Subgroups — C₁₁²×A₄×C₁₇ #37590657

Order 24684.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₁²×C₁₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×C₁₁²×C₁₇	—	4	1	no	maximal, Hall
3	C₂×C₁₁²×C₁₇	—	3	1	no	—
4	C₁₁×A₄×C₁₇	C₁₁	1	12	yes	maximal, direct product factor
5	C₁₁²×C₁₇	A₄	1	1	yes	characteristic, direct product factor, Hall
6	C₁₁²×A₄	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₁₁×C₁₇	C₃×C₁₁	1	12	yes	complemented
8	C₃×C₁₁×C₁₇	—	4	12	no	—
9	C₂²×C₁₁²	C₃×C₁₇	1	1	yes	characteristic, complemented, Hall
10	C₂×C₁₁×C₁₇	—	3	12	no	—
11	C₃×C₁₁²	—	4	1	no	Hall
12	C₂×C₁₁²	—	3	1	no	—
13	A₄×C₁₇	C₁₁²	1	1	yes	characteristic, direct product factor, Hall
14	C₁₁×C₁₇	C₁₁×A₄	1	12	yes	direct product factor
15	C₁₁×A₄	C₁₁×C₁₇	1	12	yes	direct product factor
16	C₁₁²	A₄×C₁₇	1	1	yes	characteristic, direct product factor, Sylow
17	C₂²×C₁₇	C₃×C₁₁²	1	1	yes	characteristic, complemented, Hall
18	C₃×C₁₇	—	4	1	no	Hall
19	C₂²×C₁₁	C₃×C₁₁×C₁₇	1	12	yes	complemented
20	C₂×C₁₇	—	3	1	no	—
21	C₃×C₁₁	—	4	12	no	—
22	C₂×C₁₁	—	3	12	no	—
23	C₁₇	C₁₁²×A₄	1	1	yes	characteristic, direct product factor, Sylow
24	A₄	C₁₁²×C₁₇	1	1	yes	characteristic, direct product factor, Hall
25	C₁₁	C₁₁×A₄×C₁₇	1	12	yes	direct product factor
26	C₂²	C₃×C₁₁²×C₁₇	1	1	yes	characteristic, complemented, Sylow
27	C₃	—	4	1	no	Sylow
28	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₁₁²×C₁₇	A₄
commutator, cosocle, hypocenter	C₂²	C₃×C₁₁²×C₁₇
Frattini	C₁	G
Fitting, socle	C₂²×C₁₁²×C₁₇	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₁₁²×C₁₇

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.