Subgroups — C₃³×#529 #13179984

Order 1728.

3276 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×#126	C₂	1	2	yes	maximal, complemented
2	C₃³×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃³×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃³×#123	C₂	1	2	yes	maximal, complemented
5	C₃³×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃³×#121	C₂	1	1	yes	maximal, characteristic
7	C₃³×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃³×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃³×C₄×D₄	C₂	1	2	yes	maximal, complemented
10	C₃³×#117	C₂	1	2	yes	maximal, complemented
11	C₂×C₃³×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃²×#529	C₃	1	13	yes	maximal, direct product factor
13	C₂×C₃³×Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₃³×D₄	C₂²	1	2	yes	—
15	C₂×C₃³×D₄	C₂²	1	1	yes	characteristic
16	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
17	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
18	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
19	C₂²×C₃³×C₄	C₂²	1	2	yes	—
20	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
21	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
22	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
24	C₃³×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₃³×C₄⋊C₄	C₂²	1	2	yes	—
26	C₃³×C₂²⋊C₄	C₂²	1	2	yes	—
27	C₃³×C₂²⋊C₄	C₂²	1	2	yes	—
28	C₃³×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₃³×C₂²⋊C₄	C₂²	1	2	yes	—
30	C₃³×C₂²⋊C₄	C₂²	1	2	yes	—
31	C₃³×C₄²	C₂²	1	1	yes	characteristic
32	C₃³×C₄²	C₂²	1	1	yes	characteristic
33	C₃³×C₄²	C₂²	1	1	yes	characteristic
34	C₃³×C₄²	C₂²	1	1	yes	characteristic
35	C₃³×C₄²	C₂²	1	2	yes	—
36	C₃²×#126	C₂×C₃	1	26	yes	complemented
37	C₃²×#125	C₂×C₃	1	13	yes	complemented
38	C₃²×#124	C₂×C₃	1	13	yes	complemented
39	C₃²×#123	C₂×C₃	1	26	yes	complemented
40	C₃²×C₂²⋊Q₈	C₂×C₃	1	13	yes	complemented
41	C₃²×#121	C₂×C₃	1	13	yes	—
42	C₃²×C₄×Q₈	C₂×C₃	1	13	yes	complemented
43	C₃²×C₄×D₄	C₂×C₃	1	13	yes	complemented
44	C₃²×C₄×D₄	C₂×C₃	1	26	yes	complemented
45	C₃²×#117	C₂×C₃	1	26	yes	complemented
46	C₂×C₃²×C₄²	C₂×C₃	1	13	yes	complemented
47	C₂³×C₃³	C₂³	1	2	yes	—
48	C₂³×C₃³	C₂³	1	1	yes	characteristic
49	C₃³×Q₈	—	2	2	no	—
50	C₃³×D₄	—	2	4	no	—
51	C₃³×D₄	—	2	2	no	—
52	C₂×C₃³×C₄	—	2	2	no	—
53	C₂×C₃³×C₄	—	2	2	no	—
54	C₂×C₃³×C₄	—	2	4	no	—
55	C₂×C₃³×C₄	—	2	2	no	—
56	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
57	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
58	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
59	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
60	C₂×C₃³×C₄	C₂³	1	2	yes	—
61	C₂×C₃³×C₄	C₂³	1	2	yes	—
62	C₂×C₃³×C₄	C₂³	1	2	yes	—
63	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
64	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
65	C₃×#529	C₃²	1	13	yes	direct product factor
66	C₂×C₃²×Q₈	C₂²×C₃	1	13	yes	—
67	C₂×C₃²×D₄	C₂²×C₃	1	26	yes	—
68	C₂×C₃²×D₄	C₂²×C₃	1	13	yes	—
69	C₂²×C₃²×C₄	C₂²×C₃	1	13	yes	—
70	C₂²×C₃²×C₄	C₂²×C₃	1	13	yes	—
71	C₂²×C₃²×C₄	C₂²×C₃	1	13	yes	—
72	C₂²×C₃²×C₄	C₂²×C₃	1	26	yes	—
73	C₃²×C₄⋊C₄	C₂²×C₃	1	26	yes	—
74	C₃²×C₄⋊C₄	C₂²×C₃	1	26	yes	—
75	C₃²×C₄⋊C₄	C₂²×C₃	1	13	yes	—
76	C₃²×C₄⋊C₄	C₂²×C₃	1	26	yes	—
77	C₃²×C₄⋊C₄	C₂²×C₃	1	13	yes	—
78	C₃²×C₄⋊C₄	C₂²×C₃	1	26	yes	—
79	C₃²×C₂²⋊C₄	C₂²×C₃	1	26	yes	—
80	C₃²×C₂²⋊C₄	C₂²×C₃	1	26	yes	—
81	C₃²×C₂²⋊C₄	C₂²×C₃	1	26	yes	—
82	C₃²×C₂²⋊C₄	C₂²×C₃	1	26	yes	—
83	C₃²×C₂²⋊C₄	C₂²×C₃	1	26	yes	—
84	C₃²×C₄²	C₂²×C₃	1	13	yes	—
85	C₃²×C₄²	C₂²×C₃	1	13	yes	—
86	C₃²×C₄²	C₂²×C₃	1	13	yes	—
87	C₃²×C₄²	C₂²×C₃	1	13	yes	—
88	C₃²×C₄²	C₂²×C₃	1	26	yes	—
89	C₂²×C₃³	—	2	1	no	—
90	C₂²×C₃³	—	2	1	no	—
91	C₂²×C₃³	—	2	2	no	—
92	C₂²×C₃³	—	2	2	no	—
93	C₂²×C₃³	—	2	2	no	—
94	C₂²×C₃³	#41	1	2	yes	—
95	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
96	C₃³×C₄	—	2	1	no	—
97	C₃³×C₄	—	2	1	no	—
98	C₃³×C₄	—	2	1	no	—
99	C₃³×C₄	—	2	1	no	—
100	C₃³×C₄	—	2	2	no	—
101	C₃³×C₄	—	2	2	no	—
102	C₃³×C₄	—	2	2	no	—
103	C₃³×C₄	#41	1	2	yes	—
104	C₃³×C₄	#41	1	2	yes	—
105	C₃×#126	C₂×C₃²	1	26	yes	complemented
106	C₃×#125	C₂×C₃²	1	13	yes	complemented
107	C₃×#124	C₂×C₃²	1	13	yes	complemented
108	C₃×#123	C₂×C₃²	1	26	yes	complemented
109	C₃×C₂²⋊Q₈	C₂×C₃²	1	13	yes	complemented
110	C₃×#121	C₂×C₃²	1	13	yes	—
111	C₃×C₄×Q₈	C₂×C₃²	1	13	yes	complemented
112	C₃×C₄×D₄	C₂×C₃²	1	13	yes	complemented
113	C₃×C₄×D₄	C₂×C₃²	1	26	yes	complemented
114	C₃×#117	C₂×C₃²	1	26	yes	complemented
115	C₂×C₃×C₄²	C₂×C₃²	1	13	yes	complemented
116	C₂³×C₃²	C₂³×C₃	1	26	yes	—
117	C₂³×C₃²	C₂³×C₃	1	13	yes	—
118	C₃²×Q₈	—	2	26	no	—
119	C₃²×D₄	—	2	52	no	—
120	C₃²×D₄	—	2	26	no	—
121	C₂×C₃²×C₄	—	2	26	no	—
122	C₂×C₃²×C₄	—	2	26	no	—
123	C₂×C₃²×C₄	—	2	52	no	—
124	C₂×C₃²×C₄	—	2	26	no	—
125	C₂×C₃²×C₄	C₂³×C₃	1	13	yes	—
126	C₂×C₃²×C₄	C₂³×C₃	1	13	yes	—
127	C₂×C₃²×C₄	C₂³×C₃	1	13	yes	—
128	C₂×C₃²×C₄	C₂³×C₃	1	13	yes	—
129	C₂×C₃²×C₄	C₂³×C₃	1	26	yes	—
130	C₂×C₃²×C₄	C₂³×C₃	1	26	yes	—
131	C₂×C₃²×C₄	C₂³×C₃	1	26	yes	—
132	C₂×C₃²×C₄	C₂³×C₃	1	13	yes	—
133	C₂×C₃²×C₄	C₂³×C₃	1	13	yes	—
134	#529	C₃³	1	1	yes	characteristic, direct product factor, Sylow
135	C₂×C₃³	—	2	2	no	—
136	C₂×C₃³	—	4	2	no	—
137	C₂×C₃³	C₂×#41	1	1	yes	characteristic
138	C₂×C₃³	C₂×#41	1	1	yes	characteristic
139	C₂×C₃³	C₂×#41	1	1	yes	characteristic
140	C₂×C₃×Q₈	C₂²×C₃²	1	13	yes	—
141	C₂×C₃×D₄	C₂²×C₃²	1	26	yes	—
142	C₂×C₃×D₄	C₂²×C₃²	1	13	yes	—
143	C₂²×C₃×C₄	C₂²×C₃²	1	13	yes	—
144	C₂²×C₃×C₄	C₂²×C₃²	1	13	yes	—
145	C₂²×C₃×C₄	C₂²×C₃²	1	13	yes	—
146	C₂²×C₃×C₄	C₂²×C₃²	1	26	yes	—
147	C₃×C₄⋊C₄	C₂²×C₃²	1	26	yes	—
148	C₃×C₄⋊C₄	C₂²×C₃²	1	26	yes	—
149	C₃×C₄⋊C₄	C₂²×C₃²	1	13	yes	—
150	C₃×C₄⋊C₄	C₂²×C₃²	1	26	yes	—
151	C₃×C₄⋊C₄	C₂²×C₃²	1	13	yes	—
152	C₃×C₄⋊C₄	C₂²×C₃²	1	26	yes	—
153	C₃×C₂²⋊C₄	C₂²×C₃²	1	26	yes	—
154	C₃×C₂²⋊C₄	C₂²×C₃²	1	26	yes	—
155	C₃×C₂²⋊C₄	C₂²×C₃²	1	26	yes	—
156	C₃×C₂²⋊C₄	C₂²×C₃²	1	26	yes	—
157	C₃×C₂²⋊C₄	C₂²×C₃²	1	26	yes	—
158	C₃×C₄²	C₂²×C₃²	1	13	yes	—
159	C₃×C₄²	C₂²×C₃²	1	13	yes	—
160	C₃×C₄²	C₂²×C₃²	1	13	yes	—
161	C₃×C₄²	C₂²×C₃²	1	13	yes	—
162	C₃×C₄²	C₂²×C₃²	1	26	yes	—
163	C₂²×C₃²	—	2	13	no	—
164	C₂²×C₃²	—	2	13	no	—
165	C₂²×C₃²	—	2	26	no	—
166	C₂²×C₃²	—	2	26	no	—
167	C₂²×C₃²	—	2	26	no	—
168	C₂²×C₃²	C₃×#41	1	26	yes	—
169	C₂²×C₃²	C₂⁴×C₃	1	13	yes	—
170	C₃²×C₄	—	2	13	no	—
171	C₃²×C₄	—	2	13	no	—
172	C₃²×C₄	—	2	13	no	—
173	C₃²×C₄	—	2	13	no	—
174	C₃²×C₄	—	2	26	no	—
175	C₃²×C₄	—	2	26	no	—
176	C₃²×C₄	—	2	26	no	—
177	C₃²×C₄	C₃×#41	1	26	yes	—
178	C₃²×C₄	C₃×#41	1	26	yes	—
179	#126	C₂×C₃³	1	2	yes	complemented
180	#125	C₂×C₃³	1	1	yes	characteristic, complemented
181	#124	C₂×C₃³	1	1	yes	characteristic, complemented
182	#123	C₂×C₃³	1	2	yes	complemented
183	C₂²⋊Q₈	C₂×C₃³	1	1	yes	characteristic, complemented
184	#121	C₂×C₃³	1	1	yes	characteristic
185	C₄×Q₈	C₂×C₃³	1	1	yes	characteristic, complemented
186	C₄×D₄	C₂×C₃³	1	1	yes	characteristic, complemented
187	C₄×D₄	C₂×C₃³	1	2	yes	complemented
188	#117	C₂×C₃³	1	2	yes	complemented
189	C₂×C₄²	C₂×C₃³	1	1	yes	characteristic, complemented
190	C₃³	#529	1	1	yes	characteristic, direct product factor, Sylow
191	C₂³×C₃	C₂³×C₃²	1	26	yes	—
192	C₂³×C₃	C₂³×C₃²	1	13	yes	—
193	C₃×Q₈	—	2	26	no	—
194	C₃×D₄	—	2	52	no	—
195	C₃×D₄	—	2	26	no	—
196	C₂×C₃×C₄	—	2	26	no	—
197	C₂×C₃×C₄	—	2	26	no	—
198	C₂×C₃×C₄	—	2	52	no	—
199	C₂×C₃×C₄	—	2	26	no	—
200	C₂×C₃×C₄	C₂³×C₃²	1	13	yes	—
201	C₂×C₃×C₄	C₂³×C₃²	1	13	yes	—
202	C₂×C₃×C₄	C₂³×C₃²	1	13	yes	—
203	C₂×C₃×C₄	C₂³×C₃²	1	13	yes	—
204	C₂×C₃×C₄	C₂³×C₃²	1	26	yes	—
205	C₂×C₃×C₄	C₂³×C₃²	1	26	yes	—
206	C₂×C₃×C₄	C₂³×C₃²	1	26	yes	—
207	C₂×C₃×C₄	C₂³×C₃²	1	13	yes	—
208	C₂×C₃×C₄	C₂³×C₃²	1	13	yes	—
209	C₂×C₃²	—	2	26	no	—
210	C₂×C₃²	—	4	26	no	—
211	C₂×C₃²	C₂×C₃×#41	1	13	yes	—
212	C₂×C₃²	C₂×C₃×#41	1	13	yes	—
213	C₂×C₃²	C₂×C₃×#41	1	13	yes	—
214	C₂×Q₈	C₂²×C₃³	1	1	yes	characteristic
215	C₂×D₄	C₂²×C₃³	1	2	yes	—
216	C₂×D₄	C₂²×C₃³	1	1	yes	characteristic
217	C₂²×C₄	C₂²×C₃³	1	1	yes	characteristic
218	C₂²×C₄	C₂²×C₃³	1	1	yes	characteristic
219	C₂²×C₄	C₂²×C₃³	1	1	yes	characteristic
220	C₂²×C₄	C₂²×C₃³	1	2	yes	—
221	C₄⋊C₄	C₂²×C₃³	1	2	yes	—
222	C₄⋊C₄	C₂²×C₃³	1	2	yes	—
223	C₄⋊C₄	C₂²×C₃³	1	1	yes	characteristic
224	C₄⋊C₄	C₂²×C₃³	1	2	yes	—
225	C₄⋊C₄	C₂²×C₃³	1	1	yes	characteristic
226	C₄⋊C₄	C₂²×C₃³	1	2	yes	—
227	C₂²⋊C₄	C₂²×C₃³	1	2	yes	—
228	C₂²⋊C₄	C₂²×C₃³	1	2	yes	—
229	C₂²⋊C₄	C₂²×C₃³	1	2	yes	—
230	C₂²⋊C₄	C₂²×C₃³	1	2	yes	—
231	C₂²⋊C₄	C₂²×C₃³	1	2	yes	—
232	C₄²	C₂²×C₃³	1	1	yes	characteristic
233	C₄²	C₂²×C₃³	1	1	yes	characteristic
234	C₄²	C₂²×C₃³	1	1	yes	characteristic
235	C₄²	C₂²×C₃³	1	1	yes	characteristic
236	C₄²	C₂²×C₃³	1	2	yes	—
237	C₂²×C₃	—	2	13	no	—
238	C₂²×C₃	—	2	13	no	—
239	C₂²×C₃	—	2	26	no	—
240	C₂²×C₃	—	2	26	no	—
241	C₂²×C₃	—	2	26	no	—
242	C₂²×C₃	C₃²×#41	1	26	yes	—
243	C₂²×C₃	C₂⁴×C₃²	1	13	yes	—
244	C₃×C₄	—	2	13	no	—
245	C₃×C₄	—	2	13	no	—
246	C₃×C₄	—	2	13	no	—
247	C₃×C₄	—	2	13	no	—
248	C₃×C₄	—	2	26	no	—
249	C₃×C₄	—	2	26	no	—
250	C₃×C₄	—	2	26	no	—
251	C₃×C₄	C₃²×#41	1	26	yes	—
252	C₃×C₄	C₃²×#41	1	26	yes	—
253	C₃²	C₃×#529	1	13	yes	direct product factor
254	C₂³	C₂³×C₃³	1	2	yes	—
255	C₂³	C₂³×C₃³	1	1	yes	characteristic
256	Q₈	—	2	2	no	—
257	D₄	—	2	4	no	—
258	D₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	C₂³×C₃³	1	1	yes	characteristic
264	C₂×C₄	C₂³×C₃³	1	1	yes	characteristic
265	C₂×C₄	C₂³×C₃³	1	1	yes	characteristic
266	C₂×C₄	C₂³×C₃³	1	1	yes	characteristic
267	C₂×C₄	C₂³×C₃³	1	2	yes	—
268	C₂×C₄	C₂³×C₃³	1	2	yes	—
269	C₂×C₄	C₂³×C₃³	1	2	yes	—
270	C₂×C₄	C₂³×C₃³	1	1	yes	characteristic
271	C₂×C₄	C₂³×C₃³	1	1	yes	characteristic
272	C₂×C₃	—	2	26	no	—
273	C₂×C₃	—	4	26	no	—
274	C₂×C₃	C₂×C₃²×#41	1	13	yes	—
275	C₂×C₃	C₂×C₃²×#41	1	13	yes	—
276	C₂×C₃	C₂×C₃²×#41	1	13	yes	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	C₃³×#41	1	2	yes	—
283	C₂²	C₂⁴×C₃³	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	C₃³×#41	1	2	yes	—
292	C₄	C₃³×#41	1	2	yes	—
293	C₃	C₃²×#529	1	13	yes	direct product factor
294	C₂	—	2	2	no	—
295	C₂	—	4	2	no	—
296	C₂	C₂×C₃³×#41	1	1	yes	characteristic
297	C₂	C₂×C₃³×#41	1	1	yes	characteristic
298	C₂	C₂×C₃³×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃³×C₄	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃³
socle	C₂²×C₃³	C₂⁴
prime radical	C₃³×#121	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃³×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃³×#126	C₂	semidirect
2. C₃³×#125	C₂	semidirect
3. C₃³×#124	C₂	semidirect
4. C₃³×#123	C₂	semidirect
5. C₃³×C₂²⋊Q₈	C₂	semidirect
7. C₃³×C₄×Q₈	C₂	semidirect
8. C₃³×C₄×D₄	C₂	semidirect
9. C₃³×C₄×D₄	C₂	semidirect
10. C₃³×#117	C₂	semidirect
11. C₂×C₃³×C₄²	C₂	semidirect
12. C₃²×#529	C₃	direct (N × H)
36. C₃²×#126	C₂×C₃	semidirect
37. C₃²×#125	C₂×C₃	semidirect
38. C₃²×#124	C₂×C₃	semidirect
39. C₃²×#123	C₂×C₃	semidirect
40. C₃²×C₂²⋊Q₈	C₂×C₃	semidirect
42. C₃²×C₄×Q₈	C₂×C₃	semidirect
43. C₃²×C₄×D₄	C₂×C₃	semidirect
44. C₃²×C₄×D₄	C₂×C₃	semidirect
45. C₃²×#117	C₂×C₃	semidirect
46. C₂×C₃²×C₄²	C₂×C₃	semidirect
65. C₃×#529	C₃²	direct (N × H)
105. C₃×#126	C₂×C₃²	semidirect
106. C₃×#125	C₂×C₃²	semidirect
107. C₃×#124	C₂×C₃²	semidirect
108. C₃×#123	C₂×C₃²	semidirect
109. C₃×C₂²⋊Q₈	C₂×C₃²	semidirect
111. C₃×C₄×Q₈	C₂×C₃²	semidirect
112. C₃×C₄×D₄	C₂×C₃²	semidirect
113. C₃×C₄×D₄	C₂×C₃²	semidirect
114. C₃×#117	C₂×C₃²	semidirect
115. C₂×C₃×C₄²	C₂×C₃²	semidirect
134. #529	C₃³	direct (N × H)
179. #126	C₂×C₃³	semidirect
180. #125	C₂×C₃³	semidirect
181. #124	C₂×C₃³	semidirect
182. #123	C₂×C₃³	semidirect
183. C₂²⋊Q₈	C₂×C₃³	semidirect
185. C₄×Q₈	C₂×C₃³	semidirect
186. C₄×D₄	C₂×C₃³	semidirect
187. C₄×D₄	C₂×C₃³	semidirect
188. #117	C₂×C₃³	semidirect
189. C₂×C₄²	C₂×C₃³	semidirect
190. C₃³	#529	direct (N × H)
253. C₃²	C₃×#529	direct (N × H)
293. C₃	C₃²×#529	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×#126	2
C₃³×#125	2
C₃³×#124	2
C₃³×#123	2
C₃³×C₂²⋊Q₈	2
C₃³×#121	2
C₃³×C₄×Q₈	2
C₃³×C₄×D₄	2
C₃³×#117	2
C₂×C₃³×C₄²	2
C₃²×#529	3

Subgroups — C33×#529 #13179984

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃³×#529 #13179984