Subgroups — #13034706

Order 1280.

810 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10617130	C₂	1	2	yes	maximal
2	D₅×#405	C₂	1	1	yes	maximal, characteristic
3	#10604761	C₂	1	2	yes	maximal
4	#10604723	C₂	1	1	yes	maximal, characteristic
5	#10604722	C₂	1	1	yes	maximal, characteristic
6	#10603883	C₂	1	2	yes	maximal, complemented
7	#10603881	C₂	1	2	yes	maximal, complemented
8	#10603846	C₂	1	1	yes	maximal, characteristic, complemented
9	#10603845	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#3994	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#3993	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₅×M₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
14	D₅×#117	C₂²	1	1	yes	characteristic
15	#65765	C₂²	1	2	yes	—
16	#65765	C₂²	1	2	yes	—
17	C₂×#3994	C₂²	1	2	yes	—
18	C₂×#3994	C₂²	1	2	yes	—
19	C₂×#3993	C₂²	1	2	yes	—
20	C₂×#3993	C₂²	1	2	yes	—
21	#65711	—	2	2	no	—
22	#65711	—	2	2	no	—
23	#65700	—	2	2	no	—
24	#65700	—	2	2	no	—
25	#65689	C₂²	1	2	yes	—
26	#65688	C₂²	1	2	yes	—
27	C₄×C₄⋊AGL(1,𝔽₅)	C₂²	1	2	yes	—
28	C₄×C₄⋊AGL(1,𝔽₅)	C₂²	1	2	yes	—
29	C₅×#405	C₄	1	1	yes	characteristic, complemented
30	#65411	C₄	1	1	yes	characteristic, complemented
31	#65390	C₄	1	1	yes	characteristic, complemented
32	#65286	C₄	1	1	yes	characteristic, complemented
33	#64997	C₄	1	1	yes	characteristic, complemented
34	#64997	C₄	1	1	yes	characteristic, complemented
35	D₅×#97	C₂²	1	1	yes	characteristic
36	D₅×#97	C₂²	1	1	yes	characteristic
37	#64978	C₄	1	1	yes	characteristic, complemented
38	#64978	C₄	1	1	yes	characteristic, complemented
39	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
40	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
41	#64893	C₂²	1	1	yes	characteristic
42	#64893	C₂²	1	1	yes	characteristic
43	#64893	C₂²	1	1	yes	characteristic
44	#64893	C₂²	1	1	yes	characteristic
45	#64893	C₂²	1	2	yes	—
46	#64893	C₂²	1	2	yes	—
47	#20822	—	5	1	no	maximal, Sylow
48	C₂²×C₄×D₅	C₂³	1	1	yes	characteristic
49	C₂×C₂²⋊AGL(1,𝔽₅)	—	2	2	no	—
50	C₂×C₂²⋊AGL(1,𝔽₅)	—	2	2	no	—
51	C₂×C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
52	C₂×C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
53	C₂×C₄⋊AGL(1,𝔽₅)	C₂³	1	2	yes	—
54	C₂×C₄⋊AGL(1,𝔽₅)	C₂³	1	2	yes	—
55	C₂×C₄⋊AGL(1,𝔽₅)	C₂³	1	2	yes	—
56	C₂×C₄⋊AGL(1,𝔽₅)	C₂³	1	2	yes	—
57	C₂×C₄×AGL(1,𝔽₅)	—	2	2	no	—
58	C₂×C₄×AGL(1,𝔽₅)	—	2	2	no	—
59	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
61	C₅×#117	C₂×C₄	1	1	yes	characteristic
62	#4072	C₂×C₄	1	1	yes	characteristic
63	C₂×#681	C₂×C₄	1	1	yes	characteristic
64	C₂²×#168	C₂×C₄	1	1	yes	characteristic
65	D₅×M₁₆	—	2	4	no	—
66	C₂×#676	C₂×C₄	1	1	yes	characteristic
67	C₂×#676	C₂×C₄	1	1	yes	characteristic
68	C₂×C₈×D₅	—	2	4	no	—
69	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
70	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
71	C₂×C₈×D₅	C₂³	1	1	yes	characteristic
72	C₂×C₈×D₅	C₂³	1	1	yes	characteristic
73	C₂×C₈×D₅	C₂³	1	1	yes	characteristic
74	C₂×C₈×D₅	C₂³	1	1	yes	characteristic
75	#4038	Q₈	1	1	yes	characteristic
76	#4038	Q₈	1	1	yes	characteristic
77	D₅×C₄⋊C₄	D₄	1	1	yes	characteristic
78	D₅×C₄⋊C₄	D₄	1	1	yes	characteristic
79	D₅×C₂²⋊C₄	D₄	1	1	yes	characteristic
80	D₅×C₂²⋊C₄	D₄	1	1	yes	characteristic
81	#4023	Q₈	1	1	yes	characteristic
82	#4023	Q₈	1	1	yes	characteristic
83	#4018	C₂×C₄	1	1	yes	characteristic
84	#4018	C₂×C₄	1	1	yes	characteristic
85	C₄²×D₅	C₂³	1	1	yes	characteristic
86	C₄²×D₅	C₂³	1	1	yes	characteristic
87	(C₂×AGL(1,𝔽₅))⋊C₄	—	2	2	no	—
88	(C₂×AGL(1,𝔽₅))⋊C₄	—	2	2	no	—
89	(C₂×AGL(1,𝔽₅))⋊C₄	—	2	2	no	—
90	(C₂×AGL(1,𝔽₅))⋊C₄	—	2	2	no	—
91	(C₂×AGL(1,𝔽₅))⋊C₄	—	2	2	no	—
92	(C₂×AGL(1,𝔽₅))⋊C₄	—	2	2	no	—
93	#3994	—	2	4	no	—
94	#3994	—	2	4	no	—
95	#3993	—	2	4	no	—
96	#3993	—	2	4	no	—
97	C₅×#97	C₂×C₄	1	1	yes	characteristic
98	C₅×#97	C₂×C₄	1	1	yes	characteristic
99	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
100	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
101	#3947	C₂×C₄	1	1	yes	characteristic
102	#3947	C₂×C₄	1	1	yes	characteristic
103	#3947	C₂×C₄	1	1	yes	characteristic
104	#3947	C₂×C₄	1	1	yes	characteristic
105	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
106	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
107	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
108	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
109	#3935	C₂×C₄	1	1	yes	characteristic
110	#3935	C₂×C₄	1	1	yes	characteristic
111	C₄×#168	C₂×C₄	1	1	yes	characteristic
112	C₄×#168	C₂×C₄	1	1	yes	characteristic
113	C₂×#405	—	5	1	no	—
114	#2314	—	5	2	no	—
115	#1847	—	5	2	no	—
116	#1835	—	5	1	no	—
117	#1834	—	5	1	no	—
118	#1809	—	5	2	no	—
119	#1808	—	5	1	no	—
120	#1807	—	5	1	no	—
121	#1776	—	5	2	no	—
122	C₄×#107	—	5	1	no	—
123	C₄×#106	—	5	1	no	—
124	C₂³×D₅	C₂×D₄	1	1	yes	characteristic
125	C₂²×AGL(1,𝔽₅)	—	2	2	no	—
126	C₂²×AGL(1,𝔽₅)	—	2	2	no	—
127	C₂²×AGL(1,𝔽₅)	—	2	2	no	—
128	C₂²×AGL(1,𝔽₅)	—	2	2	no	—
129	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
130	C₂²×Dic₅	C₂×Q₈	1	1	yes	characteristic
131	C₂×C₄×D₅	—	2	2	no	—
132	C₂×C₄×D₅	—	2	2	no	—
133	C₂×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₄×D₅	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×D₅	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄×D₅	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄×D₅	C₂×D₄	1	1	yes	characteristic
138	C₂×C₄×D₅	C₂×D₄	1	1	yes	characteristic
139	C₂×C₄×D₅	C₂×Q₈	1	1	yes	characteristic
140	C₂×C₄×D₅	C₂⁴	1	1	yes	characteristic
141	C₂²⋊AGL(1,𝔽₅)	—	8	2	no	—
142	C₂²⋊AGL(1,𝔽₅)	—	8	2	no	—
143	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
144	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
145	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
146	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
147	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
148	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
149	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
150	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
151	C₄⋊AGL(1,𝔽₅)	—	8	2	no	—
152	C₄⋊AGL(1,𝔽₅)	—	8	2	no	—
153	C₄×AGL(1,𝔽₅)	—	8	2	no	—
154	C₄×AGL(1,𝔽₅)	—	8	2	no	—
155	C₅×M₁₆	—	2	2	no	—
156	C₂×C₅×C₈	—	2	2	no	—
157	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
158	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
159	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
160	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
161	C₅×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
162	C₅×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
163	C₅×C₂²⋊C₄	C₄⋊C₄	1	1	yes	characteristic
164	C₅×C₂²⋊C₄	C₄⋊C₄	1	1	yes	characteristic
165	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
166	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
167	C₂²⋊Dic₅	C₄⋊C₄	1	1	yes	characteristic
168	C₂²⋊Dic₅	C₄⋊C₄	1	1	yes	characteristic
169	#685	C₄⋊C₄	1	1	yes	characteristic
170	#685	C₄⋊C₄	1	1	yes	characteristic
171	#685	C₄⋊C₄	1	1	yes	characteristic
172	#685	C₄⋊C₄	1	1	yes	characteristic
173	C₄⋊Dic₅	C₄⋊C₄	1	1	yes	characteristic
174	C₄⋊Dic₅	C₄⋊C₄	1	1	yes	characteristic
175	#683	C₄⋊C₄	1	1	yes	characteristic
176	#683	C₄⋊C₄	1	1	yes	characteristic
177	#683	C₄⋊C₄	1	1	yes	characteristic
178	#683	C₄⋊C₄	1	1	yes	characteristic
179	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
180	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
181	C₄×Dic₅	C₂×Q₈	1	1	yes	characteristic
182	C₄×Dic₅	C₂×Q₈	1	1	yes	characteristic
183	C₄×Dic₅	C₂×Q₈	1	1	yes	characteristic
184	C₄×Dic₅	C₂×Q₈	1	1	yes	characteristic
185	#681	—	2	2	no	—
186	C₂×#168	—	2	2	no	—
187	C₂×#168	C₂²×C₄	1	1	yes	characteristic
188	C₂×#168	C₂²×C₄	1	1	yes	characteristic
189	C₂×#168	C₂²×C₄	1	1	yes	characteristic
190	C₂×#168	C₂²×C₄	1	1	yes	characteristic
191	#676	—	2	2	no	—
192	#676	—	2	2	no	—
193	C₈×D₅	—	2	2	no	—
194	C₈×D₅	—	2	2	no	—
195	C₈×D₅	—	2	2	no	—
196	C₈×D₅	—	2	2	no	—
197	C₈×D₅	#41	1	4	yes	—
198	C₈×D₅	#41	1	4	yes	—
199	C₂²×M₁₆	—	5	1	no	—
200	C₂³×C₈	—	5	1	no	—
201	C₂×#117	—	5	1	no	—
202	C₂×#107	—	5	2	no	—
203	C₂×#107	—	5	2	no	—
204	C₂×#106	—	5	2	no	—
205	C₂×#106	—	5	2	no	—
206	#405	—	5	1	no	—
207	#405	—	5	1	no	—
208	#405	—	5	1	no	—
209	#405	—	5	1	no	—
210	#405	—	5	1	no	—
211	#405	—	5	1	no	—
212	#405	—	5	1	no	—
213	#405	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
214	C₂×#97	—	5	1	no	—
215	C₂×#97	—	5	1	no	—
216	C₂×C₄×C₈	—	5	1	no	—
217	C₂×C₄×C₈	—	5	1	no	—
218	#388	—	10	2	no	—
219	#388	—	10	2	no	—
220	#387	—	10	2	no	—
221	#387	—	10	2	no	—
222	#384	—	5	2	no	—
223	#382	—	5	2	no	—
224	#380	—	5	2	no	—
225	#380	—	5	2	no	—
226	C₄×C₄⋊C₄	—	5	2	no	—
227	C₄×C₄⋊C₄	—	5	2	no	—
228	#340	—	5	1	no	—
229	#340	—	5	1	no	—
230	#340	—	5	1	no	—
231	#340	—	5	1	no	—
232	#340	—	5	2	no	—
233	#340	—	5	2	no	—
234	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
235	C₂²×D₅	—	2	2	no	—
236	C₂²×D₅	—	2	1	no	—
237	C₂²×D₅	—	2	2	no	—
238	C₂²×D₅	—	2	1	no	—
239	C₂²×D₅	C₂×C₄⋊C₄	1	1	yes	characteristic
240	C₂²×D₅	C₂²×D₄	1	1	yes	characteristic
241	C₂×AGL(1,𝔽₅)	—	4	2	no	—
242	C₂×AGL(1,𝔽₅)	—	4	2	no	—
243	C₂×AGL(1,𝔽₅)	—	4	2	no	—
244	C₂×AGL(1,𝔽₅)	—	4	2	no	—
245	C₂×AGL(1,𝔽₅)	—	4	2	no	—
246	C₂×AGL(1,𝔽₅)	—	4	2	no	—
247	C₂×AGL(1,𝔽₅)	—	4	2	no	—
248	C₂×AGL(1,𝔽₅)	—	4	2	no	—
249	C₂×AGL(1,𝔽₅)	—	4	2	no	—
250	C₂×AGL(1,𝔽₅)	—	4	2	no	—
251	C₂×AGL(1,𝔽₅)	—	4	2	no	—
252	C₂×AGL(1,𝔽₅)	—	4	2	no	—
253	C₂×C₄×C₅	—	2	1	no	—
254	C₂×C₄×C₅	—	2	1	no	—
255	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
256	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
257	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
258	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
259	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
260	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
261	C₂×Dic₅	—	2	2	no	—
262	C₂×Dic₅	C₂×C₄⋊C₄	1	1	yes	characteristic
263	C₂×Dic₅	C₂×C₄⋊C₄	1	1	yes	characteristic
264	C₂×Dic₅	C₂×C₄⋊C₄	1	1	yes	characteristic
265	C₂×Dic₅	C₂×C₄⋊C₄	1	1	yes	characteristic
266	C₂×Dic₅	C₂×C₄⋊C₄	1	1	yes	characteristic
267	C₂×Dic₅	C₂²×Q₈	1	1	yes	characteristic
268	C₄×D₅	—	2	2	no	—
269	C₄×D₅	—	2	1	no	—
270	C₄×D₅	—	2	1	no	—
271	C₄×D₅	—	2	2	no	—
272	C₄×D₅	—	2	2	no	—
273	C₄×D₅	—	2	1	no	—
274	C₄×D₅	—	2	1	no	—
275	C₄×D₅	—	2	1	no	—
276	C₄×D₅	—	2	1	no	—
277	C₄×D₅	—	2	2	no	—
278	C₄×D₅	C₂×#41	1	2	yes	—
279	C₄×D₅	C₂×#41	1	2	yes	—
280	C₅×C₈	—	2	1	no	—
281	C₅×C₈	—	2	1	no	—
282	C₅×C₈	#117	1	2	yes	—
283	C₅×C₈	#117	1	2	yes	—
284	#168	—	2	1	no	—
285	#168	—	2	1	no	—
286	#168	#117	1	2	yes	—
287	#168	#117	1	2	yes	—
288	C₂³×C₄	—	5	1	no	—
289	C₂×M₁₆	—	5	1	no	—
290	C₂×M₁₆	—	5	1	no	—
291	C₂×M₁₆	—	5	1	no	—
292	C₂×M₁₆	—	10	4	no	—
293	C₂×M₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
294	C₂²×C₈	—	5	1	no	—
295	C₂²×C₈	—	5	1	no	—
296	C₂²×C₈	—	5	1	no	—
297	C₂²×C₈	—	5	1	no	—
298	C₂²×C₈	—	5	1	no	—
299	C₂²×C₈	—	5	1	no	—
300	C₂²×C₈	—	5	1	no	—
301	C₂²×C₈	—	10	4	no	—
302	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
303	#117	—	5	1	no	—
304	#117	—	5	1	no	—
305	#117	—	5	1	no	—
306	#117	—	5	1	no	—
307	#117	—	5	1	no	—
308	#117	—	5	1	no	—
309	#117	—	5	1	no	—
310	#117	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
311	C₂×C₄⋊C₄	—	5	1	no	—
312	C₂×C₄⋊C₄	—	5	1	no	—
313	C₂×C₄⋊C₄	—	5	2	no	—
314	C₂×C₄⋊C₄	—	5	2	no	—
315	C₂×C₄⋊C₄	—	5	2	no	—
316	C₂×C₄⋊C₄	—	5	2	no	—
317	C₂×C₄⋊C₄	—	10	2	no	—
318	C₂×C₄⋊C₄	—	10	2	no	—
319	C₂×C₂²⋊C₄	—	5	1	no	—
320	C₂×C₂²⋊C₄	—	5	1	no	—
321	C₂×C₂²⋊C₄	—	10	2	no	—
322	C₂×C₂²⋊C₄	—	10	2	no	—
323	C₂×C₄²	—	5	1	no	—
324	C₂×C₄²	—	5	1	no	—
325	C₂×C₄²	—	10	2	no	—
326	C₂×C₄²	—	10	2	no	—
327	#107	—	10	4	no	—
328	#107	—	10	4	no	—
329	#106	—	10	4	no	—
330	#106	—	10	4	no	—
331	#97	—	5	1	no	—
332	#97	—	5	1	no	—
333	#97	—	5	1	no	—
334	#97	—	5	1	no	—
335	#97	—	5	1	no	—
336	#97	—	5	1	no	—
337	#97	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
338	#97	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
339	C₄×C₈	—	5	1	no	—
340	C₄×C₈	—	5	1	no	—
341	C₄×C₈	—	5	1	no	—
342	C₄×C₈	—	5	1	no	—
343	C₄×C₈	—	5	1	no	—
344	C₄×C₈	—	5	1	no	—
345	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
346	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
347	#95	—	10	2	no	—
348	#95	—	10	2	no	—
349	#95	—	10	2	no	—
350	#95	—	10	2	no	—
351	#95	—	10	2	no	—
352	#95	—	10	2	no	—
353	C₂²×C₅	—	2	1	no	—
354	C₂²×C₅	—	2	1	no	—
355	C₂²×C₅	—	2	1	no	—
356	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
357	C₂×D₅	—	2	1	no	—
358	C₂×D₅	—	2	1	no	—
359	C₂×D₅	—	2	1	no	—
360	C₂×D₅	—	4	2	no	—
361	C₂×D₅	—	4	2	no	—
362	C₂×D₅	#532	1	2	yes	—
363	C₂×D₅	#576	1	1	yes	characteristic
364	C₂×D₅	#577	1	2	yes	—
365	C₂×D₅	#578	1	1	yes	characteristic
366	AGL(1,𝔽₅)	—	8	2	no	—
367	AGL(1,𝔽₅)	—	8	2	no	—
368	AGL(1,𝔽₅)	—	8	2	no	—
369	AGL(1,𝔽₅)	—	8	2	no	—
370	C₄×C₅	—	2	1	no	—
371	C₄×C₅	—	2	1	no	—
372	C₄×C₅	—	2	1	no	—
373	C₄×C₅	—	2	1	no	—
374	C₄×C₅	—	2	1	no	—
375	C₄×C₅	C₂×#117	1	1	yes	characteristic
376	C₄×C₅	C₂×#117	1	1	yes	characteristic
377	Dic₅	—	2	1	no	—
378	Dic₅	—	2	1	no	—
379	Dic₅	—	2	1	no	—
380	Dic₅	—	2	1	no	—
381	Dic₅	—	2	1	no	—
382	Dic₅	#533	1	2	yes	—
383	C₂⁴	—	5	1	no	—
384	C₂²×C₄	—	5	1	no	—
385	C₂²×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	—	10	2	no	—
394	C₂²×C₄	—	10	2	no	—
395	C₂²×C₄	—	10	2	no	—
396	C₂²×C₄	—	10	2	no	—
397	C₂²×C₄	—	10	2	no	—
398	C₂²×C₄	—	10	2	no	—
399	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
400	M₁₆	—	2	2	no	—
401	M₁₆	—	10	2	no	—
402	M₁₆	—	10	2	no	—
403	M₁₆	—	10	2	no	—
404	C₂×C₈	—	2	2	no	—
405	C₂×C₈	—	5	1	no	—
406	C₂×C₈	—	5	1	no	—
407	C₂×C₈	—	5	4	no	—
408	C₂×C₈	—	5	4	no	—
409	C₂×C₈	—	5	1	no	—
410	C₂×C₈	—	5	1	no	—
411	C₂×C₈	—	10	2	no	—
412	C₂×C₈	—	10	2	no	—
413	C₂×C₈	—	10	2	no	—
414	C₂×C₈	—	10	2	no	—
415	C₂×C₈	—	10	2	no	—
416	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
417	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
418	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
419	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
420	C₄⋊C₄	—	5	1	no	—
421	C₄⋊C₄	—	5	1	no	—
422	C₄⋊C₄	—	5	1	no	—
423	C₄⋊C₄	—	5	1	no	—
424	C₄⋊C₄	—	5	1	no	—
425	C₄⋊C₄	—	5	1	no	—
426	C₄⋊C₄	—	10	2	no	—
427	C₄⋊C₄	—	10	2	no	—
428	C₄⋊C₄	—	10	2	no	—
429	C₄⋊C₄	—	10	2	no	—
430	C₄⋊C₄	—	10	2	no	—
431	C₄⋊C₄	—	10	2	no	—
432	C₄⋊C₄	—	10	2	no	—
433	C₄⋊C₄	—	10	2	no	—
434	C₄⋊C₄	—	40	2	no	—
435	C₄⋊C₄	—	40	2	no	—
436	C₄⋊C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
437	C₄⋊C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
438	C₂²⋊C₄	—	5	1	no	—
439	C₂²⋊C₄	—	5	1	no	—
440	C₂²⋊C₄	—	5	1	no	—
441	C₂²⋊C₄	—	5	1	no	—
442	C₂²⋊C₄	—	5	1	no	—
443	C₂²⋊C₄	—	5	1	no	—
444	C₂²⋊C₄	—	40	2	no	—
445	C₂²⋊C₄	—	40	2	no	—
446	C₂²⋊C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
447	C₂²⋊C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
448	C₄²	—	5	1	no	—
449	C₄²	—	5	1	no	—
450	C₄²	—	5	1	no	—
451	C₄²	—	5	1	no	—
452	C₄²	—	5	1	no	—
453	C₄²	—	5	1	no	—
454	C₄²	—	40	2	no	—
455	C₄²	—	40	2	no	—
456	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
457	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
458	C₂×C₅	—	4	1	no	—
459	C₂×C₅	#2285	1	1	yes	characteristic
460	C₂×C₅	#2912	1	1	yes	characteristic
461	C₂×C₅	#2913	1	1	yes	characteristic
462	D₅	—	4	1	no	—
463	D₅	#3178	1	2	yes	—
464	D₅	#3179	1	2	yes	—
465	C₂³	—	5	1	no	—
466	C₂³	—	5	1	no	—
467	C₂³	—	10	1	no	—
468	C₂³	—	10	2	no	—
469	C₂³	—	10	2	no	—
470	C₂³	—	10	1	no	—
471	C₂³	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	5	1	no	—
475	C₂×C₄	—	5	2	no	—
476	C₂×C₄	—	5	2	no	—
477	C₂×C₄	—	5	1	no	—
478	C₂×C₄	—	5	1	no	—
479	C₂×C₄	—	5	1	no	—
480	C₂×C₄	—	5	1	no	—
481	C₂×C₄	—	5	1	no	—
482	C₂×C₄	—	10	2	no	—
483	C₂×C₄	—	10	1	no	—
484	C₂×C₄	—	10	2	no	—
485	C₂×C₄	—	10	1	no	—
486	C₂×C₄	—	10	2	no	—
487	C₂×C₄	—	10	2	no	—
488	C₂×C₄	—	10	1	no	—
489	C₂×C₄	—	10	1	no	—
490	C₂×C₄	—	10	2	no	—
491	C₂×C₄	—	10	1	no	—
492	C₂×C₄	—	10	1	no	—
493	C₂×C₄	—	20	2	no	—
494	C₂×C₄	—	20	2	no	—
495	C₂×C₄	—	20	2	no	—
496	C₂×C₄	—	20	2	no	—
497	C₂×C₄	—	20	2	no	—
498	C₂×C₄	—	20	2	no	—
499	C₂×C₄	—	20	2	no	—
500	C₂×C₄	—	20	2	no	—
501	C₂×C₄	—	20	2	no	—
502	C₂×C₄	—	20	2	no	—
503	C₂×C₄	—	20	2	no	—
504	C₂×C₄	—	20	2	no	—
505	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
506	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
507	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
508	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
509	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
510	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
511	C₈	—	2	1	no	—
512	C₈	—	2	1	no	—
513	C₈	—	5	2	no	—
514	C₈	—	5	2	no	—
515	C₈	—	10	1	no	—
516	C₈	—	10	1	no	—
517	C₈	#4130	1	2	yes	—
518	C₈	#4130	1	2	yes	—
519	C₅	#20822	1	1	yes	characteristic, complemented, Sylow
520	C₂²	—	2	1	no	—
521	C₂²	—	2	1	no	—
522	C₂²	—	2	1	no	—
523	C₂²	—	5	2	no	—
524	C₂²	—	5	1	no	—
525	C₂²	—	5	1	no	—
526	C₂²	—	5	2	no	—
527	C₂²	—	10	1	no	—
528	C₂²	—	10	1	no	—
529	C₂²	—	10	1	no	—
530	C₂²	—	20	2	no	—
531	C₂²	—	20	2	no	—
532	C₂²	C₂²×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
533	C₄	—	2	1	no	—
534	C₄	—	2	1	no	—
535	C₄	—	2	1	no	—
536	C₄	—	2	1	no	—
537	C₄	—	2	1	no	—
538	C₄	—	5	2	no	—
539	C₄	—	10	1	no	—
540	C₄	—	10	1	no	—
541	C₄	—	10	1	no	—
542	C₄	—	10	1	no	—
543	C₄	—	10	1	no	—
544	C₄	—	40	2	no	—
545	C₄	—	40	2	no	—
546	C₄	—	40	2	no	—
547	C₄	—	40	2	no	—
548	C₄	C₂×#4130	1	1	yes	characteristic
549	C₄	C₂×#4130	1	1	yes	characteristic
550	C₂	—	4	1	no	—
551	C₂	—	5	2	no	—
552	C₂	—	5	2	no	—
553	C₂	—	20	1	no	—
554	C₂	#10617048	1	1	yes	characteristic
555	C₂	#10617416	1	1	yes	characteristic
556	C₂	#10617417	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂³×AGL(1,𝔽₅)
Fitting	C₅×#405	C₄
socle	C₂²×C₅	C₂²×C₄⋊C₄
prime radical	C₂³×D₅	C₂×D₄
cosocle	C₂×C₄×D₅	C₂⁴
hypercenter	#405	AGL(1,𝔽₅)
hypocenter	C₅	#20822
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
6. #10603883	C₂	semidirect
7. #10603881	C₂	semidirect
8. #10603846	C₂	semidirect
9. #10603845	C₂	semidirect
10. C₄×#3994	C₂	semidirect
11. C₄×#3993	C₂	semidirect
29. C₅×#405	C₄	semidirect
30. #65411	C₄	semidirect
31. #65390	C₄	semidirect
32. #65286	C₄	semidirect
33. #64997	C₄	semidirect
34. #64997	C₄	semidirect
37. #64978	C₄	semidirect
38. #64978	C₄	semidirect
213. #405	AGL(1,𝔽₅)	semidirect
519. C₅	#20822	semidirect

Maximal subgroups

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

subgroup	index
#10617130	2
D₅×#405	2
#10604761	2
#10604723	2
#10604722	2
#10603883	2
#10603881	2
#10603846	2
#10603845	2
C₄×#3994	2
C₄×#3993	2
#20822	5