Subgroups — #14552072

Order 2187.

812 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#6887	C₃	1	1	yes	maximal, characteristic
2	#10624561	C₃	1	6	yes	maximal, complemented
3	#10624559	C₃	1	6	yes	maximal, complemented
4	#10624559	C₃	1	3	yes	maximal, complemented
5	#10624557	C₃	1	6	yes	maximal
6	#10624557	C₃	1	3	yes	maximal
7	#10624553	C₃	1	2	yes	maximal, complemented
8	#10624552	C₃	1	1	yes	maximal, characteristic, complemented
9	#10624551	C₃	1	2	yes	maximal, complemented
10	#10624550	C₃	1	1	yes	maximal, characteristic, complemented
11	#10624548	C₃	1	2	yes	maximal
12	C₉×C₉⋊C₉	C₃	1	2	yes	maximal, complemented
13	C₉×C₉⋊C₉	C₃	1	3	yes	maximal, complemented
14	C₉×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
15	#10624544	C₃	1	1	yes	maximal, characteristic, complemented
16	C₃³×C₉	C₃²	1	1	yes	characteristic
17	#6887	C₉	1	27	yes	complemented
18	C₃×C₉⋊C₉	C₃²	1	6	yes	—
19	C₃×C₉⋊C₉	C₃²	1	6	yes	—
20	C₃×C₉⋊C₉	C₃²	1	6	yes	—
21	C₃×C₉⋊C₉	C₃²	1	6	yes	—
22	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
23	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
24	C₃×C₉⋊C₉	C₃²	1	2	yes	—
25	C₃×C₉⋊C₉	C₃²	1	2	yes	—
26	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
27	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
28	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
29	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
30	C₃×C₉²	C₃²	1	3	yes	—
31	C₃×C₉²	C₃²	1	3	yes	—
32	C₃×C₉²	C₃²	1	6	yes	—
33	C₃×C₉²	C₃²	1	1	yes	characteristic
34	C₃×C₉²	C₃²	1	2	yes	—
35	#6855	C₃²	1	1	yes	characteristic
36	#6855	C₃²	1	2	yes	—
37	#6855	C₃²	1	6	yes	—
38	#6855	C₃²	1	1	yes	characteristic
39	#6855	C₃²	1	1	yes	characteristic
40	#6855	C₃²	1	3	yes	—
41	#6855	C₃²	1	2	yes	—
42	#6855	C₃²	1	2	yes	—
43	#6855	C₃²	1	3	yes	—
44	#6855	C₃²	1	6	yes	—
45	#6855	C₃²	1	3	yes	—
46	#6855	C₃²	1	6	yes	—
47	#6855	C₃²	1	6	yes	—
48	#6855	C₃²	1	6	yes	—
49	#6855	C₃²	1	6	yes	—
50	#6855	C₃²	1	6	yes	—
51	#6855	C₃²	1	6	yes	—
52	#6855	C₃²	1	6	yes	—
53	C₃⁴	C₃³	1	1	yes	characteristic
54	C₃²×C₉	—	3	3	no	—
55	C₃²×C₉	—	3	6	no	—
56	C₃²×C₉	C₃×C₉	1	9	yes	—
57	C₃²×C₉	C₃³	1	3	yes	—
58	C₃²×C₉	C₃³	1	3	yes	—
59	C₃²×C₉	C₃³	1	6	yes	—
60	C₃²×C₉	C₃³	1	6	yes	—
61	C₃²×C₉	C₃³	1	6	yes	—
62	C₃²×C₉	C₃³	1	1	yes	characteristic
63	C₃²×C₉	C₃³	1	1	yes	characteristic
64	C₃²×C₉	C₃³	1	1	yes	characteristic
65	C₃²×C₉	C₃³	1	2	yes	—
66	C₃²×C₉	C₃³	1	2	yes	—
67	C₃²×C₉	C₃³	1	2	yes	—
68	C₃²×C₉	C₃³	1	2	yes	—
69	C₃²×C₉	C₃³	1	3	yes	—
70	C₃²×C₉	C₃³	1	1	yes	characteristic
71	C₉⋊C₉	—	3	18	no	—
72	C₉⋊C₉	—	3	18	no	—
73	C₉⋊C₉	—	9	6	no	—
74	C₉⋊C₉	—	9	6	no	—
75	C₉⋊C₉	C₃×C₉	1	9	yes	—
76	C₉⋊C₉	C₃×C₉	1	18	yes	—
77	C₉⋊C₉	C₃×C₉	1	9	yes	—
78	C₉⋊C₉	C₃×C₉	1	18	yes	—
79	C₃²⋊C₉	—	3	9	no	—
80	C₃²⋊C₉	—	9	6	no	—
81	C₃²⋊C₉	C₃×C₉	1	9	yes	—
82	C₃²⋊C₉	C₃×C₉	1	18	yes	—
83	C₉²	—	3	9	no	—
84	C₉²	—	3	9	no	—
85	C₉²	—	9	6	no	—
86	C₉²	C₃×C₉	1	9	yes	—
87	C₉²	C₃×C₉	1	18	yes	—
88	C₃³	—	3	3	no	—
89	C₃³	—	3	1	no	—
90	C₃³	—	3	2	no	—
91	C₃³	—	3	6	no	—
92	C₃³	C₃²×C₉	1	3	yes	—
93	C₃³	C₃⁴	1	1	yes	characteristic
94	C₃×C₉	—	3	3	no	—
95	C₃×C₉	—	3	3	no	—
96	C₃×C₉	—	3	6	no	—
97	C₃×C₉	—	3	1	no	—
98	C₃×C₉	—	3	1	no	—
99	C₃×C₉	—	3	2	no	—
100	C₃×C₉	—	3	2	no	—
101	C₃×C₉	—	3	6	no	—
102	C₃×C₉	—	3	6	no	—
103	C₃×C₉	—	3	6	no	—
104	C₃×C₉	—	3	6	no	—
105	C₃×C₉	—	3	27	no	—
106	C₃×C₉	—	3	3	no	—
107	C₃×C₉	—	3	3	no	—
108	C₃×C₉	—	3	6	no	—
109	C₃×C₉	—	3	3	no	—
110	C₃×C₉	—	3	3	no	—
111	C₃×C₉	—	3	6	no	—
112	C₃×C₉	—	3	6	no	—
113	C₃×C₉	—	3	6	no	—
114	C₃×C₉	—	3	6	no	—
115	C₃×C₉	—	3	6	no	—
116	C₃×C₉	—	3	6	no	—
117	C₃×C₉	—	3	6	no	—
118	C₃×C₉	—	3	6	no	—
119	C₃×C₉	—	3	6	no	—
120	C₃×C₉	—	3	6	no	—
121	C₃×C₉	—	3	6	no	—
122	C₃×C₉	—	3	6	no	—
123	C₃×C₉	—	3	6	no	—
124	C₃×C₉	C₃²×C₉	1	3	yes	—
125	C₃×C₉	C₃²×C₉	1	3	yes	—
126	C₃×C₉	C₃²×C₉	1	6	yes	—
127	C₃×C₉	C₃²×C₉	1	6	yes	—
128	C₃×C₉	C₃²×C₉	1	6	yes	—
129	C₃×C₉	C₃²×C₉	1	6	yes	—
130	C₃×C₉	C₃²×C₉	1	3	yes	—
131	C₃×C₉	C₃²×C₉	1	3	yes	—
132	C₃×C₉	#737	1	9	yes	—
133	C₃×C₉	#737	1	3	yes	—
134	C₃×C₉	#737	1	6	yes	—
135	C₃²	—	3	3	no	—
136	C₃²	—	3	3	no	—
137	C₃²	—	3	6	no	—
138	C₃²	—	9	3	no	—
139	C₃²	—	9	6	no	—
140	C₃²	C₃³×C₉	1	1	yes	characteristic
141	C₃²	C₃×#737	1	1	yes	characteristic
142	C₃²	C₃×#737	1	2	yes	—
143	C₃²	C₃×#737	1	3	yes	—
144	C₃²	3₋¹⁺⁴	1	6	yes	—
145	C₉	—	3	3	no	—
146	C₉	—	3	3	no	—
147	C₉	—	3	3	no	—
148	C₉	—	3	6	no	—
149	C₉	—	3	6	no	—
150	C₉	—	3	6	no	—
151	C₉	—	3	6	no	—
152	C₉	—	3	9	no	—
153	C₉	—	9	3	no	—
154	C₉	—	9	3	no	—
155	C₉	—	9	6	no	—
156	C₉	—	9	6	no	—
157	C₉	—	9	6	no	—
158	C₉	#6887	1	9	yes	—
159	C₃	—	9	3	no	—
160	C₃	C₃×#6887	1	1	yes	characteristic
161	C₃	C₉×#737	1	1	yes	characteristic
162	C₃	#10624864	1	2	yes	—
163	C₃	#10624904	1	3	yes	—
164	C₃	#10624908	1	6	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. #10624553	C₃	semidirect
8. #10624552	C₃	semidirect
9. #10624551	C₃	semidirect
10. #10624550	C₃	semidirect
12. C₉×C₉⋊C₉	C₃	semidirect
13. C₉×C₉⋊C₉	C₃	semidirect
15. #10624544	C₃	semidirect
17. #6887	C₉	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6887	3
#10624561	3
#10624559	3
#10624557	3
#10624553	3
#10624552	3
#10624551	3
#10624550	3
#10624548	3
C₉×C₉⋊C₉	3
C₉×C₃²⋊C₉	3
#10624544	3