Subgroups — C₃×C₅×#3177 #14276211

Order 1920.

676 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#541	C₂	1	2	yes	maximal
2	C₃×C₅×#499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#494	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#491	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#470	C₂	1	2	yes	maximal
6	C₃×C₅×#467	C₂	1	2	yes	maximal
7	C₃×C₅×#441	C₂	1	2	yes	maximal, complemented
8	C₃×C₄×C₅×Q₁₆	C₂	1	2	yes	maximal, complemented
9	C₃×C₅×#405	C₂	1	1	yes	maximal, characteristic
10	C₅×#3177	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂×C₃×C₅×Q₁₆	C₂²	1	2	yes	—
12	C₂×C₃×C₅×Q₁₆	C₂²	1	2	yes	—
13	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
16	C₃×C₅×#126	—	2	4	no	—
17	C₃×C₅×#125	—	2	4	no	—
18	C₃×C₅×#125	C₂²	1	2	yes	—
19	C₃×C₅×C₂²⋊Q₈	C₂²	1	4	yes	—
20	C₃×C₄×C₅×Q₈	C₂²	1	4	yes	—
21	C₃×C₅×#117	C₂²	1	1	yes	characteristic
22	C₃×C₅×#107	C₂²	1	2	yes	—
23	C₃×C₅×#106	C₂²	1	2	yes	—
24	C₃×C₅×#103	C₂²	1	4	yes	—
25	C₃×C₅×#103	C₂²	1	4	yes	—
26	C₃×C₅×#97	C₂²	1	1	yes	characteristic
27	C₃×C₅×#97	C₂²	1	1	yes	characteristic
28	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
29	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
30	C₃×#3177	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
31	C₅×#541	C₂×C₃	1	2	yes	—
32	C₅×#499	C₂×C₃	1	1	yes	characteristic, complemented
33	C₅×#494	C₂×C₃	1	1	yes	characteristic, complemented
34	C₅×#491	C₂×C₃	1	2	yes	complemented
35	C₅×#470	C₂×C₃	1	2	yes	—
36	C₅×#467	C₂×C₃	1	2	yes	—
37	C₅×#441	C₂×C₃	1	2	yes	complemented
38	C₄×C₅×Q₁₆	C₂×C₃	1	2	yes	complemented
39	C₅×#405	C₂×C₃	1	1	yes	characteristic
40	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	—
41	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
42	C₃×C₅×Q₁₆	—	2	4	no	—
43	C₃×C₅×Q₁₆	—	2	4	no	—
44	C₃×C₅×M₁₆	—	2	2	no	—
45	C₂×C₃×C₅×C₈	—	2	2	no	—
46	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
47	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
48	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
49	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
50	C₃×C₅×C₄⋊C₄	—	2	4	no	—
51	C₃×C₅×C₄⋊C₄	—	2	4	no	—
52	C₃×C₅×C₄⋊C₄	—	2	4	no	—
53	C₃×C₅×C₄⋊C₄	—	2	4	no	—
54	C₃×C₅×C₄⋊C₄	D₄	1	2	yes	—
55	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
56	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
57	C₃×C₅×C₂²⋊C₄	D₄	1	2	yes	—
58	C₃×C₄²×C₅	—	2	4	no	—
59	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
60	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
61	C₃×#541	C₂×C₅	1	2	yes	—
62	C₃×#499	C₂×C₅	1	1	yes	characteristic, complemented
63	C₃×#494	C₂×C₅	1	1	yes	characteristic, complemented
64	C₃×#491	C₂×C₅	1	2	yes	complemented
65	C₃×#470	C₂×C₅	1	2	yes	—
66	C₃×#467	C₂×C₅	1	2	yes	—
67	C₃×#441	C₂×C₅	1	2	yes	complemented
68	C₃×C₄×Q₁₆	C₂×C₅	1	2	yes	complemented
69	C₃×#405	C₂×C₅	1	1	yes	characteristic
70	C₂×C₅×Q₁₆	C₂²×C₃	1	2	yes	—
71	C₂×C₅×Q₁₆	C₂²×C₃	1	2	yes	—
72	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
73	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
74	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
75	C₅×#126	—	2	4	no	—
76	C₅×#125	—	2	4	no	—
77	C₅×#125	C₂²×C₃	1	2	yes	—
78	C₅×C₂²⋊Q₈	C₂²×C₃	1	4	yes	—
79	C₄×C₅×Q₈	C₂²×C₃	1	4	yes	—
80	C₅×#117	C₂²×C₃	1	1	yes	characteristic
81	C₅×#107	C₂²×C₃	1	2	yes	—
82	C₅×#106	C₂²×C₃	1	2	yes	—
83	C₅×#103	C₂²×C₃	1	4	yes	—
84	C₅×#103	C₂²×C₃	1	4	yes	—
85	C₅×#97	C₂²×C₃	1	1	yes	characteristic
86	C₅×#97	C₂²×C₃	1	1	yes	characteristic
87	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
88	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
89	#3177	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
90	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
91	C₃×C₅×Q₈	—	2	4	no	—
92	C₃×C₅×Q₈	—	2	4	no	—
93	C₂×C₃×C₄×C₅	—	2	1	no	—
94	C₂×C₃×C₄×C₅	—	2	1	no	—
95	C₂×C₃×C₄×C₅	—	2	4	no	—
96	C₂×C₃×C₄×C₅	—	2	4	no	—
97	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
98	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
99	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
100	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
101	C₃×C₅×C₈	—	2	1	no	—
102	C₃×C₅×C₈	—	2	1	no	—
103	C₃×C₅×C₈	#41	1	2	yes	—
104	C₃×C₅×C₈	#41	1	2	yes	—
105	C₂×C₃×Q₁₆	C₂²×C₅	1	2	yes	—
106	C₂×C₃×Q₁₆	C₂²×C₅	1	2	yes	—
107	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
108	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
109	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
110	C₃×#126	—	2	4	no	—
111	C₃×#125	—	2	4	no	—
112	C₃×#125	C₂²×C₅	1	2	yes	—
113	C₃×C₂²⋊Q₈	C₂²×C₅	1	4	yes	—
114	C₃×C₄×Q₈	C₂²×C₅	1	4	yes	—
115	C₃×#117	C₂²×C₅	1	1	yes	characteristic
116	C₃×#107	C₂²×C₅	1	2	yes	—
117	C₃×#106	C₂²×C₅	1	2	yes	—
118	C₃×#103	C₂²×C₅	1	4	yes	—
119	C₃×#103	C₂²×C₅	1	4	yes	—
120	C₃×#97	C₂²×C₅	1	1	yes	characteristic
121	C₃×#97	C₂²×C₅	1	1	yes	characteristic
122	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
123	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
124	C₂×C₅×Q₈	C₂³×C₃	1	4	yes	—
125	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
126	C₅×Q₁₆	—	2	4	no	—
127	C₅×Q₁₆	—	2	4	no	—
128	C₅×M₁₆	—	2	2	no	—
129	C₂×C₅×C₈	—	2	2	no	—
130	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
131	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
132	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
133	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
134	C₅×C₄⋊C₄	—	2	4	no	—
135	C₅×C₄⋊C₄	—	2	4	no	—
136	C₅×C₄⋊C₄	—	2	4	no	—
137	C₅×C₄⋊C₄	—	2	4	no	—
138	C₅×C₄⋊C₄	C₃×D₄	1	2	yes	—
139	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
140	C₅×C₂²⋊C₄	—	2	4	no	—
141	C₅×C₂²⋊C₄	C₃×D₄	1	2	yes	—
142	C₄²×C₅	—	2	4	no	—
143	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
144	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
145	#541	C₂×C₃×C₅	1	2	yes	—
146	#499	C₂×C₃×C₅	1	1	yes	characteristic, complemented
147	#494	C₂×C₃×C₅	1	1	yes	characteristic, complemented
148	#491	C₂×C₃×C₅	1	2	yes	complemented
149	#470	C₂×C₃×C₅	1	2	yes	—
150	#467	C₂×C₃×C₅	1	2	yes	—
151	#441	C₂×C₃×C₅	1	2	yes	complemented
152	C₄×Q₁₆	C₂×C₃×C₅	1	2	yes	complemented
153	#405	C₂×C₃×C₅	1	1	yes	characteristic
154	C₂²×C₃×C₅	—	2	1	no	—
155	C₂²×C₃×C₅	—	2	2	no	—
156	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
157	C₃×C₄×C₅	—	2	1	no	—
158	C₃×C₄×C₅	—	2	2	no	—
159	C₃×C₄×C₅	—	2	2	no	—
160	C₃×C₄×C₅	—	4	4	no	—
161	C₃×C₄×C₅	—	4	4	no	—
162	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
163	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
164	C₂×C₃×Q₈	C₂³×C₅	1	4	yes	—
165	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
166	C₃×Q₁₆	—	2	4	no	—
167	C₃×Q₁₆	—	2	4	no	—
168	C₃×M₁₆	—	2	2	no	—
169	C₂×C₃×C₈	—	2	2	no	—
170	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
171	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
172	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
173	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
174	C₃×C₄⋊C₄	—	2	4	no	—
175	C₃×C₄⋊C₄	—	2	4	no	—
176	C₃×C₄⋊C₄	—	2	4	no	—
177	C₃×C₄⋊C₄	—	2	4	no	—
178	C₃×C₄⋊C₄	C₅×D₄	1	2	yes	—
179	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
180	C₃×C₂²⋊C₄	—	2	4	no	—
181	C₃×C₂²⋊C₄	C₅×D₄	1	2	yes	—
182	C₃×C₄²	—	2	4	no	—
183	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
184	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
185	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
186	C₅×Q₈	—	2	4	no	—
187	C₅×Q₈	—	2	4	no	—
188	C₂×C₄×C₅	—	2	1	no	—
189	C₂×C₄×C₅	—	2	1	no	—
190	C₂×C₄×C₅	—	2	4	no	—
191	C₂×C₄×C₅	—	2	4	no	—
192	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
193	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
194	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
195	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
196	C₅×C₈	—	2	1	no	—
197	C₅×C₈	—	2	1	no	—
198	C₅×C₈	C₃×#41	1	2	yes	—
199	C₅×C₈	C₃×#41	1	2	yes	—
200	C₂×Q₁₆	C₂²×C₃×C₅	1	2	yes	—
201	C₂×Q₁₆	C₂²×C₃×C₅	1	2	yes	—
202	C₂×M₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
203	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
204	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
205	#126	—	2	4	no	—
206	#125	—	2	4	no	—
207	#125	C₂²×C₃×C₅	1	2	yes	—
208	C₂²⋊Q₈	C₂²×C₃×C₅	1	4	yes	—
209	C₄×Q₈	C₂²×C₃×C₅	1	4	yes	—
210	#117	C₂²×C₃×C₅	1	1	yes	characteristic
211	#107	C₂²×C₃×C₅	1	2	yes	—
212	#106	C₂²×C₃×C₅	1	2	yes	—
213	#103	C₂²×C₃×C₅	1	4	yes	—
214	#103	C₂²×C₃×C₅	1	4	yes	—
215	#97	C₂²×C₃×C₅	1	1	yes	characteristic
216	#97	C₂²×C₃×C₅	1	1	yes	characteristic
217	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
218	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
219	C₂×C₃×C₅	—	4	1	no	—
220	C₂×C₃×C₅	#532	1	1	yes	characteristic
221	C₂×C₃×C₅	#578	1	2	yes	—
222	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
223	C₃×Q₈	—	2	4	no	—
224	C₃×Q₈	—	2	4	no	—
225	C₂×C₃×C₄	—	2	1	no	—
226	C₂×C₃×C₄	—	2	1	no	—
227	C₂×C₃×C₄	—	2	4	no	—
228	C₂×C₃×C₄	—	2	4	no	—
229	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
230	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
231	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
232	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
233	C₃×C₈	—	2	1	no	—
234	C₃×C₈	—	2	1	no	—
235	C₃×C₈	C₅×#41	1	2	yes	—
236	C₃×C₈	C₅×#41	1	2	yes	—
237	C₂²×C₅	—	2	2	no	—
238	C₂²×C₅	—	2	1	no	—
239	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
240	C₄×C₅	—	2	2	no	—
241	C₄×C₅	—	2	2	no	—
242	C₄×C₅	—	2	1	no	—
243	C₄×C₅	—	4	4	no	—
244	C₄×C₅	—	4	4	no	—
245	C₄×C₅	C₂×C₃×#41	1	1	yes	characteristic
246	C₄×C₅	C₂×C₃×#41	1	1	yes	characteristic
247	C₂×Q₈	C₂³×C₃×C₅	1	4	yes	—
248	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
249	Q₁₆	—	2	4	no	—
250	Q₁₆	—	2	4	no	—
251	M₁₆	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
254	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
255	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
256	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	C₃×C₅×D₄	1	2	yes	—
262	C₄⋊C₄	C₂³×C₃×C₅	1	4	yes	—
263	C₂²⋊C₄	—	2	4	no	—
264	C₂²⋊C₄	C₃×C₅×D₄	1	2	yes	—
265	C₄²	—	2	4	no	—
266	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
267	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
268	C₃×C₅	#3177	1	1	yes	characteristic, direct product factor, Hall
269	C₂²×C₃	—	2	1	no	—
270	C₂²×C₃	—	2	2	no	—
271	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
272	C₃×C₄	—	2	1	no	—
273	C₃×C₄	—	2	2	no	—
274	C₃×C₄	—	2	2	no	—
275	C₃×C₄	—	4	4	no	—
276	C₃×C₄	—	4	4	no	—
277	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
278	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
279	C₂×C₅	—	4	1	no	—
280	C₂×C₅	C₃×#532	1	1	yes	characteristic
281	C₂×C₅	C₃×#578	1	2	yes	—
282	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
283	Q₈	—	2	4	no	—
284	Q₈	—	2	4	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
290	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
291	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
292	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	C₃×C₅×#41	1	2	yes	—
296	C₈	C₃×C₅×#41	1	2	yes	—
297	C₂×C₃	—	4	1	no	—
298	C₂×C₃	C₅×#532	1	1	yes	characteristic
299	C₂×C₃	C₅×#578	1	2	yes	—
300	C₅	C₃×#3177	1	1	yes	characteristic, direct product factor, Sylow
301	C₂²	—	2	1	no	—
302	C₂²	—	2	2	no	—
303	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	4	4	no	—
308	C₄	—	4	4	no	—
309	C₄	C₂×C₃×C₅×#41	1	1	yes	characteristic
310	C₄	C₂×C₃×C₅×#41	1	1	yes	characteristic
311	C₃	C₅×#3177	1	1	yes	characteristic, direct product factor, Sylow
312	C₂	—	4	1	no	—
313	C₂	C₃×C₅×#532	1	1	yes	characteristic
314	C₂	C₃×C₅×#578	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
prime radical	C₂³×C₃×C₅	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₅×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#499	C₂	semidirect
3. C₃×C₅×#494	C₂	semidirect
4. C₃×C₅×#491	C₂	semidirect
7. C₃×C₅×#441	C₂	semidirect
8. C₃×C₄×C₅×Q₁₆	C₂	semidirect
10. C₅×#3177	C₃	direct (N × H)
30. C₃×#3177	C₅	direct (N × H)
32. C₅×#499	C₂×C₃	semidirect
33. C₅×#494	C₂×C₃	semidirect
34. C₅×#491	C₂×C₃	semidirect
37. C₅×#441	C₂×C₃	semidirect
38. C₄×C₅×Q₁₆	C₂×C₃	semidirect
62. C₃×#499	C₂×C₅	semidirect
63. C₃×#494	C₂×C₅	semidirect
64. C₃×#491	C₂×C₅	semidirect
67. C₃×#441	C₂×C₅	semidirect
68. C₃×C₄×Q₁₆	C₂×C₅	semidirect
89. #3177	C₃×C₅	direct (N × H)
146. #499	C₂×C₃×C₅	semidirect
147. #494	C₂×C₃×C₅	semidirect
148. #491	C₂×C₃×C₅	semidirect
151. #441	C₂×C₃×C₅	semidirect
152. C₄×Q₁₆	C₂×C₃×C₅	semidirect
268. C₃×C₅	#3177	direct (N × H)
300. C₅	C₃×#3177	direct (N × H)
311. C₃	C₅×#3177	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×#541	2
C₃×C₅×#499	2
C₃×C₅×#494	2
C₃×C₅×#491	2
C₃×C₅×#470	2
C₃×C₅×#467	2
C₃×C₅×#441	2
C₃×C₄×C₅×Q₁₆	2
C₃×C₅×#405	2
C₅×#3177	3
C₃×#3177	5

Subgroups — C3×C5×#3177 #14276211

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3177 #14276211