Subgroups — C₃×C₅×#1984 #14275018

Order 1920.

584 conjugacy classes of subgroups.

The subgroup lattice has 284 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×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#336	C₂	1	4	yes	maximal
5	C₅×#1984	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×Q₈	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
8	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
9	C₃×C₄×C₅×Q₈	—	2	2	no	—
10	C₃×C₄×C₅×Q₈	—	2	2	no	—
11	C₂×C₃×C₅×C₄⋊C₄	C₄	1	1	yes	characteristic
12	C₂×C₃×C₅×C₄⋊C₄	C₄	1	2	yes	—
13	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₃×C₅×C₄⋊C₈	—	2	2	no	—
16	C₃×C₄×C₅×C₈	—	2	2	no	—
17	C₃×#1984	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₄×C₅×Q₈	C₂×C₃	1	1	yes	characteristic
19	C₂×C₅×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
20	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic
21	C₅×#336	C₂×C₃	1	4	yes	—
22	C₂×C₃×C₅×Q₈	—	2	2	no	—
23	C₂×C₃×C₅×Q₈	—	2	2	no	—
24	C₂×C₃×C₅×Q₈	C₈	1	4	yes	complemented
25	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂×C₄	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	—	2	2	no	—
33	C₂×C₃×C₅×C₈	—	2	4	no	—
34	C₂×C₃×C₅×C₈	Q₈	1	4	yes	complemented
35	C₃×C₅×C₄⋊C₄	—	2	4	no	—
36	C₃×C₅×C₄⋊C₄	—	2	2	no	—
37	C₃×C₄²×C₅	—	2	2	no	—
38	C₃×C₄²×C₅	—	2	2	no	—
39	C₃×C₄²×C₅	D₄	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	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic
44	C₃×#336	C₂×C₅	1	4	yes	—
45	C₂²×C₅×Q₈	C₃×C₄	1	1	yes	characteristic
46	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
47	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
48	C₄×C₅×Q₈	—	2	2	no	—
49	C₄×C₅×Q₈	—	2	2	no	—
50	C₂×C₅×C₄⋊C₄	C₃×C₄	1	1	yes	characteristic
51	C₂×C₅×C₄⋊C₄	C₃×C₄	1	2	yes	—
52	C₂×C₄²×C₅	C₂²×C₃	1	2	yes	—
53	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
54	C₅×C₄⋊C₈	—	2	2	no	—
55	C₄×C₅×C₈	—	2	2	no	—
56	#1984	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
57	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₃×C₅×Q₈	—	2	8	no	—
59	C₂×C₃×C₄×C₅	—	2	2	no	—
60	C₂×C₃×C₄×C₅	—	2	2	no	—
61	C₂×C₃×C₄×C₅	—	2	2	no	—
62	C₂×C₃×C₄×C₅	—	2	2	no	—
63	C₂×C₃×C₄×C₅	—	2	4	no	—
64	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
65	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
66	C₂×C₃×C₄×C₅	C₂×C₈	1	4	yes	—
67	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
68	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
69	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
72	C₂×C₃×C₄×C₅	#41	1	2	yes	—
73	C₃×C₅×C₈	—	2	4	no	—
74	C₃×C₅×C₈	—	4	2	no	—
75	C₂²×C₃×Q₈	C₄×C₅	1	1	yes	characteristic
76	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
77	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
78	C₃×C₄×Q₈	—	2	2	no	—
79	C₃×C₄×Q₈	—	2	2	no	—
80	C₂×C₃×C₄⋊C₄	C₄×C₅	1	1	yes	characteristic
81	C₂×C₃×C₄⋊C₄	C₄×C₅	1	2	yes	—
82	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	—
83	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
84	C₃×C₄⋊C₈	—	2	2	no	—
85	C₃×C₄×C₈	—	2	2	no	—
86	C₂×C₅×Q₈	—	2	2	no	—
87	C₂×C₅×Q₈	—	2	2	no	—
88	C₂×C₅×Q₈	C₃×C₈	1	4	yes	complemented
89	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
90	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
92	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
94	C₂×C₅×C₈	—	2	2	no	—
95	C₂×C₅×C₈	—	2	2	no	—
96	C₂×C₅×C₈	—	2	2	no	—
97	C₂×C₅×C₈	—	2	4	no	—
98	C₂×C₅×C₈	C₃×Q₈	1	4	yes	complemented
99	C₅×C₄⋊C₄	—	2	4	no	—
100	C₅×C₄⋊C₄	—	2	2	no	—
101	C₄²×C₅	—	2	2	no	—
102	C₄²×C₅	—	2	2	no	—
103	C₄²×C₅	C₃×D₄	1	2	yes	—
104	C₄²×C₅	C₃×D₄	1	2	yes	—
105	C₂×C₄×Q₈	C₂×C₃×C₅	1	1	yes	characteristic
106	C₂×C₄⋊C₈	C₂×C₃×C₅	1	1	yes	characteristic
107	C₂×C₄×C₈	C₂×C₃×C₅	1	1	yes	characteristic
108	#336	C₂×C₃×C₅	1	4	yes	—
109	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂²×C₃×C₅	C₄×Q₈	1	2	yes	—
111	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
112	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
113	C₂²×C₃×C₅	#131	1	2	yes	—
114	C₃×C₄×C₅	—	2	4	no	—
115	C₃×C₄×C₅	—	2	2	no	—
116	C₃×C₄×C₅	—	4	2	no	—
117	C₃×C₄×C₅	C₂²⋊C₈	1	4	yes	—
118	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
119	C₃×C₄×C₅	#124	1	1	yes	characteristic
120	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
121	C₂×C₃×Q₈	—	2	2	no	—
122	C₂×C₃×Q₈	—	2	2	no	—
123	C₂×C₃×Q₈	C₅×C₈	1	4	yes	complemented
124	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
125	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
126	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
127	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
128	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
129	C₂×C₃×C₈	—	2	2	no	—
130	C₂×C₃×C₈	—	2	2	no	—
131	C₂×C₃×C₈	—	2	2	no	—
132	C₂×C₃×C₈	—	2	4	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×C₄²	—	2	2	no	—
137	C₃×C₄²	—	2	2	no	—
138	C₃×C₄²	C₅×D₄	1	2	yes	—
139	C₃×C₄²	C₅×D₄	1	2	yes	—
140	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
141	C₅×Q₈	—	2	8	no	—
142	C₂×C₄×C₅	—	2	2	no	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	—	2	2	no	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	2	4	no	—
147	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
148	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
149	C₂×C₄×C₅	C₂×C₃×C₈	1	4	yes	—
150	C₂×C₄×C₅	C₂×C₃×C₈	1	2	yes	—
151	C₂×C₄×C₅	C₃×M₁₆	1	2	yes	—
152	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
155	C₂×C₄×C₅	C₃×#41	1	2	yes	—
156	C₅×C₈	—	2	4	no	—
157	C₅×C₈	—	4	2	no	—
158	C₂²×Q₈	C₃×C₄×C₅	1	1	yes	characteristic
159	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
160	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
161	C₄×Q₈	—	2	2	no	—
162	C₄×Q₈	—	2	2	no	—
163	C₂×C₄⋊C₄	C₃×C₄×C₅	1	1	yes	characteristic
164	C₂×C₄⋊C₄	C₃×C₄×C₅	1	2	yes	—
165	C₂×C₄²	C₂²×C₃×C₅	1	2	yes	—
166	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
167	C₄⋊C₈	—	2	2	no	—
168	C₄×C₈	—	2	2	no	—
169	C₂×C₃×C₅	#391	1	1	yes	characteristic
170	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
171	C₂×C₃×C₅	#408	1	1	yes	characteristic
172	C₂×C₃×C₅	C₈×Q₈	1	2	yes	—
173	C₂×C₃×C₅	#446	1	2	yes	—
174	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
175	C₃×Q₈	—	2	8	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	2	4	no	—
181	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
182	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
183	C₂×C₃×C₄	C₂×C₅×C₈	1	4	yes	—
184	C₂×C₃×C₄	C₂×C₅×C₈	1	2	yes	—
185	C₂×C₃×C₄	C₅×M₁₆	1	2	yes	—
186	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
189	C₂×C₃×C₄	C₅×#41	1	2	yes	—
190	C₃×C₈	—	2	4	no	—
191	C₃×C₈	—	4	2	no	—
192	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
193	C₂²×C₅	C₃×C₄×Q₈	1	2	yes	—
194	C₂²×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
195	C₂²×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
196	C₂²×C₅	C₃×#131	1	2	yes	—
197	C₄×C₅	—	2	4	no	—
198	C₄×C₅	—	2	2	no	—
199	C₄×C₅	—	4	2	no	—
200	C₄×C₅	C₃×C₂²⋊C₈	1	4	yes	—
201	C₄×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
202	C₄×C₅	C₃×#124	1	1	yes	characteristic
203	C₄×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
204	C₂×Q₈	—	2	2	no	—
205	C₂×Q₈	—	2	2	no	—
206	C₂×Q₈	C₃×C₅×C₈	1	4	yes	complemented
207	C₂²×C₄	C₂×C₃×C₄×C₅	1	2	yes	—
208	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
209	C₂²×C₄	C₂×C₃×C₄×C₅	1	2	yes	—
210	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
211	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	C₃×C₅×Q₈	1	4	yes	complemented
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	C₃×C₅×D₄	1	2	yes	—
222	C₄²	C₃×C₅×D₄	1	2	yes	—
223	C₃×C₅	#1984	1	1	yes	characteristic, direct product factor, Hall
224	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
225	C₂²×C₃	C₄×C₅×Q₈	1	2	yes	—
226	C₂²×C₃	C₂²×C₅×C₈	1	1	yes	characteristic
227	C₂²×C₃	C₂×C₅×M₁₆	1	1	yes	characteristic
228	C₂²×C₃	C₅×#131	1	2	yes	—
229	C₃×C₄	—	2	4	no	—
230	C₃×C₄	—	2	2	no	—
231	C₃×C₄	—	4	2	no	—
232	C₃×C₄	C₅×C₂²⋊C₈	1	4	yes	—
233	C₃×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
234	C₃×C₄	C₅×#124	1	1	yes	characteristic
235	C₃×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
236	C₂×C₅	C₃×#391	1	1	yes	characteristic
237	C₂×C₅	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
238	C₂×C₅	C₃×#408	1	1	yes	characteristic
239	C₂×C₅	C₃×C₈×Q₈	1	2	yes	—
240	C₂×C₅	C₃×#446	1	2	yes	—
241	C₂³	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
242	Q₈	—	2	8	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
249	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
250	C₂×C₄	C₂×C₃×C₅×C₈	1	4	yes	—
251	C₂×C₄	C₂×C₃×C₅×C₈	1	2	yes	—
252	C₂×C₄	C₃×C₅×M₁₆	1	2	yes	—
253	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
254	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
255	C₂×C₄	C₂×C₃×C₅×Q₈	1	2	yes	—
256	C₂×C₄	C₃×C₅×#41	1	2	yes	—
257	C₈	—	2	4	no	—
258	C₈	—	4	2	no	—
259	C₂×C₃	C₅×#391	1	1	yes	characteristic
260	C₂×C₃	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
261	C₂×C₃	C₅×#408	1	1	yes	characteristic
262	C₂×C₃	C₅×C₈×Q₈	1	2	yes	—
263	C₂×C₃	C₅×#446	1	2	yes	—
264	C₅	C₃×#1984	1	1	yes	characteristic, direct product factor, Sylow
265	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
266	C₂²	C₃×C₄×C₅×Q₈	1	2	yes	—
267	C₂²	C₂²×C₃×C₅×C₈	1	1	yes	characteristic
268	C₂²	C₂×C₃×C₅×M₁₆	1	1	yes	characteristic
269	C₂²	C₃×C₅×#131	1	2	yes	—
270	C₄	—	2	4	no	—
271	C₄	—	2	2	no	—
272	C₄	—	4	2	no	—
273	C₄	C₃×C₅×C₂²⋊C₈	1	4	yes	—
274	C₄	C₃×C₅×C₂²⋊Q₈	1	2	yes	—
275	C₄	C₃×C₅×#124	1	1	yes	characteristic
276	C₄	C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
277	C₃	C₅×#1984	1	1	yes	characteristic, direct product factor, Sylow
278	C₂	C₃×C₅×#391	1	1	yes	characteristic
279	C₂	C₂×C₃×C₅×C₂²⋊C₈	1	1	yes	characteristic
280	C₂	C₃×C₅×#408	1	1	yes	characteristic
281	C₂	C₃×C₅×C₈×Q₈	1	2	yes	—
282	C₂	C₃×C₅×#446	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄×C₅	C₂³
commutator	C₂²	C₂²×C₃×C₅×C₈
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₅×#1984	C₃	direct (N × H)
17. C₃×#1984	C₅	direct (N × H)
24. C₂×C₃×C₅×Q₈	C₈	semidirect
34. C₂×C₃×C₅×C₈	Q₈	semidirect
56. #1984	C₃×C₅	direct (N × H)
88. C₂×C₅×Q₈	C₃×C₈	semidirect
98. C₂×C₅×C₈	C₃×Q₈	semidirect
123. C₂×C₃×Q₈	C₅×C₈	semidirect
133. C₂×C₃×C₈	C₅×Q₈	semidirect
206. C₂×Q₈	C₃×C₅×C₈	semidirect
216. C₂×C₈	C₃×C₅×Q₈	semidirect
223. C₃×C₅	#1984	direct (N × H)
264. C₅	C₃×#1984	direct (N × H)
277. C₃	C₅×#1984	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×Q₈	2
C₂×C₃×C₅×C₄⋊C₈	2
C₂×C₃×C₄×C₅×C₈	2
C₃×C₅×#336	2
C₅×#1984	3
C₃×#1984	5

Subgroups — C3×C5×#1984 #14275018

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1984 #14275018