Subgroups — C₅×#17724 #11940444

Order 1280.

1004 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3508	C₂	1	1	yes	maximal, characteristic
2	C₅×#2988	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2925	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1553	C₂	1	2	yes	maximal, complemented
5	C₅×#1551	C₂	1	2	yes	maximal, complemented
6	C₅×#1550	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1545	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1543	C₂	1	2	yes	maximal, complemented
9	C₅×#1542	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#1540	C₂	1	2	yes	maximal, complemented
11	C₂×C₅×#331	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#559	—	2	1	no	—
13	C₅×#559	C₄	1	2	yes	—
14	C₅×#550	—	2	1	no	—
15	C₅×#550	C₄	1	2	yes	—
16	C₅×#546	—	2	2	no	—
17	C₅×#546	C₄	1	4	yes	—
18	C₅×#540	—	2	2	no	—
19	C₅×#535	—	2	1	no	—
20	C₅×#533	C₂²	1	1	yes	characteristic
21	C₅×#532	C₂²	1	2	yes	—
22	C₅×#532	C₂²	1	2	yes	—
23	C₂×C₅×#127	C₂²	1	1	yes	characteristic
24	C₅×#518	—	2	1	no	—
25	C₄×C₅×#41	C₂²	1	1	yes	characteristic
26	C₅×C₈×Q₈	—	2	1	no	—
27	C₅×C₈×D₄	—	2	2	no	—
28	C₅×C₈×D₄	—	2	1	no	—
29	C₅×#433	—	2	2	no	—
30	C₅×#432	C₂²	1	2	yes	complemented
31	C₅×#431	C₂²	1	1	yes	characteristic
32	C₅×#431	C₂²	1	2	yes	complemented
33	C₅×#431	C₂²	1	2	yes	complemented
34	C₅×#423	C₂²	1	1	yes	characteristic
35	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
36	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	complemented
38	C₅×#408	—	2	1	no	—
39	C₅×#405	—	2	1	no	—
40	C₅×#333	C₂²	1	1	yes	characteristic, complemented
41	C₅×#333	C₂²	1	1	yes	characteristic, complemented
42	C₅×#333	C₂²	1	2	yes	complemented
43	C₅×#332	C₂²	1	1	yes	characteristic, complemented
44	C₅×#332	C₂²	1	2	yes	complemented
45	C₅×#332	C₂²	1	1	yes	characteristic, complemented
46	C₅×#331	C₂²	1	2	yes	—
47	C₅×#331	C₂²	1	2	yes	—
48	C₅×#331	C₂²	1	2	yes	—
49	C₅×#331	C₂²	1	2	yes	—
50	#17724	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂×C₅×#41	—	2	2	no	—
52	C₂×C₅×#41	C₂×C₄	1	4	yes	—
53	C₂×C₅×#41	D₄	1	1	yes	characteristic
54	C₂²×C₅×D₄	—	2	1	no	—
55	C₂²×C₅×D₄	—	2	1	no	—
56	C₂²×C₅×D₄	C₂×C₄	1	2	yes	—
57	C₂×C₅×M₁₆	—	2	1	no	—
58	C₂²×C₅×C₈	—	2	1	no	—
59	C₂²×C₅×C₈	—	2	1	no	—
60	C₂²×C₅×C₈	—	2	2	no	—
61	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
62	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
63	C₅×#127	—	2	2	no	—
64	C₅×#127	—	2	1	no	—
65	C₅×#127	—	2	1	no	—
66	C₅×#127	C₂×C₄	1	2	yes	—
67	C₅×#127	C₂³	1	2	yes	—
68	C₅×#125	—	2	1	no	—
69	C₅×#124	—	2	2	no	—
70	C₅×#124	—	2	2	no	—
71	C₅×#124	C₂×C₄	1	4	yes	—
72	C₅×#123	—	2	4	no	—
73	C₅×#123	—	2	4	no	—
74	C₅×C₂²⋊Q₈	—	2	1	no	—
75	C₅×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
76	C₅×#121	—	2	2	no	—
77	C₅×#121	—	2	2	no	—
78	C₅×#121	—	2	2	no	—
79	C₅×#121	—	2	4	no	—
80	C₅×#121	—	2	4	no	—
81	C₅×#121	C₂×C₄	1	4	yes	—
82	C₅×#120	—	2	4	no	—
83	C₄×C₅×Q₈	—	2	1	no	—
84	C₄×C₅×Q₈	C₂×C₄	1	2	yes	—
85	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
86	C₄×C₅×Q₈	D₄	1	1	yes	characteristic
87	C₄×C₅×D₄	—	2	2	no	—
88	C₄×C₅×D₄	—	2	2	no	—
89	C₄×C₅×D₄	—	2	2	no	—
90	C₄×C₅×D₄	—	2	2	no	—
91	C₄×C₅×D₄	—	2	2	no	—
92	C₄×C₅×D₄	C₂×C₄	1	4	yes	—
93	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
94	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
95	C₄×C₅×D₄	D₄	1	2	yes	—
96	C₄×C₅×D₄	D₄	1	1	yes	characteristic
97	C₅×#117	—	2	1	no	—
98	C₅×#117	D₄	1	1	yes	characteristic
99	C₅×#117	D₄	1	2	yes	—
100	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
101	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₅	C₂³	1	2	yes	—
103	C₅×C₄⋊C₈	—	2	1	no	—
104	C₅×C₄⋊C₈	—	2	1	no	—
105	C₅×C₄⋊C₈	—	2	2	no	—
106	C₅×C₄⋊C₈	—	2	1	no	—
107	C₅×C₄⋊C₈	C₂³	1	2	yes	—
108	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
109	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
110	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
111	C₅×C₄⋊C₈	C₂³	1	2	yes	—
112	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
113	C₅×C₂²⋊C₈	—	2	2	no	—
114	C₅×C₂²⋊C₈	—	2	2	no	—
115	C₅×C₂²⋊C₈	—	2	1	no	—
116	C₅×C₂²⋊C₈	—	2	2	no	—
117	C₅×#97	—	2	2	no	—
118	C₄×C₅×C₈	—	2	2	no	—
119	C₄×C₅×C₈	—	2	1	no	—
120	C₄×C₅×C₈	—	2	2	no	—
121	#3508	C₂×C₅	1	1	yes	characteristic
122	#2988	C₂×C₅	1	1	yes	characteristic, complemented
123	#2925	C₂×C₅	1	1	yes	characteristic, complemented
124	#1553	C₂×C₅	1	2	yes	complemented
125	#1551	C₂×C₅	1	2	yes	complemented
126	#1550	C₂×C₅	1	1	yes	characteristic, complemented
127	#1545	C₂×C₅	1	1	yes	characteristic, complemented
128	#1543	C₂×C₅	1	2	yes	complemented
129	#1542	C₂×C₅	1	1	yes	characteristic, complemented
130	#1540	C₂×C₅	1	2	yes	complemented
131	C₂×#331	C₂×C₅	1	1	yes	characteristic, complemented
132	C₂⁴×C₅	—	2	2	no	—
133	C₅×#41	—	2	2	no	—
134	C₅×#41	—	2	2	no	—
135	C₅×#41	—	8	2	no	—
136	C₅×#41	—	8	2	no	—
137	C₅×#41	—	8	4	no	—
138	C₂×C₅×Q₈	—	2	1	no	—
139	C₂×C₅×Q₈	C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
141	C₂×C₅×D₄	—	2	4	no	—
142	C₂×C₅×D₄	—	2	4	no	—
143	C₂×C₅×D₄	—	2	2	no	—
144	C₂×C₅×D₄	—	2	2	no	—
145	C₂×C₅×D₄	—	2	2	no	—
146	C₂×C₅×D₄	—	2	2	no	—
147	C₂×C₅×D₄	—	2	2	no	—
148	C₂×C₅×D₄	—	4	2	no	—
149	C₂×C₅×D₄	—	4	2	no	—
150	C₂×C₅×D₄	—	8	1	no	—
151	C₂×C₅×D₄	—	8	1	no	—
152	C₂×C₅×D₄	—	8	2	no	—
153	C₂×C₅×D₄	C₂²⋊C₄	1	2	yes	—
154	C₂×C₅×D₄	C₂²⋊C₄	1	1	yes	characteristic
155	C₂×C₅×D₄	C₂²×C₄	1	4	yes	—
156	C₂²×C₄×C₅	—	2	4	no	—
157	C₂²×C₄×C₅	—	2	4	no	—
158	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
159	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
160	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
161	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
162	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
164	C₅×M₁₆	—	4	2	no	—
165	C₂×C₅×C₈	—	2	2	no	—
166	C₂×C₅×C₈	—	2	1	no	—
167	C₂×C₅×C₈	—	2	1	no	—
168	C₂×C₅×C₈	—	2	1	no	—
169	C₂×C₅×C₈	—	2	2	no	—
170	C₂×C₅×C₈	—	2	1	no	—
171	C₂×C₅×C₈	—	4	2	no	—
172	C₂×C₅×C₈	—	8	1	no	—
173	C₂×C₅×C₈	—	8	2	no	—
174	C₅×C₄⋊C₄	—	2	2	no	—
175	C₅×C₄⋊C₄	—	2	2	no	—
176	C₅×C₄⋊C₄	—	2	1	no	—
177	C₅×C₄⋊C₄	—	2	1	no	—
178	C₅×C₄⋊C₄	—	2	1	no	—
179	C₅×C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
180	C₅×C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
181	C₅×C₄⋊C₄	C₂²⋊C₄	1	2	yes	—
182	C₅×C₄⋊C₄	C₂²⋊C₄	1	2	yes	—
183	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
184	C₅×C₂²⋊C₄	—	2	2	no	—
185	C₅×C₂²⋊C₄	—	2	4	no	—
186	C₅×C₂²⋊C₄	—	2	4	no	—
187	C₅×C₂²⋊C₄	—	2	4	no	—
188	C₅×C₂²⋊C₄	—	2	4	no	—
189	C₅×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	—
190	C₅×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	—
191	C₅×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	—
192	C₄²×C₅	—	2	2	no	—
193	C₄²×C₅	C₂²×C₄	1	2	yes	—
194	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
195	C₄²×C₅	C₂×D₄	1	2	yes	—
196	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
197	C₄²×C₅	C₂×D₄	1	2	yes	—
198	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
199	C₄²×C₅	C₂⁴	1	1	yes	characteristic
200	#559	—	2	1	no	—
201	#559	C₄×C₅	1	2	yes	—
202	#550	—	2	1	no	—
203	#550	C₄×C₅	1	2	yes	—
204	#546	—	2	2	no	—
205	#546	C₄×C₅	1	4	yes	—
206	#540	—	2	2	no	—
207	#535	—	2	1	no	—
208	#533	C₂²×C₅	1	1	yes	characteristic
209	#532	C₂²×C₅	1	2	yes	—
210	#532	C₂²×C₅	1	2	yes	—
211	C₂×#127	C₂²×C₅	1	1	yes	characteristic
212	#518	—	2	1	no	—
213	C₄×#41	C₂²×C₅	1	1	yes	characteristic
214	C₈×Q₈	—	2	1	no	—
215	C₈×D₄	—	2	1	no	—
216	C₈×D₄	—	2	2	no	—
217	#433	—	2	2	no	—
218	#432	C₂²×C₅	1	2	yes	complemented
219	#431	C₂²×C₅	1	1	yes	characteristic
220	#431	C₂²×C₅	1	2	yes	complemented
221	#431	C₂²×C₅	1	2	yes	complemented
222	#423	C₂²×C₅	1	1	yes	characteristic
223	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
224	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
225	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	complemented
226	#408	—	2	1	no	—
227	#405	—	2	1	no	—
228	#333	C₂²×C₅	1	1	yes	characteristic, complemented
229	#333	C₂²×C₅	1	1	yes	characteristic, complemented
230	#333	C₂²×C₅	1	2	yes	complemented
231	#332	C₂²×C₅	1	1	yes	characteristic, complemented
232	#332	C₂²×C₅	1	2	yes	complemented
233	#332	C₂²×C₅	1	1	yes	characteristic, complemented
234	#331	C₂²×C₅	1	2	yes	—
235	#331	C₂²×C₅	1	2	yes	—
236	#331	C₂²×C₅	1	2	yes	—
237	#331	C₂²×C₅	1	2	yes	—
238	C₂³×C₅	—	2	4	no	—
239	C₂³×C₅	—	4	4	no	—
240	C₂³×C₅	—	8	2	no	—
241	C₂³×C₅	—	8	2	no	—
242	C₂³×C₅	C₂×C₂²⋊C₄	1	2	yes	—
243	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
244	C₅×Q₈	—	2	2	no	—
245	C₅×Q₈	—	4	2	no	—
246	C₅×Q₈	—	4	1	no	—
247	C₅×Q₈	—	4	1	no	—
248	C₅×D₄	—	2	4	no	—
249	C₅×D₄	—	2	2	no	—
250	C₅×D₄	—	4	4	no	—
251	C₅×D₄	—	4	2	no	—
252	C₅×D₄	—	4	2	no	—
253	C₅×D₄	—	4	2	no	—
254	C₅×D₄	—	4	2	no	—
255	C₅×D₄	—	4	4	no	—
256	C₅×D₄	—	8	2	no	—
257	C₅×D₄	—	8	2	no	—
258	C₅×D₄	—	8	4	no	—
259	C₅×D₄	—	8	4	no	—
260	C₂×C₄×C₅	—	2	2	no	—
261	C₂×C₄×C₅	—	2	2	no	—
262	C₂×C₄×C₅	—	2	1	no	—
263	C₂×C₄×C₅	—	2	2	no	—
264	C₂×C₄×C₅	—	2	1	no	—
265	C₂×C₄×C₅	—	2	2	no	—
266	C₂×C₄×C₅	—	2	2	no	—
267	C₂×C₄×C₅	—	2	2	no	—
268	C₂×C₄×C₅	—	4	1	no	—
269	C₂×C₄×C₅	—	4	2	no	—
270	C₂×C₄×C₅	—	8	4	no	—
271	C₂×C₄×C₅	—	8	4	no	—
272	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
273	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
274	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
275	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
276	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
277	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
278	C₂×C₄×C₅	#135	1	2	yes	—
279	C₂×C₄×C₅	#135	1	2	yes	—
280	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
281	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
282	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
283	C₅×C₈	—	4	1	no	—
284	C₅×C₈	—	4	1	no	—
285	C₅×C₈	—	4	2	no	—
286	C₅×C₈	—	4	2	no	—
287	C₅×C₈	—	4	1	no	—
288	C₅×C₈	—	4	1	no	—
289	C₂×#41	—	2	2	no	—
290	C₂×#41	C₂×C₄×C₅	1	4	yes	—
291	C₂×#41	C₅×D₄	1	1	yes	characteristic
292	C₂²×D₄	—	2	1	no	—
293	C₂²×D₄	—	2	1	no	—
294	C₂²×D₄	C₂×C₄×C₅	1	2	yes	—
295	C₂×M₁₆	—	2	1	no	—
296	C₂²×C₈	—	2	1	no	—
297	C₂²×C₈	—	2	1	no	—
298	C₂²×C₈	—	2	2	no	—
299	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
300	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
301	#127	—	2	2	no	—
302	#127	—	2	1	no	—
303	#127	—	2	1	no	—
304	#127	C₂×C₄×C₅	1	2	yes	—
305	#127	C₂³×C₅	1	2	yes	—
306	#125	—	2	1	no	—
307	#124	—	2	2	no	—
308	#124	—	2	2	no	—
309	#124	C₂×C₄×C₅	1	4	yes	—
310	#123	—	2	4	no	—
311	#123	—	2	4	no	—
312	C₂²⋊Q₈	—	2	1	no	—
313	C₂²⋊Q₈	C₂×C₄×C₅	1	2	yes	—
314	#121	—	2	2	no	—
315	#121	—	2	2	no	—
316	#121	—	2	2	no	—
317	#121	—	2	4	no	—
318	#121	—	2	4	no	—
319	#121	C₂×C₄×C₅	1	4	yes	—
320	#120	—	2	4	no	—
321	C₄×Q₈	—	2	1	no	—
322	C₄×Q₈	C₂×C₄×C₅	1	2	yes	—
323	C₄×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
324	C₄×Q₈	C₅×D₄	1	1	yes	characteristic
325	C₄×D₄	—	2	2	no	—
326	C₄×D₄	—	2	2	no	—
327	C₄×D₄	—	2	2	no	—
328	C₄×D₄	—	2	2	no	—
329	C₄×D₄	—	2	2	no	—
330	C₄×D₄	C₂×C₄×C₅	1	4	yes	—
331	C₄×D₄	C₂×C₄×C₅	1	2	yes	—
332	C₄×D₄	C₂×C₄×C₅	1	1	yes	characteristic
333	C₄×D₄	C₅×D₄	1	2	yes	—
334	C₄×D₄	C₅×D₄	1	1	yes	characteristic
335	#117	—	2	1	no	—
336	#117	C₅×D₄	1	1	yes	characteristic
337	#117	C₅×D₄	1	2	yes	—
338	C₂×C₂²⋊C₄	—	2	4	no	—
339	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
340	C₂×C₄²	C₂³×C₅	1	2	yes	—
341	C₄⋊C₈	—	2	1	no	—
342	C₄⋊C₈	—	2	1	no	—
343	C₄⋊C₈	—	2	2	no	—
344	C₄⋊C₈	—	2	1	no	—
345	C₄⋊C₈	C₂³×C₅	1	2	yes	—
346	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
347	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
348	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
349	C₄⋊C₈	C₂³×C₅	1	2	yes	—
350	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
351	C₂²⋊C₈	—	2	2	no	—
352	C₂²⋊C₈	—	2	2	no	—
353	C₂²⋊C₈	—	2	1	no	—
354	C₂²⋊C₈	—	2	2	no	—
355	#97	—	2	2	no	—
356	C₄×C₈	—	2	2	no	—
357	C₄×C₈	—	2	1	no	—
358	C₄×C₈	—	2	2	no	—
359	C₂²×C₅	—	2	1	no	—
360	C₂²×C₅	—	2	1	no	—
361	C₂²×C₅	—	2	2	no	—
362	C₂²×C₅	—	2	2	no	—
363	C₂²×C₅	—	2	2	no	—
364	C₂²×C₅	—	4	4	no	—
365	C₂²×C₅	—	4	4	no	—
366	C₂²×C₅	—	4	4	no	—
367	C₂²×C₅	—	16	4	no	—
368	C₂²×C₅	#416	1	2	yes	—
369	C₂²×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
370	C₄×C₅	—	2	1	no	—
371	C₄×C₅	—	2	2	no	—
372	C₄×C₅	—	2	1	no	—
373	C₄×C₅	—	2	2	no	—
374	C₄×C₅	—	2	2	no	—
375	C₄×C₅	—	2	1	no	—
376	C₄×C₅	—	4	2	no	—
377	C₄×C₅	—	4	2	no	—
378	C₄×C₅	#416	1	2	yes	—
379	C₄×C₅	C₂×#135	1	1	yes	characteristic
380	C₄×C₅	C₂×#135	1	1	yes	characteristic
381	C₄×C₅	#576	1	1	yes	characteristic
382	C₄×C₅	#577	1	1	yes	characteristic
383	C₂⁴	—	2	2	no	—
384	#41	—	2	2	no	—
385	#41	—	2	2	no	—
386	#41	—	8	2	no	—
387	#41	—	8	2	no	—
388	#41	—	8	4	no	—
389	C₂×Q₈	—	2	1	no	—
390	C₂×Q₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
391	C₂×Q₈	C₂²×C₄×C₅	1	2	yes	—
392	C₂×D₄	—	2	4	no	—
393	C₂×D₄	—	2	4	no	—
394	C₂×D₄	—	2	2	no	—
395	C₂×D₄	—	2	2	no	—
396	C₂×D₄	—	2	2	no	—
397	C₂×D₄	—	2	2	no	—
398	C₂×D₄	—	2	2	no	—
399	C₂×D₄	—	4	2	no	—
400	C₂×D₄	—	4	2	no	—
401	C₂×D₄	—	8	1	no	—
402	C₂×D₄	—	8	1	no	—
403	C₂×D₄	—	8	2	no	—
404	C₂×D₄	C₅×C₂²⋊C₄	1	2	yes	—
405	C₂×D₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
406	C₂×D₄	C₂²×C₄×C₅	1	4	yes	—
407	C₂²×C₄	—	2	4	no	—
408	C₂²×C₄	—	2	4	no	—
409	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
410	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
411	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
412	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
413	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
414	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
415	M₁₆	—	4	2	no	—
416	C₂×C₈	—	2	1	no	—
417	C₂×C₈	—	2	1	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	—	2	1	no	—
420	C₂×C₈	—	2	2	no	—
421	C₂×C₈	—	2	1	no	—
422	C₂×C₈	—	4	2	no	—
423	C₂×C₈	—	8	1	no	—
424	C₂×C₈	—	8	2	no	—
425	C₄⋊C₄	—	2	2	no	—
426	C₄⋊C₄	—	2	2	no	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	2	1	no	—
429	C₄⋊C₄	—	2	1	no	—
430	C₄⋊C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
431	C₄⋊C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
432	C₄⋊C₄	C₅×C₂²⋊C₄	1	2	yes	—
433	C₄⋊C₄	C₅×C₂²⋊C₄	1	2	yes	—
434	C₄⋊C₄	C₂²×C₄×C₅	1	2	yes	—
435	C₂²⋊C₄	—	2	2	no	—
436	C₂²⋊C₄	—	2	4	no	—
437	C₂²⋊C₄	—	2	4	no	—
438	C₂²⋊C₄	—	2	4	no	—
439	C₂²⋊C₄	—	2	4	no	—
440	C₂²⋊C₄	C₅×C₂²⋊C₄	1	2	yes	—
441	C₂²⋊C₄	C₅×C₂²⋊C₄	1	2	yes	—
442	C₂²⋊C₄	C₅×C₂²⋊C₄	1	2	yes	—
443	C₄²	—	2	2	no	—
444	C₄²	C₂²×C₄×C₅	1	2	yes	—
445	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
446	C₄²	C₂×C₅×D₄	1	2	yes	—
447	C₄²	C₂×C₅×D₄	1	2	yes	—
448	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
449	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
450	C₄²	C₂⁴×C₅	1	1	yes	characteristic
451	C₂×C₅	—	2	2	no	—
452	C₂×C₅	—	4	2	no	—
453	C₂×C₅	—	8	4	no	—
454	C₂×C₅	#2888	1	1	yes	characteristic
455	C₂×C₅	C₂×#416	1	1	yes	characteristic
456	C₂×C₅	#2897	1	1	yes	characteristic
457	C₂³	—	2	4	no	—
458	C₂³	—	4	4	no	—
459	C₂³	—	8	2	no	—
460	C₂³	—	8	2	no	—
461	C₂³	C₂×C₅×C₂²⋊C₄	1	2	yes	—
462	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
463	Q₈	—	2	2	no	—
464	Q₈	—	4	2	no	—
465	Q₈	—	4	1	no	—
466	Q₈	—	4	1	no	—
467	D₄	—	2	4	no	—
468	D₄	—	2	2	no	—
469	D₄	—	4	4	no	—
470	D₄	—	4	4	no	—
471	D₄	—	4	2	no	—
472	D₄	—	4	2	no	—
473	D₄	—	4	2	no	—
474	D₄	—	4	2	no	—
475	D₄	—	8	2	no	—
476	D₄	—	8	2	no	—
477	D₄	—	8	4	no	—
478	D₄	—	8	4	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	1	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	4	1	no	—
488	C₂×C₄	—	4	2	no	—
489	C₂×C₄	—	8	4	no	—
490	C₂×C₄	—	8	4	no	—
491	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
492	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
493	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
494	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
495	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
496	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
497	C₂×C₄	C₅×#135	1	2	yes	—
498	C₂×C₄	C₅×#135	1	2	yes	—
499	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
500	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
501	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
502	C₈	—	4	1	no	—
503	C₈	—	4	1	no	—
504	C₈	—	4	2	no	—
505	C₈	—	4	1	no	—
506	C₈	—	4	2	no	—
507	C₈	—	4	1	no	—
508	C₅	#17724	1	1	yes	characteristic, direct product factor, Sylow
509	C₂²	—	2	1	no	—
510	C₂²	—	2	1	no	—
511	C₂²	—	2	2	no	—
512	C₂²	—	2	2	no	—
513	C₂²	—	2	2	no	—
514	C₂²	—	4	4	no	—
515	C₂²	—	4	4	no	—
516	C₂²	—	4	4	no	—
517	C₂²	—	16	4	no	—
518	C₂²	C₅×#416	1	2	yes	—
519	C₂²	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
520	C₄	—	2	1	no	—
521	C₄	—	2	2	no	—
522	C₄	—	2	1	no	—
523	C₄	—	2	2	no	—
524	C₄	—	2	2	no	—
525	C₄	—	2	1	no	—
526	C₄	—	4	2	no	—
527	C₄	—	4	2	no	—
528	C₄	C₅×#416	1	2	yes	—
529	C₄	C₂×C₅×#135	1	1	yes	characteristic
530	C₄	C₂×C₅×#135	1	1	yes	characteristic
531	C₄	C₅×#576	1	1	yes	characteristic
532	C₄	C₅×#577	1	1	yes	characteristic
533	C₂	—	2	2	no	—
534	C₂	—	4	2	no	—
535	C₂	—	8	4	no	—
536	C₂	C₅×#2888	1	1	yes	characteristic
537	C₂	C₂×C₅×#416	1	1	yes	characteristic
538	C₂	C₅×#2897	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₅×#3508	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄×C₅×#41 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2988	C₂	semidirect
3. C₅×#2925	C₂	semidirect
4. C₅×#1553	C₂	semidirect
5. C₅×#1551	C₂	semidirect
6. C₅×#1550	C₂	semidirect
7. C₅×#1545	C₂	semidirect
8. C₅×#1543	C₂	semidirect
9. C₅×#1542	C₂	semidirect
10. C₅×#1540	C₂	semidirect
11. C₂×C₅×#331	C₂	semidirect
30. C₅×#432	C₂²	semidirect
32. C₅×#431	C₂²	semidirect
33. C₅×#431	C₂²	semidirect
37. C₂×C₅×C₄⋊C₈	C₂²	semidirect
40. C₅×#333	C₂²	semidirect
41. C₅×#333	C₂²	semidirect
42. C₅×#333	C₂²	semidirect
43. C₅×#332	C₂²	semidirect
44. C₅×#332	C₂²	semidirect
45. C₅×#332	C₂²	semidirect
50. #17724	C₅	direct (N × H)
122. #2988	C₂×C₅	semidirect
123. #2925	C₂×C₅	semidirect
124. #1553	C₂×C₅	semidirect
125. #1551	C₂×C₅	semidirect
126. #1550	C₂×C₅	semidirect
127. #1545	C₂×C₅	semidirect
128. #1543	C₂×C₅	semidirect
129. #1542	C₂×C₅	semidirect
130. #1540	C₂×C₅	semidirect
131. C₂×#331	C₂×C₅	semidirect
218. #432	C₂²×C₅	semidirect
220. #431	C₂²×C₅	semidirect
221. #431	C₂²×C₅	semidirect
225. C₂×C₄⋊C₈	C₂²×C₅	semidirect
228. #333	C₂²×C₅	semidirect
229. #333	C₂²×C₅	semidirect
230. #333	C₂²×C₅	semidirect
231. #332	C₂²×C₅	semidirect
232. #332	C₂²×C₅	semidirect
233. #332	C₂²×C₅	semidirect
508. C₅	#17724	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×#3508	2
C₅×#2988	2
C₅×#2925	2
C₅×#1553	2
C₅×#1551	2
C₅×#1550	2
C₅×#1545	2
C₅×#1543	2
C₅×#1542	2
C₅×#1540	2
C₂×C₅×#331	2
#17724	5

Subgroups — C5×#17724 #11940444

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#17724 #11940444