Subgroups — C₅×#22783 #11945503

Order 1280.

752 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#135	C₂	1	1	yes	maximal, characteristic
2	C₅×#2925	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2906	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1937	C₂	1	2	yes	maximal, complemented
5	C₅×#1922	C₂	1	2	yes	maximal, complemented
6	C₅×#1876	C₂	1	2	yes	maximal
7	C₅×#1837	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1806	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1806	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#1764	C₂	1	2	yes	maximal, complemented
12	C₂×C₅×#135	C₂²	1	1	yes	characteristic
13	C₂×C₅×#131	C₂²	1	2	yes	—
14	C₄×C₅×#41	C₂²	1	1	yes	characteristic
15	C₄×C₅×#41	C₂²	1	1	yes	characteristic
16	C₄×C₅×Q₁₆	C₄	1	1	yes	characteristic, complemented
17	C₄×C₅×Q₁₆	C₄	1	1	yes	characteristic, complemented
18	C₄×C₅×SD₁₆	C₄	1	1	yes	characteristic, complemented
19	C₄×C₅×SD₁₆	C₄	1	1	yes	characteristic, complemented
20	C₄×C₅×SD₁₆	C₄	1	1	yes	characteristic, complemented
21	C₄×C₅×SD₁₆	C₄	1	1	yes	characteristic, complemented
22	C₄×C₅×D₈	C₄	1	1	yes	characteristic, complemented
23	C₄×C₅×D₈	C₄	1	1	yes	characteristic, complemented
24	C₅×#433	—	2	2	no	—
25	C₅×#432	—	2	1	no	—
26	C₅×#431	—	2	1	no	—
27	C₅×#427	C₂²	1	1	yes	characteristic
28	C₅×#424	—	2	2	no	—
29	C₅×#424	C₂²	1	2	yes	—
30	C₅×#423	—	2	1	no	—
31	C₅×#423	C₂²	1	1	yes	characteristic
32	C₅×#423	C₂²	1	1	yes	characteristic
33	C₂×C₅×C₄⋊C₈	—	2	1	no	—
34	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₂×C₅×#104	C₂²	1	2	yes	—
36	C₂×C₅×#104	C₂²	1	2	yes	—
37	C₅×#416	C₂²	1	1	yes	characteristic
38	C₅×#416	C₂²	1	1	yes	characteristic
39	C₅×#408	C₂²	1	2	yes	—
40	C₅×#405	C₂²	1	2	yes	—
41	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
42	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
43	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
44	C₅×#341	C₂²	1	2	yes	—
45	C₅×#341	C₂²	1	2	yes	—
46	C₅×#339	C₂²	1	2	yes	—
47	C₅×#339	C₂²	1	2	yes	—
48	C₅×#336	C₂²	1	2	yes	—
49	#22783	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂×C₅×#41	C₂³	1	1	yes	characteristic
51	C₂×C₅×#41	C₂³	1	1	yes	characteristic
52	C₅×#135	—	2	2	no	—
53	C₅×#135	—	2	2	no	—
54	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
55	C₂×C₅×SD₁₆	C₂×C₄	1	1	yes	characteristic
56	C₂×C₅×SD₁₆	C₂×C₄	1	1	yes	characteristic
57	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
58	C₅×#131	—	2	4	no	—
59	C₅×#131	—	4	2	no	—
60	C₂×C₅×M₁₆	—	2	2	no	—
61	C₂×C₅×M₁₆	C₂³	1	2	yes	—
62	C₂×C₅×M₁₆	C₂³	1	2	yes	—
63	C₂²×C₅×C₈	—	2	2	no	—
64	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
65	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
66	C₂²×C₅×C₈	C₂³	1	2	yes	—
67	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
68	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
69	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
70	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
71	C₄×C₅×D₄	—	2	1	no	—
72	C₄×C₅×D₄	—	2	1	no	—
73	C₄×C₅×D₄	—	2	1	no	—
74	C₄×C₅×D₄	—	2	1	no	—
75	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
76	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
77	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
78	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
79	C₅×#117	—	2	1	no	—
80	C₅×#117	—	2	1	no	—
81	C₅×#117	C₂³	1	1	yes	characteristic
82	C₅×#117	C₂³	1	1	yes	characteristic
83	C₂×C₅×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄²×C₅	—	2	1	no	—
85	C₂×C₄²×C₅	—	2	1	no	—
86	C₂×C₄²×C₅	C₂³	1	2	yes	—
87	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
88	C₅×#107	C₂×C₄	1	1	yes	characteristic
89	C₅×#107	C₂×C₄	1	1	yes	characteristic
90	C₅×#106	C₂×C₄	1	1	yes	characteristic
91	C₅×#106	C₂×C₄	1	1	yes	characteristic
92	C₅×C₄⋊C₈	—	2	2	no	—
93	C₅×C₄⋊C₈	—	2	2	no	—
94	C₅×C₄⋊C₈	—	2	1	no	—
95	C₅×C₄⋊C₈	—	2	1	no	—
96	C₅×C₄⋊C₈	—	2	1	no	—
97	C₅×C₄⋊C₈	—	2	1	no	—
98	C₅×C₄⋊C₈	D₄	1	1	yes	characteristic
99	C₅×C₄⋊C₈	D₄	1	1	yes	characteristic
100	C₅×C₄⋊C₈	D₄	1	2	yes	—
101	C₅×#104	—	2	4	no	—
102	C₅×#104	—	4	2	no	—
103	C₅×#103	C₂×C₄	1	1	yes	characteristic
104	C₅×#103	C₂×C₄	1	1	yes	characteristic
105	C₅×#103	C₂×C₄	1	1	yes	characteristic
106	C₅×#103	C₂×C₄	1	1	yes	characteristic
107	C₅×#102	C₂×C₄	1	1	yes	characteristic
108	C₅×#102	C₂×C₄	1	1	yes	characteristic
109	C₅×#102	C₂×C₄	1	1	yes	characteristic
110	C₅×#102	C₂×C₄	1	1	yes	characteristic
111	C₅×C₂²⋊C₈	—	2	2	no	—
112	C₅×C₂²⋊C₈	—	2	2	no	—
113	C₅×#97	—	2	2	no	—
114	C₄×C₅×C₈	—	2	2	no	—
115	C₄×C₅×C₈	—	4	2	no	—
116	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
117	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
118	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
119	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
120	C₅×#95	—	2	1	no	—
121	C₄×#135	C₂×C₅	1	1	yes	characteristic
122	#2925	C₂×C₅	1	1	yes	characteristic, complemented
123	#2906	C₂×C₅	1	1	yes	characteristic, complemented
124	#1937	C₂×C₅	1	2	yes	complemented
125	#1922	C₂×C₅	1	2	yes	complemented
126	#1876	C₂×C₅	1	2	yes	—
127	#1837	C₂×C₅	1	1	yes	characteristic, complemented
128	#1806	C₂×C₅	1	1	yes	characteristic, complemented
129	#1806	C₂×C₅	1	1	yes	characteristic, complemented
130	C₈×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
131	#1764	C₂×C₅	1	2	yes	complemented
132	C₅×#41	—	2	1	no	—
133	C₅×#41	—	2	1	no	—
134	C₅×#41	—	2	1	no	—
135	C₅×#41	—	2	1	no	—
136	C₅×#41	—	2	1	no	—
137	C₅×#41	—	2	1	no	—
138	C₅×#41	#41	1	2	yes	—
139	C₅×#41	#41	1	2	yes	—
140	C₂×C₅×Q₈	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₅×Q₈	C₂²×C₄	1	1	yes	characteristic
142	C₂×C₅×D₄	—	2	1	no	—
143	C₂×C₅×D₄	—	2	1	no	—
144	C₂×C₅×D₄	C₂²×C₄	1	1	yes	characteristic
145	C₂×C₅×D₄	C₂²×C₄	1	1	yes	characteristic
146	C₂²×C₄×C₅	—	2	1	no	—
147	C₂²×C₄×C₅	—	2	2	no	—
148	C₂²×C₄×C₅	—	2	1	no	—
149	C₂²×C₄×C₅	—	2	1	no	—
150	C₂²×C₄×C₅	—	2	1	no	—
151	C₂²×C₄×C₅	—	2	1	no	—
152	C₂²×C₄×C₅	—	2	1	no	—
153	C₂²×C₄×C₅	#41	1	1	yes	characteristic
154	C₂²×C₄×C₅	#41	1	1	yes	characteristic
155	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
156	C₅×Q₁₆	—	2	2	no	—
157	C₅×SD₁₆	—	2	2	no	—
158	C₅×SD₁₆	—	2	2	no	—
159	C₅×D₈	—	2	2	no	—
160	C₅×M₁₆	—	2	2	no	—
161	C₅×M₁₆	—	2	2	no	—
162	C₅×M₁₆	—	2	2	no	—
163	C₅×M₁₆	—	2	2	no	—
164	C₅×M₁₆	—	4	4	no	—
165	C₂×C₅×C₈	—	2	2	no	—
166	C₂×C₅×C₈	—	2	1	no	—
167	C₂×C₅×C₈	—	2	1	no	—
168	C₂×C₅×C₈	—	2	2	no	—
169	C₂×C₅×C₈	—	2	1	no	—
170	C₂×C₅×C₈	—	2	1	no	—
171	C₂×C₅×C₈	—	2	2	no	—
172	C₂×C₅×C₈	—	2	2	no	—
173	C₂×C₅×C₈	—	4	4	no	—
174	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
175	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
176	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
177	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
178	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
179	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
180	C₅×C₄⋊C₄	—	2	4	no	—
181	C₅×C₄⋊C₄	—	2	1	no	—
182	C₅×C₄⋊C₄	—	2	1	no	—
183	C₅×C₄⋊C₄	—	2	1	no	—
184	C₅×C₄⋊C₄	—	2	1	no	—
185	C₅×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
186	C₅×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
187	C₅×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
188	C₅×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
189	C₅×C₂²⋊C₄	—	2	1	no	—
190	C₅×C₂²⋊C₄	—	2	1	no	—
191	C₅×C₂²⋊C₄	—	2	1	no	—
192	C₅×C₂²⋊C₄	—	2	1	no	—
193	C₅×C₂²⋊C₄	—	2	1	no	—
194	C₅×C₂²⋊C₄	—	2	1	no	—
195	C₄²×C₅	—	2	1	no	—
196	C₄²×C₅	—	2	1	no	—
197	C₄²×C₅	—	2	1	no	—
198	C₄²×C₅	—	2	1	no	—
199	C₄²×C₅	—	2	1	no	—
200	C₄²×C₅	—	2	1	no	—
201	C₄²×C₅	—	2	2	no	—
202	C₄²×C₅	—	2	2	no	—
203	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
204	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
205	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
206	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
207	C₂×#135	C₂²×C₅	1	1	yes	characteristic
208	C₂×#131	C₂²×C₅	1	2	yes	—
209	C₄×#41	C₂²×C₅	1	1	yes	characteristic
210	C₄×#41	C₂²×C₅	1	1	yes	characteristic
211	C₄×Q₁₆	C₄×C₅	1	1	yes	characteristic, complemented
212	C₄×Q₁₆	C₄×C₅	1	1	yes	characteristic, complemented
213	C₄×SD₁₆	C₄×C₅	1	1	yes	characteristic, complemented
214	C₄×SD₁₆	C₄×C₅	1	1	yes	characteristic, complemented
215	C₄×SD₁₆	C₄×C₅	1	1	yes	characteristic, complemented
216	C₄×SD₁₆	C₄×C₅	1	1	yes	characteristic, complemented
217	C₄×D₈	C₄×C₅	1	1	yes	characteristic, complemented
218	C₄×D₈	C₄×C₅	1	1	yes	characteristic, complemented
219	#433	—	2	2	no	—
220	#432	—	2	1	no	—
221	#431	—	2	1	no	—
222	#427	C₂²×C₅	1	1	yes	characteristic
223	#424	—	2	2	no	—
224	#424	C₂²×C₅	1	2	yes	—
225	#423	—	2	1	no	—
226	#423	C₂²×C₅	1	1	yes	characteristic
227	#423	C₂²×C₅	1	1	yes	characteristic
228	C₂×C₄⋊C₈	—	2	1	no	—
229	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
230	C₂×#104	C₂²×C₅	1	2	yes	—
231	C₂×#104	C₂²×C₅	1	2	yes	—
232	#416	C₂²×C₅	1	1	yes	characteristic
233	#416	C₂²×C₅	1	1	yes	characteristic
234	#408	C₂²×C₅	1	2	yes	—
235	#405	C₂²×C₅	1	2	yes	—
236	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
237	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
238	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
239	#341	C₂²×C₅	1	2	yes	—
240	#341	C₂²×C₅	1	2	yes	—
241	#339	C₂²×C₅	1	2	yes	—
242	#339	C₂²×C₅	1	2	yes	—
243	#336	C₂²×C₅	1	2	yes	—
244	C₂³×C₅	—	2	1	no	—
245	C₂³×C₅	—	2	1	no	—
246	C₂³×C₅	C₂×#41	1	1	yes	characteristic
247	C₅×Q₈	—	2	1	no	—
248	C₅×Q₈	—	2	1	no	—
249	C₅×Q₈	—	2	1	no	—
250	C₅×Q₈	#117	1	2	yes	—
251	C₅×D₄	—	2	2	no	—
252	C₅×D₄	—	2	2	no	—
253	C₅×D₄	—	2	1	no	—
254	C₅×D₄	—	2	1	no	—
255	C₅×D₄	—	2	1	no	—
256	C₅×D₄	—	4	1	no	—
257	C₅×D₄	—	4	1	no	—
258	C₅×D₄	#117	1	2	yes	—
259	C₂×C₄×C₅	—	2	1	no	—
260	C₂×C₄×C₅	—	2	2	no	—
261	C₂×C₄×C₅	—	2	2	no	—
262	C₂×C₄×C₅	—	2	2	no	—
263	C₂×C₄×C₅	—	2	4	no	—
264	C₂×C₄×C₅	—	2	1	no	—
265	C₂×C₄×C₅	—	2	1	no	—
266	C₂×C₄×C₅	—	2	1	no	—
267	C₂×C₄×C₅	—	2	1	no	—
268	C₂×C₄×C₅	—	2	1	no	—
269	C₂×C₄×C₅	—	4	1	no	—
270	C₂×C₄×C₅	—	4	2	no	—
271	C₂×C₄×C₅	—	4	2	no	—
272	C₂×C₄×C₅	—	4	1	no	—
273	C₂×C₄×C₅	—	4	1	no	—
274	C₂×C₄×C₅	—	4	2	no	—
275	C₂×C₄×C₅	—	4	2	no	—
276	C₂×C₄×C₅	—	8	1	no	—
277	C₂×C₄×C₅	—	8	1	no	—
278	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
279	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
280	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
281	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
282	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
283	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
284	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
285	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
286	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
287	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
288	C₅×C₈	—	2	1	no	—
289	C₅×C₈	—	2	1	no	—
290	C₅×C₈	—	2	1	no	—
291	C₅×C₈	—	2	1	no	—
292	C₅×C₈	—	2	2	no	—
293	C₅×C₈	—	2	2	no	—
294	C₅×C₈	—	4	2	no	—
295	C₅×C₈	—	4	2	no	—
296	C₂×#41	C₂³×C₅	1	1	yes	characteristic
297	C₂×#41	C₂³×C₅	1	1	yes	characteristic
298	#135	—	2	2	no	—
299	#135	—	2	2	no	—
300	C₂×Q₁₆	C₂×C₄×C₅	1	1	yes	characteristic
301	C₂×SD₁₆	C₂×C₄×C₅	1	1	yes	characteristic
302	C₂×SD₁₆	C₂×C₄×C₅	1	1	yes	characteristic
303	C₂×D₈	C₂×C₄×C₅	1	1	yes	characteristic
304	#131	—	2	4	no	—
305	#131	—	4	2	no	—
306	C₂×M₁₆	—	2	2	no	—
307	C₂×M₁₆	C₂³×C₅	1	2	yes	—
308	C₂×M₁₆	C₂³×C₅	1	2	yes	—
309	C₂²×C₈	—	2	2	no	—
310	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
311	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
312	C₂²×C₈	C₂³×C₅	1	2	yes	—
313	C₄×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
314	C₄×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
315	C₄×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
316	C₄×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
317	C₄×D₄	—	2	1	no	—
318	C₄×D₄	—	2	1	no	—
319	C₄×D₄	—	2	1	no	—
320	C₄×D₄	—	2	1	no	—
321	C₄×D₄	C₂×C₄×C₅	1	1	yes	characteristic
322	C₄×D₄	C₂×C₄×C₅	1	1	yes	characteristic
323	C₄×D₄	C₂×C₄×C₅	1	1	yes	characteristic
324	C₄×D₄	C₂×C₄×C₅	1	1	yes	characteristic
325	#117	—	2	1	no	—
326	#117	—	2	1	no	—
327	#117	C₂³×C₅	1	1	yes	characteristic
328	#117	C₂³×C₅	1	1	yes	characteristic
329	C₂×C₄⋊C₄	—	2	1	no	—
330	C₂×C₄²	—	2	1	no	—
331	C₂×C₄²	—	2	1	no	—
332	C₂×C₄²	C₂³×C₅	1	2	yes	—
333	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
334	#107	C₂×C₄×C₅	1	1	yes	characteristic
335	#107	C₂×C₄×C₅	1	1	yes	characteristic
336	#106	C₂×C₄×C₅	1	1	yes	characteristic
337	#106	C₂×C₄×C₅	1	1	yes	characteristic
338	C₄⋊C₈	—	2	2	no	—
339	C₄⋊C₈	—	2	2	no	—
340	C₄⋊C₈	—	2	1	no	—
341	C₄⋊C₈	—	2	1	no	—
342	C₄⋊C₈	—	2	1	no	—
343	C₄⋊C₈	—	2	1	no	—
344	C₄⋊C₈	C₅×D₄	1	1	yes	characteristic
345	C₄⋊C₈	C₅×D₄	1	1	yes	characteristic
346	C₄⋊C₈	C₅×D₄	1	2	yes	—
347	#104	—	2	4	no	—
348	#104	—	4	2	no	—
349	#103	C₂×C₄×C₅	1	1	yes	characteristic
350	#103	C₂×C₄×C₅	1	1	yes	characteristic
351	#103	C₂×C₄×C₅	1	1	yes	characteristic
352	#103	C₂×C₄×C₅	1	1	yes	characteristic
353	#102	C₂×C₄×C₅	1	1	yes	characteristic
354	#102	C₂×C₄×C₅	1	1	yes	characteristic
355	#102	C₂×C₄×C₅	1	1	yes	characteristic
356	#102	C₂×C₄×C₅	1	1	yes	characteristic
357	C₂²⋊C₈	—	2	2	no	—
358	C₂²⋊C₈	—	2	2	no	—
359	#97	—	2	2	no	—
360	C₄×C₈	—	2	2	no	—
361	C₄×C₈	—	4	2	no	—
362	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
363	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
364	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
365	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
366	#95	—	2	1	no	—
367	C₂²×C₅	—	2	1	no	—
368	C₂²×C₅	—	2	1	no	—
369	C₂²×C₅	—	2	2	no	—
370	C₂²×C₅	—	4	1	no	—
371	C₂²×C₅	—	4	1	no	—
372	C₂²×C₅	—	4	1	no	—
373	C₂²×C₅	—	4	1	no	—
374	C₂²×C₅	—	4	1	no	—
375	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
376	C₂²×C₅	#552	1	1	yes	characteristic
377	C₂²×C₅	#553	1	1	yes	characteristic
378	C₄×C₅	—	2	2	no	—
379	C₄×C₅	—	2	1	no	—
380	C₄×C₅	—	2	1	no	—
381	C₄×C₅	—	2	1	no	—
382	C₄×C₅	—	4	4	no	—
383	C₄×C₅	—	4	1	no	—
384	C₄×C₅	—	4	1	no	—
385	C₄×C₅	—	4	1	no	—
386	C₄×C₅	—	4	1	no	—
387	C₄×C₅	—	4	1	no	—
388	C₄×C₅	#443	1	2	yes	—
389	C₄×C₅	C₂×#117	1	1	yes	characteristic
390	C₄×C₅	#520	1	1	yes	characteristic
391	C₄×C₅	#546	1	1	yes	characteristic
392	C₄×C₅	#548	1	1	yes	characteristic
393	#41	—	2	1	no	—
394	#41	—	2	1	no	—
395	#41	—	2	1	no	—
396	#41	—	2	1	no	—
397	#41	—	2	1	no	—
398	#41	—	2	1	no	—
399	#41	C₅×#41	1	2	yes	—
400	#41	C₅×#41	1	2	yes	—
401	C₂×Q₈	C₂²×C₄×C₅	1	1	yes	characteristic
402	C₂×Q₈	C₂²×C₄×C₅	1	1	yes	characteristic
403	C₂×D₄	—	2	1	no	—
404	C₂×D₄	—	2	1	no	—
405	C₂×D₄	C₂²×C₄×C₅	1	1	yes	characteristic
406	C₂×D₄	C₂²×C₄×C₅	1	1	yes	characteristic
407	C₂²×C₄	—	2	1	no	—
408	C₂²×C₄	—	2	2	no	—
409	C₂²×C₄	—	2	1	no	—
410	C₂²×C₄	—	2	1	no	—
411	C₂²×C₄	—	2	1	no	—
412	C₂²×C₄	—	2	1	no	—
413	C₂²×C₄	—	2	1	no	—
414	C₂²×C₄	C₅×#41	1	1	yes	characteristic
415	C₂²×C₄	C₅×#41	1	1	yes	characteristic
416	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
417	Q₁₆	—	2	2	no	—
418	SD₁₆	—	2	2	no	—
419	SD₁₆	—	2	2	no	—
420	D₈	—	2	2	no	—
421	M₁₆	—	2	2	no	—
422	M₁₆	—	2	2	no	—
423	M₁₆	—	2	2	no	—
424	M₁₆	—	2	2	no	—
425	M₁₆	—	4	4	no	—
426	C₂×C₈	—	2	2	no	—
427	C₂×C₈	—	2	1	no	—
428	C₂×C₈	—	2	1	no	—
429	C₂×C₈	—	2	1	no	—
430	C₂×C₈	—	2	2	no	—
431	C₂×C₈	—	2	1	no	—
432	C₂×C₈	—	2	2	no	—
433	C₂×C₈	—	2	2	no	—
434	C₂×C₈	—	4	4	no	—
435	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
436	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
437	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
438	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
439	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
440	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
441	C₄⋊C₄	—	2	4	no	—
442	C₄⋊C₄	—	2	1	no	—
443	C₄⋊C₄	—	2	1	no	—
444	C₄⋊C₄	—	2	1	no	—
445	C₄⋊C₄	—	2	1	no	—
446	C₄⋊C₄	C₂²×C₄×C₅	1	1	yes	characteristic
447	C₄⋊C₄	C₂²×C₄×C₅	1	1	yes	characteristic
448	C₄⋊C₄	C₂²×C₄×C₅	1	1	yes	characteristic
449	C₄⋊C₄	C₂²×C₄×C₅	1	1	yes	characteristic
450	C₂²⋊C₄	—	2	1	no	—
451	C₂²⋊C₄	—	2	1	no	—
452	C₂²⋊C₄	—	2	1	no	—
453	C₂²⋊C₄	—	2	1	no	—
454	C₂²⋊C₄	—	2	1	no	—
455	C₂²⋊C₄	—	2	1	no	—
456	C₄²	—	2	1	no	—
457	C₄²	—	2	1	no	—
458	C₄²	—	2	1	no	—
459	C₄²	—	2	1	no	—
460	C₄²	—	2	1	no	—
461	C₄²	—	2	1	no	—
462	C₄²	—	2	2	no	—
463	C₄²	—	2	2	no	—
464	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
465	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
466	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
467	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
468	C₂×C₅	—	2	1	no	—
469	C₂×C₅	—	2	1	no	—
470	C₂×C₅	—	4	2	no	—
471	C₂×C₅	—	8	1	no	—
472	C₂×C₅	#2352	1	1	yes	characteristic
473	C₂×C₅	C₂×#443	1	1	yes	characteristic
474	C₂×C₅	#2957	1	1	yes	characteristic
475	C₂³	—	2	1	no	—
476	C₂³	—	2	1	no	—
477	C₂³	C₂×C₅×#41	1	1	yes	characteristic
478	Q₈	—	2	1	no	—
479	Q₈	—	2	1	no	—
480	Q₈	—	2	1	no	—
481	Q₈	C₅×#117	1	2	yes	—
482	D₄	—	2	2	no	—
483	D₄	—	2	2	no	—
484	D₄	—	2	1	no	—
485	D₄	—	2	1	no	—
486	D₄	—	2	1	no	—
487	D₄	—	4	1	no	—
488	D₄	—	4	1	no	—
489	D₄	C₅×#117	1	2	yes	—
490	C₂×C₄	—	2	1	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	2	1	no	—
496	C₂×C₄	—	2	1	no	—
497	C₂×C₄	—	2	1	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	4	1	no	—
501	C₂×C₄	—	4	2	no	—
502	C₂×C₄	—	4	2	no	—
503	C₂×C₄	—	4	2	no	—
504	C₂×C₄	—	4	2	no	—
505	C₂×C₄	—	4	1	no	—
506	C₂×C₄	—	4	1	no	—
507	C₂×C₄	—	8	1	no	—
508	C₂×C₄	—	8	1	no	—
509	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
510	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
511	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
512	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
513	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
514	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
515	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
516	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
517	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
518	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
519	C₈	—	2	1	no	—
520	C₈	—	2	1	no	—
521	C₈	—	2	1	no	—
522	C₈	—	2	1	no	—
523	C₈	—	2	2	no	—
524	C₈	—	2	2	no	—
525	C₈	—	4	2	no	—
526	C₈	—	4	2	no	—
527	C₅	#22783	1	1	yes	characteristic, direct product factor, Sylow
528	C₂²	—	2	1	no	—
529	C₂²	—	2	1	no	—
530	C₂²	—	2	2	no	—
531	C₂²	—	4	1	no	—
532	C₂²	—	4	1	no	—
533	C₂²	—	4	1	no	—
534	C₂²	—	4	1	no	—
535	C₂²	—	4	1	no	—
536	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
537	C₂²	C₅×#552	1	1	yes	characteristic
538	C₂²	C₅×#553	1	1	yes	characteristic
539	C₄	—	2	2	no	—
540	C₄	—	2	1	no	—
541	C₄	—	2	1	no	—
542	C₄	—	2	1	no	—
543	C₄	—	4	4	no	—
544	C₄	—	4	1	no	—
545	C₄	—	4	1	no	—
546	C₄	—	4	1	no	—
547	C₄	—	4	1	no	—
548	C₄	—	4	1	no	—
549	C₄	C₅×#443	1	2	yes	—
550	C₄	C₂×C₅×#117	1	1	yes	characteristic
551	C₄	C₅×#520	1	1	yes	characteristic
552	C₄	C₅×#546	1	1	yes	characteristic
553	C₄	C₅×#548	1	1	yes	characteristic
554	C₂	—	2	1	no	—
555	C₂	—	2	1	no	—
556	C₂	—	4	2	no	—
557	C₂	—	8	1	no	—
558	C₂	C₅×#2352	1	1	yes	characteristic
559	C₂	C₂×C₅×#443	1	1	yes	characteristic
560	C₂	C₅×#2957	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	C₂×C₅×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₄⋊C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2925	C₂	semidirect
3. C₅×#2906	C₂	semidirect
4. C₅×#1937	C₂	semidirect
5. C₅×#1922	C₂	semidirect
7. C₅×#1837	C₂	semidirect
8. C₅×#1806	C₂	semidirect
9. C₅×#1806	C₂	semidirect
10. C₅×C₈×C₄⋊C₄	C₂	semidirect
11. C₅×#1764	C₂	semidirect
16. C₄×C₅×Q₁₆	C₄	semidirect
17. C₄×C₅×Q₁₆	C₄	semidirect
18. C₄×C₅×SD₁₆	C₄	semidirect
19. C₄×C₅×SD₁₆	C₄	semidirect
20. C₄×C₅×SD₁₆	C₄	semidirect
21. C₄×C₅×SD₁₆	C₄	semidirect
22. C₄×C₅×D₈	C₄	semidirect
23. C₄×C₅×D₈	C₄	semidirect
49. #22783	C₅	direct (N × H)
122. #2925	C₂×C₅	semidirect
123. #2906	C₂×C₅	semidirect
124. #1937	C₂×C₅	semidirect
125. #1922	C₂×C₅	semidirect
127. #1837	C₂×C₅	semidirect
128. #1806	C₂×C₅	semidirect
129. #1806	C₂×C₅	semidirect
130. C₈×C₄⋊C₄	C₂×C₅	semidirect
131. #1764	C₂×C₅	semidirect
211. C₄×Q₁₆	C₄×C₅	semidirect
212. C₄×Q₁₆	C₄×C₅	semidirect
213. C₄×SD₁₆	C₄×C₅	semidirect
214. C₄×SD₁₆	C₄×C₅	semidirect
215. C₄×SD₁₆	C₄×C₅	semidirect
216. C₄×SD₁₆	C₄×C₅	semidirect
217. C₄×D₈	C₄×C₅	semidirect
218. C₄×D₈	C₄×C₅	semidirect
527. C₅	#22783	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₅×#135	2
C₅×#2925	2
C₅×#2906	2
C₅×#1937	2
C₅×#1922	2
C₅×#1876	2
C₅×#1837	2
C₅×#1806	2
C₅×C₈×C₄⋊C₄	2
C₅×#1764	2
#22783	5

Subgroups — C5×#22783 #11945503

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22783 #11945503