Subgroups — C₃×#10601826 #14289011

Order 1920.

456 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×Q₈×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×#460	C₂	1	1	yes	maximal, characteristic
3	C₃×#65365	C₂	1	1	yes	maximal, characteristic
4	C₃×#65364	C₂	1	1	yes	maximal, characteristic
5	C₃×Q₈⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#64704	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#64701	C₂	1	1	yes	maximal, characteristic, complemented
8	#10601826	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₅×SD₁₆	—	2	1	no	—
10	C₃×C₅×#127	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
12	C₃×Q₈×Dic₅	—	2	1	no	—
13	C₂×C₃×#687	—	2	1	no	—
14	C₂×C₃×#687	—	2	2	no	—
15	C₃×#4034	—	2	2	no	—
16	C₃×#4034	—	2	1	no	—
17	C₃×#4015	—	2	1	no	—
18	C₃×#4015	C₂²	1	1	yes	characteristic
19	C₃×C₄×Dic₁₀	—	2	1	no	—
20	C₃×C₄×Dic₁₀	C₂²	1	1	yes	characteristic
21	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₃×C₅×#102	—	2	1	no	—
23	C₃×#3964	—	2	1	no	—
24	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
25	C₃×C₄×#168	C₂²	1	1	yes	characteristic
26	C₃×#1623	—	5	1	no	maximal, Hall
27	Q₈×Dic₁₀	C₂×C₃	1	1	yes	characteristic, complemented
28	C₅×#460	C₂×C₃	1	1	yes	characteristic
29	#65365	C₂×C₃	1	1	yes	characteristic
30	#65364	C₂×C₃	1	1	yes	characteristic
31	Q₈⋊#168	C₂×C₃	1	1	yes	characteristic, complemented
32	#64704	C₂×C₃	1	1	yes	characteristic, complemented
33	#64701	C₂×C₃	1	1	yes	characteristic, complemented
34	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
35	C₂×C₃×C₅×D₄	—	2	1	no	—
36	C₂×C₃×C₅×D₄	—	2	1	no	—
37	C₂×C₃×C₅×D₄	—	2	1	no	—
38	C₂×C₃×Dic₁₀	—	2	2	no	—
39	C₂×C₃×Dic₁₀	D₄	1	1	yes	characteristic
40	C₂×C₃×Dic₁₀	D₄	1	2	yes	—
41	C₃×C₅×SD₁₆	—	4	2	no	—
42	C₂×C₃×C₅×C₈	—	2	1	no	—
43	C₃×C₅×C₄⋊C₄	—	2	1	no	—
44	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
45	C₃×C₄²×C₅	—	2	1	no	—
46	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₃×#687	—	4	4	no	—
48	C₃×#687	—	4	2	no	—
49	C₃×C₄⋊Dic₅	—	2	2	no	—
50	C₃×C₄⋊Dic₅	—	2	1	no	—
51	C₃×C₄⋊Dic₅	—	2	1	no	—
52	C₃×C₄⋊Dic₅	D₄	1	1	yes	characteristic
53	C₃×#683	—	2	2	no	—
54	C₃×#683	—	2	1	no	—
55	C₃×C₄×Dic₅	—	2	2	no	—
56	C₃×C₄×Dic₅	—	2	1	no	—
57	C₂×C₃×#168	—	2	1	no	—
58	C₂×C₃×#168	—	2	1	no	—
59	C₃×Q₈²	—	5	1	no	—
60	C₃×#492	—	5	1	no	—
61	C₃×#460	—	5	1	no	—
62	C₃×#460	D₅	1	1	yes	characteristic
63	C₃×#333	—	5	1	no	—
64	C₃×Q₈⋊C₈	—	5	1	no	—
65	C₃×Q₈⋊C₈	—	5	1	no	—
66	C₂×C₅×SD₁₆	—	2	1	no	—
67	C₅×#127	C₂²×C₃	1	1	yes	characteristic
68	C₄×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
69	Q₈×Dic₅	—	2	1	no	—
70	C₂×#687	—	2	1	no	—
71	C₂×#687	—	2	2	no	—
72	#4034	—	2	2	no	—
73	#4034	—	2	1	no	—
74	#4015	—	2	1	no	—
75	#4015	C₂²×C₃	1	1	yes	characteristic
76	C₄×Dic₁₀	—	2	1	no	—
77	C₄×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
78	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
79	C₅×#102	—	2	1	no	—
80	#3964	—	2	1	no	—
81	C₄⋊#168	C₂²×C₃	1	1	yes	characteristic
82	C₄×#168	C₂²×C₃	1	1	yes	characteristic
83	#1623	—	5	1	no	Sylow
84	C₂³×C₃×C₅	—	4	1	no	—
85	C₃×C₅×Q₈	—	2	1	no	—
86	C₃×C₅×Q₈	SD₁₆	1	2	yes	complemented
87	C₃×C₅×D₄	—	4	2	no	—
88	C₃×C₅×D₄	—	4	1	no	—
89	C₃×C₅×D₄	—	4	1	no	—
90	C₃×C₅×D₄	—	4	1	no	—
91	C₃×C₅×D₄	—	4	1	no	—
92	C₂×C₃×C₄×C₅	—	2	1	no	—
93	C₂×C₃×C₄×C₅	—	2	1	no	—
94	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
95	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
96	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
97	C₂×C₃×Dic₅	—	2	1	no	—
98	C₂×C₃×Dic₅	—	2	2	no	—
99	C₂×C₃×Dic₅	—	2	1	no	—
100	C₃×Dic₁₀	—	2	1	no	—
101	C₃×Dic₁₀	—	4	2	no	—
102	C₃×Dic₁₀	—	8	2	no	—
103	C₃×Dic₁₀	SD₁₆	1	2	yes	complemented
104	C₃×C₅×C₈	—	4	1	no	—
105	C₃×#168	—	2	2	no	—
106	C₃×#168	—	4	1	no	—
107	C₂×C₃×SD₁₆	—	2	1	no	—
108	C₂×C₃×SD₁₆	—	10	1	no	—
109	C₂×C₃×SD₁₆	—	10	2	no	—
110	C₃×C₄⋊Q₈	—	5	1	no	—
111	C₃×C₄⋊Q₈	—	10	1	no	—
112	C₃×C₄⋊Q₈	—	10	2	no	—
113	C₃×C₄⋊Q₈	—	10	1	no	—
114	C₃×#127	C₂×D₅	1	1	yes	characteristic
115	C₃×C₄×Q₈	—	5	1	no	—
116	C₃×C₄×Q₈	—	10	1	no	—
117	C₃×C₄×Q₈	—	10	1	no	—
118	C₃×C₄×Q₈	C₂×D₅	1	1	yes	characteristic
119	C₃×C₄⋊C₈	—	5	1	no	—
120	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
121	C₃×#102	—	2	1	no	—
122	C₃×#102	—	10	1	no	—
123	C₃×C₄×C₈	—	5	1	no	—
124	C₂×C₅×Q₈	C₃×D₄	1	1	yes	characteristic
125	C₂×C₅×D₄	—	2	1	no	—
126	C₂×C₅×D₄	—	2	1	no	—
127	C₂×C₅×D₄	—	2	1	no	—
128	C₂×Dic₁₀	—	2	2	no	—
129	C₂×Dic₁₀	C₃×D₄	1	1	yes	characteristic
130	C₂×Dic₁₀	C₃×D₄	1	2	yes	—
131	C₅×SD₁₆	—	4	2	no	—
132	C₂×C₅×C₈	—	2	1	no	—
133	C₅×C₄⋊C₄	—	2	1	no	—
134	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
135	C₄²×C₅	—	2	1	no	—
136	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
137	#687	—	4	4	no	—
138	#687	—	4	2	no	—
139	C₄⋊Dic₅	—	2	2	no	—
140	C₄⋊Dic₅	—	2	1	no	—
141	C₄⋊Dic₅	—	2	1	no	—
142	C₄⋊Dic₅	C₃×D₄	1	1	yes	characteristic
143	#683	—	2	2	no	—
144	#683	—	2	1	no	—
145	C₄×Dic₅	—	2	2	no	—
146	C₄×Dic₅	—	2	1	no	—
147	C₂×#168	—	2	1	no	—
148	C₂×#168	—	2	1	no	—
149	Q₈²	—	5	1	no	—
150	#492	—	5	1	no	—
151	#460	—	5	1	no	—
152	#460	C₃×D₅	1	1	yes	characteristic
153	#333	—	5	1	no	—
154	Q₈⋊C₈	—	5	1	no	—
155	Q₈⋊C₈	—	5	1	no	—
156	C₂²×C₃×C₅	—	8	1	no	—
157	C₂²×C₃×C₅	—	8	1	no	—
158	C₂²×C₃×C₅	—	8	1	no	—
159	C₂²×C₃×C₅	#120	1	1	yes	characteristic
160	C₃×C₄×C₅	—	2	2	no	—
161	C₃×C₄×C₅	—	2	1	no	—
162	C₃×C₄×C₅	—	4	1	no	—
163	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
164	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
165	C₃×C₄×C₅	#137	1	1	yes	characteristic
166	C₃×C₄×C₅	#137	1	1	yes	characteristic
167	C₃×Dic₅	—	2	2	no	—
168	C₃×Dic₅	—	4	2	no	—
169	C₃×Dic₅	—	4	1	no	—
170	C₂×C₃×Q₈	—	5	1	no	—
171	C₂×C₃×Q₈	—	5	2	no	—
172	C₂×C₃×Q₈	—	10	2	no	—
173	C₂×C₃×Q₈	C₂²⋊D₅	1	1	yes	characteristic
174	C₂×C₃×D₄	—	2	1	no	—
175	C₂×C₃×D₄	—	2	1	no	—
176	C₂×C₃×D₄	—	2	1	no	—
177	C₃×SD₁₆	—	4	2	no	—
178	C₃×SD₁₆	—	20	4	no	—
179	C₃×SD₁₆	—	20	2	no	—
180	C₂×C₃×C₈	—	2	1	no	—
181	C₂×C₃×C₈	—	10	1	no	—
182	C₂×C₃×C₈	—	10	1	no	—
183	C₃×C₄⋊C₄	—	2	1	no	—
184	C₃×C₄⋊C₄	—	5	1	no	—
185	C₃×C₄⋊C₄	—	10	2	no	—
186	C₃×C₄⋊C₄	—	10	2	no	—
187	C₃×C₄⋊C₄	—	10	1	no	—
188	C₃×C₄⋊C₄	—	10	1	no	—
189	C₃×C₄⋊C₄	—	10	1	no	—
190	C₃×C₄⋊C₄	C₂²⋊D₅	1	1	yes	characteristic
191	C₃×C₄²	—	2	1	no	—
192	C₃×C₄²	—	10	2	no	—
193	C₃×C₄²	—	10	1	no	—
194	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
195	C₂³×C₅	—	4	1	no	—
196	C₅×Q₈	—	2	1	no	—
197	C₅×Q₈	C₃×SD₁₆	1	2	yes	complemented
198	C₅×D₄	—	4	2	no	—
199	C₅×D₄	—	4	1	no	—
200	C₅×D₄	—	4	1	no	—
201	C₅×D₄	—	4	1	no	—
202	C₅×D₄	—	4	1	no	—
203	C₂×C₄×C₅	—	2	1	no	—
204	C₂×C₄×C₅	—	2	1	no	—
205	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
206	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
207	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
208	C₂×Dic₅	—	2	1	no	—
209	C₂×Dic₅	—	2	2	no	—
210	C₂×Dic₅	—	2	1	no	—
211	Dic₁₀	—	2	1	no	—
212	Dic₁₀	—	4	2	no	—
213	Dic₁₀	—	8	2	no	—
214	Dic₁₀	C₃×SD₁₆	1	2	yes	complemented
215	C₅×C₈	—	4	1	no	—
216	#168	—	2	2	no	—
217	#168	—	4	1	no	—
218	C₂×SD₁₆	—	2	1	no	—
219	C₂×SD₁₆	—	10	1	no	—
220	C₂×SD₁₆	—	10	2	no	—
221	C₄⋊Q₈	—	5	1	no	—
222	C₄⋊Q₈	—	10	1	no	—
223	C₄⋊Q₈	—	10	2	no	—
224	C₄⋊Q₈	—	10	1	no	—
225	#127	C₂×C₃×D₅	1	1	yes	characteristic
226	C₄×Q₈	—	5	1	no	—
227	C₄×Q₈	—	10	1	no	—
228	C₄×Q₈	—	10	1	no	—
229	C₄×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
230	C₄⋊C₈	—	5	1	no	—
231	C₄⋊C₈	C₂×C₃×D₅	1	1	yes	characteristic
232	#102	—	2	1	no	—
233	#102	—	10	1	no	—
234	C₄×C₈	—	5	1	no	—
235	C₂×C₃×C₅	—	16	1	no	—
236	C₂×C₃×C₅	#448	1	1	yes	characteristic
237	C₂×C₃×C₅	#448	1	1	yes	characteristic
238	C₂×C₃×C₅	#453	1	1	yes	characteristic
239	C₂³×C₃	—	4	1	no	—
240	C₃×Q₈	—	2	1	no	—
241	C₃×Q₈	—	5	2	no	—
242	C₃×Q₈	—	10	1	no	—
243	C₃×Q₈	—	20	2	no	—
244	C₃×Q₈	—	40	2	no	—
245	C₃×Q₈	#687	1	2	yes	complemented
246	C₃×D₄	—	4	2	no	—
247	C₃×D₄	—	4	1	no	—
248	C₃×D₄	—	4	1	no	—
249	C₃×D₄	—	4	1	no	—
250	C₃×D₄	—	4	1	no	—
251	C₂×C₃×C₄	—	2	1	no	—
252	C₂×C₃×C₄	—	2	1	no	—
253	C₂×C₃×C₄	—	10	1	no	—
254	C₂×C₃×C₄	—	10	1	no	—
255	C₂×C₃×C₄	—	10	2	no	—
256	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
257	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
258	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
259	C₃×C₈	—	4	1	no	—
260	C₃×C₈	—	10	2	no	—
261	C₃×C₈	—	20	1	no	—
262	C₂²×C₅	—	8	1	no	—
263	C₂²×C₅	—	8	1	no	—
264	C₂²×C₅	—	8	1	no	—
265	C₂²×C₅	C₃×#120	1	1	yes	characteristic
266	C₄×C₅	—	2	2	no	—
267	C₄×C₅	—	2	1	no	—
268	C₄×C₅	—	4	1	no	—
269	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
270	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
271	C₄×C₅	C₃×#137	1	1	yes	characteristic
272	C₄×C₅	C₃×#137	1	1	yes	characteristic
273	Dic₅	—	2	2	no	—
274	Dic₅	—	4	2	no	—
275	Dic₅	—	4	1	no	—
276	C₂×Q₈	—	5	1	no	—
277	C₂×Q₈	—	5	2	no	—
278	C₂×Q₈	—	10	2	no	—
279	C₂×Q₈	C₃×C₂²⋊D₅	1	1	yes	characteristic
280	C₂×D₄	—	2	1	no	—
281	C₂×D₄	—	2	1	no	—
282	C₂×D₄	—	2	1	no	—
283	SD₁₆	—	4	2	no	—
284	SD₁₆	—	20	4	no	—
285	SD₁₆	—	20	2	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	10	1	no	—
288	C₂×C₈	—	10	1	no	—
289	C₄⋊C₄	—	2	1	no	—
290	C₄⋊C₄	—	5	1	no	—
291	C₄⋊C₄	—	10	2	no	—
292	C₄⋊C₄	—	10	2	no	—
293	C₄⋊C₄	—	10	1	no	—
294	C₄⋊C₄	—	10	1	no	—
295	C₄⋊C₄	—	10	1	no	—
296	C₄⋊C₄	C₃×C₂²⋊D₅	1	1	yes	characteristic
297	C₄²	—	2	1	no	—
298	C₄²	—	10	2	no	—
299	C₄²	—	10	1	no	—
300	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
301	C₃×C₅	#1623	1	1	yes	characteristic, complemented, Hall
302	C₂²×C₃	—	8	1	no	—
303	C₂²×C₃	—	8	1	no	—
304	C₂²×C₃	—	8	1	no	—
305	C₂²×C₃	#4083	1	1	yes	characteristic
306	C₃×C₄	—	2	2	no	—
307	C₃×C₄	—	2	1	no	—
308	C₃×C₄	—	4	1	no	—
309	C₃×C₄	—	10	2	no	—
310	C₃×C₄	—	20	2	no	—
311	C₃×C₄	—	20	1	no	—
312	C₃×C₄	D₅×SD₁₆	1	1	yes	characteristic
313	C₃×C₄	#4061	1	1	yes	characteristic
314	C₃×C₄	C₂×#687	1	1	yes	characteristic
315	C₃×C₄	#4097	1	1	yes	characteristic
316	C₂×C₅	—	16	1	no	—
317	C₂×C₅	C₃×#448	1	1	yes	characteristic
318	C₂×C₅	C₃×#448	1	1	yes	characteristic
319	C₂×C₅	C₃×#453	1	1	yes	characteristic
320	C₂³	—	4	1	no	—
321	Q₈	—	2	1	no	—
322	Q₈	—	5	2	no	—
323	Q₈	—	10	1	no	—
324	Q₈	—	20	2	no	—
325	Q₈	—	40	2	no	—
326	Q₈	C₃×#687	1	2	yes	complemented
327	D₄	—	4	2	no	—
328	D₄	—	4	1	no	—
329	D₄	—	4	1	no	—
330	D₄	—	4	1	no	—
331	D₄	—	4	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	10	1	no	—
335	C₂×C₄	—	10	1	no	—
336	C₂×C₄	—	10	2	no	—
337	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
338	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
339	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
340	C₈	—	4	1	no	—
341	C₈	—	10	2	no	—
342	C₈	—	20	1	no	—
343	C₂×C₃	—	16	1	no	—
344	C₂×C₃	#65329	1	1	yes	characteristic
345	C₂×C₃	#65459	1	1	yes	characteristic
346	C₂×C₃	#65487	1	1	yes	characteristic
347	C₅	C₃×#1623	1	1	yes	characteristic, complemented, Sylow
348	C₂²	—	8	1	no	—
349	C₂²	—	8	1	no	—
350	C₂²	—	8	1	no	—
351	C₂²	C₃×#4083	1	1	yes	characteristic
352	C₄	—	2	2	no	—
353	C₄	—	2	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	10	2	no	—
356	C₄	—	20	2	no	—
357	C₄	—	20	1	no	—
358	C₄	C₃×D₅×SD₁₆	1	1	yes	characteristic
359	C₄	C₃×#4061	1	1	yes	characteristic
360	C₄	C₂×C₃×#687	1	1	yes	characteristic
361	C₄	C₃×#4097	1	1	yes	characteristic
362	C₃	#10601826	1	1	yes	characteristic, direct product factor, Sylow
363	C₂	—	16	1	no	—
364	C₂	C₃×#65329	1	1	yes	characteristic
365	C₂	C₃×#65459	1	1	yes	characteristic
366	C₂	C₃×#65487	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	#4083
commutator, cosocle	C₄²×C₅	C₂³×C₃
Frattini	C₄²	C₂²×C₃×D₅
Fitting	C₃×C₅×#460	C₂
socle	C₂²×C₃×C₅	#120
prime radical	C₃×C₅×#127	C₂²
hypercenter	C₃×#460	D₅
hypocenter	C₅	C₃×#1623
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#460

Complements

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

N	H ≅ G/N	type
1. C₃×Q₈×Dic₁₀	C₂	semidirect
5. C₃×Q₈⋊#168	C₂	semidirect
6. C₃×#64704	C₂	semidirect
7. C₃×#64701	C₂	semidirect
8. #10601826	C₃	direct (N × H)
27. Q₈×Dic₁₀	C₂×C₃	semidirect
31. Q₈⋊#168	C₂×C₃	semidirect
32. #64704	C₂×C₃	semidirect
33. #64701	C₂×C₃	semidirect
86. C₃×C₅×Q₈	SD₁₆	semidirect
103. C₃×Dic₁₀	SD₁₆	semidirect
197. C₅×Q₈	C₃×SD₁₆	semidirect
214. Dic₁₀	C₃×SD₁₆	semidirect
245. C₃×Q₈	#687	semidirect
301. C₃×C₅	#1623	semidirect
326. Q₈	C₃×#687	semidirect
347. C₅	C₃×#1623	semidirect
362. C₃	#10601826	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×Q₈×Dic₁₀	2
C₃×C₅×#460	2
C₃×#65365	2
C₃×#65364	2
C₃×Q₈⋊#168	2
C₃×#64704	2
C₃×#64701	2
#10601826	3
C₃×#1623	5

Subgroups — C3×#10601826 #14289011

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10601826 #14289011