Subgroups — C₅×#17775 #11940495

Order 1280.

926 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3511	C₂	1	1	yes	maximal, characteristic
2	C₅×#2990	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2990	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1559	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1557	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1556	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1554	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1553	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1551	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#1551	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#1550	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#1547	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅×#1542	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×#1540	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₅×#332	C₂	1	1	yes	maximal, characteristic, complemented
16	C₅×#556	—	2	1	no	—
17	C₅×#555	—	2	1	no	—
18	C₅×#554	—	2	1	no	—
19	C₅×#553	—	2	1	no	—
20	C₅×#553	—	2	1	no	—
21	C₅×#548	—	2	1	no	—
22	C₅×#548	—	2	1	no	—
23	C₅×#547	—	2	1	no	—
24	C₅×#542	C₄	1	2	yes	—
25	C₅×#542	C₄	1	2	yes	—
26	C₅×#537	C₄	1	2	yes	—
27	C₅×#533	C₂²	1	1	yes	characteristic
28	C₅×#533	C₂²	1	1	yes	characteristic
29	C₅×#532	C₂²	1	1	yes	characteristic
30	C₅×#532	C₂²	1	1	yes	characteristic
31	C₅×#532	C₂²	1	1	yes	characteristic
32	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
33	C₅×#520	C₄	1	2	yes	—
34	C₄×C₅×#41	C₂²	1	1	yes	characteristic
35	C₅×#446	—	2	1	no	—
36	C₅×#446	—	2	1	no	—
37	C₅×#435	—	2	1	no	—
38	C₅×#435	—	2	1	no	—
39	C₅×#435	—	2	1	no	—
40	C₅×#435	—	2	1	no	—
41	C₅×C₈×D₄	—	2	1	no	—
42	C₅×C₈×D₄	—	2	1	no	—
43	C₅×#432	C₂²	1	1	yes	characteristic
44	C₅×#432	C₂²	1	1	yes	characteristic
45	C₅×#432	C₂²	1	1	yes	characteristic
46	C₅×#432	C₂²	1	1	yes	characteristic
47	C₅×#431	C₂²	1	1	yes	characteristic
48	C₅×#431	C₂²	1	1	yes	characteristic
49	C₅×#423	C₂²	1	1	yes	characteristic
50	C₅×#423	C₂²	1	1	yes	characteristic
51	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
52	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
53	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
54	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
55	C₅×#333	C₂²	1	1	yes	characteristic
56	C₅×#333	C₂²	1	1	yes	characteristic
57	C₅×#333	C₂²	1	1	yes	characteristic
58	C₅×#333	C₂²	1	1	yes	characteristic
59	C₅×#332	C₂²	1	1	yes	characteristic
60	C₅×#332	C₂²	1	1	yes	characteristic
61	C₅×#332	C₂²	1	1	yes	characteristic
62	C₅×#332	C₂²	1	1	yes	characteristic
63	C₅×#332	C₂²	1	1	yes	characteristic
64	C₅×#332	C₂²	1	1	yes	characteristic
65	C₅×#332	C₂²	1	1	yes	characteristic
66	C₅×#332	C₂²	1	1	yes	characteristic
67	C₅×#331	C₂²	1	1	yes	characteristic
68	C₅×#331	C₂²	1	1	yes	characteristic
69	C₅×#331	C₂²	1	1	yes	characteristic
70	C₅×#331	C₂²	1	1	yes	characteristic
71	#17775	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂×C₅×#41	—	2	1	no	—
73	C₂×C₅×#41	—	2	1	no	—
74	C₂×C₅×#41	C₂×C₄	1	2	yes	—
75	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
76	C₂²×C₅×Q₈	—	2	1	no	—
77	C₂×C₅×M₁₆	—	2	1	no	—
78	C₂×C₅×M₁₆	—	2	1	no	—
79	C₂²×C₅×C₈	—	2	1	no	—
80	C₂²×C₅×C₈	—	2	1	no	—
81	C₂²×C₅×C₈	—	2	1	no	—
82	C₂²×C₅×C₈	—	2	1	no	—
83	C₅×C₄⋊Q₈	—	2	1	no	—
84	C₅×C₄⋊Q₈	—	2	1	no	—
85	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
86	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
87	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
88	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
89	C₅×#127	C₂³	1	1	yes	characteristic
90	C₅×#126	—	2	2	no	—
91	C₅×#126	—	2	2	no	—
92	C₅×#125	—	2	1	no	—
93	C₅×#125	—	2	1	no	—
94	C₅×#124	—	2	2	no	—
95	C₅×#124	—	2	2	no	—
96	C₅×#124	—	2	1	no	—
97	C₅×#124	—	2	1	no	—
98	C₅×#124	C₂×C₄	1	2	yes	—
99	C₅×#123	—	2	2	no	—
100	C₅×#123	—	2	2	no	—
101	C₅×C₂²⋊Q₈	—	2	2	no	—
102	C₅×C₂²⋊Q₈	—	2	1	no	—
103	C₅×C₂²⋊Q₈	—	2	1	no	—
104	C₅×C₂²⋊Q₈	—	2	2	no	—
105	C₅×C₂²⋊Q₈	—	2	2	no	—
106	C₅×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
107	C₅×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
108	C₅×#121	—	2	2	no	—
109	C₅×#121	—	2	2	no	—
110	C₅×#121	—	2	2	no	—
111	C₅×#121	—	2	1	no	—
112	C₅×#121	—	2	1	no	—
113	C₅×#121	—	2	1	no	—
114	C₅×#121	—	2	1	no	—
115	C₅×#121	C₂×C₄	1	2	yes	—
116	C₅×#121	C₂×C₄	1	2	yes	—
117	C₄×C₅×Q₈	—	2	1	no	—
118	C₄×C₅×Q₈	—	2	1	no	—
119	C₄×C₅×Q₈	C₂×C₄	1	2	yes	—
120	C₄×C₅×Q₈	C₂×C₄	1	2	yes	—
121	C₄×C₅×Q₈	D₄	1	1	yes	characteristic
122	C₄×C₅×Q₈	D₄	1	1	yes	characteristic
123	C₄×C₅×D₄	—	2	2	no	—
124	C₄×C₅×D₄	—	2	2	no	—
125	C₄×C₅×D₄	—	2	1	no	—
126	C₄×C₅×D₄	—	2	1	no	—
127	C₄×C₅×D₄	—	2	1	no	—
128	C₄×C₅×D₄	—	2	1	no	—
129	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
130	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
131	C₄×C₅×D₄	D₄	1	1	yes	characteristic
132	C₄×C₅×D₄	D₄	1	1	yes	characteristic
133	C₄×C₅×D₄	D₄	1	1	yes	characteristic
134	C₄×C₅×D₄	D₄	1	1	yes	characteristic
135	C₄×C₅×D₄	D₄	1	1	yes	characteristic
136	C₄×C₅×D₄	D₄	1	1	yes	characteristic
137	C₅×#117	—	2	1	no	—
138	C₅×#117	—	2	1	no	—
139	C₅×#117	—	2	2	no	—
140	C₅×#117	C₂×C₄	1	1	yes	characteristic
141	C₅×#117	C₂×C₄	1	1	yes	characteristic
142	C₅×#117	C₂×C₄	1	1	yes	characteristic
143	C₂×C₅×C₄⋊C₄	—	2	1	no	—
144	C₂×C₅×C₄⋊C₄	—	2	2	no	—
145	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
146	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
147	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
148	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
149	C₅×C₄⋊C₈	—	2	1	no	—
150	C₅×C₄⋊C₈	—	2	1	no	—
151	C₅×C₄⋊C₈	—	2	1	no	—
152	C₅×C₄⋊C₈	—	2	1	no	—
153	C₅×C₄⋊C₈	—	2	1	no	—
154	C₅×C₄⋊C₈	—	2	1	no	—
155	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
156	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
157	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
158	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
159	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
160	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
161	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
162	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
163	C₅×C₂²⋊C₈	—	2	1	no	—
164	C₅×C₂²⋊C₈	—	2	1	no	—
165	C₅×C₂²⋊C₈	—	2	1	no	—
166	C₅×C₂²⋊C₈	—	2	1	no	—
167	C₅×C₂²⋊C₈	—	2	1	no	—
168	C₅×C₂²⋊C₈	—	2	1	no	—
169	C₅×#97	—	2	1	no	—
170	C₅×#97	—	2	1	no	—
171	C₅×#97	—	2	1	no	—
172	C₅×#97	—	2	1	no	—
173	C₄×C₅×C₈	—	2	1	no	—
174	C₄×C₅×C₈	—	2	1	no	—
175	#3511	C₂×C₅	1	1	yes	characteristic
176	#2990	C₂×C₅	1	1	yes	characteristic, complemented
177	#2990	C₂×C₅	1	1	yes	characteristic, complemented
178	#1559	C₂×C₅	1	1	yes	characteristic, complemented
179	#1557	C₂×C₅	1	1	yes	characteristic, complemented
180	#1556	C₂×C₅	1	1	yes	characteristic, complemented
181	#1554	C₂×C₅	1	1	yes	characteristic, complemented
182	#1553	C₂×C₅	1	1	yes	characteristic, complemented
183	#1551	C₂×C₅	1	1	yes	characteristic, complemented
184	#1551	C₂×C₅	1	1	yes	characteristic, complemented
185	#1550	C₂×C₅	1	1	yes	characteristic, complemented
186	#1547	C₂×C₅	1	1	yes	characteristic, complemented
187	#1542	C₂×C₅	1	1	yes	characteristic, complemented
188	#1540	C₂×C₅	1	1	yes	characteristic, complemented
189	C₂×#332	C₂×C₅	1	1	yes	characteristic, complemented
190	C₅×#41	—	4	2	no	—
191	C₅×#41	—	4	4	no	—
192	C₅×#41	—	8	1	no	—
193	C₅×#41	—	8	1	no	—
194	C₅×#41	—	8	1	no	—
195	C₅×#41	—	8	1	no	—
196	C₂×C₅×Q₈	—	2	2	no	—
197	C₂×C₅×Q₈	—	2	1	no	—
198	C₂×C₅×Q₈	—	2	1	no	—
199	C₂×C₅×Q₈	—	8	1	no	—
200	C₂×C₅×Q₈	—	8	1	no	—
201	C₂×C₅×Q₈	C₂²⋊C₄	1	1	yes	characteristic
202	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
203	C₂×C₅×D₄	—	2	2	no	—
204	C₂×C₅×D₄	—	2	2	no	—
205	C₂×C₅×D₄	—	2	2	no	—
206	C₂×C₅×D₄	—	2	1	no	—
207	C₂×C₅×D₄	—	2	1	no	—
208	C₂×C₅×D₄	C₂²⋊C₄	1	1	yes	characteristic
209	C₂×C₅×D₄	C₂²⋊C₄	1	1	yes	characteristic
210	C₂×C₅×D₄	C₂²⋊C₄	1	1	yes	characteristic
211	C₂×C₅×D₄	C₂²×C₄	1	2	yes	—
212	C₂²×C₄×C₅	—	2	2	no	—
213	C₂²×C₄×C₅	—	2	2	no	—
214	C₂²×C₄×C₅	—	2	2	no	—
215	C₂²×C₄×C₅	—	2	2	no	—
216	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
217	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
218	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
219	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
220	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
221	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
222	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
223	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
224	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
225	C₅×M₁₆	—	8	1	no	—
226	C₅×M₁₆	—	8	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	—	2	1	no	—
234	C₂×C₅×C₈	—	2	1	no	—
235	C₂×C₅×C₈	—	8	1	no	—
236	C₂×C₅×C₈	—	8	1	no	—
237	C₂×C₅×C₈	—	8	1	no	—
238	C₂×C₅×C₈	—	8	1	no	—
239	C₅×C₄⋊C₄	—	2	2	no	—
240	C₅×C₄⋊C₄	—	2	2	no	—
241	C₅×C₄⋊C₄	—	2	2	no	—
242	C₅×C₄⋊C₄	—	2	2	no	—
243	C₅×C₄⋊C₄	—	2	2	no	—
244	C₅×C₄⋊C₄	—	2	2	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	—	2	1	no	—
249	C₅×C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
250	C₅×C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
251	C₅×C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
252	C₅×C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
253	C₅×C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
254	C₅×C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
255	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
256	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
257	C₅×C₂²⋊C₄	—	2	2	no	—
258	C₅×C₂²⋊C₄	—	2	2	no	—
259	C₅×C₂²⋊C₄	—	2	2	no	—
260	C₅×C₂²⋊C₄	—	2	2	no	—
261	C₅×C₂²⋊C₄	—	2	2	no	—
262	C₅×C₂²⋊C₄	—	2	2	no	—
263	C₅×C₂²⋊C₄	—	2	2	no	—
264	C₅×C₂²⋊C₄	—	2	2	no	—
265	C₅×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
266	C₅×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
267	C₅×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
268	C₅×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
269	C₅×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
270	C₅×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
271	C₄²×C₅	—	2	2	no	—
272	C₄²×C₅	—	2	2	no	—
273	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
274	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
275	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
276	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
277	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
278	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
279	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
280	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
281	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
282	C₄²×C₅	C₂⁴	1	1	yes	characteristic
283	#556	—	2	1	no	—
284	#555	—	2	1	no	—
285	#554	—	2	1	no	—
286	#553	—	2	1	no	—
287	#553	—	2	1	no	—
288	#548	—	2	1	no	—
289	#548	—	2	1	no	—
290	#547	—	2	1	no	—
291	#542	C₄×C₅	1	2	yes	—
292	#542	C₄×C₅	1	2	yes	—
293	#537	C₄×C₅	1	2	yes	—
294	#533	C₂²×C₅	1	1	yes	characteristic
295	#533	C₂²×C₅	1	1	yes	characteristic
296	#532	C₂²×C₅	1	1	yes	characteristic
297	#532	C₂²×C₅	1	1	yes	characteristic
298	#532	C₂²×C₅	1	1	yes	characteristic
299	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
300	#520	C₄×C₅	1	2	yes	—
301	C₄×#41	C₂²×C₅	1	1	yes	characteristic
302	#446	—	2	1	no	—
303	#446	—	2	1	no	—
304	#435	—	2	1	no	—
305	#435	—	2	1	no	—
306	#435	—	2	1	no	—
307	#435	—	2	1	no	—
308	C₈×D₄	—	2	1	no	—
309	C₈×D₄	—	2	1	no	—
310	#432	C₂²×C₅	1	1	yes	characteristic
311	#432	C₂²×C₅	1	1	yes	characteristic
312	#432	C₂²×C₅	1	1	yes	characteristic
313	#432	C₂²×C₅	1	1	yes	characteristic
314	#431	C₂²×C₅	1	1	yes	characteristic
315	#431	C₂²×C₅	1	1	yes	characteristic
316	#423	C₂²×C₅	1	1	yes	characteristic
317	#423	C₂²×C₅	1	1	yes	characteristic
318	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
319	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
320	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
321	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
322	#333	C₂²×C₅	1	1	yes	characteristic
323	#333	C₂²×C₅	1	1	yes	characteristic
324	#333	C₂²×C₅	1	1	yes	characteristic
325	#333	C₂²×C₅	1	1	yes	characteristic
326	#332	C₂²×C₅	1	1	yes	characteristic
327	#332	C₂²×C₅	1	1	yes	characteristic
328	#332	C₂²×C₅	1	1	yes	characteristic
329	#332	C₂²×C₅	1	1	yes	characteristic
330	#332	C₂²×C₅	1	1	yes	characteristic
331	#332	C₂²×C₅	1	1	yes	characteristic
332	#332	C₂²×C₅	1	1	yes	characteristic
333	#332	C₂²×C₅	1	1	yes	characteristic
334	#331	C₂²×C₅	1	1	yes	characteristic
335	#331	C₂²×C₅	1	1	yes	characteristic
336	#331	C₂²×C₅	1	1	yes	characteristic
337	#331	C₂²×C₅	1	1	yes	characteristic
338	C₂³×C₅	—	2	2	no	—
339	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
340	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
341	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
342	C₅×Q₈	—	4	1	no	—
343	C₅×Q₈	—	4	2	no	—
344	C₅×Q₈	—	4	1	no	—
345	C₅×Q₈	—	4	1	no	—
346	C₅×Q₈	—	4	1	no	—
347	C₅×Q₈	—	4	1	no	—
348	C₅×Q₈	—	8	2	no	—
349	C₅×D₄	—	2	2	no	—
350	C₅×D₄	—	2	2	no	—
351	C₅×D₄	—	4	1	no	—
352	C₅×D₄	—	4	2	no	—
353	C₅×D₄	—	4	4	no	—
354	C₅×D₄	—	4	1	no	—
355	C₅×D₄	—	4	1	no	—
356	C₅×D₄	—	4	1	no	—
357	C₅×D₄	—	4	1	no	—
358	C₅×D₄	—	8	2	no	—
359	C₅×D₄	—	8	2	no	—
360	C₂×C₄×C₅	—	2	2	no	—
361	C₂×C₄×C₅	—	2	1	no	—
362	C₂×C₄×C₅	—	2	1	no	—
363	C₂×C₄×C₅	—	2	1	no	—
364	C₂×C₄×C₅	—	2	1	no	—
365	C₂×C₄×C₅	—	2	1	no	—
366	C₂×C₄×C₅	—	2	1	no	—
367	C₂×C₄×C₅	—	2	1	no	—
368	C₂×C₄×C₅	—	2	1	no	—
369	C₂×C₄×C₅	—	2	1	no	—
370	C₂×C₄×C₅	—	2	1	no	—
371	C₂×C₄×C₅	—	2	1	no	—
372	C₂×C₄×C₅	—	2	1	no	—
373	C₂×C₄×C₅	—	2	2	no	—
374	C₂×C₄×C₅	—	2	2	no	—
375	C₂×C₄×C₅	—	2	2	no	—
376	C₂×C₄×C₅	—	4	1	no	—
377	C₂×C₄×C₅	—	4	1	no	—
378	C₂×C₄×C₅	—	4	4	no	—
379	C₂×C₄×C₅	—	8	2	no	—
380	C₂×C₄×C₅	—	8	2	no	—
381	C₂×C₄×C₅	—	8	2	no	—
382	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
383	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
384	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
385	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
386	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
387	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
388	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
389	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
390	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
391	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
392	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
393	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
394	C₅×C₈	—	4	1	no	—
395	C₅×C₈	—	4	1	no	—
396	C₅×C₈	—	4	1	no	—
397	C₅×C₈	—	4	1	no	—
398	C₅×C₈	—	4	1	no	—
399	C₅×C₈	—	4	1	no	—
400	C₅×C₈	—	4	1	no	—
401	C₅×C₈	—	4	1	no	—
402	C₂×#41	—	2	1	no	—
403	C₂×#41	—	2	1	no	—
404	C₂×#41	C₂×C₄×C₅	1	2	yes	—
405	C₂×#41	C₂×C₄×C₅	1	1	yes	characteristic
406	C₂²×Q₈	—	2	1	no	—
407	C₂×M₁₆	—	2	1	no	—
408	C₂×M₁₆	—	2	1	no	—
409	C₂²×C₈	—	2	1	no	—
410	C₂²×C₈	—	2	1	no	—
411	C₂²×C₈	—	2	1	no	—
412	C₂²×C₈	—	2	1	no	—
413	C₄⋊Q₈	—	2	1	no	—
414	C₄⋊Q₈	—	2	1	no	—
415	C₄⋊Q₈	C₂×C₄×C₅	1	2	yes	—
416	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
417	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
418	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
419	#127	C₂³×C₅	1	1	yes	characteristic
420	#126	—	2	2	no	—
421	#126	—	2	2	no	—
422	#125	—	2	1	no	—
423	#125	—	2	1	no	—
424	#124	—	2	2	no	—
425	#124	—	2	2	no	—
426	#124	—	2	1	no	—
427	#124	—	2	1	no	—
428	#124	C₂×C₄×C₅	1	2	yes	—
429	#123	—	2	2	no	—
430	#123	—	2	2	no	—
431	C₂²⋊Q₈	—	2	2	no	—
432	C₂²⋊Q₈	—	2	1	no	—
433	C₂²⋊Q₈	—	2	1	no	—
434	C₂²⋊Q₈	—	2	2	no	—
435	C₂²⋊Q₈	—	2	2	no	—
436	C₂²⋊Q₈	C₂×C₄×C₅	1	2	yes	—
437	C₂²⋊Q₈	C₂×C₄×C₅	1	2	yes	—
438	#121	—	2	2	no	—
439	#121	—	2	2	no	—
440	#121	—	2	2	no	—
441	#121	—	2	1	no	—
442	#121	—	2	1	no	—
443	#121	—	2	1	no	—
444	#121	—	2	1	no	—
445	#121	C₂×C₄×C₅	1	2	yes	—
446	#121	C₂×C₄×C₅	1	2	yes	—
447	C₄×Q₈	—	2	1	no	—
448	C₄×Q₈	—	2	1	no	—
449	C₄×Q₈	C₂×C₄×C₅	1	2	yes	—
450	C₄×Q₈	C₂×C₄×C₅	1	2	yes	—
451	C₄×Q₈	C₅×D₄	1	1	yes	characteristic
452	C₄×Q₈	C₅×D₄	1	1	yes	characteristic
453	C₄×D₄	—	2	2	no	—
454	C₄×D₄	—	2	2	no	—
455	C₄×D₄	—	2	1	no	—
456	C₄×D₄	—	2	1	no	—
457	C₄×D₄	—	2	1	no	—
458	C₄×D₄	—	2	1	no	—
459	C₄×D₄	C₂×C₄×C₅	1	2	yes	—
460	C₄×D₄	C₂×C₄×C₅	1	2	yes	—
461	C₄×D₄	C₅×D₄	1	1	yes	characteristic
462	C₄×D₄	C₅×D₄	1	1	yes	characteristic
463	C₄×D₄	C₅×D₄	1	1	yes	characteristic
464	C₄×D₄	C₅×D₄	1	1	yes	characteristic
465	C₄×D₄	C₅×D₄	1	1	yes	characteristic
466	C₄×D₄	C₅×D₄	1	1	yes	characteristic
467	#117	—	2	1	no	—
468	#117	—	2	1	no	—
469	#117	—	2	2	no	—
470	#117	C₂×C₄×C₅	1	1	yes	characteristic
471	#117	C₂×C₄×C₅	1	1	yes	characteristic
472	#117	C₂×C₄×C₅	1	1	yes	characteristic
473	C₂×C₄⋊C₄	—	2	1	no	—
474	C₂×C₄⋊C₄	—	2	2	no	—
475	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
476	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
477	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
478	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
479	C₄⋊C₈	—	2	1	no	—
480	C₄⋊C₈	—	2	1	no	—
481	C₄⋊C₈	—	2	1	no	—
482	C₄⋊C₈	—	2	1	no	—
483	C₄⋊C₈	—	2	1	no	—
484	C₄⋊C₈	—	2	1	no	—
485	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
486	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
487	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
488	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
489	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
490	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
491	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
492	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
493	C₂²⋊C₈	—	2	1	no	—
494	C₂²⋊C₈	—	2	1	no	—
495	C₂²⋊C₈	—	2	1	no	—
496	C₂²⋊C₈	—	2	1	no	—
497	C₂²⋊C₈	—	2	1	no	—
498	C₂²⋊C₈	—	2	1	no	—
499	#97	—	2	1	no	—
500	#97	—	2	1	no	—
501	#97	—	2	1	no	—
502	#97	—	2	1	no	—
503	C₄×C₈	—	2	1	no	—
504	C₄×C₈	—	2	1	no	—
505	C₂²×C₅	—	2	1	no	—
506	C₂²×C₅	—	2	1	no	—
507	C₂²×C₅	—	2	1	no	—
508	C₂²×C₅	—	2	1	no	—
509	C₂²×C₅	—	2	1	no	—
510	C₂²×C₅	—	2	1	no	—
511	C₂²×C₅	—	2	1	no	—
512	C₂²×C₅	—	2	1	no	—
513	C₂²×C₅	—	4	2	no	—
514	C₂²×C₅	—	4	2	no	—
515	C₂²×C₅	—	4	2	no	—
516	C₂²×C₅	#413	1	2	yes	—
517	C₂²×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
518	C₄×C₅	—	2	1	no	—
519	C₄×C₅	—	2	1	no	—
520	C₄×C₅	—	2	1	no	—
521	C₄×C₅	—	2	1	no	—
522	C₄×C₅	—	2	1	no	—
523	C₄×C₅	—	2	1	no	—
524	C₄×C₅	—	2	1	no	—
525	C₄×C₅	—	2	1	no	—
526	C₄×C₅	—	2	1	no	—
527	C₄×C₅	—	2	1	no	—
528	C₄×C₅	—	4	2	no	—
529	C₄×C₅	—	4	2	no	—
530	C₄×C₅	—	4	2	no	—
531	C₄×C₅	#576	1	1	yes	characteristic
532	C₄×C₅	#577	1	1	yes	characteristic
533	C₄×C₅	#577	1	1	yes	characteristic
534	C₄×C₅	#578	1	1	yes	characteristic
535	#41	—	4	2	no	—
536	#41	—	4	4	no	—
537	#41	—	8	1	no	—
538	#41	—	8	1	no	—
539	#41	—	8	1	no	—
540	#41	—	8	1	no	—
541	C₂×Q₈	—	2	2	no	—
542	C₂×Q₈	—	2	1	no	—
543	C₂×Q₈	—	2	1	no	—
544	C₂×Q₈	—	8	1	no	—
545	C₂×Q₈	—	8	1	no	—
546	C₂×Q₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
547	C₂×Q₈	C₂²×C₄×C₅	1	2	yes	—
548	C₂×D₄	—	2	2	no	—
549	C₂×D₄	—	2	2	no	—
550	C₂×D₄	—	2	2	no	—
551	C₂×D₄	—	2	1	no	—
552	C₂×D₄	—	2	1	no	—
553	C₂×D₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
554	C₂×D₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
555	C₂×D₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
556	C₂×D₄	C₂²×C₄×C₅	1	2	yes	—
557	C₂²×C₄	—	2	2	no	—
558	C₂²×C₄	—	2	2	no	—
559	C₂²×C₄	—	2	2	no	—
560	C₂²×C₄	—	2	2	no	—
561	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
562	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
563	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
564	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
565	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
566	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
567	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
568	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
569	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
570	M₁₆	—	8	1	no	—
571	M₁₆	—	8	1	no	—
572	C₂×C₈	—	2	1	no	—
573	C₂×C₈	—	2	1	no	—
574	C₂×C₈	—	2	1	no	—
575	C₂×C₈	—	2	1	no	—
576	C₂×C₈	—	2	1	no	—
577	C₂×C₈	—	2	1	no	—
578	C₂×C₈	—	2	1	no	—
579	C₂×C₈	—	2	1	no	—
580	C₂×C₈	—	8	1	no	—
581	C₂×C₈	—	8	1	no	—
582	C₂×C₈	—	8	1	no	—
583	C₂×C₈	—	8	1	no	—
584	C₄⋊C₄	—	2	2	no	—
585	C₄⋊C₄	—	2	2	no	—
586	C₄⋊C₄	—	2	2	no	—
587	C₄⋊C₄	—	2	2	no	—
588	C₄⋊C₄	—	2	2	no	—
589	C₄⋊C₄	—	2	2	no	—
590	C₄⋊C₄	—	2	1	no	—
591	C₄⋊C₄	—	2	1	no	—
592	C₄⋊C₄	—	2	1	no	—
593	C₄⋊C₄	—	2	1	no	—
594	C₄⋊C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
595	C₄⋊C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
596	C₄⋊C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
597	C₄⋊C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
598	C₄⋊C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
599	C₄⋊C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
600	C₄⋊C₄	C₂²×C₄×C₅	1	2	yes	—
601	C₄⋊C₄	C₂²×C₄×C₅	1	2	yes	—
602	C₂²⋊C₄	—	2	2	no	—
603	C₂²⋊C₄	—	2	2	no	—
604	C₂²⋊C₄	—	2	2	no	—
605	C₂²⋊C₄	—	2	2	no	—
606	C₂²⋊C₄	—	2	2	no	—
607	C₂²⋊C₄	—	2	2	no	—
608	C₂²⋊C₄	—	2	2	no	—
609	C₂²⋊C₄	—	2	2	no	—
610	C₂²⋊C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
611	C₂²⋊C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
612	C₂²⋊C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
613	C₂²⋊C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
614	C₂²⋊C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
615	C₂²⋊C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
616	C₄²	—	2	2	no	—
617	C₄²	—	2	2	no	—
618	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
619	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
620	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
621	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
622	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
623	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
624	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
625	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
626	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
627	C₄²	C₂⁴×C₅	1	1	yes	characteristic
628	C₂×C₅	—	2	2	no	—
629	C₂×C₅	—	4	1	no	—
630	C₂×C₅	—	4	1	no	—
631	C₂×C₅	—	8	2	no	—
632	C₂×C₅	C₂×#413	1	1	yes	characteristic
633	C₂×C₅	#2897	1	1	yes	characteristic
634	C₂×C₅	#2898	1	1	yes	characteristic
635	C₂³	—	2	2	no	—
636	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
637	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
638	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
639	Q₈	—	4	1	no	—
640	Q₈	—	4	2	no	—
641	Q₈	—	4	1	no	—
642	Q₈	—	4	1	no	—
643	Q₈	—	4	1	no	—
644	Q₈	—	4	1	no	—
645	Q₈	—	8	2	no	—
646	D₄	—	2	2	no	—
647	D₄	—	2	2	no	—
648	D₄	—	4	1	no	—
649	D₄	—	4	2	no	—
650	D₄	—	4	4	no	—
651	D₄	—	4	1	no	—
652	D₄	—	4	1	no	—
653	D₄	—	4	1	no	—
654	D₄	—	4	1	no	—
655	D₄	—	8	2	no	—
656	D₄	—	8	2	no	—
657	C₂×C₄	—	2	2	no	—
658	C₂×C₄	—	2	1	no	—
659	C₂×C₄	—	2	1	no	—
660	C₂×C₄	—	2	1	no	—
661	C₂×C₄	—	2	1	no	—
662	C₂×C₄	—	2	1	no	—
663	C₂×C₄	—	2	1	no	—
664	C₂×C₄	—	2	1	no	—
665	C₂×C₄	—	2	1	no	—
666	C₂×C₄	—	2	1	no	—
667	C₂×C₄	—	2	1	no	—
668	C₂×C₄	—	2	1	no	—
669	C₂×C₄	—	2	1	no	—
670	C₂×C₄	—	2	2	no	—
671	C₂×C₄	—	2	2	no	—
672	C₂×C₄	—	2	2	no	—
673	C₂×C₄	—	4	1	no	—
674	C₂×C₄	—	4	1	no	—
675	C₂×C₄	—	4	4	no	—
676	C₂×C₄	—	8	2	no	—
677	C₂×C₄	—	8	2	no	—
678	C₂×C₄	—	8	2	no	—
679	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
680	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
681	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
682	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
683	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
684	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
685	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
686	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
687	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
688	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
689	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
690	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
691	C₈	—	4	1	no	—
692	C₈	—	4	1	no	—
693	C₈	—	4	1	no	—
694	C₈	—	4	1	no	—
695	C₈	—	4	1	no	—
696	C₈	—	4	1	no	—
697	C₈	—	4	1	no	—
698	C₈	—	4	1	no	—
699	C₅	#17775	1	1	yes	characteristic, direct product factor, Sylow
700	C₂²	—	2	1	no	—
701	C₂²	—	2	1	no	—
702	C₂²	—	2	1	no	—
703	C₂²	—	2	1	no	—
704	C₂²	—	2	1	no	—
705	C₂²	—	2	1	no	—
706	C₂²	—	2	1	no	—
707	C₂²	—	2	1	no	—
708	C₂²	—	4	2	no	—
709	C₂²	—	4	2	no	—
710	C₂²	—	4	2	no	—
711	C₂²	C₅×#413	1	2	yes	—
712	C₂²	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
713	C₄	—	2	1	no	—
714	C₄	—	2	1	no	—
715	C₄	—	2	1	no	—
716	C₄	—	2	1	no	—
717	C₄	—	2	1	no	—
718	C₄	—	2	1	no	—
719	C₄	—	2	1	no	—
720	C₄	—	2	1	no	—
721	C₄	—	2	1	no	—
722	C₄	—	2	1	no	—
723	C₄	—	4	2	no	—
724	C₄	—	4	2	no	—
725	C₄	—	4	2	no	—
726	C₄	C₅×#576	1	1	yes	characteristic
727	C₄	C₅×#577	1	1	yes	characteristic
728	C₄	C₅×#577	1	1	yes	characteristic
729	C₄	C₅×#578	1	1	yes	characteristic
730	C₂	—	2	2	no	—
731	C₂	—	4	1	no	—
732	C₂	—	4	1	no	—
733	C₂	—	8	2	no	—
734	C₂	C₂×C₅×#413	1	1	yes	characteristic
735	C₂	C₅×#2897	1	1	yes	characteristic
736	C₂	C₅×#2898	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×#3511	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×#2990	C₂	semidirect
3. C₅×#2990	C₂	semidirect
4. C₅×#1559	C₂	semidirect
5. C₅×#1557	C₂	semidirect
6. C₅×#1556	C₂	semidirect
7. C₅×#1554	C₂	semidirect
8. C₅×#1553	C₂	semidirect
9. C₅×#1551	C₂	semidirect
10. C₅×#1551	C₂	semidirect
11. C₅×#1550	C₂	semidirect
12. C₅×#1547	C₂	semidirect
13. C₅×#1542	C₂	semidirect
14. C₅×#1540	C₂	semidirect
15. C₂×C₅×#332	C₂	semidirect
71. #17775	C₅	direct (N × H)
176. #2990	C₂×C₅	semidirect
177. #2990	C₂×C₅	semidirect
178. #1559	C₂×C₅	semidirect
179. #1557	C₂×C₅	semidirect
180. #1556	C₂×C₅	semidirect
181. #1554	C₂×C₅	semidirect
182. #1553	C₂×C₅	semidirect
183. #1551	C₂×C₅	semidirect
184. #1551	C₂×C₅	semidirect
185. #1550	C₂×C₅	semidirect
186. #1547	C₂×C₅	semidirect
187. #1542	C₂×C₅	semidirect
188. #1540	C₂×C₅	semidirect
189. C₂×#332	C₂×C₅	semidirect
699. C₅	#17775	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×#3511	2
C₅×#2990	2
C₅×#1559	2
C₅×#1557	2
C₅×#1556	2
C₅×#1554	2
C₅×#1553	2
C₅×#1551	2
C₅×#1550	2
C₅×#1547	2
C₅×#1542	2
C₅×#1540	2
C₂×C₅×#332	2
#17775	5

Subgroups — C5×#17775 #11940495

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#17775 #11940495