Subgroups — D₃×#4070 #11754347

Order 960.

644 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×D₃	C₂	1	1	yes	maximal, characteristic
2	C₂×D₃×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
3	#91813	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4070	C₂	1	1	yes	maximal, characteristic, complemented
5	#91575	C₂	1	2	yes	maximal, complemented
6	D₃×C₂²⋊Dic₅	C₂	1	2	yes	maximal
7	D₃×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
8	D₃×#683	C₂	1	2	yes	maximal, complemented
9	#91369	C₂	1	1	yes	maximal, characteristic, complemented
10	#91365	C₂	1	1	yes	maximal, characteristic, complemented
11	#91352	C₂	1	2	yes	maximal, complemented
12	C₂×#4070	—	3	1	no	maximal, Hall
13	C₂³×C₅×D₃	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₀	C₂²	1	1	yes	characteristic, complemented
16	C₂²×C₅×Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×D₃	—	2	4	no	—
18	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
22	C₂×C₃×Dic₁₀	C₂²	1	1	yes	characteristic, complemented
23	C₂×#1204	C₂²	1	2	yes	complemented
24	C₂×D₃×Dic₅	C₂²	1	2	yes	—
25	C₂×D₃×Dic₅	C₂²	1	2	yes	—
26	D₃×Dic₁₀	—	2	4	no	—
27	#6719	C₂²	1	2	yes	—
28	C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic, complemented
29	#6712	C₂²	1	2	yes	complemented
30	C₃×C₂²⋊Dic₅	C₂²	1	2	yes	—
31	C₃×C₄⋊Dic₅	C₂²	1	1	yes	characteristic, complemented
32	C₃×#683	C₂²	1	2	yes	complemented
33	Dic₃⋊Dic₅	C₂²	1	2	yes	complemented
34	#6669	C₂²	1	2	yes	complemented
35	#6668	C₂²	1	2	yes	complemented
36	(C₂×D₃)⋊Dic₅	C₂²	1	2	yes	—
37	(C₂×D₃)⋊Dic₅	C₂²	1	2	yes	—
38	D₃×C₂²⋊Q₈	—	5	1	no	maximal, Hall
39	C₂³×C₄×C₅	—	3	1	no	—
40	C₂²×Dic₁₀	—	3	1	no	—
41	C₂×C₂²⋊Dic₅	—	3	2	no	—
42	C₂×C₄⋊Dic₅	—	3	1	no	—
43	#4070	—	3	1	no	—
44	#4070	—	3	1	no	—
45	#4070	—	3	2	no	—
46	#4070	—	3	1	no	—
47	#4070	—	3	2	no	—
48	#4070	D₃	1	1	yes	characteristic, direct product factor
49	C₂×#683	—	3	2	no	—
50	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
51	C₂²×C₅×D₃	—	2	2	no	—
52	C₂²×C₅×D₃	—	2	2	no	—
53	C₂²×C₅×D₃	Q₈	1	4	yes	complemented
54	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
55	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	—	2	2	no	—
57	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
59	C₂×Dic₁₅	C₂³	1	2	yes	—
60	C₂×Dic₁₅	C₂³	1	2	yes	—
61	Dic₃₀	—	2	2	no	—
62	C₂×C₅×Dic₃	—	2	2	no	—
63	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
64	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
65	C₄×C₅×D₃	—	2	2	no	—
66	C₄×C₅×D₃	—	2	2	no	—
67	C₄×C₅×D₃	—	2	2	no	—
68	C₄×C₅×D₃	D₄	1	4	yes	—
69	C₂×C₃×Dic₅	C₂³	1	2	yes	—
70	C₂×C₃×Dic₅	C₂³	1	2	yes	—
71	C₃×Dic₁₀	—	2	2	no	—
72	#1204	—	2	4	no	—
73	D₃×Dic₅	—	2	4	no	—
74	D₃×Dic₅	—	2	4	no	—
75	C₂×D₃×Q₈	—	5	1	no	—
76	C₂²×C₄×D₃	D₅	1	1	yes	characteristic
77	C₃×C₂²⋊Q₈	—	5	1	no	—
78	#946	—	5	1	no	—
79	#924	—	5	1	no	—
80	#897	—	5	1	no	—
81	#896	—	5	2	no	—
82	D₃×C₄⋊C₄	—	5	2	no	—
83	D₃×C₄⋊C₄	—	5	1	no	—
84	D₃×C₂²⋊C₄	—	5	2	no	—
85	#878	—	5	2	no	—
86	C₂⁴×C₅	—	3	1	no	—
87	C₂²×C₄×C₅	—	3	1	no	—
88	C₂²×C₄×C₅	—	3	1	no	—
89	C₂²×C₄×C₅	—	3	1	no	—
90	C₂²×C₄×C₅	—	3	1	no	—
91	C₂²×C₄×C₅	—	3	1	no	—
92	C₂²×C₄×C₅	—	6	4	no	—
93	C₂²×C₄×C₅	C₂×D₃	1	1	yes	characteristic
94	C₂²×Dic₅	—	3	2	no	—
95	C₂²×Dic₅	—	3	2	no	—
96	C₂×Dic₁₀	—	3	1	no	—
97	C₂×Dic₁₀	—	3	2	no	—
98	C₂×Dic₁₀	—	6	4	no	—
99	C₂×Dic₁₀	C₂×D₃	1	1	yes	characteristic, complemented
100	C₂²⋊Dic₅	—	3	2	no	—
101	C₂²⋊Dic₅	—	3	2	no	—
102	C₂²⋊Dic₅	—	3	2	no	—
103	C₂²⋊Dic₅	C₂×D₃	1	2	yes	—
104	C₄⋊Dic₅	—	3	1	no	—
105	C₄⋊Dic₅	—	3	2	no	—
106	C₄⋊Dic₅	C₂×D₃	1	1	yes	characteristic, complemented
107	#683	—	3	2	no	—
108	#683	—	3	2	no	—
109	#683	—	3	2	no	—
110	#683	C₂×D₃	1	2	yes	complemented
111	C₂×C₂²⋊Q₈	—	15	1	no	Sylow
112	C₂²×C₃×C₅	—	2	1	no	—
113	C₂²×C₃×C₅	—	2	1	no	—
114	C₂²×C₃×C₅	C₂×Q₈	1	2	yes	complemented
115	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
116	C₂×C₅×D₃	—	2	1	no	—
117	C₂×C₅×D₃	—	2	1	no	—
118	C₂×C₅×D₃	—	2	8	no	—
119	C₂×C₅×D₃	C₂×D₄	1	2	yes	—
120	C₂×C₅×D₃	C₂×Q₈	1	2	yes	—
121	C₂×C₅×D₃	C₂×Q₈	1	2	yes	complemented
122	C₂×C₅×D₃	#41	1	2	yes	—
123	C₃×C₄×C₅	—	2	1	no	—
124	C₃×C₄×C₅	C₂×D₄	1	2	yes	—
125	Dic₁₅	—	2	2	no	—
126	Dic₁₅	—	2	2	no	—
127	C₃×Dic₅	—	2	2	no	—
128	C₃×Dic₅	—	2	2	no	—
129	C₅×Dic₃	—	2	1	no	—
130	C₅×Dic₃	C₂×D₄	1	2	yes	—
131	C₂³×D₃	C₂×D₅	1	1	yes	characteristic
132	C₂×C₃×Q₈	—	5	1	no	—
133	C₂²×C₃×C₄	C₂×D₅	1	1	yes	characteristic
134	C₂²×Dic₃	C₂×D₅	1	1	yes	characteristic
135	D₃×Q₈	—	10	4	no	—
136	C₂×C₄×D₃	—	2	4	no	—
137	C₂×C₄×D₃	—	5	2	no	—
138	C₂×C₄×D₃	—	5	2	no	—
139	C₂×C₄×D₃	C₂×D₅	1	1	yes	characteristic
140	C₂×C₄×D₃	C₂×D₅	1	1	yes	characteristic
141	C₂×C₄×D₃	C₂×D₅	1	1	yes	characteristic
142	C₂×C₄×D₃	C₂×D₅	1	1	yes	characteristic
143	C₂×Dic₆	—	5	1	no	—
144	C₂×Dic₆	—	5	2	no	—
145	C₃×C₄⋊C₄	—	5	2	no	—
146	C₃×C₄⋊C₄	—	5	1	no	—
147	C₃×C₂²⋊C₄	—	5	2	no	—
148	#217	—	5	2	no	—
149	(C₂×D₃)⋊C₄	—	5	2	no	—
150	(C₂×D₃)⋊C₄	—	5	2	no	—
151	C₄⋊Dic₃	—	5	1	no	—
152	C₄⋊Dic₃	—	5	2	no	—
153	Dic₃⋊C₄	—	5	2	no	—
154	Dic₃⋊C₄	—	5	2	no	—
155	Dic₃⋊C₄	—	5	2	no	—
156	C₂³×C₅	—	3	1	no	—
157	C₂³×C₅	—	3	4	no	—
158	C₂³×C₅	—	3	1	no	—
159	C₂³×C₅	—	6	2	no	—
160	C₂³×C₅	—	6	2	no	—
161	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	—	3	1	no	—
164	C₂×C₄×C₅	—	3	4	no	—
165	C₂×C₄×C₅	—	3	1	no	—
166	C₂×C₄×C₅	—	6	2	no	—
167	C₂×C₄×C₅	—	6	2	no	—
168	C₂×C₄×C₅	—	6	2	no	—
169	C₂×C₄×C₅	—	6	2	no	—
170	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
172	C₂×Dic₅	—	3	2	no	—
173	C₂×Dic₅	—	3	2	no	—
174	C₂×Dic₅	—	6	4	no	—
175	C₂×Dic₅	—	6	4	no	—
176	C₂×Dic₅	C₂²×D₃	1	2	yes	—
177	C₂×Dic₅	C₂²×D₃	1	2	yes	—
178	Dic₁₀	—	2	2	no	—
179	Dic₁₀	—	6	4	no	—
180	Dic₁₀	—	6	2	no	—
181	C₂²×Q₈	—	15	1	no	—
182	C₂³×C₄	—	3	1	no	—
183	C₂²⋊Q₈	—	5	1	no	—
184	C₂²⋊Q₈	—	15	2	no	—
185	C₂²⋊Q₈	—	15	2	no	—
186	C₂²⋊Q₈	—	15	1	no	—
187	C₂²⋊Q₈	—	15	1	no	—
188	C₂²⋊Q₈	—	15	1	no	—
189	C₂×C₄⋊C₄	—	15	2	no	—
190	C₂×C₄⋊C₄	—	15	1	no	—
191	C₂×C₂²⋊C₄	—	15	2	no	—
192	C₂×C₃×C₅	—	2	2	no	—
193	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
195	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
196	C₅×D₃	—	2	2	no	—
197	C₅×D₃	C₂²⋊Q₈	1	4	yes	complemented
198	C₂³×C₃	C₂²×D₅	1	1	yes	characteristic
199	C₂²×D₃	—	2	2	no	—
200	C₂²×D₃	—	2	2	no	—
201	C₂²×D₃	Dic₁₀	1	4	yes	complemented
202	C₂²×D₃	C₂²×D₅	1	1	yes	characteristic
203	C₂²×D₃	C₂²×D₅	1	1	yes	characteristic
204	C₃×Q₈	—	10	2	no	—
205	C₂×C₃×C₄	—	2	2	no	—
206	C₂×C₃×C₄	—	5	2	no	—
207	C₂×C₃×C₄	—	5	2	no	—
208	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
210	C₂×Dic₃	—	2	2	no	—
211	C₂×Dic₃	—	5	2	no	—
212	C₂×Dic₃	—	5	2	no	—
213	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
214	C₂×Dic₃	C₂²×D₅	1	1	yes	characteristic
215	C₄×D₃	—	2	2	no	—
216	C₄×D₃	—	2	2	no	—
217	C₄×D₃	—	2	2	no	—
218	C₄×D₃	—	10	4	no	—
219	C₄×D₃	—	10	4	no	—
220	C₄×D₃	C₂²⋊D₅	1	4	yes	—
221	Dic₆	—	10	2	no	—
222	Dic₆	—	10	4	no	—
223	C₂²×C₅	—	2	1	no	—
224	C₂²×C₅	—	2	1	no	—
225	C₂²×C₅	—	3	2	no	—
226	C₂²×C₅	—	3	2	no	—
227	C₂²×C₅	—	3	2	no	—
228	C₂²×C₅	—	3	2	no	—
229	C₂²×C₅	—	6	8	no	—
230	C₂²×C₅	—	6	1	no	—
231	C₂²×C₅	—	6	1	no	—
232	C₂²×C₅	D₃×Q₈	1	2	yes	complemented
233	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
234	C₄×C₅	—	2	1	no	—
235	C₄×C₅	—	3	2	no	—
236	C₄×C₅	—	6	1	no	—
237	C₄×C₅	D₃×D₄	1	2	yes	—
238	Dic₅	—	2	2	no	—
239	Dic₅	—	2	2	no	—
240	Dic₅	—	6	2	no	—
241	Dic₅	—	6	2	no	—
242	C₂⁴	—	3	1	no	—
243	C₂×Q₈	—	5	1	no	—
244	C₂×Q₈	—	15	2	no	—
245	C₂×Q₈	—	15	1	no	—
246	C₂×Q₈	—	30	4	no	—
247	C₂²×C₄	—	3	1	no	—
248	C₂²×C₄	—	3	1	no	—
249	C₂²×C₄	—	3	1	no	—
250	C₂²×C₄	—	3	1	no	—
251	C₂²×C₄	—	3	1	no	—
252	C₂²×C₄	—	6	4	no	—
253	C₂²×C₄	—	15	2	no	—
254	C₂²×C₄	—	15	2	no	—
255	C₂²×C₄	D₃×D₅	1	1	yes	characteristic
256	C₄⋊C₄	—	5	2	no	—
257	C₄⋊C₄	—	5	1	no	—
258	C₄⋊C₄	—	15	2	no	—
259	C₄⋊C₄	—	15	2	no	—
260	C₄⋊C₄	—	15	2	no	—
261	C₄⋊C₄	—	15	2	no	—
262	C₄⋊C₄	—	15	1	no	—
263	C₂²⋊C₄	—	5	2	no	—
264	C₂²⋊C₄	—	15	2	no	—
265	C₂²⋊C₄	—	15	2	no	—
266	C₂²⋊C₄	—	15	2	no	—
267	C₃×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Hall
268	C₂²×C₃	—	2	1	no	—
269	C₂²×C₃	—	2	1	no	—
270	C₂²×C₃	C₂×Dic₁₀	1	2	yes	complemented
271	C₂²×C₃	C₂³×D₅	1	1	yes	characteristic
272	C₂×D₃	—	2	1	no	—
273	C₂×D₃	—	2	1	no	—
274	C₂×D₃	—	2	8	no	—
275	C₂×D₃	C₂×Dic₁₀	1	2	yes	—
276	C₂×D₃	C₂×Dic₁₀	1	2	yes	complemented
277	C₂×D₃	#709	1	2	yes	—
278	C₂×D₃	C₂×C₂²⋊D₅	1	2	yes	—
279	C₃×C₄	—	2	1	no	—
280	C₃×C₄	—	10	2	no	—
281	C₃×C₄	—	10	2	no	—
282	C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
283	Dic₃	—	2	1	no	—
284	Dic₃	—	10	2	no	—
285	Dic₃	—	10	2	no	—
286	Dic₃	C₂×C₂²⋊D₅	1	2	yes	—
287	C₂×C₅	—	2	2	no	—
288	C₂×C₅	—	3	4	no	—
289	C₂×C₅	—	6	2	no	—
290	C₂×C₅	C₂×D₃×D₄	1	1	yes	characteristic
291	C₂×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
292	C₂×C₅	D₃×#41	1	1	yes	characteristic
293	C₂³	—	3	1	no	—
294	C₂³	—	3	4	no	—
295	C₂³	—	3	1	no	—
296	C₂³	—	6	2	no	—
297	C₂³	—	6	2	no	—
298	C₂³	C₂×D₃×D₅	1	1	yes	characteristic
299	Q₈	—	10	2	no	—
300	Q₈	—	30	4	no	—
301	Q₈	—	30	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	3	1	no	—
304	C₂×C₄	—	3	4	no	—
305	C₂×C₄	—	3	1	no	—
306	C₂×C₄	—	5	2	no	—
307	C₂×C₄	—	5	2	no	—
308	C₂×C₄	—	6	2	no	—
309	C₂×C₄	—	6	2	no	—
310	C₂×C₄	—	6	2	no	—
311	C₂×C₄	—	6	2	no	—
312	C₂×C₄	—	15	2	no	—
313	C₂×C₄	—	15	2	no	—
314	C₂×C₄	—	30	4	no	—
315	C₂×C₄	—	30	4	no	—
316	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
317	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
318	C₂×C₃	—	2	2	no	—
319	C₂×C₃	C₂²×Dic₁₀	1	1	yes	characteristic
320	C₂×C₃	C₂×#709	1	1	yes	characteristic
321	C₂×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
322	D₃	—	2	2	no	—
323	D₃	#4070	1	4	yes	direct product factor
324	C₅	D₃×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	3	2	no	—
328	C₂²	—	3	2	no	—
329	C₂²	—	3	2	no	—
330	C₂²	—	3	2	no	—
331	C₂²	—	6	8	no	—
332	C₂²	—	6	1	no	—
333	C₂²	—	6	1	no	—
334	C₂²	D₃×Dic₁₀	1	2	yes	complemented
335	C₂²	C₂²×D₃×D₅	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	3	2	no	—
338	C₄	—	6	1	no	—
339	C₄	—	10	2	no	—
340	C₄	—	10	2	no	—
341	C₄	—	30	2	no	—
342	C₄	—	30	2	no	—
343	C₄	D₃×C₂²⋊D₅	1	2	yes	—
344	C₃	C₂×#4070	1	1	yes	characteristic, complemented, Sylow
345	C₂	—	2	2	no	—
346	C₂	—	3	4	no	—
347	C₂	—	6	2	no	—
348	C₂	C₂×D₃×Dic₁₀	1	1	yes	characteristic
349	C₂	D₃×#709	1	1	yes	characteristic
350	C₂	C₂×D₃×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×D₅
commutator, socle, cosocle	C₂²×C₃×C₅	C₂⁴
Fitting	C₂²×C₃×C₄×C₅	C₂²
prime radical	C₂³×C₅×D₃	C₂²
hypercenter	C₂²×C₄	D₃×D₅
hypocenter	C₃×C₅	C₂×C₂²⋊Q₈
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×D₃×Dic₁₀	C₂	semidirect
3. #91813	C₂	semidirect
4. C₃×#4070	C₂	semidirect
5. #91575	C₂	semidirect
7. D₃×C₄⋊Dic₅	C₂	semidirect
8. D₃×#683	C₂	semidirect
9. #91369	C₂	semidirect
10. #91365	C₂	semidirect
11. #91352	C₂	semidirect
15. C₂×Dic₃₀	C₂²	semidirect
22. C₂×C₃×Dic₁₀	C₂²	semidirect
23. C₂×#1204	C₂²	semidirect
28. C₄⋊Dic₁₅	C₂²	semidirect
29. #6712	C₂²	semidirect
31. C₃×C₄⋊Dic₅	C₂²	semidirect
32. C₃×#683	C₂²	semidirect
33. Dic₃⋊Dic₅	C₂²	semidirect
34. #6669	C₂²	semidirect
35. #6668	C₂²	semidirect
48. #4070	D₃	direct (N × H)
53. C₂²×C₅×D₃	Q₈	semidirect
99. C₂×Dic₁₀	C₂×D₃	semidirect
106. C₄⋊Dic₅	C₂×D₃	semidirect
110. #683	C₂×D₃	semidirect
114. C₂²×C₃×C₅	C₂×Q₈	semidirect
121. C₂×C₅×D₃	C₂×Q₈	semidirect
197. C₅×D₃	C₂²⋊Q₈	semidirect
201. C₂²×D₃	Dic₁₀	semidirect
232. C₂²×C₅	D₃×Q₈	semidirect
267. C₃×C₅	C₂×C₂²⋊Q₈	semidirect
270. C₂²×C₃	C₂×Dic₁₀	semidirect
276. C₂×D₃	C₂×Dic₁₀	semidirect
323. D₃	#4070	direct (N × H)
324. C₅	D₃×C₂²⋊Q₈	semidirect
334. C₂²	D₃×Dic₁₀	semidirect
344. C₃	C₂×#4070	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₅×D₃	2
C₂×D₃×Dic₁₀	2
#91813	2
C₃×#4070	2
#91575	2
D₃×C₂²⋊Dic₅	2
D₃×C₄⋊Dic₅	2
D₃×#683	2
#91369	2
#91365	2
#91352	2
C₂×#4070	3
D₃×C₂²⋊Q₈	5

Subgroups — D3×#4070 #11754347

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×#4070 #11754347