Subgroups — C₅×#19859 #11942579

Order 1280.

656 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3354	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3334	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3303	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#3283	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1723	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1693	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1683	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1671	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1669	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#1647	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#1647	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#1646	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅×#1635	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×#1581	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅×#1580	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₅×#135	—	2	1	no	—
17	C₅×#555	C₂²	1	1	yes	characteristic
18	C₅×#547	C₂²	1	1	yes	characteristic
19	C₅×#547	C₂²	1	1	yes	characteristic
20	C₅×#547	C₂²	1	1	yes	characteristic
21	C₅×#500	C₂²	1	1	yes	characteristic
22	C₅×#500	C₂²	1	1	yes	characteristic
23	C₅×#498	C₂²	1	1	yes	characteristic
24	C₅×#494	C₂²	1	1	yes	characteristic
25	C₅×#487	C₂²	1	1	yes	characteristic
26	C₅×#487	C₂²	1	1	yes	characteristic
27	C₅×#485	—	2	1	no	—
28	C₅×#482	—	2	1	no	—
29	C₅×#481	—	2	1	no	—
30	C₅×#480	—	2	1	no	—
31	C₅×#480	—	2	1	no	—
32	C₅×#476	C₂²	1	1	yes	characteristic
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×#474	C₂²	1	1	yes	characteristic
37	C₅×#469	—	2	1	no	—
38	C₅×#467	—	2	1	no	—
39	C₅×#466	—	2	1	no	—
40	C₅×#465	—	2	1	no	—
41	C₅×#461	C₂²	1	1	yes	characteristic
42	C₅×#461	C₂²	1	1	yes	characteristic
43	C₅×#461	C₂²	1	1	yes	characteristic
44	C₅×#452	—	2	1	no	—
45	C₅×#452	—	2	1	no	—
46	C₅×#449	—	2	1	no	—
47	C₄×C₅×D₈	C₂²	1	1	yes	characteristic
48	C₄×C₅×D₈	C₂²	1	1	yes	characteristic
49	C₅×#436	C₂²	1	1	yes	characteristic
50	C₅×C₈×D₄	C₂²	1	1	yes	characteristic
51	C₅×C₈×D₄	C₂²	1	1	yes	characteristic
52	C₅×C₈×D₄	C₂²	1	1	yes	characteristic
53	C₅×#428	—	2	1	no	—
54	C₂×C₅×#107	—	2	1	no	—
55	C₅×#416	—	2	1	no	—
56	C₅×#335	C₂²	1	1	yes	characteristic
57	C₅×#334	C₂²	1	1	yes	characteristic
58	C₅×#333	C₂²	1	1	yes	characteristic
59	C₅×#332	C₂²	1	1	yes	characteristic
60	C₅×#332	C₂²	1	1	yes	characteristic
61	C₅×#332	C₂²	1	1	yes	characteristic
62	C₅×D₄⋊C₈	C₂²	1	1	yes	characteristic
63	C₅×D₄⋊C₈	C₂²	1	1	yes	characteristic
64	C₅×D₄⋊C₈	C₂²	1	1	yes	characteristic
65	C₅×D₄⋊C₈	C₂²	1	1	yes	characteristic
66	C₅×#322	C₂²	1	1	yes	characteristic
67	#19859	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	C₂×C₅×#41	—	2	1	no	—
71	C₅×#135	—	4	4	no	—
72	C₂×C₅×Q₁₆	—	2	1	no	—
73	C₂×C₅×Q₁₆	—	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×D₈	D₄	1	1	yes	characteristic
78	C₂×C₅×D₈	D₄	1	1	yes	characteristic
79	C₂×C₅×M₁₆	—	2	1	no	—
80	C₂²×C₅×C₈	—	2	1	no	—
81	C₂²×C₅×C₈	—	2	1	no	—
82	C₂²×C₅×C₈	—	2	1	no	—
83	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
84	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
85	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
86	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
87	C₅×#124	—	4	1	no	—
88	C₅×#123	—	4	1	no	—
89	C₅×#123	—	4	1	no	—
90	C₅×#123	—	4	1	no	—
91	C₅×C₂²⋊Q₈	—	2	1	no	—
92	C₅×C₂²⋊Q₈	—	2	1	no	—
93	C₅×C₂²⋊Q₈	—	2	1	no	—
94	C₅×#121	—	2	1	no	—
95	C₅×#121	—	2	1	no	—
96	C₅×#121	—	2	1	no	—
97	C₅×#121	—	2	1	no	—
98	C₅×#121	—	2	1	no	—
99	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
100	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
101	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
102	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
103	C₅×#117	—	2	1	no	—
104	C₅×#117	—	2	1	no	—
105	C₂×C₅×C₄⋊C₄	—	2	1	no	—
106	C₂×C₅×C₄⋊C₄	—	2	1	no	—
107	C₂×C₅×C₄⋊C₄	—	2	1	no	—
108	C₅×#107	—	2	1	no	—
109	C₅×#107	—	2	1	no	—
110	C₅×#107	—	2	1	no	—
111	C₅×#107	—	2	1	no	—
112	C₅×#107	—	2	1	no	—
113	C₅×#107	—	4	1	no	—
114	C₅×#107	D₄	1	1	yes	characteristic
115	C₅×#107	D₄	1	1	yes	characteristic
116	C₅×#106	—	2	1	no	—
117	C₅×#106	—	2	1	no	—
118	C₅×#106	—	2	1	no	—
119	C₅×#106	—	2	1	no	—
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×#103	—	2	1	no	—
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	—	4	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	D₄	1	1	yes	characteristic
136	C₅×#102	D₄	1	1	yes	characteristic
137	C₅×#102	D₄	1	1	yes	characteristic
138	C₅×#102	D₄	1	1	yes	characteristic
139	C₅×C₂²⋊C₈	—	2	1	no	—
140	C₅×C₂²⋊C₈	—	2	1	no	—
141	C₅×C₂²⋊C₈	—	2	1	no	—
142	C₅×C₂²⋊C₈	—	2	1	no	—
143	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
144	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
145	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
146	#3354	C₂×C₅	1	1	yes	characteristic, complemented
147	#3334	C₂×C₅	1	1	yes	characteristic, complemented
148	#3303	C₂×C₅	1	1	yes	characteristic, complemented
149	#3283	C₂×C₅	1	1	yes	characteristic, complemented
150	#1723	C₂×C₅	1	1	yes	characteristic, complemented
151	#1693	C₂×C₅	1	1	yes	characteristic, complemented
152	#1683	C₂×C₅	1	1	yes	characteristic, complemented
153	#1671	C₂×C₅	1	1	yes	characteristic, complemented
154	#1669	C₂×C₅	1	1	yes	characteristic, complemented
155	#1647	C₂×C₅	1	1	yes	characteristic, complemented
156	#1647	C₂×C₅	1	1	yes	characteristic, complemented
157	#1646	C₂×C₅	1	1	yes	characteristic, complemented
158	#1635	C₂×C₅	1	1	yes	characteristic, complemented
159	#1581	C₂×C₅	1	1	yes	characteristic, complemented
160	#1580	C₂×C₅	1	1	yes	characteristic, complemented
161	C₅×#41	—	4	2	no	—
162	C₅×#41	—	4	2	no	—
163	C₅×#41	—	4	2	no	—
164	C₅×#41	—	8	1	no	—
165	C₅×#41	—	8	1	no	—
166	C₅×#41	—	8	1	no	—
167	C₂×C₅×Q₈	—	2	1	no	—
168	C₂×C₅×Q₈	—	2	1	no	—
169	C₂×C₅×Q₈	—	2	1	no	—
170	C₂×C₅×Q₈	—	2	1	no	—
171	C₂×C₅×D₄	—	4	1	no	—
172	C₂×C₅×D₄	—	4	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	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
178	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
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	—	2	1	no	—
183	C₂²×C₄×C₅	—	4	1	no	—
184	C₂²×C₄×C₅	—	4	1	no	—
185	C₂²×C₄×C₅	—	4	1	no	—
186	C₅×Q₁₆	—	4	2	no	—
187	C₅×Q₁₆	—	8	1	no	—
188	C₅×SD₁₆	—	4	2	no	—
189	C₅×SD₁₆	—	4	2	no	—
190	C₅×SD₁₆	—	8	1	no	—
191	C₅×D₈	—	2	2	no	—
192	C₅×D₈	—	2	2	no	—
193	C₅×M₁₆	—	4	2	no	—
194	C₂×C₅×C₈	—	2	1	no	—
195	C₂×C₅×C₈	—	2	1	no	—
196	C₂×C₅×C₈	—	2	1	no	—
197	C₂×C₅×C₈	—	2	1	no	—
198	C₂×C₅×C₈	—	2	1	no	—
199	C₂×C₅×C₈	—	4	2	no	—
200	C₂×C₅×C₈	—	4	2	no	—
201	C₂×C₅×C₈	—	8	1	no	—
202	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
203	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
204	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
205	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
206	C₅×C₄⋊C₄	—	2	1	no	—
207	C₅×C₄⋊C₄	—	2	1	no	—
208	C₅×C₄⋊C₄	—	2	1	no	—
209	C₅×C₄⋊C₄	—	2	1	no	—
210	C₅×C₄⋊C₄	—	2	1	no	—
211	C₅×C₄⋊C₄	—	2	1	no	—
212	C₅×C₄⋊C₄	—	2	1	no	—
213	C₅×C₄⋊C₄	—	2	1	no	—
214	C₅×C₄⋊C₄	—	4	1	no	—
215	C₅×C₄⋊C₄	—	4	1	no	—
216	C₅×C₄⋊C₄	—	4	1	no	—
217	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
218	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
219	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
220	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
221	C₅×C₂²⋊C₄	—	2	1	no	—
222	C₅×C₂²⋊C₄	—	2	1	no	—
223	C₅×C₂²⋊C₄	—	2	1	no	—
224	C₅×C₂²⋊C₄	—	2	1	no	—
225	C₅×C₂²⋊C₄	—	4	1	no	—
226	C₅×C₂²⋊C₄	—	4	1	no	—
227	C₅×C₂²⋊C₄	—	4	1	no	—
228	C₅×C₂²⋊C₄	—	4	1	no	—
229	C₅×C₂²⋊C₄	—	4	1	no	—
230	C₄²×C₅	—	4	1	no	—
231	C₄²×C₅	C₂⁴	1	1	yes	characteristic
232	C₂×#135	—	2	1	no	—
233	#555	C₂²×C₅	1	1	yes	characteristic
234	#547	C₂²×C₅	1	1	yes	characteristic
235	#547	C₂²×C₅	1	1	yes	characteristic
236	#547	C₂²×C₅	1	1	yes	characteristic
237	#500	C₂²×C₅	1	1	yes	characteristic
238	#500	C₂²×C₅	1	1	yes	characteristic
239	#498	C₂²×C₅	1	1	yes	characteristic
240	#494	C₂²×C₅	1	1	yes	characteristic
241	#487	C₂²×C₅	1	1	yes	characteristic
242	#487	C₂²×C₅	1	1	yes	characteristic
243	#485	—	2	1	no	—
244	#482	—	2	1	no	—
245	#481	—	2	1	no	—
246	#480	—	2	1	no	—
247	#480	—	2	1	no	—
248	#476	C₂²×C₅	1	1	yes	characteristic
249	#476	C₂²×C₅	1	1	yes	characteristic
250	#474	C₂²×C₅	1	1	yes	characteristic
251	#474	C₂²×C₅	1	1	yes	characteristic
252	#474	C₂²×C₅	1	1	yes	characteristic
253	#469	—	2	1	no	—
254	#467	—	2	1	no	—
255	#466	—	2	1	no	—
256	#465	—	2	1	no	—
257	#461	C₂²×C₅	1	1	yes	characteristic
258	#461	C₂²×C₅	1	1	yes	characteristic
259	#461	C₂²×C₅	1	1	yes	characteristic
260	#452	—	2	1	no	—
261	#452	—	2	1	no	—
262	#449	—	2	1	no	—
263	C₄×D₈	C₂²×C₅	1	1	yes	characteristic
264	C₄×D₈	C₂²×C₅	1	1	yes	characteristic
265	#436	C₂²×C₅	1	1	yes	characteristic
266	C₈×D₄	C₂²×C₅	1	1	yes	characteristic
267	C₈×D₄	C₂²×C₅	1	1	yes	characteristic
268	C₈×D₄	C₂²×C₅	1	1	yes	characteristic
269	#428	—	2	1	no	—
270	C₂×#107	—	2	1	no	—
271	#416	—	2	1	no	—
272	#335	C₂²×C₅	1	1	yes	characteristic
273	#334	C₂²×C₅	1	1	yes	characteristic
274	#333	C₂²×C₅	1	1	yes	characteristic
275	#332	C₂²×C₅	1	1	yes	characteristic
276	#332	C₂²×C₅	1	1	yes	characteristic
277	#332	C₂²×C₅	1	1	yes	characteristic
278	D₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
279	D₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
280	D₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
281	D₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
282	#322	C₂²×C₅	1	1	yes	characteristic
283	C₂³×C₅	—	2	1	no	—
284	C₂³×C₅	—	2	1	no	—
285	C₂³×C₅	—	2	1	no	—
286	C₂³×C₅	—	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	—	4	1	no	—
292	C₅×Q₈	—	4	1	no	—
293	C₅×Q₈	—	8	1	no	—
294	C₅×D₄	—	2	1	no	—
295	C₅×D₄	—	2	1	no	—
296	C₅×D₄	—	2	1	no	—
297	C₅×D₄	—	2	1	no	—
298	C₅×D₄	—	2	1	no	—
299	C₅×D₄	—	2	1	no	—
300	C₅×D₄	—	2	1	no	—
301	C₅×D₄	—	8	2	no	—
302	C₅×D₄	—	8	2	no	—
303	C₅×D₄	—	8	2	no	—
304	C₅×D₄	—	16	1	no	—
305	C₅×D₄	AGL(1,ℤ/8)	1	2	yes	—
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	—	2	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	—	4	1	no	—
317	C₂×C₄×C₅	—	4	1	no	—
318	C₂×C₄×C₅	—	4	1	no	—
319	C₂×C₄×C₅	—	4	1	no	—
320	C₂×C₄×C₅	—	4	1	no	—
321	C₂×C₄×C₅	—	4	1	no	—
322	C₂×C₄×C₅	—	8	2	no	—
323	C₂×C₄×C₅	—	8	2	no	—
324	C₂×C₄×C₅	—	8	2	no	—
325	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
326	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
327	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
328	C₅×C₈	—	2	1	no	—
329	C₅×C₈	—	2	1	no	—
330	C₅×C₈	—	2	1	no	—
331	C₅×C₈	—	4	1	no	—
332	C₅×C₈	—	4	1	no	—
333	C₅×C₈	—	4	1	no	—
334	C₅×C₈	—	4	1	no	—
335	C₅×C₈	—	4	1	no	—
336	C₅×C₈	#135	1	2	yes	complemented
337	C₂×#41	—	2	1	no	—
338	C₂×#41	—	2	1	no	—
339	C₂×#41	—	2	1	no	—
340	#135	—	4	4	no	—
341	C₂×Q₁₆	—	2	1	no	—
342	C₂×Q₁₆	—	2	1	no	—
343	C₂×SD₁₆	—	2	1	no	—
344	C₂×SD₁₆	—	2	1	no	—
345	C₂×SD₁₆	—	2	1	no	—
346	C₂×D₈	C₅×D₄	1	1	yes	characteristic
347	C₂×D₈	C₅×D₄	1	1	yes	characteristic
348	C₂×M₁₆	—	2	1	no	—
349	C₂²×C₈	—	2	1	no	—
350	C₂²×C₈	—	2	1	no	—
351	C₂²×C₈	—	2	1	no	—
352	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
353	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
354	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
355	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
356	#124	—	4	1	no	—
357	#123	—	4	1	no	—
358	#123	—	4	1	no	—
359	#123	—	4	1	no	—
360	C₂²⋊Q₈	—	2	1	no	—
361	C₂²⋊Q₈	—	2	1	no	—
362	C₂²⋊Q₈	—	2	1	no	—
363	#121	—	2	1	no	—
364	#121	—	2	1	no	—
365	#121	—	2	1	no	—
366	#121	—	2	1	no	—
367	#121	—	2	1	no	—
368	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
369	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
370	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
371	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
372	#117	—	2	1	no	—
373	#117	—	2	1	no	—
374	C₂×C₄⋊C₄	—	2	1	no	—
375	C₂×C₄⋊C₄	—	2	1	no	—
376	C₂×C₄⋊C₄	—	2	1	no	—
377	#107	—	2	1	no	—
378	#107	—	2	1	no	—
379	#107	—	2	1	no	—
380	#107	—	2	1	no	—
381	#107	—	2	1	no	—
382	#107	—	4	1	no	—
383	#107	C₅×D₄	1	1	yes	characteristic
384	#107	C₅×D₄	1	1	yes	characteristic
385	#106	—	2	1	no	—
386	#106	—	2	1	no	—
387	#106	—	2	1	no	—
388	#106	—	2	1	no	—
389	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
390	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
391	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
392	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
393	#103	—	2	1	no	—
394	#103	—	2	1	no	—
395	#103	—	2	1	no	—
396	#103	—	2	1	no	—
397	#103	—	2	1	no	—
398	#103	—	4	1	no	—
399	#102	—	2	1	no	—
400	#102	—	2	1	no	—
401	#102	—	2	1	no	—
402	#102	—	2	1	no	—
403	#102	—	2	1	no	—
404	#102	C₅×D₄	1	1	yes	characteristic
405	#102	C₅×D₄	1	1	yes	characteristic
406	#102	C₅×D₄	1	1	yes	characteristic
407	#102	C₅×D₄	1	1	yes	characteristic
408	C₂²⋊C₈	—	2	1	no	—
409	C₂²⋊C₈	—	2	1	no	—
410	C₂²⋊C₈	—	2	1	no	—
411	C₂²⋊C₈	—	2	1	no	—
412	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
413	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
414	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
415	C₂²×C₅	—	2	2	no	—
416	C₂²×C₅	—	4	1	no	—
417	C₂²×C₅	—	4	1	no	—
418	C₂²×C₅	—	4	1	no	—
419	C₂²×C₅	—	4	1	no	—
420	C₂²×C₅	—	4	1	no	—
421	C₂²×C₅	—	4	1	no	—
422	C₂²×C₅	—	4	1	no	—
423	C₂²×C₅	—	4	1	no	—
424	C₂²×C₅	—	4	1	no	—
425	C₂²×C₅	—	4	1	no	—
426	C₂²×C₅	—	4	1	no	—
427	C₂²×C₅	D₄²	1	1	yes	characteristic
428	C₄×C₅	—	2	1	no	—
429	C₄×C₅	—	4	1	no	—
430	C₄×C₅	—	4	1	no	—
431	C₄×C₅	—	4	1	no	—
432	C₄×C₅	—	4	1	no	—
433	C₄×C₅	—	8	1	no	—
434	C₄×C₅	—	8	1	no	—
435	C₄×C₅	—	8	1	no	—
436	C₄×C₅	—	8	1	no	—
437	C₄×C₅	C₂×#135	1	1	yes	characteristic
438	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
439	C₄×C₅	#576	1	1	yes	characteristic
440	C₄×C₅	#578	1	1	yes	characteristic
441	#41	—	4	2	no	—
442	#41	—	4	2	no	—
443	#41	—	4	2	no	—
444	#41	—	8	1	no	—
445	#41	—	8	1	no	—
446	#41	—	8	1	no	—
447	C₂×Q₈	—	2	1	no	—
448	C₂×Q₈	—	2	1	no	—
449	C₂×Q₈	—	2	1	no	—
450	C₂×Q₈	—	2	1	no	—
451	C₂×D₄	—	4	1	no	—
452	C₂×D₄	—	4	1	no	—
453	C₂×D₄	—	4	1	no	—
454	C₂×D₄	—	4	1	no	—
455	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
456	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
457	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
458	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	1	no	—
462	C₂²×C₄	—	2	1	no	—
463	C₂²×C₄	—	4	1	no	—
464	C₂²×C₄	—	4	1	no	—
465	C₂²×C₄	—	4	1	no	—
466	Q₁₆	—	4	2	no	—
467	Q₁₆	—	8	1	no	—
468	SD₁₆	—	4	2	no	—
469	SD₁₆	—	4	2	no	—
470	SD₁₆	—	8	1	no	—
471	D₈	—	2	2	no	—
472	D₈	—	2	2	no	—
473	M₁₆	—	4	2	no	—
474	C₂×C₈	—	2	1	no	—
475	C₂×C₈	—	2	1	no	—
476	C₂×C₈	—	2	1	no	—
477	C₂×C₈	—	2	1	no	—
478	C₂×C₈	—	2	1	no	—
479	C₂×C₈	—	4	2	no	—
480	C₂×C₈	—	4	2	no	—
481	C₂×C₈	—	8	1	no	—
482	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
483	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
484	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
485	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
486	C₄⋊C₄	—	2	1	no	—
487	C₄⋊C₄	—	2	1	no	—
488	C₄⋊C₄	—	2	1	no	—
489	C₄⋊C₄	—	2	1	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	1	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	4	1	no	—
495	C₄⋊C₄	—	4	1	no	—
496	C₄⋊C₄	—	4	1	no	—
497	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
498	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
499	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
500	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
501	C₂²⋊C₄	—	2	1	no	—
502	C₂²⋊C₄	—	2	1	no	—
503	C₂²⋊C₄	—	2	1	no	—
504	C₂²⋊C₄	—	2	1	no	—
505	C₂²⋊C₄	—	4	1	no	—
506	C₂²⋊C₄	—	4	1	no	—
507	C₂²⋊C₄	—	4	1	no	—
508	C₂²⋊C₄	—	4	1	no	—
509	C₂²⋊C₄	—	4	1	no	—
510	C₄²	—	4	1	no	—
511	C₄²	C₂⁴×C₅	1	1	yes	characteristic
512	C₂×C₅	—	4	2	no	—
513	C₂×C₅	—	8	1	no	—
514	C₂×C₅	—	8	1	no	—
515	C₂×C₅	—	8	1	no	—
516	C₂×C₅	#3264	1	1	yes	characteristic
517	C₂×C₅	#3283	1	1	yes	characteristic
518	C₂×C₅	#3289	1	1	yes	characteristic
519	C₂³	—	2	1	no	—
520	C₂³	—	2	1	no	—
521	C₂³	—	2	1	no	—
522	C₂³	—	2	1	no	—
523	Q₈	—	4	1	no	—
524	Q₈	—	4	1	no	—
525	Q₈	—	4	1	no	—
526	Q₈	—	4	1	no	—
527	Q₈	—	4	1	no	—
528	Q₈	—	4	1	no	—
529	Q₈	—	8	1	no	—
530	D₄	—	2	1	no	—
531	D₄	—	2	1	no	—
532	D₄	—	2	1	no	—
533	D₄	—	2	1	no	—
534	D₄	—	2	1	no	—
535	D₄	—	2	1	no	—
536	D₄	—	2	1	no	—
537	D₄	—	8	2	no	—
538	D₄	—	8	2	no	—
539	D₄	—	8	2	no	—
540	D₄	—	16	1	no	—
541	D₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
542	C₂×C₄	—	2	1	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	—	4	1	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	—	8	2	no	—
559	C₂×C₄	—	8	2	no	—
560	C₂×C₄	—	8	2	no	—
561	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
562	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
563	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
564	C₈	—	2	1	no	—
565	C₈	—	2	1	no	—
566	C₈	—	2	1	no	—
567	C₈	—	4	1	no	—
568	C₈	—	4	1	no	—
569	C₈	—	4	1	no	—
570	C₈	—	4	1	no	—
571	C₈	—	4	1	no	—
572	C₈	C₅×#135	1	2	yes	complemented
573	C₅	#19859	1	1	yes	characteristic, direct product factor, Sylow
574	C₂²	—	2	2	no	—
575	C₂²	—	4	1	no	—
576	C₂²	—	4	1	no	—
577	C₂²	—	4	1	no	—
578	C₂²	—	4	1	no	—
579	C₂²	—	4	1	no	—
580	C₂²	—	4	1	no	—
581	C₂²	—	4	1	no	—
582	C₂²	—	4	1	no	—
583	C₂²	—	4	1	no	—
584	C₂²	—	4	1	no	—
585	C₂²	—	4	1	no	—
586	C₂²	C₅×D₄²	1	1	yes	characteristic
587	C₄	—	2	1	no	—
588	C₄	—	4	1	no	—
589	C₄	—	4	1	no	—
590	C₄	—	4	1	no	—
591	C₄	—	4	1	no	—
592	C₄	—	8	1	no	—
593	C₄	—	8	1	no	—
594	C₄	—	8	1	no	—
595	C₄	—	8	1	no	—
596	C₄	C₂×C₅×#135	1	1	yes	characteristic
597	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
598	C₄	C₅×#576	1	1	yes	characteristic
599	C₄	C₅×#578	1	1	yes	characteristic
600	C₂	—	4	2	no	—
601	C₂	—	8	1	no	—
602	C₂	—	8	1	no	—
603	C₂	—	8	1	no	—
604	C₂	C₅×#3264	1	1	yes	characteristic
605	C₂	C₅×#3283	1	1	yes	characteristic
606	C₂	C₅×#3289	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
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×#3354	C₂	semidirect
2. C₅×#3334	C₂	semidirect
3. C₅×#3303	C₂	semidirect
4. C₅×#3283	C₂	semidirect
5. C₅×#1723	C₂	semidirect
6. C₅×#1693	C₂	semidirect
7. C₅×#1683	C₂	semidirect
8. C₅×#1671	C₂	semidirect
9. C₅×#1669	C₂	semidirect
10. C₅×#1647	C₂	semidirect
11. C₅×#1647	C₂	semidirect
12. C₅×#1646	C₂	semidirect
13. C₅×#1635	C₂	semidirect
14. C₅×#1581	C₂	semidirect
15. C₅×#1580	C₂	semidirect
67. #19859	C₅	direct (N × H)
146. #3354	C₂×C₅	semidirect
147. #3334	C₂×C₅	semidirect
148. #3303	C₂×C₅	semidirect
149. #3283	C₂×C₅	semidirect
150. #1723	C₂×C₅	semidirect
151. #1693	C₂×C₅	semidirect
152. #1683	C₂×C₅	semidirect
153. #1671	C₂×C₅	semidirect
154. #1669	C₂×C₅	semidirect
155. #1647	C₂×C₅	semidirect
156. #1647	C₂×C₅	semidirect
157. #1646	C₂×C₅	semidirect
158. #1635	C₂×C₅	semidirect
159. #1581	C₂×C₅	semidirect
160. #1580	C₂×C₅	semidirect
336. C₅×C₈	#135	semidirect
572. C₈	C₅×#135	semidirect
573. C₅	#19859	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×#3354	2
C₅×#3334	2
C₅×#3303	2
C₅×#3283	2
C₅×#1723	2
C₅×#1693	2
C₅×#1683	2
C₅×#1671	2
C₅×#1669	2
C₅×#1647	2
C₅×#1646	2
C₅×#1635	2
C₅×#1581	2
C₅×#1580	2
#19859	5

Subgroups — C5×#19859 #11942579

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19859 #11942579