Subgroups — C₅×#19304 #11942024

Order 1280.

710 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3182	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3110	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3036	C₂	1	1	yes	maximal, characteristic
4	C₅×D₄×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1701	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1699	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1693	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1685	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1681	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#1679	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#1673	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#1665	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅×#1569	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×#1561	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×#533	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×#500	C₂²	1	1	yes	characteristic, complemented
22	C₅×#492	C₂²	1	1	yes	characteristic, complemented
23	C₅×#487	C₂²	1	1	yes	characteristic, complemented
24	C₅×#486	C₂²	1	1	yes	characteristic, complemented
25	C₅×#484	—	2	1	no	—
26	C₅×#474	C₂²	1	1	yes	characteristic
27	C₅×#474	C₂²	1	1	yes	characteristic
28	C₅×#468	—	2	1	no	—
29	C₅×#468	—	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×#461	C₂²	1	1	yes	characteristic
35	C₅×#461	C₂²	1	1	yes	characteristic
36	C₅×#459	C₂²	1	1	yes	characteristic
37	C₅×#459	C₂²	1	1	yes	characteristic
38	C₅×#450	—	2	1	no	—
39	C₅×#440	—	2	1	no	—
40	C₅×#440	—	2	1	no	—
41	C₅×#440	—	2	1	no	—
42	C₅×#440	—	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×#332	C₂²	1	1	yes	characteristic, complemented
60	C₅×#332	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×#332	C₂²	1	1	yes	characteristic, complemented
64	C₅×#332	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	#19304	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂×C₅×#41	—	2	1	no	—
69	C₂²×C₅×D₄	D₄	1	1	yes	characteristic
70	C₂³×C₄×C₅	—	2	1	no	—
71	C₅×AGL(1,ℤ/8)	—	4	4	no	—
72	C₂×C₅×SD₁₆	—	2	1	no	—
73	C₂×C₅×SD₁₆	—	2	1	no	—
74	C₂×C₅×SD₁₆	—	2	1	no	—
75	C₂×C₅×SD₁₆	—	2	1	no	—
76	C₂×C₅×SD₁₆	—	2	1	no	—
77	C₂×C₅×SD₁₆	—	2	1	no	—
78	C₂×C₅×D₈	—	2	1	no	—
79	C₂×C₅×D₈	—	2	1	no	—
80	C₂×C₅×M₁₆	—	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	—	4	4	no	—
85	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
86	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
87	C₂²×C₅×C₈	—	2	1	no	—
88	C₂²×C₅×C₈	—	2	1	no	—
89	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
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×#125	—	2	1	no	—
94	C₅×#124	—	2	1	no	—
95	C₅×C₂²⋊Q₈	—	2	1	no	—
96	C₅×C₂²⋊Q₈	—	2	1	no	—
97	C₅×#121	—	2	1	no	—
98	C₅×#121	—	2	1	no	—
99	C₄×C₅×Q₈	—	2	1	no	—
100	C₄×C₅×Q₈	—	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	D₄	1	1	yes	characteristic
106	C₄×C₅×D₄	D₄	1	1	yes	characteristic
107	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
108	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
109	C₅×#117	—	2	1	no	—
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×#107	—	2	1	no	—
114	C₅×#107	—	2	1	no	—
115	C₅×#107	—	2	1	no	—
116	C₅×#107	—	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×#103	—	2	1	no	—
128	C₅×#103	—	2	1	no	—
129	C₅×#103	—	2	1	no	—
130	C₅×#103	—	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	#3182	C₂×C₅	1	1	yes	characteristic, complemented
148	#3110	C₂×C₅	1	1	yes	characteristic, complemented
149	#3036	C₂×C₅	1	1	yes	characteristic
150	D₄×M₁₆	C₂×C₅	1	1	yes	characteristic, complemented
151	#1701	C₂×C₅	1	1	yes	characteristic, complemented
152	#1699	C₂×C₅	1	1	yes	characteristic, complemented
153	#1693	C₂×C₅	1	1	yes	characteristic, complemented
154	#1685	C₂×C₅	1	1	yes	characteristic, complemented
155	#1681	C₂×C₅	1	1	yes	characteristic, complemented
156	#1679	C₂×C₅	1	1	yes	characteristic, complemented
157	#1673	C₂×C₅	1	1	yes	characteristic, complemented
158	#1665	C₂×C₅	1	1	yes	characteristic, complemented
159	#1569	C₂×C₅	1	1	yes	characteristic, complemented
160	#1561	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	C₂×C₅×Q₈	—	2	1	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	—	4	1	no	—
174	C₂×C₅×D₄	—	4	1	no	—
175	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
176	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₅	—	2	1	no	—
178	C₂²×C₄×C₅	—	2	1	no	—
179	C₂²×C₄×C₅	—	2	1	no	—
180	C₂²×C₄×C₅	—	2	1	no	—
181	C₂²×C₄×C₅	—	2	1	no	—
182	C₂²×C₄×C₅	—	4	2	no	—
183	C₂²×C₄×C₅	—	4	2	no	—
184	C₂²×C₄×C₅	—	4	1	no	—
185	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
186	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
187	C₂²×C₄×C₅	#41	1	1	yes	characteristic
188	C₅×SD₁₆	—	4	2	no	—
189	C₅×SD₁₆	—	4	2	no	—
190	C₅×SD₁₆	—	4	2	no	—
191	C₅×SD₁₆	—	4	2	no	—
192	C₅×SD₁₆	—	4	2	no	—
193	C₅×SD₁₆	—	4	2	no	—
194	C₅×D₈	—	4	2	no	—
195	C₅×D₈	—	4	2	no	—
196	C₅×M₁₆	—	2	2	no	—
197	C₅×M₁₆	—	4	1	no	—
198	C₅×M₁₆	—	4	2	no	—
199	C₅×M₁₆	—	4	2	no	—
200	C₅×M₁₆	—	4	2	no	—
201	C₅×M₁₆	—	4	2	no	—
202	C₂×C₅×C₈	—	2	1	no	—
203	C₂×C₅×C₈	—	2	1	no	—
204	C₂×C₅×C₈	—	2	1	no	—
205	C₂×C₅×C₈	—	2	1	no	—
206	C₂×C₅×C₈	—	2	1	no	—
207	C₂×C₅×C₈	—	2	1	no	—
208	C₂×C₅×C₈	—	4	2	no	—
209	C₂×C₅×C₈	—	4	2	no	—
210	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
211	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
212	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
213	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
214	C₅×C₄⋊C₄	—	2	1	no	—
215	C₅×C₄⋊C₄	—	2	1	no	—
216	C₅×C₄⋊C₄	—	2	1	no	—
217	C₅×C₄⋊C₄	—	2	1	no	—
218	C₅×C₄⋊C₄	—	2	1	no	—
219	C₅×C₄⋊C₄	—	2	1	no	—
220	C₅×C₄⋊C₄	—	2	1	no	—
221	C₅×C₄⋊C₄	—	4	1	no	—
222	C₅×C₄⋊C₄	—	4	1	no	—
223	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
224	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
225	C₅×C₂²⋊C₄	—	2	1	no	—
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	—	4	1	no	—
230	C₅×C₂²⋊C₄	—	4	1	no	—
231	C₅×C₂²⋊C₄	—	4	1	no	—
232	C₄²×C₅	—	4	1	no	—
233	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
234	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
235	C₄²×C₅	#41	1	1	yes	characteristic
236	C₄²×C₅	C₂⁴	1	1	yes	characteristic
237	C₂×AGL(1,ℤ/8)	—	2	1	no	—
238	C₂²×M₁₆	—	2	1	no	—
239	#533	C₂²×C₅	1	1	yes	characteristic
240	#532	C₂²×C₅	1	1	yes	characteristic
241	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
242	#500	C₂²×C₅	1	1	yes	characteristic, complemented
243	#492	C₂²×C₅	1	1	yes	characteristic, complemented
244	#487	C₂²×C₅	1	1	yes	characteristic, complemented
245	#486	C₂²×C₅	1	1	yes	characteristic, complemented
246	#484	—	2	1	no	—
247	#474	C₂²×C₅	1	1	yes	characteristic
248	#474	C₂²×C₅	1	1	yes	characteristic
249	#468	—	2	1	no	—
250	#468	—	2	1	no	—
251	#463	—	2	1	no	—
252	#463	—	2	1	no	—
253	#461	C₂²×C₅	1	1	yes	characteristic
254	#461	C₂²×C₅	1	1	yes	characteristic
255	#461	C₂²×C₅	1	1	yes	characteristic
256	#461	C₂²×C₅	1	1	yes	characteristic
257	#459	C₂²×C₅	1	1	yes	characteristic
258	#459	C₂²×C₅	1	1	yes	characteristic
259	#450	—	2	1	no	—
260	#440	—	2	1	no	—
261	#440	—	2	1	no	—
262	#440	—	2	1	no	—
263	#440	—	2	1	no	—
264	#436	C₂²×C₅	1	1	yes	characteristic
265	#436	C₂²×C₅	1	1	yes	characteristic
266	#435	—	2	1	no	—
267	#435	—	2	1	no	—
268	C₈×D₄	C₂²×C₅	1	1	yes	characteristic
269	C₈×D₄	C₂²×C₅	1	1	yes	characteristic
270	#432	C₂²×C₅	1	1	yes	characteristic
271	#423	C₂²×C₅	1	1	yes	characteristic
272	#423	C₂²×C₅	1	1	yes	characteristic
273	#417	—	2	1	no	—
274	#407	—	2	1	no	—
275	C₄×M₁₆	C₂²×C₅	1	1	yes	characteristic
276	#335	C₂²×C₅	1	1	yes	characteristic, complemented
277	#335	C₂²×C₅	1	1	yes	characteristic, complemented
278	#334	C₂²×C₅	1	1	yes	characteristic, complemented
279	#334	C₂²×C₅	1	1	yes	characteristic, complemented
280	#332	C₂²×C₅	1	1	yes	characteristic, complemented
281	#332	C₂²×C₅	1	1	yes	characteristic, complemented
282	#332	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	#332	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	C₂³×C₅	—	2	1	no	—
289	C₂³×C₅	—	2	1	no	—
290	C₂³×C₅	—	4	2	no	—
291	C₂³×C₅	—	4	2	no	—
292	C₂³×C₅	—	4	1	no	—
293	C₂³×C₅	—	8	1	no	—
294	C₂³×C₅	#121	1	1	yes	characteristic
295	C₅×Q₈	—	4	1	no	—
296	C₅×Q₈	—	4	1	no	—
297	C₅×Q₈	—	4	1	no	—
298	C₅×Q₈	—	4	1	no	—
299	C₅×Q₈	—	4	1	no	—
300	C₅×Q₈	—	4	1	no	—
301	C₅×D₄	—	2	1	no	—
302	C₅×D₄	—	2	1	no	—
303	C₅×D₄	—	2	1	no	—
304	C₅×D₄	—	4	2	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	—	8	1	no	—
310	C₅×D₄	—	8	2	no	—
311	C₅×D₄	AGL(1,ℤ/8)	1	2	yes	—
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	1	no	—
316	C₂×C₄×C₅	—	2	1	no	—
317	C₂×C₄×C₅	—	2	1	no	—
318	C₂×C₄×C₅	—	4	1	no	—
319	C₂×C₄×C₅	—	4	2	no	—
320	C₂×C₄×C₅	—	4	2	no	—
321	C₂×C₄×C₅	—	4	2	no	—
322	C₂×C₄×C₅	—	4	1	no	—
323	C₂×C₄×C₅	—	4	1	no	—
324	C₂×C₄×C₅	—	4	1	no	—
325	C₂×C₄×C₅	—	4	1	no	—
326	C₂×C₄×C₅	—	4	1	no	—
327	C₂×C₄×C₅	—	4	1	no	—
328	C₂×C₄×C₅	—	4	1	no	—
329	C₂×C₄×C₅	—	8	2	no	—
330	C₂×C₄×C₅	#121	1	1	yes	characteristic
331	C₂×C₄×C₅	#121	1	1	yes	characteristic
332	C₂×C₄×C₅	#121	1	1	yes	characteristic
333	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
334	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
335	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
336	C₅×C₈	—	2	1	no	—
337	C₅×C₈	—	2	1	no	—
338	C₅×C₈	—	2	1	no	—
339	C₅×C₈	—	4	1	no	—
340	C₅×C₈	—	4	1	no	—
341	C₅×C₈	—	4	1	no	—
342	C₅×C₈	—	4	1	no	—
343	C₅×C₈	—	4	1	no	—
344	C₅×C₈	—	4	1	no	—
345	C₅×C₈	AGL(1,ℤ/8)	1	2	yes	complemented
346	C₂×#41	—	2	1	no	—
347	C₂²×D₄	C₅×D₄	1	1	yes	characteristic
348	C₂³×C₄	—	2	1	no	—
349	AGL(1,ℤ/8)	—	4	4	no	—
350	C₂×SD₁₆	—	2	1	no	—
351	C₂×SD₁₆	—	2	1	no	—
352	C₂×SD₁₆	—	2	1	no	—
353	C₂×SD₁₆	—	2	1	no	—
354	C₂×SD₁₆	—	2	1	no	—
355	C₂×SD₁₆	—	2	1	no	—
356	C₂×D₈	—	2	1	no	—
357	C₂×D₈	—	2	1	no	—
358	C₂×M₁₆	—	2	1	no	—
359	C₂×M₁₆	—	2	1	no	—
360	C₂×M₁₆	—	2	1	no	—
361	C₂×M₁₆	—	2	1	no	—
362	C₂×M₁₆	—	4	4	no	—
363	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
364	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
365	C₂²×C₈	—	2	1	no	—
366	C₂²×C₈	—	2	1	no	—
367	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
368	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
369	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
370	#127	C₂³×C₅	1	1	yes	characteristic
371	#125	—	2	1	no	—
372	#124	—	2	1	no	—
373	C₂²⋊Q₈	—	2	1	no	—
374	C₂²⋊Q₈	—	2	1	no	—
375	#121	—	2	1	no	—
376	#121	—	2	1	no	—
377	C₄×Q₈	—	2	1	no	—
378	C₄×Q₈	—	2	1	no	—
379	C₄×D₄	—	2	1	no	—
380	C₄×D₄	—	2	1	no	—
381	C₄×D₄	—	2	1	no	—
382	C₄×D₄	—	2	1	no	—
383	C₄×D₄	C₅×D₄	1	1	yes	characteristic
384	C₄×D₄	C₅×D₄	1	1	yes	characteristic
385	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
386	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
387	#117	—	2	1	no	—
388	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
389	C₂×C₂²⋊C₄	—	2	1	no	—
390	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
391	#107	—	2	1	no	—
392	#107	—	2	1	no	—
393	#107	—	2	1	no	—
394	#107	—	2	1	no	—
395	C₄⋊C₈	—	2	1	no	—
396	C₄⋊C₈	—	2	1	no	—
397	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
398	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
399	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
400	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
401	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
402	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
403	#103	—	2	1	no	—
404	#103	—	2	1	no	—
405	#103	—	2	1	no	—
406	#103	—	2	1	no	—
407	#103	—	2	1	no	—
408	#103	—	2	1	no	—
409	#102	—	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	C₂²⋊C₈	—	2	1	no	—
416	C₂²⋊C₈	—	2	1	no	—
417	C₂²⋊C₈	—	2	1	no	—
418	C₂²⋊C₈	—	2	1	no	—
419	C₂²⋊C₈	—	2	1	no	—
420	#97	—	2	1	no	—
421	#97	C₅×D₄	1	1	yes	characteristic
422	#97	C₅×D₄	1	1	yes	characteristic
423	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
424	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
425	C₂²×C₅	—	2	1	no	—
426	C₂²×C₅	—	2	1	no	—
427	C₂²×C₅	—	2	2	no	—
428	C₂²×C₅	—	2	1	no	—
429	C₂²×C₅	—	4	1	no	—
430	C₂²×C₅	—	4	1	no	—
431	C₂²×C₅	—	4	1	no	—
432	C₂²×C₅	—	4	1	no	—
433	C₂²×C₅	—	4	1	no	—
434	C₂²×C₅	—	8	4	no	—
435	C₂²×C₅	—	8	1	no	—
436	C₂²×C₅	—	8	1	no	—
437	C₂²×C₅	—	8	1	no	—
438	C₂²×C₅	C₂×#121	1	1	yes	characteristic
439	C₄×C₅	—	2	1	no	—
440	C₄×C₅	—	2	1	no	—
441	C₄×C₅	—	2	1	no	—
442	C₄×C₅	—	2	1	no	—
443	C₄×C₅	—	4	1	no	—
444	C₄×C₅	—	4	1	no	—
445	C₄×C₅	—	8	1	no	—
446	C₄×C₅	—	8	1	no	—
447	C₄×C₅	—	8	1	no	—
448	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
449	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
450	C₄×C₅	#575	1	1	yes	characteristic
451	C₄×C₅	#575	1	1	yes	characteristic
452	C₂⁴	—	2	1	no	—
453	#41	—	4	2	no	—
454	#41	—	4	2	no	—
455	C₂×Q₈	—	2	1	no	—
456	C₂×Q₈	—	2	1	no	—
457	C₂×Q₈	—	2	1	no	—
458	C₂×D₄	—	2	2	no	—
459	C₂×D₄	—	2	1	no	—
460	C₂×D₄	—	2	1	no	—
461	C₂×D₄	—	2	1	no	—
462	C₂×D₄	—	2	1	no	—
463	C₂×D₄	—	4	1	no	—
464	C₂×D₄	—	4	1	no	—
465	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
466	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
467	C₂²×C₄	—	2	1	no	—
468	C₂²×C₄	—	2	1	no	—
469	C₂²×C₄	—	2	1	no	—
470	C₂²×C₄	—	2	1	no	—
471	C₂²×C₄	—	2	1	no	—
472	C₂²×C₄	—	4	2	no	—
473	C₂²×C₄	—	4	2	no	—
474	C₂²×C₄	—	4	1	no	—
475	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
476	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
477	C₂²×C₄	C₅×#41	1	1	yes	characteristic
478	SD₁₆	—	4	2	no	—
479	SD₁₆	—	4	2	no	—
480	SD₁₆	—	4	2	no	—
481	SD₁₆	—	4	2	no	—
482	SD₁₆	—	4	2	no	—
483	SD₁₆	—	4	2	no	—
484	D₈	—	4	2	no	—
485	D₈	—	4	2	no	—
486	M₁₆	—	2	2	no	—
487	M₁₆	—	4	1	no	—
488	M₁₆	—	4	2	no	—
489	M₁₆	—	4	2	no	—
490	M₁₆	—	4	2	no	—
491	M₁₆	—	4	2	no	—
492	C₂×C₈	—	2	1	no	—
493	C₂×C₈	—	2	1	no	—
494	C₂×C₈	—	2	1	no	—
495	C₂×C₈	—	2	1	no	—
496	C₂×C₈	—	2	1	no	—
497	C₂×C₈	—	2	1	no	—
498	C₂×C₈	—	4	2	no	—
499	C₂×C₈	—	4	2	no	—
500	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
501	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
502	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
503	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
504	C₄⋊C₄	—	2	1	no	—
505	C₄⋊C₄	—	2	1	no	—
506	C₄⋊C₄	—	2	1	no	—
507	C₄⋊C₄	—	2	1	no	—
508	C₄⋊C₄	—	2	1	no	—
509	C₄⋊C₄	—	2	1	no	—
510	C₄⋊C₄	—	2	1	no	—
511	C₄⋊C₄	—	4	1	no	—
512	C₄⋊C₄	—	4	1	no	—
513	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
514	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
515	C₂²⋊C₄	—	2	1	no	—
516	C₂²⋊C₄	—	2	1	no	—
517	C₂²⋊C₄	—	2	1	no	—
518	C₂²⋊C₄	—	2	1	no	—
519	C₂²⋊C₄	—	4	1	no	—
520	C₂²⋊C₄	—	4	1	no	—
521	C₂²⋊C₄	—	4	1	no	—
522	C₄²	—	4	1	no	—
523	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
524	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
525	C₄²	C₅×#41	1	1	yes	characteristic
526	C₄²	C₂⁴×C₅	1	1	yes	characteristic
527	C₂×C₅	—	4	1	no	—
528	C₂×C₅	—	4	2	no	—
529	C₂×C₅	—	8	1	no	—
530	C₂×C₅	—	16	1	no	—
531	C₂×C₅	#3036	1	1	yes	characteristic
532	C₂×C₅	#3055	1	1	yes	characteristic
533	C₂×C₅	#3069	1	1	yes	characteristic
534	C₂³	—	2	1	no	—
535	C₂³	—	2	1	no	—
536	C₂³	—	4	2	no	—
537	C₂³	—	4	2	no	—
538	C₂³	—	4	1	no	—
539	C₂³	—	8	1	no	—
540	C₂³	C₅×#121	1	1	yes	characteristic
541	Q₈	—	4	1	no	—
542	Q₈	—	4	1	no	—
543	Q₈	—	4	1	no	—
544	Q₈	—	4	1	no	—
545	Q₈	—	4	1	no	—
546	Q₈	—	4	1	no	—
547	D₄	—	2	1	no	—
548	D₄	—	2	1	no	—
549	D₄	—	2	1	no	—
550	D₄	—	4	2	no	—
551	D₄	—	4	1	no	—
552	D₄	—	4	1	no	—
553	D₄	—	4	1	no	—
554	D₄	—	4	1	no	—
555	D₄	—	8	1	no	—
556	D₄	—	8	2	no	—
557	D₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
558	C₂×C₄	—	2	1	no	—
559	C₂×C₄	—	2	1	no	—
560	C₂×C₄	—	2	1	no	—
561	C₂×C₄	—	2	1	no	—
562	C₂×C₄	—	2	1	no	—
563	C₂×C₄	—	2	1	no	—
564	C₂×C₄	—	4	1	no	—
565	C₂×C₄	—	4	2	no	—
566	C₂×C₄	—	4	2	no	—
567	C₂×C₄	—	4	2	no	—
568	C₂×C₄	—	4	1	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	—	4	1	no	—
571	C₂×C₄	—	4	1	no	—
572	C₂×C₄	—	4	1	no	—
573	C₂×C₄	—	4	1	no	—
574	C₂×C₄	—	4	1	no	—
575	C₂×C₄	—	8	2	no	—
576	C₂×C₄	C₅×#121	1	1	yes	characteristic
577	C₂×C₄	C₅×#121	1	1	yes	characteristic
578	C₂×C₄	C₅×#121	1	1	yes	characteristic
579	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
580	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
581	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
582	C₈	—	2	1	no	—
583	C₈	—	2	1	no	—
584	C₈	—	2	1	no	—
585	C₈	—	4	1	no	—
586	C₈	—	4	1	no	—
587	C₈	—	4	1	no	—
588	C₈	—	4	1	no	—
589	C₈	—	4	1	no	—
590	C₈	—	4	1	no	—
591	C₈	C₅×AGL(1,ℤ/8)	1	2	yes	complemented
592	C₅	#19304	1	1	yes	characteristic, direct product factor, Sylow
593	C₂²	—	2	1	no	—
594	C₂²	—	2	1	no	—
595	C₂²	—	2	2	no	—
596	C₂²	—	2	1	no	—
597	C₂²	—	4	1	no	—
598	C₂²	—	4	1	no	—
599	C₂²	—	4	1	no	—
600	C₂²	—	4	1	no	—
601	C₂²	—	4	1	no	—
602	C₂²	—	8	4	no	—
603	C₂²	—	8	1	no	—
604	C₂²	—	8	1	no	—
605	C₂²	—	8	1	no	—
606	C₂²	C₂×C₅×#121	1	1	yes	characteristic
607	C₄	—	2	1	no	—
608	C₄	—	2	1	no	—
609	C₄	—	2	1	no	—
610	C₄	—	2	1	no	—
611	C₄	—	4	1	no	—
612	C₄	—	4	1	no	—
613	C₄	—	8	1	no	—
614	C₄	—	8	1	no	—
615	C₄	—	8	1	no	—
616	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
617	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
618	C₄	C₅×#575	1	1	yes	characteristic
619	C₄	C₅×#575	1	1	yes	characteristic
620	C₂	—	4	1	no	—
621	C₂	—	4	2	no	—
622	C₂	—	8	1	no	—
623	C₂	—	16	1	no	—
624	C₂	C₅×#3036	1	1	yes	characteristic
625	C₂	C₅×#3055	1	1	yes	characteristic
626	C₂	C₅×#3069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₅×#3036	C₂
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×#3182	C₂	semidirect
2. C₅×#3110	C₂	semidirect
4. C₅×D₄×M₁₆	C₂	semidirect
5. C₅×#1701	C₂	semidirect
6. C₅×#1699	C₂	semidirect
7. C₅×#1693	C₂	semidirect
8. C₅×#1685	C₂	semidirect
9. C₅×#1681	C₂	semidirect
10. C₅×#1679	C₂	semidirect
11. C₅×#1673	C₂	semidirect
12. C₅×#1665	C₂	semidirect
13. C₅×#1569	C₂	semidirect
14. C₅×#1561	C₂	semidirect
15. C₅×#1557	C₂	semidirect
21. C₅×#500	C₂²	semidirect
22. C₅×#492	C₂²	semidirect
23. C₅×#487	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×#332	C₂²	semidirect
60. C₅×#332	C₂²	semidirect
61. C₅×#332	C₂²	semidirect
62. C₅×#332	C₂²	semidirect
63. C₅×#332	C₂²	semidirect
64. C₅×#332	C₂²	semidirect
65. C₅×#331	C₂²	semidirect
66. C₅×#331	C₂²	semidirect
67. #19304	C₅	direct (N × H)
147. #3182	C₂×C₅	semidirect
148. #3110	C₂×C₅	semidirect
150. D₄×M₁₆	C₂×C₅	semidirect
151. #1701	C₂×C₅	semidirect
152. #1699	C₂×C₅	semidirect
153. #1693	C₂×C₅	semidirect
154. #1685	C₂×C₅	semidirect
155. #1681	C₂×C₅	semidirect
156. #1679	C₂×C₅	semidirect
157. #1673	C₂×C₅	semidirect
158. #1665	C₂×C₅	semidirect
159. #1569	C₂×C₅	semidirect
160. #1561	C₂×C₅	semidirect
161. #1557	C₂×C₅	semidirect
242. #500	C₂²×C₅	semidirect
243. #492	C₂²×C₅	semidirect
244. #487	C₂²×C₅	semidirect
245. #486	C₂²×C₅	semidirect
276. #335	C₂²×C₅	semidirect
277. #335	C₂²×C₅	semidirect
278. #334	C₂²×C₅	semidirect
279. #334	C₂²×C₅	semidirect
280. #332	C₂²×C₅	semidirect
281. #332	C₂²×C₅	semidirect
282. #332	C₂²×C₅	semidirect
283. #332	C₂²×C₅	semidirect
284. #332	C₂²×C₅	semidirect
285. #332	C₂²×C₅	semidirect
286. #331	C₂²×C₅	semidirect
287. #331	C₂²×C₅	semidirect
345. C₅×C₈	AGL(1,ℤ/8)	semidirect
591. C₈	C₅×AGL(1,ℤ/8)	semidirect
592. C₅	#19304	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×#3182	2
C₅×#3110	2
C₅×#3036	2
C₅×D₄×M₁₆	2
C₅×#1701	2
C₅×#1699	2
C₅×#1693	2
C₅×#1685	2
C₅×#1681	2
C₅×#1679	2
C₅×#1673	2
C₅×#1665	2
C₅×#1569	2
C₅×#1561	2
C₅×#1557	2
#19304	5

Subgroups — C5×#19304 #11942024

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19304 #11942024