Subgroups — C₅×#19306 #11942026

Order 1280.

732 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3184	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3109	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3036	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×D₄×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1701	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1697	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1691	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1687	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1681	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#1677	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#1671	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#1667	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅×#1569	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×#1560	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅×#1557	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₅×AGL(1,ℤ/8)	—	2	1	no	—
17	C₂²×C₅×M₁₆	—	2	1	no	—
18	C₅×#532	C₂²	1	1	yes	characteristic
19	C₅×#532	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
21	C₅×#498	C₂²	1	1	yes	characteristic, complemented
22	C₅×#493	C₂²	1	1	yes	characteristic, complemented
23	C₅×#486	C₂²	1	1	yes	characteristic, complemented
24	C₅×#486	C₂²	1	1	yes	characteristic, complemented
25	C₅×#481	—	2	1	no	—
26	C₅×#476	C₂²	1	1	yes	characteristic
27	C₅×#476	C₂²	1	1	yes	characteristic
28	C₅×#469	—	2	1	no	—
29	C₅×#469	—	2	1	no	—
30	C₅×#463	—	2	1	no	—
31	C₅×#463	—	2	1	no	—
32	C₅×#461	C₂²	1	1	yes	characteristic
33	C₅×#461	C₂²	1	1	yes	characteristic
34	C₅×#459	C₂²	1	1	yes	characteristic
35	C₅×#459	C₂²	1	1	yes	characteristic
36	C₅×#459	C₂²	1	1	yes	characteristic
37	C₅×#459	C₂²	1	1	yes	characteristic
38	C₅×#447	—	2	1	no	—
39	C₅×#442	—	2	1	no	—
40	C₅×#442	—	2	1	no	—
41	C₅×#442	—	2	1	no	—
42	C₅×#442	—	2	1	no	—
43	C₅×#436	C₂²	1	1	yes	characteristic
44	C₅×#436	C₂²	1	1	yes	characteristic
45	C₅×#435	—	2	1	no	—
46	C₅×#435	—	2	1	no	—
47	C₅×C₈×D₄	C₂²	1	1	yes	characteristic
48	C₅×C₈×D₄	C₂²	1	1	yes	characteristic
49	C₅×#432	C₂²	1	1	yes	characteristic
50	C₅×#423	C₂²	1	1	yes	characteristic
51	C₅×#423	C₂²	1	1	yes	characteristic
52	C₅×#417	—	2	1	no	—
53	C₅×#407	—	2	1	no	—
54	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
55	C₅×#335	C₂²	1	1	yes	characteristic, complemented
56	C₅×#335	C₂²	1	1	yes	characteristic, complemented
57	C₅×#334	C₂²	1	1	yes	characteristic, complemented
58	C₅×#334	C₂²	1	1	yes	characteristic, complemented
59	C₅×#333	C₂²	1	1	yes	characteristic, complemented
60	C₅×#333	C₂²	1	1	yes	characteristic, complemented
61	C₅×#332	C₂²	1	1	yes	characteristic, complemented
62	C₅×#332	C₂²	1	1	yes	characteristic, complemented
63	C₅×#331	C₂²	1	1	yes	characteristic, complemented
64	C₅×#331	C₂²	1	1	yes	characteristic, complemented
65	C₅×#331	C₂²	1	1	yes	characteristic, complemented
66	C₅×#331	C₂²	1	1	yes	characteristic, complemented
67	#19306	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂×C₅×#41	—	2	1	no	—
69	C₂×C₅×#41	—	2	1	no	—
70	C₂²×C₅×D₄	D₄	1	1	yes	characteristic
71	C₂³×C₄×C₅	—	2	1	no	—
72	C₅×AGL(1,ℤ/8)	—	4	4	no	—
73	C₂×C₅×SD₁₆	—	2	1	no	—
74	C₂×C₅×SD₁₆	—	2	1	no	—
75	C₂×C₅×D₈	—	2	1	no	—
76	C₂×C₅×D₈	—	2	1	no	—
77	C₂×C₅×D₈	—	2	1	no	—
78	C₂×C₅×D₈	—	2	1	no	—
79	C₂×C₅×D₈	—	2	1	no	—
80	C₂×C₅×D₈	—	2	1	no	—
81	C₂×C₅×M₁₆	—	2	1	no	—
82	C₂×C₅×M₁₆	—	2	1	no	—
83	C₂×C₅×M₁₆	—	2	1	no	—
84	C₂×C₅×M₁₆	—	2	1	no	—
85	C₂×C₅×M₁₆	—	4	4	no	—
86	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
87	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
88	C₂²×C₅×C₈	—	2	1	no	—
89	C₂²×C₅×C₈	—	2	1	no	—
90	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
91	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
92	C₅×#127	C₂³	1	1	yes	characteristic
93	C₅×#127	C₂³	1	1	yes	characteristic
94	C₅×#124	—	2	1	no	—
95	C₅×#124	—	2	1	no	—
96	C₅×#121	—	2	1	no	—
97	C₅×#121	—	2	1	no	—
98	C₅×#121	—	2	1	no	—
99	C₅×#121	—	2	1	no	—
100	C₄×C₅×D₄	—	2	1	no	—
101	C₄×C₅×D₄	—	2	1	no	—
102	C₄×C₅×D₄	—	2	1	no	—
103	C₄×C₅×D₄	—	2	1	no	—
104	C₄×C₅×D₄	—	2	1	no	—
105	C₄×C₅×D₄	—	2	1	no	—
106	C₄×C₅×D₄	D₄	1	1	yes	characteristic
107	C₄×C₅×D₄	D₄	1	1	yes	characteristic
108	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
109	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
110	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
111	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
112	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
113	C₅×#106	—	2	1	no	—
114	C₅×#106	—	2	1	no	—
115	C₅×#106	—	2	1	no	—
116	C₅×#106	—	2	1	no	—
117	C₅×C₄⋊C₈	—	2	1	no	—
118	C₅×C₄⋊C₈	—	2	1	no	—
119	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
120	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
121	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
122	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
123	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
124	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
125	C₅×#103	—	2	1	no	—
126	C₅×#103	—	2	1	no	—
127	C₅×#102	—	2	1	no	—
128	C₅×#102	—	2	1	no	—
129	C₅×#102	—	2	1	no	—
130	C₅×#102	—	2	1	no	—
131	C₅×#102	—	2	1	no	—
132	C₅×#102	—	2	1	no	—
133	C₅×#102	—	2	1	no	—
134	C₅×#102	—	2	1	no	—
135	C₅×#102	—	2	1	no	—
136	C₅×#102	—	2	1	no	—
137	C₅×C₂²⋊C₈	—	2	1	no	—
138	C₅×C₂²⋊C₈	—	2	1	no	—
139	C₅×C₂²⋊C₈	—	2	1	no	—
140	C₅×C₂²⋊C₈	—	2	1	no	—
141	C₅×C₂²⋊C₈	—	2	1	no	—
142	C₅×#97	—	2	1	no	—
143	C₅×#97	D₄	1	1	yes	characteristic
144	C₅×#97	D₄	1	1	yes	characteristic
145	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
146	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
147	#3184	C₂×C₅	1	1	yes	characteristic, complemented
148	#3109	C₂×C₅	1	1	yes	characteristic, complemented
149	#3036	C₂×C₅	1	1	yes	characteristic, complemented
150	D₄×M₁₆	C₂×C₅	1	1	yes	characteristic, complemented
151	#1701	C₂×C₅	1	1	yes	characteristic, complemented
152	#1697	C₂×C₅	1	1	yes	characteristic, complemented
153	#1691	C₂×C₅	1	1	yes	characteristic, complemented
154	#1687	C₂×C₅	1	1	yes	characteristic, complemented
155	#1681	C₂×C₅	1	1	yes	characteristic, complemented
156	#1677	C₂×C₅	1	1	yes	characteristic, complemented
157	#1671	C₂×C₅	1	1	yes	characteristic, complemented
158	#1667	C₂×C₅	1	1	yes	characteristic, complemented
159	#1569	C₂×C₅	1	1	yes	characteristic, complemented
160	#1560	C₂×C₅	1	1	yes	characteristic, complemented
161	#1557	C₂×C₅	1	1	yes	characteristic, complemented
162	C₂⁴×C₅	—	2	1	no	—
163	C₅×#41	—	4	2	no	—
164	C₅×#41	—	4	2	no	—
165	C₅×#41	—	8	2	no	—
166	C₂×C₅×Q₈	—	2	1	no	—
167	C₂×C₅×Q₈	—	2	1	no	—
168	C₂×C₅×D₄	—	2	2	no	—
169	C₂×C₅×D₄	—	2	1	no	—
170	C₂×C₅×D₄	—	2	1	no	—
171	C₂×C₅×D₄	—	2	1	no	—
172	C₂×C₅×D₄	—	2	1	no	—
173	C₂×C₅×D₄	—	2	1	no	—
174	C₂×C₅×D₄	—	2	1	no	—
175	C₂×C₅×D₄	—	2	1	no	—
176	C₂×C₅×D₄	—	4	1	no	—
177	C₂×C₅×D₄	—	4	1	no	—
178	C₂×C₅×D₄	—	4	1	no	—
179	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
180	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₅	—	2	1	no	—
182	C₂²×C₄×C₅	—	2	1	no	—
183	C₂²×C₄×C₅	—	2	1	no	—
184	C₂²×C₄×C₅	—	2	1	no	—
185	C₂²×C₄×C₅	—	2	1	no	—
186	C₂²×C₄×C₅	—	4	2	no	—
187	C₂²×C₄×C₅	—	4	2	no	—
188	C₂²×C₄×C₅	—	4	1	no	—
189	C₂²×C₄×C₅	—	4	1	no	—
190	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
191	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
192	C₂²×C₄×C₅	#41	1	1	yes	characteristic
193	C₅×SD₁₆	—	4	2	no	—
194	C₅×SD₁₆	—	4	2	no	—
195	C₅×D₈	—	4	2	no	—
196	C₅×D₈	—	4	2	no	—
197	C₅×D₈	—	4	2	no	—
198	C₅×D₈	—	4	2	no	—
199	C₅×D₈	—	4	2	no	—
200	C₅×D₈	—	4	2	no	—
201	C₅×M₁₆	—	2	2	no	—
202	C₅×M₁₆	—	4	1	no	—
203	C₅×M₁₆	—	4	2	no	—
204	C₅×M₁₆	—	4	2	no	—
205	C₅×M₁₆	—	4	2	no	—
206	C₅×M₁₆	—	4	2	no	—
207	C₂×C₅×C₈	—	2	1	no	—
208	C₂×C₅×C₈	—	2	1	no	—
209	C₂×C₅×C₈	—	2	1	no	—
210	C₂×C₅×C₈	—	2	1	no	—
211	C₂×C₅×C₈	—	2	1	no	—
212	C₂×C₅×C₈	—	2	1	no	—
213	C₂×C₅×C₈	—	4	2	no	—
214	C₂×C₅×C₈	—	4	2	no	—
215	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
216	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
217	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
218	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
219	C₅×C₄⋊C₄	—	2	1	no	—
220	C₅×C₄⋊C₄	—	2	1	no	—
221	C₅×C₄⋊C₄	—	2	1	no	—
222	C₅×C₄⋊C₄	—	2	1	no	—
223	C₅×C₄⋊C₄	—	2	1	no	—
224	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
225	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
226	C₅×C₂²⋊C₄	—	2	1	no	—
227	C₅×C₂²⋊C₄	—	2	1	no	—
228	C₅×C₂²⋊C₄	—	2	1	no	—
229	C₅×C₂²⋊C₄	—	2	1	no	—
230	C₅×C₂²⋊C₄	—	4	1	no	—
231	C₅×C₂²⋊C₄	—	4	1	no	—
232	C₅×C₂²⋊C₄	—	4	1	no	—
233	C₅×C₂²⋊C₄	—	4	1	no	—
234	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
235	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
236	C₄²×C₅	#41	1	1	yes	characteristic
237	C₄²×C₅	C₂⁴	1	1	yes	characteristic
238	C₂×AGL(1,ℤ/8)	—	2	1	no	—
239	C₂²×M₁₆	—	2	1	no	—
240	#532	C₂²×C₅	1	1	yes	characteristic
241	#532	C₂²×C₅	1	1	yes	characteristic
242	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
243	#498	C₂²×C₅	1	1	yes	characteristic, complemented
244	#493	C₂²×C₅	1	1	yes	characteristic, complemented
245	#486	C₂²×C₅	1	1	yes	characteristic, complemented
246	#486	C₂²×C₅	1	1	yes	characteristic, complemented
247	#481	—	2	1	no	—
248	#476	C₂²×C₅	1	1	yes	characteristic
249	#476	C₂²×C₅	1	1	yes	characteristic
250	#469	—	2	1	no	—
251	#469	—	2	1	no	—
252	#463	—	2	1	no	—
253	#463	—	2	1	no	—
254	#461	C₂²×C₅	1	1	yes	characteristic
255	#461	C₂²×C₅	1	1	yes	characteristic
256	#459	C₂²×C₅	1	1	yes	characteristic
257	#459	C₂²×C₅	1	1	yes	characteristic
258	#459	C₂²×C₅	1	1	yes	characteristic
259	#459	C₂²×C₅	1	1	yes	characteristic
260	#447	—	2	1	no	—
261	#442	—	2	1	no	—
262	#442	—	2	1	no	—
263	#442	—	2	1	no	—
264	#442	—	2	1	no	—
265	#436	C₂²×C₅	1	1	yes	characteristic
266	#436	C₂²×C₅	1	1	yes	characteristic
267	#435	—	2	1	no	—
268	#435	—	2	1	no	—
269	C₈×D₄	C₂²×C₅	1	1	yes	characteristic
270	C₈×D₄	C₂²×C₅	1	1	yes	characteristic
271	#432	C₂²×C₅	1	1	yes	characteristic
272	#423	C₂²×C₅	1	1	yes	characteristic
273	#423	C₂²×C₅	1	1	yes	characteristic
274	#417	—	2	1	no	—
275	#407	—	2	1	no	—
276	C₄×M₁₆	C₂²×C₅	1	1	yes	characteristic
277	#335	C₂²×C₅	1	1	yes	characteristic, complemented
278	#335	C₂²×C₅	1	1	yes	characteristic, complemented
279	#334	C₂²×C₅	1	1	yes	characteristic, complemented
280	#334	C₂²×C₅	1	1	yes	characteristic, complemented
281	#333	C₂²×C₅	1	1	yes	characteristic, complemented
282	#333	C₂²×C₅	1	1	yes	characteristic, complemented
283	#332	C₂²×C₅	1	1	yes	characteristic, complemented
284	#332	C₂²×C₅	1	1	yes	characteristic, complemented
285	#331	C₂²×C₅	1	1	yes	characteristic, complemented
286	#331	C₂²×C₅	1	1	yes	characteristic, complemented
287	#331	C₂²×C₅	1	1	yes	characteristic, complemented
288	#331	C₂²×C₅	1	1	yes	characteristic, complemented
289	C₂³×C₅	—	2	1	no	—
290	C₂³×C₅	—	2	1	no	—
291	C₂³×C₅	—	4	2	no	—
292	C₂³×C₅	—	4	2	no	—
293	C₂³×C₅	—	4	1	no	—
294	C₂³×C₅	—	4	1	no	—
295	C₂³×C₅	—	8	1	no	—
296	C₂³×C₅	#121	1	1	yes	characteristic
297	C₅×Q₈	—	4	1	no	—
298	C₅×Q₈	—	4	1	no	—
299	C₅×Q₈	—	8	1	no	—
300	C₅×D₄	—	2	1	no	—
301	C₅×D₄	—	2	1	no	—
302	C₅×D₄	—	2	1	no	—
303	C₅×D₄	—	4	2	no	—
304	C₅×D₄	—	4	1	no	—
305	C₅×D₄	—	4	1	no	—
306	C₅×D₄	—	4	1	no	—
307	C₅×D₄	—	4	1	no	—
308	C₅×D₄	—	4	1	no	—
309	C₅×D₄	—	4	1	no	—
310	C₅×D₄	—	4	1	no	—
311	C₅×D₄	—	4	1	no	—
312	C₅×D₄	—	8	1	no	—
313	C₅×D₄	—	8	1	no	—
314	C₅×D₄	—	8	2	no	—
315	C₅×D₄	—	8	2	no	—
316	C₅×D₄	AGL(1,ℤ/8)	1	2	yes	—
317	C₂×C₄×C₅	—	2	1	no	—
318	C₂×C₄×C₅	—	2	1	no	—
319	C₂×C₄×C₅	—	2	1	no	—
320	C₂×C₄×C₅	—	2	1	no	—
321	C₂×C₄×C₅	—	2	1	no	—
322	C₂×C₄×C₅	—	2	1	no	—
323	C₂×C₄×C₅	—	4	1	no	—
324	C₂×C₄×C₅	—	4	2	no	—
325	C₂×C₄×C₅	—	4	2	no	—
326	C₂×C₄×C₅	—	4	2	no	—
327	C₂×C₄×C₅	—	4	1	no	—
328	C₂×C₄×C₅	—	4	1	no	—
329	C₂×C₄×C₅	—	4	1	no	—
330	C₂×C₄×C₅	—	4	1	no	—
331	C₂×C₄×C₅	—	4	1	no	—
332	C₂×C₄×C₅	—	4	1	no	—
333	C₂×C₄×C₅	—	8	2	no	—
334	C₂×C₄×C₅	—	8	2	no	—
335	C₂×C₄×C₅	#121	1	1	yes	characteristic
336	C₂×C₄×C₅	#121	1	1	yes	characteristic
337	C₂×C₄×C₅	#121	1	1	yes	characteristic
338	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
339	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
340	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
341	C₅×C₈	—	2	1	no	—
342	C₅×C₈	—	2	1	no	—
343	C₅×C₈	—	2	1	no	—
344	C₅×C₈	—	4	1	no	—
345	C₅×C₈	—	4	1	no	—
346	C₅×C₈	—	4	1	no	—
347	C₅×C₈	—	4	1	no	—
348	C₅×C₈	—	4	1	no	—
349	C₅×C₈	—	4	1	no	—
350	C₅×C₈	AGL(1,ℤ/8)	1	2	yes	complemented
351	C₂×#41	—	2	1	no	—
352	C₂×#41	—	2	1	no	—
353	C₂²×D₄	C₅×D₄	1	1	yes	characteristic
354	C₂³×C₄	—	2	1	no	—
355	AGL(1,ℤ/8)	—	4	4	no	—
356	C₂×SD₁₆	—	2	1	no	—
357	C₂×SD₁₆	—	2	1	no	—
358	C₂×D₈	—	2	1	no	—
359	C₂×D₈	—	2	1	no	—
360	C₂×D₈	—	2	1	no	—
361	C₂×D₈	—	2	1	no	—
362	C₂×D₈	—	2	1	no	—
363	C₂×D₈	—	2	1	no	—
364	C₂×M₁₆	—	2	1	no	—
365	C₂×M₁₆	—	2	1	no	—
366	C₂×M₁₆	—	2	1	no	—
367	C₂×M₁₆	—	2	1	no	—
368	C₂×M₁₆	—	4	4	no	—
369	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
370	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
371	C₂²×C₈	—	2	1	no	—
372	C₂²×C₈	—	2	1	no	—
373	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
374	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
375	#127	C₂³×C₅	1	1	yes	characteristic
376	#127	C₂³×C₅	1	1	yes	characteristic
377	#124	—	2	1	no	—
378	#124	—	2	1	no	—
379	#121	—	2	1	no	—
380	#121	—	2	1	no	—
381	#121	—	2	1	no	—
382	#121	—	2	1	no	—
383	C₄×D₄	—	2	1	no	—
384	C₄×D₄	—	2	1	no	—
385	C₄×D₄	—	2	1	no	—
386	C₄×D₄	—	2	1	no	—
387	C₄×D₄	—	2	1	no	—
388	C₄×D₄	—	2	1	no	—
389	C₄×D₄	C₅×D₄	1	1	yes	characteristic
390	C₄×D₄	C₅×D₄	1	1	yes	characteristic
391	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
392	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
393	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
394	C₂×C₂²⋊C₄	—	2	1	no	—
395	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
396	#106	—	2	1	no	—
397	#106	—	2	1	no	—
398	#106	—	2	1	no	—
399	#106	—	2	1	no	—
400	C₄⋊C₈	—	2	1	no	—
401	C₄⋊C₈	—	2	1	no	—
402	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
403	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
404	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
405	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
406	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
407	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
408	#103	—	2	1	no	—
409	#103	—	2	1	no	—
410	#102	—	2	1	no	—
411	#102	—	2	1	no	—
412	#102	—	2	1	no	—
413	#102	—	2	1	no	—
414	#102	—	2	1	no	—
415	#102	—	2	1	no	—
416	#102	—	2	1	no	—
417	#102	—	2	1	no	—
418	#102	—	2	1	no	—
419	#102	—	2	1	no	—
420	C₂²⋊C₈	—	2	1	no	—
421	C₂²⋊C₈	—	2	1	no	—
422	C₂²⋊C₈	—	2	1	no	—
423	C₂²⋊C₈	—	2	1	no	—
424	C₂²⋊C₈	—	2	1	no	—
425	#97	—	2	1	no	—
426	#97	C₅×D₄	1	1	yes	characteristic
427	#97	C₅×D₄	1	1	yes	characteristic
428	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
429	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
430	C₂²×C₅	—	2	1	no	—
431	C₂²×C₅	—	2	1	no	—
432	C₂²×C₅	—	2	2	no	—
433	C₂²×C₅	—	2	1	no	—
434	C₂²×C₅	—	4	1	no	—
435	C₂²×C₅	—	4	1	no	—
436	C₂²×C₅	—	4	1	no	—
437	C₂²×C₅	—	4	1	no	—
438	C₂²×C₅	—	4	1	no	—
439	C₂²×C₅	—	8	4	no	—
440	C₂²×C₅	—	8	1	no	—
441	C₂²×C₅	—	8	1	no	—
442	C₂²×C₅	—	8	1	no	—
443	C₂²×C₅	—	8	1	no	—
444	C₂²×C₅	—	8	1	no	—
445	C₂²×C₅	—	8	1	no	—
446	C₂²×C₅	C₂×#121	1	1	yes	characteristic
447	C₄×C₅	—	2	1	no	—
448	C₄×C₅	—	2	1	no	—
449	C₄×C₅	—	2	1	no	—
450	C₄×C₅	—	2	1	no	—
451	C₄×C₅	—	4	1	no	—
452	C₄×C₅	—	4	1	no	—
453	C₄×C₅	—	8	1	no	—
454	C₄×C₅	—	8	1	no	—
455	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
456	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
457	C₄×C₅	#575	1	1	yes	characteristic
458	C₄×C₅	#575	1	1	yes	characteristic
459	C₂⁴	—	2	1	no	—
460	#41	—	4	2	no	—
461	#41	—	4	2	no	—
462	#41	—	8	2	no	—
463	C₂×Q₈	—	2	1	no	—
464	C₂×Q₈	—	2	1	no	—
465	C₂×D₄	—	2	2	no	—
466	C₂×D₄	—	2	1	no	—
467	C₂×D₄	—	2	1	no	—
468	C₂×D₄	—	2	1	no	—
469	C₂×D₄	—	2	1	no	—
470	C₂×D₄	—	2	1	no	—
471	C₂×D₄	—	2	1	no	—
472	C₂×D₄	—	2	1	no	—
473	C₂×D₄	—	4	1	no	—
474	C₂×D₄	—	4	1	no	—
475	C₂×D₄	—	4	1	no	—
476	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
477	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
478	C₂²×C₄	—	2	1	no	—
479	C₂²×C₄	—	2	1	no	—
480	C₂²×C₄	—	2	1	no	—
481	C₂²×C₄	—	2	1	no	—
482	C₂²×C₄	—	2	1	no	—
483	C₂²×C₄	—	4	2	no	—
484	C₂²×C₄	—	4	2	no	—
485	C₂²×C₄	—	4	1	no	—
486	C₂²×C₄	—	4	1	no	—
487	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
488	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
489	C₂²×C₄	C₅×#41	1	1	yes	characteristic
490	SD₁₆	—	4	2	no	—
491	SD₁₆	—	4	2	no	—
492	D₈	—	4	2	no	—
493	D₈	—	4	2	no	—
494	D₈	—	4	2	no	—
495	D₈	—	4	2	no	—
496	D₈	—	4	2	no	—
497	D₈	—	4	2	no	—
498	M₁₆	—	2	2	no	—
499	M₁₆	—	4	1	no	—
500	M₁₆	—	4	2	no	—
501	M₁₆	—	4	2	no	—
502	M₁₆	—	4	2	no	—
503	M₁₆	—	4	2	no	—
504	C₂×C₈	—	2	1	no	—
505	C₂×C₈	—	2	1	no	—
506	C₂×C₈	—	2	1	no	—
507	C₂×C₈	—	2	1	no	—
508	C₂×C₈	—	2	1	no	—
509	C₂×C₈	—	2	1	no	—
510	C₂×C₈	—	4	2	no	—
511	C₂×C₈	—	4	2	no	—
512	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
513	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
514	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
515	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
516	C₄⋊C₄	—	2	1	no	—
517	C₄⋊C₄	—	2	1	no	—
518	C₄⋊C₄	—	2	1	no	—
519	C₄⋊C₄	—	2	1	no	—
520	C₄⋊C₄	—	2	1	no	—
521	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
522	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
523	C₂²⋊C₄	—	2	1	no	—
524	C₂²⋊C₄	—	2	1	no	—
525	C₂²⋊C₄	—	2	1	no	—
526	C₂²⋊C₄	—	2	1	no	—
527	C₂²⋊C₄	—	4	1	no	—
528	C₂²⋊C₄	—	4	1	no	—
529	C₂²⋊C₄	—	4	1	no	—
530	C₂²⋊C₄	—	4	1	no	—
531	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
532	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
533	C₄²	C₅×#41	1	1	yes	characteristic
534	C₄²	C₂⁴×C₅	1	1	yes	characteristic
535	C₂×C₅	—	4	1	no	—
536	C₂×C₅	—	4	2	no	—
537	C₂×C₅	—	8	1	no	—
538	C₂×C₅	—	16	1	no	—
539	C₂×C₅	—	16	1	no	—
540	C₂×C₅	#3036	1	1	yes	characteristic
541	C₂×C₅	#3055	1	1	yes	characteristic
542	C₂×C₅	#3068	1	1	yes	characteristic
543	C₂³	—	2	1	no	—
544	C₂³	—	2	1	no	—
545	C₂³	—	4	2	no	—
546	C₂³	—	4	2	no	—
547	C₂³	—	4	1	no	—
548	C₂³	—	4	1	no	—
549	C₂³	—	8	1	no	—
550	C₂³	C₅×#121	1	1	yes	characteristic
551	Q₈	—	4	1	no	—
552	Q₈	—	4	1	no	—
553	Q₈	—	8	1	no	—
554	D₄	—	2	1	no	—
555	D₄	—	2	1	no	—
556	D₄	—	2	1	no	—
557	D₄	—	4	2	no	—
558	D₄	—	4	1	no	—
559	D₄	—	4	1	no	—
560	D₄	—	4	1	no	—
561	D₄	—	4	1	no	—
562	D₄	—	4	1	no	—
563	D₄	—	4	1	no	—
564	D₄	—	4	1	no	—
565	D₄	—	4	1	no	—
566	D₄	—	8	1	no	—
567	D₄	—	8	1	no	—
568	D₄	—	8	2	no	—
569	D₄	—	8	2	no	—
570	D₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
571	C₂×C₄	—	2	1	no	—
572	C₂×C₄	—	2	1	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	1	no	—
577	C₂×C₄	—	4	1	no	—
578	C₂×C₄	—	4	2	no	—
579	C₂×C₄	—	4	2	no	—
580	C₂×C₄	—	4	2	no	—
581	C₂×C₄	—	4	1	no	—
582	C₂×C₄	—	4	1	no	—
583	C₂×C₄	—	4	1	no	—
584	C₂×C₄	—	4	1	no	—
585	C₂×C₄	—	4	1	no	—
586	C₂×C₄	—	4	1	no	—
587	C₂×C₄	—	8	2	no	—
588	C₂×C₄	—	8	2	no	—
589	C₂×C₄	C₅×#121	1	1	yes	characteristic
590	C₂×C₄	C₅×#121	1	1	yes	characteristic
591	C₂×C₄	C₅×#121	1	1	yes	characteristic
592	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
593	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
594	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
595	C₈	—	2	1	no	—
596	C₈	—	2	1	no	—
597	C₈	—	2	1	no	—
598	C₈	—	4	1	no	—
599	C₈	—	4	1	no	—
600	C₈	—	4	1	no	—
601	C₈	—	4	1	no	—
602	C₈	—	4	1	no	—
603	C₈	—	4	1	no	—
604	C₈	C₅×AGL(1,ℤ/8)	1	2	yes	complemented
605	C₅	#19306	1	1	yes	characteristic, direct product factor, Sylow
606	C₂²	—	2	1	no	—
607	C₂²	—	2	1	no	—
608	C₂²	—	2	2	no	—
609	C₂²	—	2	1	no	—
610	C₂²	—	4	1	no	—
611	C₂²	—	4	1	no	—
612	C₂²	—	4	1	no	—
613	C₂²	—	4	1	no	—
614	C₂²	—	4	1	no	—
615	C₂²	—	8	4	no	—
616	C₂²	—	8	1	no	—
617	C₂²	—	8	1	no	—
618	C₂²	—	8	1	no	—
619	C₂²	—	8	1	no	—
620	C₂²	—	8	1	no	—
621	C₂²	—	8	1	no	—
622	C₂²	C₂×C₅×#121	1	1	yes	characteristic
623	C₄	—	2	1	no	—
624	C₄	—	2	1	no	—
625	C₄	—	2	1	no	—
626	C₄	—	2	1	no	—
627	C₄	—	4	1	no	—
628	C₄	—	4	1	no	—
629	C₄	—	8	1	no	—
630	C₄	—	8	1	no	—
631	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
632	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
633	C₄	C₅×#575	1	1	yes	characteristic
634	C₄	C₅×#575	1	1	yes	characteristic
635	C₂	—	4	1	no	—
636	C₂	—	4	2	no	—
637	C₂	—	8	1	no	—
638	C₂	—	16	1	no	—
639	C₂	—	16	1	no	—
640	C₂	C₅×#3036	1	1	yes	characteristic
641	C₂	C₅×#3055	1	1	yes	characteristic
642	C₂	C₅×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#3184	C₂	semidirect
2. C₅×#3109	C₂	semidirect
3. C₅×#3036	C₂	semidirect
4. C₅×D₄×M₁₆	C₂	semidirect
5. C₅×#1701	C₂	semidirect
6. C₅×#1697	C₂	semidirect
7. C₅×#1691	C₂	semidirect
8. C₅×#1687	C₂	semidirect
9. C₅×#1681	C₂	semidirect
10. C₅×#1677	C₂	semidirect
11. C₅×#1671	C₂	semidirect
12. C₅×#1667	C₂	semidirect
13. C₅×#1569	C₂	semidirect
14. C₅×#1560	C₂	semidirect
15. C₅×#1557	C₂	semidirect
21. C₅×#498	C₂²	semidirect
22. C₅×#493	C₂²	semidirect
23. C₅×#486	C₂²	semidirect
24. C₅×#486	C₂²	semidirect
55. C₅×#335	C₂²	semidirect
56. C₅×#335	C₂²	semidirect
57. C₅×#334	C₂²	semidirect
58. C₅×#334	C₂²	semidirect
59. C₅×#333	C₂²	semidirect
60. C₅×#333	C₂²	semidirect
61. C₅×#332	C₂²	semidirect
62. C₅×#332	C₂²	semidirect
63. C₅×#331	C₂²	semidirect
64. C₅×#331	C₂²	semidirect
65. C₅×#331	C₂²	semidirect
66. C₅×#331	C₂²	semidirect
67. #19306	C₅	direct (N × H)
147. #3184	C₂×C₅	semidirect
148. #3109	C₂×C₅	semidirect
149. #3036	C₂×C₅	semidirect
150. D₄×M₁₆	C₂×C₅	semidirect
151. #1701	C₂×C₅	semidirect
152. #1697	C₂×C₅	semidirect
153. #1691	C₂×C₅	semidirect
154. #1687	C₂×C₅	semidirect
155. #1681	C₂×C₅	semidirect
156. #1677	C₂×C₅	semidirect
157. #1671	C₂×C₅	semidirect
158. #1667	C₂×C₅	semidirect
159. #1569	C₂×C₅	semidirect
160. #1560	C₂×C₅	semidirect
161. #1557	C₂×C₅	semidirect
243. #498	C₂²×C₅	semidirect
244. #493	C₂²×C₅	semidirect
245. #486	C₂²×C₅	semidirect
246. #486	C₂²×C₅	semidirect
277. #335	C₂²×C₅	semidirect
278. #335	C₂²×C₅	semidirect
279. #334	C₂²×C₅	semidirect
280. #334	C₂²×C₅	semidirect
281. #333	C₂²×C₅	semidirect
282. #333	C₂²×C₅	semidirect
283. #332	C₂²×C₅	semidirect
284. #332	C₂²×C₅	semidirect
285. #331	C₂²×C₅	semidirect
286. #331	C₂²×C₅	semidirect
287. #331	C₂²×C₅	semidirect
288. #331	C₂²×C₅	semidirect
350. C₅×C₈	AGL(1,ℤ/8)	semidirect
604. C₈	C₅×AGL(1,ℤ/8)	semidirect
605. C₅	#19306	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#3184	2
C₅×#3109	2
C₅×#3036	2
C₅×D₄×M₁₆	2
C₅×#1701	2
C₅×#1697	2
C₅×#1691	2
C₅×#1687	2
C₅×#1681	2
C₅×#1677	2
C₅×#1671	2
C₅×#1667	2
C₅×#1569	2
C₅×#1560	2
C₅×#1557	2
#19306	5

Subgroups — C5×#19306 #11942026

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19306 #11942026