Subgroups — C₅×#17715 #11940435

Order 1280.

972 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#533	C₂	1	1	yes	maximal, characteristic
2	C₅×#2920	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#431	C₂	1	1	yes	maximal, characteristic
4	C₅×#1552	C₂	1	4	yes	maximal, complemented
5	C₅×#1549	C₂	1	2	yes	maximal, complemented
6	C₅×#1546	C₂	1	2	yes	maximal, complemented
7	C₅×#1544	C₂	1	4	yes	maximal, complemented
8	C₅×#533	—	2	1	no	—
9	C₅×#533	—	2	1	no	—
10	C₅×#533	—	2	1	no	—
11	C₅×#533	—	2	1	no	—
12	C₅×#533	C₄	1	2	yes	—
13	C₅×#533	C₄	1	2	yes	—
14	C₅×#533	C₂²	1	2	yes	—
15	C₅×#533	C₂²	1	2	yes	—
16	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂×C₅×#125	C₄	1	2	yes	—
19	C₂×C₅×C₂²⋊Q₈	—	2	1	no	—
20	C₂×C₅×C₂²⋊Q₈	—	2	1	no	—
21	C₂×C₄×C₅×Q₈	—	2	1	no	—
22	C₂×C₄×C₅×Q₈	—	2	1	no	—
23	C₂×C₅×#117	C₄	1	2	yes	—
24	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
25	C₅×#432	—	2	1	no	—
26	C₅×#432	C₂²	1	2	yes	—
27	C₅×#431	—	2	1	no	—
28	C₅×#431	—	2	1	no	—
29	C₅×#431	—	2	1	no	—
30	C₅×#431	C₂²	1	2	yes	—
31	C₅×#431	C₂²	1	2	yes	—
32	C₅×#431	C₂²	1	2	yes	—
33	C₅×#423	C₂²	1	2	yes	—
34	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
35	C₅×#407	—	2	1	no	—
36	C₂×C₅×C₂²⋊C₈	—	2	1	no	—
37	C₄×C₅×M₁₆	—	2	1	no	—
38	C₂×C₄×C₅×C₈	—	2	1	no	—
39	C₅×#332	C₂²	1	8	yes	complemented
40	C₅×#332	C₂²	1	8	yes	complemented
41	#17715	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂²×C₅×Q₈	—	2	1	no	—
43	C₂²×C₅×Q₈	—	2	1	no	—
44	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
47	C₂×C₅×M₁₆	—	2	2	no	—
48	C₂²×C₅×C₈	—	2	2	no	—
49	C₅×C₄⋊Q₈	—	2	1	no	—
50	C₅×C₄⋊Q₈	—	2	1	no	—
51	C₅×C₄⋊Q₈	—	2	1	no	—
52	C₅×C₄⋊Q₈	—	2	1	no	—
53	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
54	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
55	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
56	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
57	C₅×#125	—	2	2	no	—
58	C₅×#125	—	2	2	no	—
59	C₅×#125	C₂×C₄	1	4	yes	—
60	C₅×#125	C₂×C₄	1	4	yes	—
61	C₅×C₂²⋊Q₈	—	2	2	no	—
62	C₅×C₂²⋊Q₈	—	2	2	no	—
63	C₅×C₂²⋊Q₈	—	2	2	no	—
64	C₅×C₂²⋊Q₈	—	2	2	no	—
65	C₅×C₂²⋊Q₈	—	2	2	no	—
66	C₅×C₂²⋊Q₈	—	2	2	no	—
67	C₅×C₂²⋊Q₈	—	2	2	no	—
68	C₅×C₂²⋊Q₈	—	2	2	no	—
69	C₄×C₅×Q₈	—	2	2	no	—
70	C₄×C₅×Q₈	—	2	2	no	—
71	C₄×C₅×Q₈	—	2	2	no	—
72	C₄×C₅×Q₈	—	2	2	no	—
73	C₄×C₅×Q₈	—	2	2	no	—
74	C₄×C₅×Q₈	—	2	2	no	—
75	C₄×C₅×Q₈	—	2	2	no	—
76	C₄×C₅×Q₈	—	2	2	no	—
77	C₅×#117	—	2	2	no	—
78	C₅×#117	—	2	2	no	—
79	C₅×#117	C₂×C₄	1	4	yes	—
80	C₅×#117	C₂×C₄	1	4	yes	—
81	C₂×C₅×C₄⋊C₄	—	2	2	no	—
82	C₂×C₅×C₄⋊C₄	—	2	2	no	—
83	C₂×C₅×C₄⋊C₄	—	2	1	no	—
84	C₂×C₅×C₄⋊C₄	—	2	1	no	—
85	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
86	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
87	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
88	C₂×C₄²×C₅	—	2	2	no	—
89	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
90	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
91	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
92	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
93	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
94	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
95	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
96	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
97	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
99	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
101	C₅×C₄⋊C₈	—	2	2	no	—
102	C₅×C₄⋊C₈	—	2	2	no	—
103	C₅×C₄⋊C₈	C₂³	1	4	yes	—
104	C₅×C₄⋊C₈	C₂³	1	4	yes	—
105	C₅×C₂²⋊C₈	—	2	2	no	—
106	C₅×C₂²⋊C₈	—	2	2	no	—
107	C₅×C₂²⋊C₈	—	2	2	no	—
108	C₅×C₂²⋊C₈	—	2	2	no	—
109	C₅×#97	—	2	2	no	—
110	C₄×C₅×C₈	—	2	2	no	—
111	C₄×C₅×C₈	—	2	2	no	—
112	C₄×C₅×C₈	—	2	2	no	—
113	C₂×#533	C₂×C₅	1	1	yes	characteristic
114	#2920	C₂×C₅	1	1	yes	characteristic
115	C₂×#431	C₂×C₅	1	1	yes	characteristic
116	#1552	C₂×C₅	1	4	yes	complemented
117	#1549	C₂×C₅	1	2	yes	complemented
118	#1546	C₂×C₅	1	2	yes	complemented
119	#1544	C₂×C₅	1	4	yes	complemented
120	C₂⁴×C₅	C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₅×Q₈	—	2	2	no	—
122	C₂×C₅×Q₈	—	2	2	no	—
123	C₂×C₅×Q₈	—	2	2	no	—
124	C₂×C₅×Q₈	—	2	2	no	—
125	C₂×C₅×Q₈	—	4	4	no	—
126	C₂×C₅×Q₈	—	4	4	no	—
127	C₂²×C₄×C₅	—	2	4	no	—
128	C₂²×C₄×C₅	—	2	4	no	—
129	C₂²×C₄×C₅	—	2	2	no	—
130	C₂²×C₄×C₅	—	2	2	no	—
131	C₂²×C₄×C₅	—	4	2	no	—
132	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
135	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
136	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
138	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
140	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
141	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
150	C₅×M₁₆	—	4	4	no	—
151	C₂×C₅×C₈	—	2	4	no	—
152	C₂×C₅×C₈	—	2	4	no	—
153	C₂×C₅×C₈	—	4	4	no	—
154	C₅×C₄⋊C₄	—	2	4	no	—
155	C₅×C₄⋊C₄	—	2	4	no	—
156	C₅×C₄⋊C₄	—	2	4	no	—
157	C₅×C₄⋊C₄	—	2	4	no	—
158	C₅×C₄⋊C₄	—	2	2	no	—
159	C₅×C₄⋊C₄	—	2	2	no	—
160	C₅×C₄⋊C₄	—	2	2	no	—
161	C₅×C₄⋊C₄	—	2	2	no	—
162	C₅×C₄⋊C₄	—	2	2	no	—
163	C₅×C₄⋊C₄	—	2	2	no	—
164	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
165	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
166	C₅×C₂²⋊C₄	—	2	4	no	—
167	C₅×C₂²⋊C₄	—	2	4	no	—
168	C₄²×C₅	—	2	4	no	—
169	C₄²×C₅	—	2	4	no	—
170	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
171	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
172	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
173	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
174	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
175	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
176	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
177	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
178	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
179	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
180	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
181	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
182	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
183	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
184	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
185	C₄²×C₅	C₂⁴	1	1	yes	characteristic
186	#533	—	2	1	no	—
187	#533	—	2	1	no	—
188	#533	—	2	1	no	—
189	#533	—	2	1	no	—
190	#533	C₄×C₅	1	2	yes	—
191	#533	C₄×C₅	1	2	yes	—
192	#533	C₂²×C₅	1	2	yes	—
193	#533	C₂²×C₅	1	2	yes	—
194	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
195	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
196	C₂×#125	C₄×C₅	1	2	yes	—
197	C₂×C₂²⋊Q₈	—	2	1	no	—
198	C₂×C₂²⋊Q₈	—	2	1	no	—
199	C₂×C₄×Q₈	—	2	1	no	—
200	C₂×C₄×Q₈	—	2	1	no	—
201	C₂×#117	C₄×C₅	1	2	yes	—
202	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
203	#432	—	2	1	no	—
204	#432	C₂²×C₅	1	2	yes	—
205	#431	—	2	1	no	—
206	#431	—	2	1	no	—
207	#431	—	2	1	no	—
208	#431	C₂²×C₅	1	2	yes	—
209	#431	C₂²×C₅	1	2	yes	—
210	#431	C₂²×C₅	1	2	yes	—
211	#423	C₂²×C₅	1	2	yes	—
212	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
213	#407	—	2	1	no	—
214	C₂×C₂²⋊C₈	—	2	1	no	—
215	C₄×M₁₆	—	2	1	no	—
216	C₂×C₄×C₈	—	2	1	no	—
217	#332	C₂²×C₅	1	8	yes	complemented
218	#332	C₂²×C₅	1	8	yes	complemented
219	C₂³×C₅	—	2	2	no	—
220	C₂³×C₅	—	2	2	no	—
221	C₂³×C₅	—	4	1	no	—
222	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
223	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
224	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
225	C₅×Q₈	—	4	4	no	—
226	C₅×Q₈	—	4	4	no	—
227	C₅×Q₈	—	4	4	no	—
228	C₅×Q₈	—	4	4	no	—
229	C₂×C₄×C₅	—	2	2	no	—
230	C₂×C₄×C₅	—	2	2	no	—
231	C₂×C₄×C₅	—	2	2	no	—
232	C₂×C₄×C₅	—	2	2	no	—
233	C₂×C₄×C₅	—	2	2	no	—
234	C₂×C₄×C₅	—	2	2	no	—
235	C₂×C₄×C₅	—	2	2	no	—
236	C₂×C₄×C₅	—	2	2	no	—
237	C₂×C₄×C₅	—	2	2	no	—
238	C₂×C₄×C₅	—	2	2	no	—
239	C₂×C₄×C₅	—	2	2	no	—
240	C₂×C₄×C₅	—	2	2	no	—
241	C₂×C₄×C₅	—	2	2	no	—
242	C₂×C₄×C₅	—	2	4	no	—
243	C₂×C₄×C₅	—	2	4	no	—
244	C₂×C₄×C₅	—	4	1	no	—
245	C₂×C₄×C₅	—	4	1	no	—
246	C₂×C₄×C₅	—	4	1	no	—
247	C₂×C₄×C₅	—	4	4	no	—
248	C₂×C₄×C₅	—	4	4	no	—
249	C₂×C₄×C₅	#101	1	4	yes	—
250	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
251	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
252	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
253	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
254	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
255	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
256	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
257	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
258	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
259	C₂×C₄×C₅	#135	1	4	yes	—
260	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
261	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
262	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
263	C₅×C₈	—	4	4	no	—
264	C₅×C₈	—	4	4	no	—
265	C₂²×Q₈	—	2	1	no	—
266	C₂²×Q₈	—	2	1	no	—
267	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
268	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
269	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
270	C₂×M₁₆	—	2	2	no	—
271	C₂²×C₈	—	2	2	no	—
272	C₄⋊Q₈	—	2	1	no	—
273	C₄⋊Q₈	—	2	1	no	—
274	C₄⋊Q₈	—	2	1	no	—
275	C₄⋊Q₈	—	2	1	no	—
276	C₄⋊Q₈	C₂×C₄×C₅	1	2	yes	—
277	C₄⋊Q₈	C₂×C₄×C₅	1	2	yes	—
278	C₄⋊Q₈	C₂³×C₅	1	2	yes	—
279	C₄⋊Q₈	C₂³×C₅	1	2	yes	—
280	#125	—	2	2	no	—
281	#125	—	2	2	no	—
282	#125	C₂×C₄×C₅	1	4	yes	—
283	#125	C₂×C₄×C₅	1	4	yes	—
284	C₂²⋊Q₈	—	2	2	no	—
285	C₂²⋊Q₈	—	2	2	no	—
286	C₂²⋊Q₈	—	2	2	no	—
287	C₂²⋊Q₈	—	2	2	no	—
288	C₂²⋊Q₈	—	2	2	no	—
289	C₂²⋊Q₈	—	2	2	no	—
290	C₂²⋊Q₈	—	2	2	no	—
291	C₂²⋊Q₈	—	2	2	no	—
292	C₄×Q₈	—	2	2	no	—
293	C₄×Q₈	—	2	2	no	—
294	C₄×Q₈	—	2	2	no	—
295	C₄×Q₈	—	2	2	no	—
296	C₄×Q₈	—	2	2	no	—
297	C₄×Q₈	—	2	2	no	—
298	C₄×Q₈	—	2	2	no	—
299	C₄×Q₈	—	2	2	no	—
300	#117	—	2	2	no	—
301	#117	—	2	2	no	—
302	#117	C₂×C₄×C₅	1	4	yes	—
303	#117	C₂×C₄×C₅	1	4	yes	—
304	C₂×C₄⋊C₄	—	2	2	no	—
305	C₂×C₄⋊C₄	—	2	2	no	—
306	C₂×C₄⋊C₄	—	2	1	no	—
307	C₂×C₄⋊C₄	—	2	1	no	—
308	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
309	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
310	C₂×C₂²⋊C₄	—	2	2	no	—
311	C₂×C₄²	—	2	2	no	—
312	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
313	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
314	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
315	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
316	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
317	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
318	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
319	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
320	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
321	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
322	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
323	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
324	C₄⋊C₈	—	2	2	no	—
325	C₄⋊C₈	—	2	2	no	—
326	C₄⋊C₈	C₂³×C₅	1	4	yes	—
327	C₄⋊C₈	C₂³×C₅	1	4	yes	—
328	C₂²⋊C₈	—	2	2	no	—
329	C₂²⋊C₈	—	2	2	no	—
330	C₂²⋊C₈	—	2	2	no	—
331	C₂²⋊C₈	—	2	2	no	—
332	#97	—	2	2	no	—
333	C₄×C₈	—	2	2	no	—
334	C₄×C₈	—	2	2	no	—
335	C₄×C₈	—	2	2	no	—
336	C₂²×C₅	—	2	1	no	—
337	C₂²×C₅	—	2	1	no	—
338	C₂²×C₅	—	2	1	no	—
339	C₂²×C₅	—	2	1	no	—
340	C₂²×C₅	—	2	1	no	—
341	C₂²×C₅	—	2	1	no	—
342	C₂²×C₅	—	2	1	no	—
343	C₂²×C₅	—	4	4	no	—
344	C₂²×C₅	C₂×#101	1	2	yes	—
345	C₂²×C₅	#416	1	2	yes	—
346	C₂²×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
347	C₄×C₅	—	2	1	no	—
348	C₄×C₅	—	2	1	no	—
349	C₄×C₅	—	2	1	no	—
350	C₄×C₅	—	2	1	no	—
351	C₄×C₅	—	2	1	no	—
352	C₄×C₅	—	2	1	no	—
353	C₄×C₅	—	2	1	no	—
354	C₄×C₅	—	4	4	no	—
355	C₄×C₅	—	4	4	no	—
356	C₄×C₅	C₂×#101	1	2	yes	—
357	C₄×C₅	C₂×#101	1	2	yes	—
358	C₄×C₅	#416	1	2	yes	—
359	C₄×C₅	C₂×#135	1	2	yes	—
360	C₄×C₅	#575	1	2	yes	—
361	C₂⁴	C₅×C₂²⋊C₄	1	1	yes	characteristic
362	C₂×Q₈	—	2	2	no	—
363	C₂×Q₈	—	2	2	no	—
364	C₂×Q₈	—	2	2	no	—
365	C₂×Q₈	—	2	2	no	—
366	C₂×Q₈	—	4	4	no	—
367	C₂×Q₈	—	4	4	no	—
368	C₂²×C₄	—	2	4	no	—
369	C₂²×C₄	—	2	4	no	—
370	C₂²×C₄	—	2	2	no	—
371	C₂²×C₄	—	2	2	no	—
372	C₂²×C₄	—	4	2	no	—
373	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
374	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
375	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
376	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
377	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
378	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
379	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
380	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
381	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
382	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
383	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
384	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
385	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
386	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
387	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
388	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
389	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
390	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
391	M₁₆	—	4	4	no	—
392	C₂×C₈	—	2	4	no	—
393	C₂×C₈	—	2	4	no	—
394	C₂×C₈	—	4	4	no	—
395	C₄⋊C₄	—	2	4	no	—
396	C₄⋊C₄	—	2	4	no	—
397	C₄⋊C₄	—	2	4	no	—
398	C₄⋊C₄	—	2	4	no	—
399	C₄⋊C₄	—	2	2	no	—
400	C₄⋊C₄	—	2	2	no	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	2	2	no	—
403	C₄⋊C₄	—	2	2	no	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
406	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
407	C₂²⋊C₄	—	2	4	no	—
408	C₂²⋊C₄	—	2	4	no	—
409	C₄²	—	2	4	no	—
410	C₄²	—	2	4	no	—
411	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
412	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
413	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
414	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
415	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
416	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
417	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
418	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
419	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
420	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
421	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
422	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
423	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
424	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
425	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
426	C₄²	C₂⁴×C₅	1	1	yes	characteristic
427	C₂×C₅	—	2	2	no	—
428	C₂×C₅	—	2	2	no	—
429	C₂×C₅	—	4	1	no	—
430	C₂×C₅	C₂²×#101	1	1	yes	characteristic
431	C₂×C₅	C₂×#416	1	1	yes	characteristic
432	C₂×C₅	#2896	1	1	yes	characteristic
433	C₂³	—	2	2	no	—
434	C₂³	—	2	2	no	—
435	C₂³	—	4	1	no	—
436	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
437	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
438	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
439	Q₈	—	4	4	no	—
440	Q₈	—	4	4	no	—
441	Q₈	—	4	4	no	—
442	Q₈	—	4	4	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	4	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	4	1	no	—
459	C₂×C₄	—	4	1	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	4	4	no	—
462	C₂×C₄	—	4	4	no	—
463	C₂×C₄	C₅×#101	1	4	yes	—
464	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
465	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
466	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
467	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
468	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
469	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
470	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
471	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
472	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
473	C₂×C₄	C₅×#135	1	4	yes	—
474	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
475	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
476	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
477	C₈	—	4	4	no	—
478	C₈	—	4	4	no	—
479	C₅	#17715	1	1	yes	characteristic, direct product factor, Sylow
480	C₂²	—	2	1	no	—
481	C₂²	—	2	1	no	—
482	C₂²	—	2	1	no	—
483	C₂²	—	2	1	no	—
484	C₂²	—	2	1	no	—
485	C₂²	—	2	1	no	—
486	C₂²	—	2	1	no	—
487	C₂²	—	4	4	no	—
488	C₂²	C₂×C₅×#101	1	2	yes	—
489	C₂²	C₅×#416	1	2	yes	—
490	C₂²	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
491	C₄	—	2	1	no	—
492	C₄	—	2	1	no	—
493	C₄	—	2	1	no	—
494	C₄	—	2	1	no	—
495	C₄	—	2	1	no	—
496	C₄	—	2	1	no	—
497	C₄	—	2	1	no	—
498	C₄	—	4	4	no	—
499	C₄	—	4	4	no	—
500	C₄	C₂×C₅×#101	1	2	yes	—
501	C₄	C₂×C₅×#101	1	2	yes	—
502	C₄	C₅×#416	1	2	yes	—
503	C₄	C₂×C₅×#135	1	2	yes	—
504	C₄	C₅×#575	1	2	yes	—
505	C₂	—	2	2	no	—
506	C₂	—	2	2	no	—
507	C₂	—	4	1	no	—
508	C₂	C₂²×C₅×#101	1	1	yes	characteristic
509	C₂	C₂×C₅×#416	1	1	yes	characteristic
510	C₂	C₅×#2896	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₄²	C₂⁴×C₅
socle	C₂²×C₅	C₂²×C₂²⋊C₄
prime radical	C₂⁴×C₅	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1552	C₂	semidirect
5. C₅×#1549	C₂	semidirect
6. C₅×#1546	C₂	semidirect
7. C₅×#1544	C₂	semidirect
39. C₅×#332	C₂²	semidirect
40. C₅×#332	C₂²	semidirect
41. #17715	C₅	direct (N × H)
116. #1552	C₂×C₅	semidirect
117. #1549	C₂×C₅	semidirect
118. #1546	C₂×C₅	semidirect
119. #1544	C₂×C₅	semidirect
217. #332	C₂²×C₅	semidirect
218. #332	C₂²×C₅	semidirect
479. C₅	#17715	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#533	2
C₅×#2920	2
C₂×C₅×#431	2
C₅×#1552	2
C₅×#1549	2
C₅×#1546	2
C₅×#1544	2
#17715	5

Subgroups — C5×#17715 #11940435

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#17715 #11940435