Subgroups — C₅×#78883 #14412710

Order 1920.

1452 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#1010	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#907	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#5504	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#417	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#5233	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#5120	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#810	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#4802	C₂	1	2	yes	maximal, complemented
9	C₅×D₃×#103	C₂	1	2	yes	maximal, complemented
10	C₅×#4772	C₂	1	2	yes	maximal, complemented
11	C₅×#4771	C₂	1	2	yes	maximal, complemented
12	C₅×#2898	—	3	1	no	maximal, Hall
13	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
14	C₅×#1010	—	2	4	no	—
15	C₅×#1010	C₄	1	8	yes	complemented
16	C₂×C₅×D₃×Q₈	C₂²	1	2	yes	—
17	C₂×C₅×#237	—	2	2	no	—
18	C₂×C₅×#237	C₂²	1	2	yes	—
19	C₂×C₅×#235	—	2	1	no	—
20	C₂×C₅×#235	C₂²	1	1	yes	characteristic
21	C₂²×C₅×Dic₆	—	2	1	no	—
22	C₂²×C₅×Dic₆	C₂²	1	1	yes	characteristic
23	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
24	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₄×C₅×#67	—	2	2	no	—
26	C₅×#926	C₂²	1	1	yes	characteristic
27	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
28	C₅×#907	—	2	4	no	—
29	C₂×C₅×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	—
30	C₂×C₅×D₃×C₈	C₂²	1	2	yes	—
31	C₅×#893	—	2	1	no	—
32	C₅×#892	C₂²	1	2	yes	—
33	C₅×D₃×C₄⋊C₄	C₂²	1	2	yes	—
34	C₅×#887	—	2	1	no	—
35	C₃×C₅×#103	C₂²	1	2	yes	—
36	C₃×C₅×#102	C₂²	1	2	yes	—
37	C₅×#835	C₂²	1	1	yes	characteristic
38	C₅×#835	C₂²	1	1	yes	characteristic
39	C₅×#832	C₂²	1	1	yes	characteristic
40	C₅×#832	C₂²	1	1	yes	characteristic
41	C₅×#816	C₂²	1	2	yes	—
42	C₅×#816	C₂²	1	2	yes	—
43	C₅×#810	C₂²	1	2	yes	—
44	C₅×#810	C₂²	1	2	yes	—
45	#78883	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
46	C₂×C₅×2₋¹⁺⁴	—	3	1	no	—
47	C₂×C₅×#131	—	3	1	no	—
48	C₅×#520	—	3	1	no	—
49	C₅×#419	—	3	2	no	—
50	C₅×#419	—	3	1	no	—
51	C₅×#417	—	3	2	no	—
52	C₅×#417	D₃	1	1	yes	characteristic, complemented
53	C₅×#416	—	3	2	no	—
54	C₅×#416	—	3	1	no	—
55	C₂×C₅×#103	—	3	2	no	—
56	C₂×C₅×#103	—	3	1	no	—
57	C₃×C₅×#41	—	2	2	no	—
58	C₃×C₅×#41	C₂×C₄	1	4	yes	complemented
59	C₂×C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
60	C₂×C₃×C₅×D₄	—	2	1	no	—
61	C₂×C₃×C₅×D₄	C₂³	1	1	yes	characteristic
62	C₂²×C₃×C₄×C₅	—	2	1	no	—
63	C₂²×C₃×C₄×C₅	—	2	1	no	—
64	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
65	C₂×C₅×#67	—	2	2	no	—
66	C₂×C₅×#67	D₄	1	2	yes	—
67	C₂²×C₅×Dic₃	—	2	1	no	—
68	C₂²×C₅×Dic₃	—	2	2	no	—
69	C₅×D₃×Q₈	—	2	4	no	—
70	C₅×D₃×Q₈	C₂×C₄	1	8	yes	complemented
71	C₅×#237	—	2	16	no	—
72	C₅×#237	—	2	4	no	—
73	C₅×#237	C₂×C₄	1	8	yes	complemented
74	C₅×#235	—	2	8	no	—
75	C₅×#235	C₂×C₄	1	4	yes	—
76	C₅×#235	D₄	1	4	yes	—
77	C₂×C₅×D₁₂	D₄	1	1	yes	characteristic
78	C₂×C₄×C₅×D₃	—	2	2	no	—
79	C₂×C₄×C₅×D₃	—	2	2	no	—
80	C₂×C₄×C₅×D₃	C₂³	1	2	yes	—
81	C₂×C₅×Dic₆	—	2	1	no	—
82	C₂×C₅×Dic₆	—	2	2	no	—
83	C₂×C₅×Dic₆	—	2	2	no	—
84	C₂×C₅×Dic₆	—	2	8	no	—
85	C₂×C₅×Dic₆	C₂×C₄	1	4	yes	complemented
86	C₂×C₅×Dic₆	D₄	1	1	yes	characteristic
87	C₂×C₅×Dic₆	C₂³	1	2	yes	—
88	C₃×C₅×M₁₆	—	2	2	no	—
89	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
90	C₃×C₅×C₄⋊C₄	—	2	1	no	—
91	C₃×C₅×C₄⋊C₄	C₂³	1	2	yes	—
92	C₅×#217	—	2	1	no	—
93	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
94	C₅×C₄⋊Dic₃	C₂³	1	1	yes	characteristic
95	C₅×C₄⋊Dic₃	C₂³	1	1	yes	characteristic
96	C₅×Dic₃⋊C₄	—	2	2	no	—
97	C₄×C₅×Dic₃	—	2	1	no	—
98	C₄×C₅×Dic₃	—	2	2	no	—
99	C₂×C₅×C₃⋊C₈	—	2	2	no	—
100	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
101	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
102	C₅×ΓL(1,𝔽₂₅)	—	2	4	no	—
103	C₅×D₃×C₈	—	2	4	no	—
104	C₂×#1010	C₂×C₅	1	1	yes	characteristic
105	C₂×#907	C₂×C₅	1	1	yes	characteristic, complemented
106	#5504	C₂×C₅	1	1	yes	characteristic, complemented
107	C₃×#417	C₂×C₅	1	1	yes	characteristic, complemented
108	#5233	C₂×C₅	1	1	yes	characteristic, complemented
109	#5120	C₂×C₅	1	1	yes	characteristic, complemented
110	C₂×#810	C₂×C₅	1	1	yes	characteristic, complemented
111	#4802	C₂×C₅	1	2	yes	complemented
112	D₃×#103	C₂×C₅	1	2	yes	complemented
113	#4772	C₂×C₅	1	2	yes	complemented
114	#4771	C₂×C₅	1	2	yes	complemented
115	C₅×2₋¹⁺⁴	—	3	8	no	—
116	C₅×2₋¹⁺⁴	—	6	4	no	—
117	C₂×C₅×#41	—	3	1	no	—
118	C₂×C₅×#41	—	3	2	no	—
119	C₂×C₅×#41	—	6	1	no	—
120	C₂×C₅×#41	—	6	2	no	—
121	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic
122	C₂²×C₅×Q₈	—	3	2	no	—
123	C₂²×C₅×Q₈	—	3	1	no	—
124	C₂²×C₅×Q₈	—	6	1	no	—
125	C₅×#131	—	6	4	no	—
126	C₂×C₅×M₁₆	—	3	2	no	—
127	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
128	C₂²×C₅×C₈	—	3	1	no	—
129	C₂²×C₅×C₈	—	3	2	no	—
130	C₄×C₅×Q₈	—	6	1	no	—
131	C₄×C₅×D₄	—	6	1	no	—
132	C₄×C₅×D₄	—	6	2	no	—
133	C₅×#117	—	3	1	no	—
134	C₅×#117	—	3	2	no	—
135	C₂×C₅×C₄⋊C₄	—	3	2	no	—
136	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
137	C₅×#103	—	3	1	no	—
138	C₅×#103	—	3	1	no	—
139	C₅×#103	—	3	2	no	—
140	C₅×#103	—	3	2	no	—
141	C₅×#103	—	3	2	no	—
142	C₅×#103	—	3	2	no	—
143	C₅×#103	C₂×D₃	1	2	yes	—
144	C₅×#102	—	3	1	no	—
145	C₅×#102	—	3	1	no	—
146	C₅×#102	C₂×D₃	1	2	yes	—
147	#2898	—	3	1	no	Sylow
148	C₂³×C₃×C₅	—	2	1	no	—
149	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₅×D₃	C₂×D₄	1	2	yes	—
151	C₃×C₅×Q₈	—	2	1	no	—
152	C₃×C₅×Q₈	C₂²×C₄	1	2	yes	—
153	C₃×C₅×D₄	—	2	4	no	—
154	C₃×C₅×D₄	—	2	1	no	—
155	C₃×C₅×D₄	C₂²×C₄	1	2	yes	—
156	C₂×C₃×C₄×C₅	—	2	1	no	—
157	C₂×C₃×C₄×C₅	—	2	4	no	—
158	C₂×C₃×C₄×C₅	—	2	2	no	—
159	C₂×C₃×C₄×C₅	—	8	1	no	—
160	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
161	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
162	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
163	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
164	C₅×#67	—	2	8	no	—
165	C₅×#67	C₂²⋊C₄	1	8	yes	—
166	C₂×C₅×Dic₃	—	2	4	no	—
167	C₂×C₅×Dic₃	—	2	2	no	—
168	C₂×C₅×Dic₃	—	2	8	no	—
169	C₂×C₅×Dic₃	—	2	1	no	—
170	C₂×C₅×Dic₃	—	2	1	no	—
171	C₂×C₅×Dic₃	C₂×D₄	1	2	yes	—
172	C₅×D₁₂	C₂²⋊C₄	1	4	yes	—
173	C₄×C₅×D₃	—	2	8	no	—
174	C₄×C₅×D₃	—	8	2	no	—
175	C₄×C₅×D₃	C₂²×C₄	1	4	yes	—
176	C₄×C₅×D₃	C₂×D₄	1	4	yes	—
177	C₅×Dic₆	—	2	4	no	—
178	C₅×Dic₆	—	2	2	no	—
179	C₅×Dic₆	—	2	8	no	—
180	C₅×Dic₆	C₂²⋊C₄	1	4	yes	—
181	C₅×Dic₆	C₂²×C₄	1	4	yes	—
182	C₃×C₅×C₈	—	2	2	no	—
183	C₅×C₃⋊C₈	—	2	1	no	—
184	C₅×C₃⋊C₈	—	2	1	no	—
185	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
186	#1010	—	2	4	no	—
187	#1010	C₄×C₅	1	8	yes	complemented
188	C₂×D₃×Q₈	C₂²×C₅	1	2	yes	—
189	C₂×#237	—	2	2	no	—
190	C₂×#237	C₂²×C₅	1	2	yes	—
191	C₂×#235	—	2	1	no	—
192	C₂×#235	C₂²×C₅	1	1	yes	characteristic
193	C₂²×Dic₆	—	2	1	no	—
194	C₂²×Dic₆	C₂²×C₅	1	1	yes	characteristic
195	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
196	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
197	C₄×#67	—	2	2	no	—
198	#926	C₂²×C₅	1	1	yes	characteristic
199	C₂²×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
200	#907	—	2	4	no	—
201	C₂×ΓL(1,𝔽₂₅)	C₂²×C₅	1	2	yes	—
202	C₂×D₃×C₈	C₂²×C₅	1	2	yes	—
203	#893	—	2	1	no	—
204	#892	C₂²×C₅	1	2	yes	—
205	D₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
206	#887	—	2	1	no	—
207	C₃×#103	C₂²×C₅	1	2	yes	—
208	C₃×#102	C₂²×C₅	1	2	yes	—
209	#835	C₂²×C₅	1	1	yes	characteristic
210	#835	C₂²×C₅	1	1	yes	characteristic
211	#832	C₂²×C₅	1	1	yes	characteristic
212	#832	C₂²×C₅	1	1	yes	characteristic
213	#816	C₂²×C₅	1	2	yes	—
214	#816	C₂²×C₅	1	2	yes	—
215	#810	C₂²×C₅	1	2	yes	—
216	#810	C₂²×C₅	1	2	yes	—
217	C₅×#41	—	2	2	no	—
218	C₅×#41	—	3	4	no	—
219	C₅×#41	—	3	4	no	—
220	C₅×#41	—	3	8	no	—
221	C₅×#41	—	6	8	no	—
222	C₅×#41	—	6	4	no	—
223	C₅×#41	—	6	16	no	—
224	C₅×#41	C₄×D₃	1	4	yes	complemented
225	C₂×C₅×Q₈	—	3	1	no	—
226	C₂×C₅×Q₈	—	3	8	no	—
227	C₂×C₅×Q₈	—	3	4	no	—
228	C₂×C₅×Q₈	—	3	2	no	—
229	C₂×C₅×Q₈	—	6	2	no	—
230	C₂×C₅×Q₈	—	6	8	no	—
231	C₂×C₅×Q₈	—	6	4	no	—
232	C₂×C₅×Q₈	—	6	1	no	—
233	C₂×C₅×Q₈	—	6	2	no	—
234	C₂×C₅×Q₈	C₂²×D₃	1	1	yes	characteristic
235	C₂×C₅×D₄	—	2	1	no	—
236	C₂×C₅×D₄	—	3	2	no	—
237	C₂×C₅×D₄	—	3	1	no	—
238	C₂×C₅×D₄	—	6	2	no	—
239	C₂×C₅×D₄	C₂²×D₃	1	1	yes	characteristic
240	C₂²×C₄×C₅	—	2	1	no	—
241	C₂²×C₄×C₅	—	2	1	no	—
242	C₂²×C₄×C₅	—	3	2	no	—
243	C₂²×C₄×C₅	—	6	2	no	—
244	C₂²×C₄×C₅	—	6	2	no	—
245	C₂²×C₄×C₅	—	6	2	no	—
246	C₂²×C₄×C₅	—	6	1	no	—
247	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
248	C₅×M₁₆	—	2	2	no	—
249	C₅×M₁₆	—	6	4	no	—
250	C₂×C₅×C₈	—	3	1	no	—
251	C₂×C₅×C₈	—	3	1	no	—
252	C₂×C₅×C₈	—	6	4	no	—
253	C₂×C₅×C₈	—	6	2	no	—
254	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
255	C₅×C₄⋊C₄	—	2	1	no	—
256	C₅×C₄⋊C₄	—	3	1	no	—
257	C₅×C₄⋊C₄	—	3	1	no	—
258	C₅×C₄⋊C₄	—	6	2	no	—
259	C₅×C₄⋊C₄	C₂²×D₃	1	2	yes	—
260	C₅×C₂²⋊C₄	—	6	2	no	—
261	C₅×C₂²⋊C₄	—	6	1	no	—
262	C₄²×C₅	—	6	2	no	—
263	C₄²×C₅	—	6	1	no	—
264	C₂×2₋¹⁺⁴	—	3	1	no	—
265	C₂×#131	—	3	1	no	—
266	#520	—	3	1	no	—
267	#419	—	3	1	no	—
268	#419	—	3	2	no	—
269	#417	—	3	2	no	—
270	#417	C₅×D₃	1	1	yes	characteristic, complemented
271	#416	—	3	1	no	—
272	#416	—	3	2	no	—
273	C₂×#103	—	3	1	no	—
274	C₂×#103	—	3	2	no	—
275	C₂²×C₃×C₅	—	2	1	no	—
276	C₂²×C₃×C₅	—	2	1	no	—
277	C₂²×C₃×C₅	—	2	2	no	—
278	C₂²×C₃×C₅	—	4	1	no	—
279	C₂²×C₃×C₅	—	4	1	no	—
280	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	2	yes	—
281	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
282	C₂×C₅×D₃	—	2	2	no	—
283	C₂×C₅×D₃	—	2	2	no	—
284	C₂×C₅×D₃	C₂×C₂²⋊C₄	1	4	yes	—
285	C₃×C₄×C₅	—	2	2	no	—
286	C₃×C₄×C₅	—	4	2	no	—
287	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
288	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
289	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
290	C₅×Dic₃	—	2	4	no	—
291	C₅×Dic₃	—	4	1	no	—
292	C₅×Dic₃	—	4	1	no	—
293	C₅×Dic₃	C₂×C₂²⋊C₄	1	4	yes	—
294	C₃×#41	—	2	2	no	—
295	C₃×#41	C₂×C₄×C₅	1	4	yes	complemented
296	C₂×C₃×Q₈	C₂³×C₅	1	1	yes	characteristic
297	C₂×C₃×D₄	—	2	1	no	—
298	C₂×C₃×D₄	C₂³×C₅	1	1	yes	characteristic
299	C₂²×C₃×C₄	—	2	1	no	—
300	C₂²×C₃×C₄	—	2	1	no	—
301	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
302	C₂×#67	—	2	2	no	—
303	C₂×#67	C₅×D₄	1	2	yes	—
304	C₂²×Dic₃	—	2	1	no	—
305	C₂²×Dic₃	—	2	2	no	—
306	D₃×Q₈	—	2	4	no	—
307	D₃×Q₈	C₂×C₄×C₅	1	8	yes	complemented
308	#237	—	2	16	no	—
309	#237	—	2	4	no	—
310	#237	C₂×C₄×C₅	1	8	yes	complemented
311	#235	—	2	8	no	—
312	#235	C₂×C₄×C₅	1	4	yes	—
313	#235	C₅×D₄	1	4	yes	—
314	C₂×D₁₂	C₅×D₄	1	1	yes	characteristic
315	C₂×C₄×D₃	—	2	2	no	—
316	C₂×C₄×D₃	—	2	2	no	—
317	C₂×C₄×D₃	C₂³×C₅	1	2	yes	—
318	C₂×Dic₆	—	2	1	no	—
319	C₂×Dic₆	—	2	2	no	—
320	C₂×Dic₆	—	2	2	no	—
321	C₂×Dic₆	—	2	8	no	—
322	C₂×Dic₆	C₂×C₄×C₅	1	4	yes	complemented
323	C₂×Dic₆	C₅×D₄	1	1	yes	characteristic
324	C₂×Dic₆	C₂³×C₅	1	2	yes	—
325	C₃×M₁₆	—	2	2	no	—
326	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
327	C₃×C₄⋊C₄	—	2	1	no	—
328	C₃×C₄⋊C₄	C₂³×C₅	1	2	yes	—
329	#217	—	2	1	no	—
330	(C₂×D₃)⋊C₄	—	2	2	no	—
331	C₄⋊Dic₃	C₂³×C₅	1	1	yes	characteristic
332	C₄⋊Dic₃	C₂³×C₅	1	1	yes	characteristic
333	Dic₃⋊C₄	—	2	2	no	—
334	C₄×Dic₃	—	2	1	no	—
335	C₄×Dic₃	—	2	2	no	—
336	C₂×C₃⋊C₈	—	2	2	no	—
337	C₂×C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
338	C₂×C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
339	ΓL(1,𝔽₂₅)	—	2	4	no	—
340	D₃×C₈	—	2	4	no	—
341	C₂³×C₅	—	2	1	no	—
342	C₂³×C₅	—	3	2	no	—
343	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
344	C₅×Q₈	—	2	1	no	—
345	C₅×Q₈	—	3	4	no	—
346	C₅×Q₈	—	3	4	no	—
347	C₅×Q₈	—	6	8	no	—
348	C₅×Q₈	—	6	4	no	—
349	C₅×Q₈	—	6	2	no	—
350	C₅×Q₈	C₂×C₄×D₃	1	2	yes	—
351	C₅×D₄	—	2	4	no	—
352	C₅×D₄	—	2	1	no	—
353	C₅×D₄	—	3	8	no	—
354	C₅×D₄	—	3	4	no	—
355	C₅×D₄	—	6	8	no	—
356	C₅×D₄	C₂×C₄×D₃	1	2	yes	—
357	C₂×C₄×C₅	—	2	1	no	—
358	C₂×C₄×C₅	—	2	4	no	—
359	C₂×C₄×C₅	—	2	2	no	—
360	C₂×C₄×C₅	—	3	2	no	—
361	C₂×C₄×C₅	—	3	4	no	—
362	C₂×C₄×C₅	—	3	4	no	—
363	C₂×C₄×C₅	—	6	8	no	—
364	C₂×C₄×C₅	—	6	8	no	—
365	C₂×C₄×C₅	—	6	4	no	—
366	C₂×C₄×C₅	—	6	2	no	—
367	C₂×C₄×C₅	—	6	1	no	—
368	C₂×C₄×C₅	—	6	1	no	—
369	C₂×C₄×C₅	—	8	1	no	—
370	C₂×C₄×C₅	—	24	2	no	—
371	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
372	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
373	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
374	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
375	C₅×C₈	—	2	2	no	—
376	C₅×C₈	—	6	1	no	—
377	C₅×C₈	—	6	1	no	—
378	2₋¹⁺⁴	—	3	8	no	—
379	2₋¹⁺⁴	—	6	4	no	—
380	C₂×#41	—	3	1	no	—
381	C₂×#41	—	3	2	no	—
382	C₂×#41	—	6	1	no	—
383	C₂×#41	—	6	2	no	—
384	C₂×#41	C₂×C₅×D₃	1	1	yes	characteristic
385	C₂²×Q₈	—	3	2	no	—
386	C₂²×Q₈	—	3	1	no	—
387	C₂²×Q₈	—	6	1	no	—
388	#131	—	6	4	no	—
389	C₂×M₁₆	—	3	2	no	—
390	C₂×M₁₆	C₂×C₅×D₃	1	1	yes	characteristic
391	C₂²×C₈	—	3	1	no	—
392	C₂²×C₈	—	3	2	no	—
393	C₄×Q₈	—	6	1	no	—
394	C₄×D₄	—	6	1	no	—
395	C₄×D₄	—	6	2	no	—
396	#117	—	3	1	no	—
397	#117	—	3	2	no	—
398	C₂×C₄⋊C₄	—	3	2	no	—
399	C₂×C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
400	#103	—	3	1	no	—
401	#103	—	3	1	no	—
402	#103	—	3	2	no	—
403	#103	—	3	2	no	—
404	#103	—	3	2	no	—
405	#103	—	3	2	no	—
406	#103	C₂×C₅×D₃	1	2	yes	—
407	#102	—	3	1	no	—
408	#102	—	3	1	no	—
409	#102	C₂×C₅×D₃	1	2	yes	—
410	C₂×C₃×C₅	—	2	2	no	—
411	C₂×C₃×C₅	—	4	2	no	—
412	C₂×C₃×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
413	C₂×C₃×C₅	#577	1	1	yes	characteristic
414	C₂×C₃×C₅	#578	1	1	yes	characteristic
415	C₅×D₃	—	2	4	no	—
416	C₂³×C₃	—	2	1	no	—
417	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
418	C₂²×D₃	C₂×C₅×D₄	1	2	yes	—
419	C₃×Q₈	—	2	1	no	—
420	C₃×Q₈	C₂²×C₄×C₅	1	2	yes	—
421	C₃×D₄	—	2	4	no	—
422	C₃×D₄	—	2	1	no	—
423	C₃×D₄	C₂²×C₄×C₅	1	2	yes	—
424	C₂×C₃×C₄	—	2	1	no	—
425	C₂×C₃×C₄	—	2	4	no	—
426	C₂×C₃×C₄	—	2	2	no	—
427	C₂×C₃×C₄	—	8	1	no	—
428	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
429	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
430	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
431	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
432	#67	—	2	8	no	—
433	#67	C₅×C₂²⋊C₄	1	8	yes	—
434	C₂×Dic₃	—	2	4	no	—
435	C₂×Dic₃	—	2	2	no	—
436	C₂×Dic₃	—	2	8	no	—
437	C₂×Dic₃	—	2	1	no	—
438	C₂×Dic₃	—	2	1	no	—
439	C₂×Dic₃	C₂×C₅×D₄	1	2	yes	—
440	D₁₂	C₅×C₂²⋊C₄	1	4	yes	—
441	C₄×D₃	—	2	8	no	—
442	C₄×D₃	—	8	2	no	—
443	C₄×D₃	C₂²×C₄×C₅	1	4	yes	—
444	C₄×D₃	C₂×C₅×D₄	1	4	yes	—
445	Dic₆	—	2	4	no	—
446	Dic₆	—	2	2	no	—
447	Dic₆	—	2	8	no	—
448	Dic₆	C₅×C₂²⋊C₄	1	4	yes	—
449	Dic₆	C₂²×C₄×C₅	1	4	yes	—
450	C₃×C₈	—	2	2	no	—
451	C₃⋊C₈	—	2	1	no	—
452	C₃⋊C₈	—	2	1	no	—
453	C₂²×C₅	—	2	1	no	—
454	C₂²×C₅	—	2	1	no	—
455	C₂²×C₅	—	2	2	no	—
456	C₂²×C₅	—	3	4	no	—
457	C₂²×C₅	—	4	1	no	—
458	C₂²×C₅	—	4	1	no	—
459	C₂²×C₅	—	6	2	no	—
460	C₂²×C₅	—	6	2	no	—
461	C₂²×C₅	D₃×C₂²⋊C₄	1	2	yes	—
462	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
463	C₄×C₅	—	2	2	no	—
464	C₄×C₅	—	3	4	no	—
465	C₄×C₅	—	4	2	no	—
466	C₄×C₅	—	6	4	no	—
467	C₄×C₅	—	12	1	no	—
468	C₄×C₅	—	12	1	no	—
469	C₄×C₅	D₃×C₂²⋊C₄	1	2	yes	—
470	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
471	C₄×C₅	C₂×D₃×D₄	1	1	yes	characteristic
472	#41	—	2	2	no	—
473	#41	—	3	4	no	—
474	#41	—	3	4	no	—
475	#41	—	3	8	no	—
476	#41	—	6	8	no	—
477	#41	—	6	4	no	—
478	#41	—	6	16	no	—
479	#41	C₄×C₅×D₃	1	4	yes	complemented
480	C₂×Q₈	—	3	1	no	—
481	C₂×Q₈	—	3	8	no	—
482	C₂×Q₈	—	3	4	no	—
483	C₂×Q₈	—	3	2	no	—
484	C₂×Q₈	—	6	2	no	—
485	C₂×Q₈	—	6	8	no	—
486	C₂×Q₈	—	6	4	no	—
487	C₂×Q₈	—	6	1	no	—
488	C₂×Q₈	—	6	2	no	—
489	C₂×Q₈	C₂²×C₅×D₃	1	1	yes	characteristic
490	C₂×D₄	—	2	1	no	—
491	C₂×D₄	—	3	2	no	—
492	C₂×D₄	—	3	1	no	—
493	C₂×D₄	—	6	2	no	—
494	C₂×D₄	C₂²×C₅×D₃	1	1	yes	characteristic
495	C₂²×C₄	—	2	1	no	—
496	C₂²×C₄	—	2	1	no	—
497	C₂²×C₄	—	3	2	no	—
498	C₂²×C₄	—	6	2	no	—
499	C₂²×C₄	—	6	2	no	—
500	C₂²×C₄	—	6	2	no	—
501	C₂²×C₄	—	6	1	no	—
502	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
503	M₁₆	—	2	2	no	—
504	M₁₆	—	6	4	no	—
505	C₂×C₈	—	3	1	no	—
506	C₂×C₈	—	3	1	no	—
507	C₂×C₈	—	6	4	no	—
508	C₂×C₈	—	6	2	no	—
509	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
510	C₄⋊C₄	—	2	1	no	—
511	C₄⋊C₄	—	3	1	no	—
512	C₄⋊C₄	—	3	1	no	—
513	C₄⋊C₄	—	6	2	no	—
514	C₄⋊C₄	C₂²×C₅×D₃	1	2	yes	—
515	C₂²⋊C₄	—	6	2	no	—
516	C₂²⋊C₄	—	6	1	no	—
517	C₄²	—	6	2	no	—
518	C₄²	—	6	1	no	—
519	C₃×C₅	#2898	1	1	yes	characteristic, complemented, Hall
520	C₂²×C₃	—	2	1	no	—
521	C₂²×C₃	—	2	1	no	—
522	C₂²×C₃	—	2	2	no	—
523	C₂²×C₃	—	4	1	no	—
524	C₂²×C₃	—	4	1	no	—
525	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	2	yes	—
526	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
527	C₂×D₃	—	2	2	no	—
528	C₂×D₃	—	2	2	no	—
529	C₂×D₃	C₂×C₅×C₂²⋊C₄	1	4	yes	—
530	C₃×C₄	—	2	2	no	—
531	C₃×C₄	—	4	2	no	—
532	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
533	C₃×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
534	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
535	Dic₃	—	2	4	no	—
536	Dic₃	—	4	1	no	—
537	Dic₃	—	4	1	no	—
538	Dic₃	C₂×C₅×C₂²⋊C₄	1	4	yes	—
539	C₂×C₅	—	2	2	no	—
540	C₂×C₅	—	4	2	no	—
541	C₂×C₅	—	6	4	no	—
542	C₂×C₅	C₂×D₃×C₂²⋊C₄	1	1	yes	characteristic
543	C₂×C₅	#5775	1	1	yes	characteristic
544	C₂×C₅	#5779	1	1	yes	characteristic
545	C₂³	—	2	1	no	—
546	C₂³	—	3	2	no	—
547	C₂³	C₅×D₃×D₄	1	1	yes	characteristic
548	Q₈	—	2	1	no	—
549	Q₈	—	3	4	no	—
550	Q₈	—	3	4	no	—
551	Q₈	—	6	8	no	—
552	Q₈	—	6	4	no	—
553	Q₈	—	6	2	no	—
554	Q₈	C₂×C₄×C₅×D₃	1	2	yes	—
555	D₄	—	2	4	no	—
556	D₄	—	2	1	no	—
557	D₄	—	3	8	no	—
558	D₄	—	3	4	no	—
559	D₄	—	6	8	no	—
560	D₄	C₂×C₄×C₅×D₃	1	2	yes	—
561	C₂×C₄	—	2	1	no	—
562	C₂×C₄	—	2	4	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	3	2	no	—
565	C₂×C₄	—	3	4	no	—
566	C₂×C₄	—	3	4	no	—
567	C₂×C₄	—	6	8	no	—
568	C₂×C₄	—	6	8	no	—
569	C₂×C₄	—	6	4	no	—
570	C₂×C₄	—	6	2	no	—
571	C₂×C₄	—	6	1	no	—
572	C₂×C₄	—	6	1	no	—
573	C₂×C₄	—	8	1	no	—
574	C₂×C₄	—	24	2	no	—
575	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
576	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
577	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
578	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
579	C₈	—	2	2	no	—
580	C₈	—	6	1	no	—
581	C₈	—	6	1	no	—
582	C₂×C₃	—	2	2	no	—
583	C₂×C₃	—	4	2	no	—
584	C₂×C₃	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
585	C₂×C₃	C₅×#577	1	1	yes	characteristic
586	C₂×C₃	C₅×#578	1	1	yes	characteristic
587	D₃	—	2	4	no	—
588	C₅	#78883	1	1	yes	characteristic, direct product factor, Sylow
589	C₂²	—	2	1	no	—
590	C₂²	—	2	1	no	—
591	C₂²	—	2	2	no	—
592	C₂²	—	3	4	no	—
593	C₂²	—	4	1	no	—
594	C₂²	—	4	1	no	—
595	C₂²	—	6	2	no	—
596	C₂²	—	6	2	no	—
597	C₂²	C₅×D₃×C₂²⋊C₄	1	2	yes	—
598	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
599	C₄	—	2	2	no	—
600	C₄	—	3	4	no	—
601	C₄	—	4	2	no	—
602	C₄	—	6	4	no	—
603	C₄	—	12	1	no	—
604	C₄	—	12	1	no	—
605	C₄	C₅×D₃×C₂²⋊C₄	1	2	yes	—
606	C₄	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
607	C₄	C₂×C₅×D₃×D₄	1	1	yes	characteristic
608	C₃	C₅×#2898	1	1	yes	characteristic, complemented, Sylow
609	C₂	—	2	2	no	—
610	C₂	—	4	2	no	—
611	C₂	—	6	4	no	—
612	C₂	C₂×C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic
613	C₂	C₅×#5775	1	1	yes	characteristic
614	C₂	C₅×#5779	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×#417	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂×C₅×#1010	C₂
cosocle	C₂×C₃×C₄	C₂⁴×C₅
hypercenter	C₅×#417	D₃
hypocenter	C₃	C₅×#2898
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#417

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#907	C₂	semidirect
3. C₅×#5504	C₂	semidirect
4. C₃×C₅×#417	C₂	semidirect
5. C₅×#5233	C₂	semidirect
6. C₅×#5120	C₂	semidirect
7. C₂×C₅×#810	C₂	semidirect
8. C₅×#4802	C₂	semidirect
9. C₅×D₃×#103	C₂	semidirect
10. C₅×#4772	C₂	semidirect
11. C₅×#4771	C₂	semidirect
15. C₅×#1010	C₄	semidirect
45. #78883	C₅	direct (N × H)
52. C₅×#417	D₃	semidirect
58. C₃×C₅×#41	C₂×C₄	semidirect
70. C₅×D₃×Q₈	C₂×C₄	semidirect
73. C₅×#237	C₂×C₄	semidirect
85. C₂×C₅×Dic₆	C₂×C₄	semidirect
105. C₂×#907	C₂×C₅	semidirect
106. #5504	C₂×C₅	semidirect
107. C₃×#417	C₂×C₅	semidirect
108. #5233	C₂×C₅	semidirect
109. #5120	C₂×C₅	semidirect
110. C₂×#810	C₂×C₅	semidirect
111. #4802	C₂×C₅	semidirect
112. D₃×#103	C₂×C₅	semidirect
113. #4772	C₂×C₅	semidirect
114. #4771	C₂×C₅	semidirect
187. #1010	C₄×C₅	semidirect
224. C₅×#41	C₄×D₃	semidirect
270. #417	C₅×D₃	semidirect
295. C₃×#41	C₂×C₄×C₅	semidirect
307. D₃×Q₈	C₂×C₄×C₅	semidirect
310. #237	C₂×C₄×C₅	semidirect
322. C₂×Dic₆	C₂×C₄×C₅	semidirect
479. #41	C₄×C₅×D₃	semidirect
519. C₃×C₅	#2898	semidirect
588. C₅	#78883	direct (N × H)
608. C₃	C₅×#2898	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#1010	2
C₂×C₅×#907	2
C₅×#5504	2
C₃×C₅×#417	2
C₅×#5233	2
C₅×#5120	2
C₂×C₅×#810	2
C₅×#4802	2
C₅×D₃×#103	2
C₅×#4772	2
C₅×#4771	2
C₅×#2898	3
#78883	5

Subgroups — C5×#78883 #14412710

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#78883 #14412710