Subgroups — C₅×#34275 #11956995

Order 1280.

3410 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#535	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#520	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#518	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2363	C₂	1	4	yes	maximal, complemented
5	C₅×#2361	C₂	1	4	yes	maximal, complemented
6	C₅×#2358	C₂	1	8	yes	maximal, complemented
7	C₅×#2328	C₂	1	2	yes	maximal, complemented
8	C₅×#2326	C₂	1	2	yes	maximal, complemented
9	C₅×#2317	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₅×#127	C₂	1	2	yes	maximal, complemented
11	C₄×C₅×#124	C₂	1	2	yes	maximal, complemented
12	C₅×#2297	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₅×#390	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₅×#117	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂²×C₅×#41	C₂²	1	1	yes	characteristic
16	C₂³×C₅×D₄	C₂²	1	1	yes	characteristic
17	C₅×#535	C₄	1	16	yes	complemented
18	C₂×C₅×#127	C₂²	1	2	yes	—
19	C₂×C₅×#124	C₂²	1	2	yes	—
20	C₂×C₅×#121	C₂²	1	4	yes	—
21	C₂×C₅×#120	C₂²	1	4	yes	—
22	C₅×#520	—	2	8	no	—
23	C₅×#518	—	2	8	no	—
24	C₂×C₄×C₅×Q₈	C₂²	1	2	yes	complemented
25	C₂×C₄×C₅×D₄	C₂²	1	2	yes	—
26	C₂×C₄×C₅×D₄	C₂²	1	2	yes	—
27	C₂×C₄×C₅×D₄	C₂²	1	2	yes	—
28	C₂×C₄×C₅×D₄	C₂²	1	2	yes	—
29	C₂×C₄×C₅×D₄	C₂²	1	4	yes	—
30	C₂×C₄×C₅×D₄	C₂²	1	2	yes	complemented
31	C₂×C₄×C₅×D₄	C₂²	1	8	yes	complemented
32	C₂×C₅×#117	C₂²	1	1	yes	characteristic
33	C₂×C₅×#117	C₂²	1	1	yes	characteristic
34	C₂×C₅×#117	C₂²	1	1	yes	characteristic
35	C₂×C₅×#117	C₂²	1	2	yes	complemented
36	C₂²×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
37	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	complemented
38	C₂²×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
39	C₂²×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
40	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
41	C₅×#391	C₂²	1	1	yes	characteristic, complemented
42	C₅×#391	C₂²	1	1	yes	characteristic, complemented
43	C₅×#390	C₂²	1	2	yes	—
44	C₅×#390	C₂²	1	2	yes	—
45	C₅×#390	C₂²	1	4	yes	—
46	C₅×#390	C₂²	1	1	yes	characteristic, complemented
47	C₅×#390	C₂²	1	1	yes	characteristic, complemented
48	C₅×#390	C₂²	1	8	yes	complemented
49	C₅×#389	C₂²	1	4	yes	complemented
50	C₅×#388	C₂²	1	8	yes	complemented
51	C₅×#388	C₂²	1	8	yes	complemented
52	C₅×#386	C₂²	1	4	yes	—
53	C₅×#386	C₂²	1	8	yes	complemented
54	C₅×#386	C₂²	1	8	yes	complemented
55	C₅×#385	C₂²	1	4	yes	—
56	C₅×#385	C₂²	1	8	yes	complemented
57	C₅×#384	C₂²	1	2	yes	complemented
58	C₅×#382	C₂²	1	2	yes	complemented
59	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	complemented
60	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	complemented
61	C₄×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
62	C₄×C₅×C₂²⋊C₄	C₂²	1	4	yes	complemented
63	C₄×C₅×C₂²⋊C₄	C₂²	1	8	yes	complemented
64	C₅×#376	C₂²	1	2	yes	complemented
65	C₄³×C₅	C₂²	1	2	yes	complemented
66	#34275	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂⁵×C₅	C₂³	1	2	yes	—
68	C₂×C₅×#41	—	2	4	no	—
69	C₂×C₅×#41	—	2	4	no	—
70	C₂×C₅×#41	C₂×C₄	1	8	yes	complemented
71	C₂²×C₅×Q₈	C₂³	1	1	yes	characteristic
72	C₂²×C₅×D₄	—	2	4	no	—
73	C₂²×C₅×D₄	—	2	4	no	—
74	C₂²×C₅×D₄	C₂×C₄	1	8	yes	complemented
75	C₂²×C₅×D₄	C₂³	1	2	yes	—
76	C₂²×C₅×D₄	C₂³	1	2	yes	—
77	C₂²×C₅×D₄	C₂³	1	2	yes	—
78	C₂²×C₅×D₄	C₂³	1	1	yes	characteristic
79	C₂²×C₅×D₄	C₂³	1	8	yes	—
80	C₂³×C₄×C₅	C₂³	1	2	yes	—
81	C₂³×C₄×C₅	C₂³	1	2	yes	—
82	C₂³×C₄×C₅	C₂³	1	4	yes	—
83	C₂³×C₄×C₅	C₂³	1	2	yes	—
84	C₂³×C₄×C₅	C₂³	1	4	yes	—
85	C₂³×C₄×C₅	C₂³	1	4	yes	—
86	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
87	C₅×#127	C₂×C₄	1	16	yes	complemented
88	C₅×#124	C₂×C₄	1	16	yes	complemented
89	C₅×#121	C₂×C₄	1	32	yes	complemented
90	C₅×#120	C₂×C₄	1	32	yes	complemented
91	C₄×C₅×Q₈	—	2	8	no	—
92	C₄×C₅×D₄	—	2	8	no	—
93	C₄×C₅×D₄	—	2	8	no	—
94	C₄×C₅×D₄	—	2	16	no	—
95	C₄×C₅×D₄	—	2	8	no	—
96	C₄×C₅×D₄	—	2	8	no	—
97	C₄×C₅×D₄	—	2	8	no	—
98	C₄×C₅×D₄	—	4	16	no	—
99	C₅×#117	—	2	4	no	—
100	C₅×#117	—	2	8	no	—
101	C₅×#117	C₂×C₄	1	8	yes	complemented
102	C₅×#117	D₄	1	8	yes	complemented
103	C₂×C₅×C₄⋊C₄	—	2	4	no	—
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	2	yes	—
107	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
108	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
109	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
110	C₂×C₅×C₄⋊C₄	C₂³	1	8	yes	—
111	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
112	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
113	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
114	C₂×C₅×C₂²⋊C₄	—	2	16	no	—
115	C₂×C₅×C₂²⋊C₄	—	4	8	no	—
116	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
117	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
118	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
119	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
120	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
121	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
122	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
123	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
124	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
125	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
126	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
127	C₂×C₄²×C₅	—	2	4	no	—
128	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
129	C₂×C₄²×C₅	C₂³	1	4	yes	—
130	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
131	C₂×C₄²×C₅	C₂³	1	2	yes	—
132	C₂×C₄²×C₅	C₂³	1	4	yes	—
133	C₂×C₄²×C₅	C₂³	1	2	yes	—
134	C₂×C₄²×C₅	C₂³	1	4	yes	—
135	C₂×C₄²×C₅	C₂³	1	2	yes	—
136	C₂×C₄²×C₅	C₂³	1	2	yes	—
137	C₅×#95	C₂³	1	4	yes	—
138	C₅×#95	C₂³	1	4	yes	—
139	C₅×#95	C₂³	1	4	yes	—
140	C₅×#95	C₂³	1	8	yes	—
141	C₅×#95	C₂³	1	4	yes	—
142	C₂×#535	C₂×C₅	1	1	yes	characteristic
143	C₂×#520	C₂×C₅	1	1	yes	characteristic, complemented
144	C₂×#518	C₂×C₅	1	1	yes	characteristic, complemented
145	#2363	C₂×C₅	1	4	yes	complemented
146	#2361	C₂×C₅	1	4	yes	complemented
147	#2358	C₂×C₅	1	8	yes	complemented
148	#2328	C₂×C₅	1	2	yes	complemented
149	#2326	C₂×C₅	1	2	yes	complemented
150	#2317	C₂×C₅	1	1	yes	characteristic, complemented
151	C₄×#127	C₂×C₅	1	2	yes	complemented
152	C₄×#124	C₂×C₅	1	2	yes	complemented
153	#2297	C₂×C₅	1	1	yes	characteristic, complemented
154	C₂×#390	C₂×C₅	1	1	yes	characteristic, complemented
155	C₄×#117	C₂×C₅	1	1	yes	characteristic, complemented
156	C₂⁴×C₅	—	2	4	no	—
157	C₂⁴×C₅	—	2	4	no	—
158	C₂⁴×C₅	—	4	4	no	—
159	C₂⁴×C₅	—	4	4	no	—
160	C₂⁴×C₅	C₂²×C₄	1	8	yes	—
161	C₂⁴×C₅	C₂⁴	1	4	yes	—
162	C₂⁴×C₅	C₂⁴	1	2	yes	—
163	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
164	C₅×#41	—	2	32	no	—
165	C₂×C₅×Q₈	—	2	2	no	—
166	C₂×C₅×Q₈	—	2	2	no	—
167	C₂×C₅×Q₈	C₂²×C₄	1	4	yes	—
168	C₂×C₅×D₄	—	2	4	no	—
169	C₂×C₅×D₄	—	2	4	no	—
170	C₂×C₅×D₄	—	2	32	no	—
171	C₂×C₅×D₄	—	2	4	no	—
172	C₂×C₅×D₄	—	2	4	no	—
173	C₂×C₅×D₄	—	2	2	no	—
174	C₂×C₅×D₄	—	2	4	no	—
175	C₂×C₅×D₄	—	2	4	no	—
176	C₂×C₅×D₄	—	2	2	no	—
177	C₂×C₅×D₄	—	4	8	no	—
178	C₂×C₅×D₄	—	4	8	no	—
179	C₂×C₅×D₄	C₂²×C₄	1	8	yes	—
180	C₂×C₅×D₄	C₂²×C₄	1	8	yes	—
181	C₂×C₅×D₄	C₂²×C₄	1	8	yes	—
182	C₂×C₅×D₄	C₂²×C₄	1	4	yes	—
183	C₂×C₅×D₄	C₂²×C₄	1	32	yes	—
184	C₂²×C₄×C₅	—	2	4	no	—
185	C₂²×C₄×C₅	—	2	4	no	—
186	C₂²×C₄×C₅	—	2	4	no	—
187	C₂²×C₄×C₅	—	2	4	no	—
188	C₂²×C₄×C₅	—	2	4	no	—
189	C₂²×C₄×C₅	—	2	8	no	—
190	C₂²×C₄×C₅	—	2	8	no	—
191	C₂²×C₄×C₅	—	2	8	no	—
192	C₂²×C₄×C₅	—	4	2	no	—
193	C₂²×C₄×C₅	—	4	4	no	—
194	C₂²×C₄×C₅	—	4	8	no	—
195	C₂²×C₄×C₅	—	4	8	no	—
196	C₂²×C₄×C₅	—	4	8	no	—
197	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
198	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
199	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
200	C₂²×C₄×C₅	#41	1	4	yes	—
201	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
202	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
203	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
204	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
205	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
206	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
207	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
208	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
209	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
210	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
211	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
212	C₅×C₄⋊C₄	—	2	4	no	—
213	C₅×C₄⋊C₄	—	2	8	no	—
214	C₅×C₄⋊C₄	—	2	4	no	—
215	C₅×C₄⋊C₄	—	2	2	no	—
216	C₅×C₄⋊C₄	—	2	2	no	—
217	C₅×C₄⋊C₄	—	4	8	no	—
218	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
219	C₅×C₄⋊C₄	C₂×D₄	1	4	yes	—
220	C₅×C₄⋊C₄	C₂×D₄	1	4	yes	—
221	C₅×C₂²⋊C₄	—	2	4	no	—
222	C₅×C₂²⋊C₄	—	2	8	no	—
223	C₅×C₂²⋊C₄	—	2	8	no	—
224	C₅×C₂²⋊C₄	—	2	8	no	—
225	C₅×C₂²⋊C₄	—	2	16	no	—
226	C₅×C₂²⋊C₄	—	2	8	no	—
227	C₅×C₂²⋊C₄	—	4	8	no	—
228	C₅×C₂²⋊C₄	—	4	4	no	—
229	C₅×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
230	C₅×C₂²⋊C₄	C₂²×C₄	1	32	yes	—
231	C₅×C₂²⋊C₄	C₂×D₄	1	8	yes	—
232	C₄²×C₅	—	2	4	no	—
233	C₄²×C₅	—	2	8	no	—
234	C₄²×C₅	—	2	2	no	—
235	C₄²×C₅	—	2	4	no	—
236	C₄²×C₅	—	2	2	no	—
237	C₄²×C₅	—	2	8	no	—
238	C₄²×C₅	—	4	4	no	—
239	C₄²×C₅	C₂²×C₄	1	8	yes	—
240	C₄²×C₅	C₂×D₄	1	8	yes	—
241	C₂²×#41	C₂²×C₅	1	1	yes	characteristic
242	C₂³×D₄	C₂²×C₅	1	1	yes	characteristic
243	#535	C₄×C₅	1	16	yes	complemented
244	C₂×#127	C₂²×C₅	1	2	yes	—
245	C₂×#124	C₂²×C₅	1	2	yes	—
246	C₂×#121	C₂²×C₅	1	4	yes	—
247	C₂×#120	C₂²×C₅	1	4	yes	—
248	#520	—	2	8	no	—
249	#518	—	2	8	no	—
250	C₂×C₄×Q₈	C₂²×C₅	1	2	yes	complemented
251	C₂×C₄×D₄	C₂²×C₅	1	2	yes	—
252	C₂×C₄×D₄	C₂²×C₅	1	2	yes	—
253	C₂×C₄×D₄	C₂²×C₅	1	4	yes	—
254	C₂×C₄×D₄	C₂²×C₅	1	2	yes	—
255	C₂×C₄×D₄	C₂²×C₅	1	2	yes	—
256	C₂×C₄×D₄	C₂²×C₅	1	2	yes	complemented
257	C₂×C₄×D₄	C₂²×C₅	1	8	yes	complemented
258	C₂×#117	C₂²×C₅	1	1	yes	characteristic
259	C₂×#117	C₂²×C₅	1	1	yes	characteristic
260	C₂×#117	C₂²×C₅	1	1	yes	characteristic
261	C₂×#117	C₂²×C₅	1	2	yes	complemented
262	C₂²×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
263	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	complemented
264	C₂²×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
265	C₂²×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
266	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
267	#391	C₂²×C₅	1	1	yes	characteristic, complemented
268	#391	C₂²×C₅	1	1	yes	characteristic, complemented
269	#390	C₂²×C₅	1	2	yes	—
270	#390	C₂²×C₅	1	2	yes	—
271	#390	C₂²×C₅	1	4	yes	—
272	#390	C₂²×C₅	1	1	yes	characteristic, complemented
273	#390	C₂²×C₅	1	8	yes	complemented
274	#390	C₂²×C₅	1	1	yes	characteristic, complemented
275	#389	C₂²×C₅	1	4	yes	complemented
276	#388	C₂²×C₅	1	8	yes	complemented
277	#388	C₂²×C₅	1	8	yes	complemented
278	#386	C₂²×C₅	1	4	yes	—
279	#386	C₂²×C₅	1	8	yes	complemented
280	#386	C₂²×C₅	1	8	yes	complemented
281	#385	C₂²×C₅	1	4	yes	—
282	#385	C₂²×C₅	1	8	yes	complemented
283	#384	C₂²×C₅	1	2	yes	complemented
284	#382	C₂²×C₅	1	2	yes	complemented
285	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	complemented
286	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	complemented
287	C₄×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
288	C₄×C₂²⋊C₄	C₂²×C₅	1	8	yes	complemented
289	C₄×C₂²⋊C₄	C₂²×C₅	1	4	yes	complemented
290	#376	C₂²×C₅	1	2	yes	complemented
291	C₄³	C₂²×C₅	1	2	yes	complemented
292	C₂³×C₅	—	2	2	no	—
293	C₂³×C₅	—	2	2	no	—
294	C₂³×C₅	—	2	2	no	—
295	C₂³×C₅	—	2	4	no	—
296	C₂³×C₅	—	2	2	no	—
297	C₂³×C₅	—	2	4	no	—
298	C₂³×C₅	—	2	8	no	—
299	C₂³×C₅	—	2	4	no	—
300	C₂³×C₅	—	2	4	no	—
301	C₂³×C₅	—	2	8	no	—
302	C₂³×C₅	—	2	16	no	—
303	C₂³×C₅	—	4	16	no	—
304	C₂³×C₅	—	8	8	no	—
305	C₂³×C₅	—	8	4	no	—
306	C₂³×C₅	—	8	4	no	—
307	C₂³×C₅	C₂³×C₄	1	8	yes	—
308	C₂³×C₅	C₂³×C₄	1	4	yes	—
309	C₂³×C₅	C₂³×C₄	1	2	yes	—
310	C₂³×C₅	C₂²×D₄	1	2	yes	—
311	C₂³×C₅	C₂×#41	1	2	yes	—
312	C₂³×C₅	C₂⁵	1	1	yes	characteristic
313	C₅×Q₈	—	2	8	no	—
314	C₅×D₄	—	2	16	no	—
315	C₅×D₄	—	2	16	no	—
316	C₅×D₄	—	2	16	no	—
317	C₅×D₄	—	2	8	no	—
318	C₅×D₄	—	4	32	no	—
319	C₂×C₄×C₅	—	2	2	no	—
320	C₂×C₄×C₅	—	2	2	no	—
321	C₂×C₄×C₅	—	2	2	no	—
322	C₂×C₄×C₅	—	2	4	no	—
323	C₂×C₄×C₅	—	2	4	no	—
324	C₂×C₄×C₅	—	2	2	no	—
325	C₂×C₄×C₅	—	2	4	no	—
326	C₂×C₄×C₅	—	2	4	no	—
327	C₂×C₄×C₅	—	2	16	no	—
328	C₂×C₄×C₅	—	2	2	no	—
329	C₂×C₄×C₅	—	2	2	no	—
330	C₂×C₄×C₅	—	2	4	no	—
331	C₂×C₄×C₅	—	2	2	no	—
332	C₂×C₄×C₅	—	2	8	no	—
333	C₂×C₄×C₅	—	2	4	no	—
334	C₂×C₄×C₅	—	2	2	no	—
335	C₂×C₄×C₅	—	2	2	no	—
336	C₂×C₄×C₅	—	2	4	no	—
337	C₂×C₄×C₅	—	4	8	no	—
338	C₂×C₄×C₅	—	8	8	no	—
339	C₂×C₄×C₅	—	8	8	no	—
340	C₂×C₄×C₅	—	8	8	no	—
341	C₂×C₄×C₅	—	8	4	no	—
342	C₂×C₄×C₅	C₄×D₄	1	16	yes	complemented
343	C₂×C₄×C₅	C₂³×C₄	1	8	yes	—
344	C₂×C₄×C₅	C₂³×C₄	1	4	yes	—
345	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
346	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
347	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
348	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
349	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
350	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
351	C₂×C₄×C₅	C₂×#41	1	2	yes	—
352	C₂×C₄×C₅	C₂×#41	1	2	yes	—
353	C₂×C₄×C₅	H(5,𝔽₂)	1	4	yes	—
354	C₂⁵	C₂³×C₅	1	2	yes	—
355	C₂×#41	—	2	4	no	—
356	C₂×#41	—	2	4	no	—
357	C₂×#41	C₂×C₄×C₅	1	8	yes	complemented
358	C₂²×Q₈	C₂³×C₅	1	1	yes	characteristic
359	C₂²×D₄	—	2	4	no	—
360	C₂²×D₄	—	2	4	no	—
361	C₂²×D₄	C₂×C₄×C₅	1	8	yes	complemented
362	C₂²×D₄	C₂³×C₅	1	2	yes	—
363	C₂²×D₄	C₂³×C₅	1	2	yes	—
364	C₂²×D₄	C₂³×C₅	1	2	yes	—
365	C₂²×D₄	C₂³×C₅	1	1	yes	characteristic
366	C₂²×D₄	C₂³×C₅	1	8	yes	—
367	C₂³×C₄	C₂³×C₅	1	2	yes	—
368	C₂³×C₄	C₂³×C₅	1	2	yes	—
369	C₂³×C₄	C₂³×C₅	1	2	yes	—
370	C₂³×C₄	C₂³×C₅	1	4	yes	—
371	C₂³×C₄	C₂³×C₅	1	4	yes	—
372	C₂³×C₄	C₂³×C₅	1	4	yes	—
373	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
374	#127	C₂×C₄×C₅	1	16	yes	complemented
375	#124	C₂×C₄×C₅	1	16	yes	complemented
376	#121	C₂×C₄×C₅	1	32	yes	complemented
377	#120	C₂×C₄×C₅	1	32	yes	complemented
378	C₄×Q₈	—	2	8	no	—
379	C₄×D₄	—	2	8	no	—
380	C₄×D₄	—	2	8	no	—
381	C₄×D₄	—	2	16	no	—
382	C₄×D₄	—	2	8	no	—
383	C₄×D₄	—	2	8	no	—
384	C₄×D₄	—	2	8	no	—
385	C₄×D₄	—	4	16	no	—
386	#117	—	2	4	no	—
387	#117	—	2	8	no	—
388	#117	C₂×C₄×C₅	1	8	yes	complemented
389	#117	C₅×D₄	1	8	yes	complemented
390	C₂×C₄⋊C₄	—	2	4	no	—
391	C₂×C₄⋊C₄	—	2	8	no	—
392	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
393	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
394	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
395	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
396	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
397	C₂×C₄⋊C₄	C₂³×C₅	1	8	yes	—
398	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
399	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
400	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
401	C₂×C₂²⋊C₄	—	2	16	no	—
402	C₂×C₂²⋊C₄	—	4	8	no	—
403	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
404	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
405	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
406	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
407	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
408	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
409	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
410	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
411	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
412	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
413	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
414	C₂×C₄²	—	2	4	no	—
415	C₂×C₄²	C₂³×C₅	1	2	yes	—
416	C₂×C₄²	C₂³×C₅	1	4	yes	—
417	C₂×C₄²	C₂³×C₅	1	4	yes	—
418	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
419	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
420	C₂×C₄²	C₂³×C₅	1	2	yes	—
421	C₂×C₄²	C₂³×C₅	1	4	yes	—
422	C₂×C₄²	C₂³×C₅	1	2	yes	—
423	C₂×C₄²	C₂³×C₅	1	2	yes	—
424	#95	C₂³×C₅	1	4	yes	—
425	#95	C₂³×C₅	1	4	yes	—
426	#95	C₂³×C₅	1	4	yes	—
427	#95	C₂³×C₅	1	8	yes	—
428	#95	C₂³×C₅	1	4	yes	—
429	C₂²×C₅	—	2	4	no	—
430	C₂²×C₅	—	2	4	no	—
431	C₂²×C₅	—	2	2	no	—
432	C₂²×C₅	—	2	2	no	—
433	C₂²×C₅	—	2	4	no	—
434	C₂²×C₅	—	2	8	no	—
435	C₂²×C₅	—	2	8	no	—
436	C₂²×C₅	—	2	16	no	—
437	C₂²×C₅	—	4	4	no	—
438	C₂²×C₅	—	4	8	no	—
439	C₂²×C₅	—	4	2	no	—
440	C₂²×C₅	—	4	4	no	—
441	C₂²×C₅	—	4	4	no	—
442	C₂²×C₅	—	4	2	no	—
443	C₂²×C₅	—	8	16	no	—
444	C₂²×C₅	C₂×C₄×D₄	1	4	yes	—
445	C₂²×C₅	C₂⁴×C₄	1	1	yes	characteristic
446	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
447	C₂²×C₅	C₂²×#41	1	1	yes	characteristic
448	C₂²×C₅	C₂×H(5,𝔽₂)	1	2	yes	—
449	C₂²×C₅	#585	1	2	yes	—
450	C₄×C₅	—	2	4	no	—
451	C₄×C₅	—	2	8	no	—
452	C₄×C₅	—	2	4	no	—
453	C₄×C₅	—	2	4	no	—
454	C₄×C₅	—	2	4	no	—
455	C₄×C₅	—	4	2	no	—
456	C₄×C₅	—	4	2	no	—
457	C₄×C₅	—	4	4	no	—
458	C₄×C₅	—	4	2	no	—
459	C₄×C₅	C₂×C₄×D₄	1	4	yes	—
460	C₄×C₅	C₂×C₄×D₄	1	4	yes	—
461	C₂⁴	—	2	4	no	—
462	C₂⁴	—	2	4	no	—
463	C₂⁴	—	4	4	no	—
464	C₂⁴	—	4	4	no	—
465	C₂⁴	C₂²×C₄×C₅	1	8	yes	—
466	C₂⁴	C₂⁴×C₅	1	4	yes	—
467	C₂⁴	C₂⁴×C₅	1	2	yes	—
468	C₂⁴	C₂⁴×C₅	1	1	yes	characteristic
469	#41	—	2	32	no	—
470	C₂×Q₈	—	2	2	no	—
471	C₂×Q₈	—	2	2	no	—
472	C₂×Q₈	C₂²×C₄×C₅	1	4	yes	—
473	C₂×D₄	—	2	4	no	—
474	C₂×D₄	—	2	4	no	—
475	C₂×D₄	—	2	32	no	—
476	C₂×D₄	—	2	4	no	—
477	C₂×D₄	—	2	4	no	—
478	C₂×D₄	—	2	2	no	—
479	C₂×D₄	—	2	4	no	—
480	C₂×D₄	—	2	4	no	—
481	C₂×D₄	—	2	2	no	—
482	C₂×D₄	—	4	8	no	—
483	C₂×D₄	—	4	8	no	—
484	C₂×D₄	C₂²×C₄×C₅	1	8	yes	—
485	C₂×D₄	C₂²×C₄×C₅	1	8	yes	—
486	C₂×D₄	C₂²×C₄×C₅	1	8	yes	—
487	C₂×D₄	C₂²×C₄×C₅	1	4	yes	—
488	C₂×D₄	C₂²×C₄×C₅	1	32	yes	—
489	C₂²×C₄	—	2	4	no	—
490	C₂²×C₄	—	2	4	no	—
491	C₂²×C₄	—	2	4	no	—
492	C₂²×C₄	—	2	4	no	—
493	C₂²×C₄	—	2	4	no	—
494	C₂²×C₄	—	2	8	no	—
495	C₂²×C₄	—	2	8	no	—
496	C₂²×C₄	—	2	8	no	—
497	C₂²×C₄	—	4	2	no	—
498	C₂²×C₄	—	4	4	no	—
499	C₂²×C₄	—	4	8	no	—
500	C₂²×C₄	—	4	8	no	—
501	C₂²×C₄	—	4	8	no	—
502	C₂²×C₄	C₂²×C₄×C₅	1	8	yes	—
503	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
504	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
505	C₂²×C₄	C₅×#41	1	4	yes	—
506	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
507	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
508	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
509	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
510	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
511	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
512	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
513	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
514	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
515	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
516	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
517	C₄⋊C₄	—	2	4	no	—
518	C₄⋊C₄	—	2	8	no	—
519	C₄⋊C₄	—	2	4	no	—
520	C₄⋊C₄	—	2	2	no	—
521	C₄⋊C₄	—	2	2	no	—
522	C₄⋊C₄	—	4	8	no	—
523	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
524	C₄⋊C₄	C₂×C₅×D₄	1	4	yes	—
525	C₄⋊C₄	C₂×C₅×D₄	1	4	yes	—
526	C₂²⋊C₄	—	2	4	no	—
527	C₂²⋊C₄	—	2	8	no	—
528	C₂²⋊C₄	—	2	8	no	—
529	C₂²⋊C₄	—	2	8	no	—
530	C₂²⋊C₄	—	2	16	no	—
531	C₂²⋊C₄	—	2	8	no	—
532	C₂²⋊C₄	—	4	8	no	—
533	C₂²⋊C₄	—	4	4	no	—
534	C₂²⋊C₄	C₂²×C₄×C₅	1	8	yes	—
535	C₂²⋊C₄	C₂²×C₄×C₅	1	32	yes	—
536	C₂²⋊C₄	C₂×C₅×D₄	1	8	yes	—
537	C₄²	—	2	4	no	—
538	C₄²	—	2	8	no	—
539	C₄²	—	2	2	no	—
540	C₄²	—	2	4	no	—
541	C₄²	—	2	2	no	—
542	C₄²	—	2	8	no	—
543	C₄²	—	4	4	no	—
544	C₄²	C₂²×C₄×C₅	1	8	yes	—
545	C₄²	C₂×C₅×D₄	1	8	yes	—
546	C₂×C₅	—	2	4	no	—
547	C₂×C₅	—	4	4	no	—
548	C₂×C₅	—	4	8	no	—
549	C₂×C₅	C₂²×C₄×D₄	1	1	yes	characteristic
550	C₂×C₅	C₄×H(5,𝔽₂)	1	2	yes	—
551	C₂×C₅	#3494	1	2	yes	—
552	C₂×C₅	#3498	1	1	yes	characteristic
553	C₂×C₅	#3500	1	1	yes	characteristic
554	C₂³	—	2	2	no	—
555	C₂³	—	2	2	no	—
556	C₂³	—	2	2	no	—
557	C₂³	—	2	4	no	—
558	C₂³	—	2	2	no	—
559	C₂³	—	2	4	no	—
560	C₂³	—	2	8	no	—
561	C₂³	—	2	4	no	—
562	C₂³	—	2	4	no	—
563	C₂³	—	2	8	no	—
564	C₂³	—	2	16	no	—
565	C₂³	—	4	16	no	—
566	C₂³	—	8	8	no	—
567	C₂³	—	8	4	no	—
568	C₂³	—	8	4	no	—
569	C₂³	C₂³×C₄×C₅	1	8	yes	—
570	C₂³	C₂³×C₄×C₅	1	4	yes	—
571	C₂³	C₂³×C₄×C₅	1	2	yes	—
572	C₂³	C₂²×C₅×D₄	1	2	yes	—
573	C₂³	C₂×C₅×#41	1	2	yes	—
574	C₂³	C₂⁵×C₅	1	1	yes	characteristic
575	Q₈	—	2	8	no	—
576	D₄	—	2	16	no	—
577	D₄	—	2	16	no	—
578	D₄	—	2	16	no	—
579	D₄	—	2	8	no	—
580	D₄	—	4	32	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	2	4	no	—
585	C₂×C₄	—	2	4	no	—
586	C₂×C₄	—	2	2	no	—
587	C₂×C₄	—	2	4	no	—
588	C₂×C₄	—	2	4	no	—
589	C₂×C₄	—	2	16	no	—
590	C₂×C₄	—	2	2	no	—
591	C₂×C₄	—	2	2	no	—
592	C₂×C₄	—	2	4	no	—
593	C₂×C₄	—	2	2	no	—
594	C₂×C₄	—	2	8	no	—
595	C₂×C₄	—	2	4	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	2	4	no	—
599	C₂×C₄	—	4	8	no	—
600	C₂×C₄	—	8	8	no	—
601	C₂×C₄	—	8	8	no	—
602	C₂×C₄	—	8	8	no	—
603	C₂×C₄	—	8	4	no	—
604	C₂×C₄	C₄×C₅×D₄	1	16	yes	complemented
605	C₂×C₄	C₂³×C₄×C₅	1	8	yes	—
606	C₂×C₄	C₂³×C₄×C₅	1	4	yes	—
607	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
608	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
609	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
610	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
611	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
612	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
613	C₂×C₄	C₂×C₅×#41	1	2	yes	—
614	C₂×C₄	C₂×C₅×#41	1	2	yes	—
615	C₂×C₄	C₅×H(5,𝔽₂)	1	4	yes	—
616	C₅	#34275	1	1	yes	characteristic, direct product factor, Sylow
617	C₂²	—	2	4	no	—
618	C₂²	—	2	4	no	—
619	C₂²	—	2	2	no	—
620	C₂²	—	2	2	no	—
621	C₂²	—	2	4	no	—
622	C₂²	—	2	8	no	—
623	C₂²	—	2	8	no	—
624	C₂²	—	2	16	no	—
625	C₂²	—	4	8	no	—
626	C₂²	—	4	4	no	—
627	C₂²	—	4	2	no	—
628	C₂²	—	4	4	no	—
629	C₂²	—	4	4	no	—
630	C₂²	—	4	2	no	—
631	C₂²	—	8	16	no	—
632	C₂²	C₂×C₄×C₅×D₄	1	4	yes	—
633	C₂²	C₂⁴×C₄×C₅	1	1	yes	characteristic
634	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
635	C₂²	C₂²×C₅×#41	1	1	yes	characteristic
636	C₂²	C₂×C₅×H(5,𝔽₂)	1	2	yes	—
637	C₂²	C₅×#585	1	2	yes	—
638	C₄	—	2	4	no	—
639	C₄	—	2	8	no	—
640	C₄	—	2	4	no	—
641	C₄	—	2	4	no	—
642	C₄	—	2	4	no	—
643	C₄	—	4	2	no	—
644	C₄	—	4	2	no	—
645	C₄	—	4	4	no	—
646	C₄	—	4	2	no	—
647	C₄	C₂×C₄×C₅×D₄	1	4	yes	—
648	C₄	C₂×C₄×C₅×D₄	1	4	yes	—
649	C₂	—	2	4	no	—
650	C₂	—	4	8	no	—
651	C₂	—	4	4	no	—
652	C₂	C₂²×C₄×C₅×D₄	1	1	yes	characteristic
653	C₂	C₄×C₅×H(5,𝔽₂)	1	2	yes	—
654	C₂	C₅×#3494	1	2	yes	—
655	C₂	C₅×#3498	1	1	yes	characteristic
656	C₂	C₅×#3500	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₅
Frattini, cosocle	C₂³	C₂⁵×C₅
prime radical	C₂×C₅×#535	C₂
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. C₂×C₅×#520	C₂	semidirect
3. C₂×C₅×#518	C₂	semidirect
4. C₅×#2363	C₂	semidirect
5. C₅×#2361	C₂	semidirect
6. C₅×#2358	C₂	semidirect
7. C₅×#2328	C₂	semidirect
8. C₅×#2326	C₂	semidirect
9. C₅×#2317	C₂	semidirect
10. C₄×C₅×#127	C₂	semidirect
11. C₄×C₅×#124	C₂	semidirect
12. C₅×#2297	C₂	semidirect
13. C₂×C₅×#390	C₂	semidirect
14. C₄×C₅×#117	C₂	semidirect
17. C₅×#535	C₄	semidirect
24. C₂×C₄×C₅×Q₈	C₂²	semidirect
30. C₂×C₄×C₅×D₄	C₂²	semidirect
31. C₂×C₄×C₅×D₄	C₂²	semidirect
35. C₂×C₅×#117	C₂²	semidirect
37. C₂²×C₅×C₄⋊C₄	C₂²	semidirect
41. C₅×#391	C₂²	semidirect
42. C₅×#391	C₂²	semidirect
46. C₅×#390	C₂²	semidirect
47. C₅×#390	C₂²	semidirect
48. C₅×#390	C₂²	semidirect
49. C₅×#389	C₂²	semidirect
50. C₅×#388	C₂²	semidirect
51. C₅×#388	C₂²	semidirect
53. C₅×#386	C₂²	semidirect
54. C₅×#386	C₂²	semidirect
56. C₅×#385	C₂²	semidirect
57. C₅×#384	C₂²	semidirect
58. C₅×#382	C₂²	semidirect
59. C₄×C₅×C₄⋊C₄	C₂²	semidirect
60. C₄×C₅×C₄⋊C₄	C₂²	semidirect
62. C₄×C₅×C₂²⋊C₄	C₂²	semidirect
63. C₄×C₅×C₂²⋊C₄	C₂²	semidirect
64. C₅×#376	C₂²	semidirect
65. C₄³×C₅	C₂²	semidirect
66. #34275	C₅	direct (N × H)
70. C₂×C₅×#41	C₂×C₄	semidirect
74. C₂²×C₅×D₄	C₂×C₄	semidirect
87. C₅×#127	C₂×C₄	semidirect
88. C₅×#124	C₂×C₄	semidirect
89. C₅×#121	C₂×C₄	semidirect
90. C₅×#120	C₂×C₄	semidirect
101. C₅×#117	C₂×C₄	semidirect
102. C₅×#117	D₄	semidirect
143. C₂×#520	C₂×C₅	semidirect
144. C₂×#518	C₂×C₅	semidirect
145. #2363	C₂×C₅	semidirect
146. #2361	C₂×C₅	semidirect
147. #2358	C₂×C₅	semidirect
148. #2328	C₂×C₅	semidirect
149. #2326	C₂×C₅	semidirect
150. #2317	C₂×C₅	semidirect
151. C₄×#127	C₂×C₅	semidirect
152. C₄×#124	C₂×C₅	semidirect
153. #2297	C₂×C₅	semidirect
154. C₂×#390	C₂×C₅	semidirect
155. C₄×#117	C₂×C₅	semidirect
243. #535	C₄×C₅	semidirect
250. C₂×C₄×Q₈	C₂²×C₅	semidirect
256. C₂×C₄×D₄	C₂²×C₅	semidirect
257. C₂×C₄×D₄	C₂²×C₅	semidirect
261. C₂×#117	C₂²×C₅	semidirect
263. C₂²×C₄⋊C₄	C₂²×C₅	semidirect
267. #391	C₂²×C₅	semidirect
268. #391	C₂²×C₅	semidirect
272. #390	C₂²×C₅	semidirect
273. #390	C₂²×C₅	semidirect
274. #390	C₂²×C₅	semidirect
275. #389	C₂²×C₅	semidirect
276. #388	C₂²×C₅	semidirect
277. #388	C₂²×C₅	semidirect
279. #386	C₂²×C₅	semidirect
280. #386	C₂²×C₅	semidirect
282. #385	C₂²×C₅	semidirect
283. #384	C₂²×C₅	semidirect
284. #382	C₂²×C₅	semidirect
285. C₄×C₄⋊C₄	C₂²×C₅	semidirect
286. C₄×C₄⋊C₄	C₂²×C₅	semidirect
288. C₄×C₂²⋊C₄	C₂²×C₅	semidirect
289. C₄×C₂²⋊C₄	C₂²×C₅	semidirect
290. #376	C₂²×C₅	semidirect
291. C₄³	C₂²×C₅	semidirect
342. C₂×C₄×C₅	C₄×D₄	semidirect
357. C₂×#41	C₂×C₄×C₅	semidirect
361. C₂²×D₄	C₂×C₄×C₅	semidirect
374. #127	C₂×C₄×C₅	semidirect
375. #124	C₂×C₄×C₅	semidirect
376. #121	C₂×C₄×C₅	semidirect
377. #120	C₂×C₄×C₅	semidirect
388. #117	C₂×C₄×C₅	semidirect
389. #117	C₅×D₄	semidirect
604. C₂×C₄	C₄×C₅×D₄	semidirect
616. C₅	#34275	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×#535	2
C₂×C₅×#520	2
C₂×C₅×#518	2
C₅×#2363	2
C₅×#2361	2
C₅×#2358	2
C₅×#2328	2
C₅×#2326	2
C₅×#2317	2
C₄×C₅×#127	2
C₄×C₅×#124	2
C₅×#2297	2
C₂×C₅×#390	2
C₄×C₅×#117	2
#34275	5

Subgroups — C5×#34275 #11956995

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#34275 #11956995