Subgroups — C₇²×D₂₃₇ #37551177

Order 23226.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇²×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇²×D₇₉	—	3	1	no	maximal, Hall
3	C₇²×C₇₉	D₃	1	1	yes	characteristic, complemented, Hall
4	C₇×D₂₃₇	C₇	1	8	yes	maximal, direct product factor
5	C₃×C₇×C₇₉	C₂×C₇	1	8	yes	complemented
6	C₇×D₇₉	—	3	8	no	—
7	C₇×C₇₉	D₃×C₇	1	8	yes	complemented
8	D₂₃₇	C₇²	1	1	yes	characteristic, direct product factor, Hall
9	D₃×C₇²	—	79	1	no	maximal, Hall
10	C₃×C₇₉	C₂×C₇²	1	1	yes	characteristic, complemented, Hall
11	D₇₉	—	3	1	no	Hall
12	C₃×C₇²	D₇₉	1	1	yes	characteristic, complemented, Hall
13	C₂×C₇²	—	237	1	no	Hall
14	C₇₉	D₃×C₇²	1	1	yes	characteristic, complemented, Sylow
15	C₇²	D₂₃₇	1	1	yes	characteristic, direct product factor, Sylow
16	D₃×C₇	—	79	8	no	—
17	C₃×C₇	C₇×D₇₉	1	8	yes	complemented
18	C₂×C₇	—	237	8	no	—
19	C₇	C₇×D₂₃₇	1	8	yes	direct product factor
20	D₃	—	79	1	no	Hall
21	C₃	C₇²×D₇₉	1	1	yes	characteristic, complemented, Sylow
22	C₂	—	237	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₇²	D₂₃₇
commutator, cosocle, hypocenter	C₃×C₇₉	C₂×C₇²
Frattini	C₁	G
Fitting, socle	C₃×C₇²×C₇₉	C₂
LCS	G ▷ C₃×C₇₉
UCS	C₁ ◁ C₇²

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.