Subgroups — #14552149

Order 2187.

722 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#6898	C₃	1	1	yes	maximal, characteristic
2	#10624561	C₃	1	8	yes	maximal, complemented
3	#10624559	C₃	1	8	yes	maximal, complemented
4	#10624559	C₃	1	8	yes	maximal, complemented
5	#10624557	C₃	1	8	yes	maximal
6	#10624556	C₃	1	4	yes	maximal
7	#10624552	C₃	1	1	yes	maximal, characteristic, complemented
8	#10624550	C₃	1	2	yes	maximal, complemented
9	C₃³×C₉	C₃²	1	1	yes	characteristic
10	#6898	C₉	1	27	yes	complemented
11	C₃×C₉⋊C₉	C₃²	1	4	yes	—
12	C₃×C₉⋊C₉	C₃²	1	8	yes	—
13	C₃×C₉⋊C₉	C₃²	1	8	yes	—
14	C₃×C₉⋊C₉	C₃²	1	8	yes	—
15	C₃×C₉⋊C₉	C₃²	1	8	yes	—
16	C₃×C₃²⋊C₉	C₃²	1	8	yes	—
17	C₃×C₃²⋊C₉	C₃²	1	4	yes	—
18	C₃×C₉²	C₃²	1	8	yes	—
19	C₃×C₉²	C₃²	1	1	yes	characteristic
20	#6855	C₃²	1	4	yes	—
21	#6855	C₃²	1	8	yes	—
22	#6855	C₃²	1	4	yes	—
23	#6855	C₃²	1	8	yes	—
24	#6855	C₃²	1	8	yes	—
25	#6855	C₃²	1	8	yes	—
26	#6855	C₃²	1	16	yes	—
27	#6855	C₃²	1	8	yes	—
28	#6855	C₃²	1	8	yes	—
29	C₃⁴	C₃³	1	1	yes	characteristic
30	C₃²×C₉	—	9	4	no	—
31	C₃²×C₉	C₃³	1	1	yes	characteristic
32	C₃²×C₉	C₃³	1	2	yes	—
33	C₃²×C₉	C₃³	1	8	yes	—
34	C₃²×C₉	C₃³	1	8	yes	—
35	C₃²×C₉	C₃³	1	8	yes	—
36	C₃²×C₉	C₃³	1	8	yes	—
37	C₃²×C₉	C₃³	1	4	yes	—
38	C₉⋊C₉	—	9	8	no	—
39	C₉⋊C₉	—	9	8	no	—
40	C₉⋊C₉	—	9	4	no	—
41	C₉⋊C₉	—	9	8	no	—
42	C₉⋊C₉	C₃×C₉	1	72	yes	complemented
43	C₃²⋊C₉	—	9	8	no	—
44	C₃²⋊C₉	C₃×C₉	1	36	yes	—
45	C₉²	—	9	8	no	—
46	C₉²	C₃×C₉	1	9	yes	complemented
47	C₃³	—	3	8	no	—
48	C₃³	—	3	4	no	—
49	C₃³	C₃²×C₉	1	3	yes	—
50	C₃³	C₃⁴	1	1	yes	characteristic
51	C₃×C₉	—	3	8	no	—
52	C₃×C₉	—	3	8	no	—
53	C₃×C₉	—	3	4	no	—
54	C₃×C₉	—	3	8	no	—
55	C₃×C₉	—	3	8	no	—
56	C₃×C₉	—	3	8	no	—
57	C₃×C₉	—	3	8	no	—
58	C₃×C₉	—	3	8	no	—
59	C₃×C₉	—	3	8	no	—
60	C₃×C₉	—	3	8	no	—
61	C₃×C₉	—	3	4	no	—
62	C₃×C₉	—	3	8	no	—
63	C₃×C₉	—	3	8	no	—
64	C₃×C₉	—	3	8	no	—
65	C₃×C₉	—	3	8	no	—
66	C₃×C₉	—	3	8	no	—
67	C₃×C₉	—	3	8	no	—
68	C₃×C₉	—	3	8	no	—
69	C₃×C₉	—	3	4	no	—
70	C₃×C₉	—	3	4	no	—
71	C₃×C₉	—	9	9	no	—
72	C₃×C₉	C₃²×C₉	1	12	yes	—
73	C₃×C₉	C₃²×C₉	1	24	yes	—
74	C₃²	—	3	12	no	—
75	C₃²	—	9	1	no	—
76	C₃²	—	9	8	no	—
77	C₃²	C₃³×C₉	1	1	yes	characteristic
78	C₃²	3₋¹⁺⁴	1	4	yes	—
79	C₃²	3₋¹⁺⁴	1	8	yes	—
80	C₉	—	3	24	no	—
81	C₉	—	3	12	no	—
82	C₉	—	9	2	no	—
83	C₉	—	9	1	no	—
84	C₉	—	9	8	no	—
85	C₉	—	9	8	no	—
86	C₉	—	9	8	no	—
87	C₃	—	9	3	no	—
88	C₃	#10624864	1	4	yes	—
89	C₃	#10624913	1	1	yes	characteristic
90	C₃	#10624914	1	8	yes	—

	subgroup	G/H
center, Frattini, socle, cosocle	C₃³	C₃⁴
commutator	C₃²	C₃³×C₉
prime radical	C₃⁴	C₃³
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃³ ◁ G
LECS	G ▷ C₃³ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #10624561	C₃	semidirect
3. #10624559	C₃	semidirect
4. #10624559	C₃	semidirect
7. #10624552	C₃	semidirect
8. #10624550	C₃	semidirect
10. #6898	C₉	semidirect
42. C₉⋊C₉	C₃×C₉	semidirect
46. C₉²	C₃×C₉	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6898	3
#10624561	3
#10624559	3
#10624557	3
#10624556	3
#10624552	3
#10624550	3