Subgroups — C₃×C₅×#1990 #14275024

Order 1920.

600 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#384	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#340	C₂	1	4	yes	maximal
5	C₅×#1990	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×Q₈	—	2	1	no	—
7	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
8	C₃×C₅×C₄⋊Q₈	—	2	2	no	—
9	C₃×C₅×C₄⋊Q₈	C₄	1	4	yes	complemented
10	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
11	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
12	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
13	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
16	C₃×C₄×C₅×C₈	—	2	2	no	—
17	C₃×#1990	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
19	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic
20	C₅×#384	C₂×C₃	1	1	yes	characteristic
21	C₅×#340	C₂×C₃	1	4	yes	—
22	C₂×C₃×C₅×Q₈	—	2	4	no	—
23	C₂×C₃×C₅×Q₈	—	4	2	no	—
24	C₂×C₃×C₅×Q₈	—	4	2	no	—
25	C₂²×C₃×C₄×C₅	—	2	2	no	—
26	C₂²×C₃×C₄×C₅	—	2	2	no	—
27	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₂×C₃×C₅×C₈	—	2	2	no	—
31	C₂×C₃×C₅×C₈	—	2	2	no	—
32	C₂×C₃×C₅×C₈	Q₈	1	4	yes	complemented
33	C₃×C₅×C₄⋊C₄	—	2	4	no	—
34	C₃×C₅×C₄⋊C₄	—	2	2	no	—
35	C₃×C₅×C₄⋊C₄	—	2	4	no	—
36	C₃×C₅×C₄⋊C₄	—	4	2	no	—
37	C₃×C₅×C₄⋊C₄	—	4	2	no	—
38	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
39	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
40	C₃×C₄²×C₅	D₄	1	2	yes	—
41	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
42	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic
43	C₃×#384	C₂×C₅	1	1	yes	characteristic
44	C₃×#340	C₂×C₅	1	4	yes	—
45	C₂²×C₅×Q₈	—	2	1	no	—
46	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
47	C₅×C₄⋊Q₈	—	2	2	no	—
48	C₅×C₄⋊Q₈	C₃×C₄	1	4	yes	complemented
49	C₂×C₅×C₄⋊C₄	—	2	1	no	—
50	C₂×C₅×C₄⋊C₄	—	2	1	no	—
51	C₂×C₅×C₄⋊C₄	—	2	1	no	—
52	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
53	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
54	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
55	C₄×C₅×C₈	—	2	2	no	—
56	#1990	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
57	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
58	C₃×C₅×Q₈	—	4	8	no	—
59	C₂×C₃×C₄×C₅	—	2	4	no	—
60	C₂×C₃×C₄×C₅	—	4	2	no	—
61	C₂×C₃×C₄×C₅	—	4	2	no	—
62	C₂×C₃×C₄×C₅	—	4	2	no	—
63	C₂×C₃×C₄×C₅	—	4	2	no	—
64	C₂×C₃×C₄×C₅	—	4	2	no	—
65	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
66	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
67	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
68	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
69	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
70	C₂×C₃×C₄×C₅	Q₁₆	1	2	yes	—
71	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
74	C₂×C₃×C₄×C₅	#41	1	2	yes	—
75	C₃×C₅×C₈	—	2	4	no	—
76	C₂²×C₃×Q₈	—	2	1	no	—
77	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
78	C₃×C₄⋊Q₈	—	2	2	no	—
79	C₃×C₄⋊Q₈	C₄×C₅	1	4	yes	complemented
80	C₂×C₃×C₄⋊C₄	—	2	1	no	—
81	C₂×C₃×C₄⋊C₄	—	2	1	no	—
82	C₂×C₃×C₄⋊C₄	—	2	1	no	—
83	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
84	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
85	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
86	C₃×C₄×C₈	—	2	2	no	—
87	C₂×C₅×Q₈	—	2	4	no	—
88	C₂×C₅×Q₈	—	4	2	no	—
89	C₂×C₅×Q₈	—	4	2	no	—
90	C₂²×C₄×C₅	—	2	2	no	—
91	C₂²×C₄×C₅	—	2	2	no	—
92	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
95	C₂×C₅×C₈	—	2	2	no	—
96	C₂×C₅×C₈	—	2	2	no	—
97	C₂×C₅×C₈	C₃×Q₈	1	4	yes	complemented
98	C₅×C₄⋊C₄	—	2	4	no	—
99	C₅×C₄⋊C₄	—	2	2	no	—
100	C₅×C₄⋊C₄	—	2	4	no	—
101	C₅×C₄⋊C₄	—	4	2	no	—
102	C₅×C₄⋊C₄	—	4	2	no	—
103	C₅×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
104	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
105	C₄²×C₅	C₃×D₄	1	2	yes	—
106	C₂×C₄⋊Q₈	C₂×C₃×C₅	1	1	yes	characteristic
107	C₂×C₄×C₈	C₂×C₃×C₅	1	1	yes	characteristic
108	#384	C₂×C₃×C₅	1	1	yes	characteristic
109	#340	C₂×C₃×C₅	1	4	yes	—
110	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂²×C₃×C₅	#124	1	1	yes	characteristic
112	C₂²×C₃×C₅	C₄⋊Q₈	1	1	yes	characteristic
113	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
114	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
115	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
116	C₂²×C₃×C₅	C₂×Q₁₆	1	1	yes	characteristic
117	C₃×C₄×C₅	—	4	4	no	—
118	C₃×C₄×C₅	—	8	2	no	—
119	C₃×C₄×C₅	#102	1	4	yes	—
120	C₃×C₄×C₅	#103	1	4	yes	—
121	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
122	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₃×Q₈	—	2	4	no	—
124	C₂×C₃×Q₈	—	4	2	no	—
125	C₂×C₃×Q₈	—	4	2	no	—
126	C₂²×C₃×C₄	—	2	2	no	—
127	C₂²×C₃×C₄	—	2	2	no	—
128	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
129	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
130	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
131	C₂×C₃×C₈	—	2	2	no	—
132	C₂×C₃×C₈	—	2	2	no	—
133	C₂×C₃×C₈	C₅×Q₈	1	4	yes	complemented
134	C₃×C₄⋊C₄	—	2	4	no	—
135	C₃×C₄⋊C₄	—	2	2	no	—
136	C₃×C₄⋊C₄	—	2	4	no	—
137	C₃×C₄⋊C₄	—	4	2	no	—
138	C₃×C₄⋊C₄	—	4	2	no	—
139	C₃×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
140	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
141	C₃×C₄²	C₅×D₄	1	2	yes	—
142	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
143	C₅×Q₈	—	4	8	no	—
144	C₂×C₄×C₅	—	2	4	no	—
145	C₂×C₄×C₅	—	4	2	no	—
146	C₂×C₄×C₅	—	4	2	no	—
147	C₂×C₄×C₅	—	4	2	no	—
148	C₂×C₄×C₅	—	4	2	no	—
149	C₂×C₄×C₅	—	4	2	no	—
150	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
151	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
152	C₂×C₄×C₅	C₃×D₈	1	2	yes	—
153	C₂×C₄×C₅	C₃×SD₁₆	1	2	yes	—
154	C₂×C₄×C₅	C₃×SD₁₆	1	2	yes	—
155	C₂×C₄×C₅	C₃×Q₁₆	1	2	yes	—
156	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
159	C₂×C₄×C₅	C₃×#41	1	2	yes	—
160	C₅×C₈	—	2	4	no	—
161	C₂²×Q₈	—	2	1	no	—
162	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
163	C₄⋊Q₈	—	2	2	no	—
164	C₄⋊Q₈	C₃×C₄×C₅	1	4	yes	complemented
165	C₂×C₄⋊C₄	—	2	1	no	—
166	C₂×C₄⋊C₄	—	2	1	no	—
167	C₂×C₄⋊C₄	—	2	1	no	—
168	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
169	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
170	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
171	C₄×C₈	—	2	2	no	—
172	C₂×C₃×C₅	#391	1	1	yes	characteristic
173	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
174	C₂×C₃×C₅	C₂×#103	1	1	yes	characteristic
175	C₂×C₃×C₅	#486	1	1	yes	characteristic
176	C₂×C₃×C₅	#487	1	1	yes	characteristic
177	C₂×C₃×C₅	#498	1	1	yes	characteristic
178	C₂×C₃×C₅	#500	1	1	yes	characteristic
179	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
180	C₃×Q₈	—	4	8	no	—
181	C₂×C₃×C₄	—	2	4	no	—
182	C₂×C₃×C₄	—	4	2	no	—
183	C₂×C₃×C₄	—	4	2	no	—
184	C₂×C₃×C₄	—	4	2	no	—
185	C₂×C₃×C₄	—	4	2	no	—
186	C₂×C₃×C₄	—	4	2	no	—
187	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
188	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
189	C₂×C₃×C₄	C₅×D₈	1	2	yes	—
190	C₂×C₃×C₄	C₅×SD₁₆	1	2	yes	—
191	C₂×C₃×C₄	C₅×SD₁₆	1	2	yes	—
192	C₂×C₃×C₄	C₅×Q₁₆	1	2	yes	—
193	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
196	C₂×C₃×C₄	C₅×#41	1	2	yes	—
197	C₃×C₈	—	2	4	no	—
198	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
199	C₂²×C₅	C₃×#124	1	1	yes	characteristic
200	C₂²×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
201	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
202	C₂²×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
203	C₂²×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
204	C₂²×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
205	C₄×C₅	—	4	4	no	—
206	C₄×C₅	—	8	2	no	—
207	C₄×C₅	C₃×#102	1	4	yes	—
208	C₄×C₅	C₃×#103	1	4	yes	—
209	C₄×C₅	C₃×C₄×Q₈	1	2	yes	—
210	C₄×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
211	C₂×Q₈	—	2	4	no	—
212	C₂×Q₈	—	4	2	no	—
213	C₂×Q₈	—	4	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
217	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
218	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	C₃×C₅×Q₈	1	4	yes	complemented
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	4	2	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₄⋊C₄	C₂×C₃×C₄×C₅	1	4	yes	—
228	C₄²	C₂×C₃×C₄×C₅	1	2	yes	—
229	C₄²	C₃×C₅×D₄	1	2	yes	—
230	C₃×C₅	#1990	1	1	yes	characteristic, direct product factor, Hall
231	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
232	C₂²×C₃	C₅×#124	1	1	yes	characteristic
233	C₂²×C₃	C₅×C₄⋊Q₈	1	1	yes	characteristic
234	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
235	C₂²×C₃	C₂×C₅×SD₁₆	1	1	yes	characteristic
236	C₂²×C₃	C₂×C₅×SD₁₆	1	1	yes	characteristic
237	C₂²×C₃	C₂×C₅×Q₁₆	1	1	yes	characteristic
238	C₃×C₄	—	4	4	no	—
239	C₃×C₄	—	8	2	no	—
240	C₃×C₄	C₅×#102	1	4	yes	—
241	C₃×C₄	C₅×#103	1	4	yes	—
242	C₃×C₄	C₄×C₅×Q₈	1	2	yes	—
243	C₃×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
244	C₂×C₅	C₃×#391	1	1	yes	characteristic
245	C₂×C₅	C₂×C₃×#102	1	1	yes	characteristic
246	C₂×C₅	C₂×C₃×#103	1	1	yes	characteristic
247	C₂×C₅	C₃×#486	1	1	yes	characteristic
248	C₂×C₅	C₃×#487	1	1	yes	characteristic
249	C₂×C₅	C₃×#498	1	1	yes	characteristic
250	C₂×C₅	C₃×#500	1	1	yes	characteristic
251	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
252	Q₈	—	4	8	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
260	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
261	C₂×C₄	C₃×C₅×D₈	1	2	yes	—
262	C₂×C₄	C₃×C₅×SD₁₆	1	2	yes	—
263	C₂×C₄	C₃×C₅×SD₁₆	1	2	yes	—
264	C₂×C₄	C₃×C₅×Q₁₆	1	2	yes	—
265	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
266	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
267	C₂×C₄	C₂×C₃×C₅×Q₈	1	2	yes	—
268	C₂×C₄	C₃×C₅×#41	1	2	yes	—
269	C₈	—	2	4	no	—
270	C₂×C₃	C₅×#391	1	1	yes	characteristic
271	C₂×C₃	C₂×C₅×#102	1	1	yes	characteristic
272	C₂×C₃	C₂×C₅×#103	1	1	yes	characteristic
273	C₂×C₃	C₅×#486	1	1	yes	characteristic
274	C₂×C₃	C₅×#487	1	1	yes	characteristic
275	C₂×C₃	C₅×#498	1	1	yes	characteristic
276	C₂×C₃	C₅×#500	1	1	yes	characteristic
277	C₅	C₃×#1990	1	1	yes	characteristic, direct product factor, Sylow
278	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
279	C₂²	C₃×C₅×#124	1	1	yes	characteristic
280	C₂²	C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
281	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
282	C₂²	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
283	C₂²	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
284	C₂²	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
285	C₄	—	4	4	no	—
286	C₄	—	8	2	no	—
287	C₄	C₃×C₅×#102	1	4	yes	—
288	C₄	C₃×C₅×#103	1	4	yes	—
289	C₄	C₃×C₄×C₅×Q₈	1	2	yes	—
290	C₄	C₃×C₅×C₂²⋊Q₈	1	2	yes	—
291	C₃	C₅×#1990	1	1	yes	characteristic, direct product factor, Sylow
292	C₂	C₃×C₅×#391	1	1	yes	characteristic
293	C₂	C₂×C₃×C₅×#102	1	1	yes	characteristic
294	C₂	C₂×C₃×C₅×#103	1	1	yes	characteristic
295	C₂	C₃×C₅×#486	1	1	yes	characteristic
296	C₂	C₃×C₅×#487	1	1	yes	characteristic
297	C₂	C₃×C₅×#498	1	1	yes	characteristic
298	C₂	C₃×C₅×#500	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃×C₅	C₂×D₄
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₅×#1990	C₃	direct (N × H)
9. C₃×C₅×C₄⋊Q₈	C₄	semidirect
17. C₃×#1990	C₅	direct (N × H)
32. C₂×C₃×C₅×C₈	Q₈	semidirect
48. C₅×C₄⋊Q₈	C₃×C₄	semidirect
56. #1990	C₃×C₅	direct (N × H)
79. C₃×C₄⋊Q₈	C₄×C₅	semidirect
97. C₂×C₅×C₈	C₃×Q₈	semidirect
133. C₂×C₃×C₈	C₅×Q₈	semidirect
164. C₄⋊Q₈	C₃×C₄×C₅	semidirect
221. C₂×C₈	C₃×C₅×Q₈	semidirect
230. C₃×C₅	#1990	direct (N × H)
277. C₅	C₃×#1990	direct (N × H)
291. C₃	C₅×#1990	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×C₅×C₄⋊Q₈	2
C₂×C₃×C₄×C₅×C₈	2
C₃×C₅×#384	2
C₃×C₅×#340	2
C₅×#1990	3
C₃×#1990	5

Subgroups — C3×C5×#1990 #14275024

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1990 #14275024