Subgroups — C₅×#21724 #11944444

Order 1280.

750 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#498	C₂	1	1	yes	maximal, characteristic
2	C₅×#2331	C₂	1	2	yes	maximal
3	C₅×#1993	C₂	1	2	yes	maximal
4	C₅×#1944	C₂	1	2	yes	maximal
5	C₅×#1941	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×#106	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×#498	C₄	1	8	yes	complemented
12	C₂×C₅×#107	C₂²	1	2	yes	—
13	C₂×C₅×#106	C₂²	1	2	yes	—
14	C₂×C₅×#106	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	#21724	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×#106	—	2	4	no	—
48	C₅×#106	C₂×C₄	1	8	yes	—
49	C₅×#106	C₂×C₄	1	8	yes	—
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	4	yes	—
61	C₂×#498	C₂×C₅	1	1	yes	characteristic
62	#2331	C₂×C₅	1	2	yes	—
63	#1993	C₂×C₅	1	2	yes	—
64	#1944	C₂×C₅	1	2	yes	—
65	#1941	C₂×C₅	1	2	yes	—
66	#1846	C₂×C₅	1	1	yes	characteristic
67	#1776	C₂×C₅	1	2	yes	—
68	C₄×#106	C₂×C₅	1	2	yes	—
69	#1726	C₂×C₅	1	1	yes	characteristic
70	C₂×C₅×Q₈	—	2	8	no	—
71	C₂×C₅×Q₈	—	8	4	no	—
72	C₂²×C₄×C₅	—	2	4	no	—
73	C₂²×C₄×C₅	—	2	4	no	—
74	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
78	C₂×C₅×C₈	—	2	1	no	—
79	C₂×C₅×C₈	—	2	1	no	—
80	C₂×C₅×C₈	—	2	1	no	—
81	C₂×C₅×C₈	—	2	1	no	—
82	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
83	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
84	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
85	C₂×C₅×C₈	C₂×Q₈	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	#41	1	2	yes	—
89	C₂×C₅×C₈	#41	1	2	yes	—
90	C₅×C₄⋊C₄	—	2	8	no	—
91	C₅×C₄⋊C₄	—	2	4	no	—
92	C₅×C₄⋊C₄	—	2	4	no	—
93	C₅×C₄⋊C₄	—	2	4	no	—
94	C₅×C₄⋊C₄	—	8	4	no	—
95	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
96	C₄²×C₅	—	2	1	no	—
97	C₄²×C₅	—	2	1	no	—
98	C₄²×C₅	—	8	4	no	—
99	C₄²×C₅	C₂²×C₄	1	2	yes	—
100	C₄²×C₅	C₂×Q₈	1	2	yes	—
101	C₄²×C₅	C₂×Q₈	1	2	yes	—
102	C₄²×C₅	#41	1	2	yes	—
103	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
104	#498	C₄×C₅	1	8	yes	complemented
105	C₂×#107	C₂²×C₅	1	2	yes	—
106	C₂×#106	C₂²×C₅	1	2	yes	—
107	C₂×#106	C₂²×C₅	1	2	yes	—
108	C₂×#106	C₂²×C₅	1	2	yes	—
109	C₂×#97	C₂²×C₅	1	1	yes	characteristic
110	C₂×#97	C₂²×C₅	1	1	yes	characteristic
111	C₂×#97	C₂²×C₅	1	1	yes	characteristic
112	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
113	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
114	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
115	#391	—	2	4	no	—
116	#389	C₂²×C₅	1	4	yes	—
117	#387	—	2	4	no	—
118	#382	C₂²×C₅	1	2	yes	—
119	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
120	#376	C₂²×C₅	1	1	yes	characteristic
121	#340	C₂²×C₅	1	4	yes	—
122	#340	C₂²×C₅	1	4	yes	—
123	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
124	C₅×Q₈	—	8	8	no	—
125	C₂×C₄×C₅	—	2	4	no	—
126	C₂×C₄×C₅	—	2	4	no	—
127	C₂×C₄×C₅	—	2	8	no	—
128	C₂×C₄×C₅	—	2	2	no	—
129	C₂×C₄×C₅	—	2	4	no	—
130	C₂×C₄×C₅	—	4	4	no	—
131	C₂×C₄×C₅	—	4	4	no	—
132	C₂×C₄×C₅	—	4	4	no	—
133	C₂×C₄×C₅	—	4	4	no	—
134	C₂×C₄×C₅	—	4	4	no	—
135	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
136	C₂×C₄×C₅	C₄⋊Q₈	1	4	yes	—
137	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
143	C₅×C₈	—	2	2	no	—
144	C₅×C₈	—	2	2	no	—
145	C₅×C₈	C₄×Q₈	1	4	yes	—
146	C₅×C₈	C₄×Q₈	1	4	yes	—
147	C₂²×Q₈	—	2	2	no	—
148	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
149	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
150	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
151	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
152	C₄⋊Q₈	—	2	4	no	—
153	C₄⋊Q₈	C₂×C₄×C₅	1	8	yes	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	—	2	4	no	—
156	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
157	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
158	C₂×C₄²	—	2	4	no	—
159	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
160	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
161	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
162	#107	—	2	4	no	—
163	#106	—	2	4	no	—
164	#106	C₂×C₄×C₅	1	8	yes	—
165	#106	C₂×C₄×C₅	1	8	yes	—
166	#97	—	2	2	no	—
167	#97	—	2	2	no	—
168	#97	C₅×Q₈	1	4	yes	complemented
169	C₄×C₈	—	2	2	no	—
170	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
171	C₄×C₈	C₅×Q₈	1	4	yes	complemented
172	#95	—	2	4	no	—
173	#95	—	2	2	no	—
174	#95	—	2	2	no	—
175	#95	—	2	4	no	—
176	#95	C₅×D₄	1	4	yes	—
177	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
178	C₂²×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
179	C₂²×C₅	#532	1	1	yes	characteristic
180	C₂²×C₅	#577	1	4	yes	—
181	C₄×C₅	—	2	4	no	—
182	C₄×C₅	—	2	4	no	—
183	C₄×C₅	—	4	2	no	—
184	C₄×C₅	—	4	8	no	—
185	C₄×C₅	—	8	4	no	—
186	C₄×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
187	C₄×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
188	C₄×C₅	#533	1	1	yes	characteristic
189	C₄×C₅	#533	1	1	yes	characteristic
190	C₂×Q₈	—	2	8	no	—
191	C₂×Q₈	—	8	4	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
195	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
196	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
197	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
203	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
204	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
205	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
206	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
207	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
208	C₂×C₈	C₅×#41	1	2	yes	—
209	C₂×C₈	C₅×#41	1	2	yes	—
210	C₄⋊C₄	—	2	8	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	8	4	no	—
215	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
216	C₄²	—	2	1	no	—
217	C₄²	—	2	1	no	—
218	C₄²	—	8	4	no	—
219	C₄²	C₂²×C₄×C₅	1	2	yes	—
220	C₄²	C₂×C₅×Q₈	1	2	yes	—
221	C₄²	C₂×C₅×Q₈	1	2	yes	—
222	C₄²	C₅×#41	1	2	yes	—
223	C₂×C₅	C₄×C₄⋊Q₈	1	1	yes	characteristic
224	C₂×C₅	#2949	1	2	yes	—
225	C₂×C₅	#3163	1	2	yes	—
226	C₂×C₅	#3173	1	2	yes	—
227	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
228	Q₈	—	8	8	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
240	C₂×C₄	C₅×C₄⋊Q₈	1	4	yes	—
241	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
242	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
243	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
244	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
245	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
246	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	—	2	2	no	—
249	C₈	C₄×C₅×Q₈	1	4	yes	—
250	C₈	C₄×C₅×Q₈	1	4	yes	—
251	C₅	#21724	1	1	yes	characteristic, direct product factor, Sylow
252	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
253	C₂²	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
254	C₂²	C₅×#532	1	1	yes	characteristic
255	C₂²	C₅×#577	1	4	yes	—
256	C₄	—	2	4	no	—
257	C₄	—	2	4	no	—
258	C₄	—	4	2	no	—
259	C₄	—	4	8	no	—
260	C₄	—	8	4	no	—
261	C₄	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
262	C₄	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
263	C₄	C₅×#533	1	1	yes	characteristic
264	C₄	C₅×#533	1	1	yes	characteristic
265	C₂	C₄×C₅×C₄⋊Q₈	1	1	yes	characteristic
266	C₂	C₅×#2949	1	2	yes	—
267	C₂	C₅×#3163	1	2	yes	—
268	C₂	C₅×#3173	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×#498	C₄	semidirect
30. #21724	C₅	direct (N × H)
52. C₅×#97	Q₈	semidirect
55. C₄×C₅×C₈	Q₈	semidirect
104. #498	C₄×C₅	semidirect
168. #97	C₅×Q₈	semidirect
171. C₄×C₈	C₅×Q₈	semidirect
251. C₅	#21724	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×#498	2
C₅×#2331	2
C₅×#1993	2
C₅×#1944	2
C₅×#1941	2
C₅×#1846	2
C₅×#1776	2
C₄×C₅×#106	2
C₅×#1726	2
#21724	5

Subgroups — C5×#21724 #11944444

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21724 #11944444