Subgroups — C₅×#80941 #14458170

Order 1920.

952 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	C₅×#5774	C₂	1	2	yes	maximal, complemented
2	C₅×#5757	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#5752	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#472	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#5198	C₂	1	2	yes	maximal, complemented
6	C₅×#5151	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#5144	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#4897	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#4895	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#4749	C₂	1	2	yes	maximal, complemented
11	C₅×#4747	C₂	1	2	yes	maximal, complemented
12	C₅×#3292	—	3	1	no	maximal, Hall
13	C₅×#1009	C₂²	1	2	yes	complemented
14	C₅×#1004	C₂²	1	2	yes	complemented
15	C₃×C₅×AGL(1,ℤ/8)	C₂²	1	2	yes	complemented
16	C₂×C₃×C₅×D₈	C₂²	1	1	yes	characteristic, complemented
17	C₃×C₅×#131	C₂²	1	1	yes	characteristic, complemented
18	C₅×#950	C₂²	1	1	yes	characteristic
19	C₅×#940	C₂²	1	2	yes	complemented
20	C₅×#932	C₂²	1	2	yes	complemented
21	C₂×C₅×#213	C₂²	1	2	yes	complemented
22	C₅×#916	—	2	2	no	—
23	C₅×#914	—	2	2	no	—
24	C₅×#912	—	2	1	no	—
25	C₅×#911	—	2	2	no	—
26	C₅×#911	—	2	2	no	—
27	C₅×D₃×D₈	—	2	2	no	—
28	C₅×D₃×D₈	—	2	1	no	—
29	C₅×#908	—	2	2	no	—
30	C₅×#908	C₂²	1	2	yes	complemented
31	C₅×#908	C₂²	1	2	yes	complemented
32	C₅×#907	C₂²	1	2	yes	complemented
33	C₅×#904	—	2	1	no	—
34	C₅×#904	C₂²	1	1	yes	characteristic
35	C₂×C₅×D₂₄	—	2	1	no	—
36	C₂×C₅×D₂₄	C₂²	1	1	yes	characteristic, complemented
37	C₅×#901	C₂²	1	1	yes	characteristic, complemented
38	C₃×C₅×#108	C₂²	1	1	yes	characteristic, complemented
39	C₃×C₅×#100	C₂²	1	2	yes	complemented
40	C₅×#837	C₂²	1	1	yes	characteristic, complemented
41	C₅×#825	C₂²	1	2	yes	complemented
42	C₅×#825	C₂²	1	1	yes	characteristic, complemented
43	C₅×#823	C₂²	1	2	yes	complemented
44	C₅×#822	C₂²	1	1	yes	characteristic, complemented
45	C₅×C₈×Dic₃	C₂²	1	1	yes	characteristic, complemented
46	#80941	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
47	C₅×#576	—	3	2	no	—
48	C₅×#576	—	3	1	no	—
49	C₅×#576	—	3	1	no	—
50	C₅×#493	—	3	1	no	—
51	C₅×#472	—	3	1	no	—
52	C₅×#472	—	3	2	no	—
53	C₅×#472	D₃	1	1	yes	characteristic, complemented
54	C₅×#453	—	3	2	no	—
55	C₅×#453	—	3	2	no	—
56	C₅×#443	—	3	1	no	—
57	C₅×#443	—	3	1	no	—
58	C₃×C₅×#41	C₂³	1	1	yes	characteristic
59	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
60	C₂×C₅×#67	—	4	2	no	—
61	C₂×C₅×#67	—	4	2	no	—
62	C₅×#239	—	2	2	no	—
63	C₅×#237	—	2	2	no	—
64	C₅×#237	—	2	2	no	—
65	C₅×D₃×D₄	—	2	2	no	—
66	C₅×D₃×D₄	—	2	2	no	—
67	C₅×D₃×D₄	—	2	2	no	—
68	C₅×D₃×D₄	—	4	2	no	—
69	C₅×#235	—	2	2	no	—
70	C₅×#235	C₂³	1	1	yes	characteristic
71	C₅×#235	C₂³	1	2	yes	—
72	C₂×C₅×D₁₂	—	2	2	no	—
73	C₂×C₅×D₁₂	C₂³	1	2	yes	—
74	C₃×C₅×SD₁₆	—	2	2	no	—
75	C₃×C₅×D₈	—	2	2	no	—
76	C₃×C₅×D₈	—	2	1	no	—
77	C₃×C₅×D₈	—	2	1	no	—
78	C₃×C₅×M₁₆	—	2	1	no	—
79	C₃×C₅×M₁₆	C₂³	1	2	yes	—
80	C₃×C₅×M₁₆	C₂³	1	1	yes	characteristic
81	C₂×C₃×C₅×C₈	—	2	1	no	—
82	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
83	C₅×#216	D₄	1	1	yes	characteristic, complemented
84	C₅×#215	D₄	1	1	yes	characteristic, complemented
85	C₅×#214	—	2	2	no	—
86	C₅×#214	D₄	1	1	yes	characteristic, complemented
87	C₅×#213	—	2	2	no	—
88	C₅×#213	—	2	2	no	—
89	C₅×#213	—	2	2	no	—
90	C₅×#213	D₄	1	1	yes	characteristic, complemented
91	C₄×C₅×Dic₃	C₂³	1	1	yes	characteristic
92	C₅×#208	C₂³	1	1	yes	characteristic
93	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
94	C₅×Dic₁₂	D₄	1	1	yes	characteristic, complemented
95	C₅×D₂₄	—	2	2	no	—
96	C₅×D₂₄	—	2	2	no	—
97	C₅×D₂₄	—	2	1	no	—
98	C₅×D₂₄	—	2	1	no	—
99	C₅×D₂₄	D₄	1	1	yes	characteristic, complemented
100	C₅×#204	—	2	2	no	—
101	C₅×#204	—	2	2	no	—
102	C₅×#204	D₄	1	2	yes	complemented
103	C₅×ΓL(1,𝔽₂₅)	—	2	1	no	—
104	C₅×ΓL(1,𝔽₂₅)	—	2	2	no	—
105	C₅×D₃×C₈	—	2	1	no	—
106	C₅×D₃×C₈	—	2	2	no	—
107	#5774	C₂×C₅	1	2	yes	complemented
108	#5757	C₂×C₅	1	1	yes	characteristic, complemented
109	#5752	C₂×C₅	1	1	yes	characteristic, complemented
110	C₃×#472	C₂×C₅	1	1	yes	characteristic, complemented
111	#5198	C₂×C₅	1	2	yes	complemented
112	#5151	C₂×C₅	1	1	yes	characteristic, complemented
113	#5144	C₂×C₅	1	1	yes	characteristic, complemented
114	#4897	C₂×C₅	1	1	yes	characteristic, complemented
115	#4895	C₂×C₅	1	1	yes	characteristic, complemented
116	#4749	C₂×C₅	1	2	yes	complemented
117	#4747	C₂×C₅	1	2	yes	complemented
118	C₅×H(5,𝔽₂)	—	3	2	no	—
119	C₅×H(5,𝔽₂)	—	3	2	no	—
120	C₅×AGL(1,ℤ/8)	—	3	2	no	—
121	C₅×AGL(1,ℤ/8)	—	3	2	no	—
122	C₅×AGL(1,ℤ/8)	—	3	2	no	—
123	C₅×AGL(1,ℤ/8)	—	6	2	no	—
124	C₅×AGL(1,ℤ/8)	—	6	2	no	—
125	C₅×AGL(1,ℤ/8)	—	6	2	no	—
126	C₅×AGL(1,ℤ/8)	—	6	2	no	—
127	C₅×AGL(1,ℤ/8)	C₂×D₃	1	2	yes	complemented
128	C₅×#135	—	3	1	no	—
129	C₅×#135	—	3	1	no	—
130	C₅×#135	—	6	1	no	—
131	C₅×#135	—	6	1	no	—
132	C₅×#135	—	6	2	no	—
133	C₂×C₅×D₈	—	3	2	no	—
134	C₂×C₅×D₈	—	3	1	no	—
135	C₂×C₅×D₈	—	6	1	no	—
136	C₂×C₅×D₈	—	6	2	no	—
137	C₂×C₅×D₈	—	6	1	no	—
138	C₂×C₅×D₈	C₂×D₃	1	1	yes	characteristic, complemented
139	C₅×#131	—	3	2	no	—
140	C₅×#131	—	3	1	no	—
141	C₅×#131	C₂×D₃	1	1	yes	characteristic, complemented
142	C₅×#127	—	3	2	no	—
143	C₅×#108	—	3	1	no	—
144	C₅×#108	C₂×D₃	1	1	yes	characteristic, complemented
145	C₅×#104	—	3	1	no	—
146	C₅×#104	—	3	2	no	—
147	C₅×#104	—	3	1	no	—
148	C₅×#100	—	3	2	no	—
149	C₅×#100	C₂×D₃	1	2	yes	complemented
150	C₄×C₅×C₈	—	3	1	no	—
151	#3292	—	3	1	no	Sylow
152	C₂³×C₃×C₅	—	2	2	no	—
153	C₂²×C₅×D₃	—	2	2	no	—
154	C₂²×C₅×D₃	—	4	2	no	—
155	C₂²×C₅×D₃	—	4	2	no	—
156	C₃×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
157	C₃×C₅×D₄	—	2	1	no	—
158	C₃×C₅×D₄	—	2	2	no	—
159	C₃×C₅×D₄	—	2	2	no	—
160	C₃×C₅×D₄	C₂×D₄	1	1	yes	characteristic
161	C₂×C₃×C₄×C₅	—	2	1	no	—
162	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
163	C₅×#67	—	2	2	no	—
164	C₅×#67	—	2	1	no	—
165	C₅×#67	—	4	2	no	—
166	C₅×#67	—	4	2	no	—
167	C₅×#67	—	4	2	no	—
168	C₅×#67	—	8	4	no	—
169	C₂×C₅×Dic₃	—	2	1	no	—
170	C₂×C₅×Dic₃	—	4	2	no	—
171	C₅×D₁₂	—	2	2	no	—
172	C₅×D₁₂	—	2	2	no	—
173	C₅×D₁₂	—	4	2	no	—
174	C₅×D₁₂	C₂×D₄	1	2	yes	—
175	C₅×D₁₂	C₂×D₄	1	1	yes	characteristic
176	C₄×C₅×D₃	—	2	1	no	—
177	C₄×C₅×D₃	—	2	2	no	—
178	C₄×C₅×D₃	—	4	2	no	—
179	C₅×Dic₆	C₂×D₄	1	1	yes	characteristic
180	C₅×Dic₆	C₂×D₄	1	2	yes	—
181	C₃×C₅×C₈	—	2	2	no	—
182	C₃×C₅×C₈	—	2	1	no	—
183	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
184	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
185	C₅×C₃⋊C₈	—	2	1	no	—
186	C₅×C₃⋊C₈	C₂×D₄	1	1	yes	characteristic
187	C₅×C₃⋊C₈	C₂×D₄	1	1	yes	characteristic
188	#1009	C₂²×C₅	1	2	yes	complemented
189	#1004	C₂²×C₅	1	2	yes	complemented
190	C₃×AGL(1,ℤ/8)	C₂²×C₅	1	2	yes	complemented
191	C₂×C₃×D₈	C₂²×C₅	1	1	yes	characteristic, complemented
192	C₃×#131	C₂²×C₅	1	1	yes	characteristic, complemented
193	#950	C₂²×C₅	1	1	yes	characteristic
194	#940	C₂²×C₅	1	2	yes	complemented
195	#932	C₂²×C₅	1	2	yes	complemented
196	C₂×#213	C₂²×C₅	1	2	yes	complemented
197	#916	—	2	2	no	—
198	#914	—	2	2	no	—
199	#912	—	2	1	no	—
200	#911	—	2	2	no	—
201	#911	—	2	2	no	—
202	D₃×D₈	—	2	2	no	—
203	D₃×D₈	—	2	1	no	—
204	#908	—	2	2	no	—
205	#908	C₂²×C₅	1	2	yes	complemented
206	#908	C₂²×C₅	1	2	yes	complemented
207	#907	C₂²×C₅	1	2	yes	complemented
208	#904	—	2	1	no	—
209	#904	C₂²×C₅	1	1	yes	characteristic
210	C₂×D₂₄	—	2	1	no	—
211	C₂×D₂₄	C₂²×C₅	1	1	yes	characteristic, complemented
212	#901	C₂²×C₅	1	1	yes	characteristic, complemented
213	C₃×#108	C₂²×C₅	1	1	yes	characteristic, complemented
214	C₃×#100	C₂²×C₅	1	2	yes	complemented
215	#837	C₂²×C₅	1	1	yes	characteristic, complemented
216	#825	C₂²×C₅	1	1	yes	characteristic, complemented
217	#825	C₂²×C₅	1	2	yes	complemented
218	#823	C₂²×C₅	1	2	yes	complemented
219	#822	C₂²×C₅	1	1	yes	characteristic, complemented
220	C₈×Dic₃	C₂²×C₅	1	1	yes	characteristic, complemented
221	C₅×#41	—	3	2	no	—
222	C₅×#41	—	3	1	no	—
223	C₅×#41	—	6	2	no	—
224	C₅×#41	—	6	2	no	—
225	C₅×#41	—	6	2	no	—
226	C₅×#41	—	6	2	no	—
227	C₅×#41	C₂²×D₃	1	1	yes	characteristic
228	C₂×C₅×D₄	—	3	2	no	—
229	C₂×C₅×D₄	—	6	2	no	—
230	C₂×C₅×D₄	—	6	2	no	—
231	C₂×C₅×D₄	—	6	2	no	—
232	C₂×C₅×D₄	—	6	2	no	—
233	C₂×C₅×D₄	—	12	2	no	—
234	C₂×C₅×D₄	—	12	2	no	—
235	C₂×C₅×D₄	—	12	2	no	—
236	C₂×C₅×D₄	C₂²×D₃	1	2	yes	—
237	C₅×Q₁₆	—	3	1	no	—
238	C₅×Q₁₆	—	3	1	no	—
239	C₅×SD₁₆	—	2	2	no	—
240	C₅×SD₁₆	—	3	1	no	—
241	C₅×SD₁₆	—	3	1	no	—
242	C₅×SD₁₆	—	3	2	no	—
243	C₅×SD₁₆	—	6	2	no	—
244	C₅×SD₁₆	—	6	2	no	—
245	C₅×SD₁₆	—	6	2	no	—
246	C₅×D₈	—	2	2	no	—
247	C₅×D₈	—	2	1	no	—
248	C₅×D₈	—	2	1	no	—
249	C₅×D₈	—	3	1	no	—
250	C₅×D₈	—	3	1	no	—
251	C₅×D₈	—	6	2	no	—
252	C₅×D₈	—	6	2	no	—
253	C₅×D₈	—	6	2	no	—
254	C₅×D₈	—	6	2	no	—
255	C₅×D₈	—	6	2	no	—
256	C₅×D₈	—	6	1	no	—
257	C₅×D₈	—	6	1	no	—
258	C₅×M₁₆	—	2	1	no	—
259	C₅×M₁₆	—	3	1	no	—
260	C₅×M₁₆	—	6	2	no	—
261	C₅×M₁₆	—	6	1	no	—
262	C₅×M₁₆	C₂²×D₃	1	2	yes	—
263	C₅×M₁₆	C₂²×D₃	1	1	yes	characteristic
264	C₂×C₅×C₈	—	2	1	no	—
265	C₂×C₅×C₈	—	3	1	no	—
266	C₂×C₅×C₈	—	6	2	no	—
267	C₂×C₅×C₈	—	6	1	no	—
268	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
269	C₄²×C₅	—	3	1	no	—
270	#576	—	3	2	no	—
271	#576	—	3	1	no	—
272	#576	—	3	1	no	—
273	#493	—	3	1	no	—
274	#472	—	3	1	no	—
275	#472	—	3	2	no	—
276	#472	C₅×D₃	1	1	yes	characteristic, complemented
277	#453	—	3	2	no	—
278	#453	—	3	2	no	—
279	#443	—	3	1	no	—
280	#443	—	3	1	no	—
281	C₂²×C₃×C₅	—	2	1	no	—
282	C₂²×C₃×C₅	—	4	2	no	—
283	C₂²×C₃×C₅	—	8	2	no	—
284	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
285	C₂×C₅×D₃	—	2	2	no	—
286	C₂×C₅×D₃	—	2	1	no	—
287	C₂×C₅×D₃	—	4	2	no	—
288	C₂×C₅×D₃	—	8	2	no	—
289	C₂×C₅×D₃	—	16	2	no	—
290	C₂×C₅×D₃	—	16	2	no	—
291	C₃×C₄×C₅	—	2	1	no	—
292	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
293	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
294	C₅×Dic₃	—	2	2	no	—
295	C₅×Dic₃	—	2	2	no	—
296	C₅×Dic₃	—	2	1	no	—
297	C₃×#41	C₂³×C₅	1	1	yes	characteristic
298	C₂×C₃×D₄	C₂³×C₅	1	2	yes	—
299	C₂×#67	—	4	2	no	—
300	C₂×#67	—	4	2	no	—
301	#239	—	2	2	no	—
302	#237	—	2	2	no	—
303	#237	—	2	2	no	—
304	D₃×D₄	—	2	2	no	—
305	D₃×D₄	—	2	2	no	—
306	D₃×D₄	—	2	2	no	—
307	D₃×D₄	—	4	2	no	—
308	#235	—	2	2	no	—
309	#235	C₂³×C₅	1	1	yes	characteristic
310	#235	C₂³×C₅	1	2	yes	—
311	C₂×D₁₂	—	2	2	no	—
312	C₂×D₁₂	C₂³×C₅	1	2	yes	—
313	C₃×SD₁₆	—	2	2	no	—
314	C₃×D₈	—	2	2	no	—
315	C₃×D₈	—	2	1	no	—
316	C₃×D₈	—	2	1	no	—
317	C₃×M₁₆	—	2	1	no	—
318	C₃×M₁₆	C₂³×C₅	1	1	yes	characteristic
319	C₃×M₁₆	C₂³×C₅	1	2	yes	—
320	C₂×C₃×C₈	—	2	1	no	—
321	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
322	#216	C₅×D₄	1	1	yes	characteristic, complemented
323	#215	C₅×D₄	1	1	yes	characteristic, complemented
324	#214	—	2	2	no	—
325	#214	C₅×D₄	1	1	yes	characteristic, complemented
326	#213	—	2	2	no	—
327	#213	—	2	2	no	—
328	#213	—	2	2	no	—
329	#213	C₅×D₄	1	1	yes	characteristic, complemented
330	C₄×Dic₃	C₂³×C₅	1	1	yes	characteristic
331	#208	C₂³×C₅	1	1	yes	characteristic
332	C₂×C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
333	Dic₁₂	C₅×D₄	1	1	yes	characteristic, complemented
334	D₂₄	—	2	2	no	—
335	D₂₄	—	2	2	no	—
336	D₂₄	—	2	1	no	—
337	D₂₄	—	2	1	no	—
338	D₂₄	C₅×D₄	1	1	yes	characteristic, complemented
339	#204	—	2	2	no	—
340	#204	—	2	2	no	—
341	#204	C₅×D₄	1	2	yes	complemented
342	ΓL(1,𝔽₂₅)	—	2	1	no	—
343	ΓL(1,𝔽₂₅)	—	2	2	no	—
344	D₃×C₈	—	2	1	no	—
345	D₃×C₈	—	2	2	no	—
346	C₂³×C₅	—	2	2	no	—
347	C₂³×C₅	—	6	2	no	—
348	C₂³×C₅	—	12	2	no	—
349	C₂³×C₅	—	12	2	no	—
350	C₅×Q₈	—	3	1	no	—
351	C₅×Q₈	—	3	2	no	—
352	C₅×Q₈	D₃×D₄	1	1	yes	characteristic
353	C₅×D₄	—	2	1	no	—
354	C₅×D₄	—	2	2	no	—
355	C₅×D₄	—	2	2	no	—
356	C₅×D₄	—	3	1	no	—
357	C₅×D₄	—	3	2	no	—
358	C₅×D₄	—	6	2	no	—
359	C₅×D₄	—	6	1	no	—
360	C₅×D₄	—	6	2	no	—
361	C₅×D₄	—	6	2	no	—
362	C₅×D₄	—	12	2	no	—
363	C₅×D₄	—	12	2	no	—
364	C₅×D₄	—	12	2	no	—
365	C₅×D₄	—	12	2	no	—
366	C₅×D₄	—	24	4	no	—
367	C₅×D₄	D₃×D₄	1	1	yes	characteristic
368	C₂×C₄×C₅	—	2	1	no	—
369	C₂×C₄×C₅	—	6	1	no	—
370	C₂×C₄×C₅	—	6	2	no	—
371	C₂×C₄×C₅	—	6	1	no	—
372	C₂×C₄×C₅	—	12	2	no	—
373	C₂×C₄×C₅	—	12	2	no	—
374	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
375	C₅×C₈	—	2	1	no	—
376	C₅×C₈	—	2	2	no	—
377	C₅×C₈	—	3	1	no	—
378	C₅×C₈	—	3	1	no	—
379	C₅×C₈	—	6	1	no	—
380	C₅×C₈	D₃×D₄	1	1	yes	characteristic
381	C₅×C₈	D₃×D₄	1	1	yes	characteristic
382	H(5,𝔽₂)	—	3	2	no	—
383	H(5,𝔽₂)	—	3	2	no	—
384	AGL(1,ℤ/8)	—	3	2	no	—
385	AGL(1,ℤ/8)	—	3	2	no	—
386	AGL(1,ℤ/8)	—	3	2	no	—
387	AGL(1,ℤ/8)	—	6	2	no	—
388	AGL(1,ℤ/8)	—	6	2	no	—
389	AGL(1,ℤ/8)	—	6	2	no	—
390	AGL(1,ℤ/8)	—	6	2	no	—
391	AGL(1,ℤ/8)	C₂×C₅×D₃	1	2	yes	complemented
392	#135	—	3	1	no	—
393	#135	—	3	1	no	—
394	#135	—	6	1	no	—
395	#135	—	6	1	no	—
396	#135	—	6	2	no	—
397	C₂×D₈	—	3	2	no	—
398	C₂×D₈	—	3	1	no	—
399	C₂×D₈	—	6	1	no	—
400	C₂×D₈	—	6	2	no	—
401	C₂×D₈	—	6	1	no	—
402	C₂×D₈	C₂×C₅×D₃	1	1	yes	characteristic, complemented
403	#131	—	3	2	no	—
404	#131	—	3	1	no	—
405	#131	C₂×C₅×D₃	1	1	yes	characteristic, complemented
406	#127	—	3	2	no	—
407	#108	—	3	1	no	—
408	#108	C₂×C₅×D₃	1	1	yes	characteristic, complemented
409	#104	—	3	1	no	—
410	#104	—	3	2	no	—
411	#104	—	3	1	no	—
412	#100	—	3	2	no	—
413	#100	C₂×C₅×D₃	1	2	yes	complemented
414	C₄×C₈	—	3	1	no	—
415	C₂×C₃×C₅	—	2	1	no	—
416	C₂×C₃×C₅	—	4	1	no	—
417	C₂×C₃×C₅	—	8	2	no	—
418	C₂×C₃×C₅	D₄²	1	1	yes	characteristic
419	C₅×D₃	—	4	2	no	—
420	C₅×D₃	—	4	1	no	—
421	C₅×D₃	—	8	2	no	—
422	C₂³×C₃	—	2	2	no	—
423	C₂²×D₃	—	2	2	no	—
424	C₂²×D₃	—	4	2	no	—
425	C₂²×D₃	—	4	2	no	—
426	C₃×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
427	C₃×D₄	—	2	1	no	—
428	C₃×D₄	—	2	2	no	—
429	C₃×D₄	—	2	2	no	—
430	C₃×D₄	C₂×C₅×D₄	1	1	yes	characteristic
431	C₂×C₃×C₄	—	2	1	no	—
432	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
433	#67	—	2	2	no	—
434	#67	—	2	1	no	—
435	#67	—	4	2	no	—
436	#67	—	4	2	no	—
437	#67	—	4	2	no	—
438	#67	—	8	4	no	—
439	C₂×Dic₃	—	2	1	no	—
440	C₂×Dic₃	—	4	2	no	—
441	D₁₂	—	2	2	no	—
442	D₁₂	—	2	2	no	—
443	D₁₂	—	4	2	no	—
444	D₁₂	C₂×C₅×D₄	1	2	yes	—
445	D₁₂	C₂×C₅×D₄	1	1	yes	characteristic
446	C₄×D₃	—	2	1	no	—
447	C₄×D₃	—	2	2	no	—
448	C₄×D₃	—	4	2	no	—
449	Dic₆	C₂×C₅×D₄	1	1	yes	characteristic
450	Dic₆	C₂×C₅×D₄	1	2	yes	—
451	C₃×C₈	—	2	2	no	—
452	C₃×C₈	—	2	1	no	—
453	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
454	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
455	C₃⋊C₈	—	2	1	no	—
456	C₃⋊C₈	C₂×C₅×D₄	1	1	yes	characteristic
457	C₃⋊C₈	C₂×C₅×D₄	1	1	yes	characteristic
458	C₂²×C₅	—	2	1	no	—
459	C₂²×C₅	—	4	2	no	—
460	C₂²×C₅	—	6	2	no	—
461	C₂²×C₅	—	6	1	no	—
462	C₂²×C₅	—	8	2	no	—
463	C₂²×C₅	—	12	2	no	—
464	C₂²×C₅	—	24	2	no	—
465	C₂²×C₅	—	48	2	no	—
466	C₂²×C₅	—	48	2	no	—
467	C₂²×C₅	#1009	1	1	yes	characteristic
468	C₄×C₅	—	2	1	no	—
469	C₄×C₅	—	6	2	no	—
470	C₄×C₅	—	6	2	no	—
471	C₄×C₅	—	6	1	no	—
472	C₄×C₅	C₂×D₃×D₄	1	1	yes	characteristic
473	C₄×C₅	C₂×D₃×D₄	1	1	yes	characteristic
474	#41	—	3	2	no	—
475	#41	—	3	1	no	—
476	#41	—	6	2	no	—
477	#41	—	6	2	no	—
478	#41	—	6	2	no	—
479	#41	—	6	2	no	—
480	#41	C₂²×C₅×D₃	1	1	yes	characteristic
481	C₂×D₄	—	3	2	no	—
482	C₂×D₄	—	6	2	no	—
483	C₂×D₄	—	6	2	no	—
484	C₂×D₄	—	6	2	no	—
485	C₂×D₄	—	6	2	no	—
486	C₂×D₄	—	12	2	no	—
487	C₂×D₄	—	12	2	no	—
488	C₂×D₄	—	12	2	no	—
489	C₂×D₄	C₂²×C₅×D₃	1	2	yes	—
490	Q₁₆	—	3	1	no	—
491	Q₁₆	—	3	1	no	—
492	SD₁₆	—	2	2	no	—
493	SD₁₆	—	3	1	no	—
494	SD₁₆	—	3	1	no	—
495	SD₁₆	—	3	2	no	—
496	SD₁₆	—	6	2	no	—
497	SD₁₆	—	6	2	no	—
498	SD₁₆	—	6	2	no	—
499	D₈	—	2	2	no	—
500	D₈	—	2	1	no	—
501	D₈	—	2	1	no	—
502	D₈	—	3	1	no	—
503	D₈	—	3	1	no	—
504	D₈	—	6	2	no	—
505	D₈	—	6	2	no	—
506	D₈	—	6	2	no	—
507	D₈	—	6	2	no	—
508	D₈	—	6	2	no	—
509	D₈	—	6	1	no	—
510	D₈	—	6	1	no	—
511	M₁₆	—	2	1	no	—
512	M₁₆	—	3	1	no	—
513	M₁₆	—	6	2	no	—
514	M₁₆	—	6	1	no	—
515	M₁₆	C₂²×C₅×D₃	1	2	yes	—
516	M₁₆	C₂²×C₅×D₃	1	1	yes	characteristic
517	C₂×C₈	—	2	1	no	—
518	C₂×C₈	—	3	1	no	—
519	C₂×C₈	—	6	2	no	—
520	C₂×C₈	—	6	1	no	—
521	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
522	C₄²	—	3	1	no	—
523	C₃×C₅	#3292	1	1	yes	characteristic, complemented, Hall
524	C₂²×C₃	—	2	1	no	—
525	C₂²×C₃	—	4	2	no	—
526	C₂²×C₃	—	8	2	no	—
527	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
528	C₂×D₃	—	2	2	no	—
529	C₂×D₃	—	2	1	no	—
530	C₂×D₃	—	4	2	no	—
531	C₂×D₃	—	8	2	no	—
532	C₂×D₃	—	16	2	no	—
533	C₂×D₃	—	16	2	no	—
534	C₃×C₄	—	2	1	no	—
535	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
536	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
537	Dic₃	—	2	2	no	—
538	Dic₃	—	2	2	no	—
539	Dic₃	—	2	1	no	—
540	C₂×C₅	—	2	1	no	—
541	C₂×C₅	—	4	1	no	—
542	C₂×C₅	—	8	2	no	—
543	C₂×C₅	—	12	2	no	—
544	C₂×C₅	—	12	1	no	—
545	C₂×C₅	—	24	2	no	—
546	C₂×C₅	#5609	1	1	yes	characteristic
547	C₂³	—	2	2	no	—
548	C₂³	—	6	2	no	—
549	C₂³	—	12	2	no	—
550	C₂³	—	12	2	no	—
551	Q₈	—	3	1	no	—
552	Q₈	—	3	2	no	—
553	Q₈	C₅×D₃×D₄	1	1	yes	characteristic
554	D₄	—	2	1	no	—
555	D₄	—	2	2	no	—
556	D₄	—	2	2	no	—
557	D₄	—	3	1	no	—
558	D₄	—	3	2	no	—
559	D₄	—	6	2	no	—
560	D₄	—	6	1	no	—
561	D₄	—	6	2	no	—
562	D₄	—	6	2	no	—
563	D₄	—	12	2	no	—
564	D₄	—	12	2	no	—
565	D₄	—	12	2	no	—
566	D₄	—	12	2	no	—
567	D₄	—	24	4	no	—
568	D₄	C₅×D₃×D₄	1	1	yes	characteristic
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	6	1	no	—
571	C₂×C₄	—	6	2	no	—
572	C₂×C₄	—	6	1	no	—
573	C₂×C₄	—	12	2	no	—
574	C₂×C₄	—	12	2	no	—
575	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
576	C₈	—	2	1	no	—
577	C₈	—	2	2	no	—
578	C₈	—	3	1	no	—
579	C₈	—	3	1	no	—
580	C₈	—	6	1	no	—
581	C₈	C₅×D₃×D₄	1	1	yes	characteristic
582	C₈	C₅×D₃×D₄	1	1	yes	characteristic
583	C₂×C₃	—	2	1	no	—
584	C₂×C₃	—	4	1	no	—
585	C₂×C₃	—	8	2	no	—
586	C₂×C₃	C₅×D₄²	1	1	yes	characteristic
587	D₃	—	4	2	no	—
588	D₃	—	4	1	no	—
589	D₃	—	8	2	no	—
590	C₅	#80941	1	1	yes	characteristic, direct product factor, Sylow
591	C₂²	—	2	1	no	—
592	C₂²	—	4	2	no	—
593	C₂²	—	6	2	no	—
594	C₂²	—	6	1	no	—
595	C₂²	—	8	2	no	—
596	C₂²	—	12	2	no	—
597	C₂²	—	24	2	no	—
598	C₂²	—	48	2	no	—
599	C₂²	—	48	2	no	—
600	C₂²	C₅×#1009	1	1	yes	characteristic
601	C₄	—	2	1	no	—
602	C₄	—	6	2	no	—
603	C₄	—	6	2	no	—
604	C₄	—	6	1	no	—
605	C₄	C₂×C₅×D₃×D₄	1	1	yes	characteristic
606	C₄	C₂×C₅×D₃×D₄	1	1	yes	characteristic
607	C₃	C₅×#3292	1	1	yes	characteristic, complemented, Sylow
608	C₂	—	2	1	no	—
609	C₂	—	4	1	no	—
610	C₂	—	8	2	no	—
611	C₂	—	12	2	no	—
612	C₂	—	12	1	no	—
613	C₂	—	24	2	no	—
614	C₂	C₅×#5609	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	#5609
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×#472	C₂
socle	C₂×C₃×C₅	D₄²
hypercenter	C₅×#472	D₃
hypocenter	C₃	C₅×#3292
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#472

Complements

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

N	H ≅ G/N	type
1. C₅×#5774	C₂	semidirect
2. C₅×#5757	C₂	semidirect
3. C₅×#5752	C₂	semidirect
4. C₃×C₅×#472	C₂	semidirect
5. C₅×#5198	C₂	semidirect
6. C₅×#5151	C₂	semidirect
7. C₅×#5144	C₂	semidirect
8. C₅×#4897	C₂	semidirect
9. C₅×#4895	C₂	semidirect
10. C₅×#4749	C₂	semidirect
11. C₅×#4747	C₂	semidirect
13. C₅×#1009	C₂²	semidirect
14. C₅×#1004	C₂²	semidirect
15. C₃×C₅×AGL(1,ℤ/8)	C₂²	semidirect
16. C₂×C₃×C₅×D₈	C₂²	semidirect
17. C₃×C₅×#131	C₂²	semidirect
19. C₅×#940	C₂²	semidirect
20. C₅×#932	C₂²	semidirect
21. C₂×C₅×#213	C₂²	semidirect
30. C₅×#908	C₂²	semidirect
31. C₅×#908	C₂²	semidirect
32. C₅×#907	C₂²	semidirect
36. C₂×C₅×D₂₄	C₂²	semidirect
37. C₅×#901	C₂²	semidirect
38. C₃×C₅×#108	C₂²	semidirect
39. C₃×C₅×#100	C₂²	semidirect
40. C₅×#837	C₂²	semidirect
41. C₅×#825	C₂²	semidirect
42. C₅×#825	C₂²	semidirect
43. C₅×#823	C₂²	semidirect
44. C₅×#822	C₂²	semidirect
45. C₅×C₈×Dic₃	C₂²	semidirect
46. #80941	C₅	direct (N × H)
53. C₅×#472	D₃	semidirect
83. C₅×#216	D₄	semidirect
84. C₅×#215	D₄	semidirect
86. C₅×#214	D₄	semidirect
90. C₅×#213	D₄	semidirect
94. C₅×Dic₁₂	D₄	semidirect
99. C₅×D₂₄	D₄	semidirect
102. C₅×#204	D₄	semidirect
107. #5774	C₂×C₅	semidirect
108. #5757	C₂×C₅	semidirect
109. #5752	C₂×C₅	semidirect
110. C₃×#472	C₂×C₅	semidirect
111. #5198	C₂×C₅	semidirect
112. #5151	C₂×C₅	semidirect
113. #5144	C₂×C₅	semidirect
114. #4897	C₂×C₅	semidirect
115. #4895	C₂×C₅	semidirect
116. #4749	C₂×C₅	semidirect
117. #4747	C₂×C₅	semidirect
127. C₅×AGL(1,ℤ/8)	C₂×D₃	semidirect
138. C₂×C₅×D₈	C₂×D₃	semidirect
141. C₅×#131	C₂×D₃	semidirect
144. C₅×#108	C₂×D₃	semidirect
149. C₅×#100	C₂×D₃	semidirect
188. #1009	C₂²×C₅	semidirect
189. #1004	C₂²×C₅	semidirect
190. C₃×AGL(1,ℤ/8)	C₂²×C₅	semidirect
191. C₂×C₃×D₈	C₂²×C₅	semidirect
192. C₃×#131	C₂²×C₅	semidirect
194. #940	C₂²×C₅	semidirect
195. #932	C₂²×C₅	semidirect
196. C₂×#213	C₂²×C₅	semidirect
205. #908	C₂²×C₅	semidirect
206. #908	C₂²×C₅	semidirect
207. #907	C₂²×C₅	semidirect
211. C₂×D₂₄	C₂²×C₅	semidirect
212. #901	C₂²×C₅	semidirect
213. C₃×#108	C₂²×C₅	semidirect
214. C₃×#100	C₂²×C₅	semidirect
215. #837	C₂²×C₅	semidirect
216. #825	C₂²×C₅	semidirect
217. #825	C₂²×C₅	semidirect
218. #823	C₂²×C₅	semidirect
219. #822	C₂²×C₅	semidirect
220. C₈×Dic₃	C₂²×C₅	semidirect
276. #472	C₅×D₃	semidirect
322. #216	C₅×D₄	semidirect
323. #215	C₅×D₄	semidirect
325. #214	C₅×D₄	semidirect
329. #213	C₅×D₄	semidirect
333. Dic₁₂	C₅×D₄	semidirect
338. D₂₄	C₅×D₄	semidirect
341. #204	C₅×D₄	semidirect
391. AGL(1,ℤ/8)	C₂×C₅×D₃	semidirect
402. C₂×D₈	C₂×C₅×D₃	semidirect
405. #131	C₂×C₅×D₃	semidirect
408. #108	C₂×C₅×D₃	semidirect
413. #100	C₂×C₅×D₃	semidirect
523. C₃×C₅	#3292	semidirect
590. C₅	#80941	direct (N × H)
607. C₃	C₅×#3292	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5774	2
C₅×#5757	2
C₅×#5752	2
C₃×C₅×#472	2
C₅×#5198	2
C₅×#5151	2
C₅×#5144	2
C₅×#4897	2
C₅×#4895	2
C₅×#4749	2
C₅×#4747	2
C₅×#3292	3
#80941	5

Subgroups — C5×#80941 #14458170

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#80941 #14458170