Subgroups — C₃×C₄×#3948 #14295746

Order 1920.

664 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×Dic₁₀	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#3948	C₂	1	2	yes	maximal
4	C₃×C₄×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic
5	C₃×#64771	C₂	1	2	yes	maximal
6	C₄×#3948	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×Dic₁₀	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
12	C₂×C₃×#683	—	2	1	no	—
13	C₂×C₃×C₄×Dic₅	—	2	1	no	—
14	C₂×C₃×C₄×Dic₅	—	2	1	no	—
15	C₃×C₄×Dic₁₀	—	2	2	no	—
16	C₃×C₄×Dic₁₀	C₄	1	4	yes	complemented
17	C₃×C₄×C₅×C₈	—	2	2	no	—
18	C₃×#3963	—	2	1	no	—
19	C₃×#3948	C₄	1	8	yes	direct product factor
20	C₃×C₄×#103	—	5	1	no	maximal, Hall
21	C₂×C₄×Dic₁₀	C₂×C₃	1	1	yes	characteristic
22	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic
23	C₂×#3948	C₂×C₃	1	2	yes	—
24	C₄×C₄⋊Dic₅	C₂×C₃	1	1	yes	characteristic
25	#64771	C₂×C₃	1	2	yes	—
26	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂²×C₃×Dic₅	—	2	1	no	—
30	C₂²×C₃×Dic₅	—	2	2	no	—
31	C₂²×C₃×Dic₅	—	2	1	no	—
32	C₂×C₃×Dic₁₀	—	2	1	no	—
33	C₂×C₃×Dic₁₀	—	2	2	no	—
34	C₂×C₃×Dic₁₀	C₂×C₄	1	4	yes	—
35	C₂×C₃×Dic₁₀	C₂×C₄	1	2	yes	—
36	C₂×C₃×C₅×C₈	—	2	2	no	—
37	C₂×C₃×C₅×C₈	—	2	2	no	—
38	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	—
39	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
40	C₃×C₄²×C₅	D₄	1	2	yes	—
41	C₃×C₄⋊Dic₅	—	2	1	no	—
42	C₃×C₄⋊Dic₅	—	2	2	no	—
43	C₃×C₄⋊Dic₅	C₂×C₄	1	4	yes	—
44	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
45	C₃×#683	—	2	4	no	—
46	C₃×C₄×Dic₅	—	2	4	no	—
47	C₃×C₄×Dic₅	—	4	2	no	—
48	C₂×C₃×C₄×Q₈	—	5	1	no	—
49	C₂×C₃×#103	—	5	2	no	—
50	C₂×C₃×C₄×C₈	D₅	1	1	yes	characteristic
51	C₃×C₄×C₄⋊C₄	—	5	1	no	—
52	C₃×#340	—	5	2	no	—
53	C₂²×Dic₁₀	C₂²×C₃	1	1	yes	characteristic
54	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
55	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₄⋊Dic₅	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₄⋊Dic₅	C₂²×C₃	1	2	yes	—
58	C₂×#683	—	2	1	no	—
59	C₂×C₄×Dic₅	—	2	1	no	—
60	C₂×C₄×Dic₅	—	2	1	no	—
61	C₄×Dic₁₀	—	2	2	no	—
62	C₄×Dic₁₀	C₃×C₄	1	4	yes	complemented
63	C₄×C₅×C₈	—	2	2	no	—
64	#3963	—	2	1	no	—
65	#3948	C₃×C₄	1	8	yes	direct product factor
66	C₄×#103	—	5	1	no	Sylow
67	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	—	2	2	no	—
69	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
70	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
71	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
72	C₂×C₃×C₄×C₅	Q₁₆	1	2	yes	—
73	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	#41	1	2	yes	—
77	C₂×C₃×Dic₅	—	2	2	no	—
78	C₂×C₃×Dic₅	—	2	4	no	—
79	C₂×C₃×Dic₅	—	2	2	no	—
80	C₂×C₃×Dic₅	—	2	4	no	—
81	C₂×C₃×Dic₅	—	4	2	no	—
82	C₂×C₃×Dic₅	—	4	2	no	—
83	C₂×C₃×Dic₅	—	4	2	no	—
84	C₃×Dic₁₀	—	2	2	no	—
85	C₃×Dic₁₀	—	2	4	no	—
86	C₃×Dic₁₀	C₄²	1	4	yes	complemented
87	C₃×C₅×C₈	—	2	4	no	—
88	C₂²×C₃×Q₈	—	5	1	no	—
89	C₂²×C₃×C₈	C₂×D₅	1	2	yes	—
90	C₃×C₄×Q₈	—	5	4	no	—
91	C₃×C₄×Q₈	—	10	2	no	—
92	C₂×C₃×C₄⋊C₄	—	5	2	no	—
93	C₂×C₃×C₄⋊C₄	—	5	1	no	—
94	C₂×C₃×C₄⋊C₄	—	10	1	no	—
95	C₂×C₃×C₄²	—	10	1	no	—
96	C₂×C₃×C₄²	—	10	1	no	—
97	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
98	C₃×#103	—	5	8	no	—
99	C₃×C₄×C₈	—	2	2	no	—
100	C₃×#95	—	10	1	no	—
101	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
104	C₂²×Dic₅	—	2	1	no	—
105	C₂²×Dic₅	—	2	2	no	—
106	C₂²×Dic₅	—	2	1	no	—
107	C₂×Dic₁₀	—	2	1	no	—
108	C₂×Dic₁₀	—	2	2	no	—
109	C₂×Dic₁₀	C₂×C₃×C₄	1	4	yes	—
110	C₂×Dic₁₀	C₂×C₃×C₄	1	2	yes	—
111	C₂×C₅×C₈	—	2	2	no	—
112	C₂×C₅×C₈	—	2	2	no	—
113	C₂×C₅×C₈	C₂×C₃×C₄	1	4	yes	—
114	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
115	C₄²×C₅	C₃×D₄	1	2	yes	—
116	C₄⋊Dic₅	—	2	1	no	—
117	C₄⋊Dic₅	—	2	2	no	—
118	C₄⋊Dic₅	C₂×C₃×C₄	1	4	yes	—
119	C₄⋊Dic₅	C₂×C₃×C₄	1	2	yes	—
120	#683	—	2	4	no	—
121	C₄×Dic₅	—	2	4	no	—
122	C₄×Dic₅	—	4	2	no	—
123	C₂×C₄×Q₈	—	5	1	no	—
124	C₂×#103	—	5	2	no	—
125	C₂×C₄×C₈	C₃×D₅	1	1	yes	characteristic
126	C₄×C₄⋊C₄	—	5	1	no	—
127	#340	—	5	2	no	—
128	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
130	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
131	C₂²×C₃×C₅	C₂×Q₁₆	1	1	yes	characteristic
132	C₂²×C₃×C₅	#135	1	2	yes	—
133	C₃×C₄×C₅	—	2	2	no	—
134	C₃×C₄×C₅	#103	1	4	yes	complemented
135	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
136	C₃×C₄×C₅	#117	1	1	yes	characteristic
137	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
138	C₃×Dic₅	—	2	4	no	—
139	C₃×Dic₅	—	4	2	no	—
140	C₃×Dic₅	—	4	4	no	—
141	C₂×C₃×Q₈	—	5	4	no	—
142	C₂×C₃×Q₈	—	5	2	no	—
143	C₂×C₃×Q₈	—	10	2	no	—
144	C₂×C₃×Q₈	—	10	1	no	—
145	C₂²×C₃×C₄	—	10	1	no	—
146	C₂²×C₃×C₄	—	10	2	no	—
147	C₂²×C₃×C₄	—	10	1	no	—
148	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
149	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
150	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
151	C₂×C₃×C₈	—	2	2	no	—
152	C₂×C₃×C₈	—	2	2	no	—
153	C₂×C₃×C₈	C₄×D₅	1	4	yes	—
154	C₃×C₄⋊C₄	—	5	4	no	—
155	C₃×C₄⋊C₄	—	5	2	no	—
156	C₃×C₄⋊C₄	—	10	4	no	—
157	C₃×C₄⋊C₄	—	10	2	no	—
158	C₃×C₄⋊C₄	—	10	1	no	—
159	C₃×C₄²	—	10	4	no	—
160	C₃×C₄²	—	20	2	no	—
161	C₃×C₄²	C₄×D₅	1	2	yes	—
162	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
163	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
164	C₂×C₄×C₅	—	2	2	no	—
165	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
166	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
167	C₂×C₄×C₅	C₃×SD₁₆	1	2	yes	—
168	C₂×C₄×C₅	C₃×Q₁₆	1	2	yes	—
169	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
171	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
172	C₂×C₄×C₅	C₃×#41	1	2	yes	—
173	C₂×Dic₅	—	2	2	no	—
174	C₂×Dic₅	—	2	4	no	—
175	C₂×Dic₅	—	2	2	no	—
176	C₂×Dic₅	—	2	4	no	—
177	C₂×Dic₅	—	4	2	no	—
178	C₂×Dic₅	—	4	2	no	—
179	C₂×Dic₅	—	4	2	no	—
180	Dic₁₀	—	2	2	no	—
181	Dic₁₀	—	2	4	no	—
182	Dic₁₀	C₃×C₄²	1	4	yes	complemented
183	C₅×C₈	—	2	4	no	—
184	C₂²×Q₈	—	5	1	no	—
185	C₂²×C₈	C₂×C₃×D₅	1	2	yes	—
186	C₄×Q₈	—	5	4	no	—
187	C₄×Q₈	—	10	2	no	—
188	C₂×C₄⋊C₄	—	5	1	no	—
189	C₂×C₄⋊C₄	—	5	2	no	—
190	C₂×C₄⋊C₄	—	10	1	no	—
191	C₂×C₄²	—	10	1	no	—
192	C₂×C₄²	—	10	1	no	—
193	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
194	#103	—	5	8	no	—
195	C₄×C₈	—	2	2	no	—
196	#95	—	10	1	no	—
197	C₂×C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
198	C₂×C₃×C₅	C₂×#103	1	1	yes	characteristic
199	C₂×C₃×C₅	#416	1	1	yes	characteristic
200	C₂×C₃×C₅	C₄×SD₁₆	1	2	yes	—
201	C₂×C₃×C₅	C₄×Q₁₆	1	2	yes	—
202	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
203	C₃×Q₈	—	5	4	no	—
204	C₃×Q₈	—	10	2	no	—
205	C₃×Q₈	—	10	4	no	—
206	C₂×C₃×C₄	—	2	2	no	—
207	C₂×C₃×C₄	—	10	4	no	—
208	C₂×C₃×C₄	—	10	2	no	—
209	C₂×C₃×C₄	—	10	4	no	—
210	C₂×C₃×C₄	—	10	2	no	—
211	C₂×C₃×C₄	—	20	2	no	—
212	C₂×C₃×C₄	—	20	2	no	—
213	C₂×C₃×C₄	—	20	2	no	—
214	C₂×C₃×C₄	#677	1	2	yes	—
215	C₂×C₃×C₄	Dic₂₀	1	2	yes	—
216	C₂×C₃×C₄	#685	1	2	yes	—
217	C₂×C₃×C₄	#685	1	2	yes	—
218	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
220	C₂×C₃×C₄	#709	1	2	yes	—
221	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
222	C₃×C₈	—	2	4	no	—
223	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
224	C₂²×C₅	C₃×C₄×D₄	1	2	yes	—
225	C₂²×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
226	C₂²×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
227	C₂²×C₅	C₃×#135	1	2	yes	—
228	C₄×C₅	—	2	2	no	—
229	C₄×C₅	C₃×#103	1	4	yes	complemented
230	C₄×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
231	C₄×C₅	C₃×#117	1	1	yes	characteristic
232	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
233	Dic₅	—	2	4	no	—
234	Dic₅	—	4	2	no	—
235	Dic₅	—	4	4	no	—
236	C₂×Q₈	—	5	4	no	—
237	C₂×Q₈	—	5	2	no	—
238	C₂×Q₈	—	10	2	no	—
239	C₂×Q₈	—	10	1	no	—
240	C₂²×C₄	—	10	1	no	—
241	C₂²×C₄	—	10	2	no	—
242	C₂²×C₄	—	10	1	no	—
243	C₂²×C₄	C₃×D₂₀	1	1	yes	characteristic
244	C₂²×C₄	C₃×D₂₀	1	1	yes	characteristic
245	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	C₃×C₄×D₅	1	4	yes	—
249	C₄⋊C₄	—	5	4	no	—
250	C₄⋊C₄	—	5	2	no	—
251	C₄⋊C₄	—	10	4	no	—
252	C₄⋊C₄	—	10	2	no	—
253	C₄⋊C₄	—	10	1	no	—
254	C₄²	—	10	4	no	—
255	C₄²	—	20	2	no	—
256	C₄²	C₃×C₄×D₅	1	2	yes	—
257	C₄²	C₃×C₂²⋊D₅	1	2	yes	—
258	C₃×C₅	C₄×#103	1	1	yes	characteristic, complemented, Hall
259	C₂²×C₃	C₄×D₂₀	1	2	yes	—
260	C₂²×C₃	C₂×#677	1	1	yes	characteristic
261	C₂²×C₃	#4050	1	2	yes	—
262	C₂²×C₃	C₂×Dic₂₀	1	1	yes	characteristic
263	C₂²×C₃	C₂×#685	1	1	yes	characteristic
264	C₃×C₄	—	2	2	no	—
265	C₃×C₄	—	10	4	no	—
266	C₃×C₄	—	20	2	no	—
267	C₃×C₄	—	20	4	no	—
268	C₃×C₄	#3948	1	4	yes	direct product factor
269	C₃×C₄	C₄²×D₅	1	1	yes	characteristic
270	C₃×C₄	#4018	1	1	yes	characteristic
271	C₃×C₄	C₄×C₂²⋊D₅	1	2	yes	—
272	C₂×C₅	C₃×C₄×C₂²⋊C₄	1	1	yes	characteristic
273	C₂×C₅	C₂×C₃×#103	1	1	yes	characteristic
274	C₂×C₅	C₃×#416	1	1	yes	characteristic
275	C₂×C₅	C₃×C₄×SD₁₆	1	2	yes	—
276	C₂×C₅	C₃×C₄×Q₁₆	1	2	yes	—
277	C₂³	C₂×C₃×D₂₀	1	1	yes	characteristic
278	Q₈	—	5	4	no	—
279	Q₈	—	10	2	no	—
280	Q₈	—	10	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	10	4	no	—
283	C₂×C₄	—	10	2	no	—
284	C₂×C₄	—	10	4	no	—
285	C₂×C₄	—	10	2	no	—
286	C₂×C₄	—	20	2	no	—
287	C₂×C₄	—	20	2	no	—
288	C₂×C₄	—	20	2	no	—
289	C₂×C₄	C₃×#677	1	2	yes	—
290	C₂×C₄	C₃×Dic₂₀	1	2	yes	—
291	C₂×C₄	C₃×#685	1	2	yes	—
292	C₂×C₄	C₃×#685	1	2	yes	—
293	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
294	C₂×C₄	C₂×C₃×C₄×D₅	1	2	yes	—
295	C₂×C₄	C₃×#709	1	2	yes	—
296	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
297	C₈	—	2	4	no	—
298	C₂×C₃	C₄×#677	1	2	yes	—
299	C₂×C₃	C₄×Dic₂₀	1	2	yes	—
300	C₂×C₃	C₄×#685	1	1	yes	characteristic
301	C₂×C₃	C₂×#3948	1	1	yes	characteristic
302	C₂×C₃	#65402	1	1	yes	characteristic
303	C₅	C₃×C₄×#103	1	1	yes	characteristic, complemented, Sylow
304	C₂²	C₃×C₄×D₂₀	1	2	yes	—
305	C₂²	C₂×C₃×#677	1	1	yes	characteristic
306	C₂²	C₃×#4050	1	2	yes	—
307	C₂²	C₂×C₃×Dic₂₀	1	1	yes	characteristic
308	C₂²	C₂×C₃×#685	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	10	4	no	—
311	C₄	—	20	2	no	—
312	C₄	—	20	4	no	—
313	C₄	C₃×#3948	1	4	yes	direct product factor
314	C₄	C₃×C₄²×D₅	1	1	yes	characteristic
315	C₄	C₃×#4018	1	1	yes	characteristic
316	C₄	C₃×C₄×C₂²⋊D₅	1	2	yes	—
317	C₃	C₄×#3948	1	1	yes	characteristic, direct product factor, Sylow
318	C₂	C₃×C₄×#677	1	2	yes	—
319	C₂	C₃×C₄×Dic₂₀	1	2	yes	—
320	C₂	C₃×C₄×#685	1	1	yes	characteristic
321	C₂	C₂×C₃×#3948	1	1	yes	characteristic
322	C₂	C₃×#65402	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	D₂₀
commutator	C₄×C₅	C₂×C₃×C₄²
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₂×C₃×C₄×C₈	D₅
hypocenter	C₅	C₃×C₄×#103
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. C₄×#3948	C₃	direct (N × H)
16. C₃×C₄×Dic₁₀	C₄	semidirect
19. C₃×#3948	C₄	direct (N × H)
62. C₄×Dic₁₀	C₃×C₄	semidirect
65. #3948	C₃×C₄	direct (N × H)
86. C₃×Dic₁₀	C₄²	semidirect
134. C₃×C₄×C₅	#103	semidirect
182. Dic₁₀	C₃×C₄²	semidirect
229. C₄×C₅	C₃×#103	semidirect
258. C₃×C₅	C₄×#103	semidirect
268. C₃×C₄	#3948	direct (N × H)
303. C₅	C₃×C₄×#103	semidirect
313. C₄	C₃×#3948	direct (N × H)
317. C₃	C₄×#3948	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×Dic₁₀	2
C₂×C₃×C₄×C₅×C₈	2
C₂×C₃×#3948	2
C₃×C₄×C₄⋊Dic₅	2
C₃×#64771	2
C₄×#3948	3
C₃×C₄×#103	5

Subgroups — C3×C4×#3948 #14295746

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×#3948 #14295746