Subgroups — C₅×#22224 #11944944

Order 1280.

842 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#489	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×#124	C₂	1	1	yes	maximal, characteristic
4	C₅×#1993	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1959	C₂	1	1	yes	maximal, characteristic
6	C₅×#1927	C₂	1	2	yes	maximal, complemented
7	C₅×#1925	C₂	1	2	yes	maximal
8	C₅×#1864	C₂	1	2	yes	maximal
9	C₅×#1863	C₂	1	2	yes	maximal, complemented
10	C₅×#1847	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#1745	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₅×#124	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×Q₈	—	2	1	no	—
15	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
17	C₅×#489	C₄	1	8	yes	complemented
18	C₂×C₅×#107	C₂²	1	2	yes	—
19	C₂×C₅×#106	C₂²	1	2	yes	—
20	C₂×C₅×#103	C₂²	1	2	yes	—
21	C₂×C₅×#103	C₂²	1	2	yes	—
22	C₂×C₅×#102	C₂²	1	2	yes	—
23	C₂×C₅×#102	C₂²	1	2	yes	—
24	C₂×C₅×#97	C₂²	1	1	yes	characteristic
25	C₂×C₅×#97	C₂²	1	1	yes	characteristic
26	C₅×#391	—	2	1	no	—
27	C₅×#391	C₂²	1	1	yes	characteristic
28	C₅×#390	C₂²	1	1	yes	characteristic
29	C₅×#389	—	2	1	no	—
30	C₅×#389	C₂²	1	2	yes	—
31	C₅×#388	—	2	1	no	—
32	C₅×#388	—	2	1	no	—
33	C₅×#384	C₂²	1	1	yes	characteristic
34	C₅×#382	C₂²	1	1	yes	characteristic
35	C₄×C₅×C₄⋊C₄	—	2	1	no	—
36	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
37	C₄×C₅×C₂²⋊C₄	—	2	1	no	—
38	C₄×C₅×C₂²⋊C₄	—	2	1	no	—
39	C₄³×C₅	C₂²	1	1	yes	characteristic
40	C₅×#340	C₂²	1	2	yes	—
41	C₅×#340	C₂²	1	2	yes	—
42	C₅×#336	C₂²	1	2	yes	—
43	C₅×#336	C₂²	1	2	yes	—
44	#22224	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂²×C₅×Q₈	—	2	1	no	—
46	C₂²×C₅×Q₈	C₂³	1	1	yes	characteristic
47	C₂²×C₅×D₄	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₅	—	2	1	no	—
49	C₂²×C₅×C₈	C₂³	1	2	yes	—
50	C₂²×C₅×C₈	C₂³	1	2	yes	—
51	C₅×C₄⋊Q₈	—	2	2	no	—
52	C₅×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
53	C₅×#124	—	2	2	no	—
54	C₅×#124	C₂×C₄	1	4	yes	—
55	C₄×C₅×Q₈	—	4	2	no	—
56	C₄×C₅×Q₈	—	4	4	no	—
57	C₄×C₅×D₄	—	4	2	no	—
58	C₂×C₅×C₄⋊C₄	—	2	1	no	—
59	C₂×C₅×C₄⋊C₄	—	2	1	no	—
60	C₂×C₅×C₄⋊C₄	—	2	1	no	—
61	C₂×C₅×C₄⋊C₄	—	2	1	no	—
62	C₂×C₅×C₄⋊C₄	—	2	2	no	—
63	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
64	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
65	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
66	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
67	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
68	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
69	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
70	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
71	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
72	C₂×C₄²×C₅	—	2	1	no	—
73	C₂×C₄²×C₅	—	2	2	no	—
74	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
75	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
77	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
78	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
81	C₅×#107	—	4	2	no	—
82	C₅×#106	—	4	2	no	—
83	C₅×#103	—	4	2	no	—
84	C₅×#103	C₂×C₄	1	8	yes	complemented
85	C₅×#102	—	4	2	no	—
86	C₅×#102	C₂×C₄	1	8	yes	—
87	C₅×#97	—	2	2	no	—
88	C₅×#97	C₂×C₄	1	4	yes	complemented
89	C₅×#95	—	2	1	no	—
90	C₅×#95	—	2	1	no	—
91	C₅×#95	—	2	1	no	—
92	C₅×#95	—	2	1	no	—
93	C₅×#95	—	2	2	no	—
94	C₅×#95	—	2	2	no	—
95	C₂×#489	C₂×C₅	1	1	yes	characteristic
96	C₄×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic, complemented
97	C₄×#124	C₂×C₅	1	1	yes	characteristic
98	#1993	C₂×C₅	1	1	yes	characteristic, complemented
99	#1959	C₂×C₅	1	1	yes	characteristic
100	#1927	C₂×C₅	1	2	yes	complemented
101	#1925	C₂×C₅	1	2	yes	—
102	#1864	C₂×C₅	1	2	yes	—
103	#1863	C₂×C₅	1	2	yes	complemented
104	#1847	C₂×C₅	1	1	yes	characteristic, complemented
105	#1745	C₂×C₅	1	1	yes	characteristic, complemented
106	C₂⁴×C₅	—	2	1	no	—
107	C₂×C₅×Q₈	—	2	4	no	—
108	C₂×C₅×Q₈	—	2	1	no	—
109	C₂×C₅×Q₈	—	2	2	no	—
110	C₂×C₅×Q₈	—	2	2	no	—
111	C₂×C₅×Q₈	—	4	2	no	—
112	C₂×C₅×Q₈	—	4	2	no	—
113	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
114	C₂×C₅×D₄	—	2	1	no	—
115	C₂×C₅×D₄	—	2	2	no	—
116	C₂×C₅×D₄	—	2	2	no	—
117	C₂×C₅×D₄	C₂²×C₄	1	2	yes	—
118	C₂²×C₄×C₅	—	2	1	no	—
119	C₂²×C₄×C₅	—	2	2	no	—
120	C₂²×C₄×C₅	—	2	1	no	—
121	C₂²×C₄×C₅	—	2	2	no	—
122	C₂²×C₄×C₅	—	2	1	no	—
123	C₂²×C₄×C₅	—	4	2	no	—
124	C₂²×C₄×C₅	—	4	2	no	—
125	C₂²×C₄×C₅	—	4	2	no	—
126	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
133	C₂×C₅×C₈	—	2	2	no	—
134	C₂×C₅×C₈	—	2	2	no	—
135	C₂×C₅×C₈	—	2	2	no	—
136	C₂×C₅×C₈	—	2	2	no	—
137	C₂×C₅×C₈	—	2	2	no	—
138	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
139	C₅×C₄⋊C₄	—	2	4	no	—
140	C₅×C₄⋊C₄	—	2	2	no	—
141	C₅×C₄⋊C₄	—	2	2	no	—
142	C₅×C₄⋊C₄	—	2	2	no	—
143	C₅×C₄⋊C₄	—	2	2	no	—
144	C₅×C₄⋊C₄	—	2	2	no	—
145	C₅×C₄⋊C₄	—	4	2	no	—
146	C₅×C₄⋊C₄	—	4	2	no	—
147	C₅×C₄⋊C₄	—	4	2	no	—
148	C₅×C₄⋊C₄	—	4	4	no	—
149	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
150	C₅×C₂²⋊C₄	—	2	4	no	—
151	C₅×C₂²⋊C₄	—	2	4	no	—
152	C₅×C₂²⋊C₄	—	4	2	no	—
153	C₅×C₂²⋊C₄	—	4	2	no	—
154	C₅×C₂²⋊C₄	—	4	2	no	—
155	C₄²×C₅	—	2	2	no	—
156	C₄²×C₅	—	2	2	no	—
157	C₄²×C₅	—	2	2	no	—
158	C₄²×C₅	—	2	2	no	—
159	C₄²×C₅	—	4	1	no	—
160	C₄²×C₅	—	4	1	no	—
161	C₄²×C₅	—	4	2	no	—
162	C₄²×C₅	—	4	4	no	—
163	C₄²×C₅	C₂²×C₄	1	2	yes	—
164	C₄²×C₅	#41	1	2	yes	—
165	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
166	C₂×#124	C₂²×C₅	1	1	yes	characteristic
167	C₂×C₄×Q₈	—	2	1	no	—
168	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
169	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
170	#489	C₄×C₅	1	8	yes	complemented
171	C₂×#107	C₂²×C₅	1	2	yes	—
172	C₂×#106	C₂²×C₅	1	2	yes	—
173	C₂×#103	C₂²×C₅	1	2	yes	—
174	C₂×#103	C₂²×C₅	1	2	yes	—
175	C₂×#102	C₂²×C₅	1	2	yes	—
176	C₂×#102	C₂²×C₅	1	2	yes	—
177	C₂×#97	C₂²×C₅	1	1	yes	characteristic
178	C₂×#97	C₂²×C₅	1	1	yes	characteristic
179	#391	—	2	1	no	—
180	#391	C₂²×C₅	1	1	yes	characteristic
181	#390	C₂²×C₅	1	1	yes	characteristic
182	#389	—	2	1	no	—
183	#389	C₂²×C₅	1	2	yes	—
184	#388	—	2	1	no	—
185	#388	—	2	1	no	—
186	#384	C₂²×C₅	1	1	yes	characteristic
187	#382	C₂²×C₅	1	1	yes	characteristic
188	C₄×C₄⋊C₄	—	2	1	no	—
189	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
190	C₄×C₂²⋊C₄	—	2	1	no	—
191	C₄×C₂²⋊C₄	—	2	1	no	—
192	C₄³	C₂²×C₅	1	1	yes	characteristic
193	#340	C₂²×C₅	1	2	yes	—
194	#340	C₂²×C₅	1	2	yes	—
195	#336	C₂²×C₅	1	2	yes	—
196	#336	C₂²×C₅	1	2	yes	—
197	C₂³×C₅	—	2	2	no	—
198	C₂³×C₅	—	4	1	no	—
199	C₂³×C₅	—	4	1	no	—
200	C₂³×C₅	—	4	1	no	—
201	C₂³×C₅	—	4	1	no	—
202	C₂³×C₅	—	4	1	no	—
203	C₂³×C₅	—	4	1	no	—
204	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
205	C₅×Q₈	—	2	4	no	—
206	C₅×Q₈	—	4	8	no	—
207	C₅×Q₈	—	4	2	no	—
208	C₅×D₄	—	2	4	no	—
209	C₅×D₄	—	4	2	no	—
210	C₂×C₄×C₅	—	2	1	no	—
211	C₂×C₄×C₅	—	2	1	no	—
212	C₂×C₄×C₅	—	2	1	no	—
213	C₂×C₄×C₅	—	2	1	no	—
214	C₂×C₄×C₅	—	2	1	no	—
215	C₂×C₄×C₅	—	2	1	no	—
216	C₂×C₄×C₅	—	2	1	no	—
217	C₂×C₄×C₅	—	2	1	no	—
218	C₂×C₄×C₅	—	2	2	no	—
219	C₂×C₄×C₅	—	2	4	no	—
220	C₂×C₄×C₅	—	2	1	no	—
221	C₂×C₄×C₅	—	2	1	no	—
222	C₂×C₄×C₅	—	2	1	no	—
223	C₂×C₄×C₅	—	2	1	no	—
224	C₂×C₄×C₅	—	2	1	no	—
225	C₂×C₄×C₅	—	2	1	no	—
226	C₂×C₄×C₅	—	4	1	no	—
227	C₂×C₄×C₅	—	4	1	no	—
228	C₂×C₄×C₅	—	4	2	no	—
229	C₂×C₄×C₅	—	4	2	no	—
230	C₂×C₄×C₅	—	4	1	no	—
231	C₂×C₄×C₅	—	4	1	no	—
232	C₂×C₄×C₅	—	4	2	no	—
233	C₂×C₄×C₅	—	4	2	no	—
234	C₂×C₄×C₅	—	4	1	no	—
235	C₂×C₄×C₅	—	4	1	no	—
236	C₂×C₄×C₅	—	4	1	no	—
237	C₂×C₄×C₅	—	4	2	no	—
238	C₂×C₄×C₅	—	4	1	no	—
239	C₂×C₄×C₅	—	8	4	no	—
240	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
241	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
242	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
243	C₂×C₄×C₅	#137	1	2	yes	—
244	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
245	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
246	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
247	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
248	C₅×C₈	—	2	4	no	—
249	C₅×C₈	—	4	2	no	—
250	C₂²×Q₈	—	2	1	no	—
251	C₂²×Q₈	C₂³×C₅	1	1	yes	characteristic
252	C₂²×D₄	C₂³×C₅	1	1	yes	characteristic
253	C₂³×C₄	—	2	1	no	—
254	C₂²×C₈	C₂³×C₅	1	2	yes	—
255	C₂²×C₈	C₂³×C₅	1	2	yes	—
256	C₄⋊Q₈	—	2	2	no	—
257	C₄⋊Q₈	C₂×C₄×C₅	1	4	yes	complemented
258	#124	—	2	2	no	—
259	#124	C₂×C₄×C₅	1	4	yes	—
260	C₄×Q₈	—	4	2	no	—
261	C₄×Q₈	—	4	4	no	—
262	C₄×D₄	—	4	2	no	—
263	C₂×C₄⋊C₄	—	2	1	no	—
264	C₂×C₄⋊C₄	—	2	1	no	—
265	C₂×C₄⋊C₄	—	2	2	no	—
266	C₂×C₄⋊C₄	—	2	1	no	—
267	C₂×C₄⋊C₄	—	2	1	no	—
268	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
269	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
270	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
271	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
272	C₂×C₂²⋊C₄	—	2	1	no	—
273	C₂×C₂²⋊C₄	—	2	1	no	—
274	C₂×C₂²⋊C₄	—	2	1	no	—
275	C₂×C₂²⋊C₄	—	2	1	no	—
276	C₂×C₂²⋊C₄	—	2	1	no	—
277	C₂×C₄²	—	2	2	no	—
278	C₂×C₄²	—	2	1	no	—
279	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
280	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
281	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
282	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
283	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
284	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
285	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
286	#107	—	4	2	no	—
287	#106	—	4	2	no	—
288	#103	—	4	2	no	—
289	#103	C₂×C₄×C₅	1	8	yes	complemented
290	#102	—	4	2	no	—
291	#102	C₂×C₄×C₅	1	8	yes	—
292	#97	—	2	2	no	—
293	#97	C₂×C₄×C₅	1	4	yes	complemented
294	#95	—	2	1	no	—
295	#95	—	2	1	no	—
296	#95	—	2	1	no	—
297	#95	—	2	1	no	—
298	#95	—	2	2	no	—
299	#95	—	2	2	no	—
300	C₂²×C₅	—	4	2	no	—
301	C₂²×C₅	—	4	2	no	—
302	C₂²×C₅	—	4	2	no	—
303	C₂²×C₅	—	8	1	no	—
304	C₂²×C₅	—	8	1	no	—
305	C₂²×C₅	—	8	1	no	—
306	C₂²×C₅	—	8	1	no	—
307	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
308	C₂²×C₅	#535	1	1	yes	characteristic
309	C₂²×C₅	#536	1	1	yes	characteristic
310	C₂²×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
311	C₂²×C₅	C₂×#137	1	1	yes	characteristic
312	C₂²×C₅	#575	1	1	yes	characteristic
313	C₂²×C₅	#575	1	1	yes	characteristic
314	C₄×C₅	—	2	2	no	—
315	C₄×C₅	—	2	2	no	—
316	C₄×C₅	—	2	2	no	—
317	C₄×C₅	—	2	2	no	—
318	C₄×C₅	—	4	2	no	—
319	C₄×C₅	—	4	4	no	—
320	C₄×C₅	—	4	1	no	—
321	C₄×C₅	—	4	1	no	—
322	C₄×C₅	—	8	1	no	—
323	C₄×C₅	—	8	2	no	—
324	C₄×C₅	—	8	1	no	—
325	C₄×C₅	#520	1	2	yes	—
326	C₄×C₅	#542	1	2	yes	—
327	C₂⁴	—	2	1	no	—
328	C₂×Q₈	—	2	4	no	—
329	C₂×Q₈	—	2	1	no	—
330	C₂×Q₈	—	2	2	no	—
331	C₂×Q₈	—	2	2	no	—
332	C₂×Q₈	—	4	2	no	—
333	C₂×Q₈	—	4	2	no	—
334	C₂×Q₈	C₂²×C₄×C₅	1	2	yes	—
335	C₂×D₄	—	2	1	no	—
336	C₂×D₄	—	2	2	no	—
337	C₂×D₄	—	2	2	no	—
338	C₂×D₄	C₂²×C₄×C₅	1	2	yes	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	2	1	no	—
342	C₂²×C₄	—	2	1	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	4	2	no	—
345	C₂²×C₄	—	4	2	no	—
346	C₂²×C₄	—	4	2	no	—
347	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
348	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
349	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
350	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
351	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
352	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
353	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	—	2	2	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
360	C₄⋊C₄	—	2	4	no	—
361	C₄⋊C₄	—	2	2	no	—
362	C₄⋊C₄	—	2	2	no	—
363	C₄⋊C₄	—	2	2	no	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	4	2	no	—
367	C₄⋊C₄	—	4	2	no	—
368	C₄⋊C₄	—	4	4	no	—
369	C₄⋊C₄	—	4	2	no	—
370	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
371	C₂²⋊C₄	—	2	4	no	—
372	C₂²⋊C₄	—	2	4	no	—
373	C₂²⋊C₄	—	4	2	no	—
374	C₂²⋊C₄	—	4	2	no	—
375	C₂²⋊C₄	—	4	2	no	—
376	C₄²	—	2	2	no	—
377	C₄²	—	2	2	no	—
378	C₄²	—	2	2	no	—
379	C₄²	—	2	2	no	—
380	C₄²	—	4	1	no	—
381	C₄²	—	4	1	no	—
382	C₄²	—	4	2	no	—
383	C₄²	—	4	4	no	—
384	C₄²	C₂²×C₄×C₅	1	2	yes	—
385	C₄²	C₅×#41	1	2	yes	—
386	C₂×C₅	—	8	2	no	—
387	C₂×C₅	#2326	1	1	yes	characteristic
388	C₂×C₅	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
389	C₂×C₅	C₄×#137	1	1	yes	characteristic
390	C₂×C₅	#3214	1	1	yes	characteristic
391	C₂×C₅	#3215	1	1	yes	characteristic
392	C₂×C₅	#3238	1	1	yes	characteristic
393	C₂×C₅	#3240	1	1	yes	characteristic
394	C₂³	—	2	2	no	—
395	C₂³	—	4	1	no	—
396	C₂³	—	4	1	no	—
397	C₂³	—	4	1	no	—
398	C₂³	—	4	1	no	—
399	C₂³	—	4	1	no	—
400	C₂³	—	4	1	no	—
401	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
402	Q₈	—	2	4	no	—
403	Q₈	—	4	8	no	—
404	Q₈	—	4	2	no	—
405	D₄	—	2	4	no	—
406	D₄	—	4	2	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	4	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	4	1	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	1	no	—
428	C₂×C₄	—	4	1	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	1	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	4	1	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	8	4	no	—
437	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
438	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
439	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
440	C₂×C₄	C₅×#137	1	2	yes	—
441	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
442	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
443	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
444	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
445	C₈	—	2	4	no	—
446	C₈	—	4	2	no	—
447	C₅	#22224	1	1	yes	characteristic, direct product factor, Sylow
448	C₂²	—	4	2	no	—
449	C₂²	—	4	2	no	—
450	C₂²	—	4	2	no	—
451	C₂²	—	8	1	no	—
452	C₂²	—	8	1	no	—
453	C₂²	—	8	1	no	—
454	C₂²	—	8	1	no	—
455	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
456	C₂²	C₅×#535	1	1	yes	characteristic
457	C₂²	C₅×#536	1	1	yes	characteristic
458	C₂²	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
459	C₂²	C₂×C₅×#137	1	1	yes	characteristic
460	C₂²	C₅×#575	1	1	yes	characteristic
461	C₂²	C₅×#575	1	1	yes	characteristic
462	C₄	—	2	2	no	—
463	C₄	—	2	2	no	—
464	C₄	—	2	2	no	—
465	C₄	—	2	2	no	—
466	C₄	—	4	1	no	—
467	C₄	—	4	2	no	—
468	C₄	—	4	4	no	—
469	C₄	—	4	1	no	—
470	C₄	—	8	1	no	—
471	C₄	—	8	2	no	—
472	C₄	—	8	1	no	—
473	C₄	C₅×#520	1	2	yes	—
474	C₄	C₅×#542	1	2	yes	—
475	C₂	—	8	2	no	—
476	C₂	C₅×#2326	1	1	yes	characteristic
477	C₂	C₄×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
478	C₂	C₄×C₅×#137	1	1	yes	characteristic
479	C₂	C₅×#3214	1	1	yes	characteristic
480	C₂	C₅×#3215	1	1	yes	characteristic
481	C₂	C₅×#3238	1	1	yes	characteristic
482	C₂	C₅×#3240	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂²×C₅×D₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₅×C₄⋊Q₈	C₂	semidirect
4. C₅×#1993	C₂	semidirect
6. C₅×#1927	C₂	semidirect
9. C₅×#1863	C₂	semidirect
10. C₅×#1847	C₂	semidirect
11. C₅×#1745	C₂	semidirect
17. C₅×#489	C₄	semidirect
44. #22224	C₅	direct (N × H)
52. C₅×C₄⋊Q₈	C₂×C₄	semidirect
84. C₅×#103	C₂×C₄	semidirect
88. C₅×#97	C₂×C₄	semidirect
96. C₄×C₄⋊Q₈	C₂×C₅	semidirect
98. #1993	C₂×C₅	semidirect
100. #1927	C₂×C₅	semidirect
103. #1863	C₂×C₅	semidirect
104. #1847	C₂×C₅	semidirect
105. #1745	C₂×C₅	semidirect
170. #489	C₄×C₅	semidirect
257. C₄⋊Q₈	C₂×C₄×C₅	semidirect
289. #103	C₂×C₄×C₅	semidirect
293. #97	C₂×C₄×C₅	semidirect
447. C₅	#22224	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#489	2
C₄×C₅×C₄⋊Q₈	2
C₄×C₅×#124	2
C₅×#1993	2
C₅×#1959	2
C₅×#1927	2
C₅×#1925	2
C₅×#1864	2
C₅×#1863	2
C₅×#1847	2
C₅×#1745	2
#22224	5

Subgroups — C5×#22224 #11944944

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22224 #11944944