Subgroups — #27270830

Order 13568.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20272779	C₂	1	2	yes	maximal
2	C₅₃×#1311	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11719908	—	2	2	no	—
4	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic
6	C₈²×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×#11719908	—	4	2	no	—
8	C₂×C₁₆×C₅₃	—	2	1	no	—
9	C₂×C₁₆×C₅₃	C₈	1	1	yes	characteristic
10	C₂×C₁₆×C₅₃	C₈	1	1	yes	characteristic
11	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
13	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
14	C₂×C₈×C₅₃	—	2	1	no	—
15	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
16	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
17	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
18	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
19	C₁₆×C₅₃	—	2	2	no	—
20	C₁₆×C₅₃	—	2	1	no	—
21	C₁₆×C₅₃	C₁₆	1	2	yes	complemented
22	#11719908	—	8	2	no	—
23	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
24	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
25	C₂×C₄×C₅₃	#108	1	1	yes	characteristic
26	C₈×C₅₃	—	2	1	no	—
27	C₈×C₅₃	—	2	2	no	—
28	C₈×C₅₃	—	4	1	no	—
29	C₈×C₅₃	C₂×C₁₆	1	1	yes	characteristic
30	C₈×C₅₃	M₃₂	1	1	yes	characteristic
31	#7099	—	53	1	no	maximal, Sylow
32	C₂²×C₅₃	#334	1	1	yes	characteristic
33	C₄×C₅₃	—	2	1	no	—
34	C₄×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
35	C₄×C₅₃	#364	1	1	yes	characteristic
36	C₄×C₅₃	#365	1	2	yes	—
37	#1367	—	53	2	no	—
38	#1311	D₅₃	1	1	yes	characteristic
39	C₂×C₅₃	#1367	1	1	yes	characteristic
40	C₂×C₅₃	#1368	1	1	yes	characteristic
41	C₂×C₅₃	#1369	1	1	yes	characteristic
42	C₄⋊C₁₆	—	106	2	no	—
43	C₄×C₁₆	Dic₅₃	1	1	yes	characteristic
44	C₄×C₁₆	Dic₅₃	1	1	yes	characteristic
45	C₈²	C₂×D₅₃	1	1	yes	characteristic
46	C₅₃	#7099	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₁₆	—	2	1	no	—
48	C₂×C₁₆	—	212	2	no	—
49	C₂×C₁₆	#88322	1	1	yes	characteristic
50	C₂×C₁₆	#88322	1	1	yes	characteristic
51	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
52	C₄×C₈	D₂₁₂	1	1	yes	characteristic
53	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
54	C₂×C₈	—	2	1	no	—
55	C₂×C₈	#11719913	1	2	yes	—
56	C₂×C₈	C₂×#88322	1	1	yes	characteristic
57	C₂×C₈	#11719917	1	1	yes	characteristic
58	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
59	C₁₆	—	2	2	no	—
60	C₁₆	—	2	1	no	—
61	C₁₆	—	424	2	no	—
62	C₁₆	#11719908	1	2	yes	complemented
63	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
64	C₂×C₄	#13133149	1	1	yes	characteristic
65	C₂×C₄	#13133153	1	1	yes	characteristic
66	C₈	—	2	1	no	—
67	C₈	—	2	2	no	—
68	C₈	—	4	1	no	—
69	C₈	C₂×#11719908	1	1	yes	characteristic
70	C₈	#13133157	1	1	yes	characteristic
71	C₂²	#16810374	1	1	yes	characteristic
72	C₄	—	2	1	no	—
73	C₄	C₄⋊#11719908	1	1	yes	characteristic
74	C₄	#16810444	1	1	yes	characteristic
75	C₄	#16810446	1	2	yes	—
76	C₂	#20272779	1	1	yes	characteristic
77	C₂	#20272781	1	1	yes	characteristic
78	C₂	#20272783	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133149
commutator	C₈×C₅₃	C₂×C₁₆
Frattini	C₈²	C₂×D₅₃
Fitting	C₅₃×#1311	C₂
socle, prime radical	C₂²×C₅₃	#334
cosocle	C₈²×C₅₃	C₂²
hypercenter	#1311	D₅₃
hypocenter	C₅₃	#7099
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1311

Complements

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

N	H ≅ G/N	type
21. C₁₆×C₅₃	C₁₆	semidirect
46. C₅₃	#7099	semidirect
62. C₁₆	#11719908	semidirect

Maximal subgroups

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

subgroup	index
#20272779	2
C₅₃×#1311	2
#7099	53