Subgroups — C₉×#5698 #13153172

Order 1728.

632 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#898	C₂	1	4	yes	maximal
3	C₉×#897	C₂	1	2	yes	maximal
4	C₉×#890	C₂	1	4	yes	maximal, complemented
5	C₉×#887	C₂	1	2	yes	maximal, complemented
6	C₉×#872	C₂	1	1	yes	maximal, characteristic
7	C₉×#869	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#5698	C₃	1	1	yes	maximal, characteristic
9	C₉×#541	—	3	1	no	maximal
10	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₉×Dic₆	C₂²	1	2	yes	—
12	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
13	C₃×C₉×C₄⋊C₄	C₂²	1	4	yes	—
14	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
15	C₉×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
16	C₉×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
17	C₉×C₄⋊Dic₃	C₂²	1	4	yes	—
18	C₉×C₄⋊Dic₃	C₂²	1	4	yes	—
19	C₉×Dic₃⋊C₄	C₂²	1	2	yes	—
20	C₉×Dic₃⋊C₄	C₂²	1	4	yes	—
21	C₄×C₉×Dic₃	C₂²	1	1	yes	characteristic
22	C₄×C₉×Dic₃	C₂²	1	4	yes	—
23	C₃²×#125	C₂×C₃	1	1	yes	characteristic
24	C₃×#898	C₂×C₃	1	4	yes	—
25	C₃×#897	C₂×C₃	1	2	yes	—
26	C₃×#890	C₂×C₃	1	4	yes	—
27	C₃×#887	C₂×C₃	1	2	yes	—
28	C₃×#872	C₂×C₃	1	1	yes	characteristic
29	C₃×#869	C₂×C₃	1	1	yes	characteristic
30	C₉×C₄⋊Q₈	—	3	1	no	—
31	C₉×#126	—	3	4	no	—
32	C₉×#125	—	2	1	no	—
33	C₉×#125	—	3	4	no	—
34	C₉×#125	D₃	1	1	yes	characteristic, complemented
35	C₉×C₂²⋊Q₈	—	3	2	no	—
36	C₄×Q₈×C₉	—	3	2	no	—
37	C₉×#117	—	3	1	no	—
38	C₂²×D₃×C₉	C₂³	1	1	yes	characteristic
39	C₂×C₃×C₄×C₉	C₂³	1	4	yes	—
40	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
41	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
42	C₂×C₉×Dic₃	C₂³	1	4	yes	—
43	C₂×C₉×Dic₃	C₂³	1	1	yes	characteristic
44	C₂×C₉×Dic₃	C₂³	1	2	yes	—
45	C₄×D₃×C₉	—	2	2	no	—
46	C₉×Dic₆	—	2	4	no	—
47	C₃×#541	—	3	1	no	—
48	#5698	C₉	1	1	yes	characteristic, direct product factor
49	C₂×C₃×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
50	C₂×C₃×Dic₆	C₂²×C₃	1	2	yes	—
51	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
52	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
53	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
54	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	4	yes	—
55	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
56	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
57	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
58	C₃×Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
59	C₃×Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
60	C₃×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
61	C₃×C₄×Dic₃	C₂²×C₃	1	4	yes	—
62	C₂×Q₈×C₉	—	3	2	no	—
63	C₂²×C₄×C₉	—	3	1	no	—
64	C₉×C₄⋊C₄	—	2	2	no	—
65	C₉×C₄⋊C₄	—	2	4	no	—
66	C₉×C₄⋊C₄	—	3	2	no	—
67	C₉×C₄⋊C₄	—	3	4	no	—
68	C₉×C₄⋊C₄	—	3	4	no	—
69	C₉×C₄⋊C₄	—	3	4	no	—
70	C₉×C₄⋊C₄	C₂×D₃	1	2	yes	—
71	C₉×C₄⋊C₄	C₂×D₃	1	4	yes	—
72	C₉×C₂²⋊C₄	—	3	2	no	—
73	C₉×C₂²⋊C₄	—	3	4	no	—
74	C₄²×C₉	—	2	1	no	—
75	C₄²×C₉	—	3	1	no	—
76	C₄²×C₉	—	3	4	no	—
77	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
78	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
79	C₂×D₃×C₉	—	2	1	no	—
80	C₂×D₃×C₉	—	2	2	no	—
81	C₃×C₄×C₉	—	2	4	no	—
82	C₃×C₄×C₉	—	2	2	no	—
83	C₃×C₄×C₉	#41	1	2	yes	—
84	C₉×Dic₃	—	2	4	no	—
85	C₉×Dic₃	—	2	1	no	—
86	C₉×Dic₃	—	2	2	no	—
87	C₃×C₄⋊Q₈	—	3	1	no	—
88	C₃×#126	—	3	4	no	—
89	C₃×#125	—	2	1	no	—
90	C₃×#125	—	3	4	no	—
91	C₃×#125	C₂×C₉	1	1	yes	characteristic, complemented
92	C₃×#125	C₃×D₃	1	1	yes	characteristic
93	C₃×C₂²⋊Q₈	—	3	2	no	—
94	C₃×C₄×Q₈	—	3	2	no	—
95	C₃×#117	—	3	1	no	—
96	#898	C₂×C₉	1	4	yes	—
97	#897	C₂×C₉	1	2	yes	—
98	#890	C₂×C₉	1	4	yes	complemented
99	#887	C₂×C₉	1	2	yes	complemented
100	#872	C₂×C₉	1	1	yes	characteristic
101	#869	C₂×C₉	1	1	yes	characteristic, complemented
102	C₂²×C₃×D₃	C₂³×C₃	1	1	yes	characteristic
103	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
104	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
105	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
106	C₂×C₃×Dic₃	C₂³×C₃	1	2	yes	—
107	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
108	C₂×C₃×Dic₃	C₂³×C₃	1	1	yes	characteristic
109	C₃×C₄×D₃	—	2	2	no	—
110	C₃×Dic₆	—	2	4	no	—
111	C₂³×C₉	—	3	1	no	—
112	Q₈×C₉	—	6	4	no	—
113	C₂×C₄×C₉	—	2	1	no	—
114	C₂×C₄×C₉	—	2	2	no	—
115	C₂×C₄×C₉	—	2	4	no	—
116	C₂×C₄×C₉	—	3	2	no	—
117	C₂×C₄×C₉	—	3	4	no	—
118	C₂×C₄×C₉	—	3	1	no	—
119	C₂×C₄×C₉	—	6	2	no	—
120	C₂×C₄×C₉	C₂²×D₃	1	4	yes	—
121	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
122	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
123	#541	—	3	1	no	Sylow
124	C₂×C₃×C₉	C₂×#41	1	1	yes	characteristic
125	C₂×C₃×C₉	2₋¹⁺⁴	1	2	yes	—
126	D₃×C₉	—	4	1	no	—
127	C₂×C₃×Q₈	—	3	2	no	—
128	C₂²×C₃×C₄	—	3	1	no	—
129	C₂×C₄×D₃	C₂²×C₉	1	1	yes	characteristic
130	C₂×Dic₆	C₂²×C₉	1	2	yes	—
131	C₃×C₄⋊C₄	—	2	2	no	—
132	C₃×C₄⋊C₄	—	2	4	no	—
133	C₃×C₄⋊C₄	—	3	2	no	—
134	C₃×C₄⋊C₄	—	3	4	no	—
135	C₃×C₄⋊C₄	—	3	4	no	—
136	C₃×C₄⋊C₄	—	3	4	no	—
137	C₃×C₄⋊C₄	C₂²×C₉	1	2	yes	—
138	C₃×C₄⋊C₄	C₂²×C₉	1	4	yes	—
139	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
140	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
141	C₃×C₂²⋊C₄	—	3	2	no	—
142	C₃×C₂²⋊C₄	—	3	4	no	—
143	C₃×C₄²	—	2	1	no	—
144	C₃×C₄²	—	3	1	no	—
145	C₃×C₄²	—	3	4	no	—
146	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
147	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
148	(C₂×D₃)⋊C₄	C₂²×C₉	1	2	yes	—
149	(C₂×D₃)⋊C₄	C₂²×C₉	1	4	yes	—
150	C₄⋊Dic₃	C₂²×C₉	1	4	yes	—
151	C₄⋊Dic₃	C₂²×C₉	1	4	yes	—
152	Dic₃⋊C₄	C₂²×C₉	1	2	yes	—
153	Dic₃⋊C₄	C₂²×C₉	1	4	yes	—
154	C₄×Dic₃	C₂²×C₉	1	1	yes	characteristic
155	C₄×Dic₃	C₂²×C₉	1	4	yes	—
156	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
157	C₂×C₃×D₃	—	2	1	no	—
158	C₂×C₃×D₃	—	2	2	no	—
159	C₃²×C₄	—	2	4	no	—
160	C₃²×C₄	—	2	2	no	—
161	C₃²×C₄	C₃×#41	1	2	yes	—
162	C₃×Dic₃	—	2	4	no	—
163	C₃×Dic₃	—	2	1	no	—
164	C₃×Dic₃	—	2	2	no	—
165	C₂²×C₉	—	2	1	no	—
166	C₂²×C₉	—	6	1	no	—
167	C₂²×C₉	—	6	2	no	—
168	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
169	C₄×C₉	—	2	4	no	—
170	C₄×C₉	—	2	2	no	—
171	C₄×C₉	—	2	2	no	—
172	C₄×C₉	—	4	2	no	—
173	C₄×C₉	—	4	4	no	—
174	C₄×C₉	—	6	1	no	—
175	C₄×C₉	—	6	4	no	—
176	C₄×C₉	—	6	2	no	—
177	C₄×C₉	#239	1	2	yes	—
178	C₄⋊Q₈	—	3	1	no	—
179	#126	—	3	4	no	—
180	#125	—	3	4	no	—
181	#125	D₃×C₉	1	1	yes	characteristic, complemented
182	C₂²⋊Q₈	—	3	2	no	—
183	C₄×Q₈	—	3	2	no	—
184	#117	—	3	1	no	—
185	C₃×C₉	#541	1	1	yes	characteristic, complemented, Sylow
186	C₂³×C₃	—	3	1	no	—
187	C₂²×D₃	C₂³×C₉	1	1	yes	characteristic
188	C₃×Q₈	—	6	4	no	—
189	C₂×C₃×C₄	—	2	1	no	—
190	C₂×C₃×C₄	—	2	2	no	—
191	C₂×C₃×C₄	—	2	4	no	—
192	C₂×C₃×C₄	—	3	2	no	—
193	C₂×C₃×C₄	—	3	4	no	—
194	C₂×C₃×C₄	—	3	1	no	—
195	C₂×C₃×C₄	—	6	2	no	—
196	C₂×C₃×C₄	C₂³×C₉	1	4	yes	—
197	C₂×C₃×C₄	C₂³×C₉	1	2	yes	—
198	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
200	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
201	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
202	C₂×Dic₃	C₂³×C₉	1	2	yes	—
203	C₂×Dic₃	C₂³×C₉	1	4	yes	—
204	C₂×Dic₃	C₂³×C₉	1	1	yes	characteristic
205	C₄×D₃	—	2	2	no	—
206	Dic₆	—	2	4	no	—
207	C₂×C₃²	C₂×C₃×#41	1	1	yes	characteristic
208	C₂×C₃²	C₃×2₋¹⁺⁴	1	2	yes	—
209	C₃×D₃	—	4	1	no	—
210	C₂×C₉	—	2	1	no	—
211	C₂×C₉	—	2	2	no	—
212	C₂×C₉	—	12	1	no	—
213	C₂×C₉	C₂×#239	1	1	yes	characteristic
214	C₂×C₉	#1010	1	2	yes	—
215	C₂×Q₈	—	3	2	no	—
216	C₂²×C₄	—	3	1	no	—
217	C₄⋊C₄	—	3	2	no	—
218	C₄⋊C₄	—	3	4	no	—
219	C₄⋊C₄	—	3	4	no	—
220	C₄⋊C₄	—	3	4	no	—
221	C₄⋊C₄	C₂×D₃×C₉	1	2	yes	—
222	C₄⋊C₄	C₂×D₃×C₉	1	4	yes	—
223	C₂²⋊C₄	—	3	2	no	—
224	C₂²⋊C₄	—	3	4	no	—
225	C₄²	—	3	1	no	—
226	C₄²	—	3	4	no	—
227	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	—	6	1	no	—
230	C₂²×C₃	—	6	2	no	—
231	C₂²×C₃	C₂⁴×C₉	1	1	yes	characteristic
232	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
233	C₂×D₃	—	2	1	no	—
234	C₂×D₃	—	2	2	no	—
235	C₃×C₄	—	2	4	no	—
236	C₃×C₄	—	2	2	no	—
237	C₃×C₄	—	2	4	no	—
238	C₃×C₄	—	2	2	no	—
239	C₃×C₄	—	2	2	no	—
240	C₃×C₄	—	4	2	no	—
241	C₃×C₄	—	4	4	no	—
242	C₃×C₄	—	6	1	no	—
243	C₃×C₄	—	6	4	no	—
244	C₃×C₄	—	6	2	no	—
245	C₃×C₄	C₉×#41	1	2	yes	—
246	C₃×C₄	C₃×#239	1	2	yes	—
247	Dic₃	—	2	4	no	—
248	Dic₃	—	2	1	no	—
249	Dic₃	—	2	2	no	—
250	C₃²	C₃×#541	1	1	yes	characteristic
251	C₉	—	2	1	no	—
252	C₉	#5698	1	1	yes	characteristic, direct product factor
253	C₂³	—	3	1	no	—
254	Q₈	—	6	4	no	—
255	C₂×C₄	—	3	2	no	—
256	C₂×C₄	—	3	4	no	—
257	C₂×C₄	—	3	1	no	—
258	C₂×C₄	—	6	2	no	—
259	C₂×C₄	C₂²×D₃×C₉	1	4	yes	—
260	C₂×C₄	C₂²×D₃×C₉	1	2	yes	—
261	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
262	C₂×C₃	—	2	1	no	—
263	C₂×C₃	—	2	2	no	—
264	C₂×C₃	—	12	1	no	—
265	C₂×C₃	C₂×C₉×#41	1	1	yes	characteristic
266	C₂×C₃	C₉×2₋¹⁺⁴	1	2	yes	—
267	C₂×C₃	C₂×C₃×#239	1	1	yes	characteristic
268	C₂×C₃	C₃×#1010	1	2	yes	—
269	D₃	—	4	1	no	—
270	C₂²	—	6	1	no	—
271	C₂²	—	6	2	no	—
272	C₂²	C₂³×D₃×C₉	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	—	2	2	no	—
275	C₄	—	6	1	no	—
276	C₄	—	6	4	no	—
277	C₄	—	6	2	no	—
278	C₄	C₉×#239	1	2	yes	—
279	C₃	—	2	1	no	—
280	C₃	C₉×#541	1	1	yes	characteristic, complemented
281	C₃	C₃×#5698	1	1	yes	characteristic
282	C₂	—	12	1	no	—
283	C₂	C₂×C₉×#239	1	1	yes	characteristic
284	C₂	C₉×#1010	1	2	yes	—

	subgroup	G/H
center	C₂²×C₉	C₂³×D₃
commutator	C₂²×C₃	C₂⁴×C₉
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₃×C₉×#125	C₂
socle, cosocle	C₂²×C₃²	C₂⁴×C₃
prime radical	C₂²×C₃×D₃	C₂³×C₃
hypercenter	C₉×#125	D₃
hypocenter	C₃	C₉×#541
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₉×#125

Complements

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

N	H ≅ G/N	type
1. C₃×C₉×#125	C₂	semidirect
4. C₉×#890	C₂	semidirect
5. C₉×#887	C₂	semidirect
7. C₉×#869	C₂	semidirect
34. C₉×#125	D₃	semidirect
48. #5698	C₉	direct (N × H)
91. C₃×#125	C₂×C₉	semidirect
98. #890	C₂×C₉	semidirect
99. #887	C₂×C₉	semidirect
101. #869	C₂×C₉	semidirect
181. #125	D₃×C₉	semidirect
185. C₃×C₉	#541	semidirect
252. C₉	#5698	direct (N × H)
280. C₃	C₉×#541	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×#125	2
C₉×#898	2
C₉×#897	2
C₉×#890	2
C₉×#887	2
C₉×#872	2
C₉×#869	2
C₃×#5698	3
C₉×#541	3

Subgroups — C9×#5698 #13153172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5698 #13153172