Subgroups — C₁₁²×D₁₁₅ #42123744

Order 27830.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₁₁²×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₁₁²×D₂₃	—	5	1	no	maximal, Hall
3	C₁₁²×C₂₃	D₅	1	1	yes	characteristic, complemented, Hall
4	C₁₁×D₁₁₅	C₁₁	1	12	yes	maximal, direct product factor
5	C₅×C₁₁×C₂₃	C₂×C₁₁	1	12	yes	complemented
6	D₅×C₁₁²	—	23	1	no	maximal, Hall
7	C₅×C₁₁²	D₂₃	1	1	yes	characteristic, complemented, Hall
8	C₁₁×D₂₃	—	5	12	no	—
9	C₁₁×C₂₃	D₅×C₁₁	1	12	yes	complemented
10	C₂×C₁₁²	—	115	1	no	Hall
11	D₁₁₅	C₁₁²	1	1	yes	characteristic, direct product factor, Hall
12	C₁₁²	D₁₁₅	1	1	yes	characteristic, direct product factor, Sylow
13	C₅×C₂₃	C₂×C₁₁²	1	1	yes	characteristic, complemented, Hall
14	D₅×C₁₁	—	23	12	no	—
15	C₅×C₁₁	C₁₁×D₂₃	1	12	yes	complemented
16	D₂₃	—	5	1	no	Hall
17	C₂₃	D₅×C₁₁²	1	1	yes	characteristic, complemented, Sylow
18	C₂×C₁₁	—	115	12	no	—
19	C₁₁	C₁₁×D₁₁₅	1	12	yes	direct product factor
20	D₅	—	23	1	no	Hall
21	C₅	C₁₁²×D₂₃	1	1	yes	characteristic, complemented, Sylow
22	C₂	—	115	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁₁²	D₁₁₅
commutator, cosocle, hypocenter	C₅×C₂₃	C₂×C₁₁²
Frattini	C₁	G
Fitting, socle	C₅×C₁₁²×C₂₃	C₂
LCS	G ▷ C₅×C₂₃
UCS	C₁ ◁ C₁₁²

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.