Subgroups — C₅×#83126 #14419146

Order 1920.

760 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#529	C₂	1	1	yes	maximal, characteristic
2	C₅×#5024	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×Q₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5012	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#5011	C₂	1	2	yes	maximal, complemented
6	C₅×D₄×C₃⋊C₈	C₂	1	2	yes	maximal, complemented
7	C₅×D₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#5000	C₂	1	2	yes	maximal, complemented
9	C₅×#4997	C₂	1	2	yes	maximal, complemented
10	C₅×#4926	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#2966	—	3	1	no	maximal, Hall
13	C₃×C₅×#126	C₄	1	2	yes	—
14	C₃×C₅×#125	C₄	1	1	yes	characteristic
15	C₃×C₅×#124	C₄	1	1	yes	characteristic
16	C₃×C₅×#123	C₄	1	2	yes	—
17	C₃×C₅×C₂²⋊Q₈	C₄	1	1	yes	characteristic
18	C₃×C₅×#121	C₄	1	1	yes	characteristic
19	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
20	C₃×C₄×C₅×D₄	C₂²	1	2	yes	—
21	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
22	C₃×C₅×#117	C₂²	1	2	yes	—
23	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
24	C₂×C₅×#208	C₂²	1	2	yes	—
25	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
26	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
27	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
28	C₅×#830	C₂²	1	2	yes	—
29	C₅×#830	C₂²	1	2	yes	—
30	C₅×#830	C₂²	1	2	yes	—
31	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
32	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
33	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
34	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
35	C₅×#803	C₂²	1	2	yes	—
36	C₅×#803	C₂²	1	2	yes	—
37	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
38	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
39	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
40	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
41	C₄×C₅×C₃⋊C₈	C₂²	1	2	yes	—
42	#83126	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
43	C₅×#529	D₃	1	1	yes	characteristic
44	C₅×#446	—	3	1	no	—
45	C₅×C₈×Q₈	—	3	1	no	—
46	C₅×#436	—	3	1	no	—
47	C₅×#435	—	3	2	no	—
48	C₅×C₈×D₄	—	3	2	no	—
49	C₅×C₈×D₄	—	3	1	no	—
50	C₅×#433	—	3	2	no	—
51	C₅×#432	—	3	1	no	—
52	C₅×#405	—	3	2	no	—
53	C₂×C₄×C₅×C₈	—	3	1	no	—
54	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
55	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
56	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
60	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
61	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
62	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
63	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
64	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
65	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
66	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
67	C₃×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
68	C₃×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
69	C₃×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
70	C₃×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
71	C₃×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
72	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
73	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
74	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
75	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
76	C₃×C₄²×C₅	C₂³	1	2	yes	—
77	C₅×#208	—	2	4	no	—
78	C₂×C₅×C₃⋊C₈	—	2	2	no	—
79	C₂×C₅×C₃⋊C₈	—	2	4	no	—
80	C₂×C₅×C₃⋊C₈	—	2	2	no	—
81	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
82	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
83	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
84	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
85	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
86	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
87	C₃×#529	C₂×C₅	1	1	yes	characteristic
88	#5024	C₂×C₅	1	1	yes	characteristic, complemented
89	Q₈×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
90	#5012	C₂×C₅	1	1	yes	characteristic, complemented
91	#5011	C₂×C₅	1	2	yes	complemented
92	D₄×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
93	D₄×C₃⋊C₈	C₂×C₅	1	2	yes	complemented
94	#5000	C₂×C₅	1	2	yes	complemented
95	#4997	C₂×C₅	1	2	yes	complemented
96	#4926	C₂×C₅	1	1	yes	characteristic, complemented
97	C₂×C₄×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
98	C₂×C₅×M₁₆	—	3	2	no	—
99	C₂²×C₅×C₈	—	3	1	no	—
100	C₂²×C₅×C₈	—	3	1	no	—
101	C₂²×C₅×C₈	—	3	2	no	—
102	C₅×#126	Dic₃	1	2	yes	—
103	C₅×#125	Dic₃	1	1	yes	characteristic
104	C₅×#124	Dic₃	1	1	yes	characteristic
105	C₅×#123	Dic₃	1	2	yes	—
106	C₅×C₂²⋊Q₈	Dic₃	1	1	yes	characteristic
107	C₅×#121	Dic₃	1	1	yes	characteristic
108	C₄×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
109	C₄×C₅×D₄	C₂×D₃	1	2	yes	—
110	C₄×C₅×D₄	C₂×D₃	1	1	yes	characteristic
111	C₅×#117	C₂×D₃	1	2	yes	—
112	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
113	C₅×C₄⋊C₈	—	3	1	no	—
114	C₅×C₄⋊C₈	—	3	1	no	—
115	C₅×C₄⋊C₈	—	3	2	no	—
116	C₅×C₄⋊C₈	—	3	2	no	—
117	C₅×C₂²⋊C₈	—	3	2	no	—
118	C₅×C₂²⋊C₈	—	3	2	no	—
119	C₅×C₂²⋊C₈	—	3	2	no	—
120	C₅×#97	—	3	2	no	—
121	C₅×#97	—	3	2	no	—
122	C₄×C₅×C₈	—	3	1	no	—
123	C₄×C₅×C₈	—	3	1	no	—
124	C₄×C₅×C₈	—	3	1	no	—
125	C₄×C₅×C₈	—	3	1	no	—
126	C₄×C₅×C₈	—	3	2	no	—
127	#2966	—	3	1	no	Sylow
128	C₂³×C₃×C₅	C₂²×C₄	1	2	yes	—
129	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
130	C₃×C₅×Q₈	—	2	2	no	—
131	C₃×C₅×D₄	—	2	4	no	—
132	C₃×C₅×D₄	—	2	2	no	—
133	C₂×C₃×C₄×C₅	—	2	2	no	—
134	C₂×C₃×C₄×C₅	—	2	2	no	—
135	C₂×C₃×C₄×C₅	—	2	1	no	—
136	C₂×C₃×C₄×C₅	—	2	1	no	—
137	C₂×C₃×C₄×C₅	—	2	2	no	—
138	C₂×C₃×C₄×C₅	—	2	2	no	—
139	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
142	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
143	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
144	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
145	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
146	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
147	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
148	C₅×C₃⋊C₈	—	2	1	no	—
149	C₅×C₃⋊C₈	—	2	1	no	—
150	C₅×C₃⋊C₈	—	2	2	no	—
151	C₅×C₃⋊C₈	—	2	2	no	—
152	C₅×C₃⋊C₈	#41	1	2	yes	—
153	C₅×C₃⋊C₈	#41	1	2	yes	—
154	C₃×#126	C₄×C₅	1	2	yes	—
155	C₃×#125	C₄×C₅	1	1	yes	characteristic
156	C₃×#124	C₄×C₅	1	1	yes	characteristic
157	C₃×#123	C₄×C₅	1	2	yes	—
158	C₃×C₂²⋊Q₈	C₄×C₅	1	1	yes	characteristic
159	C₃×#121	C₄×C₅	1	1	yes	characteristic
160	C₃×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
161	C₃×C₄×D₄	C₂²×C₅	1	2	yes	—
162	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
163	C₃×#117	C₂²×C₅	1	2	yes	—
164	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
165	C₂×#208	C₂²×C₅	1	2	yes	—
166	C₂²×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
167	C₂²×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
168	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
169	#830	C₂²×C₅	1	2	yes	—
170	#830	C₂²×C₅	1	2	yes	—
171	#830	C₂²×C₅	1	2	yes	—
172	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
173	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
174	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	—
175	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	—
176	#803	C₂²×C₅	1	2	yes	—
177	#803	C₂²×C₅	1	2	yes	—
178	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
179	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
180	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
181	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
182	C₄×C₃⋊C₈	C₂²×C₅	1	2	yes	—
183	C₂×C₅×Q₈	C₂×Dic₃	1	1	yes	characteristic
184	C₂×C₅×D₄	C₂×Dic₃	1	2	yes	—
185	C₂×C₅×D₄	C₂×Dic₃	1	1	yes	characteristic
186	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
187	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
188	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
189	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
190	C₅×M₁₆	—	6	4	no	—
191	C₂×C₅×C₈	—	3	1	no	—
192	C₂×C₅×C₈	—	3	1	no	—
193	C₂×C₅×C₈	—	3	1	no	—
194	C₂×C₅×C₈	—	3	1	no	—
195	C₂×C₅×C₈	—	3	2	no	—
196	C₂×C₅×C₈	—	3	2	no	—
197	C₂×C₅×C₈	—	6	2	no	—
198	C₂×C₅×C₈	—	6	4	no	—
199	C₂×C₅×C₈	—	6	2	no	—
200	C₅×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
201	C₅×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
202	C₅×C₄⋊C₄	C₂×Dic₃	1	1	yes	characteristic
203	C₅×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
204	C₅×C₄⋊C₄	C₂×Dic₃	1	1	yes	characteristic
205	C₅×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
206	C₅×C₂²⋊C₄	C₂×Dic₃	1	2	yes	—
207	C₅×C₂²⋊C₄	C₂×Dic₃	1	2	yes	—
208	C₅×C₂²⋊C₄	C₂×Dic₃	1	2	yes	—
209	C₅×C₂²⋊C₄	C₂×Dic₃	1	2	yes	—
210	C₅×C₂²⋊C₄	C₂×Dic₃	1	2	yes	—
211	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
212	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
213	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
214	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
215	C₄²×C₅	C₂²×D₃	1	2	yes	—
216	#529	C₅×D₃	1	1	yes	characteristic
217	#446	—	3	1	no	—
218	C₈×Q₈	—	3	1	no	—
219	#436	—	3	1	no	—
220	#435	—	3	2	no	—
221	C₈×D₄	—	3	1	no	—
222	C₈×D₄	—	3	2	no	—
223	#433	—	3	2	no	—
224	#432	—	3	1	no	—
225	#405	—	3	2	no	—
226	C₂×C₄×C₈	—	3	1	no	—
227	C₂²×C₃×C₅	—	2	1	no	—
228	C₂²×C₃×C₅	—	2	1	no	—
229	C₂²×C₃×C₅	—	2	2	no	—
230	C₂²×C₃×C₅	—	2	2	no	—
231	C₂²×C₃×C₅	—	2	2	no	—
232	C₂²×C₃×C₅	#131	1	2	yes	—
233	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
234	C₃×C₄×C₅	—	2	1	no	—
235	C₃×C₄×C₅	—	2	2	no	—
236	C₃×C₄×C₅	—	2	1	no	—
237	C₃×C₄×C₅	—	2	1	no	—
238	C₃×C₄×C₅	—	2	1	no	—
239	C₃×C₄×C₅	—	2	2	no	—
240	C₃×C₄×C₅	—	2	2	no	—
241	C₃×C₄×C₅	#131	1	2	yes	—
242	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
243	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
244	C₂×C₃×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
245	C₂×C₃×D₄	C₂×C₄×C₅	1	2	yes	—
246	C₂×C₃×D₄	C₂×C₄×C₅	1	1	yes	characteristic
247	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
248	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
249	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
250	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
251	C₃×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
252	C₃×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
253	C₃×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
254	C₃×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
255	C₃×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
256	C₃×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
257	C₃×C₂²⋊C₄	C₂×C₄×C₅	1	2	yes	—
258	C₃×C₂²⋊C₄	C₂×C₄×C₅	1	2	yes	—
259	C₃×C₂²⋊C₄	C₂×C₄×C₅	1	2	yes	—
260	C₃×C₂²⋊C₄	C₂×C₄×C₅	1	2	yes	—
261	C₃×C₂²⋊C₄	C₂×C₄×C₅	1	2	yes	—
262	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
263	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
264	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
265	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
266	C₃×C₄²	C₂³×C₅	1	2	yes	—
267	#208	—	2	4	no	—
268	C₂×C₃⋊C₈	—	2	2	no	—
269	C₂×C₃⋊C₈	—	2	4	no	—
270	C₂×C₃⋊C₈	—	2	2	no	—
271	C₂×C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
272	C₂×C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
273	C₂×C₃⋊C₈	C₂³×C₅	1	2	yes	—
274	C₂×C₃⋊C₈	C₂³×C₅	1	2	yes	—
275	C₂×C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
276	C₂×C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
277	C₂³×C₅	C₂²×Dic₃	1	2	yes	—
278	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
279	C₅×Q₈	—	2	2	no	—
280	C₅×D₄	—	2	4	no	—
281	C₅×D₄	—	2	2	no	—
282	C₂×C₄×C₅	—	2	2	no	—
283	C₂×C₄×C₅	—	2	2	no	—
284	C₂×C₄×C₅	—	2	1	no	—
285	C₂×C₄×C₅	—	2	1	no	—
286	C₂×C₄×C₅	—	2	2	no	—
287	C₂×C₄×C₅	—	2	2	no	—
288	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
289	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
290	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
291	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
292	C₂×C₄×C₅	C₂²×Dic₃	1	2	yes	—
293	C₂×C₄×C₅	C₂²×Dic₃	1	2	yes	—
294	C₂×C₄×C₅	C₂²×Dic₃	1	2	yes	—
295	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
296	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
297	C₅×C₈	—	3	2	no	—
298	C₅×C₈	—	3	2	no	—
299	C₅×C₈	—	6	1	no	—
300	C₅×C₈	—	6	1	no	—
301	C₅×C₈	—	6	2	no	—
302	C₅×C₈	—	6	2	no	—
303	C₂×M₁₆	—	3	2	no	—
304	C₂²×C₈	—	3	1	no	—
305	C₂²×C₈	—	3	1	no	—
306	C₂²×C₈	—	3	2	no	—
307	#126	C₅×Dic₃	1	2	yes	—
308	#125	C₅×Dic₃	1	1	yes	characteristic
309	#124	C₅×Dic₃	1	1	yes	characteristic
310	#123	C₅×Dic₃	1	2	yes	—
311	C₂²⋊Q₈	C₅×Dic₃	1	1	yes	characteristic
312	#121	C₅×Dic₃	1	1	yes	characteristic
313	C₄×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
314	C₄×D₄	C₂×C₅×D₃	1	2	yes	—
315	C₄×D₄	C₂×C₅×D₃	1	1	yes	characteristic
316	#117	C₂×C₅×D₃	1	2	yes	—
317	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
318	C₄⋊C₈	—	3	1	no	—
319	C₄⋊C₈	—	3	1	no	—
320	C₄⋊C₈	—	3	2	no	—
321	C₄⋊C₈	—	3	2	no	—
322	C₂²⋊C₈	—	3	2	no	—
323	C₂²⋊C₈	—	3	2	no	—
324	C₂²⋊C₈	—	3	2	no	—
325	#97	—	3	2	no	—
326	#97	—	3	2	no	—
327	C₄×C₈	—	3	1	no	—
328	C₄×C₈	—	3	1	no	—
329	C₄×C₈	—	3	1	no	—
330	C₄×C₈	—	3	1	no	—
331	C₄×C₈	—	3	2	no	—
332	C₂×C₃×C₅	—	2	2	no	—
333	C₂×C₃×C₅	—	4	2	no	—
334	C₂×C₃×C₅	C₄×#41	1	1	yes	characteristic
335	C₂×C₃×C₅	C₂×#131	1	1	yes	characteristic
336	C₂×C₃×C₅	C₂×#131	1	1	yes	characteristic
337	C₂³×C₃	C₂²×C₄×C₅	1	2	yes	—
338	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
339	C₃×Q₈	—	2	2	no	—
340	C₃×D₄	—	2	4	no	—
341	C₃×D₄	—	2	2	no	—
342	C₂×C₃×C₄	—	2	2	no	—
343	C₂×C₃×C₄	—	2	2	no	—
344	C₂×C₃×C₄	—	2	1	no	—
345	C₂×C₃×C₄	—	2	1	no	—
346	C₂×C₃×C₄	—	2	2	no	—
347	C₂×C₃×C₄	—	2	2	no	—
348	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
349	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
350	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
351	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
352	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
353	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
354	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
355	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
356	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
357	C₃⋊C₈	—	2	1	no	—
358	C₃⋊C₈	—	2	1	no	—
359	C₃⋊C₈	—	2	2	no	—
360	C₃⋊C₈	—	2	2	no	—
361	C₃⋊C₈	C₅×#41	1	2	yes	—
362	C₃⋊C₈	C₅×#41	1	2	yes	—
363	C₂²×C₅	—	2	1	no	—
364	C₂²×C₅	—	2	1	no	—
365	C₂²×C₅	—	2	2	no	—
366	C₂²×C₅	—	2	2	no	—
367	C₂²×C₅	—	2	2	no	—
368	C₂²×C₅	#948	1	2	yes	—
369	C₂²×C₅	C₂³×Dic₃	1	1	yes	characteristic
370	C₄×C₅	—	2	1	no	—
371	C₄×C₅	—	2	1	no	—
372	C₄×C₅	—	2	1	no	—
373	C₄×C₅	—	2	2	no	—
374	C₄×C₅	—	2	2	no	—
375	C₄×C₅	—	2	1	no	—
376	C₄×C₅	—	2	2	no	—
377	C₄×C₅	#948	1	2	yes	—
378	C₄×C₅	D₃×#41	1	1	yes	characteristic
379	C₄×C₅	D₃×#41	1	1	yes	characteristic
380	C₂×Q₈	C₂×C₅×Dic₃	1	1	yes	characteristic
381	C₂×D₄	C₂×C₅×Dic₃	1	2	yes	—
382	C₂×D₄	C₂×C₅×Dic₃	1	1	yes	characteristic
383	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
384	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
385	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
386	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
387	M₁₆	—	6	4	no	—
388	C₂×C₈	—	3	1	no	—
389	C₂×C₈	—	3	1	no	—
390	C₂×C₈	—	3	1	no	—
391	C₂×C₈	—	3	1	no	—
392	C₂×C₈	—	3	2	no	—
393	C₂×C₈	—	3	2	no	—
394	C₂×C₈	—	6	2	no	—
395	C₂×C₈	—	6	4	no	—
396	C₂×C₈	—	6	2	no	—
397	C₄⋊C₄	C₂×C₅×Dic₃	1	2	yes	—
398	C₄⋊C₄	C₂×C₅×Dic₃	1	2	yes	—
399	C₄⋊C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
400	C₄⋊C₄	C₂×C₅×Dic₃	1	2	yes	—
401	C₄⋊C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
402	C₄⋊C₄	C₂×C₅×Dic₃	1	2	yes	—
403	C₂²⋊C₄	C₂×C₅×Dic₃	1	2	yes	—
404	C₂²⋊C₄	C₂×C₅×Dic₃	1	2	yes	—
405	C₂²⋊C₄	C₂×C₅×Dic₃	1	2	yes	—
406	C₂²⋊C₄	C₂×C₅×Dic₃	1	2	yes	—
407	C₂²⋊C₄	C₂×C₅×Dic₃	1	2	yes	—
408	C₄²	C₂×C₅×Dic₃	1	1	yes	characteristic
409	C₄²	C₂×C₅×Dic₃	1	1	yes	characteristic
410	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
411	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
412	C₄²	C₂²×C₅×D₃	1	2	yes	—
413	C₃×C₅	#2966	1	1	yes	characteristic, complemented, Hall
414	C₂²×C₃	—	2	1	no	—
415	C₂²×C₃	—	2	1	no	—
416	C₂²×C₃	—	2	2	no	—
417	C₂²×C₃	—	2	2	no	—
418	C₂²×C₃	—	2	2	no	—
419	C₂²×C₃	C₅×#131	1	2	yes	—
420	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
421	C₃×C₄	—	2	1	no	—
422	C₃×C₄	—	2	2	no	—
423	C₃×C₄	—	2	1	no	—
424	C₃×C₄	—	2	1	no	—
425	C₃×C₄	—	2	1	no	—
426	C₃×C₄	—	2	2	no	—
427	C₃×C₄	—	2	2	no	—
428	C₃×C₄	C₅×#131	1	2	yes	—
429	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
430	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
431	C₂×C₅	—	2	2	no	—
432	C₂×C₅	—	4	2	no	—
433	C₂×C₅	C₂×#948	1	1	yes	characteristic
434	C₂×C₅	C₂×#948	1	1	yes	characteristic
435	C₂×C₅	Dic₃×#41	1	1	yes	characteristic
436	C₂³	C₂²×C₅×Dic₃	1	2	yes	—
437	C₂³	C₂²×C₅×Dic₃	1	1	yes	characteristic
438	Q₈	—	2	2	no	—
439	D₄	—	2	4	no	—
440	D₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
448	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
449	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
450	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
451	C₂×C₄	C₂²×C₅×Dic₃	1	2	yes	—
452	C₂×C₄	C₂²×C₅×Dic₃	1	2	yes	—
453	C₂×C₄	C₂²×C₅×Dic₃	1	2	yes	—
454	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
455	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
456	C₈	—	3	2	no	—
457	C₈	—	3	2	no	—
458	C₈	—	6	1	no	—
459	C₈	—	6	1	no	—
460	C₈	—	6	2	no	—
461	C₈	—	6	2	no	—
462	C₂×C₃	—	2	2	no	—
463	C₂×C₃	—	4	2	no	—
464	C₂×C₃	C₄×C₅×#41	1	1	yes	characteristic
465	C₂×C₃	C₂×C₅×#131	1	1	yes	characteristic
466	C₂×C₃	C₂×C₅×#131	1	1	yes	characteristic
467	C₅	#83126	1	1	yes	characteristic, direct product factor, Sylow
468	C₂²	—	2	1	no	—
469	C₂²	—	2	1	no	—
470	C₂²	—	2	2	no	—
471	C₂²	—	2	2	no	—
472	C₂²	—	2	2	no	—
473	C₂²	C₅×#948	1	2	yes	—
474	C₂²	C₂³×C₅×Dic₃	1	1	yes	characteristic
475	C₄	—	2	1	no	—
476	C₄	—	2	1	no	—
477	C₄	—	2	1	no	—
478	C₄	—	2	2	no	—
479	C₄	—	2	2	no	—
480	C₄	—	2	1	no	—
481	C₄	—	2	2	no	—
482	C₄	C₅×#948	1	2	yes	—
483	C₄	C₅×D₃×#41	1	1	yes	characteristic
484	C₄	C₅×D₃×#41	1	1	yes	characteristic
485	C₃	C₅×#2966	1	1	yes	characteristic, complemented, Sylow
486	C₂	—	2	2	no	—
487	C₂	—	4	2	no	—
488	C₂	C₂×C₅×#948	1	1	yes	characteristic
489	C₂	C₂×C₅×#948	1	1	yes	characteristic
490	C₂	C₅×Dic₃×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₄×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×#529	C₂
socle	C₂²×C₃×C₅	C₂³×C₄
prime radical	C₃×C₅×#121	C₄
cosocle	C₂×C₃×C₄	C₂⁴×C₅
hypercenter	C₅×#529	D₃
hypocenter	C₃	C₅×#2966
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₅×#529

Complements

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

N	H ≅ G/N	type
2. C₅×#5024	C₂	semidirect
3. C₅×Q₈×C₃⋊C₈	C₂	semidirect
4. C₅×#5012	C₂	semidirect
5. C₅×#5011	C₂	semidirect
6. C₅×D₄×C₃⋊C₈	C₂	semidirect
7. C₅×D₄×C₃⋊C₈	C₂	semidirect
8. C₅×#5000	C₂	semidirect
9. C₅×#4997	C₂	semidirect
10. C₅×#4926	C₂	semidirect
11. C₂×C₄×C₅×C₃⋊C₈	C₂	semidirect
42. #83126	C₅	direct (N × H)
88. #5024	C₂×C₅	semidirect
89. Q₈×C₃⋊C₈	C₂×C₅	semidirect
90. #5012	C₂×C₅	semidirect
91. #5011	C₂×C₅	semidirect
92. D₄×C₃⋊C₈	C₂×C₅	semidirect
93. D₄×C₃⋊C₈	C₂×C₅	semidirect
94. #5000	C₂×C₅	semidirect
95. #4997	C₂×C₅	semidirect
96. #4926	C₂×C₅	semidirect
97. C₂×C₄×C₃⋊C₈	C₂×C₅	semidirect
413. C₃×C₅	#2966	semidirect
467. C₅	#83126	direct (N × H)
485. C₃	C₅×#2966	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#529	2
C₅×#5024	2
C₅×Q₈×C₃⋊C₈	2
C₅×#5012	2
C₅×#5011	2
C₅×D₄×C₃⋊C₈	2
C₅×#5000	2
C₅×#4997	2
C₅×#4926	2
C₂×C₄×C₅×C₃⋊C₈	2
C₅×#2966	3
#83126	5

Subgroups — C5×#83126 #14419146

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#83126 #14419146