Subgroups — C₅×#26085 #11948805

Order 1280.

678 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3163	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3084	C₂	1	1	yes	maximal, characteristic
3	C₅×#3083	C₂	1	1	yes	maximal, characteristic
4	C₅×#2913	C₂	1	1	yes	maximal, characteristic
5	C₅×#2090	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2089	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#2061	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2060	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1952	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#1946	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#1860	C₂	1	1	yes	maximal, characteristic
12	C₅×#1859	C₂	1	1	yes	maximal, characteristic
13	C₅×#1847	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×#1815	C₂	1	1	yes	maximal, characteristic
15	C₅×#1765	C₂	1	1	yes	maximal, characteristic
16	C₂×C₅×#131	C₂²	1	1	yes	characteristic
17	C₅×#544	C₂²	1	1	yes	characteristic
18	C₅×#544	C₂²	1	1	yes	characteristic
19	C₅×#520	C₂²	1	1	yes	characteristic
20	C₅×#520	C₂²	1	1	yes	characteristic
21	C₅×#501	—	2	1	no	—
22	C₅×#501	—	2	1	no	—
23	C₅×#499	—	2	1	no	—
24	C₅×#498	—	2	1	no	—
25	C₅×#479	—	2	1	no	—
26	C₅×#479	—	2	1	no	—
27	C₅×#478	—	2	1	no	—
28	C₅×#478	—	2	1	no	—
29	C₅×#477	—	2	1	no	—
30	C₅×#476	—	2	1	no	—
31	C₅×#475	—	2	1	no	—
32	C₅×#474	—	2	1	no	—
33	C₂×C₅×#108	C₂²	1	1	yes	characteristic
34	C₅×#428	—	2	1	no	—
35	C₅×#428	—	2	1	no	—
36	C₅×#428	C₂²	1	1	yes	characteristic
37	C₅×#428	C₂²	1	1	yes	characteristic
38	C₅×#428	C₂²	1	1	yes	characteristic
39	C₅×#428	C₂²	1	1	yes	characteristic
40	C₅×#428	C₂²	1	1	yes	characteristic
41	C₅×#428	C₂²	1	1	yes	characteristic
42	C₅×#427	—	2	1	no	—
43	C₅×#427	C₂²	1	1	yes	characteristic
44	C₂×C₅×#107	—	2	1	no	—
45	C₂×C₅×#107	C₂²	1	1	yes	characteristic
46	C₂×C₅×#107	C₂²	1	1	yes	characteristic
47	C₂×C₅×#106	C₂²	1	1	yes	characteristic
48	C₂×C₅×#106	C₂²	1	1	yes	characteristic
49	C₅×#424	C₂²	1	1	yes	characteristic
50	C₅×#424	C₂²	1	1	yes	characteristic
51	C₂×C₅×#104	C₂²	1	1	yes	characteristic
52	C₅×#417	C₂²	1	1	yes	characteristic
53	C₅×#417	C₂²	1	1	yes	characteristic
54	C₅×#416	C₂²	1	1	yes	characteristic
55	C₅×#405	C₂²	1	1	yes	characteristic
56	C₂×C₅×#97	C₂²	1	1	yes	characteristic
57	C₅×#384	C₂²	1	1	yes	characteristic
58	C₅×#382	C₂²	1	1	yes	characteristic
59	C₅×#343	C₂²	1	1	yes	characteristic
60	C₅×#343	C₂²	1	1	yes	characteristic
61	C₅×#343	C₂²	1	1	yes	characteristic
62	C₅×#343	C₂²	1	1	yes	characteristic
63	C₅×#341	C₂²	1	1	yes	characteristic
64	C₅×#339	C₂²	1	1	yes	characteristic
65	C₅×#339	C₂²	1	1	yes	characteristic
66	C₅×#339	C₂²	1	1	yes	characteristic
67	#26085	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂×C₅×#41	C₂³	1	1	yes	characteristic
69	C₅×#131	—	2	2	no	—
70	C₅×#131	Q₈	1	4	yes	complemented
71	C₂×C₅×M₁₆	—	2	1	no	—
72	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
73	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
74	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
75	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
76	C₂²×C₅×C₈	—	2	1	no	—
77	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
78	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
79	C₅×C₄⋊Q₈	—	2	1	no	—
80	C₅×C₄⋊Q₈	—	2	1	no	—
81	C₅×C₄⋊Q₈	—	2	1	no	—
82	C₅×C₄⋊Q₈	—	2	1	no	—
83	C₅×#125	—	2	1	no	—
84	C₅×#125	—	2	1	no	—
85	C₅×#125	—	2	1	no	—
86	C₅×#125	—	2	1	no	—
87	C₅×C₂²⋊Q₈	—	4	1	no	—
88	C₅×C₂²⋊Q₈	—	4	1	no	—
89	C₄×C₅×Q₈	—	2	1	no	—
90	C₄×C₅×Q₈	—	2	1	no	—
91	C₄×C₅×D₄	—	2	1	no	—
92	C₄×C₅×D₄	—	2	1	no	—
93	C₄×C₅×D₄	—	4	1	no	—
94	C₄×C₅×D₄	—	4	1	no	—
95	C₅×#117	—	2	1	no	—
96	C₅×#117	—	2	1	no	—
97	C₅×#117	C₂³	1	1	yes	characteristic
98	C₅×#117	C₂³	1	1	yes	characteristic
99	C₅×#117	C₂³	1	1	yes	characteristic
100	C₂×C₅×C₄⋊C₄	—	2	1	no	—
101	C₂×C₅×C₄⋊C₄	—	2	1	no	—
102	C₂×C₅×C₄⋊C₄	—	4	1	no	—
103	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
104	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
105	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
106	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
107	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
108	C₅×#108	—	2	2	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	—	2	1	no	—
114	C₅×#107	—	2	1	no	—
115	C₅×#107	—	2	1	no	—
116	C₅×#107	—	2	1	no	—
117	C₅×#107	—	2	1	no	—
118	C₅×#107	—	2	1	no	—
119	C₅×#107	D₄	1	1	yes	characteristic
120	C₅×#107	D₄	1	1	yes	characteristic
121	C₅×#106	—	2	1	no	—
122	C₅×#106	—	2	1	no	—
123	C₅×#106	—	2	1	no	—
124	C₅×#106	—	2	1	no	—
125	C₅×#106	—	2	1	no	—
126	C₅×#106	—	2	1	no	—
127	C₅×#106	—	2	1	no	—
128	C₅×#106	—	2	1	no	—
129	C₅×#106	—	2	1	no	—
130	C₅×#106	—	2	1	no	—
131	C₅×#106	D₄	1	1	yes	characteristic
132	C₅×#106	D₄	1	1	yes	characteristic
133	C₅×C₄⋊C₈	—	2	1	no	—
134	C₅×C₄⋊C₈	—	2	1	no	—
135	C₅×C₄⋊C₈	—	2	1	no	—
136	C₅×C₄⋊C₈	—	2	1	no	—
137	C₅×#104	Q₈	1	4	yes	complemented
138	C₅×#103	—	2	1	no	—
139	C₅×#103	—	2	1	no	—
140	C₅×#103	D₄	1	1	yes	characteristic
141	C₅×#103	D₄	1	1	yes	characteristic
142	C₅×#102	—	2	1	no	—
143	C₅×#102	—	2	1	no	—
144	C₅×#102	D₄	1	1	yes	characteristic
145	C₅×#102	D₄	1	1	yes	characteristic
146	C₅×#97	—	2	1	no	—
147	C₅×#97	—	2	2	no	—
148	C₄×C₅×C₈	—	2	1	no	—
149	C₅×#95	—	4	1	no	—
150	#3163	C₂×C₅	1	1	yes	characteristic, complemented
151	#3084	C₂×C₅	1	1	yes	characteristic
152	#3083	C₂×C₅	1	1	yes	characteristic
153	#2913	C₂×C₅	1	1	yes	characteristic
154	#2090	C₂×C₅	1	1	yes	characteristic, complemented
155	#2089	C₂×C₅	1	1	yes	characteristic, complemented
156	#2061	C₂×C₅	1	1	yes	characteristic, complemented
157	#2060	C₂×C₅	1	1	yes	characteristic, complemented
158	#1952	C₂×C₅	1	1	yes	characteristic, complemented
159	#1946	C₂×C₅	1	1	yes	characteristic, complemented
160	#1860	C₂×C₅	1	1	yes	characteristic
161	#1859	C₂×C₅	1	1	yes	characteristic
162	#1847	C₂×C₅	1	1	yes	characteristic, complemented
163	#1815	C₂×C₅	1	1	yes	characteristic
164	#1765	C₂×C₅	1	1	yes	characteristic
165	C₅×#41	—	2	1	no	—
166	C₅×#41	—	2	1	no	—
167	C₅×#41	C₂×Q₈	1	2	yes	—
168	C₅×#41	C₂×Q₈	1	2	yes	—
169	C₂×C₅×Q₈	—	4	1	no	—
170	C₂×C₅×Q₈	—	4	1	no	—
171	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
172	C₂×C₅×D₄	—	2	1	no	—
173	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₅	—	2	1	no	—
175	C₂²×C₄×C₅	—	2	1	no	—
176	C₂²×C₄×C₅	—	2	1	no	—
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	—	4	1	no	—
181	C₂²×C₄×C₅	—	4	1	no	—
182	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
183	C₅×M₁₆	—	2	1	no	—
184	C₅×M₁₆	—	2	1	no	—
185	C₅×M₁₆	—	2	4	no	—
186	C₅×M₁₆	—	2	1	no	—
187	C₅×M₁₆	—	2	1	no	—
188	C₅×M₁₆	—	2	1	no	—
189	C₅×M₁₆	—	2	1	no	—
190	C₅×M₁₆	C₂×Q₈	1	2	yes	—
191	C₅×M₁₆	C₂×Q₈	1	2	yes	—
192	C₂×C₅×C₈	—	2	1	no	—
193	C₂×C₅×C₈	—	2	1	no	—
194	C₂×C₅×C₈	—	2	1	no	—
195	C₂×C₅×C₈	—	2	1	no	—
196	C₂×C₅×C₈	—	2	4	no	—
197	C₂×C₅×C₈	—	2	1	no	—
198	C₂×C₅×C₈	—	2	1	no	—
199	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
200	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
201	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
202	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
203	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
204	C₂×C₅×C₈	#41	1	2	yes	—
205	C₅×C₄⋊C₄	—	2	1	no	—
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	—	4	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	—	4	1	no	—
218	C₅×C₄⋊C₄	—	4	1	no	—
219	C₅×C₄⋊C₄	—	4	1	no	—
220	C₅×C₄⋊C₄	—	4	1	no	—
221	C₅×C₄⋊C₄	—	8	2	no	—
222	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
223	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
224	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
225	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
226	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
227	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
228	C₅×C₂²⋊C₄	—	2	1	no	—
229	C₅×C₂²⋊C₄	—	2	1	no	—
230	C₅×C₂²⋊C₄	—	2	1	no	—
231	C₅×C₂²⋊C₄	—	4	1	no	—
232	C₅×C₂²⋊C₄	—	4	1	no	—
233	C₄²×C₅	—	2	1	no	—
234	C₄²×C₅	—	2	1	no	—
235	C₄²×C₅	—	2	1	no	—
236	C₄²×C₅	—	2	1	no	—
237	C₄²×C₅	—	4	1	no	—
238	C₄²×C₅	—	4	1	no	—
239	C₄²×C₅	C₂×Q₈	1	2	yes	—
240	C₂×#131	C₂²×C₅	1	1	yes	characteristic
241	#544	C₂²×C₅	1	1	yes	characteristic
242	#544	C₂²×C₅	1	1	yes	characteristic
243	#520	C₂²×C₅	1	1	yes	characteristic
244	#520	C₂²×C₅	1	1	yes	characteristic
245	#501	—	2	1	no	—
246	#501	—	2	1	no	—
247	#499	—	2	1	no	—
248	#498	—	2	1	no	—
249	#479	—	2	1	no	—
250	#479	—	2	1	no	—
251	#478	—	2	1	no	—
252	#478	—	2	1	no	—
253	#477	—	2	1	no	—
254	#476	—	2	1	no	—
255	#475	—	2	1	no	—
256	#474	—	2	1	no	—
257	C₂×#108	C₂²×C₅	1	1	yes	characteristic
258	#428	—	2	1	no	—
259	#428	—	2	1	no	—
260	#428	C₂²×C₅	1	1	yes	characteristic
261	#428	C₂²×C₅	1	1	yes	characteristic
262	#428	C₂²×C₅	1	1	yes	characteristic
263	#428	C₂²×C₅	1	1	yes	characteristic
264	#428	C₂²×C₅	1	1	yes	characteristic
265	#428	C₂²×C₅	1	1	yes	characteristic
266	#427	—	2	1	no	—
267	#427	C₂²×C₅	1	1	yes	characteristic
268	C₂×#107	—	2	1	no	—
269	C₂×#107	C₂²×C₅	1	1	yes	characteristic
270	C₂×#107	C₂²×C₅	1	1	yes	characteristic
271	C₂×#106	C₂²×C₅	1	1	yes	characteristic
272	C₂×#106	C₂²×C₅	1	1	yes	characteristic
273	#424	C₂²×C₅	1	1	yes	characteristic
274	#424	C₂²×C₅	1	1	yes	characteristic
275	C₂×#104	C₂²×C₅	1	1	yes	characteristic
276	#417	C₂²×C₅	1	1	yes	characteristic
277	#417	C₂²×C₅	1	1	yes	characteristic
278	#416	C₂²×C₅	1	1	yes	characteristic
279	#405	C₂²×C₅	1	1	yes	characteristic
280	C₂×#97	C₂²×C₅	1	1	yes	characteristic
281	#384	C₂²×C₅	1	1	yes	characteristic
282	#382	C₂²×C₅	1	1	yes	characteristic
283	#343	C₂²×C₅	1	1	yes	characteristic
284	#343	C₂²×C₅	1	1	yes	characteristic
285	#343	C₂²×C₅	1	1	yes	characteristic
286	#343	C₂²×C₅	1	1	yes	characteristic
287	#341	C₂²×C₅	1	1	yes	characteristic
288	#339	C₂²×C₅	1	1	yes	characteristic
289	#339	C₂²×C₅	1	1	yes	characteristic
290	#339	C₂²×C₅	1	1	yes	characteristic
291	C₂³×C₅	—	2	1	no	—
292	C₂³×C₅	H(5,𝔽₂)	1	1	yes	characteristic
293	C₅×Q₈	—	2	1	no	—
294	C₅×Q₈	—	16	1	no	—
295	C₅×Q₈	—	16	1	no	—
296	C₅×Q₈	C₄⋊Q₈	1	2	yes	—
297	C₅×D₄	—	2	2	no	—
298	C₅×D₄	—	2	2	no	—
299	C₅×D₄	—	2	1	no	—
300	C₅×D₄	C₄⋊Q₈	1	2	yes	—
301	C₂×C₄×C₅	—	2	1	no	—
302	C₂×C₄×C₅	—	2	2	no	—
303	C₂×C₄×C₅	—	2	2	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	—	2	1	no	—
310	C₂×C₄×C₅	—	2	2	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	—	8	1	no	—
318	C₂×C₄×C₅	—	8	1	no	—
319	C₂×C₄×C₅	—	8	1	no	—
320	C₂×C₄×C₅	—	8	1	no	—
321	C₂×C₄×C₅	—	16	1	no	—
322	C₂×C₄×C₅	—	16	1	no	—
323	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
324	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
325	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
326	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
327	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
328	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
329	C₅×C₈	—	2	2	no	—
330	C₅×C₈	—	2	1	no	—
331	C₅×C₈	—	2	1	no	—
332	C₅×C₈	—	2	2	no	—
333	C₅×C₈	—	4	1	no	—
334	C₅×C₈	—	4	1	no	—
335	C₅×C₈	C₂²⋊Q₈	1	2	yes	—
336	C₅×C₈	C₂²⋊Q₈	1	2	yes	—
337	C₂×#41	C₂³×C₅	1	1	yes	characteristic
338	#131	—	2	2	no	—
339	#131	C₅×Q₈	1	4	yes	complemented
340	C₂×M₁₆	—	2	1	no	—
341	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
342	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
343	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
344	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
345	C₂²×C₈	—	2	1	no	—
346	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
347	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
348	C₄⋊Q₈	—	2	1	no	—
349	C₄⋊Q₈	—	2	1	no	—
350	C₄⋊Q₈	—	2	1	no	—
351	C₄⋊Q₈	—	2	1	no	—
352	#125	—	2	1	no	—
353	#125	—	2	1	no	—
354	#125	—	2	1	no	—
355	#125	—	2	1	no	—
356	C₂²⋊Q₈	—	4	1	no	—
357	C₂²⋊Q₈	—	4	1	no	—
358	C₄×Q₈	—	2	1	no	—
359	C₄×Q₈	—	2	1	no	—
360	C₄×D₄	—	2	1	no	—
361	C₄×D₄	—	2	1	no	—
362	C₄×D₄	—	4	1	no	—
363	C₄×D₄	—	4	1	no	—
364	#117	—	2	1	no	—
365	#117	—	2	1	no	—
366	#117	C₂³×C₅	1	1	yes	characteristic
367	#117	C₂³×C₅	1	1	yes	characteristic
368	#117	C₂³×C₅	1	1	yes	characteristic
369	C₂×C₄⋊C₄	—	2	1	no	—
370	C₂×C₄⋊C₄	—	2	1	no	—
371	C₂×C₄⋊C₄	—	4	1	no	—
372	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
373	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
374	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
375	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
376	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
377	#108	—	2	2	no	—
378	#107	—	2	1	no	—
379	#107	—	2	1	no	—
380	#107	—	2	1	no	—
381	#107	—	2	1	no	—
382	#107	—	2	1	no	—
383	#107	—	2	1	no	—
384	#107	—	2	1	no	—
385	#107	—	2	1	no	—
386	#107	—	2	1	no	—
387	#107	—	2	1	no	—
388	#107	C₅×D₄	1	1	yes	characteristic
389	#107	C₅×D₄	1	1	yes	characteristic
390	#106	—	2	1	no	—
391	#106	—	2	1	no	—
392	#106	—	2	1	no	—
393	#106	—	2	1	no	—
394	#106	—	2	1	no	—
395	#106	—	2	1	no	—
396	#106	—	2	1	no	—
397	#106	—	2	1	no	—
398	#106	—	2	1	no	—
399	#106	—	2	1	no	—
400	#106	C₅×D₄	1	1	yes	characteristic
401	#106	C₅×D₄	1	1	yes	characteristic
402	C₄⋊C₈	—	2	1	no	—
403	C₄⋊C₈	—	2	1	no	—
404	C₄⋊C₈	—	2	1	no	—
405	C₄⋊C₈	—	2	1	no	—
406	#104	C₅×Q₈	1	4	yes	complemented
407	#103	—	2	1	no	—
408	#103	—	2	1	no	—
409	#103	C₅×D₄	1	1	yes	characteristic
410	#103	C₅×D₄	1	1	yes	characteristic
411	#102	—	2	1	no	—
412	#102	—	2	1	no	—
413	#102	C₅×D₄	1	1	yes	characteristic
414	#102	C₅×D₄	1	1	yes	characteristic
415	#97	—	2	1	no	—
416	#97	—	2	2	no	—
417	C₄×C₈	—	2	1	no	—
418	#95	—	4	1	no	—
419	C₂²×C₅	—	2	1	no	—
420	C₂²×C₅	—	2	1	no	—
421	C₂²×C₅	—	2	2	no	—
422	C₂²×C₅	—	4	1	no	—
423	C₂²×C₅	—	4	1	no	—
424	C₂²×C₅	#544	1	1	yes	characteristic
425	C₂²×C₅	D₄²	1	1	yes	characteristic
426	C₂²×C₅	#554	1	1	yes	characteristic
427	C₄×C₅	—	2	2	no	—
428	C₄×C₅	—	4	2	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	—	4	1	no	—
434	C₄×C₅	—	4	1	no	—
435	C₄×C₅	—	8	1	no	—
436	C₄×C₅	—	8	1	no	—
437	C₄×C₅	—	8	1	no	—
438	C₄×C₅	—	8	1	no	—
439	C₄×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
440	C₄×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
441	C₄×C₅	#547	1	1	yes	characteristic
442	C₄×C₅	D₄×Q₈	1	1	yes	characteristic
443	#41	—	2	1	no	—
444	#41	—	2	1	no	—
445	#41	C₂×C₅×Q₈	1	2	yes	—
446	#41	C₂×C₅×Q₈	1	2	yes	—
447	C₂×Q₈	—	4	1	no	—
448	C₂×Q₈	—	4	1	no	—
449	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
450	C₂×D₄	—	2	1	no	—
451	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
452	C₂²×C₄	—	2	1	no	—
453	C₂²×C₄	—	2	1	no	—
454	C₂²×C₄	—	2	1	no	—
455	C₂²×C₄	—	2	1	no	—
456	C₂²×C₄	—	2	1	no	—
457	C₂²×C₄	—	2	1	no	—
458	C₂²×C₄	—	4	1	no	—
459	C₂²×C₄	—	4	1	no	—
460	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
461	M₁₆	—	2	1	no	—
462	M₁₆	—	2	1	no	—
463	M₁₆	—	2	4	no	—
464	M₁₆	—	2	1	no	—
465	M₁₆	—	2	1	no	—
466	M₁₆	—	2	1	no	—
467	M₁₆	—	2	1	no	—
468	M₁₆	C₂×C₅×Q₈	1	2	yes	—
469	M₁₆	C₂×C₅×Q₈	1	2	yes	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	—	2	1	no	—
472	C₂×C₈	—	2	1	no	—
473	C₂×C₈	—	2	1	no	—
474	C₂×C₈	—	2	4	no	—
475	C₂×C₈	—	2	1	no	—
476	C₂×C₈	—	2	1	no	—
477	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
478	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
479	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
480	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
481	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
482	C₂×C₈	C₅×#41	1	2	yes	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	2	1	no	—
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	—	4	1	no	—
492	C₄⋊C₄	—	4	1	no	—
493	C₄⋊C₄	—	4	1	no	—
494	C₄⋊C₄	—	4	1	no	—
495	C₄⋊C₄	—	4	1	no	—
496	C₄⋊C₄	—	4	1	no	—
497	C₄⋊C₄	—	4	1	no	—
498	C₄⋊C₄	—	4	1	no	—
499	C₄⋊C₄	—	8	2	no	—
500	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
501	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
502	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
503	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
504	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
505	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
506	C₂²⋊C₄	—	2	1	no	—
507	C₂²⋊C₄	—	2	1	no	—
508	C₂²⋊C₄	—	2	1	no	—
509	C₂²⋊C₄	—	4	1	no	—
510	C₂²⋊C₄	—	4	1	no	—
511	C₄²	—	2	1	no	—
512	C₄²	—	2	1	no	—
513	C₄²	—	2	1	no	—
514	C₄²	—	2	1	no	—
515	C₄²	—	4	1	no	—
516	C₄²	—	4	1	no	—
517	C₄²	C₂×C₅×Q₈	1	2	yes	—
518	C₂×C₅	—	2	1	no	—
519	C₂×C₅	—	2	1	no	—
520	C₂×C₅	—	4	2	no	—
521	C₂×C₅	#2440	1	1	yes	characteristic
522	C₂×C₅	#3288	1	1	yes	characteristic
523	C₂×C₅	#3289	1	1	yes	characteristic
524	C₂³	—	2	1	no	—
525	C₂³	C₅×H(5,𝔽₂)	1	1	yes	characteristic
526	Q₈	—	2	1	no	—
527	Q₈	—	16	1	no	—
528	Q₈	—	16	1	no	—
529	Q₈	C₅×C₄⋊Q₈	1	2	yes	—
530	D₄	—	2	2	no	—
531	D₄	—	2	2	no	—
532	D₄	—	2	1	no	—
533	D₄	C₅×C₄⋊Q₈	1	2	yes	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	1	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	—	4	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	—	8	1	no	—
551	C₂×C₄	—	8	1	no	—
552	C₂×C₄	—	8	1	no	—
553	C₂×C₄	—	8	1	no	—
554	C₂×C₄	—	16	1	no	—
555	C₂×C₄	—	16	1	no	—
556	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
557	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
558	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
559	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
560	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
561	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
562	C₈	—	2	2	no	—
563	C₈	—	2	1	no	—
564	C₈	—	2	1	no	—
565	C₈	—	2	2	no	—
566	C₈	—	4	1	no	—
567	C₈	—	4	1	no	—
568	C₈	C₅×C₂²⋊Q₈	1	2	yes	—
569	C₈	C₅×C₂²⋊Q₈	1	2	yes	—
570	C₅	#26085	1	1	yes	characteristic, direct product factor, Sylow
571	C₂²	—	2	1	no	—
572	C₂²	—	2	1	no	—
573	C₂²	—	2	2	no	—
574	C₂²	—	4	1	no	—
575	C₂²	—	4	1	no	—
576	C₂²	C₅×#544	1	1	yes	characteristic
577	C₂²	C₅×D₄²	1	1	yes	characteristic
578	C₂²	C₅×#554	1	1	yes	characteristic
579	C₄	—	2	2	no	—
580	C₄	—	4	2	no	—
581	C₄	—	4	1	no	—
582	C₄	—	4	1	no	—
583	C₄	—	4	1	no	—
584	C₄	—	4	1	no	—
585	C₄	—	4	1	no	—
586	C₄	—	4	1	no	—
587	C₄	—	8	1	no	—
588	C₄	—	8	1	no	—
589	C₄	—	8	1	no	—
590	C₄	—	8	1	no	—
591	C₄	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
592	C₄	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
593	C₄	C₅×#547	1	1	yes	characteristic
594	C₄	C₅×D₄×Q₈	1	1	yes	characteristic
595	C₂	—	2	1	no	—
596	C₂	—	2	1	no	—
597	C₂	—	4	2	no	—
598	C₂	C₅×#2440	1	1	yes	characteristic
599	C₂	C₅×#3288	1	1	yes	characteristic
600	C₂	C₅×#3289	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂×C₅×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#3163	C₂	semidirect
5. C₅×#2090	C₂	semidirect
6. C₅×#2089	C₂	semidirect
7. C₅×#2061	C₂	semidirect
8. C₅×#2060	C₂	semidirect
9. C₅×#1952	C₂	semidirect
10. C₅×#1946	C₂	semidirect
13. C₅×#1847	C₂	semidirect
67. #26085	C₅	direct (N × H)
70. C₅×#131	Q₈	semidirect
137. C₅×#104	Q₈	semidirect
150. #3163	C₂×C₅	semidirect
154. #2090	C₂×C₅	semidirect
155. #2089	C₂×C₅	semidirect
156. #2061	C₂×C₅	semidirect
157. #2060	C₂×C₅	semidirect
158. #1952	C₂×C₅	semidirect
159. #1946	C₂×C₅	semidirect
162. #1847	C₂×C₅	semidirect
339. #131	C₅×Q₈	semidirect
406. #104	C₅×Q₈	semidirect
570. C₅	#26085	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×#3163	2
C₅×#3084	2
C₅×#3083	2
C₅×#2913	2
C₅×#2090	2
C₅×#2089	2
C₅×#2061	2
C₅×#2060	2
C₅×#1952	2
C₅×#1946	2
C₅×#1860	2
C₅×#1859	2
C₅×#1847	2
C₅×#1815	2
C₅×#1765	2
#26085	5

Subgroups — C5×#26085 #11948805

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#26085 #11948805