Subgroups — C₅×#22383 #11945103

Order 1280.

614 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#125	C₂	1	1	yes	maximal, characteristic
2	C₅×#1940	C₂	1	2	yes	maximal
3	C₅×#1917	C₂	1	2	yes	maximal
4	C₅×#1917	C₂	1	2	yes	maximal
5	C₅×#1916	C₂	1	2	yes	maximal
6	C₅×#1916	C₂	1	2	yes	maximal
7	C₅×#1839	C₂	1	1	yes	maximal, characteristic
8	C₅×#1770	C₂	1	2	yes	maximal
9	C₅×#1767	C₂	1	1	yes	maximal, characteristic
10	C₂×C₅×#125	C₄	1	1	yes	characteristic
11	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₅×#97	C₂²	1	1	yes	characteristic
15	C₂×C₅×#97	C₂²	1	1	yes	characteristic
16	C₂×C₅×#97	C₂²	1	2	yes	—
17	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
18	C₅×#389	C₄	1	1	yes	characteristic
19	C₅×#389	C₄	1	1	yes	characteristic
20	C₅×#387	C₄	1	2	yes	—
21	C₅×#387	C₄	1	2	yes	—
22	C₅×#382	C₄	1	1	yes	characteristic
23	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
24	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
26	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₄³×C₅	C₂²	1	1	yes	characteristic
28	C₅×#336	C₂²	1	2	yes	—
29	C₅×#336	C₂²	1	2	yes	—
30	C₅×#336	C₂²	1	2	yes	—
31	C₅×#336	C₂²	1	1	yes	characteristic
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	2	yes	—
37	#22383	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂²×C₅×C₈	C₂³	1	2	yes	—
39	C₂²×C₅×C₈	C₂³	1	2	yes	—
40	C₂²×C₅×C₈	C₂³	1	2	yes	—
41	C₂²×C₅×C₈	C₂³	1	2	yes	—
42	C₅×#125	—	2	4	no	—
43	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
44	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
45	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
46	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
47	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
48	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
49	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
50	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
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	C₅×C₄⋊C₈	—	2	4	no	—
61	C₅×C₄⋊C₈	—	2	4	no	—
62	C₅×C₄⋊C₈	—	4	2	no	—
63	C₅×#97	—	4	1	no	—
64	C₅×#97	—	4	1	no	—
65	C₅×#97	—	4	2	no	—
66	C₄×C₅×C₈	—	2	4	no	—
67	C₅×#95	C₂×C₄	1	2	yes	—
68	C₅×#95	C₂×C₄	1	2	yes	—
69	C₅×#95	C₂×C₄	1	2	yes	—
70	C₅×#95	C₂×C₄	1	2	yes	—
71	C₅×#95	C₂×C₄	1	2	yes	—
72	C₅×#95	C₂×C₄	1	2	yes	—
73	C₄×#125	C₂×C₅	1	1	yes	characteristic
74	#1940	C₂×C₅	1	2	yes	—
75	#1917	C₂×C₅	1	2	yes	—
76	#1917	C₂×C₅	1	2	yes	—
77	#1916	C₂×C₅	1	2	yes	—
78	#1916	C₂×C₅	1	2	yes	—
79	#1839	C₂×C₅	1	1	yes	characteristic
80	#1770	C₂×C₅	1	2	yes	—
81	#1767	C₂×C₅	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
93	C₂×C₅×C₈	—	2	2	no	—
94	C₂×C₅×C₈	—	2	2	no	—
95	C₂×C₅×C₈	—	2	2	no	—
96	C₂×C₅×C₈	—	2	2	no	—
97	C₂×C₅×C₈	—	2	2	no	—
98	C₂×C₅×C₈	—	2	2	no	—
99	C₂×C₅×C₈	—	2	2	no	—
100	C₂×C₅×C₈	—	2	2	no	—
101	C₂×C₅×C₈	—	2	2	no	—
102	C₂×C₅×C₈	—	2	2	no	—
103	C₂×C₅×C₈	—	2	2	no	—
104	C₂×C₅×C₈	#41	1	4	yes	—
105	C₅×C₄⋊C₄	—	2	4	no	—
106	C₅×C₄⋊C₄	—	2	4	no	—
107	C₅×C₄⋊C₄	—	2	2	no	—
108	C₅×C₄⋊C₄	—	2	4	no	—
109	C₅×C₄⋊C₄	—	4	1	no	—
110	C₅×C₄⋊C₄	—	4	1	no	—
111	C₅×C₄⋊C₄	—	4	2	no	—
112	C₅×C₄⋊C₄	M₁₆	1	4	yes	—
113	C₄²×C₅	—	2	2	no	—
114	C₄²×C₅	—	2	2	no	—
115	C₄²×C₅	—	2	2	no	—
116	C₄²×C₅	—	2	2	no	—
117	C₄²×C₅	—	2	2	no	—
118	C₄²×C₅	—	2	1	no	—
119	C₄²×C₅	—	2	2	no	—
120	C₄²×C₅	—	2	2	no	—
121	C₄²×C₅	—	2	1	no	—
122	C₄²×C₅	—	2	2	no	—
123	C₄²×C₅	—	4	1	no	—
124	C₄²×C₅	—	4	1	no	—
125	C₂×#125	C₄×C₅	1	1	yes	characteristic
126	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
127	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
128	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
129	C₂×#97	C₂²×C₅	1	1	yes	characteristic
130	C₂×#97	C₂²×C₅	1	2	yes	—
131	C₂×#97	C₂²×C₅	1	1	yes	characteristic
132	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
133	#389	C₄×C₅	1	1	yes	characteristic
134	#389	C₄×C₅	1	1	yes	characteristic
135	#387	C₄×C₅	1	2	yes	—
136	#387	C₄×C₅	1	2	yes	—
137	#382	C₄×C₅	1	1	yes	characteristic
138	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
139	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
140	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
141	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
142	C₄³	C₂²×C₅	1	1	yes	characteristic
143	#336	C₂²×C₅	1	1	yes	characteristic
144	#336	C₂²×C₅	1	2	yes	—
145	#336	C₂²×C₅	1	2	yes	—
146	#336	C₂²×C₅	1	1	yes	characteristic
147	#336	C₂²×C₅	1	2	yes	—
148	#336	C₂²×C₅	1	2	yes	—
149	#336	C₂²×C₅	1	2	yes	—
150	#336	C₂²×C₅	1	2	yes	—
151	#336	C₂²×C₅	1	2	yes	—
152	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₄×C₅	—	2	1	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	—	2	1	no	—
156	C₂×C₄×C₅	—	2	1	no	—
157	C₂×C₄×C₅	—	2	1	no	—
158	C₂×C₄×C₅	—	2	2	no	—
159	C₂×C₄×C₅	—	2	2	no	—
160	C₂×C₄×C₅	—	2	1	no	—
161	C₂×C₄×C₅	—	2	1	no	—
162	C₂×C₄×C₅	—	2	1	no	—
163	C₂×C₄×C₅	—	2	2	no	—
164	C₂×C₄×C₅	—	2	1	no	—
165	C₂×C₄×C₅	—	2	2	no	—
166	C₂×C₄×C₅	—	2	1	no	—
167	C₂×C₄×C₅	—	2	1	no	—
168	C₂×C₄×C₅	—	2	2	no	—
169	C₂×C₄×C₅	—	2	1	no	—
170	C₂×C₄×C₅	—	2	1	no	—
171	C₂×C₄×C₅	—	2	2	no	—
172	C₂×C₄×C₅	—	2	2	no	—
173	C₂×C₄×C₅	—	2	1	no	—
174	C₂×C₄×C₅	—	2	1	no	—
175	C₂×C₄×C₅	—	2	1	no	—
176	C₂×C₄×C₅	—	2	1	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	C₂×M₁₆	1	2	yes	—
181	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
182	C₂×C₄×C₅	#131	1	2	yes	—
183	C₂×C₄×C₅	C₂×#41	1	2	yes	—
184	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
185	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
186	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
187	C₅×C₈	—	2	4	no	—
188	C₅×C₈	—	4	2	no	—
189	C₅×C₈	—	4	2	no	—
190	C₅×C₈	—	4	2	no	—
191	C₂²×C₈	C₂³×C₅	1	2	yes	—
192	C₂²×C₈	C₂³×C₅	1	2	yes	—
193	C₂²×C₈	C₂³×C₅	1	2	yes	—
194	C₂²×C₈	C₂³×C₅	1	2	yes	—
195	#125	—	2	4	no	—
196	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
197	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
198	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
199	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
200	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
201	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
202	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
203	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
204	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
205	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
206	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
207	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
208	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
209	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
210	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
211	C₂×C₄²	C₂³×C₅	1	2	yes	—
212	C₂×C₄²	C₂³×C₅	1	2	yes	—
213	C₄⋊C₈	—	2	4	no	—
214	C₄⋊C₈	—	2	4	no	—
215	C₄⋊C₈	—	4	2	no	—
216	#97	—	4	1	no	—
217	#97	—	4	1	no	—
218	#97	—	4	2	no	—
219	C₄×C₈	—	2	4	no	—
220	#95	C₂×C₄×C₅	1	2	yes	—
221	#95	C₂×C₄×C₅	1	2	yes	—
222	#95	C₂×C₄×C₅	1	2	yes	—
223	#95	C₂×C₄×C₅	1	2	yes	—
224	#95	C₂×C₄×C₅	1	2	yes	—
225	#95	C₂×C₄×C₅	1	2	yes	—
226	C₂²×C₅	C₄×#41	1	1	yes	characteristic
227	C₂²×C₅	#519	1	1	yes	characteristic
228	C₂²×C₅	#520	1	1	yes	characteristic
229	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
230	C₂²×C₅	C₂×#131	1	1	yes	characteristic
231	C₂²×C₅	#568	1	1	yes	characteristic
232	C₂²×C₅	#568	1	1	yes	characteristic
233	C₄×C₅	—	2	2	no	—
234	C₄×C₅	—	2	4	no	—
235	C₄×C₅	—	2	2	no	—
236	C₄×C₅	—	4	2	no	—
237	C₄×C₅	—	4	1	no	—
238	C₄×C₅	—	4	1	no	—
239	C₄×C₅	—	4	1	no	—
240	C₄×C₅	—	4	2	no	—
241	C₄×C₅	—	4	1	no	—
242	C₄×C₅	—	4	2	no	—
243	C₄×C₅	#539	1	2	yes	—
244	C₄×C₅	#541	1	2	yes	—
245	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
246	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
247	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
248	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
249	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
250	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
251	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
252	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
253	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
254	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
255	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	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	C₅×#41	1	4	yes	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	4	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	C₅×M₁₆	1	4	yes	—
276	C₄²	—	2	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	2	1	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	4	1	no	—
287	C₄²	—	4	1	no	—
288	C₂×C₅	#2336	1	1	yes	characteristic
289	C₂×C₅	#2968	1	1	yes	characteristic
290	C₂×C₅	#2970	1	1	yes	characteristic
291	C₂×C₅	#2983	1	1	yes	characteristic
292	C₂×C₅	#2985	1	1	yes	characteristic
293	C₂×C₅	#2991	1	1	yes	characteristic
294	C₂×C₅	#2995	1	1	yes	characteristic
295	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
324	C₂×C₄	C₂×C₅×M₁₆	1	4	yes	—
325	C₂×C₄	C₅×#131	1	2	yes	—
326	C₂×C₄	C₂×C₅×#41	1	2	yes	—
327	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
328	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
329	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
330	C₈	—	2	4	no	—
331	C₈	—	4	2	no	—
332	C₈	—	4	2	no	—
333	C₈	—	4	2	no	—
334	C₅	#22383	1	1	yes	characteristic, direct product factor, Sylow
335	C₂²	C₄×C₅×#41	1	1	yes	characteristic
336	C₂²	C₅×#519	1	1	yes	characteristic
337	C₂²	C₅×#520	1	1	yes	characteristic
338	C₂²	C₂²×C₅×M₁₆	1	1	yes	characteristic
339	C₂²	C₂×C₅×#131	1	1	yes	characteristic
340	C₂²	C₅×#568	1	1	yes	characteristic
341	C₂²	C₅×#568	1	1	yes	characteristic
342	C₄	—	2	2	no	—
343	C₄	—	2	2	no	—
344	C₄	—	2	4	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	4	2	no	—
348	C₄	—	4	1	no	—
349	C₄	—	4	2	no	—
350	C₄	—	4	2	no	—
351	C₄	—	4	1	no	—
352	C₄	C₅×#539	1	2	yes	—
353	C₄	C₅×#541	1	2	yes	—
354	C₂	C₅×#2336	1	1	yes	characteristic
355	C₂	C₅×#2968	1	1	yes	characteristic
356	C₂	C₅×#2970	1	1	yes	characteristic
357	C₂	C₅×#2983	1	1	yes	characteristic
358	C₂	C₅×#2985	1	1	yes	characteristic
359	C₂	C₅×#2991	1	1	yes	characteristic
360	C₂	C₅×#2995	1	1	yes	characteristic

	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
37. #22383	C₅	direct (N × H)
334. C₅	#22383	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×#125	2
C₅×#1940	2
C₅×#1917	2
C₅×#1916	2
C₅×#1839	2
C₅×#1770	2
C₅×#1767	2
#22383	5

Subgroups — C5×#22383 #11945103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22383 #11945103