Subgroups — C₅×#34393 #11957113

Order 1280.

2510 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#532	C₂	1	1	yes	maximal, characteristic
2	C₅×#2377	C₂	1	4	yes	maximal, complemented
3	C₅×#2366	C₂	1	2	yes	maximal, complemented
4	C₅×#2365	C₂	1	2	yes	maximal, complemented
5	C₅×#2357	C₂	1	2	yes	maximal, complemented
6	C₅×#2349	C₂	1	2	yes	maximal, complemented
7	C₅×#2341	C₂	1	2	yes	maximal, complemented
8	C₅×#2339	C₂	1	2	yes	maximal, complemented
9	C₅×#2326	C₂	1	4	yes	maximal, complemented
10	C₅×#2324	C₂	1	4	yes	maximal, complemented
11	C₅×#2318	C₂	1	4	yes	maximal, complemented
12	C₅×#2285	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅×#2283	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²×C₅×#41	C₂²	1	2	yes	—
15	C₅×#532	C₄	1	16	yes	complemented
16	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₅×#127	C₂²	1	1	yes	characteristic
18	C₂×C₅×#124	C₂²	1	2	yes	—
19	C₂×C₅×#121	C₂²	1	2	yes	—
20	C₂×C₅×#121	C₂²	1	2	yes	—
21	C₂×C₄×C₅×Q₈	C₂²	1	4	yes	—
22	C₂×C₄×C₅×D₄	C₂²	1	4	yes	—
23	C₂×C₄×C₅×D₄	C₂²	1	4	yes	—
24	C₂×C₄×C₅×D₄	C₂²	1	2	yes	—
25	C₂×C₄×C₅×D₄	C₂²	1	2	yes	—
26	C₂×C₅×#117	C₂²	1	2	yes	—
27	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
28	C₅×#391	C₂²	1	4	yes	—
29	C₅×#390	C₂²	1	8	yes	—
30	C₅×#390	C₂²	1	4	yes	—
31	C₅×#390	C₂²	1	4	yes	—
32	C₅×#389	C₂²	1	4	yes	—
33	C₅×#388	C₂²	1	8	yes	—
34	C₅×#388	C₂²	1	8	yes	—
35	C₅×#387	C₂²	1	8	yes	—
36	C₅×#386	C₂²	1	8	yes	—
37	C₅×#386	C₂²	1	8	yes	—
38	C₅×#385	C₂²	1	8	yes	—
39	C₅×#385	C₂²	1	8	yes	—
40	C₅×#384	C₂²	1	2	yes	—
41	C₅×#383	C₂²	1	2	yes	—
42	C₅×#382	C₂²	1	1	yes	characteristic
43	C₅×#382	C₂²	1	1	yes	characteristic
44	C₅×#382	C₂²	1	2	yes	—
45	C₅×#381	C₂²	1	2	yes	—
46	C₅×#381	C₂²	1	4	yes	—
47	C₅×#380	C₂²	1	2	yes	—
48	C₅×#380	C₂²	1	1	yes	characteristic
49	C₅×#380	C₂²	1	1	yes	characteristic
50	C₅×#380	C₂²	1	2	yes	—
51	C₅×#380	C₂²	1	4	yes	—
52	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
53	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
54	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
55	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
56	C₄×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
57	C₄×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
58	C₅×#376	C₂²	1	2	yes	—
59	C₂×C₅×#95	C₂²	1	4	yes	—
60	#34393	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂×C₅×#41	—	4	8	no	—
62	C₂×C₅×#41	C₂×C₄	1	16	yes	complemented
63	C₂²×C₅×Q₈	C₂³	1	2	yes	—
64	C₂²×C₅×D₄	C₂³	1	2	yes	—
65	C₂²×C₅×D₄	C₂³	1	2	yes	—
66	C₂²×C₅×D₄	C₂³	1	2	yes	—
67	C₂²×C₅×D₄	C₂³	1	4	yes	—
68	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₅	C₂³	1	8	yes	—
71	C₂³×C₄×C₅	C₂³	1	4	yes	—
72	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
73	C₅×C₄⋊Q₈	C₂×C₄	1	8	yes	complemented
74	C₅×#127	C₂×C₄	1	8	yes	—
75	C₅×#124	C₂×C₄	1	16	yes	complemented
76	C₅×#121	C₂×C₄	1	16	yes	complemented
77	C₅×#121	C₂×C₄	1	16	yes	complemented
78	C₄×C₅×Q₈	—	4	8	no	—
79	C₄×C₅×D₄	—	4	8	no	—
80	C₄×C₅×D₄	—	4	8	no	—
81	C₄×C₅×D₄	C₂×C₄	1	16	yes	complemented
82	C₄×C₅×D₄	C₂×C₄	1	16	yes	complemented
83	C₅×#117	—	4	4	no	—
84	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
85	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
86	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
87	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
88	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
89	C₂×C₅×C₄⋊C₄	C₂³	1	8	yes	—
90	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
91	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
92	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
93	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
94	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
95	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
96	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
97	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
98	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
99	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
100	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
101	C₂×C₄²×C₅	C₂×C₄	1	8	yes	complemented
102	C₂×C₄²×C₅	C₂³	1	4	yes	—
103	C₂×C₄²×C₅	C₂³	1	4	yes	—
104	C₂×C₄²×C₅	C₂³	1	8	yes	—
105	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
107	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
108	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
109	C₅×#95	C₂³	1	2	yes	—
110	C₅×#95	C₂³	1	2	yes	—
111	C₅×#95	C₂³	1	8	yes	—
112	C₅×#95	C₂³	1	4	yes	—
113	C₅×#95	C₂³	1	2	yes	—
114	C₅×#95	C₂³	1	2	yes	—
115	C₅×#95	C₂³	1	2	yes	—
116	C₅×#95	C₂³	1	2	yes	—
117	C₅×#95	C₂³	1	4	yes	—
118	C₅×#95	C₂³	1	4	yes	—
119	C₅×#95	C₂³	1	8	yes	—
120	C₂×#532	C₂×C₅	1	1	yes	characteristic
121	#2377	C₂×C₅	1	4	yes	complemented
122	#2366	C₂×C₅	1	2	yes	complemented
123	#2365	C₂×C₅	1	2	yes	complemented
124	#2357	C₂×C₅	1	2	yes	complemented
125	#2349	C₂×C₅	1	2	yes	complemented
126	#2341	C₂×C₅	1	2	yes	complemented
127	#2339	C₂×C₅	1	2	yes	complemented
128	#2326	C₂×C₅	1	4	yes	complemented
129	#2324	C₂×C₅	1	4	yes	complemented
130	#2318	C₂×C₅	1	4	yes	complemented
131	#2285	C₂×C₅	1	1	yes	characteristic, complemented
132	#2283	C₂×C₅	1	1	yes	characteristic, complemented
133	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
134	C₂⁴×C₅	C₂⁴	1	4	yes	—
135	C₅×#41	—	4	32	no	—
136	C₂×C₅×Q₈	—	2	8	no	—
137	C₂×C₅×Q₈	C₂²×C₄	1	8	yes	—
138	C₂×C₅×D₄	—	2	4	no	—
139	C₂×C₅×D₄	—	2	8	no	—
140	C₂×C₅×D₄	—	2	4	no	—
141	C₂×C₅×D₄	—	4	4	no	—
142	C₂×C₅×D₄	—	4	8	no	—
143	C₂×C₅×D₄	C₂²×C₄	1	8	yes	—
144	C₂×C₅×D₄	C₂²×C₄	1	8	yes	—
145	C₂×C₅×D₄	C₂²×C₄	1	8	yes	—
146	C₂×C₅×D₄	C₂²×C₄	1	16	yes	—
147	C₂²×C₄×C₅	—	2	2	no	—
148	C₂²×C₄×C₅	—	2	4	no	—
149	C₂²×C₄×C₅	—	2	2	no	—
150	C₂²×C₄×C₅	—	4	2	no	—
151	C₂²×C₄×C₅	—	4	8	no	—
152	C₂²×C₄×C₅	—	4	8	no	—
153	C₂²×C₄×C₅	—	4	8	no	—
154	C₂²×C₄×C₅	—	4	8	no	—
155	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
156	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
157	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
158	C₂²×C₄×C₅	C₂²×C₄	1	16	yes	—
159	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
160	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
161	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
162	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
163	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
164	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
165	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
166	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
167	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
168	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
169	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
170	C₅×C₄⋊C₄	—	2	4	no	—
171	C₅×C₄⋊C₄	—	2	4	no	—
172	C₅×C₄⋊C₄	—	2	4	no	—
173	C₅×C₄⋊C₄	—	2	4	no	—
174	C₅×C₄⋊C₄	—	4	4	no	—
175	C₅×C₄⋊C₄	—	4	4	no	—
176	C₅×C₄⋊C₄	—	4	8	no	—
177	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
178	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
179	C₅×C₂²⋊C₄	—	4	8	no	—
180	C₅×C₂²⋊C₄	—	4	4	no	—
181	C₅×C₂²⋊C₄	—	4	2	no	—
182	C₅×C₂²⋊C₄	—	4	2	no	—
183	C₅×C₂²⋊C₄	—	4	8	no	—
184	C₅×C₂²⋊C₄	—	4	8	no	—
185	C₅×C₂²⋊C₄	C₂²×C₄	1	16	yes	—
186	C₅×C₂²⋊C₄	C₂²×C₄	1	16	yes	—
187	C₄²×C₅	—	2	8	no	—
188	C₄²×C₅	—	2	8	no	—
189	C₄²×C₅	—	4	8	no	—
190	C₄²×C₅	C₂²×C₄	1	4	yes	—
191	C₄²×C₅	C₂²×C₄	1	4	yes	—
192	C₄²×C₅	C₂²×C₄	1	4	yes	—
193	C₄²×C₅	C₂²×C₄	1	4	yes	—
194	C₂²×#41	C₂²×C₅	1	2	yes	—
195	#532	C₄×C₅	1	16	yes	complemented
196	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
197	C₂×#127	C₂²×C₅	1	1	yes	characteristic
198	C₂×#124	C₂²×C₅	1	2	yes	—
199	C₂×#121	C₂²×C₅	1	2	yes	—
200	C₂×#121	C₂²×C₅	1	2	yes	—
201	C₂×C₄×Q₈	C₂²×C₅	1	4	yes	—
202	C₂×C₄×D₄	C₂²×C₅	1	4	yes	—
203	C₂×C₄×D₄	C₂²×C₅	1	4	yes	—
204	C₂×C₄×D₄	C₂²×C₅	1	2	yes	—
205	C₂×C₄×D₄	C₂²×C₅	1	2	yes	—
206	C₂×#117	C₂²×C₅	1	2	yes	—
207	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
208	#391	C₂²×C₅	1	4	yes	—
209	#390	C₂²×C₅	1	8	yes	—
210	#390	C₂²×C₅	1	4	yes	—
211	#390	C₂²×C₅	1	4	yes	—
212	#389	C₂²×C₅	1	4	yes	—
213	#388	C₂²×C₅	1	8	yes	—
214	#388	C₂²×C₅	1	8	yes	—
215	#387	C₂²×C₅	1	8	yes	—
216	#386	C₂²×C₅	1	8	yes	—
217	#386	C₂²×C₅	1	8	yes	—
218	#385	C₂²×C₅	1	8	yes	—
219	#385	C₂²×C₅	1	8	yes	—
220	#384	C₂²×C₅	1	2	yes	—
221	#383	C₂²×C₅	1	2	yes	—
222	#382	C₂²×C₅	1	2	yes	—
223	#382	C₂²×C₅	1	1	yes	characteristic
224	#382	C₂²×C₅	1	1	yes	characteristic
225	#381	C₂²×C₅	1	4	yes	—
226	#381	C₂²×C₅	1	2	yes	—
227	#380	C₂²×C₅	1	4	yes	—
228	#380	C₂²×C₅	1	2	yes	—
229	#380	C₂²×C₅	1	1	yes	characteristic
230	#380	C₂²×C₅	1	1	yes	characteristic
231	#380	C₂²×C₅	1	2	yes	—
232	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
233	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
234	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
235	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
236	C₄×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
237	C₄×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
238	#376	C₂²×C₅	1	2	yes	—
239	C₂×#95	C₂²×C₅	1	4	yes	—
240	C₂³×C₅	—	2	1	no	—
241	C₂³×C₅	—	2	4	no	—
242	C₂³×C₅	—	2	8	no	—
243	C₂³×C₅	—	2	2	no	—
244	C₂³×C₅	—	2	1	no	—
245	C₂³×C₅	—	2	4	no	—
246	C₂³×C₅	—	2	2	no	—
247	C₂³×C₅	—	2	8	no	—
248	C₂³×C₅	C₂³×C₄	1	8	yes	—
249	C₂³×C₅	C₂³×C₄	1	2	yes	—
250	C₂³×C₅	C₂⁵	1	1	yes	characteristic
251	C₅×Q₈	—	2	16	no	—
252	C₅×D₄	—	2	16	no	—
253	C₅×D₄	—	2	16	no	—
254	C₅×D₄	—	4	8	no	—
255	C₅×D₄	—	4	16	no	—
256	C₂×C₄×C₅	—	2	8	no	—
257	C₂×C₄×C₅	—	2	1	no	—
258	C₂×C₄×C₅	—	2	2	no	—
259	C₂×C₄×C₅	—	2	1	no	—
260	C₂×C₄×C₅	—	2	2	no	—
261	C₂×C₄×C₅	—	2	2	no	—
262	C₂×C₄×C₅	—	2	8	no	—
263	C₂×C₄×C₅	—	2	8	no	—
264	C₂×C₄×C₅	—	2	4	no	—
265	C₂×C₄×C₅	—	2	4	no	—
266	C₂×C₄×C₅	—	2	4	no	—
267	C₂×C₄×C₅	—	2	4	no	—
268	C₂×C₄×C₅	—	2	4	no	—
269	C₂×C₄×C₅	—	2	4	no	—
270	C₂×C₄×C₅	—	2	4	no	—
271	C₂×C₄×C₅	—	2	4	no	—
272	C₂×C₄×C₅	—	2	4	no	—
273	C₂×C₄×C₅	—	2	4	no	—
274	C₂×C₄×C₅	—	2	4	no	—
275	C₂×C₄×C₅	—	2	4	no	—
276	C₂×C₄×C₅	—	4	4	no	—
277	C₂×C₄×C₅	—	4	16	no	—
278	C₂×C₄×C₅	—	8	16	no	—
279	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
280	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
281	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
282	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
283	C₂×C₄×C₅	C₂³×C₄	1	8	yes	—
284	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
285	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
286	C₂×C₄×C₅	H(5,𝔽₂)	1	4	yes	—
287	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
288	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
289	C₂×#41	—	4	8	no	—
290	C₂×#41	C₂×C₄×C₅	1	16	yes	complemented
291	C₂²×Q₈	C₂³×C₅	1	2	yes	—
292	C₂²×D₄	C₂³×C₅	1	2	yes	—
293	C₂²×D₄	C₂³×C₅	1	2	yes	—
294	C₂²×D₄	C₂³×C₅	1	2	yes	—
295	C₂²×D₄	C₂³×C₅	1	4	yes	—
296	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
297	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
298	C₂³×C₄	C₂³×C₅	1	8	yes	—
299	C₂³×C₄	C₂³×C₅	1	4	yes	—
300	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
301	C₄⋊Q₈	C₂×C₄×C₅	1	8	yes	complemented
302	#127	C₂×C₄×C₅	1	8	yes	—
303	#124	C₂×C₄×C₅	1	16	yes	complemented
304	#121	C₂×C₄×C₅	1	16	yes	complemented
305	#121	C₂×C₄×C₅	1	16	yes	complemented
306	C₄×Q₈	—	4	8	no	—
307	C₄×D₄	—	4	8	no	—
308	C₄×D₄	—	4	8	no	—
309	C₄×D₄	C₂×C₄×C₅	1	16	yes	complemented
310	C₄×D₄	C₂×C₄×C₅	1	16	yes	complemented
311	#117	—	4	4	no	—
312	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
313	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
314	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
315	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
316	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
317	C₂×C₄⋊C₄	C₂³×C₅	1	8	yes	—
318	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
319	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
320	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
321	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
322	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
323	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
324	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
325	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
326	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
327	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
328	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
329	C₂×C₄²	C₂×C₄×C₅	1	8	yes	complemented
330	C₂×C₄²	C₂³×C₅	1	4	yes	—
331	C₂×C₄²	C₂³×C₅	1	4	yes	—
332	C₂×C₄²	C₂³×C₅	1	8	yes	—
333	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
334	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
335	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
336	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
337	#95	C₂³×C₅	1	2	yes	—
338	#95	C₂³×C₅	1	2	yes	—
339	#95	C₂³×C₅	1	8	yes	—
340	#95	C₂³×C₅	1	4	yes	—
341	#95	C₂³×C₅	1	2	yes	—
342	#95	C₂³×C₅	1	2	yes	—
343	#95	C₂³×C₅	1	2	yes	—
344	#95	C₂³×C₅	1	2	yes	—
345	#95	C₂³×C₅	1	4	yes	—
346	#95	C₂³×C₅	1	4	yes	—
347	#95	C₂³×C₅	1	8	yes	—
348	C₂²×C₅	—	2	2	no	—
349	C₂²×C₅	—	2	8	no	—
350	C₂²×C₅	—	2	2	no	—
351	C₂²×C₅	—	2	8	no	—
352	C₂²×C₅	—	2	2	no	—
353	C₂²×C₅	—	2	8	no	—
354	C₂²×C₅	—	4	8	no	—
355	C₂²×C₅	—	4	2	no	—
356	C₂²×C₅	—	4	8	no	—
357	C₂²×C₅	—	4	2	no	—
358	C₂²×C₅	C₂⁴×C₄	1	1	yes	characteristic
359	C₂²×C₅	C₂×H(5,𝔽₂)	1	2	yes	—
360	C₂²×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
361	C₂²×C₅	C₂×2₋¹⁺⁴	1	1	yes	characteristic
362	C₂²×C₅	#585	1	2	yes	—
363	C₄×C₅	—	2	2	no	—
364	C₄×C₅	—	2	8	no	—
365	C₄×C₅	—	2	2	no	—
366	C₄×C₅	—	2	2	no	—
367	C₄×C₅	—	2	2	no	—
368	C₄×C₅	—	4	4	no	—
369	C₄×C₅	—	4	4	no	—
370	C₄×C₅	—	4	4	no	—
371	C₄×C₅	—	4	4	no	—
372	C₄×C₅	#518	1	4	yes	—
373	C₄×C₅	#520	1	4	yes	—
374	C₂⁴	C₂⁴×C₅	1	1	yes	characteristic
375	C₂⁴	C₂⁴×C₅	1	4	yes	—
376	#41	—	4	32	no	—
377	C₂×Q₈	—	2	8	no	—
378	C₂×Q₈	C₂²×C₄×C₅	1	8	yes	—
379	C₂×D₄	—	2	4	no	—
380	C₂×D₄	—	2	8	no	—
381	C₂×D₄	—	2	4	no	—
382	C₂×D₄	—	4	8	no	—
383	C₂×D₄	—	4	4	no	—
384	C₂×D₄	C₂²×C₄×C₅	1	8	yes	—
385	C₂×D₄	C₂²×C₄×C₅	1	8	yes	—
386	C₂×D₄	C₂²×C₄×C₅	1	8	yes	—
387	C₂×D₄	C₂²×C₄×C₅	1	16	yes	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	4	no	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	—	4	8	no	—
392	C₂²×C₄	—	4	2	no	—
393	C₂²×C₄	—	4	8	no	—
394	C₂²×C₄	—	4	8	no	—
395	C₂²×C₄	—	4	8	no	—
396	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
397	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
398	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
399	C₂²×C₄	C₂²×C₄×C₅	1	16	yes	—
400	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
401	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
402	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
403	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
404	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
405	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
406	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
407	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
408	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
409	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
410	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
411	C₄⋊C₄	—	2	4	no	—
412	C₄⋊C₄	—	2	4	no	—
413	C₄⋊C₄	—	2	4	no	—
414	C₄⋊C₄	—	2	4	no	—
415	C₄⋊C₄	—	4	4	no	—
416	C₄⋊C₄	—	4	8	no	—
417	C₄⋊C₄	—	4	4	no	—
418	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
419	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
420	C₂²⋊C₄	—	4	2	no	—
421	C₂²⋊C₄	—	4	2	no	—
422	C₂²⋊C₄	—	4	8	no	—
423	C₂²⋊C₄	—	4	8	no	—
424	C₂²⋊C₄	—	4	8	no	—
425	C₂²⋊C₄	—	4	4	no	—
426	C₂²⋊C₄	C₂²×C₄×C₅	1	16	yes	—
427	C₂²⋊C₄	C₂²×C₄×C₅	1	16	yes	—
428	C₄²	—	2	8	no	—
429	C₄²	—	2	8	no	—
430	C₄²	—	4	8	no	—
431	C₄²	C₂²×C₄×C₅	1	4	yes	—
432	C₄²	C₂²×C₄×C₅	1	4	yes	—
433	C₄²	C₂²×C₄×C₅	1	4	yes	—
434	C₄²	C₂²×C₄×C₅	1	4	yes	—
435	C₂×C₅	—	4	2	no	—
436	C₂×C₅	—	4	8	no	—
437	C₂×C₅	C₂×#518	1	1	yes	characteristic
438	C₂×C₅	C₂×#520	1	1	yes	characteristic
439	C₂×C₅	#3428	1	1	yes	characteristic
440	C₂×C₅	#3546	1	2	yes	—
441	C₂×C₅	#3554	1	2	yes	—
442	C₂³	—	2	1	no	—
443	C₂³	—	2	4	no	—
444	C₂³	—	2	8	no	—
445	C₂³	—	2	2	no	—
446	C₂³	—	2	1	no	—
447	C₂³	—	2	4	no	—
448	C₂³	—	2	2	no	—
449	C₂³	—	2	8	no	—
450	C₂³	C₂³×C₄×C₅	1	8	yes	—
451	C₂³	C₂³×C₄×C₅	1	2	yes	—
452	C₂³	C₂⁵×C₅	1	1	yes	characteristic
453	Q₈	—	2	16	no	—
454	D₄	—	2	16	no	—
455	D₄	—	2	16	no	—
456	D₄	—	4	16	no	—
457	D₄	—	4	8	no	—
458	C₂×C₄	—	2	8	no	—
459	C₂×C₄	—	2	1	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	8	no	—
465	C₂×C₄	—	2	8	no	—
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	2	4	no	—
471	C₂×C₄	—	2	4	no	—
472	C₂×C₄	—	2	4	no	—
473	C₂×C₄	—	2	4	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	2	4	no	—
477	C₂×C₄	—	2	4	no	—
478	C₂×C₄	—	4	16	no	—
479	C₂×C₄	—	4	4	no	—
480	C₂×C₄	—	8	16	no	—
481	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
482	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
483	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
484	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
485	C₂×C₄	C₂³×C₄×C₅	1	8	yes	—
486	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
487	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
488	C₂×C₄	C₅×H(5,𝔽₂)	1	4	yes	—
489	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
490	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
491	C₅	#34393	1	1	yes	characteristic, direct product factor, Sylow
492	C₂²	—	2	2	no	—
493	C₂²	—	2	8	no	—
494	C₂²	—	2	2	no	—
495	C₂²	—	2	8	no	—
496	C₂²	—	2	2	no	—
497	C₂²	—	2	8	no	—
498	C₂²	—	4	8	no	—
499	C₂²	—	4	2	no	—
500	C₂²	—	4	8	no	—
501	C₂²	—	4	2	no	—
502	C₂²	C₂⁴×C₄×C₅	1	1	yes	characteristic
503	C₂²	C₂×C₅×H(5,𝔽₂)	1	2	yes	—
504	C₂²	C₂×C₅×H(5,𝔽₂)	1	1	yes	characteristic
505	C₂²	C₂×C₅×2₋¹⁺⁴	1	1	yes	characteristic
506	C₂²	C₅×#585	1	2	yes	—
507	C₄	—	2	2	no	—
508	C₄	—	2	8	no	—
509	C₄	—	2	2	no	—
510	C₄	—	2	2	no	—
511	C₄	—	2	2	no	—
512	C₄	—	4	4	no	—
513	C₄	—	4	4	no	—
514	C₄	—	4	4	no	—
515	C₄	—	4	4	no	—
516	C₄	C₅×#518	1	4	yes	—
517	C₄	C₅×#520	1	4	yes	—
518	C₂	—	4	2	no	—
519	C₂	—	4	8	no	—
520	C₂	C₂×C₅×#518	1	1	yes	characteristic
521	C₂	C₂×C₅×#520	1	1	yes	characteristic
522	C₂	C₅×#3428	1	1	yes	characteristic
523	C₂	C₅×#3546	1	2	yes	—
524	C₂	C₅×#3554	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×#532	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. C₅×#2377	C₂	semidirect
3. C₅×#2366	C₂	semidirect
4. C₅×#2365	C₂	semidirect
5. C₅×#2357	C₂	semidirect
6. C₅×#2349	C₂	semidirect
7. C₅×#2341	C₂	semidirect
8. C₅×#2339	C₂	semidirect
9. C₅×#2326	C₂	semidirect
10. C₅×#2324	C₂	semidirect
11. C₅×#2318	C₂	semidirect
12. C₅×#2285	C₂	semidirect
13. C₅×#2283	C₂	semidirect
15. C₅×#532	C₄	semidirect
60. #34393	C₅	direct (N × H)
62. C₂×C₅×#41	C₂×C₄	semidirect
73. C₅×C₄⋊Q₈	C₂×C₄	semidirect
75. C₅×#124	C₂×C₄	semidirect
76. C₅×#121	C₂×C₄	semidirect
77. C₅×#121	C₂×C₄	semidirect
81. C₄×C₅×D₄	C₂×C₄	semidirect
82. C₄×C₅×D₄	C₂×C₄	semidirect
101. C₂×C₄²×C₅	C₂×C₄	semidirect
121. #2377	C₂×C₅	semidirect
122. #2366	C₂×C₅	semidirect
123. #2365	C₂×C₅	semidirect
124. #2357	C₂×C₅	semidirect
125. #2349	C₂×C₅	semidirect
126. #2341	C₂×C₅	semidirect
127. #2339	C₂×C₅	semidirect
128. #2326	C₂×C₅	semidirect
129. #2324	C₂×C₅	semidirect
130. #2318	C₂×C₅	semidirect
131. #2285	C₂×C₅	semidirect
132. #2283	C₂×C₅	semidirect
195. #532	C₄×C₅	semidirect
290. C₂×#41	C₂×C₄×C₅	semidirect
301. C₄⋊Q₈	C₂×C₄×C₅	semidirect
303. #124	C₂×C₄×C₅	semidirect
304. #121	C₂×C₄×C₅	semidirect
305. #121	C₂×C₄×C₅	semidirect
309. C₄×D₄	C₂×C₄×C₅	semidirect
310. C₄×D₄	C₂×C₄×C₅	semidirect
329. C₂×C₄²	C₂×C₄×C₅	semidirect
491. C₅	#34393	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×#532	2
C₅×#2377	2
C₅×#2366	2
C₅×#2365	2
C₅×#2357	2
C₅×#2349	2
C₅×#2341	2
C₅×#2339	2
C₅×#2326	2
C₅×#2324	2
C₅×#2318	2
C₅×#2285	2
C₅×#2283	2
#34393	5

Subgroups — C5×#34393 #11957113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#34393 #11957113