Subgroups — #27305131

Order 13568.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20272926	C₂	1	3	yes	maximal, complemented
2	C₂×#16810342	C₂	1	3	yes	maximal, complemented
3	C₅₃×#2258	C₂	1	1	yes	maximal, characteristic
4	#16810342	C₂²	1	1	yes	characteristic
5	#16810342	C₂²	1	3	yes	—
6	C₅₃×#504	C₄	1	1	yes	characteristic
7	C₅₃×M₆₄	C₄	1	3	yes	—
8	C₂×C₃₂×C₅₃	C₂²	1	3	yes	—
9	#131×C₅₃	C₈	1	1	yes	characteristic
10	M₃₂×C₅₃	C₈	1	3	yes	—
11	C₂×C₁₆×C₅₃	C₂×C₄	1	3	yes	—
12	C₃₂×C₅₃	C₂×C₄	1	3	yes	—
13	C₃₂×C₅₃	C₂³	1	1	yes	characteristic
14	#41×C₅₃	C₁₆	1	1	yes	characteristic
15	M₁₆×C₅₃	C₁₆	1	3	yes	—
16	C₂×C₈×C₅₃	C₂×C₈	1	3	yes	—
17	C₁₆×C₅₃	C₂×C₈	1	3	yes	—
18	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
19	Q₈×C₅₃	C₃₂	1	1	yes	characteristic
20	D₄×C₅₃	C₃₂	1	3	yes	—
21	C₂×C₄×C₅₃	C₂×C₁₆	1	3	yes	—
22	C₈×C₅₃	C₂×C₁₆	1	3	yes	—
23	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
24	#13737	—	53	1	no	maximal, Sylow
25	C₂²×C₅₃	C₂×C₃₂	1	3	yes	—
26	C₄×C₅₃	C₂×C₃₂	1	3	yes	—
27	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
28	#2258	D₅₃	1	1	yes	characteristic
29	M₁₂₈	—	53	3	no	—
30	C₂×C₆₄	—	53	3	no	—
31	C₂×C₅₃	—	2	3	no	—
32	C₂×C₅₃	C₂²×C₃₂	1	1	yes	characteristic
33	#504	Dic₅₃	1	1	yes	characteristic
34	M₆₄	Dic₅₃	1	3	yes	—
35	C₂×C₃₂	C₂×D₅₃	1	3	yes	—
36	C₆₄	—	53	1	no	—
37	C₆₄	—	53	3	no	—
38	C₅₃	#13737	1	1	yes	characteristic, complemented, Sylow
39	#131	#88322	1	1	yes	characteristic
40	M₃₂	#88322	1	3	yes	—
41	C₂×C₁₆	C₂×Dic₅₃	1	3	yes	—
42	C₃₂	C₂×Dic₅₃	1	3	yes	—
43	C₃₂	C₂²×D₅₃	1	1	yes	characteristic
44	#41	#11719908	1	1	yes	characteristic
45	M₁₆	#11719908	1	3	yes	—
46	C₂×C₈	C₂×#88322	1	3	yes	—
47	C₁₆	C₂×#88322	1	3	yes	—
48	C₁₆	C₂²×Dic₅₃	1	1	yes	characteristic
49	Q₈	#13133123	1	1	yes	characteristic
50	D₄	#13133123	1	3	yes	—
51	C₂×C₄	C₂×#11719908	1	3	yes	—
52	C₈	C₂×#11719908	1	3	yes	—
53	C₈	C₂²×#88322	1	1	yes	characteristic
54	C₂²	C₂×#13133123	1	3	yes	—
55	C₄	C₂×#13133123	1	3	yes	—
56	C₄	C₂²×#11719908	1	1	yes	characteristic
57	C₂	—	2	3	no	—
58	C₂	C₂²×#13133123	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃₂	C₂²×D₅₃
commutator, socle	C₂×C₅₃	C₂²×C₃₂
Fitting	C₅₃×#2258	C₂
prime radical	#41×C₅₃	C₁₆
cosocle	C₃₂×C₅₃	C₂³
hypercenter	#2258	D₅₃
hypocenter	C₅₃	#13737
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₃₂ ◁ #2258

Complements

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

N	H ≅ G/N	type
1. #20272926	C₂	semidirect
2. C₂×#16810342	C₂	semidirect
38. C₅₃	#13737	semidirect

Maximal subgroups

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

subgroup	index
#20272926	2
C₂×#16810342	2
C₅₃×#2258	2
#13737	53