Subgroups — C₅×#21922 #11944642

Order 1280.

798 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	C₄×C₅×#117	C₂	1	1	yes	maximal, characteristic
2	C₅×#1923	C₂	1	4	yes	maximal
3	C₅×#1916	C₂	1	4	yes	maximal, complemented
4	C₅×#1842	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1831	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1797	C₂	1	2	yes	maximal
7	C₄×C₅×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
8	C₂×C₅×#117	C₄	1	1	yes	characteristic
9	C₂×C₅×#117	C₄	1	1	yes	characteristic
10	C₂²×C₄²×C₅	C₂²	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	4	yes	—
14	C₂×C₅×C₂²⋊C₈	C₂²	1	4	yes	—
15	C₂×C₄×C₅×C₈	C₂²	1	4	yes	—
16	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	—
17	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
18	C₄×C₅×C₂²⋊C₄	C₄	1	2	yes	—
19	C₄×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₅×#376	C₄	1	1	yes	characteristic
21	C₅×#376	C₄	1	1	yes	characteristic
22	C₄³×C₅	C₂²	1	1	yes	characteristic
23	C₄³×C₅	C₂²	1	1	yes	characteristic
24	C₅×#336	C₂²	1	4	yes	—
25	C₅×#336	C₂²	1	4	yes	—
26	C₅×#336	C₂²	1	4	yes	—
27	#21922	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂²×C₅×C₈	C₂³	1	4	yes	—
32	C₂²×C₅×C₈	C₂³	1	4	yes	—
33	C₅×#117	—	2	4	no	—
34	C₅×#117	C₈	1	8	yes	complemented
35	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
36	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
37	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
38	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅	—	2	2	no	—
40	C₂×C₄²×C₅	—	2	2	no	—
41	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₅	C₂³	1	2	yes	—
50	C₂×C₄²×C₅	C₂³	1	2	yes	—
51	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
53	C₅×C₄⋊C₈	—	2	4	no	—
54	C₅×C₄⋊C₈	—	2	4	no	—
55	C₅×C₄⋊C₈	—	4	4	no	—
56	C₅×C₂²⋊C₈	—	4	4	no	—
57	C₄×C₅×C₈	—	4	4	no	—
58	C₅×#95	C₂×C₄	1	2	yes	—
59	C₅×#95	C₂×C₄	1	2	yes	—
60	C₅×#95	C₂×C₄	1	2	yes	—
61	C₅×#95	C₂×C₄	1	2	yes	—
62	C₄×#117	C₂×C₅	1	1	yes	characteristic
63	#1923	C₂×C₅	1	4	yes	—
64	#1916	C₂×C₅	1	4	yes	complemented
65	#1842	C₂×C₅	1	1	yes	characteristic, complemented
66	#1831	C₂×C₅	1	1	yes	characteristic, complemented
67	#1797	C₂×C₅	1	2	yes	—
68	C₄×C₄⋊C₈	C₂×C₅	1	2	yes	complemented
69	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	—	2	2	no	—
71	C₂²×C₄×C₅	—	2	2	no	—
72	C₂²×C₄×C₅	—	2	2	no	—
73	C₂²×C₄×C₅	—	2	1	no	—
74	C₂²×C₄×C₅	—	2	1	no	—
75	C₂²×C₄×C₅	—	2	2	no	—
76	C₂²×C₄×C₅	—	2	2	no	—
77	C₂²×C₄×C₅	—	4	2	no	—
78	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
79	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
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	1	yes	characteristic
88	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
89	C₂×C₅×C₈	—	2	4	no	—
90	C₂×C₅×C₈	—	2	4	no	—
91	C₂×C₅×C₈	—	2	4	no	—
92	C₂×C₅×C₈	—	2	4	no	—
93	C₂×C₅×C₈	—	2	4	no	—
94	C₂×C₅×C₈	—	2	4	no	—
95	C₅×C₄⋊C₄	—	2	4	no	—
96	C₅×C₄⋊C₄	C₂×C₈	1	8	yes	—
97	C₅×C₂²⋊C₄	—	2	4	no	—
98	C₅×C₂²⋊C₄	C₂×C₈	1	8	yes	—
99	C₄²×C₅	—	2	2	no	—
100	C₄²×C₅	—	2	2	no	—
101	C₄²×C₅	—	2	2	no	—
102	C₄²×C₅	—	2	1	no	—
103	C₄²×C₅	—	2	1	no	—
104	C₄²×C₅	—	2	2	no	—
105	C₄²×C₅	—	2	2	no	—
106	C₄²×C₅	—	2	2	no	—
107	C₄²×C₅	—	2	2	no	—
108	C₄²×C₅	—	4	2	no	—
109	C₄²×C₅	—	4	2	no	—
110	C₄²×C₅	C₂×C₈	1	4	yes	—
111	C₄²×C₅	C₂×C₈	1	4	yes	—
112	C₄²×C₅	M₁₆	1	4	yes	—
113	C₄²×C₅	#41	1	2	yes	—
114	C₄²×C₅	#41	1	2	yes	—
115	C₂×#117	C₄×C₅	1	1	yes	characteristic
116	C₂×#117	C₄×C₅	1	1	yes	characteristic
117	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
118	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
119	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
120	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
121	C₂×C₂²⋊C₈	C₂²×C₅	1	4	yes	—
122	C₂×C₄×C₈	C₂²×C₅	1	4	yes	—
123	C₄×C₄⋊C₄	C₄×C₅	1	2	yes	—
124	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
125	C₄×C₂²⋊C₄	C₄×C₅	1	2	yes	—
126	C₄×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
127	#376	C₄×C₅	1	1	yes	characteristic
128	#376	C₄×C₅	1	1	yes	characteristic
129	C₄³	C₂²×C₅	1	1	yes	characteristic
130	C₄³	C₂²×C₅	1	1	yes	characteristic
131	#336	C₂²×C₅	1	4	yes	—
132	#336	C₂²×C₅	1	4	yes	—
133	#336	C₂²×C₅	1	4	yes	—
134	C₂³×C₅	—	2	1	no	—
135	C₂³×C₅	—	2	2	no	—
136	C₂³×C₅	—	2	1	no	—
137	C₂³×C₅	—	2	2	no	—
138	C₂³×C₅	C₂²×C₈	1	2	yes	—
139	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₅	—	2	1	no	—
141	C₂×C₄×C₅	—	2	2	no	—
142	C₂×C₄×C₅	—	2	2	no	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	—	2	1	no	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	2	2	no	—
147	C₂×C₄×C₅	—	2	2	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	2	8	no	—
150	C₂×C₄×C₅	—	2	1	no	—
151	C₂×C₄×C₅	—	2	1	no	—
152	C₂×C₄×C₅	—	2	2	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	—	2	2	no	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	—	4	4	no	—
158	C₂×C₄×C₅	—	4	1	no	—
159	C₂×C₄×C₅	—	4	1	no	—
160	C₂×C₄×C₅	—	4	2	no	—
161	C₂×C₄×C₅	#117	1	2	yes	—
162	C₂×C₄×C₅	#117	1	2	yes	—
163	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
164	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
165	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
166	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
167	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
168	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
169	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
171	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
172	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
173	C₅×C₈	—	4	4	no	—
174	C₅×C₈	—	4	4	no	—
175	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
176	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
177	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
178	C₂²×C₈	C₂³×C₅	1	4	yes	—
179	C₂²×C₈	C₂³×C₅	1	4	yes	—
180	#117	—	2	4	no	—
181	#117	C₅×C₈	1	8	yes	complemented
182	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
183	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
184	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	2	yes	—
185	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	2	yes	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
189	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
190	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
191	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
192	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
193	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
194	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
195	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
196	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
197	C₂×C₄²	C₂³×C₅	1	2	yes	—
198	C₂×C₄²	C₂³×C₅	1	2	yes	—
199	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
200	C₄⋊C₈	—	2	4	no	—
201	C₄⋊C₈	—	2	4	no	—
202	C₄⋊C₈	—	4	4	no	—
203	C₂²⋊C₈	—	4	4	no	—
204	C₄×C₈	—	4	4	no	—
205	#95	C₂×C₄×C₅	1	2	yes	—
206	#95	C₂×C₄×C₅	1	2	yes	—
207	#95	C₂×C₄×C₅	1	2	yes	—
208	#95	C₂×C₄×C₅	1	2	yes	—
209	C₂²×C₅	—	2	2	no	—
210	C₂²×C₅	—	2	4	no	—
211	C₂²×C₅	—	4	1	no	—
212	C₂²×C₅	—	4	1	no	—
213	C₂²×C₅	—	4	2	no	—
214	C₂²×C₅	C₂×#117	1	1	yes	characteristic
215	C₂²×C₅	#520	1	2	yes	—
216	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
217	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
218	C₂²×C₅	#568	1	2	yes	—
219	C₄×C₅	—	2	2	no	—
220	C₄×C₅	—	2	2	no	—
221	C₄×C₅	—	2	4	no	—
222	C₄×C₅	—	2	4	no	—
223	C₄×C₅	—	2	4	no	—
224	C₄×C₅	—	4	1	no	—
225	C₄×C₅	—	4	1	no	—
226	C₄×C₅	—	4	2	no	—
227	C₄×C₅	#431	1	4	yes	—
228	C₄×C₅	#540	1	1	yes	characteristic
229	C₄×C₅	#541	1	1	yes	characteristic
230	C₄×C₅	#542	1	2	yes	—
231	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	C₂×C₅×C₈	1	4	yes	—
241	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
242	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
243	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
244	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
245	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
246	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
247	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
248	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
249	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
250	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	—	2	4	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	C₂×C₅×C₈	1	8	yes	—
259	C₂²⋊C₄	—	2	4	no	—
260	C₂²⋊C₄	C₂×C₅×C₈	1	8	yes	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	4	2	no	—
271	C₄²	—	4	2	no	—
272	C₄²	C₂×C₅×C₈	1	4	yes	—
273	C₄²	C₂×C₅×C₈	1	4	yes	—
274	C₄²	C₅×M₁₆	1	4	yes	—
275	C₄²	C₅×#41	1	2	yes	—
276	C₄²	C₅×#41	1	2	yes	—
277	C₂×C₅	—	4	2	no	—
278	C₂×C₅	#2314	1	1	yes	characteristic
279	C₂×C₅	C₂×#431	1	1	yes	characteristic
280	C₂×C₅	#2921	1	1	yes	characteristic
281	C₂×C₅	#2988	1	2	yes	—
282	C₂×C₅	#2989	1	2	yes	—
283	C₂³	—	2	1	no	—
284	C₂³	—	2	2	no	—
285	C₂³	—	2	1	no	—
286	C₂³	—	2	2	no	—
287	C₂³	C₂²×C₅×C₈	1	2	yes	—
288	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	8	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	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	C₅×#117	1	2	yes	—
311	C₂×C₄	C₅×#117	1	2	yes	—
312	C₂×C₄	C₂²×C₅×C₈	1	4	yes	—
313	C₂×C₄	C₂²×C₅×C₈	1	4	yes	—
314	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
315	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
316	C₂×C₄	C₂×C₅×M₁₆	1	4	yes	—
317	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
318	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
319	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
320	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
321	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
322	C₈	—	4	4	no	—
323	C₈	—	4	4	no	—
324	C₅	#21922	1	1	yes	characteristic, direct product factor, Sylow
325	C₂²	—	2	4	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	4	2	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	C₂×C₅×#117	1	1	yes	characteristic
331	C₂²	C₅×#520	1	2	yes	—
332	C₂²	C₂³×C₅×C₈	1	1	yes	characteristic
333	C₂²	C₂²×C₅×M₁₆	1	1	yes	characteristic
334	C₂²	C₅×#568	1	2	yes	—
335	C₄	—	2	4	no	—
336	C₄	—	2	4	no	—
337	C₄	—	2	4	no	—
338	C₄	—	2	2	no	—
339	C₄	—	2	2	no	—
340	C₄	—	4	2	no	—
341	C₄	—	4	1	no	—
342	C₄	—	4	1	no	—
343	C₄	C₅×#431	1	4	yes	—
344	C₄	C₅×#540	1	1	yes	characteristic
345	C₄	C₅×#541	1	1	yes	characteristic
346	C₄	C₅×#542	1	2	yes	—
347	C₂	—	4	2	no	—
348	C₂	C₅×#2314	1	1	yes	characteristic
349	C₂	C₂×C₅×#431	1	1	yes	characteristic
350	C₂	C₅×#2921	1	1	yes	characteristic
351	C₂	C₅×#2988	1	2	yes	—
352	C₂	C₅×#2989	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂⁴
commutator	C₂²	C₂³×C₅×C₈
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂³×C₅	C₂³×C₄
prime radical	C₂⁴×C₅	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1916	C₂	semidirect
4. C₅×#1842	C₂	semidirect
5. C₅×#1831	C₂	semidirect
7. C₄×C₅×C₄⋊C₈	C₂	semidirect
27. #21922	C₅	direct (N × H)
34. C₅×#117	C₈	semidirect
64. #1916	C₂×C₅	semidirect
65. #1842	C₂×C₅	semidirect
66. #1831	C₂×C₅	semidirect
68. C₄×C₄⋊C₈	C₂×C₅	semidirect
181. #117	C₅×C₈	semidirect
324. C₅	#21922	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×#117	2
C₅×#1923	2
C₅×#1916	2
C₅×#1842	2
C₅×#1831	2
C₅×#1797	2
C₄×C₅×C₄⋊C₈	2
#21922	5

Subgroups — C5×#21922 #11944642

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21922 #11944642