Subgroups — #36381204

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	C₉×#11719689	C₃	1	1	yes	maximal, characteristic
3	(C₃₁⋊C₉)⋊C₂₇	C₃	1	6	yes	maximal, complemented
4	C₃²⋊(C₃₁⋊C₂₇)	C₃	1	1	yes	maximal, characteristic
5	C₂₇×C₃₁⋊C₉	C₃	1	3	yes	maximal, complemented
6	C₃₁×C₃²⋊C₂₇	C₃	1	1	yes	maximal, characteristic
7	C₃×#11719689	C₉	1	1	yes	characteristic
8	C₃×C₃₁⋊C₂₇	—	3	3	no	—
9	C₃×C₃₁⋊C₂₇	C₃²	1	2	yes	—
10	C₃×C₃₁⋊C₂₇	C₃²	1	3	yes	—
11	C₃×C₃₁⋊C₂₇	C₃²	1	1	yes	characteristic
12	(C₃₁⋊C₉)⋊C₉	C₉	1	6	yes	—
13	C₃²⋊(C₃₁⋊C₉)	C₉	1	2	yes	—
14	C₉×C₃₁⋊C₉	C₃²	1	3	yes	—
15	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
16	C₃×C₂₇×C₃₁	C₃²	1	3	yes	—
17	C₃³×C₃₁	C₃×C₉	1	1	yes	characteristic
18	C₃×C₉×C₃₁	—	3	1	no	—
19	C₃×C₉×C₃₁	—	3	1	no	—
20	C₃×C₉×C₃₁	—	3	1	no	—
21	C₃×C₉×C₃₁	C₃×C₉	1	2	yes	—
22	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
23	#11719689	C₂₇	1	9	yes	complemented
24	C₃×C₃₁⋊C₉	C₃×C₉	1	6	yes	—
25	C₃×C₃₁⋊C₉	C₃×C₉	1	3	yes	—
26	C₂₇×C₃₁	—	3	3	no	—
27	C₃₁⋊C₂₇	—	3	2	no	—
28	C₃₁⋊C₂₇	—	3	3	no	—
29	C₃₁⋊C₂₇	3₋¹⁺²	1	3	yes	complemented
30	C₂₇×3₋¹⁺²	—	31	1	no	maximal, Sylow
31	C₃²×C₃₁	—	3	1	no	—
32	C₃²×C₃₁	—	3	2	no	—
33	C₃²×C₃₁	C₃×C₂₇	1	3	yes	—
34	C₃²×C₃₁	C₃²×C₉	1	1	yes	characteristic
35	C₉×C₃₁	—	3	2	no	—
36	C₉×C₃₁	C₃×3₋¹⁺²	1	1	yes	characteristic
37	C₉×C₃₁	#737	1	1	yes	characteristic
38	C₉×C₃₁	#737	1	1	yes	characteristic
39	C₃₁⋊C₉	—	3	6	no	—
40	C₃₁⋊C₉	C₃×C₂₇	1	9	yes	complemented
41	C₃²×C₂₇	—	31	1	no	—
42	C₉×3₋¹⁺²	—	31	1	no	—
43	C₉⋊C₂₇	—	31	6	no	—
44	C₃²⋊C₂₇	—	31	1	no	—
45	C₃²⋊C₂₇	C₃₁⋊C₃	1	1	yes	characteristic
46	C₉×C₂₇	—	31	3	no	—
47	C₃×C₃₁	—	3	3	no	—
48	C₃×C₃₁	C₉×3₋¹⁺²	1	1	yes	characteristic
49	C₃×C₃₁	C₃²×C₂₇	1	1	yes	characteristic
50	C₃×C₃₁	#6903	1	2	yes	—
51	C₃×3₋¹⁺²	—	31	1	no	—
52	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
53	C₃×C₂₇	—	31	1	no	—
54	C₃×C₂₇	—	31	3	no	—
55	C₃×C₂₇	—	31	2	no	—
56	C₃×C₂₇	—	93	3	no	—
57	C₃×C₂₇	C₃×C₃₁⋊C₃	1	3	yes	—
58	C₉⋊C₉	—	31	6	no	—
59	C₃²⋊C₉	—	31	2	no	—
60	C₉²	—	31	3	no	—
61	C₃₁	C₂₇×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
62	C₃³	C₉×C₃₁⋊C₃	1	1	yes	characteristic
63	3₋¹⁺²	—	31	9	no	—
64	C₃×C₉	—	3	1	no	—
65	C₃×C₉	—	3	1	no	—
66	C₃×C₉	—	3	1	no	—
67	C₃×C₉	—	31	3	no	—
68	C₃×C₉	—	31	6	no	—
69	C₃×C₉	C₉×C₃₁⋊C₃	1	2	yes	—
70	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
71	C₂₇	—	3	3	no	—
72	C₂₇	—	31	3	no	—
73	C₂₇	—	93	2	no	—
74	C₂₇	—	93	3	no	—
75	C₃²	—	3	1	no	—
76	C₃²	—	3	2	no	—
77	C₃²	C₂₇×C₃₁⋊C₃	1	3	yes	—
78	C₃²	C₃×C₉×C₃₁⋊C₃	1	1	yes	characteristic
79	C₉	—	3	2	no	—
80	C₉	—	31	9	no	—
81	C₉	—	93	6	no	—
82	C₉	3₋¹⁺²×C₃₁⋊C₃	1	1	yes	characteristic
83	C₉	#14578911	1	1	yes	characteristic
84	C₉	#14578913	1	1	yes	characteristic
85	C₃	—	3	3	no	—
86	C₃	#21419306	1	1	yes	characteristic
87	C₃	C₃×C₂₇×C₃₁⋊C₃	1	1	yes	characteristic
88	C₃	#21419372	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. (C₃₁⋊C₉)⋊C₂₇	C₃	semidirect
5. C₂₇×C₃₁⋊C₉	C₃	semidirect
23. #11719689	C₂₇	semidirect
29. C₃₁⋊C₂₇	3₋¹⁺²	semidirect
40. C₃₁⋊C₉	C₃×C₂₇	semidirect
61. C₃₁	C₂₇×3₋¹⁺²	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
C₉×#11719689	3
(C₃₁⋊C₉)⋊C₂₇	3
C₃²⋊(C₃₁⋊C₂₇)	3
C₂₇×C₃₁⋊C₉	3
C₃₁×C₃²⋊C₂₇	3
C₂₇×3₋¹⁺²	31