Subgroups — C₅₃×C₂²⋊#3861 #44528478

Order 31164.

100 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×#3861	—	4	1	no	maximal, Hall
3	C₂×C₇²×C₅₃	—	3	1	no	—
4	C₅₃×C₂²⋊ΓL(1,𝔽₈)	—	7	8	no	maximal
5	C₇²×C₅₃	A₄	1	1	yes	characteristic, complemented, Hall
6	C₂²×C₇×C₅₃	ΓL(1,𝔽₈)	1	8	yes	complemented
7	ΓL(1,𝔽₈)×C₅₃	—	28	8	no	—
8	C₂×C₇×C₅₃	—	3	8	no	—
9	A₄×C₅₃	—	49	1	no	Hall
10	C₂²⋊#3861	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₇×C₅₃	C₂²⋊ΓL(1,𝔽₈)	1	8	yes	complemented
12	C₂²×C₅₃	#3861	1	1	yes	characteristic, complemented, Hall
13	C₂²×C₇²	C₃×C₅₃	1	1	yes	characteristic, complemented, Hall
14	C₃×C₅₃	—	196	1	no	Hall
15	#3861	—	4	1	no	Hall
16	C₂×C₅₃	—	3	1	no	—
17	C₂×C₇²	—	3	1	no	—
18	C₂²⋊ΓL(1,𝔽₈)	—	7	8	no	—
19	C₅₃	C₂²⋊#3861	1	1	yes	characteristic, direct product factor, Sylow
20	C₇²	A₄×C₅₃	1	1	yes	characteristic, complemented, Sylow
21	C₂²×C₇	ΓL(1,𝔽₈)×C₅₃	1	8	yes	complemented
22	ΓL(1,𝔽₈)	—	28	8	no	—
23	C₂×C₇	—	3	8	no	—
24	A₄	—	49	1	no	Hall
25	C₇	C₅₃×C₂²⋊ΓL(1,𝔽₈)	1	8	yes	complemented
26	C₂²	C₅₃×#3861	1	1	yes	characteristic, complemented, Sylow
27	C₃	—	196	1	no	Sylow
28	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₅₃	C₂²⋊#3861
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.