Subgroups — C₅×#19763 #11942483

Order 1280.

756 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3384	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3295	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×D₄×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×D₄×D₈	C₂	1	1	yes	maximal, characteristic
5	C₅×#1715	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1688	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1677	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1671	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1665	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#1639	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#1629	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#1627	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅×#1622	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×#1582	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅×#1580	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₅×SD₁₆	—	2	1	no	—
17	C₂²×C₅×D₈	—	2	1	no	—
18	C₅×#554	C₂²	1	1	yes	characteristic, complemented
19	C₅×D₄×Q₈	C₂²	1	1	yes	characteristic, complemented
20	C₅×D₄²	C₂²	1	1	yes	characteristic
21	C₅×D₄²	C₂²	1	1	yes	characteristic
22	C₅×#501	—	2	1	no	—
23	C₅×#498	C₂²	1	1	yes	characteristic, complemented
24	C₅×#493	C₂²	1	1	yes	characteristic
25	C₅×#492	C₂²	1	1	yes	characteristic, complemented
26	C₅×#492	C₂²	1	1	yes	characteristic, complemented
27	C₅×#486	C₂²	1	1	yes	characteristic, complemented
28	C₅×#486	C₂²	1	1	yes	characteristic, complemented
29	C₅×#481	—	2	1	no	—
30	C₅×#478	—	2	1	no	—
31	C₅×#476	—	2	1	no	—
32	C₅×#476	C₂²	1	1	yes	characteristic
33	C₅×#476	C₂²	1	1	yes	characteristic, complemented
34	C₅×#474	C₂²	1	1	yes	characteristic, complemented
35	C₅×#466	—	2	1	no	—
36	C₅×#465	—	2	1	no	—
37	C₅×#465	—	2	1	no	—
38	C₅×#461	C₂²	1	1	yes	characteristic
39	C₅×#460	C₂²	1	1	yes	characteristic, complemented
40	C₅×#460	C₂²	1	1	yes	characteristic, complemented
41	C₅×#459	C₂²	1	1	yes	characteristic
42	C₅×#459	C₂²	1	1	yes	characteristic
43	C₅×#450	—	2	1	no	—
44	C₅×#450	—	2	1	no	—
45	C₅×#448	—	2	1	no	—
46	C₅×#447	—	2	1	no	—
47	C₅×#447	—	2	1	no	—
48	C₅×#446	C₂²	1	1	yes	characteristic, complemented
49	C₅×#442	—	2	1	no	—
50	C₄×C₅×SD₁₆	C₂²	1	1	yes	characteristic
51	C₄×C₅×D₈	C₂²	1	1	yes	characteristic
52	C₅×C₈×D₄	C₂²	1	1	yes	characteristic
53	C₅×C₈×D₄	C₂²	1	1	yes	characteristic, complemented
54	C₅×C₈×D₄	C₂²	1	1	yes	characteristic, complemented
55	C₂×C₅×#102	—	2	1	no	—
56	C₅×#334	C₂²	1	1	yes	characteristic, complemented
57	C₅×#334	C₂²	1	1	yes	characteristic, complemented
58	C₅×#333	C₂²	1	1	yes	characteristic, complemented
59	C₅×#332	C₂²	1	1	yes	characteristic, complemented
60	C₅×#331	C₂²	1	1	yes	characteristic, complemented
61	C₅×#331	C₂²	1	1	yes	characteristic, complemented
62	C₅×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
63	C₅×D₄⋊C₈	C₂²	1	1	yes	characteristic
64	C₅×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
65	C₅×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
66	C₅×#322	C₂²	1	1	yes	characteristic, complemented
67	#19763	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂²×C₅×Q₈	—	2	1	no	—
69	C₂²×C₅×D₄	—	2	1	no	—
70	C₂²×C₅×D₄	—	2	1	no	—
71	C₂²×C₅×D₄	—	2	1	no	—
72	C₂²×C₅×D₄	—	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×SD₁₆	—	4	1	no	—
79	C₂×C₅×SD₁₆	—	4	4	no	—
80	C₂×C₅×SD₁₆	D₄	1	1	yes	characteristic
81	C₂×C₅×D₈	—	2	1	no	—
82	C₂×C₅×D₈	—	2	1	no	—
83	C₂×C₅×D₈	—	2	1	no	—
84	C₂×C₅×D₈	—	2	1	no	—
85	C₂×C₅×D₈	—	4	4	no	—
86	C₂×C₅×D₈	D₄	1	1	yes	characteristic
87	C₂²×C₅×C₈	—	2	1	no	—
88	C₂²×C₅×C₈	—	2	1	no	—
89	C₂²×C₅×C₈	—	2	1	no	—
90	C₅×C₄⋊Q₈	—	2	1	no	—
91	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
92	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
93	C₅×#127	C₂³	1	1	yes	characteristic
94	C₅×#127	C₂³	1	1	yes	characteristic
95	C₅×#125	—	2	1	no	—
96	C₅×C₂²⋊Q₈	—	2	1	no	—
97	C₅×C₂²⋊Q₈	—	2	1	no	—
98	C₅×C₂²⋊Q₈	—	4	1	no	—
99	C₅×C₂²⋊Q₈	—	4	1	no	—
100	C₅×#121	—	2	1	no	—
101	C₅×#121	—	2	1	no	—
102	C₅×#121	—	2	1	no	—
103	C₅×#121	—	2	1	no	—
104	C₅×#120	—	4	1	no	—
105	C₅×#120	—	4	1	no	—
106	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
107	C₄×C₅×D₄	—	2	1	no	—
108	C₄×C₅×D₄	—	2	1	no	—
109	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
110	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
111	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
112	C₂×C₅×C₄⋊C₄	—	2	1	no	—
113	C₅×#107	—	2	1	no	—
114	C₅×#107	D₄	1	1	yes	characteristic
115	C₅×#106	—	2	1	no	—
116	C₅×#106	—	2	1	no	—
117	C₅×#106	—	2	1	no	—
118	C₅×#106	—	2	1	no	—
119	C₅×#106	D₄	1	1	yes	characteristic
120	C₅×C₄⋊C₈	—	2	1	no	—
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	D₄	1	1	yes	characteristic
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	—	4	1	no	—
137	C₅×#102	D₄	1	1	yes	characteristic
138	C₅×#102	D₄	1	1	yes	characteristic
139	C₅×#102	D₄	1	1	yes	characteristic
140	C₅×C₂²⋊C₈	—	2	1	no	—
141	C₅×C₂²⋊C₈	—	2	1	no	—
142	C₅×C₂²⋊C₈	—	2	1	no	—
143	C₅×#97	—	2	1	no	—
144	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
145	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
146	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
147	#3384	C₂×C₅	1	1	yes	characteristic, complemented
148	#3295	C₂×C₅	1	1	yes	characteristic, complemented
149	D₄×SD₁₆	C₂×C₅	1	1	yes	characteristic, complemented
150	D₄×D₈	C₂×C₅	1	1	yes	characteristic
151	#1715	C₂×C₅	1	1	yes	characteristic, complemented
152	#1688	C₂×C₅	1	1	yes	characteristic, complemented
153	#1677	C₂×C₅	1	1	yes	characteristic, complemented
154	#1671	C₂×C₅	1	1	yes	characteristic, complemented
155	#1665	C₂×C₅	1	1	yes	characteristic, complemented
156	#1639	C₂×C₅	1	1	yes	characteristic, complemented
157	#1629	C₂×C₅	1	1	yes	characteristic, complemented
158	#1627	C₂×C₅	1	1	yes	characteristic, complemented
159	#1622	C₂×C₅	1	1	yes	characteristic, complemented
160	#1582	C₂×C₅	1	1	yes	characteristic, complemented
161	#1580	C₂×C₅	1	1	yes	characteristic, complemented
162	C₂⁴×C₅	—	4	1	no	—
163	C₂⁴×C₅	—	4	1	no	—
164	C₂×C₅×Q₈	—	2	1	no	—
165	C₂×C₅×Q₈	—	2	1	no	—
166	C₂×C₅×Q₈	—	4	1	no	—
167	C₂×C₅×Q₈	—	4	2	no	—
168	C₂×C₅×Q₈	—	8	1	no	—
169	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
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	—	4	1	no	—
180	C₂×C₅×D₄	—	4	2	no	—
181	C₂×C₅×D₄	—	4	2	no	—
182	C₂×C₅×D₄	—	4	2	no	—
183	C₂×C₅×D₄	—	4	2	no	—
184	C₂×C₅×D₄	—	8	1	no	—
185	C₂×C₅×D₄	—	8	1	no	—
186	C₂×C₅×D₄	—	8	1	no	—
187	C₂×C₅×D₄	—	8	1	no	—
188	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
189	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
190	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
191	C₂²×C₄×C₅	—	2	1	no	—
192	C₂²×C₄×C₅	—	2	1	no	—
193	C₂²×C₄×C₅	—	2	1	no	—
194	C₂²×C₄×C₅	—	4	1	no	—
195	C₂²×C₄×C₅	—	4	1	no	—
196	C₅×SD₁₆	—	2	2	no	—
197	C₅×SD₁₆	—	4	2	no	—
198	C₅×SD₁₆	—	4	2	no	—
199	C₅×SD₁₆	—	4	2	no	—
200	C₅×SD₁₆	—	8	1	no	—
201	C₅×SD₁₆	—	8	1	no	—
202	C₅×SD₁₆	—	8	2	no	—
203	C₅×D₈	—	4	2	no	—
204	C₅×D₈	—	4	2	no	—
205	C₅×D₈	—	4	2	no	—
206	C₅×D₈	—	8	1	no	—
207	C₅×D₈	SD₁₆	1	4	yes	complemented
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	—	8	1	no	—
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	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
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	—	2	1	no	—
225	C₅×C₄⋊C₄	—	4	1	no	—
226	C₅×C₄⋊C₄	—	4	1	no	—
227	C₅×C₄⋊C₄	—	4	1	no	—
228	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
229	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
230	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
231	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
232	C₅×C₂²⋊C₄	—	2	1	no	—
233	C₅×C₂²⋊C₄	—	2	1	no	—
234	C₅×C₂²⋊C₄	—	2	1	no	—
235	C₅×C₂²⋊C₄	—	4	1	no	—
236	C₅×C₂²⋊C₄	—	4	1	no	—
237	C₅×C₂²⋊C₄	—	4	1	no	—
238	C₅×C₂²⋊C₄	—	4	1	no	—
239	C₄²×C₅	—	2	1	no	—
240	C₄²×C₅	C₂⁴	1	1	yes	characteristic
241	C₂²×SD₁₆	—	2	1	no	—
242	C₂²×D₈	—	2	1	no	—
243	#554	C₂²×C₅	1	1	yes	characteristic, complemented
244	D₄×Q₈	C₂²×C₅	1	1	yes	characteristic, complemented
245	D₄²	C₂²×C₅	1	1	yes	characteristic
246	D₄²	C₂²×C₅	1	1	yes	characteristic
247	#501	—	2	1	no	—
248	#498	C₂²×C₅	1	1	yes	characteristic, complemented
249	#493	C₂²×C₅	1	1	yes	characteristic
250	#492	C₂²×C₅	1	1	yes	characteristic, complemented
251	#492	C₂²×C₅	1	1	yes	characteristic, complemented
252	#486	C₂²×C₅	1	1	yes	characteristic, complemented
253	#486	C₂²×C₅	1	1	yes	characteristic, complemented
254	#481	—	2	1	no	—
255	#478	—	2	1	no	—
256	#476	—	2	1	no	—
257	#476	C₂²×C₅	1	1	yes	characteristic
258	#476	C₂²×C₅	1	1	yes	characteristic, complemented
259	#474	C₂²×C₅	1	1	yes	characteristic, complemented
260	#466	—	2	1	no	—
261	#465	—	2	1	no	—
262	#465	—	2	1	no	—
263	#461	C₂²×C₅	1	1	yes	characteristic
264	#460	C₂²×C₅	1	1	yes	characteristic, complemented
265	#460	C₂²×C₅	1	1	yes	characteristic, complemented
266	#459	C₂²×C₅	1	1	yes	characteristic
267	#459	C₂²×C₅	1	1	yes	characteristic
268	#450	—	2	1	no	—
269	#450	—	2	1	no	—
270	#448	—	2	1	no	—
271	#447	—	2	1	no	—
272	#447	—	2	1	no	—
273	#446	C₂²×C₅	1	1	yes	characteristic, complemented
274	#442	—	2	1	no	—
275	C₄×SD₁₆	C₂²×C₅	1	1	yes	characteristic
276	C₄×D₈	C₂²×C₅	1	1	yes	characteristic
277	C₈×D₄	C₂²×C₅	1	1	yes	characteristic
278	C₈×D₄	C₂²×C₅	1	1	yes	characteristic, complemented
279	C₈×D₄	C₂²×C₅	1	1	yes	characteristic, complemented
280	C₂×#102	—	2	1	no	—
281	#334	C₂²×C₅	1	1	yes	characteristic, complemented
282	#334	C₂²×C₅	1	1	yes	characteristic, complemented
283	#333	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	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic, complemented
288	D₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
289	D₄⋊C₈	C₂²×C₅	1	1	yes	characteristic, complemented
290	D₄⋊C₈	C₂²×C₅	1	1	yes	characteristic, complemented
291	#322	C₂²×C₅	1	1	yes	characteristic, complemented
292	C₂³×C₅	—	2	1	no	—
293	C₂³×C₅	—	2	1	no	—
294	C₂³×C₅	—	2	1	no	—
295	C₂³×C₅	—	4	1	no	—
296	C₂³×C₅	—	4	1	no	—
297	C₂³×C₅	—	8	2	no	—
298	C₂³×C₅	—	8	2	no	—
299	C₂³×C₅	—	8	2	no	—
300	C₂³×C₅	—	8	2	no	—
301	C₂³×C₅	—	16	1	no	—
302	C₂³×C₅	—	16	1	no	—
303	C₅×Q₈	—	2	1	no	—
304	C₅×Q₈	—	2	1	no	—
305	C₅×Q₈	—	4	1	no	—
306	C₅×Q₈	—	4	1	no	—
307	C₅×Q₈	—	8	1	no	—
308	C₅×Q₈	—	8	2	no	—
309	C₅×D₄	—	2	1	no	—
310	C₅×D₄	—	2	1	no	—
311	C₅×D₄	—	2	1	no	—
312	C₅×D₄	—	4	1	no	—
313	C₅×D₄	—	4	1	no	—
314	C₅×D₄	—	4	1	no	—
315	C₅×D₄	—	4	1	no	—
316	C₅×D₄	—	4	1	no	—
317	C₅×D₄	—	4	1	no	—
318	C₅×D₄	—	4	1	no	—
319	C₅×D₄	—	4	1	no	—
320	C₅×D₄	—	8	1	no	—
321	C₅×D₄	—	8	1	no	—
322	C₅×D₄	—	8	2	no	—
323	C₅×D₄	—	8	2	no	—
324	C₅×D₄	—	8	2	no	—
325	C₅×D₄	—	8	2	no	—
326	C₅×D₄	C₂×SD₁₆	1	2	yes	—
327	C₅×D₄	C₂×SD₁₆	1	2	yes	complemented
328	C₅×D₄	AGL(1,ℤ/8)	1	2	yes	—
329	C₂×C₄×C₅	—	2	1	no	—
330	C₂×C₄×C₅	—	2	1	no	—
331	C₂×C₄×C₅	—	2	1	no	—
332	C₂×C₄×C₅	—	2	1	no	—
333	C₂×C₄×C₅	—	2	1	no	—
334	C₂×C₄×C₅	—	4	1	no	—
335	C₂×C₄×C₅	—	4	1	no	—
336	C₂×C₄×C₅	—	4	1	no	—
337	C₂×C₄×C₅	—	4	1	no	—
338	C₂×C₄×C₅	—	4	1	no	—
339	C₂×C₄×C₅	—	4	1	no	—
340	C₂×C₄×C₅	—	4	1	no	—
341	C₂×C₄×C₅	—	4	1	no	—
342	C₂×C₄×C₅	—	8	2	no	—
343	C₂×C₄×C₅	—	16	1	no	—
344	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
345	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
346	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
347	C₅×C₈	—	2	1	no	—
348	C₅×C₈	—	2	1	no	—
349	C₅×C₈	—	2	1	no	—
350	C₅×C₈	—	4	1	no	—
351	C₅×C₈	—	4	1	no	—
352	C₅×C₈	—	4	1	no	—
353	C₅×C₈	—	4	1	no	—
354	C₅×C₈	—	4	1	no	—
355	C₅×C₈	C₂×SD₁₆	1	2	yes	complemented
356	C₂²×Q₈	—	2	1	no	—
357	C₂²×D₄	—	2	1	no	—
358	C₂²×D₄	—	2	1	no	—
359	C₂²×D₄	—	2	1	no	—
360	C₂²×D₄	—	2	1	no	—
361	C₂×SD₁₆	—	2	1	no	—
362	C₂×SD₁₆	—	2	1	no	—
363	C₂×SD₁₆	—	2	1	no	—
364	C₂×SD₁₆	—	2	1	no	—
365	C₂×SD₁₆	—	2	1	no	—
366	C₂×SD₁₆	—	4	1	no	—
367	C₂×SD₁₆	—	4	4	no	—
368	C₂×SD₁₆	C₅×D₄	1	1	yes	characteristic
369	C₂×D₈	—	2	1	no	—
370	C₂×D₈	—	2	1	no	—
371	C₂×D₈	—	2	1	no	—
372	C₂×D₈	—	2	1	no	—
373	C₂×D₈	—	4	4	no	—
374	C₂×D₈	C₅×D₄	1	1	yes	characteristic
375	C₂²×C₈	—	2	1	no	—
376	C₂²×C₈	—	2	1	no	—
377	C₂²×C₈	—	2	1	no	—
378	C₄⋊Q₈	—	2	1	no	—
379	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
380	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
381	#127	C₂³×C₅	1	1	yes	characteristic
382	#127	C₂³×C₅	1	1	yes	characteristic
383	#125	—	2	1	no	—
384	C₂²⋊Q₈	—	2	1	no	—
385	C₂²⋊Q₈	—	2	1	no	—
386	C₂²⋊Q₈	—	4	1	no	—
387	C₂²⋊Q₈	—	4	1	no	—
388	#121	—	2	1	no	—
389	#121	—	2	1	no	—
390	#121	—	2	1	no	—
391	#121	—	2	1	no	—
392	#120	—	4	1	no	—
393	#120	—	4	1	no	—
394	C₄×Q₈	C₂³×C₅	1	1	yes	characteristic
395	C₄×D₄	—	2	1	no	—
396	C₄×D₄	—	2	1	no	—
397	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
398	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
399	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
400	C₂×C₄⋊C₄	—	2	1	no	—
401	#107	—	2	1	no	—
402	#107	C₅×D₄	1	1	yes	characteristic
403	#106	—	2	1	no	—
404	#106	—	2	1	no	—
405	#106	—	2	1	no	—
406	#106	—	2	1	no	—
407	#106	C₅×D₄	1	1	yes	characteristic
408	C₄⋊C₈	—	2	1	no	—
409	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
410	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
411	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
412	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
413	#103	—	2	1	no	—
414	#103	C₅×D₄	1	1	yes	characteristic
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	#102	—	2	1	no	—
421	#102	—	2	1	no	—
422	#102	—	2	1	no	—
423	#102	—	2	1	no	—
424	#102	—	4	1	no	—
425	#102	C₅×D₄	1	1	yes	characteristic
426	#102	C₅×D₄	1	1	yes	characteristic
427	#102	C₅×D₄	1	1	yes	characteristic
428	C₂²⋊C₈	—	2	1	no	—
429	C₂²⋊C₈	—	2	1	no	—
430	C₂²⋊C₈	—	2	1	no	—
431	#97	—	2	1	no	—
432	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
433	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
434	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
435	C₂²×C₅	—	2	2	no	—
436	C₂²×C₅	—	2	2	no	—
437	C₂²×C₅	—	2	2	no	—
438	C₂²×C₅	—	4	1	no	—
439	C₂²×C₅	—	4	1	no	—
440	C₂²×C₅	—	4	1	no	—
441	C₂²×C₅	—	4	1	no	—
442	C₂²×C₅	—	4	1	no	—
443	C₂²×C₅	—	4	1	no	—
444	C₂²×C₅	—	8	1	no	—
445	C₂²×C₅	—	8	1	no	—
446	C₂²×C₅	—	8	1	no	—
447	C₂²×C₅	—	8	1	no	—
448	C₂²×C₅	—	8	1	no	—
449	C₂²×C₅	—	8	1	no	—
450	C₂²×C₅	—	16	4	no	—
451	C₂²×C₅	—	16	4	no	—
452	C₂²×C₅	D₄²	1	1	yes	characteristic
453	C₄×C₅	—	2	1	no	—
454	C₄×C₅	—	4	1	no	—
455	C₄×C₅	—	4	1	no	—
456	C₄×C₅	—	4	1	no	—
457	C₄×C₅	—	4	1	no	—
458	C₄×C₅	—	4	1	no	—
459	C₄×C₅	—	8	1	no	—
460	C₄×C₅	—	8	1	no	—
461	C₄×C₅	C₂²×SD₁₆	1	1	yes	characteristic
462	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
463	C₄×C₅	#576	1	1	yes	characteristic
464	C₄×C₅	#577	1	1	yes	characteristic
465	C₂⁴	—	4	1	no	—
466	C₂⁴	—	4	1	no	—
467	C₂×Q₈	—	2	1	no	—
468	C₂×Q₈	—	2	1	no	—
469	C₂×Q₈	—	4	1	no	—
470	C₂×Q₈	—	4	2	no	—
471	C₂×Q₈	—	8	1	no	—
472	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
473	C₂×D₄	—	2	1	no	—
474	C₂×D₄	—	2	1	no	—
475	C₂×D₄	—	2	1	no	—
476	C₂×D₄	—	2	1	no	—
477	C₂×D₄	—	2	1	no	—
478	C₂×D₄	—	2	1	no	—
479	C₂×D₄	—	4	1	no	—
480	C₂×D₄	—	4	1	no	—
481	C₂×D₄	—	4	1	no	—
482	C₂×D₄	—	4	1	no	—
483	C₂×D₄	—	4	2	no	—
484	C₂×D₄	—	4	2	no	—
485	C₂×D₄	—	4	2	no	—
486	C₂×D₄	—	4	2	no	—
487	C₂×D₄	—	8	1	no	—
488	C₂×D₄	—	8	1	no	—
489	C₂×D₄	—	8	1	no	—
490	C₂×D₄	—	8	1	no	—
491	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
492	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
493	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
494	C₂²×C₄	—	2	1	no	—
495	C₂²×C₄	—	2	1	no	—
496	C₂²×C₄	—	2	1	no	—
497	C₂²×C₄	—	4	1	no	—
498	C₂²×C₄	—	4	1	no	—
499	SD₁₆	—	2	2	no	—
500	SD₁₆	—	4	2	no	—
501	SD₁₆	—	4	2	no	—
502	SD₁₆	—	4	2	no	—
503	SD₁₆	—	8	1	no	—
504	SD₁₆	—	8	1	no	—
505	SD₁₆	—	8	2	no	—
506	D₈	—	4	2	no	—
507	D₈	—	4	2	no	—
508	D₈	—	4	2	no	—
509	D₈	—	8	1	no	—
510	D₈	C₅×SD₁₆	1	4	yes	complemented
511	C₂×C₈	—	2	1	no	—
512	C₂×C₈	—	2	1	no	—
513	C₂×C₈	—	2	1	no	—
514	C₂×C₈	—	2	1	no	—
515	C₂×C₈	—	2	1	no	—
516	C₂×C₈	—	4	2	no	—
517	C₂×C₈	—	4	2	no	—
518	C₂×C₈	—	8	1	no	—
519	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
520	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
521	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
522	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
523	C₄⋊C₄	—	2	1	no	—
524	C₄⋊C₄	—	2	1	no	—
525	C₄⋊C₄	—	2	1	no	—
526	C₄⋊C₄	—	2	1	no	—
527	C₄⋊C₄	—	2	1	no	—
528	C₄⋊C₄	—	4	1	no	—
529	C₄⋊C₄	—	4	1	no	—
530	C₄⋊C₄	—	4	1	no	—
531	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
532	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
533	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
534	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
535	C₂²⋊C₄	—	2	1	no	—
536	C₂²⋊C₄	—	2	1	no	—
537	C₂²⋊C₄	—	2	1	no	—
538	C₂²⋊C₄	—	4	1	no	—
539	C₂²⋊C₄	—	4	1	no	—
540	C₂²⋊C₄	—	4	1	no	—
541	C₂²⋊C₄	—	4	1	no	—
542	C₄²	—	2	1	no	—
543	C₄²	C₂⁴×C₅	1	1	yes	characteristic
544	C₂×C₅	—	4	2	no	—
545	C₂×C₅	—	4	2	no	—
546	C₂×C₅	—	4	2	no	—
547	C₂×C₅	—	16	1	no	—
548	C₂×C₅	—	16	1	no	—
549	C₂×C₅	#3264	1	1	yes	characteristic
550	C₂×C₅	D₄×SD₁₆	1	1	yes	characteristic
551	C₂×C₅	#3288	1	1	yes	characteristic
552	C₂³	—	2	1	no	—
553	C₂³	—	2	1	no	—
554	C₂³	—	2	1	no	—
555	C₂³	—	4	1	no	—
556	C₂³	—	4	1	no	—
557	C₂³	—	8	2	no	—
558	C₂³	—	8	2	no	—
559	C₂³	—	8	2	no	—
560	C₂³	—	8	2	no	—
561	C₂³	—	16	1	no	—
562	C₂³	—	16	1	no	—
563	Q₈	—	2	1	no	—
564	Q₈	—	2	1	no	—
565	Q₈	—	4	1	no	—
566	Q₈	—	4	1	no	—
567	Q₈	—	8	1	no	—
568	Q₈	—	8	2	no	—
569	D₄	—	2	1	no	—
570	D₄	—	2	1	no	—
571	D₄	—	2	1	no	—
572	D₄	—	4	1	no	—
573	D₄	—	4	1	no	—
574	D₄	—	4	1	no	—
575	D₄	—	4	1	no	—
576	D₄	—	4	1	no	—
577	D₄	—	4	1	no	—
578	D₄	—	4	1	no	—
579	D₄	—	4	1	no	—
580	D₄	—	8	1	no	—
581	D₄	—	8	1	no	—
582	D₄	—	8	2	no	—
583	D₄	—	8	2	no	—
584	D₄	—	8	2	no	—
585	D₄	—	8	2	no	—
586	D₄	C₂×C₅×SD₁₆	1	2	yes	—
587	D₄	C₂×C₅×SD₁₆	1	2	yes	complemented
588	D₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
589	C₂×C₄	—	2	1	no	—
590	C₂×C₄	—	2	1	no	—
591	C₂×C₄	—	2	1	no	—
592	C₂×C₄	—	2	1	no	—
593	C₂×C₄	—	2	1	no	—
594	C₂×C₄	—	4	1	no	—
595	C₂×C₄	—	4	1	no	—
596	C₂×C₄	—	4	1	no	—
597	C₂×C₄	—	4	1	no	—
598	C₂×C₄	—	4	1	no	—
599	C₂×C₄	—	4	1	no	—
600	C₂×C₄	—	4	1	no	—
601	C₂×C₄	—	4	1	no	—
602	C₂×C₄	—	8	2	no	—
603	C₂×C₄	—	16	1	no	—
604	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
605	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
606	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
607	C₈	—	2	1	no	—
608	C₈	—	2	1	no	—
609	C₈	—	2	1	no	—
610	C₈	—	4	1	no	—
611	C₈	—	4	1	no	—
612	C₈	—	4	1	no	—
613	C₈	—	4	1	no	—
614	C₈	—	4	1	no	—
615	C₈	C₂×C₅×SD₁₆	1	2	yes	complemented
616	C₅	#19763	1	1	yes	characteristic, direct product factor, Sylow
617	C₂²	—	2	2	no	—
618	C₂²	—	2	2	no	—
619	C₂²	—	2	2	no	—
620	C₂²	—	4	1	no	—
621	C₂²	—	4	1	no	—
622	C₂²	—	4	1	no	—
623	C₂²	—	4	1	no	—
624	C₂²	—	4	1	no	—
625	C₂²	—	4	1	no	—
626	C₂²	—	8	1	no	—
627	C₂²	—	8	1	no	—
628	C₂²	—	8	1	no	—
629	C₂²	—	8	1	no	—
630	C₂²	—	8	1	no	—
631	C₂²	—	8	1	no	—
632	C₂²	—	16	4	no	—
633	C₂²	—	16	4	no	—
634	C₂²	C₅×D₄²	1	1	yes	characteristic
635	C₄	—	2	1	no	—
636	C₄	—	4	1	no	—
637	C₄	—	4	1	no	—
638	C₄	—	4	1	no	—
639	C₄	—	4	1	no	—
640	C₄	—	4	1	no	—
641	C₄	—	8	1	no	—
642	C₄	—	8	1	no	—
643	C₄	C₂²×C₅×SD₁₆	1	1	yes	characteristic
644	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
645	C₄	C₅×#576	1	1	yes	characteristic
646	C₄	C₅×#577	1	1	yes	characteristic
647	C₂	—	4	2	no	—
648	C₂	—	4	2	no	—
649	C₂	—	4	2	no	—
650	C₂	—	16	1	no	—
651	C₂	—	16	1	no	—
652	C₂	C₅×#3264	1	1	yes	characteristic
653	C₂	C₅×D₄×SD₁₆	1	1	yes	characteristic
654	C₂	C₅×#3288	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₅×D₄×D₈	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#3384	C₂	semidirect
2. C₅×#3295	C₂	semidirect
3. C₅×D₄×SD₁₆	C₂	semidirect
5. C₅×#1715	C₂	semidirect
6. C₅×#1688	C₂	semidirect
7. C₅×#1677	C₂	semidirect
8. C₅×#1671	C₂	semidirect
9. C₅×#1665	C₂	semidirect
10. C₅×#1639	C₂	semidirect
11. C₅×#1629	C₂	semidirect
12. C₅×#1627	C₂	semidirect
13. C₅×#1622	C₂	semidirect
14. C₅×#1582	C₂	semidirect
15. C₅×#1580	C₂	semidirect
18. C₅×#554	C₂²	semidirect
19. C₅×D₄×Q₈	C₂²	semidirect
23. C₅×#498	C₂²	semidirect
25. C₅×#492	C₂²	semidirect
26. C₅×#492	C₂²	semidirect
27. C₅×#486	C₂²	semidirect
28. C₅×#486	C₂²	semidirect
33. C₅×#476	C₂²	semidirect
34. C₅×#474	C₂²	semidirect
39. C₅×#460	C₂²	semidirect
40. C₅×#460	C₂²	semidirect
48. C₅×#446	C₂²	semidirect
53. C₅×C₈×D₄	C₂²	semidirect
54. C₅×C₈×D₄	C₂²	semidirect
56. C₅×#334	C₂²	semidirect
57. C₅×#334	C₂²	semidirect
58. C₅×#333	C₂²	semidirect
59. C₅×#332	C₂²	semidirect
60. C₅×#331	C₂²	semidirect
61. C₅×#331	C₂²	semidirect
62. C₅×Q₈⋊C₈	C₂²	semidirect
64. C₅×D₄⋊C₈	C₂²	semidirect
65. C₅×D₄⋊C₈	C₂²	semidirect
66. C₅×#322	C₂²	semidirect
67. #19763	C₅	direct (N × H)
147. #3384	C₂×C₅	semidirect
148. #3295	C₂×C₅	semidirect
149. D₄×SD₁₆	C₂×C₅	semidirect
151. #1715	C₂×C₅	semidirect
152. #1688	C₂×C₅	semidirect
153. #1677	C₂×C₅	semidirect
154. #1671	C₂×C₅	semidirect
155. #1665	C₂×C₅	semidirect
156. #1639	C₂×C₅	semidirect
157. #1629	C₂×C₅	semidirect
158. #1627	C₂×C₅	semidirect
159. #1622	C₂×C₅	semidirect
160. #1582	C₂×C₅	semidirect
161. #1580	C₂×C₅	semidirect
207. C₅×D₈	SD₁₆	semidirect
243. #554	C₂²×C₅	semidirect
244. D₄×Q₈	C₂²×C₅	semidirect
248. #498	C₂²×C₅	semidirect
250. #492	C₂²×C₅	semidirect
251. #492	C₂²×C₅	semidirect
252. #486	C₂²×C₅	semidirect
253. #486	C₂²×C₅	semidirect
258. #476	C₂²×C₅	semidirect
259. #474	C₂²×C₅	semidirect
264. #460	C₂²×C₅	semidirect
265. #460	C₂²×C₅	semidirect
273. #446	C₂²×C₅	semidirect
278. C₈×D₄	C₂²×C₅	semidirect
279. C₈×D₄	C₂²×C₅	semidirect
281. #334	C₂²×C₅	semidirect
282. #334	C₂²×C₅	semidirect
283. #333	C₂²×C₅	semidirect
284. #332	C₂²×C₅	semidirect
285. #331	C₂²×C₅	semidirect
286. #331	C₂²×C₅	semidirect
287. Q₈⋊C₈	C₂²×C₅	semidirect
289. D₄⋊C₈	C₂²×C₅	semidirect
290. D₄⋊C₈	C₂²×C₅	semidirect
291. #322	C₂²×C₅	semidirect
327. C₅×D₄	C₂×SD₁₆	semidirect
355. C₅×C₈	C₂×SD₁₆	semidirect
510. D₈	C₅×SD₁₆	semidirect
587. D₄	C₂×C₅×SD₁₆	semidirect
615. C₈	C₂×C₅×SD₁₆	semidirect
616. C₅	#19763	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×#3384	2
C₅×#3295	2
C₅×D₄×SD₁₆	2
C₅×D₄×D₈	2
C₅×#1715	2
C₅×#1688	2
C₅×#1677	2
C₅×#1671	2
C₅×#1665	2
C₅×#1639	2
C₅×#1629	2
C₅×#1627	2
C₅×#1622	2
C₅×#1582	2
C₅×#1580	2
#19763	5

Subgroups — C5×#19763 #11942483

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19763 #11942483