Subgroups — C₃×C₅₃×#3861 #37552617

Order 23373.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3861	C₃	1	3	yes	maximal, direct product factor
2	C₃×C₇²×C₅₃	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×ΓL(1,𝔽₈)×C₅₃	—	7	8	no	maximal
4	C₇²×C₅₃	C₃²	1	1	yes	characteristic, complemented, Hall
5	C₃×C₇×C₅₃	ΓL(1,𝔽₈)	1	8	yes	complemented
6	ΓL(1,𝔽₈)×C₅₃	—	7	24	no	—
7	C₃²×C₅₃	—	49	1	no	Hall
8	C₃×#3861	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₇×C₅₃	C₃×ΓL(1,𝔽₈)	1	8	yes	complemented
10	C₃×C₅₃	—	49	3	no	—
11	C₃×C₅₃	#3861	1	1	yes	characteristic, direct product factor
12	C₃×C₇²	C₃×C₅₃	1	1	yes	characteristic, complemented
13	#3861	C₃×C₅₃	1	3	yes	direct product factor
14	C₃×ΓL(1,𝔽₈)	—	7	8	no	—
15	C₅₃	C₃×#3861	1	1	yes	characteristic, direct product factor, Sylow
16	C₇²	C₃²×C₅₃	1	1	yes	characteristic, complemented, Sylow
17	C₃×C₇	ΓL(1,𝔽₈)×C₅₃	1	8	yes	complemented
18	ΓL(1,𝔽₈)	—	7	24	no	—
19	C₃²	—	49	1	no	Sylow
20	C₇	C₃×ΓL(1,𝔽₈)×C₅₃	1	8	yes	complemented
21	C₃	—	49	3	no	—
22	C₃	C₅₃×#3861	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₃×C₅₃	#3861
commutator, cosocle, hypocenter	C₇²	C₃²×C₅₃
Frattini	C₁	G
Fitting, socle	C₃×C₇²×C₅₃	C₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₃×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.