Subgroups — C₅×#22060 #11944780

Order 1280.

926 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3002	C₂	1	1	yes	maximal, characteristic
2	C₅×#2929	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#408	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1974	C₂	1	2	yes	maximal, complemented
5	C₅×#1937	C₂	1	2	yes	maximal, complemented
6	C₅×#1929	C₂	1	2	yes	maximal, complemented
7	C₅×#1876	C₂	1	2	yes	maximal, complemented
8	C₅×#1871	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1855	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#1789	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×C₈×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₅×#41	C₂²	1	1	yes	characteristic
13	C₂×C₅×#135	C₂²	1	2	yes	—
14	C₂×C₅×#131	C₂²	1	1	yes	characteristic, complemented
15	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
16	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₅×#525	C₂²	1	1	yes	characteristic
18	C₅×#452	C₄	1	2	yes	complemented
19	C₅×#451	C₄	1	1	yes	characteristic, complemented
20	C₅×#450	C₄	1	1	yes	characteristic, complemented
21	C₅×#449	C₄	1	2	yes	complemented
22	C₅×#448	C₄	1	1	yes	characteristic, complemented
23	C₅×#447	C₄	1	1	yes	characteristic, complemented
24	C₅×#435	—	2	1	no	—
25	C₅×#435	—	2	1	no	—
26	C₅×C₈×D₄	—	2	1	no	—
27	C₅×C₈×D₄	—	2	1	no	—
28	C₂×C₅×#108	C₂²	1	2	yes	complemented
29	C₅×#424	C₂²	1	1	yes	characteristic, complemented
30	C₂×C₅×#104	C₂²	1	2	yes	—
31	C₂×C₅×#104	C₂²	1	2	yes	—
32	C₂×C₅×#104	C₂²	1	2	yes	complemented
33	C₅×#416	C₂²	1	2	yes	—
34	C₅×#408	—	2	1	no	—
35	C₅×#408	—	2	1	no	—
36	C₅×#408	C₂²	1	2	yes	—
37	C₅×#408	C₂²	1	2	yes	—
38	C₅×#408	C₂²	1	1	yes	characteristic, complemented
39	C₅×#407	C₂²	1	1	yes	characteristic
40	C₂×C₅×C₂²⋊C₈	—	2	1	no	—
41	C₂×C₅×C₂²⋊C₈	—	2	1	no	—
42	C₂×C₅×C₂²⋊C₈	C₂²	1	1	yes	characteristic
43	C₅×#405	C₂²	1	1	yes	characteristic, complemented
44	C₂×C₄×C₅×C₈	C₂²	1	2	yes	complemented
45	C₄×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
46	C₅×#341	C₂²	1	2	yes	complemented
47	C₅×#339	C₂²	1	2	yes	complemented
48	C₅×#336	C₂²	1	2	yes	complemented
49	#22060	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂×C₅×#41	—	2	1	no	—
51	C₂×C₅×#41	—	2	1	no	—
52	C₂×C₅×#41	—	4	2	no	—
53	C₂×C₅×#41	—	4	2	no	—
54	C₂×C₅×#41	C₂×C₄	1	2	yes	—
55	C₂×C₅×#41	C₂³	1	2	yes	—
56	C₂×C₅×#41	C₂³	1	1	yes	characteristic
57	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₅×D₄	—	2	1	no	—
59	C₂²×C₅×D₄	C₂×C₄	1	1	yes	characteristic
60	C₂³×C₄×C₅	—	2	1	no	—
61	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
62	C₅×#135	—	4	2	no	—
63	C₅×#135	—	4	2	no	—
64	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
65	C₂×C₅×SD₁₆	C₂×C₄	1	2	yes	—
66	C₂×C₅×SD₁₆	C₂×C₄	1	2	yes	—
67	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
68	C₅×#131	—	4	2	no	—
69	C₂×C₅×M₁₆	—	2	1	no	—
70	C₂×C₅×M₁₆	—	2	1	no	—
71	C₂×C₅×M₁₆	—	4	2	no	—
72	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
73	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
74	C₂×C₅×M₁₆	C₂³	1	2	yes	—
75	C₂²×C₅×C₈	—	2	1	no	—
76	C₂²×C₅×C₈	—	2	1	no	—
77	C₂²×C₅×C₈	—	4	2	no	—
78	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
79	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
80	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
81	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
82	C₂²×C₅×C₈	C₂³	1	2	yes	—
83	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
84	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
85	C₅×#123	—	2	1	no	—
86	C₅×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
87	C₅×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
88	C₅×#121	C₂×C₄	1	1	yes	characteristic
89	C₅×#121	C₂×C₄	1	1	yes	characteristic
90	C₅×#120	—	2	1	no	—
91	C₄×C₅×D₄	—	2	1	no	—
92	C₄×C₅×D₄	—	2	1	no	—
93	C₅×#117	C₂³	1	1	yes	characteristic
94	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
95	C₂×C₄²×C₅	C₂³	1	2	yes	—
96	C₅×#108	—	4	2	no	—
97	C₅×C₄⋊C₈	—	2	1	no	—
98	C₅×C₄⋊C₈	—	2	1	no	—
99	C₅×#104	—	4	2	no	—
100	C₅×#104	—	4	2	no	—
101	C₅×#104	—	4	2	no	—
102	C₅×#103	C₂×C₄	1	2	yes	—
103	C₅×#103	C₂×C₄	1	2	yes	—
104	C₅×#102	C₂×C₄	1	2	yes	—
105	C₅×#102	C₂×C₄	1	2	yes	—
106	C₅×C₂²⋊C₈	—	2	1	no	—
107	C₅×C₂²⋊C₈	—	2	1	no	—
108	C₅×C₂²⋊C₈	—	2	2	no	—
109	C₅×C₂²⋊C₈	—	2	2	no	—
110	C₅×C₂²⋊C₈	—	2	2	no	—
111	C₅×C₂²⋊C₈	—	2	2	no	—
112	C₅×C₂²⋊C₈	—	2	1	no	—
113	C₅×C₂²⋊C₈	—	2	1	no	—
114	C₅×C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
115	C₅×C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
116	C₅×C₂²⋊C₈	C₂×C₄	1	2	yes	—
117	C₅×#97	—	2	1	no	—
118	C₄×C₅×C₈	—	2	1	no	—
119	C₄×C₅×C₈	—	4	2	no	—
120	C₅×#95	—	2	1	no	—
121	#3002	C₂×C₅	1	1	yes	characteristic
122	#2929	C₂×C₅	1	1	yes	characteristic, complemented
123	C₂×#408	C₂×C₅	1	1	yes	characteristic, complemented
124	#1974	C₂×C₅	1	2	yes	complemented
125	#1937	C₂×C₅	1	2	yes	complemented
126	#1929	C₂×C₅	1	2	yes	complemented
127	#1876	C₂×C₅	1	2	yes	complemented
128	#1871	C₂×C₅	1	1	yes	characteristic, complemented
129	#1855	C₂×C₅	1	1	yes	characteristic, complemented
130	#1789	C₂×C₅	1	1	yes	characteristic, complemented
131	C₈×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
132	C₂⁴×C₅	—	2	1	no	—
133	C₂⁴×C₅	#41	1	1	yes	characteristic
134	C₅×#41	—	2	2	no	—
135	C₅×#41	—	2	2	no	—
136	C₅×#41	—	2	2	no	—
137	C₅×#41	—	2	2	no	—
138	C₅×#41	—	2	2	no	—
139	C₅×#41	—	2	2	no	—
140	C₅×#41	—	4	2	no	—
141	C₅×#41	—	4	2	no	—
142	C₅×#41	—	4	4	no	—
143	C₅×#41	—	4	2	no	—
144	C₅×#41	—	4	2	no	—
145	C₂×C₅×Q₈	—	2	1	no	—
146	C₂×C₅×Q₈	—	4	2	no	—
147	C₂×C₅×Q₈	C₂²×C₄	1	1	yes	characteristic
148	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
149	C₂×C₅×D₄	—	2	1	no	—
150	C₂×C₅×D₄	—	2	1	no	—
151	C₂×C₅×D₄	—	2	2	no	—
152	C₂×C₅×D₄	—	2	1	no	—
153	C₂×C₅×D₄	—	2	2	no	—
154	C₂×C₅×D₄	—	2	1	no	—
155	C₂×C₅×D₄	—	4	2	no	—
156	C₂×C₅×D₄	—	4	4	no	—
157	C₂×C₅×D₄	C₂²×C₄	1	1	yes	characteristic
158	C₂×C₅×D₄	C₂²×C₄	1	2	yes	—
159	C₂²×C₄×C₅	—	2	2	no	—
160	C₂²×C₄×C₅	—	2	2	no	—
161	C₂²×C₄×C₅	—	2	1	no	—
162	C₂²×C₄×C₅	—	2	2	no	—
163	C₂²×C₄×C₅	—	2	2	no	—
164	C₂²×C₄×C₅	—	2	1	no	—
165	C₂²×C₄×C₅	—	2	2	no	—
166	C₂²×C₄×C₅	—	4	2	no	—
167	C₂²×C₄×C₅	—	4	2	no	—
168	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
169	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
170	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₅	#41	1	1	yes	characteristic
173	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
174	C₅×Q₁₆	—	4	2	no	—
175	C₅×SD₁₆	—	4	2	no	—
176	C₅×SD₁₆	—	4	2	no	—
177	C₅×D₈	—	4	2	no	—
178	C₅×M₁₆	—	2	2	no	—
179	C₅×M₁₆	—	2	2	no	—
180	C₅×M₁₆	—	2	2	no	—
181	C₅×M₁₆	—	2	2	no	—
182	C₅×M₁₆	—	4	2	no	—
183	C₅×M₁₆	—	4	2	no	—
184	C₂×C₅×C₈	—	2	2	no	—
185	C₂×C₅×C₈	—	2	2	no	—
186	C₂×C₅×C₈	—	2	2	no	—
187	C₂×C₅×C₈	—	2	2	no	—
188	C₂×C₅×C₈	—	2	2	no	—
189	C₂×C₅×C₈	—	2	1	no	—
190	C₂×C₅×C₈	—	2	1	no	—
191	C₂×C₅×C₈	—	4	1	no	—
192	C₂×C₅×C₈	—	4	1	no	—
193	C₂×C₅×C₈	—	4	1	no	—
194	C₂×C₅×C₈	—	4	1	no	—
195	C₂×C₅×C₈	—	4	2	no	—
196	C₂×C₅×C₈	—	4	2	no	—
197	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
198	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
199	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
200	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
201	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
202	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
203	C₅×C₄⋊C₄	—	2	1	no	—
204	C₅×C₄⋊C₄	—	2	1	no	—
205	C₅×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
206	C₅×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
207	C₅×C₂²⋊C₄	—	2	4	no	—
208	C₅×C₂²⋊C₄	—	2	1	no	—
209	C₅×C₂²⋊C₄	—	2	1	no	—
210	C₅×C₂²⋊C₄	—	2	1	no	—
211	C₅×C₂²⋊C₄	—	2	1	no	—
212	C₅×C₂²⋊C₄	—	2	1	no	—
213	C₄²×C₅	—	2	1	no	—
214	C₄²×C₅	—	2	2	no	—
215	C₄²×C₅	—	2	2	no	—
216	C₂²×#41	C₂²×C₅	1	1	yes	characteristic
217	C₂×#135	C₂²×C₅	1	2	yes	—
218	C₂×#131	C₂²×C₅	1	1	yes	characteristic, complemented
219	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
220	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
221	#525	C₂²×C₅	1	1	yes	characteristic
222	#452	C₄×C₅	1	2	yes	complemented
223	#451	C₄×C₅	1	1	yes	characteristic, complemented
224	#450	C₄×C₅	1	1	yes	characteristic, complemented
225	#449	C₄×C₅	1	2	yes	complemented
226	#448	C₄×C₅	1	1	yes	characteristic, complemented
227	#447	C₄×C₅	1	1	yes	characteristic, complemented
228	#435	—	2	1	no	—
229	#435	—	2	1	no	—
230	C₈×D₄	—	2	1	no	—
231	C₈×D₄	—	2	1	no	—
232	C₂×#108	C₂²×C₅	1	2	yes	complemented
233	#424	C₂²×C₅	1	1	yes	characteristic, complemented
234	C₂×#104	C₂²×C₅	1	2	yes	—
235	C₂×#104	C₂²×C₅	1	2	yes	—
236	C₂×#104	C₂²×C₅	1	2	yes	complemented
237	#416	C₂²×C₅	1	2	yes	—
238	#408	—	2	1	no	—
239	#408	—	2	1	no	—
240	#408	C₂²×C₅	1	2	yes	—
241	#408	C₂²×C₅	1	2	yes	—
242	#408	C₂²×C₅	1	1	yes	characteristic, complemented
243	#407	C₂²×C₅	1	1	yes	characteristic
244	C₂×C₂²⋊C₈	—	2	1	no	—
245	C₂×C₂²⋊C₈	—	2	1	no	—
246	C₂×C₂²⋊C₈	C₂²×C₅	1	1	yes	characteristic
247	#405	C₂²×C₅	1	1	yes	characteristic, complemented
248	C₂×C₄×C₈	C₂²×C₅	1	2	yes	complemented
249	C₄×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
250	#341	C₂²×C₅	1	2	yes	complemented
251	#339	C₂²×C₅	1	2	yes	complemented
252	#336	C₂²×C₅	1	2	yes	complemented
253	C₂³×C₅	—	2	2	no	—
254	C₂³×C₅	—	2	2	no	—
255	C₂³×C₅	—	2	1	no	—
256	C₂³×C₅	—	2	2	no	—
257	C₂³×C₅	—	4	1	no	—
258	C₂³×C₅	—	4	2	no	—
259	C₂³×C₅	—	4	2	no	—
260	C₂³×C₅	—	4	2	no	—
261	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
262	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
263	C₂³×C₅	C₂×#41	1	1	yes	characteristic
264	C₅×Q₈	—	2	2	no	—
265	C₅×Q₈	—	2	2	no	—
266	C₅×Q₈	—	2	2	no	—
267	C₅×Q₈	—	4	2	no	—
268	C₅×D₄	—	2	2	no	—
269	C₅×D₄	—	2	2	no	—
270	C₅×D₄	—	2	2	no	—
271	C₅×D₄	—	4	2	no	—
272	C₅×D₄	—	4	2	no	—
273	C₅×D₄	—	4	4	no	—
274	C₅×D₄	—	4	2	no	—
275	C₅×D₄	—	4	2	no	—
276	C₅×D₄	—	4	2	no	—
277	C₅×D₄	—	4	2	no	—
278	C₂×C₄×C₅	—	2	2	no	—
279	C₂×C₄×C₅	—	2	2	no	—
280	C₂×C₄×C₅	—	2	2	no	—
281	C₂×C₄×C₅	—	2	1	no	—
282	C₂×C₄×C₅	—	2	1	no	—
283	C₂×C₄×C₅	—	2	1	no	—
284	C₂×C₄×C₅	—	2	1	no	—
285	C₂×C₄×C₅	—	2	4	no	—
286	C₂×C₄×C₅	—	2	1	no	—
287	C₂×C₄×C₅	—	2	1	no	—
288	C₂×C₄×C₅	—	2	1	no	—
289	C₂×C₄×C₅	—	2	2	no	—
290	C₂×C₄×C₅	—	4	4	no	—
291	C₂×C₄×C₅	—	4	2	no	—
292	C₂×C₄×C₅	—	4	2	no	—
293	C₂×C₄×C₅	—	4	2	no	—
294	C₂×C₄×C₅	—	4	1	no	—
295	C₂×C₄×C₅	—	4	2	no	—
296	C₂×C₄×C₅	—	4	2	no	—
297	C₂×C₄×C₅	—	4	1	no	—
298	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
299	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
300	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
301	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
302	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
303	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
304	C₅×C₈	—	2	2	no	—
305	C₅×C₈	—	2	2	no	—
306	C₅×C₈	—	2	1	no	—
307	C₅×C₈	—	2	1	no	—
308	C₅×C₈	—	2	1	no	—
309	C₅×C₈	—	2	1	no	—
310	C₅×C₈	—	4	1	no	—
311	C₅×C₈	—	4	2	no	—
312	C₅×C₈	—	4	1	no	—
313	C₂×#41	—	2	1	no	—
314	C₂×#41	—	2	1	no	—
315	C₂×#41	—	4	2	no	—
316	C₂×#41	—	4	2	no	—
317	C₂×#41	C₂×C₄×C₅	1	2	yes	—
318	C₂×#41	C₂³×C₅	1	1	yes	characteristic
319	C₂×#41	C₂³×C₅	1	2	yes	—
320	C₂²×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
321	C₂²×D₄	—	2	1	no	—
322	C₂²×D₄	C₂×C₄×C₅	1	1	yes	characteristic
323	C₂³×C₄	—	2	1	no	—
324	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
325	#135	—	4	2	no	—
326	#135	—	4	2	no	—
327	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
328	C₂×SD₁₆	C₂×C₄×C₅	1	2	yes	—
329	C₂×SD₁₆	C₂×C₄×C₅	1	2	yes	—
330	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
331	#131	—	4	2	no	—
332	C₂×M₁₆	—	2	1	no	—
333	C₂×M₁₆	—	2	1	no	—
334	C₂×M₁₆	—	4	2	no	—
335	C₂×M₁₆	C₂³×C₅	1	2	yes	—
336	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
337	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
338	C₂²×C₈	—	2	1	no	—
339	C₂²×C₈	—	2	1	no	—
340	C₂²×C₈	—	4	2	no	—
341	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
342	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
343	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
344	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
345	C₂²×C₈	C₂³×C₅	1	2	yes	—
346	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
347	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
348	#123	—	2	1	no	—
349	C₂²⋊Q₈	C₂×C₄×C₅	1	1	yes	characteristic
350	C₂²⋊Q₈	C₂×C₄×C₅	1	1	yes	characteristic
351	#121	C₂×C₄×C₅	1	1	yes	characteristic
352	#121	C₂×C₄×C₅	1	1	yes	characteristic
353	#120	—	2	1	no	—
354	C₄×D₄	—	2	1	no	—
355	C₄×D₄	—	2	1	no	—
356	#117	C₂³×C₅	1	1	yes	characteristic
357	C₂×C₂²⋊C₄	—	2	1	no	—
358	C₂×C₄²	C₂³×C₅	1	2	yes	—
359	#108	—	4	2	no	—
360	C₄⋊C₈	—	2	1	no	—
361	C₄⋊C₈	—	2	1	no	—
362	#104	—	4	2	no	—
363	#104	—	4	2	no	—
364	#104	—	4	2	no	—
365	#103	C₂×C₄×C₅	1	2	yes	—
366	#103	C₂×C₄×C₅	1	2	yes	—
367	#102	C₂×C₄×C₅	1	2	yes	—
368	#102	C₂×C₄×C₅	1	2	yes	—
369	C₂²⋊C₈	—	2	1	no	—
370	C₂²⋊C₈	—	2	1	no	—
371	C₂²⋊C₈	—	2	2	no	—
372	C₂²⋊C₈	—	2	2	no	—
373	C₂²⋊C₈	—	2	2	no	—
374	C₂²⋊C₈	—	2	2	no	—
375	C₂²⋊C₈	—	2	1	no	—
376	C₂²⋊C₈	—	2	1	no	—
377	C₂²⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
378	C₂²⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
379	C₂²⋊C₈	C₂×C₄×C₅	1	2	yes	—
380	#97	—	2	1	no	—
381	C₄×C₈	—	2	1	no	—
382	C₄×C₈	—	4	2	no	—
383	#95	—	2	1	no	—
384	C₂²×C₅	—	2	1	no	—
385	C₂²×C₅	—	2	1	no	—
386	C₂²×C₅	—	2	1	no	—
387	C₂²×C₅	—	2	1	no	—
388	C₂²×C₅	—	2	1	no	—
389	C₂²×C₅	—	2	1	no	—
390	C₂²×C₅	—	2	1	no	—
391	C₂²×C₅	—	4	4	no	—
392	C₂²×C₅	—	4	2	no	—
393	C₂²×C₅	—	4	1	no	—
394	C₂²×C₅	—	4	1	no	—
395	C₂²×C₅	—	4	2	no	—
396	C₂²×C₅	—	4	1	no	—
397	C₂²×C₅	—	4	2	no	—
398	C₂²×C₅	—	8	4	no	—
399	C₂²×C₅	#443	1	2	yes	—
400	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
401	C₂²×C₅	#538	1	2	yes	—
402	C₄×C₅	—	2	1	no	—
403	C₄×C₅	—	2	1	no	—
404	C₄×C₅	—	4	4	no	—
405	C₄×C₅	—	4	2	no	—
406	C₄×C₅	—	4	1	no	—
407	C₄×C₅	—	4	1	no	—
408	C₄×C₅	—	4	1	no	—
409	C₄×C₅	#518	1	2	yes	—
410	C₄×C₅	#534	1	1	yes	characteristic
411	C₄×C₅	#535	1	1	yes	characteristic
412	C₂⁴	—	2	1	no	—
413	C₂⁴	C₅×#41	1	1	yes	characteristic
414	#41	—	2	2	no	—
415	#41	—	2	2	no	—
416	#41	—	2	2	no	—
417	#41	—	2	2	no	—
418	#41	—	2	2	no	—
419	#41	—	2	2	no	—
420	#41	—	4	2	no	—
421	#41	—	4	2	no	—
422	#41	—	4	4	no	—
423	#41	—	4	2	no	—
424	#41	—	4	2	no	—
425	C₂×Q₈	—	2	1	no	—
426	C₂×Q₈	—	4	2	no	—
427	C₂×Q₈	C₂²×C₄×C₅	1	2	yes	—
428	C₂×Q₈	C₂²×C₄×C₅	1	1	yes	characteristic
429	C₂×D₄	—	2	1	no	—
430	C₂×D₄	—	2	1	no	—
431	C₂×D₄	—	2	2	no	—
432	C₂×D₄	—	2	1	no	—
433	C₂×D₄	—	2	2	no	—
434	C₂×D₄	—	2	1	no	—
435	C₂×D₄	—	4	2	no	—
436	C₂×D₄	—	4	4	no	—
437	C₂×D₄	C₂²×C₄×C₅	1	2	yes	—
438	C₂×D₄	C₂²×C₄×C₅	1	1	yes	characteristic
439	C₂²×C₄	—	2	2	no	—
440	C₂²×C₄	—	2	2	no	—
441	C₂²×C₄	—	2	1	no	—
442	C₂²×C₄	—	2	2	no	—
443	C₂²×C₄	—	2	2	no	—
444	C₂²×C₄	—	2	2	no	—
445	C₂²×C₄	—	2	1	no	—
446	C₂²×C₄	—	4	2	no	—
447	C₂²×C₄	—	4	2	no	—
448	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
449	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
450	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
451	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
452	C₂²×C₄	C₅×#41	1	1	yes	characteristic
453	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
454	Q₁₆	—	4	2	no	—
455	SD₁₆	—	4	2	no	—
456	SD₁₆	—	4	2	no	—
457	D₈	—	4	2	no	—
458	M₁₆	—	2	2	no	—
459	M₁₆	—	2	2	no	—
460	M₁₆	—	2	2	no	—
461	M₁₆	—	2	2	no	—
462	M₁₆	—	4	2	no	—
463	M₁₆	—	4	2	no	—
464	C₂×C₈	—	2	2	no	—
465	C₂×C₈	—	2	2	no	—
466	C₂×C₈	—	2	2	no	—
467	C₂×C₈	—	2	2	no	—
468	C₂×C₈	—	2	1	no	—
469	C₂×C₈	—	2	2	no	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	—	4	1	no	—
472	C₂×C₈	—	4	1	no	—
473	C₂×C₈	—	4	1	no	—
474	C₂×C₈	—	4	1	no	—
475	C₂×C₈	—	4	2	no	—
476	C₂×C₈	—	4	2	no	—
477	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
478	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
479	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
480	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
481	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
482	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	C₂²×C₄×C₅	1	1	yes	characteristic
486	C₄⋊C₄	C₂²×C₄×C₅	1	1	yes	characteristic
487	C₂²⋊C₄	—	2	4	no	—
488	C₂²⋊C₄	—	2	1	no	—
489	C₂²⋊C₄	—	2	1	no	—
490	C₂²⋊C₄	—	2	1	no	—
491	C₂²⋊C₄	—	2	1	no	—
492	C₂²⋊C₄	—	2	1	no	—
493	C₄²	—	2	1	no	—
494	C₄²	—	2	2	no	—
495	C₄²	—	2	2	no	—
496	C₂×C₅	—	2	2	no	—
497	C₂×C₅	—	2	2	no	—
498	C₂×C₅	—	4	1	no	—
499	C₂×C₅	—	8	1	no	—
500	C₂×C₅	—	8	2	no	—
501	C₂×C₅	#2321	1	1	yes	characteristic
502	C₂×C₅	C₂×#443	1	1	yes	characteristic
503	C₂×C₅	#2955	1	1	yes	characteristic
504	C₂³	—	2	2	no	—
505	C₂³	—	2	2	no	—
506	C₂³	—	2	1	no	—
507	C₂³	—	2	2	no	—
508	C₂³	—	4	1	no	—
509	C₂³	—	4	2	no	—
510	C₂³	—	4	2	no	—
511	C₂³	—	4	2	no	—
512	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
513	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
514	C₂³	C₂×C₅×#41	1	1	yes	characteristic
515	Q₈	—	2	2	no	—
516	Q₈	—	2	2	no	—
517	Q₈	—	2	2	no	—
518	Q₈	—	4	2	no	—
519	D₄	—	2	2	no	—
520	D₄	—	2	2	no	—
521	D₄	—	2	2	no	—
522	D₄	—	4	2	no	—
523	D₄	—	4	4	no	—
524	D₄	—	4	2	no	—
525	D₄	—	4	2	no	—
526	D₄	—	4	2	no	—
527	D₄	—	4	2	no	—
528	D₄	—	4	2	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	4	4	no	—
542	C₂×C₄	—	4	2	no	—
543	C₂×C₄	—	4	2	no	—
544	C₂×C₄	—	4	2	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	2	no	—
547	C₂×C₄	—	4	2	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
550	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
551	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
552	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
553	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
554	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
555	C₈	—	2	2	no	—
556	C₈	—	2	2	no	—
557	C₈	—	2	1	no	—
558	C₈	—	2	1	no	—
559	C₈	—	2	1	no	—
560	C₈	—	2	1	no	—
561	C₈	—	4	1	no	—
562	C₈	—	4	2	no	—
563	C₈	—	4	1	no	—
564	C₅	#22060	1	1	yes	characteristic, direct product factor, Sylow
565	C₂²	—	2	1	no	—
566	C₂²	—	2	1	no	—
567	C₂²	—	2	1	no	—
568	C₂²	—	2	1	no	—
569	C₂²	—	2	1	no	—
570	C₂²	—	2	1	no	—
571	C₂²	—	2	1	no	—
572	C₂²	—	4	4	no	—
573	C₂²	—	4	2	no	—
574	C₂²	—	4	1	no	—
575	C₂²	—	4	1	no	—
576	C₂²	—	4	2	no	—
577	C₂²	—	4	1	no	—
578	C₂²	—	4	2	no	—
579	C₂²	—	8	4	no	—
580	C₂²	C₅×#443	1	2	yes	—
581	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
582	C₂²	C₅×#538	1	2	yes	—
583	C₄	—	2	1	no	—
584	C₄	—	2	1	no	—
585	C₄	—	4	4	no	—
586	C₄	—	4	2	no	—
587	C₄	—	4	1	no	—
588	C₄	—	4	1	no	—
589	C₄	—	4	1	no	—
590	C₄	C₅×#518	1	2	yes	—
591	C₄	C₅×#534	1	1	yes	characteristic
592	C₄	C₅×#535	1	1	yes	characteristic
593	C₂	—	2	2	no	—
594	C₂	—	2	2	no	—
595	C₂	—	4	1	no	—
596	C₂	—	8	2	no	—
597	C₂	—	8	1	no	—
598	C₂	C₅×#2321	1	1	yes	characteristic
599	C₂	C₂×C₅×#443	1	1	yes	characteristic
600	C₂	C₅×#2955	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	C₂×C₄×D₄
prime radical	C₅×#3002	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2929	C₂	semidirect
3. C₂×C₅×#408	C₂	semidirect
4. C₅×#1974	C₂	semidirect
5. C₅×#1937	C₂	semidirect
6. C₅×#1929	C₂	semidirect
7. C₅×#1876	C₂	semidirect
8. C₅×#1871	C₂	semidirect
9. C₅×#1855	C₂	semidirect
10. C₅×#1789	C₂	semidirect
11. C₅×C₈×C₂²⋊C₄	C₂	semidirect
14. C₂×C₅×#131	C₂²	semidirect
18. C₅×#452	C₄	semidirect
19. C₅×#451	C₄	semidirect
20. C₅×#450	C₄	semidirect
21. C₅×#449	C₄	semidirect
22. C₅×#448	C₄	semidirect
23. C₅×#447	C₄	semidirect
28. C₂×C₅×#108	C₂²	semidirect
29. C₅×#424	C₂²	semidirect
32. C₂×C₅×#104	C₂²	semidirect
38. C₅×#408	C₂²	semidirect
43. C₅×#405	C₂²	semidirect
44. C₂×C₄×C₅×C₈	C₂²	semidirect
46. C₅×#341	C₂²	semidirect
47. C₅×#339	C₂²	semidirect
48. C₅×#336	C₂²	semidirect
49. #22060	C₅	direct (N × H)
122. #2929	C₂×C₅	semidirect
123. C₂×#408	C₂×C₅	semidirect
124. #1974	C₂×C₅	semidirect
125. #1937	C₂×C₅	semidirect
126. #1929	C₂×C₅	semidirect
127. #1876	C₂×C₅	semidirect
128. #1871	C₂×C₅	semidirect
129. #1855	C₂×C₅	semidirect
130. #1789	C₂×C₅	semidirect
131. C₈×C₂²⋊C₄	C₂×C₅	semidirect
218. C₂×#131	C₂²×C₅	semidirect
222. #452	C₄×C₅	semidirect
223. #451	C₄×C₅	semidirect
224. #450	C₄×C₅	semidirect
225. #449	C₄×C₅	semidirect
226. #448	C₄×C₅	semidirect
227. #447	C₄×C₅	semidirect
232. C₂×#108	C₂²×C₅	semidirect
233. #424	C₂²×C₅	semidirect
236. C₂×#104	C₂²×C₅	semidirect
242. #408	C₂²×C₅	semidirect
247. #405	C₂²×C₅	semidirect
248. C₂×C₄×C₈	C₂²×C₅	semidirect
250. #341	C₂²×C₅	semidirect
251. #339	C₂²×C₅	semidirect
252. #336	C₂²×C₅	semidirect
564. C₅	#22060	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×#3002	2
C₅×#2929	2
C₂×C₅×#408	2
C₅×#1974	2
C₅×#1937	2
C₅×#1929	2
C₅×#1876	2
C₅×#1871	2
C₅×#1855	2
C₅×#1789	2
C₅×C₈×C₂²⋊C₄	2
#22060	5

Subgroups — C5×#22060 #11944780

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22060 #11944780