Subgroups — C₅×#21856 #11944576

Order 1280.

1200 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3444	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#423	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1994	C₂	1	2	yes	maximal, complemented
5	C₅×#1962	C₂	1	2	yes	maximal, complemented
6	C₅×#1871	C₂	1	4	yes	maximal, complemented
7	C₅×#1806	C₂	1	4	yes	maximal, complemented
8	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
9	C₅×#533	C₄	1	1	yes	characteristic, complemented
10	C₅×#533	C₄	1	1	yes	characteristic, complemented
11	C₅×#533	C₂²	1	1	yes	characteristic
12	C₅×#532	C₄	1	1	yes	characteristic, complemented
13	C₅×#532	C₄	1	1	yes	characteristic, complemented
14	C₅×#532	C₂²	1	1	yes	characteristic
15	C₅×#529	—	2	1	no	—
16	C₅×#529	—	2	1	no	—
17	C₅×#529	—	2	1	no	—
18	C₅×#529	—	2	1	no	—
19	C₅×#529	C₄	1	2	yes	complemented
20	C₅×#529	C₄	1	2	yes	complemented
21	C₅×#525	—	2	1	no	—
22	C₅×#525	—	2	1	no	—
23	C₅×#525	C₂²	1	2	yes	—
24	C₄×C₅×#41	—	2	1	no	—
25	C₄×C₅×#41	—	2	1	no	—
26	C₄×C₅×#41	C₂²	1	2	yes	—
27	C₂×C₅×#117	—	2	1	no	—
28	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
29	C₅×#423	—	2	1	no	—
30	C₅×#423	—	2	1	no	—
31	C₅×#423	C₂²	1	2	yes	—
32	C₅×#423	C₂²	1	2	yes	—
33	C₂×C₅×C₄⋊C₈	—	2	1	no	—
34	C₂×C₅×C₄⋊C₈	—	2	1	no	—
35	C₂×C₅×#104	C₂²	1	8	yes	—
36	C₄×C₅×C₄⋊C₄	—	2	1	no	—
37	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
38	C₄×C₅×C₂²⋊C₄	—	2	1	no	—
39	C₄×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
40	C₅×#376	—	2	1	no	—
41	C₄³×C₅	C₂²	1	2	yes	—
42	C₅×#339	C₂²	1	8	yes	—
43	#21856	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂×C₅×#41	—	2	1	no	—
45	C₂×C₅×#41	—	2	1	no	—
46	C₂×C₅×#41	C₂³	1	2	yes	—
47	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
50	C₂×C₅×M₁₆	—	2	2	no	—
51	C₂×C₅×M₁₆	—	4	4	no	—
52	C₂×C₅×M₁₆	C₂³	1	4	yes	—
53	C₂²×C₅×C₈	—	2	2	no	—
54	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
55	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
56	C₅×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
57	C₅×#127	C₂×C₄	1	1	yes	characteristic
58	C₅×#126	—	2	2	no	—
59	C₅×#126	—	2	2	no	—
60	C₅×#125	—	2	1	no	—
61	C₅×#125	—	2	1	no	—
62	C₅×#125	C₂×C₄	1	2	yes	—
63	C₅×#124	—	2	1	no	—
64	C₅×#124	—	2	1	no	—
65	C₅×#124	C₂×C₄	1	2	yes	—
66	C₅×#123	—	2	2	no	—
67	C₅×#123	—	2	2	no	—
68	C₅×#123	—	2	2	no	—
69	C₅×C₂²⋊Q₈	—	2	1	no	—
70	C₅×C₂²⋊Q₈	—	2	1	no	—
71	C₅×C₂²⋊Q₈	—	2	1	no	—
72	C₅×C₂²⋊Q₈	—	2	1	no	—
73	C₅×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
74	C₅×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
75	C₅×#121	—	2	1	no	—
76	C₅×#121	—	2	1	no	—
77	C₅×#121	—	2	1	no	—
78	C₅×#121	—	2	1	no	—
79	C₅×#121	C₂×C₄	1	2	yes	—
80	C₅×#121	C₂×C₄	1	2	yes	—
81	C₅×#120	—	2	2	no	—
82	C₄×C₅×Q₈	—	2	1	no	—
83	C₄×C₅×Q₈	—	2	1	no	—
84	C₄×C₅×Q₈	—	2	1	no	—
85	C₄×C₅×Q₈	—	2	1	no	—
86	C₄×C₅×Q₈	C₂×C₄	1	2	yes	—
87	C₄×C₅×Q₈	C₂×C₄	1	2	yes	—
88	C₄×C₅×D₄	—	2	2	no	—
89	C₄×C₅×D₄	—	2	2	no	—
90	C₄×C₅×D₄	—	2	2	no	—
91	C₄×C₅×D₄	—	2	2	no	—
92	C₄×C₅×D₄	—	2	2	no	—
93	C₄×C₅×D₄	—	2	2	no	—
94	C₄×C₅×D₄	—	2	1	no	—
95	C₄×C₅×D₄	—	2	1	no	—
96	C₄×C₅×D₄	—	2	1	no	—
97	C₄×C₅×D₄	—	2	1	no	—
98	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
99	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
100	C₅×#117	—	2	8	no	—
101	C₅×#117	—	2	1	no	—
102	C₅×#117	—	2	1	no	—
103	C₅×#117	—	2	2	no	—
104	C₅×#117	—	2	2	no	—
105	C₅×#117	—	2	2	no	—
106	C₅×#117	C₂³	1	2	yes	—
107	C₂×C₅×C₄⋊C₄	—	2	2	no	—
108	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
109	C₂×C₄²×C₅	—	2	2	no	—
110	C₂×C₄²×C₅	—	2	2	no	—
111	C₂×C₄²×C₅	—	2	2	no	—
112	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
113	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
114	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
115	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
116	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
117	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
118	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
119	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
120	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
121	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
122	C₂×C₄²×C₅	C₂³	1	4	yes	—
123	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
124	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
125	C₅×C₄⋊C₈	—	2	2	no	—
126	C₅×C₄⋊C₈	—	2	2	no	—
127	C₅×C₄⋊C₈	—	2	2	no	—
128	C₅×C₄⋊C₈	—	2	2	no	—
129	C₅×#104	—	4	8	no	—
130	C₅×#95	—	2	2	no	—
131	C₅×#95	—	2	2	no	—
132	#3444	C₂×C₅	1	1	yes	characteristic
133	C₂×#423	C₂×C₅	1	1	yes	characteristic, complemented
134	C₄×#117	C₂×C₅	1	1	yes	characteristic, complemented
135	#1994	C₂×C₅	1	2	yes	complemented
136	#1962	C₂×C₅	1	2	yes	complemented
137	#1871	C₂×C₅	1	4	yes	complemented
138	#1806	C₂×C₅	1	4	yes	complemented
139	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
140	C₅×#41	—	2	4	no	—
141	C₅×#41	—	2	4	no	—
142	C₅×#41	—	4	4	no	—
143	C₅×#41	—	4	4	no	—
144	C₂×C₅×Q₈	—	2	1	no	—
145	C₂×C₅×Q₈	—	2	1	no	—
146	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
147	C₂×C₅×D₄	—	2	2	no	—
148	C₂×C₅×D₄	—	2	2	no	—
149	C₂×C₅×D₄	—	2	2	no	—
150	C₂×C₅×D₄	—	2	1	no	—
151	C₂×C₅×D₄	—	2	1	no	—
152	C₂×C₅×D₄	C₂²×C₄	1	2	yes	—
153	C₂²×C₄×C₅	—	2	4	no	—
154	C₂²×C₄×C₅	—	2	2	no	—
155	C₂²×C₄×C₅	—	2	2	no	—
156	C₂²×C₄×C₅	—	2	4	no	—
157	C₂²×C₄×C₅	—	2	2	no	—
158	C₂²×C₄×C₅	—	2	2	no	—
159	C₂²×C₄×C₅	—	2	2	no	—
160	C₂²×C₄×C₅	—	4	2	no	—
161	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
162	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
163	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
164	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
165	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
166	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
167	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
168	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
169	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
170	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
171	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
179	C₅×M₁₆	—	2	8	no	—
180	C₅×M₁₆	—	4	4	no	—
181	C₂×C₅×C₈	—	2	4	no	—
182	C₂×C₅×C₈	—	4	4	no	—
183	C₅×C₄⋊C₄	—	2	8	no	—
184	C₅×C₄⋊C₄	—	2	1	no	—
185	C₅×C₄⋊C₄	—	2	1	no	—
186	C₅×C₄⋊C₄	—	2	1	no	—
187	C₅×C₄⋊C₄	—	2	1	no	—
188	C₅×C₄⋊C₄	—	2	2	no	—
189	C₅×C₄⋊C₄	—	2	2	no	—
190	C₅×C₄⋊C₄	—	2	2	no	—
191	C₅×C₄⋊C₄	—	2	2	no	—
192	C₅×C₄⋊C₄	—	2	2	no	—
193	C₅×C₄⋊C₄	—	2	2	no	—
194	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
195	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
196	C₅×C₂²⋊C₄	—	2	8	no	—
197	C₅×C₂²⋊C₄	—	2	2	no	—
198	C₅×C₂²⋊C₄	—	2	2	no	—
199	C₅×C₂²⋊C₄	—	2	2	no	—
200	C₅×C₂²⋊C₄	—	2	2	no	—
201	C₅×C₂²⋊C₄	—	2	2	no	—
202	C₅×C₂²⋊C₄	—	2	2	no	—
203	C₅×C₂²⋊C₄	—	2	2	no	—
204	C₅×C₂²⋊C₄	—	2	2	no	—
205	C₅×C₂²⋊C₄	—	2	2	no	—
206	C₄²×C₅	—	2	8	no	—
207	C₄²×C₅	—	2	2	no	—
208	C₄²×C₅	—	2	2	no	—
209	C₄²×C₅	—	2	2	no	—
210	C₄²×C₅	—	2	4	no	—
211	C₄²×C₅	—	2	4	no	—
212	C₄²×C₅	—	4	4	no	—
213	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
214	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
215	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
216	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
217	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
218	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
219	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
220	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
221	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
222	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
223	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
224	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
225	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
226	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
227	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
228	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
229	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
230	#533	C₄×C₅	1	1	yes	characteristic, complemented
231	#533	C₄×C₅	1	1	yes	characteristic, complemented
232	#533	C₂²×C₅	1	1	yes	characteristic
233	#532	C₄×C₅	1	1	yes	characteristic, complemented
234	#532	C₄×C₅	1	1	yes	characteristic, complemented
235	#532	C₂²×C₅	1	1	yes	characteristic
236	#529	—	2	1	no	—
237	#529	—	2	1	no	—
238	#529	—	2	1	no	—
239	#529	—	2	1	no	—
240	#529	C₄×C₅	1	2	yes	complemented
241	#529	C₄×C₅	1	2	yes	complemented
242	#525	—	2	1	no	—
243	#525	—	2	1	no	—
244	#525	C₂²×C₅	1	2	yes	—
245	C₄×#41	—	2	1	no	—
246	C₄×#41	—	2	1	no	—
247	C₄×#41	C₂²×C₅	1	2	yes	—
248	C₂×#117	—	2	1	no	—
249	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
250	#423	—	2	1	no	—
251	#423	—	2	1	no	—
252	#423	C₂²×C₅	1	2	yes	—
253	#423	C₂²×C₅	1	2	yes	—
254	C₂×C₄⋊C₈	—	2	1	no	—
255	C₂×C₄⋊C₈	—	2	1	no	—
256	C₂×#104	C₂²×C₅	1	8	yes	—
257	C₄×C₄⋊C₄	—	2	1	no	—
258	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
259	C₄×C₂²⋊C₄	—	2	1	no	—
260	C₄×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
261	#376	—	2	1	no	—
262	C₄³	C₂²×C₅	1	2	yes	—
263	#339	C₂²×C₅	1	8	yes	—
264	C₂³×C₅	—	2	2	no	—
265	C₂³×C₅	—	2	2	no	—
266	C₂³×C₅	—	2	2	no	—
267	C₂³×C₅	—	4	1	no	—
268	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
269	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
270	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
271	C₅×Q₈	—	2	4	no	—
272	C₅×Q₈	—	4	2	no	—
273	C₅×Q₈	—	4	2	no	—
274	C₅×D₄	—	2	4	no	—
275	C₅×D₄	—	4	4	no	—
276	C₅×D₄	—	4	4	no	—
277	C₅×D₄	—	4	4	no	—
278	C₅×D₄	—	4	2	no	—
279	C₅×D₄	—	4	2	no	—
280	C₂×C₄×C₅	—	2	2	no	—
281	C₂×C₄×C₅	—	2	2	no	—
282	C₂×C₄×C₅	—	2	2	no	—
283	C₂×C₄×C₅	—	2	2	no	—
284	C₂×C₄×C₅	—	2	1	no	—
285	C₂×C₄×C₅	—	2	1	no	—
286	C₂×C₄×C₅	—	2	1	no	—
287	C₂×C₄×C₅	—	2	1	no	—
288	C₂×C₄×C₅	—	2	2	no	—
289	C₂×C₄×C₅	—	2	2	no	—
290	C₂×C₄×C₅	—	2	2	no	—
291	C₂×C₄×C₅	—	2	2	no	—
292	C₂×C₄×C₅	—	2	2	no	—
293	C₂×C₄×C₅	—	2	2	no	—
294	C₂×C₄×C₅	—	2	2	no	—
295	C₂×C₄×C₅	—	2	8	no	—
296	C₂×C₄×C₅	—	2	2	no	—
297	C₂×C₄×C₅	—	2	2	no	—
298	C₂×C₄×C₅	—	2	2	no	—
299	C₂×C₄×C₅	—	4	1	no	—
300	C₂×C₄×C₅	—	4	1	no	—
301	C₂×C₄×C₅	—	4	1	no	—
302	C₂×C₄×C₅	—	4	4	no	—
303	C₂×C₄×C₅	—	4	4	no	—
304	C₂×C₄×C₅	—	4	4	no	—
305	C₂×C₄×C₅	—	4	2	no	—
306	C₂×C₄×C₅	—	4	2	no	—
307	C₂×C₄×C₅	—	4	4	no	—
308	C₂×C₄×C₅	#104	1	4	yes	complemented
309	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
310	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
311	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
312	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
313	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
314	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
315	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
316	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
317	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
318	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
319	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
320	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
321	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
322	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
323	C₅×C₈	—	4	4	no	—
324	C₂×#41	—	2	1	no	—
325	C₂×#41	—	2	1	no	—
326	C₂×#41	C₂³×C₅	1	2	yes	—
327	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
328	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
329	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
330	C₂×M₁₆	—	2	2	no	—
331	C₂×M₁₆	—	4	4	no	—
332	C₂×M₁₆	C₂³×C₅	1	4	yes	—
333	C₂²×C₈	—	2	2	no	—
334	C₄⋊Q₈	C₂×C₄×C₅	1	1	yes	characteristic
335	C₄⋊Q₈	C₂×C₄×C₅	1	1	yes	characteristic
336	C₄⋊Q₈	C₂×C₄×C₅	1	1	yes	characteristic
337	#127	C₂×C₄×C₅	1	1	yes	characteristic
338	#126	—	2	2	no	—
339	#126	—	2	2	no	—
340	#125	—	2	1	no	—
341	#125	—	2	1	no	—
342	#125	C₂×C₄×C₅	1	2	yes	—
343	#124	—	2	1	no	—
344	#124	—	2	1	no	—
345	#124	C₂×C₄×C₅	1	2	yes	—
346	#123	—	2	2	no	—
347	#123	—	2	2	no	—
348	#123	—	2	2	no	—
349	C₂²⋊Q₈	—	2	1	no	—
350	C₂²⋊Q₈	—	2	1	no	—
351	C₂²⋊Q₈	—	2	1	no	—
352	C₂²⋊Q₈	—	2	1	no	—
353	C₂²⋊Q₈	C₂×C₄×C₅	1	2	yes	—
354	C₂²⋊Q₈	C₂×C₄×C₅	1	2	yes	—
355	#121	—	2	1	no	—
356	#121	—	2	1	no	—
357	#121	—	2	1	no	—
358	#121	—	2	1	no	—
359	#121	C₂×C₄×C₅	1	2	yes	—
360	#121	C₂×C₄×C₅	1	2	yes	—
361	#120	—	2	2	no	—
362	C₄×Q₈	—	2	1	no	—
363	C₄×Q₈	—	2	1	no	—
364	C₄×Q₈	—	2	1	no	—
365	C₄×Q₈	—	2	1	no	—
366	C₄×Q₈	C₂×C₄×C₅	1	2	yes	—
367	C₄×Q₈	C₂×C₄×C₅	1	2	yes	—
368	C₄×D₄	—	2	2	no	—
369	C₄×D₄	—	2	2	no	—
370	C₄×D₄	—	2	2	no	—
371	C₄×D₄	—	2	2	no	—
372	C₄×D₄	—	2	2	no	—
373	C₄×D₄	—	2	2	no	—
374	C₄×D₄	—	2	1	no	—
375	C₄×D₄	—	2	1	no	—
376	C₄×D₄	—	2	1	no	—
377	C₄×D₄	—	2	1	no	—
378	C₄×D₄	C₂×C₄×C₅	1	2	yes	—
379	C₄×D₄	C₂×C₄×C₅	1	2	yes	—
380	#117	—	2	8	no	—
381	#117	—	2	1	no	—
382	#117	—	2	1	no	—
383	#117	—	2	2	no	—
384	#117	—	2	2	no	—
385	#117	—	2	2	no	—
386	#117	C₂³×C₅	1	2	yes	—
387	C₂×C₄⋊C₄	—	2	2	no	—
388	C₂×C₂²⋊C₄	—	2	2	no	—
389	C₂×C₄²	—	2	2	no	—
390	C₂×C₄²	—	2	2	no	—
391	C₂×C₄²	—	2	2	no	—
392	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
393	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
394	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
395	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
396	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
397	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
398	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
399	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
400	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
401	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
402	C₂×C₄²	C₂³×C₅	1	4	yes	—
403	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
404	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
405	C₄⋊C₈	—	2	2	no	—
406	C₄⋊C₈	—	2	2	no	—
407	C₄⋊C₈	—	2	2	no	—
408	C₄⋊C₈	—	2	2	no	—
409	#104	—	4	8	no	—
410	#95	—	2	2	no	—
411	#95	—	2	2	no	—
412	C₂²×C₅	—	2	1	no	—
413	C₂²×C₅	—	2	1	no	—
414	C₂²×C₅	—	2	1	no	—
415	C₂²×C₅	—	2	1	no	—
416	C₂²×C₅	—	2	1	no	—
417	C₂²×C₅	—	2	1	no	—
418	C₂²×C₅	—	2	1	no	—
419	C₂²×C₅	—	4	4	no	—
420	C₂²×C₅	—	4	2	no	—
421	C₂²×C₅	—	4	2	no	—
422	C₂²×C₅	—	4	2	no	—
423	C₂²×C₅	C₂×#104	1	2	yes	—
424	C₂²×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
425	C₂²×C₅	#537	1	2	yes	—
426	C₄×C₅	—	2	1	no	—
427	C₄×C₅	—	2	1	no	—
428	C₄×C₅	—	2	1	no	—
429	C₄×C₅	—	2	1	no	—
430	C₄×C₅	—	2	1	no	—
431	C₄×C₅	—	2	1	no	—
432	C₄×C₅	—	2	1	no	—
433	C₄×C₅	—	4	8	no	—
434	C₄×C₅	—	4	2	no	—
435	C₄×C₅	—	4	2	no	—
436	C₄×C₅	—	4	2	no	—
437	C₄×C₅	C₂×#104	1	2	yes	—
438	C₄×C₅	C₂×#104	1	2	yes	—
439	C₄×C₅	#421	1	2	yes	—
440	C₄×C₅	#520	1	2	yes	—
441	C₄×C₅	#535	1	1	yes	characteristic
442	C₄×C₅	#536	1	1	yes	characteristic
443	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
444	#41	—	2	4	no	—
445	#41	—	2	4	no	—
446	#41	—	4	4	no	—
447	#41	—	4	4	no	—
448	C₂×Q₈	—	2	1	no	—
449	C₂×Q₈	—	2	1	no	—
450	C₂×Q₈	C₂²×C₄×C₅	1	2	yes	—
451	C₂×D₄	—	2	2	no	—
452	C₂×D₄	—	2	2	no	—
453	C₂×D₄	—	2	2	no	—
454	C₂×D₄	—	2	1	no	—
455	C₂×D₄	—	2	1	no	—
456	C₂×D₄	C₂²×C₄×C₅	1	2	yes	—
457	C₂²×C₄	—	2	2	no	—
458	C₂²×C₄	—	2	2	no	—
459	C₂²×C₄	—	2	4	no	—
460	C₂²×C₄	—	2	4	no	—
461	C₂²×C₄	—	2	2	no	—
462	C₂²×C₄	—	2	2	no	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	4	2	no	—
465	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
466	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
467	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
468	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
469	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
470	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
471	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
472	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
473	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
474	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
475	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
476	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
477	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
478	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
479	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
480	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
481	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
482	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
483	M₁₆	—	2	8	no	—
484	M₁₆	—	4	4	no	—
485	C₂×C₈	—	2	4	no	—
486	C₂×C₈	—	4	4	no	—
487	C₄⋊C₄	—	2	8	no	—
488	C₄⋊C₄	—	2	1	no	—
489	C₄⋊C₄	—	2	1	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	1	no	—
492	C₄⋊C₄	—	2	2	no	—
493	C₄⋊C₄	—	2	2	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	—	2	2	no	—
496	C₄⋊C₄	—	2	2	no	—
497	C₄⋊C₄	—	2	2	no	—
498	C₄⋊C₄	C₂²×C₄×C₅	1	2	yes	—
499	C₄⋊C₄	C₂²×C₄×C₅	1	2	yes	—
500	C₂²⋊C₄	—	2	8	no	—
501	C₂²⋊C₄	—	2	2	no	—
502	C₂²⋊C₄	—	2	2	no	—
503	C₂²⋊C₄	—	2	2	no	—
504	C₂²⋊C₄	—	2	2	no	—
505	C₂²⋊C₄	—	2	2	no	—
506	C₂²⋊C₄	—	2	2	no	—
507	C₂²⋊C₄	—	2	2	no	—
508	C₂²⋊C₄	—	2	2	no	—
509	C₂²⋊C₄	—	2	2	no	—
510	C₄²	—	2	8	no	—
511	C₄²	—	2	2	no	—
512	C₄²	—	2	2	no	—
513	C₄²	—	2	2	no	—
514	C₄²	—	2	4	no	—
515	C₄²	—	2	4	no	—
516	C₄²	—	4	4	no	—
517	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
518	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
519	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
520	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
521	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
522	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
523	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
524	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
525	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
526	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
527	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
528	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
529	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
530	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
531	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
532	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
533	C₂×C₅	—	2	2	no	—
534	C₂×C₅	—	2	2	no	—
535	C₂×C₅	—	4	1	no	—
536	C₂×C₅	—	8	2	no	—
537	C₂×C₅	#2311	1	1	yes	characteristic
538	C₂×C₅	C₂²×#104	1	1	yes	characteristic
539	C₂×C₅	C₂×#421	1	1	yes	characteristic
540	C₂³	—	2	2	no	—
541	C₂³	—	2	2	no	—
542	C₂³	—	2	2	no	—
543	C₂³	—	4	1	no	—
544	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
545	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
546	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
547	Q₈	—	2	4	no	—
548	Q₈	—	4	2	no	—
549	Q₈	—	4	2	no	—
550	D₄	—	2	4	no	—
551	D₄	—	4	4	no	—
552	D₄	—	4	4	no	—
553	D₄	—	4	4	no	—
554	D₄	—	4	2	no	—
555	D₄	—	4	2	no	—
556	C₂×C₄	—	2	2	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	1	no	—
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	8	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	4	1	no	—
576	C₂×C₄	—	4	1	no	—
577	C₂×C₄	—	4	1	no	—
578	C₂×C₄	—	4	4	no	—
579	C₂×C₄	—	4	4	no	—
580	C₂×C₄	—	4	4	no	—
581	C₂×C₄	—	4	4	no	—
582	C₂×C₄	—	4	2	no	—
583	C₂×C₄	—	4	2	no	—
584	C₂×C₄	C₅×#104	1	4	yes	complemented
585	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
586	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
587	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
588	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
589	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
590	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
591	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
592	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
593	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
594	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
595	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
596	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
597	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
598	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
599	C₈	—	4	4	no	—
600	C₅	#21856	1	1	yes	characteristic, direct product factor, Sylow
601	C₂²	—	2	1	no	—
602	C₂²	—	2	1	no	—
603	C₂²	—	2	1	no	—
604	C₂²	—	2	1	no	—
605	C₂²	—	2	1	no	—
606	C₂²	—	2	1	no	—
607	C₂²	—	2	1	no	—
608	C₂²	—	4	4	no	—
609	C₂²	—	4	2	no	—
610	C₂²	—	4	2	no	—
611	C₂²	—	4	2	no	—
612	C₂²	C₂×C₅×#104	1	2	yes	—
613	C₂²	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
614	C₂²	C₅×#537	1	2	yes	—
615	C₄	—	2	1	no	—
616	C₄	—	2	1	no	—
617	C₄	—	2	1	no	—
618	C₄	—	2	1	no	—
619	C₄	—	2	1	no	—
620	C₄	—	2	1	no	—
621	C₄	—	2	1	no	—
622	C₄	—	4	8	no	—
623	C₄	—	4	2	no	—
624	C₄	—	4	2	no	—
625	C₄	—	4	2	no	—
626	C₄	C₂×C₅×#104	1	2	yes	—
627	C₄	C₂×C₅×#104	1	2	yes	—
628	C₄	C₅×#421	1	2	yes	—
629	C₄	C₅×#520	1	2	yes	—
630	C₄	C₅×#535	1	1	yes	characteristic
631	C₄	C₅×#536	1	1	yes	characteristic
632	C₂	—	2	2	no	—
633	C₂	—	2	2	no	—
634	C₂	—	4	1	no	—
635	C₂	—	8	2	no	—
636	C₂	C₅×#2311	1	1	yes	characteristic
637	C₂	C₂²×C₅×#104	1	1	yes	characteristic
638	C₂	C₂×C₅×#421	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂²×C₅	C₂²×C₂²⋊C₄
prime radical	C₅×#3444	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#423	C₂	semidirect
3. C₄×C₅×#117	C₂	semidirect
4. C₅×#1994	C₂	semidirect
5. C₅×#1962	C₂	semidirect
6. C₅×#1871	C₂	semidirect
7. C₅×#1806	C₂	semidirect
9. C₅×#533	C₄	semidirect
10. C₅×#533	C₄	semidirect
12. C₅×#532	C₄	semidirect
13. C₅×#532	C₄	semidirect
19. C₅×#529	C₄	semidirect
20. C₅×#529	C₄	semidirect
43. #21856	C₅	direct (N × H)
133. C₂×#423	C₂×C₅	semidirect
134. C₄×#117	C₂×C₅	semidirect
135. #1994	C₂×C₅	semidirect
136. #1962	C₂×C₅	semidirect
137. #1871	C₂×C₅	semidirect
138. #1806	C₂×C₅	semidirect
230. #533	C₄×C₅	semidirect
231. #533	C₄×C₅	semidirect
233. #532	C₄×C₅	semidirect
234. #532	C₄×C₅	semidirect
240. #529	C₄×C₅	semidirect
241. #529	C₄×C₅	semidirect
308. C₂×C₄×C₅	#104	semidirect
584. C₂×C₄	C₅×#104	semidirect
600. C₅	#21856	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#3444	2
C₂×C₅×#423	2
C₄×C₅×#117	2
C₅×#1994	2
C₅×#1962	2
C₅×#1871	2
C₅×#1806	2
#21856	5

Subgroups — C5×#21856 #11944576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21856 #11944576