Subgroups — #11757432

Order 972.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#66436	C₃	1	1	yes	maximal, characteristic
2	C₂²×C₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
3	#66353	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₂²⋊C₂₇	C₃	1	1	yes	maximal, characteristic, complemented
5	#6873	—	4	1	no	maximal, Sylow
6	C₂×C₂₇⋊C₃	—	3	1	no	—
7	C₂²×C₃×C₉	C₃²	1	1	yes	characteristic
8	C₂²⋊H(3,𝔽₃)	C₉	1	1	yes	characteristic
9	#1094	C₉	1	2	yes	—
10	C₉⋊A₄	—	3	2	no	—
11	C₉×A₄	—	3	1	no	—
12	C₂²×C₂₇	—	3	1	no	—
13	C₂²⋊C₂₇	—	3	1	no	—
14	C₂²⋊C₂₇	—	3	1	no	—
15	#737	—	4	1	no	—
16	C₂₇⋊C₃	—	4	1	no	—
17	C₂₇⋊C₃	A₄	1	1	yes	characteristic, complemented
18	C₃×C₂₇	—	4	1	no	—
19	C₂×C₃×C₉	—	3	1	no	—
20	C₂×C₂₇	—	3	3	no	—
21	C₂²×C₃²	C₃×C₉	1	1	yes	characteristic
22	C₃×A₄	—	3	1	no	—
23	C₂²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
24	C₂²×C₉	3₋¹⁺²	1	2	yes	—
25	C₂²⋊C₉	—	3	2	no	—
26	3₋¹⁺²	—	4	2	no	—
27	3₋¹⁺²	—	12	2	no	—
28	H(3,𝔽₃)	—	4	1	no	—
29	C₃×C₉	—	12	1	no	—
30	C₃×C₉	C₃×A₄	1	1	yes	characteristic
31	C₂₇	—	3	1	no	—
32	C₂₇	—	12	1	no	—
33	C₂₇	—	12	1	no	—
34	C₂×C₃²	—	3	1	no	—
35	C₂×C₉	—	3	2	no	—
36	C₂×C₉	—	3	1	no	—
37	C₂²×C₃	—	3	1	no	—
38	C₂²×C₃	C₃²⋊C₉	1	1	yes	characteristic
39	A₄	—	9	1	no	—
40	C₃²	—	12	1	no	—
41	C₃²	C₉×A₄	1	1	yes	characteristic
42	C₉	—	12	2	no	—
43	C₉	C₉⋊A₄	1	2	yes	—
44	C₉	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
45	C₂×C₃	—	3	1	no	—
46	C₂×C₃	—	9	1	no	—
47	C₂²	#6873	1	1	yes	characteristic, complemented, Sylow
48	C₃	—	3	1	no	—
49	C₃	—	36	1	no	—
50	C₃	(C₃×A₄)⋊C₉	1	1	yes	characteristic
51	C₂	—	3	1	no	—

	subgroup	G/H
center	C₉	C₂²⋊H(3,𝔽₃)
commutator	C₂²×C₃²	C₃×C₉
Frattini	C₃×C₉	C₃×A₄
Fitting	C₂²×C₂₇⋊C₃	C₃
socle	C₂²×C₃	C₃²⋊C₉
prime radical	C₂²⋊H(3,𝔽₃)	C₉
cosocle	C₂²×C₃×C₉	C₃²
hypercenter	C₂₇⋊C₃	A₄
hypocenter	C₂²	#6873
LCS	G ▷ C₂²×C₃² ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₉ ◁ C₃×C₉ ◁ C₂₇⋊C₃

Complements

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

N	H ≅ G/N	type
2. C₂²×C₂₇⋊C₃	C₃	semidirect
3. #66353	C₃	semidirect
4. C₃×C₂²⋊C₂₇	C₃	semidirect
17. C₂₇⋊C₃	A₄	semidirect
47. C₂²	#6873	semidirect

Maximal subgroups

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

subgroup	index
#66436	3
C₂²×C₂₇⋊C₃	3
#66353	3
C₃×C₂²⋊C₂₇	3
#6873	4