Subgroups — C₅×#17704 #11940424

Order 1280.

1298 conjugacy classes of subgroups.

The subgroup lattice has 588 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	C₂×C₅×#431	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#423	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1547	C₂	1	4	yes	maximal, complemented
5	C₅×#1546	C₂	1	2	yes	maximal, complemented
6	C₅×#1545	C₂	1	2	yes	maximal, complemented
7	C₅×#1540	C₂	1	4	yes	maximal, complemented
8	C₂²×C₅×#41	—	2	1	no	—
9	C₂²×C₅×M₁₆	—	2	1	no	—
10	C₅×#532	—	2	1	no	—
11	C₅×#532	—	2	1	no	—
12	C₅×#532	—	2	1	no	—
13	C₅×#532	—	2	1	no	—
14	C₅×#532	C₄	1	2	yes	—
15	C₅×#532	C₄	1	2	yes	—
16	C₅×#532	C₂²	1	2	yes	—
17	C₅×#532	C₂²	1	2	yes	—
18	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂×C₅×#127	C₂²	1	1	yes	characteristic
20	C₂×C₅×#124	—	2	1	no	—
21	C₂×C₅×#121	—	2	1	no	—
22	C₂×C₅×#121	C₄	1	2	yes	—
23	C₂×C₄×C₅×D₄	—	2	1	no	—
24	C₂×C₄×C₅×D₄	C₄	1	2	yes	—
25	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
26	C₅×#431	—	2	1	no	—
27	C₅×#431	—	2	1	no	—
28	C₅×#431	C₂²	1	2	yes	—
29	C₅×#431	C₂²	1	2	yes	complemented
30	C₅×#423	—	2	1	no	—
31	C₅×#423	—	2	1	no	—
32	C₅×#423	C₂²	1	2	yes	—
33	C₅×#423	C₂²	1	2	yes	complemented
34	C₂×C₅×C₄⋊C₈	—	2	1	no	—
35	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	complemented
36	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	complemented
37	C₂×C₅×C₂²⋊C₈	—	2	1	no	—
38	C₂×C₄×C₅×C₈	—	2	1	no	—
39	C₅×#332	C₂²	1	8	yes	complemented
40	C₅×#331	C₂²	1	8	yes	complemented
41	#17704	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂×C₅×#41	—	2	2	no	—
43	C₂×C₅×#41	—	2	2	no	—
44	C₂×C₅×#41	—	2	2	no	—
45	C₂×C₅×#41	—	2	2	no	—
46	C₂×C₅×#41	—	4	4	no	—
47	C₂×C₅×#41	—	4	4	no	—
48	C₂²×C₅×Q₈	—	2	1	no	—
49	C₂²×C₅×D₄	—	2	2	no	—
50	C₂²×C₅×D₄	—	2	1	no	—
51	C₂²×C₅×D₄	—	2	1	no	—
52	C₂²×C₅×D₄	C₂×C₄	1	2	yes	—
53	C₂³×C₄×C₅	—	2	2	no	—
54	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
55	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
56	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
57	C₂×C₅×M₁₆	—	2	2	no	—
58	C₂×C₅×M₁₆	—	2	2	no	—
59	C₂×C₅×M₁₆	—	4	4	no	—
60	C₂²×C₅×C₈	—	2	2	no	—
61	C₂²×C₅×C₈	—	2	2	no	—
62	C₅×C₄⋊Q₈	—	2	1	no	—
63	C₅×C₄⋊Q₈	—	2	1	no	—
64	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
65	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
66	C₅×#127	—	2	1	no	—
67	C₅×#127	—	2	1	no	—
68	C₅×#127	C₂×C₄	1	2	yes	—
69	C₅×#127	C₂³	1	2	yes	—
70	C₅×#124	—	2	2	no	—
71	C₅×#124	—	2	2	no	—
72	C₅×#124	—	2	2	no	—
73	C₅×#124	—	2	2	no	—
74	C₅×#121	—	2	2	no	—
75	C₅×#121	—	2	2	no	—
76	C₅×#121	—	2	2	no	—
77	C₅×#121	—	2	2	no	—
78	C₅×#121	—	2	2	no	—
79	C₅×#121	—	2	2	no	—
80	C₅×#121	C₂×C₄	1	4	yes	—
81	C₅×#121	C₂×C₄	1	4	yes	—
82	C₄×C₅×D₄	—	2	2	no	—
83	C₄×C₅×D₄	—	2	2	no	—
84	C₄×C₅×D₄	—	2	2	no	—
85	C₄×C₅×D₄	—	2	2	no	—
86	C₄×C₅×D₄	—	2	2	no	—
87	C₄×C₅×D₄	—	2	2	no	—
88	C₄×C₅×D₄	C₂×C₄	1	4	yes	—
89	C₄×C₅×D₄	C₂×C₄	1	4	yes	—
90	C₂×C₅×C₄⋊C₄	—	2	1	no	—
91	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
92	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
93	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
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	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
106	C₅×C₄⋊C₈	—	2	2	no	—
107	C₅×C₄⋊C₈	—	2	2	no	—
108	C₅×C₄⋊C₈	—	2	2	no	—
109	C₅×C₄⋊C₈	—	2	2	no	—
110	C₅×C₄⋊C₈	C₂³	1	4	yes	—
111	C₅×C₄⋊C₈	C₂³	1	4	yes	—
112	C₅×C₂²⋊C₈	—	2	2	no	—
113	C₅×C₂²⋊C₈	—	2	2	no	—
114	C₄×C₅×C₈	—	2	2	no	—
115	C₄×C₅×C₈	—	2	2	no	—
116	C₂×#532	C₂×C₅	1	1	yes	characteristic
117	C₂×#431	C₂×C₅	1	1	yes	characteristic, complemented
118	C₂×#423	C₂×C₅	1	1	yes	characteristic, complemented
119	#1547	C₂×C₅	1	4	yes	complemented
120	#1546	C₂×C₅	1	2	yes	complemented
121	#1545	C₂×C₅	1	2	yes	complemented
122	#1540	C₂×C₅	1	4	yes	complemented
123	C₂⁴×C₅	—	2	2	no	—
124	C₂⁴×C₅	C₂²⋊C₄	1	1	yes	characteristic
125	C₅×#41	—	4	8	no	—
126	C₅×#41	—	4	8	no	—
127	C₅×#41	—	4	4	no	—
128	C₅×#41	—	4	4	no	—
129	C₅×#41	—	4	4	no	—
130	C₅×#41	—	4	4	no	—
131	C₂×C₅×Q₈	—	2	2	no	—
132	C₂×C₅×Q₈	—	2	2	no	—
133	C₂×C₅×Q₈	—	4	2	no	—
134	C₂×C₅×Q₈	—	4	2	no	—
135	C₂×C₅×D₄	—	2	4	no	—
136	C₂×C₅×D₄	—	2	4	no	—
137	C₂×C₅×D₄	—	2	2	no	—
138	C₂×C₅×D₄	—	2	2	no	—
139	C₂×C₅×D₄	—	2	2	no	—
140	C₂×C₅×D₄	—	2	2	no	—
141	C₂×C₅×D₄	—	2	2	no	—
142	C₂×C₅×D₄	—	4	4	no	—
143	C₂×C₅×D₄	—	4	4	no	—
144	C₂×C₅×D₄	—	4	8	no	—
145	C₂×C₅×D₄	—	4	2	no	—
146	C₂×C₅×D₄	—	4	2	no	—
147	C₂×C₅×D₄	C₂²×C₄	1	4	yes	—
148	C₂²×C₄×C₅	—	2	2	no	—
149	C₂²×C₄×C₅	—	2	4	no	—
150	C₂²×C₄×C₅	—	2	4	no	—
151	C₂²×C₄×C₅	—	4	2	no	—
152	C₂²×C₄×C₅	—	4	2	no	—
153	C₂²×C₄×C₅	—	4	8	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
159	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
160	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
161	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
162	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
163	C₂²×C₄×C₅	D₈	1	4	yes	—
164	C₂²×C₄×C₅	SD₁₆	1	4	yes	—
165	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
166	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
167	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
168	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
174	C₅×M₁₆	—	4	4	no	—
175	C₅×M₁₆	—	4	4	no	—
176	C₂×C₅×C₈	—	2	4	no	—
177	C₂×C₅×C₈	—	2	4	no	—
178	C₂×C₅×C₈	—	4	4	no	—
179	C₂×C₅×C₈	—	4	4	no	—
180	C₅×C₄⋊C₄	—	2	2	no	—
181	C₅×C₄⋊C₄	—	2	2	no	—
182	C₅×C₄⋊C₄	—	2	2	no	—
183	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
184	C₅×C₂²⋊C₄	—	2	4	no	—
185	C₅×C₂²⋊C₄	—	2	4	no	—
186	C₅×C₂²⋊C₄	—	2	4	no	—
187	C₅×C₂²⋊C₄	—	2	4	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
194	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
195	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
196	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
197	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
198	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
199	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
200	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
201	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
202	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
203	C₄²×C₅	C₂⁴	1	1	yes	characteristic
204	C₂²×#41	—	2	1	no	—
205	C₂²×M₁₆	—	2	1	no	—
206	#532	—	2	1	no	—
207	#532	—	2	1	no	—
208	#532	—	2	1	no	—
209	#532	—	2	1	no	—
210	#532	C₄×C₅	1	2	yes	—
211	#532	C₄×C₅	1	2	yes	—
212	#532	C₂²×C₅	1	2	yes	—
213	#532	C₂²×C₅	1	2	yes	—
214	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
215	C₂×#127	C₂²×C₅	1	1	yes	characteristic
216	C₂×#124	—	2	1	no	—
217	C₂×#121	—	2	1	no	—
218	C₂×#121	C₄×C₅	1	2	yes	—
219	C₂×C₄×D₄	—	2	1	no	—
220	C₂×C₄×D₄	C₄×C₅	1	2	yes	—
221	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
222	#431	—	2	1	no	—
223	#431	—	2	1	no	—
224	#431	C₂²×C₅	1	2	yes	—
225	#431	C₂²×C₅	1	2	yes	complemented
226	#423	—	2	1	no	—
227	#423	—	2	1	no	—
228	#423	C₂²×C₅	1	2	yes	—
229	#423	C₂²×C₅	1	2	yes	complemented
230	C₂×C₄⋊C₈	—	2	1	no	—
231	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	complemented
232	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	complemented
233	C₂×C₂²⋊C₈	—	2	1	no	—
234	C₂×C₄×C₈	—	2	1	no	—
235	#332	C₂²×C₅	1	8	yes	complemented
236	#331	C₂²×C₅	1	8	yes	complemented
237	C₂³×C₅	—	2	4	no	—
238	C₂³×C₅	—	4	1	no	—
239	C₂³×C₅	—	4	1	no	—
240	C₂³×C₅	—	4	4	no	—
241	C₂³×C₅	—	4	4	no	—
242	C₂³×C₅	—	4	4	no	—
243	C₂³×C₅	#102	1	4	yes	—
244	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
245	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
246	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
247	C₅×Q₈	—	4	2	no	—
248	C₅×Q₈	—	4	2	no	—
249	C₅×Q₈	—	4	4	no	—
250	C₅×D₄	—	4	4	no	—
251	C₅×D₄	—	4	4	no	—
252	C₅×D₄	—	4	8	no	—
253	C₅×D₄	—	4	8	no	—
254	C₅×D₄	—	4	2	no	—
255	C₅×D₄	—	4	2	no	—
256	C₅×D₄	—	4	4	no	—
257	C₅×D₄	—	4	4	no	—
258	C₅×D₄	—	4	4	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	2	no	—
263	C₂×C₄×C₅	—	2	2	no	—
264	C₂×C₄×C₅	—	2	2	no	—
265	C₂×C₄×C₅	—	2	4	no	—
266	C₂×C₄×C₅	—	4	1	no	—
267	C₂×C₄×C₅	—	4	1	no	—
268	C₂×C₄×C₅	—	4	1	no	—
269	C₂×C₄×C₅	—	4	1	no	—
270	C₂×C₄×C₅	—	4	1	no	—
271	C₂×C₄×C₅	—	4	1	no	—
272	C₂×C₄×C₅	—	4	4	no	—
273	C₂×C₄×C₅	—	4	8	no	—
274	C₂×C₄×C₅	—	4	8	no	—
275	C₂×C₄×C₅	#102	1	4	yes	—
276	C₂×C₄×C₅	#102	1	4	yes	—
277	C₂×C₄×C₅	#102	1	4	yes	—
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	C₂×C₂²⋊C₄	1	1	yes	characteristic
284	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
285	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
286	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
287	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
288	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
289	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
290	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
291	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
292	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
293	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
294	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
295	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
296	C₅×C₈	—	4	4	no	—
297	C₅×C₈	—	4	4	no	—
298	C₂×#41	—	2	2	no	—
299	C₂×#41	—	2	2	no	—
300	C₂×#41	—	2	2	no	—
301	C₂×#41	—	2	2	no	—
302	C₂×#41	—	4	4	no	—
303	C₂×#41	—	4	4	no	—
304	C₂²×Q₈	—	2	1	no	—
305	C₂²×D₄	—	2	2	no	—
306	C₂²×D₄	—	2	1	no	—
307	C₂²×D₄	—	2	1	no	—
308	C₂²×D₄	C₂×C₄×C₅	1	2	yes	—
309	C₂³×C₄	—	2	2	no	—
310	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
311	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
312	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
313	C₂×M₁₆	—	2	2	no	—
314	C₂×M₁₆	—	2	2	no	—
315	C₂×M₁₆	—	4	4	no	—
316	C₂²×C₈	—	2	2	no	—
317	C₂²×C₈	—	2	2	no	—
318	C₄⋊Q₈	—	2	1	no	—
319	C₄⋊Q₈	—	2	1	no	—
320	C₄⋊Q₈	C₂×C₄×C₅	1	2	yes	—
321	C₄⋊Q₈	C₂³×C₅	1	2	yes	—
322	#127	—	2	1	no	—
323	#127	—	2	1	no	—
324	#127	C₂×C₄×C₅	1	2	yes	—
325	#127	C₂³×C₅	1	2	yes	—
326	#124	—	2	2	no	—
327	#124	—	2	2	no	—
328	#124	—	2	2	no	—
329	#124	—	2	2	no	—
330	#121	—	2	2	no	—
331	#121	—	2	2	no	—
332	#121	—	2	2	no	—
333	#121	—	2	2	no	—
334	#121	—	2	2	no	—
335	#121	—	2	2	no	—
336	#121	C₂×C₄×C₅	1	4	yes	—
337	#121	C₂×C₄×C₅	1	4	yes	—
338	C₄×D₄	—	2	2	no	—
339	C₄×D₄	—	2	2	no	—
340	C₄×D₄	—	2	2	no	—
341	C₄×D₄	—	2	2	no	—
342	C₄×D₄	—	2	2	no	—
343	C₄×D₄	—	2	2	no	—
344	C₄×D₄	C₂×C₄×C₅	1	4	yes	—
345	C₄×D₄	C₂×C₄×C₅	1	4	yes	—
346	C₂×C₄⋊C₄	—	2	1	no	—
347	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
348	C₂×C₂²⋊C₄	—	2	2	no	—
349	C₂×C₂²⋊C₄	—	2	2	no	—
350	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
351	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
352	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
353	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
354	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
355	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
356	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
357	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
358	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
359	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
360	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
361	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
362	C₄⋊C₈	—	2	2	no	—
363	C₄⋊C₈	—	2	2	no	—
364	C₄⋊C₈	—	2	2	no	—
365	C₄⋊C₈	—	2	2	no	—
366	C₄⋊C₈	C₂³×C₅	1	4	yes	—
367	C₄⋊C₈	C₂³×C₅	1	4	yes	—
368	C₂²⋊C₈	—	2	2	no	—
369	C₂²⋊C₈	—	2	2	no	—
370	C₄×C₈	—	2	2	no	—
371	C₄×C₈	—	2	2	no	—
372	C₂²×C₅	—	2	1	no	—
373	C₂²×C₅	—	2	1	no	—
374	C₂²×C₅	—	2	1	no	—
375	C₂²×C₅	—	2	1	no	—
376	C₂²×C₅	—	2	1	no	—
377	C₂²×C₅	—	2	1	no	—
378	C₂²×C₅	—	4	4	no	—
379	C₂²×C₅	—	4	4	no	—
380	C₂²×C₅	—	4	4	no	—
381	C₂²×C₅	—	4	4	no	—
382	C₂²×C₅	—	8	8	no	—
383	C₂²×C₅	C₂×#102	1	2	yes	—
384	C₂²×C₅	C₂×#102	1	2	yes	—
385	C₂²×C₅	C₂×#102	1	2	yes	—
386	C₂²×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
387	C₄×C₅	—	2	1	no	—
388	C₄×C₅	—	2	1	no	—
389	C₄×C₅	—	2	1	no	—
390	C₄×C₅	—	2	1	no	—
391	C₄×C₅	—	2	1	no	—
392	C₄×C₅	—	2	1	no	—
393	C₄×C₅	—	4	4	no	—
394	C₄×C₅	#413	1	2	yes	—
395	C₄×C₅	C₂×#102	1	2	yes	—
396	C₄×C₅	C₂×#102	1	2	yes	—
397	C₄×C₅	C₂×#102	1	2	yes	—
398	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
399	C₄×C₅	C₂²×SD₁₆	1	1	yes	characteristic
400	C₄×C₅	#575	1	2	yes	—
401	C₂⁴	—	2	2	no	—
402	C₂⁴	C₅×C₂²⋊C₄	1	1	yes	characteristic
403	#41	—	4	4	no	—
404	#41	—	4	4	no	—
405	#41	—	4	4	no	—
406	#41	—	4	4	no	—
407	#41	—	4	8	no	—
408	#41	—	4	8	no	—
409	C₂×Q₈	—	2	2	no	—
410	C₂×Q₈	—	2	2	no	—
411	C₂×Q₈	—	4	2	no	—
412	C₂×Q₈	—	4	2	no	—
413	C₂×D₄	—	2	4	no	—
414	C₂×D₄	—	2	4	no	—
415	C₂×D₄	—	2	2	no	—
416	C₂×D₄	—	2	2	no	—
417	C₂×D₄	—	2	2	no	—
418	C₂×D₄	—	2	2	no	—
419	C₂×D₄	—	2	2	no	—
420	C₂×D₄	—	4	4	no	—
421	C₂×D₄	—	4	4	no	—
422	C₂×D₄	—	4	8	no	—
423	C₂×D₄	—	4	2	no	—
424	C₂×D₄	—	4	2	no	—
425	C₂×D₄	C₂²×C₄×C₅	1	4	yes	—
426	C₂²×C₄	—	2	2	no	—
427	C₂²×C₄	—	2	4	no	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	4	2	no	—
430	C₂²×C₄	—	4	2	no	—
431	C₂²×C₄	—	4	8	no	—
432	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
433	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
434	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
435	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
436	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
437	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
438	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
439	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
440	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
441	C₂²×C₄	C₅×D₈	1	4	yes	—
442	C₂²×C₄	C₅×SD₁₆	1	4	yes	—
443	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
444	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
445	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
446	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
447	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
448	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
449	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
450	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
451	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
452	M₁₆	—	4	4	no	—
453	M₁₆	—	4	4	no	—
454	C₂×C₈	—	2	4	no	—
455	C₂×C₈	—	2	4	no	—
456	C₂×C₈	—	4	4	no	—
457	C₂×C₈	—	4	4	no	—
458	C₄⋊C₄	—	2	2	no	—
459	C₄⋊C₄	—	2	2	no	—
460	C₄⋊C₄	—	2	2	no	—
461	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
462	C₂²⋊C₄	—	2	4	no	—
463	C₂²⋊C₄	—	2	4	no	—
464	C₂²⋊C₄	—	2	4	no	—
465	C₂²⋊C₄	—	2	4	no	—
466	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
467	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
468	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
469	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
470	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
471	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
472	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
473	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
474	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
475	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
476	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
477	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
478	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
479	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
480	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
481	C₄²	C₂⁴×C₅	1	1	yes	characteristic
482	C₂×C₅	—	2	2	no	—
483	C₂×C₅	—	2	2	no	—
484	C₂×C₅	—	2	2	no	—
485	C₂×C₅	—	8	4	no	—
486	C₂×C₅	C₂×#413	1	1	yes	characteristic
487	C₂×C₅	C₂²×#102	1	1	yes	characteristic
488	C₂×C₅	#2896	1	1	yes	characteristic
489	C₂³	—	2	4	no	—
490	C₂³	—	4	1	no	—
491	C₂³	—	4	1	no	—
492	C₂³	—	4	4	no	—
493	C₂³	—	4	4	no	—
494	C₂³	—	4	4	no	—
495	C₂³	C₅×#102	1	4	yes	—
496	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
497	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
498	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
499	Q₈	—	4	4	no	—
500	Q₈	—	4	2	no	—
501	Q₈	—	4	2	no	—
502	D₄	—	4	4	no	—
503	D₄	—	4	4	no	—
504	D₄	—	4	8	no	—
505	D₄	—	4	8	no	—
506	D₄	—	4	4	no	—
507	D₄	—	4	4	no	—
508	D₄	—	4	2	no	—
509	D₄	—	4	2	no	—
510	D₄	—	4	4	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	4	1	no	—
519	C₂×C₄	—	4	1	no	—
520	C₂×C₄	—	4	1	no	—
521	C₂×C₄	—	4	1	no	—
522	C₂×C₄	—	4	1	no	—
523	C₂×C₄	—	4	1	no	—
524	C₂×C₄	—	4	4	no	—
525	C₂×C₄	—	4	8	no	—
526	C₂×C₄	—	4	8	no	—
527	C₂×C₄	C₅×#102	1	4	yes	—
528	C₂×C₄	C₅×#102	1	4	yes	—
529	C₂×C₄	C₅×#102	1	4	yes	—
530	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
531	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
532	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
533	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
534	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
535	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
536	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
537	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
538	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
539	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
540	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
541	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
542	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
543	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
544	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
545	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
546	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
547	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
548	C₈	—	4	4	no	—
549	C₈	—	4	4	no	—
550	C₅	#17704	1	1	yes	characteristic, direct product factor, Sylow
551	C₂²	—	2	1	no	—
552	C₂²	—	2	1	no	—
553	C₂²	—	2	1	no	—
554	C₂²	—	2	1	no	—
555	C₂²	—	2	1	no	—
556	C₂²	—	2	1	no	—
557	C₂²	—	4	4	no	—
558	C₂²	—	4	4	no	—
559	C₂²	—	4	4	no	—
560	C₂²	—	4	4	no	—
561	C₂²	—	8	8	no	—
562	C₂²	C₂×C₅×#102	1	2	yes	—
563	C₂²	C₂×C₅×#102	1	2	yes	—
564	C₂²	C₂×C₅×#102	1	2	yes	—
565	C₂²	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
566	C₄	—	2	1	no	—
567	C₄	—	2	1	no	—
568	C₄	—	2	1	no	—
569	C₄	—	2	1	no	—
570	C₄	—	2	1	no	—
571	C₄	—	2	1	no	—
572	C₄	—	4	4	no	—
573	C₄	C₅×#413	1	2	yes	—
574	C₄	C₂×C₅×#102	1	2	yes	—
575	C₄	C₂×C₅×#102	1	2	yes	—
576	C₄	C₂×C₅×#102	1	2	yes	—
577	C₄	C₂²×C₅×D₈	1	1	yes	characteristic
578	C₄	C₂²×C₅×SD₁₆	1	1	yes	characteristic
579	C₄	C₅×#575	1	2	yes	—
580	C₂	—	2	2	no	—
581	C₂	—	2	2	no	—
582	C₂	—	2	2	no	—
583	C₂	—	8	4	no	—
584	C₂	C₂×C₅×#413	1	1	yes	characteristic
585	C₂	C₂²×C₅×#102	1	1	yes	characteristic
586	C₂	C₅×#2896	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₄²	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. C₂×C₅×#431	C₂	semidirect
3. C₂×C₅×#423	C₂	semidirect
4. C₅×#1547	C₂	semidirect
5. C₅×#1546	C₂	semidirect
6. C₅×#1545	C₂	semidirect
7. C₅×#1540	C₂	semidirect
29. C₅×#431	C₂²	semidirect
33. C₅×#423	C₂²	semidirect
35. C₂×C₅×C₄⋊C₈	C₂²	semidirect
36. C₂×C₅×C₄⋊C₈	C₂²	semidirect
39. C₅×#332	C₂²	semidirect
40. C₅×#331	C₂²	semidirect
41. #17704	C₅	direct (N × H)
117. C₂×#431	C₂×C₅	semidirect
118. C₂×#423	C₂×C₅	semidirect
119. #1547	C₂×C₅	semidirect
120. #1546	C₂×C₅	semidirect
121. #1545	C₂×C₅	semidirect
122. #1540	C₂×C₅	semidirect
225. #431	C₂²×C₅	semidirect
229. #423	C₂²×C₅	semidirect
231. C₂×C₄⋊C₈	C₂²×C₅	semidirect
232. C₂×C₄⋊C₈	C₂²×C₅	semidirect
235. #332	C₂²×C₅	semidirect
236. #331	C₂²×C₅	semidirect
550. C₅	#17704	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
C₂×C₅×#431	2
C₂×C₅×#423	2
C₅×#1547	2
C₅×#1546	2
C₅×#1545	2
C₅×#1540	2
#17704	5

Subgroups — C5×#17704 #11940424

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#17704 #11940424