Subgroups — C₅×#21697 #11944417

Order 1280.

1238 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#442	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×C₅×D₈	C₂	1	2	yes	maximal, complemented
4	C₅×#2328	C₂	1	2	yes	maximal, complemented
5	C₅×#1972	C₂	1	2	yes	maximal, complemented
6	C₅×#1970	C₂	1	2	yes	maximal, complemented
7	C₅×#1961	C₂	1	2	yes	maximal, complemented
8	C₅×#1846	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1726	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₅×D₈	C₂²	1	2	yes	—
11	C₂²×C₅×D₈	C₂²	1	2	yes	—
12	C₂×C₅×#127	C₂²	1	2	yes	—
13	C₂×C₄×C₅×D₄	C₂²	1	4	yes	—
14	C₅×#493	C₄	1	8	yes	complemented
15	C₅×#442	—	2	8	no	—
16	C₄×C₅×D₈	—	2	8	no	—
17	C₂×C₅×#107	C₂²	1	2	yes	—
18	C₂×C₅×#106	C₂²	1	2	yes	—
19	C₂×C₅×#102	C₂²	1	4	yes	—
20	C₂×C₅×#102	C₂²	1	4	yes	—
21	C₂×C₅×#97	C₂²	1	1	yes	characteristic
22	C₂×C₅×#97	C₂²	1	1	yes	characteristic
23	C₂×C₅×#97	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
27	C₅×#390	C₂²	1	4	yes	—
28	C₅×#386	—	2	8	no	—
29	C₅×#382	C₂²	1	2	yes	—
30	C₅×#376	C₂²	1	1	yes	characteristic
31	#21697	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂²×C₅×D₄	—	2	4	no	—
33	C₂²×C₅×D₄	C₂³	1	4	yes	—
34	C₂³×C₄×C₅	—	2	4	no	—
35	C₂×C₅×D₈	—	2	4	no	—
36	C₂×C₅×D₈	—	2	4	no	—
37	C₂×C₅×D₈	—	2	4	no	—
38	C₂×C₅×D₈	—	2	4	no	—
39	C₂×C₅×D₈	C₂×C₄	1	8	yes	complemented
40	C₂×C₅×D₈	C₂×C₄	1	8	yes	complemented
41	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
45	C₅×#127	—	2	4	no	—
46	C₅×#127	C₂×C₄	1	8	yes	complemented
47	C₄×C₅×D₄	—	2	8	no	—
48	C₄×C₅×D₄	—	2	8	no	—
49	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
50	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
51	C₂×C₅×C₂²⋊C₄	—	2	8	no	—
52	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
55	C₅×#107	—	2	4	no	—
56	C₅×#106	—	2	4	no	—
57	C₅×#102	—	2	8	no	—
58	C₅×#102	—	2	8	no	—
59	C₅×#97	—	2	2	no	—
60	C₅×#97	—	2	2	no	—
61	C₅×#97	D₄	1	4	yes	complemented
62	C₄×C₅×C₈	—	2	2	no	—
63	C₄×C₅×C₈	C₂×C₄	1	4	yes	complemented
64	C₄×C₅×C₈	D₄	1	4	yes	complemented
65	C₅×#95	—	2	4	no	—
66	C₅×#95	D₄	1	4	yes	—
67	C₂×#493	C₂×C₅	1	1	yes	characteristic
68	C₂×#442	C₂×C₅	1	2	yes	complemented
69	C₂×C₄×D₈	C₂×C₅	1	2	yes	complemented
70	#2328	C₂×C₅	1	2	yes	complemented
71	#1972	C₂×C₅	1	2	yes	complemented
72	#1970	C₂×C₅	1	2	yes	complemented
73	#1961	C₂×C₅	1	2	yes	complemented
74	#1846	C₂×C₅	1	1	yes	characteristic, complemented
75	#1726	C₂×C₅	1	1	yes	characteristic, complemented
76	C₂⁴×C₅	—	2	4	no	—
77	C₂×C₅×D₄	—	2	16	no	—
78	C₂×C₅×D₄	—	2	4	no	—
79	C₂×C₅×D₄	—	2	4	no	—
80	C₂×C₅×D₄	—	2	4	no	—
81	C₂×C₅×D₄	—	2	4	no	—
82	C₂×C₅×D₄	—	2	4	no	—
83	C₂×C₅×D₄	—	8	8	no	—
84	C₂×C₅×D₄	C₂²×C₄	1	8	yes	—
85	C₂²×C₄×C₅	—	2	4	no	—
86	C₂²×C₄×C₅	—	4	8	no	—
87	C₂²×C₄×C₅	—	8	8	no	—
88	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
92	C₅×D₈	—	2	16	no	—
93	C₅×D₈	—	2	16	no	—
94	C₂×C₅×C₈	—	2	1	no	—
95	C₂×C₅×C₈	—	2	1	no	—
96	C₂×C₅×C₈	—	2	1	no	—
97	C₂×C₅×C₈	—	2	1	no	—
98	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
99	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
100	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
101	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
102	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
103	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
104	C₂×C₅×C₈	#41	1	2	yes	—
105	C₂×C₅×C₈	#41	1	2	yes	—
106	C₅×C₄⋊C₄	—	2	4	no	—
107	C₅×C₄⋊C₄	—	2	4	no	—
108	C₅×C₂²⋊C₄	—	4	8	no	—
109	C₅×C₂²⋊C₄	—	8	8	no	—
110	C₄²×C₅	—	2	1	no	—
111	C₄²×C₅	—	2	1	no	—
112	C₄²×C₅	C₂²×C₄	1	2	yes	—
113	C₄²×C₅	C₂×D₄	1	2	yes	—
114	C₄²×C₅	C₂×D₄	1	2	yes	—
115	C₄²×C₅	#41	1	2	yes	—
116	C₂²×D₈	C₂²×C₅	1	2	yes	—
117	C₂²×D₈	C₂²×C₅	1	2	yes	—
118	C₂×#127	C₂²×C₅	1	2	yes	—
119	C₂×C₄×D₄	C₂²×C₅	1	4	yes	—
120	#493	C₄×C₅	1	8	yes	complemented
121	#442	—	2	8	no	—
122	C₄×D₈	—	2	8	no	—
123	C₂×#107	C₂²×C₅	1	2	yes	—
124	C₂×#106	C₂²×C₅	1	2	yes	—
125	C₂×#102	C₂²×C₅	1	4	yes	—
126	C₂×#102	C₂²×C₅	1	4	yes	—
127	C₂×#97	C₂²×C₅	1	1	yes	characteristic
128	C₂×#97	C₂²×C₅	1	1	yes	characteristic
129	C₂×#97	C₂²×C₅	1	1	yes	characteristic
130	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
131	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
132	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
133	#390	C₂²×C₅	1	4	yes	—
134	#386	—	2	8	no	—
135	#382	C₂²×C₅	1	2	yes	—
136	#376	C₂²×C₅	1	1	yes	characteristic
137	C₂³×C₅	—	2	8	no	—
138	C₂³×C₅	—	4	4	no	—
139	C₂³×C₅	—	4	4	no	—
140	C₂³×C₅	—	4	8	no	—
141	C₂³×C₅	—	4	8	no	—
142	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
143	C₅×D₄	—	2	8	no	—
144	C₅×D₄	—	2	8	no	—
145	C₅×D₄	—	4	4	no	—
146	C₅×D₄	—	4	4	no	—
147	C₅×D₄	—	8	16	no	—
148	C₂×C₄×C₅	—	2	4	no	—
149	C₂×C₄×C₅	—	2	4	no	—
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	—	2	4	no	—
152	C₂×C₄×C₅	—	4	4	no	—
153	C₂×C₄×C₅	—	4	8	no	—
154	C₂×C₄×C₅	—	8	16	no	—
155	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
156	C₂×C₄×C₅	#127	1	4	yes	—
157	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
163	C₅×C₈	—	2	2	no	—
164	C₅×C₈	—	2	2	no	—
165	C₅×C₈	C₄×D₄	1	4	yes	—
166	C₅×C₈	C₄×D₄	1	4	yes	—
167	C₂²×D₄	—	2	4	no	—
168	C₂²×D₄	C₂³×C₅	1	4	yes	—
169	C₂³×C₄	—	2	4	no	—
170	C₂×D₈	—	2	4	no	—
171	C₂×D₈	—	2	4	no	—
172	C₂×D₈	—	2	4	no	—
173	C₂×D₈	—	2	4	no	—
174	C₂×D₈	C₂×C₄×C₅	1	8	yes	complemented
175	C₂×D₈	C₂×C₄×C₅	1	8	yes	complemented
176	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
177	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
178	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
179	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
180	#127	—	2	4	no	—
181	#127	C₂×C₄×C₅	1	8	yes	complemented
182	C₄×D₄	—	2	8	no	—
183	C₄×D₄	—	2	8	no	—
184	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
185	C₂×C₂²⋊C₄	—	2	4	no	—
186	C₂×C₂²⋊C₄	—	2	8	no	—
187	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
188	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
189	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
190	#107	—	2	4	no	—
191	#106	—	2	4	no	—
192	#102	—	2	8	no	—
193	#102	—	2	8	no	—
194	#97	—	2	2	no	—
195	#97	—	2	2	no	—
196	#97	C₅×D₄	1	4	yes	complemented
197	C₄×C₈	—	2	2	no	—
198	C₄×C₈	C₂×C₄×C₅	1	4	yes	complemented
199	C₄×C₈	C₅×D₄	1	4	yes	complemented
200	#95	—	2	4	no	—
201	#95	C₅×D₄	1	4	yes	—
202	C₂²×C₅	—	4	8	no	—
203	C₂²×C₅	—	4	16	no	—
204	C₂²×C₅	—	8	4	no	—
205	C₂²×C₅	—	8	4	no	—
206	C₂²×C₅	—	8	8	no	—
207	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
208	C₂²×C₅	C₂×#127	1	1	yes	characteristic
209	C₂²×C₅	#532	1	1	yes	characteristic
210	C₂²×C₅	#576	1	4	yes	—
211	C₄×C₅	—	2	4	no	—
212	C₄×C₅	—	2	4	no	—
213	C₄×C₅	—	4	2	no	—
214	C₄×C₅	—	8	4	no	—
215	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
216	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
217	C₄×C₅	#532	1	1	yes	characteristic
218	C₄×C₅	#532	1	1	yes	characteristic
219	C₂⁴	—	2	4	no	—
220	C₂×D₄	—	2	16	no	—
221	C₂×D₄	—	2	4	no	—
222	C₂×D₄	—	2	4	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	2	4	no	—
225	C₂×D₄	—	2	4	no	—
226	C₂×D₄	—	8	8	no	—
227	C₂×D₄	C₂²×C₄×C₅	1	8	yes	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	4	8	no	—
230	C₂²×C₄	—	8	8	no	—
231	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
232	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
233	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
234	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
235	D₈	—	2	16	no	—
236	D₈	—	2	16	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
242	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
243	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
244	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
245	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
246	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
247	C₂×C₈	C₅×#41	1	2	yes	—
248	C₂×C₈	C₅×#41	1	2	yes	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₂²⋊C₄	—	4	8	no	—
252	C₂²⋊C₄	—	8	8	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₄²	C₂²×C₄×C₅	1	2	yes	—
256	C₄²	C₂×C₅×D₄	1	2	yes	—
257	C₄²	C₂×C₅×D₄	1	2	yes	—
258	C₄²	C₅×#41	1	2	yes	—
259	C₂×C₅	—	8	8	no	—
260	C₂×C₅	C₄×#127	1	1	yes	characteristic
261	C₂×C₅	#2948	1	2	yes	—
262	C₂×C₅	#3155	1	2	yes	—
263	C₂×C₅	#3176	1	2	yes	—
264	C₂³	—	2	8	no	—
265	C₂³	—	4	4	no	—
266	C₂³	—	4	4	no	—
267	C₂³	—	4	8	no	—
268	C₂³	—	4	8	no	—
269	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
270	D₄	—	2	8	no	—
271	D₄	—	2	8	no	—
272	D₄	—	4	4	no	—
273	D₄	—	4	4	no	—
274	D₄	—	8	16	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	8	no	—
281	C₂×C₄	—	8	16	no	—
282	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
283	C₂×C₄	C₅×#127	1	4	yes	—
284	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
285	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
286	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
287	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
288	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
289	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
290	C₈	—	2	2	no	—
291	C₈	—	2	2	no	—
292	C₈	C₄×C₅×D₄	1	4	yes	—
293	C₈	C₄×C₅×D₄	1	4	yes	—
294	C₅	#21697	1	1	yes	characteristic, direct product factor, Sylow
295	C₂²	—	4	8	no	—
296	C₂²	—	4	16	no	—
297	C₂²	—	8	4	no	—
298	C₂²	—	8	4	no	—
299	C₂²	—	8	8	no	—
300	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
301	C₂²	C₂×C₅×#127	1	1	yes	characteristic
302	C₂²	C₅×#532	1	1	yes	characteristic
303	C₂²	C₅×#576	1	4	yes	—
304	C₄	—	2	4	no	—
305	C₄	—	2	4	no	—
306	C₄	—	4	2	no	—
307	C₄	—	8	4	no	—
308	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
309	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
310	C₄	C₅×#532	1	1	yes	characteristic
311	C₄	C₅×#532	1	1	yes	characteristic
312	C₂	—	8	8	no	—
313	C₂	C₄×C₅×#127	1	1	yes	characteristic
314	C₂	C₅×#2948	1	2	yes	—
315	C₂	C₅×#3155	1	2	yes	—
316	C₂	C₅×#3176	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂×C₅×#493	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₅×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#442	C₂	semidirect
3. C₂×C₄×C₅×D₈	C₂	semidirect
4. C₅×#2328	C₂	semidirect
5. C₅×#1972	C₂	semidirect
6. C₅×#1970	C₂	semidirect
7. C₅×#1961	C₂	semidirect
8. C₅×#1846	C₂	semidirect
9. C₅×#1726	C₂	semidirect
14. C₅×#493	C₄	semidirect
31. #21697	C₅	direct (N × H)
39. C₂×C₅×D₈	C₂×C₄	semidirect
40. C₂×C₅×D₈	C₂×C₄	semidirect
46. C₅×#127	C₂×C₄	semidirect
61. C₅×#97	D₄	semidirect
63. C₄×C₅×C₈	C₂×C₄	semidirect
64. C₄×C₅×C₈	D₄	semidirect
68. C₂×#442	C₂×C₅	semidirect
69. C₂×C₄×D₈	C₂×C₅	semidirect
70. #2328	C₂×C₅	semidirect
71. #1972	C₂×C₅	semidirect
72. #1970	C₂×C₅	semidirect
73. #1961	C₂×C₅	semidirect
74. #1846	C₂×C₅	semidirect
75. #1726	C₂×C₅	semidirect
120. #493	C₄×C₅	semidirect
174. C₂×D₈	C₂×C₄×C₅	semidirect
175. C₂×D₈	C₂×C₄×C₅	semidirect
181. #127	C₂×C₄×C₅	semidirect
196. #97	C₅×D₄	semidirect
198. C₄×C₈	C₂×C₄×C₅	semidirect
199. C₄×C₈	C₅×D₄	semidirect
294. C₅	#21697	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×#493	2
C₂×C₅×#442	2
C₂×C₄×C₅×D₈	2
C₅×#2328	2
C₅×#1972	2
C₅×#1970	2
C₅×#1961	2
C₅×#1846	2
C₅×#1726	2
#21697	5

Subgroups — C5×#21697 #11944417

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21697 #11944417