Subgroups — C₅×#65829 #13106649

Order 1600.

628 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#4099	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×C₂²⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4075	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₅×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4072	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4070	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#4032	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#4031	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#4029	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#4027	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#4025	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅×#4025	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×#4024	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅×#4023	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂⁴×C₅²	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
19	C₂×C₅×C₂²⋊D₅	C₂²	1	1	yes	characteristic
20	C₂×C₅×C₂²⋊D₅	C₂²	1	1	yes	characteristic
21	C₂×C₅×C₂²⋊D₅	C₂²	1	1	yes	characteristic
22	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
23	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic, complemented
25	C₂×C₅×Dic₁₀	C₂²	1	1	yes	characteristic, complemented
26	C₅²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
27	C₅²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
28	C₅²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
29	C₅²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
30	C₅×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
31	C₅×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
32	C₅×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
33	C₅×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
34	C₅×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
35	C₅×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
36	C₅×C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
37	C₅×#685	C₂²	1	1	yes	characteristic, complemented
38	C₅×#685	C₂²	1	1	yes	characteristic, complemented
39	C₅×#685	C₂²	1	1	yes	characteristic, complemented
40	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic, complemented
41	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic, complemented
42	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic, complemented
43	C₅×#683	C₂²	1	1	yes	characteristic, complemented
44	C₅×#683	C₂²	1	1	yes	characteristic, complemented
45	C₅×#683	C₂²	1	1	yes	characteristic, complemented
46	C₅×#683	C₂²	1	1	yes	characteristic, complemented
47	C₅×#683	C₂²	1	1	yes	characteristic, complemented
48	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic, complemented
49	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic, complemented
50	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic, complemented
51	C₅×#551	—	5	1	no	maximal
52	#65829	C₅	1	1	yes	maximal, characteristic, direct product factor
53	C₂³×C₅²	—	2	2	no	—
54	C₂³×C₅²	—	2	2	no	—
55	C₂³×C₅²	—	4	1	no	—
56	C₂³×C₅²	C₂³	1	1	yes	characteristic
57	C₂³×C₅²	C₂³	1	1	yes	characteristic
58	C₂³×C₅²	C₂³	1	1	yes	characteristic
59	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
60	C₂×C₄×C₅²	—	2	2	no	—
61	C₂×C₄×C₅²	—	4	1	no	—
62	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
63	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
64	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
65	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
66	C₅×C₂²⋊D₅	—	2	2	no	—
67	C₅×C₂²⋊D₅	—	2	2	no	—
68	C₅×C₂²⋊D₅	—	4	1	no	—
69	C₂×C₅×Dic₅	—	2	2	no	—
70	C₂×C₅×Dic₅	—	4	1	no	—
71	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
72	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
73	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
74	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
75	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
76	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
77	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
78	C₄×C₅×D₅	—	4	1	no	—
79	C₅×Dic₁₀	—	4	1	no	—
80	C₅×#126	—	5	1	no	—
81	C₅×#126	—	5	1	no	—
82	C₅×#124	—	5	1	no	—
83	C₅×#123	—	5	1	no	—
84	C₅×#123	—	5	1	no	—
85	C₅×#123	—	5	1	no	—
86	C₅×C₂²⋊Q₈	—	5	1	no	—
87	C₅×C₂²⋊Q₈	—	5	1	no	—
88	C₅×#120	—	5	1	no	—
89	C₄×C₅×D₄	—	5	1	no	—
90	C₄×C₅×D₄	—	5	1	no	—
91	C₅×#117	—	5	1	no	—
92	C₅×#117	—	5	1	no	—
93	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
94	C₂×C₅×C₂²⋊C₄	—	5	1	no	—
95	C₂×C₅×C₂²⋊C₄	D₅	1	1	yes	characteristic, complemented
96	C₂×C₅×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
97	#4099	C₂×C₅	1	1	yes	characteristic
98	C₂×C₂²⋊Dic₅	C₂×C₅	1	1	yes	characteristic, complemented
99	#4075	C₂×C₅	1	1	yes	characteristic, complemented
100	C₄×C₂²⋊D₅	C₂×C₅	1	1	yes	characteristic, complemented
101	#4072	C₂×C₅	1	1	yes	characteristic, complemented
102	#4070	C₂×C₅	1	1	yes	characteristic, complemented
103	#4032	C₂×C₅	1	1	yes	characteristic, complemented
104	#4031	C₂×C₅	1	1	yes	characteristic, complemented
105	#4029	C₂×C₅	1	1	yes	characteristic, complemented
106	#4027	C₂×C₅	1	1	yes	characteristic, complemented
107	#4025	C₂×C₅	1	1	yes	characteristic, complemented
108	#4025	C₂×C₅	1	1	yes	characteristic, complemented
109	#4024	C₂×C₅	1	1	yes	characteristic, complemented
110	#4023	C₂×C₅	1	1	yes	characteristic, complemented
111	C₂²×C₅²	—	2	1	no	—
112	C₂²×C₅²	—	2	1	no	—
113	C₂²×C₅²	—	2	1	no	—
114	C₂²×C₅²	—	2	1	no	—
115	C₂²×C₅²	—	2	1	no	—
116	C₂²×C₅²	—	2	1	no	—
117	C₂²×C₅²	—	2	1	no	—
118	C₂²×C₅²	—	4	4	no	—
119	C₂²×C₅²	#41	1	2	yes	—
120	C₂²×C₅²	#41	1	2	yes	—
121	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
122	C₂×C₅×D₅	—	2	1	no	—
123	C₂×C₅×D₅	—	2	1	no	—
124	C₂×C₅×D₅	—	2	1	no	—
125	C₄×C₅²	—	2	1	no	—
126	C₄×C₅²	—	2	1	no	—
127	C₄×C₅²	—	2	1	no	—
128	C₄×C₅²	—	2	1	no	—
129	C₅×Dic₅	—	2	1	no	—
130	C₅×Dic₅	—	2	1	no	—
131	C₅×Dic₅	—	2	1	no	—
132	C₅×Dic₅	—	2	1	no	—
133	C₅×Dic₅	—	2	1	no	—
134	C₅×Dic₅	—	2	1	no	—
135	C₅×Dic₅	—	2	1	no	—
136	C₂⁴×C₅	—	2	2	no	—
137	C₂⁴×C₅	C₂×D₅	1	1	yes	characteristic
138	C₂⁴×C₅	C₂²×C₅	1	1	yes	characteristic
139	C₂×C₅×Q₈	—	5	1	no	—
140	C₂×C₅×D₄	—	5	1	no	—
141	C₂×C₅×D₄	—	5	1	no	—
142	C₂×C₅×D₄	—	5	1	no	—
143	C₂²×C₄×C₅	—	2	2	no	—
144	C₂²×C₄×C₅	—	2	2	no	—
145	C₂²×C₄×C₅	—	5	1	no	—
146	C₂²×C₄×C₅	—	5	1	no	—
147	C₂²×C₄×C₅	—	5	1	no	—
148	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
149	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
150	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
151	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
152	C₂×C₂²⋊D₅	C₂²×C₅	1	1	yes	characteristic
153	C₂×C₂²⋊D₅	C₂²×C₅	1	1	yes	characteristic
154	C₂×C₂²⋊D₅	C₂²×C₅	1	1	yes	characteristic
155	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
156	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
157	C₂×C₄×D₅	C₂²×C₅	1	1	yes	characteristic, complemented
158	C₂×Dic₁₀	C₂²×C₅	1	1	yes	characteristic, complemented
159	C₅×C₄⋊C₄	—	5	1	no	—
160	C₅×C₄⋊C₄	—	5	1	no	—
161	C₅×C₄⋊C₄	—	5	1	no	—
162	C₅×C₄⋊C₄	—	5	1	no	—
163	C₅×C₄⋊C₄	—	5	1	no	—
164	C₅×C₄⋊C₄	—	5	1	no	—
165	C₅×C₄⋊C₄	—	5	1	no	—
166	C₅×C₄⋊C₄	—	5	1	no	—
167	C₅×C₂²⋊C₄	—	2	2	no	—
168	C₅×C₂²⋊C₄	—	2	2	no	—
169	C₅×C₂²⋊C₄	—	2	2	no	—
170	C₅×C₂²⋊C₄	—	2	2	no	—
171	C₅×C₂²⋊C₄	—	5	1	no	—
172	C₅×C₂²⋊C₄	—	5	1	no	—
173	C₅×C₂²⋊C₄	—	5	1	no	—
174	C₅×C₂²⋊C₄	—	5	1	no	—
175	C₅×C₂²⋊C₄	—	5	1	no	—
176	C₅×C₂²⋊C₄	—	5	1	no	—
177	C₅×C₂²⋊C₄	—	5	1	no	—
178	C₅×C₂²⋊C₄	—	5	1	no	—
179	C₅×C₂²⋊C₄	—	5	1	no	—
180	C₅×C₂²⋊C₄	—	5	1	no	—
181	C₅×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
182	C₅×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
183	C₅×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
184	C₅×C₂²⋊C₄	C₂×D₅	1	1	yes	characteristic
185	C₅×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
186	C₅×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
187	C₅×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
188	C₅×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
189	C₄²×C₅	—	5	1	no	—
190	C₄²×C₅	—	5	1	no	—
191	C₄²×C₅	—	5	1	no	—
192	C₂²⋊Dic₅	C₂²×C₅	1	1	yes	characteristic
193	C₂²⋊Dic₅	C₂²×C₅	1	1	yes	characteristic
194	C₂²⋊Dic₅	C₂²×C₅	1	1	yes	characteristic
195	C₂²⋊Dic₅	C₂²×C₅	1	1	yes	characteristic
196	C₂²⋊Dic₅	C₂²×C₅	1	1	yes	characteristic
197	C₂²⋊Dic₅	C₂²×C₅	1	1	yes	characteristic
198	C₂²⋊Dic₅	C₂²×C₅	1	1	yes	characteristic
199	#685	C₂²×C₅	1	1	yes	characteristic, complemented
200	#685	C₂²×C₅	1	1	yes	characteristic, complemented
201	#685	C₂²×C₅	1	1	yes	characteristic, complemented
202	C₄⋊Dic₅	C₂²×C₅	1	1	yes	characteristic, complemented
203	C₄⋊Dic₅	C₂²×C₅	1	1	yes	characteristic, complemented
204	C₄⋊Dic₅	C₂²×C₅	1	1	yes	characteristic, complemented
205	#683	C₂²×C₅	1	1	yes	characteristic, complemented
206	#683	C₂²×C₅	1	1	yes	characteristic, complemented
207	#683	C₂²×C₅	1	1	yes	characteristic, complemented
208	#683	C₂²×C₅	1	1	yes	characteristic, complemented
209	#683	C₂²×C₅	1	1	yes	characteristic, complemented
210	C₄×Dic₅	C₂²×C₅	1	1	yes	characteristic, complemented
211	C₄×Dic₅	C₂²×C₅	1	1	yes	characteristic, complemented
212	C₄×Dic₅	C₂²×C₅	1	1	yes	characteristic, complemented
213	#551	—	5	1	no	Sylow
214	C₂×C₅²	—	2	2	no	—
215	C₂×C₅²	—	2	2	no	—
216	C₂×C₅²	—	4	1	no	—
217	C₂×C₅²	C₂×#41	1	1	yes	characteristic
218	C₂×C₅²	C₂×#41	1	1	yes	characteristic
219	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
220	C₅×D₅	—	4	1	no	—
221	C₂³×C₅	—	2	8	no	—
222	C₂³×C₅	—	2	2	no	—
223	C₂³×C₅	—	2	2	no	—
224	C₂³×C₅	—	2	2	no	—
225	C₂³×C₅	—	2	2	no	—
226	C₂³×C₅	—	2	2	no	—
227	C₂³×C₅	—	2	2	no	—
228	C₂³×C₅	—	2	2	no	—
229	C₂³×C₅	—	4	4	no	—
230	C₂³×C₅	—	4	1	no	—
231	C₂³×C₅	—	4	1	no	—
232	C₂³×C₅	—	4	4	no	—
233	C₂³×C₅	—	5	1	no	—
234	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
235	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
236	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
237	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
238	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
239	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
240	C₂²×D₅	C₂³×C₅	1	1	yes	characteristic
241	C₅×Q₈	—	20	1	no	—
242	C₅×D₄	—	10	2	no	—
243	C₅×D₄	—	10	2	no	—
244	C₅×D₄	—	20	1	no	—
245	C₂×C₄×C₅	—	2	2	no	—
246	C₂×C₄×C₅	—	2	2	no	—
247	C₂×C₄×C₅	—	2	2	no	—
248	C₂×C₄×C₅	—	2	2	no	—
249	C₂×C₄×C₅	—	2	2	no	—
250	C₂×C₄×C₅	—	2	2	no	—
251	C₂×C₄×C₅	—	4	4	no	—
252	C₂×C₄×C₅	—	4	1	no	—
253	C₂×C₄×C₅	—	4	1	no	—
254	C₂×C₄×C₅	—	4	4	no	—
255	C₂×C₄×C₅	—	5	1	no	—
256	C₂×C₄×C₅	—	5	1	no	—
257	C₂×C₄×C₅	—	5	1	no	—
258	C₂×C₄×C₅	—	5	1	no	—
259	C₂×C₄×C₅	—	5	1	no	—
260	C₂×C₄×C₅	—	5	1	no	—
261	C₂×C₄×C₅	—	5	1	no	—
262	C₂×C₄×C₅	—	10	2	no	—
263	C₂×C₄×C₅	—	20	1	no	—
264	C₂×C₄×C₅	—	20	1	no	—
265	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
266	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
267	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
268	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
269	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
270	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
271	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
272	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
273	C₂²⋊D₅	—	2	2	no	—
274	C₂²⋊D₅	—	2	2	no	—
275	C₂²⋊D₅	—	4	1	no	—
276	C₂×Dic₅	—	2	2	no	—
277	C₂×Dic₅	—	4	1	no	—
278	C₂×Dic₅	C₂³×C₅	1	1	yes	characteristic
279	C₂×Dic₅	C₂³×C₅	1	1	yes	characteristic
280	C₂×Dic₅	C₂³×C₅	1	1	yes	characteristic
281	C₂×Dic₅	C₂³×C₅	1	1	yes	characteristic
282	C₂×Dic₅	C₂³×C₅	1	1	yes	characteristic
283	C₂×Dic₅	C₂³×C₅	1	1	yes	characteristic
284	C₂×Dic₅	C₂³×C₅	1	1	yes	characteristic
285	C₄×D₅	—	4	1	no	—
286	Dic₁₀	—	4	1	no	—
287	#126	—	5	1	no	—
288	#126	—	5	1	no	—
289	#124	—	5	1	no	—
290	#123	—	5	1	no	—
291	#123	—	5	1	no	—
292	#123	—	5	1	no	—
293	C₂²⋊Q₈	—	5	1	no	—
294	C₂²⋊Q₈	—	5	1	no	—
295	#120	—	5	1	no	—
296	C₄×D₄	—	5	1	no	—
297	C₄×D₄	—	5	1	no	—
298	#117	—	5	1	no	—
299	#117	—	5	1	no	—
300	C₂×C₂²⋊C₄	—	5	1	no	—
301	C₂×C₂²⋊C₄	C₅×D₅	1	1	yes	characteristic, complemented
302	C₅²	#551	1	1	yes	characteristic, complemented, Sylow
303	C₂²×C₅	—	2	4	no	—
304	C₂²×C₅	—	2	4	no	—
305	C₂²×C₅	—	2	1	no	—
306	C₂²×C₅	—	2	1	no	—
307	C₂²×C₅	—	2	1	no	—
308	C₂²×C₅	—	2	1	no	—
309	C₂²×C₅	—	2	1	no	—
310	C₂²×C₅	—	2	1	no	—
311	C₂²×C₅	—	2	1	no	—
312	C₂²×C₅	—	2	1	no	—
313	C₂²×C₅	—	2	4	no	—
314	C₂²×C₅	—	2	1	no	—
315	C₂²×C₅	—	2	1	no	—
316	C₂²×C₅	—	2	1	no	—
317	C₂²×C₅	—	2	1	no	—
318	C₂²×C₅	—	2	1	no	—
319	C₂²×C₅	—	2	1	no	—
320	C₂²×C₅	—	2	4	no	—
321	C₂²×C₅	—	2	4	no	—
322	C₂²×C₅	—	2	2	no	—
323	C₂²×C₅	—	4	2	no	—
324	C₂²×C₅	—	4	2	no	—
325	C₂²×C₅	—	4	2	no	—
326	C₂²×C₅	—	4	2	no	—
327	C₂²×C₅	—	4	4	no	—
328	C₂²×C₅	—	4	4	no	—
329	C₂²×C₅	—	4	16	no	—
330	C₂²×C₅	—	10	1	no	—
331	C₂²×C₅	—	10	1	no	—
332	C₂²×C₅	—	10	1	no	—
333	C₂²×C₅	#709	1	2	yes	—
334	C₂²×C₅	#711	1	2	yes	—
335	C₂²×C₅	C₅×#41	1	2	yes	—
336	C₂²×C₅	C₅×#41	1	2	yes	—
337	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
338	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
339	C₂×D₅	—	2	1	no	—
340	C₂×D₅	—	2	1	no	—
341	C₂×D₅	—	2	1	no	—
342	C₄×C₅	—	2	1	no	—
343	C₄×C₅	—	2	1	no	—
344	C₄×C₅	—	2	1	no	—
345	C₄×C₅	—	2	1	no	—
346	C₄×C₅	—	2	1	no	—
347	C₄×C₅	—	2	1	no	—
348	C₄×C₅	—	2	1	no	—
349	C₄×C₅	—	2	1	no	—
350	C₄×C₅	—	4	2	no	—
351	C₄×C₅	—	4	2	no	—
352	C₄×C₅	—	4	2	no	—
353	C₄×C₅	—	4	2	no	—
354	C₄×C₅	—	10	1	no	—
355	C₄×C₅	—	10	1	no	—
356	C₄×C₅	—	10	1	no	—
357	C₄×C₅	—	10	1	no	—
358	C₄×C₅	—	10	1	no	—
359	C₄×C₅	—	10	1	no	—
360	C₄×C₅	—	10	1	no	—
361	Dic₅	—	2	1	no	—
362	Dic₅	—	2	1	no	—
363	Dic₅	—	2	1	no	—
364	Dic₅	—	2	1	no	—
365	Dic₅	—	2	1	no	—
366	Dic₅	—	2	1	no	—
367	Dic₅	—	2	1	no	—
368	C₂⁴	C₂×C₅×D₅	1	1	yes	characteristic
369	C₂×Q₈	—	5	1	no	—
370	C₂×D₄	—	5	1	no	—
371	C₂×D₄	—	5	1	no	—
372	C₂×D₄	—	5	1	no	—
373	C₂²×C₄	—	5	1	no	—
374	C₂²×C₄	—	5	1	no	—
375	C₂²×C₄	—	5	1	no	—
376	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
377	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
378	C₄⋊C₄	—	5	1	no	—
379	C₄⋊C₄	—	5	1	no	—
380	C₄⋊C₄	—	5	1	no	—
381	C₄⋊C₄	—	5	1	no	—
382	C₄⋊C₄	—	5	1	no	—
383	C₄⋊C₄	—	5	1	no	—
384	C₄⋊C₄	—	5	1	no	—
385	C₄⋊C₄	—	5	1	no	—
386	C₂²⋊C₄	—	5	1	no	—
387	C₂²⋊C₄	—	5	1	no	—
388	C₂²⋊C₄	—	5	1	no	—
389	C₂²⋊C₄	—	5	1	no	—
390	C₂²⋊C₄	—	5	1	no	—
391	C₂²⋊C₄	—	5	1	no	—
392	C₂²⋊C₄	—	5	1	no	—
393	C₂²⋊C₄	—	5	1	no	—
394	C₂²⋊C₄	—	5	1	no	—
395	C₂²⋊C₄	—	5	1	no	—
396	C₂²⋊C₄	C₂×C₅×D₅	1	1	yes	characteristic
397	C₂²⋊C₄	C₂×C₅×D₅	1	1	yes	characteristic
398	C₂²⋊C₄	C₂×C₅×D₅	1	1	yes	characteristic
399	C₂²⋊C₄	C₂×C₅×D₅	1	1	yes	characteristic
400	C₄²	—	5	1	no	—
401	C₄²	—	5	1	no	—
402	C₄²	—	5	1	no	—
403	C₂×C₅	—	2	8	no	—
404	C₂×C₅	—	2	2	no	—
405	C₂×C₅	—	2	2	no	—
406	C₂×C₅	—	2	2	no	—
407	C₂×C₅	—	2	2	no	—
408	C₂×C₅	—	2	2	no	—
409	C₂×C₅	—	2	2	no	—
410	C₂×C₅	—	2	2	no	—
411	C₂×C₅	—	4	4	no	—
412	C₂×C₅	—	4	4	no	—
413	C₂×C₅	—	4	1	no	—
414	C₂×C₅	—	4	1	no	—
415	C₂×C₅	—	20	1	no	—
416	C₂×C₅	C₂×#709	1	1	yes	characteristic
417	C₂×C₅	C₂×#711	1	1	yes	characteristic
418	C₂×C₅	#4144	1	1	yes	characteristic
419	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
420	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
421	C₂×C₅	C₅×H(5,𝔽₂)	1	1	yes	characteristic
422	D₅	—	4	1	no	—
423	C₂³	—	2	2	no	—
424	C₂³	—	2	2	no	—
425	C₂³	—	4	1	no	—
426	C₂³	—	5	1	no	—
427	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
428	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
429	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
430	Q₈	—	20	1	no	—
431	D₄	—	10	2	no	—
432	D₄	—	10	2	no	—
433	D₄	—	20	1	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	5	1	no	—
437	C₂×C₄	—	5	1	no	—
438	C₂×C₄	—	5	1	no	—
439	C₂×C₄	—	5	1	no	—
440	C₂×C₄	—	5	1	no	—
441	C₂×C₄	—	5	1	no	—
442	C₂×C₄	—	5	1	no	—
443	C₂×C₄	—	10	2	no	—
444	C₂×C₄	—	20	1	no	—
445	C₂×C₄	—	20	1	no	—
446	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
447	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
448	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
449	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
450	C₅	—	2	2	no	—
451	C₅	#65829	1	1	yes	characteristic, direct product factor
452	C₅	C₅×#551	1	1	yes	characteristic, complemented
453	C₂²	—	2	1	no	—
454	C₂²	—	2	1	no	—
455	C₂²	—	2	1	no	—
456	C₂²	—	2	1	no	—
457	C₂²	—	2	1	no	—
458	C₂²	—	2	1	no	—
459	C₂²	—	2	1	no	—
460	C₂²	—	4	4	no	—
461	C₂²	—	10	1	no	—
462	C₂²	—	10	1	no	—
463	C₂²	—	10	1	no	—
464	C₂²	C₅×#709	1	2	yes	—
465	C₂²	C₅×#711	1	2	yes	—
466	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
467	C₄	—	2	1	no	—
468	C₄	—	2	1	no	—
469	C₄	—	2	1	no	—
470	C₄	—	2	1	no	—
471	C₄	—	10	1	no	—
472	C₄	—	10	1	no	—
473	C₄	—	10	1	no	—
474	C₄	—	10	1	no	—
475	C₄	—	10	1	no	—
476	C₄	—	10	1	no	—
477	C₄	—	10	1	no	—
478	C₂	—	2	2	no	—
479	C₂	—	2	2	no	—
480	C₂	—	4	1	no	—
481	C₂	—	20	1	no	—
482	C₂	C₂×C₅×#709	1	1	yes	characteristic
483	C₂	C₂×C₅×#711	1	1	yes	characteristic
484	C₂	C₅×#4144	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₅
Fitting	C₂×C₅²×C₂²⋊C₄	C₂
socle	C₂²×C₅²	C₂⁴
prime radical	C₅×#4099	C₂
hypercenter	C₂×C₅×C₂²⋊C₄	D₅
hypocenter	C₅	C₅×#551
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₅×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₅²×C₂²⋊C₄	C₂	semidirect
3. C₂×C₅×C₂²⋊Dic₅	C₂	semidirect
4. C₅×#4075	C₂	semidirect
5. C₄×C₅×C₂²⋊D₅	C₂	semidirect
6. C₅×#4072	C₂	semidirect
7. C₅×#4070	C₂	semidirect
8. C₅×#4032	C₂	semidirect
9. C₅×#4031	C₂	semidirect
10. C₅×#4029	C₂	semidirect
11. C₅×#4027	C₂	semidirect
12. C₅×#4025	C₂	semidirect
13. C₅×#4025	C₂	semidirect
14. C₅×#4024	C₂	semidirect
15. C₅×#4023	C₂	semidirect
24. C₂×C₄×C₅×D₅	C₂²	semidirect
25. C₂×C₅×Dic₁₀	C₂²	semidirect
37. C₅×#685	C₂²	semidirect
38. C₅×#685	C₂²	semidirect
39. C₅×#685	C₂²	semidirect
40. C₅×C₄⋊Dic₅	C₂²	semidirect
41. C₅×C₄⋊Dic₅	C₂²	semidirect
42. C₅×C₄⋊Dic₅	C₂²	semidirect
43. C₅×#683	C₂²	semidirect
44. C₅×#683	C₂²	semidirect
45. C₅×#683	C₂²	semidirect
46. C₅×#683	C₂²	semidirect
47. C₅×#683	C₂²	semidirect
48. C₄×C₅×Dic₅	C₂²	semidirect
49. C₄×C₅×Dic₅	C₂²	semidirect
50. C₄×C₅×Dic₅	C₂²	semidirect
52. #65829	C₅	direct (N × H)
95. C₂×C₅×C₂²⋊C₄	D₅	semidirect
96. C₂×C₅×C₂²⋊C₄	C₂×C₅	semidirect
98. C₂×C₂²⋊Dic₅	C₂×C₅	semidirect
99. #4075	C₂×C₅	semidirect
100. C₄×C₂²⋊D₅	C₂×C₅	semidirect
101. #4072	C₂×C₅	semidirect
102. #4070	C₂×C₅	semidirect
103. #4032	C₂×C₅	semidirect
104. #4031	C₂×C₅	semidirect
105. #4029	C₂×C₅	semidirect
106. #4027	C₂×C₅	semidirect
107. #4025	C₂×C₅	semidirect
108. #4025	C₂×C₅	semidirect
109. #4024	C₂×C₅	semidirect
110. #4023	C₂×C₅	semidirect
157. C₂×C₄×D₅	C₂²×C₅	semidirect
158. C₂×Dic₁₀	C₂²×C₅	semidirect
199. #685	C₂²×C₅	semidirect
200. #685	C₂²×C₅	semidirect
201. #685	C₂²×C₅	semidirect
202. C₄⋊Dic₅	C₂²×C₅	semidirect
203. C₄⋊Dic₅	C₂²×C₅	semidirect
204. C₄⋊Dic₅	C₂²×C₅	semidirect
205. #683	C₂²×C₅	semidirect
206. #683	C₂²×C₅	semidirect
207. #683	C₂²×C₅	semidirect
208. #683	C₂²×C₅	semidirect
209. #683	C₂²×C₅	semidirect
210. C₄×Dic₅	C₂²×C₅	semidirect
211. C₄×Dic₅	C₂²×C₅	semidirect
212. C₄×Dic₅	C₂²×C₅	semidirect
301. C₂×C₂²⋊C₄	C₅×D₅	semidirect
302. C₅²	#551	semidirect
451. C₅	#65829	direct (N × H)
452. C₅	C₅×#551	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅²×C₂²⋊C₄	2
C₅×#4099	2
C₂×C₅×C₂²⋊Dic₅	2
C₅×#4075	2
C₄×C₅×C₂²⋊D₅	2
C₅×#4072	2
C₅×#4070	2
C₅×#4032	2
C₅×#4031	2
C₅×#4029	2
C₅×#4027	2
C₅×#4025	2
C₅×#4024	2
C₅×#4023	2
C₅×#551	5
#65829	5

Subgroups — C5×#65829 #13106649

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#65829 #13106649