Subgroups — C₃×#10599224 #14285448

Order 1920.

420 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#66062	C₂	1	1	yes	maximal, characteristic
2	C₃×#65126	C₂	1	1	yes	maximal, characteristic
3	C₃×#65126	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#332	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64767	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#64706	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#64702	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599224	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₃×#4092	—	2	1	no	—
11	C₃×Q₈×Dic₅	—	2	1	no	—
12	C₂×C₃×#676	—	2	1	no	—
13	C₂×C₃×#676	—	2	1	no	—
14	C₃×#4043	—	2	1	no	—
15	C₃×#4038	C₄	1	2	yes	—
16	C₃×#4036	C₄	1	2	yes	—
17	C₃×#4033	—	2	1	no	—
18	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
19	C₃×#4015	C₂²	1	1	yes	characteristic
20	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₃×#3946	—	2	1	no	—
23	C₃×#3946	—	2	1	no	—
24	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
25	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
26	C₃×#1561	—	5	1	no	maximal, Hall
27	#66062	C₂×C₃	1	1	yes	characteristic
28	#65126	C₂×C₃	1	1	yes	characteristic
29	#65126	C₂×C₃	1	1	yes	characteristic
30	C₅×#332	C₂×C₃	1	1	yes	characteristic, complemented
31	#64767	C₂×C₃	1	1	yes	characteristic, complemented
32	#64706	C₂×C₃	1	1	yes	characteristic, complemented
33	#64702	C₂×C₃	1	1	yes	characteristic, complemented
34	C₂×C₃×C₅×Q₈	—	2	1	no	—
35	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
37	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
38	C₂×C₃×Dic₁₀	—	2	1	no	—
39	C₂×C₃×C₅×C₈	—	2	1	no	—
40	C₂×C₃×C₅×C₈	—	2	1	no	—
41	C₃×C₅×C₄⋊C₄	—	2	1	no	—
42	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
43	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₃×#685	—	2	2	no	—
45	C₃×C₄⋊Dic₅	—	2	1	no	—
46	C₃×C₄⋊Dic₅	—	2	2	no	—
47	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
48	C₃×#683	—	2	2	no	—
49	C₃×C₄×Dic₅	—	2	2	no	—
50	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
51	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
52	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
53	C₂×C₃×#168	—	2	1	no	—
54	C₂×C₃×#168	—	2	1	no	—
55	C₃×#676	—	4	2	no	—
56	C₃×#676	—	4	2	no	—
57	C₃×#533	—	5	1	no	—
58	C₃×#423	—	5	1	no	—
59	C₃×#423	—	5	1	no	—
60	C₃×#332	—	5	1	no	—
61	C₃×#332	—	5	1	no	—
62	C₃×#332	—	5	1	no	—
63	C₃×#332	D₅	1	1	yes	characteristic, complemented
64	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
65	#4092	—	2	1	no	—
66	Q₈×Dic₅	—	2	1	no	—
67	C₂×#676	—	2	1	no	—
68	C₂×#676	—	2	1	no	—
69	#4043	—	2	1	no	—
70	#4038	C₃×C₄	1	2	yes	—
71	#4036	C₃×C₄	1	2	yes	—
72	#4033	—	2	1	no	—
73	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
74	#4015	C₂²×C₃	1	1	yes	characteristic
75	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
76	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
77	#3946	—	2	1	no	—
78	#3946	—	2	1	no	—
79	C₄⋊#168	C₂²×C₃	1	1	yes	characteristic
80	C₄⋊#168	C₂²×C₃	1	1	yes	characteristic
81	#1561	—	5	1	no	Sylow
82	C₂²×C₃×D₅	C₂²⋊C₄	1	1	yes	characteristic
83	C₃×C₅×Q₈	—	4	1	no	—
84	C₃×C₅×Q₈	—	4	1	no	—
85	C₂×C₃×C₄×C₅	—	2	2	no	—
86	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×Dic₅	—	2	2	no	—
90	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
93	C₃×C₄×D₅	—	2	1	no	—
94	C₃×C₄×D₅	—	2	1	no	—
95	C₃×C₄×D₅	—	2	1	no	—
96	C₃×C₄×D₅	—	2	1	no	—
97	C₃×C₄×D₅	—	2	2	no	—
98	C₃×Dic₁₀	—	4	1	no	—
99	C₃×Dic₁₀	—	4	1	no	—
100	C₃×C₅×C₈	—	4	1	no	—
101	C₃×C₅×C₈	—	4	1	no	—
102	C₃×#168	—	4	1	no	—
103	C₃×#168	—	4	1	no	—
104	C₂×C₃×M₁₆	—	10	1	no	—
105	C₂×C₃×M₁₆	—	10	1	no	—
106	C₃×C₄⋊Q₈	—	5	1	no	—
107	C₃×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
108	C₃×#125	—	5	2	no	—
109	C₃×C₂²⋊Q₈	—	10	1	no	—
110	C₃×C₂²⋊Q₈	—	10	1	no	—
111	C₃×C₄×Q₈	—	10	1	no	—
112	C₃×C₄×Q₈	—	10	1	no	—
113	C₃×#117	—	5	2	no	—
114	C₂×C₃×C₄²	—	5	1	no	—
115	C₃×C₄⋊C₈	—	5	1	no	—
116	C₃×C₄⋊C₈	—	5	1	no	—
117	C₃×C₄⋊C₈	—	10	1	no	—
118	C₃×C₄⋊C₈	—	10	1	no	—
119	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
120	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
121	C₂×C₅×Q₈	—	2	1	no	—
122	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
123	C₂×C₄×D₅	C₃×D₄	1	1	yes	characteristic
124	C₂×C₄×D₅	C₃×D₄	1	1	yes	characteristic
125	C₂×Dic₁₀	—	2	1	no	—
126	C₂×C₅×C₈	—	2	1	no	—
127	C₂×C₅×C₈	—	2	1	no	—
128	C₅×C₄⋊C₄	—	2	1	no	—
129	C₅×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
130	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
131	#685	—	2	2	no	—
132	C₄⋊Dic₅	—	2	1	no	—
133	C₄⋊Dic₅	—	2	2	no	—
134	C₄⋊Dic₅	C₂×C₃×C₄	1	2	yes	—
135	#683	—	2	2	no	—
136	C₄×Dic₅	—	2	2	no	—
137	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
138	C₄×Dic₅	C₃×D₄	1	1	yes	characteristic
139	C₄×Dic₅	C₃×D₄	1	1	yes	characteristic
140	C₂×#168	—	2	1	no	—
141	C₂×#168	—	2	1	no	—
142	#676	—	4	2	no	—
143	#676	—	4	2	no	—
144	#533	—	5	1	no	—
145	#423	—	5	1	no	—
146	#423	—	5	1	no	—
147	#332	—	5	1	no	—
148	#332	—	5	1	no	—
149	#332	—	5	1	no	—
150	#332	C₃×D₅	1	1	yes	characteristic, complemented
151	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₂×C₃×D₅	—	2	1	no	—
153	C₂×C₃×D₅	—	2	1	no	—
154	C₂×C₃×D₅	—	2	1	no	—
155	C₃×C₄×C₅	—	2	1	no	—
156	C₃×C₄×C₅	—	4	2	no	—
157	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
158	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
159	C₃×C₄×C₅	#137	1	1	yes	characteristic
160	C₃×C₄×C₅	#137	1	1	yes	characteristic
161	C₃×Dic₅	—	2	1	no	—
162	C₃×Dic₅	—	2	1	no	—
163	C₃×Dic₅	—	4	2	no	—
164	C₃×Dic₅	#101	1	2	yes	—
165	C₂×C₃×Q₈	—	2	1	no	—
166	C₂×C₃×Q₈	—	10	1	no	—
167	C₂²×C₃×C₄	—	5	1	no	—
168	C₂²×C₃×C₄	—	5	1	no	—
169	C₂²×C₃×C₄	—	5	1	no	—
170	C₃×M₁₆	—	20	2	no	—
171	C₃×M₁₆	—	20	2	no	—
172	C₂×C₃×C₈	—	2	1	no	—
173	C₂×C₃×C₈	—	2	1	no	—
174	C₂×C₃×C₈	—	10	1	no	—
175	C₂×C₃×C₈	—	10	1	no	—
176	C₃×C₄⋊C₄	—	2	1	no	—
177	C₃×C₄⋊C₄	—	5	2	no	—
178	C₃×C₄⋊C₄	—	10	2	no	—
179	C₃×C₄⋊C₄	—	10	2	no	—
180	C₃×C₄⋊C₄	—	10	1	no	—
181	C₃×C₄⋊C₄	C₄×D₅	1	2	yes	—
182	C₃×C₂²⋊C₄	—	10	2	no	—
183	C₃×C₄²	—	5	1	no	—
184	C₃×C₄²	—	5	1	no	—
185	C₃×C₄²	—	5	1	no	—
186	C₃×C₄²	—	10	2	no	—
187	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
188	C₂²×D₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
189	C₅×Q₈	—	4	1	no	—
190	C₅×Q₈	—	4	1	no	—
191	C₂×C₄×C₅	—	2	2	no	—
192	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
193	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
194	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
195	C₂×Dic₅	—	2	2	no	—
196	C₂×Dic₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
197	C₂×Dic₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
198	C₂×Dic₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
199	C₄×D₅	—	2	1	no	—
200	C₄×D₅	—	2	1	no	—
201	C₄×D₅	—	2	1	no	—
202	C₄×D₅	—	2	1	no	—
203	C₄×D₅	—	2	2	no	—
204	Dic₁₀	—	4	1	no	—
205	Dic₁₀	—	4	1	no	—
206	C₅×C₈	—	4	1	no	—
207	C₅×C₈	—	4	1	no	—
208	#168	—	4	1	no	—
209	#168	—	4	1	no	—
210	C₂×M₁₆	—	10	1	no	—
211	C₂×M₁₆	—	10	1	no	—
212	C₄⋊Q₈	—	5	1	no	—
213	C₄⋊Q₈	C₂×C₃×D₅	1	1	yes	characteristic
214	#125	—	5	2	no	—
215	C₂²⋊Q₈	—	10	1	no	—
216	C₂²⋊Q₈	—	10	1	no	—
217	C₄×Q₈	—	10	1	no	—
218	C₄×Q₈	—	10	1	no	—
219	#117	—	5	2	no	—
220	C₂×C₄²	—	5	1	no	—
221	C₄⋊C₈	—	5	1	no	—
222	C₄⋊C₈	—	5	1	no	—
223	C₄⋊C₈	—	10	1	no	—
224	C₄⋊C₈	—	10	1	no	—
225	C₄⋊C₈	C₂×C₃×D₅	1	1	yes	characteristic
226	C₄⋊C₈	C₂×C₃×D₅	1	1	yes	characteristic
227	C₂×C₃×C₅	C₂×#101	1	1	yes	characteristic
228	C₂×C₃×C₅	#417	1	1	yes	characteristic
229	C₂×C₃×C₅	#417	1	1	yes	characteristic
230	C₃×D₅	—	4	1	no	—
231	C₂³×C₃	—	5	1	no	—
232	C₃×Q₈	—	4	1	no	—
233	C₃×Q₈	—	4	1	no	—
234	C₃×Q₈	—	20	1	no	—
235	C₃×Q₈	—	20	1	no	—
236	C₂×C₃×C₄	—	2	2	no	—
237	C₂×C₃×C₄	—	5	1	no	—
238	C₂×C₃×C₄	—	5	1	no	—
239	C₂×C₃×C₄	—	5	1	no	—
240	C₂×C₃×C₄	—	10	2	no	—
241	C₂×C₃×C₄	—	10	1	no	—
242	C₂×C₃×C₄	—	10	1	no	—
243	C₂×C₃×C₄	—	10	1	no	—
244	C₂×C₃×C₄	—	10	1	no	—
245	C₂×C₃×C₄	—	10	2	no	—
246	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
247	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
248	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
249	C₃×C₈	—	4	1	no	—
250	C₃×C₈	—	4	1	no	—
251	C₃×C₈	—	20	1	no	—
252	C₃×C₈	—	20	1	no	—
253	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
254	C₂×D₅	—	2	1	no	—
255	C₂×D₅	—	2	1	no	—
256	C₂×D₅	—	2	1	no	—
257	C₄×C₅	—	2	1	no	—
258	C₄×C₅	—	4	2	no	—
259	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
260	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
261	C₄×C₅	C₃×#137	1	1	yes	characteristic
262	C₄×C₅	C₃×#137	1	1	yes	characteristic
263	Dic₅	—	2	1	no	—
264	Dic₅	—	2	1	no	—
265	Dic₅	—	4	2	no	—
266	Dic₅	C₃×#101	1	2	yes	—
267	C₂×Q₈	—	2	1	no	—
268	C₂×Q₈	—	10	1	no	—
269	C₂²×C₄	—	5	1	no	—
270	C₂²×C₄	—	5	1	no	—
271	C₂²×C₄	—	5	1	no	—
272	M₁₆	—	20	2	no	—
273	M₁₆	—	20	2	no	—
274	C₂×C₈	—	2	1	no	—
275	C₂×C₈	—	2	1	no	—
276	C₂×C₈	—	10	1	no	—
277	C₂×C₈	—	10	1	no	—
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	5	2	no	—
280	C₄⋊C₄	—	10	2	no	—
281	C₄⋊C₄	—	10	2	no	—
282	C₄⋊C₄	—	10	1	no	—
283	C₄⋊C₄	C₃×C₄×D₅	1	2	yes	—
284	C₂²⋊C₄	—	10	2	no	—
285	C₄²	—	5	1	no	—
286	C₄²	—	5	1	no	—
287	C₄²	—	5	1	no	—
288	C₄²	—	10	2	no	—
289	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
290	C₃×C₅	#1561	1	1	yes	characteristic, complemented, Hall
291	C₂²×C₃	—	10	1	no	—
292	C₂²×C₃	—	10	1	no	—
293	C₂²×C₃	—	10	1	no	—
294	C₂²×C₃	D₅×C₂²⋊C₄	1	1	yes	characteristic
295	C₃×C₄	—	2	1	no	—
296	C₃×C₄	—	4	2	no	—
297	C₃×C₄	—	5	2	no	—
298	C₃×C₄	—	10	1	no	—
299	C₃×C₄	—	10	1	no	—
300	C₃×C₄	—	20	2	no	—
301	C₃×C₄	#4057	1	1	yes	characteristic
302	C₃×C₄	#4060	1	1	yes	characteristic
303	C₃×C₄	#4061	1	1	yes	characteristic
304	C₃×C₄	#4064	1	1	yes	characteristic
305	C₂×C₅	C₂×C₃×#101	1	1	yes	characteristic
306	C₂×C₅	C₃×#417	1	1	yes	characteristic
307	C₂×C₅	C₃×#417	1	1	yes	characteristic
308	D₅	—	4	1	no	—
309	C₂³	—	5	1	no	—
310	Q₈	—	4	1	no	—
311	Q₈	—	4	1	no	—
312	Q₈	—	20	1	no	—
313	Q₈	—	20	1	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	5	1	no	—
316	C₂×C₄	—	5	1	no	—
317	C₂×C₄	—	5	1	no	—
318	C₂×C₄	—	10	2	no	—
319	C₂×C₄	—	10	1	no	—
320	C₂×C₄	—	10	1	no	—
321	C₂×C₄	—	10	1	no	—
322	C₂×C₄	—	10	1	no	—
323	C₂×C₄	—	10	2	no	—
324	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
325	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
326	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
327	C₈	—	4	1	no	—
328	C₈	—	4	1	no	—
329	C₈	—	20	1	no	—
330	C₈	—	20	1	no	—
331	C₂×C₃	—	20	1	no	—
332	C₂×C₃	D₅×#101	1	1	yes	characteristic
333	C₂×C₃	#65060	1	1	yes	characteristic
334	C₂×C₃	#65092	1	1	yes	characteristic
335	C₅	C₃×#1561	1	1	yes	characteristic, complemented, Sylow
336	C₂²	—	10	1	no	—
337	C₂²	—	10	1	no	—
338	C₂²	—	10	1	no	—
339	C₂²	C₃×D₅×C₂²⋊C₄	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	4	2	no	—
342	C₄	—	5	2	no	—
343	C₄	—	10	1	no	—
344	C₄	—	10	1	no	—
345	C₄	—	20	2	no	—
346	C₄	C₃×#4057	1	1	yes	characteristic
347	C₄	C₃×#4060	1	1	yes	characteristic
348	C₄	C₃×#4061	1	1	yes	characteristic
349	C₄	C₃×#4064	1	1	yes	characteristic
350	C₃	#10599224	1	1	yes	characteristic, direct product factor, Sylow
351	C₂	—	20	1	no	—
352	C₂	C₃×D₅×#101	1	1	yes	characteristic
353	C₂	C₃×#65060	1	1	yes	characteristic
354	C₂	C₃×#65092	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	D₅×C₂²⋊C₄
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₄²	C₂²×C₃×D₅
Fitting	C₃×C₅×#332	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₃×D₅	C₂²⋊C₄
cosocle	C₄²×C₅	C₂³×C₃
hypercenter	C₃×#332	D₅
hypocenter	C₅	C₃×#1561
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#332

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×#332	C₂	semidirect
5. C₃×#64767	C₂	semidirect
6. C₃×#64706	C₂	semidirect
7. C₃×#64702	C₂	semidirect
8. #10599224	C₃	direct (N × H)
30. C₅×#332	C₂×C₃	semidirect
31. #64767	C₂×C₃	semidirect
32. #64706	C₂×C₃	semidirect
33. #64702	C₂×C₃	semidirect
63. C₃×#332	D₅	semidirect
150. #332	C₃×D₅	semidirect
290. C₃×C₅	#1561	semidirect
335. C₅	C₃×#1561	semidirect
350. C₃	#10599224	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×#66062	2
C₃×#65126	2
C₃×C₅×#332	2
C₃×#64767	2
C₃×#64706	2
C₃×#64702	2
#10599224	3
C₃×#1561	5

Subgroups — C3×#10599224 #14285448

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10599224 #14285448