Subgroups — C₅×#19913 #11942633

Order 1280.

618 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3372	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3364	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3324	C₂	1	1	yes	maximal, characteristic
4	C₅×#3295	C₂	1	1	yes	maximal, characteristic
5	C₅×#1711	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1703	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1702	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1679	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1678	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#1644	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#1639	C₂	1	1	yes	maximal, characteristic
12	C₅×#1638	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅×#1626	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×#1590	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅×C₈×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
16	C₅×#557	C₂²	1	1	yes	characteristic, complemented
17	C₅×#556	C₂²	1	1	yes	characteristic
18	C₅×#554	C₂²	1	1	yes	characteristic
19	C₅×D₄²	C₂²	1	1	yes	characteristic
20	C₅×#491	—	2	1	no	—
21	C₅×#488	—	2	1	no	—
22	C₅×#487	C₂²	1	1	yes	characteristic, complemented
23	C₅×#487	C₂²	1	1	yes	characteristic, complemented
24	C₅×#487	C₂²	1	1	yes	characteristic, complemented
25	C₅×#486	C₂²	1	1	yes	characteristic
26	C₅×#486	C₂²	1	1	yes	characteristic
27	C₅×#486	C₂²	1	1	yes	characteristic
28	C₅×#482	—	2	1	no	—
29	C₅×#481	—	2	1	no	—
30	C₅×#479	—	2	1	no	—
31	C₅×#479	—	2	1	no	—
32	C₅×#477	C₂²	1	1	yes	characteristic, complemented
33	C₅×#476	C₂²	1	1	yes	characteristic
34	C₅×#474	C₂²	1	1	yes	characteristic
35	C₅×#474	C₂²	1	1	yes	characteristic
36	C₅×#468	—	2	1	no	—
37	C₅×#465	—	2	1	no	—
38	C₅×#464	—	2	1	no	—
39	C₅×#462	C₂²	1	1	yes	characteristic, complemented
40	C₅×#461	—	2	1	no	—
41	C₅×#461	C₂²	1	1	yes	characteristic
42	C₅×#460	C₂²	1	1	yes	characteristic
43	C₅×#460	C₂²	1	1	yes	characteristic
44	C₅×#450	—	2	1	no	—
45	C₅×#450	—	2	1	no	—
46	C₅×#446	C₂²	1	1	yes	characteristic, complemented
47	C₅×C₈×Q₈	C₂²	1	1	yes	characteristic, complemented
48	C₅×#440	—	2	1	no	—
49	C₄×C₅×Q₁₆	—	2	1	no	—
50	C₄×C₅×SD₁₆	C₂²	1	1	yes	characteristic
51	C₄×C₅×SD₁₆	C₂²	1	1	yes	characteristic
52	C₅×#436	C₂²	1	1	yes	characteristic
53	C₅×C₈×D₄	C₂²	1	1	yes	characteristic
54	C₅×#418	—	2	1	no	—
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×#332	C₂²	1	1	yes	characteristic, complemented
61	C₅×#331	C₂²	1	1	yes	characteristic
62	C₅×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
63	C₅×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
64	C₅×D₄⋊C₈	C₂²	1	1	yes	characteristic
65	C₅×D₄⋊C₈	C₂²	1	1	yes	characteristic
66	C₅×C₈²	C₂²	1	1	yes	characteristic, complemented
67	#19913	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂²×C₅×D₄	—	2	1	no	—
69	C₂²×C₅×D₄	—	2	1	no	—
70	C₂×C₅×Q₁₆	—	2	1	no	—
71	C₂×C₅×SD₁₆	—	2	1	no	—
72	C₂×C₅×SD₁₆	—	2	1	no	—
73	C₂×C₅×SD₁₆	—	2	1	no	—
74	C₂×C₅×SD₁₆	D₄	1	1	yes	characteristic
75	C₂×C₅×SD₁₆	D₄	1	1	yes	characteristic
76	C₂×C₅×M₁₆	—	2	1	no	—
77	C₂²×C₅×C₈	—	2	1	no	—
78	C₅×C₄⋊Q₈	—	2	1	no	—
79	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
80	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
81	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
82	C₅×#127	C₂³	1	1	yes	characteristic
83	C₅×#126	—	4	1	no	—
84	C₅×#125	—	2	1	no	—
85	C₅×#125	—	2	1	no	—
86	C₅×#125	—	4	1	no	—
87	C₅×#124	—	2	1	no	—
88	C₅×C₂²⋊Q₈	—	2	1	no	—
89	C₅×C₂²⋊Q₈	—	2	1	no	—
90	C₅×C₂²⋊Q₈	—	4	1	no	—
91	C₅×#121	—	2	1	no	—
92	C₅×#121	—	2	1	no	—
93	C₅×#120	—	4	1	no	—
94	C₄×C₅×Q₈	—	2	1	no	—
95	C₄×C₅×Q₈	—	2	1	no	—
96	C₄×C₅×Q₈	—	2	1	no	—
97	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
98	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
99	C₄×C₅×D₄	—	2	1	no	—
100	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
101	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
102	C₅×#117	—	2	1	no	—
103	C₂×C₅×C₄⋊C₄	—	2	1	no	—
104	C₅×#107	—	2	1	no	—
105	C₅×#107	—	2	1	no	—
106	C₅×#107	—	2	1	no	—
107	C₅×#106	—	2	1	no	—
108	C₅×#106	D₄	1	1	yes	characteristic
109	C₅×#106	D₄	1	1	yes	characteristic
110	C₅×C₄⋊C₈	—	2	1	no	—
111	C₅×C₄⋊C₈	—	2	1	no	—
112	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
113	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
114	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
115	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
116	C₅×#103	—	2	1	no	—
117	C₅×#103	—	2	1	no	—
118	C₅×#103	—	2	1	no	—
119	C₅×#103	—	2	1	no	—
120	C₅×#103	—	2	1	no	—
121	C₅×#103	—	2	1	no	—
122	C₅×#103	—	2	1	no	—
123	C₅×#103	—	4	1	no	—
124	C₅×#103	D₄	1	1	yes	characteristic
125	C₅×#103	D₄	1	1	yes	characteristic
126	C₅×#102	—	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	—	4	1	no	—
136	C₅×#102	D₄	1	1	yes	characteristic
137	C₅×#102	D₄	1	1	yes	characteristic
138	C₅×C₂²⋊C₈	—	2	1	no	—
139	C₅×C₂²⋊C₈	—	2	1	no	—
140	C₅×#97	—	2	1	no	—
141	C₄×C₅×C₈	—	2	1	no	—
142	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
143	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
144	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
145	#3372	C₂×C₅	1	1	yes	characteristic, complemented
146	#3364	C₂×C₅	1	1	yes	characteristic, complemented
147	#3324	C₂×C₅	1	1	yes	characteristic
148	#3295	C₂×C₅	1	1	yes	characteristic
149	#1711	C₂×C₅	1	1	yes	characteristic, complemented
150	#1703	C₂×C₅	1	1	yes	characteristic, complemented
151	#1702	C₂×C₅	1	1	yes	characteristic, complemented
152	#1679	C₂×C₅	1	1	yes	characteristic, complemented
153	#1678	C₂×C₅	1	1	yes	characteristic, complemented
154	#1644	C₂×C₅	1	1	yes	characteristic, complemented
155	#1639	C₂×C₅	1	1	yes	characteristic
156	#1638	C₂×C₅	1	1	yes	characteristic, complemented
157	#1626	C₂×C₅	1	1	yes	characteristic, complemented
158	#1590	C₂×C₅	1	1	yes	characteristic, complemented
159	C₈×SD₁₆	C₂×C₅	1	1	yes	characteristic, complemented
160	C₂⁴×C₅	—	4	1	no	—
161	C₂×C₅×Q₈	—	2	1	no	—
162	C₂×C₅×Q₈	—	2	1	no	—
163	C₂×C₅×Q₈	—	2	1	no	—
164	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
165	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
166	C₂×C₅×D₄	—	2	1	no	—
167	C₂×C₅×D₄	—	2	1	no	—
168	C₂×C₅×D₄	—	2	1	no	—
169	C₂×C₅×D₄	—	4	1	no	—
170	C₂×C₅×D₄	—	4	1	no	—
171	C₂×C₅×D₄	—	4	2	no	—
172	C₂×C₅×D₄	—	4	2	no	—
173	C₂×C₅×D₄	—	8	1	no	—
174	C₂×C₅×D₄	—	8	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	—	4	1	no	—
180	C₅×Q₁₆	—	4	2	no	—
181	C₅×SD₁₆	—	2	2	no	—
182	C₅×SD₁₆	—	4	1	no	—
183	C₅×SD₁₆	—	4	2	no	—
184	C₅×SD₁₆	—	8	1	no	—
185	C₅×SD₁₆	—	8	1	no	—
186	C₅×M₁₆	—	8	1	no	—
187	C₂×C₅×C₈	—	2	1	no	—
188	C₂×C₅×C₈	—	2	1	no	—
189	C₂×C₅×C₈	—	2	1	no	—
190	C₂×C₅×C₈	—	2	1	no	—
191	C₂×C₅×C₈	—	2	1	no	—
192	C₂×C₅×C₈	—	8	1	no	—
193	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
194	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
195	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
196	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
197	C₅×C₄⋊C₄	—	2	1	no	—
198	C₅×C₄⋊C₄	—	2	1	no	—
199	C₅×C₄⋊C₄	—	2	1	no	—
200	C₅×C₄⋊C₄	—	2	1	no	—
201	C₅×C₄⋊C₄	—	2	1	no	—
202	C₅×C₄⋊C₄	—	2	1	no	—
203	C₅×C₄⋊C₄	—	2	1	no	—
204	C₅×C₄⋊C₄	—	2	1	no	—
205	C₅×C₄⋊C₄	—	4	1	no	—
206	C₅×C₄⋊C₄	—	4	1	no	—
207	C₅×C₄⋊C₄	—	4	1	no	—
208	C₅×C₄⋊C₄	—	4	1	no	—
209	C₅×C₄⋊C₄	—	4	1	no	—
210	C₅×C₄⋊C₄	—	4	1	no	—
211	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
212	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
213	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
214	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
215	C₅×C₂²⋊C₄	—	2	1	no	—
216	C₅×C₂²⋊C₄	—	2	1	no	—
217	C₅×C₂²⋊C₄	—	4	1	no	—
218	C₅×C₂²⋊C₄	—	4	1	no	—
219	C₅×C₂²⋊C₄	—	4	1	no	—
220	C₅×C₂²⋊C₄	—	4	1	no	—
221	C₄²×C₅	—	2	1	no	—
222	C₄²×C₅	—	2	1	no	—
223	C₄²×C₅	—	4	1	no	—
224	C₄²×C₅	—	4	1	no	—
225	C₄²×C₅	C₂⁴	1	1	yes	characteristic
226	#557	C₂²×C₅	1	1	yes	characteristic, complemented
227	#556	C₂²×C₅	1	1	yes	characteristic
228	#554	C₂²×C₅	1	1	yes	characteristic
229	D₄²	C₂²×C₅	1	1	yes	characteristic
230	#491	—	2	1	no	—
231	#488	—	2	1	no	—
232	#487	C₂²×C₅	1	1	yes	characteristic, complemented
233	#487	C₂²×C₅	1	1	yes	characteristic, complemented
234	#487	C₂²×C₅	1	1	yes	characteristic, complemented
235	#486	C₂²×C₅	1	1	yes	characteristic
236	#486	C₂²×C₅	1	1	yes	characteristic
237	#486	C₂²×C₅	1	1	yes	characteristic
238	#482	—	2	1	no	—
239	#481	—	2	1	no	—
240	#479	—	2	1	no	—
241	#479	—	2	1	no	—
242	#477	C₂²×C₅	1	1	yes	characteristic, complemented
243	#476	C₂²×C₅	1	1	yes	characteristic
244	#474	C₂²×C₅	1	1	yes	characteristic
245	#474	C₂²×C₅	1	1	yes	characteristic
246	#468	—	2	1	no	—
247	#465	—	2	1	no	—
248	#464	—	2	1	no	—
249	#462	C₂²×C₅	1	1	yes	characteristic, complemented
250	#461	—	2	1	no	—
251	#461	C₂²×C₅	1	1	yes	characteristic
252	#460	C₂²×C₅	1	1	yes	characteristic
253	#460	C₂²×C₅	1	1	yes	characteristic
254	#450	—	2	1	no	—
255	#450	—	2	1	no	—
256	#446	C₂²×C₅	1	1	yes	characteristic, complemented
257	C₈×Q₈	C₂²×C₅	1	1	yes	characteristic, complemented
258	#440	—	2	1	no	—
259	C₄×Q₁₆	—	2	1	no	—
260	C₄×SD₁₆	C₂²×C₅	1	1	yes	characteristic
261	C₄×SD₁₆	C₂²×C₅	1	1	yes	characteristic
262	#436	C₂²×C₅	1	1	yes	characteristic
263	C₈×D₄	C₂²×C₅	1	1	yes	characteristic
264	#418	—	2	1	no	—
265	C₂×#102	—	2	1	no	—
266	#334	C₂²×C₅	1	1	yes	characteristic, complemented
267	#334	C₂²×C₅	1	1	yes	characteristic, complemented
268	#333	C₂²×C₅	1	1	yes	characteristic, complemented
269	#332	C₂²×C₅	1	1	yes	characteristic, complemented
270	#332	C₂²×C₅	1	1	yes	characteristic, complemented
271	#331	C₂²×C₅	1	1	yes	characteristic
272	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic, complemented
273	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic, complemented
274	D₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
275	D₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
276	C₈²	C₂²×C₅	1	1	yes	characteristic, complemented
277	C₂³×C₅	—	2	1	no	—
278	C₂³×C₅	—	2	1	no	—
279	C₂³×C₅	—	4	1	no	—
280	C₂³×C₅	—	8	2	no	—
281	C₂³×C₅	—	8	2	no	—
282	C₂³×C₅	—	16	1	no	—
283	C₅×Q₈	—	2	1	no	—
284	C₅×Q₈	—	2	1	no	—
285	C₅×Q₈	—	2	1	no	—
286	C₅×Q₈	—	2	1	no	—
287	C₅×Q₈	—	4	1	no	—
288	C₅×Q₈	—	4	1	no	—
289	C₅×Q₈	—	4	1	no	—
290	C₅×Q₈	—	4	1	no	—
291	C₅×Q₈	—	8	1	no	—
292	C₅×D₄	—	2	1	no	—
293	C₅×D₄	—	2	1	no	—
294	C₅×D₄	—	2	1	no	—
295	C₅×D₄	—	4	1	no	—
296	C₅×D₄	—	4	1	no	—
297	C₅×D₄	—	4	1	no	—
298	C₅×D₄	—	4	1	no	—
299	C₅×D₄	—	8	1	no	—
300	C₅×D₄	—	8	2	no	—
301	C₅×D₄	—	8	2	no	—
302	C₅×D₄	#135	1	2	yes	—
303	C₂×C₄×C₅	—	2	1	no	—
304	C₂×C₄×C₅	—	2	1	no	—
305	C₂×C₄×C₅	—	2	1	no	—
306	C₂×C₄×C₅	—	2	1	no	—
307	C₂×C₄×C₅	—	2	1	no	—
308	C₂×C₄×C₅	—	2	1	no	—
309	C₂×C₄×C₅	—	4	1	no	—
310	C₂×C₄×C₅	—	4	1	no	—
311	C₂×C₄×C₅	—	4	1	no	—
312	C₂×C₄×C₅	—	4	1	no	—
313	C₂×C₄×C₅	—	4	1	no	—
314	C₂×C₄×C₅	—	4	1	no	—
315	C₂×C₄×C₅	—	4	1	no	—
316	C₂×C₄×C₅	—	16	1	no	—
317	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
318	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
319	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
320	C₅×C₈	—	2	1	no	—
321	C₅×C₈	—	2	1	no	—
322	C₅×C₈	—	2	1	no	—
323	C₅×C₈	—	2	1	no	—
324	C₅×C₈	—	4	1	no	—
325	C₅×C₈	—	4	1	no	—
326	C₅×C₈	—	4	1	no	—
327	C₅×C₈	—	4	1	no	—
328	C₅×C₈	—	4	1	no	—
329	C₂²×D₄	—	2	1	no	—
330	C₂²×D₄	—	2	1	no	—
331	C₂×Q₁₆	—	2	1	no	—
332	C₂×SD₁₆	—	2	1	no	—
333	C₂×SD₁₆	—	2	1	no	—
334	C₂×SD₁₆	—	2	1	no	—
335	C₂×SD₁₆	C₅×D₄	1	1	yes	characteristic
336	C₂×SD₁₆	C₅×D₄	1	1	yes	characteristic
337	C₂×M₁₆	—	2	1	no	—
338	C₂²×C₈	—	2	1	no	—
339	C₄⋊Q₈	—	2	1	no	—
340	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
341	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
342	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
343	#127	C₂³×C₅	1	1	yes	characteristic
344	#126	—	4	1	no	—
345	#125	—	2	1	no	—
346	#125	—	2	1	no	—
347	#125	—	4	1	no	—
348	#124	—	2	1	no	—
349	C₂²⋊Q₈	—	2	1	no	—
350	C₂²⋊Q₈	—	2	1	no	—
351	C₂²⋊Q₈	—	4	1	no	—
352	#121	—	2	1	no	—
353	#121	—	2	1	no	—
354	#120	—	4	1	no	—
355	C₄×Q₈	—	2	1	no	—
356	C₄×Q₈	—	2	1	no	—
357	C₄×Q₈	—	2	1	no	—
358	C₄×Q₈	C₂³×C₅	1	1	yes	characteristic
359	C₄×Q₈	C₂³×C₅	1	1	yes	characteristic
360	C₄×D₄	—	2	1	no	—
361	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
362	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
363	#117	—	2	1	no	—
364	C₂×C₄⋊C₄	—	2	1	no	—
365	#107	—	2	1	no	—
366	#107	—	2	1	no	—
367	#107	—	2	1	no	—
368	#106	—	2	1	no	—
369	#106	C₅×D₄	1	1	yes	characteristic
370	#106	C₅×D₄	1	1	yes	characteristic
371	C₄⋊C₈	—	2	1	no	—
372	C₄⋊C₈	—	2	1	no	—
373	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
374	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
375	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
376	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
377	#103	—	2	1	no	—
378	#103	—	2	1	no	—
379	#103	—	2	1	no	—
380	#103	—	2	1	no	—
381	#103	—	2	1	no	—
382	#103	—	2	1	no	—
383	#103	—	2	1	no	—
384	#103	—	4	1	no	—
385	#103	C₅×D₄	1	1	yes	characteristic
386	#103	C₅×D₄	1	1	yes	characteristic
387	#102	—	2	1	no	—
388	#102	—	2	1	no	—
389	#102	—	2	1	no	—
390	#102	—	2	1	no	—
391	#102	—	2	1	no	—
392	#102	—	2	1	no	—
393	#102	—	2	1	no	—
394	#102	—	2	1	no	—
395	#102	—	2	1	no	—
396	#102	—	4	1	no	—
397	#102	C₅×D₄	1	1	yes	characteristic
398	#102	C₅×D₄	1	1	yes	characteristic
399	C₂²⋊C₈	—	2	1	no	—
400	C₂²⋊C₈	—	2	1	no	—
401	#97	—	2	1	no	—
402	C₄×C₈	—	2	1	no	—
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	C₂²×C₅	—	2	2	no	—
407	C₂²×C₅	—	4	1	no	—
408	C₂²×C₅	—	4	1	no	—
409	C₂²×C₅	—	4	1	no	—
410	C₂²×C₅	—	4	1	no	—
411	C₂²×C₅	—	4	1	no	—
412	C₂²×C₅	—	8	1	no	—
413	C₂²×C₅	—	8	1	no	—
414	C₂²×C₅	—	8	1	no	—
415	C₂²×C₅	—	16	4	no	—
416	C₂²×C₅	D₄²	1	1	yes	characteristic
417	C₄×C₅	—	2	1	no	—
418	C₄×C₅	—	4	1	no	—
419	C₄×C₅	—	4	1	no	—
420	C₄×C₅	—	4	1	no	—
421	C₄×C₅	—	4	1	no	—
422	C₄×C₅	—	4	1	no	—
423	C₄×C₅	—	4	1	no	—
424	C₄×C₅	—	8	1	no	—
425	C₄×C₅	—	8	1	no	—
426	C₄×C₅	—	8	1	no	—
427	C₄×C₅	C₂×#135	1	1	yes	characteristic
428	C₄×C₅	#575	1	1	yes	characteristic
429	C₄×C₅	#577	1	1	yes	characteristic
430	C₄×C₅	#577	1	1	yes	characteristic
431	C₂⁴	—	4	1	no	—
432	C₂×Q₈	—	2	1	no	—
433	C₂×Q₈	—	2	1	no	—
434	C₂×Q₈	—	2	1	no	—
435	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
436	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
437	C₂×D₄	—	2	1	no	—
438	C₂×D₄	—	2	1	no	—
439	C₂×D₄	—	2	1	no	—
440	C₂×D₄	—	4	1	no	—
441	C₂×D₄	—	4	1	no	—
442	C₂×D₄	—	4	2	no	—
443	C₂×D₄	—	4	2	no	—
444	C₂×D₄	—	8	1	no	—
445	C₂×D₄	—	8	1	no	—
446	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
447	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	—	2	1	no	—
450	C₂²×C₄	—	4	1	no	—
451	Q₁₆	—	4	2	no	—
452	SD₁₆	—	2	2	no	—
453	SD₁₆	—	4	1	no	—
454	SD₁₆	—	4	2	no	—
455	SD₁₆	—	8	1	no	—
456	SD₁₆	—	8	1	no	—
457	M₁₆	—	8	1	no	—
458	C₂×C₈	—	2	1	no	—
459	C₂×C₈	—	2	1	no	—
460	C₂×C₈	—	2	1	no	—
461	C₂×C₈	—	2	1	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	—	8	1	no	—
464	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
465	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
466	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
467	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	2	1	no	—
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	—	2	1	no	—
473	C₄⋊C₄	—	2	1	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	4	1	no	—
477	C₄⋊C₄	—	4	1	no	—
478	C₄⋊C₄	—	4	1	no	—
479	C₄⋊C₄	—	4	1	no	—
480	C₄⋊C₄	—	4	1	no	—
481	C₄⋊C₄	—	4	1	no	—
482	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
483	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
484	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
485	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
486	C₂²⋊C₄	—	2	1	no	—
487	C₂²⋊C₄	—	2	1	no	—
488	C₂²⋊C₄	—	4	1	no	—
489	C₂²⋊C₄	—	4	1	no	—
490	C₂²⋊C₄	—	4	1	no	—
491	C₂²⋊C₄	—	4	1	no	—
492	C₄²	—	2	1	no	—
493	C₄²	—	2	1	no	—
494	C₄²	—	4	1	no	—
495	C₄²	—	4	1	no	—
496	C₄²	C₂⁴×C₅	1	1	yes	characteristic
497	C₂×C₅	—	4	2	no	—
498	C₂×C₅	—	8	1	no	—
499	C₂×C₅	—	16	1	no	—
500	C₂×C₅	#3275	1	1	yes	characteristic
501	C₂×C₅	#3282	1	1	yes	characteristic
502	C₂×C₅	#3290	1	1	yes	characteristic
503	C₂³	—	2	1	no	—
504	C₂³	—	2	1	no	—
505	C₂³	—	4	1	no	—
506	C₂³	—	8	2	no	—
507	C₂³	—	8	2	no	—
508	C₂³	—	16	1	no	—
509	Q₈	—	2	1	no	—
510	Q₈	—	2	1	no	—
511	Q₈	—	2	1	no	—
512	Q₈	—	2	1	no	—
513	Q₈	—	4	1	no	—
514	Q₈	—	4	1	no	—
515	Q₈	—	4	1	no	—
516	Q₈	—	4	1	no	—
517	Q₈	—	8	1	no	—
518	D₄	—	2	1	no	—
519	D₄	—	2	1	no	—
520	D₄	—	2	1	no	—
521	D₄	—	4	1	no	—
522	D₄	—	4	1	no	—
523	D₄	—	4	1	no	—
524	D₄	—	4	1	no	—
525	D₄	—	8	1	no	—
526	D₄	—	8	2	no	—
527	D₄	—	8	2	no	—
528	D₄	C₅×#135	1	2	yes	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	4	1	no	—
540	C₂×C₄	—	4	1	no	—
541	C₂×C₄	—	4	1	no	—
542	C₂×C₄	—	16	1	no	—
543	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
544	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
545	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
546	C₈	—	2	1	no	—
547	C₈	—	2	1	no	—
548	C₈	—	2	1	no	—
549	C₈	—	2	1	no	—
550	C₈	—	4	1	no	—
551	C₈	—	4	1	no	—
552	C₈	—	4	1	no	—
553	C₈	—	4	1	no	—
554	C₈	—	4	1	no	—
555	C₅	#19913	1	1	yes	characteristic, direct product factor, Sylow
556	C₂²	—	2	2	no	—
557	C₂²	—	4	1	no	—
558	C₂²	—	4	1	no	—
559	C₂²	—	4	1	no	—
560	C₂²	—	4	1	no	—
561	C₂²	—	4	1	no	—
562	C₂²	—	8	1	no	—
563	C₂²	—	8	1	no	—
564	C₂²	—	8	1	no	—
565	C₂²	—	16	4	no	—
566	C₂²	C₅×D₄²	1	1	yes	characteristic
567	C₄	—	2	1	no	—
568	C₄	—	4	1	no	—
569	C₄	—	4	1	no	—
570	C₄	—	4	1	no	—
571	C₄	—	4	1	no	—
572	C₄	—	4	1	no	—
573	C₄	—	4	1	no	—
574	C₄	—	8	1	no	—
575	C₄	—	8	1	no	—
576	C₄	—	8	1	no	—
577	C₄	C₂×C₅×#135	1	1	yes	characteristic
578	C₄	C₅×#575	1	1	yes	characteristic
579	C₄	C₅×#577	1	1	yes	characteristic
580	C₄	C₅×#577	1	1	yes	characteristic
581	C₂	—	4	2	no	—
582	C₂	—	8	1	no	—
583	C₂	—	16	1	no	—
584	C₂	C₅×#3275	1	1	yes	characteristic
585	C₂	C₅×#3282	1	1	yes	characteristic
586	C₂	C₅×#3290	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₅×D₄²	C₂²
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×#3372	C₂	semidirect
2. C₅×#3364	C₂	semidirect
5. C₅×#1711	C₂	semidirect
6. C₅×#1703	C₂	semidirect
7. C₅×#1702	C₂	semidirect
8. C₅×#1679	C₂	semidirect
9. C₅×#1678	C₂	semidirect
10. C₅×#1644	C₂	semidirect
12. C₅×#1638	C₂	semidirect
13. C₅×#1626	C₂	semidirect
14. C₅×#1590	C₂	semidirect
15. C₅×C₈×SD₁₆	C₂	semidirect
16. C₅×#557	C₂²	semidirect
22. C₅×#487	C₂²	semidirect
23. C₅×#487	C₂²	semidirect
24. C₅×#487	C₂²	semidirect
32. C₅×#477	C₂²	semidirect
39. C₅×#462	C₂²	semidirect
46. C₅×#446	C₂²	semidirect
47. C₅×C₈×Q₈	C₂²	semidirect
56. C₅×#334	C₂²	semidirect
57. C₅×#334	C₂²	semidirect
58. C₅×#333	C₂²	semidirect
59. C₅×#332	C₂²	semidirect
60. C₅×#332	C₂²	semidirect
62. C₅×Q₈⋊C₈	C₂²	semidirect
63. C₅×Q₈⋊C₈	C₂²	semidirect
66. C₅×C₈²	C₂²	semidirect
67. #19913	C₅	direct (N × H)
145. #3372	C₂×C₅	semidirect
146. #3364	C₂×C₅	semidirect
149. #1711	C₂×C₅	semidirect
150. #1703	C₂×C₅	semidirect
151. #1702	C₂×C₅	semidirect
152. #1679	C₂×C₅	semidirect
153. #1678	C₂×C₅	semidirect
154. #1644	C₂×C₅	semidirect
156. #1638	C₂×C₅	semidirect
157. #1626	C₂×C₅	semidirect
158. #1590	C₂×C₅	semidirect
159. C₈×SD₁₆	C₂×C₅	semidirect
226. #557	C₂²×C₅	semidirect
232. #487	C₂²×C₅	semidirect
233. #487	C₂²×C₅	semidirect
234. #487	C₂²×C₅	semidirect
242. #477	C₂²×C₅	semidirect
249. #462	C₂²×C₅	semidirect
256. #446	C₂²×C₅	semidirect
257. C₈×Q₈	C₂²×C₅	semidirect
266. #334	C₂²×C₅	semidirect
267. #334	C₂²×C₅	semidirect
268. #333	C₂²×C₅	semidirect
269. #332	C₂²×C₅	semidirect
270. #332	C₂²×C₅	semidirect
272. Q₈⋊C₈	C₂²×C₅	semidirect
273. Q₈⋊C₈	C₂²×C₅	semidirect
276. C₈²	C₂²×C₅	semidirect
555. C₅	#19913	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×#3372	2
C₅×#3364	2
C₅×#3324	2
C₅×#3295	2
C₅×#1711	2
C₅×#1703	2
C₅×#1702	2
C₅×#1679	2
C₅×#1678	2
C₅×#1644	2
C₅×#1639	2
C₅×#1638	2
C₅×#1626	2
C₅×#1590	2
C₅×C₈×SD₁₆	2
#19913	5

Subgroups — C5×#19913 #11942633

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19913 #11942633