Subgroups — C₅×#21725 #11944445

Order 1280.

750 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#500	C₂	1	1	yes	maximal, characteristic
2	C₅×#2331	C₂	1	2	yes	maximal
3	C₅×#1993	C₂	1	2	yes	maximal
4	C₅×#1943	C₂	1	2	yes	maximal
5	C₅×#1942	C₂	1	2	yes	maximal
6	C₅×#1846	C₂	1	1	yes	maximal, characteristic
7	C₅×#1776	C₂	1	2	yes	maximal
8	C₄×C₅×#107	C₂	1	2	yes	maximal
9	C₅×#1726	C₂	1	1	yes	maximal, characteristic
10	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₅×#500	C₄	1	8	yes	complemented
12	C₂×C₅×#107	C₂²	1	2	yes	—
13	C₂×C₅×#107	C₂²	1	2	yes	—
14	C₂×C₅×#107	C₂²	1	2	yes	—
15	C₂×C₅×#106	C₂²	1	2	yes	—
16	C₂×C₅×#97	C₂²	1	1	yes	characteristic
17	C₂×C₅×#97	C₂²	1	1	yes	characteristic
18	C₂×C₅×#97	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₅×#391	—	2	4	no	—
23	C₅×#389	C₂²	1	4	yes	—
24	C₅×#387	—	2	4	no	—
25	C₅×#382	C₂²	1	2	yes	—
26	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
27	C₅×#376	C₂²	1	1	yes	characteristic
28	C₅×#340	C₂²	1	4	yes	—
29	C₅×#340	C₂²	1	4	yes	—
30	#21725	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂²×C₅×Q₈	—	2	2	no	—
32	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₅×C₄⋊Q₈	—	2	4	no	—
37	C₅×C₄⋊Q₈	C₂×C₄	1	8	yes	—
38	C₂×C₅×C₄⋊C₄	—	2	2	no	—
39	C₂×C₅×C₄⋊C₄	—	2	4	no	—
40	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
41	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
42	C₂×C₄²×C₅	—	2	4	no	—
43	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₅×#107	—	2	4	no	—
47	C₅×#107	C₂×C₄	1	8	yes	—
48	C₅×#107	C₂×C₄	1	8	yes	—
49	C₅×#106	—	2	4	no	—
50	C₅×#97	—	2	2	no	—
51	C₅×#97	—	2	2	no	—
52	C₅×#97	Q₈	1	4	yes	complemented
53	C₄×C₅×C₈	—	2	2	no	—
54	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
55	C₄×C₅×C₈	Q₈	1	4	yes	complemented
56	C₅×#95	—	2	4	no	—
57	C₅×#95	—	2	2	no	—
58	C₅×#95	—	2	2	no	—
59	C₅×#95	—	2	4	no	—
60	C₅×#95	D₄	1	2	yes	—
61	C₅×#95	D₄	1	2	yes	—
62	C₂×#500	C₂×C₅	1	1	yes	characteristic
63	#2331	C₂×C₅	1	2	yes	—
64	#1993	C₂×C₅	1	2	yes	—
65	#1943	C₂×C₅	1	2	yes	—
66	#1942	C₂×C₅	1	2	yes	—
67	#1846	C₂×C₅	1	1	yes	characteristic
68	#1776	C₂×C₅	1	2	yes	—
69	C₄×#107	C₂×C₅	1	2	yes	—
70	#1726	C₂×C₅	1	1	yes	characteristic
71	C₂×C₅×Q₈	—	2	8	no	—
72	C₂×C₅×Q₈	—	8	4	no	—
73	C₂²×C₄×C₅	—	2	4	no	—
74	C₂²×C₄×C₅	—	2	4	no	—
75	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
80	C₂×C₅×C₈	—	2	1	no	—
81	C₂×C₅×C₈	—	2	1	no	—
82	C₂×C₅×C₈	—	2	1	no	—
83	C₂×C₅×C₈	—	2	1	no	—
84	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
85	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
86	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
87	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
88	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
89	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
90	C₂×C₅×C₈	#41	1	2	yes	—
91	C₂×C₅×C₈	#41	1	2	yes	—
92	C₅×C₄⋊C₄	—	2	8	no	—
93	C₅×C₄⋊C₄	—	2	4	no	—
94	C₅×C₄⋊C₄	—	2	4	no	—
95	C₅×C₄⋊C₄	—	2	4	no	—
96	C₅×C₄⋊C₄	—	8	4	no	—
97	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
98	C₄²×C₅	—	2	1	no	—
99	C₄²×C₅	—	2	1	no	—
100	C₄²×C₅	—	8	4	no	—
101	C₄²×C₅	C₂²×C₄	1	2	yes	—
102	C₄²×C₅	C₂×Q₈	1	2	yes	—
103	C₄²×C₅	C₂×Q₈	1	2	yes	—
104	C₄²×C₅	#41	1	2	yes	—
105	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
106	#500	C₄×C₅	1	8	yes	complemented
107	C₂×#107	C₂²×C₅	1	2	yes	—
108	C₂×#107	C₂²×C₅	1	2	yes	—
109	C₂×#107	C₂²×C₅	1	2	yes	—
110	C₂×#106	C₂²×C₅	1	2	yes	—
111	C₂×#97	C₂²×C₅	1	1	yes	characteristic
112	C₂×#97	C₂²×C₅	1	1	yes	characteristic
113	C₂×#97	C₂²×C₅	1	1	yes	characteristic
114	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
115	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
116	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
117	#391	—	2	4	no	—
118	#389	C₂²×C₅	1	4	yes	—
119	#387	—	2	4	no	—
120	#382	C₂²×C₅	1	2	yes	—
121	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
122	#376	C₂²×C₅	1	1	yes	characteristic
123	#340	C₂²×C₅	1	4	yes	—
124	#340	C₂²×C₅	1	4	yes	—
125	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
126	C₅×Q₈	—	8	8	no	—
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	—	2	2	no	—
129	C₂×C₄×C₅	—	2	2	no	—
130	C₂×C₄×C₅	—	2	2	no	—
131	C₂×C₄×C₅	—	2	8	no	—
132	C₂×C₄×C₅	—	2	2	no	—
133	C₂×C₄×C₅	—	2	2	no	—
134	C₂×C₄×C₅	—	2	2	no	—
135	C₂×C₄×C₅	—	4	4	no	—
136	C₂×C₄×C₅	—	4	4	no	—
137	C₂×C₄×C₅	—	4	4	no	—
138	C₂×C₄×C₅	—	4	4	no	—
139	C₂×C₄×C₅	—	4	4	no	—
140	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
141	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
142	C₂×C₄×C₅	C₄⋊Q₈	1	4	yes	—
143	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
149	C₅×C₈	—	2	2	no	—
150	C₅×C₈	—	2	2	no	—
151	C₅×C₈	C₄×Q₈	1	4	yes	—
152	C₅×C₈	C₄×Q₈	1	4	yes	—
153	C₂²×Q₈	—	2	2	no	—
154	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
155	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
156	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
157	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
158	C₄⋊Q₈	—	2	4	no	—
159	C₄⋊Q₈	C₂×C₄×C₅	1	8	yes	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	4	no	—
162	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
163	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
164	C₂×C₄²	—	2	4	no	—
165	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
166	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
167	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
168	#107	—	2	4	no	—
169	#107	C₂×C₄×C₅	1	8	yes	—
170	#107	C₂×C₄×C₅	1	8	yes	—
171	#106	—	2	4	no	—
172	#97	—	2	2	no	—
173	#97	—	2	2	no	—
174	#97	C₅×Q₈	1	4	yes	complemented
175	C₄×C₈	—	2	2	no	—
176	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
177	C₄×C₈	C₅×Q₈	1	4	yes	complemented
178	#95	—	2	4	no	—
179	#95	—	2	2	no	—
180	#95	—	2	2	no	—
181	#95	—	2	4	no	—
182	#95	C₅×D₄	1	2	yes	—
183	#95	C₅×D₄	1	2	yes	—
184	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
185	C₂²×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
186	C₂²×C₅	#532	1	1	yes	characteristic
187	C₂²×C₅	#576	1	2	yes	—
188	C₂²×C₅	#578	1	2	yes	—
189	C₄×C₅	—	2	4	no	—
190	C₄×C₅	—	2	2	no	—
191	C₄×C₅	—	2	2	no	—
192	C₄×C₅	—	4	2	no	—
193	C₄×C₅	—	4	8	no	—
194	C₄×C₅	—	8	4	no	—
195	C₄×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
196	C₄×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
197	C₄×C₅	#533	1	1	yes	characteristic
198	C₄×C₅	#533	1	1	yes	characteristic
199	C₂×Q₈	—	2	8	no	—
200	C₂×Q₈	—	8	4	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
204	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
207	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
213	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
214	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
215	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
216	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
217	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
218	C₂×C₈	C₅×#41	1	2	yes	—
219	C₂×C₈	C₅×#41	1	2	yes	—
220	C₄⋊C₄	—	2	8	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	8	4	no	—
225	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
226	C₄²	—	2	1	no	—
227	C₄²	—	2	1	no	—
228	C₄²	—	8	4	no	—
229	C₄²	C₂²×C₄×C₅	1	2	yes	—
230	C₄²	C₂×C₅×Q₈	1	2	yes	—
231	C₄²	C₂×C₅×Q₈	1	2	yes	—
232	C₄²	C₅×#41	1	2	yes	—
233	C₂×C₅	C₄×C₄⋊Q₈	1	1	yes	characteristic
234	C₂×C₅	#2948	1	1	yes	characteristic
235	C₂×C₅	#2950	1	1	yes	characteristic
236	C₂×C₅	#3164	1	2	yes	—
237	C₂×C₅	#3175	1	2	yes	—
238	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
239	Q₈	—	8	8	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
254	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
255	C₂×C₄	C₅×C₄⋊Q₈	1	4	yes	—
256	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
257	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
258	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
259	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
260	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
261	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
262	C₈	—	2	2	no	—
263	C₈	—	2	2	no	—
264	C₈	C₄×C₅×Q₈	1	4	yes	—
265	C₈	C₄×C₅×Q₈	1	4	yes	—
266	C₅	#21725	1	1	yes	characteristic, direct product factor, Sylow
267	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
268	C₂²	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
269	C₂²	C₅×#532	1	1	yes	characteristic
270	C₂²	C₅×#576	1	2	yes	—
271	C₂²	C₅×#578	1	2	yes	—
272	C₄	—	2	4	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	4	2	no	—
276	C₄	—	4	8	no	—
277	C₄	—	8	4	no	—
278	C₄	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
279	C₄	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
280	C₄	C₅×#533	1	1	yes	characteristic
281	C₄	C₅×#533	1	1	yes	characteristic
282	C₂	C₄×C₅×C₄⋊Q₈	1	1	yes	characteristic
283	C₂	C₅×#2948	1	1	yes	characteristic
284	C₂	C₅×#2950	1	1	yes	characteristic
285	C₂	C₅×#3164	1	2	yes	—
286	C₂	C₅×#3175	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
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
11. C₅×#500	C₄	semidirect
30. #21725	C₅	direct (N × H)
52. C₅×#97	Q₈	semidirect
55. C₄×C₅×C₈	Q₈	semidirect
106. #500	C₄×C₅	semidirect
174. #97	C₅×Q₈	semidirect
177. C₄×C₈	C₅×Q₈	semidirect
266. C₅	#21725	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×#500	2
C₅×#2331	2
C₅×#1993	2
C₅×#1943	2
C₅×#1942	2
C₅×#1846	2
C₅×#1776	2
C₄×C₅×#107	2
C₅×#1726	2
#21725	5

Subgroups — C5×#21725 #11944445

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21725 #11944445