Subgroups — C₃×C₂²⋊(C₂²⋊C₂₇) #13048166

Order 1296.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃×C₉	C₃	1	1	yes	maximal, characteristic
2	C₂²⋊(C₂²⋊C₂₇)	C₃	1	3	yes	maximal, direct product factor
3	C₃×C₂²⋊C₂₇	—	4	5	no	maximal
4	C₂³×C₃×C₉	—	3	5	no	—
5	C₂⁴×C₃²	C₉	1	1	yes	characteristic
6	C₂⁴×C₉	C₉	1	2	yes	—
7	C₂⁴×C₉	C₃²	1	1	yes	characteristic
8	C₂²×C₃×C₉	—	3	10	no	—
9	C₂²×C₃×C₉	A₄	1	5	yes	—
10	C₂²⋊C₂₇	—	4	15	no	—
11	C₃×C₂₇	—	16	1	no	Sylow
12	C₂³×C₃²	—	3	5	no	—
13	C₂³×C₉	—	3	10	no	—
14	C₂³×C₉	—	3	5	no	—
15	C₂×C₃×C₉	—	3	5	no	—
16	C₂⁴×C₃	C₂₇	1	3	yes	complemented
17	C₂⁴×C₃	C₃×C₉	1	1	yes	characteristic
18	C₂²×C₃²	—	3	10	no	—
19	C₂²×C₃²	C₂²⋊C₉	1	5	yes	—
20	C₂²×C₉	—	3	10	no	—
21	C₂²×C₉	—	3	20	no	—
22	C₂²×C₉	C₂²⋊C₉	1	10	yes	—
23	C₂²×C₉	C₃×A₄	1	5	yes	—
24	C₃×C₉	C₂²⋊A₄	1	1	yes	characteristic
25	C₂₇	—	16	3	no	—
26	C₂³×C₃	—	3	15	no	—
27	C₂³×C₃	—	3	5	no	—
28	C₂×C₃²	—	3	5	no	—
29	C₂×C₉	—	3	5	no	—
30	C₂×C₉	—	3	10	no	—
31	C₂⁴	C₃×C₂₇	1	1	yes	characteristic, complemented, Sylow
32	C₂²×C₃	—	3	10	no	—
33	C₂²×C₃	—	3	30	no	—
34	C₂²×C₃	C₂²⋊C₂₇	1	15	yes	complemented
35	C₂²×C₃	C₃×C₂²⋊C₉	1	5	yes	—
36	C₃²	C₂²⋊(C₂²⋊C₉)	1	1	yes	characteristic
37	C₉	C₂²⋊(C₂²⋊C₉)	1	2	yes	—
38	C₉	C₃×C₂²⋊A₄	1	1	yes	characteristic
39	C₂³	—	3	5	no	—
40	C₂×C₃	—	3	15	no	—
41	C₂×C₃	—	3	5	no	—
42	C₂²	—	3	10	no	—
43	C₂²	C₃×C₂²⋊C₂₇	1	5	yes	complemented
44	C₃	C₂²⋊(C₂²⋊C₂₇)	1	3	yes	direct product factor
45	C₃	C₃×C₂²⋊(C₂²⋊C₉)	1	1	yes	characteristic
46	C₂	—	3	5	no	—

	subgroup	G/H
center, hypercenter	C₃×C₉	C₂²⋊A₄
commutator, hypocenter	C₂⁴	C₃×C₂₇
Frattini	C₉	C₃×C₂²⋊A₄
Fitting	C₂⁴×C₃×C₉	C₃
socle, prime radical	C₂⁴×C₃²	C₉
cosocle	C₂⁴×C₉	C₃²
LCS	G ▷ C₂⁴
UCS	C₁ ◁ C₃×C₉

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂²⋊(C₂²⋊C₂₇)	C₃	direct (N × H)
16. C₂⁴×C₃	C₂₇	semidirect
31. C₂⁴	C₃×C₂₇	semidirect
34. C₂²×C₃	C₂²⋊C₂₇	semidirect
43. C₂²	C₃×C₂²⋊C₂₇	semidirect
44. C₃	C₂²⋊(C₂²⋊C₂₇)	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁴×C₃×C₉	3
C₂²⋊(C₂²⋊C₂₇)	3
C₃×C₂²⋊C₂₇	4

Subgroups — C3×C22⋊(C22⋊C27) #13048166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₂²⋊(C₂²⋊C₂₇) #13048166