Subgroups — C₇²×A₄×C₅₃ #44528476

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

	subgroup	G/H
center, hypercenter	C₇²×C₅₃	A₄
commutator, cosocle, hypocenter	C₂²	C₃×C₇²×C₅₃
Frattini	C₁	G
Fitting, socle	C₂²×C₇²×C₅₃	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₇²×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.