	yes	characteristic
27	C₂×C₄×C₅₃	—	2	4	no	—
28	C₂×C₄×C₅₃	—	2	2	no	—
29	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
30	C₂×C₄×C₅₃	C₄²	1	2	yes	—
31	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
32	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₅₃	—	2	2	no	—
34	C₈×C₅₃	M₁₆	1	4	yes	complemented
35	#88322	—	2	4	no	—
36	#88322	M₁₆	1	8	yes	complemented
37	C₂²×C₅₃	—	2	2	no	—
38	C₂²×C₅₃	—	2	1	no	—
39	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
40	C₄×C₅₃	—	2	2	no	—
41	C₄×C₅₃	—	2	1	no	—
42	C₄×C₅₃	C₂×M₁₆	1	4	yes	—
43	C₄×C₅₃	C₂×M₁₆	1	2	yes	—
44	#1455	—	53	1	no	maximal, Sylow
45	C₂×C₅₃	—	4	1	no	—
46	C₂×C₅₃	C₄×M₁₆	1	2	yes	—
47	C₂×C₅₃	C₄×M₁₆	1	1	yes	characteristic
48	C₄×M₁₆	—	53	2	no	—
49	C₄×M₁₆	D₅₃	1	1	yes	characteristic
50	#322	—	53	1	no	—
51	#322	—	53	2	no	—
52	C₈²	—	53	1	no	—
53	C₅₃	#1455	1	1	yes	characteristic, complemented, Sylow
54	C₂×M₁₆	—	53	4	no	—
55	C₂×M₁₆	Dic₅₃	1	2	yes	—
56	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
57	#97	—	53	4	no	—
58	#97	Dic₅₃	1	2	yes	—
59	C₄×C₈	—	53	2	no	—
60	C₄×C₈	—	53	2	no	—
61	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
62	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
63	C₂²×C₄	C₄×D₅₃	1	2	yes	—
64	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
65	M₁₆	—	2	4	no	—
66	M₁₆	—	106	8	no	—
67	C₂×C₈	—	53	4	no	—
68	C₂×C₈	—	53	4	no	—
69	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
70	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
71	C₄²	C₄×D₅₃	1	2	yes	—
72	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
73	C₄²	C₂²×D₅₃	1	1	yes	characteristic
74	C₂³	C₄×Dic₅₃	1	1	yes	characteristic
75	C₂×C₄	—	2	4	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
78	C₂×C₄	C₄×Dic₅₃	1	2	yes	—
79	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
80	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
81	C₈	—	2	2	no	—
82	C₈	—	53	8	no	—
83	C₈	—	106	4	no	—
84	C₈	#11719917	1	4	yes	complemented
85	C₂²	—	2	2	no	—
86	C₂²	—	2	1	no	—
87	C₂²	C₂×C₄×Dic₅₃	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	2	1	no	—
90	C₄	M₁₆×D₅₃	1	4	yes	—
91	C₄	C₂×#11719917	1	2	yes	—
92	C₂	—	4	1	no	—
93	C₂	M₁₆×Dic₅₃	1	2	yes	—
94	C₂	C₄×#11719917	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂×C₄²
Fitting	C₄×M₁₆×C₅₃	C₂
hypercenter	C₄×M₁₆	D₅₃
hypocenter	C₅₃	#1455
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #16810345	C₂	semidirect
3. #16810344	C₂	semidirect
4. C₈×#88322	C₂	semidirect
34. C₈×C₅₃	M₁₆	semidirect
36. #88322	M₁₆	semidirect
53. C₅₃	#1455	semidirect
84. C₈	#11719917	semidirect