Subgroups — #27271765

Order 13568.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#13133123	C₂	1	2	yes	maximal
2	C₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133123	—	2	2	no	—
4	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
6	C₈²×C₅₃	C₄	1	1	yes	characteristic
7	#13133123	—	4	2	no	—
8	C₂×C₁₆×C₅₃	—	2	1	no	—
9	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
10	C₂×C₁₆×C₅₃	Q₈	1	1	yes	characteristic
11	C₄×C₈×C₅₃	C₈	1	1	yes	characteristic
12	C₄×C₈×C₅₃	C₈	1	1	yes	characteristic
13	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₅₃	—	2	1	no	—
15	C₂×C₈×C₅₃	C₁₆	1	1	yes	characteristic
16	C₂×C₈×C₅₃	C₁₆	1	1	yes	characteristic
17	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
19	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
20	C₁₆×C₅₃	—	2	2	no	—
21	C₁₆×C₅₃	—	2	1	no	—
22	C₁₆×C₅₃	D₈	1	1	yes	characteristic
23	C₁₆×C₅₃	Q₁₆	1	1	yes	characteristic
24	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
25	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
26	C₂×C₄×C₅₃	M₃₂	1	1	yes	characteristic
27	C₈×C₅₃	—	2	2	no	—
28	C₈×C₅₃	—	2	1	no	—
29	C₈×C₅₃	—	2	1	no	—
30	C₈×C₅₃	C₃₂	1	2	yes	complemented
31	C₈×C₅₃	#107	1	1	yes	characteristic
32	C₈×C₅₃	#108	1	1	yes	characteristic
33	#7458	—	53	1	no	maximal, Sylow
34	C₂²×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
35	C₄×C₅₃	—	2	1	no	—
36	C₄×C₅₃	#335	1	1	yes	characteristic
37	C₄×C₅₃	#364	1	1	yes	characteristic
38	C₄×C₅₃	C₂×C₃₂	1	1	yes	characteristic
39	C₄×C₅₃	M₆₄	1	1	yes	characteristic
40	C₄⋊C₃₂	—	53	2	no	—
41	C₈×C₁₆	D₅₃	1	1	yes	characteristic
42	C₂×C₅₃	#1370	1	1	yes	characteristic
43	C₂×C₅₃	C₄⋊C₃₂	1	1	yes	characteristic
44	C₂×C₅₃	#1422	1	1	yes	characteristic
45	C₂×C₃₂	—	106	2	no	—
46	C₄×C₁₆	Dic₅₃	1	1	yes	characteristic
47	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
48	C₈²	Dic₅₃	1	1	yes	characteristic
49	C₅₃	#7458	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₁₆	—	2	1	no	—
51	C₂×C₁₆	Dic₁₀₆	1	1	yes	characteristic
52	C₂×C₁₆	D₂₁₂	1	1	yes	characteristic
53	C₄×C₈	#88322	1	1	yes	characteristic
54	C₄×C₈	#88322	1	1	yes	characteristic
55	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
56	C₃₂	—	212	2	no	—
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	#11719908	1	1	yes	characteristic
59	C₂×C₈	#11719908	1	1	yes	characteristic
60	C₂×C₈	#11719917	1	1	yes	characteristic
61	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
62	C₄²	C₂×#88322	1	1	yes	characteristic
63	C₁₆	—	2	2	no	—
64	C₁₆	—	2	1	no	—
65	C₁₆	D₄₂₄	1	1	yes	characteristic
66	C₁₆	Dic₂₁₂	1	1	yes	characteristic
67	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
68	C₂×C₄	C₂×#11719908	1	1	yes	characteristic
69	C₂×C₄	#13133157	1	1	yes	characteristic
70	C₈	—	2	2	no	—
71	C₈	—	2	1	no	—
72	C₈	—	2	1	no	—
73	C₈	#13133123	1	2	yes	complemented
74	C₈	#13133151	1	1	yes	characteristic
75	C₈	#13133153	1	1	yes	characteristic
76	C₂²	C₄⋊#11719908	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	#16810376	1	1	yes	characteristic
79	C₄	#16810444	1	1	yes	characteristic
80	C₄	C₂×#13133123	1	1	yes	characteristic
81	C₄	#16810457	1	1	yes	characteristic
82	C₂	#20272785	1	1	yes	characteristic
83	C₂	C₄⋊#13133123	1	1	yes	characteristic
84	C₂	#20272913	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	D₂₁₂
commutator	C₄×C₅₃	C₂×C₃₂
Frattini	C₄×C₁₆	C₂×D₅₃
Fitting	C₈×C₁₆×C₅₃	C₂
socle, prime radical	C₂²×C₅₃	C₄⋊C₁₆
cosocle	C₄×C₁₆×C₅₃	C₂²
hypercenter	C₈×C₁₆	D₅₃
hypocenter	C₅₃	#7458
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₁₆ ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
30. C₈×C₅₃	C₃₂	semidirect
49. C₅₃	#7458	semidirect
73. C₈	#13133123	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄⋊#13133123	2
C₈×C₁₆×C₅₃	2
#7458	53