Subgroups — C₅×#22302 #11945022

Order 1280.

598 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2402	C₂	1	1	yes	maximal, characteristic
2	C₅×#1940	C₂	1	1	yes	maximal, characteristic
3	C₅×#1940	C₂	1	1	yes	maximal, characteristic
4	C₅×#1940	C₂	1	2	yes	maximal
5	C₅×#1939	C₂	1	2	yes	maximal
6	C₅×#1917	C₂	1	2	yes	maximal
7	C₅×#1916	C₂	1	2	yes	maximal
8	C₅×#1843	C₂	1	1	yes	maximal, characteristic
9	C₅×#1770	C₂	1	1	yes	maximal, characteristic
10	C₅×C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
11	C₅×#1746	C₂	1	1	yes	maximal, characteristic
12	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₅×#97	C₂²	1	1	yes	characteristic
19	C₂×C₅×#97	C₂²	1	2	yes	—
20	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
21	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₅×#400	C₄	1	2	yes	—
23	C₅×#400	C₄	1	2	yes	—
24	C₅×#398	C₄	1	2	yes	—
25	C₅×#398	C₄	1	2	yes	—
26	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
28	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
29	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
30	C₅×#376	C₂²	1	1	yes	characteristic
31	C₅×#336	C₂²	1	2	yes	—
32	C₅×#336	C₂²	1	2	yes	—
33	C₅×#336	C₂²	1	1	yes	characteristic
34	C₅×#336	C₂²	1	2	yes	—
35	C₅×#336	C₂²	1	2	yes	—
36	C₅×#336	C₂²	1	1	yes	characteristic
37	C₅×#336	C₂²	1	1	yes	characteristic
38	C₅×#336	C₂²	1	1	yes	characteristic
39	#22302	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂²×C₅×C₈	C₂³	1	2	yes	—
41	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₅×C₈	C₂³	1	2	yes	—
44	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
45	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
46	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
47	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
48	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
49	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
50	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
51	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₅	C₂³	1	2	yes	—
55	C₂×C₄²×C₅	C₂³	1	2	yes	—
56	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
57	C₅×C₄⋊C₈	—	2	2	no	—
58	C₅×C₄⋊C₈	—	2	2	no	—
59	C₅×C₄⋊C₈	—	4	2	no	—
60	C₅×C₄⋊C₈	—	4	2	no	—
61	C₅×C₄⋊C₈	—	4	2	no	—
62	C₅×C₄⋊C₈	—	4	2	no	—
63	C₅×#97	—	2	2	no	—
64	C₅×#97	—	4	2	no	—
65	C₄×C₅×C₈	—	4	2	no	—
66	C₄×C₅×C₈	Q₈	1	4	yes	—
67	C₅×#95	C₂×C₄	1	2	yes	—
68	C₅×#95	C₂×C₄	1	2	yes	—
69	C₅×#95	C₂×C₄	1	1	yes	characteristic
70	C₅×#95	C₂×C₄	1	1	yes	characteristic
71	C₅×#95	C₂×C₄	1	2	yes	—
72	C₅×#95	C₂×C₄	1	2	yes	—
73	C₅×#95	C₂×C₄	1	2	yes	—
74	C₅×#95	C₂×C₄	1	2	yes	—
75	C₅×#95	C₂×C₄	1	1	yes	characteristic
76	C₅×#95	C₂×C₄	1	1	yes	characteristic
77	#2402	C₂×C₅	1	1	yes	characteristic
78	#1940	C₂×C₅	1	1	yes	characteristic
79	#1940	C₂×C₅	1	1	yes	characteristic
80	#1940	C₂×C₅	1	2	yes	—
81	#1939	C₂×C₅	1	2	yes	—
82	#1917	C₂×C₅	1	2	yes	—
83	#1916	C₂×C₅	1	2	yes	—
84	#1843	C₂×C₅	1	1	yes	characteristic
85	#1770	C₂×C₅	1	1	yes	characteristic
86	C₈×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
87	#1746	C₂×C₅	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
97	C₂×C₅×C₈	—	2	1	no	—
98	C₂×C₅×C₈	—	2	1	no	—
99	C₂×C₅×C₈	—	2	1	no	—
100	C₂×C₅×C₈	—	2	1	no	—
101	C₂×C₅×C₈	—	2	1	no	—
102	C₂×C₅×C₈	—	2	1	no	—
103	C₂×C₅×C₈	—	2	2	no	—
104	C₂×C₅×C₈	—	2	2	no	—
105	C₂×C₅×C₈	—	2	2	no	—
106	C₂×C₅×C₈	—	2	2	no	—
107	C₂×C₅×C₈	—	2	2	no	—
108	C₂×C₅×C₈	—	2	1	no	—
109	C₂×C₅×C₈	—	2	1	no	—
110	C₂×C₅×C₈	—	2	2	no	—
111	C₂×C₅×C₈	—	2	1	no	—
112	C₂×C₅×C₈	—	2	1	no	—
113	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
114	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
115	C₅×C₄⋊C₄	—	2	4	no	—
116	C₅×C₄⋊C₄	—	2	4	no	—
117	C₅×C₄⋊C₄	—	2	4	no	—
118	C₅×C₄⋊C₄	—	4	2	no	—
119	C₅×C₄⋊C₄	—	4	2	no	—
120	C₅×C₄⋊C₄	—	4	2	no	—
121	C₄²×C₅	—	2	2	no	—
122	C₄²×C₅	—	2	2	no	—
123	C₄²×C₅	—	2	1	no	—
124	C₄²×C₅	—	2	1	no	—
125	C₄²×C₅	—	2	2	no	—
126	C₄²×C₅	—	2	1	no	—
127	C₄²×C₅	—	2	1	no	—
128	C₄²×C₅	—	2	2	no	—
129	C₄²×C₅	C₂×Q₈	1	2	yes	—
130	C₄²×C₅	#41	1	2	yes	—
131	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
132	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
133	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
134	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
135	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
136	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
137	C₂×#97	C₂²×C₅	1	1	yes	characteristic
138	C₂×#97	C₂²×C₅	1	2	yes	—
139	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
140	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
141	#400	C₄×C₅	1	2	yes	—
142	#400	C₄×C₅	1	2	yes	—
143	#398	C₄×C₅	1	2	yes	—
144	#398	C₄×C₅	1	2	yes	—
145	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
146	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
147	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
148	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
149	#376	C₂²×C₅	1	1	yes	characteristic
150	#336	C₂²×C₅	1	1	yes	characteristic
151	#336	C₂²×C₅	1	2	yes	—
152	#336	C₂²×C₅	1	2	yes	—
153	#336	C₂²×C₅	1	2	yes	—
154	#336	C₂²×C₅	1	1	yes	characteristic
155	#336	C₂²×C₅	1	2	yes	—
156	#336	C₂²×C₅	1	1	yes	characteristic
157	#336	C₂²×C₅	1	1	yes	characteristic
158	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
159	C₂×C₄×C₅	—	2	2	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	2	2	no	—
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	—	2	2	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	—	2	2	no	—
173	C₂×C₄×C₅	—	2	2	no	—
174	C₂×C₄×C₅	—	2	2	no	—
175	C₂×C₄×C₅	—	2	2	no	—
176	C₂×C₄×C₅	—	2	2	no	—
177	C₂×C₄×C₅	—	2	2	no	—
178	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
179	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
180	C₂×C₄×C₅	#131	1	4	yes	—
181	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
183	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
184	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
185	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
186	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
187	C₅×C₈	—	2	2	no	—
188	C₅×C₈	—	2	2	no	—
189	C₅×C₈	—	4	1	no	—
190	C₅×C₈	—	4	2	no	—
191	C₅×C₈	—	4	2	no	—
192	C₅×C₈	—	4	1	no	—
193	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
194	C₂²×C₈	C₂³×C₅	1	2	yes	—
195	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
196	C₂²×C₈	C₂³×C₅	1	2	yes	—
197	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
199	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
200	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
201	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
202	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
203	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
204	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
205	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
206	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
207	C₂×C₄²	C₂³×C₅	1	2	yes	—
208	C₂×C₄²	C₂³×C₅	1	2	yes	—
209	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
210	C₄⋊C₈	—	2	2	no	—
211	C₄⋊C₈	—	2	2	no	—
212	C₄⋊C₈	—	4	2	no	—
213	C₄⋊C₈	—	4	2	no	—
214	C₄⋊C₈	—	4	2	no	—
215	C₄⋊C₈	—	4	2	no	—
216	#97	—	2	2	no	—
217	#97	—	4	2	no	—
218	C₄×C₈	—	4	2	no	—
219	C₄×C₈	C₅×Q₈	1	4	yes	—
220	#95	C₂×C₄×C₅	1	2	yes	—
221	#95	C₂×C₄×C₅	1	2	yes	—
222	#95	C₂×C₄×C₅	1	1	yes	characteristic
223	#95	C₂×C₄×C₅	1	1	yes	characteristic
224	#95	C₂×C₄×C₅	1	2	yes	—
225	#95	C₂×C₄×C₅	1	2	yes	—
226	#95	C₂×C₄×C₅	1	2	yes	—
227	#95	C₂×C₄×C₅	1	2	yes	—
228	#95	C₂×C₄×C₅	1	1	yes	characteristic
229	#95	C₂×C₄×C₅	1	1	yes	characteristic
230	C₂²×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
231	C₂²×C₅	#520	1	1	yes	characteristic
232	C₂²×C₅	#520	1	1	yes	characteristic
233	C₂²×C₅	C₂×#131	1	2	yes	—
234	C₂²×C₅	#568	1	2	yes	—
235	C₄×C₅	—	2	2	no	—
236	C₄×C₅	—	2	2	no	—
237	C₄×C₅	—	2	4	no	—
238	C₄×C₅	—	4	2	no	—
239	C₄×C₅	—	4	2	no	—
240	C₄×C₅	—	4	2	no	—
241	C₄×C₅	—	4	2	no	—
242	C₄×C₅	—	4	2	no	—
243	C₄×C₅	#540	1	1	yes	characteristic
244	C₄×C₅	#541	1	1	yes	characteristic
245	C₄×C₅	#544	1	1	yes	characteristic
246	C₄×C₅	#544	1	1	yes	characteristic
247	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
248	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
249	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
250	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
251	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
252	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
253	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
254	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
255	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
273	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
274	C₄⋊C₄	—	2	4	no	—
275	C₄⋊C₄	—	2	4	no	—
276	C₄⋊C₄	—	2	4	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	2	1	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	2	2	no	—
288	C₄²	C₂×C₅×Q₈	1	2	yes	—
289	C₄²	C₅×#41	1	2	yes	—
290	C₂×C₅	#2330	1	1	yes	characteristic
291	C₂×C₅	#2960	1	1	yes	characteristic
292	C₂×C₅	#2961	1	1	yes	characteristic
293	C₂×C₅	#2992	1	2	yes	—
294	C₂×C₅	#2993	1	2	yes	—
295	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	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	C₄×C₅×Q₈	1	2	yes	—
316	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
317	C₂×C₄	C₅×#131	1	4	yes	—
318	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
319	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
320	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
321	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
322	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
323	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
324	C₈	—	2	2	no	—
325	C₈	—	2	2	no	—
326	C₈	—	4	1	no	—
327	C₈	—	4	2	no	—
328	C₈	—	4	2	no	—
329	C₈	—	4	1	no	—
330	C₅	#22302	1	1	yes	characteristic, direct product factor, Sylow
331	C₂²	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
332	C₂²	C₅×#520	1	1	yes	characteristic
333	C₂²	C₅×#520	1	1	yes	characteristic
334	C₂²	C₂×C₅×#131	1	2	yes	—
335	C₂²	C₅×#568	1	2	yes	—
336	C₄	—	2	4	no	—
337	C₄	—	2	2	no	—
338	C₄	—	2	2	no	—
339	C₄	—	4	2	no	—
340	C₄	—	4	2	no	—
341	C₄	—	4	2	no	—
342	C₄	—	4	2	no	—
343	C₄	—	4	2	no	—
344	C₄	C₅×#540	1	1	yes	characteristic
345	C₄	C₅×#541	1	1	yes	characteristic
346	C₄	C₅×#544	1	1	yes	characteristic
347	C₄	C₅×#544	1	1	yes	characteristic
348	C₂	C₅×#2330	1	1	yes	characteristic
349	C₂	C₅×#2960	1	1	yes	characteristic
350	C₂	C₅×#2961	1	1	yes	characteristic
351	C₂	C₅×#2992	1	2	yes	—
352	C₂	C₅×#2993	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
39. #22302	C₅	direct (N × H)
330. C₅	#22302	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×#2402	2
C₅×#1940	2
C₅×#1939	2
C₅×#1917	2
C₅×#1916	2
C₅×#1843	2
C₅×#1770	2
C₅×C₈×C₄⋊C₄	2
C₅×#1746	2
#22302	5

Subgroups — C5×#22302 #11945022

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22302 #11945022