Subgroups — 3₋¹⁺²×C₃₁⋊C₂₇ #36381203

Order 22599.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₃₁⋊C₂₇	C₃	1	1	yes	maximal, characteristic
2	#21419248	C₃	1	3	yes	maximal, complemented
3	#21419247	C₃	1	3	yes	maximal, complemented
4	C₃²⋊(C₃₁⋊C₂₇)	C₃	1	2	yes	maximal
5	C₉×C₃₁⋊C₂₇	C₃	1	3	yes	maximal, complemented
6	C₉×3₋¹⁺²×C₃₁	C₃	1	1	yes	maximal, characteristic
7	C₃×C₃₁⋊C₂₇	—	3	3	no	—
8	C₃×C₃₁⋊C₂₇	C₃²	1	1	yes	characteristic
9	C₃×C₃₁⋊C₂₇	C₃²	1	1	yes	characteristic
10	C₃×C₃₁⋊C₂₇	C₃²	1	6	yes	—
11	C₃×C₃₁⋊C₂₇	C₃²	1	1	yes	characteristic
12	C₃×3₋¹⁺²×C₃₁	C₉	1	1	yes	characteristic
13	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
14	C₃₁×C₉⋊C₉	C₉	1	6	yes	—
15	C₃₁×C₃²⋊C₉	C₉	1	2	yes	—
16	C₉²×C₃₁	C₃²	1	3	yes	—
17	C₃³×C₃₁	C₃×C₉	1	1	yes	characteristic
18	3₋¹⁺²×C₃₁	C₂₇	1	9	yes	complemented
19	C₃×C₉×C₃₁	—	3	1	no	—
20	C₃×C₉×C₃₁	—	3	2	no	—
21	C₃×C₉×C₃₁	C₃×C₉	1	6	yes	—
22	C₃×C₉×C₃₁	C₃×C₉	1	2	yes	—
23	C₃×C₉×C₃₁	C₃×C₉	1	3	yes	—
24	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
25	C₃₁⋊C₂₇	—	3	1	no	—
26	C₃₁⋊C₂₇	—	3	1	no	—
27	C₃₁⋊C₂₇	—	3	6	no	—
28	C₃₁⋊C₂₇	3₋¹⁺²	1	3	yes	direct product factor
29	C₂₇×3₋¹⁺²	—	31	1	no	maximal, Sylow
30	C₃²×C₃₁	—	3	1	no	—
31	C₃²×C₃₁	—	3	2	no	—
32	C₃²×C₃₁	C₃×C₂₇	1	3	yes	—
33	C₃²×C₃₁	C₃²×C₉	1	1	yes	characteristic
34	C₉×C₃₁	—	3	2	no	—
35	C₉×C₃₁	—	3	6	no	—
36	C₉×C₃₁	C₃×C₂₇	1	9	yes	complemented
37	C₉×C₃₁	C₃×3₋¹⁺²	1	1	yes	characteristic
38	C₉×C₃₁	#737	1	2	yes	—
39	C₃²×C₂₇	—	31	1	no	—
40	C₉×3₋¹⁺²	C₃₁⋊C₃	1	1	yes	characteristic
41	C₉⋊C₂₇	—	31	3	no	—
42	C₉⋊C₂₇	—	31	3	no	—
43	C₃²⋊C₂₇	—	31	2	no	—
44	C₉×C₂₇	—	31	3	no	—
45	C₃×C₃₁	—	3	3	no	—
46	C₃×C₃₁	C₉×3₋¹⁺²	1	1	yes	characteristic
47	C₃×C₃₁	C₃²×C₂₇	1	1	yes	characteristic
48	C₃×C₃₁	#6903	1	2	yes	—
49	C₃×3₋¹⁺²	C₃₁⋊C₉	1	1	yes	characteristic
50	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
51	C₃×C₂₇	—	31	1	no	—
52	C₃×C₂₇	—	31	6	no	—
53	C₃×C₂₇	—	31	1	no	—
54	C₃×C₂₇	—	31	1	no	—
55	C₃×C₂₇	—	93	3	no	—
56	C₉⋊C₉	C₃₁⋊C₉	1	6	yes	—
57	C₃²⋊C₉	C₃₁⋊C₉	1	2	yes	—
58	C₉²	C₃×C₃₁⋊C₃	1	3	yes	—
59	C₃₁	C₂₇×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
60	C₃³	C₃×C₃₁⋊C₉	1	1	yes	characteristic
61	3₋¹⁺²	C₃₁⋊C₂₇	1	9	yes	direct product factor
62	C₃×C₉	—	3	1	no	—
63	C₃×C₉	—	3	2	no	—
64	C₃×C₉	C₃×C₃₁⋊C₉	1	6	yes	—
65	C₃×C₉	C₃×C₃₁⋊C₉	1	2	yes	—
66	C₃×C₉	C₃×C₃₁⋊C₉	1	3	yes	—
67	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
68	C₂₇	—	31	3	no	—
69	C₂₇	—	93	1	no	—
70	C₂₇	—	93	1	no	—
71	C₂₇	—	93	6	no	—
72	C₃²	—	3	1	no	—
73	C₃²	—	3	2	no	—
74	C₃²	C₃×C₃₁⋊C₂₇	1	3	yes	—
75	C₃²	C₃²×C₃₁⋊C₉	1	1	yes	characteristic
76	C₉	—	3	2	no	—
77	C₉	—	3	6	no	—
78	C₉	C₃×C₃₁⋊C₂₇	1	9	yes	complemented
79	C₉	3₋¹⁺²×C₃₁⋊C₃	1	1	yes	characteristic
80	C₉	#14578913	1	2	yes	—
81	C₃	—	3	3	no	—
82	C₃	3₋¹⁺²×C₃₁⋊C₉	1	1	yes	characteristic
83	C₃	C₃²×C₃₁⋊C₂₇	1	1	yes	characteristic
84	C₃	#21419371	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃×C₉	C₃²×C₃₁⋊C₃
commutator	C₃×C₃₁	C₃²×C₂₇
Fitting	C₉×3₋¹⁺²×C₃₁	C₃
socle	C₃²×C₃₁	C₃²×C₉
prime radical	C₃³×C₃₁	C₃×C₉
cosocle	C₃×C₉×C₃₁	C₃³
hypercenter	C₉×3₋¹⁺²	C₃₁⋊C₃
hypocenter	C₃₁	C₂₇×3₋¹⁺²
LCS	G ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃×C₉ ◁ C₉×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
2. #21419248	C₃	semidirect
3. #21419247	C₃	semidirect
5. C₉×C₃₁⋊C₂₇	C₃	semidirect
18. 3₋¹⁺²×C₃₁	C₂₇	semidirect
28. C₃₁⋊C₂₇	3₋¹⁺²	direct (N × H)
36. C₉×C₃₁	C₃×C₂₇	semidirect
59. C₃₁	C₂₇×3₋¹⁺²	semidirect
61. 3₋¹⁺²	C₃₁⋊C₂₇	direct (N × H)
78. C₉	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
#21419248	3
#21419247	3
C₃²⋊(C₃₁⋊C₂₇)	3
C₉×C₃₁⋊C₂₇	3
C₉×3₋¹⁺²×C₃₁	3
C₂₇×3₋¹⁺²	31

Subgroups — 3−1+2×C31⋊C27 #36381203

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×C₃₁⋊C₂₇ #36381203