Subgroups — C₅×#10619178 #72223275

Order 3200.

3826 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#66283	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#66170	C₂	1	8	yes	maximal, complemented
4	C₅×#66169	C₂	1	8	yes	maximal, complemented
5	C₅×#66156	C₂	1	6	yes	maximal, complemented
6	C₅×#66153	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#4078	C₂	1	6	yes	maximal, complemented
8	C₅²×H(5,𝔽₂)	—	2	4	no	—
9	C₅²×H(5,𝔽₂)	C₂²	1	8	yes	complemented
10	C₂×C₅²×#41	—	2	1	no	—
11	C₂×C₅²×#41	C₂²	1	1	yes	characteristic
12	C₂×C₅²×#41	C₂²	1	3	yes	complemented
13	C₂²×C₅²×D₄	—	2	3	no	—
14	C₂²×C₅²×D₄	C₂²	1	3	yes	complemented
15	C₅×#4150	C₂²	1	4	yes	complemented
16	C₅×#4149	C₂²	1	4	yes	complemented
17	C₅×D₅×#41	—	2	4	no	—
18	C₅×#4147	—	2	1	no	—
19	C₅×#4144	—	2	3	no	—
20	C₂×C₅×#709	C₂²	1	6	yes	complemented
21	C₅×#4097	C₂²	1	12	yes	complemented
22	C₅×#4096	C₂²	1	24	yes	complemented
23	C₅×#4095	C₂²	1	12	yes	complemented
24	C₅×#4094	C₂²	1	8	yes	complemented
25	C₅×#4088	C₂²	1	4	yes	complemented
26	C₂×C₅×#687	C₂²	1	12	yes	complemented
27	C₅×#4078	C₂²	1	12	yes	complemented
28	C₅×#4078	C₂²	1	24	yes	complemented
29	C₂×C₅×#686	C₂²	1	12	yes	complemented
30	C₂×C₅×#681	C₂²	1	6	yes	complemented
31	C₅×#3585	—	5	1	no	maximal
32	#10619178	C₅	1	1	yes	maximal, characteristic, direct product factor
33	C₂⁴×C₅²	—	2	3	no	—
34	C₅²×#41	—	2	8	no	—
35	C₅²×#41	—	2	6	no	—
36	C₅²×#41	D₄	1	4	yes	—
37	C₅²×#41	C₂³	1	12	yes	—
38	C₅²×#41	C₂³	1	4	yes	—
39	C₂×C₅²×Q₈	D₄	1	1	yes	characteristic
40	C₂×C₅²×Q₈	C₂³	1	1	yes	characteristic
41	C₂×C₅²×D₄	—	2	3	no	—
42	C₂×C₅²×D₄	—	2	24	no	—
43	C₂×C₅²×D₄	—	2	3	no	—
44	C₂×C₅²×D₄	—	2	6	no	—
45	C₂×C₅²×D₄	D₄	1	3	yes	—
46	C₂×C₅²×D₄	C₂³	1	12	yes	—
47	C₂×C₅²×D₄	C₂³	1	3	yes	—
48	C₂²×C₄×C₅²	—	2	3	no	—
49	C₂²×C₄×C₅²	C₂³	1	3	yes	—
50	C₂×C₅×C₂²⋊D₅	—	2	6	no	—
51	C₅×#713	—	2	4	no	—
52	C₅×Q₈×D₅	—	2	4	no	—
53	C₅×#711	—	2	12	no	—
54	C₅×D₄×D₅	—	2	12	no	—
55	C₅×#709	—	2	6	no	—
56	C₅×#709	—	2	6	no	—
57	C₅×#709	C₂³	1	4	yes	—
58	C₅×#709	C₂³	1	12	yes	complemented
59	C₂×C₅×D₂₀	C₂³	1	6	yes	complemented
60	C₂×C₄×C₅×D₅	—	2	6	no	—
61	C₂×C₅×Dic₁₀	C₂³	1	6	yes	complemented
62	C₅×#689	C₂³	1	8	yes	complemented
63	C₅×#688	C₂³	1	8	yes	complemented
64	C₅×#687	C₂³	1	24	yes	complemented
65	C₅×#686	C₂³	1	24	yes	complemented
66	C₅×#681	C₂³	1	4	yes	—
67	C₅×#681	C₂³	1	12	yes	complemented
68	C₂×C₅×#168	C₂³	1	12	yes	complemented
69	C₅×#585	—	5	1	no	—
70	C₂×C₅×H(5,𝔽₂)	—	2	2	no	—
71	C₂×C₅×H(5,𝔽₂)	D₅	1	1	yes	characteristic, complemented
72	C₂×C₅×H(5,𝔽₂)	C₂×C₅	1	1	yes	characteristic, complemented
73	#66283	C₂×C₅	1	1	yes	characteristic, complemented
74	C₅×#577	—	5	8	no	—
75	C₅×#576	—	5	8	no	—
76	C₅×#575	—	5	6	no	—
77	C₂×C₅×AGL(1,ℤ/8)	—	5	6	no	—
78	C₅×#568	—	5	1	no	—
79	#66170	C₂×C₅	1	8	yes	complemented
80	#66169	C₂×C₅	1	8	yes	complemented
81	#66156	C₂×C₅	1	6	yes	complemented
82	#66153	C₂×C₅	1	1	yes	characteristic, complemented
83	C₂×#4078	C₂×C₅	1	6	yes	complemented
84	C₂³×C₅²	—	2	3	no	—
85	C₂³×C₅²	—	2	12	no	—
86	C₂³×C₅²	—	8	6	no	—
87	C₂³×C₅²	C₂×D₄	1	3	yes	—
88	C₂²×C₅×D₅	—	2	6	no	—
89	C₅²×Q₈	—	2	1	no	—
90	C₅²×Q₈	C₂×D₄	1	4	yes	—
91	C₅²×Q₈	C₂⁴	1	2	yes	—
92	C₅²×D₄	—	2	12	no	—
93	C₅²×D₄	—	2	12	no	—
94	C₅²×D₄	—	2	3	no	—
95	C₅²×D₄	C₂×D₄	1	12	yes	—
96	C₅²×D₄	C₂⁴	1	6	yes	—
97	C₂×C₄×C₅²	—	2	1	no	—
98	C₂×C₄×C₅²	—	2	12	no	—
99	C₂×C₄×C₅²	C₂×D₄	1	6	yes	—
100	C₂×C₄×C₅²	C₂×D₄	1	3	yes	—
101	C₂×C₄×C₅²	C₂⁴	1	6	yes	—
102	C₂×C₄×C₅²	C₂⁴	1	1	yes	characteristic
103	C₅×C₂²⋊D₅	—	2	4	no	—
104	C₅×C₂²⋊D₅	—	2	12	no	—
105	C₂×C₅×Dic₅	—	2	6	no	—
106	C₅×D₂₀	—	2	6	no	—
107	C₅×D₂₀	C₂⁴	1	4	yes	—
108	C₄×C₅×D₅	—	2	4	no	—
109	C₄×C₅×D₅	—	2	12	no	—
110	C₅×Dic₁₀	—	2	6	no	—
111	C₅×Dic₁₀	C₂⁴	1	4	yes	—
112	C₅×#168	C₂⁴	1	8	yes	—
113	C₅×2₋¹⁺⁴	—	5	4	no	—
114	C₅×2₋¹⁺⁴	—	10	1	no	—
115	C₅×H(5,𝔽₂)	—	2	16	no	—
116	C₅×H(5,𝔽₂)	—	2	4	no	—
117	C₅×H(5,𝔽₂)	—	2	4	no	—
118	C₅×H(5,𝔽₂)	—	2	16	no	—
119	C₅×H(5,𝔽₂)	—	5	4	no	—
120	C₅×H(5,𝔽₂)	—	10	3	no	—
121	C₅×H(5,𝔽₂)	C₂×D₅	1	8	yes	complemented
122	C₅×H(5,𝔽₂)	C₂²×C₅	1	8	yes	complemented
123	C₂×C₅×#41	—	2	4	no	—
124	C₂×C₅×#41	—	2	1	no	—
125	C₂×C₅×#41	—	2	1	no	—
126	C₂×C₅×#41	—	2	2	no	—
127	C₂×C₅×#41	—	2	6	no	—
128	C₂×C₅×#41	—	5	6	no	—
129	C₂×C₅×#41	—	10	4	no	—
130	C₂×C₅×#41	C₂×D₅	1	1	yes	characteristic
131	C₂×C₅×#41	C₂×D₅	1	3	yes	complemented
132	C₂×C₅×#41	C₂²×C₅	1	1	yes	characteristic
133	C₂×C₅×#41	C₂²×C₅	1	3	yes	complemented
134	C₂²×C₅×D₄	—	2	12	no	—
135	C₂²×C₅×D₄	—	2	3	no	—
136	C₂²×C₅×D₄	—	2	3	no	—
137	C₂²×C₅×D₄	—	2	6	no	—
138	C₂²×C₅×D₄	C₂×D₅	1	3	yes	complemented
139	C₂²×C₅×D₄	C₂²×C₅	1	3	yes	complemented
140	#4150	C₂²×C₅	1	4	yes	complemented
141	#4149	C₂²×C₅	1	4	yes	complemented
142	D₅×#41	—	2	4	no	—
143	#4147	—	2	1	no	—
144	#4144	—	2	3	no	—
145	C₂×#709	C₂²×C₅	1	6	yes	complemented
146	C₅×#137	—	5	12	no	—
147	C₅×#137	—	5	4	no	—
148	C₅×AGL(1,ℤ/8)	—	5	12	no	—
149	C₅×AGL(1,ℤ/8)	—	5	12	no	—
150	C₅×AGL(1,ℤ/8)	—	5	24	no	—
151	C₅×#135	—	5	24	no	—
152	C₂×C₅×SD₁₆	—	5	12	no	—
153	C₂×C₅×D₈	—	5	12	no	—
154	C₅×#131	—	5	8	no	—
155	C₂×C₅×M₁₆	—	5	6	no	—
156	#4097	C₂²×C₅	1	12	yes	complemented
157	#4096	C₂²×C₅	1	24	yes	complemented
158	#4095	C₂²×C₅	1	12	yes	complemented
159	#4094	C₂²×C₅	1	8	yes	complemented
160	#4088	C₂²×C₅	1	4	yes	complemented
161	C₂×#687	C₂²×C₅	1	12	yes	complemented
162	#4078	C₂²×C₅	1	12	yes	complemented
163	#4078	C₂²×C₅	1	24	yes	complemented
164	C₂×#686	C₂²×C₅	1	12	yes	complemented
165	C₂×#681	C₂²×C₅	1	6	yes	complemented
166	#3585	—	5	1	no	Sylow
167	C₂²×C₅²	—	2	6	no	—
168	C₂²×C₅²	—	4	3	no	—
169	C₂²×C₅²	—	4	6	no	—
170	C₂²×C₅²	—	8	12	no	—
171	C₂²×C₅²	C₂²×D₄	1	6	yes	—
172	C₂²×C₅²	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₅×D₅	—	2	4	no	—
174	C₂×C₅×D₅	—	8	6	no	—
175	C₄×C₅²	—	2	2	no	—
176	C₄×C₅²	C₂²×D₄	1	6	yes	—
177	C₄×C₅²	C₂²×D₄	1	1	yes	characteristic
178	C₄×C₅²	C₂⁵	1	1	yes	characteristic
179	C₅×Dic₅	—	2	4	no	—
180	C₂⁴×C₅	—	2	12	no	—
181	C₂⁴×C₅	—	2	3	no	—
182	C₂⁴×C₅	—	2	3	no	—
183	C₅×#41	—	2	32	no	—
184	C₅×#41	—	2	24	no	—
185	C₅×#41	—	2	8	no	—
186	C₅×#41	—	2	8	no	—
187	C₅×#41	—	2	6	no	—
188	C₅×#41	—	2	6	no	—
189	C₅×#41	—	2	8	no	—
190	C₅×#41	—	2	24	no	—
191	C₅×#41	—	2	8	no	—
192	C₅×#41	—	5	12	no	—
193	C₅×#41	—	5	4	no	—
194	C₅×#41	—	10	4	no	—
195	C₅×#41	—	10	12	no	—
196	C₅×#41	—	10	6	no	—
197	C₅×#41	—	10	6	no	—
198	C₅×#41	C₂²⋊D₅	1	4	yes	—
199	C₅×#41	C₅×D₄	1	4	yes	—
200	C₅×#41	C₂²×D₅	1	12	yes	—
201	C₅×#41	C₂²×D₅	1	4	yes	—
202	C₅×#41	C₂³×C₅	1	12	yes	—
203	C₅×#41	C₂³×C₅	1	4	yes	—
204	C₂×C₅×Q₈	—	2	2	no	—
205	C₂×C₅×Q₈	—	2	2	no	—
206	C₂×C₅×Q₈	—	5	6	no	—
207	C₂×C₅×Q₈	—	10	4	no	—
208	C₂×C₅×Q₈	C₂²⋊D₅	1	1	yes	characteristic
209	C₂×C₅×Q₈	C₅×D₄	1	1	yes	characteristic
210	C₂×C₅×Q₈	C₂²×D₅	1	1	yes	characteristic
211	C₂×C₅×Q₈	C₂³×C₅	1	1	yes	characteristic
212	C₂×C₅×D₄	—	2	12	no	—
213	C₂×C₅×D₄	—	2	12	no	—
214	C₂×C₅×D₄	—	2	96	no	—
215	C₂×C₅×D₄	—	2	24	no	—
216	C₂×C₅×D₄	—	2	3	no	—
217	C₂×C₅×D₄	—	2	3	no	—
218	C₂×C₅×D₄	—	2	24	no	—
219	C₂×C₅×D₄	—	2	24	no	—
220	C₂×C₅×D₄	—	2	3	no	—
221	C₂×C₅×D₄	—	2	6	no	—
222	C₂×C₅×D₄	—	2	3	no	—
223	C₂×C₅×D₄	—	2	6	no	—
224	C₂×C₅×D₄	—	2	6	no	—
225	C₂×C₅×D₄	—	2	6	no	—
226	C₂×C₅×D₄	—	2	24	no	—
227	C₂×C₅×D₄	—	5	6	no	—
228	C₂×C₅×D₄	—	10	6	no	—
229	C₂×C₅×D₄	—	10	12	no	—
230	C₂×C₅×D₄	C₂²⋊D₅	1	3	yes	—
231	C₂×C₅×D₄	C₅×D₄	1	3	yes	—
232	C₂×C₅×D₄	C₂²×D₅	1	12	yes	—
233	C₂×C₅×D₄	C₂²×D₅	1	3	yes	—
234	C₂×C₅×D₄	C₂³×C₅	1	3	yes	—
235	C₂×C₅×D₄	C₂³×C₅	1	12	yes	—
236	C₂²×C₄×C₅	—	2	12	no	—
237	C₂²×C₄×C₅	—	2	3	no	—
238	C₂²×C₄×C₅	—	2	3	no	—
239	C₂²×C₄×C₅	—	2	6	no	—
240	C₂²×C₄×C₅	—	10	6	no	—
241	C₂²×C₄×C₅	C₂²×D₅	1	3	yes	—
242	C₂²×C₄×C₅	C₂³×C₅	1	3	yes	—
243	C₂×C₂²⋊D₅	—	2	6	no	—
244	#713	—	2	4	no	—
245	Q₈×D₅	—	2	4	no	—
246	#711	—	2	12	no	—
247	D₄×D₅	—	2	12	no	—
248	#709	—	2	6	no	—
249	#709	—	2	6	no	—
250	#709	C₂³×C₅	1	4	yes	—
251	#709	C₂³×C₅	1	12	yes	complemented
252	C₂×D₂₀	C₂³×C₅	1	6	yes	complemented
253	C₂×C₄×D₅	—	2	6	no	—
254	C₂×Dic₁₀	C₂³×C₅	1	6	yes	complemented
255	C₅×Q₁₆	—	5	8	no	—
256	C₅×SD₁₆	—	5	8	no	—
257	C₅×SD₁₆	—	5	24	no	—
258	C₅×D₈	—	5	24	no	—
259	C₅×M₁₆	—	5	4	no	—
260	C₅×M₁₆	—	5	12	no	—
261	C₂×C₅×C₈	—	5	12	no	—
262	#689	C₂³×C₅	1	8	yes	complemented
263	#688	C₂³×C₅	1	8	yes	complemented
264	#687	C₂³×C₅	1	24	yes	complemented
265	#686	C₂³×C₅	1	24	yes	complemented
266	#681	C₂³×C₅	1	4	yes	—
267	#681	C₂³×C₅	1	12	yes	complemented
268	C₂×#168	C₂³×C₅	1	12	yes	complemented
269	#585	—	5	1	no	—
270	C₂×H(5,𝔽₂)	C₅×D₅	1	1	yes	characteristic, complemented
271	#577	—	5	8	no	—
272	#576	—	5	8	no	—
273	#575	—	5	6	no	—
274	C₂×AGL(1,ℤ/8)	—	5	6	no	—
275	#568	—	5	1	no	—
276	C₂×C₅²	—	2	1	no	—
277	C₂×C₅²	—	2	6	no	—
278	C₂×C₅²	—	4	6	no	—
279	C₂×C₅²	C₂³×D₄	1	1	yes	characteristic
280	C₅×D₅	—	4	4	no	—
281	C₂³×C₅	—	2	12	no	—
282	C₂³×C₅	—	2	48	no	—
283	C₂³×C₅	—	2	3	no	—
284	C₂³×C₅	—	2	3	no	—
285	C₂³×C₅	—	2	12	no	—
286	C₂³×C₅	—	2	12	no	—
287	C₂³×C₅	—	2	6	no	—
288	C₂³×C₅	—	8	6	no	—
289	C₂³×C₅	—	8	6	no	—
290	C₂³×C₅	—	8	24	no	—
291	C₂³×C₅	—	10	6	no	—
292	C₂³×C₅	C₂×C₂²⋊D₅	1	3	yes	—
293	C₂³×C₅	C₂×C₅×D₄	1	3	yes	—
294	C₂²×D₅	—	2	6	no	—
295	C₅×Q₈	—	2	4	no	—
296	C₅×Q₈	—	2	1	no	—
297	C₅×Q₈	—	2	1	no	—
298	C₅×Q₈	—	2	8	no	—
299	C₅×Q₈	—	2	4	no	—
300	C₅×Q₈	—	5	4	no	—
301	C₅×Q₈	—	10	6	no	—
302	C₅×Q₈	C₂×C₂²⋊D₅	1	4	yes	—
303	C₅×Q₈	C₂×C₅×D₄	1	4	yes	—
304	C₅×Q₈	C₂³×D₅	1	2	yes	—
305	C₅×Q₈	C₂⁴×C₅	1	2	yes	—
306	C₅×D₄	—	2	48	no	—
307	C₅×D₄	—	2	48	no	—
308	C₅×D₄	—	2	12	no	—
309	C₅×D₄	—	2	12	no	—
310	C₅×D₄	—	2	12	no	—
311	C₅×D₄	—	2	12	no	—
312	C₅×D₄	—	2	3	no	—
313	C₅×D₄	—	2	12	no	—
314	C₅×D₄	—	2	3	no	—
315	C₅×D₄	—	2	24	no	—
316	C₅×D₄	—	2	12	no	—
317	C₅×D₄	—	5	4	no	—
318	C₅×D₄	—	10	4	no	—
319	C₅×D₄	—	10	12	no	—
320	C₅×D₄	—	10	6	no	—
321	C₅×D₄	C₂×C₂²⋊D₅	1	12	yes	—
322	C₅×D₄	C₂×C₅×D₄	1	12	yes	—
323	C₅×D₄	C₂³×D₅	1	6	yes	—
324	C₅×D₄	C₂⁴×C₅	1	6	yes	—
325	C₂×C₄×C₅	—	2	4	no	—
326	C₂×C₄×C₅	—	2	48	no	—
327	C₂×C₄×C₅	—	2	1	no	—
328	C₂×C₄×C₅	—	2	1	no	—
329	C₂×C₄×C₅	—	2	12	no	—
330	C₂×C₄×C₅	—	2	12	no	—
331	C₂×C₄×C₅	—	2	6	no	—
332	C₂×C₄×C₅	—	2	12	no	—
333	C₂×C₄×C₅	—	2	2	no	—
334	C₂×C₄×C₅	—	2	12	no	—
335	C₂×C₄×C₅	—	10	6	no	—
336	C₂×C₄×C₅	—	10	4	no	—
337	C₂×C₄×C₅	—	10	12	no	—
338	C₂×C₄×C₅	C₂×C₂²⋊D₅	1	3	yes	—
339	C₂×C₄×C₅	C₂×C₂²⋊D₅	1	6	yes	—
340	C₂×C₄×C₅	C₂×C₅×D₄	1	6	yes	—
341	C₂×C₄×C₅	C₂×C₅×D₄	1	3	yes	—
342	C₂×C₄×C₅	C₂³×D₅	1	6	yes	—
343	C₂×C₄×C₅	C₂³×D₅	1	1	yes	characteristic
344	C₂×C₄×C₅	C₂⁴×C₅	1	6	yes	—
345	C₂×C₄×C₅	C₂⁴×C₅	1	1	yes	characteristic
346	C₂²⋊D₅	—	2	4	no	—
347	C₂²⋊D₅	—	2	12	no	—
348	C₂×Dic₅	—	2	6	no	—
349	D₂₀	—	2	6	no	—
350	D₂₀	C₂⁴×C₅	1	4	yes	—
351	C₄×D₅	—	2	4	no	—
352	C₄×D₅	—	2	12	no	—
353	Dic₁₀	—	2	6	no	—
354	Dic₁₀	C₂⁴×C₅	1	4	yes	—
355	C₅×C₈	—	5	8	no	—
356	#168	C₂⁴×C₅	1	8	yes	—
357	2₋¹⁺⁴	—	5	4	no	—
358	2₋¹⁺⁴	—	10	1	no	—
359	H(5,𝔽₂)	—	2	4	no	—
360	H(5,𝔽₂)	—	5	4	no	—
361	H(5,𝔽₂)	—	10	3	no	—
362	H(5,𝔽₂)	C₂×C₅×D₅	1	8	yes	complemented
363	C₂×#41	—	2	1	no	—
364	C₂×#41	—	5	6	no	—
365	C₂×#41	—	10	4	no	—
366	C₂×#41	C₂×C₅×D₅	1	1	yes	characteristic
367	C₂×#41	C₂×C₅×D₅	1	3	yes	complemented
368	C₂²×D₄	—	2	3	no	—
369	C₂²×D₄	C₂×C₅×D₅	1	3	yes	complemented
370	#137	—	5	12	no	—
371	#137	—	5	4	no	—
372	AGL(1,ℤ/8)	—	5	12	no	—
373	AGL(1,ℤ/8)	—	5	12	no	—
374	AGL(1,ℤ/8)	—	5	24	no	—
375	#135	—	5	24	no	—
376	C₂×SD₁₆	—	5	12	no	—
377	C₂×D₈	—	5	12	no	—
378	#131	—	5	8	no	—
379	C₂×M₁₆	—	5	6	no	—
380	C₅²	#3585	1	1	yes	characteristic, complemented, Sylow
381	C₂²×C₅	—	2	24	no	—
382	C₂²×C₅	—	2	6	no	—
383	C₂²×C₅	—	2	6	no	—
384	C₂²×C₅	—	2	2	no	—
385	C₂²×C₅	—	2	12	no	—
386	C₂²×C₅	—	4	3	no	—
387	C₂²×C₅	—	4	3	no	—
388	C₂²×C₅	—	4	12	no	—
389	C₂²×C₅	—	4	24	no	—
390	C₂²×C₅	—	4	6	no	—
391	C₂²×C₅	—	4	6	no	—
392	C₂²×C₅	—	8	12	no	—
393	C₂²×C₅	—	8	12	no	—
394	C₂²×C₅	—	8	48	no	—
395	C₂²×C₅	—	10	4	no	—
396	C₂²×C₅	—	40	6	no	—
397	C₂²×C₅	C₂²×C₂²⋊D₅	1	1	yes	characteristic
398	C₂²×C₅	C₂²×C₂²⋊D₅	1	6	yes	—
399	C₂²×C₅	C₂²×C₅×D₄	1	6	yes	—
400	C₂²×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
401	C₂×D₅	—	2	4	no	—
402	C₂×D₅	—	8	6	no	—
403	C₄×C₅	—	2	8	no	—
404	C₄×C₅	—	2	2	no	—
405	C₄×C₅	—	2	2	no	—
406	C₄×C₅	—	2	12	no	—
407	C₄×C₅	—	2	2	no	—
408	C₄×C₅	—	2	2	no	—
409	C₄×C₅	—	10	4	no	—
410	C₄×C₅	C₂²×C₂²⋊D₅	1	6	yes	—
411	C₄×C₅	C₂²×C₂²⋊D₅	1	1	yes	characteristic
412	C₄×C₅	C₂²×C₅×D₄	1	6	yes	—
413	C₄×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
414	C₄×C₅	C₂⁴×D₅	1	1	yes	characteristic
415	C₄×C₅	C₂⁵×C₅	1	1	yes	characteristic
416	Dic₅	—	2	4	no	—
417	C₂⁴	—	2	3	no	—
418	#41	—	2	8	no	—
419	#41	—	2	6	no	—
420	#41	—	5	12	no	—
421	#41	—	5	4	no	—
422	#41	—	10	4	no	—
423	#41	—	10	12	no	—
424	#41	—	10	6	no	—
425	#41	—	10	6	no	—
426	#41	C₅×C₂²⋊D₅	1	4	yes	—
427	#41	C₂²×C₅×D₅	1	12	yes	—
428	#41	C₂²×C₅×D₅	1	4	yes	—
429	C₂×Q₈	—	5	6	no	—
430	C₂×Q₈	—	10	4	no	—
431	C₂×Q₈	C₅×C₂²⋊D₅	1	1	yes	characteristic
432	C₂×Q₈	C₂²×C₅×D₅	1	1	yes	characteristic
433	C₂×D₄	—	2	3	no	—
434	C₂×D₄	—	2	24	no	—
435	C₂×D₄	—	2	3	no	—
436	C₂×D₄	—	2	6	no	—
437	C₂×D₄	—	5	6	no	—
438	C₂×D₄	—	10	6	no	—
439	C₂×D₄	—	10	12	no	—
440	C₂×D₄	C₅×C₂²⋊D₅	1	3	yes	—
441	C₂×D₄	C₂²×C₅×D₅	1	12	yes	—
442	C₂×D₄	C₂²×C₅×D₅	1	3	yes	—
443	C₂²×C₄	—	2	3	no	—
444	C₂²×C₄	—	10	6	no	—
445	C₂²×C₄	C₂²×C₅×D₅	1	3	yes	—
446	Q₁₆	—	5	8	no	—
447	SD₁₆	—	5	8	no	—
448	SD₁₆	—	5	24	no	—
449	D₈	—	5	24	no	—
450	M₁₆	—	5	4	no	—
451	M₁₆	—	5	12	no	—
452	C₂×C₈	—	5	12	no	—
453	C₂×C₅	—	2	4	no	—
454	C₂×C₅	—	2	1	no	—
455	C₂×C₅	—	2	6	no	—
456	C₂×C₅	—	2	6	no	—
457	C₂×C₅	—	2	1	no	—
458	C₂×C₅	—	2	2	no	—
459	C₂×C₅	—	4	12	no	—
460	C₂×C₅	—	4	24	no	—
461	C₂×C₅	—	4	6	no	—
462	C₂×C₅	—	4	6	no	—
463	C₂×C₅	—	20	4	no	—
464	C₂×C₅	C₂³×C₂²⋊D₅	1	1	yes	characteristic
465	C₂×C₅	C₂³×C₅×D₄	1	1	yes	characteristic
466	D₅	—	4	4	no	—
467	C₂³	—	2	3	no	—
468	C₂³	—	2	12	no	—
469	C₂³	—	8	6	no	—
470	C₂³	—	10	6	no	—
471	C₂³	C₂×C₅×C₂²⋊D₅	1	3	yes	—
472	Q₈	—	2	1	no	—
473	Q₈	—	5	4	no	—
474	Q₈	—	10	6	no	—
475	Q₈	C₂×C₅×C₂²⋊D₅	1	4	yes	—
476	Q₈	C₂³×C₅×D₅	1	2	yes	—
477	D₄	—	2	12	no	—
478	D₄	—	2	12	no	—
479	D₄	—	2	3	no	—
480	D₄	—	5	4	no	—
481	D₄	—	10	4	no	—
482	D₄	—	10	12	no	—
483	D₄	—	10	6	no	—
484	D₄	C₂×C₅×C₂²⋊D₅	1	12	yes	—
485	D₄	C₂³×C₅×D₅	1	6	yes	—
486	C₂×C₄	—	2	1	no	—
487	C₂×C₄	—	2	12	no	—
488	C₂×C₄	—	10	6	no	—
489	C₂×C₄	—	10	4	no	—
490	C₂×C₄	—	10	12	no	—
491	C₂×C₄	C₂×C₅×C₂²⋊D₅	1	3	yes	—
492	C₂×C₄	C₂×C₅×C₂²⋊D₅	1	6	yes	—
493	C₂×C₄	C₂³×C₅×D₅	1	6	yes	—
494	C₂×C₄	C₂³×C₅×D₅	1	1	yes	characteristic
495	C₈	—	5	8	no	—
496	C₅	—	2	2	no	—
497	C₅	#10619178	1	1	yes	characteristic, direct product factor
498	C₅	C₅×#3585	1	1	yes	characteristic, complemented
499	C₂²	—	2	6	no	—
500	C₂²	—	4	3	no	—
501	C₂²	—	4	6	no	—
502	C₂²	—	8	12	no	—
503	C₂²	—	10	4	no	—
504	C₂²	—	40	6	no	—
505	C₂²	C₂²×C₅×C₂²⋊D₅	1	1	yes	characteristic
506	C₂²	C₂²×C₅×C₂²⋊D₅	1	6	yes	—
507	C₄	—	2	2	no	—
508	C₄	—	10	4	no	—
509	C₄	C₂²×C₅×C₂²⋊D₅	1	6	yes	—
510	C₄	C₂²×C₅×C₂²⋊D₅	1	1	yes	characteristic
511	C₄	C₂⁴×C₅×D₅	1	1	yes	characteristic
512	C₂	—	2	1	no	—
513	C₂	—	2	6	no	—
514	C₂	—	4	6	no	—
515	C₂	—	20	4	no	—
516	C₂	C₂³×C₅×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	C₂³×C₂²⋊D₅
commutator, cosocle	C₄×C₅	C₂⁵×C₅
Frattini	C₄	C₂⁴×C₅×D₅
Fitting	C₂×C₅²×H(5,𝔽₂)	C₂
socle	C₂×C₅²	C₂³×D₄
hypercenter	C₂×C₅×H(5,𝔽₂)	D₅
hypocenter	C₅	C₅×#3585
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₅×#41 ◁ C₂×C₅×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂×C₅²×H(5,𝔽₂)	C₂	semidirect
2. C₅×#66283	C₂	semidirect
3. C₅×#66170	C₂	semidirect
4. C₅×#66169	C₂	semidirect
5. C₅×#66156	C₂	semidirect
6. C₅×#66153	C₂	semidirect
7. C₂×C₅×#4078	C₂	semidirect
9. C₅²×H(5,𝔽₂)	C₂²	semidirect
12. C₂×C₅²×#41	C₂²	semidirect
14. C₂²×C₅²×D₄	C₂²	semidirect
15. C₅×#4150	C₂²	semidirect
16. C₅×#4149	C₂²	semidirect
20. C₂×C₅×#709	C₂²	semidirect
21. C₅×#4097	C₂²	semidirect
22. C₅×#4096	C₂²	semidirect
23. C₅×#4095	C₂²	semidirect
24. C₅×#4094	C₂²	semidirect
25. C₅×#4088	C₂²	semidirect
26. C₂×C₅×#687	C₂²	semidirect
27. C₅×#4078	C₂²	semidirect
28. C₅×#4078	C₂²	semidirect
29. C₂×C₅×#686	C₂²	semidirect
30. C₂×C₅×#681	C₂²	semidirect
32. #10619178	C₅	direct (N × H)
58. C₅×#709	C₂³	semidirect
59. C₂×C₅×D₂₀	C₂³	semidirect
61. C₂×C₅×Dic₁₀	C₂³	semidirect
62. C₅×#689	C₂³	semidirect
63. C₅×#688	C₂³	semidirect
64. C₅×#687	C₂³	semidirect
65. C₅×#686	C₂³	semidirect
67. C₅×#681	C₂³	semidirect
68. C₂×C₅×#168	C₂³	semidirect
71. C₂×C₅×H(5,𝔽₂)	D₅	semidirect
72. C₂×C₅×H(5,𝔽₂)	C₂×C₅	semidirect
73. #66283	C₂×C₅	semidirect
79. #66170	C₂×C₅	semidirect
80. #66169	C₂×C₅	semidirect
81. #66156	C₂×C₅	semidirect
82. #66153	C₂×C₅	semidirect
83. C₂×#4078	C₂×C₅	semidirect
121. C₅×H(5,𝔽₂)	C₂×D₅	semidirect
122. C₅×H(5,𝔽₂)	C₂²×C₅	semidirect
131. C₂×C₅×#41	C₂×D₅	semidirect
133. C₂×C₅×#41	C₂²×C₅	semidirect
138. C₂²×C₅×D₄	C₂×D₅	semidirect
139. C₂²×C₅×D₄	C₂²×C₅	semidirect
140. #4150	C₂²×C₅	semidirect
141. #4149	C₂²×C₅	semidirect
145. C₂×#709	C₂²×C₅	semidirect
156. #4097	C₂²×C₅	semidirect
157. #4096	C₂²×C₅	semidirect
158. #4095	C₂²×C₅	semidirect
159. #4094	C₂²×C₅	semidirect
160. #4088	C₂²×C₅	semidirect
161. C₂×#687	C₂²×C₅	semidirect
162. #4078	C₂²×C₅	semidirect
163. #4078	C₂²×C₅	semidirect
164. C₂×#686	C₂²×C₅	semidirect
165. C₂×#681	C₂²×C₅	semidirect
251. #709	C₂³×C₅	semidirect
252. C₂×D₂₀	C₂³×C₅	semidirect
254. C₂×Dic₁₀	C₂³×C₅	semidirect
262. #689	C₂³×C₅	semidirect
263. #688	C₂³×C₅	semidirect
264. #687	C₂³×C₅	semidirect
265. #686	C₂³×C₅	semidirect
267. #681	C₂³×C₅	semidirect
268. C₂×#168	C₂³×C₅	semidirect
270. C₂×H(5,𝔽₂)	C₅×D₅	semidirect
362. H(5,𝔽₂)	C₂×C₅×D₅	semidirect
367. C₂×#41	C₂×C₅×D₅	semidirect
369. C₂²×D₄	C₂×C₅×D₅	semidirect
380. C₅²	#3585	semidirect
497. C₅	#10619178	direct (N × H)
498. C₅	C₅×#3585	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅²×H(5,𝔽₂)	2
C₅×#66283	2
C₅×#66170	2
C₅×#66169	2
C₅×#66156	2
C₅×#66153	2
C₂×C₅×#4078	2
C₅×#3585	5
#10619178	5

Subgroups — C5×#10619178 #72223275

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10619178 #72223275