Subgroups — C₃×C₄×#3965 #14295417

Order 1920.

720 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3965	C₂	1	2	yes	maximal
3	C₃×C₄×#681	C₂	1	2	yes	maximal, complemented
4	C₃×#64750	C₂	1	2	yes	maximal, complemented
5	C₄×#3965	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
7	C₂³×C₃×C₄×C₅	C₄	1	2	yes	—
8	C₃×C₄×C₅×D₄	—	2	2	no	—
9	C₃×C₄×C₅×D₄	—	2	2	no	—
10	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×C₂²⋊C₄	C₄	1	2	yes	—
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₃×#681	C₂²	1	4	yes	—
14	C₃×#3965	C₄	1	8	yes	direct product factor
15	C₃×#3935	—	2	2	no	—
16	C₃×C₄×#168	—	2	2	no	—
17	C₃×C₄×#100	—	5	1	no	maximal, Hall
18	C₂×C₄×C₅×D₄	C₂×C₃	1	1	yes	characteristic
19	C₂×#3965	C₂×C₃	1	2	yes	—
20	C₄×#681	C₂×C₃	1	2	yes	complemented
21	#64750	C₂×C₃	1	2	yes	complemented
22	C₂⁴×C₃×C₅	C₂×C₄	1	2	yes	—
23	C₂×C₃×C₅×D₄	—	2	2	no	—
24	C₂×C₃×C₅×D₄	—	2	2	no	—
25	C₂×C₃×C₅×D₄	C₂×C₄	1	4	yes	—
26	C₂²×C₃×C₄×C₅	—	2	2	no	—
27	C₂²×C₃×C₄×C₅	—	2	2	no	—
28	C₂²×C₃×C₄×C₅	—	4	4	no	—
29	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
31	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₃×C₅×C₄⋊C₄	—	2	2	no	—
34	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
35	C₃×C₄²×C₅	D₄	1	2	yes	—
36	C₃×C₄²×C₅	D₄	1	2	yes	—
37	C₃×#681	—	2	4	no	—
38	C₃×#681	C₂×C₄	1	8	yes	complemented
39	C₂×C₃×#168	—	2	4	no	—
40	C₂×C₃×C₄×D₄	D₅	1	1	yes	characteristic
41	C₂×C₃×#100	—	5	2	no	—
42	C₃×C₄×M₁₆	—	5	2	no	—
43	C₃×#343	—	5	2	no	—
44	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
45	C₂³×C₄×C₅	C₃×C₄	1	2	yes	—
46	C₄×C₅×D₄	—	2	2	no	—
47	C₄×C₅×D₄	—	2	2	no	—
48	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
49	C₂×C₅×C₂²⋊C₄	C₃×C₄	1	2	yes	—
50	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
51	C₂×#681	C₂²×C₃	1	4	yes	—
52	#3965	C₃×C₄	1	8	yes	direct product factor
53	#3935	—	2	2	no	—
54	C₄×#168	—	2	2	no	—
55	C₄×#100	—	5	1	no	Sylow
56	C₂³×C₃×C₅	—	2	2	no	—
57	C₂³×C₃×C₅	—	2	2	no	—
58	C₂³×C₃×C₅	—	4	2	no	—
59	C₂³×C₃×C₅	—	4	2	no	—
60	C₂³×C₃×C₅	C₄²	1	4	yes	—
61	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₃×C₅×D₄	—	2	8	no	—
63	C₂×C₃×C₄×C₅	—	2	2	no	—
64	C₂×C₃×C₄×C₅	—	2	2	no	—
65	C₂×C₃×C₄×C₅	—	4	1	no	—
66	C₂×C₃×C₄×C₅	—	4	4	no	—
67	C₂×C₃×C₄×C₅	—	4	4	no	—
68	C₂×C₃×C₄×C₅	—	4	2	no	—
69	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
70	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
73	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
74	C₂×C₃×C₄×C₅	#41	1	2	yes	—
75	C₃×#168	—	2	8	no	—
76	C₂²×C₃×D₄	C₂×D₅	1	1	yes	characteristic
77	C₂³×C₃×C₄	Dic₅	1	2	yes	—
78	C₂×C₃×M₁₆	—	5	4	no	—
79	C₃×C₄×D₄	—	2	2	no	—
80	C₃×C₄×D₄	—	2	2	no	—
81	C₂×C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
82	C₂×C₃×C₂²⋊C₄	Dic₅	1	2	yes	—
83	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
84	C₃×#100	—	5	8	no	—
85	C₃×#97	—	10	2	no	—
86	C₃×C₄×C₈	—	10	2	no	—
87	C₂⁴×C₅	C₂×C₃×C₄	1	2	yes	—
88	C₂×C₅×D₄	—	2	2	no	—
89	C₂×C₅×D₄	—	2	2	no	—
90	C₂×C₅×D₄	C₂×C₃×C₄	1	4	yes	—
91	C₂²×C₄×C₅	—	2	2	no	—
92	C₂²×C₄×C₅	—	2	2	no	—
93	C₂²×C₄×C₅	—	4	4	no	—
94	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
96	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
98	C₅×C₄⋊C₄	—	2	2	no	—
99	C₅×C₂²⋊C₄	—	2	4	no	—
100	C₄²×C₅	C₃×D₄	1	2	yes	—
101	C₄²×C₅	C₃×D₄	1	2	yes	—
102	#681	—	2	4	no	—
103	#681	C₂×C₃×C₄	1	8	yes	complemented
104	C₂×#168	—	2	4	no	—
105	C₂×C₄×D₄	C₃×D₅	1	1	yes	characteristic
106	C₂×#100	—	5	2	no	—
107	C₄×M₁₆	—	5	2	no	—
108	#343	—	5	2	no	—
109	C₂²×C₃×C₅	—	2	1	no	—
110	C₂²×C₃×C₅	—	2	1	no	—
111	C₂²×C₃×C₅	—	2	4	no	—
112	C₂²×C₃×C₅	—	4	4	no	—
113	C₂²×C₃×C₅	—	4	4	no	—
114	C₂²×C₃×C₅	—	4	2	no	—
115	C₂²×C₃×C₅	—	4	2	no	—
116	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
117	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂²×C₃×C₅	#117	1	1	yes	characteristic
119	C₃×C₄×C₅	—	2	1	no	—
120	C₃×C₄×C₅	—	2	2	no	—
121	C₃×C₄×C₅	—	4	2	no	—
122	C₃×C₄×C₅	#100	1	2	yes	complemented
123	C₃×C₄×C₅	C₄×D₄	1	4	yes	—
124	C₂⁴×C₃	C₂×Dic₅	1	2	yes	—
125	C₂×C₃×D₄	—	2	2	no	—
126	C₂×C₃×D₄	—	2	2	no	—
127	C₂×C₃×D₄	C₄×D₅	1	4	yes	—
128	C₂²×C₃×C₄	—	2	2	no	—
129	C₂²×C₃×C₄	—	2	2	no	—
130	C₂²×C₃×C₄	—	4	4	no	—
131	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
132	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
133	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
134	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
135	C₃×M₁₆	—	5	8	no	—
136	C₃×M₁₆	—	10	4	no	—
137	C₂×C₃×C₈	—	10	4	no	—
138	C₃×C₄⋊C₄	—	2	2	no	—
139	C₃×C₂²⋊C₄	—	2	4	no	—
140	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
141	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
142	C₂³×C₅	—	2	2	no	—
143	C₂³×C₅	—	2	2	no	—
144	C₂³×C₅	—	4	2	no	—
145	C₂³×C₅	—	4	2	no	—
146	C₂³×C₅	C₃×C₄²	1	4	yes	—
147	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
148	C₅×D₄	—	2	8	no	—
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	—	4	1	no	—
152	C₂×C₄×C₅	—	4	4	no	—
153	C₂×C₄×C₅	—	4	4	no	—
154	C₂×C₄×C₅	—	4	2	no	—
155	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
156	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
157	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
159	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
160	C₂×C₄×C₅	C₃×#41	1	2	yes	—
161	#168	—	2	8	no	—
162	C₂²×D₄	C₂×C₃×D₅	1	1	yes	characteristic
163	C₂³×C₄	C₃×Dic₅	1	2	yes	—
164	C₂×M₁₆	—	5	4	no	—
165	C₄×D₄	—	2	2	no	—
166	C₄×D₄	—	2	2	no	—
167	C₂×C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
168	C₂×C₂²⋊C₄	C₃×Dic₅	1	2	yes	—
169	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
170	#100	—	5	8	no	—
171	#97	—	10	2	no	—
172	C₄×C₈	—	10	2	no	—
173	C₂×C₃×C₅	—	2	1	no	—
174	C₂×C₃×C₅	—	2	1	no	—
175	C₂×C₃×C₅	—	4	4	no	—
176	C₂×C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
177	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
178	C₂×C₃×C₅	#413	1	1	yes	characteristic
179	C₂³×C₃	—	2	2	no	—
180	C₂³×C₃	—	2	2	no	—
181	C₂³×C₃	—	4	2	no	—
182	C₂³×C₃	—	4	2	no	—
183	C₂³×C₃	C₄×Dic₅	1	4	yes	—
184	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
185	C₃×D₄	—	2	8	no	—
186	C₂×C₃×C₄	—	2	2	no	—
187	C₂×C₃×C₄	—	2	2	no	—
188	C₂×C₃×C₄	—	4	1	no	—
189	C₂×C₃×C₄	—	4	4	no	—
190	C₂×C₃×C₄	—	4	4	no	—
191	C₂×C₃×C₄	—	4	2	no	—
192	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
193	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
196	C₂×C₃×C₄	#709	1	2	yes	—
197	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
198	C₃×C₈	—	10	8	no	—
199	C₂²×C₅	—	2	1	no	—
200	C₂²×C₅	—	2	1	no	—
201	C₂²×C₅	—	2	4	no	—
202	C₂²×C₅	—	4	4	no	—
203	C₂²×C₅	—	4	4	no	—
204	C₂²×C₅	—	4	2	no	—
205	C₂²×C₅	—	4	2	no	—
206	C₂²×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
207	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
208	C₂²×C₅	C₃×#117	1	1	yes	characteristic
209	C₄×C₅	—	2	1	no	—
210	C₄×C₅	—	2	2	no	—
211	C₄×C₅	—	4	2	no	—
212	C₄×C₅	C₃×#100	1	2	yes	complemented
213	C₄×C₅	C₃×C₄×D₄	1	4	yes	—
214	C₂⁴	C₂×C₃×Dic₅	1	2	yes	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	C₃×C₄×D₅	1	4	yes	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	4	4	no	—
221	C₂²×C₄	C₂×C₃×Dic₅	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₃×Dic₅	1	2	yes	—
223	C₂²×C₄	C₂×C₃×Dic₅	1	1	yes	characteristic
224	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
225	M₁₆	—	5	8	no	—
226	M₁₆	—	10	4	no	—
227	C₂×C₈	—	10	4	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	2	4	no	—
230	C₄²	C₃×C₂²⋊D₅	1	2	yes	—
231	C₄²	C₃×C₂²⋊D₅	1	2	yes	—
232	C₃×C₅	C₄×#100	1	1	yes	characteristic, complemented, Hall
233	C₂²×C₃	—	2	1	no	—
234	C₂²×C₃	—	2	1	no	—
235	C₂²×C₃	—	2	4	no	—
236	C₂²×C₃	—	4	4	no	—
237	C₂²×C₃	—	4	4	no	—
238	C₂²×C₃	—	4	2	no	—
239	C₂²×C₃	—	4	2	no	—
240	C₂²×C₃	C₂×C₄×Dic₅	1	1	yes	characteristic
241	C₂²×C₃	#4072	1	1	yes	characteristic
242	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
243	C₃×C₄	—	2	1	no	—
244	C₃×C₄	—	2	2	no	—
245	C₃×C₄	—	4	2	no	—
246	C₃×C₄	#3965	1	2	yes	direct product factor
247	C₃×C₄	C₄×C₂²⋊D₅	1	4	yes	—
248	C₂×C₅	—	2	1	no	—
249	C₂×C₅	—	2	1	no	—
250	C₂×C₅	—	4	4	no	—
251	C₂×C₅	C₃×C₄×C₂²⋊C₄	1	1	yes	characteristic
252	C₂×C₅	C₂×C₃×#100	1	1	yes	characteristic
253	C₂×C₅	C₃×#413	1	1	yes	characteristic
254	C₂³	—	2	2	no	—
255	C₂³	—	2	2	no	—
256	C₂³	—	4	2	no	—
257	C₂³	—	4	2	no	—
258	C₂³	C₃×C₄×Dic₅	1	4	yes	—
259	C₂³	C₂²×C₃×Dic₅	1	1	yes	characteristic
260	D₄	—	2	8	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	C₃×C₂²⋊Dic₅	1	2	yes	—
268	C₂×C₄	C₃×C₂²⋊Dic₅	1	1	yes	characteristic
269	C₂×C₄	C₃×C₂²⋊Dic₅	1	1	yes	characteristic
270	C₂×C₄	C₂×C₃×C₄×D₅	1	2	yes	—
271	C₂×C₄	C₃×#709	1	2	yes	—
272	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	2	yes	—
273	C₈	—	10	8	no	—
274	C₂×C₃	—	2	1	no	—
275	C₂×C₃	—	2	1	no	—
276	C₂×C₃	—	4	4	no	—
277	C₂×C₃	C₄×C₂²⋊Dic₅	1	1	yes	characteristic
278	C₂×C₃	C₂×#3965	1	1	yes	characteristic
279	C₂×C₃	#65526	1	1	yes	characteristic
280	C₅	C₃×C₄×#100	1	1	yes	characteristic, complemented, Sylow
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	4	4	no	—
285	C₂²	—	4	4	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	C₂×C₃×C₄×Dic₅	1	1	yes	characteristic
289	C₂²	C₃×#4072	1	1	yes	characteristic
290	C₂²	C₂×C₃×C₂²⋊Dic₅	1	1	yes	characteristic
291	C₄	—	2	1	no	—
292	C₄	—	2	2	no	—
293	C₄	—	4	2	no	—
294	C₄	C₃×#3965	1	2	yes	direct product factor
295	C₄	C₃×C₄×C₂²⋊D₅	1	4	yes	—
296	C₃	C₄×#3965	1	1	yes	characteristic, direct product factor, Sylow
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	4	4	no	—
300	C₂	C₃×C₄×C₂²⋊Dic₅	1	1	yes	characteristic
301	C₂	C₂×C₃×#3965	1	1	yes	characteristic
302	C₂	C₃×#65526	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²⋊Dic₅
commutator	C₂²×C₅	C₂×C₃×C₄²
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₂×C₃×C₄×C₅×D₄	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₃×C₅×D₄	C₂²
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₂×C₃×C₄×D₄	D₅
hypocenter	C₅	C₃×C₄×#100
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×D₄

Complements

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

N	H ≅ G/N	type
3. C₃×C₄×#681	C₂	semidirect
4. C₃×#64750	C₂	semidirect
5. C₄×#3965	C₃	direct (N × H)
14. C₃×#3965	C₄	direct (N × H)
20. C₄×#681	C₂×C₃	semidirect
21. #64750	C₂×C₃	semidirect
38. C₃×#681	C₂×C₄	semidirect
52. #3965	C₃×C₄	direct (N × H)
103. #681	C₂×C₃×C₄	semidirect
122. C₃×C₄×C₅	#100	semidirect
212. C₄×C₅	C₃×#100	semidirect
232. C₃×C₅	C₄×#100	semidirect
246. C₃×C₄	#3965	direct (N × H)
280. C₅	C₃×C₄×#100	semidirect
294. C₄	C₃×#3965	direct (N × H)
296. C₃	C₄×#3965	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₅×D₄	2
C₂×C₃×#3965	2
C₃×C₄×#681	2
C₃×#64750	2
C₄×#3965	3
C₃×C₄×#100	5

Subgroups — C3×C4×#3965 #14295417

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×#3965 #14295417