Subgroups — C₅×#81614 #14459946

Order 1920.

772 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5732	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#5726	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#487	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5090	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#5087	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4815	C₂	1	2	yes	maximal, complemented
7	C₅×#4813	C₂	1	2	yes	maximal, complemented
8	C₅×#4808	C₂	1	2	yes	maximal, complemented
9	C₅×#4804	C₂	1	2	yes	maximal, complemented
10	C₄×C₅×#204	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×C₈×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#3302	—	3	1	no	maximal, Hall
13	C₂×C₅×#239	C₂²	1	2	yes	—
14	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₅×#947	—	2	1	no	—
17	C₅×#946	C₂²	1	2	yes	—
18	C₅×Q₈×Dic₃	—	2	1	no	—
19	C₂×C₅×#216	C₂²	1	2	yes	—
20	C₂×C₅×#204	C₂²	1	1	yes	characteristic
21	C₂×C₅×D₃×C₈	C₂²	1	2	yes	—
22	C₅×#898	—	2	2	no	—
23	C₅×#897	C₂²	1	2	yes	—
24	C₅×#895	—	2	1	no	—
25	C₅×#894	—	2	2	no	—
26	C₅×#892	C₂²	1	2	yes	—
27	C₅×D₃×C₄⋊C₄	—	2	1	no	—
28	C₄×C₅×D₁₂	C₂²	1	1	yes	characteristic
29	C₄×C₅×D₁₂	C₂²	1	1	yes	characteristic
30	C₄×C₅×Dic₆	C₂²	1	1	yes	characteristic
31	C₃×C₅×#103	C₂²	1	2	yes	—
32	C₃×C₅×#103	C₂²	1	2	yes	—
33	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
34	C₅×#835	C₂²	1	2	yes	—
35	C₅×#821	C₂²	1	1	yes	characteristic
36	C₅×(C₂×D₃)⋊C₈	C₂²	1	2	yes	—
37	C₅×#817	C₂²	1	1	yes	characteristic
38	C₅×#816	C₂²	1	1	yes	characteristic
39	C₅×#809	C₂²	1	2	yes	—
40	C₅×#807	C₂²	1	2	yes	—
41	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
42	#81614	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
43	C₅×#556	—	3	1	no	—
44	C₅×#547	—	3	1	no	—
45	C₅×#487	D₃	1	1	yes	characteristic, complemented
46	C₅×#485	—	3	2	no	—
47	C₅×#474	—	3	1	no	—
48	C₅×#467	—	3	2	no	—
49	C₅×#462	—	3	1	no	—
50	C₅×#452	—	3	2	no	—
51	C₄×C₅×SD₁₆	—	3	1	no	—
52	C₅×C₈×D₄	—	3	1	no	—
53	C₅×#416	—	3	2	no	—
54	C₂×C₃×C₅×Q₈	—	2	1	no	—
55	C₂×C₃×C₅×Q₈	C₂³	1	2	yes	—
56	C₅×#239	—	2	4	no	—
57	C₅×#239	—	4	2	no	—
58	C₂×C₅×D₁₂	—	2	2	no	—
59	C₂×C₅×D₁₂	D₄	1	1	yes	characteristic
60	C₂×C₅×D₁₂	C₂³	1	1	yes	characteristic
61	C₂×C₄×C₅×D₃	—	2	1	no	—
62	C₂×C₄×C₅×D₃	—	2	2	no	—
63	C₂×C₄×C₅×D₃	C₂³	1	2	yes	—
64	C₂×C₅×Dic₆	C₂³	1	1	yes	characteristic
65	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
66	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
67	C₃×C₅×C₄⋊C₄	—	2	1	no	—
68	C₃×C₅×C₄⋊C₄	—	2	1	no	—
69	C₃×C₅×C₄⋊C₄	—	2	1	no	—
70	C₃×C₅×C₄⋊C₄	C₂³	1	2	yes	—
71	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
72	C₅×#216	—	4	2	no	—
73	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
74	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
75	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
76	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
77	C₅×(C₂×D₃)⋊C₄	D₄	1	2	yes	—
78	C₅×C₄⋊Dic₃	—	2	2	no	—
79	C₅×C₄⋊Dic₃	D₄	1	1	yes	characteristic
80	C₅×C₄⋊Dic₃	C₂³	1	1	yes	characteristic
81	C₅×C₄⋊Dic₃	C₂³	1	1	yes	characteristic
82	C₅×Dic₃⋊C₄	—	2	1	no	—
83	C₅×Dic₃⋊C₄	—	2	2	no	—
84	C₄×C₅×Dic₃	—	2	1	no	—
85	C₄×C₅×Dic₃	—	2	2	no	—
86	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
87	C₅×#204	—	2	2	no	—
88	C₅×D₃×C₈	—	4	2	no	—
89	#5732	C₂×C₅	1	1	yes	characteristic, complemented
90	#5726	C₂×C₅	1	1	yes	characteristic
91	C₃×#487	C₂×C₅	1	1	yes	characteristic, complemented
92	#5090	C₂×C₅	1	1	yes	characteristic, complemented
93	#5087	C₂×C₅	1	1	yes	characteristic, complemented
94	#4815	C₂×C₅	1	2	yes	complemented
95	#4813	C₂×C₅	1	2	yes	complemented
96	#4808	C₂×C₅	1	2	yes	complemented
97	#4804	C₂×C₅	1	2	yes	complemented
98	C₄×#204	C₂×C₅	1	1	yes	characteristic, complemented
99	C₈×D₁₂	C₂×C₅	1	1	yes	characteristic, complemented
100	C₂×C₅×#41	—	3	2	no	—
101	C₂×C₅×Q₁₆	—	3	2	no	—
102	C₂×C₅×SD₁₆	—	3	1	no	—
103	C₂²×C₅×C₈	—	3	2	no	—
104	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
105	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
106	C₅×#126	—	6	2	no	—
107	C₅×#124	—	6	1	no	—
108	C₅×#123	—	6	2	no	—
109	C₅×C₂²⋊Q₈	—	3	2	no	—
110	C₅×C₂²⋊Q₈	—	3	2	no	—
111	C₅×#121	—	6	1	no	—
112	C₄×C₅×Q₈	—	3	1	no	—
113	C₄×C₅×Q₈	—	6	1	no	—
114	C₄×C₅×D₄	—	3	1	no	—
115	C₄×C₅×D₄	—	3	1	no	—
116	C₅×#117	—	3	2	no	—
117	C₂×C₅×C₄⋊C₄	—	6	1	no	—
118	C₅×#107	—	3	2	no	—
119	C₅×#106	—	3	1	no	—
120	C₅×C₄⋊C₈	—	3	1	no	—
121	C₅×#103	—	3	2	no	—
122	C₅×#103	—	3	1	no	—
123	C₅×#103	C₂×D₃	1	2	yes	—
124	C₅×#103	C₂×D₃	1	2	yes	—
125	C₅×#102	—	3	1	no	—
126	C₅×#102	—	3	2	no	—
127	C₅×C₂²⋊C₈	—	3	2	no	—
128	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
129	#3302	—	3	1	no	Sylow
130	C₂²×C₅×D₃	—	2	1	no	—
131	C₂²×C₅×D₃	C₂×D₄	1	2	yes	—
132	C₃×C₅×Q₈	—	2	2	no	—
133	C₃×C₅×Q₈	—	2	2	no	—
134	C₃×C₅×Q₈	—	4	2	no	—
135	C₂×C₃×C₄×C₅	—	2	2	no	—
136	C₂×C₃×C₄×C₅	—	2	2	no	—
137	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
138	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
139	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
140	C₂×C₅×Dic₃	—	2	1	no	—
141	C₂×C₅×Dic₃	—	2	1	no	—
142	C₂×C₅×Dic₃	—	2	1	no	—
143	C₂×C₅×Dic₃	C₂×D₄	1	2	yes	—
144	C₅×D₁₂	—	2	2	no	—
145	C₅×D₁₂	—	2	1	no	—
146	C₅×D₁₂	—	4	4	no	—
147	C₅×D₁₂	#41	1	2	yes	—
148	C₄×C₅×D₃	—	2	2	no	—
149	C₄×C₅×D₃	—	2	2	no	—
150	C₄×C₅×D₃	—	4	4	no	—
151	C₄×C₅×D₃	—	4	2	no	—
152	C₅×Dic₆	—	2	1	no	—
153	C₅×Dic₆	—	2	1	no	—
154	C₃×C₅×C₈	—	2	1	no	—
155	C₃×C₅×C₈	—	2	1	no	—
156	C₅×C₃⋊C₈	—	2	2	no	—
157	C₂×#239	C₂²×C₅	1	2	yes	—
158	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
159	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
160	#947	—	2	1	no	—
161	#946	C₂²×C₅	1	2	yes	—
162	Q₈×Dic₃	—	2	1	no	—
163	C₂×#216	C₂²×C₅	1	2	yes	—
164	C₂×#204	C₂²×C₅	1	1	yes	characteristic
165	C₂×D₃×C₈	C₂²×C₅	1	2	yes	—
166	#898	—	2	2	no	—
167	#897	C₂²×C₅	1	2	yes	—
168	#895	—	2	1	no	—
169	#894	—	2	2	no	—
170	#892	C₂²×C₅	1	2	yes	—
171	D₃×C₄⋊C₄	—	2	1	no	—
172	C₄×D₁₂	C₂²×C₅	1	1	yes	characteristic
173	C₄×D₁₂	C₂²×C₅	1	1	yes	characteristic
174	C₄×Dic₆	C₂²×C₅	1	1	yes	characteristic
175	C₃×#103	C₂²×C₅	1	2	yes	—
176	C₃×#103	C₂²×C₅	1	2	yes	—
177	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
178	#835	C₂²×C₅	1	2	yes	—
179	#821	C₂²×C₅	1	1	yes	characteristic
180	(C₂×D₃)⋊C₈	C₂²×C₅	1	2	yes	—
181	#817	C₂²×C₅	1	1	yes	characteristic
182	#816	C₂²×C₅	1	1	yes	characteristic
183	#809	C₂²×C₅	1	2	yes	—
184	#807	C₂²×C₅	1	2	yes	—
185	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
186	C₅×#41	—	6	4	no	—
187	C₅×#41	—	12	2	no	—
188	C₂×C₅×Q₈	—	2	1	no	—
189	C₂×C₅×Q₈	—	3	1	no	—
190	C₂×C₅×Q₈	C₂²×D₃	1	2	yes	—
191	C₂×C₅×D₄	—	3	1	no	—
192	C₂×C₅×D₄	—	3	1	no	—
193	C₂×C₅×D₄	—	6	2	no	—
194	C₂²×C₄×C₅	—	3	2	no	—
195	C₂²×C₄×C₅	—	6	2	no	—
196	C₂²×C₄×C₅	—	6	1	no	—
197	C₅×Q₁₆	—	12	2	no	—
198	C₅×SD₁₆	—	6	2	no	—
199	C₂×C₅×C₈	—	3	2	no	—
200	C₂×C₅×C₈	—	12	2	no	—
201	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
202	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
203	C₅×C₄⋊C₄	—	2	1	no	—
204	C₅×C₄⋊C₄	—	2	1	no	—
205	C₅×C₄⋊C₄	—	2	1	no	—
206	C₅×C₄⋊C₄	—	3	1	no	—
207	C₅×C₄⋊C₄	—	3	1	no	—
208	C₅×C₄⋊C₄	—	3	1	no	—
209	C₅×C₄⋊C₄	—	6	2	no	—
210	C₅×C₄⋊C₄	—	6	2	no	—
211	C₅×C₄⋊C₄	—	6	1	no	—
212	C₅×C₄⋊C₄	C₂²×D₃	1	2	yes	—
213	C₅×C₂²⋊C₄	—	3	2	no	—
214	C₅×C₂²⋊C₄	—	6	2	no	—
215	C₅×C₂²⋊C₄	—	6	2	no	—
216	C₅×C₂²⋊C₄	—	6	2	no	—
217	C₅×C₂²⋊C₄	—	6	1	no	—
218	C₄²×C₅	—	6	2	no	—
219	C₄²×C₅	—	6	1	no	—
220	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
221	#556	—	3	1	no	—
222	#547	—	3	1	no	—
223	#487	C₅×D₃	1	1	yes	characteristic, complemented
224	#485	—	3	2	no	—
225	#474	—	3	1	no	—
226	#467	—	3	2	no	—
227	#462	—	3	1	no	—
228	#452	—	3	2	no	—
229	C₄×SD₁₆	—	3	1	no	—
230	C₈×D₄	—	3	1	no	—
231	#416	—	3	2	no	—
232	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
233	C₂×C₅×D₃	—	2	2	no	—
234	C₂×C₅×D₃	—	2	2	no	—
235	C₂×C₅×D₃	—	2	2	no	—
236	C₂×C₅×D₃	—	2	2	no	—
237	C₂×C₅×D₃	—	4	1	no	—
238	C₂×C₅×D₃	—	4	1	no	—
239	C₃×C₄×C₅	—	2	1	no	—
240	C₃×C₄×C₅	—	4	2	no	—
241	C₃×C₄×C₅	—	4	2	no	—
242	C₃×C₄×C₅	#135	1	2	yes	—
243	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
244	C₃×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
245	C₅×Dic₃	—	2	2	no	—
246	C₅×Dic₃	—	4	1	no	—
247	C₅×Dic₃	—	4	1	no	—
248	C₅×Dic₃	—	4	1	no	—
249	C₂×C₃×Q₈	—	2	1	no	—
250	C₂×C₃×Q₈	C₂³×C₅	1	2	yes	—
251	#239	—	2	4	no	—
252	#239	—	4	2	no	—
253	C₂×D₁₂	—	2	2	no	—
254	C₂×D₁₂	C₅×D₄	1	1	yes	characteristic
255	C₂×D₁₂	C₂³×C₅	1	1	yes	characteristic
256	C₂×C₄×D₃	—	2	1	no	—
257	C₂×C₄×D₃	—	2	2	no	—
258	C₂×C₄×D₃	C₂³×C₅	1	2	yes	—
259	C₂×Dic₆	C₂³×C₅	1	1	yes	characteristic
260	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
261	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
262	C₃×C₄⋊C₄	—	2	1	no	—
263	C₃×C₄⋊C₄	—	2	1	no	—
264	C₃×C₄⋊C₄	—	2	1	no	—
265	C₃×C₄⋊C₄	C₂³×C₅	1	2	yes	—
266	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
267	#216	—	4	2	no	—
268	(C₂×D₃)⋊C₄	—	2	2	no	—
269	(C₂×D₃)⋊C₄	—	2	2	no	—
270	(C₂×D₃)⋊C₄	—	2	2	no	—
271	(C₂×D₃)⋊C₄	—	2	1	no	—
272	(C₂×D₃)⋊C₄	C₅×D₄	1	2	yes	—
273	C₄⋊Dic₃	—	2	2	no	—
274	C₄⋊Dic₃	C₅×D₄	1	1	yes	characteristic
275	C₄⋊Dic₃	C₂³×C₅	1	1	yes	characteristic
276	C₄⋊Dic₃	C₂³×C₅	1	1	yes	characteristic
277	Dic₃⋊C₄	—	2	1	no	—
278	Dic₃⋊C₄	—	2	2	no	—
279	C₄×Dic₃	—	2	1	no	—
280	C₄×Dic₃	—	2	2	no	—
281	C₂×C₃⋊C₈	C₂³×C₅	1	2	yes	—
282	#204	—	2	2	no	—
283	D₃×C₈	—	4	2	no	—
284	C₂³×C₅	—	3	2	no	—
285	C₂³×C₅	—	6	1	no	—
286	C₅×Q₈	—	2	2	no	—
287	C₅×Q₈	—	2	2	no	—
288	C₅×Q₈	—	4	2	no	—
289	C₅×Q₈	—	6	1	no	—
290	C₅×Q₈	—	6	1	no	—
291	C₅×D₄	—	3	2	no	—
292	C₅×D₄	—	6	2	no	—
293	C₅×D₄	—	6	1	no	—
294	C₅×D₄	—	12	4	no	—
295	C₂×C₄×C₅	—	2	2	no	—
296	C₂×C₄×C₅	—	2	2	no	—
297	C₂×C₄×C₅	—	3	2	no	—
298	C₂×C₄×C₅	—	6	2	no	—
299	C₂×C₄×C₅	—	6	2	no	—
300	C₂×C₄×C₅	—	6	1	no	—
301	C₂×C₄×C₅	—	6	1	no	—
302	C₂×C₄×C₅	—	6	1	no	—
303	C₂×C₄×C₅	—	12	4	no	—
304	C₂×C₄×C₅	—	12	2	no	—
305	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
306	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
307	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
308	C₅×C₈	—	2	1	no	—
309	C₅×C₈	—	2	1	no	—
310	C₅×C₈	—	6	2	no	—
311	C₂×#41	—	3	2	no	—
312	C₂×Q₁₆	—	3	2	no	—
313	C₂×SD₁₆	—	3	1	no	—
314	C₂²×C₈	—	3	2	no	—
315	C₄⋊Q₈	C₂×C₅×D₃	1	1	yes	characteristic
316	C₄⋊Q₈	C₂×C₅×D₃	1	1	yes	characteristic
317	#126	—	6	2	no	—
318	#124	—	6	1	no	—
319	#123	—	6	2	no	—
320	C₂²⋊Q₈	—	3	2	no	—
321	C₂²⋊Q₈	—	3	2	no	—
322	#121	—	6	1	no	—
323	C₄×Q₈	—	3	1	no	—
324	C₄×Q₈	—	6	1	no	—
325	C₄×D₄	—	3	1	no	—
326	C₄×D₄	—	3	1	no	—
327	#117	—	3	2	no	—
328	C₂×C₄⋊C₄	—	6	1	no	—
329	#107	—	3	2	no	—
330	#106	—	3	1	no	—
331	C₄⋊C₈	—	3	1	no	—
332	#103	—	3	2	no	—
333	#103	—	3	1	no	—
334	#103	C₂×C₅×D₃	1	2	yes	—
335	#103	C₂×C₅×D₃	1	2	yes	—
336	#102	—	3	1	no	—
337	#102	—	3	2	no	—
338	C₂²⋊C₈	—	3	2	no	—
339	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
340	C₂×C₃×C₅	#546	1	1	yes	characteristic
341	C₂×C₃×C₅	C₂×#135	1	1	yes	characteristic
342	C₂×C₃×C₅	#578	1	1	yes	characteristic
343	C₅×D₃	—	4	2	no	—
344	C₅×D₃	—	4	2	no	—
345	C₂²×D₃	—	2	1	no	—
346	C₂²×D₃	C₂×C₅×D₄	1	2	yes	—
347	C₃×Q₈	—	2	2	no	—
348	C₃×Q₈	—	2	2	no	—
349	C₃×Q₈	—	4	2	no	—
350	C₂×C₃×C₄	—	2	2	no	—
351	C₂×C₃×C₄	—	2	2	no	—
352	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
353	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
354	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
355	C₂×Dic₃	—	2	1	no	—
356	C₂×Dic₃	—	2	1	no	—
357	C₂×Dic₃	—	2	1	no	—
358	C₂×Dic₃	C₂×C₅×D₄	1	2	yes	—
359	D₁₂	—	2	2	no	—
360	D₁₂	—	2	1	no	—
361	D₁₂	—	4	4	no	—
362	D₁₂	C₅×#41	1	2	yes	—
363	C₄×D₃	—	2	2	no	—
364	C₄×D₃	—	2	2	no	—
365	C₄×D₃	—	4	4	no	—
366	C₄×D₃	—	4	2	no	—
367	Dic₆	—	2	1	no	—
368	Dic₆	—	2	1	no	—
369	C₃×C₈	—	2	1	no	—
370	C₃×C₈	—	2	1	no	—
371	C₃⋊C₈	—	2	2	no	—
372	C₂²×C₅	—	6	2	no	—
373	C₂²×C₅	—	6	2	no	—
374	C₂²×C₅	—	6	2	no	—
375	C₂²×C₅	—	6	2	no	—
376	C₂²×C₅	—	12	1	no	—
377	C₂²×C₅	—	12	1	no	—
378	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
379	C₄×C₅	—	2	1	no	—
380	C₄×C₅	—	4	2	no	—
381	C₄×C₅	—	4	2	no	—
382	C₄×C₅	—	6	2	no	—
383	C₄×C₅	—	12	1	no	—
384	C₄×C₅	—	12	1	no	—
385	C₄×C₅	—	12	1	no	—
386	C₄×C₅	#916	1	2	yes	—
387	C₄×C₅	D₃×#41	1	1	yes	characteristic
388	C₄×C₅	#1009	1	1	yes	characteristic
389	#41	—	6	4	no	—
390	#41	—	12	2	no	—
391	C₂×Q₈	—	2	1	no	—
392	C₂×Q₈	—	3	1	no	—
393	C₂×Q₈	C₂²×C₅×D₃	1	2	yes	—
394	C₂×D₄	—	3	1	no	—
395	C₂×D₄	—	3	1	no	—
396	C₂×D₄	—	6	2	no	—
397	C₂²×C₄	—	3	2	no	—
398	C₂²×C₄	—	6	2	no	—
399	C₂²×C₄	—	6	1	no	—
400	Q₁₆	—	12	2	no	—
401	SD₁₆	—	6	2	no	—
402	C₂×C₈	—	3	2	no	—
403	C₂×C₈	—	12	2	no	—
404	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
405	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
406	C₄⋊C₄	—	2	1	no	—
407	C₄⋊C₄	—	2	1	no	—
408	C₄⋊C₄	—	2	1	no	—
409	C₄⋊C₄	—	3	1	no	—
410	C₄⋊C₄	—	3	1	no	—
411	C₄⋊C₄	—	3	1	no	—
412	C₄⋊C₄	—	6	2	no	—
413	C₄⋊C₄	—	6	2	no	—
414	C₄⋊C₄	—	6	1	no	—
415	C₄⋊C₄	C₂²×C₅×D₃	1	2	yes	—
416	C₂²⋊C₄	—	3	2	no	—
417	C₂²⋊C₄	—	6	2	no	—
418	C₂²⋊C₄	—	6	2	no	—
419	C₂²⋊C₄	—	6	2	no	—
420	C₂²⋊C₄	—	6	1	no	—
421	C₄²	—	6	2	no	—
422	C₄²	—	6	1	no	—
423	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
424	C₃×C₅	#3302	1	1	yes	characteristic, complemented, Hall
425	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
426	C₂×D₃	—	2	2	no	—
427	C₂×D₃	—	2	2	no	—
428	C₂×D₃	—	2	2	no	—
429	C₂×D₃	—	2	2	no	—
430	C₂×D₃	—	4	1	no	—
431	C₂×D₃	—	4	1	no	—
432	C₃×C₄	—	2	1	no	—
433	C₃×C₄	—	4	2	no	—
434	C₃×C₄	—	4	2	no	—
435	C₃×C₄	C₅×#135	1	2	yes	—
436	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
437	C₃×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
438	Dic₃	—	2	2	no	—
439	Dic₃	—	4	1	no	—
440	Dic₃	—	4	1	no	—
441	Dic₃	—	4	1	no	—
442	C₂×C₅	—	12	2	no	—
443	C₂×C₅	—	12	2	no	—
444	C₂×C₅	#5676	1	1	yes	characteristic
445	C₂×C₅	C₂×#916	1	1	yes	characteristic
446	C₂×C₅	#5766	1	1	yes	characteristic
447	C₂³	—	3	2	no	—
448	C₂³	—	6	1	no	—
449	Q₈	—	2	2	no	—
450	Q₈	—	2	2	no	—
451	Q₈	—	4	2	no	—
452	Q₈	—	6	1	no	—
453	Q₈	—	6	1	no	—
454	D₄	—	3	2	no	—
455	D₄	—	6	2	no	—
456	D₄	—	6	1	no	—
457	D₄	—	12	4	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	3	2	no	—
461	C₂×C₄	—	6	2	no	—
462	C₂×C₄	—	6	2	no	—
463	C₂×C₄	—	6	1	no	—
464	C₂×C₄	—	6	1	no	—
465	C₂×C₄	—	6	1	no	—
466	C₂×C₄	—	12	4	no	—
467	C₂×C₄	—	12	2	no	—
468	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
469	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
470	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
471	C₈	—	2	1	no	—
472	C₈	—	2	1	no	—
473	C₈	—	6	2	no	—
474	C₂×C₃	C₅×#546	1	1	yes	characteristic
475	C₂×C₃	C₂×C₅×#135	1	1	yes	characteristic
476	C₂×C₃	C₅×#578	1	1	yes	characteristic
477	D₃	—	4	2	no	—
478	D₃	—	4	2	no	—
479	C₅	#81614	1	1	yes	characteristic, direct product factor, Sylow
480	C₂²	—	6	2	no	—
481	C₂²	—	6	2	no	—
482	C₂²	—	6	2	no	—
483	C₂²	—	6	2	no	—
484	C₂²	—	12	1	no	—
485	C₂²	—	12	1	no	—
486	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
487	C₄	—	2	1	no	—
488	C₄	—	4	2	no	—
489	C₄	—	4	2	no	—
490	C₄	—	6	2	no	—
491	C₄	—	12	1	no	—
492	C₄	—	12	1	no	—
493	C₄	—	12	1	no	—
494	C₄	C₅×#916	1	2	yes	—
495	C₄	C₅×D₃×#41	1	1	yes	characteristic
496	C₄	C₅×#1009	1	1	yes	characteristic
497	C₃	C₅×#3302	1	1	yes	characteristic, complemented, Sylow
498	C₂	—	12	2	no	—
499	C₂	—	12	2	no	—
500	C₂	C₅×#5676	1	1	yes	characteristic
501	C₂	C₂×C₅×#916	1	1	yes	characteristic
502	C₂	C₅×#5766	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×#487	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₅×#5726	C₂
hypercenter	C₅×#487	D₃
hypocenter	C₃	C₅×#3302
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#487

Complements

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

N	H ≅ G/N	type
1. C₅×#5732	C₂	semidirect
3. C₃×C₅×#487	C₂	semidirect
4. C₅×#5090	C₂	semidirect
5. C₅×#5087	C₂	semidirect
6. C₅×#4815	C₂	semidirect
7. C₅×#4813	C₂	semidirect
8. C₅×#4808	C₂	semidirect
9. C₅×#4804	C₂	semidirect
10. C₄×C₅×#204	C₂	semidirect
11. C₅×C₈×D₁₂	C₂	semidirect
42. #81614	C₅	direct (N × H)
45. C₅×#487	D₃	semidirect
89. #5732	C₂×C₅	semidirect
91. C₃×#487	C₂×C₅	semidirect
92. #5090	C₂×C₅	semidirect
93. #5087	C₂×C₅	semidirect
94. #4815	C₂×C₅	semidirect
95. #4813	C₂×C₅	semidirect
96. #4808	C₂×C₅	semidirect
97. #4804	C₂×C₅	semidirect
98. C₄×#204	C₂×C₅	semidirect
99. C₈×D₁₂	C₂×C₅	semidirect
223. #487	C₅×D₃	semidirect
424. C₃×C₅	#3302	semidirect
479. C₅	#81614	direct (N × H)
497. C₃	C₅×#3302	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5732	2
C₅×#5726	2
C₃×C₅×#487	2
C₅×#5090	2
C₅×#5087	2
C₅×#4815	2
C₅×#4813	2
C₅×#4808	2
C₅×#4804	2
C₄×C₅×#204	2
C₅×C₈×D₁₂	2
C₅×#3302	3
#81614	5

Subgroups — C5×#81614 #14459946

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#81614 #14459946