Subgroups — C₅×#23029 #11945749

Order 1280.

1138 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#466	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#442	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₅×#106	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2361	C₂	1	2	yes	maximal, complemented
5	C₅×#1972	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1943	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1926	C₂	1	2	yes	maximal, complemented
8	C₅×#1874	C₂	1	2	yes	maximal, complemented
9	C₅×#1864	C₂	1	2	yes	maximal, complemented
10	C₅×#1776	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#1752	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
13	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₅×#121	C₂²	1	2	yes	—
15	C₂×C₄×C₅×D₄	—	2	2	no	—
16	C₂×C₄×C₅×D₄	C₂²	1	2	yes	—
17	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
18	C₅×#466	C₄	1	8	yes	complemented
19	C₅×#442	—	2	4	no	—
20	C₂×C₅×#107	C₂²	1	1	yes	characteristic
21	C₂×C₅×#106	—	2	4	no	—
22	C₂×C₅×#106	C₂²	1	2	yes	—
23	C₂×C₅×#106	C₂²	1	1	yes	characteristic
24	C₂×C₅×#106	C₂²	1	1	yes	characteristic
25	C₂×C₅×#102	C₂²	1	2	yes	—
26	C₂×C₅×#102	C₂²	1	2	yes	—
27	C₂×C₅×#102	C₂²	1	2	yes	—
28	C₂×C₅×C₂²⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₅×#97	C₂²	1	1	yes	characteristic
30	C₂×C₅×#97	C₂²	1	1	yes	characteristic
31	C₅×#390	C₂²	1	2	yes	—
32	C₅×#389	C₂²	1	2	yes	—
33	C₅×#388	—	2	2	no	—
34	C₅×#386	—	2	2	no	—
35	C₅×#385	—	2	2	no	—
36	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
37	C₄×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
38	C₅×#340	C₂²	1	2	yes	—
39	C₅×#340	C₂²	1	2	yes	—
40	C₅×#336	C₂²	1	1	yes	characteristic
41	C₅×#336	C₂²	1	1	yes	characteristic
42	#23029	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂²×C₅×D₄	—	2	2	no	—
44	C₂²×C₅×D₄	C₂³	1	2	yes	—
45	C₂³×C₄×C₅	—	2	2	no	—
46	C₂³×C₄×C₅	—	2	2	no	—
47	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
48	C₂×C₅×D₈	—	2	2	no	—
49	C₂×C₅×D₈	—	2	2	no	—
50	C₂×C₅×D₈	C₂×C₄	1	4	yes	complemented
51	C₂²×C₅×C₈	—	2	2	no	—
52	C₂²×C₅×C₈	—	2	2	no	—
53	C₂²×C₅×C₈	C₂×C₄	1	4	yes	complemented
54	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
55	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
56	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
57	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
58	C₅×#121	—	2	4	no	—
59	C₅×#121	C₂×C₄	1	8	yes	complemented
60	C₄×C₅×D₄	—	2	4	no	—
61	C₄×C₅×D₄	—	4	2	no	—
62	C₄×C₅×D₄	—	4	8	no	—
63	C₂×C₅×C₄⋊C₄	—	2	4	no	—
64	C₂×C₅×C₄⋊C₄	—	2	2	no	—
65	C₂×C₅×C₄⋊C₄	—	2	2	no	—
66	C₂×C₅×C₄⋊C₄	—	4	2	no	—
67	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
68	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
69	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
70	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
71	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
72	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
73	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
74	C₂×C₅×C₂²⋊C₄	D₄	1	2	yes	—
75	C₂×C₄²×C₅	—	2	2	no	—
76	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
78	C₅×#107	C₂×C₄	1	4	yes	complemented
79	C₅×#106	—	2	2	no	—
80	C₅×#106	—	2	4	no	—
81	C₅×#106	D₄	1	4	yes	complemented
82	C₅×#102	—	2	4	no	—
83	C₅×#102	—	4	2	no	—
84	C₅×#102	C₂×C₄	1	8	yes	complemented
85	C₅×C₂²⋊C₈	—	2	2	no	—
86	C₅×#97	—	2	2	no	—
87	C₅×#97	—	2	2	no	—
88	C₅×#95	—	2	2	no	—
89	C₅×#95	—	2	2	no	—
90	C₅×#95	D₄	1	2	yes	—
91	C₂×#466	C₂×C₅	1	1	yes	characteristic
92	C₂×#442	C₂×C₅	1	1	yes	characteristic, complemented
93	C₂²×#106	C₂×C₅	1	1	yes	characteristic, complemented
94	#2361	C₂×C₅	1	2	yes	complemented
95	#1972	C₂×C₅	1	1	yes	characteristic, complemented
96	#1943	C₂×C₅	1	1	yes	characteristic, complemented
97	#1926	C₂×C₅	1	2	yes	complemented
98	#1874	C₂×C₅	1	2	yes	complemented
99	#1864	C₂×C₅	1	2	yes	complemented
100	#1776	C₂×C₅	1	1	yes	characteristic, complemented
101	#1752	C₂×C₅	1	1	yes	characteristic, complemented
102	C₂⁴×C₅	—	2	2	no	—
103	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
104	C₂×C₅×D₄	—	2	8	no	—
105	C₂×C₅×D₄	—	2	2	no	—
106	C₂×C₅×D₄	—	2	2	no	—
107	C₂×C₅×D₄	—	2	2	no	—
108	C₂×C₅×D₄	—	2	2	no	—
109	C₂×C₅×D₄	—	2	2	no	—
110	C₂×C₅×D₄	—	4	4	no	—
111	C₂×C₅×D₄	—	4	4	no	—
112	C₂×C₅×D₄	C₂²×C₄	1	4	yes	—
113	C₂²×C₄×C₅	—	2	1	no	—
114	C₂²×C₄×C₅	—	2	1	no	—
115	C₂²×C₄×C₅	—	2	1	no	—
116	C₂²×C₄×C₅	—	2	1	no	—
117	C₂²×C₄×C₅	—	2	2	no	—
118	C₂²×C₄×C₅	—	2	2	no	—
119	C₂²×C₄×C₅	—	2	2	no	—
120	C₂²×C₄×C₅	—	4	4	no	—
121	C₂²×C₄×C₅	—	4	4	no	—
122	C₂²×C₄×C₅	—	4	4	no	—
123	C₂²×C₄×C₅	—	4	4	no	—
124	C₂²×C₄×C₅	—	8	2	no	—
125	C₂²×C₄×C₅	—	8	2	no	—
126	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
127	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
128	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
129	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₅	#41	1	2	yes	—
131	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
132	C₅×D₈	—	2	8	no	—
133	C₂×C₅×C₈	—	2	1	no	—
134	C₂×C₅×C₈	—	2	1	no	—
135	C₂×C₅×C₈	—	2	8	no	—
136	C₂×C₅×C₈	—	2	1	no	—
137	C₂×C₅×C₈	—	2	1	no	—
138	C₂×C₅×C₈	—	2	1	no	—
139	C₂×C₅×C₈	—	2	1	no	—
140	C₂×C₅×C₈	—	2	1	no	—
141	C₂×C₅×C₈	—	2	1	no	—
142	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
143	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
144	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
145	C₂×C₅×C₈	#41	1	2	yes	—
146	C₅×C₄⋊C₄	—	2	2	no	—
147	C₅×C₄⋊C₄	—	2	2	no	—
148	C₅×C₄⋊C₄	—	2	2	no	—
149	C₅×C₄⋊C₄	—	2	2	no	—
150	C₅×C₄⋊C₄	—	4	4	no	—
151	C₅×C₄⋊C₄	—	4	4	no	—
152	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
153	C₅×C₄⋊C₄	C₂×D₄	1	4	yes	—
154	C₅×C₂²⋊C₄	—	2	4	no	—
155	C₅×C₂²⋊C₄	—	2	8	no	—
156	C₅×C₂²⋊C₄	—	4	4	no	—
157	C₅×C₂²⋊C₄	—	4	4	no	—
158	C₅×C₂²⋊C₄	—	8	2	no	—
159	C₄²×C₅	—	2	1	no	—
160	C₄²×C₅	—	2	1	no	—
161	C₄²×C₅	—	2	1	no	—
162	C₄²×C₅	—	2	1	no	—
163	C₄²×C₅	—	4	4	no	—
164	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
165	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
166	C₂×#121	C₂²×C₅	1	2	yes	—
167	C₂×C₄×D₄	—	2	2	no	—
168	C₂×C₄×D₄	C₂²×C₅	1	2	yes	—
169	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
170	#466	C₄×C₅	1	8	yes	complemented
171	#442	—	2	4	no	—
172	C₂×#107	C₂²×C₅	1	1	yes	characteristic
173	C₂×#106	—	2	4	no	—
174	C₂×#106	C₂²×C₅	1	2	yes	—
175	C₂×#106	C₂²×C₅	1	1	yes	characteristic
176	C₂×#106	C₂²×C₅	1	1	yes	characteristic
177	C₂×#102	C₂²×C₅	1	2	yes	—
178	C₂×#102	C₂²×C₅	1	2	yes	—
179	C₂×#102	C₂²×C₅	1	2	yes	—
180	C₂×C₂²⋊C₈	C₂²×C₅	1	1	yes	characteristic
181	C₂×#97	C₂²×C₅	1	1	yes	characteristic
182	C₂×#97	C₂²×C₅	1	1	yes	characteristic
183	#390	C₂²×C₅	1	2	yes	—
184	#389	C₂²×C₅	1	2	yes	—
185	#388	—	2	2	no	—
186	#386	—	2	2	no	—
187	#385	—	2	2	no	—
188	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
189	C₄×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
190	#340	C₂²×C₅	1	2	yes	—
191	#340	C₂²×C₅	1	2	yes	—
192	#336	C₂²×C₅	1	1	yes	characteristic
193	#336	C₂²×C₅	1	1	yes	characteristic
194	C₂³×C₅	—	2	1	no	—
195	C₂³×C₅	—	2	1	no	—
196	C₂³×C₅	—	2	2	no	—
197	C₂³×C₅	—	2	4	no	—
198	C₂³×C₅	—	4	2	no	—
199	C₂³×C₅	—	4	2	no	—
200	C₂³×C₅	—	4	2	no	—
201	C₂³×C₅	—	4	2	no	—
202	C₂³×C₅	—	4	2	no	—
203	C₂³×C₅	—	4	2	no	—
204	C₂³×C₅	C₄×D₄	1	2	yes	—
205	C₂³×C₅	AGL(1,ℤ/8)	1	4	yes	—
206	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
207	C₅×D₄	—	2	4	no	—
208	C₅×D₄	—	2	4	no	—
209	C₅×D₄	—	4	16	no	—
210	C₅×D₄	—	4	2	no	—
211	C₅×D₄	—	4	2	no	—
212	C₂×C₄×C₅	—	2	1	no	—
213	C₂×C₄×C₅	—	2	1	no	—
214	C₂×C₄×C₅	—	2	2	no	—
215	C₂×C₄×C₅	—	2	2	no	—
216	C₂×C₄×C₅	—	2	2	no	—
217	C₂×C₄×C₅	—	2	2	no	—
218	C₂×C₄×C₅	—	2	4	no	—
219	C₂×C₄×C₅	—	2	2	no	—
220	C₂×C₄×C₅	—	2	4	no	—
221	C₂×C₄×C₅	—	2	2	no	—
222	C₂×C₄×C₅	—	2	2	no	—
223	C₂×C₄×C₅	—	2	2	no	—
224	C₂×C₄×C₅	—	2	2	no	—
225	C₂×C₄×C₅	—	2	2	no	—
226	C₂×C₄×C₅	—	4	2	no	—
227	C₂×C₄×C₅	—	4	2	no	—
228	C₂×C₄×C₅	—	4	2	no	—
229	C₂×C₄×C₅	—	4	2	no	—
230	C₂×C₄×C₅	—	4	2	no	—
231	C₂×C₄×C₅	—	4	2	no	—
232	C₂×C₄×C₅	—	4	2	no	—
233	C₂×C₄×C₅	—	8	8	no	—
234	C₂×C₄×C₅	—	8	8	no	—
235	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
236	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
237	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
238	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
239	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
240	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
241	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
242	C₅×C₈	—	2	2	no	—
243	C₅×C₈	—	4	1	no	—
244	C₅×C₈	—	4	1	no	—
245	C₅×C₈	C₄×D₄	1	4	yes	complemented
246	C₂²×D₄	—	2	2	no	—
247	C₂²×D₄	C₂³×C₅	1	2	yes	—
248	C₂³×C₄	—	2	2	no	—
249	C₂³×C₄	—	2	2	no	—
250	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
251	C₂×D₈	—	2	2	no	—
252	C₂×D₈	—	2	2	no	—
253	C₂×D₈	C₂×C₄×C₅	1	4	yes	complemented
254	C₂²×C₈	—	2	2	no	—
255	C₂²×C₈	—	2	2	no	—
256	C₂²×C₈	C₂×C₄×C₅	1	4	yes	complemented
257	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
258	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
259	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
260	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
261	#121	—	2	4	no	—
262	#121	C₂×C₄×C₅	1	8	yes	complemented
263	C₄×D₄	—	2	4	no	—
264	C₄×D₄	—	4	2	no	—
265	C₄×D₄	—	4	8	no	—
266	C₂×C₄⋊C₄	—	2	4	no	—
267	C₂×C₄⋊C₄	—	2	2	no	—
268	C₂×C₄⋊C₄	—	2	2	no	—
269	C₂×C₄⋊C₄	—	4	2	no	—
270	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
271	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
272	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
273	C₂×C₂²⋊C₄	—	2	2	no	—
274	C₂×C₂²⋊C₄	—	2	2	no	—
275	C₂×C₂²⋊C₄	—	2	2	no	—
276	C₂×C₂²⋊C₄	—	2	2	no	—
277	C₂×C₂²⋊C₄	C₅×D₄	1	2	yes	—
278	C₂×C₄²	—	2	2	no	—
279	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
280	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
281	#107	C₂×C₄×C₅	1	4	yes	complemented
282	#106	—	2	2	no	—
283	#106	—	2	4	no	—
284	#106	C₅×D₄	1	4	yes	complemented
285	#102	—	2	4	no	—
286	#102	—	4	2	no	—
287	#102	C₂×C₄×C₅	1	8	yes	complemented
288	C₂²⋊C₈	—	2	2	no	—
289	#97	—	2	2	no	—
290	#97	—	2	2	no	—
291	#95	—	2	2	no	—
292	#95	—	2	2	no	—
293	#95	C₅×D₄	1	2	yes	—
294	C₂²×C₅	—	2	4	no	—
295	C₂²×C₅	—	2	2	no	—
296	C₂²×C₅	—	2	2	no	—
297	C₂²×C₅	—	2	4	no	—
298	C₂²×C₅	—	4	4	no	—
299	C₂²×C₅	—	4	4	no	—
300	C₂²×C₅	—	4	4	no	—
301	C₂²×C₅	—	8	2	no	—
302	C₂²×C₅	—	8	2	no	—
303	C₂²×C₅	—	8	2	no	—
304	C₂²×C₅	—	8	2	no	—
305	C₂²×C₅	#442	1	4	yes	complemented
306	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
307	C₂²×C₅	D₄²	1	1	yes	characteristic
308	C₂²×C₅	#547	1	1	yes	characteristic
309	C₂²×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
310	C₂²×C₅	#577	1	2	yes	—
311	C₄×C₅	—	2	2	no	—
312	C₄×C₅	—	4	2	no	—
313	C₄×C₅	—	4	4	no	—
314	C₄×C₅	—	4	2	no	—
315	C₄×C₅	—	4	4	no	—
316	C₄×C₅	—	8	2	no	—
317	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
318	C₄×C₅	#520	1	1	yes	characteristic
319	C₄×C₅	#548	1	1	yes	characteristic
320	C₄×C₅	#553	1	1	yes	characteristic
321	C₂⁴	—	2	2	no	—
322	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
323	C₂×D₄	—	2	8	no	—
324	C₂×D₄	—	2	2	no	—
325	C₂×D₄	—	2	2	no	—
326	C₂×D₄	—	2	2	no	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	—	2	2	no	—
329	C₂×D₄	—	4	4	no	—
330	C₂×D₄	—	4	4	no	—
331	C₂×D₄	C₂²×C₄×C₅	1	4	yes	—
332	C₂²×C₄	—	2	1	no	—
333	C₂²×C₄	—	2	1	no	—
334	C₂²×C₄	—	2	1	no	—
335	C₂²×C₄	—	2	1	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	2	no	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	4	4	no	—
340	C₂²×C₄	—	4	4	no	—
341	C₂²×C₄	—	4	4	no	—
342	C₂²×C₄	—	4	4	no	—
343	C₂²×C₄	—	8	2	no	—
344	C₂²×C₄	—	8	2	no	—
345	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
346	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
347	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
348	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
349	C₂²×C₄	C₅×#41	1	2	yes	—
350	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
351	D₈	—	2	8	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	—	2	1	no	—
354	C₂×C₈	—	2	8	no	—
355	C₂×C₈	—	2	1	no	—
356	C₂×C₈	—	2	1	no	—
357	C₂×C₈	—	2	1	no	—
358	C₂×C₈	—	2	1	no	—
359	C₂×C₈	—	2	1	no	—
360	C₂×C₈	—	2	1	no	—
361	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
362	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
363	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
364	C₂×C₈	C₅×#41	1	2	yes	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	4	4	no	—
370	C₄⋊C₄	—	4	4	no	—
371	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
372	C₄⋊C₄	C₂×C₅×D₄	1	4	yes	—
373	C₂²⋊C₄	—	2	4	no	—
374	C₂²⋊C₄	—	2	8	no	—
375	C₂²⋊C₄	—	4	4	no	—
376	C₂²⋊C₄	—	4	4	no	—
377	C₂²⋊C₄	—	8	2	no	—
378	C₄²	—	2	1	no	—
379	C₄²	—	2	1	no	—
380	C₄²	—	2	1	no	—
381	C₄²	—	2	1	no	—
382	C₄²	—	4	4	no	—
383	C₂×C₅	—	2	4	no	—
384	C₂×C₅	—	8	4	no	—
385	C₂×C₅	#2361	1	1	yes	characteristic
386	C₂×C₅	C₂×#442	1	1	yes	characteristic
387	C₂×C₅	#2952	1	1	yes	characteristic
388	C₂×C₅	#3265	1	2	yes	—
389	C₂×C₅	#3345	1	2	yes	—
390	C₂³	—	2	1	no	—
391	C₂³	—	2	1	no	—
392	C₂³	—	2	2	no	—
393	C₂³	—	2	4	no	—
394	C₂³	—	4	2	no	—
395	C₂³	—	4	2	no	—
396	C₂³	—	4	2	no	—
397	C₂³	—	4	2	no	—
398	C₂³	—	4	2	no	—
399	C₂³	—	4	2	no	—
400	C₂³	C₄×C₅×D₄	1	2	yes	—
401	C₂³	C₅×AGL(1,ℤ/8)	1	4	yes	—
402	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
403	D₄	—	2	4	no	—
404	D₄	—	2	4	no	—
405	D₄	—	4	16	no	—
406	D₄	—	4	2	no	—
407	D₄	—	4	2	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	4	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	8	8	no	—
430	C₂×C₄	—	8	8	no	—
431	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
432	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
433	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
434	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
435	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
436	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
437	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
438	C₈	—	2	2	no	—
439	C₈	—	4	1	no	—
440	C₈	—	4	1	no	—
441	C₈	C₄×C₅×D₄	1	4	yes	complemented
442	C₅	#23029	1	1	yes	characteristic, direct product factor, Sylow
443	C₂²	—	2	4	no	—
444	C₂²	—	2	4	no	—
445	C₂²	—	2	2	no	—
446	C₂²	—	2	2	no	—
447	C₂²	—	4	4	no	—
448	C₂²	—	4	4	no	—
449	C₂²	—	4	4	no	—
450	C₂²	—	8	2	no	—
451	C₂²	—	8	2	no	—
452	C₂²	—	8	2	no	—
453	C₂²	—	8	2	no	—
454	C₂²	C₅×#442	1	4	yes	complemented
455	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
456	C₂²	C₅×D₄²	1	1	yes	characteristic
457	C₂²	C₅×#547	1	1	yes	characteristic
458	C₂²	C₂×C₅×AGL(1,ℤ/8)	1	2	yes	—
459	C₂²	C₅×#577	1	2	yes	—
460	C₄	—	2	2	no	—
461	C₄	—	4	2	no	—
462	C₄	—	4	4	no	—
463	C₄	—	4	2	no	—
464	C₄	—	4	4	no	—
465	C₄	—	8	2	no	—
466	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
467	C₄	C₅×#520	1	1	yes	characteristic
468	C₄	C₅×#548	1	1	yes	characteristic
469	C₄	C₅×#553	1	1	yes	characteristic
470	C₂	—	2	4	no	—
471	C₂	—	8	4	no	—
472	C₂	C₅×#2361	1	1	yes	characteristic
473	C₂	C₂×C₅×#442	1	1	yes	characteristic
474	C₂	C₅×#2952	1	1	yes	characteristic
475	C₂	C₅×#3265	1	2	yes	—
476	C₂	C₅×#3345	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂×C₅×#466	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄×C₅×C₂²⋊C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#442	C₂	semidirect
3. C₂²×C₅×#106	C₂	semidirect
4. C₅×#2361	C₂	semidirect
5. C₅×#1972	C₂	semidirect
6. C₅×#1943	C₂	semidirect
7. C₅×#1926	C₂	semidirect
8. C₅×#1874	C₂	semidirect
9. C₅×#1864	C₂	semidirect
10. C₅×#1776	C₂	semidirect
11. C₅×#1752	C₂	semidirect
18. C₅×#466	C₄	semidirect
42. #23029	C₅	direct (N × H)
50. C₂×C₅×D₈	C₂×C₄	semidirect
53. C₂²×C₅×C₈	C₂×C₄	semidirect
59. C₅×#121	C₂×C₄	semidirect
78. C₅×#107	C₂×C₄	semidirect
81. C₅×#106	D₄	semidirect
84. C₅×#102	C₂×C₄	semidirect
92. C₂×#442	C₂×C₅	semidirect
93. C₂²×#106	C₂×C₅	semidirect
94. #2361	C₂×C₅	semidirect
95. #1972	C₂×C₅	semidirect
96. #1943	C₂×C₅	semidirect
97. #1926	C₂×C₅	semidirect
98. #1874	C₂×C₅	semidirect
99. #1864	C₂×C₅	semidirect
100. #1776	C₂×C₅	semidirect
101. #1752	C₂×C₅	semidirect
170. #466	C₄×C₅	semidirect
245. C₅×C₈	C₄×D₄	semidirect
253. C₂×D₈	C₂×C₄×C₅	semidirect
256. C₂²×C₈	C₂×C₄×C₅	semidirect
262. #121	C₂×C₄×C₅	semidirect
281. #107	C₂×C₄×C₅	semidirect
284. #106	C₅×D₄	semidirect
287. #102	C₂×C₄×C₅	semidirect
305. C₂²×C₅	#442	semidirect
441. C₈	C₄×C₅×D₄	semidirect
442. C₅	#23029	direct (N × H)
454. C₂²	C₅×#442	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#466	2
C₂×C₅×#442	2
C₂²×C₅×#106	2
C₅×#2361	2
C₅×#1972	2
C₅×#1943	2
C₅×#1926	2
C₅×#1874	2
C₅×#1864	2
C₅×#1776	2
C₅×#1752	2
#23029	5

Subgroups — C5×#23029 #11945749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#23029 #11945749