Subgroups — C₃×C₅×#3231 #14276265

Order 1920.

652 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#533	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#533	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#500	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#498	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#487	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₅×#487	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₅×#485	C₂	1	4	yes	maximal
8	C₃×C₅×#479	C₂	1	4	yes	maximal, complemented
9	C₃×C₅×#431	C₂	1	1	yes	maximal, characteristic
10	C₅×#3231	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₃×C₅×#125	—	2	1	no	—
16	C₃×C₅×#125	C₂²	1	2	yes	—
17	C₃×C₅×C₂²⋊Q₈	—	2	1	no	—
18	C₃×C₅×C₂²⋊Q₈	—	2	1	no	—
19	C₃×C₅×C₂²⋊Q₈	—	2	1	no	—
20	C₃×C₅×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₃×C₄×C₅×Q₈	—	2	1	no	—
22	C₃×C₄×C₅×Q₈	—	2	1	no	—
23	C₃×C₄×C₅×Q₈	—	2	1	no	—
24	C₃×C₄×C₅×Q₈	C₂²	1	2	yes	—
25	C₃×C₅×#117	—	2	1	no	—
26	C₃×C₅×#117	C₂²	1	2	yes	—
27	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
28	C₃×C₅×#107	C₂²	1	4	yes	—
29	C₃×C₅×#106	C₂²	1	4	yes	—
30	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
31	C₃×C₅×#103	C₂²	1	4	yes	—
32	C₃×C₅×#103	C₂²	1	4	yes	—
33	C₃×C₅×C₂²⋊C₈	C₂²	1	2	yes	—
34	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
35	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
36	C₃×#3231	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
37	C₅×#533	C₂×C₃	1	1	yes	characteristic
38	C₅×#533	C₂×C₃	1	1	yes	characteristic
39	C₅×#500	C₂×C₃	1	1	yes	characteristic, complemented
40	C₅×#498	C₂×C₃	1	1	yes	characteristic, complemented
41	C₅×#487	C₂×C₃	1	1	yes	characteristic, complemented
42	C₅×#487	C₂×C₃	1	1	yes	characteristic, complemented
43	C₅×#485	C₂×C₃	1	4	yes	—
44	C₅×#479	C₂×C₃	1	4	yes	complemented
45	C₅×#431	C₂×C₃	1	1	yes	characteristic
46	C₂×C₃×C₅×Q₈	—	2	1	no	—
47	C₂×C₃×C₅×Q₈	—	2	1	no	—
48	C₂×C₃×C₅×Q₈	—	2	1	no	—
49	C₂×C₃×C₅×Q₈	C₂³	1	2	yes	—
50	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
51	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
52	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
53	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
54	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
55	C₃×C₅×C₄⋊C₄	—	2	1	no	—
56	C₃×C₅×C₄⋊C₄	—	2	1	no	—
57	C₃×C₅×C₄⋊C₄	—	2	2	no	—
58	C₃×C₅×C₄⋊C₄	—	2	2	no	—
59	C₃×C₅×C₄⋊C₄	—	2	1	no	—
60	C₃×C₅×C₄⋊C₄	—	2	1	no	—
61	C₃×C₅×C₄⋊C₄	—	2	1	no	—
62	C₃×C₅×C₄⋊C₄	—	2	2	no	—
63	C₃×C₅×C₄⋊C₄	—	2	2	no	—
64	C₃×C₅×C₄⋊C₄	C₂³	1	2	yes	—
65	C₃×C₅×C₄⋊C₄	C₂³	1	2	yes	—
66	C₃×C₅×C₄⋊C₄	C₂³	1	2	yes	—
67	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
68	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
69	C₃×C₄²×C₅	—	2	2	no	—
70	C₃×C₄²×C₅	—	2	2	no	—
71	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
72	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
73	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
74	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
75	C₃×#533	C₂×C₅	1	1	yes	characteristic
76	C₃×#533	C₂×C₅	1	1	yes	characteristic
77	C₃×#500	C₂×C₅	1	1	yes	characteristic, complemented
78	C₃×#498	C₂×C₅	1	1	yes	characteristic, complemented
79	C₃×#487	C₂×C₅	1	1	yes	characteristic, complemented
80	C₃×#487	C₂×C₅	1	1	yes	characteristic, complemented
81	C₃×#485	C₂×C₅	1	4	yes	—
82	C₃×#479	C₂×C₅	1	4	yes	complemented
83	C₃×#431	C₂×C₅	1	1	yes	characteristic
84	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
85	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
86	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
87	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
88	C₅×#125	—	2	1	no	—
89	C₅×#125	C₂²×C₃	1	2	yes	—
90	C₅×C₂²⋊Q₈	—	2	1	no	—
91	C₅×C₂²⋊Q₈	—	2	1	no	—
92	C₅×C₂²⋊Q₈	—	2	1	no	—
93	C₅×C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
94	C₄×C₅×Q₈	—	2	1	no	—
95	C₄×C₅×Q₈	—	2	1	no	—
96	C₄×C₅×Q₈	—	2	1	no	—
97	C₄×C₅×Q₈	C₂²×C₃	1	2	yes	—
98	C₅×#117	—	2	1	no	—
99	C₅×#117	C₂²×C₃	1	2	yes	—
100	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
101	C₅×#107	C₂²×C₃	1	4	yes	—
102	C₅×#106	C₂²×C₃	1	4	yes	—
103	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
104	C₅×#103	C₂²×C₃	1	4	yes	—
105	C₅×#103	C₂²×C₃	1	4	yes	—
106	C₅×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
107	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
108	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
109	#3231	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
110	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
111	C₃×C₅×Q₈	—	2	4	no	—
112	C₃×C₅×Q₈	—	4	2	no	—
113	C₃×C₅×Q₈	—	4	2	no	—
114	C₃×C₅×Q₈	—	4	2	no	—
115	C₂×C₃×C₄×C₅	—	2	2	no	—
116	C₂×C₃×C₄×C₅	—	2	2	no	—
117	C₂×C₃×C₄×C₅	—	2	2	no	—
118	C₂×C₃×C₄×C₅	—	2	2	no	—
119	C₂×C₃×C₄×C₅	—	2	2	no	—
120	C₂×C₃×C₄×C₅	—	2	2	no	—
121	C₂×C₃×C₄×C₅	—	2	2	no	—
122	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
123	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
124	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
125	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
126	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
127	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
128	C₃×C₅×C₈	—	2	2	no	—
129	C₃×C₅×C₈	—	2	2	no	—
130	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
131	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
132	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
133	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
134	C₃×#125	—	2	1	no	—
135	C₃×#125	C₂²×C₅	1	2	yes	—
136	C₃×C₂²⋊Q₈	—	2	1	no	—
137	C₃×C₂²⋊Q₈	—	2	1	no	—
138	C₃×C₂²⋊Q₈	—	2	1	no	—
139	C₃×C₂²⋊Q₈	C₂²×C₅	1	2	yes	—
140	C₃×C₄×Q₈	—	2	1	no	—
141	C₃×C₄×Q₈	—	2	1	no	—
142	C₃×C₄×Q₈	—	2	1	no	—
143	C₃×C₄×Q₈	C₂²×C₅	1	2	yes	—
144	C₃×#117	—	2	1	no	—
145	C₃×#117	C₂²×C₅	1	2	yes	—
146	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
147	C₃×#107	C₂²×C₅	1	4	yes	—
148	C₃×#106	C₂²×C₅	1	4	yes	—
149	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
150	C₃×#103	C₂²×C₅	1	4	yes	—
151	C₃×#103	C₂²×C₅	1	4	yes	—
152	C₃×C₂²⋊C₈	C₂²×C₅	1	2	yes	—
153	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
154	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
155	C₂×C₅×Q₈	—	2	1	no	—
156	C₂×C₅×Q₈	—	2	1	no	—
157	C₂×C₅×Q₈	—	2	1	no	—
158	C₂×C₅×Q₈	C₂³×C₃	1	2	yes	—
159	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
162	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
163	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
164	C₅×C₄⋊C₄	—	2	1	no	—
165	C₅×C₄⋊C₄	—	2	1	no	—
166	C₅×C₄⋊C₄	—	2	2	no	—
167	C₅×C₄⋊C₄	—	2	2	no	—
168	C₅×C₄⋊C₄	—	2	1	no	—
169	C₅×C₄⋊C₄	—	2	1	no	—
170	C₅×C₄⋊C₄	—	2	1	no	—
171	C₅×C₄⋊C₄	—	2	2	no	—
172	C₅×C₄⋊C₄	—	2	2	no	—
173	C₅×C₄⋊C₄	C₂³×C₃	1	2	yes	—
174	C₅×C₄⋊C₄	C₂³×C₃	1	2	yes	—
175	C₅×C₄⋊C₄	C₂³×C₃	1	2	yes	—
176	C₅×C₂²⋊C₄	—	2	2	no	—
177	C₅×C₂²⋊C₄	—	2	2	no	—
178	C₄²×C₅	—	2	2	no	—
179	C₄²×C₅	—	2	2	no	—
180	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
181	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
182	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
183	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
184	#533	C₂×C₃×C₅	1	1	yes	characteristic
185	#533	C₂×C₃×C₅	1	1	yes	characteristic
186	#500	C₂×C₃×C₅	1	1	yes	characteristic, complemented
187	#498	C₂×C₃×C₅	1	1	yes	characteristic, complemented
188	#487	C₂×C₃×C₅	1	1	yes	characteristic, complemented
189	#487	C₂×C₃×C₅	1	1	yes	characteristic, complemented
190	#485	C₂×C₃×C₅	1	4	yes	—
191	#479	C₂×C₃×C₅	1	4	yes	complemented
192	#431	C₂×C₃×C₅	1	1	yes	characteristic
193	C₂²×C₃×C₅	—	2	1	no	—
194	C₂²×C₃×C₅	—	2	1	no	—
195	C₂²×C₃×C₅	—	2	1	no	—
196	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
197	C₃×C₄×C₅	—	2	1	no	—
198	C₃×C₄×C₅	—	2	1	no	—
199	C₃×C₄×C₅	—	2	1	no	—
200	C₃×C₄×C₅	—	4	2	no	—
201	C₃×C₄×C₅	—	4	2	no	—
202	C₃×C₄×C₅	—	4	2	no	—
203	C₃×C₄×C₅	—	4	2	no	—
204	C₃×C₄×C₅	#135	1	2	yes	—
205	C₃×C₄×C₅	#137	1	2	yes	—
206	C₃×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
207	C₂×C₃×Q₈	—	2	1	no	—
208	C₂×C₃×Q₈	—	2	1	no	—
209	C₂×C₃×Q₈	—	2	1	no	—
210	C₂×C₃×Q₈	C₂³×C₅	1	2	yes	—
211	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
212	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
213	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
214	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
215	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
216	C₃×C₄⋊C₄	—	2	1	no	—
217	C₃×C₄⋊C₄	—	2	1	no	—
218	C₃×C₄⋊C₄	—	2	2	no	—
219	C₃×C₄⋊C₄	—	2	2	no	—
220	C₃×C₄⋊C₄	—	2	1	no	—
221	C₃×C₄⋊C₄	—	2	1	no	—
222	C₃×C₄⋊C₄	—	2	1	no	—
223	C₃×C₄⋊C₄	—	2	2	no	—
224	C₃×C₄⋊C₄	—	2	2	no	—
225	C₃×C₄⋊C₄	C₂³×C₅	1	2	yes	—
226	C₃×C₄⋊C₄	C₂³×C₅	1	2	yes	—
227	C₃×C₄⋊C₄	C₂³×C₅	1	2	yes	—
228	C₃×C₂²⋊C₄	—	2	2	no	—
229	C₃×C₂²⋊C₄	—	2	2	no	—
230	C₃×C₄²	—	2	2	no	—
231	C₃×C₄²	—	2	2	no	—
232	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
233	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
234	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
235	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
236	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
237	C₅×Q₈	—	2	4	no	—
238	C₅×Q₈	—	4	2	no	—
239	C₅×Q₈	—	4	2	no	—
240	C₅×Q₈	—	4	2	no	—
241	C₂×C₄×C₅	—	2	2	no	—
242	C₂×C₄×C₅	—	2	2	no	—
243	C₂×C₄×C₅	—	2	2	no	—
244	C₂×C₄×C₅	—	2	2	no	—
245	C₂×C₄×C₅	—	2	2	no	—
246	C₂×C₄×C₅	—	2	2	no	—
247	C₂×C₄×C₅	—	2	2	no	—
248	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
249	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
250	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
251	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
252	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
253	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
254	C₅×C₈	—	2	2	no	—
255	C₅×C₈	—	2	2	no	—
256	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
257	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
258	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
259	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
260	#125	—	2	1	no	—
261	#125	C₂²×C₃×C₅	1	2	yes	—
262	C₂²⋊Q₈	—	2	1	no	—
263	C₂²⋊Q₈	—	2	1	no	—
264	C₂²⋊Q₈	—	2	1	no	—
265	C₂²⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
266	C₄×Q₈	—	2	1	no	—
267	C₄×Q₈	—	2	1	no	—
268	C₄×Q₈	—	2	1	no	—
269	C₄×Q₈	C₂²×C₃×C₅	1	2	yes	—
270	#117	—	2	1	no	—
271	#117	C₂²×C₃×C₅	1	2	yes	—
272	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
273	#107	C₂²×C₃×C₅	1	4	yes	—
274	#106	C₂²×C₃×C₅	1	4	yes	—
275	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
276	#103	C₂²×C₃×C₅	1	4	yes	—
277	#103	C₂²×C₃×C₅	1	4	yes	—
278	C₂²⋊C₈	C₂²×C₃×C₅	1	2	yes	—
279	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
280	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
281	C₂×C₃×C₅	—	4	1	no	—
282	C₂×C₃×C₅	#536	1	1	yes	characteristic
283	C₂×C₃×C₅	C₂×#135	1	1	yes	characteristic
284	C₂×C₃×C₅	C₂×#137	1	1	yes	characteristic
285	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
286	C₃×Q₈	—	2	4	no	—
287	C₃×Q₈	—	4	2	no	—
288	C₃×Q₈	—	4	2	no	—
289	C₃×Q₈	—	4	2	no	—
290	C₂×C₃×C₄	—	2	2	no	—
291	C₂×C₃×C₄	—	2	2	no	—
292	C₂×C₃×C₄	—	2	2	no	—
293	C₂×C₃×C₄	—	2	2	no	—
294	C₂×C₃×C₄	—	2	2	no	—
295	C₂×C₃×C₄	—	2	2	no	—
296	C₂×C₃×C₄	—	2	2	no	—
297	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
298	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
299	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
300	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
301	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
302	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
303	C₃×C₈	—	2	2	no	—
304	C₃×C₈	—	2	2	no	—
305	C₂²×C₅	—	2	1	no	—
306	C₂²×C₅	—	2	1	no	—
307	C₂²×C₅	—	2	1	no	—
308	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
309	C₄×C₅	—	2	1	no	—
310	C₄×C₅	—	2	1	no	—
311	C₄×C₅	—	2	1	no	—
312	C₄×C₅	—	4	2	no	—
313	C₄×C₅	—	4	2	no	—
314	C₄×C₅	—	4	2	no	—
315	C₄×C₅	—	4	2	no	—
316	C₄×C₅	C₃×#135	1	2	yes	—
317	C₄×C₅	C₃×#137	1	2	yes	—
318	C₄×C₅	C₃×2₋¹⁺⁴	1	2	yes	—
319	C₂×Q₈	—	2	1	no	—
320	C₂×Q₈	—	2	1	no	—
321	C₂×Q₈	—	2	1	no	—
322	C₂×Q₈	C₂³×C₃×C₅	1	2	yes	—
323	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
324	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
325	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
326	C₂×C₈	C₂³×C₃×C₅	1	2	yes	—
327	C₂×C₈	C₂³×C₃×C₅	1	2	yes	—
328	C₄⋊C₄	—	2	1	no	—
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	2	1	no	—
333	C₄⋊C₄	—	2	1	no	—
334	C₄⋊C₄	—	2	1	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	C₂³×C₃×C₅	1	2	yes	—
338	C₄⋊C₄	C₂³×C₃×C₅	1	2	yes	—
339	C₄⋊C₄	C₂³×C₃×C₅	1	2	yes	—
340	C₂²⋊C₄	—	2	2	no	—
341	C₂²⋊C₄	—	2	2	no	—
342	C₄²	—	2	2	no	—
343	C₄²	—	2	2	no	—
344	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
345	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
346	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
347	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
348	C₃×C₅	#3231	1	1	yes	characteristic, direct product factor, Hall
349	C₂²×C₃	—	2	1	no	—
350	C₂²×C₃	—	2	1	no	—
351	C₂²×C₃	—	2	1	no	—
352	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
353	C₃×C₄	—	2	1	no	—
354	C₃×C₄	—	2	1	no	—
355	C₃×C₄	—	2	1	no	—
356	C₃×C₄	—	4	2	no	—
357	C₃×C₄	—	4	2	no	—
358	C₃×C₄	—	4	2	no	—
359	C₃×C₄	—	4	2	no	—
360	C₃×C₄	C₅×#135	1	2	yes	—
361	C₃×C₄	C₅×#137	1	2	yes	—
362	C₃×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
363	C₂×C₅	—	4	1	no	—
364	C₂×C₅	C₃×#536	1	1	yes	characteristic
365	C₂×C₅	C₂×C₃×#135	1	1	yes	characteristic
366	C₂×C₅	C₂×C₃×#137	1	1	yes	characteristic
367	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
368	Q₈	—	2	4	no	—
369	Q₈	—	4	2	no	—
370	Q₈	—	4	2	no	—
371	Q₈	—	4	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
380	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
381	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
382	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
383	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
384	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
385	C₈	—	2	2	no	—
386	C₈	—	2	2	no	—
387	C₂×C₃	—	4	1	no	—
388	C₂×C₃	C₅×#536	1	1	yes	characteristic
389	C₂×C₃	C₂×C₅×#135	1	1	yes	characteristic
390	C₂×C₃	C₂×C₅×#137	1	1	yes	characteristic
391	C₅	C₃×#3231	1	1	yes	characteristic, direct product factor, Sylow
392	C₂²	—	2	1	no	—
393	C₂²	—	2	1	no	—
394	C₂²	—	2	1	no	—
395	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
396	C₄	—	2	1	no	—
397	C₄	—	2	1	no	—
398	C₄	—	2	1	no	—
399	C₄	—	4	2	no	—
400	C₄	—	4	2	no	—
401	C₄	—	4	2	no	—
402	C₄	—	4	2	no	—
403	C₄	C₃×C₅×#135	1	2	yes	—
404	C₄	C₃×C₅×#137	1	2	yes	—
405	C₄	C₃×C₅×2₋¹⁺⁴	1	2	yes	—
406	C₃	C₅×#3231	1	1	yes	characteristic, direct product factor, Sylow
407	C₂	—	4	1	no	—
408	C₂	C₃×C₅×#536	1	1	yes	characteristic
409	C₂	C₂×C₃×C₅×#135	1	1	yes	characteristic
410	C₂	C₂×C₃×C₅×#137	1	1	yes	characteristic

	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 ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#500	C₂	semidirect
4. C₃×C₅×#498	C₂	semidirect
5. C₃×C₅×#487	C₂	semidirect
6. C₃×C₅×#487	C₂	semidirect
8. C₃×C₅×#479	C₂	semidirect
10. C₅×#3231	C₃	direct (N × H)
36. C₃×#3231	C₅	direct (N × H)
39. C₅×#500	C₂×C₃	semidirect
40. C₅×#498	C₂×C₃	semidirect
41. C₅×#487	C₂×C₃	semidirect
42. C₅×#487	C₂×C₃	semidirect
44. C₅×#479	C₂×C₃	semidirect
77. C₃×#500	C₂×C₅	semidirect
78. C₃×#498	C₂×C₅	semidirect
79. C₃×#487	C₂×C₅	semidirect
80. C₃×#487	C₂×C₅	semidirect
82. C₃×#479	C₂×C₅	semidirect
109. #3231	C₃×C₅	direct (N × H)
186. #500	C₂×C₃×C₅	semidirect
187. #498	C₂×C₃×C₅	semidirect
188. #487	C₂×C₃×C₅	semidirect
189. #487	C₂×C₃×C₅	semidirect
191. #479	C₂×C₃×C₅	semidirect
348. C₃×C₅	#3231	direct (N × H)
391. C₅	C₃×#3231	direct (N × H)
406. C₃	C₅×#3231	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×#533	2
C₃×C₅×#500	2
C₃×C₅×#498	2
C₃×C₅×#487	2
C₃×C₅×#485	2
C₃×C₅×#479	2
C₃×C₅×#431	2
C₅×#3231	3
C₃×#3231	5

Subgroups — C3×C5×#3231 #14276265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3231 #14276265