Subgroups — C₃×C₅×#3184 #14276218

Order 1920.

892 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#532	C₂	1	2	yes	maximal, complemented
2	C₃×C₅×#498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#486	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#469	C₂	1	4	yes	maximal, complemented
6	C₃×C₅×#442	C₂	1	4	yes	maximal, complemented
7	C₃×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#3184	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₅×#41	—	2	2	no	—
10	C₂×C₃×C₅×D₈	C₂²	1	4	yes	—
11	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
12	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₃×C₅×#127	C₂²	1	2	yes	—
14	C₃×C₅×#124	—	2	2	no	—
15	C₃×C₅×#121	—	2	2	no	—
16	C₃×C₅×#121	C₂²	1	4	yes	—
17	C₃×C₄×C₅×D₄	—	2	2	no	—
18	C₃×C₄×C₅×D₄	C₂²	1	4	yes	—
19	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
20	C₃×C₅×#106	C₂²	1	4	yes	—
21	C₃×C₅×#102	C₂²	1	8	yes	—
22	C₃×C₅×#97	C₂²	1	2	yes	—
23	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
24	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
25	C₃×#3184	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
26	C₅×#532	C₂×C₃	1	2	yes	complemented
27	C₅×#498	C₂×C₃	1	1	yes	characteristic, complemented
28	C₅×#493	C₂×C₃	1	1	yes	characteristic, complemented
29	C₅×#486	C₂×C₃	1	2	yes	complemented
30	C₅×#469	C₂×C₃	1	4	yes	complemented
31	C₅×#442	C₂×C₃	1	4	yes	complemented
32	C₄×C₅×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
33	C₃×C₅×#41	—	4	8	no	—
34	C₂×C₃×C₅×Q₈	—	2	2	no	—
35	C₂×C₃×C₅×D₄	—	2	4	no	—
36	C₂×C₃×C₅×D₄	—	2	2	no	—
37	C₂×C₃×C₅×D₄	—	2	2	no	—
38	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
39	C₂²×C₃×C₄×C₅	—	2	4	no	—
40	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
41	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
42	C₃×C₅×D₈	—	2	8	no	—
43	C₃×C₅×M₁₆	—	2	4	no	—
44	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
45	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
46	C₃×C₅×C₄⋊C₄	—	2	2	no	—
47	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
48	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
49	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
50	C₃×C₄²×C₅	D₄	1	2	yes	—
51	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
53	C₃×#532	C₂×C₅	1	2	yes	complemented
54	C₃×#498	C₂×C₅	1	1	yes	characteristic, complemented
55	C₃×#493	C₂×C₅	1	1	yes	characteristic, complemented
56	C₃×#486	C₂×C₅	1	2	yes	complemented
57	C₃×#469	C₂×C₅	1	4	yes	complemented
58	C₃×#442	C₂×C₅	1	4	yes	complemented
59	C₃×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic, complemented
60	C₂×C₅×#41	—	2	2	no	—
61	C₂×C₅×D₈	C₂²×C₃	1	4	yes	—
62	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
63	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
64	C₅×#127	C₂²×C₃	1	2	yes	—
65	C₅×#124	—	2	2	no	—
66	C₅×#121	—	2	2	no	—
67	C₅×#121	C₂²×C₃	1	4	yes	—
68	C₄×C₅×D₄	—	2	2	no	—
69	C₄×C₅×D₄	C₂²×C₃	1	4	yes	—
70	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
71	C₅×#106	C₂²×C₃	1	4	yes	—
72	C₅×#102	C₂²×C₃	1	8	yes	—
73	C₅×#97	C₂²×C₃	1	2	yes	—
74	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
75	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
76	#3184	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
77	C₂³×C₃×C₅	—	2	4	no	—
78	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
79	C₃×C₅×Q₈	—	4	4	no	—
80	C₃×C₅×D₄	—	2	8	no	—
81	C₃×C₅×D₄	—	4	8	no	—
82	C₃×C₅×D₄	—	4	4	no	—
83	C₃×C₅×D₄	—	4	4	no	—
84	C₂×C₃×C₄×C₅	—	2	4	no	—
85	C₂×C₃×C₄×C₅	—	2	2	no	—
86	C₂×C₃×C₄×C₅	—	2	4	no	—
87	C₂×C₃×C₄×C₅	—	4	8	no	—
88	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
90	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
91	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
92	C₃×C₅×C₈	—	2	2	no	—
93	C₃×C₅×C₈	#41	1	4	yes	complemented
94	C₂×C₃×#41	—	2	2	no	—
95	C₂×C₃×D₈	C₂²×C₅	1	4	yes	—
96	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
97	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
98	C₃×#127	C₂²×C₅	1	2	yes	—
99	C₃×#124	—	2	2	no	—
100	C₃×#121	—	2	2	no	—
101	C₃×#121	C₂²×C₅	1	4	yes	—
102	C₃×C₄×D₄	—	2	2	no	—
103	C₃×C₄×D₄	C₂²×C₅	1	4	yes	—
104	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
105	C₃×#106	C₂²×C₅	1	4	yes	—
106	C₃×#102	C₂²×C₅	1	8	yes	—
107	C₃×#97	C₂²×C₅	1	2	yes	—
108	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
109	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
110	C₅×#41	—	4	8	no	—
111	C₂×C₅×Q₈	—	2	2	no	—
112	C₂×C₅×D₄	—	2	4	no	—
113	C₂×C₅×D₄	—	2	2	no	—
114	C₂×C₅×D₄	—	2	2	no	—
115	C₂×C₅×D₄	C₂³×C₃	1	4	yes	—
116	C₂²×C₄×C₅	—	2	4	no	—
117	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
118	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
119	C₅×D₈	—	2	8	no	—
120	C₅×M₁₆	—	2	4	no	—
121	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
122	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
123	C₅×C₄⋊C₄	—	2	2	no	—
124	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
125	C₅×C₂²⋊C₄	—	2	4	no	—
126	C₅×C₂²⋊C₄	—	2	4	no	—
127	C₄²×C₅	C₃×D₄	1	2	yes	—
128	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
129	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
130	#532	C₂×C₃×C₅	1	2	yes	complemented
131	#498	C₂×C₃×C₅	1	1	yes	characteristic, complemented
132	#493	C₂×C₃×C₅	1	1	yes	characteristic, complemented
133	#486	C₂×C₃×C₅	1	2	yes	complemented
134	#469	C₂×C₃×C₅	1	4	yes	complemented
135	#442	C₂×C₃×C₅	1	4	yes	complemented
136	C₄×M₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
137	C₂²×C₃×C₅	—	2	1	no	—
138	C₂²×C₃×C₅	—	2	2	no	—
139	C₂²×C₃×C₅	—	4	4	no	—
140	C₂²×C₃×C₅	—	4	4	no	—
141	C₂²×C₃×C₅	—	4	4	no	—
142	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
143	C₃×C₄×C₅	—	2	1	no	—
144	C₃×C₄×C₅	—	2	2	no	—
145	C₃×C₄×C₅	—	4	4	no	—
146	C₃×C₄×C₅	AGL(1,ℤ/8)	1	4	yes	—
147	C₃×C₄×C₅	C₂×#41	1	2	yes	—
148	C₃×#41	—	4	8	no	—
149	C₂×C₃×Q₈	—	2	2	no	—
150	C₂×C₃×D₄	—	2	4	no	—
151	C₂×C₃×D₄	—	2	2	no	—
152	C₂×C₃×D₄	—	2	2	no	—
153	C₂×C₃×D₄	C₂³×C₅	1	4	yes	—
154	C₂²×C₃×C₄	—	2	4	no	—
155	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
156	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
157	C₃×D₈	—	2	8	no	—
158	C₃×M₁₆	—	2	4	no	—
159	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
160	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
161	C₃×C₄⋊C₄	—	2	2	no	—
162	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
163	C₃×C₂²⋊C₄	—	2	4	no	—
164	C₃×C₂²⋊C₄	—	2	4	no	—
165	C₃×C₄²	C₅×D₄	1	2	yes	—
166	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
167	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
168	C₂³×C₅	—	2	4	no	—
169	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
170	C₅×Q₈	—	4	4	no	—
171	C₅×D₄	—	2	8	no	—
172	C₅×D₄	—	4	8	no	—
173	C₅×D₄	—	4	4	no	—
174	C₅×D₄	—	4	4	no	—
175	C₂×C₄×C₅	—	2	4	no	—
176	C₂×C₄×C₅	—	2	2	no	—
177	C₂×C₄×C₅	—	2	4	no	—
178	C₂×C₄×C₅	—	4	8	no	—
179	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
181	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
182	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
183	C₅×C₈	—	2	2	no	—
184	C₅×C₈	C₃×#41	1	4	yes	complemented
185	C₂×#41	—	2	2	no	—
186	C₂×D₈	C₂²×C₃×C₅	1	4	yes	—
187	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
188	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
189	#127	C₂²×C₃×C₅	1	2	yes	—
190	#124	—	2	2	no	—
191	#121	—	2	2	no	—
192	#121	C₂²×C₃×C₅	1	4	yes	—
193	C₄×D₄	—	2	2	no	—
194	C₄×D₄	C₂²×C₃×C₅	1	4	yes	—
195	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
196	#106	C₂²×C₃×C₅	1	4	yes	—
197	#102	C₂²×C₃×C₅	1	8	yes	—
198	#97	C₂²×C₃×C₅	1	2	yes	—
199	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
200	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
201	C₂×C₃×C₅	—	4	1	no	—
202	C₂×C₃×C₅	—	8	4	no	—
203	C₂×C₃×C₅	#532	1	1	yes	characteristic
204	C₂×C₃×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
205	C₂³×C₃	—	2	4	no	—
206	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
207	C₃×Q₈	—	4	4	no	—
208	C₃×D₄	—	2	8	no	—
209	C₃×D₄	—	4	8	no	—
210	C₃×D₄	—	4	4	no	—
211	C₃×D₄	—	4	4	no	—
212	C₂×C₃×C₄	—	2	4	no	—
213	C₂×C₃×C₄	—	2	2	no	—
214	C₂×C₃×C₄	—	2	4	no	—
215	C₂×C₃×C₄	—	4	8	no	—
216	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
218	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
219	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
220	C₃×C₈	—	2	2	no	—
221	C₃×C₈	C₅×#41	1	4	yes	complemented
222	C₂²×C₅	—	2	2	no	—
223	C₂²×C₅	—	2	1	no	—
224	C₂²×C₅	—	4	4	no	—
225	C₂²×C₅	—	4	4	no	—
226	C₂²×C₅	—	4	4	no	—
227	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
228	C₄×C₅	—	2	2	no	—
229	C₄×C₅	—	2	1	no	—
230	C₄×C₅	—	4	4	no	—
231	C₄×C₅	C₃×AGL(1,ℤ/8)	1	4	yes	—
232	C₄×C₅	C₂×C₃×#41	1	2	yes	—
233	#41	—	4	8	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×D₄	—	2	4	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	2	2	no	—
238	C₂×D₄	C₂³×C₃×C₅	1	4	yes	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
241	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
242	D₈	—	2	8	no	—
243	M₁₆	—	2	4	no	—
244	C₂×C₈	C₂³×C₃×C₅	1	2	yes	—
245	C₂×C₈	C₂³×C₃×C₅	1	2	yes	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	C₂³×C₃×C₅	1	4	yes	—
248	C₂²⋊C₄	—	2	4	no	—
249	C₂²⋊C₄	—	2	4	no	—
250	C₄²	C₃×C₅×D₄	1	2	yes	—
251	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
252	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
253	C₃×C₅	#3184	1	1	yes	characteristic, direct product factor, Hall
254	C₂²×C₃	—	2	1	no	—
255	C₂²×C₃	—	2	2	no	—
256	C₂²×C₃	—	4	4	no	—
257	C₂²×C₃	—	4	4	no	—
258	C₂²×C₃	—	4	4	no	—
259	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	—	2	2	no	—
262	C₃×C₄	—	4	4	no	—
263	C₃×C₄	C₅×AGL(1,ℤ/8)	1	4	yes	—
264	C₃×C₄	C₂×C₅×#41	1	2	yes	—
265	C₂×C₅	—	4	1	no	—
266	C₂×C₅	—	8	4	no	—
267	C₂×C₅	C₃×#532	1	1	yes	characteristic
268	C₂×C₅	C₂×C₃×AGL(1,ℤ/8)	1	2	yes	—
269	C₂³	—	2	4	no	—
270	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
271	Q₈	—	4	4	no	—
272	D₄	—	2	8	no	—
273	D₄	—	4	8	no	—
274	D₄	—	4	4	no	—
275	D₄	—	4	4	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	4	8	no	—
280	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
281	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
282	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
283	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
284	C₈	—	2	2	no	—
285	C₈	C₃×C₅×#41	1	4	yes	complemented
286	C₂×C₃	—	4	1	no	—
287	C₂×C₃	—	8	4	no	—
288	C₂×C₃	C₅×#532	1	1	yes	characteristic
289	C₂×C₃	C₂×C₅×AGL(1,ℤ/8)	1	2	yes	—
290	C₅	C₃×#3184	1	1	yes	characteristic, direct product factor, Sylow
291	C₂²	—	2	1	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	4	4	no	—
294	C₂²	—	4	4	no	—
295	C₂²	—	4	4	no	—
296	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	2	2	no	—
299	C₄	—	4	4	no	—
300	C₄	C₃×C₅×AGL(1,ℤ/8)	1	4	yes	—
301	C₄	C₂×C₃×C₅×#41	1	2	yes	—
302	C₃	C₅×#3184	1	1	yes	characteristic, direct product factor, Sylow
303	C₂	—	4	1	no	—
304	C₂	—	8	4	no	—
305	C₂	C₃×C₅×#532	1	1	yes	characteristic
306	C₂	C₂×C₃×C₅×AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#532	C₂	semidirect
2. C₃×C₅×#498	C₂	semidirect
3. C₃×C₅×#493	C₂	semidirect
4. C₃×C₅×#486	C₂	semidirect
5. C₃×C₅×#469	C₂	semidirect
6. C₃×C₅×#442	C₂	semidirect
7. C₃×C₄×C₅×M₁₆	C₂	semidirect
8. C₅×#3184	C₃	direct (N × H)
25. C₃×#3184	C₅	direct (N × H)
26. C₅×#532	C₂×C₃	semidirect
27. C₅×#498	C₂×C₃	semidirect
28. C₅×#493	C₂×C₃	semidirect
29. C₅×#486	C₂×C₃	semidirect
30. C₅×#469	C₂×C₃	semidirect
31. C₅×#442	C₂×C₃	semidirect
32. C₄×C₅×M₁₆	C₂×C₃	semidirect
53. C₃×#532	C₂×C₅	semidirect
54. C₃×#498	C₂×C₅	semidirect
55. C₃×#493	C₂×C₅	semidirect
56. C₃×#486	C₂×C₅	semidirect
57. C₃×#469	C₂×C₅	semidirect
58. C₃×#442	C₂×C₅	semidirect
59. C₃×C₄×M₁₆	C₂×C₅	semidirect
76. #3184	C₃×C₅	direct (N × H)
93. C₃×C₅×C₈	#41	semidirect
130. #532	C₂×C₃×C₅	semidirect
131. #498	C₂×C₃×C₅	semidirect
132. #493	C₂×C₃×C₅	semidirect
133. #486	C₂×C₃×C₅	semidirect
134. #469	C₂×C₃×C₅	semidirect
135. #442	C₂×C₃×C₅	semidirect
136. C₄×M₁₆	C₂×C₃×C₅	semidirect
184. C₅×C₈	C₃×#41	semidirect
221. C₃×C₈	C₅×#41	semidirect
253. C₃×C₅	#3184	direct (N × H)
285. C₈	C₃×C₅×#41	semidirect
290. C₅	C₃×#3184	direct (N × H)
302. C₃	C₅×#3184	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×#532	2
C₃×C₅×#498	2
C₃×C₅×#493	2
C₃×C₅×#486	2
C₃×C₅×#469	2
C₃×C₅×#442	2
C₃×C₄×C₅×M₁₆	2
C₅×#3184	3
C₃×#3184	5

Subgroups — C3×C5×#3184 #14276218

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3184 #14276218