Subgroups — C₅×#29830 #11952550

Order 1280.

766 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#477	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#462	C₂	1	1	yes	maximal, characteristic
3	C₅×#2862	C₂	1	1	yes	maximal, characteristic
4	C₅×#2295	C₂	1	1	yes	maximal, characteristic
5	C₅×#2068	C₂	1	4	yes	maximal
6	C₅×#2056	C₂	1	4	yes	maximal
7	C₅×#1987	C₂	1	1	yes	maximal, characteristic
8	C₅×#1981	C₂	1	1	yes	maximal, characteristic
9	C₅×#1842	C₂	1	1	yes	maximal, characteristic
10	C₂²×C₅×Q₁₆	C₂²	1	2	yes	—
11	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
14	C₅×#477	—	2	4	no	—
15	C₅×#462	—	2	4	no	—
16	C₂×C₅×#107	C₂²	1	2	yes	—
17	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
20	C₂×C₅×#103	C₂²	1	2	yes	—
21	C₂×C₅×#103	C₂²	1	2	yes	—
22	C₂×C₅×#103	C₂²	1	4	yes	—
23	C₅×#391	—	2	2	no	—
24	C₅×#391	C₂²	1	2	yes	—
25	C₅×#391	C₂²	1	4	yes	—
26	C₅×#389	C₂²	1	2	yes	—
27	C₅×#387	—	2	2	no	—
28	C₅×#382	—	2	1	no	—
29	C₅×#382	—	2	1	no	—
30	C₄×C₅×C₄⋊C₄	—	2	1	no	—
31	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
32	C₅×#376	—	2	1	no	—
33	C₄³×C₅	C₂²	1	1	yes	characteristic
34	C₅×#340	C₂²	1	4	yes	—
35	#29830	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂²×C₅×Q₈	—	2	1	no	—
37	C₂²×C₅×Q₈	C₂³	1	2	yes	—
38	C₂²×C₅×Q₈	C₂³	1	1	yes	characteristic
39	C₂×C₅×Q₁₆	—	2	4	no	—
40	C₂×C₅×Q₁₆	—	4	4	no	—
41	C₂²×C₅×C₈	C₂³	1	2	yes	—
42	C₂²×C₅×C₈	C₂³	1	2	yes	—
43	C₅×C₄⋊Q₈	—	2	2	no	—
44	C₅×C₄⋊Q₈	—	2	2	no	—
45	C₅×C₄⋊Q₈	—	2	2	no	—
46	C₅×C₄⋊Q₈	—	2	2	no	—
47	C₄×C₅×Q₈	—	2	2	no	—
48	C₄×C₅×Q₈	D₄	1	4	yes	—
49	C₂×C₅×C₄⋊C₄	—	2	1	no	—
50	C₂×C₅×C₄⋊C₄	—	2	2	no	—
51	C₂×C₅×C₄⋊C₄	—	2	1	no	—
52	C₂×C₅×C₄⋊C₄	—	2	1	no	—
53	C₂×C₅×C₄⋊C₄	—	2	1	no	—
54	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
55	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
56	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₅	—	2	1	no	—
58	C₂×C₄²×C₅	—	2	1	no	—
59	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
60	C₂×C₄²×C₅	D₄	1	2	yes	—
61	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
62	C₂×C₄²×C₅	C₂³	1	2	yes	—
63	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
64	C₅×#107	—	2	4	no	—
65	C₅×C₄⋊C₈	—	2	2	no	—
66	C₅×C₄⋊C₈	—	2	2	no	—
67	C₅×C₄⋊C₈	—	4	4	no	—
68	C₅×#103	—	2	4	no	—
69	C₅×#103	—	2	4	no	—
70	C₅×#103	—	4	4	no	—
71	C₅×#95	—	2	2	no	—
72	C₅×#95	—	2	2	no	—
73	C₅×#95	—	2	2	no	—
74	C₅×#95	—	2	1	no	—
75	C₅×#95	—	2	4	no	—
76	C₅×#95	—	2	4	no	—
77	C₂×#477	C₂×C₅	1	1	yes	characteristic
78	C₂×#462	C₂×C₅	1	1	yes	characteristic
79	#2862	C₂×C₅	1	1	yes	characteristic
80	#2295	C₂×C₅	1	1	yes	characteristic
81	#2068	C₂×C₅	1	4	yes	—
82	#2056	C₂×C₅	1	4	yes	—
83	#1987	C₂×C₅	1	1	yes	characteristic
84	#1981	C₂×C₅	1	1	yes	characteristic
85	#1842	C₂×C₅	1	1	yes	characteristic
86	C₂×C₅×Q₈	—	2	1	no	—
87	C₂×C₅×Q₈	—	2	2	no	—
88	C₂×C₅×Q₈	—	2	2	no	—
89	C₂×C₅×Q₈	—	2	2	no	—
90	C₂×C₅×Q₈	—	2	4	no	—
91	C₂×C₅×Q₈	—	2	2	no	—
92	C₂×C₅×Q₈	—	2	4	no	—
93	C₂×C₅×Q₈	—	4	4	no	—
94	C₂×C₅×Q₈	—	4	2	no	—
95	C₂×C₅×Q₈	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₅	—	2	1	no	—
97	C₂²×C₄×C₅	—	2	1	no	—
98	C₂²×C₄×C₅	—	2	2	no	—
99	C₂²×C₄×C₅	—	2	2	no	—
100	C₂²×C₄×C₅	—	2	2	no	—
101	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
106	C₅×Q₁₆	—	4	8	no	—
107	C₂×C₅×C₈	—	2	2	no	—
108	C₂×C₅×C₈	—	2	2	no	—
109	C₂×C₅×C₈	—	2	4	no	—
110	C₂×C₅×C₈	—	2	4	no	—
111	C₅×C₄⋊C₄	—	2	4	no	—
112	C₅×C₄⋊C₄	—	2	1	no	—
113	C₅×C₄⋊C₄	—	2	2	no	—
114	C₅×C₄⋊C₄	—	2	2	no	—
115	C₅×C₄⋊C₄	—	2	2	no	—
116	C₅×C₄⋊C₄	—	2	2	no	—
117	C₅×C₄⋊C₄	—	4	4	no	—
118	C₅×C₄⋊C₄	—	4	2	no	—
119	C₅×C₄⋊C₄	—	4	2	no	—
120	C₅×C₄⋊C₄	—	4	2	no	—
121	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
122	C₄²×C₅	—	2	4	no	—
123	C₄²×C₅	—	2	2	no	—
124	C₄²×C₅	—	2	2	no	—
125	C₄²×C₅	—	2	2	no	—
126	C₄²×C₅	—	4	2	no	—
127	C₄²×C₅	—	4	2	no	—
128	C₄²×C₅	Q₁₆	1	4	yes	complemented
129	C₄²×C₅	C₂×D₄	1	2	yes	—
130	C₄²×C₅	#41	1	2	yes	—
131	C₂²×Q₁₆	C₂²×C₅	1	2	yes	—
132	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
133	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
134	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
135	#477	—	2	4	no	—
136	#462	—	2	4	no	—
137	C₂×#107	C₂²×C₅	1	2	yes	—
138	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
139	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
140	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
141	C₂×#103	C₂²×C₅	1	2	yes	—
142	C₂×#103	C₂²×C₅	1	2	yes	—
143	C₂×#103	C₂²×C₅	1	4	yes	—
144	#391	—	2	2	no	—
145	#391	C₂²×C₅	1	2	yes	—
146	#391	C₂²×C₅	1	4	yes	—
147	#389	C₂²×C₅	1	2	yes	—
148	#387	—	2	2	no	—
149	#382	—	2	1	no	—
150	#382	—	2	1	no	—
151	C₄×C₄⋊C₄	—	2	1	no	—
152	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
153	#376	—	2	1	no	—
154	C₄³	C₂²×C₅	1	1	yes	characteristic
155	#340	C₂²×C₅	1	4	yes	—
156	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
157	C₅×Q₈	—	2	2	no	—
158	C₅×Q₈	—	2	2	no	—
159	C₅×Q₈	—	4	2	no	—
160	C₅×Q₈	—	4	2	no	—
161	C₅×Q₈	—	4	2	no	—
162	C₅×Q₈	—	4	4	no	—
163	C₅×Q₈	—	8	4	no	—
164	C₂×C₄×C₅	—	2	2	no	—
165	C₂×C₄×C₅	—	2	2	no	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	—	2	2	no	—
168	C₂×C₄×C₅	—	2	2	no	—
169	C₂×C₄×C₅	—	2	2	no	—
170	C₂×C₄×C₅	—	2	2	no	—
171	C₂×C₄×C₅	—	2	2	no	—
172	C₂×C₄×C₅	—	4	2	no	—
173	C₂×C₄×C₅	—	4	2	no	—
174	C₂×C₄×C₅	—	4	2	no	—
175	C₂×C₄×C₅	—	4	2	no	—
176	C₂×C₄×C₅	—	4	2	no	—
177	C₂×C₄×C₅	—	4	4	no	—
178	C₂×C₄×C₅	—	4	2	no	—
179	C₂×C₄×C₅	—	4	2	no	—
180	C₂×C₄×C₅	—	4	4	no	—
181	C₂×C₄×C₅	#121	1	2	yes	—
182	C₂×C₄×C₅	#121	1	2	yes	—
183	C₂×C₄×C₅	C₂×Q₁₆	1	4	yes	—
184	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
185	C₂×C₄×C₅	#137	1	2	yes	—
186	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
188	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
189	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
190	C₅×C₈	—	4	2	no	—
191	C₅×C₈	—	4	2	no	—
192	C₂²×Q₈	—	2	1	no	—
193	C₂²×Q₈	C₂³×C₅	1	2	yes	—
194	C₂²×Q₈	C₂³×C₅	1	1	yes	characteristic
195	C₂×Q₁₆	—	2	4	no	—
196	C₂×Q₁₆	—	4	4	no	—
197	C₂²×C₈	C₂³×C₅	1	2	yes	—
198	C₂²×C₈	C₂³×C₅	1	2	yes	—
199	C₄⋊Q₈	—	2	2	no	—
200	C₄⋊Q₈	—	2	2	no	—
201	C₄⋊Q₈	—	2	2	no	—
202	C₄⋊Q₈	—	2	2	no	—
203	C₄×Q₈	—	2	2	no	—
204	C₄×Q₈	C₅×D₄	1	4	yes	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
211	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
212	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
213	C₂×C₄²	—	2	1	no	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
216	C₂×C₄²	C₅×D₄	1	2	yes	—
217	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
218	C₂×C₄²	C₂³×C₅	1	2	yes	—
219	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
220	#107	—	2	4	no	—
221	C₄⋊C₈	—	2	2	no	—
222	C₄⋊C₈	—	2	2	no	—
223	C₄⋊C₈	—	4	4	no	—
224	#103	—	2	4	no	—
225	#103	—	2	4	no	—
226	#103	—	4	4	no	—
227	#95	—	2	2	no	—
228	#95	—	2	2	no	—
229	#95	—	2	2	no	—
230	#95	—	2	1	no	—
231	#95	—	2	4	no	—
232	#95	—	2	4	no	—
233	C₂²×C₅	C₂×#121	1	1	yes	characteristic
234	C₂²×C₅	#537	1	2	yes	—
235	C₂²×C₅	C₂²×Q₁₆	1	1	yes	characteristic
236	C₂²×C₅	C₂×#137	1	1	yes	characteristic
237	C₂²×C₅	#575	1	2	yes	—
238	C₄×C₅	—	2	4	no	—
239	C₄×C₅	—	2	2	no	—
240	C₄×C₅	—	4	2	no	—
241	C₄×C₅	—	4	2	no	—
242	C₄×C₅	—	4	2	no	—
243	C₄×C₅	—	8	2	no	—
244	C₄×C₅	—	8	2	no	—
245	C₄×C₅	—	8	2	no	—
246	C₄×C₅	#462	1	4	yes	—
247	C₄×C₅	#535	1	1	yes	characteristic
248	C₄×C₅	#536	1	1	yes	characteristic
249	C₄×C₅	#542	1	2	yes	—
250	C₂×Q₈	—	2	1	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	2	4	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂×Q₈	—	2	4	no	—
257	C₂×Q₈	—	4	4	no	—
258	C₂×Q₈	—	4	2	no	—
259	C₂×Q₈	C₂×C₅×D₄	1	2	yes	—
260	C₂²×C₄	—	2	1	no	—
261	C₂²×C₄	—	2	1	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
266	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
267	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
268	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
269	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
270	Q₁₆	—	4	8	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	4	no	—
274	C₂×C₈	—	2	4	no	—
275	C₄⋊C₄	—	2	4	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	4	4	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	—	4	2	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
286	C₄²	—	2	4	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	4	2	no	—
291	C₄²	—	4	2	no	—
292	C₄²	C₅×Q₁₆	1	4	yes	complemented
293	C₄²	C₂×C₅×D₄	1	2	yes	—
294	C₄²	C₅×#41	1	2	yes	—
295	C₂×C₅	#2620	1	1	yes	characteristic
296	C₂×C₅	C₂×#462	1	1	yes	characteristic
297	C₂×C₅	#3040	1	1	yes	characteristic
298	C₂×C₅	#3249	1	2	yes	—
299	C₂×C₅	#3252	1	2	yes	—
300	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
301	Q₈	—	2	2	no	—
302	Q₈	—	2	2	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	4	2	no	—
305	Q₈	—	4	2	no	—
306	Q₈	—	4	4	no	—
307	Q₈	—	8	4	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	4	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	4	no	—
325	C₂×C₄	C₅×#121	1	2	yes	—
326	C₂×C₄	C₅×#121	1	2	yes	—
327	C₂×C₄	C₂×C₅×Q₁₆	1	4	yes	—
328	C₂×C₄	C₂×C₅×Q₁₆	1	2	yes	—
329	C₂×C₄	C₅×#137	1	2	yes	—
330	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
331	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
332	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
333	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
334	C₈	—	4	2	no	—
335	C₈	—	4	2	no	—
336	C₅	#29830	1	1	yes	characteristic, direct product factor, Sylow
337	C₂²	C₂×C₅×#121	1	1	yes	characteristic
338	C₂²	C₅×#537	1	2	yes	—
339	C₂²	C₂²×C₅×Q₁₆	1	1	yes	characteristic
340	C₂²	C₂×C₅×#137	1	1	yes	characteristic
341	C₂²	C₅×#575	1	2	yes	—
342	C₄	—	2	4	no	—
343	C₄	—	2	2	no	—
344	C₄	—	4	2	no	—
345	C₄	—	4	2	no	—
346	C₄	—	4	2	no	—
347	C₄	—	8	2	no	—
348	C₄	—	8	2	no	—
349	C₄	—	8	2	no	—
350	C₄	C₅×#462	1	4	yes	—
351	C₄	C₅×#535	1	1	yes	characteristic
352	C₄	C₅×#536	1	1	yes	characteristic
353	C₄	C₅×#542	1	2	yes	—
354	C₂	C₅×#2620	1	1	yes	characteristic
355	C₂	C₂×C₅×#462	1	1	yes	characteristic
356	C₂	C₅×#3040	1	1	yes	characteristic
357	C₂	C₅×#3249	1	2	yes	—
358	C₂	C₅×#3252	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
35. #29830	C₅	direct (N × H)
128. C₄²×C₅	Q₁₆	semidirect
292. C₄²	C₅×Q₁₆	semidirect
336. C₅	#29830	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#477	2
C₂×C₅×#462	2
C₅×#2862	2
C₅×#2295	2
C₅×#2068	2
C₅×#2056	2
C₅×#1987	2
C₅×#1981	2
C₅×#1842	2
#29830	5

Subgroups — C5×#29830 #11952550

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#29830 #11952550