Subgroups — C₅×#17548 #11940268

Order 1280.

1102 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3520	C₂	1	1	yes	maximal, characteristic
2	C₅×#2974	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2275	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1509	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1508	C₂	1	2	yes	maximal, complemented
6	C₅×#1506	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1504	C₂	1	2	yes	maximal, complemented
8	C₅×#1503	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1501	C₂	1	2	yes	maximal, complemented
10	C₅×#1500	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅×#328	C₂	1	2	yes	maximal, complemented
12	C₅×#556	C₄	1	1	yes	characteristic, complemented
13	C₅×#554	—	2	1	no	—
14	C₅×#552	C₄	1	2	yes	complemented
15	C₅×#551	—	2	2	no	—
16	C₅×D₄×Q₈	C₄	1	1	yes	characteristic, complemented
17	C₅×#548	C₄	1	2	yes	complemented
18	C₅×#547	C₄	1	1	yes	characteristic, complemented
19	C₅×#547	C₄	1	1	yes	characteristic, complemented
20	C₅×#546	—	2	1	no	—
21	C₅×#542	—	2	1	no	—
22	C₅×#539	—	2	2	no	—
23	C₅×#533	C₂²	1	1	yes	characteristic
24	C₅×#529	C₂²	1	1	yes	characteristic
25	C₅×#525	C₂²	1	2	yes	—
26	C₂×C₅×C₂²⋊Q₈	C₂²	1	2	yes	—
27	C₅×#520	—	2	1	no	—
28	C₂×C₄×C₅×D₄	—	2	1	no	—
29	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
30	C₅×#436	—	2	1	no	—
31	C₅×#435	—	2	2	no	—
32	C₅×C₈×D₄	—	2	1	no	—
33	C₅×#432	C₂²	1	1	yes	characteristic
34	C₅×#431	C₂²	1	1	yes	characteristic
35	C₅×#407	C₂²	1	2	yes	—
36	C₂×C₅×C₂²⋊C₈	C₂²	1	2	yes	—
37	C₄×C₅×C₄⋊C₄	—	2	1	no	—
38	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
39	C₄×C₅×C₂²⋊C₄	—	2	2	no	—
40	C₄×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
41	C₅×#376	C₂²	1	1	yes	characteristic
42	C₂×C₅×#95	C₂²	1	2	yes	—
43	C₅×#328	C₂²	1	1	yes	characteristic
44	C₅×#328	C₂²	1	1	yes	characteristic
45	C₅×#328	C₂²	1	2	yes	—
46	C₅×#328	C₂²	1	2	yes	—
47	C₅×#328	C₂²	1	2	yes	—
48	C₅×#328	C₂²	1	2	yes	—
49	C₅×#328	C₂²	1	2	yes	—
50	C₅×#327	C₂²	1	1	yes	characteristic
51	C₅×#327	C₂²	1	1	yes	characteristic
52	C₅×#327	C₂²	1	2	yes	—
53	#17548	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂×C₅×#41	—	2	1	no	—
55	C₂×C₅×#41	C₂×C₄	1	2	yes	complemented
56	C₂²×C₅×Q₈	C₂×C₄	1	2	yes	complemented
57	C₂²×C₅×D₄	D₄	1	1	yes	characteristic
58	C₂³×C₄×C₅	—	2	2	no	—
59	C₂³×C₄×C₅	C₂³	1	2	yes	—
60	C₂×C₅×M₁₆	—	2	2	no	—
61	C₂²×C₅×C₈	—	2	2	no	—
62	C₅×C₄⋊Q₈	—	2	1	no	—
63	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
64	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
65	C₅×#126	—	2	2	no	—
66	C₅×#126	—	2	2	no	—
67	C₅×#126	—	2	1	no	—
68	C₅×#125	—	2	2	no	—
69	C₅×#125	—	2	1	no	—
70	C₅×#125	C₂×C₄	1	1	yes	characteristic, complemented
71	C₅×#124	—	2	1	no	—
72	C₅×#124	—	2	2	no	—
73	C₅×#124	C₂×C₄	1	1	yes	characteristic, complemented
74	C₅×#123	—	2	2	no	—
75	C₅×#123	—	2	2	no	—
76	C₅×#123	—	2	1	no	—
77	C₅×#123	—	2	2	no	—
78	C₅×#123	—	2	2	no	—
79	C₅×C₂²⋊Q₈	—	2	2	no	—
80	C₅×C₂²⋊Q₈	—	2	2	no	—
81	C₅×C₂²⋊Q₈	—	2	1	no	—
82	C₅×C₂²⋊Q₈	—	2	2	no	—
83	C₅×C₂²⋊Q₈	—	2	1	no	—
84	C₅×C₂²⋊Q₈	—	2	2	no	—
85	C₅×C₂²⋊Q₈	C₂×C₄	1	2	yes	complemented
86	C₅×C₂²⋊Q₈	C₂×C₄	1	2	yes	complemented
87	C₅×C₂²⋊Q₈	C₂×C₄	1	2	yes	complemented
88	C₅×C₂²⋊Q₈	C₂³	1	1	yes	characteristic
89	C₅×C₂²⋊Q₈	C₂³	1	2	yes	—
90	C₅×#121	—	2	2	no	—
91	C₅×#121	—	2	1	no	—
92	C₅×#121	C₂×C₄	1	2	yes	complemented
93	C₅×#121	C₂³	1	1	yes	characteristic
94	C₅×#120	—	2	2	no	—
95	C₄×C₅×Q₈	—	2	1	no	—
96	C₄×C₅×Q₈	—	2	1	no	—
97	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic, complemented
98	C₄×C₅×Q₈	C₂×C₄	1	2	yes	complemented
99	C₄×C₅×D₄	—	2	8	no	—
100	C₄×C₅×D₄	—	2	1	no	—
101	C₄×C₅×D₄	—	2	2	no	—
102	C₄×C₅×D₄	—	2	2	no	—
103	C₄×C₅×D₄	—	2	1	no	—
104	C₄×C₅×D₄	—	2	2	no	—
105	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
106	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
107	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
108	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic, complemented
109	C₄×C₅×D₄	D₄	1	1	yes	characteristic
110	C₄×C₅×D₄	D₄	1	2	yes	—
111	C₄×C₅×D₄	D₄	1	1	yes	characteristic
112	C₅×#117	—	2	1	no	—
113	C₅×#117	—	2	2	no	—
114	C₅×#117	—	2	1	no	—
115	C₅×#117	C₂×C₄	1	2	yes	complemented
116	C₂×C₅×C₄⋊C₄	—	2	1	no	—
117	C₂×C₅×C₄⋊C₄	—	2	2	no	—
118	C₂×C₅×C₄⋊C₄	—	2	1	no	—
119	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	complemented
120	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
121	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
122	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
123	C₂×C₅×C₂²⋊C₄	D₄	1	2	yes	—
124	C₂×C₄²×C₅	—	2	1	no	—
125	C₂×C₄²×C₅	—	2	1	no	—
126	C₂×C₄²×C₅	—	2	2	no	—
127	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
128	C₅×C₄⋊C₈	—	2	1	no	—
129	C₅×C₄⋊C₈	—	2	1	no	—
130	C₅×C₂²⋊C₈	—	2	2	no	—
131	C₅×C₂²⋊C₈	—	2	2	no	—
132	C₅×C₂²⋊C₈	C₂³	1	1	yes	characteristic
133	C₅×C₂²⋊C₈	C₂³	1	2	yes	—
134	C₅×C₂²⋊C₈	C₂³	1	1	yes	characteristic
135	C₅×#97	—	2	1	no	—
136	C₄×C₅×C₈	—	2	1	no	—
137	C₅×#95	—	2	2	no	—
138	C₅×#95	—	2	1	no	—
139	C₅×#95	C₂³	1	1	yes	characteristic
140	C₅×#95	C₂³	1	2	yes	—
141	C₅×#95	C₂³	1	1	yes	characteristic
142	#3520	C₂×C₅	1	1	yes	characteristic
143	#2974	C₂×C₅	1	1	yes	characteristic, complemented
144	#2275	C₂×C₅	1	1	yes	characteristic, complemented
145	#1509	C₂×C₅	1	1	yes	characteristic, complemented
146	#1508	C₂×C₅	1	2	yes	complemented
147	#1506	C₂×C₅	1	1	yes	characteristic, complemented
148	#1504	C₂×C₅	1	2	yes	complemented
149	#1503	C₂×C₅	1	1	yes	characteristic, complemented
150	#1501	C₂×C₅	1	2	yes	complemented
151	#1500	C₂×C₅	1	1	yes	characteristic, complemented
152	C₂×#328	C₂×C₅	1	2	yes	complemented
153	C₂⁴×C₅	C₂×D₄	1	2	yes	—
154	C₅×#41	—	4	4	no	—
155	C₅×#41	—	8	2	no	—
156	C₂×C₅×Q₈	—	2	1	no	—
157	C₂×C₅×Q₈	—	2	2	no	—
158	C₂×C₅×Q₈	—	8	2	no	—
159	C₂×C₅×Q₈	C₂²×C₄	1	1	yes	characteristic
160	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
161	C₂×C₅×D₄	—	2	4	no	—
162	C₂×C₅×D₄	—	2	2	no	—
163	C₂×C₅×D₄	—	2	1	no	—
164	C₂×C₅×D₄	—	2	2	no	—
165	C₂×C₅×D₄	C₂²⋊C₄	1	2	yes	—
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	2	no	—
170	C₂²×C₄×C₅	—	2	1	no	—
171	C₂²×C₄×C₅	—	2	2	no	—
172	C₂²×C₄×C₅	—	2	1	no	—
173	C₂²×C₄×C₅	—	2	8	no	—
174	C₂²×C₄×C₅	—	2	2	no	—
175	C₂²×C₄×C₅	—	2	1	no	—
176	C₂²×C₄×C₅	—	2	2	no	—
177	C₂²×C₄×C₅	—	8	2	no	—
178	C₂²×C₄×C₅	—	8	2	no	—
179	C₂²×C₄×C₅	—	8	2	no	—
180	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
181	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
182	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
183	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
184	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
185	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
186	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
187	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
188	C₅×M₁₆	—	8	2	no	—
189	C₂×C₅×C₈	—	2	1	no	—
190	C₂×C₅×C₈	—	2	2	no	—
191	C₂×C₅×C₈	—	2	1	no	—
192	C₂×C₅×C₈	—	8	2	no	—
193	C₅×C₄⋊C₄	—	2	4	no	—
194	C₅×C₄⋊C₄	—	2	2	no	—
195	C₅×C₄⋊C₄	—	2	1	no	—
196	C₅×C₄⋊C₄	—	2	2	no	—
197	C₅×C₄⋊C₄	—	2	2	no	—
198	C₅×C₄⋊C₄	—	2	1	no	—
199	C₅×C₄⋊C₄	—	2	1	no	—
200	C₅×C₄⋊C₄	—	2	1	no	—
201	C₅×C₄⋊C₄	—	2	2	no	—
202	C₅×C₄⋊C₄	—	2	1	no	—
203	C₅×C₄⋊C₄	—	2	2	no	—
204	C₅×C₄⋊C₄	—	2	1	no	—
205	C₅×C₄⋊C₄	—	2	1	no	—
206	C₅×C₄⋊C₄	—	2	2	no	—
207	C₅×C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
208	C₅×C₄⋊C₄	C₂²⋊C₄	1	2	yes	—
209	C₅×C₄⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
210	C₅×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
211	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
212	C₅×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
213	C₅×C₂²⋊C₄	—	2	8	no	—
214	C₅×C₂²⋊C₄	—	2	2	no	—
215	C₅×C₂²⋊C₄	—	2	2	no	—
216	C₅×C₂²⋊C₄	—	2	2	no	—
217	C₅×C₂²⋊C₄	—	2	2	no	—
218	C₅×C₂²⋊C₄	—	2	1	no	—
219	C₅×C₂²⋊C₄	—	2	2	no	—
220	C₅×C₂²⋊C₄	—	2	1	no	—
221	C₅×C₂²⋊C₄	—	2	1	no	—
222	C₅×C₂²⋊C₄	—	2	2	no	—
223	C₅×C₂²⋊C₄	—	2	1	no	—
224	C₅×C₂²⋊C₄	—	2	1	no	—
225	C₅×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	—
226	C₅×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	—
227	C₅×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	—
228	C₅×C₂²⋊C₄	C₂²⋊C₄	1	2	yes	—
229	C₄²×C₅	—	2	4	no	—
230	C₄²×C₅	—	2	2	no	—
231	C₄²×C₅	—	2	2	no	—
232	C₄²×C₅	—	2	1	no	—
233	C₄²×C₅	—	8	1	no	—
234	C₄²×C₅	—	8	2	no	—
235	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
236	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
237	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
238	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
239	#556	C₄×C₅	1	1	yes	characteristic, complemented
240	#554	—	2	1	no	—
241	#552	C₄×C₅	1	2	yes	complemented
242	#551	—	2	2	no	—
243	D₄×Q₈	C₄×C₅	1	1	yes	characteristic, complemented
244	#548	C₄×C₅	1	2	yes	complemented
245	#547	C₄×C₅	1	1	yes	characteristic, complemented
246	#547	C₄×C₅	1	1	yes	characteristic, complemented
247	#546	—	2	1	no	—
248	#542	—	2	1	no	—
249	#539	—	2	2	no	—
250	#533	C₂²×C₅	1	1	yes	characteristic
251	#529	C₂²×C₅	1	1	yes	characteristic
252	#525	C₂²×C₅	1	2	yes	—
253	C₂×C₂²⋊Q₈	C₂²×C₅	1	2	yes	—
254	#520	—	2	1	no	—
255	C₂×C₄×D₄	—	2	1	no	—
256	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
257	#436	—	2	1	no	—
258	#435	—	2	2	no	—
259	C₈×D₄	—	2	1	no	—
260	#432	C₂²×C₅	1	1	yes	characteristic
261	#431	C₂²×C₅	1	1	yes	characteristic
262	#407	C₂²×C₅	1	2	yes	—
263	C₂×C₂²⋊C₈	C₂²×C₅	1	2	yes	—
264	C₄×C₄⋊C₄	—	2	1	no	—
265	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
266	C₄×C₂²⋊C₄	—	2	2	no	—
267	C₄×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
268	#376	C₂²×C₅	1	1	yes	characteristic
269	C₂×#95	C₂²×C₅	1	2	yes	—
270	#328	C₂²×C₅	1	1	yes	characteristic
271	#328	C₂²×C₅	1	2	yes	—
272	#328	C₂²×C₅	1	2	yes	—
273	#328	C₂²×C₅	1	2	yes	—
274	#328	C₂²×C₅	1	1	yes	characteristic
275	#328	C₂²×C₅	1	2	yes	—
276	#328	C₂²×C₅	1	2	yes	—
277	#327	C₂²×C₅	1	1	yes	characteristic
278	#327	C₂²×C₅	1	1	yes	characteristic
279	#327	C₂²×C₅	1	2	yes	—
280	C₂³×C₅	—	2	2	no	—
281	C₂³×C₅	—	2	2	no	—
282	C₂³×C₅	—	2	4	no	—
283	C₂³×C₅	—	2	1	no	—
284	C₂³×C₅	—	4	2	no	—
285	C₂³×C₅	C₂×C₂²⋊C₄	1	2	yes	—
286	C₂³×C₅	C₂×C₂²⋊C₄	1	2	yes	—
287	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
288	C₅×Q₈	—	4	1	no	—
289	C₅×Q₈	—	4	2	no	—
290	C₅×Q₈	—	4	2	no	—
291	C₅×Q₈	—	8	2	no	—
292	C₅×D₄	—	2	4	no	—
293	C₅×D₄	—	2	2	no	—
294	C₅×D₄	—	4	1	no	—
295	C₅×D₄	—	4	1	no	—
296	C₅×D₄	—	4	1	no	—
297	C₅×D₄	—	4	4	no	—
298	C₅×D₄	—	8	2	no	—
299	C₅×D₄	#104	1	4	yes	—
300	C₂×C₄×C₅	—	2	1	no	—
301	C₂×C₄×C₅	—	2	1	no	—
302	C₂×C₄×C₅	—	2	2	no	—
303	C₂×C₄×C₅	—	2	2	no	—
304	C₂×C₄×C₅	—	2	2	no	—
305	C₂×C₄×C₅	—	2	4	no	—
306	C₂×C₄×C₅	—	2	1	no	—
307	C₂×C₄×C₅	—	2	2	no	—
308	C₂×C₄×C₅	—	2	1	no	—
309	C₂×C₄×C₅	—	2	2	no	—
310	C₂×C₄×C₅	—	2	1	no	—
311	C₂×C₄×C₅	—	4	2	no	—
312	C₂×C₄×C₅	—	4	2	no	—
313	C₂×C₄×C₅	—	4	2	no	—
314	C₂×C₄×C₅	—	4	1	no	—
315	C₂×C₄×C₅	—	4	2	no	—
316	C₂×C₄×C₅	—	4	2	no	—
317	C₂×C₄×C₅	—	4	1	no	—
318	C₂×C₄×C₅	—	4	2	no	—
319	C₂×C₄×C₅	—	4	1	no	—
320	C₂×C₄×C₅	—	4	2	no	—
321	C₂×C₄×C₅	—	4	1	no	—
322	C₂×C₄×C₅	—	4	1	no	—
323	C₂×C₄×C₅	—	4	4	no	—
324	C₂×C₄×C₅	—	4	2	no	—
325	C₂×C₄×C₅	—	8	2	no	—
326	C₂×C₄×C₅	—	8	2	no	—
327	C₂×C₄×C₅	—	8	8	no	—
328	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
329	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
330	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
331	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
332	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
333	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
334	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
335	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
336	C₅×C₈	—	4	1	no	—
337	C₅×C₈	—	4	2	no	—
338	C₅×C₈	—	4	1	no	—
339	C₂×#41	—	2	1	no	—
340	C₂×#41	C₂×C₄×C₅	1	2	yes	complemented
341	C₂²×Q₈	C₂×C₄×C₅	1	2	yes	complemented
342	C₂²×D₄	C₅×D₄	1	1	yes	characteristic
343	C₂³×C₄	—	2	2	no	—
344	C₂³×C₄	C₂³×C₅	1	2	yes	—
345	C₂×M₁₆	—	2	2	no	—
346	C₂²×C₈	—	2	2	no	—
347	C₄⋊Q₈	—	2	1	no	—
348	C₄⋊Q₈	C₂×C₄×C₅	1	1	yes	characteristic, complemented
349	C₄⋊Q₈	C₂×C₄×C₅	1	1	yes	characteristic, complemented
350	#126	—	2	2	no	—
351	#126	—	2	2	no	—
352	#126	—	2	1	no	—
353	#125	—	2	2	no	—
354	#125	—	2	1	no	—
355	#125	C₂×C₄×C₅	1	1	yes	characteristic, complemented
356	#124	—	2	1	no	—
357	#124	—	2	2	no	—
358	#124	C₂×C₄×C₅	1	1	yes	characteristic, complemented
359	#123	—	2	2	no	—
360	#123	—	2	2	no	—
361	#123	—	2	1	no	—
362	#123	—	2	2	no	—
363	#123	—	2	2	no	—
364	C₂²⋊Q₈	—	2	2	no	—
365	C₂²⋊Q₈	—	2	2	no	—
366	C₂²⋊Q₈	—	2	1	no	—
367	C₂²⋊Q₈	—	2	2	no	—
368	C₂²⋊Q₈	—	2	1	no	—
369	C₂²⋊Q₈	—	2	2	no	—
370	C₂²⋊Q₈	C₂×C₄×C₅	1	2	yes	complemented
371	C₂²⋊Q₈	C₂×C₄×C₅	1	2	yes	complemented
372	C₂²⋊Q₈	C₂×C₄×C₅	1	2	yes	complemented
373	C₂²⋊Q₈	C₂³×C₅	1	1	yes	characteristic
374	C₂²⋊Q₈	C₂³×C₅	1	2	yes	—
375	#121	—	2	2	no	—
376	#121	—	2	1	no	—
377	#121	C₂×C₄×C₅	1	2	yes	complemented
378	#121	C₂³×C₅	1	1	yes	characteristic
379	#120	—	2	2	no	—
380	C₄×Q₈	—	2	1	no	—
381	C₄×Q₈	—	2	1	no	—
382	C₄×Q₈	C₂×C₄×C₅	1	1	yes	characteristic, complemented
383	C₄×Q₈	C₂×C₄×C₅	1	2	yes	complemented
384	C₄×D₄	—	2	8	no	—
385	C₄×D₄	—	2	1	no	—
386	C₄×D₄	—	2	2	no	—
387	C₄×D₄	—	2	2	no	—
388	C₄×D₄	—	2	1	no	—
389	C₄×D₄	—	2	2	no	—
390	C₄×D₄	C₂×C₄×C₅	1	1	yes	characteristic
391	C₄×D₄	C₂×C₄×C₅	1	1	yes	characteristic
392	C₄×D₄	C₂×C₄×C₅	1	2	yes	—
393	C₄×D₄	C₂×C₄×C₅	1	1	yes	characteristic, complemented
394	C₄×D₄	C₅×D₄	1	1	yes	characteristic
395	C₄×D₄	C₅×D₄	1	2	yes	—
396	C₄×D₄	C₅×D₄	1	1	yes	characteristic
397	#117	—	2	1	no	—
398	#117	—	2	2	no	—
399	#117	—	2	1	no	—
400	#117	C₂×C₄×C₅	1	2	yes	complemented
401	C₂×C₄⋊C₄	—	2	1	no	—
402	C₂×C₄⋊C₄	—	2	2	no	—
403	C₂×C₄⋊C₄	—	2	1	no	—
404	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	complemented
405	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
406	C₂×C₂²⋊C₄	—	2	2	no	—
407	C₂×C₂²⋊C₄	—	2	2	no	—
408	C₂×C₂²⋊C₄	C₅×D₄	1	2	yes	—
409	C₂×C₄²	—	2	1	no	—
410	C₂×C₄²	—	2	1	no	—
411	C₂×C₄²	—	2	2	no	—
412	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
413	C₄⋊C₈	—	2	1	no	—
414	C₄⋊C₈	—	2	1	no	—
415	C₂²⋊C₈	—	2	2	no	—
416	C₂²⋊C₈	—	2	2	no	—
417	C₂²⋊C₈	C₂³×C₅	1	1	yes	characteristic
418	C₂²⋊C₈	C₂³×C₅	1	2	yes	—
419	C₂²⋊C₈	C₂³×C₅	1	1	yes	characteristic
420	#97	—	2	1	no	—
421	C₄×C₈	—	2	1	no	—
422	#95	—	2	2	no	—
423	#95	—	2	1	no	—
424	#95	C₂³×C₅	1	1	yes	characteristic
425	#95	C₂³×C₅	1	2	yes	—
426	#95	C₂³×C₅	1	1	yes	characteristic
427	C₂²×C₅	—	2	1	no	—
428	C₂²×C₅	—	2	2	no	—
429	C₂²×C₅	—	2	2	no	—
430	C₂²×C₅	—	2	1	no	—
431	C₂²×C₅	—	2	2	no	—
432	C₂²×C₅	—	2	2	no	—
433	C₂²×C₅	—	2	2	no	—
434	C₂²×C₅	—	4	4	no	—
435	C₂²×C₅	—	4	4	no	—
436	C₂²×C₅	—	4	1	no	—
437	C₂²×C₅	—	4	1	no	—
438	C₂²×C₅	—	4	1	no	—
439	C₂²×C₅	C₂×#104	1	4	yes	—
440	C₂²×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
441	C₂²×C₅	#577	1	1	yes	characteristic
442	C₂²×C₅	#578	1	1	yes	characteristic
443	C₄×C₅	—	2	1	no	—
444	C₄×C₅	—	2	1	no	—
445	C₄×C₅	—	2	2	no	—
446	C₄×C₅	—	2	2	no	—
447	C₄×C₅	—	2	1	no	—
448	C₄×C₅	—	2	1	no	—
449	C₄×C₅	—	2	1	no	—
450	C₄×C₅	—	4	2	no	—
451	C₄×C₅	—	4	4	no	—
452	C₄×C₅	—	4	2	no	—
453	C₄×C₅	—	4	2	no	—
454	C₄×C₅	—	4	1	no	—
455	C₄×C₅	—	4	1	no	—
456	C₄×C₅	—	4	1	no	—
457	C₄×C₅	—	4	2	no	—
458	C₄×C₅	C₂×#104	1	2	yes	—
459	C₂⁴	C₂×C₅×D₄	1	2	yes	—
460	#41	—	4	4	no	—
461	#41	—	8	2	no	—
462	C₂×Q₈	—	2	1	no	—
463	C₂×Q₈	—	2	2	no	—
464	C₂×Q₈	—	8	2	no	—
465	C₂×Q₈	C₂²×C₄×C₅	1	1	yes	characteristic
466	C₂×Q₈	C₂²×C₄×C₅	1	2	yes	—
467	C₂×D₄	—	2	4	no	—
468	C₂×D₄	—	2	2	no	—
469	C₂×D₄	—	2	1	no	—
470	C₂×D₄	—	2	2	no	—
471	C₂×D₄	C₅×C₂²⋊C₄	1	2	yes	—
472	C₂×D₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
473	C₂×D₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
474	C₂×D₄	C₂²×C₄×C₅	1	1	yes	characteristic
475	C₂²×C₄	—	2	2	no	—
476	C₂²×C₄	—	2	1	no	—
477	C₂²×C₄	—	2	1	no	—
478	C₂²×C₄	—	2	2	no	—
479	C₂²×C₄	—	2	8	no	—
480	C₂²×C₄	—	2	2	no	—
481	C₂²×C₄	—	2	1	no	—
482	C₂²×C₄	—	2	2	no	—
483	C₂²×C₄	—	8	2	no	—
484	C₂²×C₄	—	8	2	no	—
485	C₂²×C₄	—	8	2	no	—
486	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
487	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
488	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
489	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
490	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
491	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
492	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
493	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
494	M₁₆	—	8	2	no	—
495	C₂×C₈	—	2	1	no	—
496	C₂×C₈	—	2	1	no	—
497	C₂×C₈	—	2	2	no	—
498	C₂×C₈	—	8	2	no	—
499	C₄⋊C₄	—	2	4	no	—
500	C₄⋊C₄	—	2	2	no	—
501	C₄⋊C₄	—	2	1	no	—
502	C₄⋊C₄	—	2	2	no	—
503	C₄⋊C₄	—	2	2	no	—
504	C₄⋊C₄	—	2	1	no	—
505	C₄⋊C₄	—	2	1	no	—
506	C₄⋊C₄	—	2	1	no	—
507	C₄⋊C₄	—	2	2	no	—
508	C₄⋊C₄	—	2	1	no	—
509	C₄⋊C₄	—	2	2	no	—
510	C₄⋊C₄	—	2	1	no	—
511	C₄⋊C₄	—	2	1	no	—
512	C₄⋊C₄	—	2	2	no	—
513	C₄⋊C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
514	C₄⋊C₄	C₅×C₂²⋊C₄	1	2	yes	—
515	C₄⋊C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
516	C₄⋊C₄	C₂²×C₄×C₅	1	1	yes	characteristic
517	C₄⋊C₄	C₂²×C₄×C₅	1	2	yes	—
518	C₄⋊C₄	C₂²×C₄×C₅	1	1	yes	characteristic
519	C₂²⋊C₄	—	2	8	no	—
520	C₂²⋊C₄	—	2	2	no	—
521	C₂²⋊C₄	—	2	2	no	—
522	C₂²⋊C₄	—	2	2	no	—
523	C₂²⋊C₄	—	2	2	no	—
524	C₂²⋊C₄	—	2	1	no	—
525	C₂²⋊C₄	—	2	2	no	—
526	C₂²⋊C₄	—	2	1	no	—
527	C₂²⋊C₄	—	2	1	no	—
528	C₂²⋊C₄	—	2	2	no	—
529	C₂²⋊C₄	—	2	1	no	—
530	C₂²⋊C₄	—	2	1	no	—
531	C₂²⋊C₄	C₅×C₂²⋊C₄	1	2	yes	—
532	C₂²⋊C₄	C₅×C₂²⋊C₄	1	2	yes	—
533	C₂²⋊C₄	C₅×C₂²⋊C₄	1	2	yes	—
534	C₂²⋊C₄	C₅×C₂²⋊C₄	1	2	yes	—
535	C₄²	—	2	4	no	—
536	C₄²	—	2	2	no	—
537	C₄²	—	2	2	no	—
538	C₄²	—	2	1	no	—
539	C₄²	—	8	1	no	—
540	C₄²	—	8	2	no	—
541	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
542	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
543	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
544	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
545	C₂×C₅	—	2	1	no	—
546	C₂×C₅	—	2	1	no	—
547	C₂×C₅	—	2	4	no	—
548	C₂×C₅	—	4	2	no	—
549	C₂×C₅	—	8	1	no	—
550	C₂×C₅	#2883	1	1	yes	characteristic
551	C₂×C₅	#2898	1	1	yes	characteristic
552	C₂×C₅	C₂²×#104	1	1	yes	characteristic
553	C₂³	—	2	4	no	—
554	C₂³	—	2	2	no	—
555	C₂³	—	2	2	no	—
556	C₂³	—	2	1	no	—
557	C₂³	—	4	2	no	—
558	C₂³	C₂×C₅×C₂²⋊C₄	1	2	yes	—
559	C₂³	C₂×C₅×C₂²⋊C₄	1	2	yes	—
560	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
561	Q₈	—	4	1	no	—
562	Q₈	—	4	2	no	—
563	Q₈	—	4	2	no	—
564	Q₈	—	8	2	no	—
565	D₄	—	2	4	no	—
566	D₄	—	2	2	no	—
567	D₄	—	4	1	no	—
568	D₄	—	4	4	no	—
569	D₄	—	4	1	no	—
570	D₄	—	4	1	no	—
571	D₄	—	8	2	no	—
572	D₄	C₅×#104	1	4	yes	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	2	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	4	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	2	1	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	2	1	no	—
584	C₂×C₄	—	4	2	no	—
585	C₂×C₄	—	4	2	no	—
586	C₂×C₄	—	4	2	no	—
587	C₂×C₄	—	4	1	no	—
588	C₂×C₄	—	4	2	no	—
589	C₂×C₄	—	4	2	no	—
590	C₂×C₄	—	4	1	no	—
591	C₂×C₄	—	4	2	no	—
592	C₂×C₄	—	4	1	no	—
593	C₂×C₄	—	4	4	no	—
594	C₂×C₄	—	4	2	no	—
595	C₂×C₄	—	4	1	no	—
596	C₂×C₄	—	4	1	no	—
597	C₂×C₄	—	4	2	no	—
598	C₂×C₄	—	8	2	no	—
599	C₂×C₄	—	8	2	no	—
600	C₂×C₄	—	8	8	no	—
601	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
602	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
603	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
604	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
605	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
606	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
607	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
608	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
609	C₈	—	4	1	no	—
610	C₈	—	4	2	no	—
611	C₈	—	4	1	no	—
612	C₅	#17548	1	1	yes	characteristic, direct product factor, Sylow
613	C₂²	—	2	1	no	—
614	C₂²	—	2	2	no	—
615	C₂²	—	2	2	no	—
616	C₂²	—	2	1	no	—
617	C₂²	—	2	2	no	—
618	C₂²	—	2	2	no	—
619	C₂²	—	2	2	no	—
620	C₂²	—	4	4	no	—
621	C₂²	—	4	4	no	—
622	C₂²	—	4	1	no	—
623	C₂²	—	4	1	no	—
624	C₂²	—	4	1	no	—
625	C₂²	C₂×C₅×#104	1	4	yes	—
626	C₂²	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
627	C₂²	C₅×#577	1	1	yes	characteristic
628	C₂²	C₅×#578	1	1	yes	characteristic
629	C₄	—	2	1	no	—
630	C₄	—	2	1	no	—
631	C₄	—	2	2	no	—
632	C₄	—	2	2	no	—
633	C₄	—	2	1	no	—
634	C₄	—	2	1	no	—
635	C₄	—	2	1	no	—
636	C₄	—	4	2	no	—
637	C₄	—	4	4	no	—
638	C₄	—	4	2	no	—
639	C₄	—	4	2	no	—
640	C₄	—	4	1	no	—
641	C₄	—	4	1	no	—
642	C₄	—	4	1	no	—
643	C₄	—	4	2	no	—
644	C₄	C₂×C₅×#104	1	2	yes	—
645	C₂	—	2	1	no	—
646	C₂	—	2	1	no	—
647	C₂	—	2	4	no	—
648	C₂	—	4	2	no	—
649	C₂	—	8	1	no	—
650	C₂	C₅×#2883	1	1	yes	characteristic
651	C₂	C₅×#2898	1	1	yes	characteristic
652	C₂	C₂²×C₅×#104	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₅×#3520	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2974	C₂	semidirect
3. C₅×#2275	C₂	semidirect
4. C₅×#1509	C₂	semidirect
5. C₅×#1508	C₂	semidirect
6. C₅×#1506	C₂	semidirect
7. C₅×#1504	C₂	semidirect
8. C₅×#1503	C₂	semidirect
9. C₅×#1501	C₂	semidirect
10. C₅×#1500	C₂	semidirect
11. C₂×C₅×#328	C₂	semidirect
12. C₅×#556	C₄	semidirect
14. C₅×#552	C₄	semidirect
16. C₅×D₄×Q₈	C₄	semidirect
17. C₅×#548	C₄	semidirect
18. C₅×#547	C₄	semidirect
19. C₅×#547	C₄	semidirect
53. #17548	C₅	direct (N × H)
55. C₂×C₅×#41	C₂×C₄	semidirect
56. C₂²×C₅×Q₈	C₂×C₄	semidirect
63. C₅×C₄⋊Q₈	C₂×C₄	semidirect
64. C₅×C₄⋊Q₈	C₂×C₄	semidirect
70. C₅×#125	C₂×C₄	semidirect
73. C₅×#124	C₂×C₄	semidirect
85. C₅×C₂²⋊Q₈	C₂×C₄	semidirect
86. C₅×C₂²⋊Q₈	C₂×C₄	semidirect
87. C₅×C₂²⋊Q₈	C₂×C₄	semidirect
92. C₅×#121	C₂×C₄	semidirect
97. C₄×C₅×Q₈	C₂×C₄	semidirect
98. C₄×C₅×Q₈	C₂×C₄	semidirect
108. C₄×C₅×D₄	C₂×C₄	semidirect
115. C₅×#117	C₂×C₄	semidirect
119. C₂×C₅×C₄⋊C₄	C₂×C₄	semidirect
143. #2974	C₂×C₅	semidirect
144. #2275	C₂×C₅	semidirect
145. #1509	C₂×C₅	semidirect
146. #1508	C₂×C₅	semidirect
147. #1506	C₂×C₅	semidirect
148. #1504	C₂×C₅	semidirect
149. #1503	C₂×C₅	semidirect
150. #1501	C₂×C₅	semidirect
151. #1500	C₂×C₅	semidirect
152. C₂×#328	C₂×C₅	semidirect
239. #556	C₄×C₅	semidirect
241. #552	C₄×C₅	semidirect
243. D₄×Q₈	C₄×C₅	semidirect
244. #548	C₄×C₅	semidirect
245. #547	C₄×C₅	semidirect
246. #547	C₄×C₅	semidirect
340. C₂×#41	C₂×C₄×C₅	semidirect
341. C₂²×Q₈	C₂×C₄×C₅	semidirect
348. C₄⋊Q₈	C₂×C₄×C₅	semidirect
349. C₄⋊Q₈	C₂×C₄×C₅	semidirect
355. #125	C₂×C₄×C₅	semidirect
358. #124	C₂×C₄×C₅	semidirect
370. C₂²⋊Q₈	C₂×C₄×C₅	semidirect
371. C₂²⋊Q₈	C₂×C₄×C₅	semidirect
372. C₂²⋊Q₈	C₂×C₄×C₅	semidirect
377. #121	C₂×C₄×C₅	semidirect
382. C₄×Q₈	C₂×C₄×C₅	semidirect
383. C₄×Q₈	C₂×C₄×C₅	semidirect
393. C₄×D₄	C₂×C₄×C₅	semidirect
400. #117	C₂×C₄×C₅	semidirect
404. C₂×C₄⋊C₄	C₂×C₄×C₅	semidirect
612. C₅	#17548	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×#3520	2
C₅×#2974	2
C₅×#2275	2
C₅×#1509	2
C₅×#1508	2
C₅×#1506	2
C₅×#1504	2
C₅×#1503	2
C₅×#1501	2
C₅×#1500	2
C₂×C₅×#328	2
#17548	5

Subgroups — C5×#17548 #11940268

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#17548 #11940268