Subgroups — C₂×D₃×#3990 #14478102

Order 1920.

800 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	C₂×D₃×C₈×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×#91219	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3990	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#91166	C₂	1	2	yes	maximal, complemented
5	C₂×D₃×#674	C₂	1	2	yes	maximal, complemented
6	D₃×#3990	C₂	1	8	yes	maximal, direct product factor
7	C₂²×#3990	—	3	1	no	maximal, Hall
8	C₂×C₄×D₃×D₅	C₄	1	1	yes	characteristic
9	C₂×C₈×D₁₅	C₄	1	1	yes	characteristic
10	C₂×C₅×D₃×C₈	C₄	1	1	yes	characteristic
11	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
12	C₂×#6648	C₂²	1	1	yes	characteristic
13	C₂×D₃×#168	C₂²	1	1	yes	characteristic
14	C₂×D₅×C₃⋊C₈	C₄	1	1	yes	characteristic
15	D₃×C₈×D₅	C₄	1	4	yes	—
16	D₃×C₈×D₅	C₂²	1	4	yes	—
17	C₂×C₃⋊#674	C₂²	1	2	yes	complemented
18	#91219	C₂²	1	4	yes	complemented
19	C₂×C₃×#674	C₂²	1	2	yes	complemented
20	C₃×#3990	C₂²	1	4	yes	complemented
21	#91166	C₂²	1	8	yes	complemented
22	D₃×#674	C₂²	1	8	yes	complemented
23	C₂×D₃×M₃₂	—	5	1	no	maximal, Hall
24	C₂²×C₈×D₅	—	3	1	no	—
25	C₂²×#674	—	3	2	no	—
26	C₂×#3990	—	3	1	no	—
27	C₂×#3990	—	3	2	no	—
28	C₂×#3990	—	3	8	no	—
29	C₂×#3990	D₃	1	1	yes	characteristic, direct product factor
30	C₂²×D₃×D₅	C₈	1	1	yes	characteristic
31	C₂×C₄×D₁₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
34	C₂×#1200	C₈	1	1	yes	characteristic
35	C₂×D₃×Dic₅	C₈	1	1	yes	characteristic
36	C₂×D₅×Dic₃	C₈	1	1	yes	characteristic
37	C₄×D₃×D₅	C₈	1	4	yes	—
38	C₄×D₃×D₅	C₂×C₄	1	4	yes	—
39	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
41	C₈×D₁₅	C₂×C₄	1	4	yes	—
42	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
43	C₅×D₃×C₈	C₂×C₄	1	4	yes	—
44	C₂×C₃×#168	C₂³	1	1	yes	characteristic
45	C₃×C₈×D₅	C₂×C₄	1	2	yes	—
46	C₃×C₈×D₅	C₂³	1	2	yes	—
47	#6648	C₂×C₄	1	2	yes	—
48	#6648	C₂³	1	2	yes	—
49	D₃×#168	C₂×C₄	1	2	yes	—
50	D₃×#168	C₂³	1	2	yes	—
51	D₅×C₃⋊C₈	C₂×C₄	1	4	yes	—
52	C₃⋊#674	C₂³	1	4	yes	complemented
53	C₃×#674	C₂³	1	4	yes	complemented
54	C₂²×D₃×C₈	—	5	1	no	—
55	C₂×C₃×M₃₂	—	5	1	no	—
56	C₂×#812	—	5	1	no	—
57	D₃×M₃₂	—	5	8	no	—
58	C₂×#798	—	5	2	no	—
59	C₂×D₃×C₁₆	—	5	2	no	—
60	C₂²×C₄×D₅	—	3	1	no	—
61	C₂²×C₅×C₈	—	3	1	no	—
62	C₂²×#168	—	3	1	no	—
63	C₂×C₈×D₅	—	3	1	no	—
64	C₂×C₈×D₅	—	3	1	no	—
65	C₂×C₈×D₅	—	3	1	no	—
66	C₂×C₈×D₅	—	3	4	no	—
67	C₂×C₈×D₅	—	3	4	no	—
68	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
69	C₂×#674	—	3	2	no	—
70	C₂×#674	—	3	8	no	—
71	C₂×#674	C₂×D₃	1	2	yes	complemented
72	#3990	—	3	4	no	—
73	#3990	—	3	8	no	—
74	#3990	C₂×D₃	1	4	yes	direct product factor
75	C₂²×M₃₂	—	15	1	no	Sylow
76	C₂²×D₁₅	C₂×C₈	1	1	yes	characteristic
77	C₂²×C₅×D₃	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₃×D₅	C₂×C₈	1	1	yes	characteristic
79	C₂×D₃×D₅	—	2	4	no	—
80	C₂×D₃×D₅	C₂×C₈	1	4	yes	—
81	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₂×Dic₁₅	C₂×C₈	1	1	yes	characteristic
83	C₄×D₁₅	C₂×C₈	1	2	yes	—
84	C₄×D₁₅	C₂²×C₄	1	2	yes	—
85	C₂×C₅×Dic₃	C₂×C₈	1	1	yes	characteristic
86	C₄×C₅×D₃	C₂×C₈	1	2	yes	—
87	C₄×C₅×D₃	C₂²×C₄	1	2	yes	—
88	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
89	C₃×C₄×D₅	C₂×C₈	1	2	yes	—
90	C₃×C₄×D₅	C₂²×C₄	1	2	yes	—
91	#1200	C₂×C₈	1	4	yes	—
92	D₃×Dic₅	C₂×C₈	1	4	yes	—
93	D₅×Dic₃	C₂×C₈	1	4	yes	—
94	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
95	C₃⋊#168	C₂²×C₄	1	2	yes	—
96	C₃×#168	C₂²×C₄	1	1	yes	characteristic
97	C₃×#168	C₂⁴	1	1	yes	characteristic
98	C₅×C₃⋊C₈	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×D₃	—	5	1	no	—
100	C₂²×C₃×C₈	—	5	1	no	—
101	C₂²×C₃⋊C₈	—	5	1	no	—
102	C₂×D₃×C₈	—	5	1	no	—
103	C₂×D₃×C₈	—	5	1	no	—
104	C₂×D₃×C₈	—	5	1	no	—
105	C₂×D₃×C₈	—	5	4	no	—
106	C₂×D₃×C₈	—	5	4	no	—
107	C₂×D₃×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
108	C₃×M₃₂	—	5	4	no	—
109	C₂×C₃×C₁₆	—	5	2	no	—
110	#812	—	5	4	no	—
111	C₂×C₃⋊C₁₆	—	5	2	no	—
112	#798	—	5	8	no	—
113	D₃×C₁₆	—	5	8	no	—
114	C₂³×D₅	—	3	1	no	—
115	C₂²×C₄×C₅	—	3	1	no	—
116	C₂²×Dic₅	—	3	1	no	—
117	C₂×C₄×D₅	—	3	1	no	—
118	C₂×C₄×D₅	—	3	1	no	—
119	C₂×C₄×D₅	—	3	1	no	—
120	C₂×C₄×D₅	—	3	4	no	—
121	C₂×C₄×D₅	—	3	4	no	—
122	C₂×C₄×D₅	C₄×D₃	1	1	yes	characteristic
123	C₂×C₅×C₈	—	3	1	no	—
124	C₂×C₅×C₈	—	3	4	no	—
125	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
126	C₂×#168	—	3	1	no	—
127	C₂×#168	—	3	2	no	—
128	C₂×#168	—	3	2	no	—
129	C₂×#168	C₂²×D₃	1	1	yes	characteristic
130	C₈×D₅	—	3	4	no	—
131	C₈×D₅	—	3	4	no	—
132	C₈×D₅	—	3	2	no	—
133	C₈×D₅	—	3	2	no	—
134	C₈×D₅	C₄×D₃	1	2	yes	—
135	C₈×D₅	C₂²×D₃	1	2	yes	—
136	#674	—	3	4	no	—
137	#674	C₂²×D₃	1	4	yes	complemented
138	C₂³×C₈	—	15	1	no	—
139	C₂×M₃₂	—	5	1	no	—
140	C₂×M₃₂	—	15	2	no	—
141	C₂×M₃₂	—	15	8	no	—
142	C₂×M₃₂	—	15	1	no	—
143	C₂²×C₁₆	—	15	2	no	—
144	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
145	C₂×D₁₅	—	2	2	no	—
146	C₂×D₁₅	C₂²×C₈	1	2	yes	—
147	C₂×C₅×D₃	M₃₂	1	4	yes	complemented
148	C₂×C₅×D₃	C₂²×C₈	1	2	yes	—
149	C₂×C₃×D₅	—	2	2	no	—
150	C₂×C₃×D₅	C₂²×C₈	1	2	yes	—
151	D₃×D₅	—	2	8	no	—
152	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
153	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
154	Dic₁₅	C₂²×C₈	1	2	yes	—
155	C₃×Dic₅	C₂²×C₈	1	2	yes	—
156	C₅×Dic₃	C₂²×C₈	1	2	yes	—
157	C₂³×D₃	—	5	1	no	—
158	C₂²×C₃×C₄	—	5	1	no	—
159	C₂²×Dic₃	—	5	1	no	—
160	C₂×C₄×D₃	—	5	1	no	—
161	C₂×C₄×D₃	—	5	1	no	—
162	C₂×C₄×D₃	—	5	1	no	—
163	C₂×C₄×D₃	—	5	4	no	—
164	C₂×C₄×D₃	—	5	4	no	—
165	C₂×C₄×D₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
166	C₂×C₃×C₈	—	5	1	no	—
167	C₂×C₃×C₈	—	5	2	no	—
168	C₂×C₃×C₈	—	5	2	no	—
169	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
170	C₂×C₃⋊C₈	—	5	1	no	—
171	C₂×C₃⋊C₈	—	5	4	no	—
172	C₂×C₃⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
173	D₃×C₈	—	5	2	no	—
174	D₃×C₈	—	5	2	no	—
175	D₃×C₈	—	5	4	no	—
176	D₃×C₈	—	5	2	no	—
177	D₃×C₈	—	5	2	no	—
178	D₃×C₈	C₂×AGL(1,𝔽₅)	1	4	yes	—
179	C₃×C₁₆	—	5	4	no	—
180	C₃⋊C₁₆	—	5	4	no	—
181	C₂³×C₅	—	3	1	no	—
182	C₂²×D₅	—	3	1	no	—
183	C₂²×D₅	—	3	4	no	—
184	C₂²×D₅	—	6	4	no	—
185	C₂²×D₅	D₃×C₈	1	1	yes	characteristic
186	C₂×C₄×C₅	—	3	1	no	—
187	C₂×C₄×C₅	—	3	2	no	—
188	C₂×C₄×C₅	—	3	2	no	—
189	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
190	C₂×Dic₅	—	3	1	no	—
191	C₂×Dic₅	—	3	4	no	—
192	C₂×Dic₅	D₃×C₈	1	1	yes	characteristic
193	C₄×D₅	—	3	4	no	—
194	C₄×D₅	—	3	2	no	—
195	C₄×D₅	—	3	2	no	—
196	C₄×D₅	—	3	4	no	—
197	C₄×D₅	D₃×C₈	1	2	yes	—
198	C₄×D₅	C₂×C₄×D₃	1	2	yes	—
199	C₅×C₈	—	3	2	no	—
200	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
201	#168	—	3	2	no	—
202	#168	C₂×C₄×D₃	1	1	yes	characteristic
203	#168	C₂³×D₃	1	1	yes	characteristic
204	C₂³×C₄	—	15	1	no	—
205	C₂²×C₈	—	3	1	no	—
206	C₂²×C₈	—	5	1	no	—
207	C₂²×C₈	—	15	4	no	—
208	C₂²×C₈	—	15	1	no	—
209	C₂²×C₈	—	15	4	no	—
210	C₂²×C₈	—	15	1	no	—
211	C₂²×C₈	—	15	1	no	—
212	C₂²×C₈	—	15	1	no	—
213	M₃₂	—	5	4	no	—
214	M₃₂	—	15	8	no	—
215	M₃₂	—	15	4	no	—
216	C₂×C₁₆	—	5	2	no	—
217	C₂×C₁₆	—	15	8	no	—
218	C₂×C₁₆	—	15	2	no	—
219	C₂×C₃×C₅	C₂×M₃₂	1	2	yes	complemented
220	C₂×C₃×C₅	C₂³×C₈	1	1	yes	characteristic
221	D₁₅	—	2	2	no	—
222	C₃×D₅	—	2	2	no	—
223	C₅×D₃	C₂×M₃₂	1	4	yes	complemented
224	C₂³×C₃	—	5	1	no	—
225	C₂²×D₃	—	5	1	no	—
226	C₂²×D₃	—	5	4	no	—
227	C₂²×D₃	—	10	4	no	—
228	C₂²×D₃	D₅⋊C₈	1	1	yes	characteristic
229	C₂×C₃×C₄	—	5	1	no	—
230	C₂×C₃×C₄	—	5	2	no	—
231	C₂×C₃×C₄	—	5	2	no	—
232	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
233	C₂×Dic₃	—	5	1	no	—
234	C₂×Dic₃	—	5	4	no	—
235	C₂×Dic₃	D₅⋊C₈	1	1	yes	characteristic
236	C₄×D₃	—	5	4	no	—
237	C₄×D₃	—	5	2	no	—
238	C₄×D₃	—	5	2	no	—
239	C₄×D₃	—	5	4	no	—
240	C₄×D₃	D₅⋊C₈	1	2	yes	—
241	C₄×D₃	C₂²×AGL(1,𝔽₅)	1	2	yes	—
242	C₃×C₈	—	5	1	no	—
243	C₃×C₈	—	5	1	no	—
244	C₃×C₈	C₂²×AGL(1,𝔽₅)	1	2	yes	—
245	C₃⋊C₈	—	5	2	no	—
246	C₃⋊C₈	C₂²×AGL(1,𝔽₅)	1	2	yes	—
247	C₂²×C₅	—	3	4	no	—
248	C₂²×C₅	—	3	2	no	—
249	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
250	C₂×D₅	—	2	2	no	—
251	C₂×D₅	—	3	2	no	—
252	C₂×D₅	—	6	8	no	—
253	C₂×D₅	—	6	2	no	—
254	C₂×D₅	C₂×D₃×C₈	1	2	yes	—
255	C₄×C₅	—	3	2	no	—
256	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
257	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
258	Dic₅	—	3	2	no	—
259	Dic₅	C₂×D₃×C₈	1	2	yes	—
260	C₂⁴	—	15	1	no	—
261	C₂²×C₄	—	3	1	no	—
262	C₂²×C₄	—	5	1	no	—
263	C₂²×C₄	—	15	4	no	—
264	C₂²×C₄	—	15	1	no	—
265	C₂²×C₄	—	15	4	no	—
266	C₂²×C₄	—	15	1	no	—
267	C₂²×C₄	—	15	1	no	—
268	C₂²×C₄	—	15	1	no	—
269	C₂×C₈	—	3	1	no	—
270	C₂×C₈	—	3	4	no	—
271	C₂×C₈	—	5	1	no	—
272	C₂×C₈	—	5	2	no	—
273	C₂×C₈	—	5	2	no	—
274	C₂×C₈	—	15	2	no	—
275	C₂×C₈	—	15	2	no	—
276	C₂×C₈	—	15	4	no	—
277	C₂×C₈	—	15	2	no	—
278	C₂×C₈	—	15	2	no	—
279	C₂×C₈	—	15	1	no	—
280	C₂×C₈	—	15	4	no	—
281	C₂×C₈	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
282	C₁₆	—	5	4	no	—
283	C₁₆	—	15	4	no	—
284	C₃×C₅	C₂²×M₃₂	1	1	yes	characteristic, complemented, Hall
285	C₂²×C₃	—	5	2	no	—
286	C₂²×C₃	—	10	2	no	—
287	C₂²×C₃	C₂×D₅⋊C₈	1	1	yes	characteristic
288	C₂×D₃	—	5	2	no	—
289	C₂×D₃	—	10	2	no	—
290	C₂×D₃	—	10	8	no	—
291	C₂×D₃	#3990	1	4	yes	direct product factor
292	C₂×D₃	C₂×D₅⋊C₈	1	2	yes	—
293	C₃×C₄	—	5	2	no	—
294	C₃×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
295	C₃×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
296	Dic₃	—	5	2	no	—
297	Dic₃	C₂×D₅⋊C₈	1	2	yes	—
298	C₂×C₅	—	3	4	no	—
299	C₂×C₅	D₃×M₃₂	1	2	yes	complemented
300	C₂×C₅	C₂²×D₃×C₈	1	1	yes	characteristic
301	D₅	—	2	2	no	—
302	D₅	—	6	2	no	—
303	C₂³	—	3	1	no	—
304	C₂³	—	5	1	no	—
305	C₂³	—	15	4	no	—
306	C₂³	—	15	1	no	—
307	C₂³	—	30	4	no	—
308	C₂×C₄	—	3	1	no	—
309	C₂×C₄	—	3	2	no	—
310	C₂×C₄	—	3	2	no	—
311	C₂×C₄	—	5	1	no	—
312	C₂×C₄	—	5	2	no	—
313	C₂×C₄	—	5	2	no	—
314	C₂×C₄	—	15	4	no	—
315	C₂×C₄	—	15	4	no	—
316	C₂×C₄	—	15	4	no	—
317	C₂×C₄	—	15	1	no	—
318	C₂×C₄	—	15	2	no	—
319	C₂×C₄	—	15	2	no	—
320	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
321	C₈	—	3	2	no	—
322	C₈	—	5	1	no	—
323	C₈	—	5	1	no	—
324	C₈	—	15	2	no	—
325	C₈	C₂×D₃×AGL(1,𝔽₅)	1	2	yes	—
326	C₂×C₃	—	10	2	no	—
327	C₂×C₃	C₂×#3990	1	2	yes	complemented
328	C₂×C₃	C₂²×D₅⋊C₈	1	1	yes	characteristic
329	D₃	—	10	2	no	—
330	D₃	C₂×#3990	1	4	yes	direct product factor
331	C₅	C₂×D₃×M₃₂	1	1	yes	characteristic, complemented, Sylow
332	C₂²	—	3	2	no	—
333	C₂²	—	3	4	no	—
334	C₂²	—	5	2	no	—
335	C₂²	—	10	2	no	—
336	C₂²	—	15	2	no	—
337	C₂²	—	30	8	no	—
338	C₂²	—	30	2	no	—
339	C₂²	D₃×D₅⋊C₈	1	1	yes	characteristic
340	C₄	—	3	2	no	—
341	C₄	—	5	2	no	—
342	C₄	—	15	2	no	—
343	C₄	D₃×D₅⋊C₈	1	1	yes	characteristic
344	C₄	C₂²×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
345	C₃	C₂²×#3990	1	1	yes	characteristic, complemented, Sylow
346	C₂	—	3	4	no	—
347	C₂	—	10	2	no	—
348	C₂	—	30	2	no	—
349	C₂	D₃×#3990	1	2	yes	direct product factor
350	C₂	C₂×D₃×D₅⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)
commutator	C₂×C₃×C₅	C₂³×C₈
Frattini	C₄	C₂²×D₃×AGL(1,𝔽₅)
Fitting	C₂×C₃×C₅×C₈	C₂×C₄
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂²×D₃×D₅	C₈
cosocle	C₃×#168	C₂⁴
hypercenter	C₂×C₈	D₃×AGL(1,𝔽₅)
hypocenter	C₃×C₅	C₂²×M₃₂
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#91219	C₂	semidirect
3. C₂×C₃×#3990	C₂	semidirect
4. C₂×#91166	C₂	semidirect
5. C₂×D₃×#674	C₂	semidirect
6. D₃×#3990	C₂	direct (N × H)
17. C₂×C₃⋊#674	C₂²	semidirect
18. #91219	C₂²	semidirect
19. C₂×C₃×#674	C₂²	semidirect
20. C₃×#3990	C₂²	semidirect
21. #91166	C₂²	semidirect
22. D₃×#674	C₂²	semidirect
29. C₂×#3990	D₃	direct (N × H)
52. C₃⋊#674	C₂³	semidirect
53. C₃×#674	C₂³	semidirect
71. C₂×#674	C₂×D₃	semidirect
74. #3990	C₂×D₃	direct (N × H)
137. #674	C₂²×D₃	semidirect
147. C₂×C₅×D₃	M₃₂	semidirect
219. C₂×C₃×C₅	C₂×M₃₂	semidirect
223. C₅×D₃	C₂×M₃₂	semidirect
284. C₃×C₅	C₂²×M₃₂	semidirect
291. C₂×D₃	#3990	direct (N × H)
299. C₂×C₅	D₃×M₃₂	semidirect
327. C₂×C₃	C₂×#3990	semidirect
330. D₃	C₂×#3990	direct (N × H)
331. C₅	C₂×D₃×M₃₂	semidirect
345. C₃	C₂²×#3990	semidirect
349. C₂	D₃×#3990	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×D₃×C₈×D₅	2
C₂×#91219	2
C₂×C₃×#3990	2
C₂×#91166	2
C₂×D₃×#674	2
D₃×#3990	2
C₂²×#3990	3
C₂×D₃×M₃₂	5

Subgroups — C2×D3×#3990 #14478102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₃×#3990 #14478102