Subgroups — C₃×C₄×#3957 #14295581

Order 1920.

720 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×#709	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3957	C₂	1	2	yes	maximal
4	C₃×C₄²×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64775	C₂	1	2	yes	maximal, complemented
6	C₄×#3957	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×#709	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₃×C₄×C₂²⋊D₅	—	2	2	no	—
11	C₃×#4072	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄×Dic₅	—	2	2	no	—
13	C₂×C₃×C₄×Dic₅	C₂²	1	2	yes	—
14	C₃×C₄×D₂₀	C₄	1	2	yes	complemented
15	C₃×#4018	—	2	1	no	—
16	C₃×C₄²×D₅	—	2	1	no	—
17	C₃×C₄×Dic₁₀	C₄	1	2	yes	complemented
18	C₃×C₅×#97	—	2	1	no	—
19	C₃×C₄×C₅×C₈	—	2	1	no	—
20	C₃×#3957	C₄	1	8	yes	direct product factor
21	C₃×C₄×#104	—	5	1	no	maximal, Hall
22	C₄×#709	C₂×C₃	1	1	yes	characteristic
23	C₄×C₅×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×#3957	C₂×C₃	1	2	yes	—
25	C₄²×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
26	#64775	C₂×C₃	1	2	yes	complemented
27	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂×C₃×C₂²⋊D₅	—	2	1	no	—
30	C₂²×C₃×Dic₅	—	2	2	no	—
31	C₃×#709	—	2	2	no	—
32	C₃×#709	C₂×C₄	1	4	yes	—
33	C₂×C₃×D₂₀	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×C₄×D₅	—	2	1	no	—
35	C₂×C₃×C₄×D₅	—	2	2	no	—
36	C₂×C₃×Dic₁₀	C₂×C₄	1	1	yes	characteristic
37	C₃×C₅×M₁₆	—	2	2	no	—
38	C₃×C₅×M₁₆	C₂×C₄	1	4	yes	complemented
39	C₂×C₃×C₅×C₈	—	2	2	no	—
40	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
41	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
43	C₃×C₂²⋊Dic₅	—	2	1	no	—
44	C₃×#685	—	2	2	no	—
45	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
46	C₃×#683	—	2	2	no	—
47	C₃×C₄×Dic₅	—	2	8	no	—
48	C₃×C₄×Dic₅	—	2	2	no	—
49	C₃×C₄×Dic₅	—	2	1	no	—
50	C₃×C₄×Dic₅	—	2	2	no	—
51	C₃×C₄×Dic₅	C₂×C₄	1	4	yes	complemented
52	C₃×C₄×#41	—	5	1	no	—
53	C₂×C₃×#104	—	5	2	no	—
54	C₃×C₄×M₁₆	D₅	1	1	yes	characteristic, complemented
55	C₃×C₄³	—	5	1	no	—
56	C₃×#339	—	5	2	no	—
57	C₂×#709	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
59	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
60	C₄×C₂²⋊D₅	—	2	2	no	—
61	#4072	C₂²×C₃	1	1	yes	characteristic
62	C₂×C₄×Dic₅	—	2	2	no	—
63	C₂×C₄×Dic₅	C₂²×C₃	1	2	yes	—
64	C₄×D₂₀	C₃×C₄	1	2	yes	complemented
65	#4018	—	2	1	no	—
66	C₄²×D₅	—	2	1	no	—
67	C₄×Dic₁₀	C₃×C₄	1	2	yes	complemented
68	C₅×#97	—	2	1	no	—
69	C₄×C₅×C₈	—	2	1	no	—
70	#3957	C₃×C₄	1	8	yes	direct product factor
71	C₄×#104	—	5	1	no	Sylow
72	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₃×D₅	—	2	1	no	—
74	C₂×C₃×C₄×C₅	—	2	4	no	—
75	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
76	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
79	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	#41	1	2	yes	—
81	C₃×C₂²⋊D₅	—	2	4	no	—
82	C₂×C₃×Dic₅	—	2	8	no	—
83	C₂×C₃×Dic₅	—	2	1	no	—
84	C₂×C₃×Dic₅	—	2	4	no	—
85	C₂×C₃×Dic₅	—	2	2	no	—
86	C₃×D₂₀	—	2	1	no	—
87	C₃×D₂₀	C₄²	1	2	yes	complemented
88	C₃×C₄×D₅	—	2	4	no	—
89	C₃×C₄×D₅	—	4	4	no	—
90	C₃×Dic₁₀	—	2	1	no	—
91	C₃×Dic₁₀	C₄²	1	2	yes	complemented
92	C₃×C₅×C₈	—	2	4	no	—
93	C₂×C₃×#41	—	5	1	no	—
94	C₂×C₃×M₁₆	C₂×D₅	1	2	yes	—
95	C₃×C₄×Q₈	—	5	2	no	—
96	C₃×C₄×D₄	—	5	2	no	—
97	C₃×C₄×D₄	—	10	2	no	—
98	C₃×#117	—	5	1	no	—
99	C₃×#117	—	10	1	no	—
100	C₂×C₃×C₄²	—	5	2	no	—
101	C₂×C₃×C₄²	—	10	2	no	—
102	C₂×C₃×C₄²	—	10	1	no	—
103	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
104	C₃×#104	—	5	8	no	—
105	C₃×#97	—	2	1	no	—
106	C₃×C₄×C₈	—	2	1	no	—
107	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
108	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
109	C₂×C₂²⋊D₅	—	2	1	no	—
110	C₂²×Dic₅	—	2	2	no	—
111	#709	—	2	2	no	—
112	#709	C₂×C₃×C₄	1	4	yes	—
113	C₂×D₂₀	C₂×C₃×C₄	1	1	yes	characteristic
114	C₂×C₄×D₅	—	2	1	no	—
115	C₂×C₄×D₅	—	2	2	no	—
116	C₂×Dic₁₀	C₂×C₃×C₄	1	1	yes	characteristic
117	C₅×M₁₆	—	2	2	no	—
118	C₅×M₁₆	C₂×C₃×C₄	1	4	yes	complemented
119	C₂×C₅×C₈	—	2	2	no	—
120	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
121	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
122	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
123	C₂²⋊Dic₅	—	2	1	no	—
124	#685	—	2	2	no	—
125	C₄⋊Dic₅	C₂×C₃×C₄	1	2	yes	—
126	#683	—	2	2	no	—
127	C₄×Dic₅	—	2	8	no	—
128	C₄×Dic₅	—	2	2	no	—
129	C₄×Dic₅	—	2	1	no	—
130	C₄×Dic₅	—	2	2	no	—
131	C₄×Dic₅	C₂×C₃×C₄	1	4	yes	complemented
132	C₄×#41	—	5	1	no	—
133	C₂×#104	—	5	2	no	—
134	C₄×M₁₆	C₃×D₅	1	1	yes	characteristic, complemented
135	C₄³	—	5	1	no	—
136	#339	—	5	2	no	—
137	C₂²×C₃×C₅	—	2	2	no	—
138	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
140	C₂×C₃×D₅	—	2	2	no	—
141	C₂×C₃×D₅	—	4	2	no	—
142	C₃×C₄×C₅	—	2	2	no	—
143	C₃×C₄×C₅	#104	1	4	yes	complemented
144	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
145	C₃×C₄×C₅	#117	1	1	yes	characteristic
146	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
147	C₃×Dic₅	—	2	8	no	—
148	C₃×Dic₅	—	2	2	no	—
149	C₃×Dic₅	—	4	2	no	—
150	C₃×#41	—	5	4	no	—
151	C₃×#41	—	10	2	no	—
152	C₂×C₃×Q₈	—	5	1	no	—
153	C₂×C₃×D₄	—	5	1	no	—
154	C₂×C₃×D₄	—	10	1	no	—
155	C₂²×C₃×C₄	—	10	2	no	—
156	C₂²×C₃×C₄	—	10	1	no	—
157	C₂²×C₃×C₄	—	10	2	no	—
158	C₂²×C₃×C₄	D₂₀	1	2	yes	—
159	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
160	C₃×M₁₆	—	2	2	no	—
161	C₃×M₁₆	C₄×D₅	1	4	yes	complemented
162	C₂×C₃×C₈	—	2	2	no	—
163	C₃×C₄⋊C₄	—	5	2	no	—
164	C₃×C₄⋊C₄	—	10	2	no	—
165	C₃×C₂²⋊C₄	—	10	1	no	—
166	C₃×C₂²⋊C₄	—	10	2	no	—
167	C₃×C₄²	—	5	4	no	—
168	C₃×C₄²	—	10	8	no	—
169	C₃×C₄²	—	10	2	no	—
170	C₃×C₄²	—	10	2	no	—
171	C₃×C₄²	—	10	1	no	—
172	C₃×C₄²	C₄×D₅	1	2	yes	—
173	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
174	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
175	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
176	C₂²×D₅	—	2	1	no	—
177	C₂×C₄×C₅	—	2	4	no	—
178	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
179	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
180	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
182	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
183	C₂×C₄×C₅	C₃×#41	1	2	yes	—
184	C₂²⋊D₅	—	2	4	no	—
185	C₂×Dic₅	—	2	8	no	—
186	C₂×Dic₅	—	2	1	no	—
187	C₂×Dic₅	—	2	4	no	—
188	C₂×Dic₅	—	2	2	no	—
189	D₂₀	—	2	1	no	—
190	D₂₀	C₃×C₄²	1	2	yes	complemented
191	C₄×D₅	—	2	4	no	—
192	C₄×D₅	—	4	4	no	—
193	Dic₁₀	—	2	1	no	—
194	Dic₁₀	C₃×C₄²	1	2	yes	complemented
195	C₅×C₈	—	2	4	no	—
196	C₂×#41	—	5	1	no	—
197	C₂×M₁₆	C₂×C₃×D₅	1	2	yes	—
198	C₄×Q₈	—	5	2	no	—
199	C₄×D₄	—	5	2	no	—
200	C₄×D₄	—	10	2	no	—
201	#117	—	5	1	no	—
202	#117	—	10	1	no	—
203	C₂×C₄²	—	5	2	no	—
204	C₂×C₄²	—	10	2	no	—
205	C₂×C₄²	—	10	1	no	—
206	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
207	#104	—	5	8	no	—
208	#97	—	2	1	no	—
209	C₄×C₈	—	2	1	no	—
210	C₂×C₃×C₅	—	2	2	no	—
211	C₂×C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
212	C₂×C₃×C₅	C₂×#104	1	2	yes	—
213	C₃×D₅	—	4	2	no	—
214	C₂³×C₃	—	10	1	no	—
215	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
216	C₃×Q₈	—	5	2	no	—
217	C₃×Q₈	—	10	1	no	—
218	C₃×D₄	—	5	2	no	—
219	C₃×D₄	—	10	4	no	—
220	C₃×D₄	—	10	1	no	—
221	C₂×C₃×C₄	—	2	4	no	—
222	C₂×C₃×C₄	—	10	8	no	—
223	C₂×C₃×C₄	—	10	4	no	—
224	C₂×C₃×C₄	—	10	1	no	—
225	C₂×C₃×C₄	—	10	4	no	—
226	C₂×C₃×C₄	—	10	2	no	—
227	C₂×C₃×C₄	—	20	4	no	—
228	C₂×C₃×C₄	#685	1	2	yes	—
229	C₂×C₃×C₄	#685	1	2	yes	—
230	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
231	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
232	C₂×C₃×C₄	#709	1	2	yes	—
233	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
234	C₃×C₈	—	2	4	no	—
235	C₂²×C₅	—	2	2	no	—
236	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
237	C₂²×C₅	C₃×C₄×D₄	1	2	yes	—
238	C₂×D₅	—	2	2	no	—
239	C₂×D₅	—	4	2	no	—
240	C₄×C₅	—	2	2	no	—
241	C₄×C₅	C₃×#104	1	4	yes	complemented
242	C₄×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
243	C₄×C₅	C₃×#117	1	1	yes	characteristic
244	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
245	Dic₅	—	2	8	no	—
246	Dic₅	—	2	2	no	—
247	Dic₅	—	4	2	no	—
248	#41	—	5	4	no	—
249	#41	—	10	2	no	—
250	C₂×Q₈	—	5	1	no	—
251	C₂×D₄	—	5	1	no	—
252	C₂×D₄	—	10	1	no	—
253	C₂²×C₄	—	10	2	no	—
254	C₂²×C₄	—	10	1	no	—
255	C₂²×C₄	—	10	2	no	—
256	C₂²×C₄	C₃×D₂₀	1	2	yes	—
257	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
258	M₁₆	—	2	2	no	—
259	M₁₆	C₃×C₄×D₅	1	4	yes	complemented
260	C₂×C₈	—	2	2	no	—
261	C₄⋊C₄	—	5	2	no	—
262	C₄⋊C₄	—	10	2	no	—
263	C₂²⋊C₄	—	10	1	no	—
264	C₂²⋊C₄	—	10	2	no	—
265	C₄²	—	5	4	no	—
266	C₄²	—	10	8	no	—
267	C₄²	—	10	2	no	—
268	C₄²	—	10	2	no	—
269	C₄²	—	10	1	no	—
270	C₄²	C₃×C₄×D₅	1	2	yes	—
271	C₄²	C₃×C₂²⋊D₅	1	1	yes	characteristic
272	C₄²	C₃×C₂²⋊D₅	1	1	yes	characteristic
273	C₃×C₅	C₄×#104	1	1	yes	characteristic, complemented, Hall
274	C₂²×C₃	—	2	2	no	—
275	C₂²×C₃	—	10	2	no	—
276	C₂²×C₃	—	20	2	no	—
277	C₂²×C₃	C₄×D₂₀	1	2	yes	—
278	C₂²×C₃	C₂×#685	1	1	yes	characteristic
279	C₃×C₄	—	2	2	no	—
280	C₃×C₄	—	10	8	no	—
281	C₃×C₄	—	10	2	no	—
282	C₃×C₄	—	20	2	no	—
283	C₃×C₄	#3957	1	4	yes	direct product factor
284	C₃×C₄	C₄²×D₅	1	1	yes	characteristic
285	C₃×C₄	#4018	1	1	yes	characteristic
286	C₃×C₄	C₄×C₂²⋊D₅	1	2	yes	—
287	C₂×C₅	—	2	2	no	—
288	C₂×C₅	C₃×C₄×C₂²⋊C₄	1	1	yes	characteristic
289	C₂×C₅	C₂×C₃×#104	1	2	yes	—
290	D₅	—	4	2	no	—
291	C₂³	—	10	1	no	—
292	C₂³	C₂×C₃×D₂₀	1	1	yes	characteristic
293	Q₈	—	5	2	no	—
294	Q₈	—	10	1	no	—
295	D₄	—	5	2	no	—
296	D₄	—	10	4	no	—
297	D₄	—	10	1	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	10	8	no	—
300	C₂×C₄	—	10	4	no	—
301	C₂×C₄	—	10	1	no	—
302	C₂×C₄	—	10	4	no	—
303	C₂×C₄	—	10	2	no	—
304	C₂×C₄	—	20	4	no	—
305	C₂×C₄	C₃×#685	1	2	yes	—
306	C₂×C₄	C₃×#685	1	2	yes	—
307	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
308	C₂×C₄	C₂×C₃×C₄×D₅	1	2	yes	—
309	C₂×C₄	C₃×#709	1	2	yes	—
310	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
311	C₈	—	2	4	no	—
312	C₂×C₃	—	2	2	no	—
313	C₂×C₃	—	20	2	no	—
314	C₂×C₃	C₄×#685	1	1	yes	characteristic
315	C₂×C₃	C₂×#3957	1	2	yes	—
316	C₅	C₃×C₄×#104	1	1	yes	characteristic, complemented, Sylow
317	C₂²	—	2	2	no	—
318	C₂²	—	10	2	no	—
319	C₂²	—	20	2	no	—
320	C₂²	C₃×C₄×D₂₀	1	2	yes	—
321	C₂²	C₂×C₃×#685	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	10	8	no	—
324	C₄	—	10	2	no	—
325	C₄	—	20	2	no	—
326	C₄	C₃×#3957	1	4	yes	direct product factor
327	C₄	C₃×C₄²×D₅	1	1	yes	characteristic
328	C₄	C₃×#4018	1	1	yes	characteristic
329	C₄	C₃×C₄×C₂²⋊D₅	1	2	yes	—
330	C₃	C₄×#3957	1	1	yes	characteristic, direct product factor, Sylow
331	C₂	—	2	2	no	—
332	C₂	—	20	2	no	—
333	C₂	C₃×C₄×#685	1	1	yes	characteristic
334	C₂	C₂×C₃×#3957	1	2	yes	—

	subgroup	G/H
center	C₃×C₄²	C₂²⋊D₅
commutator	C₄×C₅	C₂×C₃×C₄²
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₃×C₄×C₅×M₁₆	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂×C₃×#709	C₂²
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₃×C₄×M₁₆	D₅
hypocenter	C₅	C₃×C₄×#104
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₄² ◁ C₂×C₃×C₄² ◁ C₃×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₅×M₁₆	C₂	semidirect
4. C₃×C₄²×Dic₅	C₂	semidirect
5. C₃×#64775	C₂	semidirect
6. C₄×#3957	C₃	direct (N × H)
14. C₃×C₄×D₂₀	C₄	semidirect
17. C₃×C₄×Dic₁₀	C₄	semidirect
20. C₃×#3957	C₄	direct (N × H)
23. C₄×C₅×M₁₆	C₂×C₃	semidirect
25. C₄²×Dic₅	C₂×C₃	semidirect
26. #64775	C₂×C₃	semidirect
38. C₃×C₅×M₁₆	C₂×C₄	semidirect
51. C₃×C₄×Dic₅	C₂×C₄	semidirect
54. C₃×C₄×M₁₆	D₅	semidirect
64. C₄×D₂₀	C₃×C₄	semidirect
67. C₄×Dic₁₀	C₃×C₄	semidirect
70. #3957	C₃×C₄	direct (N × H)
87. C₃×D₂₀	C₄²	semidirect
91. C₃×Dic₁₀	C₄²	semidirect
118. C₅×M₁₆	C₂×C₃×C₄	semidirect
131. C₄×Dic₅	C₂×C₃×C₄	semidirect
134. C₄×M₁₆	C₃×D₅	semidirect
143. C₃×C₄×C₅	#104	semidirect
161. C₃×M₁₆	C₄×D₅	semidirect
190. D₂₀	C₃×C₄²	semidirect
194. Dic₁₀	C₃×C₄²	semidirect
241. C₄×C₅	C₃×#104	semidirect
259. M₁₆	C₃×C₄×D₅	semidirect
273. C₃×C₅	C₄×#104	semidirect
283. C₃×C₄	#3957	direct (N × H)
316. C₅	C₃×C₄×#104	semidirect
326. C₄	C₃×#3957	direct (N × H)
330. C₃	C₄×#3957	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×C₄×#709	2
C₃×C₄×C₅×M₁₆	2
C₂×C₃×#3957	2
C₃×C₄²×Dic₅	2
C₃×#64775	2
C₄×#3957	3
C₃×C₄×#104	5

Subgroups — C3×C4×#3957 #14295581

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×#3957 #14295581