Subgroups — C₅×#34265 #11956985

Order 1280.

3586 conjugacy classes of subgroups.

The subgroup lattice has 632 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	C₂²×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2367	C₂	1	4	yes	maximal, complemented
5	C₅×#2366	C₂	1	4	yes	maximal, complemented
6	C₅×#2363	C₂	1	4	yes	maximal, complemented
7	C₅×#2361	C₂	1	4	yes	maximal, complemented
8	C₅×#2321	C₂	1	4	yes	maximal, complemented
9	C₅×#2309	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₅×#120	C₂	1	4	yes	maximal, complemented
11	C₅×#2297	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#2296	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅×#2274	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²×C₅×#41	C₂²	1	1	yes	characteristic
15	C₂³×C₅×D₄	C₂²	1	1	yes	characteristic
16	C₂⁴×C₄×C₅	C₂²	1	2	yes	—
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	C₂×C₄×C₅×Q₈	C₂²	1	2	yes	complemented
24	C₂×C₄×C₅×D₄	—	2	8	no	—
25	C₂×C₄×C₅×D₄	C₂²	1	4	yes	—
26	C₂×C₄×C₅×D₄	C₂²	1	4	yes	—
27	C₂×C₄×C₅×D₄	C₂²	1	4	yes	—
28	C₂×C₄×C₅×D₄	C₂²	1	2	yes	complemented
29	C₂×C₄×C₅×D₄	C₂²	1	8	yes	complemented
30	C₂×C₅×#117	C₂²	1	1	yes	characteristic
31	C₂×C₅×#117	C₂²	1	1	yes	characteristic
32	C₂×C₅×#117	C₂²	1	1	yes	characteristic
33	C₂×C₅×#117	C₂²	1	2	yes	complemented
34	C₂²×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
35	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	complemented
36	C₂²×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
37	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
38	C₅×#391	C₂²	1	2	yes	complemented
39	C₅×#390	C₂²	1	2	yes	—
40	C₅×#390	C₂²	1	2	yes	—
41	C₅×#390	C₂²	1	4	yes	—
42	C₅×#390	C₂²	1	2	yes	complemented
43	C₅×#390	C₂²	1	8	yes	complemented
44	C₅×#389	C₂²	1	2	yes	complemented
45	C₅×#389	C₂²	1	2	yes	complemented
46	C₅×#388	C₂²	1	8	yes	complemented
47	C₅×#388	C₂²	1	8	yes	complemented
48	C₅×#386	C₂²	1	2	yes	—
49	C₅×#386	C₂²	1	2	yes	—
50	C₅×#386	C₂²	1	8	yes	complemented
51	C₅×#386	C₂²	1	8	yes	complemented
52	C₅×#385	C₂²	1	2	yes	—
53	C₅×#385	C₂²	1	2	yes	—
54	C₅×#385	C₂²	1	8	yes	complemented
55	C₅×#379	C₂²	1	4	yes	—
56	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	complemented
57	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	complemented
58	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	complemented
59	C₄×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
60	C₄×C₅×C₂²⋊C₄	C₂²	1	4	yes	complemented
61	C₄×C₅×C₂²⋊C₄	C₂²	1	8	yes	complemented
62	#34265	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂⁵×C₅	C₂³	1	2	yes	—
64	C₂×C₅×#41	—	2	4	no	—
65	C₂×C₅×#41	—	2	4	no	—
66	C₂×C₅×#41	C₂×C₄	1	8	yes	complemented
67	C₂²×C₅×Q₈	C₂³	1	1	yes	characteristic
68	C₂²×C₅×D₄	—	2	4	no	—
69	C₂²×C₅×D₄	—	2	4	no	—
70	C₂²×C₅×D₄	C₂×C₄	1	8	yes	complemented
71	C₂²×C₅×D₄	C₂³	1	2	yes	—
72	C₂²×C₅×D₄	C₂³	1	2	yes	—
73	C₂²×C₅×D₄	C₂³	1	2	yes	—
74	C₂²×C₅×D₄	C₂³	1	1	yes	characteristic
75	C₂²×C₅×D₄	C₂³	1	8	yes	—
76	C₂³×C₄×C₅	—	2	8	no	—
77	C₂³×C₄×C₅	—	4	8	no	—
78	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
79	C₂³×C₄×C₅	C₂³	1	8	yes	—
80	C₂³×C₄×C₅	C₂³	1	2	yes	—
81	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
82	C₂³×C₄×C₅	C₂³	1	4	yes	—
83	C₂³×C₄×C₅	C₂³	1	2	yes	—
84	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
85	C₅×#127	C₂×C₄	1	16	yes	complemented
86	C₅×#124	C₂×C₄	1	16	yes	complemented
87	C₅×#121	C₂×C₄	1	32	yes	complemented
88	C₅×#120	C₂×C₄	1	32	yes	complemented
89	C₄×C₅×Q₈	—	2	8	no	—
90	C₄×C₅×D₄	—	2	16	no	—
91	C₄×C₅×D₄	—	2	16	no	—
92	C₄×C₅×D₄	—	2	16	no	—
93	C₄×C₅×D₄	—	2	8	no	—
94	C₄×C₅×D₄	—	4	16	no	—
95	C₅×#117	—	2	4	no	—
96	C₅×#117	—	2	4	no	—
97	C₅×#117	—	2	8	no	—
98	C₅×#117	C₂×C₄	1	8	yes	complemented
99	C₂×C₅×C₄⋊C₄	—	2	8	no	—
100	C₂×C₅×C₄⋊C₄	D₄	1	8	yes	complemented
101	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
102	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
103	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
104	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
105	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
106	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
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	2	yes	—
110	C₂×C₅×C₂²⋊C₄	—	2	8	no	—
111	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
112	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
113	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
114	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
115	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
116	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
117	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
118	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
119	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
120	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
121	C₂×C₄²×C₅	—	2	4	no	—
122	C₂×C₄²×C₅	C₂³	1	2	yes	—
123	C₂×C₄²×C₅	C₂³	1	4	yes	—
124	C₂×C₄²×C₅	C₂³	1	2	yes	—
125	C₂×C₄²×C₅	C₂³	1	2	yes	—
126	C₂×C₄²×C₅	C₂³	1	2	yes	—
127	C₂×C₄²×C₅	C₂³	1	8	yes	—
128	C₂×C₄²×C₅	C₂³	1	2	yes	—
129	C₅×#95	C₂³	1	8	yes	—
130	C₅×#95	C₂³	1	4	yes	—
131	C₅×#95	C₂³	1	4	yes	—
132	C₅×#95	C₂³	1	4	yes	—
133	C₅×#95	C₂³	1	4	yes	—
134	C₂×#535	C₂×C₅	1	1	yes	characteristic
135	C₂×#520	C₂×C₅	1	1	yes	characteristic, complemented
136	C₂²×C₄×D₄	C₂×C₅	1	1	yes	characteristic, complemented
137	#2367	C₂×C₅	1	4	yes	complemented
138	#2366	C₂×C₅	1	4	yes	complemented
139	#2363	C₂×C₅	1	4	yes	complemented
140	#2361	C₂×C₅	1	4	yes	complemented
141	#2321	C₂×C₅	1	4	yes	complemented
142	#2309	C₂×C₅	1	1	yes	characteristic, complemented
143	C₄×#120	C₂×C₅	1	4	yes	complemented
144	#2297	C₂×C₅	1	1	yes	characteristic, complemented
145	#2296	C₂×C₅	1	1	yes	characteristic, complemented
146	#2274	C₂×C₅	1	1	yes	characteristic, complemented
147	C₂⁴×C₅	—	2	4	no	—
148	C₂⁴×C₅	—	2	4	no	—
149	C₂⁴×C₅	—	4	4	no	—
150	C₂⁴×C₅	—	4	4	no	—
151	C₂⁴×C₅	C₂²×C₄	1	8	yes	—
152	C₂⁴×C₅	C₂⁴	1	4	yes	—
153	C₂⁴×C₅	C₂⁴	1	2	yes	—
154	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
155	C₅×#41	—	2	32	no	—
156	C₂×C₅×Q₈	—	2	2	no	—
157	C₂×C₅×Q₈	—	2	2	no	—
158	C₂×C₅×Q₈	C₂²×C₄	1	4	yes	complemented
159	C₂×C₅×D₄	—	2	4	no	—
160	C₂×C₅×D₄	—	2	4	no	—
161	C₂×C₅×D₄	—	2	32	no	—
162	C₂×C₅×D₄	—	2	4	no	—
163	C₂×C₅×D₄	—	2	4	no	—
164	C₂×C₅×D₄	—	2	2	no	—
165	C₂×C₅×D₄	—	2	4	no	—
166	C₂×C₅×D₄	—	2	4	no	—
167	C₂×C₅×D₄	—	2	2	no	—
168	C₂×C₅×D₄	—	4	8	no	—
169	C₂×C₅×D₄	—	4	8	no	—
170	C₂×C₅×D₄	C₂²×C₄	1	8	yes	—
171	C₂×C₅×D₄	C₂²×C₄	1	8	yes	—
172	C₂×C₅×D₄	C₂²×C₄	1	8	yes	—
173	C₂×C₅×D₄	C₂²×C₄	1	4	yes	complemented
174	C₂×C₅×D₄	C₂²×C₄	1	32	yes	complemented
175	C₂²×C₄×C₅	—	2	2	no	—
176	C₂²×C₄×C₅	—	2	4	no	—
177	C₂²×C₄×C₅	—	2	4	no	—
178	C₂²×C₄×C₅	—	2	4	no	—
179	C₂²×C₄×C₅	—	2	2	no	—
180	C₂²×C₄×C₅	—	2	16	no	—
181	C₂²×C₄×C₅	—	2	16	no	—
182	C₂²×C₄×C₅	—	2	8	no	—
183	C₂²×C₄×C₅	—	4	4	no	—
184	C₂²×C₄×C₅	—	4	8	no	—
185	C₂²×C₄×C₅	—	4	2	no	—
186	C₂²×C₄×C₅	—	4	8	no	—
187	C₂²×C₄×C₅	—	4	32	no	—
188	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
189	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	complemented
190	C₂²×C₄×C₅	C₂×D₄	1	8	yes	—
191	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
192	C₂²×C₄×C₅	#41	1	4	yes	—
193	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
194	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
195	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
196	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
197	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
198	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
199	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
200	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
201	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
202	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
203	C₅×C₄⋊C₄	—	2	2	no	—
204	C₅×C₄⋊C₄	—	2	8	no	—
205	C₅×C₄⋊C₄	—	2	2	no	—
206	C₅×C₄⋊C₄	—	2	4	no	—
207	C₅×C₄⋊C₄	—	2	8	no	—
208	C₅×C₄⋊C₄	—	4	4	no	—
209	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	complemented
210	C₅×C₄⋊C₄	C₂×D₄	1	8	yes	—
211	C₅×C₄⋊C₄	C₂×D₄	1	8	yes	—
212	C₅×C₂²⋊C₄	—	2	4	no	—
213	C₅×C₂²⋊C₄	—	2	4	no	—
214	C₅×C₂²⋊C₄	—	2	16	no	—
215	C₅×C₂²⋊C₄	—	2	8	no	—
216	C₅×C₂²⋊C₄	—	2	8	no	—
217	C₅×C₂²⋊C₄	—	4	8	no	—
218	C₅×C₂²⋊C₄	—	4	4	no	—
219	C₅×C₂²⋊C₄	—	4	8	no	—
220	C₅×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
221	C₅×C₂²⋊C₄	C₂²×C₄	1	32	yes	complemented
222	C₄²×C₅	—	2	4	no	—
223	C₄²×C₅	—	2	4	no	—
224	C₄²×C₅	—	2	8	no	—
225	C₄²×C₅	—	2	4	no	—
226	C₄²×C₅	—	2	4	no	—
227	C₄²×C₅	—	4	8	no	—
228	C₄²×C₅	C₂²×C₄	1	8	yes	complemented
229	C₂²×#41	C₂²×C₅	1	1	yes	characteristic
230	C₂³×D₄	C₂²×C₅	1	1	yes	characteristic
231	C₂⁴×C₄	C₂²×C₅	1	2	yes	—
232	#535	C₄×C₅	1	16	yes	complemented
233	C₂×#127	C₂²×C₅	1	2	yes	—
234	C₂×#124	C₂²×C₅	1	2	yes	—
235	C₂×#121	C₂²×C₅	1	4	yes	—
236	C₂×#120	C₂²×C₅	1	4	yes	—
237	#520	—	2	8	no	—
238	C₂×C₄×Q₈	C₂²×C₅	1	2	yes	complemented
239	C₂×C₄×D₄	—	2	8	no	—
240	C₂×C₄×D₄	C₂²×C₅	1	4	yes	—
241	C₂×C₄×D₄	C₂²×C₅	1	4	yes	—
242	C₂×C₄×D₄	C₂²×C₅	1	4	yes	—
243	C₂×C₄×D₄	C₂²×C₅	1	2	yes	complemented
244	C₂×C₄×D₄	C₂²×C₅	1	8	yes	complemented
245	C₂×#117	C₂²×C₅	1	1	yes	characteristic
246	C₂×#117	C₂²×C₅	1	1	yes	characteristic
247	C₂×#117	C₂²×C₅	1	1	yes	characteristic
248	C₂×#117	C₂²×C₅	1	2	yes	complemented
249	C₂²×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
250	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	complemented
251	C₂²×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
252	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
253	#391	C₂²×C₅	1	2	yes	complemented
254	#390	C₂²×C₅	1	2	yes	—
255	#390	C₂²×C₅	1	2	yes	—
256	#390	C₂²×C₅	1	4	yes	—
257	#390	C₂²×C₅	1	8	yes	complemented
258	#390	C₂²×C₅	1	2	yes	complemented
259	#389	C₂²×C₅	1	2	yes	complemented
260	#389	C₂²×C₅	1	2	yes	complemented
261	#388	C₂²×C₅	1	8	yes	complemented
262	#388	C₂²×C₅	1	8	yes	complemented
263	#386	C₂²×C₅	1	2	yes	—
264	#386	C₂²×C₅	1	2	yes	—
265	#386	C₂²×C₅	1	8	yes	complemented
266	#386	C₂²×C₅	1	8	yes	complemented
267	#385	C₂²×C₅	1	2	yes	—
268	#385	C₂²×C₅	1	2	yes	—
269	#385	C₂²×C₅	1	8	yes	complemented
270	#379	C₂²×C₅	1	4	yes	—
271	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	complemented
272	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	complemented
273	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	complemented
274	C₄×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
275	C₄×C₂²⋊C₄	C₂²×C₅	1	4	yes	complemented
276	C₄×C₂²⋊C₄	C₂²×C₅	1	8	yes	complemented
277	C₂³×C₅	—	2	2	no	—
278	C₂³×C₅	—	2	2	no	—
279	C₂³×C₅	—	2	16	no	—
280	C₂³×C₅	—	2	2	no	—
281	C₂³×C₅	—	2	4	no	—
282	C₂³×C₅	—	2	2	no	—
283	C₂³×C₅	—	2	4	no	—
284	C₂³×C₅	—	2	4	no	—
285	C₂³×C₅	—	2	8	no	—
286	C₂³×C₅	—	2	4	no	—
287	C₂³×C₅	—	2	8	no	—
288	C₂³×C₅	—	4	16	no	—
289	C₂³×C₅	—	4	8	no	—
290	C₂³×C₅	—	4	8	no	—
291	C₂³×C₅	—	4	16	no	—
292	C₂³×C₅	C₂³×C₄	1	8	yes	—
293	C₂³×C₅	C₂³×C₄	1	4	yes	—
294	C₂³×C₅	C₂³×C₄	1	2	yes	—
295	C₂³×C₅	C₂²×D₄	1	2	yes	—
296	C₂³×C₅	C₂×#41	1	2	yes	—
297	C₂³×C₅	C₂⁵	1	1	yes	characteristic
298	C₅×Q₈	—	2	8	no	—
299	C₅×D₄	—	2	16	no	—
300	C₅×D₄	—	2	16	no	—
301	C₅×D₄	—	2	16	no	—
302	C₅×D₄	—	2	8	no	—
303	C₅×D₄	—	4	32	no	—
304	C₂×C₄×C₅	—	2	8	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	4	no	—
309	C₂×C₄×C₅	—	2	8	no	—
310	C₂×C₄×C₅	—	2	2	no	—
311	C₂×C₄×C₅	—	2	4	no	—
312	C₂×C₄×C₅	—	2	16	no	—
313	C₂×C₄×C₅	—	2	2	no	—
314	C₂×C₄×C₅	—	2	2	no	—
315	C₂×C₄×C₅	—	2	2	no	—
316	C₂×C₄×C₅	—	2	4	no	—
317	C₂×C₄×C₅	—	2	4	no	—
318	C₂×C₄×C₅	—	2	4	no	—
319	C₂×C₄×C₅	—	2	4	no	—
320	C₂×C₄×C₅	—	2	4	no	—
321	C₂×C₄×C₅	—	2	2	no	—
322	C₂×C₄×C₅	—	2	2	no	—
323	C₂×C₄×C₅	—	4	8	no	—
324	C₂×C₄×C₅	—	4	32	no	—
325	C₂×C₄×C₅	—	4	4	no	—
326	C₂×C₄×C₅	—	8	8	no	—
327	C₂×C₄×C₅	C₄×D₄	1	16	yes	complemented
328	C₂×C₄×C₅	C₂³×C₄	1	8	yes	—
329	C₂×C₄×C₅	C₂³×C₄	1	4	yes	—
330	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
331	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
332	C₂×C₄×C₅	C₂²×D₄	1	8	yes	—
333	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
334	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
335	C₂×C₄×C₅	C₂×#41	1	2	yes	—
336	C₂×C₄×C₅	C₂×#41	1	2	yes	—
337	C₂×C₄×C₅	H(5,𝔽₂)	1	4	yes	—
338	C₂⁵	C₂³×C₅	1	2	yes	—
339	C₂×#41	—	2	4	no	—
340	C₂×#41	—	2	4	no	—
341	C₂×#41	C₂×C₄×C₅	1	8	yes	complemented
342	C₂²×Q₈	C₂³×C₅	1	1	yes	characteristic
343	C₂²×D₄	—	2	4	no	—
344	C₂²×D₄	—	2	4	no	—
345	C₂²×D₄	C₂×C₄×C₅	1	8	yes	complemented
346	C₂²×D₄	C₂³×C₅	1	2	yes	—
347	C₂²×D₄	C₂³×C₅	1	2	yes	—
348	C₂²×D₄	C₂³×C₅	1	2	yes	—
349	C₂²×D₄	C₂³×C₅	1	1	yes	characteristic
350	C₂²×D₄	C₂³×C₅	1	8	yes	—
351	C₂³×C₄	—	2	8	no	—
352	C₂³×C₄	—	4	8	no	—
353	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
354	C₂³×C₄	C₂³×C₅	1	8	yes	—
355	C₂³×C₄	C₂³×C₅	1	2	yes	—
356	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
357	C₂³×C₄	C₂³×C₅	1	4	yes	—
358	C₂³×C₄	C₂³×C₅	1	2	yes	—
359	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
360	#127	C₂×C₄×C₅	1	16	yes	complemented
361	#124	C₂×C₄×C₅	1	16	yes	complemented
362	#121	C₂×C₄×C₅	1	32	yes	complemented
363	#120	C₂×C₄×C₅	1	32	yes	complemented
364	C₄×Q₈	—	2	8	no	—
365	C₄×D₄	—	2	16	no	—
366	C₄×D₄	—	2	16	no	—
367	C₄×D₄	—	2	16	no	—
368	C₄×D₄	—	2	8	no	—
369	C₄×D₄	—	4	16	no	—
370	#117	—	2	4	no	—
371	#117	—	2	4	no	—
372	#117	—	2	8	no	—
373	#117	C₂×C₄×C₅	1	8	yes	complemented
374	C₂×C₄⋊C₄	—	2	8	no	—
375	C₂×C₄⋊C₄	C₅×D₄	1	8	yes	complemented
376	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
377	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
378	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
379	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
380	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
381	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
382	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
383	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
384	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
385	C₂×C₂²⋊C₄	—	2	8	no	—
386	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
387	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
388	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
389	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
390	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
391	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
392	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
393	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
394	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
395	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
396	C₂×C₄²	—	2	4	no	—
397	C₂×C₄²	C₂³×C₅	1	2	yes	—
398	C₂×C₄²	C₂³×C₅	1	2	yes	—
399	C₂×C₄²	C₂³×C₅	1	4	yes	—
400	C₂×C₄²	C₂³×C₅	1	2	yes	—
401	C₂×C₄²	C₂³×C₅	1	2	yes	—
402	C₂×C₄²	C₂³×C₅	1	8	yes	—
403	C₂×C₄²	C₂³×C₅	1	2	yes	—
404	#95	C₂³×C₅	1	8	yes	—
405	#95	C₂³×C₅	1	4	yes	—
406	#95	C₂³×C₅	1	4	yes	—
407	#95	C₂³×C₅	1	4	yes	—
408	#95	C₂³×C₅	1	4	yes	—
409	C₂²×C₅	—	2	4	no	—
410	C₂²×C₅	—	2	2	no	—
411	C₂²×C₅	—	2	2	no	—
412	C₂²×C₅	—	2	4	no	—
413	C₂²×C₅	—	2	4	no	—
414	C₂²×C₅	—	2	8	no	—
415	C₂²×C₅	—	2	8	no	—
416	C₂²×C₅	—	2	16	no	—
417	C₂²×C₅	—	4	32	no	—
418	C₂²×C₅	—	4	4	no	—
419	C₂²×C₅	—	4	2	no	—
420	C₂²×C₅	—	4	2	no	—
421	C₂²×C₅	—	4	4	no	—
422	C₂²×C₅	—	4	4	no	—
423	C₂²×C₅	—	4	8	no	—
424	C₂²×C₅	C₂×C₄×D₄	1	4	yes	—
425	C₂²×C₅	C₂⁴×C₄	1	1	yes	characteristic
426	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
427	C₂²×C₅	C₂²×#41	1	1	yes	characteristic
428	C₂²×C₅	C₂×H(5,𝔽₂)	1	2	yes	—
429	C₂²×C₅	#585	1	2	yes	—
430	C₄×C₅	—	2	4	no	—
431	C₄×C₅	—	2	8	no	—
432	C₄×C₅	—	2	4	no	—
433	C₄×C₅	—	2	8	no	—
434	C₄×C₅	—	4	2	no	—
435	C₄×C₅	—	4	2	no	—
436	C₄×C₅	—	4	2	no	—
437	C₄×C₅	—	4	4	no	—
438	C₄×C₅	C₂×C₄×D₄	1	4	yes	—
439	C₄×C₅	C₂×C₄×D₄	1	4	yes	—
440	C₂⁴	—	2	4	no	—
441	C₂⁴	—	2	4	no	—
442	C₂⁴	—	4	4	no	—
443	C₂⁴	—	4	4	no	—
444	C₂⁴	C₂²×C₄×C₅	1	8	yes	—
445	C₂⁴	C₂⁴×C₅	1	4	yes	—
446	C₂⁴	C₂⁴×C₅	1	2	yes	—
447	C₂⁴	C₂⁴×C₅	1	1	yes	characteristic
448	#41	—	2	32	no	—
449	C₂×Q₈	—	2	2	no	—
450	C₂×Q₈	—	2	2	no	—
451	C₂×Q₈	C₂²×C₄×C₅	1	4	yes	complemented
452	C₂×D₄	—	2	4	no	—
453	C₂×D₄	—	2	4	no	—
454	C₂×D₄	—	2	32	no	—
455	C₂×D₄	—	2	4	no	—
456	C₂×D₄	—	2	4	no	—
457	C₂×D₄	—	2	2	no	—
458	C₂×D₄	—	2	4	no	—
459	C₂×D₄	—	2	4	no	—
460	C₂×D₄	—	2	2	no	—
461	C₂×D₄	—	4	8	no	—
462	C₂×D₄	—	4	8	no	—
463	C₂×D₄	C₂²×C₄×C₅	1	8	yes	—
464	C₂×D₄	C₂²×C₄×C₅	1	8	yes	—
465	C₂×D₄	C₂²×C₄×C₅	1	8	yes	—
466	C₂×D₄	C₂²×C₄×C₅	1	4	yes	complemented
467	C₂×D₄	C₂²×C₄×C₅	1	32	yes	complemented
468	C₂²×C₄	—	2	2	no	—
469	C₂²×C₄	—	2	4	no	—
470	C₂²×C₄	—	2	4	no	—
471	C₂²×C₄	—	2	4	no	—
472	C₂²×C₄	—	2	2	no	—
473	C₂²×C₄	—	2	16	no	—
474	C₂²×C₄	—	2	16	no	—
475	C₂²×C₄	—	2	8	no	—
476	C₂²×C₄	—	4	4	no	—
477	C₂²×C₄	—	4	8	no	—
478	C₂²×C₄	—	4	2	no	—
479	C₂²×C₄	—	4	8	no	—
480	C₂²×C₄	—	4	32	no	—
481	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
482	C₂²×C₄	C₂²×C₄×C₅	1	8	yes	complemented
483	C₂²×C₄	C₂×C₅×D₄	1	8	yes	—
484	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
485	C₂²×C₄	C₅×#41	1	4	yes	—
486	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
487	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
488	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
489	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
490	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
491	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
492	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
493	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
494	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
495	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
496	C₄⋊C₄	—	2	2	no	—
497	C₄⋊C₄	—	2	8	no	—
498	C₄⋊C₄	—	2	2	no	—
499	C₄⋊C₄	—	2	4	no	—
500	C₄⋊C₄	—	2	8	no	—
501	C₄⋊C₄	—	4	4	no	—
502	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	complemented
503	C₄⋊C₄	C₂×C₅×D₄	1	8	yes	—
504	C₄⋊C₄	C₂×C₅×D₄	1	8	yes	—
505	C₂²⋊C₄	—	2	4	no	—
506	C₂²⋊C₄	—	2	4	no	—
507	C₂²⋊C₄	—	2	16	no	—
508	C₂²⋊C₄	—	2	8	no	—
509	C₂²⋊C₄	—	2	8	no	—
510	C₂²⋊C₄	—	4	8	no	—
511	C₂²⋊C₄	—	4	8	no	—
512	C₂²⋊C₄	—	4	4	no	—
513	C₂²⋊C₄	C₂²×C₄×C₅	1	8	yes	—
514	C₂²⋊C₄	C₂²×C₄×C₅	1	32	yes	complemented
515	C₄²	—	2	4	no	—
516	C₄²	—	2	4	no	—
517	C₄²	—	2	8	no	—
518	C₄²	—	2	4	no	—
519	C₄²	—	2	4	no	—
520	C₄²	—	4	8	no	—
521	C₄²	C₂²×C₄×C₅	1	8	yes	complemented
522	C₂×C₅	—	2	4	no	—
523	C₂×C₅	—	4	4	no	—
524	C₂×C₅	—	4	8	no	—
525	C₂×C₅	C₂²×C₄×D₄	1	1	yes	characteristic
526	C₂×C₅	C₄×H(5,𝔽₂)	1	2	yes	—
527	C₂×C₅	#3484	1	1	yes	characteristic
528	C₂×C₅	#3487	1	1	yes	characteristic
529	C₂×C₅	#3506	1	2	yes	—
530	C₂³	—	2	2	no	—
531	C₂³	—	2	2	no	—
532	C₂³	—	2	16	no	—
533	C₂³	—	2	2	no	—
534	C₂³	—	2	4	no	—
535	C₂³	—	2	2	no	—
536	C₂³	—	2	4	no	—
537	C₂³	—	2	4	no	—
538	C₂³	—	2	8	no	—
539	C₂³	—	2	4	no	—
540	C₂³	—	2	8	no	—
541	C₂³	—	4	16	no	—
542	C₂³	—	4	8	no	—
543	C₂³	—	4	8	no	—
544	C₂³	—	4	16	no	—
545	C₂³	C₂³×C₄×C₅	1	8	yes	—
546	C₂³	C₂³×C₄×C₅	1	4	yes	—
547	C₂³	C₂³×C₄×C₅	1	2	yes	—
548	C₂³	C₂²×C₅×D₄	1	2	yes	—
549	C₂³	C₂×C₅×#41	1	2	yes	—
550	C₂³	C₂⁵×C₅	1	1	yes	characteristic
551	Q₈	—	2	8	no	—
552	D₄	—	2	16	no	—
553	D₄	—	2	16	no	—
554	D₄	—	2	16	no	—
555	D₄	—	2	8	no	—
556	D₄	—	4	32	no	—
557	C₂×C₄	—	2	8	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	4	no	—
562	C₂×C₄	—	2	8	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	4	no	—
565	C₂×C₄	—	2	16	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	4	no	—
570	C₂×C₄	—	2	4	no	—
571	C₂×C₄	—	2	4	no	—
572	C₂×C₄	—	2	4	no	—
573	C₂×C₄	—	2	4	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	2	2	no	—
576	C₂×C₄	—	4	8	no	—
577	C₂×C₄	—	4	32	no	—
578	C₂×C₄	—	4	4	no	—
579	C₂×C₄	—	8	8	no	—
580	C₂×C₄	C₄×C₅×D₄	1	16	yes	complemented
581	C₂×C₄	C₂³×C₄×C₅	1	8	yes	—
582	C₂×C₄	C₂³×C₄×C₅	1	4	yes	—
583	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
584	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
585	C₂×C₄	C₂²×C₅×D₄	1	8	yes	—
586	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
587	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
588	C₂×C₄	C₂×C₅×#41	1	2	yes	—
589	C₂×C₄	C₂×C₅×#41	1	2	yes	—
590	C₂×C₄	C₅×H(5,𝔽₂)	1	4	yes	—
591	C₅	#34265	1	1	yes	characteristic, direct product factor, Sylow
592	C₂²	—	2	4	no	—
593	C₂²	—	2	2	no	—
594	C₂²	—	2	2	no	—
595	C₂²	—	2	4	no	—
596	C₂²	—	2	4	no	—
597	C₂²	—	2	8	no	—
598	C₂²	—	2	8	no	—
599	C₂²	—	2	16	no	—
600	C₂²	—	4	32	no	—
601	C₂²	—	4	4	no	—
602	C₂²	—	4	2	no	—
603	C₂²	—	4	4	no	—
604	C₂²	—	4	2	no	—
605	C₂²	—	4	8	no	—
606	C₂²	—	4	4	no	—
607	C₂²	C₂×C₄×C₅×D₄	1	4	yes	—
608	C₂²	C₂⁴×C₄×C₅	1	1	yes	characteristic
609	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
610	C₂²	C₂²×C₅×#41	1	1	yes	characteristic
611	C₂²	C₂×C₅×H(5,𝔽₂)	1	2	yes	—
612	C₂²	C₅×#585	1	2	yes	—
613	C₄	—	2	4	no	—
614	C₄	—	2	8	no	—
615	C₄	—	2	4	no	—
616	C₄	—	2	8	no	—
617	C₄	—	4	2	no	—
618	C₄	—	4	2	no	—
619	C₄	—	4	2	no	—
620	C₄	—	4	4	no	—
621	C₄	C₂×C₄×C₅×D₄	1	4	yes	—
622	C₄	C₂×C₄×C₅×D₄	1	4	yes	—
623	C₂	—	2	4	no	—
624	C₂	—	4	8	no	—
625	C₂	—	4	4	no	—
626	C₂	C₂²×C₄×C₅×D₄	1	1	yes	characteristic
627	C₂	C₄×C₅×H(5,𝔽₂)	1	2	yes	—
628	C₂	C₅×#3484	1	1	yes	characteristic
629	C₂	C₅×#3487	1	1	yes	characteristic
630	C₂	C₅×#3506	1	2	yes	—

	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. C₂²×C₄×C₅×D₄	C₂	semidirect
4. C₅×#2367	C₂	semidirect
5. C₅×#2366	C₂	semidirect
6. C₅×#2363	C₂	semidirect
7. C₅×#2361	C₂	semidirect
8. C₅×#2321	C₂	semidirect
9. C₅×#2309	C₂	semidirect
10. C₄×C₅×#120	C₂	semidirect
11. C₅×#2297	C₂	semidirect
12. C₅×#2296	C₂	semidirect
13. C₅×#2274	C₂	semidirect
17. C₅×#535	C₄	semidirect
23. C₂×C₄×C₅×Q₈	C₂²	semidirect
28. C₂×C₄×C₅×D₄	C₂²	semidirect
29. C₂×C₄×C₅×D₄	C₂²	semidirect
33. C₂×C₅×#117	C₂²	semidirect
35. C₂²×C₅×C₄⋊C₄	C₂²	semidirect
38. C₅×#391	C₂²	semidirect
42. C₅×#390	C₂²	semidirect
43. C₅×#390	C₂²	semidirect
44. C₅×#389	C₂²	semidirect
45. C₅×#389	C₂²	semidirect
46. C₅×#388	C₂²	semidirect
47. C₅×#388	C₂²	semidirect
50. C₅×#386	C₂²	semidirect
51. C₅×#386	C₂²	semidirect
54. C₅×#385	C₂²	semidirect
56. C₄×C₅×C₄⋊C₄	C₂²	semidirect
57. C₄×C₅×C₄⋊C₄	C₂²	semidirect
58. C₄×C₅×C₄⋊C₄	C₂²	semidirect
60. C₄×C₅×C₂²⋊C₄	C₂²	semidirect
61. C₄×C₅×C₂²⋊C₄	C₂²	semidirect
62. #34265	C₅	direct (N × H)
66. C₂×C₅×#41	C₂×C₄	semidirect
70. C₂²×C₅×D₄	C₂×C₄	semidirect
85. C₅×#127	C₂×C₄	semidirect
86. C₅×#124	C₂×C₄	semidirect
87. C₅×#121	C₂×C₄	semidirect
88. C₅×#120	C₂×C₄	semidirect
98. C₅×#117	C₂×C₄	semidirect
100. C₂×C₅×C₄⋊C₄	D₄	semidirect
135. C₂×#520	C₂×C₅	semidirect
136. C₂²×C₄×D₄	C₂×C₅	semidirect
137. #2367	C₂×C₅	semidirect
138. #2366	C₂×C₅	semidirect
139. #2363	C₂×C₅	semidirect
140. #2361	C₂×C₅	semidirect
141. #2321	C₂×C₅	semidirect
142. #2309	C₂×C₅	semidirect
143. C₄×#120	C₂×C₅	semidirect
144. #2297	C₂×C₅	semidirect
145. #2296	C₂×C₅	semidirect
146. #2274	C₂×C₅	semidirect
158. C₂×C₅×Q₈	C₂²×C₄	semidirect
173. C₂×C₅×D₄	C₂²×C₄	semidirect
174. C₂×C₅×D₄	C₂²×C₄	semidirect
189. C₂²×C₄×C₅	C₂²×C₄	semidirect
209. C₅×C₄⋊C₄	C₂²×C₄	semidirect
221. C₅×C₂²⋊C₄	C₂²×C₄	semidirect
228. C₄²×C₅	C₂²×C₄	semidirect
232. #535	C₄×C₅	semidirect
238. C₂×C₄×Q₈	C₂²×C₅	semidirect
243. C₂×C₄×D₄	C₂²×C₅	semidirect
244. C₂×C₄×D₄	C₂²×C₅	semidirect
248. C₂×#117	C₂²×C₅	semidirect
250. C₂²×C₄⋊C₄	C₂²×C₅	semidirect
253. #391	C₂²×C₅	semidirect
257. #390	C₂²×C₅	semidirect
258. #390	C₂²×C₅	semidirect
259. #389	C₂²×C₅	semidirect
260. #389	C₂²×C₅	semidirect
261. #388	C₂²×C₅	semidirect
262. #388	C₂²×C₅	semidirect
265. #386	C₂²×C₅	semidirect
266. #386	C₂²×C₅	semidirect
269. #385	C₂²×C₅	semidirect
271. C₄×C₄⋊C₄	C₂²×C₅	semidirect
272. C₄×C₄⋊C₄	C₂²×C₅	semidirect
273. C₄×C₄⋊C₄	C₂²×C₅	semidirect
275. C₄×C₂²⋊C₄	C₂²×C₅	semidirect
276. C₄×C₂²⋊C₄	C₂²×C₅	semidirect
327. C₂×C₄×C₅	C₄×D₄	semidirect
341. C₂×#41	C₂×C₄×C₅	semidirect
345. C₂²×D₄	C₂×C₄×C₅	semidirect
360. #127	C₂×C₄×C₅	semidirect
361. #124	C₂×C₄×C₅	semidirect
362. #121	C₂×C₄×C₅	semidirect
363. #120	C₂×C₄×C₅	semidirect
373. #117	C₂×C₄×C₅	semidirect
375. C₂×C₄⋊C₄	C₅×D₄	semidirect
451. C₂×Q₈	C₂²×C₄×C₅	semidirect
466. C₂×D₄	C₂²×C₄×C₅	semidirect
467. C₂×D₄	C₂²×C₄×C₅	semidirect
482. C₂²×C₄	C₂²×C₄×C₅	semidirect
502. C₄⋊C₄	C₂²×C₄×C₅	semidirect
514. C₂²⋊C₄	C₂²×C₄×C₅	semidirect
521. C₄²	C₂²×C₄×C₅	semidirect
580. C₂×C₄	C₄×C₅×D₄	semidirect
591. C₅	#34265	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
C₂²×C₄×C₅×D₄	2
C₅×#2367	2
C₅×#2366	2
C₅×#2363	2
C₅×#2361	2
C₅×#2321	2
C₅×#2309	2
C₄×C₅×#120	2
C₅×#2297	2
C₅×#2296	2
C₅×#2274	2
#34265	5

Subgroups — C5×#34265 #11956985

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#34265 #11956985