Subgroups — C₅×#22310 #11945030

Order 1280.

630 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2399	C₂	1	1	yes	maximal, characteristic
2	C₅×#1985	C₂	1	2	yes	maximal
3	C₅×#1984	C₂	1	2	yes	maximal
4	C₅×#1940	C₂	1	1	yes	maximal, characteristic
5	C₅×#1940	C₂	1	1	yes	maximal, characteristic
6	C₅×#1917	C₂	1	2	yes	maximal
7	C₅×#1916	C₂	1	2	yes	maximal
8	C₅×#1832	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×C₄×C₅×Q₈	C₂²	1	2	yes	—
13	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₅×#97	C₂²	1	1	yes	characteristic
18	C₂×C₅×#97	C₂²	1	2	yes	—
19	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
20	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₅×#398	C₄	1	2	yes	—
22	C₅×#398	C₄	1	2	yes	—
23	C₅×#395	C₄	1	2	yes	—
24	C₅×#395	C₄	1	2	yes	—
25	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
28	C₅×#376	C₂²	1	1	yes	characteristic
29	C₅×#336	C₂²	1	2	yes	—
30	C₅×#336	C₂²	1	2	yes	—
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	1	yes	characteristic
36	C₅×#336	C₂²	1	2	yes	—
37	C₅×#336	C₂²	1	1	yes	characteristic
38	C₅×#336	C₂²	1	1	yes	characteristic
39	#22310	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂²×C₅×Q₈	C₂×C₄	1	2	yes	—
41	C₂²×C₅×C₈	C₂³	1	2	yes	—
42	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₅×C₈	C₂³	1	2	yes	—
45	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
46	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
47	C₄×C₅×Q₈	—	4	2	no	—
48	C₄×C₅×Q₈	—	4	2	no	—
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×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
53	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
54	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
55	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₅	C₂³	1	2	yes	—
59	C₂×C₄²×C₅	C₂³	1	2	yes	—
60	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
61	C₅×C₄⋊C₈	—	2	2	no	—
62	C₅×C₄⋊C₈	—	2	2	no	—
63	C₅×C₄⋊C₈	—	4	2	no	—
64	C₅×C₄⋊C₈	—	4	2	no	—
65	C₅×#97	—	2	2	no	—
66	C₅×#97	—	4	2	no	—
67	C₄×C₅×C₈	—	4	2	no	—
68	C₄×C₅×C₈	Q₈	1	4	yes	complemented
69	C₅×#95	C₂×C₄	1	2	yes	—
70	C₅×#95	C₂×C₄	1	2	yes	—
71	C₅×#95	C₂×C₄	1	1	yes	characteristic
72	C₅×#95	C₂×C₄	1	2	yes	—
73	C₅×#95	C₂×C₄	1	1	yes	characteristic
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	#2399	C₂×C₅	1	1	yes	characteristic
78	#1985	C₂×C₅	1	2	yes	—
79	#1984	C₂×C₅	1	2	yes	—
80	#1940	C₂×C₅	1	1	yes	characteristic
81	#1940	C₂×C₅	1	1	yes	characteristic
82	#1917	C₂×C₅	1	2	yes	—
83	#1916	C₂×C₅	1	2	yes	—
84	#1832	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	C₂×C₅×Q₈	—	2	4	no	—
89	C₂×C₅×Q₈	—	4	2	no	—
90	C₂×C₅×Q₈	—	4	2	no	—
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	2	yes	—
94	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
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	1	no	—
104	C₂×C₅×C₈	—	2	1	no	—
105	C₂×C₅×C₈	—	2	1	no	—
106	C₂×C₅×C₈	—	2	2	no	—
107	C₂×C₅×C₈	—	2	2	no	—
108	C₂×C₅×C₈	—	2	2	no	—
109	C₂×C₅×C₈	—	2	2	no	—
110	C₂×C₅×C₈	—	2	1	no	—
111	C₂×C₅×C₈	—	2	1	no	—
112	C₂×C₅×C₈	—	2	2	no	—
113	C₂×C₅×C₈	—	2	1	no	—
114	C₂×C₅×C₈	—	2	1	no	—
115	C₂×C₅×C₈	—	2	2	no	—
116	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
117	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
118	C₅×C₄⋊C₄	—	2	4	no	—
119	C₅×C₄⋊C₄	—	2	4	no	—
120	C₅×C₄⋊C₄	—	4	2	no	—
121	C₅×C₄⋊C₄	—	4	2	no	—
122	C₅×C₄⋊C₄	—	4	2	no	—
123	C₅×C₄⋊C₄	—	4	2	no	—
124	C₅×C₄⋊C₄	—	4	2	no	—
125	C₄²×C₅	—	2	1	no	—
126	C₄²×C₅	—	2	1	no	—
127	C₄²×C₅	—	2	2	no	—
128	C₄²×C₅	—	2	2	no	—
129	C₄²×C₅	—	2	2	no	—
130	C₄²×C₅	—	2	2	no	—
131	C₄²×C₅	—	2	1	no	—
132	C₄²×C₅	—	2	1	no	—
133	C₄²×C₅	C₂×Q₈	1	2	yes	—
134	C₄²×C₅	#41	1	2	yes	—
135	C₂×C₄×Q₈	C₂²×C₅	1	2	yes	—
136	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
137	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
138	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
139	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
140	C₂×#97	C₂²×C₅	1	1	yes	characteristic
141	C₂×#97	C₂²×C₅	1	2	yes	—
142	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
143	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
144	#398	C₄×C₅	1	2	yes	—
145	#398	C₄×C₅	1	2	yes	—
146	#395	C₄×C₅	1	2	yes	—
147	#395	C₄×C₅	1	2	yes	—
148	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
149	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
150	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
151	#376	C₂²×C₅	1	1	yes	characteristic
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	2	yes	—
157	#336	C₂²×C₅	1	1	yes	characteristic
158	#336	C₂²×C₅	1	2	yes	—
159	#336	C₂²×C₅	1	2	yes	—
160	#336	C₂²×C₅	1	1	yes	characteristic
161	#336	C₂²×C₅	1	1	yes	characteristic
162	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
163	C₅×Q₈	—	4	8	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	—	2	2	no	—
179	C₂×C₄×C₅	—	2	2	no	—
180	C₂×C₄×C₅	—	2	2	no	—
181	C₂×C₄×C₅	—	2	2	no	—
182	C₂×C₄×C₅	—	2	2	no	—
183	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
184	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
185	C₂×C₄×C₅	#131	1	4	yes	—
186	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
187	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
188	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
189	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
190	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
191	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
192	C₅×C₈	—	2	2	no	—
193	C₅×C₈	—	2	2	no	—
194	C₅×C₈	—	4	1	no	—
195	C₅×C₈	—	4	1	no	—
196	C₅×C₈	—	4	2	no	—
197	C₅×C₈	—	4	2	no	—
198	C₂²×Q₈	C₂×C₄×C₅	1	2	yes	—
199	C₂²×C₈	C₂³×C₅	1	2	yes	—
200	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
201	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
202	C₂²×C₈	C₂³×C₅	1	2	yes	—
203	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
204	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
205	C₄×Q₈	—	4	2	no	—
206	C₄×Q₈	—	4	2	no	—
207	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
208	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
209	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
210	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
211	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
212	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
213	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
214	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
215	C₂×C₄²	C₂³×C₅	1	2	yes	—
216	C₂×C₄²	C₂³×C₅	1	2	yes	—
217	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
218	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
219	C₄⋊C₈	—	2	2	no	—
220	C₄⋊C₈	—	2	2	no	—
221	C₄⋊C₈	—	4	2	no	—
222	C₄⋊C₈	—	4	2	no	—
223	#97	—	2	2	no	—
224	#97	—	4	2	no	—
225	C₄×C₈	—	4	2	no	—
226	C₄×C₈	C₅×Q₈	1	4	yes	complemented
227	#95	C₂×C₄×C₅	1	2	yes	—
228	#95	C₂×C₄×C₅	1	2	yes	—
229	#95	C₂×C₄×C₅	1	2	yes	—
230	#95	C₂×C₄×C₅	1	1	yes	characteristic
231	#95	C₂×C₄×C₅	1	1	yes	characteristic
232	#95	C₂×C₄×C₅	1	2	yes	—
233	#95	C₂×C₄×C₅	1	1	yes	characteristic
234	#95	C₂×C₄×C₅	1	1	yes	characteristic
235	C₂²×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
236	C₂²×C₅	#518	1	1	yes	characteristic
237	C₂²×C₅	#519	1	1	yes	characteristic
238	C₂²×C₅	C₂×#131	1	2	yes	—
239	C₂²×C₅	#568	1	2	yes	—
240	C₄×C₅	—	2	2	no	—
241	C₄×C₅	—	2	2	no	—
242	C₄×C₅	—	2	4	no	—
243	C₄×C₅	—	4	2	no	—
244	C₄×C₅	—	4	2	no	—
245	C₄×C₅	—	4	2	no	—
246	C₄×C₅	—	4	2	no	—
247	C₄×C₅	—	4	2	no	—
248	C₄×C₅	#542	1	1	yes	characteristic
249	C₄×C₅	#542	1	1	yes	characteristic
250	C₄×C₅	#544	1	1	yes	characteristic
251	C₄×C₅	#544	1	1	yes	characteristic
252	C₂×Q₈	—	2	4	no	—
253	C₂×Q₈	—	4	2	no	—
254	C₂×Q₈	—	4	2	no	—
255	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
256	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
257	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
258	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
259	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
260	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
261	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
262	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
263	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	1	no	—
275	C₂×C₈	—	2	1	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	2	1	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
281	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
282	C₄⋊C₄	—	2	4	no	—
283	C₄⋊C₄	—	2	4	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄⋊C₄	—	4	2	no	—
286	C₄⋊C₄	—	4	2	no	—
287	C₄⋊C₄	—	4	2	no	—
288	C₄⋊C₄	—	4	2	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	2	no	—
292	C₄²	—	2	2	no	—
293	C₄²	—	2	1	no	—
294	C₄²	—	2	1	no	—
295	C₄²	—	2	1	no	—
296	C₄²	—	2	1	no	—
297	C₄²	C₂×C₅×Q₈	1	2	yes	—
298	C₄²	C₅×#41	1	2	yes	—
299	C₂×C₅	#2331	1	1	yes	characteristic
300	C₂×C₅	#2960	1	1	yes	characteristic
301	C₂×C₅	#2961	1	1	yes	characteristic
302	C₂×C₅	#2976	1	2	yes	—
303	C₂×C₅	#2985	1	2	yes	—
304	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
305	Q₈	—	4	8	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	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
326	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
327	C₂×C₄	C₅×#131	1	4	yes	—
328	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
329	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
330	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
331	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
332	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
333	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	2	2	no	—
336	C₈	—	4	1	no	—
337	C₈	—	4	1	no	—
338	C₈	—	4	2	no	—
339	C₈	—	4	2	no	—
340	C₅	#22310	1	1	yes	characteristic, direct product factor, Sylow
341	C₂²	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
342	C₂²	C₅×#518	1	1	yes	characteristic
343	C₂²	C₅×#519	1	1	yes	characteristic
344	C₂²	C₂×C₅×#131	1	2	yes	—
345	C₂²	C₅×#568	1	2	yes	—
346	C₄	—	2	2	no	—
347	C₄	—	2	2	no	—
348	C₄	—	2	4	no	—
349	C₄	—	4	2	no	—
350	C₄	—	4	2	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	2	no	—
353	C₄	—	4	2	no	—
354	C₄	C₅×#542	1	1	yes	characteristic
355	C₄	C₅×#542	1	1	yes	characteristic
356	C₄	C₅×#544	1	1	yes	characteristic
357	C₄	C₅×#544	1	1	yes	characteristic
358	C₂	C₅×#2331	1	1	yes	characteristic
359	C₂	C₅×#2960	1	1	yes	characteristic
360	C₂	C₅×#2961	1	1	yes	characteristic
361	C₂	C₅×#2976	1	2	yes	—
362	C₂	C₅×#2985	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. #22310	C₅	direct (N × H)
68. C₄×C₅×C₈	Q₈	semidirect
226. C₄×C₈	C₅×Q₈	semidirect
340. C₅	#22310	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×#2399	2
C₅×#1985	2
C₅×#1984	2
C₅×#1940	2
C₅×#1917	2
C₅×#1916	2
C₅×#1832	2
C₅×#1770	2
C₅×C₈×C₄⋊C₄	2
C₅×#1746	2
#22310	5

Subgroups — C5×#22310 #11945030

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22310 #11945030