Subgroups — C₉×#5000 #13142675

Order 1728.

364 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#117	C₂	1	1	yes	maximal, characteristic
2	C₉×#830	C₂	1	2	yes	maximal
3	C₉×C₄⋊(C₃⋊C₈)	C₂	1	2	yes	maximal, complemented
4	C₉×#803	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₉×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#5000	C₃	1	1	yes	maximal, characteristic
7	C₉×#433	—	3	1	no	maximal
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₃×C₉×C₄⋊C₄	C₄	1	2	yes	—
10	C₃×C₉×C₂²⋊C₄	C₄	1	2	yes	—
11	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
13	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
14	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
15	C₃²×#117	C₂×C₃	1	1	yes	characteristic
16	C₃×#830	C₂×C₃	1	2	yes	—
17	C₃×C₄⋊(C₃⋊C₈)	C₂×C₃	1	2	yes	—
18	C₃×#803	C₂×C₃	1	1	yes	characteristic
19	C₃×C₄×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic
20	C₉×#117	—	2	1	no	—
21	C₉×#117	D₃	1	1	yes	characteristic
22	C₉×C₄⋊C₈	—	3	2	no	—
23	C₉×C₂²⋊C₈	—	3	2	no	—
24	C₉×#97	—	3	1	no	—
25	C₄×C₈×C₉	—	3	1	no	—
26	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₄×C₉	—	2	2	no	—
28	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
32	C₉×C₃⋊C₈	—	2	2	no	—
33	C₉×C₃⋊C₈	—	2	2	no	—
34	C₃×#433	—	3	1	no	—
35	#5000	C₉	1	1	yes	characteristic, direct product factor
36	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
37	C₃²×C₄⋊C₄	C₃×C₄	1	2	yes	—
38	C₃²×C₂²⋊C₄	C₃×C₄	1	2	yes	—
39	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
40	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
41	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	2	yes	—
42	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	2	yes	—
43	C₂²×C₄×C₉	—	2	1	no	—
44	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
45	C₂×C₈×C₉	—	3	2	no	—
46	C₂×C₈×C₉	—	3	2	no	—
47	C₉×C₄⋊C₄	—	2	2	no	—
48	C₉×C₄⋊C₄	Dic₃	1	2	yes	—
49	C₉×C₂²⋊C₄	—	2	2	no	—
50	C₉×C₂²⋊C₄	Dic₃	1	2	yes	—
51	C₄²×C₉	—	2	1	no	—
52	C₄²×C₉	—	2	1	no	—
53	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
54	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
55	C₂²×C₃×C₉	—	2	1	no	—
56	C₂²×C₃×C₉	—	2	2	no	—
57	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
58	C₃×C₄×C₉	—	2	1	no	—
59	C₃×C₄×C₉	—	2	2	no	—
60	C₃×C₄×C₉	—	2	2	no	—
61	C₃×C₄×C₉	#41	1	2	yes	—
62	C₃×#117	—	2	1	no	—
63	C₃×#117	C₂×C₉	1	1	yes	characteristic
64	C₃×#117	C₃×D₃	1	1	yes	characteristic
65	C₃×C₄⋊C₈	—	3	2	no	—
66	C₃×C₂²⋊C₈	—	3	2	no	—
67	C₃×#97	—	3	1	no	—
68	C₃×C₄×C₈	—	3	1	no	—
69	#830	C₂×C₉	1	2	yes	—
70	C₄⋊(C₃⋊C₈)	C₂×C₉	1	2	yes	complemented
71	#803	C₂×C₉	1	1	yes	characteristic, complemented
72	C₄×C₃⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
73	C₂³×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
74	C₂×C₃²×C₄	—	2	2	no	—
75	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
76	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
77	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
78	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
79	C₂³×C₉	—	2	1	no	—
80	C₂³×C₉	C₂×Dic₃	1	1	yes	characteristic
81	C₃×C₃⋊C₈	—	2	2	no	—
82	C₃×C₃⋊C₈	—	2	2	no	—
83	C₂×C₄×C₉	—	2	4	no	—
84	C₂×C₄×C₉	—	2	2	no	—
85	C₂×C₄×C₉	—	2	1	no	—
86	C₂×C₄×C₉	—	2	2	no	—
87	C₂×C₄×C₉	—	2	2	no	—
88	C₂×C₄×C₉	—	2	1	no	—
89	C₂×C₄×C₉	C₂×Dic₃	1	2	yes	—
90	C₂×C₄×C₉	C₂×Dic₃	1	2	yes	—
91	C₂×C₄×C₉	C₂×Dic₃	1	1	yes	characteristic
92	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
93	C₈×C₉	—	6	2	no	—
94	C₈×C₉	—	6	2	no	—
95	#433	—	3	1	no	Sylow
96	C₂×C₃×C₉	—	4	1	no	—
97	C₂×C₃×C₉	#117	1	1	yes	characteristic
98	C₂×C₃×C₉	#131	1	2	yes	—
99	C₂²×C₃×C₄	—	2	1	no	—
100	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
101	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
102	C₂×C₃×C₈	—	3	2	no	—
103	C₂×C₃×C₈	—	3	2	no	—
104	C₃×C₄⋊C₄	—	2	2	no	—
105	C₃×C₄⋊C₄	C₄×C₉	1	2	yes	—
106	C₃×C₄⋊C₄	C₃×Dic₃	1	2	yes	—
107	C₃×C₂²⋊C₄	—	2	2	no	—
108	C₃×C₂²⋊C₄	C₄×C₉	1	2	yes	—
109	C₃×C₂²⋊C₄	C₃×Dic₃	1	2	yes	—
110	C₃×C₄²	—	2	1	no	—
111	C₃×C₄²	—	2	1	no	—
112	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
113	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
114	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
115	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
116	C₂×C₃⋊C₈	C₂²×C₉	1	2	yes	—
117	C₂×C₃⋊C₈	C₂²×C₉	1	2	yes	—
118	C₂²×C₃²	—	2	1	no	—
119	C₂²×C₃²	—	2	2	no	—
120	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
121	C₃²×C₄	—	2	1	no	—
122	C₃²×C₄	—	2	2	no	—
123	C₃²×C₄	—	2	2	no	—
124	C₃²×C₄	C₃×#41	1	2	yes	—
125	C₂²×C₉	—	2	2	no	—
126	C₂²×C₉	—	2	1	no	—
127	C₂²×C₉	—	2	2	no	—
128	C₂²×C₉	—	2	1	no	—
129	C₂²×C₉	—	4	2	no	—
130	C₂²×C₉	C₂²×Dic₃	1	1	yes	characteristic
131	C₄×C₉	—	2	2	no	—
132	C₄×C₉	—	2	1	no	—
133	C₄×C₉	—	2	2	no	—
134	C₄×C₉	—	2	2	no	—
135	C₄×C₉	—	2	2	no	—
136	C₄×C₉	—	4	2	no	—
137	C₄×C₉	—	4	2	no	—
138	C₄×C₉	#235	1	2	yes	—
139	#117	D₃×C₉	1	1	yes	characteristic
140	C₄⋊C₈	—	3	2	no	—
141	C₂²⋊C₈	—	3	2	no	—
142	#97	—	3	1	no	—
143	C₄×C₈	—	3	1	no	—
144	C₃×C₉	#433	1	1	yes	characteristic, complemented, Sylow
145	C₂³×C₃	—	2	1	no	—
146	C₂³×C₃	C₂×C₄×C₉	1	1	yes	characteristic
147	C₂³×C₃	C₂×C₃×Dic₃	1	1	yes	characteristic
148	C₂×C₃×C₄	—	2	4	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	1	no	—
152	C₂×C₃×C₄	—	2	2	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	1	no	—
155	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
156	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
157	C₂×C₃×C₄	C₂×C₄×C₉	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
159	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
160	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
161	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
163	C₃×C₈	—	6	2	no	—
164	C₃×C₈	—	6	2	no	—
165	C₃⋊C₈	—	2	2	no	—
166	C₃⋊C₈	—	2	2	no	—
167	C₂×C₃²	—	4	1	no	—
168	C₂×C₃²	C₃×#117	1	1	yes	characteristic
169	C₂×C₃²	C₃×#131	1	2	yes	—
170	C₂×C₉	—	2	1	no	—
171	C₂×C₉	—	2	2	no	—
172	C₂×C₉	—	4	2	no	—
173	C₂×C₉	—	4	1	no	—
174	C₂×C₉	#926	1	1	yes	characteristic
175	C₂×C₉	#948	1	2	yes	—
176	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
177	C₂×C₈	—	3	2	no	—
178	C₂×C₈	—	3	2	no	—
179	C₄⋊C₄	C₉×Dic₃	1	2	yes	—
180	C₂²⋊C₄	C₉×Dic₃	1	2	yes	—
181	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
182	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
183	C₂²×C₃	—	2	2	no	—
184	C₂²×C₃	—	2	1	no	—
185	C₂²×C₃	—	2	2	no	—
186	C₂²×C₃	—	2	1	no	—
187	C₂²×C₃	—	2	2	no	—
188	C₂²×C₃	—	2	1	no	—
189	C₂²×C₃	—	4	2	no	—
190	C₂²×C₃	C₂²×C₄×C₉	1	1	yes	characteristic
191	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
192	C₃×C₄	—	2	2	no	—
193	C₃×C₄	—	2	1	no	—
194	C₃×C₄	—	2	2	no	—
195	C₃×C₄	—	2	2	no	—
196	C₃×C₄	—	2	1	no	—
197	C₃×C₄	—	2	2	no	—
198	C₃×C₄	—	2	2	no	—
199	C₃×C₄	—	2	2	no	—
200	C₃×C₄	—	4	2	no	—
201	C₃×C₄	—	4	2	no	—
202	C₃×C₄	C₉×#41	1	2	yes	—
203	C₃×C₄	C₃×#235	1	2	yes	—
204	C₃²	C₃×#433	1	1	yes	characteristic
205	C₉	—	2	1	no	—
206	C₉	#5000	1	1	yes	characteristic, direct product factor
207	C₂³	C₂×C₉×Dic₃	1	1	yes	characteristic
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	C₂×C₉×Dic₃	1	2	yes	—
210	C₂×C₄	C₂×C₉×Dic₃	1	2	yes	—
211	C₂×C₄	C₂×C₉×Dic₃	1	1	yes	characteristic
212	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
213	C₈	—	6	2	no	—
214	C₈	—	6	2	no	—
215	C₂×C₃	—	2	1	no	—
216	C₂×C₃	—	2	2	no	—
217	C₂×C₃	—	4	2	no	—
218	C₂×C₃	—	4	1	no	—
219	C₂×C₃	—	4	1	no	—
220	C₂×C₃	C₉×#117	1	1	yes	characteristic
221	C₂×C₃	C₉×#131	1	2	yes	—
222	C₂×C₃	C₃×#926	1	1	yes	characteristic
223	C₂×C₃	C₃×#948	1	2	yes	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	2	no	—
226	C₂²	C₂²×C₉×Dic₃	1	1	yes	characteristic
227	C₄	—	2	1	no	—
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	C₉×#235	1	2	yes	—
231	C₃	—	2	1	no	—
232	C₃	C₉×#433	1	1	yes	characteristic, complemented
233	C₃	C₃×#5000	1	1	yes	characteristic
234	C₂	—	4	1	no	—
235	C₂	C₉×#926	1	1	yes	characteristic
236	C₂	C₉×#948	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₉	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₄×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#117	C₂
socle	C₂²×C₃²	C₂²×C₃×C₄
prime radical	C₂³×C₃²	C₂×C₃×C₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#117	D₃
hypocenter	C₃	C₉×#433
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₉ ◁ C₉×#117

Complements

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

N	H ≅ G/N	type
3. C₉×C₄⋊(C₃⋊C₈)	C₂	semidirect
4. C₉×#803	C₂	semidirect
5. C₄×C₉×C₃⋊C₈	C₂	semidirect
35. #5000	C₉	direct (N × H)
70. C₄⋊(C₃⋊C₈)	C₂×C₉	semidirect
71. #803	C₂×C₉	semidirect
72. C₄×C₃⋊C₈	C₂×C₉	semidirect
144. C₃×C₉	#433	semidirect
206. C₉	#5000	direct (N × H)
232. C₃	C₉×#433	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×#117	2
C₉×#830	2
C₉×C₄⋊(C₃⋊C₈)	2
C₉×#803	2
C₄×C₉×C₃⋊C₈	2
C₃×#5000	3
C₉×#433	3

Subgroups — C9×#5000 #13142675

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5000 #13142675