Subgroups — C₅×#34322 #11957042

Order 1280.

2228 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#556	C₂	1	1	yes	maximal, characteristic
2	C₅×#2373	C₂	1	4	yes	maximal, complemented
3	C₅×#2367	C₂	1	2	yes	maximal
4	C₅×#2365	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#2360	C₂	1	2	yes	maximal, complemented
6	C₅×#2355	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#2351	C₂	1	2	yes	maximal, complemented
8	C₅×D₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₅×#2346	C₂	1	2	yes	maximal, complemented
10	C₅×#2334	C₂	1	4	yes	maximal, complemented
11	C₅×#2323	C₂	1	2	yes	maximal, complemented
12	C₄×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×C₅×#124	C₂	1	2	yes	maximal
14	C₅×#2295	C₂	1	2	yes	maximal, complemented
15	C₅×#2293	C₂	1	1	yes	maximal, characteristic
16	C₅×#2274	C₂	1	2	yes	maximal, complemented
17	C₄²×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
18	C₅×#556	C₄	1	16	yes	complemented
19	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₅×#126	C₂²	1	4	yes	—
21	C₂×C₅×#124	C₂²	1	2	yes	—
22	C₂×C₅×C₂²⋊Q₈	C₂²	1	2	yes	—
23	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₅×Q₈	C₂²	1	2	yes	—
26	C₂×C₄×C₅×Q₈	C₂²	1	2	yes	—
27	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₅×Q₈	C₂²	1	2	yes	—
29	C₂×C₄×C₅×D₄	C₂²	1	2	yes	—
30	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
31	C₂×C₅×#117	C₂²	1	2	yes	—
32	C₂×C₅×#117	C₂²	1	2	yes	—
33	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
34	C₅×#391	C₂²	1	2	yes	—
35	C₅×#391	C₂²	1	2	yes	—
36	C₅×#391	C₂²	1	2	yes	—
37	C₅×#390	C₂²	1	1	yes	characteristic
38	C₅×#390	C₂²	1	1	yes	characteristic
39	C₅×#389	C₂²	1	1	yes	characteristic
40	C₅×#389	C₂²	1	1	yes	characteristic
41	C₅×#389	C₂²	1	2	yes	—
42	C₅×#389	C₂²	1	2	yes	—
43	C₅×#389	C₂²	1	2	yes	—
44	C₅×#389	C₂²	1	2	yes	—
45	C₅×#389	C₂²	1	4	yes	—
46	C₅×#389	C₂²	1	4	yes	—
47	C₅×#388	C₂²	1	2	yes	—
48	C₅×#388	C₂²	1	2	yes	—
49	C₅×#388	C₂²	1	4	yes	—
50	C₅×#388	C₂²	1	4	yes	—
51	C₅×#388	C₂²	1	4	yes	—
52	C₅×#388	C₂²	1	4	yes	—
53	C₅×#387	C₂²	1	4	yes	—
54	C₅×#387	C₂²	1	4	yes	—
55	C₅×#387	C₂²	1	4	yes	—
56	C₅×#387	C₂²	1	4	yes	—
57	C₅×#385	C₂²	1	4	yes	—
58	C₅×#385	C₂²	1	4	yes	—
59	C₅×#384	C₂²	1	1	yes	characteristic
60	C₅×#382	C₂²	1	2	yes	—
61	C₅×#382	C₂²	1	4	yes	—
62	C₅×#380	C₂²	1	2	yes	—
63	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
64	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
65	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
66	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
67	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
68	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
69	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
70	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
71	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
72	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
73	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
74	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
75	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
76	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
77	C₄×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
78	C₄×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
79	C₄×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
80	C₄×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
81	C₅×#376	C₂²	1	4	yes	—
82	C₄³×C₅	C₂²	1	1	yes	characteristic
83	C₄³×C₅	C₂²	1	2	yes	—
84	#34322	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂²×C₅×Q₈	C₂³	1	1	yes	characteristic
86	C₂²×C₅×Q₈	C₂³	1	2	yes	—
87	C₂²×C₅×D₄	C₂³	1	1	yes	characteristic
88	C₂³×C₄×C₅	C₂³	1	2	yes	—
89	C₂³×C₄×C₅	C₂³	1	4	yes	—
90	C₅×C₄⋊Q₈	C₂×C₄	1	8	yes	complemented
91	C₅×#126	C₂×C₄	1	32	yes	complemented
92	C₅×#124	C₂×C₄	1	16	yes	—
93	C₅×C₂²⋊Q₈	C₂×C₄	1	16	yes	complemented
94	C₄×C₅×Q₈	—	2	4	no	—
95	C₄×C₅×Q₈	—	2	4	no	—
96	C₄×C₅×Q₈	—	4	4	no	—
97	C₄×C₅×Q₈	—	4	4	no	—
98	C₄×C₅×Q₈	C₂×C₄	1	16	yes	complemented
99	C₄×C₅×Q₈	C₂×C₄	1	8	yes	complemented
100	C₄×C₅×D₄	—	4	4	no	—
101	C₄×C₅×D₄	C₂×C₄	1	8	yes	—
102	C₅×#117	—	2	8	no	—
103	C₅×#117	C₂×C₄	1	16	yes	complemented
104	C₂×C₅×C₄⋊C₄	—	2	8	no	—
105	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
106	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
107	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
108	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
109	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
110	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
111	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
112	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
113	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
114	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
115	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
116	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
117	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
118	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
119	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
120	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
121	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
122	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
123	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
124	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
125	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
126	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
127	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
128	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
129	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
130	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
131	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
132	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
133	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
134	C₂×C₄²×C₅	C₂³	1	2	yes	—
135	C₂×C₄²×C₅	C₂³	1	2	yes	—
136	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
137	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
138	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
139	C₂×C₄²×C₅	C₂³	1	2	yes	—
140	C₂×C₄²×C₅	C₂³	1	2	yes	—
141	C₂×C₄²×C₅	C₂³	1	2	yes	—
142	C₂×C₄²×C₅	C₂³	1	2	yes	—
143	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
144	C₂×C₄²×C₅	C₂³	1	4	yes	—
145	C₂×C₄²×C₅	C₂³	1	4	yes	—
146	C₂×C₄²×C₅	C₂³	1	4	yes	—
147	C₂×C₄²×C₅	C₂³	1	2	yes	—
148	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
149	C₂×C₄²×C₅	C₂³	1	4	yes	—
150	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
151	C₅×#95	C₂³	1	2	yes	—
152	C₅×#95	C₂³	1	4	yes	—
153	C₅×#95	C₂³	1	4	yes	—
154	C₅×#95	C₂³	1	4	yes	—
155	C₅×#95	C₂³	1	2	yes	—
156	C₅×#95	C₂³	1	4	yes	—
157	C₅×#95	C₂³	1	4	yes	—
158	C₅×#95	C₂³	1	4	yes	—
159	C₅×#95	C₂³	1	2	yes	—
160	C₅×#95	C₂³	1	4	yes	—
161	C₅×#95	C₂³	1	2	yes	—
162	C₅×#95	C₂³	1	2	yes	—
163	C₅×#95	C₂³	1	4	yes	—
164	C₅×#95	C₂³	1	2	yes	—
165	C₂×#556	C₂×C₅	1	1	yes	characteristic
166	#2373	C₂×C₅	1	4	yes	complemented
167	#2367	C₂×C₅	1	2	yes	—
168	#2365	C₂×C₅	1	1	yes	characteristic, complemented
169	#2360	C₂×C₅	1	2	yes	complemented
170	#2355	C₂×C₅	1	1	yes	characteristic, complemented
171	#2351	C₂×C₅	1	2	yes	complemented
172	D₄×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
173	#2346	C₂×C₅	1	2	yes	complemented
174	#2334	C₂×C₅	1	4	yes	complemented
175	#2323	C₂×C₅	1	2	yes	complemented
176	C₄×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic, complemented
177	C₄×#124	C₂×C₅	1	2	yes	—
178	#2295	C₂×C₅	1	2	yes	complemented
179	#2293	C₂×C₅	1	1	yes	characteristic
180	#2274	C₂×C₅	1	2	yes	complemented
181	C₄²×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
182	C₂⁴×C₅	C₂⁴	1	2	yes	—
183	C₂×C₅×Q₈	—	2	2	no	—
184	C₂×C₅×Q₈	—	2	2	no	—
185	C₂×C₅×Q₈	—	2	4	no	—
186	C₂×C₅×Q₈	—	2	4	no	—
187	C₂×C₅×Q₈	C₂²×C₄	1	4	yes	—
188	C₂×C₅×Q₈	C₂²×C₄	1	8	yes	—
189	C₂×C₅×D₄	—	2	2	no	—
190	C₂×C₅×D₄	—	2	2	no	—
191	C₂×C₅×D₄	C₂²×C₄	1	4	yes	—
192	C₂²×C₄×C₅	—	2	4	no	—
193	C₂²×C₄×C₅	—	2	4	no	—
194	C₂²×C₄×C₅	—	2	8	no	—
195	C₂²×C₄×C₅	—	4	4	no	—
196	C₂²×C₄×C₅	—	4	4	no	—
197	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
198	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
199	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
200	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
201	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
202	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
203	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
204	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
205	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
206	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
207	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
208	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
209	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
210	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
211	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
212	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
213	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
214	C₅×C₄⋊C₄	—	2	8	no	—
215	C₅×C₄⋊C₄	—	2	4	no	—
216	C₅×C₄⋊C₄	—	2	4	no	—
217	C₅×C₄⋊C₄	—	2	4	no	—
218	C₅×C₄⋊C₄	—	2	4	no	—
219	C₅×C₄⋊C₄	—	2	4	no	—
220	C₅×C₄⋊C₄	—	2	2	no	—
221	C₅×C₄⋊C₄	—	2	4	no	—
222	C₅×C₄⋊C₄	—	2	2	no	—
223	C₅×C₄⋊C₄	—	2	2	no	—
224	C₅×C₄⋊C₄	—	4	2	no	—
225	C₅×C₄⋊C₄	—	4	4	no	—
226	C₅×C₄⋊C₄	—	4	4	no	—
227	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
228	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
229	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
230	C₅×C₄⋊C₄	C₂²×C₄	1	16	yes	—
231	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
232	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
233	C₅×C₄⋊C₄	#41	1	4	yes	—
234	C₅×C₂²⋊C₄	—	2	4	no	—
235	C₅×C₂²⋊C₄	—	2	4	no	—
236	C₅×C₂²⋊C₄	—	2	4	no	—
237	C₅×C₂²⋊C₄	—	4	2	no	—
238	C₅×C₂²⋊C₄	—	4	4	no	—
239	C₅×C₂²⋊C₄	C₂²×C₄	1	16	yes	—
240	C₅×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
241	C₅×C₂²⋊C₄	C₂²×C₄	1	16	yes	—
242	C₄²×C₅	—	2	4	no	—
243	C₄²×C₅	—	2	4	no	—
244	C₄²×C₅	—	2	2	no	—
245	C₄²×C₅	—	2	4	no	—
246	C₄²×C₅	—	2	4	no	—
247	C₄²×C₅	—	2	2	no	—
248	C₄²×C₅	—	2	2	no	—
249	C₄²×C₅	—	2	2	no	—
250	C₄²×C₅	—	2	4	no	—
251	C₄²×C₅	—	4	2	no	—
252	C₄²×C₅	—	4	1	no	—
253	C₄²×C₅	—	4	4	no	—
254	C₄²×C₅	—	4	4	no	—
255	C₄²×C₅	—	4	4	no	—
256	C₄²×C₅	C₂²×C₄	1	16	yes	—
257	C₄²×C₅	C₂²×C₄	1	4	yes	—
258	C₄²×C₅	C₂²×C₄	1	8	yes	—
259	C₄²×C₅	#41	1	4	yes	—
260	#556	C₄×C₅	1	16	yes	complemented
261	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
262	C₂×#126	C₂²×C₅	1	4	yes	—
263	C₂×#124	C₂²×C₅	1	2	yes	—
264	C₂×C₂²⋊Q₈	C₂²×C₅	1	2	yes	—
265	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
266	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
267	C₂×C₄×Q₈	C₂²×C₅	1	2	yes	—
268	C₂×C₄×Q₈	C₂²×C₅	1	2	yes	—
269	C₂×C₄×Q₈	C₂²×C₅	1	2	yes	—
270	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
271	C₂×C₄×D₄	C₂²×C₅	1	2	yes	—
272	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
273	C₂×#117	C₂²×C₅	1	2	yes	—
274	C₂×#117	C₂²×C₅	1	2	yes	—
275	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
276	#391	C₂²×C₅	1	2	yes	—
277	#391	C₂²×C₅	1	2	yes	—
278	#391	C₂²×C₅	1	2	yes	—
279	#390	C₂²×C₅	1	1	yes	characteristic
280	#390	C₂²×C₅	1	1	yes	characteristic
281	#389	C₂²×C₅	1	2	yes	—
282	#389	C₂²×C₅	1	2	yes	—
283	#389	C₂²×C₅	1	2	yes	—
284	#389	C₂²×C₅	1	4	yes	—
285	#389	C₂²×C₅	1	4	yes	—
286	#389	C₂²×C₅	1	2	yes	—
287	#389	C₂²×C₅	1	1	yes	characteristic
288	#389	C₂²×C₅	1	1	yes	characteristic
289	#388	C₂²×C₅	1	4	yes	—
290	#388	C₂²×C₅	1	4	yes	—
291	#388	C₂²×C₅	1	2	yes	—
292	#388	C₂²×C₅	1	4	yes	—
293	#388	C₂²×C₅	1	4	yes	—
294	#388	C₂²×C₅	1	2	yes	—
295	#387	C₂²×C₅	1	4	yes	—
296	#387	C₂²×C₅	1	4	yes	—
297	#387	C₂²×C₅	1	4	yes	—
298	#387	C₂²×C₅	1	4	yes	—
299	#385	C₂²×C₅	1	4	yes	—
300	#385	C₂²×C₅	1	4	yes	—
301	#384	C₂²×C₅	1	1	yes	characteristic
302	#382	C₂²×C₅	1	4	yes	—
303	#382	C₂²×C₅	1	2	yes	—
304	#380	C₂²×C₅	1	2	yes	—
305	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
306	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
307	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
308	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
309	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
310	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
311	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
312	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
313	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
314	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
315	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
316	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
317	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
318	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
319	C₄×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
320	C₄×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
321	C₄×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
322	C₄×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
323	#376	C₂²×C₅	1	4	yes	—
324	C₄³	C₂²×C₅	1	1	yes	characteristic
325	C₄³	C₂²×C₅	1	2	yes	—
326	C₂³×C₅	—	2	2	no	—
327	C₂³×C₅	—	2	2	no	—
328	C₂³×C₅	—	2	2	no	—
329	C₂³×C₅	—	2	2	no	—
330	C₂³×C₅	—	2	2	no	—
331	C₂³×C₅	—	2	2	no	—
332	C₂³×C₅	C₂³×C₄	1	4	yes	—
333	C₂³×C₅	C₂⁵	1	1	yes	characteristic
334	C₅×Q₈	—	2	8	no	—
335	C₅×Q₈	—	2	16	no	—
336	C₅×D₄	—	2	8	no	—
337	C₂×C₄×C₅	—	2	1	no	—
338	C₂×C₄×C₅	—	2	2	no	—
339	C₂×C₄×C₅	—	2	2	no	—
340	C₂×C₄×C₅	—	2	2	no	—
341	C₂×C₄×C₅	—	2	2	no	—
342	C₂×C₄×C₅	—	2	1	no	—
343	C₂×C₄×C₅	—	2	2	no	—
344	C₂×C₄×C₅	—	2	4	no	—
345	C₂×C₄×C₅	—	2	2	no	—
346	C₂×C₄×C₅	—	2	1	no	—
347	C₂×C₄×C₅	—	2	1	no	—
348	C₂×C₄×C₅	—	2	1	no	—
349	C₂×C₄×C₅	—	2	1	no	—
350	C₂×C₄×C₅	—	2	4	no	—
351	C₂×C₄×C₅	—	2	2	no	—
352	C₂×C₄×C₅	—	2	1	no	—
353	C₂×C₄×C₅	—	2	1	no	—
354	C₂×C₄×C₅	—	2	2	no	—
355	C₂×C₄×C₅	—	2	16	no	—
356	C₂×C₄×C₅	—	2	2	no	—
357	C₂×C₄×C₅	—	2	2	no	—
358	C₂×C₄×C₅	—	2	2	no	—
359	C₂×C₄×C₅	—	2	1	no	—
360	C₂×C₄×C₅	—	2	4	no	—
361	C₂×C₄×C₅	—	2	2	no	—
362	C₂×C₄×C₅	—	2	1	no	—
363	C₂×C₄×C₅	—	2	2	no	—
364	C₂×C₄×C₅	—	2	4	no	—
365	C₂×C₄×C₅	—	2	2	no	—
366	C₂×C₄×C₅	—	2	2	no	—
367	C₂×C₄×C₅	—	2	2	no	—
368	C₂×C₄×C₅	—	2	2	no	—
369	C₂×C₄×C₅	—	2	1	no	—
370	C₂×C₄×C₅	—	2	4	no	—
371	C₂×C₄×C₅	—	8	8	no	—
372	C₂×C₄×C₅	C₂³×C₄	1	4	yes	—
373	C₂×C₄×C₅	C₂³×C₄	1	4	yes	—
374	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
375	C₂×C₄×C₅	C₂³×C₄	1	4	yes	—
376	C₂×C₄×C₅	C₂³×C₄	1	8	yes	—
377	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
378	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
379	C₂×C₄×C₅	C₂×#41	1	2	yes	—
380	C₂×C₄×C₅	C₂×#41	1	4	yes	—
381	C₂×C₄×C₅	C₂×#41	1	2	yes	—
382	C₂×C₄×C₅	C₂×#41	1	2	yes	—
383	C₂×C₄×C₅	C₂×#41	1	2	yes	—
384	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
385	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
386	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
387	C₂²×Q₈	C₂³×C₅	1	1	yes	characteristic
388	C₂²×Q₈	C₂³×C₅	1	2	yes	—
389	C₂²×D₄	C₂³×C₅	1	1	yes	characteristic
390	C₂³×C₄	C₂³×C₅	1	2	yes	—
391	C₂³×C₄	C₂³×C₅	1	4	yes	—
392	C₄⋊Q₈	C₂×C₄×C₅	1	8	yes	complemented
393	#126	C₂×C₄×C₅	1	32	yes	complemented
394	#124	C₂×C₄×C₅	1	16	yes	—
395	C₂²⋊Q₈	C₂×C₄×C₅	1	16	yes	complemented
396	C₄×Q₈	—	2	4	no	—
397	C₄×Q₈	—	2	4	no	—
398	C₄×Q₈	—	4	4	no	—
399	C₄×Q₈	—	4	4	no	—
400	C₄×Q₈	C₂×C₄×C₅	1	16	yes	complemented
401	C₄×Q₈	C₂×C₄×C₅	1	8	yes	complemented
402	C₄×D₄	—	4	4	no	—
403	C₄×D₄	C₂×C₄×C₅	1	8	yes	—
404	#117	—	2	8	no	—
405	#117	C₂×C₄×C₅	1	16	yes	complemented
406	C₂×C₄⋊C₄	—	2	8	no	—
407	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
408	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
409	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
410	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
411	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
412	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
413	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
414	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
415	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
416	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
417	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
418	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
419	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
420	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
421	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
422	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
423	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
424	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
425	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
426	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
427	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
428	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
429	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
430	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
431	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
432	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
433	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
434	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
435	C₂×C₄²	C₂³×C₅	1	2	yes	—
436	C₂×C₄²	C₂³×C₅	1	2	yes	—
437	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
438	C₂×C₄²	C₂³×C₅	1	2	yes	—
439	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
440	C₂×C₄²	C₂³×C₅	1	2	yes	—
441	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
442	C₂×C₄²	C₂³×C₅	1	2	yes	—
443	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
444	C₂×C₄²	C₂³×C₅	1	4	yes	—
445	C₂×C₄²	C₂³×C₅	1	4	yes	—
446	C₂×C₄²	C₂³×C₅	1	2	yes	—
447	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
448	C₂×C₄²	C₂³×C₅	1	4	yes	—
449	C₂×C₄²	C₂³×C₅	1	4	yes	—
450	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
451	C₂×C₄²	C₂³×C₅	1	2	yes	—
452	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
453	#95	C₂³×C₅	1	2	yes	—
454	#95	C₂³×C₅	1	4	yes	—
455	#95	C₂³×C₅	1	4	yes	—
456	#95	C₂³×C₅	1	4	yes	—
457	#95	C₂³×C₅	1	2	yes	—
458	#95	C₂³×C₅	1	4	yes	—
459	#95	C₂³×C₅	1	4	yes	—
460	#95	C₂³×C₅	1	4	yes	—
461	#95	C₂³×C₅	1	2	yes	—
462	#95	C₂³×C₅	1	4	yes	—
463	#95	C₂³×C₅	1	2	yes	—
464	#95	C₂³×C₅	1	2	yes	—
465	#95	C₂³×C₅	1	4	yes	—
466	#95	C₂³×C₅	1	2	yes	—
467	C₂²×C₅	—	2	4	no	—
468	C₂²×C₅	—	2	4	no	—
469	C₂²×C₅	—	2	4	no	—
470	C₂²×C₅	—	4	2	no	—
471	C₂²×C₅	—	4	2	no	—
472	C₂²×C₅	—	4	2	no	—
473	C₂²×C₅	—	4	2	no	—
474	C₂²×C₅	C₂⁴×C₄	1	1	yes	characteristic
475	C₂²×C₅	C₂²×#41	1	1	yes	characteristic
476	C₂²×C₅	C₂²×#41	1	1	yes	characteristic
477	C₂²×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
478	C₂²×C₅	C₂×2₋¹⁺⁴	1	1	yes	characteristic
479	C₂²×C₅	C₂×2₋¹⁺⁴	1	1	yes	characteristic
480	C₂²×C₅	#585	1	1	yes	characteristic
481	C₄×C₅	—	2	4	no	—
482	C₄×C₅	—	2	4	no	—
483	C₄×C₅	—	2	8	no	—
484	C₄×C₅	—	2	4	no	—
485	C₄×C₅	—	2	2	no	—
486	C₄×C₅	—	2	2	no	—
487	C₄×C₅	—	2	2	no	—
488	C₄×C₅	—	2	4	no	—
489	C₄×C₅	—	2	2	no	—
490	C₄×C₅	—	4	2	no	—
491	C₄×C₅	—	4	2	no	—
492	C₄×C₅	—	4	2	no	—
493	C₄×C₅	—	4	1	no	—
494	C₄×C₅	—	4	4	no	—
495	C₄×C₅	C₄×#41	1	4	yes	—
496	C₄×C₅	#520	1	4	yes	—
497	C₂⁴	C₂⁴×C₅	1	2	yes	—
498	C₂×Q₈	—	2	2	no	—
499	C₂×Q₈	—	2	2	no	—
500	C₂×Q₈	—	2	4	no	—
501	C₂×Q₈	—	2	4	no	—
502	C₂×Q₈	C₂²×C₄×C₅	1	4	yes	—
503	C₂×Q₈	C₂²×C₄×C₅	1	8	yes	—
504	C₂×D₄	—	2	2	no	—
505	C₂×D₄	—	2	2	no	—
506	C₂×D₄	C₂²×C₄×C₅	1	4	yes	—
507	C₂²×C₄	—	2	4	no	—
508	C₂²×C₄	—	2	4	no	—
509	C₂²×C₄	—	2	8	no	—
510	C₂²×C₄	—	4	4	no	—
511	C₂²×C₄	—	4	4	no	—
512	C₂²×C₄	C₂²×C₄×C₅	1	8	yes	—
513	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
514	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
515	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
516	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
517	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
518	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
519	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
520	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
521	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
522	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
523	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
524	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
525	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
526	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
527	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
528	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
529	C₄⋊C₄	—	2	8	no	—
530	C₄⋊C₄	—	2	4	no	—
531	C₄⋊C₄	—	2	4	no	—
532	C₄⋊C₄	—	2	4	no	—
533	C₄⋊C₄	—	2	4	no	—
534	C₄⋊C₄	—	2	4	no	—
535	C₄⋊C₄	—	2	2	no	—
536	C₄⋊C₄	—	2	4	no	—
537	C₄⋊C₄	—	2	2	no	—
538	C₄⋊C₄	—	2	2	no	—
539	C₄⋊C₄	—	4	4	no	—
540	C₄⋊C₄	—	4	4	no	—
541	C₄⋊C₄	—	4	2	no	—
542	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
543	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
544	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
545	C₄⋊C₄	C₂²×C₄×C₅	1	16	yes	—
546	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
547	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
548	C₄⋊C₄	C₅×#41	1	4	yes	—
549	C₂²⋊C₄	—	2	4	no	—
550	C₂²⋊C₄	—	2	4	no	—
551	C₂²⋊C₄	—	2	4	no	—
552	C₂²⋊C₄	—	4	4	no	—
553	C₂²⋊C₄	—	4	2	no	—
554	C₂²⋊C₄	C₂²×C₄×C₅	1	16	yes	—
555	C₂²⋊C₄	C₂²×C₄×C₅	1	8	yes	—
556	C₂²⋊C₄	C₂²×C₄×C₅	1	16	yes	—
557	C₄²	—	2	4	no	—
558	C₄²	—	2	4	no	—
559	C₄²	—	2	2	no	—
560	C₄²	—	2	4	no	—
561	C₄²	—	2	4	no	—
562	C₄²	—	2	2	no	—
563	C₄²	—	2	2	no	—
564	C₄²	—	2	2	no	—
565	C₄²	—	2	4	no	—
566	C₄²	—	4	4	no	—
567	C₄²	—	4	4	no	—
568	C₄²	—	4	2	no	—
569	C₄²	—	4	1	no	—
570	C₄²	—	4	4	no	—
571	C₄²	C₂²×C₄×C₅	1	16	yes	—
572	C₄²	C₂²×C₄×C₅	1	4	yes	—
573	C₄²	C₂²×C₄×C₅	1	8	yes	—
574	C₄²	C₅×#41	1	4	yes	—
575	C₂×C₅	—	4	4	no	—
576	C₂×C₅	C₂×C₄×#41	1	1	yes	characteristic
577	C₂×C₅	C₂×#520	1	1	yes	characteristic
578	C₂×C₅	C₄×2₋¹⁺⁴	1	1	yes	characteristic
579	C₂×C₅	#3507	1	1	yes	characteristic
580	C₂×C₅	#3511	1	1	yes	characteristic
581	C₂×C₅	#3514	1	1	yes	characteristic
582	C₂×C₅	#3523	1	1	yes	characteristic
583	C₂³	—	2	2	no	—
584	C₂³	—	2	2	no	—
585	C₂³	—	2	2	no	—
586	C₂³	—	2	2	no	—
587	C₂³	—	2	2	no	—
588	C₂³	—	2	2	no	—
589	C₂³	C₂³×C₄×C₅	1	4	yes	—
590	C₂³	C₂⁵×C₅	1	1	yes	characteristic
591	Q₈	—	2	8	no	—
592	Q₈	—	2	16	no	—
593	D₄	—	2	8	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	2	2	no	—
599	C₂×C₄	—	2	1	no	—
600	C₂×C₄	—	2	2	no	—
601	C₂×C₄	—	2	4	no	—
602	C₂×C₄	—	2	2	no	—
603	C₂×C₄	—	2	1	no	—
604	C₂×C₄	—	2	1	no	—
605	C₂×C₄	—	2	1	no	—
606	C₂×C₄	—	2	1	no	—
607	C₂×C₄	—	2	4	no	—
608	C₂×C₄	—	2	2	no	—
609	C₂×C₄	—	2	1	no	—
610	C₂×C₄	—	2	1	no	—
611	C₂×C₄	—	2	2	no	—
612	C₂×C₄	—	2	16	no	—
613	C₂×C₄	—	2	2	no	—
614	C₂×C₄	—	2	2	no	—
615	C₂×C₄	—	2	2	no	—
616	C₂×C₄	—	2	1	no	—
617	C₂×C₄	—	2	4	no	—
618	C₂×C₄	—	2	2	no	—
619	C₂×C₄	—	2	1	no	—
620	C₂×C₄	—	2	2	no	—
621	C₂×C₄	—	2	4	no	—
622	C₂×C₄	—	2	2	no	—
623	C₂×C₄	—	2	2	no	—
624	C₂×C₄	—	2	2	no	—
625	C₂×C₄	—	2	2	no	—
626	C₂×C₄	—	2	1	no	—
627	C₂×C₄	—	2	4	no	—
628	C₂×C₄	—	8	8	no	—
629	C₂×C₄	C₂³×C₄×C₅	1	4	yes	—
630	C₂×C₄	C₂³×C₄×C₅	1	4	yes	—
631	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
632	C₂×C₄	C₂³×C₄×C₅	1	4	yes	—
633	C₂×C₄	C₂³×C₄×C₅	1	8	yes	—
634	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
635	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
636	C₂×C₄	C₂×C₅×#41	1	2	yes	—
637	C₂×C₄	C₂×C₅×#41	1	4	yes	—
638	C₂×C₄	C₂×C₅×#41	1	2	yes	—
639	C₂×C₄	C₂×C₅×#41	1	2	yes	—
640	C₂×C₄	C₂×C₅×#41	1	2	yes	—
641	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
642	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
643	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
644	C₅	#34322	1	1	yes	characteristic, direct product factor, Sylow
645	C₂²	—	2	4	no	—
646	C₂²	—	2	4	no	—
647	C₂²	—	2	4	no	—
648	C₂²	—	4	2	no	—
649	C₂²	—	4	2	no	—
650	C₂²	—	4	2	no	—
651	C₂²	—	4	2	no	—
652	C₂²	C₂⁴×C₄×C₅	1	1	yes	characteristic
653	C₂²	C₂²×C₅×#41	1	1	yes	characteristic
654	C₂²	C₂²×C₅×#41	1	1	yes	characteristic
655	C₂²	C₂×C₅×H(5,𝔽₂)	1	1	yes	characteristic
656	C₂²	C₂×C₅×2₋¹⁺⁴	1	1	yes	characteristic
657	C₂²	C₂×C₅×2₋¹⁺⁴	1	1	yes	characteristic
658	C₂²	C₅×#585	1	1	yes	characteristic
659	C₄	—	2	4	no	—
660	C₄	—	2	4	no	—
661	C₄	—	2	8	no	—
662	C₄	—	2	4	no	—
663	C₄	—	2	2	no	—
664	C₄	—	2	2	no	—
665	C₄	—	2	2	no	—
666	C₄	—	2	4	no	—
667	C₄	—	2	2	no	—
668	C₄	—	4	2	no	—
669	C₄	—	4	2	no	—
670	C₄	—	4	2	no	—
671	C₄	—	4	4	no	—
672	C₄	—	4	1	no	—
673	C₄	C₄×C₅×#41	1	4	yes	—
674	C₄	C₅×#520	1	4	yes	—
675	C₂	—	4	4	no	—
676	C₂	C₂×C₄×C₅×#41	1	1	yes	characteristic
677	C₂	C₂×C₅×#520	1	1	yes	characteristic
678	C₂	C₄×C₅×2₋¹⁺⁴	1	1	yes	characteristic
679	C₂	C₅×#3507	1	1	yes	characteristic
680	C₂	C₅×#3511	1	1	yes	characteristic
681	C₂	C₅×#3514	1	1	yes	characteristic
682	C₂	C₅×#3523	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₅
Frattini, cosocle	C₂³	C₂⁵×C₅
prime radical	C₂²×C₅×D₄	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2373	C₂	semidirect
4. C₅×#2365	C₂	semidirect
5. C₅×#2360	C₂	semidirect
6. C₅×#2355	C₂	semidirect
7. C₅×#2351	C₂	semidirect
9. C₅×#2346	C₂	semidirect
10. C₅×#2334	C₂	semidirect
11. C₅×#2323	C₂	semidirect
12. C₄×C₅×C₄⋊Q₈	C₂	semidirect
14. C₅×#2295	C₂	semidirect
16. C₅×#2274	C₂	semidirect
17. C₄²×C₅×Q₈	C₂	semidirect
18. C₅×#556	C₄	semidirect
84. #34322	C₅	direct (N × H)
90. C₅×C₄⋊Q₈	C₂×C₄	semidirect
91. C₅×#126	C₂×C₄	semidirect
93. C₅×C₂²⋊Q₈	C₂×C₄	semidirect
98. C₄×C₅×Q₈	C₂×C₄	semidirect
99. C₄×C₅×Q₈	C₂×C₄	semidirect
103. C₅×#117	C₂×C₄	semidirect
166. #2373	C₂×C₅	semidirect
168. #2365	C₂×C₅	semidirect
169. #2360	C₂×C₅	semidirect
170. #2355	C₂×C₅	semidirect
171. #2351	C₂×C₅	semidirect
173. #2346	C₂×C₅	semidirect
174. #2334	C₂×C₅	semidirect
175. #2323	C₂×C₅	semidirect
176. C₄×C₄⋊Q₈	C₂×C₅	semidirect
178. #2295	C₂×C₅	semidirect
180. #2274	C₂×C₅	semidirect
181. C₄²×Q₈	C₂×C₅	semidirect
260. #556	C₄×C₅	semidirect
392. C₄⋊Q₈	C₂×C₄×C₅	semidirect
393. #126	C₂×C₄×C₅	semidirect
395. C₂²⋊Q₈	C₂×C₄×C₅	semidirect
400. C₄×Q₈	C₂×C₄×C₅	semidirect
401. C₄×Q₈	C₂×C₄×C₅	semidirect
405. #117	C₂×C₄×C₅	semidirect
644. C₅	#34322	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×#556	2
C₅×#2373	2
C₅×#2367	2
C₅×#2365	2
C₅×#2360	2
C₅×#2355	2
C₅×#2351	2
C₅×D₄×C₄⋊C₄	2
C₅×#2346	2
C₅×#2334	2
C₅×#2323	2
C₄×C₅×C₄⋊Q₈	2
C₄×C₅×#124	2
C₅×#2295	2
C₅×#2293	2
C₅×#2274	2
C₄²×C₅×Q₈	2
#34322	5

Subgroups — C5×#34322 #11957042

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#34322 #11957042