Subgroups — C₅×#25266 #11947986

Order 1280.

750 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3092	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#484	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3056	C₂	1	1	yes	maximal, characteristic
4	C₅×#2933	C₂	1	1	yes	maximal, characteristic
5	C₅×#2089	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2078	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#2064	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2052	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1952	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#1930	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#1908	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#1901	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅×#1855	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×#1816	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₅×#343	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₅×#137	C₂²	1	1	yes	characteristic
17	C₂×C₅×#131	C₂²	1	1	yes	characteristic
18	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
19	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
20	C₅×#525	C₂²	1	1	yes	characteristic
21	C₂×C₅×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
22	C₂²×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₅×#484	—	2	1	no	—
24	C₅×#484	—	2	1	no	—
25	C₅×#484	—	2	1	no	—
26	C₅×#484	—	2	1	no	—
27	C₅×#484	—	2	1	no	—
28	C₅×#483	—	2	1	no	—
29	C₅×#481	—	2	1	no	—
30	C₅×#480	—	2	1	no	—
31	C₅×#470	—	2	1	no	—
32	C₅×#468	—	2	1	no	—
33	C₅×#467	—	2	1	no	—
34	C₅×#465	—	2	1	no	—
35	C₅×#436	—	2	1	no	—
36	C₅×#435	—	2	1	no	—
37	C₅×#435	—	2	1	no	—
38	C₅×C₈×D₄	—	2	1	no	—
39	C₂×C₅×#108	C₂²	1	1	yes	characteristic
40	C₂×C₅×#108	C₂²	1	1	yes	characteristic
41	C₅×#428	C₂²	1	1	yes	characteristic
42	C₅×#428	C₂²	1	1	yes	characteristic
43	C₅×#428	C₂²	1	1	yes	characteristic
44	C₂×C₅×#107	C₂²	1	1	yes	characteristic
45	C₂×C₅×#107	C₂²	1	1	yes	characteristic
46	C₅×#424	C₂²	1	1	yes	characteristic
47	C₅×#419	C₂²	1	1	yes	characteristic
48	C₅×#417	C₂²	1	1	yes	characteristic
49	C₅×#417	C₂²	1	1	yes	characteristic
50	C₅×#417	C₂²	1	1	yes	characteristic
51	C₂×C₅×#103	C₂²	1	1	yes	characteristic
52	C₂×C₅×#103	C₂²	1	1	yes	characteristic
53	C₂×C₅×#101	C₂²	1	1	yes	characteristic
54	C₂×C₅×#101	C₂²	1	1	yes	characteristic
55	C₅×#408	C₂²	1	1	yes	characteristic
56	C₅×#407	C₂²	1	1	yes	characteristic
57	C₂×C₅×C₂²⋊C₈	C₂²	1	1	yes	characteristic
58	C₅×#405	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×#343	C₂²	1	1	yes	characteristic
64	C₅×#343	C₂²	1	1	yes	characteristic
65	C₅×#341	C₂²	1	1	yes	characteristic
66	C₅×#341	C₂²	1	1	yes	characteristic
67	#25266	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂×C₅×#41	C₂³	1	1	yes	characteristic
69	C₂²×C₅×Q₈	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₅	—	2	1	no	—
71	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
72	C₅×#137	—	2	2	no	—
73	C₅×#137	—	4	1	no	—
74	C₂×C₅×Q₁₆	—	2	1	no	—
75	C₂×C₅×SD₁₆	—	2	1	no	—
76	C₅×#131	—	2	2	no	—
77	C₅×#131	—	4	1	no	—
78	C₂×C₅×M₁₆	—	2	1	no	—
79	C₂×C₅×M₁₆	—	2	1	no	—
80	C₂×C₅×M₁₆	—	2	1	no	—
81	C₂×C₅×M₁₆	—	2	2	no	—
82	C₂×C₅×M₁₆	—	2	2	no	—
83	C₂×C₅×M₁₆	—	4	1	no	—
84	C₂×C₅×M₁₆	—	4	1	no	—
85	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
86	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
87	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
88	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
89	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
90	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
91	C₂²×C₅×C₈	—	2	1	no	—
92	C₂²×C₅×C₈	—	2	1	no	—
93	C₂²×C₅×C₈	—	2	1	no	—
94	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
95	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
96	C₅×#123	—	2	1	no	—
97	C₅×C₂²⋊Q₈	—	2	1	no	—
98	C₅×C₂²⋊Q₈	—	2	1	no	—
99	C₅×C₂²⋊Q₈	—	2	1	no	—
100	C₅×C₂²⋊Q₈	—	2	1	no	—
101	C₅×C₂²⋊Q₈	D₄	1	1	yes	characteristic
102	C₅×C₂²⋊Q₈	D₄	1	1	yes	characteristic
103	C₅×#121	D₄	1	1	yes	characteristic
104	C₅×#121	D₄	1	1	yes	characteristic
105	C₅×#120	—	2	1	no	—
106	C₄×C₅×D₄	—	2	1	no	—
107	C₄×C₅×D₄	—	2	1	no	—
108	C₅×#117	C₂³	1	1	yes	characteristic
109	C₂×C₅×C₄⋊C₄	—	2	1	no	—
110	C₂×C₅×C₄⋊C₄	—	2	1	no	—
111	C₂×C₅×C₄⋊C₄	—	2	1	no	—
112	C₂×C₅×C₄⋊C₄	—	2	1	no	—
113	C₂×C₅×C₄⋊C₄	—	2	1	no	—
114	C₂×C₅×C₄⋊C₄	—	2	1	no	—
115	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
116	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
117	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
118	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
119	C₅×#108	—	2	2	no	—
120	C₅×#108	—	4	1	no	—
121	C₅×#107	—	2	1	no	—
122	C₅×#107	—	2	1	no	—
123	C₅×#107	—	2	1	no	—
124	C₅×#107	—	2	1	no	—
125	C₅×#107	—	2	1	no	—
126	C₅×#107	—	2	1	no	—
127	C₅×#107	—	2	1	no	—
128	C₅×#106	—	2	1	no	—
129	C₅×#106	—	2	1	no	—
130	C₅×#106	—	2	1	no	—
131	C₅×C₄⋊C₈	—	2	1	no	—
132	C₅×C₄⋊C₈	—	2	1	no	—
133	C₅×#103	—	2	1	no	—
134	C₅×#103	—	2	1	no	—
135	C₅×#103	—	2	1	no	—
136	C₅×#103	—	2	1	no	—
137	C₅×#103	—	2	1	no	—
138	C₅×#103	—	2	1	no	—
139	C₅×#103	—	2	1	no	—
140	C₅×#103	—	2	1	no	—
141	C₅×#103	—	2	1	no	—
142	C₅×#102	—	2	1	no	—
143	C₅×#102	—	2	1	no	—
144	C₅×#102	—	2	1	no	—
145	C₅×#101	—	2	2	no	—
146	C₅×#101	—	4	1	no	—
147	C₅×C₂²⋊C₈	—	2	1	no	—
148	C₅×C₂²⋊C₈	—	2	1	no	—
149	C₅×C₂²⋊C₈	—	2	1	no	—
150	C₅×C₂²⋊C₈	—	2	1	no	—
151	C₅×C₂²⋊C₈	D₄	1	1	yes	characteristic
152	C₅×C₂²⋊C₈	D₄	1	1	yes	characteristic
153	C₅×C₂²⋊C₈	D₄	1	1	yes	characteristic
154	C₅×C₂²⋊C₈	D₄	1	1	yes	characteristic
155	C₅×#97	—	2	1	no	—
156	C₄×C₅×C₈	—	2	1	no	—
157	#3092	C₂×C₅	1	1	yes	characteristic
158	C₂×#484	C₂×C₅	1	1	yes	characteristic, complemented
159	#3056	C₂×C₅	1	1	yes	characteristic
160	#2933	C₂×C₅	1	1	yes	characteristic
161	#2089	C₂×C₅	1	1	yes	characteristic, complemented
162	#2078	C₂×C₅	1	1	yes	characteristic, complemented
163	#2064	C₂×C₅	1	1	yes	characteristic, complemented
164	#2052	C₂×C₅	1	1	yes	characteristic, complemented
165	#1952	C₂×C₅	1	1	yes	characteristic, complemented
166	#1930	C₂×C₅	1	1	yes	characteristic, complemented
167	#1908	C₂×C₅	1	1	yes	characteristic, complemented
168	#1901	C₂×C₅	1	1	yes	characteristic, complemented
169	#1855	C₂×C₅	1	1	yes	characteristic, complemented
170	#1816	C₂×C₅	1	1	yes	characteristic, complemented
171	C₂×#343	C₂×C₅	1	1	yes	characteristic, complemented
172	C₂⁴×C₅	#41	1	1	yes	characteristic
173	C₅×#41	—	2	1	no	—
174	C₅×#41	—	2	1	no	—
175	C₅×#41	—	2	1	no	—
176	C₅×#41	—	2	1	no	—
177	C₂×C₅×Q₈	—	2	1	no	—
178	C₂×C₅×Q₈	—	2	1	no	—
179	C₂×C₅×Q₈	—	2	1	no	—
180	C₂×C₅×Q₈	—	2	1	no	—
181	C₂×C₅×Q₈	—	2	1	no	—
182	C₂×C₅×Q₈	—	2	1	no	—
183	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
184	C₂×C₅×D₄	—	2	1	no	—
185	C₂×C₅×D₄	—	2	1	no	—
186	C₂×C₅×D₄	—	2	1	no	—
187	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
188	C₂²×C₄×C₅	—	2	1	no	—
189	C₂²×C₄×C₅	—	2	1	no	—
190	C₂²×C₄×C₅	—	2	1	no	—
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	—	2	1	no	—
195	C₂²×C₄×C₅	—	2	1	no	—
196	C₂²×C₄×C₅	—	2	1	no	—
197	C₂²×C₄×C₅	—	4	1	no	—
198	C₂²×C₄×C₅	—	4	1	no	—
199	C₂²×C₄×C₅	—	8	2	no	—
200	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
201	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
202	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
203	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
204	C₂²×C₄×C₅	#41	1	1	yes	characteristic
205	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
206	C₅×Q₁₆	—	4	2	no	—
207	C₅×SD₁₆	—	4	2	no	—
208	C₅×M₁₆	—	2	1	no	—
209	C₅×M₁₆	—	2	1	no	—
210	C₅×M₁₆	—	2	1	no	—
211	C₅×M₁₆	—	2	1	no	—
212	C₅×M₁₆	—	2	1	no	—
213	C₅×M₁₆	—	2	1	no	—
214	C₅×M₁₆	—	2	1	no	—
215	C₅×M₁₆	—	2	1	no	—
216	C₅×M₁₆	—	2	1	no	—
217	C₅×M₁₆	—	2	1	no	—
218	C₅×M₁₆	—	4	2	no	—
219	C₅×M₁₆	—	4	2	no	—
220	C₅×M₁₆	—	4	2	no	—
221	C₅×M₁₆	#41	1	2	yes	—
222	C₅×M₁₆	#41	1	2	yes	—
223	C₂×C₅×C₈	—	2	1	no	—
224	C₂×C₅×C₈	—	2	1	no	—
225	C₂×C₅×C₈	—	2	1	no	—
226	C₂×C₅×C₈	—	2	1	no	—
227	C₂×C₅×C₈	—	2	1	no	—
228	C₂×C₅×C₈	—	2	1	no	—
229	C₂×C₅×C₈	—	2	1	no	—
230	C₂×C₅×C₈	—	2	1	no	—
231	C₂×C₅×C₈	—	2	1	no	—
232	C₂×C₅×C₈	—	2	1	no	—
233	C₂×C₅×C₈	—	4	2	no	—
234	C₂×C₅×C₈	—	4	2	no	—
235	C₂×C₅×C₈	—	4	2	no	—
236	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
237	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
238	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
239	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
240	C₅×C₄⋊C₄	—	2	1	no	—
241	C₅×C₄⋊C₄	—	2	1	no	—
242	C₅×C₄⋊C₄	—	2	1	no	—
243	C₅×C₄⋊C₄	—	2	1	no	—
244	C₅×C₄⋊C₄	—	2	1	no	—
245	C₅×C₄⋊C₄	—	2	1	no	—
246	C₅×C₄⋊C₄	—	2	1	no	—
247	C₅×C₄⋊C₄	—	2	1	no	—
248	C₅×C₄⋊C₄	—	4	1	no	—
249	C₅×C₄⋊C₄	—	4	1	no	—
250	C₅×C₄⋊C₄	—	4	1	no	—
251	C₅×C₄⋊C₄	—	4	1	no	—
252	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
253	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
254	C₅×C₂²⋊C₄	—	2	1	no	—
255	C₅×C₂²⋊C₄	—	2	1	no	—
256	C₅×C₂²⋊C₄	—	2	1	no	—
257	C₅×C₂²⋊C₄	—	2	1	no	—
258	C₅×C₂²⋊C₄	—	2	1	no	—
259	C₅×C₂²⋊C₄	—	4	1	no	—
260	C₅×C₂²⋊C₄	—	4	1	no	—
261	C₄²×C₅	—	2	1	no	—
262	C₂×#137	C₂²×C₅	1	1	yes	characteristic
263	C₂×#131	C₂²×C₅	1	1	yes	characteristic
264	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
265	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
266	#525	C₂²×C₅	1	1	yes	characteristic
267	C₂×C₂²⋊Q₈	C₂²×C₅	1	1	yes	characteristic
268	C₂²×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
269	#484	—	2	1	no	—
270	#484	—	2	1	no	—
271	#484	—	2	1	no	—
272	#484	—	2	1	no	—
273	#484	—	2	1	no	—
274	#483	—	2	1	no	—
275	#481	—	2	1	no	—
276	#480	—	2	1	no	—
277	#470	—	2	1	no	—
278	#468	—	2	1	no	—
279	#467	—	2	1	no	—
280	#465	—	2	1	no	—
281	#436	—	2	1	no	—
282	#435	—	2	1	no	—
283	#435	—	2	1	no	—
284	C₈×D₄	—	2	1	no	—
285	C₂×#108	C₂²×C₅	1	1	yes	characteristic
286	C₂×#108	C₂²×C₅	1	1	yes	characteristic
287	#428	C₂²×C₅	1	1	yes	characteristic
288	#428	C₂²×C₅	1	1	yes	characteristic
289	#428	C₂²×C₅	1	1	yes	characteristic
290	C₂×#107	C₂²×C₅	1	1	yes	characteristic
291	C₂×#107	C₂²×C₅	1	1	yes	characteristic
292	#424	C₂²×C₅	1	1	yes	characteristic
293	#419	C₂²×C₅	1	1	yes	characteristic
294	#417	C₂²×C₅	1	1	yes	characteristic
295	#417	C₂²×C₅	1	1	yes	characteristic
296	#417	C₂²×C₅	1	1	yes	characteristic
297	C₂×#103	C₂²×C₅	1	1	yes	characteristic
298	C₂×#103	C₂²×C₅	1	1	yes	characteristic
299	C₂×#101	C₂²×C₅	1	1	yes	characteristic
300	C₂×#101	C₂²×C₅	1	1	yes	characteristic
301	#408	C₂²×C₅	1	1	yes	characteristic
302	#407	C₂²×C₅	1	1	yes	characteristic
303	C₂×C₂²⋊C₈	C₂²×C₅	1	1	yes	characteristic
304	#405	C₂²×C₅	1	1	yes	characteristic
305	#343	C₂²×C₅	1	1	yes	characteristic
306	#343	C₂²×C₅	1	1	yes	characteristic
307	#343	C₂²×C₅	1	1	yes	characteristic
308	#343	C₂²×C₅	1	1	yes	characteristic
309	#343	C₂²×C₅	1	1	yes	characteristic
310	#343	C₂²×C₅	1	1	yes	characteristic
311	#341	C₂²×C₅	1	1	yes	characteristic
312	#341	C₂²×C₅	1	1	yes	characteristic
313	C₂³×C₅	—	2	1	no	—
314	C₂³×C₅	—	2	1	no	—
315	C₂³×C₅	—	2	2	no	—
316	C₂³×C₅	—	2	1	no	—
317	C₂³×C₅	—	4	1	no	—
318	C₂³×C₅	#121	1	1	yes	characteristic
319	C₂³×C₅	#121	1	1	yes	characteristic
320	C₂³×C₅	C₂×#41	1	1	yes	characteristic
321	C₅×Q₈	—	2	1	no	—
322	C₅×Q₈	—	2	1	no	—
323	C₅×Q₈	—	4	1	no	—
324	C₅×Q₈	—	4	1	no	—
325	C₅×Q₈	—	4	1	no	—
326	C₅×Q₈	—	4	1	no	—
327	C₅×D₄	—	2	1	no	—
328	C₅×D₄	—	2	1	no	—
329	C₅×D₄	—	4	1	no	—
330	C₅×D₄	—	4	1	no	—
331	C₅×D₄	—	4	2	no	—
332	C₅×D₄	—	8	1	no	—
333	C₂×C₄×C₅	—	2	1	no	—
334	C₂×C₄×C₅	—	2	1	no	—
335	C₂×C₄×C₅	—	2	1	no	—
336	C₂×C₄×C₅	—	2	1	no	—
337	C₂×C₄×C₅	—	2	1	no	—
338	C₂×C₄×C₅	—	2	1	no	—
339	C₂×C₄×C₅	—	2	1	no	—
340	C₂×C₄×C₅	—	2	1	no	—
341	C₂×C₄×C₅	—	2	1	no	—
342	C₂×C₄×C₅	—	2	1	no	—
343	C₂×C₄×C₅	—	2	1	no	—
344	C₂×C₄×C₅	—	2	1	no	—
345	C₂×C₄×C₅	—	2	1	no	—
346	C₂×C₄×C₅	—	4	1	no	—
347	C₂×C₄×C₅	—	4	1	no	—
348	C₂×C₄×C₅	—	4	1	no	—
349	C₂×C₄×C₅	—	4	1	no	—
350	C₂×C₄×C₅	—	4	1	no	—
351	C₂×C₄×C₅	—	4	1	no	—
352	C₂×C₄×C₅	—	4	1	no	—
353	C₂×C₄×C₅	—	4	1	no	—
354	C₂×C₄×C₅	—	8	2	no	—
355	C₂×C₄×C₅	—	8	2	no	—
356	C₂×C₄×C₅	—	8	1	no	—
357	C₂×C₄×C₅	—	8	1	no	—
358	C₂×C₄×C₅	—	8	1	no	—
359	C₂×C₄×C₅	#121	1	1	yes	characteristic
360	C₂×C₄×C₅	#121	1	1	yes	characteristic
361	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
362	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
363	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
364	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
365	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
366	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
367	C₅×C₈	—	2	1	no	—
368	C₅×C₈	—	2	1	no	—
369	C₅×C₈	—	2	2	no	—
370	C₅×C₈	—	2	2	no	—
371	C₅×C₈	—	2	1	no	—
372	C₅×C₈	—	2	1	no	—
373	C₅×C₈	—	4	1	no	—
374	C₅×C₈	—	4	1	no	—
375	C₅×C₈	—	4	1	no	—
376	C₅×C₈	—	4	1	no	—
377	C₂×#41	C₂³×C₅	1	1	yes	characteristic
378	C₂²×Q₈	C₂³×C₅	1	1	yes	characteristic
379	C₂³×C₄	—	2	1	no	—
380	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
381	#137	—	2	2	no	—
382	#137	—	4	1	no	—
383	C₂×Q₁₆	—	2	1	no	—
384	C₂×SD₁₆	—	2	1	no	—
385	#131	—	2	2	no	—
386	#131	—	4	1	no	—
387	C₂×M₁₆	—	2	1	no	—
388	C₂×M₁₆	—	2	1	no	—
389	C₂×M₁₆	—	2	1	no	—
390	C₂×M₁₆	—	2	2	no	—
391	C₂×M₁₆	—	2	2	no	—
392	C₂×M₁₆	—	4	1	no	—
393	C₂×M₁₆	—	4	1	no	—
394	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
395	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
396	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
397	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
398	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
399	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
400	C₂²×C₈	—	2	1	no	—
401	C₂²×C₈	—	2	1	no	—
402	C₂²×C₈	—	2	1	no	—
403	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
404	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
405	#123	—	2	1	no	—
406	C₂²⋊Q₈	—	2	1	no	—
407	C₂²⋊Q₈	—	2	1	no	—
408	C₂²⋊Q₈	—	2	1	no	—
409	C₂²⋊Q₈	—	2	1	no	—
410	C₂²⋊Q₈	C₅×D₄	1	1	yes	characteristic
411	C₂²⋊Q₈	C₅×D₄	1	1	yes	characteristic
412	#121	C₅×D₄	1	1	yes	characteristic
413	#121	C₅×D₄	1	1	yes	characteristic
414	#120	—	2	1	no	—
415	C₄×D₄	—	2	1	no	—
416	C₄×D₄	—	2	1	no	—
417	#117	C₂³×C₅	1	1	yes	characteristic
418	C₂×C₄⋊C₄	—	2	1	no	—
419	C₂×C₄⋊C₄	—	2	1	no	—
420	C₂×C₄⋊C₄	—	2	1	no	—
421	C₂×C₄⋊C₄	—	2	1	no	—
422	C₂×C₄⋊C₄	—	2	1	no	—
423	C₂×C₄⋊C₄	—	2	1	no	—
424	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
425	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
426	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
427	C₂×C₂²⋊C₄	—	2	1	no	—
428	#108	—	2	2	no	—
429	#108	—	4	1	no	—
430	#107	—	2	1	no	—
431	#107	—	2	1	no	—
432	#107	—	2	1	no	—
433	#107	—	2	1	no	—
434	#107	—	2	1	no	—
435	#107	—	2	1	no	—
436	#107	—	2	1	no	—
437	#106	—	2	1	no	—
438	#106	—	2	1	no	—
439	#106	—	2	1	no	—
440	C₄⋊C₈	—	2	1	no	—
441	C₄⋊C₈	—	2	1	no	—
442	#103	—	2	1	no	—
443	#103	—	2	1	no	—
444	#103	—	2	1	no	—
445	#103	—	2	1	no	—
446	#103	—	2	1	no	—
447	#103	—	2	1	no	—
448	#103	—	2	1	no	—
449	#103	—	2	1	no	—
450	#103	—	2	1	no	—
451	#102	—	2	1	no	—
452	#102	—	2	1	no	—
453	#102	—	2	1	no	—
454	#101	—	2	2	no	—
455	#101	—	4	1	no	—
456	C₂²⋊C₈	—	2	1	no	—
457	C₂²⋊C₈	—	2	1	no	—
458	C₂²⋊C₈	—	2	1	no	—
459	C₂²⋊C₈	—	2	1	no	—
460	C₂²⋊C₈	C₅×D₄	1	1	yes	characteristic
461	C₂²⋊C₈	C₅×D₄	1	1	yes	characteristic
462	C₂²⋊C₈	C₅×D₄	1	1	yes	characteristic
463	C₂²⋊C₈	C₅×D₄	1	1	yes	characteristic
464	#97	—	2	1	no	—
465	C₄×C₈	—	2	1	no	—
466	C₂²×C₅	—	2	1	no	—
467	C₂²×C₅	—	2	1	no	—
468	C₂²×C₅	—	2	1	no	—
469	C₂²×C₅	—	2	1	no	—
470	C₂²×C₅	—	2	1	no	—
471	C₂²×C₅	—	2	1	no	—
472	C₂²×C₅	—	2	1	no	—
473	C₂²×C₅	—	4	2	no	—
474	C₂²×C₅	—	4	2	no	—
475	C₂²×C₅	—	4	1	no	—
476	C₂²×C₅	—	4	1	no	—
477	C₂²×C₅	—	4	1	no	—
478	C₂²×C₅	#473	1	2	yes	—
479	C₂²×C₅	C₂×#121	1	1	yes	characteristic
480	C₂²×C₅	#529	1	1	yes	characteristic
481	C₂²×C₅	#539	1	1	yes	characteristic
482	C₄×C₅	—	2	1	no	—
483	C₄×C₅	—	2	1	no	—
484	C₄×C₅	—	4	1	no	—
485	C₄×C₅	—	4	1	no	—
486	C₄×C₅	—	4	1	no	—
487	C₄×C₅	—	8	1	no	—
488	C₄×C₅	—	8	1	no	—
489	C₄×C₅	—	8	1	no	—
490	C₄×C₅	—	8	1	no	—
491	C₄×C₅	#546	1	1	yes	characteristic
492	C₄×C₅	#546	1	1	yes	characteristic
493	C₄×C₅	#547	1	1	yes	characteristic
494	C₄×C₅	#547	1	1	yes	characteristic
495	C₂⁴	C₅×#41	1	1	yes	characteristic
496	#41	—	2	1	no	—
497	#41	—	2	1	no	—
498	#41	—	2	1	no	—
499	#41	—	2	1	no	—
500	C₂×Q₈	—	2	1	no	—
501	C₂×Q₈	—	2	1	no	—
502	C₂×Q₈	—	2	1	no	—
503	C₂×Q₈	—	2	1	no	—
504	C₂×Q₈	—	2	1	no	—
505	C₂×Q₈	—	2	1	no	—
506	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
507	C₂×D₄	—	2	1	no	—
508	C₂×D₄	—	2	1	no	—
509	C₂×D₄	—	2	1	no	—
510	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
511	C₂²×C₄	—	2	1	no	—
512	C₂²×C₄	—	2	1	no	—
513	C₂²×C₄	—	2	1	no	—
514	C₂²×C₄	—	2	1	no	—
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	—	2	1	no	—
520	C₂²×C₄	—	4	1	no	—
521	C₂²×C₄	—	4	1	no	—
522	C₂²×C₄	—	8	2	no	—
523	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
524	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
525	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
526	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
527	C₂²×C₄	C₅×#41	1	1	yes	characteristic
528	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
529	Q₁₆	—	4	2	no	—
530	SD₁₆	—	4	2	no	—
531	M₁₆	—	2	1	no	—
532	M₁₆	—	2	1	no	—
533	M₁₆	—	2	1	no	—
534	M₁₆	—	2	1	no	—
535	M₁₆	—	2	1	no	—
536	M₁₆	—	2	1	no	—
537	M₁₆	—	2	1	no	—
538	M₁₆	—	2	1	no	—
539	M₁₆	—	2	1	no	—
540	M₁₆	—	2	1	no	—
541	M₁₆	—	4	2	no	—
542	M₁₆	—	4	2	no	—
543	M₁₆	—	4	2	no	—
544	M₁₆	C₅×#41	1	2	yes	—
545	M₁₆	C₅×#41	1	2	yes	—
546	C₂×C₈	—	2	1	no	—
547	C₂×C₈	—	2	1	no	—
548	C₂×C₈	—	2	1	no	—
549	C₂×C₈	—	2	1	no	—
550	C₂×C₈	—	2	1	no	—
551	C₂×C₈	—	2	1	no	—
552	C₂×C₈	—	2	1	no	—
553	C₂×C₈	—	2	1	no	—
554	C₂×C₈	—	2	1	no	—
555	C₂×C₈	—	2	1	no	—
556	C₂×C₈	—	4	2	no	—
557	C₂×C₈	—	4	2	no	—
558	C₂×C₈	—	4	2	no	—
559	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
560	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
561	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
562	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
563	C₄⋊C₄	—	2	1	no	—
564	C₄⋊C₄	—	2	1	no	—
565	C₄⋊C₄	—	2	1	no	—
566	C₄⋊C₄	—	2	1	no	—
567	C₄⋊C₄	—	2	1	no	—
568	C₄⋊C₄	—	2	1	no	—
569	C₄⋊C₄	—	2	1	no	—
570	C₄⋊C₄	—	2	1	no	—
571	C₄⋊C₄	—	4	1	no	—
572	C₄⋊C₄	—	4	1	no	—
573	C₄⋊C₄	—	4	1	no	—
574	C₄⋊C₄	—	4	1	no	—
575	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
576	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
577	C₂²⋊C₄	—	2	1	no	—
578	C₂²⋊C₄	—	2	1	no	—
579	C₂²⋊C₄	—	2	1	no	—
580	C₂²⋊C₄	—	2	1	no	—
581	C₂²⋊C₄	—	2	1	no	—
582	C₂²⋊C₄	—	4	1	no	—
583	C₂²⋊C₄	—	4	1	no	—
584	C₄²	—	2	1	no	—
585	C₂×C₅	—	2	1	no	—
586	C₂×C₅	—	2	1	no	—
587	C₂×C₅	—	2	2	no	—
588	C₂×C₅	—	4	1	no	—
589	C₂×C₅	—	8	1	no	—
590	C₂×C₅	#2425	1	1	yes	characteristic
591	C₂×C₅	C₂×#473	1	1	yes	characteristic
592	C₂×C₅	#3068	1	1	yes	characteristic
593	C₂³	—	2	2	no	—
594	C₂³	—	2	1	no	—
595	C₂³	—	2	1	no	—
596	C₂³	—	2	1	no	—
597	C₂³	—	4	1	no	—
598	C₂³	C₅×#121	1	1	yes	characteristic
599	C₂³	C₅×#121	1	1	yes	characteristic
600	C₂³	C₂×C₅×#41	1	1	yes	characteristic
601	Q₈	—	2	1	no	—
602	Q₈	—	2	1	no	—
603	Q₈	—	4	1	no	—
604	Q₈	—	4	1	no	—
605	Q₈	—	4	1	no	—
606	Q₈	—	4	1	no	—
607	D₄	—	2	1	no	—
608	D₄	—	2	1	no	—
609	D₄	—	4	2	no	—
610	D₄	—	4	1	no	—
611	D₄	—	4	1	no	—
612	D₄	—	8	1	no	—
613	C₂×C₄	—	2	1	no	—
614	C₂×C₄	—	2	1	no	—
615	C₂×C₄	—	2	1	no	—
616	C₂×C₄	—	2	1	no	—
617	C₂×C₄	—	2	1	no	—
618	C₂×C₄	—	2	1	no	—
619	C₂×C₄	—	2	1	no	—
620	C₂×C₄	—	2	1	no	—
621	C₂×C₄	—	2	1	no	—
622	C₂×C₄	—	2	1	no	—
623	C₂×C₄	—	2	1	no	—
624	C₂×C₄	—	2	1	no	—
625	C₂×C₄	—	2	1	no	—
626	C₂×C₄	—	4	1	no	—
627	C₂×C₄	—	4	1	no	—
628	C₂×C₄	—	4	1	no	—
629	C₂×C₄	—	4	1	no	—
630	C₂×C₄	—	4	1	no	—
631	C₂×C₄	—	4	1	no	—
632	C₂×C₄	—	4	1	no	—
633	C₂×C₄	—	4	1	no	—
634	C₂×C₄	—	8	2	no	—
635	C₂×C₄	—	8	2	no	—
636	C₂×C₄	—	8	1	no	—
637	C₂×C₄	—	8	1	no	—
638	C₂×C₄	—	8	1	no	—
639	C₂×C₄	C₅×#121	1	1	yes	characteristic
640	C₂×C₄	C₅×#121	1	1	yes	characteristic
641	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
642	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
643	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
644	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
645	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
646	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
647	C₈	—	2	1	no	—
648	C₈	—	2	1	no	—
649	C₈	—	2	2	no	—
650	C₈	—	2	2	no	—
651	C₈	—	2	1	no	—
652	C₈	—	2	1	no	—
653	C₈	—	4	1	no	—
654	C₈	—	4	1	no	—
655	C₈	—	4	1	no	—
656	C₈	—	4	1	no	—
657	C₅	#25266	1	1	yes	characteristic, direct product factor, Sylow
658	C₂²	—	2	1	no	—
659	C₂²	—	2	1	no	—
660	C₂²	—	2	1	no	—
661	C₂²	—	2	1	no	—
662	C₂²	—	2	1	no	—
663	C₂²	—	2	1	no	—
664	C₂²	—	2	1	no	—
665	C₂²	—	4	2	no	—
666	C₂²	—	4	2	no	—
667	C₂²	—	4	1	no	—
668	C₂²	—	4	1	no	—
669	C₂²	—	4	1	no	—
670	C₂²	C₅×#473	1	2	yes	—
671	C₂²	C₂×C₅×#121	1	1	yes	characteristic
672	C₂²	C₅×#529	1	1	yes	characteristic
673	C₂²	C₅×#539	1	1	yes	characteristic
674	C₄	—	2	1	no	—
675	C₄	—	2	1	no	—
676	C₄	—	4	1	no	—
677	C₄	—	4	1	no	—
678	C₄	—	4	1	no	—
679	C₄	—	8	1	no	—
680	C₄	—	8	1	no	—
681	C₄	—	8	1	no	—
682	C₄	—	8	1	no	—
683	C₄	C₅×#546	1	1	yes	characteristic
684	C₄	C₅×#546	1	1	yes	characteristic
685	C₄	C₅×#547	1	1	yes	characteristic
686	C₄	C₅×#547	1	1	yes	characteristic
687	C₂	—	2	2	no	—
688	C₂	—	2	1	no	—
689	C₂	—	2	1	no	—
690	C₂	—	4	1	no	—
691	C₂	—	8	1	no	—
692	C₂	C₅×#2425	1	1	yes	characteristic
693	C₂	C₂×C₅×#473	1	1	yes	characteristic
694	C₂	C₅×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₅×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#484	C₂	semidirect
5. C₅×#2089	C₂	semidirect
6. C₅×#2078	C₂	semidirect
7. C₅×#2064	C₂	semidirect
8. C₅×#2052	C₂	semidirect
9. C₅×#1952	C₂	semidirect
10. C₅×#1930	C₂	semidirect
11. C₅×#1908	C₂	semidirect
12. C₅×#1901	C₂	semidirect
13. C₅×#1855	C₂	semidirect
14. C₅×#1816	C₂	semidirect
15. C₂×C₅×#343	C₂	semidirect
67. #25266	C₅	direct (N × H)
158. C₂×#484	C₂×C₅	semidirect
161. #2089	C₂×C₅	semidirect
162. #2078	C₂×C₅	semidirect
163. #2064	C₂×C₅	semidirect
164. #2052	C₂×C₅	semidirect
165. #1952	C₂×C₅	semidirect
166. #1930	C₂×C₅	semidirect
167. #1908	C₂×C₅	semidirect
168. #1901	C₂×C₅	semidirect
169. #1855	C₂×C₅	semidirect
170. #1816	C₂×C₅	semidirect
171. C₂×#343	C₂×C₅	semidirect
657. C₅	#25266	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×#3092	2
C₂×C₅×#484	2
C₅×#3056	2
C₅×#2933	2
C₅×#2089	2
C₅×#2078	2
C₅×#2064	2
C₅×#2052	2
C₅×#1952	2
C₅×#1930	2
C₅×#1908	2
C₅×#1901	2
C₅×#1855	2
C₅×#1816	2
C₂×C₅×#343	2
#25266	5

Subgroups — C5×#25266 #11947986

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#25266 #11947986