Subgroups — C₅×#17946 #11940666

Order 1280.

616 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3105	C₂	1	1	yes	maximal, characteristic
2	C₅×#2993	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2975	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1594	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1592	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1590	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1589	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1588	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1586	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#1583	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#1581	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#1574	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅×#1486	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×#1482	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅×#1454	C₂	1	1	yes	maximal, characteristic, complemented
16	C₅×#529	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
18	C₅×#484	C₄	1	1	yes	characteristic
19	C₅×#481	C₄	1	1	yes	characteristic
20	C₅×#478	C₄	1	1	yes	characteristic
21	C₅×#478	C₄	1	1	yes	characteristic
22	C₅×#463	C₄	1	1	yes	characteristic
23	C₅×#463	C₄	1	1	yes	characteristic
24	C₅×#450	C₄	1	1	yes	characteristic
25	C₅×#447	C₄	1	1	yes	characteristic
26	C₅×#446	C₂²	1	1	yes	characteristic, complemented
27	C₅×C₈×Q₈	C₂²	1	1	yes	characteristic, complemented
28	C₄×C₅×SD₁₆	C₂²	1	1	yes	characteristic
29	C₄×C₅×SD₁₆	C₂²	1	1	yes	characteristic
30	C₄×C₅×D₈	C₂²	1	1	yes	characteristic
31	C₄×C₅×D₈	C₂²	1	1	yes	characteristic
32	C₅×#436	—	2	1	no	—
33	C₅×#436	—	2	1	no	—
34	C₅×#436	—	2	1	no	—
35	C₅×#436	C₂²	1	1	yes	characteristic
36	C₅×#436	C₂²	1	1	yes	characteristic
37	C₅×#436	C₂²	1	1	yes	characteristic
38	C₅×#436	C₂²	1	1	yes	characteristic
39	C₅×#436	C₂²	1	1	yes	characteristic, complemented
40	C₅×#435	—	2	1	no	—
41	C₅×C₈×D₄	C₂²	1	1	yes	characteristic, complemented
42	C₅×#433	—	2	1	no	—
43	C₅×#431	C₂²	1	1	yes	characteristic
44	C₅×#431	C₂²	1	1	yes	characteristic
45	C₅×#431	C₂²	1	1	yes	characteristic
46	C₅×#431	C₂²	1	1	yes	characteristic
47	C₅×#424	—	2	1	no	—
48	C₅×#423	C₂²	1	1	yes	characteristic
49	C₅×#407	—	2	1	no	—
50	C₅×#407	—	2	1	no	—
51	C₅×#335	C₂²	1	1	yes	characteristic, complemented
52	C₅×#335	C₂²	1	1	yes	characteristic, complemented
53	C₅×#334	C₂²	1	1	yes	characteristic, complemented
54	C₅×#334	C₂²	1	1	yes	characteristic, complemented
55	C₅×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
56	C₅×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
57	C₅×D₄⋊C₈	C₂²	1	1	yes	characteristic
58	C₅×D₄⋊C₈	C₂²	1	1	yes	characteristic
59	C₅×D₄⋊C₈	C₂²	1	1	yes	characteristic
60	C₅×D₄⋊C₈	C₂²	1	1	yes	characteristic
61	C₅×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
62	C₅×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
63	C₅×#322	C₂²	1	1	yes	characteristic, complemented
64	C₅×#322	C₂²	1	1	yes	characteristic, complemented
65	C₅×#322	C₂²	1	1	yes	characteristic, complemented
66	C₅×C₈²	C₂²	1	1	yes	characteristic, complemented
67	#17946	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂²×C₅×D₄	C₂×C₄	1	1	yes	characteristic
69	C₂³×C₄×C₅	—	2	1	no	—
70	C₂×C₅×SD₁₆	C₂×C₄	1	1	yes	characteristic
71	C₂×C₅×SD₁₆	C₂×C₄	1	1	yes	characteristic
72	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
73	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
74	C₂×C₅×M₁₆	—	2	1	no	—
75	C₂×C₅×M₁₆	—	2	1	no	—
76	C₂×C₅×M₁₆	—	2	1	no	—
77	C₂×C₅×M₁₆	—	2	1	no	—
78	C₂×C₅×M₁₆	—	2	1	no	—
79	C₂×C₅×M₁₆	—	2	1	no	—
80	C₂²×C₅×C₈	—	2	1	no	—
81	C₅×#126	—	2	1	no	—
82	C₅×#125	C₂×C₄	1	1	yes	characteristic
83	C₅×#124	C₂×C₄	1	1	yes	characteristic
84	C₅×#123	—	2	1	no	—
85	C₅×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
86	C₅×#121	C₂×C₄	1	1	yes	characteristic
87	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
88	C₄×C₅×D₄	—	2	1	no	—
89	C₄×C₅×D₄	—	2	1	no	—
90	C₄×C₅×D₄	—	2	1	no	—
91	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
92	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
93	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
94	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
95	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
96	C₅×#117	—	2	1	no	—
97	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
98	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
99	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
100	C₅×#107	C₂×C₄	1	1	yes	characteristic
101	C₅×#107	C₂×C₄	1	1	yes	characteristic
102	C₅×#106	C₂×C₄	1	1	yes	characteristic
103	C₅×#106	C₂×C₄	1	1	yes	characteristic
104	C₅×C₄⋊C₈	—	2	1	no	—
105	C₅×C₄⋊C₈	—	2	1	no	—
106	C₅×C₄⋊C₈	—	2	1	no	—
107	C₅×C₄⋊C₈	—	2	1	no	—
108	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
109	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
110	C₅×C₄⋊C₈	D₄	1	1	yes	characteristic
111	C₅×C₄⋊C₈	D₄	1	1	yes	characteristic
112	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
113	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
114	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
115	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
116	C₅×#103	C₂×C₄	1	1	yes	characteristic
117	C₅×#103	C₂×C₄	1	1	yes	characteristic
118	C₅×#102	C₂×C₄	1	1	yes	characteristic
119	C₅×#102	C₂×C₄	1	1	yes	characteristic
120	C₅×#102	C₂×C₄	1	1	yes	characteristic
121	C₅×#102	C₂×C₄	1	1	yes	characteristic
122	C₅×#102	C₂×C₄	1	1	yes	characteristic
123	C₅×#102	C₂×C₄	1	1	yes	characteristic
124	C₅×C₂²⋊C₈	—	2	1	no	—
125	C₅×C₂²⋊C₈	—	2	1	no	—
126	C₅×C₂²⋊C₈	—	2	1	no	—
127	C₅×C₂²⋊C₈	—	2	1	no	—
128	C₅×C₂²⋊C₈	—	2	1	no	—
129	C₅×C₂²⋊C₈	—	2	1	no	—
130	C₅×C₂²⋊C₈	—	2	1	no	—
131	C₅×C₂²⋊C₈	—	2	1	no	—
132	C₅×C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
133	C₅×C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
134	C₅×C₂²⋊C₈	D₄	1	1	yes	characteristic
135	C₅×C₂²⋊C₈	D₄	1	1	yes	characteristic
136	C₅×#97	—	2	1	no	—
137	C₄×C₅×C₈	—	2	1	no	—
138	C₄×C₅×C₈	—	2	1	no	—
139	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
140	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
141	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
142	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
143	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
144	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
145	#3105	C₂×C₅	1	1	yes	characteristic
146	#2993	C₂×C₅	1	1	yes	characteristic, complemented
147	#2975	C₂×C₅	1	1	yes	characteristic, complemented
148	#1594	C₂×C₅	1	1	yes	characteristic, complemented
149	#1592	C₂×C₅	1	1	yes	characteristic, complemented
150	#1590	C₂×C₅	1	1	yes	characteristic, complemented
151	#1589	C₂×C₅	1	1	yes	characteristic, complemented
152	#1588	C₂×C₅	1	1	yes	characteristic, complemented
153	#1586	C₂×C₅	1	1	yes	characteristic, complemented
154	#1583	C₂×C₅	1	1	yes	characteristic, complemented
155	#1581	C₂×C₅	1	1	yes	characteristic, complemented
156	#1574	C₂×C₅	1	1	yes	characteristic, complemented
157	#1486	C₂×C₅	1	1	yes	characteristic, complemented
158	#1482	C₂×C₅	1	1	yes	characteristic, complemented
159	#1454	C₂×C₅	1	1	yes	characteristic, complemented
160	C₂⁴×C₅	—	2	1	no	—
161	C₂×C₅×Q₈	C₂²×C₄	1	1	yes	characteristic
162	C₂×C₅×D₄	—	2	1	no	—
163	C₂×C₅×D₄	—	2	1	no	—
164	C₂×C₅×D₄	—	4	1	no	—
165	C₂×C₅×D₄	—	4	1	no	—
166	C₂×C₅×D₄	C₂²×C₄	1	1	yes	characteristic
167	C₂×C₅×D₄	C₂²×C₄	1	1	yes	characteristic
168	C₂×C₅×D₄	C₂²×C₄	1	1	yes	characteristic
169	C₂²×C₄×C₅	—	2	1	no	—
170	C₂²×C₄×C₅	—	2	1	no	—
171	C₂²×C₄×C₅	—	2	1	no	—
172	C₂²×C₄×C₅	—	2	1	no	—
173	C₂²×C₄×C₅	—	2	1	no	—
174	C₂²×C₄×C₅	—	2	1	no	—
175	C₂²×C₄×C₅	—	8	1	no	—
176	C₂²×C₄×C₅	—	8	1	no	—
177	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
178	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₅	#41	1	1	yes	characteristic
180	C₅×SD₁₆	—	4	1	no	—
181	C₅×SD₁₆	—	4	1	no	—
182	C₅×D₈	—	4	1	no	—
183	C₅×D₈	—	4	1	no	—
184	C₅×M₁₆	—	8	1	no	—
185	C₅×M₁₆	—	8	1	no	—
186	C₅×M₁₆	—	8	1	no	—
187	C₅×M₁₆	—	8	1	no	—
188	C₅×M₁₆	—	8	1	no	—
189	C₅×M₁₆	—	8	1	no	—
190	C₂×C₅×C₈	—	2	1	no	—
191	C₂×C₅×C₈	—	2	1	no	—
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	—	8	1	no	—
197	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
198	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
199	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
200	C₂×C₅×C₈	C₂²×C₄	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×D₄	1	1	yes	characteristic
204	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
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	C₂²×C₄	1	1	yes	characteristic
211	C₅×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
212	C₅×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
213	C₅×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
214	C₅×C₂²⋊C₄	—	2	1	no	—
215	C₅×C₂²⋊C₄	—	2	1	no	—
216	C₅×C₂²⋊C₄	—	2	1	no	—
217	C₅×C₂²⋊C₄	—	2	1	no	—
218	C₅×C₂²⋊C₄	—	2	1	no	—
219	C₅×C₂²⋊C₄	—	2	1	no	—
220	C₅×C₂²⋊C₄	—	2	1	no	—
221	C₅×C₂²⋊C₄	—	2	1	no	—
222	C₅×C₂²⋊C₄	—	2	1	no	—
223	C₄²×C₅	—	2	1	no	—
224	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
225	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
226	C₄²×C₅	#41	1	1	yes	characteristic
227	C₄²×C₅	C₂⁴	1	1	yes	characteristic
228	#529	C₂²×C₅	1	1	yes	characteristic
229	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
230	#484	C₄×C₅	1	1	yes	characteristic
231	#481	C₄×C₅	1	1	yes	characteristic
232	#478	C₄×C₅	1	1	yes	characteristic
233	#478	C₄×C₅	1	1	yes	characteristic
234	#463	C₄×C₅	1	1	yes	characteristic
235	#463	C₄×C₅	1	1	yes	characteristic
236	#450	C₄×C₅	1	1	yes	characteristic
237	#447	C₄×C₅	1	1	yes	characteristic
238	#446	C₂²×C₅	1	1	yes	characteristic, complemented
239	C₈×Q₈	C₂²×C₅	1	1	yes	characteristic, complemented
240	C₄×SD₁₆	C₂²×C₅	1	1	yes	characteristic
241	C₄×SD₁₆	C₂²×C₅	1	1	yes	characteristic
242	C₄×D₈	C₂²×C₅	1	1	yes	characteristic
243	C₄×D₈	C₂²×C₅	1	1	yes	characteristic
244	#436	—	2	1	no	—
245	#436	—	2	1	no	—
246	#436	—	2	1	no	—
247	#436	C₂²×C₅	1	1	yes	characteristic
248	#436	C₂²×C₅	1	1	yes	characteristic
249	#436	C₂²×C₅	1	1	yes	characteristic
250	#436	C₂²×C₅	1	1	yes	characteristic
251	#436	C₂²×C₅	1	1	yes	characteristic, complemented
252	#435	—	2	1	no	—
253	C₈×D₄	C₂²×C₅	1	1	yes	characteristic, complemented
254	#433	—	2	1	no	—
255	#431	C₂²×C₅	1	1	yes	characteristic
256	#431	C₂²×C₅	1	1	yes	characteristic
257	#431	C₂²×C₅	1	1	yes	characteristic
258	#431	C₂²×C₅	1	1	yes	characteristic
259	#424	—	2	1	no	—
260	#423	C₂²×C₅	1	1	yes	characteristic
261	#407	—	2	1	no	—
262	#407	—	2	1	no	—
263	#335	C₂²×C₅	1	1	yes	characteristic, complemented
264	#335	C₂²×C₅	1	1	yes	characteristic, complemented
265	#334	C₂²×C₅	1	1	yes	characteristic, complemented
266	#334	C₂²×C₅	1	1	yes	characteristic, complemented
267	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic, complemented
268	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic, complemented
269	D₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
270	D₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
271	D₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
272	D₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
273	D₄⋊C₈	C₂²×C₅	1	1	yes	characteristic, complemented
274	D₄⋊C₈	C₂²×C₅	1	1	yes	characteristic, complemented
275	#322	C₂²×C₅	1	1	yes	characteristic, complemented
276	#322	C₂²×C₅	1	1	yes	characteristic, complemented
277	#322	C₂²×C₅	1	1	yes	characteristic, complemented
278	C₈²	C₂²×C₅	1	1	yes	characteristic, complemented
279	C₂³×C₅	—	2	1	no	—
280	C₂³×C₅	—	2	1	no	—
281	C₂³×C₅	—	2	1	no	—
282	C₂³×C₅	—	4	2	no	—
283	C₂³×C₅	—	8	1	no	—
284	C₂³×C₅	—	8	1	no	—
285	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
286	C₅×Q₈	—	2	1	no	—
287	C₅×Q₈	—	2	1	no	—
288	C₅×D₄	—	2	1	no	—
289	C₅×D₄	—	2	1	no	—
290	C₅×D₄	—	2	1	no	—
291	C₅×D₄	—	2	1	no	—
292	C₅×D₄	—	2	1	no	—
293	C₅×D₄	—	2	1	no	—
294	C₅×D₄	—	4	2	no	—
295	C₅×D₄	—	8	1	no	—
296	C₂×C₄×C₅	—	2	1	no	—
297	C₂×C₄×C₅	—	2	1	no	—
298	C₂×C₄×C₅	—	2	1	no	—
299	C₂×C₄×C₅	—	2	1	no	—
300	C₂×C₄×C₅	—	2	1	no	—
301	C₂×C₄×C₅	—	2	1	no	—
302	C₂×C₄×C₅	—	2	1	no	—
303	C₂×C₄×C₅	—	2	1	no	—
304	C₂×C₄×C₅	—	2	1	no	—
305	C₂×C₄×C₅	—	4	1	no	—
306	C₂×C₄×C₅	—	4	1	no	—
307	C₂×C₄×C₅	—	4	1	no	—
308	C₂×C₄×C₅	—	4	1	no	—
309	C₂×C₄×C₅	—	4	1	no	—
310	C₂×C₄×C₅	—	4	1	no	—
311	C₂×C₄×C₅	—	4	1	no	—
312	C₂×C₄×C₅	—	8	1	no	—
313	C₂×C₄×C₅	—	8	1	no	—
314	C₂×C₄×C₅	—	8	1	no	—
315	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
316	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
317	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
318	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
319	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
320	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
321	C₅×C₈	—	2	1	no	—
322	C₅×C₈	—	2	1	no	—
323	C₅×C₈	—	2	1	no	—
324	C₅×C₈	—	2	1	no	—
325	C₅×C₈	—	2	1	no	—
326	C₅×C₈	—	2	1	no	—
327	C₅×C₈	—	2	1	no	—
328	C₅×C₈	—	2	1	no	—
329	C₅×C₈	—	4	1	no	—
330	C₅×C₈	—	4	1	no	—
331	C₅×C₈	—	4	1	no	—
332	C₅×C₈	—	4	1	no	—
333	C₅×C₈	—	4	1	no	—
334	C₅×C₈	—	4	1	no	—
335	C₂²×D₄	C₂×C₄×C₅	1	1	yes	characteristic
336	C₂³×C₄	—	2	1	no	—
337	C₂×SD₁₆	C₂×C₄×C₅	1	1	yes	characteristic
338	C₂×SD₁₆	C₂×C₄×C₅	1	1	yes	characteristic
339	C₂×D₈	C₂×C₄×C₅	1	1	yes	characteristic
340	C₂×D₈	C₂×C₄×C₅	1	1	yes	characteristic
341	C₂×M₁₆	—	2	1	no	—
342	C₂×M₁₆	—	2	1	no	—
343	C₂×M₁₆	—	2	1	no	—
344	C₂×M₁₆	—	2	1	no	—
345	C₂×M₁₆	—	2	1	no	—
346	C₂×M₁₆	—	2	1	no	—
347	C₂²×C₈	—	2	1	no	—
348	#126	—	2	1	no	—
349	#125	C₂×C₄×C₅	1	1	yes	characteristic
350	#124	C₂×C₄×C₅	1	1	yes	characteristic
351	#123	—	2	1	no	—
352	C₂²⋊Q₈	C₂×C₄×C₅	1	1	yes	characteristic
353	#121	C₂×C₄×C₅	1	1	yes	characteristic
354	C₄×Q₈	C₂³×C₅	1	1	yes	characteristic
355	C₄×D₄	—	2	1	no	—
356	C₄×D₄	—	2	1	no	—
357	C₄×D₄	—	2	1	no	—
358	C₄×D₄	C₂×C₄×C₅	1	1	yes	characteristic
359	C₄×D₄	C₂×C₄×C₅	1	1	yes	characteristic
360	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
361	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
362	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
363	#117	—	2	1	no	—
364	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
365	C₂×C₂²⋊C₄	—	2	1	no	—
366	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
367	#107	C₂×C₄×C₅	1	1	yes	characteristic
368	#107	C₂×C₄×C₅	1	1	yes	characteristic
369	#106	C₂×C₄×C₅	1	1	yes	characteristic
370	#106	C₂×C₄×C₅	1	1	yes	characteristic
371	C₄⋊C₈	—	2	1	no	—
372	C₄⋊C₈	—	2	1	no	—
373	C₄⋊C₈	—	2	1	no	—
374	C₄⋊C₈	—	2	1	no	—
375	C₄⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
376	C₄⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
377	C₄⋊C₈	C₅×D₄	1	1	yes	characteristic
378	C₄⋊C₈	C₅×D₄	1	1	yes	characteristic
379	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
380	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
381	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
382	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
383	#103	C₂×C₄×C₅	1	1	yes	characteristic
384	#103	C₂×C₄×C₅	1	1	yes	characteristic
385	#102	C₂×C₄×C₅	1	1	yes	characteristic
386	#102	C₂×C₄×C₅	1	1	yes	characteristic
387	#102	C₂×C₄×C₅	1	1	yes	characteristic
388	#102	C₂×C₄×C₅	1	1	yes	characteristic
389	#102	C₂×C₄×C₅	1	1	yes	characteristic
390	#102	C₂×C₄×C₅	1	1	yes	characteristic
391	C₂²⋊C₈	—	2	1	no	—
392	C₂²⋊C₈	—	2	1	no	—
393	C₂²⋊C₈	—	2	1	no	—
394	C₂²⋊C₈	—	2	1	no	—
395	C₂²⋊C₈	—	2	1	no	—
396	C₂²⋊C₈	—	2	1	no	—
397	C₂²⋊C₈	—	2	1	no	—
398	C₂²⋊C₈	—	2	1	no	—
399	C₂²⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
400	C₂²⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
401	C₂²⋊C₈	C₅×D₄	1	1	yes	characteristic
402	C₂²⋊C₈	C₅×D₄	1	1	yes	characteristic
403	#97	—	2	1	no	—
404	C₄×C₈	—	2	1	no	—
405	C₄×C₈	—	2	1	no	—
406	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
407	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
408	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
409	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
410	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
411	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
412	C₂²×C₅	—	2	1	no	—
413	C₂²×C₅	—	2	1	no	—
414	C₂²×C₅	—	2	1	no	—
415	C₂²×C₅	—	4	1	no	—
416	C₂²×C₅	—	4	1	no	—
417	C₂²×C₅	—	4	1	no	—
418	C₂²×C₅	—	4	1	no	—
419	C₂²×C₅	—	4	1	no	—
420	C₂²×C₅	—	4	1	no	—
421	C₂²×C₅	—	4	1	no	—
422	C₂²×C₅	—	4	1	no	—
423	C₂²×C₅	—	4	1	no	—
424	C₂²×C₅	—	8	4	no	—
425	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
426	C₄×C₅	—	2	1	no	—
427	C₄×C₅	—	2	1	no	—
428	C₄×C₅	—	2	1	no	—
429	C₄×C₅	—	2	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	#568	1	1	yes	characteristic
436	C₄×C₅	#568	1	1	yes	characteristic
437	C₄×C₅	#576	1	1	yes	characteristic
438	C₄×C₅	#577	1	1	yes	characteristic
439	C₂⁴	—	2	1	no	—
440	C₂×Q₈	C₂²×C₄×C₅	1	1	yes	characteristic
441	C₂×D₄	—	2	1	no	—
442	C₂×D₄	—	2	1	no	—
443	C₂×D₄	—	4	1	no	—
444	C₂×D₄	—	4	1	no	—
445	C₂×D₄	C₂²×C₄×C₅	1	1	yes	characteristic
446	C₂×D₄	C₂²×C₄×C₅	1	1	yes	characteristic
447	C₂×D₄	C₂²×C₄×C₅	1	1	yes	characteristic
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	—	2	1	no	—
450	C₂²×C₄	—	2	1	no	—
451	C₂²×C₄	—	2	1	no	—
452	C₂²×C₄	—	2	1	no	—
453	C₂²×C₄	—	2	1	no	—
454	C₂²×C₄	—	8	1	no	—
455	C₂²×C₄	—	8	1	no	—
456	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
457	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
458	C₂²×C₄	C₅×#41	1	1	yes	characteristic
459	SD₁₆	—	4	1	no	—
460	SD₁₆	—	4	1	no	—
461	D₈	—	4	1	no	—
462	D₈	—	4	1	no	—
463	M₁₆	—	8	1	no	—
464	M₁₆	—	8	1	no	—
465	M₁₆	—	8	1	no	—
466	M₁₆	—	8	1	no	—
467	M₁₆	—	8	1	no	—
468	M₁₆	—	8	1	no	—
469	C₂×C₈	—	2	1	no	—
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	1	no	—
475	C₂×C₈	—	8	1	no	—
476	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
477	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
478	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
479	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
480	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
481	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
482	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
483	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
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	C₂²×C₄×C₅	1	1	yes	characteristic
490	C₄⋊C₄	C₂²×C₄×C₅	1	1	yes	characteristic
491	C₄⋊C₄	C₂²×C₄×C₅	1	1	yes	characteristic
492	C₄⋊C₄	C₂²×C₄×C₅	1	1	yes	characteristic
493	C₂²⋊C₄	—	2	1	no	—
494	C₂²⋊C₄	—	2	1	no	—
495	C₂²⋊C₄	—	2	1	no	—
496	C₂²⋊C₄	—	2	1	no	—
497	C₂²⋊C₄	—	2	1	no	—
498	C₂²⋊C₄	—	2	1	no	—
499	C₂²⋊C₄	—	2	1	no	—
500	C₂²⋊C₄	—	2	1	no	—
501	C₂²⋊C₄	—	2	1	no	—
502	C₄²	—	2	1	no	—
503	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
504	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
505	C₄²	C₅×#41	1	1	yes	characteristic
506	C₄²	C₂⁴×C₅	1	1	yes	characteristic
507	C₂×C₅	—	4	1	no	—
508	C₂×C₅	—	8	1	no	—
509	C₂×C₅	—	8	1	no	—
510	C₂×C₅	—	8	1	no	—
511	C₂×C₅	#2930	1	1	yes	characteristic
512	C₂×C₅	#2946	1	1	yes	characteristic
513	C₂×C₅	#2957	1	1	yes	characteristic
514	C₂³	—	2	1	no	—
515	C₂³	—	2	1	no	—
516	C₂³	—	2	1	no	—
517	C₂³	—	4	2	no	—
518	C₂³	—	8	1	no	—
519	C₂³	—	8	1	no	—
520	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
521	Q₈	—	2	1	no	—
522	Q₈	—	2	1	no	—
523	D₄	—	2	1	no	—
524	D₄	—	2	1	no	—
525	D₄	—	2	1	no	—
526	D₄	—	2	1	no	—
527	D₄	—	2	1	no	—
528	D₄	—	2	1	no	—
529	D₄	—	4	2	no	—
530	D₄	—	8	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	4	1	no	—
541	C₂×C₄	—	4	1	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	8	1	no	—
548	C₂×C₄	—	8	1	no	—
549	C₂×C₄	—	8	1	no	—
550	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
551	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
552	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
553	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
554	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
555	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
556	C₈	—	2	1	no	—
557	C₈	—	2	1	no	—
558	C₈	—	2	1	no	—
559	C₈	—	2	1	no	—
560	C₈	—	2	1	no	—
561	C₈	—	2	1	no	—
562	C₈	—	2	1	no	—
563	C₈	—	2	1	no	—
564	C₈	—	4	1	no	—
565	C₈	—	4	1	no	—
566	C₈	—	4	1	no	—
567	C₈	—	4	1	no	—
568	C₈	—	4	1	no	—
569	C₈	—	4	1	no	—
570	C₅	#17946	1	1	yes	characteristic, direct product factor, Sylow
571	C₂²	—	2	1	no	—
572	C₂²	—	2	1	no	—
573	C₂²	—	2	1	no	—
574	C₂²	—	4	1	no	—
575	C₂²	—	4	1	no	—
576	C₂²	—	4	1	no	—
577	C₂²	—	4	1	no	—
578	C₂²	—	4	1	no	—
579	C₂²	—	4	1	no	—
580	C₂²	—	4	1	no	—
581	C₂²	—	4	1	no	—
582	C₂²	—	4	1	no	—
583	C₂²	—	8	4	no	—
584	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
585	C₄	—	2	1	no	—
586	C₄	—	2	1	no	—
587	C₄	—	2	1	no	—
588	C₄	—	2	1	no	—
589	C₄	—	4	1	no	—
590	C₄	—	4	1	no	—
591	C₄	—	4	1	no	—
592	C₄	—	4	1	no	—
593	C₄	—	4	1	no	—
594	C₄	C₅×#568	1	1	yes	characteristic
595	C₄	C₅×#568	1	1	yes	characteristic
596	C₄	C₅×#576	1	1	yes	characteristic
597	C₄	C₅×#577	1	1	yes	characteristic
598	C₂	—	4	1	no	—
599	C₂	—	8	1	no	—
600	C₂	—	8	1	no	—
601	C₂	—	8	1	no	—
602	C₂	C₅×#2930	1	1	yes	characteristic
603	C₂	C₅×#2946	1	1	yes	characteristic
604	C₂	C₅×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₄²	C₂⁴×C₅
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₅×#447	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₅×#431 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2993	C₂	semidirect
3. C₅×#2975	C₂	semidirect
4. C₅×#1594	C₂	semidirect
5. C₅×#1592	C₂	semidirect
6. C₅×#1590	C₂	semidirect
7. C₅×#1589	C₂	semidirect
8. C₅×#1588	C₂	semidirect
9. C₅×#1586	C₂	semidirect
10. C₅×#1583	C₂	semidirect
11. C₅×#1581	C₂	semidirect
12. C₅×#1574	C₂	semidirect
13. C₅×#1486	C₂	semidirect
14. C₅×#1482	C₂	semidirect
15. C₅×#1454	C₂	semidirect
26. C₅×#446	C₂²	semidirect
27. C₅×C₈×Q₈	C₂²	semidirect
39. C₅×#436	C₂²	semidirect
41. C₅×C₈×D₄	C₂²	semidirect
51. C₅×#335	C₂²	semidirect
52. C₅×#335	C₂²	semidirect
53. C₅×#334	C₂²	semidirect
54. C₅×#334	C₂²	semidirect
55. C₅×Q₈⋊C₈	C₂²	semidirect
56. C₅×Q₈⋊C₈	C₂²	semidirect
61. C₅×D₄⋊C₈	C₂²	semidirect
62. C₅×D₄⋊C₈	C₂²	semidirect
63. C₅×#322	C₂²	semidirect
64. C₅×#322	C₂²	semidirect
65. C₅×#322	C₂²	semidirect
66. C₅×C₈²	C₂²	semidirect
67. #17946	C₅	direct (N × H)
146. #2993	C₂×C₅	semidirect
147. #2975	C₂×C₅	semidirect
148. #1594	C₂×C₅	semidirect
149. #1592	C₂×C₅	semidirect
150. #1590	C₂×C₅	semidirect
151. #1589	C₂×C₅	semidirect
152. #1588	C₂×C₅	semidirect
153. #1586	C₂×C₅	semidirect
154. #1583	C₂×C₅	semidirect
155. #1581	C₂×C₅	semidirect
156. #1574	C₂×C₅	semidirect
157. #1486	C₂×C₅	semidirect
158. #1482	C₂×C₅	semidirect
159. #1454	C₂×C₅	semidirect
238. #446	C₂²×C₅	semidirect
239. C₈×Q₈	C₂²×C₅	semidirect
251. #436	C₂²×C₅	semidirect
253. C₈×D₄	C₂²×C₅	semidirect
263. #335	C₂²×C₅	semidirect
264. #335	C₂²×C₅	semidirect
265. #334	C₂²×C₅	semidirect
266. #334	C₂²×C₅	semidirect
267. Q₈⋊C₈	C₂²×C₅	semidirect
268. Q₈⋊C₈	C₂²×C₅	semidirect
273. D₄⋊C₈	C₂²×C₅	semidirect
274. D₄⋊C₈	C₂²×C₅	semidirect
275. #322	C₂²×C₅	semidirect
276. #322	C₂²×C₅	semidirect
277. #322	C₂²×C₅	semidirect
278. C₈²	C₂²×C₅	semidirect
570. C₅	#17946	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×#3105	2
C₅×#2993	2
C₅×#2975	2
C₅×#1594	2
C₅×#1592	2
C₅×#1590	2
C₅×#1589	2
C₅×#1588	2
C₅×#1586	2
C₅×#1583	2
C₅×#1581	2
C₅×#1574	2
C₅×#1486	2
C₅×#1482	2
C₅×#1454	2
#17946	5

Subgroups — C5×#17946 #11940666

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#17946 #11940666