Subgroups — C₃×C₅×#3136 #14276170

Order 1920.

808 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#544	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×#535	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#490	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#489	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#488	C₂	1	2	yes	maximal, complemented
6	C₃×C₅×#486	C₂	1	2	yes	maximal, complemented
7	C₃×C₅×#418	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₅×#417	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×C₅×#416	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃×C₅×#102	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×C₅×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#3136	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
18	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	complemented
19	C₃×C₅×#127	C₂²	1	2	yes	—
20	C₃×C₅×#125	—	2	1	no	—
21	C₃×C₅×#125	C₂²	1	2	yes	complemented
22	C₃×C₅×#124	C₂²	1	2	yes	—
23	C₃×C₅×C₂²⋊Q₈	—	2	2	no	—
24	C₃×C₅×#121	—	2	1	no	—
25	C₃×C₅×#121	—	2	1	no	—
26	C₃×C₅×#120	—	2	2	no	—
27	C₃×C₅×#117	C₂²	1	1	yes	characteristic
28	C₃×C₅×#117	C₂²	1	1	yes	characteristic
29	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
30	C₃×C₅×#103	C₂²	1	1	yes	characteristic, complemented
31	C₃×C₅×#103	C₂²	1	2	yes	complemented
32	C₃×C₅×#103	C₂²	1	1	yes	characteristic, complemented
33	C₃×C₅×#102	C₂²	1	2	yes	—
34	C₃×C₅×#102	C₂²	1	2	yes	—
35	C₃×C₅×#102	C₂²	1	2	yes	—
36	C₃×C₅×#102	C₂²	1	2	yes	—
37	C₃×C₅×#102	C₂²	1	1	yes	characteristic, complemented
38	C₃×C₅×#102	C₂²	1	2	yes	complemented
39	C₃×C₅×#102	C₂²	1	1	yes	characteristic, complemented
40	C₃×C₅×#97	C₂²	1	2	yes	complemented
41	C₃×C₄×C₅×C₈	C₂²	1	2	yes	complemented
42	C₃×#3136	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
43	C₅×#544	C₂×C₃	1	1	yes	characteristic, complemented
44	C₅×#535	C₂×C₃	1	1	yes	characteristic
45	C₅×#490	C₂×C₃	1	2	yes	complemented
46	C₅×#489	C₂×C₃	1	2	yes	complemented
47	C₅×#488	C₂×C₃	1	2	yes	complemented
48	C₅×#486	C₂×C₃	1	2	yes	complemented
49	C₅×#418	C₂×C₃	1	1	yes	characteristic, complemented
50	C₅×#417	C₂×C₃	1	1	yes	characteristic, complemented
51	C₅×#416	C₂×C₃	1	1	yes	characteristic, complemented
52	C₂×C₅×#102	C₂×C₃	1	1	yes	characteristic, complemented
53	C₅×#405	C₂×C₃	1	1	yes	characteristic, complemented
54	C₂⁴×C₃×C₅	—	2	1	no	—
55	C₃×C₅×#41	—	2	2	no	—
56	C₃×C₅×#41	—	2	2	no	—
57	C₂×C₃×C₅×Q₈	—	2	2	no	—
58	C₂×C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
59	C₂×C₃×C₅×D₄	—	2	1	no	—
60	C₂×C₃×C₅×D₄	—	2	1	no	—
61	C₂×C₃×C₅×D₄	—	2	2	no	—
62	C₂×C₃×C₅×D₄	—	2	2	no	—
63	C₂×C₃×C₅×D₄	—	2	2	no	—
64	C₂×C₃×C₅×D₄	—	2	2	no	—
65	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
66	C₂×C₃×C₅×D₄	C₂³	1	1	yes	characteristic
67	C₂²×C₃×C₄×C₅	—	2	1	no	—
68	C₂²×C₃×C₄×C₅	—	2	1	no	—
69	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
70	C₃×C₅×M₁₆	—	2	2	no	—
71	C₂×C₃×C₅×C₈	—	2	2	no	—
72	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
73	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
74	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
75	C₃×C₅×C₄⋊C₄	—	2	1	no	—
76	C₃×C₅×C₄⋊C₄	—	2	2	no	—
77	C₃×C₅×C₄⋊C₄	—	2	2	no	—
78	C₃×C₅×C₄⋊C₄	—	2	2	no	—
79	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
80	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
81	C₃×C₅×C₄⋊C₄	C₂³	1	2	yes	—
82	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
83	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
84	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
85	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
86	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
87	C₃×C₅×C₂²⋊C₄	D₄	1	2	yes	—
88	C₃×C₄²×C₅	—	2	1	no	—
89	C₃×C₄²×C₅	C₂³	1	2	yes	—
90	C₃×#544	C₂×C₅	1	1	yes	characteristic, complemented
91	C₃×#535	C₂×C₅	1	1	yes	characteristic
92	C₃×#490	C₂×C₅	1	2	yes	complemented
93	C₃×#489	C₂×C₅	1	2	yes	complemented
94	C₃×#488	C₂×C₅	1	2	yes	complemented
95	C₃×#486	C₂×C₅	1	2	yes	complemented
96	C₃×#418	C₂×C₅	1	1	yes	characteristic, complemented
97	C₃×#417	C₂×C₅	1	1	yes	characteristic, complemented
98	C₃×#416	C₂×C₅	1	1	yes	characteristic, complemented
99	C₂×C₃×#102	C₂×C₅	1	1	yes	characteristic, complemented
100	C₃×#405	C₂×C₅	1	1	yes	characteristic, complemented
101	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
102	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
103	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
104	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
105	C₅×C₄⋊Q₈	—	2	1	no	—
106	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	complemented
107	C₅×#127	C₂²×C₃	1	2	yes	—
108	C₅×#125	—	2	1	no	—
109	C₅×#125	C₂²×C₃	1	2	yes	complemented
110	C₅×#124	C₂²×C₃	1	2	yes	—
111	C₅×C₂²⋊Q₈	—	2	2	no	—
112	C₅×#121	—	2	1	no	—
113	C₅×#121	—	2	1	no	—
114	C₅×#120	—	2	2	no	—
115	C₅×#117	C₂²×C₃	1	1	yes	characteristic
116	C₅×#117	C₂²×C₃	1	1	yes	characteristic
117	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
118	C₅×#103	C₂²×C₃	1	1	yes	characteristic, complemented
119	C₅×#103	C₂²×C₃	1	1	yes	characteristic, complemented
120	C₅×#103	C₂²×C₃	1	2	yes	complemented
121	C₅×#102	C₂²×C₃	1	2	yes	—
122	C₅×#102	C₂²×C₃	1	2	yes	—
123	C₅×#102	C₂²×C₃	1	2	yes	—
124	C₅×#102	C₂²×C₃	1	2	yes	—
125	C₅×#102	C₂²×C₃	1	1	yes	characteristic, complemented
126	C₅×#102	C₂²×C₃	1	1	yes	characteristic, complemented
127	C₅×#102	C₂²×C₃	1	2	yes	complemented
128	C₅×#97	C₂²×C₃	1	2	yes	complemented
129	C₄×C₅×C₈	C₂²×C₃	1	2	yes	complemented
130	#3136	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
131	C₂³×C₃×C₅	—	2	2	no	—
132	C₂³×C₃×C₅	—	2	1	no	—
133	C₂³×C₃×C₅	—	4	2	no	—
134	C₂³×C₃×C₅	—	8	1	no	—
135	C₂³×C₃×C₅	—	8	1	no	—
136	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
137	C₃×C₅×Q₈	—	2	1	no	—
138	C₃×C₅×Q₈	—	2	1	no	—
139	C₃×C₅×Q₈	—	8	2	no	—
140	C₃×C₅×D₄	—	2	2	no	—
141	C₃×C₅×D₄	—	2	2	no	—
142	C₃×C₅×D₄	—	2	1	no	—
143	C₃×C₅×D₄	—	2	1	no	—
144	C₃×C₅×D₄	—	4	2	no	—
145	C₃×C₅×D₄	—	4	2	no	—
146	C₃×C₅×D₄	—	8	2	no	—
147	C₃×C₅×D₄	—	8	2	no	—
148	C₂×C₃×C₄×C₅	—	2	1	no	—
149	C₂×C₃×C₄×C₅	—	2	1	no	—
150	C₂×C₃×C₄×C₅	—	2	1	no	—
151	C₂×C₃×C₄×C₅	—	2	2	no	—
152	C₂×C₃×C₄×C₅	—	4	2	no	—
153	C₂×C₃×C₄×C₅	—	8	1	no	—
154	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
155	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
156	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
157	C₂×C₃×C₄×C₅	#41	1	2	yes	—
158	C₂×C₃×C₄×C₅	#41	1	2	yes	—
159	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
160	C₃×C₅×C₈	—	2	1	no	—
161	C₃×C₅×C₈	—	2	1	no	—
162	C₃×C₅×C₈	—	2	2	no	—
163	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
164	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
165	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
166	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
167	C₃×C₄⋊Q₈	—	2	1	no	—
168	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	complemented
169	C₃×#127	C₂²×C₅	1	2	yes	—
170	C₃×#125	—	2	1	no	—
171	C₃×#125	C₂²×C₅	1	2	yes	complemented
172	C₃×#124	C₂²×C₅	1	2	yes	—
173	C₃×C₂²⋊Q₈	—	2	2	no	—
174	C₃×#121	—	2	1	no	—
175	C₃×#121	—	2	1	no	—
176	C₃×#120	—	2	2	no	—
177	C₃×#117	C₂²×C₅	1	1	yes	characteristic
178	C₃×#117	C₂²×C₅	1	1	yes	characteristic
179	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
180	C₃×#103	C₂²×C₅	1	1	yes	characteristic, complemented
181	C₃×#103	C₂²×C₅	1	1	yes	characteristic, complemented
182	C₃×#103	C₂²×C₅	1	2	yes	complemented
183	C₃×#102	C₂²×C₅	1	2	yes	—
184	C₃×#102	C₂²×C₅	1	2	yes	—
185	C₃×#102	C₂²×C₅	1	2	yes	—
186	C₃×#102	C₂²×C₅	1	2	yes	—
187	C₃×#102	C₂²×C₅	1	1	yes	characteristic, complemented
188	C₃×#102	C₂²×C₅	1	1	yes	characteristic, complemented
189	C₃×#102	C₂²×C₅	1	2	yes	complemented
190	C₃×#97	C₂²×C₅	1	2	yes	complemented
191	C₃×C₄×C₈	C₂²×C₅	1	2	yes	complemented
192	C₂⁴×C₅	—	2	1	no	—
193	C₅×#41	—	2	2	no	—
194	C₅×#41	—	2	2	no	—
195	C₂×C₅×Q₈	—	2	2	no	—
196	C₂×C₅×Q₈	C₂³×C₃	1	1	yes	characteristic
197	C₂×C₅×D₄	—	2	1	no	—
198	C₂×C₅×D₄	—	2	1	no	—
199	C₂×C₅×D₄	—	2	2	no	—
200	C₂×C₅×D₄	—	2	2	no	—
201	C₂×C₅×D₄	—	2	2	no	—
202	C₂×C₅×D₄	—	2	2	no	—
203	C₂×C₅×D₄	C₂³×C₃	1	2	yes	—
204	C₂×C₅×D₄	C₂³×C₃	1	1	yes	characteristic
205	C₂²×C₄×C₅	—	2	1	no	—
206	C₂²×C₄×C₅	—	2	1	no	—
207	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
208	C₅×M₁₆	—	2	2	no	—
209	C₂×C₅×C₈	—	2	2	no	—
210	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
211	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
212	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
213	C₅×C₄⋊C₄	—	2	1	no	—
214	C₅×C₄⋊C₄	—	2	2	no	—
215	C₅×C₄⋊C₄	—	2	2	no	—
216	C₅×C₄⋊C₄	—	2	2	no	—
217	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
218	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
219	C₅×C₄⋊C₄	C₂³×C₃	1	2	yes	—
220	C₅×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
221	C₅×C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
222	C₅×C₂²⋊C₄	—	2	1	no	—
223	C₅×C₂²⋊C₄	—	2	2	no	—
224	C₅×C₂²⋊C₄	—	2	2	no	—
225	C₅×C₂²⋊C₄	C₃×D₄	1	2	yes	—
226	C₄²×C₅	—	2	1	no	—
227	C₄²×C₅	C₂³×C₃	1	2	yes	—
228	#544	C₂×C₃×C₅	1	1	yes	characteristic, complemented
229	#535	C₂×C₃×C₅	1	1	yes	characteristic
230	#490	C₂×C₃×C₅	1	2	yes	complemented
231	#489	C₂×C₃×C₅	1	2	yes	complemented
232	#488	C₂×C₃×C₅	1	2	yes	complemented
233	#486	C₂×C₃×C₅	1	2	yes	complemented
234	#418	C₂×C₃×C₅	1	1	yes	characteristic, complemented
235	#417	C₂×C₃×C₅	1	1	yes	characteristic, complemented
236	#416	C₂×C₃×C₅	1	1	yes	characteristic, complemented
237	C₂×#102	C₂×C₃×C₅	1	1	yes	characteristic, complemented
238	#405	C₂×C₃×C₅	1	1	yes	characteristic, complemented
239	C₂²×C₃×C₅	—	2	1	no	—
240	C₂²×C₃×C₅	—	2	1	no	—
241	C₂²×C₃×C₅	—	4	1	no	—
242	C₂²×C₃×C₅	—	4	2	no	—
243	C₂²×C₃×C₅	—	4	1	no	—
244	C₂²×C₃×C₅	—	4	2	no	—
245	C₂²×C₃×C₅	—	4	1	no	—
246	C₂²×C₃×C₅	—	4	2	no	—
247	C₂²×C₃×C₅	—	8	4	no	—
248	C₂²×C₃×C₅	#124	1	2	yes	—
249	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
250	C₃×C₄×C₅	—	2	2	no	—
251	C₃×C₄×C₅	—	2	2	no	—
252	C₃×C₄×C₅	—	4	1	no	—
253	C₃×C₄×C₅	—	4	2	no	—
254	C₃×C₄×C₅	—	4	1	no	—
255	C₃×C₄×C₅	—	4	1	no	—
256	C₃×C₄×C₅	#124	1	2	yes	—
257	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
258	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
259	C₂⁴×C₃	—	2	1	no	—
260	C₃×#41	—	2	2	no	—
261	C₃×#41	—	2	2	no	—
262	C₂×C₃×Q₈	—	2	2	no	—
263	C₂×C₃×Q₈	C₂³×C₅	1	1	yes	characteristic
264	C₂×C₃×D₄	—	2	1	no	—
265	C₂×C₃×D₄	—	2	1	no	—
266	C₂×C₃×D₄	—	2	2	no	—
267	C₂×C₃×D₄	—	2	2	no	—
268	C₂×C₃×D₄	—	2	2	no	—
269	C₂×C₃×D₄	—	2	2	no	—
270	C₂×C₃×D₄	C₂³×C₅	1	2	yes	—
271	C₂×C₃×D₄	C₂³×C₅	1	1	yes	characteristic
272	C₂²×C₃×C₄	—	2	1	no	—
273	C₂²×C₃×C₄	—	2	1	no	—
274	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
275	C₃×M₁₆	—	2	2	no	—
276	C₂×C₃×C₈	—	2	2	no	—
277	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
278	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
279	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
280	C₃×C₄⋊C₄	—	2	1	no	—
281	C₃×C₄⋊C₄	—	2	2	no	—
282	C₃×C₄⋊C₄	—	2	2	no	—
283	C₃×C₄⋊C₄	—	2	2	no	—
284	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
285	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
286	C₃×C₄⋊C₄	C₂³×C₅	1	2	yes	—
287	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
288	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
289	C₃×C₂²⋊C₄	—	2	1	no	—
290	C₃×C₂²⋊C₄	—	2	2	no	—
291	C₃×C₂²⋊C₄	—	2	2	no	—
292	C₃×C₂²⋊C₄	C₅×D₄	1	2	yes	—
293	C₃×C₄²	—	2	1	no	—
294	C₃×C₄²	C₂³×C₅	1	2	yes	—
295	C₂³×C₅	—	2	2	no	—
296	C₂³×C₅	—	2	1	no	—
297	C₂³×C₅	—	4	2	no	—
298	C₂³×C₅	—	8	1	no	—
299	C₂³×C₅	—	8	1	no	—
300	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
301	C₅×Q₈	—	2	1	no	—
302	C₅×Q₈	—	2	1	no	—
303	C₅×Q₈	—	8	2	no	—
304	C₅×D₄	—	2	2	no	—
305	C₅×D₄	—	2	2	no	—
306	C₅×D₄	—	2	1	no	—
307	C₅×D₄	—	2	1	no	—
308	C₅×D₄	—	4	2	no	—
309	C₅×D₄	—	4	2	no	—
310	C₅×D₄	—	8	2	no	—
311	C₅×D₄	—	8	2	no	—
312	C₂×C₄×C₅	—	2	1	no	—
313	C₂×C₄×C₅	—	2	1	no	—
314	C₂×C₄×C₅	—	2	1	no	—
315	C₂×C₄×C₅	—	2	2	no	—
316	C₂×C₄×C₅	—	4	2	no	—
317	C₂×C₄×C₅	—	8	1	no	—
318	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
319	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
320	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
321	C₂×C₄×C₅	C₃×#41	1	2	yes	—
322	C₂×C₄×C₅	C₃×#41	1	2	yes	—
323	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
324	C₅×C₈	—	2	1	no	—
325	C₅×C₈	—	2	1	no	—
326	C₅×C₈	—	2	2	no	—
327	C₂×#41	C₂²×C₃×C₅	1	1	yes	characteristic
328	C₂²×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
329	C₂×M₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
330	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
331	C₄⋊Q₈	—	2	1	no	—
332	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	complemented
333	#127	C₂²×C₃×C₅	1	2	yes	—
334	#125	—	2	1	no	—
335	#125	C₂²×C₃×C₅	1	2	yes	complemented
336	#124	C₂²×C₃×C₅	1	2	yes	—
337	C₂²⋊Q₈	—	2	2	no	—
338	#121	—	2	1	no	—
339	#121	—	2	1	no	—
340	#120	—	2	2	no	—
341	#117	C₂²×C₃×C₅	1	1	yes	characteristic
342	#117	C₂²×C₃×C₅	1	1	yes	characteristic
343	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
344	#103	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
345	#103	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
346	#103	C₂²×C₃×C₅	1	2	yes	complemented
347	#102	C₂²×C₃×C₅	1	2	yes	—
348	#102	C₂²×C₃×C₅	1	2	yes	—
349	#102	C₂²×C₃×C₅	1	2	yes	—
350	#102	C₂²×C₃×C₅	1	2	yes	—
351	#102	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
352	#102	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
353	#102	C₂²×C₃×C₅	1	2	yes	complemented
354	#97	C₂²×C₃×C₅	1	2	yes	complemented
355	C₄×C₈	C₂²×C₃×C₅	1	2	yes	complemented
356	C₂×C₃×C₅	—	2	2	no	—
357	C₂×C₃×C₅	—	8	1	no	—
358	C₂×C₃×C₅	—	8	2	no	—
359	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
360	C₂×C₃×C₅	#576	1	1	yes	characteristic
361	C₂×C₃×C₅	#577	1	1	yes	characteristic
362	C₂³×C₃	—	2	2	no	—
363	C₂³×C₃	—	2	1	no	—
364	C₂³×C₃	—	4	2	no	—
365	C₂³×C₃	—	8	1	no	—
366	C₂³×C₃	—	8	1	no	—
367	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
368	C₃×Q₈	—	2	1	no	—
369	C₃×Q₈	—	2	1	no	—
370	C₃×Q₈	—	8	2	no	—
371	C₃×D₄	—	2	2	no	—
372	C₃×D₄	—	2	2	no	—
373	C₃×D₄	—	2	1	no	—
374	C₃×D₄	—	2	1	no	—
375	C₃×D₄	—	4	2	no	—
376	C₃×D₄	—	4	2	no	—
377	C₃×D₄	—	8	2	no	—
378	C₃×D₄	—	8	2	no	—
379	C₂×C₃×C₄	—	2	1	no	—
380	C₂×C₃×C₄	—	2	1	no	—
381	C₂×C₃×C₄	—	2	1	no	—
382	C₂×C₃×C₄	—	2	2	no	—
383	C₂×C₃×C₄	—	4	2	no	—
384	C₂×C₃×C₄	—	8	1	no	—
385	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
386	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
387	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
388	C₂×C₃×C₄	C₅×#41	1	2	yes	—
389	C₂×C₃×C₄	C₅×#41	1	2	yes	—
390	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
391	C₃×C₈	—	2	1	no	—
392	C₃×C₈	—	2	1	no	—
393	C₃×C₈	—	2	2	no	—
394	C₂²×C₅	—	2	1	no	—
395	C₂²×C₅	—	2	1	no	—
396	C₂²×C₅	—	4	1	no	—
397	C₂²×C₅	—	4	2	no	—
398	C₂²×C₅	—	4	1	no	—
399	C₂²×C₅	—	4	2	no	—
400	C₂²×C₅	—	4	1	no	—
401	C₂²×C₅	—	4	2	no	—
402	C₂²×C₅	—	8	4	no	—
403	C₂²×C₅	C₃×#124	1	2	yes	—
404	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
405	C₄×C₅	—	2	2	no	—
406	C₄×C₅	—	2	2	no	—
407	C₄×C₅	—	4	2	no	—
408	C₄×C₅	—	4	1	no	—
409	C₄×C₅	—	4	1	no	—
410	C₄×C₅	—	4	1	no	—
411	C₄×C₅	C₃×#124	1	2	yes	—
412	C₄×C₅	C₂×C₃×#41	1	1	yes	characteristic
413	C₄×C₅	C₂×C₃×#41	1	1	yes	characteristic
414	C₂⁴	—	2	1	no	—
415	#41	—	2	2	no	—
416	#41	—	2	2	no	—
417	C₂×Q₈	—	2	2	no	—
418	C₂×Q₈	C₂³×C₃×C₅	1	1	yes	characteristic
419	C₂×D₄	—	2	1	no	—
420	C₂×D₄	—	2	1	no	—
421	C₂×D₄	—	2	2	no	—
422	C₂×D₄	—	2	2	no	—
423	C₂×D₄	—	2	2	no	—
424	C₂×D₄	—	2	2	no	—
425	C₂×D₄	C₂³×C₃×C₅	1	2	yes	—
426	C₂×D₄	C₂³×C₃×C₅	1	1	yes	characteristic
427	C₂²×C₄	—	2	1	no	—
428	C₂²×C₄	—	2	1	no	—
429	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
430	M₁₆	—	2	2	no	—
431	C₂×C₈	—	2	2	no	—
432	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
433	C₂×C₈	C₂³×C₃×C₅	1	2	yes	—
434	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
435	C₄⋊C₄	—	2	1	no	—
436	C₄⋊C₄	—	2	2	no	—
437	C₄⋊C₄	—	2	2	no	—
438	C₄⋊C₄	—	2	2	no	—
439	C₄⋊C₄	C₃×C₅×D₄	1	1	yes	characteristic
440	C₄⋊C₄	C₃×C₅×D₄	1	1	yes	characteristic
441	C₄⋊C₄	C₂³×C₃×C₅	1	2	yes	—
442	C₄⋊C₄	C₂³×C₃×C₅	1	1	yes	characteristic
443	C₄⋊C₄	C₂³×C₃×C₅	1	1	yes	characteristic
444	C₂²⋊C₄	—	2	1	no	—
445	C₂²⋊C₄	—	2	2	no	—
446	C₂²⋊C₄	—	2	2	no	—
447	C₂²⋊C₄	C₃×C₅×D₄	1	2	yes	—
448	C₄²	—	2	1	no	—
449	C₄²	C₂³×C₃×C₅	1	2	yes	—
450	C₃×C₅	#3136	1	1	yes	characteristic, direct product factor, Hall
451	C₂²×C₃	—	2	1	no	—
452	C₂²×C₃	—	2	1	no	—
453	C₂²×C₃	—	4	1	no	—
454	C₂²×C₃	—	4	2	no	—
455	C₂²×C₃	—	4	1	no	—
456	C₂²×C₃	—	4	2	no	—
457	C₂²×C₃	—	4	1	no	—
458	C₂²×C₃	—	4	2	no	—
459	C₂²×C₃	—	8	4	no	—
460	C₂²×C₃	C₅×#124	1	2	yes	—
461	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
462	C₃×C₄	—	2	2	no	—
463	C₃×C₄	—	2	2	no	—
464	C₃×C₄	—	4	1	no	—
465	C₃×C₄	—	4	2	no	—
466	C₃×C₄	—	4	1	no	—
467	C₃×C₄	—	4	1	no	—
468	C₃×C₄	C₅×#124	1	2	yes	—
469	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
470	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
471	C₂×C₅	—	2	2	no	—
472	C₂×C₅	—	8	2	no	—
473	C₂×C₅	—	8	1	no	—
474	C₂×C₅	C₂×C₃×#124	1	1	yes	characteristic
475	C₂×C₅	C₃×#576	1	1	yes	characteristic
476	C₂×C₅	C₃×#577	1	1	yes	characteristic
477	C₂³	—	2	2	no	—
478	C₂³	—	2	1	no	—
479	C₂³	—	4	2	no	—
480	C₂³	—	8	1	no	—
481	C₂³	—	8	1	no	—
482	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
483	Q₈	—	2	1	no	—
484	Q₈	—	2	1	no	—
485	Q₈	—	8	2	no	—
486	D₄	—	2	2	no	—
487	D₄	—	2	2	no	—
488	D₄	—	2	1	no	—
489	D₄	—	2	1	no	—
490	D₄	—	4	2	no	—
491	D₄	—	4	2	no	—
492	D₄	—	8	2	no	—
493	D₄	—	8	2	no	—
494	C₂×C₄	—	2	1	no	—
495	C₂×C₄	—	2	1	no	—
496	C₂×C₄	—	2	1	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	4	2	no	—
499	C₂×C₄	—	8	1	no	—
500	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
501	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
502	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
503	C₂×C₄	C₃×C₅×#41	1	2	yes	—
504	C₂×C₄	C₃×C₅×#41	1	2	yes	—
505	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
506	C₈	—	2	1	no	—
507	C₈	—	2	1	no	—
508	C₈	—	2	2	no	—
509	C₂×C₃	—	2	2	no	—
510	C₂×C₃	—	8	2	no	—
511	C₂×C₃	—	8	1	no	—
512	C₂×C₃	C₂×C₅×#124	1	1	yes	characteristic
513	C₂×C₃	C₅×#576	1	1	yes	characteristic
514	C₂×C₃	C₅×#577	1	1	yes	characteristic
515	C₅	C₃×#3136	1	1	yes	characteristic, direct product factor, Sylow
516	C₂²	—	2	1	no	—
517	C₂²	—	2	1	no	—
518	C₂²	—	4	1	no	—
519	C₂²	—	4	2	no	—
520	C₂²	—	4	1	no	—
521	C₂²	—	4	2	no	—
522	C₂²	—	4	1	no	—
523	C₂²	—	4	2	no	—
524	C₂²	—	8	4	no	—
525	C₂²	C₃×C₅×#124	1	2	yes	—
526	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
527	C₄	—	2	2	no	—
528	C₄	—	2	2	no	—
529	C₄	—	4	1	no	—
530	C₄	—	4	2	no	—
531	C₄	—	4	1	no	—
532	C₄	—	4	1	no	—
533	C₄	C₃×C₅×#124	1	2	yes	—
534	C₄	C₂×C₃×C₅×#41	1	1	yes	characteristic
535	C₄	C₂×C₃×C₅×#41	1	1	yes	characteristic
536	C₃	C₅×#3136	1	1	yes	characteristic, direct product factor, Sylow
537	C₂	—	2	2	no	—
538	C₂	—	8	2	no	—
539	C₂	—	8	1	no	—
540	C₂	C₂×C₃×C₅×#124	1	1	yes	characteristic
541	C₂	C₃×C₅×#576	1	1	yes	characteristic
542	C₂	C₃×C₅×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
prime radical	C₃×C₅×#535	C₂
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
1. C₃×C₅×#544	C₂	semidirect
3. C₃×C₅×#490	C₂	semidirect
4. C₃×C₅×#489	C₂	semidirect
5. C₃×C₅×#488	C₂	semidirect
6. C₃×C₅×#486	C₂	semidirect
7. C₃×C₅×#418	C₂	semidirect
8. C₃×C₅×#417	C₂	semidirect
9. C₃×C₅×#416	C₂	semidirect
10. C₂×C₃×C₅×#102	C₂	semidirect
11. C₃×C₅×#405	C₂	semidirect
12. C₅×#3136	C₃	direct (N × H)
18. C₃×C₅×C₄⋊Q₈	C₂²	semidirect
21. C₃×C₅×#125	C₂²	semidirect
30. C₃×C₅×#103	C₂²	semidirect
31. C₃×C₅×#103	C₂²	semidirect
32. C₃×C₅×#103	C₂²	semidirect
37. C₃×C₅×#102	C₂²	semidirect
38. C₃×C₅×#102	C₂²	semidirect
39. C₃×C₅×#102	C₂²	semidirect
40. C₃×C₅×#97	C₂²	semidirect
41. C₃×C₄×C₅×C₈	C₂²	semidirect
42. C₃×#3136	C₅	direct (N × H)
43. C₅×#544	C₂×C₃	semidirect
45. C₅×#490	C₂×C₃	semidirect
46. C₅×#489	C₂×C₃	semidirect
47. C₅×#488	C₂×C₃	semidirect
48. C₅×#486	C₂×C₃	semidirect
49. C₅×#418	C₂×C₃	semidirect
50. C₅×#417	C₂×C₃	semidirect
51. C₅×#416	C₂×C₃	semidirect
52. C₂×C₅×#102	C₂×C₃	semidirect
53. C₅×#405	C₂×C₃	semidirect
90. C₃×#544	C₂×C₅	semidirect
92. C₃×#490	C₂×C₅	semidirect
93. C₃×#489	C₂×C₅	semidirect
94. C₃×#488	C₂×C₅	semidirect
95. C₃×#486	C₂×C₅	semidirect
96. C₃×#418	C₂×C₅	semidirect
97. C₃×#417	C₂×C₅	semidirect
98. C₃×#416	C₂×C₅	semidirect
99. C₂×C₃×#102	C₂×C₅	semidirect
100. C₃×#405	C₂×C₅	semidirect
106. C₅×C₄⋊Q₈	C₂²×C₃	semidirect
109. C₅×#125	C₂²×C₃	semidirect
118. C₅×#103	C₂²×C₃	semidirect
119. C₅×#103	C₂²×C₃	semidirect
120. C₅×#103	C₂²×C₃	semidirect
125. C₅×#102	C₂²×C₃	semidirect
126. C₅×#102	C₂²×C₃	semidirect
127. C₅×#102	C₂²×C₃	semidirect
128. C₅×#97	C₂²×C₃	semidirect
129. C₄×C₅×C₈	C₂²×C₃	semidirect
130. #3136	C₃×C₅	direct (N × H)
168. C₃×C₄⋊Q₈	C₂²×C₅	semidirect
171. C₃×#125	C₂²×C₅	semidirect
180. C₃×#103	C₂²×C₅	semidirect
181. C₃×#103	C₂²×C₅	semidirect
182. C₃×#103	C₂²×C₅	semidirect
187. C₃×#102	C₂²×C₅	semidirect
188. C₃×#102	C₂²×C₅	semidirect
189. C₃×#102	C₂²×C₅	semidirect
190. C₃×#97	C₂²×C₅	semidirect
191. C₃×C₄×C₈	C₂²×C₅	semidirect
228. #544	C₂×C₃×C₅	semidirect
230. #490	C₂×C₃×C₅	semidirect
231. #489	C₂×C₃×C₅	semidirect
232. #488	C₂×C₃×C₅	semidirect
233. #486	C₂×C₃×C₅	semidirect
234. #418	C₂×C₃×C₅	semidirect
235. #417	C₂×C₃×C₅	semidirect
236. #416	C₂×C₃×C₅	semidirect
237. C₂×#102	C₂×C₃×C₅	semidirect
238. #405	C₂×C₃×C₅	semidirect
332. C₄⋊Q₈	C₂²×C₃×C₅	semidirect
335. #125	C₂²×C₃×C₅	semidirect
344. #103	C₂²×C₃×C₅	semidirect
345. #103	C₂²×C₃×C₅	semidirect
346. #103	C₂²×C₃×C₅	semidirect
351. #102	C₂²×C₃×C₅	semidirect
352. #102	C₂²×C₃×C₅	semidirect
353. #102	C₂²×C₃×C₅	semidirect
354. #97	C₂²×C₃×C₅	semidirect
355. C₄×C₈	C₂²×C₃×C₅	semidirect
450. C₃×C₅	#3136	direct (N × H)
515. C₅	C₃×#3136	direct (N × H)
536. C₃	C₅×#3136	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×#544	2
C₃×C₅×#535	2
C₃×C₅×#490	2
C₃×C₅×#489	2
C₃×C₅×#488	2
C₃×C₅×#486	2
C₃×C₅×#418	2
C₃×C₅×#417	2
C₃×C₅×#416	2
C₂×C₃×C₅×#102	2
C₃×C₅×#405	2
C₅×#3136	3
C₃×#3136	5

Subgroups — C3×C5×#3136 #14276170

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3136 #14276170