Subgroups — C₅×#17719 #11940439

Order 1280.

1050 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3444	C₂	1	1	yes	maximal, characteristic
2	C₅×#2920	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1552	C₂	1	2	yes	maximal, complemented
5	C₅×#1551	C₂	1	2	yes	maximal, complemented
6	C₅×#1550	C₂	1	2	yes	maximal, complemented
7	C₅×#1547	C₂	1	2	yes	maximal, complemented
8	C₅×#1542	C₂	1	2	yes	maximal, complemented
9	C₅×#1540	C₂	1	2	yes	maximal, complemented
10	C₂³×C₅×C₈	—	2	1	no	—
11	C₅×#533	C₂²	1	1	yes	characteristic
12	C₅×#533	C₂²	1	1	yes	characteristic
13	C₅×#533	C₂²	1	1	yes	characteristic
14	C₅×#532	C₂²	1	1	yes	characteristic
15	C₅×#532	C₂²	1	1	yes	characteristic
16	C₅×#532	C₂²	1	1	yes	characteristic
17	C₅×#529	—	2	1	no	—
18	C₅×#529	—	2	1	no	—
19	C₅×#529	—	2	1	no	—
20	C₅×#529	—	2	1	no	—
21	C₅×#529	C₄	1	2	yes	—
22	C₅×#529	C₄	1	2	yes	—
23	C₅×#525	—	2	1	no	—
24	C₅×#525	—	2	1	no	—
25	C₅×#525	C₄	1	2	yes	—
26	C₄×C₅×#41	—	2	1	no	—
27	C₄×C₅×#41	—	2	1	no	—
28	C₄×C₅×#41	C₄	1	2	yes	—
29	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
30	C₅×#432	—	2	1	no	—
31	C₅×#432	C₂²	1	2	yes	—
32	C₅×#431	—	2	1	no	—
33	C₅×#431	C₂²	1	2	yes	—
34	C₅×#423	C₂²	1	2	yes	—
35	C₂×C₅×C₄⋊C₈	—	2	1	no	—
36	C₂×C₅×C₄⋊C₈	—	2	1	no	—
37	C₂×C₅×C₄⋊C₈	—	2	1	no	—
38	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
39	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
40	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
41	C₅×#407	—	2	1	no	—
42	C₄×C₅×M₁₆	—	2	1	no	—
43	C₅×#333	C₂²	1	4	yes	complemented
44	C₅×#332	C₂²	1	4	yes	complemented
45	C₅×#332	C₂²	1	4	yes	complemented
46	C₅×#331	C₂²	1	4	yes	complemented
47	#17719	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂×C₅×#41	—	2	1	no	—
49	C₂×C₅×#41	—	2	1	no	—
50	C₂×C₅×#41	C₂×C₄	1	2	yes	—
51	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
52	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
53	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
54	C₂×C₅×M₁₆	—	2	2	no	—
55	C₂²×C₅×C₈	—	2	2	no	—
56	C₂²×C₅×C₈	—	2	2	no	—
57	C₂²×C₅×C₈	—	2	2	no	—
58	C₂²×C₅×C₈	—	4	4	no	—
59	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
60	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
61	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
62	C₅×#127	C₂³	1	1	yes	characteristic
63	C₅×#126	—	2	2	no	—
64	C₅×#126	—	2	2	no	—
65	C₅×#125	—	2	1	no	—
66	C₅×#125	—	2	1	no	—
67	C₅×#125	C₂×C₄	1	2	yes	—
68	C₅×#124	—	2	1	no	—
69	C₅×#124	—	2	1	no	—
70	C₅×#124	C₂×C₄	1	2	yes	—
71	C₅×#123	—	2	2	no	—
72	C₅×#123	—	2	2	no	—
73	C₅×#123	—	2	2	no	—
74	C₅×C₂²⋊Q₈	—	2	1	no	—
75	C₅×C₂²⋊Q₈	—	2	1	no	—
76	C₅×C₂²⋊Q₈	—	2	1	no	—
77	C₅×C₂²⋊Q₈	—	2	1	no	—
78	C₅×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
79	C₅×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
80	C₅×#121	—	2	1	no	—
81	C₅×#121	—	2	1	no	—
82	C₅×#121	—	2	1	no	—
83	C₅×#121	—	2	1	no	—
84	C₅×#121	C₂×C₄	1	2	yes	—
85	C₅×#121	C₂×C₄	1	2	yes	—
86	C₅×#120	—	2	2	no	—
87	C₄×C₅×Q₈	—	2	1	no	—
88	C₄×C₅×Q₈	—	2	1	no	—
89	C₄×C₅×Q₈	—	2	1	no	—
90	C₄×C₅×Q₈	—	2	1	no	—
91	C₄×C₅×Q₈	C₂×C₄	1	2	yes	—
92	C₄×C₅×Q₈	C₂×C₄	1	2	yes	—
93	C₄×C₅×D₄	—	2	2	no	—
94	C₄×C₅×D₄	—	2	2	no	—
95	C₄×C₅×D₄	—	2	2	no	—
96	C₄×C₅×D₄	—	2	2	no	—
97	C₄×C₅×D₄	—	2	2	no	—
98	C₄×C₅×D₄	—	2	2	no	—
99	C₄×C₅×D₄	—	2	1	no	—
100	C₄×C₅×D₄	—	2	1	no	—
101	C₄×C₅×D₄	—	2	1	no	—
102	C₄×C₅×D₄	—	2	1	no	—
103	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
104	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
105	C₅×#117	—	2	1	no	—
106	C₅×#117	—	2	1	no	—
107	C₅×#117	—	2	2	no	—
108	C₅×#117	—	2	2	no	—
109	C₅×#117	—	2	2	no	—
110	C₅×#117	C₂×C₄	1	2	yes	—
111	C₂×C₄²×C₅	—	2	2	no	—
112	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
113	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
114	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
115	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
116	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
117	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
118	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
119	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
120	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
121	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
122	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
123	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
124	C₅×C₄⋊C₈	—	2	2	no	—
125	C₅×C₄⋊C₈	—	2	2	no	—
126	C₅×C₄⋊C₈	—	2	2	no	—
127	C₅×C₄⋊C₈	—	2	2	no	—
128	C₅×C₄⋊C₈	C₂³	1	4	yes	—
129	C₅×C₄⋊C₈	C₂³	1	4	yes	—
130	C₅×C₂²⋊C₈	—	2	2	no	—
131	C₅×C₂²⋊C₈	—	2	2	no	—
132	C₅×#97	—	2	2	no	—
133	C₄×C₅×C₈	—	2	2	no	—
134	#3444	C₂×C₅	1	1	yes	characteristic
135	#2920	C₂×C₅	1	1	yes	characteristic, complemented
136	C₂²×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
137	#1552	C₂×C₅	1	2	yes	complemented
138	#1551	C₂×C₅	1	2	yes	complemented
139	#1550	C₂×C₅	1	2	yes	complemented
140	#1547	C₂×C₅	1	2	yes	complemented
141	#1542	C₂×C₅	1	2	yes	complemented
142	#1540	C₂×C₅	1	2	yes	complemented
143	C₂⁴×C₅	C₂²⋊C₄	1	1	yes	characteristic
144	C₅×#41	—	4	4	no	—
145	C₅×#41	—	4	4	no	—
146	C₅×#41	—	4	2	no	—
147	C₅×#41	—	4	2	no	—
148	C₂×C₅×Q₈	—	2	1	no	—
149	C₂×C₅×Q₈	—	2	1	no	—
150	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
151	C₂×C₅×D₄	—	2	2	no	—
152	C₂×C₅×D₄	—	2	2	no	—
153	C₂×C₅×D₄	—	2	2	no	—
154	C₂×C₅×D₄	—	2	1	no	—
155	C₂×C₅×D₄	—	2	1	no	—
156	C₂×C₅×D₄	C₂²×C₄	1	2	yes	—
157	C₂²×C₄×C₅	—	2	4	no	—
158	C₂²×C₄×C₅	—	2	2	no	—
159	C₂²×C₄×C₅	—	2	2	no	—
160	C₂²×C₄×C₅	—	2	2	no	—
161	C₂²×C₄×C₅	—	2	2	no	—
162	C₂²×C₄×C₅	—	2	2	no	—
163	C₂²×C₄×C₅	—	4	2	no	—
164	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
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	C₂²⋊C₄	1	1	yes	characteristic
169	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
170	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
171	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
172	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
173	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
174	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
175	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
176	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
182	C₅×M₁₆	—	4	4	no	—
183	C₂×C₅×C₈	—	2	4	no	—
184	C₂×C₅×C₈	—	2	4	no	—
185	C₂×C₅×C₈	—	4	4	no	—
186	C₂×C₅×C₈	—	4	4	no	—
187	C₂×C₅×C₈	—	4	4	no	—
188	C₅×C₄⋊C₄	—	2	2	no	—
189	C₅×C₄⋊C₄	—	2	2	no	—
190	C₅×C₄⋊C₄	—	2	2	no	—
191	C₅×C₄⋊C₄	—	2	2	no	—
192	C₅×C₄⋊C₄	—	2	2	no	—
193	C₅×C₄⋊C₄	—	2	2	no	—
194	C₅×C₄⋊C₄	—	2	1	no	—
195	C₅×C₄⋊C₄	—	2	1	no	—
196	C₅×C₄⋊C₄	—	2	1	no	—
197	C₅×C₄⋊C₄	—	2	1	no	—
198	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
199	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
200	C₅×C₂²⋊C₄	—	2	2	no	—
201	C₅×C₂²⋊C₄	—	2	2	no	—
202	C₅×C₂²⋊C₄	—	2	2	no	—
203	C₅×C₂²⋊C₄	—	2	2	no	—
204	C₅×C₂²⋊C₄	—	2	2	no	—
205	C₅×C₂²⋊C₄	—	2	2	no	—
206	C₅×C₂²⋊C₄	—	2	2	no	—
207	C₅×C₂²⋊C₄	—	2	2	no	—
208	C₅×C₂²⋊C₄	—	2	2	no	—
209	C₄²×C₅	—	2	2	no	—
210	C₄²×C₅	—	2	2	no	—
211	C₄²×C₅	—	2	2	no	—
212	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
213	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
214	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
215	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
216	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
217	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
218	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
219	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
220	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
221	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
222	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
223	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
224	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
225	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
226	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
227	C₄²×C₅	C₂⁴	1	1	yes	characteristic
228	C₂³×C₈	—	2	1	no	—
229	#533	C₂²×C₅	1	1	yes	characteristic
230	#533	C₂²×C₅	1	1	yes	characteristic
231	#533	C₂²×C₅	1	1	yes	characteristic
232	#532	C₂²×C₅	1	1	yes	characteristic
233	#532	C₂²×C₅	1	1	yes	characteristic
234	#532	C₂²×C₅	1	1	yes	characteristic
235	#529	—	2	1	no	—
236	#529	—	2	1	no	—
237	#529	—	2	1	no	—
238	#529	—	2	1	no	—
239	#529	C₄×C₅	1	2	yes	—
240	#529	C₄×C₅	1	2	yes	—
241	#525	—	2	1	no	—
242	#525	—	2	1	no	—
243	#525	C₄×C₅	1	2	yes	—
244	C₄×#41	—	2	1	no	—
245	C₄×#41	—	2	1	no	—
246	C₄×#41	C₄×C₅	1	2	yes	—
247	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
248	#432	—	2	1	no	—
249	#432	C₂²×C₅	1	2	yes	—
250	#431	—	2	1	no	—
251	#431	C₂²×C₅	1	2	yes	—
252	#423	C₂²×C₅	1	2	yes	—
253	C₂×C₄⋊C₈	—	2	1	no	—
254	C₂×C₄⋊C₈	—	2	1	no	—
255	C₂×C₄⋊C₈	—	2	1	no	—
256	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
257	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
258	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
259	#407	—	2	1	no	—
260	C₄×M₁₆	—	2	1	no	—
261	#333	C₂²×C₅	1	4	yes	complemented
262	#332	C₂²×C₅	1	4	yes	complemented
263	#332	C₂²×C₅	1	4	yes	complemented
264	#331	C₂²×C₅	1	4	yes	complemented
265	C₂³×C₅	—	2	2	no	—
266	C₂³×C₅	—	2	2	no	—
267	C₂³×C₅	—	2	2	no	—
268	C₂³×C₅	—	2	2	no	—
269	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
270	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
271	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
272	C₅×Q₈	—	4	2	no	—
273	C₅×Q₈	—	4	2	no	—
274	C₅×Q₈	—	4	1	no	—
275	C₅×Q₈	—	4	1	no	—
276	C₅×D₄	—	4	2	no	—
277	C₅×D₄	—	4	4	no	—
278	C₅×D₄	—	4	4	no	—
279	C₅×D₄	—	4	4	no	—
280	C₅×D₄	—	4	2	no	—
281	C₅×D₄	—	4	1	no	—
282	C₅×D₄	—	4	1	no	—
283	C₂×C₄×C₅	—	2	2	no	—
284	C₂×C₄×C₅	—	2	2	no	—
285	C₂×C₄×C₅	—	2	2	no	—
286	C₂×C₄×C₅	—	2	2	no	—
287	C₂×C₄×C₅	—	2	2	no	—
288	C₂×C₄×C₅	—	2	2	no	—
289	C₂×C₄×C₅	—	2	1	no	—
290	C₂×C₄×C₅	—	2	1	no	—
291	C₂×C₄×C₅	—	2	1	no	—
292	C₂×C₄×C₅	—	2	1	no	—
293	C₂×C₄×C₅	—	2	2	no	—
294	C₂×C₄×C₅	—	2	2	no	—
295	C₂×C₄×C₅	—	2	2	no	—
296	C₂×C₄×C₅	—	2	2	no	—
297	C₂×C₄×C₅	—	2	2	no	—
298	C₂×C₄×C₅	—	2	2	no	—
299	C₂×C₄×C₅	—	2	2	no	—
300	C₂×C₄×C₅	—	4	1	no	—
301	C₂×C₄×C₅	—	4	1	no	—
302	C₂×C₄×C₅	—	4	1	no	—
303	C₂×C₄×C₅	—	4	1	no	—
304	C₂×C₄×C₅	—	4	4	no	—
305	C₂×C₄×C₅	—	4	4	no	—
306	C₂×C₄×C₅	—	4	4	no	—
307	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
308	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
309	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
310	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
311	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
312	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
313	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
314	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
315	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
316	C₂×C₄×C₅	#135	1	4	yes	—
317	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
318	C₂×C₄×C₅	#137	1	2	yes	—
319	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
320	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
321	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
322	C₅×C₈	—	4	4	no	—
323	C₅×C₈	—	4	4	no	—
324	C₂×#41	—	2	1	no	—
325	C₂×#41	—	2	1	no	—
326	C₂×#41	C₂×C₄×C₅	1	2	yes	—
327	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
328	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
329	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
330	C₂×M₁₆	—	2	2	no	—
331	C₂²×C₈	—	2	2	no	—
332	C₂²×C₈	—	2	2	no	—
333	C₂²×C₈	—	2	2	no	—
334	C₂²×C₈	—	4	4	no	—
335	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
336	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
337	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
338	#127	C₂³×C₅	1	1	yes	characteristic
339	#126	—	2	2	no	—
340	#126	—	2	2	no	—
341	#125	—	2	1	no	—
342	#125	—	2	1	no	—
343	#125	C₂×C₄×C₅	1	2	yes	—
344	#124	—	2	1	no	—
345	#124	—	2	1	no	—
346	#124	C₂×C₄×C₅	1	2	yes	—
347	#123	—	2	2	no	—
348	#123	—	2	2	no	—
349	#123	—	2	2	no	—
350	C₂²⋊Q₈	—	2	1	no	—
351	C₂²⋊Q₈	—	2	1	no	—
352	C₂²⋊Q₈	—	2	1	no	—
353	C₂²⋊Q₈	—	2	1	no	—
354	C₂²⋊Q₈	C₂×C₄×C₅	1	2	yes	—
355	C₂²⋊Q₈	C₂×C₄×C₅	1	2	yes	—
356	#121	—	2	1	no	—
357	#121	—	2	1	no	—
358	#121	—	2	1	no	—
359	#121	—	2	1	no	—
360	#121	C₂×C₄×C₅	1	2	yes	—
361	#121	C₂×C₄×C₅	1	2	yes	—
362	#120	—	2	2	no	—
363	C₄×Q₈	—	2	1	no	—
364	C₄×Q₈	—	2	1	no	—
365	C₄×Q₈	—	2	1	no	—
366	C₄×Q₈	—	2	1	no	—
367	C₄×Q₈	C₂×C₄×C₅	1	2	yes	—
368	C₄×Q₈	C₂×C₄×C₅	1	2	yes	—
369	C₄×D₄	—	2	2	no	—
370	C₄×D₄	—	2	2	no	—
371	C₄×D₄	—	2	2	no	—
372	C₄×D₄	—	2	2	no	—
373	C₄×D₄	—	2	2	no	—
374	C₄×D₄	—	2	2	no	—
375	C₄×D₄	—	2	1	no	—
376	C₄×D₄	—	2	1	no	—
377	C₄×D₄	—	2	1	no	—
378	C₄×D₄	—	2	1	no	—
379	C₄×D₄	C₂×C₄×C₅	1	2	yes	—
380	C₄×D₄	C₂×C₄×C₅	1	2	yes	—
381	#117	—	2	1	no	—
382	#117	—	2	1	no	—
383	#117	—	2	2	no	—
384	#117	—	2	2	no	—
385	#117	—	2	2	no	—
386	#117	C₂×C₄×C₅	1	2	yes	—
387	C₂×C₄²	—	2	2	no	—
388	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
389	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
390	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
391	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
392	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
393	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
394	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
395	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
396	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
397	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
398	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
399	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
400	C₄⋊C₈	—	2	2	no	—
401	C₄⋊C₈	—	2	2	no	—
402	C₄⋊C₈	—	2	2	no	—
403	C₄⋊C₈	—	2	2	no	—
404	C₄⋊C₈	C₂³×C₅	1	4	yes	—
405	C₄⋊C₈	C₂³×C₅	1	4	yes	—
406	C₂²⋊C₈	—	2	2	no	—
407	C₂²⋊C₈	—	2	2	no	—
408	#97	—	2	2	no	—
409	C₄×C₈	—	2	2	no	—
410	C₂²×C₅	—	2	8	no	—
411	C₂²×C₅	—	2	1	no	—
412	C₂²×C₅	—	2	1	no	—
413	C₂²×C₅	—	2	1	no	—
414	C₂²×C₅	—	2	1	no	—
415	C₂²×C₅	—	2	1	no	—
416	C₂²×C₅	—	2	1	no	—
417	C₂²×C₅	—	4	2	no	—
418	C₂²×C₅	—	4	2	no	—
419	C₂²×C₅	—	4	2	no	—
420	C₂²×C₅	#413	1	2	yes	—
421	C₂²×C₅	#416	1	2	yes	—
422	C₂²×C₅	#417	1	2	yes	—
423	C₂²×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
424	C₄×C₅	—	2	1	no	—
425	C₄×C₅	—	2	1	no	—
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	—	2	1	no	—
431	C₄×C₅	—	2	1	no	—
432	C₄×C₅	—	4	2	no	—
433	C₄×C₅	—	4	2	no	—
434	C₄×C₅	—	4	2	no	—
435	C₄×C₅	#416	1	2	yes	—
436	C₄×C₅	#417	1	2	yes	—
437	C₄×C₅	C₂×#135	1	2	yes	—
438	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
439	C₄×C₅	C₂×#137	1	1	yes	characteristic
440	C₂⁴	C₅×C₂²⋊C₄	1	1	yes	characteristic
441	#41	—	4	4	no	—
442	#41	—	4	4	no	—
443	#41	—	4	2	no	—
444	#41	—	4	2	no	—
445	C₂×Q₈	—	2	1	no	—
446	C₂×Q₈	—	2	1	no	—
447	C₂×Q₈	C₂²×C₄×C₅	1	2	yes	—
448	C₂×D₄	—	2	2	no	—
449	C₂×D₄	—	2	2	no	—
450	C₂×D₄	—	2	2	no	—
451	C₂×D₄	—	2	1	no	—
452	C₂×D₄	—	2	1	no	—
453	C₂×D₄	C₂²×C₄×C₅	1	2	yes	—
454	C₂²×C₄	—	2	4	no	—
455	C₂²×C₄	—	2	2	no	—
456	C₂²×C₄	—	2	2	no	—
457	C₂²×C₄	—	2	2	no	—
458	C₂²×C₄	—	2	2	no	—
459	C₂²×C₄	—	2	2	no	—
460	C₂²×C₄	—	4	2	no	—
461	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
462	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
463	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
464	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
465	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
466	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
467	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
468	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
469	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
470	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
471	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
472	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
473	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
474	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
475	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
476	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
477	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
478	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
479	M₁₆	—	4	4	no	—
480	C₂×C₈	—	2	4	no	—
481	C₂×C₈	—	2	4	no	—
482	C₂×C₈	—	4	4	no	—
483	C₂×C₈	—	4	4	no	—
484	C₂×C₈	—	4	4	no	—
485	C₄⋊C₄	—	2	2	no	—
486	C₄⋊C₄	—	2	2	no	—
487	C₄⋊C₄	—	2	2	no	—
488	C₄⋊C₄	—	2	2	no	—
489	C₄⋊C₄	—	2	2	no	—
490	C₄⋊C₄	—	2	2	no	—
491	C₄⋊C₄	—	2	1	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	C₂²×C₄×C₅	1	2	yes	—
496	C₄⋊C₄	C₂²×C₄×C₅	1	2	yes	—
497	C₂²⋊C₄	—	2	2	no	—
498	C₂²⋊C₄	—	2	2	no	—
499	C₂²⋊C₄	—	2	2	no	—
500	C₂²⋊C₄	—	2	2	no	—
501	C₂²⋊C₄	—	2	2	no	—
502	C₂²⋊C₄	—	2	2	no	—
503	C₂²⋊C₄	—	2	2	no	—
504	C₂²⋊C₄	—	2	2	no	—
505	C₂²⋊C₄	—	2	2	no	—
506	C₄²	—	2	2	no	—
507	C₄²	—	2	2	no	—
508	C₄²	—	2	2	no	—
509	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
510	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
511	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
512	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
513	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
514	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
515	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
516	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
517	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
518	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
519	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
520	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
521	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
522	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
523	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
524	C₄²	C₂⁴×C₅	1	1	yes	characteristic
525	C₂×C₅	—	2	2	no	—
526	C₂×C₅	—	2	2	no	—
527	C₂×C₅	—	2	2	no	—
528	C₂×C₅	—	8	2	no	—
529	C₂×C₅	C₂×#413	1	1	yes	characteristic
530	C₂×C₅	C₂×#416	1	1	yes	characteristic
531	C₂×C₅	C₂×#417	1	1	yes	characteristic
532	C₂³	—	2	2	no	—
533	C₂³	—	2	2	no	—
534	C₂³	—	2	2	no	—
535	C₂³	—	2	2	no	—
536	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
537	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
538	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
539	Q₈	—	4	2	no	—
540	Q₈	—	4	2	no	—
541	Q₈	—	4	1	no	—
542	Q₈	—	4	1	no	—
543	D₄	—	4	2	no	—
544	D₄	—	4	4	no	—
545	D₄	—	4	4	no	—
546	D₄	—	4	4	no	—
547	D₄	—	4	2	no	—
548	D₄	—	4	1	no	—
549	D₄	—	4	1	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	2	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	1	no	—
561	C₂×C₄	—	2	1	no	—
562	C₂×C₄	—	2	1	no	—
563	C₂×C₄	—	2	1	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	4	1	no	—
568	C₂×C₄	—	4	1	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	—	4	1	no	—
571	C₂×C₄	—	4	4	no	—
572	C₂×C₄	—	4	4	no	—
573	C₂×C₄	—	4	4	no	—
574	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
575	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
576	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
577	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
578	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
579	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
580	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
581	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
582	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
583	C₂×C₄	C₅×#135	1	4	yes	—
584	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
585	C₂×C₄	C₅×#137	1	2	yes	—
586	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
587	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
588	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
589	C₈	—	4	4	no	—
590	C₈	—	4	4	no	—
591	C₅	#17719	1	1	yes	characteristic, direct product factor, Sylow
592	C₂²	—	2	8	no	—
593	C₂²	—	2	1	no	—
594	C₂²	—	2	1	no	—
595	C₂²	—	2	1	no	—
596	C₂²	—	2	1	no	—
597	C₂²	—	2	1	no	—
598	C₂²	—	2	1	no	—
599	C₂²	—	4	2	no	—
600	C₂²	—	4	2	no	—
601	C₂²	—	4	2	no	—
602	C₂²	C₅×#413	1	2	yes	—
603	C₂²	C₅×#416	1	2	yes	—
604	C₂²	C₅×#417	1	2	yes	—
605	C₂²	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
606	C₄	—	2	1	no	—
607	C₄	—	2	1	no	—
608	C₄	—	2	1	no	—
609	C₄	—	2	1	no	—
610	C₄	—	2	1	no	—
611	C₄	—	2	1	no	—
612	C₄	—	2	1	no	—
613	C₄	—	2	1	no	—
614	C₄	—	4	2	no	—
615	C₄	—	4	2	no	—
616	C₄	—	4	2	no	—
617	C₄	C₅×#416	1	2	yes	—
618	C₄	C₅×#417	1	2	yes	—
619	C₄	C₂×C₅×#135	1	2	yes	—
620	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
621	C₄	C₂×C₅×#137	1	1	yes	characteristic
622	C₂	—	2	2	no	—
623	C₂	—	2	2	no	—
624	C₂	—	2	2	no	—
625	C₂	—	8	2	no	—
626	C₂	C₂×C₅×#413	1	1	yes	characteristic
627	C₂	C₂×C₅×#416	1	1	yes	characteristic
628	C₂	C₂×C₅×#417	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	C₅×#3444	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. C₂²×C₅×C₄⋊C₈	C₂	semidirect
4. C₅×#1552	C₂	semidirect
5. C₅×#1551	C₂	semidirect
6. C₅×#1550	C₂	semidirect
7. C₅×#1547	C₂	semidirect
8. C₅×#1542	C₂	semidirect
9. C₅×#1540	C₂	semidirect
43. C₅×#333	C₂²	semidirect
44. C₅×#332	C₂²	semidirect
45. C₅×#332	C₂²	semidirect
46. C₅×#331	C₂²	semidirect
47. #17719	C₅	direct (N × H)
135. #2920	C₂×C₅	semidirect
136. C₂²×C₄⋊C₈	C₂×C₅	semidirect
137. #1552	C₂×C₅	semidirect
138. #1551	C₂×C₅	semidirect
139. #1550	C₂×C₅	semidirect
140. #1547	C₂×C₅	semidirect
141. #1542	C₂×C₅	semidirect
142. #1540	C₂×C₅	semidirect
261. #333	C₂²×C₅	semidirect
262. #332	C₂²×C₅	semidirect
263. #332	C₂²×C₅	semidirect
264. #331	C₂²×C₅	semidirect
591. C₅	#17719	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×#3444	2
C₅×#2920	2
C₂²×C₅×C₄⋊C₈	2
C₅×#1552	2
C₅×#1551	2
C₅×#1550	2
C₅×#1547	2
C₅×#1542	2
C₅×#1540	2
#17719	5

Subgroups — C5×#17719 #11940439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#17719 #11940439