Subgroups — C₅×#17544 #11940264

Order 1280.

1364 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#532	C₂	1	1	yes	maximal, characteristic
2	C₅×#2920	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1507	C₂	1	4	yes	maximal, complemented
5	C₅×#1505	C₂	1	4	yes	maximal, complemented
6	C₅×#1502	C₂	1	4	yes	maximal, complemented
7	C₂²×C₅×#41	—	2	1	no	—
8	C₅×#532	—	2	1	no	—
9	C₅×#532	—	2	1	no	—
10	C₅×#532	—	2	1	no	—
11	C₅×#532	—	2	1	no	—
12	C₅×#532	C₄	1	2	yes	complemented
13	C₅×#532	C₄	1	2	yes	complemented
14	C₅×#532	C₂²	1	2	yes	—
15	C₅×#532	C₂²	1	2	yes	—
16	C₂×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
17	C₂×C₅×#127	C₄	1	1	yes	characteristic, complemented
18	C₂×C₅×#124	—	2	1	no	—
19	C₂×C₅×#121	—	2	1	no	—
20	C₂×C₅×#121	C₂²	1	2	yes	—
21	C₂×C₄×C₅×D₄	—	2	1	no	—
22	C₂×C₄×C₅×D₄	C₄	1	2	yes	complemented
23	C₂×C₅×#117	—	2	1	no	—
24	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
25	C₅×#432	—	2	1	no	—
26	C₅×#432	C₂²	1	2	yes	complemented
27	C₅×#431	—	2	1	no	—
28	C₅×#431	C₂²	1	2	yes	complemented
29	C₅×#423	—	2	1	no	—
30	C₅×#407	C₂²	1	2	yes	—
31	C₄×C₅×M₁₆	—	2	1	no	—
32	C₄×C₅×C₄⋊C₄	—	2	1	no	—
33	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	complemented
34	C₄×C₅×C₂²⋊C₄	—	2	1	no	—
35	C₄×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
36	C₅×#376	C₂²	1	2	yes	complemented
37	C₄³×C₅	—	2	1	no	—
38	C₅×#327	C₂²	1	16	yes	complemented
39	#17544	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂×C₅×#41	—	2	2	no	—
41	C₂×C₅×#41	—	2	2	no	—
42	C₂×C₅×#41	—	2	2	no	—
43	C₂×C₅×#41	—	2	2	no	—
44	C₂×C₅×#41	—	4	8	no	—
45	C₂²×C₅×Q₈	—	2	1	no	—
46	C₂²×C₅×D₄	—	2	2	no	—
47	C₂²×C₅×D₄	—	2	1	no	—
48	C₂²×C₅×D₄	—	2	1	no	—
49	C₂²×C₅×D₄	C₂×C₄	1	2	yes	—
50	C₂³×C₄×C₅	—	2	2	no	—
51	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
52	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
53	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
54	C₂×C₅×M₁₆	—	2	2	no	—
55	C₅×C₄⋊Q₈	—	2	1	no	—
56	C₅×C₄⋊Q₈	—	2	1	no	—
57	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
58	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
59	C₅×#127	—	2	1	no	—
60	C₅×#127	—	2	1	no	—
61	C₅×#127	C₂×C₄	1	2	yes	—
62	C₅×#127	C₂×C₄	1	2	yes	—
63	C₅×#124	—	2	2	no	—
64	C₅×#124	—	2	2	no	—
65	C₅×#124	—	2	2	no	—
66	C₅×#124	—	2	2	no	—
67	C₅×#121	—	2	2	no	—
68	C₅×#121	—	2	2	no	—
69	C₅×#121	—	2	2	no	—
70	C₅×#121	—	2	2	no	—
71	C₅×#121	—	2	2	no	—
72	C₅×#121	—	2	2	no	—
73	C₅×#121	C₂×C₄	1	4	yes	—
74	C₅×#121	C₂³	1	4	yes	—
75	C₄×C₅×D₄	—	2	2	no	—
76	C₄×C₅×D₄	—	2	2	no	—
77	C₄×C₅×D₄	—	2	2	no	—
78	C₄×C₅×D₄	—	2	2	no	—
79	C₄×C₅×D₄	—	2	2	no	—
80	C₄×C₅×D₄	—	2	2	no	—
81	C₄×C₅×D₄	C₂×C₄	1	4	yes	—
82	C₄×C₅×D₄	C₂×C₄	1	4	yes	—
83	C₅×#117	—	2	8	no	—
84	C₂×C₅×C₄⋊C₄	—	2	1	no	—
85	C₂×C₅×C₄⋊C₄	—	2	2	no	—
86	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
87	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
88	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
89	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
90	C₂×C₄²×C₅	—	2	2	no	—
91	C₂×C₄²×C₅	—	2	2	no	—
92	C₂×C₄²×C₅	—	2	2	no	—
93	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
94	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
95	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
96	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
97	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
100	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
101	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
102	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
103	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
105	C₅×C₄⋊C₈	—	2	2	no	—
106	C₅×C₄⋊C₈	—	2	2	no	—
107	C₅×C₂²⋊C₈	—	2	2	no	—
108	C₅×C₂²⋊C₈	C₂³	1	4	yes	—
109	C₅×#97	—	2	2	no	—
110	C₄×C₅×C₈	—	2	2	no	—
111	C₅×#95	—	2	2	no	—
112	C₅×#95	C₂³	1	4	yes	—
113	C₂×#532	C₂×C₅	1	1	yes	characteristic
114	#2920	C₂×C₅	1	1	yes	characteristic, complemented
115	C₄×#117	C₂×C₅	1	1	yes	characteristic, complemented
116	#1507	C₂×C₅	1	4	yes	complemented
117	#1505	C₂×C₅	1	4	yes	complemented
118	#1502	C₂×C₅	1	4	yes	complemented
119	C₂⁴×C₅	—	2	2	no	—
120	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
121	C₅×#41	—	4	8	no	—
122	C₅×#41	—	4	8	no	—
123	C₅×#41	—	4	8	no	—
124	C₅×#41	—	4	8	no	—
125	C₂×C₅×Q₈	—	2	2	no	—
126	C₂×C₅×Q₈	—	2	2	no	—
127	C₂×C₅×Q₈	—	4	4	no	—
128	C₂×C₅×D₄	—	2	4	no	—
129	C₂×C₅×D₄	—	2	4	no	—
130	C₂×C₅×D₄	—	2	2	no	—
131	C₂×C₅×D₄	—	2	2	no	—
132	C₂×C₅×D₄	—	2	2	no	—
133	C₂×C₅×D₄	—	2	2	no	—
134	C₂×C₅×D₄	—	2	2	no	—
135	C₂×C₅×D₄	—	4	4	no	—
136	C₂×C₅×D₄	—	4	4	no	—
137	C₂×C₅×D₄	—	4	8	no	—
138	C₂×C₅×D₄	—	4	4	no	—
139	C₂×C₅×D₄	C₂²×C₄	1	4	yes	—
140	C₂²×C₄×C₅	—	2	4	no	—
141	C₂²×C₄×C₅	—	2	4	no	—
142	C₂²×C₄×C₅	—	2	2	no	—
143	C₂²×C₄×C₅	—	2	4	no	—
144	C₂²×C₄×C₅	—	2	4	no	—
145	C₂²×C₄×C₅	—	2	4	no	—
146	C₂²×C₄×C₅	—	4	2	no	—
147	C₂²×C₄×C₅	—	4	8	no	—
148	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
151	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
153	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
154	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
156	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
158	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
166	C₅×M₁₆	—	4	4	no	—
167	C₂×C₅×C₈	—	2	4	no	—
168	C₅×C₄⋊C₄	—	2	8	no	—
169	C₅×C₄⋊C₄	—	2	2	no	—
170	C₅×C₄⋊C₄	—	2	2	no	—
171	C₅×C₄⋊C₄	—	2	2	no	—
172	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
173	C₅×C₂²⋊C₄	—	2	8	no	—
174	C₅×C₂²⋊C₄	—	2	4	no	—
175	C₅×C₂²⋊C₄	—	2	4	no	—
176	C₅×C₂²⋊C₄	—	2	4	no	—
177	C₅×C₂²⋊C₄	—	2	4	no	—
178	C₄²×C₅	—	2	8	no	—
179	C₄²×C₅	—	4	4	no	—
180	C₄²×C₅	—	4	4	no	—
181	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
182	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
183	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
184	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
185	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
186	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
187	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
188	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
189	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
190	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
191	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
192	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
193	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
194	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
195	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
196	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
197	C₂²×#41	—	2	1	no	—
198	#532	—	2	1	no	—
199	#532	—	2	1	no	—
200	#532	—	2	1	no	—
201	#532	—	2	1	no	—
202	#532	C₄×C₅	1	2	yes	complemented
203	#532	C₄×C₅	1	2	yes	complemented
204	#532	C₂²×C₅	1	2	yes	—
205	#532	C₂²×C₅	1	2	yes	—
206	C₂×C₄⋊Q₈	C₄×C₅	1	1	yes	characteristic, complemented
207	C₂×#127	C₄×C₅	1	1	yes	characteristic, complemented
208	C₂×#124	—	2	1	no	—
209	C₂×#121	—	2	1	no	—
210	C₂×#121	C₂²×C₅	1	2	yes	—
211	C₂×C₄×D₄	—	2	1	no	—
212	C₂×C₄×D₄	C₄×C₅	1	2	yes	complemented
213	C₂×#117	—	2	1	no	—
214	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
215	#432	—	2	1	no	—
216	#432	C₂²×C₅	1	2	yes	complemented
217	#431	—	2	1	no	—
218	#431	C₂²×C₅	1	2	yes	complemented
219	#423	—	2	1	no	—
220	#407	C₂²×C₅	1	2	yes	—
221	C₄×M₁₆	—	2	1	no	—
222	C₄×C₄⋊C₄	—	2	1	no	—
223	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	complemented
224	C₄×C₂²⋊C₄	—	2	1	no	—
225	C₄×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
226	#376	C₂²×C₅	1	2	yes	complemented
227	C₄³	—	2	1	no	—
228	#327	C₂²×C₅	1	16	yes	complemented
229	C₂³×C₅	—	2	2	no	—
230	C₂³×C₅	—	2	2	no	—
231	C₂³×C₅	—	2	4	no	—
232	C₂³×C₅	—	4	1	no	—
233	C₂³×C₅	—	4	4	no	—
234	C₂³×C₅	—	4	4	no	—
235	C₂³×C₅	—	4	4	no	—
236	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
237	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
238	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
239	C₅×Q₈	—	4	4	no	—
240	C₅×Q₈	—	4	4	no	—
241	C₅×D₄	—	4	4	no	—
242	C₅×D₄	—	4	4	no	—
243	C₅×D₄	—	4	8	no	—
244	C₅×D₄	—	4	8	no	—
245	C₅×D₄	—	4	4	no	—
246	C₅×D₄	—	4	4	no	—
247	C₅×D₄	—	4	4	no	—
248	C₅×D₄	—	4	4	no	—
249	C₂×C₄×C₅	—	2	2	no	—
250	C₂×C₄×C₅	—	2	2	no	—
251	C₂×C₄×C₅	—	2	2	no	—
252	C₂×C₄×C₅	—	2	2	no	—
253	C₂×C₄×C₅	—	2	2	no	—
254	C₂×C₄×C₅	—	2	2	no	—
255	C₂×C₄×C₅	—	2	2	no	—
256	C₂×C₄×C₅	—	2	2	no	—
257	C₂×C₄×C₅	—	2	2	no	—
258	C₂×C₄×C₅	—	2	2	no	—
259	C₂×C₄×C₅	—	2	2	no	—
260	C₂×C₄×C₅	—	2	2	no	—
261	C₂×C₄×C₅	—	2	2	no	—
262	C₂×C₄×C₅	—	2	8	no	—
263	C₂×C₄×C₅	—	2	4	no	—
264	C₂×C₄×C₅	—	4	1	no	—
265	C₂×C₄×C₅	—	4	1	no	—
266	C₂×C₄×C₅	—	4	1	no	—
267	C₂×C₄×C₅	—	4	8	no	—
268	C₂×C₄×C₅	—	4	4	no	—
269	C₂×C₄×C₅	—	4	4	no	—
270	C₂×C₄×C₅	—	4	4	no	—
271	C₂×C₄×C₅	—	4	8	no	—
272	C₂×C₄×C₅	#104	1	4	yes	—
273	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
274	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
275	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	C₂×C₂²⋊C₄	1	1	yes	characteristic
279	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
280	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
281	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
282	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
283	C₂×C₄×C₅	AGL(1,ℤ/8)	1	4	yes	—
284	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
285	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
286	C₅×C₈	—	4	4	no	—
287	C₂×#41	—	2	2	no	—
288	C₂×#41	—	2	2	no	—
289	C₂×#41	—	2	2	no	—
290	C₂×#41	—	2	2	no	—
291	C₂×#41	—	4	8	no	—
292	C₂²×Q₈	—	2	1	no	—
293	C₂²×D₄	—	2	2	no	—
294	C₂²×D₄	—	2	1	no	—
295	C₂²×D₄	—	2	1	no	—
296	C₂²×D₄	C₂×C₄×C₅	1	2	yes	—
297	C₂³×C₄	—	2	2	no	—
298	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
299	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
300	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
301	C₂×M₁₆	—	2	2	no	—
302	C₄⋊Q₈	—	2	1	no	—
303	C₄⋊Q₈	—	2	1	no	—
304	C₄⋊Q₈	C₂×C₄×C₅	1	2	yes	—
305	C₄⋊Q₈	C₂×C₄×C₅	1	2	yes	—
306	#127	—	2	1	no	—
307	#127	—	2	1	no	—
308	#127	C₂×C₄×C₅	1	2	yes	—
309	#127	C₂×C₄×C₅	1	2	yes	—
310	#124	—	2	2	no	—
311	#124	—	2	2	no	—
312	#124	—	2	2	no	—
313	#124	—	2	2	no	—
314	#121	—	2	2	no	—
315	#121	—	2	2	no	—
316	#121	—	2	2	no	—
317	#121	—	2	2	no	—
318	#121	—	2	2	no	—
319	#121	—	2	2	no	—
320	#121	C₂×C₄×C₅	1	4	yes	—
321	#121	C₂³×C₅	1	4	yes	—
322	C₄×D₄	—	2	2	no	—
323	C₄×D₄	—	2	2	no	—
324	C₄×D₄	—	2	2	no	—
325	C₄×D₄	—	2	2	no	—
326	C₄×D₄	—	2	2	no	—
327	C₄×D₄	—	2	2	no	—
328	C₄×D₄	C₂×C₄×C₅	1	4	yes	—
329	C₄×D₄	C₂×C₄×C₅	1	4	yes	—
330	#117	—	2	8	no	—
331	C₂×C₄⋊C₄	—	2	1	no	—
332	C₂×C₄⋊C₄	—	2	2	no	—
333	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
334	C₂×C₂²⋊C₄	—	2	2	no	—
335	C₂×C₂²⋊C₄	—	2	2	no	—
336	C₂×C₂²⋊C₄	—	2	2	no	—
337	C₂×C₄²	—	2	2	no	—
338	C₂×C₄²	—	2	2	no	—
339	C₂×C₄²	—	2	2	no	—
340	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
341	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
342	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
343	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
344	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
345	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
346	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
347	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
348	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
349	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
350	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
351	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
352	C₄⋊C₈	—	2	2	no	—
353	C₄⋊C₈	—	2	2	no	—
354	C₂²⋊C₈	—	2	2	no	—
355	C₂²⋊C₈	C₂³×C₅	1	4	yes	—
356	#97	—	2	2	no	—
357	C₄×C₈	—	2	2	no	—
358	#95	—	2	2	no	—
359	#95	C₂³×C₅	1	4	yes	—
360	C₂²×C₅	—	2	1	no	—
361	C₂²×C₅	—	2	1	no	—
362	C₂²×C₅	—	2	1	no	—
363	C₂²×C₅	—	2	1	no	—
364	C₂²×C₅	—	2	1	no	—
365	C₂²×C₅	—	2	1	no	—
366	C₂²×C₅	—	2	1	no	—
367	C₂²×C₅	—	4	4	no	—
368	C₂²×C₅	—	4	4	no	—
369	C₂²×C₅	—	4	4	no	—
370	C₂²×C₅	—	4	4	no	—
371	C₂²×C₅	—	8	8	no	—
372	C₂²×C₅	C₂×#104	1	2	yes	—
373	C₂²×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
374	C₂²×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
375	C₄×C₅	—	2	1	no	—
376	C₄×C₅	—	2	1	no	—
377	C₄×C₅	—	2	1	no	—
378	C₄×C₅	—	2	1	no	—
379	C₄×C₅	—	2	1	no	—
380	C₄×C₅	—	2	1	no	—
381	C₄×C₅	—	2	1	no	—
382	C₄×C₅	—	4	8	no	—
383	C₄×C₅	—	4	4	no	—
384	C₄×C₅	#410	1	2	yes	—
385	C₄×C₅	#418	1	2	yes	—
386	C₄×C₅	#418	1	2	yes	—
387	C₄×C₅	C₂×#104	1	2	yes	—
388	C₄×C₅	C₂×#104	1	2	yes	—
389	C₂⁴	—	2	2	no	—
390	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
391	#41	—	4	8	no	—
392	#41	—	4	8	no	—
393	#41	—	4	8	no	—
394	#41	—	4	8	no	—
395	C₂×Q₈	—	2	2	no	—
396	C₂×Q₈	—	2	2	no	—
397	C₂×Q₈	—	4	4	no	—
398	C₂×D₄	—	2	4	no	—
399	C₂×D₄	—	2	4	no	—
400	C₂×D₄	—	2	2	no	—
401	C₂×D₄	—	2	2	no	—
402	C₂×D₄	—	2	2	no	—
403	C₂×D₄	—	2	2	no	—
404	C₂×D₄	—	2	2	no	—
405	C₂×D₄	—	4	4	no	—
406	C₂×D₄	—	4	4	no	—
407	C₂×D₄	—	4	8	no	—
408	C₂×D₄	—	4	4	no	—
409	C₂×D₄	C₂²×C₄×C₅	1	4	yes	—
410	C₂²×C₄	—	2	4	no	—
411	C₂²×C₄	—	2	4	no	—
412	C₂²×C₄	—	2	2	no	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	4	no	—
415	C₂²×C₄	—	2	4	no	—
416	C₂²×C₄	—	4	2	no	—
417	C₂²×C₄	—	4	8	no	—
418	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
419	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
420	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
421	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
422	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
423	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
424	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
425	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
426	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
427	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
428	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
429	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
430	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
431	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
432	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
433	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
434	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
435	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
436	M₁₆	—	4	4	no	—
437	C₂×C₈	—	2	4	no	—
438	C₄⋊C₄	—	2	8	no	—
439	C₄⋊C₄	—	2	2	no	—
440	C₄⋊C₄	—	2	2	no	—
441	C₄⋊C₄	—	2	2	no	—
442	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
443	C₂²⋊C₄	—	2	8	no	—
444	C₂²⋊C₄	—	2	4	no	—
445	C₂²⋊C₄	—	2	4	no	—
446	C₂²⋊C₄	—	2	4	no	—
447	C₂²⋊C₄	—	2	4	no	—
448	C₄²	—	2	8	no	—
449	C₄²	—	4	4	no	—
450	C₄²	—	4	4	no	—
451	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
452	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
453	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
454	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
455	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
456	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
457	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
458	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
459	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
460	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
461	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
462	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
463	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
464	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
465	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
466	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
467	C₂×C₅	—	2	2	no	—
468	C₂×C₅	—	2	2	no	—
469	C₂×C₅	—	4	1	no	—
470	C₂×C₅	—	8	4	no	—
471	C₂×C₅	C₂×#410	1	1	yes	characteristic
472	C₂×C₅	C₂×#418	1	1	yes	characteristic
473	C₂×C₅	C₂²×#104	1	1	yes	characteristic
474	C₂³	—	2	2	no	—
475	C₂³	—	2	2	no	—
476	C₂³	—	2	4	no	—
477	C₂³	—	4	1	no	—
478	C₂³	—	4	4	no	—
479	C₂³	—	4	4	no	—
480	C₂³	—	4	4	no	—
481	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
482	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
483	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
484	Q₈	—	4	4	no	—
485	Q₈	—	4	4	no	—
486	D₄	—	4	4	no	—
487	D₄	—	4	4	no	—
488	D₄	—	4	8	no	—
489	D₄	—	4	8	no	—
490	D₄	—	4	4	no	—
491	D₄	—	4	4	no	—
492	D₄	—	4	4	no	—
493	D₄	—	4	4	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	8	no	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	4	1	no	—
510	C₂×C₄	—	4	1	no	—
511	C₂×C₄	—	4	1	no	—
512	C₂×C₄	—	4	8	no	—
513	C₂×C₄	—	4	4	no	—
514	C₂×C₄	—	4	4	no	—
515	C₂×C₄	—	4	4	no	—
516	C₂×C₄	—	4	8	no	—
517	C₂×C₄	C₅×#104	1	4	yes	—
518	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
519	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
520	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
521	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
522	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
523	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
524	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
525	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
526	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
527	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
528	C₂×C₄	C₅×AGL(1,ℤ/8)	1	4	yes	—
529	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
530	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
531	C₈	—	4	4	no	—
532	C₅	#17544	1	1	yes	characteristic, direct product factor, Sylow
533	C₂²	—	2	1	no	—
534	C₂²	—	2	1	no	—
535	C₂²	—	2	1	no	—
536	C₂²	—	2	1	no	—
537	C₂²	—	2	1	no	—
538	C₂²	—	2	1	no	—
539	C₂²	—	2	1	no	—
540	C₂²	—	4	4	no	—
541	C₂²	—	4	4	no	—
542	C₂²	—	4	4	no	—
543	C₂²	—	4	4	no	—
544	C₂²	—	8	8	no	—
545	C₂²	C₂×C₅×#104	1	2	yes	—
546	C₂²	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
547	C₂²	C₂×C₅×AGL(1,ℤ/8)	1	2	yes	—
548	C₄	—	2	1	no	—
549	C₄	—	2	1	no	—
550	C₄	—	2	1	no	—
551	C₄	—	2	1	no	—
552	C₄	—	2	1	no	—
553	C₄	—	2	1	no	—
554	C₄	—	2	1	no	—
555	C₄	—	4	8	no	—
556	C₄	—	4	4	no	—
557	C₄	C₅×#410	1	2	yes	—
558	C₄	C₅×#418	1	2	yes	—
559	C₄	C₅×#418	1	2	yes	—
560	C₄	C₂×C₅×#104	1	2	yes	—
561	C₄	C₂×C₅×#104	1	2	yes	—
562	C₂	—	2	2	no	—
563	C₂	—	2	2	no	—
564	C₂	—	4	1	no	—
565	C₂	—	8	4	no	—
566	C₂	C₂×C₅×#410	1	1	yes	characteristic
567	C₂	C₂×C₅×#418	1	1	yes	characteristic
568	C₂	C₂²×C₅×#104	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂²×C₅	C₂²×C₂²⋊C₄
prime radical	C₂×C₅×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2920	C₂	semidirect
3. C₄×C₅×#117	C₂	semidirect
4. C₅×#1507	C₂	semidirect
5. C₅×#1505	C₂	semidirect
6. C₅×#1502	C₂	semidirect
12. C₅×#532	C₄	semidirect
13. C₅×#532	C₄	semidirect
16. C₂×C₅×C₄⋊Q₈	C₄	semidirect
17. C₂×C₅×#127	C₄	semidirect
22. C₂×C₄×C₅×D₄	C₄	semidirect
26. C₅×#432	C₂²	semidirect
28. C₅×#431	C₂²	semidirect
33. C₄×C₅×C₄⋊C₄	C₂²	semidirect
36. C₅×#376	C₂²	semidirect
38. C₅×#327	C₂²	semidirect
39. #17544	C₅	direct (N × H)
114. #2920	C₂×C₅	semidirect
115. C₄×#117	C₂×C₅	semidirect
116. #1507	C₂×C₅	semidirect
117. #1505	C₂×C₅	semidirect
118. #1502	C₂×C₅	semidirect
202. #532	C₄×C₅	semidirect
203. #532	C₄×C₅	semidirect
206. C₂×C₄⋊Q₈	C₄×C₅	semidirect
207. C₂×#127	C₄×C₅	semidirect
212. C₂×C₄×D₄	C₄×C₅	semidirect
216. #432	C₂²×C₅	semidirect
218. #431	C₂²×C₅	semidirect
223. C₄×C₄⋊C₄	C₂²×C₅	semidirect
226. #376	C₂²×C₅	semidirect
228. #327	C₂²×C₅	semidirect
532. C₅	#17544	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₅×#532	2
C₅×#2920	2
C₄×C₅×#117	2
C₅×#1507	2
C₅×#1505	2
C₅×#1502	2
#17544	5

Subgroups — C5×#17544 #11940264

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#17544 #11940264