Subgroups — #27271906

Order 13568.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20277008	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810342	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×M₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811039	C₄	1	1	yes	characteristic
5	#16810457	C₄	1	1	yes	characteristic
6	#16810456	—	2	1	no	—
7	C₃₂×D₅₃	—	2	1	no	—
8	#16810342	—	2	1	no	—
9	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×C₆₄	—	2	1	no	—
11	#13133223	C₈	1	1	yes	characteristic
12	#13133157	C₈	1	1	yes	characteristic
13	#13133156	—	2	1	no	—
14	C₁₆×D₅₃	—	2	1	no	—
15	#13133123	—	2	1	no	—
16	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₃₂×C₅₃	D₄	1	1	yes	characteristic
18	C₃₂×C₅₃	D₄	1	1	yes	characteristic
19	#11719945	C₁₆	1	1	yes	characteristic
20	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
21	#11719917	C₁₆	1	1	yes	characteristic
22	#11719912	—	2	1	no	—
23	C₈×D₅₃	—	2	1	no	—
24	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₁₆×C₅₃	M₁₆	1	1	yes	characteristic
26	#11719908	—	2	1	no	—
27	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
28	C₂²⋊D₅₃	—	2	1	no	—
29	D₂₁₂	C₃₂	1	1	yes	characteristic
30	C₄×D₅₃	—	2	1	no	—
31	Dic₁₀₆	C₃₂	1	1	yes	characteristic
32	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
33	C₈×C₅₃	M₃₂	1	1	yes	characteristic
34	#88322	—	2	1	no	—
35	#7514	—	53	1	no	maximal, Sylow
36	C₂²×C₅₃	M₆₄	1	1	yes	characteristic
37	C₂×D₅₃	—	2	1	no	—
38	C₄×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
39	C₄×C₅₃	C₂×C₃₂	1	1	yes	characteristic
40	Dic₅₃	—	2	1	no	—
41	#2258	—	53	1	no	—
42	M₁₂₈	D₅₃	1	1	yes	characteristic, complemented
43	C₂×C₆₄	—	53	1	no	—
44	C₂×C₅₃	—	2	1	no	—
45	C₂×C₅₃	C₂²⋊C₃₂	1	1	yes	characteristic
46	D₅₃	—	4	1	no	—
47	#504	—	53	1	no	—
48	M₆₄	—	53	1	no	—
49	M₆₄	—	106	1	no	—
50	C₂×C₃₂	—	106	1	no	—
51	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
52	C₆₄	—	2	1	no	—
53	C₆₄	—	106	1	no	—
54	C₅₃	#7514	1	1	yes	characteristic, complemented, Sylow
55	#131	—	53	1	no	—
56	M₃₂	—	53	1	no	—
57	M₃₂	—	106	1	no	—
58	C₂×C₁₆	—	106	1	no	—
59	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
60	C₃₂	—	106	1	no	—
61	C₃₂	D₂₁₂	1	1	yes	characteristic
62	C₃₂	C₂²⋊D₅₃	1	1	yes	characteristic
63	#41	—	53	1	no	—
64	M₁₆	—	53	1	no	—
65	M₁₆	—	106	1	no	—
66	C₂×C₈	—	106	1	no	—
67	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
68	C₁₆	—	106	1	no	—
69	C₁₆	#11719912	1	1	yes	characteristic
70	C₁₆	#11719921	1	1	yes	characteristic
71	Q₈	—	53	1	no	—
72	D₄	—	53	1	no	—
73	D₄	—	106	1	no	—
74	C₂×C₄	—	106	1	no	—
75	C₂×C₄	C₁₆×D₅₃	1	1	yes	characteristic
76	C₈	—	106	1	no	—
77	C₈	#13133129	1	1	yes	characteristic
78	C₈	#13133156	1	1	yes	characteristic
79	C₂²	—	106	1	no	—
80	C₂²	#16810456	1	1	yes	characteristic
81	C₄	—	106	1	no	—
82	C₄	#16810402	1	1	yes	characteristic
83	C₄	C₃₂×D₅₃	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	212	1	no	—
86	C₂	#20272852	1	1	yes	characteristic

	subgroup	G/H
center	C₃₂	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂×C₃₂
Frattini	C₂×C₃₂	C₂×D₅₃
Fitting	C₅₃×M₁₂₈	C₂
socle	C₂×C₅₃	C₂²⋊C₃₂
prime radical	#11719945	C₁₆
cosocle	C₂×C₃₂×C₅₃	C₂²
hypercenter	M₁₂₈	D₅₃
hypocenter	C₅₃	#7514
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₃₂ ◁ C₂×C₃₂ ◁ M₁₂₈

Complements

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

N	H ≅ G/N	type
2. C₂×#16810342	C₂	semidirect
3. C₅₃×M₁₂₈	C₂	semidirect
42. M₁₂₈	D₅₃	semidirect
54. C₅₃	#7514	semidirect

Maximal subgroups

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

subgroup	index
#20277008	2
C₂×#16810342	2
C₅₃×M₁₂₈	2
#7514	53