Subgroups — C₅×#79199 #14416652

Order 1920.

1096 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#908	C₂	1	1	yes	maximal, characteristic
2	C₅×#5522	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#893	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#428	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×Dic₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#5001	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4966	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#4889	C₂	1	2	yes	maximal, complemented
9	C₅×#4872	C₂	1	2	yes	maximal, complemented
10	C₅×#4869	C₂	1	2	yes	maximal, complemented
11	C₅×#4852	C₂	1	2	yes	maximal, complemented
12	C₄×C₅×AGL(1,ℤ/8)	—	3	1	no	maximal, Hall
13	C₂×C₅×#235	C₂²	1	1	yes	characteristic
14	C₂²×C₅×D₁₂	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₅×D₃	—	2	1	no	—
16	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
17	C₃×C₅×#117	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
20	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
21	C₂×C₅×#208	C₂²	1	1	yes	characteristic
22	C₅×#908	C₄	1	8	yes	complemented
23	C₂×C₅×D₂₄	C₂²	1	2	yes	—
24	C₂×C₅×#204	C₂²	1	2	yes	—
25	C₅×#893	—	2	1	no	—
26	C₅×#893	—	2	1	no	—
27	C₅×#893	C₂²	1	2	yes	—
28	C₅×#893	C₂²	1	2	yes	—
29	C₅×#893	C₂²	1	1	yes	characteristic, complemented
30	C₅×#893	C₂²	1	1	yes	characteristic, complemented
31	C₅×#887	C₂²	1	1	yes	characteristic, complemented
32	C₅×#887	C₂²	1	1	yes	characteristic, complemented
33	C₅×#67⋊C₄	—	2	1	no	—
34	C₅×#67⋊C₄	—	2	1	no	—
35	C₅×#872	—	2	1	no	—
36	C₄²×C₅×D₃	—	2	1	no	—
37	C₃×C₅×#107	C₂²	1	2	yes	complemented
38	C₃×C₅×#106	C₂²	1	2	yes	complemented
39	C₅×#815	C₂²	1	2	yes	complemented
40	C₅×C₈×Dic₃	C₂²	1	2	yes	complemented
41	C₅×#810	C₂²	1	2	yes	complemented
42	C₅×#809	C₂²	1	2	yes	—
43	C₅×#809	C₂²	1	2	yes	—
44	C₅×#809	C₂²	1	2	yes	complemented
45	#79199	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
46	C₂×C₅×AGL(1,ℤ/8)	—	3	1	no	—
47	C₄×C₅×#41	—	3	1	no	—
48	C₂×C₄×C₅×D₄	—	3	1	no	—
49	C₅×#442	—	3	2	no	—
50	C₅×#440	—	3	2	no	—
51	C₄×C₅×SD₁₆	—	3	2	no	—
52	C₄×C₅×D₈	—	3	2	no	—
53	C₅×#428	D₃	1	1	yes	characteristic, complemented
54	C₅×#418	—	3	1	no	—
55	C₅×#417	—	3	1	no	—
56	C₄×C₅×M₁₆	—	3	1	no	—
57	C₂³×C₅×D₃	—	2	1	no	—
58	C₂²×C₃×C₄×C₅	—	2	1	no	—
59	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
60	C₂×C₅×#67	—	2	1	no	—
61	C₂²×C₅×Dic₃	D₄	1	1	yes	characteristic
62	C₂²×C₅×Dic₃	D₄	1	1	yes	characteristic
63	C₅×#235	—	2	2	no	—
64	C₅×#235	C₂×C₄	1	4	yes	complemented
65	C₂×C₅×D₁₂	—	2	1	no	—
66	C₂×C₅×D₁₂	—	2	2	no	—
67	C₂×C₅×D₁₂	C₂×C₄	1	4	yes	complemented
68	C₂×C₅×D₁₂	C₂³	1	2	yes	—
69	C₂×C₅×D₁₂	C₂³	1	1	yes	characteristic
70	C₂×C₄×C₅×D₃	—	2	1	no	—
71	C₂×C₄×C₅×D₃	—	2	1	no	—
72	C₂×C₄×C₅×D₃	—	2	1	no	—
73	C₂×C₄×C₅×D₃	—	2	2	no	—
74	C₂×C₄×C₅×D₃	—	2	2	no	—
75	C₂×C₄×C₅×D₃	—	4	4	no	—
76	C₂×C₅×Dic₆	C₂³	1	1	yes	characteristic
77	C₃×C₅×M₁₆	C₂×C₄	1	4	yes	complemented
78	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
79	C₃×C₅×C₄⋊C₄	—	2	1	no	—
80	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
81	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
82	C₃×C₅×C₄⋊C₄	C₂³	1	2	yes	—
83	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
84	C₃×C₄²×C₅	—	2	1	no	—
85	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
86	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
87	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
88	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
89	C₅×Dic₃⋊C₄	—	2	1	no	—
90	C₅×Dic₃⋊C₄	—	2	1	no	—
91	C₄×C₅×Dic₃	—	2	1	no	—
92	C₄×C₅×Dic₃	—	2	1	no	—
93	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
94	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
95	C₄×C₅×Dic₃	C₂³	1	1	yes	characteristic
96	C₄×C₅×Dic₃	C₂³	1	1	yes	characteristic
97	C₅×#208	—	2	2	no	—
98	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
99	C₅×D₂₄	C₂×C₄	1	8	yes	complemented
100	C₅×#204	C₂×C₄	1	8	yes	complemented
101	C₂×#908	C₂×C₅	1	1	yes	characteristic
102	#5522	C₂×C₅	1	1	yes	characteristic, complemented
103	C₂×#893	C₂×C₅	1	1	yes	characteristic, complemented
104	C₃×#428	C₂×C₅	1	1	yes	characteristic, complemented
105	Dic₃×M₁₆	C₂×C₅	1	1	yes	characteristic, complemented
106	#5001	C₂×C₅	1	1	yes	characteristic, complemented
107	#4966	C₂×C₅	1	1	yes	characteristic, complemented
108	#4889	C₂×C₅	1	2	yes	complemented
109	#4872	C₂×C₅	1	2	yes	complemented
110	#4869	C₂×C₅	1	2	yes	complemented
111	#4852	C₂×C₅	1	2	yes	complemented
112	C₂×C₅×#41	—	3	1	no	—
113	C₂²×C₅×D₄	—	3	1	no	—
114	C₂³×C₄×C₅	—	6	1	no	—
115	C₅×AGL(1,ℤ/8)	—	3	8	no	—
116	C₂×C₅×SD₁₆	—	3	2	no	—
117	C₂×C₅×D₈	—	3	2	no	—
118	C₂×C₅×M₁₆	—	3	1	no	—
119	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
120	C₄×C₅×Q₈	—	3	1	no	—
121	C₄×C₅×Q₈	—	3	1	no	—
122	C₄×C₅×D₄	—	3	2	no	—
123	C₄×C₅×D₄	—	3	2	no	—
124	C₄×C₅×D₄	—	3	1	no	—
125	C₄×C₅×D₄	—	3	1	no	—
126	C₄×C₅×D₄	—	6	1	no	—
127	C₄×C₅×D₄	—	6	1	no	—
128	C₄×C₅×D₄	—	6	1	no	—
129	C₄×C₅×D₄	—	6	1	no	—
130	C₅×#117	—	6	1	no	—
131	C₅×#117	C₂×D₃	1	1	yes	characteristic
132	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
133	C₂×C₅×C₂²⋊C₄	—	6	1	no	—
134	C₂×C₄²×C₅	—	3	1	no	—
135	C₂×C₄²×C₅	—	6	1	no	—
136	C₅×#107	C₂×D₃	1	2	yes	complemented
137	C₅×#106	C₂×D₃	1	2	yes	complemented
138	C₅×#103	—	3	2	no	—
139	C₅×#102	—	3	2	no	—
140	C₅×#102	—	3	2	no	—
141	C₅×#102	—	3	2	no	—
142	C₅×#97	—	3	2	no	—
143	C₄×C₅×C₈	—	3	2	no	—
144	C₄×AGL(1,ℤ/8)	—	3	1	no	Sylow
145	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₅×D₃	—	2	4	no	—
147	C₂²×C₅×D₃	—	2	1	no	—
148	C₂²×C₅×D₃	—	2	2	no	—
149	C₂²×C₅×D₃	—	4	2	no	—
150	C₂²×C₅×D₃	—	4	2	no	—
151	C₂×C₃×C₄×C₅	—	2	1	no	—
152	C₂×C₃×C₄×C₅	—	2	1	no	—
153	C₂×C₃×C₄×C₅	—	2	2	no	—
154	C₂×C₃×C₄×C₅	—	4	2	no	—
155	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
156	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
157	C₂×C₃×C₄×C₅	#41	1	2	yes	—
158	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
159	C₅×#67	—	2	4	no	—
160	C₂×C₅×Dic₃	—	2	1	no	—
161	C₂×C₅×Dic₃	—	2	2	no	—
162	C₂×C₅×Dic₃	—	4	1	no	—
163	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
164	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
165	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
166	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
167	C₅×D₁₂	—	2	4	no	—
168	C₅×D₁₂	—	2	2	no	—
169	C₅×D₁₂	—	2	1	no	—
170	C₅×D₁₂	C₂²×C₄	1	4	yes	—
171	C₅×D₁₂	C₂²×C₄	1	2	yes	complemented
172	C₄×C₅×D₃	—	2	4	no	—
173	C₄×C₅×D₃	—	4	2	no	—
174	C₄×C₅×D₃	—	4	4	no	—
175	C₄×C₅×D₃	—	4	2	no	—
176	C₄×C₅×D₃	—	4	4	no	—
177	C₅×Dic₆	—	2	1	no	—
178	C₅×Dic₆	C₂²×C₄	1	2	yes	complemented
179	C₃×C₅×C₈	C₂²×C₄	1	4	yes	complemented
180	C₅×C₃⋊C₈	—	2	2	no	—
181	C₂×#235	C₂²×C₅	1	1	yes	characteristic
182	C₂²×D₁₂	C₂²×C₅	1	1	yes	characteristic
183	C₂²×C₄×D₃	—	2	1	no	—
184	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
185	C₃×#117	C₂²×C₅	1	1	yes	characteristic
186	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
187	C₂×(C₂×D₃)⋊C₄	—	2	1	no	—
188	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
189	C₂×#208	C₂²×C₅	1	1	yes	characteristic
190	#908	C₄×C₅	1	8	yes	complemented
191	C₂×D₂₄	C₂²×C₅	1	2	yes	—
192	C₂×#204	C₂²×C₅	1	2	yes	—
193	#893	—	2	1	no	—
194	#893	—	2	1	no	—
195	#893	C₂²×C₅	1	2	yes	—
196	#893	C₂²×C₅	1	2	yes	—
197	#893	C₂²×C₅	1	1	yes	characteristic, complemented
198	#893	C₂²×C₅	1	1	yes	characteristic, complemented
199	#887	C₂²×C₅	1	1	yes	characteristic, complemented
200	#887	C₂²×C₅	1	1	yes	characteristic, complemented
201	#67⋊C₄	—	2	1	no	—
202	#67⋊C₄	—	2	1	no	—
203	#872	—	2	1	no	—
204	C₄²×D₃	—	2	1	no	—
205	C₃×#107	C₂²×C₅	1	2	yes	complemented
206	C₃×#106	C₂²×C₅	1	2	yes	complemented
207	#815	C₂²×C₅	1	2	yes	complemented
208	C₈×Dic₃	C₂²×C₅	1	2	yes	complemented
209	#810	C₂²×C₅	1	2	yes	complemented
210	#809	C₂²×C₅	1	2	yes	—
211	#809	C₂²×C₅	1	2	yes	—
212	#809	C₂²×C₅	1	2	yes	complemented
213	C₂⁴×C₅	—	6	1	no	—
214	C₅×#41	—	3	4	no	—
215	C₅×#41	—	6	2	no	—
216	C₂×C₅×Q₈	—	3	1	no	—
217	C₂×C₅×D₄	—	3	4	no	—
218	C₂×C₅×D₄	—	3	1	no	—
219	C₂×C₅×D₄	—	3	2	no	—
220	C₂×C₅×D₄	—	6	1	no	—
221	C₂×C₅×D₄	—	6	1	no	—
222	C₂×C₅×D₄	—	6	2	no	—
223	C₂²×C₄×C₅	—	2	1	no	—
224	C₂²×C₄×C₅	—	3	1	no	—
225	C₂²×C₄×C₅	—	3	1	no	—
226	C₂²×C₄×C₅	—	6	1	no	—
227	C₂²×C₄×C₅	—	6	1	no	—
228	C₂²×C₄×C₅	—	6	2	no	—
229	C₂²×C₄×C₅	—	6	2	no	—
230	C₂²×C₄×C₅	—	6	1	no	—
231	C₂²×C₄×C₅	—	12	4	no	—
232	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
233	C₅×SD₁₆	—	3	8	no	—
234	C₅×D₈	—	3	8	no	—
235	C₅×M₁₆	—	6	2	no	—
236	C₅×M₁₆	C₄×D₃	1	4	yes	complemented
237	C₂×C₅×C₈	—	3	2	no	—
238	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
239	C₅×C₄⋊C₄	—	2	1	no	—
240	C₅×C₄⋊C₄	—	6	1	no	—
241	C₅×C₄⋊C₄	—	6	1	no	—
242	C₅×C₄⋊C₄	C₂²×D₃	1	2	yes	—
243	C₅×C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
244	C₅×C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
245	C₅×C₂²⋊C₄	—	2	1	no	—
246	C₅×C₂²⋊C₄	—	6	2	no	—
247	C₅×C₂²⋊C₄	—	6	2	no	—
248	C₅×C₂²⋊C₄	—	6	1	no	—
249	C₅×C₂²⋊C₄	—	6	1	no	—
250	C₄²×C₅	—	2	1	no	—
251	C₄²×C₅	—	3	1	no	—
252	C₄²×C₅	—	3	1	no	—
253	C₄²×C₅	—	3	1	no	—
254	C₄²×C₅	—	3	1	no	—
255	C₄²×C₅	—	6	1	no	—
256	C₄²×C₅	—	6	1	no	—
257	C₂×AGL(1,ℤ/8)	—	3	1	no	—
258	C₄×#41	—	3	1	no	—
259	C₂×C₄×D₄	—	3	1	no	—
260	#442	—	3	2	no	—
261	#440	—	3	2	no	—
262	C₄×SD₁₆	—	3	2	no	—
263	C₄×D₈	—	3	2	no	—
264	#428	C₅×D₃	1	1	yes	characteristic, complemented
265	#418	—	3	1	no	—
266	#417	—	3	1	no	—
267	C₄×M₁₆	—	3	1	no	—
268	C₂²×C₃×C₅	—	2	1	no	—
269	C₂²×C₃×C₅	—	2	1	no	—
270	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
271	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
272	C₂×C₅×D₃	—	2	2	no	—
273	C₂×C₅×D₃	—	2	4	no	—
274	C₂×C₅×D₃	—	4	8	no	—
275	C₂×C₅×D₃	—	4	2	no	—
276	C₂×C₅×D₃	—	4	1	no	—
277	C₂×C₅×D₃	—	4	1	no	—
278	C₂×C₅×D₃	—	4	2	no	—
279	C₃×C₄×C₅	—	4	2	no	—
280	C₃×C₄×C₅	—	4	1	no	—
281	C₃×C₄×C₅	—	4	1	no	—
282	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
283	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
284	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
285	C₅×Dic₃	—	2	2	no	—
286	C₅×Dic₃	—	2	1	no	—
287	C₅×Dic₃	—	2	1	no	—
288	C₅×Dic₃	—	2	1	no	—
289	C₅×Dic₃	AGL(1,ℤ/8)	1	2	yes	complemented
290	C₂³×D₃	—	2	1	no	—
291	C₂²×C₃×C₄	—	2	1	no	—
292	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
293	C₂×#67	—	2	1	no	—
294	C₂²×Dic₃	C₅×D₄	1	1	yes	characteristic
295	C₂²×Dic₃	C₅×D₄	1	1	yes	characteristic
296	#235	—	2	2	no	—
297	#235	C₂×C₄×C₅	1	4	yes	complemented
298	C₂×D₁₂	—	2	1	no	—
299	C₂×D₁₂	—	2	2	no	—
300	C₂×D₁₂	C₂×C₄×C₅	1	4	yes	complemented
301	C₂×D₁₂	C₂³×C₅	1	2	yes	—
302	C₂×D₁₂	C₂³×C₅	1	1	yes	characteristic
303	C₂×C₄×D₃	—	2	1	no	—
304	C₂×C₄×D₃	—	2	1	no	—
305	C₂×C₄×D₃	—	2	1	no	—
306	C₂×C₄×D₃	—	2	2	no	—
307	C₂×C₄×D₃	—	2	2	no	—
308	C₂×C₄×D₃	—	4	4	no	—
309	C₂×Dic₆	C₂³×C₅	1	1	yes	characteristic
310	C₃×M₁₆	C₂×C₄×C₅	1	4	yes	complemented
311	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
312	C₃×C₄⋊C₄	—	2	1	no	—
313	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
314	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
315	C₃×C₄⋊C₄	C₂³×C₅	1	2	yes	—
316	C₃×C₂²⋊C₄	—	2	1	no	—
317	C₃×C₄²	—	2	1	no	—
318	(C₂×D₃)⋊C₄	—	2	1	no	—
319	(C₂×D₃)⋊C₄	—	2	1	no	—
320	(C₂×D₃)⋊C₄	—	2	2	no	—
321	(C₂×D₃)⋊C₄	—	2	2	no	—
322	Dic₃⋊C₄	—	2	1	no	—
323	Dic₃⋊C₄	—	2	1	no	—
324	C₄×Dic₃	—	2	1	no	—
325	C₄×Dic₃	—	2	1	no	—
326	C₄×Dic₃	C₅×D₄	1	1	yes	characteristic
327	C₄×Dic₃	C₅×D₄	1	1	yes	characteristic
328	C₄×Dic₃	C₂³×C₅	1	1	yes	characteristic
329	C₄×Dic₃	C₂³×C₅	1	1	yes	characteristic
330	#208	—	2	2	no	—
331	C₂×C₃⋊C₈	C₂³×C₅	1	2	yes	—
332	D₂₄	C₂×C₄×C₅	1	8	yes	complemented
333	#204	C₂×C₄×C₅	1	8	yes	complemented
334	C₂³×C₅	—	6	4	no	—
335	C₂³×C₅	—	6	1	no	—
336	C₂³×C₅	—	6	2	no	—
337	C₂³×C₅	—	12	2	no	—
338	C₂³×C₅	—	12	2	no	—
339	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
340	C₅×Q₈	—	3	2	no	—
341	C₅×Q₈	—	6	1	no	—
342	C₅×D₄	—	3	2	no	—
343	C₅×D₄	—	3	4	no	—
344	C₅×D₄	—	6	4	no	—
345	C₅×D₄	—	6	4	no	—
346	C₅×D₄	—	6	2	no	—
347	C₅×D₄	—	6	1	no	—
348	C₂×C₄×C₅	—	2	1	no	—
349	C₂×C₄×C₅	—	2	1	no	—
350	C₂×C₄×C₅	—	2	2	no	—
351	C₂×C₄×C₅	—	3	1	no	—
352	C₂×C₄×C₅	—	3	1	no	—
353	C₂×C₄×C₅	—	3	1	no	—
354	C₂×C₄×C₅	—	3	1	no	—
355	C₂×C₄×C₅	—	4	2	no	—
356	C₂×C₄×C₅	—	6	2	no	—
357	C₂×C₄×C₅	—	6	1	no	—
358	C₂×C₄×C₅	—	6	4	no	—
359	C₂×C₄×C₅	—	12	2	no	—
360	C₂×C₄×C₅	—	12	4	no	—
361	C₂×C₄×C₅	—	12	2	no	—
362	C₂×C₄×C₅	—	12	4	no	—
363	C₂×C₄×C₅	—	12	1	no	—
364	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
365	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
366	C₂×C₄×C₅	#239	1	2	yes	—
367	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
368	C₅×C₈	—	6	2	no	—
369	C₅×C₈	C₂×C₄×D₃	1	4	yes	complemented
370	C₂×#41	—	3	1	no	—
371	C₂²×D₄	—	3	1	no	—
372	C₂³×C₄	—	6	1	no	—
373	AGL(1,ℤ/8)	—	3	8	no	—
374	C₂×SD₁₆	—	3	2	no	—
375	C₂×D₈	—	3	2	no	—
376	C₂×M₁₆	—	3	1	no	—
377	C₂×M₁₆	C₂×C₅×D₃	1	1	yes	characteristic
378	C₄×Q₈	—	3	1	no	—
379	C₄×Q₈	—	3	1	no	—
380	C₄×D₄	—	3	2	no	—
381	C₄×D₄	—	3	2	no	—
382	C₄×D₄	—	3	1	no	—
383	C₄×D₄	—	3	1	no	—
384	C₄×D₄	—	6	1	no	—
385	C₄×D₄	—	6	1	no	—
386	C₄×D₄	—	6	1	no	—
387	C₄×D₄	—	6	1	no	—
388	#117	—	6	1	no	—
389	#117	C₂×C₅×D₃	1	1	yes	characteristic
390	C₂×C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
391	C₂×C₂²⋊C₄	—	6	1	no	—
392	C₂×C₄²	—	3	1	no	—
393	C₂×C₄²	—	6	1	no	—
394	#107	C₂×C₅×D₃	1	2	yes	complemented
395	#106	C₂×C₅×D₃	1	2	yes	complemented
396	#103	—	3	2	no	—
397	#102	—	3	2	no	—
398	#102	—	3	2	no	—
399	#102	—	3	2	no	—
400	#97	—	3	2	no	—
401	C₄×C₈	—	3	2	no	—
402	C₂×C₃×C₅	—	2	2	no	—
403	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
404	C₂×C₃×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
405	C₂×C₃×C₅	#575	1	1	yes	characteristic
406	C₅×D₃	—	4	2	no	—
407	C₅×D₃	—	4	4	no	—
408	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
409	C₂²×D₃	—	2	4	no	—
410	C₂²×D₃	—	2	1	no	—
411	C₂²×D₃	—	2	2	no	—
412	C₂²×D₃	—	4	2	no	—
413	C₂²×D₃	—	4	2	no	—
414	C₂×C₃×C₄	—	2	1	no	—
415	C₂×C₃×C₄	—	2	1	no	—
416	C₂×C₃×C₄	—	2	2	no	—
417	C₂×C₃×C₄	—	4	2	no	—
418	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
419	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
420	C₂×C₃×C₄	C₅×#41	1	2	yes	—
421	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
422	#67	—	2	4	no	—
423	C₂×Dic₃	—	2	1	no	—
424	C₂×Dic₃	—	2	2	no	—
425	C₂×Dic₃	—	4	1	no	—
426	C₂×Dic₃	C₂×C₅×D₄	1	1	yes	characteristic
427	C₂×Dic₃	C₂×C₅×D₄	1	1	yes	characteristic
428	C₂×Dic₃	C₂×C₅×D₄	1	1	yes	characteristic
429	C₂×Dic₃	C₂×C₅×D₄	1	1	yes	characteristic
430	D₁₂	—	2	4	no	—
431	D₁₂	—	2	2	no	—
432	D₁₂	—	2	1	no	—
433	D₁₂	C₂²×C₄×C₅	1	4	yes	—
434	D₁₂	C₂²×C₄×C₅	1	2	yes	complemented
435	C₄×D₃	—	2	4	no	—
436	C₄×D₃	—	4	2	no	—
437	C₄×D₃	—	4	4	no	—
438	C₄×D₃	—	4	2	no	—
439	C₄×D₃	—	4	4	no	—
440	Dic₆	—	2	1	no	—
441	Dic₆	C₂²×C₄×C₅	1	2	yes	complemented
442	C₃×C₈	C₂²×C₄×C₅	1	4	yes	complemented
443	C₃⋊C₈	—	2	2	no	—
444	C₂²×C₅	—	2	1	no	—
445	C₂²×C₅	—	2	1	no	—
446	C₂²×C₅	—	6	2	no	—
447	C₂²×C₅	—	6	4	no	—
448	C₂²×C₅	—	12	8	no	—
449	C₂²×C₅	—	12	2	no	—
450	C₂²×C₅	—	12	1	no	—
451	C₂²×C₅	—	12	1	no	—
452	C₂²×C₅	—	12	2	no	—
453	C₂²×C₅	#893	1	2	yes	—
454	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
455	C₄×C₅	—	3	2	no	—
456	C₄×C₅	—	4	2	no	—
457	C₄×C₅	—	4	1	no	—
458	C₄×C₅	—	4	1	no	—
459	C₄×C₅	—	6	2	no	—
460	C₄×C₅	—	6	1	no	—
461	C₄×C₅	—	6	1	no	—
462	C₄×C₅	—	6	1	no	—
463	C₄×C₅	#893	1	2	yes	—
464	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
465	C₄×C₅	C₂×#239	1	1	yes	characteristic
466	C₂⁴	—	6	1	no	—
467	#41	—	3	4	no	—
468	#41	—	6	2	no	—
469	C₂×Q₈	—	3	1	no	—
470	C₂×D₄	—	3	4	no	—
471	C₂×D₄	—	3	1	no	—
472	C₂×D₄	—	3	2	no	—
473	C₂×D₄	—	6	1	no	—
474	C₂×D₄	—	6	1	no	—
475	C₂×D₄	—	6	2	no	—
476	C₂²×C₄	—	2	1	no	—
477	C₂²×C₄	—	3	1	no	—
478	C₂²×C₄	—	3	1	no	—
479	C₂²×C₄	—	6	1	no	—
480	C₂²×C₄	—	6	1	no	—
481	C₂²×C₄	—	6	2	no	—
482	C₂²×C₄	—	6	2	no	—
483	C₂²×C₄	—	6	1	no	—
484	C₂²×C₄	—	12	4	no	—
485	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
486	SD₁₆	—	3	8	no	—
487	D₈	—	3	8	no	—
488	M₁₆	—	6	2	no	—
489	M₁₆	C₄×C₅×D₃	1	4	yes	complemented
490	C₂×C₈	—	3	2	no	—
491	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	6	1	no	—
494	C₄⋊C₄	—	6	1	no	—
495	C₄⋊C₄	C₂²×C₅×D₃	1	1	yes	characteristic
496	C₄⋊C₄	C₂²×C₅×D₃	1	1	yes	characteristic
497	C₄⋊C₄	C₂²×C₅×D₃	1	2	yes	—
498	C₂²⋊C₄	—	2	1	no	—
499	C₂²⋊C₄	—	6	2	no	—
500	C₂²⋊C₄	—	6	2	no	—
501	C₂²⋊C₄	—	6	1	no	—
502	C₂²⋊C₄	—	6	1	no	—
503	C₄²	—	2	1	no	—
504	C₄²	—	3	1	no	—
505	C₄²	—	3	1	no	—
506	C₄²	—	3	1	no	—
507	C₄²	—	3	1	no	—
508	C₄²	—	6	1	no	—
509	C₄²	—	6	1	no	—
510	C₃×C₅	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Hall
511	C₂²×C₃	—	2	1	no	—
512	C₂²×C₃	—	2	1	no	—
513	C₂²×C₃	C₄×C₅×D₄	1	2	yes	—
514	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
515	C₂×D₃	—	2	2	no	—
516	C₂×D₃	—	2	4	no	—
517	C₂×D₃	—	4	8	no	—
518	C₂×D₃	—	4	2	no	—
519	C₂×D₃	—	4	1	no	—
520	C₂×D₃	—	4	1	no	—
521	C₂×D₃	—	4	2	no	—
522	C₃×C₄	—	4	2	no	—
523	C₃×C₄	—	4	1	no	—
524	C₃×C₄	—	4	1	no	—
525	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
526	C₃×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
527	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
528	Dic₃	—	2	2	no	—
529	Dic₃	—	2	1	no	—
530	Dic₃	—	2	1	no	—
531	Dic₃	—	2	1	no	—
532	Dic₃	C₅×AGL(1,ℤ/8)	1	2	yes	complemented
533	C₂×C₅	—	2	2	no	—
534	C₂×C₅	—	12	4	no	—
535	C₂×C₅	—	12	2	no	—
536	C₂×C₅	C₂×#893	1	1	yes	characteristic
537	C₂×C₅	D₃×AGL(1,ℤ/8)	1	1	yes	characteristic
538	C₂×C₅	#5777	1	1	yes	characteristic
539	C₂³	—	6	4	no	—
540	C₂³	—	6	1	no	—
541	C₂³	—	6	2	no	—
542	C₂³	—	12	2	no	—
543	C₂³	—	12	2	no	—
544	C₂³	C₅×D₃×D₄	1	1	yes	characteristic
545	Q₈	—	3	2	no	—
546	Q₈	—	6	1	no	—
547	D₄	—	3	2	no	—
548	D₄	—	3	4	no	—
549	D₄	—	6	4	no	—
550	D₄	—	6	4	no	—
551	D₄	—	6	2	no	—
552	D₄	—	6	1	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	3	1	no	—
557	C₂×C₄	—	3	1	no	—
558	C₂×C₄	—	3	1	no	—
559	C₂×C₄	—	3	1	no	—
560	C₂×C₄	—	4	2	no	—
561	C₂×C₄	—	6	2	no	—
562	C₂×C₄	—	6	1	no	—
563	C₂×C₄	—	6	4	no	—
564	C₂×C₄	—	12	4	no	—
565	C₂×C₄	—	12	2	no	—
566	C₂×C₄	—	12	4	no	—
567	C₂×C₄	—	12	2	no	—
568	C₂×C₄	—	12	1	no	—
569	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
570	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
571	C₂×C₄	C₅×#239	1	2	yes	—
572	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
573	C₈	—	6	2	no	—
574	C₈	C₂×C₄×C₅×D₃	1	4	yes	complemented
575	C₂×C₃	—	2	2	no	—
576	C₂×C₃	C₂×C₄×C₅×D₄	1	1	yes	characteristic
577	C₂×C₃	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
578	C₂×C₃	C₅×#575	1	1	yes	characteristic
579	D₃	—	4	4	no	—
580	D₃	—	4	2	no	—
581	C₅	#79199	1	1	yes	characteristic, direct product factor, Sylow
582	C₂²	—	2	1	no	—
583	C₂²	—	2	1	no	—
584	C₂²	—	6	2	no	—
585	C₂²	—	6	4	no	—
586	C₂²	—	12	8	no	—
587	C₂²	—	12	2	no	—
588	C₂²	—	12	1	no	—
589	C₂²	—	12	1	no	—
590	C₂²	—	12	2	no	—
591	C₂²	C₅×#893	1	2	yes	—
592	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
593	C₄	—	3	2	no	—
594	C₄	—	4	2	no	—
595	C₄	—	4	1	no	—
596	C₄	—	4	1	no	—
597	C₄	—	6	2	no	—
598	C₄	—	6	1	no	—
599	C₄	—	6	1	no	—
600	C₄	—	6	1	no	—
601	C₄	C₅×#893	1	2	yes	—
602	C₄	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
603	C₄	C₂×C₅×#239	1	1	yes	characteristic
604	C₃	C₄×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
605	C₂	—	2	2	no	—
606	C₂	—	12	4	no	—
607	C₂	—	12	2	no	—
608	C₂	C₂×C₅×#893	1	1	yes	characteristic
609	C₂	C₅×D₃×AGL(1,ℤ/8)	1	1	yes	characteristic
610	C₂	C₅×#5777	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×D₃×D₄
commutator	C₃×C₄	C₂³×C₄×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×#428	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂×C₅×#908	C₂
cosocle	C₂×C₃×C₄	C₂⁴×C₅
hypercenter	C₅×#428	D₃
hypocenter	C₃	C₄×C₅×AGL(1,ℤ/8)
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#428

Complements

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

N	H ≅ G/N	type
2. C₅×#5522	C₂	semidirect
3. C₂×C₅×#893	C₂	semidirect
4. C₃×C₅×#428	C₂	semidirect
5. C₅×Dic₃×M₁₆	C₂	semidirect
6. C₅×#5001	C₂	semidirect
7. C₅×#4966	C₂	semidirect
8. C₅×#4889	C₂	semidirect
9. C₅×#4872	C₂	semidirect
10. C₅×#4869	C₂	semidirect
11. C₅×#4852	C₂	semidirect
22. C₅×#908	C₄	semidirect
29. C₅×#893	C₂²	semidirect
30. C₅×#893	C₂²	semidirect
31. C₅×#887	C₂²	semidirect
32. C₅×#887	C₂²	semidirect
37. C₃×C₅×#107	C₂²	semidirect
38. C₃×C₅×#106	C₂²	semidirect
39. C₅×#815	C₂²	semidirect
40. C₅×C₈×Dic₃	C₂²	semidirect
41. C₅×#810	C₂²	semidirect
44. C₅×#809	C₂²	semidirect
45. #79199	C₅	direct (N × H)
53. C₅×#428	D₃	semidirect
64. C₅×#235	C₂×C₄	semidirect
67. C₂×C₅×D₁₂	C₂×C₄	semidirect
77. C₃×C₅×M₁₆	C₂×C₄	semidirect
99. C₅×D₂₄	C₂×C₄	semidirect
100. C₅×#204	C₂×C₄	semidirect
102. #5522	C₂×C₅	semidirect
103. C₂×#893	C₂×C₅	semidirect
104. C₃×#428	C₂×C₅	semidirect
105. Dic₃×M₁₆	C₂×C₅	semidirect
106. #5001	C₂×C₅	semidirect
107. #4966	C₂×C₅	semidirect
108. #4889	C₂×C₅	semidirect
109. #4872	C₂×C₅	semidirect
110. #4869	C₂×C₅	semidirect
111. #4852	C₂×C₅	semidirect
136. C₅×#107	C₂×D₃	semidirect
137. C₅×#106	C₂×D₃	semidirect
171. C₅×D₁₂	C₂²×C₄	semidirect
178. C₅×Dic₆	C₂²×C₄	semidirect
179. C₃×C₅×C₈	C₂²×C₄	semidirect
190. #908	C₄×C₅	semidirect
197. #893	C₂²×C₅	semidirect
198. #893	C₂²×C₅	semidirect
199. #887	C₂²×C₅	semidirect
200. #887	C₂²×C₅	semidirect
205. C₃×#107	C₂²×C₅	semidirect
206. C₃×#106	C₂²×C₅	semidirect
207. #815	C₂²×C₅	semidirect
208. C₈×Dic₃	C₂²×C₅	semidirect
209. #810	C₂²×C₅	semidirect
212. #809	C₂²×C₅	semidirect
236. C₅×M₁₆	C₄×D₃	semidirect
264. #428	C₅×D₃	semidirect
289. C₅×Dic₃	AGL(1,ℤ/8)	semidirect
297. #235	C₂×C₄×C₅	semidirect
300. C₂×D₁₂	C₂×C₄×C₅	semidirect
310. C₃×M₁₆	C₂×C₄×C₅	semidirect
332. D₂₄	C₂×C₄×C₅	semidirect
333. #204	C₂×C₄×C₅	semidirect
369. C₅×C₈	C₂×C₄×D₃	semidirect
394. #107	C₂×C₅×D₃	semidirect
395. #106	C₂×C₅×D₃	semidirect
434. D₁₂	C₂²×C₄×C₅	semidirect
441. Dic₆	C₂²×C₄×C₅	semidirect
442. C₃×C₈	C₂²×C₄×C₅	semidirect
489. M₁₆	C₄×C₅×D₃	semidirect
510. C₃×C₅	C₄×AGL(1,ℤ/8)	semidirect
532. Dic₃	C₅×AGL(1,ℤ/8)	semidirect
574. C₈	C₂×C₄×C₅×D₃	semidirect
581. C₅	#79199	direct (N × H)
604. C₃	C₄×C₅×AGL(1,ℤ/8)	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₅×#908	2
C₅×#5522	2
C₂×C₅×#893	2
C₃×C₅×#428	2
C₅×Dic₃×M₁₆	2
C₅×#5001	2
C₅×#4966	2
C₅×#4889	2
C₅×#4872	2
C₅×#4869	2
C₅×#4852	2
C₄×C₅×AGL(1,ℤ/8)	3
#79199	5

Subgroups — C5×#79199 #14416652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#79199 #14416652