Subgroups — H(3,𝔽₃)×C₃₁⋊C₂₇ #36381111

Order 22599.

292 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₃₁⋊C₂₇	C₃	1	4	yes	maximal, complemented
2	C₃²⋊(C₃₁⋊C₂₇)	C₃	1	8	yes	maximal, complemented
3	C₉×H(3,𝔽₃)×C₃₁	C₃	1	1	yes	maximal, characteristic
4	C₃×C₃₁⋊C₂₇	—	3	12	no	—
5	C₃×C₃₁⋊C₂₇	C₃²	1	8	yes	—
6	C₃×C₃₁⋊C₂₇	C₃²	1	1	yes	characteristic
7	C₃×H(3,𝔽₃)×C₃₁	C₉	1	1	yes	characteristic
8	C₃²×C₉×C₃₁	C₃²	1	4	yes	—
9	C₃₁×C₃²⋊C₉	C₉	1	8	yes	—
10	C₃³×C₃₁	C₃×C₉	1	4	yes	—
11	H(3,𝔽₃)×C₃₁	C₂₇	1	9	yes	complemented
12	C₃×C₉×C₃₁	—	3	4	no	—
13	C₃×C₉×C₃₁	—	3	8	no	—
14	C₃×C₉×C₃₁	C₃×C₉	1	8	yes	—
15	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
16	C₃₁⋊C₂₇	—	3	8	no	—
17	C₃₁⋊C₂₇	H(3,𝔽₃)	1	3	yes	direct product factor
18	C₂₇×H(3,𝔽₃)	—	31	1	no	maximal, Sylow
19	C₃²×C₃₁	—	3	4	no	—
20	C₃²×C₃₁	—	3	8	no	—
21	C₃²×C₃₁	C₃×C₂₇	1	12	yes	complemented
22	C₃²×C₃₁	C₃²×C₉	1	1	yes	characteristic
23	C₉×C₃₁	—	3	8	no	—
24	C₉×C₃₁	C₃×H(3,𝔽₃)	1	1	yes	characteristic
25	C₉×C₃₁	#737	1	2	yes	—
26	C₃²×C₂₇	—	31	4	no	—
27	C₉×H(3,𝔽₃)	C₃₁⋊C₃	1	1	yes	characteristic
28	C₃²⋊C₂₇	—	31	8	no	—
29	C₃×C₃₁	—	3	12	no	—
30	C₃×C₃₁	C₉×H(3,𝔽₃)	1	1	yes	characteristic
31	C₃×C₃₁	C₃²×C₂₇	1	1	yes	characteristic
32	C₃×C₃₁	#6903	1	2	yes	—
33	C₃×H(3,𝔽₃)	C₃₁⋊C₉	1	1	yes	characteristic
34	C₃²×C₉	C₃×C₃₁⋊C₃	1	4	yes	—
35	C₃×C₂₇	—	31	1	no	—
36	C₃×C₂₇	—	31	8	no	—
37	C₃×C₂₇	—	93	12	no	—
38	C₃²⋊C₉	C₃₁⋊C₉	1	8	yes	—
39	C₃₁	C₂₇×H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
40	C₃³	C₃×C₃₁⋊C₉	1	4	yes	—
41	H(3,𝔽₃)	C₃₁⋊C₂₇	1	9	yes	direct product factor
42	C₃×C₉	—	3	4	no	—
43	C₃×C₉	—	3	8	no	—
44	C₃×C₉	C₃×C₃₁⋊C₉	1	8	yes	—
45	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
46	C₂₇	—	31	3	no	—
47	C₂₇	—	93	8	no	—
48	C₃²	—	3	4	no	—
49	C₃²	—	3	8	no	—
50	C₃²	C₃×C₃₁⋊C₂₇	1	12	yes	complemented
51	C₃²	C₃²×C₃₁⋊C₉	1	1	yes	characteristic
52	C₉	—	3	8	no	—
53	C₉	H(3,𝔽₃)×C₃₁⋊C₃	1	1	yes	characteristic
54	C₉	#14578911	1	2	yes	—
55	C₃	—	3	12	no	—
56	C₃	H(3,𝔽₃)×C₃₁⋊C₉	1	1	yes	characteristic
57	C₃	C₃²×C₃₁⋊C₂₇	1	1	yes	characteristic
58	C₃	#21419371	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃×C₉	C₃²×C₃₁⋊C₃
commutator	C₃×C₃₁	C₃²×C₂₇
Fitting	C₉×H(3,𝔽₃)×C₃₁	C₃
socle	C₃²×C₃₁	C₃²×C₉
prime radical	C₃×H(3,𝔽₃)×C₃₁	C₉
cosocle	C₃×C₉×C₃₁	C₃³
hypercenter	C₉×H(3,𝔽₃)	C₃₁⋊C₃
hypocenter	C₃₁	C₂₇×H(3,𝔽₃)
LCS	G ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃×C₉ ◁ C₉×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃²×C₃₁⋊C₂₇	C₃	semidirect
2. C₃²⋊(C₃₁⋊C₂₇)	C₃	semidirect
11. H(3,𝔽₃)×C₃₁	C₂₇	semidirect
17. C₃₁⋊C₂₇	H(3,𝔽₃)	direct (N × H)
21. C₃²×C₃₁	C₃×C₂₇	semidirect
39. C₃₁	C₂₇×H(3,𝔽₃)	semidirect
41. H(3,𝔽₃)	C₃₁⋊C₂₇	direct (N × H)
50. C₃²	C₃×C₃₁⋊C₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₃₁⋊C₂₇	3
C₃²⋊(C₃₁⋊C₂₇)	3
C₉×H(3,𝔽₃)×C₃₁	3
C₂₇×H(3,𝔽₃)	31

Subgroups — H(3,𝔽3)×C31⋊C27 #36381111

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽₃)×C₃₁⋊C₂₇ #36381111