Subgroups — #27271986

Order 13568.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₈₄₈	C₂	1	1	yes	maximal, characteristic
2	#20272919	C₂	1	1	yes	maximal, characteristic
3	C₅₃×M₁₂₈	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₄₂₄	—	2	1	no	—
5	Dic₈₄₈	—	2	1	no	—
6	Dic₈₄₈	C₄	1	2	yes	—
7	#16810453	—	2	1	no	—
8	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×C₆₄	—	2	1	no	—
10	C₂×Dic₂₁₂	—	4	1	no	—
11	Dic₄₂₄	—	2	2	no	—
12	Dic₄₂₄	—	4	1	no	—
13	#13133153	—	4	1	no	—
14	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
15	C₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₅₃	D₄	1	1	yes	characteristic
17	C₂×Dic₁₀₆	—	8	1	no	—
18	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
19	#11719917	—	8	1	no	—
20	Dic₂₁₂	—	4	2	no	—
21	Dic₂₁₂	—	8	1	no	—
22	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₁₆×C₅₃	SD₁₆	1	1	yes	characteristic
24	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
25	C₂×Dic₅₃	—	16	1	no	—
26	Dic₁₀₆	—	8	2	no	—
27	Dic₁₀₆	—	16	1	no	—
28	C₈×C₅₃	#102	1	1	yes	characteristic
29	C₈×C₅₃	SD₃₂	1	1	yes	characteristic
30	#88322	—	16	1	no	—
31	#7542	—	53	1	no	maximal, Sylow
32	C₂²×C₅₃	SD₆₄	1	1	yes	characteristic
33	C₄×C₅₃	#357	1	1	yes	characteristic
34	C₄×C₅₃	D₃₂	1	1	yes	characteristic
35	Dic₅₃	—	16	2	no	—
36	C₂×Q₆₄	—	53	1	no	—
37	M₁₂₈	D₅₃	1	1	yes	characteristic
38	#1425	—	53	1	no	—
39	C₂×C₅₃	—	2	1	no	—
40	C₂×C₅₃	#1415	1	1	yes	characteristic
41	C₂×Q₃₂	—	106	1	no	—
42	Q₆₄	—	53	2	no	—
43	Q₆₄	—	106	1	no	—
44	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
45	#368	—	106	1	no	—
46	C₆₄	—	2	1	no	—
47	C₅₃	#7542	1	1	yes	characteristic, complemented, Sylow
48	C₂×Q₁₆	—	212	1	no	—
49	Q₃₂	—	106	2	no	—
50	Q₃₂	—	212	1	no	—
51	C₂×C₁₆	D₂₁₂	1	1	yes	characteristic
52	#108	—	212	1	no	—
53	C₃₂	C₄×D₅₃	1	1	yes	characteristic
54	C₃₂	C₂²⋊D₅₃	1	1	yes	characteristic
55	C₂×Q₈	—	424	1	no	—
56	Q₁₆	—	424	1	no	—
57	Q₁₆	—	212	2	no	—
58	M₁₆	—	424	1	no	—
59	C₂×C₈	D₄₂₄	1	1	yes	characteristic
60	C₁₆	#11719913	1	1	yes	characteristic
61	C₁₆	#11719921	1	1	yes	characteristic
62	Q₈	—	848	1	no	—
63	Q₈	—	424	2	no	—
64	C₂×C₄	—	848	1	no	—
65	C₂×C₄	D₈₄₈	1	1	yes	characteristic
66	C₈	—	848	1	no	—
67	C₈	#13133139	1	1	yes	characteristic
68	C₈	#13133162	1	1	yes	characteristic
69	C₂²	#16810462	1	1	yes	characteristic
70	C₄	—	848	2	no	—
71	C₄	#16810425	1	1	yes	characteristic
72	C₄	D₁₆₉₆	1	1	yes	characteristic
73	C₂	—	2	1	no	—
74	C₂	#20272894	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20272894
commutator	C₃₂×C₅₃	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₅₃
Fitting	C₅₃×M₁₂₈	C₂
socle	C₂×C₅₃	#1415
prime radical	C₂²×C₅₃	SD₆₄
cosocle	C₂×C₃₂×C₅₃	C₂²
hypercenter	M₁₂₈	D₅₃
hypocenter	C₅₃	#7542
LCS	G ▷ C₃₂×C₅₃ ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ M₁₂₈

Complements

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

N	H ≅ G/N	type
47. C₅₃	#7542	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₈₄₈	2
#20272919	2
C₅₃×M₁₂₈	2
#7542	53