Subgroups — C₅×#22194 #11944914

Order 1280.

878 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#465	C₂	1	1	yes	maximal, characteristic
2	C₅×#2324	C₂	1	1	yes	maximal, characteristic
3	C₅×#2323	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1928	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1925	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1920	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1919	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1895	C₂	1	1	yes	maximal, characteristic
9	C₅×#1894	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#1880	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#1877	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#1863	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅×#1862	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×#1808	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅×#1729	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₅×SD₁₆	C₂²	1	1	yes	characteristic
17	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
18	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₅×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₅×#121	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
23	C₂×C₅×#117	C₂²	1	1	yes	characteristic
24	C₅×#465	C₄	1	8	yes	complemented
25	C₂×C₅×#107	C₂²	1	1	yes	characteristic
26	C₂×C₅×#107	C₂²	1	1	yes	characteristic
27	C₂×C₅×#106	C₂²	1	1	yes	characteristic
28	C₂×C₅×#103	C₂²	1	1	yes	characteristic
29	C₂×C₅×#103	C₂²	1	1	yes	characteristic
30	C₂×C₅×#103	C₂²	1	1	yes	characteristic
31	C₂×C₅×#102	C₂²	1	1	yes	characteristic
32	C₂×C₅×#102	C₂²	1	1	yes	characteristic
33	C₂×C₅×#102	C₂²	1	1	yes	characteristic
34	C₅×#391	C₂²	1	1	yes	characteristic
35	C₅×#390	—	2	1	no	—
36	C₅×#390	C₂²	1	1	yes	characteristic
37	C₅×#389	—	2	1	no	—
38	C₅×#389	C₂²	1	1	yes	characteristic
39	C₅×#389	C₂²	1	1	yes	characteristic
40	C₅×#388	—	2	1	no	—
41	C₅×#388	—	2	1	no	—
42	C₅×#388	—	2	1	no	—
43	C₅×#387	—	2	1	no	—
44	C₅×#386	—	2	1	no	—
45	C₅×#385	—	2	1	no	—
46	C₅×#380	C₂²	1	1	yes	characteristic
47	C₅×#380	C₂²	1	1	yes	characteristic
48	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
49	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
50	C₅×#340	C₂²	1	1	yes	characteristic
51	C₅×#340	C₂²	1	1	yes	characteristic
52	C₅×#340	C₂²	1	1	yes	characteristic
53	C₅×#340	C₂²	1	1	yes	characteristic
54	C₅×#340	C₂²	1	1	yes	characteristic
55	C₅×#340	C₂²	1	1	yes	characteristic
56	C₅×#336	C₂²	1	1	yes	characteristic
57	C₅×#336	C₂²	1	1	yes	characteristic
58	C₅×#336	C₂²	1	1	yes	characteristic
59	C₅×#336	C₂²	1	1	yes	characteristic
60	#22194	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂²×C₅×Q₈	C₂³	1	1	yes	characteristic
62	C₂²×C₅×D₄	—	2	1	no	—
63	C₂²×C₅×D₄	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₅	—	2	1	no	—
65	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
66	C₂×C₅×SD₁₆	—	4	1	no	—
67	C₂×C₅×SD₁₆	—	4	1	no	—
68	C₂×C₅×SD₁₆	C₂×C₄	1	4	yes	—
69	C₂×C₅×M₁₆	—	2	2	no	—
70	C₂×C₅×M₁₆	—	4	1	no	—
71	C₂×C₅×M₁₆	—	4	1	no	—
72	C₂²×C₅×C₈	—	4	1	no	—
73	C₂²×C₅×C₈	—	4	1	no	—
74	C₂²×C₅×C₈	C₂×C₄	1	4	yes	complemented
75	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
76	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
77	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
78	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
79	C₅×C₂²⋊Q₈	—	2	2	no	—
80	C₅×C₂²⋊Q₈	C₂×C₄	1	4	yes	complemented
81	C₅×#121	—	2	2	no	—
82	C₅×#121	C₂×C₄	1	4	yes	—
83	C₄×C₅×Q₈	—	4	1	no	—
84	C₄×C₅×Q₈	—	4	1	no	—
85	C₄×C₅×D₄	—	4	1	no	—
86	C₄×C₅×D₄	—	4	1	no	—
87	C₅×#117	—	4	1	no	—
88	C₅×#117	—	4	1	no	—
89	C₂×C₅×C₄⋊C₄	—	2	1	no	—
90	C₂×C₅×C₄⋊C₄	—	2	1	no	—
91	C₂×C₅×C₄⋊C₄	—	2	1	no	—
92	C₂×C₅×C₄⋊C₄	—	2	1	no	—
93	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
94	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
95	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
96	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
97	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
98	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
99	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
100	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
101	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
102	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
103	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
104	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
105	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
106	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
107	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
108	C₂×C₅×C₂²⋊C₄	D₄	1	1	yes	characteristic
109	C₂×C₅×C₂²⋊C₄	D₄	1	1	yes	characteristic
110	C₂×C₄²×C₅	—	2	1	no	—
111	C₂×C₄²×C₅	—	2	1	no	—
112	C₂×C₄²×C₅	—	2	1	no	—
113	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
115	C₅×#107	—	4	1	no	—
116	C₅×#107	—	4	1	no	—
117	C₅×#106	C₂×C₄	1	4	yes	complemented
118	C₅×#103	—	4	1	no	—
119	C₅×#103	—	4	1	no	—
120	C₅×#103	C₂×C₄	1	4	yes	complemented
121	C₅×#102	—	4	1	no	—
122	C₅×#102	—	4	1	no	—
123	C₅×#102	C₂×C₄	1	4	yes	—
124	C₅×#95	—	2	1	no	—
125	C₅×#95	—	2	1	no	—
126	C₅×#95	—	2	1	no	—
127	C₅×#95	—	2	1	no	—
128	C₅×#95	—	2	1	no	—
129	C₅×#95	—	2	1	no	—
130	C₅×#95	—	2	1	no	—
131	C₅×#95	—	2	1	no	—
132	C₂×#465	C₂×C₅	1	1	yes	characteristic
133	#2324	C₂×C₅	1	1	yes	characteristic
134	#2323	C₂×C₅	1	1	yes	characteristic, complemented
135	#1928	C₂×C₅	1	1	yes	characteristic, complemented
136	#1925	C₂×C₅	1	1	yes	characteristic, complemented
137	#1920	C₂×C₅	1	1	yes	characteristic, complemented
138	#1919	C₂×C₅	1	1	yes	characteristic, complemented
139	#1895	C₂×C₅	1	1	yes	characteristic
140	#1894	C₂×C₅	1	1	yes	characteristic, complemented
141	#1880	C₂×C₅	1	1	yes	characteristic, complemented
142	#1877	C₂×C₅	1	1	yes	characteristic, complemented
143	#1863	C₂×C₅	1	1	yes	characteristic, complemented
144	#1862	C₂×C₅	1	1	yes	characteristic, complemented
145	#1808	C₂×C₅	1	1	yes	characteristic, complemented
146	#1729	C₂×C₅	1	1	yes	characteristic, complemented
147	C₂⁴×C₅	—	2	1	no	—
148	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
149	C₂×C₅×Q₈	—	2	1	no	—
150	C₂×C₅×Q₈	—	2	1	no	—
151	C₂×C₅×Q₈	—	2	1	no	—
152	C₂×C₅×Q₈	—	2	1	no	—
153	C₂×C₅×Q₈	—	2	1	no	—
154	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
155	C₂×C₅×D₄	—	2	4	no	—
156	C₂×C₅×D₄	—	2	1	no	—
157	C₂×C₅×D₄	—	2	1	no	—
158	C₂×C₅×D₄	—	2	1	no	—
159	C₂×C₅×D₄	—	2	1	no	—
160	C₂×C₅×D₄	—	2	1	no	—
161	C₂×C₅×D₄	—	8	1	no	—
162	C₂×C₅×D₄	—	8	1	no	—
163	C₂×C₅×D₄	C₂²×C₄	1	2	yes	—
164	C₂²×C₄×C₅	—	2	1	no	—
165	C₂²×C₄×C₅	—	2	1	no	—
166	C₂²×C₄×C₅	—	2	1	no	—
167	C₂²×C₄×C₅	—	2	1	no	—
168	C₂²×C₄×C₅	—	2	1	no	—
169	C₂²×C₄×C₅	—	2	1	no	—
170	C₂²×C₄×C₅	—	2	1	no	—
171	C₂²×C₄×C₅	—	2	1	no	—
172	C₂²×C₄×C₅	—	2	1	no	—
173	C₂²×C₄×C₅	—	2	1	no	—
174	C₂²×C₄×C₅	—	2	1	no	—
175	C₂²×C₄×C₅	—	4	2	no	—
176	C₂²×C₄×C₅	—	8	1	no	—
177	C₂²×C₄×C₅	—	8	1	no	—
178	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
179	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
182	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
183	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
184	C₂²×C₄×C₅	#41	1	2	yes	—
185	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
186	C₅×SD₁₆	—	4	4	no	—
187	C₅×M₁₆	—	4	4	no	—
188	C₂×C₅×C₈	—	2	1	no	—
189	C₂×C₅×C₈	—	2	1	no	—
190	C₂×C₅×C₈	—	2	1	no	—
191	C₂×C₅×C₈	—	2	1	no	—
192	C₂×C₅×C₈	—	2	1	no	—
193	C₂×C₅×C₈	—	2	1	no	—
194	C₂×C₅×C₈	—	2	1	no	—
195	C₂×C₅×C₈	—	2	1	no	—
196	C₂×C₅×C₈	—	2	1	no	—
197	C₂×C₅×C₈	—	2	1	no	—
198	C₂×C₅×C₈	—	4	4	no	—
199	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
200	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
201	C₅×C₄⋊C₄	—	2	4	no	—
202	C₅×C₄⋊C₄	—	2	1	no	—
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	—	2	1	no	—
207	C₅×C₄⋊C₄	—	2	1	no	—
208	C₅×C₄⋊C₄	—	2	1	no	—
209	C₅×C₄⋊C₄	—	2	1	no	—
210	C₅×C₄⋊C₄	—	2	1	no	—
211	C₅×C₄⋊C₄	—	2	1	no	—
212	C₅×C₄⋊C₄	—	4	1	no	—
213	C₅×C₄⋊C₄	—	4	1	no	—
214	C₅×C₄⋊C₄	—	4	2	no	—
215	C₅×C₄⋊C₄	—	8	1	no	—
216	C₅×C₄⋊C₄	—	8	1	no	—
217	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
218	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
219	C₅×C₂²⋊C₄	—	2	4	no	—
220	C₅×C₂²⋊C₄	—	2	4	no	—
221	C₅×C₂²⋊C₄	—	4	1	no	—
222	C₅×C₂²⋊C₄	—	4	1	no	—
223	C₅×C₂²⋊C₄	—	4	2	no	—
224	C₅×C₂²⋊C₄	—	8	1	no	—
225	C₅×C₂²⋊C₄	—	8	1	no	—
226	C₅×C₂²⋊C₄	—	8	1	no	—
227	C₅×C₂²⋊C₄	—	8	1	no	—
228	C₄²×C₅	—	2	1	no	—
229	C₄²×C₅	—	2	1	no	—
230	C₄²×C₅	—	2	1	no	—
231	C₄²×C₅	—	2	1	no	—
232	C₄²×C₅	—	4	2	no	—
233	C₄²×C₅	—	8	1	no	—
234	C₄²×C₅	—	8	1	no	—
235	C₂²×SD₁₆	C₂²×C₅	1	1	yes	characteristic
236	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
237	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
238	C₂×C₂²⋊Q₈	C₂²×C₅	1	1	yes	characteristic
239	C₂×#121	C₂²×C₅	1	1	yes	characteristic
240	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
241	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
242	C₂×#117	C₂²×C₅	1	1	yes	characteristic
243	#465	C₄×C₅	1	8	yes	complemented
244	C₂×#107	C₂²×C₅	1	1	yes	characteristic
245	C₂×#107	C₂²×C₅	1	1	yes	characteristic
246	C₂×#106	C₂²×C₅	1	1	yes	characteristic
247	C₂×#103	C₂²×C₅	1	1	yes	characteristic
248	C₂×#103	C₂²×C₅	1	1	yes	characteristic
249	C₂×#103	C₂²×C₅	1	1	yes	characteristic
250	C₂×#102	C₂²×C₅	1	1	yes	characteristic
251	C₂×#102	C₂²×C₅	1	1	yes	characteristic
252	C₂×#102	C₂²×C₅	1	1	yes	characteristic
253	#391	C₂²×C₅	1	1	yes	characteristic
254	#390	—	2	1	no	—
255	#390	C₂²×C₅	1	1	yes	characteristic
256	#389	—	2	1	no	—
257	#389	C₂²×C₅	1	1	yes	characteristic
258	#389	C₂²×C₅	1	1	yes	characteristic
259	#388	—	2	1	no	—
260	#388	—	2	1	no	—
261	#388	—	2	1	no	—
262	#387	—	2	1	no	—
263	#386	—	2	1	no	—
264	#385	—	2	1	no	—
265	#380	C₂²×C₅	1	1	yes	characteristic
266	#380	C₂²×C₅	1	1	yes	characteristic
267	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
268	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
269	#340	C₂²×C₅	1	1	yes	characteristic
270	#340	C₂²×C₅	1	1	yes	characteristic
271	#340	C₂²×C₅	1	1	yes	characteristic
272	#340	C₂²×C₅	1	1	yes	characteristic
273	#340	C₂²×C₅	1	1	yes	characteristic
274	#340	C₂²×C₅	1	1	yes	characteristic
275	#336	C₂²×C₅	1	1	yes	characteristic
276	#336	C₂²×C₅	1	1	yes	characteristic
277	#336	C₂²×C₅	1	1	yes	characteristic
278	#336	C₂²×C₅	1	1	yes	characteristic
279	C₂³×C₅	—	2	1	no	—
280	C₂³×C₅	—	2	1	no	—
281	C₂³×C₅	—	2	1	no	—
282	C₂³×C₅	—	2	1	no	—
283	C₂³×C₅	—	2	1	no	—
284	C₂³×C₅	—	2	1	no	—
285	C₂³×C₅	—	2	2	no	—
286	C₂³×C₅	—	4	1	no	—
287	C₂³×C₅	—	4	1	no	—
288	C₂³×C₅	—	4	1	no	—
289	C₂³×C₅	—	4	1	no	—
290	C₂³×C₅	—	4	1	no	—
291	C₂³×C₅	—	4	1	no	—
292	C₂³×C₅	C₄×D₄	1	2	yes	—
293	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
294	C₅×Q₈	—	2	2	no	—
295	C₅×Q₈	—	2	2	no	—
296	C₅×Q₈	—	4	1	no	—
297	C₅×Q₈	—	4	1	no	—
298	C₅×D₄	—	2	2	no	—
299	C₅×D₄	—	2	2	no	—
300	C₅×D₄	—	4	1	no	—
301	C₅×D₄	—	4	1	no	—
302	C₅×D₄	—	8	4	no	—
303	C₂×C₄×C₅	—	2	1	no	—
304	C₂×C₄×C₅	—	2	1	no	—
305	C₂×C₄×C₅	—	2	2	no	—
306	C₂×C₄×C₅	—	2	2	no	—
307	C₂×C₄×C₅	—	2	2	no	—
308	C₂×C₄×C₅	—	2	2	no	—
309	C₂×C₄×C₅	—	2	2	no	—
310	C₂×C₄×C₅	—	2	2	no	—
311	C₂×C₄×C₅	—	2	2	no	—
312	C₂×C₄×C₅	—	2	1	no	—
313	C₂×C₄×C₅	—	2	1	no	—
314	C₂×C₄×C₅	—	2	1	no	—
315	C₂×C₄×C₅	—	2	1	no	—
316	C₂×C₄×C₅	—	2	1	no	—
317	C₂×C₄×C₅	—	2	1	no	—
318	C₂×C₄×C₅	—	2	1	no	—
319	C₂×C₄×C₅	—	2	1	no	—
320	C₂×C₄×C₅	—	2	1	no	—
321	C₂×C₄×C₅	—	2	1	no	—
322	C₂×C₄×C₅	—	2	1	no	—
323	C₂×C₄×C₅	—	2	1	no	—
324	C₂×C₄×C₅	—	4	1	no	—
325	C₂×C₄×C₅	—	4	1	no	—
326	C₂×C₄×C₅	—	4	1	no	—
327	C₂×C₄×C₅	—	4	1	no	—
328	C₂×C₄×C₅	—	4	1	no	—
329	C₂×C₄×C₅	—	4	1	no	—
330	C₂×C₄×C₅	—	4	1	no	—
331	C₂×C₄×C₅	—	4	1	no	—
332	C₂×C₄×C₅	—	4	1	no	—
333	C₂×C₄×C₅	—	4	1	no	—
334	C₂×C₄×C₅	—	4	1	no	—
335	C₂×C₄×C₅	—	4	1	no	—
336	C₂×C₄×C₅	—	4	1	no	—
337	C₂×C₄×C₅	—	4	1	no	—
338	C₂×C₄×C₅	—	4	1	no	—
339	C₂×C₄×C₅	—	4	1	no	—
340	C₂×C₄×C₅	—	4	1	no	—
341	C₂×C₄×C₅	—	4	1	no	—
342	C₂×C₄×C₅	—	8	4	no	—
343	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
344	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
345	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
346	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
347	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
348	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
349	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
350	C₅×C₈	—	2	2	no	—
351	C₅×C₈	—	2	2	no	—
352	C₅×C₈	—	4	1	no	—
353	C₅×C₈	—	4	1	no	—
354	C₂²×Q₈	C₂³×C₅	1	1	yes	characteristic
355	C₂²×D₄	—	2	1	no	—
356	C₂²×D₄	C₂³×C₅	1	1	yes	characteristic
357	C₂³×C₄	—	2	1	no	—
358	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
359	C₂×SD₁₆	—	4	1	no	—
360	C₂×SD₁₆	—	4	1	no	—
361	C₂×SD₁₆	C₂×C₄×C₅	1	4	yes	—
362	C₂×M₁₆	—	2	2	no	—
363	C₂×M₁₆	—	4	1	no	—
364	C₂×M₁₆	—	4	1	no	—
365	C₂²×C₈	—	4	1	no	—
366	C₂²×C₈	—	4	1	no	—
367	C₂²×C₈	C₂×C₄×C₅	1	4	yes	complemented
368	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
369	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
370	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
371	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
372	C₂²⋊Q₈	—	2	2	no	—
373	C₂²⋊Q₈	C₂×C₄×C₅	1	4	yes	complemented
374	#121	—	2	2	no	—
375	#121	C₂×C₄×C₅	1	4	yes	—
376	C₄×Q₈	—	4	1	no	—
377	C₄×Q₈	—	4	1	no	—
378	C₄×D₄	—	4	1	no	—
379	C₄×D₄	—	4	1	no	—
380	#117	—	4	1	no	—
381	#117	—	4	1	no	—
382	C₂×C₄⋊C₄	—	2	1	no	—
383	C₂×C₄⋊C₄	—	2	1	no	—
384	C₂×C₄⋊C₄	—	2	1	no	—
385	C₂×C₄⋊C₄	—	2	1	no	—
386	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
387	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
388	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
389	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
390	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
391	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
392	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
393	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
394	C₂×C₂²⋊C₄	—	2	1	no	—
395	C₂×C₂²⋊C₄	—	2	1	no	—
396	C₂×C₂²⋊C₄	—	2	1	no	—
397	C₂×C₂²⋊C₄	—	2	1	no	—
398	C₂×C₂²⋊C₄	—	2	1	no	—
399	C₂×C₂²⋊C₄	—	2	1	no	—
400	C₂×C₂²⋊C₄	—	2	1	no	—
401	C₂×C₂²⋊C₄	C₅×D₄	1	1	yes	characteristic
402	C₂×C₂²⋊C₄	C₅×D₄	1	1	yes	characteristic
403	C₂×C₄²	—	2	1	no	—
404	C₂×C₄²	—	2	1	no	—
405	C₂×C₄²	—	2	1	no	—
406	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
407	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
408	#107	—	4	1	no	—
409	#107	—	4	1	no	—
410	#106	C₂×C₄×C₅	1	4	yes	complemented
411	#103	—	4	1	no	—
412	#103	—	4	1	no	—
413	#103	C₂×C₄×C₅	1	4	yes	complemented
414	#102	—	4	1	no	—
415	#102	—	4	1	no	—
416	#102	C₂×C₄×C₅	1	4	yes	—
417	#95	—	2	1	no	—
418	#95	—	2	1	no	—
419	#95	—	2	1	no	—
420	#95	—	2	1	no	—
421	#95	—	2	1	no	—
422	#95	—	2	1	no	—
423	#95	—	2	1	no	—
424	#95	—	2	1	no	—
425	C₂²×C₅	—	2	2	no	—
426	C₂²×C₅	—	2	2	no	—
427	C₂²×C₅	—	2	2	no	—
428	C₂²×C₅	—	4	1	no	—
429	C₂²×C₅	—	4	1	no	—
430	C₂²×C₅	—	4	1	no	—
431	C₂²×C₅	—	4	1	no	—
432	C₂²×C₅	—	4	2	no	—
433	C₂²×C₅	—	4	2	no	—
434	C₂²×C₅	—	4	2	no	—
435	C₂²×C₅	—	8	1	no	—
436	C₂²×C₅	—	8	1	no	—
437	C₂²×C₅	—	8	1	no	—
438	C₂²×C₅	—	8	1	no	—
439	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
440	C₂²×C₅	#534	1	1	yes	characteristic
441	C₂²×C₅	#537	1	1	yes	characteristic
442	C₂²×C₅	#576	1	1	yes	characteristic
443	C₂²×C₅	#577	1	1	yes	characteristic
444	C₂²×C₅	#577	1	1	yes	characteristic
445	C₂²×C₅	#578	1	1	yes	characteristic
446	C₄×C₅	—	2	2	no	—
447	C₄×C₅	—	4	2	no	—
448	C₄×C₅	—	4	1	no	—
449	C₄×C₅	—	4	1	no	—
450	C₄×C₅	—	4	1	no	—
451	C₄×C₅	—	4	1	no	—
452	C₄×C₅	—	4	2	no	—
453	C₄×C₅	—	4	2	no	—
454	C₄×C₅	—	8	1	no	—
455	C₄×C₅	—	8	1	no	—
456	C₄×C₅	—	8	1	no	—
457	C₄×C₅	—	8	1	no	—
458	C₄×C₅	#518	1	1	yes	characteristic
459	C₄×C₅	#520	1	1	yes	characteristic
460	C₄×C₅	#538	1	1	yes	characteristic
461	C₄×C₅	#539	1	1	yes	characteristic
462	C₂⁴	—	2	1	no	—
463	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
464	C₂×Q₈	—	2	1	no	—
465	C₂×Q₈	—	2	1	no	—
466	C₂×Q₈	—	2	1	no	—
467	C₂×Q₈	—	2	1	no	—
468	C₂×Q₈	—	2	1	no	—
469	C₂×Q₈	C₂²×C₄×C₅	1	2	yes	—
470	C₂×D₄	—	2	4	no	—
471	C₂×D₄	—	2	1	no	—
472	C₂×D₄	—	2	1	no	—
473	C₂×D₄	—	2	1	no	—
474	C₂×D₄	—	2	1	no	—
475	C₂×D₄	—	2	1	no	—
476	C₂×D₄	—	8	1	no	—
477	C₂×D₄	—	8	1	no	—
478	C₂×D₄	C₂²×C₄×C₅	1	2	yes	—
479	C₂²×C₄	—	2	1	no	—
480	C₂²×C₄	—	2	1	no	—
481	C₂²×C₄	—	2	1	no	—
482	C₂²×C₄	—	2	1	no	—
483	C₂²×C₄	—	2	1	no	—
484	C₂²×C₄	—	2	1	no	—
485	C₂²×C₄	—	2	1	no	—
486	C₂²×C₄	—	2	1	no	—
487	C₂²×C₄	—	2	1	no	—
488	C₂²×C₄	—	2	1	no	—
489	C₂²×C₄	—	2	1	no	—
490	C₂²×C₄	—	4	2	no	—
491	C₂²×C₄	—	8	1	no	—
492	C₂²×C₄	—	8	1	no	—
493	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
494	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
495	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
496	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
497	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
498	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
499	C₂²×C₄	C₅×#41	1	2	yes	—
500	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
501	SD₁₆	—	4	4	no	—
502	M₁₆	—	4	4	no	—
503	C₂×C₈	—	2	1	no	—
504	C₂×C₈	—	2	1	no	—
505	C₂×C₈	—	2	1	no	—
506	C₂×C₈	—	2	1	no	—
507	C₂×C₈	—	2	1	no	—
508	C₂×C₈	—	2	1	no	—
509	C₂×C₈	—	2	1	no	—
510	C₂×C₈	—	2	1	no	—
511	C₂×C₈	—	2	1	no	—
512	C₂×C₈	—	2	1	no	—
513	C₂×C₈	—	4	4	no	—
514	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
515	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
516	C₄⋊C₄	—	2	4	no	—
517	C₄⋊C₄	—	2	1	no	—
518	C₄⋊C₄	—	2	1	no	—
519	C₄⋊C₄	—	2	1	no	—
520	C₄⋊C₄	—	2	1	no	—
521	C₄⋊C₄	—	2	1	no	—
522	C₄⋊C₄	—	2	1	no	—
523	C₄⋊C₄	—	2	1	no	—
524	C₄⋊C₄	—	2	1	no	—
525	C₄⋊C₄	—	2	1	no	—
526	C₄⋊C₄	—	2	1	no	—
527	C₄⋊C₄	—	4	1	no	—
528	C₄⋊C₄	—	4	1	no	—
529	C₄⋊C₄	—	4	2	no	—
530	C₄⋊C₄	—	8	1	no	—
531	C₄⋊C₄	—	8	1	no	—
532	C₄⋊C₄	C₂²×C₄×C₅	1	2	yes	—
533	C₄⋊C₄	C₂²×C₄×C₅	1	2	yes	—
534	C₂²⋊C₄	—	2	4	no	—
535	C₂²⋊C₄	—	2	4	no	—
536	C₂²⋊C₄	—	4	1	no	—
537	C₂²⋊C₄	—	4	1	no	—
538	C₂²⋊C₄	—	4	2	no	—
539	C₂²⋊C₄	—	8	1	no	—
540	C₂²⋊C₄	—	8	1	no	—
541	C₂²⋊C₄	—	8	1	no	—
542	C₂²⋊C₄	—	8	1	no	—
543	C₄²	—	2	1	no	—
544	C₄²	—	2	1	no	—
545	C₄²	—	2	1	no	—
546	C₄²	—	2	1	no	—
547	C₄²	—	4	2	no	—
548	C₄²	—	8	1	no	—
549	C₄²	—	8	1	no	—
550	C₂×C₅	—	4	2	no	—
551	C₂×C₅	—	8	2	no	—
552	C₂×C₅	#2325	1	1	yes	characteristic
553	C₂×C₅	#2949	1	1	yes	characteristic
554	C₂×C₅	#2951	1	1	yes	characteristic
555	C₂×C₅	#3198	1	1	yes	characteristic
556	C₂×C₅	#3201	1	1	yes	characteristic
557	C₂×C₅	#3257	1	1	yes	characteristic
558	C₂×C₅	#3258	1	1	yes	characteristic
559	C₂³	—	2	1	no	—
560	C₂³	—	2	1	no	—
561	C₂³	—	2	1	no	—
562	C₂³	—	2	1	no	—
563	C₂³	—	2	1	no	—
564	C₂³	—	2	1	no	—
565	C₂³	—	2	2	no	—
566	C₂³	—	4	1	no	—
567	C₂³	—	4	1	no	—
568	C₂³	—	4	1	no	—
569	C₂³	—	4	1	no	—
570	C₂³	—	4	1	no	—
571	C₂³	—	4	1	no	—
572	C₂³	C₄×C₅×D₄	1	2	yes	—
573	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
574	Q₈	—	2	2	no	—
575	Q₈	—	2	2	no	—
576	Q₈	—	4	1	no	—
577	Q₈	—	4	1	no	—
578	D₄	—	2	2	no	—
579	D₄	—	2	2	no	—
580	D₄	—	4	1	no	—
581	D₄	—	4	1	no	—
582	D₄	—	8	4	no	—
583	C₂×C₄	—	2	1	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	2	2	no	—
586	C₂×C₄	—	2	2	no	—
587	C₂×C₄	—	2	2	no	—
588	C₂×C₄	—	2	2	no	—
589	C₂×C₄	—	2	2	no	—
590	C₂×C₄	—	2	2	no	—
591	C₂×C₄	—	2	2	no	—
592	C₂×C₄	—	2	1	no	—
593	C₂×C₄	—	2	1	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	2	1	no	—
596	C₂×C₄	—	2	1	no	—
597	C₂×C₄	—	2	1	no	—
598	C₂×C₄	—	2	1	no	—
599	C₂×C₄	—	2	1	no	—
600	C₂×C₄	—	2	1	no	—
601	C₂×C₄	—	2	1	no	—
602	C₂×C₄	—	2	1	no	—
603	C₂×C₄	—	2	1	no	—
604	C₂×C₄	—	4	1	no	—
605	C₂×C₄	—	4	1	no	—
606	C₂×C₄	—	4	1	no	—
607	C₂×C₄	—	4	1	no	—
608	C₂×C₄	—	4	1	no	—
609	C₂×C₄	—	4	1	no	—
610	C₂×C₄	—	4	1	no	—
611	C₂×C₄	—	4	1	no	—
612	C₂×C₄	—	4	1	no	—
613	C₂×C₄	—	4	1	no	—
614	C₂×C₄	—	4	1	no	—
615	C₂×C₄	—	4	1	no	—
616	C₂×C₄	—	4	1	no	—
617	C₂×C₄	—	4	1	no	—
618	C₂×C₄	—	4	1	no	—
619	C₂×C₄	—	4	1	no	—
620	C₂×C₄	—	4	1	no	—
621	C₂×C₄	—	4	1	no	—
622	C₂×C₄	—	8	4	no	—
623	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
624	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
625	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
626	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
627	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
628	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
629	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
630	C₈	—	2	2	no	—
631	C₈	—	2	2	no	—
632	C₈	—	4	1	no	—
633	C₈	—	4	1	no	—
634	C₅	#22194	1	1	yes	characteristic, direct product factor, Sylow
635	C₂²	—	2	2	no	—
636	C₂²	—	2	2	no	—
637	C₂²	—	2	2	no	—
638	C₂²	—	4	1	no	—
639	C₂²	—	4	1	no	—
640	C₂²	—	4	1	no	—
641	C₂²	—	4	1	no	—
642	C₂²	—	4	2	no	—
643	C₂²	—	4	2	no	—
644	C₂²	—	4	2	no	—
645	C₂²	—	8	1	no	—
646	C₂²	—	8	1	no	—
647	C₂²	—	8	1	no	—
648	C₂²	—	8	1	no	—
649	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
650	C₂²	C₅×#534	1	1	yes	characteristic
651	C₂²	C₅×#537	1	1	yes	characteristic
652	C₂²	C₅×#576	1	1	yes	characteristic
653	C₂²	C₅×#577	1	1	yes	characteristic
654	C₂²	C₅×#577	1	1	yes	characteristic
655	C₂²	C₅×#578	1	1	yes	characteristic
656	C₄	—	2	2	no	—
657	C₄	—	4	2	no	—
658	C₄	—	4	1	no	—
659	C₄	—	4	1	no	—
660	C₄	—	4	1	no	—
661	C₄	—	4	1	no	—
662	C₄	—	4	2	no	—
663	C₄	—	4	2	no	—
664	C₄	—	8	1	no	—
665	C₄	—	8	1	no	—
666	C₄	—	8	1	no	—
667	C₄	—	8	1	no	—
668	C₄	C₅×#518	1	1	yes	characteristic
669	C₄	C₅×#520	1	1	yes	characteristic
670	C₄	C₅×#538	1	1	yes	characteristic
671	C₄	C₅×#539	1	1	yes	characteristic
672	C₂	—	4	2	no	—
673	C₂	—	8	2	no	—
674	C₂	C₅×#2325	1	1	yes	characteristic
675	C₂	C₅×#2949	1	1	yes	characteristic
676	C₂	C₅×#2951	1	1	yes	characteristic
677	C₂	C₅×#3198	1	1	yes	characteristic
678	C₂	C₅×#3201	1	1	yes	characteristic
679	C₂	C₅×#3257	1	1	yes	characteristic
680	C₂	C₅×#3258	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	C₂×C₅×#121	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₅×#117 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#2323	C₂	semidirect
4. C₅×#1928	C₂	semidirect
5. C₅×#1925	C₂	semidirect
6. C₅×#1920	C₂	semidirect
7. C₅×#1919	C₂	semidirect
9. C₅×#1894	C₂	semidirect
10. C₅×#1880	C₂	semidirect
11. C₅×#1877	C₂	semidirect
12. C₅×#1863	C₂	semidirect
13. C₅×#1862	C₂	semidirect
14. C₅×#1808	C₂	semidirect
15. C₅×#1729	C₂	semidirect
24. C₅×#465	C₄	semidirect
60. #22194	C₅	direct (N × H)
74. C₂²×C₅×C₈	C₂×C₄	semidirect
80. C₅×C₂²⋊Q₈	C₂×C₄	semidirect
117. C₅×#106	C₂×C₄	semidirect
120. C₅×#103	C₂×C₄	semidirect
134. #2323	C₂×C₅	semidirect
135. #1928	C₂×C₅	semidirect
136. #1925	C₂×C₅	semidirect
137. #1920	C₂×C₅	semidirect
138. #1919	C₂×C₅	semidirect
140. #1894	C₂×C₅	semidirect
141. #1880	C₂×C₅	semidirect
142. #1877	C₂×C₅	semidirect
143. #1863	C₂×C₅	semidirect
144. #1862	C₂×C₅	semidirect
145. #1808	C₂×C₅	semidirect
146. #1729	C₂×C₅	semidirect
243. #465	C₄×C₅	semidirect
367. C₂²×C₈	C₂×C₄×C₅	semidirect
373. C₂²⋊Q₈	C₂×C₄×C₅	semidirect
410. #106	C₂×C₄×C₅	semidirect
413. #103	C₂×C₄×C₅	semidirect
634. C₅	#22194	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×#465	2
C₅×#2324	2
C₅×#2323	2
C₅×#1928	2
C₅×#1925	2
C₅×#1920	2
C₅×#1919	2
C₅×#1895	2
C₅×#1894	2
C₅×#1880	2
C₅×#1877	2
C₅×#1863	2
C₅×#1862	2
C₅×#1808	2
C₅×#1729	2
#22194	5

Subgroups — C5×#22194 #11944914

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22194 #11944914