Subgroups — C₅×#65923 #13106744

Order 1600.

822 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×D₄×D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅²×#120	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4099	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4084	C₂	1	2	yes	maximal, complemented
6	C₅×#4083	C₂	1	2	yes	maximal, complemented
7	C₅×#4081	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#4032	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#4030	C₂	1	2	yes	maximal, complemented
10	C₅×#4029	C₂	1	2	yes	maximal, complemented
11	C₅×D₅×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂⁴×C₅²	C₂²	1	1	yes	characteristic, complemented
13	C₂³×C₅×D₅	C₂²	1	1	yes	characteristic
14	C₂³×C₅×D₅	C₂²	1	1	yes	characteristic, complemented
15	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
16	C₂×C₅²×D₄	C₂²	1	2	yes	complemented
17	C₂×C₅×C₂²⋊D₅	—	2	4	no	—
18	C₂×C₅×C₂²⋊D₅	C₂²	1	1	yes	characteristic
19	C₂×C₅×C₂²⋊D₅	C₂²	1	2	yes	—
20	C₂×C₅×C₂²⋊D₅	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₅×C₂²⋊D₅	C₂²	1	2	yes	complemented
22	C₂×C₅×C₂²⋊D₅	C₂²	1	2	yes	complemented
23	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic, complemented
24	C₅×D₄×D₅	—	2	4	no	—
25	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic, complemented
26	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic, complemented
27	C₂×C₄×C₅×D₅	C₂²	1	2	yes	complemented
28	C₅²×C₂²⋊C₄	C₂²	1	1	yes	characteristic, complemented
29	C₅²×C₂²⋊C₄	C₂²	1	2	yes	complemented
30	C₅×C₂²⋊Dic₅	C₂²	1	2	yes	complemented
31	C₅×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic, complemented
32	C₅×C₂²⋊Dic₅	C₂²	1	2	yes	complemented
33	C₅×#685	C₂²	1	2	yes	complemented
34	C₅×#685	C₂²	1	2	yes	complemented
35	C₅×C₄⋊Dic₅	C₂²	1	2	yes	complemented
36	C₅×#683	C₂²	1	2	yes	complemented
37	C₅×#534	—	5	1	no	maximal
38	#65923	C₅	1	1	yes	maximal, characteristic, direct product factor
39	C₂³×C₅²	—	2	2	no	—
40	C₂³×C₅²	—	4	2	no	—
41	C₂³×C₅²	C₂³	1	2	yes	—
42	C₂³×C₅²	C₂³	1	1	yes	characteristic
43	C₂³×C₅²	C₂³	1	1	yes	characteristic
44	C₂²×C₅×D₅	—	2	2	no	—
45	C₂²×C₅×D₅	—	4	2	no	—
46	C₂²×C₅×D₅	—	4	2	no	—
47	C₂²×C₅×D₅	D₄	1	4	yes	complemented
48	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
49	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
50	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
51	C₅²×D₄	—	2	2	no	—
52	C₅²×D₄	—	4	2	no	—
53	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
54	C₂×C₄×C₅²	C₂³	1	2	yes	—
55	C₅×C₂²⋊D₅	—	2	2	no	—
56	C₅×C₂²⋊D₅	—	2	2	no	—
57	C₅×C₂²⋊D₅	—	2	4	no	—
58	C₅×C₂²⋊D₅	—	2	4	no	—
59	C₅×C₂²⋊D₅	—	4	2	no	—
60	C₂×C₅×Dic₅	—	2	2	no	—
61	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
62	C₂×C₅×Dic₅	C₂³	1	2	yes	—
63	C₂×C₅×Dic₅	C₂³	1	2	yes	—
64	C₅×D₂₀	—	2	2	no	—
65	C₄×C₅×D₅	—	2	2	no	—
66	C₄×C₅×D₅	—	4	2	no	—
67	C₂²×C₅×D₄	—	5	1	no	—
68	C₂²×C₅×D₄	—	5	1	no	—
69	C₂²×C₂²⋊D₅	C₂×C₅	1	1	yes	characteristic, complemented
70	C₂×D₄×D₅	C₂×C₅	1	1	yes	characteristic, complemented
71	C₅×#123	—	5	1	no	—
72	C₅×#123	—	5	1	no	—
73	C₅×#123	—	5	2	no	—
74	C₅×#121	—	5	2	no	—
75	C₅×#121	—	5	2	no	—
76	C₅×#120	—	2	2	no	—
77	C₅×#120	—	5	1	no	—
78	C₅×#120	—	5	2	no	—
79	C₅×#120	D₅	1	1	yes	characteristic, complemented
80	C₅×#120	C₂×C₅	1	1	yes	characteristic, complemented
81	C₂×C₅×C₂²⋊C₄	—	5	1	no	—
82	#4099	C₂×C₅	1	1	yes	characteristic, complemented
83	#4084	C₂×C₅	1	2	yes	complemented
84	#4083	C₂×C₅	1	2	yes	complemented
85	#4081	C₂×C₅	1	1	yes	characteristic, complemented
86	#4032	C₂×C₅	1	1	yes	characteristic, complemented
87	#4030	C₂×C₅	1	2	yes	complemented
88	#4029	C₂×C₅	1	2	yes	complemented
89	D₅×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
90	C₂²×C₅²	—	2	2	no	—
91	C₂²×C₅²	—	2	2	no	—
92	C₂²×C₅²	—	2	1	no	—
93	C₂²×C₅²	—	2	2	no	—
94	C₂²×C₅²	—	2	2	no	—
95	C₂²×C₅²	—	2	2	no	—
96	C₂²×C₅²	—	4	4	no	—
97	C₂²×C₅²	C₂×D₄	1	2	yes	—
98	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
99	C₂×C₅×D₅	—	2	1	no	—
100	C₂×C₅×D₅	—	2	2	no	—
101	C₂×C₅×D₅	—	2	2	no	—
102	C₂×C₅×D₅	—	2	2	no	—
103	C₂×C₅×D₅	—	4	4	no	—
104	C₂×C₅×D₅	—	4	4	no	—
105	C₂×C₅×D₅	C₂×D₄	1	2	yes	—
106	C₂×C₅×D₅	C₂×D₄	1	2	yes	—
107	C₄×C₅²	—	2	1	no	—
108	C₄×C₅²	—	2	2	no	—
109	C₅×Dic₅	—	2	1	no	—
110	C₅×Dic₅	—	2	2	no	—
111	C₅×Dic₅	—	2	2	no	—
112	C₂⁴×C₅	—	2	2	no	—
113	C₂⁴×C₅	—	5	1	no	—
114	C₂⁴×C₅	—	5	1	no	—
115	C₂⁴×C₅	C₂×D₅	1	1	yes	characteristic, complemented
116	C₂⁴×C₅	C₂²×C₅	1	1	yes	characteristic, complemented
117	C₂³×D₅	C₂²×C₅	1	1	yes	characteristic
118	C₂³×D₅	C₂²×C₅	1	1	yes	characteristic, complemented
119	C₂×C₅×D₄	—	2	2	no	—
120	C₂×C₅×D₄	—	2	4	no	—
121	C₂×C₅×D₄	—	5	2	no	—
122	C₂×C₅×D₄	—	5	1	no	—
123	C₂×C₅×D₄	—	5	1	no	—
124	C₂×C₅×D₄	—	5	2	no	—
125	C₂×C₅×D₄	—	5	2	no	—
126	C₂×C₅×D₄	—	5	1	no	—
127	C₂×C₅×D₄	—	10	4	no	—
128	C₂×C₅×D₄	—	10	4	no	—
129	C₂×C₅×D₄	C₂×D₅	1	1	yes	characteristic
130	C₂×C₅×D₄	C₂×D₅	1	2	yes	complemented
131	C₂×C₅×D₄	C₂²×C₅	1	1	yes	characteristic
132	C₂×C₅×D₄	C₂²×C₅	1	2	yes	complemented
133	C₂²×C₄×C₅	—	5	1	no	—
134	C₂²×C₄×C₅	—	5	1	no	—
135	C₂²×C₄×C₅	—	5	2	no	—
136	C₂×C₂²⋊D₅	—	2	4	no	—
137	C₂×C₂²⋊D₅	C₂²×C₅	1	1	yes	characteristic
138	C₂×C₂²⋊D₅	C₂²×C₅	1	2	yes	—
139	C₂×C₂²⋊D₅	C₂²×C₅	1	1	yes	characteristic, complemented
140	C₂×C₂²⋊D₅	C₂²×C₅	1	2	yes	complemented
141	C₂×C₂²⋊D₅	C₂²×C₅	1	2	yes	complemented
142	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic, complemented
143	D₄×D₅	—	2	4	no	—
144	C₂×D₂₀	C₂²×C₅	1	1	yes	characteristic, complemented
145	C₂×C₄×D₅	C₂²×C₅	1	1	yes	characteristic, complemented
146	C₂×C₄×D₅	C₂²×C₅	1	2	yes	complemented
147	C₅×C₄⋊C₄	—	5	2	no	—
148	C₅×C₄⋊C₄	—	5	2	no	—
149	C₅×C₂²⋊C₄	—	2	2	no	—
150	C₅×C₂²⋊C₄	—	2	4	no	—
151	C₅×C₂²⋊C₄	—	5	1	no	—
152	C₅×C₂²⋊C₄	—	5	2	no	—
153	C₅×C₂²⋊C₄	—	5	2	no	—
154	C₅×C₂²⋊C₄	—	5	2	no	—
155	C₅×C₂²⋊C₄	—	5	2	no	—
156	C₅×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic, complemented
157	C₅×C₂²⋊C₄	C₂×D₅	1	2	yes	complemented
158	C₅×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic, complemented
159	C₅×C₂²⋊C₄	C₂²×C₅	1	2	yes	complemented
160	C₂²⋊Dic₅	C₂²×C₅	1	1	yes	characteristic, complemented
161	C₂²⋊Dic₅	C₂²×C₅	1	2	yes	complemented
162	C₂²⋊Dic₅	C₂²×C₅	1	2	yes	complemented
163	#685	C₂²×C₅	1	2	yes	complemented
164	#685	C₂²×C₅	1	2	yes	complemented
165	C₄⋊Dic₅	C₂²×C₅	1	2	yes	complemented
166	#683	C₂²×C₅	1	2	yes	complemented
167	#534	—	5	1	no	Sylow
168	C₂×C₅²	—	2	2	no	—
169	C₂×C₅²	—	4	2	no	—
170	C₂×C₅²	—	4	1	no	—
171	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₅²	H(5,𝔽₂)	1	2	yes	—
173	C₅×D₅	—	2	2	no	—
174	C₅×D₅	—	2	2	no	—
175	C₅×D₅	—	4	1	no	—
176	C₂³×C₅	—	2	2	no	—
177	C₂³×C₅	—	2	2	no	—
178	C₂³×C₅	—	2	4	no	—
179	C₂³×C₅	—	2	2	no	—
180	C₂³×C₅	—	2	2	no	—
181	C₂³×C₅	—	4	4	no	—
182	C₂³×C₅	—	4	2	no	—
183	C₂³×C₅	—	4	2	no	—
184	C₂³×C₅	—	4	8	no	—
185	C₂³×C₅	—	5	1	no	—
186	C₂³×C₅	—	5	1	no	—
187	C₂³×C₅	—	5	4	no	—
188	C₂³×C₅	—	5	1	no	—
189	C₂³×C₅	—	10	2	no	—
190	C₂³×C₅	—	20	2	no	—
191	C₂³×C₅	—	20	2	no	—
192	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
193	C₂³×C₅	C₂²×D₅	1	2	yes	—
194	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
195	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
196	C₂³×C₅	C₂³×C₅	1	2	yes	—
197	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
198	C₂²×D₅	—	2	2	no	—
199	C₂²×D₅	—	4	2	no	—
200	C₂²×D₅	—	4	2	no	—
201	C₂²×D₅	C₅×D₄	1	4	yes	complemented
202	C₂²×D₅	C₂³×C₅	1	1	yes	characteristic
203	C₂²×D₅	C₂³×C₅	1	1	yes	characteristic
204	C₂²×D₅	C₂³×C₅	1	1	yes	characteristic
205	C₅×D₄	—	2	2	no	—
206	C₅×D₄	—	2	2	no	—
207	C₅×D₄	—	4	4	no	—
208	C₅×D₄	—	4	2	no	—
209	C₅×D₄	—	4	2	no	—
210	C₅×D₄	—	4	8	no	—
211	C₅×D₄	—	10	2	no	—
212	C₅×D₄	—	10	2	no	—
213	C₅×D₄	—	10	4	no	—
214	C₅×D₄	—	10	4	no	—
215	C₅×D₄	—	10	2	no	—
216	C₅×D₄	—	20	2	no	—
217	C₂×C₄×C₅	—	2	4	no	—
218	C₂×C₄×C₅	—	2	2	no	—
219	C₂×C₄×C₅	—	5	2	no	—
220	C₂×C₄×C₅	—	5	1	no	—
221	C₂×C₄×C₅	—	5	2	no	—
222	C₂×C₄×C₅	—	10	2	no	—
223	C₂×C₄×C₅	—	10	2	no	—
224	C₂×C₄×C₅	—	20	2	no	—
225	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
226	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
227	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
228	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
229	C₂²⋊D₅	—	2	2	no	—
230	C₂²⋊D₅	—	2	2	no	—
231	C₂²⋊D₅	—	2	4	no	—
232	C₂²⋊D₅	—	2	4	no	—
233	C₂²⋊D₅	—	4	2	no	—
234	C₂×Dic₅	—	2	2	no	—
235	C₂×Dic₅	C₂³×C₅	1	2	yes	—
236	C₂×Dic₅	C₂³×C₅	1	2	yes	—
237	C₂×Dic₅	C₂³×C₅	1	1	yes	characteristic
238	D₂₀	—	2	2	no	—
239	C₄×D₅	—	2	2	no	—
240	C₄×D₅	—	4	2	no	—
241	C₂²×D₄	—	5	1	no	—
242	C₂²×D₄	—	5	1	no	—
243	#123	—	5	1	no	—
244	#123	—	5	2	no	—
245	#123	—	5	1	no	—
246	#121	—	5	2	no	—
247	#121	—	5	2	no	—
248	#120	—	5	1	no	—
249	#120	—	5	2	no	—
250	#120	C₅×D₅	1	1	yes	characteristic, complemented
251	C₂×C₂²⋊C₄	—	5	1	no	—
252	C₅²	#534	1	1	yes	characteristic, complemented, Sylow
253	C₂²×C₅	—	2	2	no	—
254	C₂²×C₅	—	2	2	no	—
255	C₂²×C₅	—	2	8	no	—
256	C₂²×C₅	—	2	1	no	—
257	C₂²×C₅	—	2	2	no	—
258	C₂²×C₅	—	2	2	no	—
259	C₂²×C₅	—	2	2	no	—
260	C₂²×C₅	—	2	2	no	—
261	C₂²×C₅	—	2	2	no	—
262	C₂²×C₅	—	2	1	no	—
263	C₂²×C₅	—	2	2	no	—
264	C₂²×C₅	—	2	2	no	—
265	C₂²×C₅	—	2	2	no	—
266	C₂²×C₅	—	2	4	no	—
267	C₂²×C₅	—	2	2	no	—
268	C₂²×C₅	—	4	4	no	—
269	C₂²×C₅	—	4	4	no	—
270	C₂²×C₅	—	4	4	no	—
271	C₂²×C₅	—	4	2	no	—
272	C₂²×C₅	—	4	4	no	—
273	C₂²×C₅	—	4	4	no	—
274	C₂²×C₅	—	4	4	no	—
275	C₂²×C₅	—	4	16	no	—
276	C₂²×C₅	—	5	2	no	—
277	C₂²×C₅	—	5	2	no	—
278	C₂²×C₅	—	10	2	no	—
279	C₂²×C₅	—	10	2	no	—
280	C₂²×C₅	—	10	1	no	—
281	C₂²×C₅	—	10	2	no	—
282	C₂²×C₅	—	20	4	no	—
283	C₂²×C₅	—	20	4	no	—
284	C₂²×C₅	D₄×D₅	1	2	yes	—
285	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
286	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
287	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
288	C₂×D₅	—	2	2	no	—
289	C₂×D₅	—	2	2	no	—
290	C₂×D₅	—	2	1	no	—
291	C₂×D₅	—	2	2	no	—
292	C₂×D₅	—	4	4	no	—
293	C₂×D₅	—	4	4	no	—
294	C₂×D₅	C₂×C₅×D₄	1	2	yes	—
295	C₂×D₅	C₂×C₅×D₄	1	2	yes	—
296	C₄×C₅	—	2	1	no	—
297	C₄×C₅	—	2	2	no	—
298	C₄×C₅	—	2	1	no	—
299	C₄×C₅	—	2	2	no	—
300	C₄×C₅	—	4	2	no	—
301	C₄×C₅	—	4	4	no	—
302	C₄×C₅	—	10	1	no	—
303	C₄×C₅	—	10	2	no	—
304	C₄×C₅	—	10	2	no	—
305	Dic₅	—	2	1	no	—
306	Dic₅	—	2	2	no	—
307	Dic₅	—	2	2	no	—
308	C₂⁴	—	5	1	no	—
309	C₂⁴	—	5	1	no	—
310	C₂⁴	C₂×C₅×D₅	1	1	yes	characteristic, complemented
311	C₂×D₄	—	5	2	no	—
312	C₂×D₄	—	5	1	no	—
313	C₂×D₄	—	5	1	no	—
314	C₂×D₄	—	5	2	no	—
315	C₂×D₄	—	5	2	no	—
316	C₂×D₄	—	5	1	no	—
317	C₂×D₄	—	10	4	no	—
318	C₂×D₄	—	10	4	no	—
319	C₂×D₄	C₂×C₅×D₅	1	1	yes	characteristic
320	C₂×D₄	C₂×C₅×D₅	1	2	yes	complemented
321	C₂²×C₄	—	5	1	no	—
322	C₂²×C₄	—	5	1	no	—
323	C₂²×C₄	—	5	2	no	—
324	C₄⋊C₄	—	5	2	no	—
325	C₄⋊C₄	—	5	2	no	—
326	C₂²⋊C₄	—	5	1	no	—
327	C₂²⋊C₄	—	5	2	no	—
328	C₂²⋊C₄	—	5	2	no	—
329	C₂²⋊C₄	—	5	2	no	—
330	C₂²⋊C₄	—	5	2	no	—
331	C₂²⋊C₄	C₂×C₅×D₅	1	1	yes	characteristic, complemented
332	C₂²⋊C₄	C₂×C₅×D₅	1	2	yes	complemented
333	C₂×C₅	—	2	2	no	—
334	C₂×C₅	—	2	2	no	—
335	C₂×C₅	—	2	2	no	—
336	C₂×C₅	—	2	4	no	—
337	C₂×C₅	—	4	4	no	—
338	C₂×C₅	—	4	8	no	—
339	C₂×C₅	—	4	2	no	—
340	C₂×C₅	—	4	1	no	—
341	C₂×C₅	—	4	1	no	—
342	C₂×C₅	—	4	2	no	—
343	C₂×C₅	—	8	2	no	—
344	C₂×C₅	—	10	2	no	—
345	C₂×C₅	—	10	2	no	—
346	C₂×C₅	—	20	1	no	—
347	C₂×C₅	C₂×D₄×D₅	1	1	yes	characteristic
348	C₂×C₅	#4144	1	2	yes	—
349	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
350	C₂×C₅	C₅×H(5,𝔽₂)	1	2	yes	—
351	D₅	—	2	2	no	—
352	D₅	—	2	2	no	—
353	D₅	—	4	1	no	—
354	C₂³	—	2	2	no	—
355	C₂³	—	4	2	no	—
356	C₂³	—	5	1	no	—
357	C₂³	—	5	1	no	—
358	C₂³	—	5	4	no	—
359	C₂³	—	5	1	no	—
360	C₂³	—	10	2	no	—
361	C₂³	—	20	2	no	—
362	C₂³	—	20	2	no	—
363	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
364	C₂³	C₂²×C₅×D₅	1	2	yes	—
365	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
366	D₄	—	2	2	no	—
367	D₄	—	4	2	no	—
368	D₄	—	10	2	no	—
369	D₄	—	10	2	no	—
370	D₄	—	10	4	no	—
371	D₄	—	10	4	no	—
372	D₄	—	10	2	no	—
373	D₄	—	20	2	no	—
374	C₂×C₄	—	5	2	no	—
375	C₂×C₄	—	5	1	no	—
376	C₂×C₄	—	5	2	no	—
377	C₂×C₄	—	10	2	no	—
378	C₂×C₄	—	10	2	no	—
379	C₂×C₄	—	20	2	no	—
380	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
381	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
382	C₅	—	2	2	no	—
383	C₅	#65923	1	1	yes	characteristic, direct product factor
384	C₅	C₅×#534	1	1	yes	characteristic, complemented
385	C₂²	—	2	2	no	—
386	C₂²	—	2	1	no	—
387	C₂²	—	2	2	no	—
388	C₂²	—	2	2	no	—
389	C₂²	—	2	2	no	—
390	C₂²	—	2	2	no	—
391	C₂²	—	4	4	no	—
392	C₂²	—	5	2	no	—
393	C₂²	—	5	2	no	—
394	C₂²	—	10	2	no	—
395	C₂²	—	10	2	no	—
396	C₂²	—	10	1	no	—
397	C₂²	—	10	2	no	—
398	C₂²	—	20	4	no	—
399	C₂²	—	20	4	no	—
400	C₂²	C₅×D₄×D₅	1	2	yes	—
401	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
402	C₄	—	2	1	no	—
403	C₄	—	2	2	no	—
404	C₄	—	10	1	no	—
405	C₄	—	10	2	no	—
406	C₄	—	10	2	no	—
407	C₂	—	2	2	no	—
408	C₂	—	4	2	no	—
409	C₂	—	4	1	no	—
410	C₂	—	10	2	no	—
411	C₂	—	10	2	no	—
412	C₂	—	20	1	no	—
413	C₂	C₂×C₅×D₄×D₅	1	1	yes	characteristic
414	C₂	C₅×#4144	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₅
Fitting	C₅²×#120	C₂
socle	C₂²×C₅²	C₂⁴
hypercenter	C₅×#120	D₅
hypocenter	C₅	C₅×#534
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#120

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×C₂²⋊D₅	C₂	semidirect
2. C₂×C₅×D₄×D₅	C₂	semidirect
3. C₅²×#120	C₂	semidirect
4. C₅×#4099	C₂	semidirect
5. C₅×#4084	C₂	semidirect
6. C₅×#4083	C₂	semidirect
7. C₅×#4081	C₂	semidirect
8. C₅×#4032	C₂	semidirect
9. C₅×#4030	C₂	semidirect
10. C₅×#4029	C₂	semidirect
11. C₅×D₅×C₂²⋊C₄	C₂	semidirect
12. C₂⁴×C₅²	C₂²	semidirect
14. C₂³×C₅×D₅	C₂²	semidirect
16. C₂×C₅²×D₄	C₂²	semidirect
20. C₂×C₅×C₂²⋊D₅	C₂²	semidirect
21. C₂×C₅×C₂²⋊D₅	C₂²	semidirect
22. C₂×C₅×C₂²⋊D₅	C₂²	semidirect
23. C₂²×C₅×Dic₅	C₂²	semidirect
25. C₂×C₅×D₂₀	C₂²	semidirect
26. C₂×C₄×C₅×D₅	C₂²	semidirect
27. C₂×C₄×C₅×D₅	C₂²	semidirect
28. C₅²×C₂²⋊C₄	C₂²	semidirect
29. C₅²×C₂²⋊C₄	C₂²	semidirect
30. C₅×C₂²⋊Dic₅	C₂²	semidirect
31. C₅×C₂²⋊Dic₅	C₂²	semidirect
32. C₅×C₂²⋊Dic₅	C₂²	semidirect
33. C₅×#685	C₂²	semidirect
34. C₅×#685	C₂²	semidirect
35. C₅×C₄⋊Dic₅	C₂²	semidirect
36. C₅×#683	C₂²	semidirect
38. #65923	C₅	direct (N × H)
47. C₂²×C₅×D₅	D₄	semidirect
69. C₂²×C₂²⋊D₅	C₂×C₅	semidirect
70. C₂×D₄×D₅	C₂×C₅	semidirect
79. C₅×#120	D₅	semidirect
80. C₅×#120	C₂×C₅	semidirect
82. #4099	C₂×C₅	semidirect
83. #4084	C₂×C₅	semidirect
84. #4083	C₂×C₅	semidirect
85. #4081	C₂×C₅	semidirect
86. #4032	C₂×C₅	semidirect
87. #4030	C₂×C₅	semidirect
88. #4029	C₂×C₅	semidirect
89. D₅×C₂²⋊C₄	C₂×C₅	semidirect
115. C₂⁴×C₅	C₂×D₅	semidirect
116. C₂⁴×C₅	C₂²×C₅	semidirect
118. C₂³×D₅	C₂²×C₅	semidirect
130. C₂×C₅×D₄	C₂×D₅	semidirect
132. C₂×C₅×D₄	C₂²×C₅	semidirect
139. C₂×C₂²⋊D₅	C₂²×C₅	semidirect
140. C₂×C₂²⋊D₅	C₂²×C₅	semidirect
141. C₂×C₂²⋊D₅	C₂²×C₅	semidirect
142. C₂²×Dic₅	C₂²×C₅	semidirect
144. C₂×D₂₀	C₂²×C₅	semidirect
145. C₂×C₄×D₅	C₂²×C₅	semidirect
146. C₂×C₄×D₅	C₂²×C₅	semidirect
156. C₅×C₂²⋊C₄	C₂×D₅	semidirect
157. C₅×C₂²⋊C₄	C₂×D₅	semidirect
158. C₅×C₂²⋊C₄	C₂²×C₅	semidirect
159. C₅×C₂²⋊C₄	C₂²×C₅	semidirect
160. C₂²⋊Dic₅	C₂²×C₅	semidirect
161. C₂²⋊Dic₅	C₂²×C₅	semidirect
162. C₂²⋊Dic₅	C₂²×C₅	semidirect
163. #685	C₂²×C₅	semidirect
164. #685	C₂²×C₅	semidirect
165. C₄⋊Dic₅	C₂²×C₅	semidirect
166. #683	C₂²×C₅	semidirect
201. C₂²×D₅	C₅×D₄	semidirect
250. #120	C₅×D₅	semidirect
252. C₅²	#534	semidirect
310. C₂⁴	C₂×C₅×D₅	semidirect
320. C₂×D₄	C₂×C₅×D₅	semidirect
331. C₂²⋊C₄	C₂×C₅×D₅	semidirect
332. C₂²⋊C₄	C₂×C₅×D₅	semidirect
383. C₅	#65923	direct (N × H)
384. C₅	C₅×#534	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₅×C₂²⋊D₅	2
C₂×C₅×D₄×D₅	2
C₅²×#120	2
C₅×#4099	2
C₅×#4084	2
C₅×#4083	2
C₅×#4081	2
C₅×#4032	2
C₅×#4030	2
C₅×#4029	2
C₅×D₅×C₂²⋊C₄	2
C₅×#534	5
#65923	5

Subgroups — C5×#65923 #13106744

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65923 #13106744