Subgroups — C₅₃×C₂²⋊#3862 #44528479

Order 31164.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇²×C₅₃	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₅₃×#3862	—	4	1	no	maximal, Hall
3	C₂×C₇²×C₅₃	—	3	1	no	—
4	C₅₃×C₂²⋊ΓL(1,𝔽₈)	—	7	2	no	maximal
5	C₇²×C₅₃	A₄	1	1	yes	characteristic, complemented, Hall
6	C₂²×C₇×C₅₃	—	3	2	no	—
7	C₂²×C₇×C₅₃	ΓL(1,𝔽₈)	1	2	yes	complemented
8	ΓL(1,𝔽₈)×C₅₃	—	28	2	no	—
9	C₂×C₇×C₅₃	—	3	6	no	—
10	C₂×C₇×C₅₃	—	3	2	no	—
11	A₄×C₅₃	—	49	1	no	Hall
12	C₂²⋊#3862	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₇×C₅₃	—	3	2	no	—
14	C₇×C₅₃	C₂²⋊ΓL(1,𝔽₈)	1	2	yes	complemented
15	C₂²×C₅₃	#3862	1	1	yes	characteristic, complemented, Hall
16	C₂²×C₇²	C₃×C₅₃	1	1	yes	characteristic, complemented, Hall
17	C₃×C₅₃	—	196	1	no	Hall
18	#3862	—	4	1	no	Hall
19	C₂×C₅₃	—	3	1	no	—
20	C₂×C₇²	—	3	1	no	—
21	C₂²⋊ΓL(1,𝔽₈)	—	7	2	no	—
22	C₅₃	C₂²⋊#3862	1	1	yes	characteristic, direct product factor, Sylow
23	C₇²	A₄×C₅₃	1	1	yes	characteristic, complemented, Sylow
24	C₂²×C₇	—	3	2	no	—
25	C₂²×C₇	ΓL(1,𝔽₈)×C₅₃	1	2	yes	complemented
26	ΓL(1,𝔽₈)	—	28	2	no	—
27	C₂×C₇	—	3	6	no	—
28	C₂×C₇	—	3	2	no	—
29	A₄	—	49	1	no	Hall
30	C₇	—	3	2	no	—
31	C₇	C₅₃×C₂²⋊ΓL(1,𝔽₈)	1	2	yes	complemented
32	C₂²	C₅₃×#3862	1	1	yes	characteristic, complemented, Sylow
33	C₃	—	196	1	no	Sylow
34	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₅₃	C₂²⋊#3862
commutator, cosocle, hypocenter	C₂²×C₇²	C₃×C₅₃
Frattini	C₁	G
Fitting, socle	C₂²×C₇²×C₅₃	C₃
LCS	G ▷ C₂²×C₇²
UCS	C₁ ◁ C₅₃

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

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