Subgroups — C₅×#17815 #11940535

Order 1280.

638 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2920	C₂	1	2	yes	maximal
2	C₂×C₅×#431	C₂	1	1	yes	maximal, characteristic
3	C₅×#1572	C₂	1	8	yes	maximal, complemented
4	C₅×#1567	C₂	1	2	yes	maximal, complemented
5	C₅×#1566	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₅×#432	—	2	2	no	—
8	C₅×#432	C₂²	1	4	yes	—
9	C₅×#431	—	2	2	no	—
10	C₅×#431	C₄	1	4	yes	—
11	C₅×#431	C₂²	1	4	yes	—
12	C₅×#431	C₂²	1	4	yes	—
13	C₅×#423	C₂²	1	4	yes	—
14	C₂×C₅×C₄⋊C₈	C₄	1	2	yes	—
15	C₅×#407	—	2	2	no	—
16	C₂×C₅×C₂²⋊C₈	C₄	1	2	yes	—
17	C₄×C₅×M₁₆	—	2	2	no	—
18	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₅×#335	C₂²	1	8	yes	complemented
21	C₅×#334	C₂²	1	8	yes	complemented
22	#17815	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₅	Q₈	1	1	yes	characteristic
26	C₂×C₅×M₁₆	—	2	4	no	—
27	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₅	D₄	1	2	yes	—
33	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
34	C₂×C₄²×C₅	Q₈	1	2	yes	—
35	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₅	C₂³	1	2	yes	—
37	C₅×C₄⋊C₈	—	2	4	no	—
38	C₅×C₄⋊C₈	C₂×C₄	1	8	yes	—
39	C₅×C₄⋊C₈	C₂³	1	8	yes	—
40	C₅×C₂²⋊C₈	—	2	4	no	—
41	C₅×C₂²⋊C₈	—	2	4	no	—
42	C₅×C₂²⋊C₈	C₂×C₄	1	8	yes	—
43	C₅×#97	—	2	4	no	—
44	C₄×C₅×C₈	—	2	4	no	—
45	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
46	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
47	C₄×C₅×C₈	C₂³	1	2	yes	—
48	C₄×C₅×C₈	C₂³	1	2	yes	—
49	#2920	C₂×C₅	1	2	yes	—
50	C₂×#431	C₂×C₅	1	1	yes	characteristic
51	#1572	C₂×C₅	1	8	yes	complemented
52	#1567	C₂×C₅	1	2	yes	complemented
53	#1566	C₂×C₅	1	2	yes	complemented
54	C₂⁴×C₅	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅	—	2	4	no	—
56	C₂²×C₄×C₅	—	4	2	no	—
57	C₂²×C₄×C₅	—	4	2	no	—
58	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
70	C₅×M₁₆	—	4	8	no	—
71	C₂×C₅×C₈	—	2	8	no	—
72	C₂×C₅×C₈	—	4	2	no	—
73	C₂×C₅×C₈	—	4	2	no	—
74	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
75	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
76	C₄²×C₅	C₄⋊C₄	1	2	yes	—
77	C₄²×C₅	C₄⋊C₄	1	2	yes	—
78	C₄²×C₅	C₄⋊C₄	1	2	yes	—
79	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₅	C₂²×C₄	1	2	yes	—
81	C₄²×C₅	C₂×D₄	1	2	yes	—
82	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₅	C₂×Q₈	1	2	yes	—
84	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
85	C₄²×C₅	C₂⁴	1	1	yes	characteristic
86	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
87	#432	—	2	2	no	—
88	#432	C₂²×C₅	1	4	yes	—
89	#431	—	2	2	no	—
90	#431	C₄×C₅	1	4	yes	—
91	#431	C₂²×C₅	1	4	yes	—
92	#431	C₂²×C₅	1	4	yes	—
93	#423	C₂²×C₅	1	4	yes	—
94	C₂×C₄⋊C₈	C₄×C₅	1	2	yes	—
95	#407	—	2	2	no	—
96	C₂×C₂²⋊C₈	C₄×C₅	1	2	yes	—
97	C₄×M₁₆	—	2	2	no	—
98	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
99	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
100	#335	C₂²×C₅	1	8	yes	complemented
101	#334	C₂²×C₅	1	8	yes	complemented
102	C₂³×C₅	—	2	2	no	—
103	C₂³×C₅	—	4	1	no	—
104	C₂³×C₅	—	4	1	no	—
105	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
106	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	—	2	2	no	—
109	C₂×C₄×C₅	—	2	4	no	—
110	C₂×C₄×C₅	—	2	4	no	—
111	C₂×C₄×C₅	—	2	4	no	—
112	C₂×C₄×C₅	—	2	4	no	—
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	4	1	no	—
115	C₂×C₄×C₅	—	4	2	no	—
116	C₂×C₄×C₅	—	4	1	no	—
117	C₂×C₄×C₅	—	4	2	no	—
118	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅	#135	1	4	yes	—
125	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
128	C₅×C₈	—	2	4	no	—
129	C₅×C₈	—	2	4	no	—
130	C₅×C₈	—	4	8	no	—
131	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
132	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
133	C₂³×C₄	C₅×Q₈	1	1	yes	characteristic
134	C₂×M₁₆	—	2	4	no	—
135	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
136	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
137	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
138	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
139	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
140	C₂×C₄²	C₅×D₄	1	2	yes	—
141	C₂×C₄²	C₅×Q₈	1	1	yes	characteristic
142	C₂×C₄²	C₅×Q₈	1	2	yes	—
143	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
144	C₂×C₄²	C₂³×C₅	1	2	yes	—
145	C₄⋊C₈	—	2	4	no	—
146	C₄⋊C₈	C₂×C₄×C₅	1	8	yes	—
147	C₄⋊C₈	C₂³×C₅	1	8	yes	—
148	C₂²⋊C₈	—	2	4	no	—
149	C₂²⋊C₈	—	2	4	no	—
150	C₂²⋊C₈	C₂×C₄×C₅	1	8	yes	—
151	#97	—	2	4	no	—
152	C₄×C₈	—	2	4	no	—
153	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
154	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
155	C₄×C₈	C₂³×C₅	1	2	yes	—
156	C₄×C₈	C₂³×C₅	1	2	yes	—
157	C₂²×C₅	—	2	1	no	—
158	C₂²×C₅	—	2	1	no	—
159	C₂²×C₅	—	2	2	no	—
160	C₂²×C₅	—	2	2	no	—
161	C₂²×C₅	—	2	2	no	—
162	C₂²×C₅	—	4	4	no	—
163	C₂²×C₅	#427	1	2	yes	—
164	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
165	C₄×C₅	—	2	1	no	—
166	C₄×C₅	—	2	2	no	—
167	C₄×C₅	—	2	2	no	—
168	C₄×C₅	—	2	2	no	—
169	C₄×C₅	—	2	1	no	—
170	C₄×C₅	#427	1	2	yes	—
171	C₄×C₅	#430	1	2	yes	—
172	C₄×C₅	#568	1	2	yes	—
173	C₄×C₅	C₂×#135	1	2	yes	—
174	C₂⁴	C₅×C₄⋊C₄	1	1	yes	characteristic
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	—	4	2	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
179	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
180	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
181	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
182	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
183	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
184	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
185	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
187	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
190	M₁₆	—	4	8	no	—
191	C₂×C₈	—	2	8	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
195	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
196	C₄²	C₅×C₄⋊C₄	1	2	yes	—
197	C₄²	C₅×C₄⋊C₄	1	2	yes	—
198	C₄²	C₅×C₄⋊C₄	1	2	yes	—
199	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
200	C₄²	C₂²×C₄×C₅	1	2	yes	—
201	C₄²	C₂×C₅×D₄	1	2	yes	—
202	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
203	C₄²	C₂×C₅×Q₈	1	2	yes	—
204	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
205	C₄²	C₂⁴×C₅	1	1	yes	characteristic
206	C₂×C₅	—	2	2	no	—
207	C₂×C₅	—	4	2	no	—
208	C₂×C₅	#2905	1	1	yes	characteristic
209	C₂×C₅	C₂×#427	1	1	yes	characteristic
210	C₂×C₅	C₂×#430	1	1	yes	characteristic
211	C₂³	—	2	2	no	—
212	C₂³	—	4	1	no	—
213	C₂³	—	4	1	no	—
214	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
215	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	4	1	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
228	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
229	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
230	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
231	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
232	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
233	C₂×C₄	C₅×#135	1	4	yes	—
234	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
235	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
236	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
237	C₈	—	2	4	no	—
238	C₈	—	2	4	no	—
239	C₈	—	4	8	no	—
240	C₅	#17815	1	1	yes	characteristic, direct product factor, Sylow
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	4	4	no	—
247	C₂²	C₅×#427	1	2	yes	—
248	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
249	C₄	—	2	1	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	C₅×#427	1	2	yes	—
255	C₄	C₅×#430	1	2	yes	—
256	C₄	C₅×#568	1	2	yes	—
257	C₄	C₂×C₅×#135	1	2	yes	—
258	C₂	—	2	2	no	—
259	C₂	—	4	2	no	—
260	C₂	C₅×#2905	1	1	yes	characteristic
261	C₂	C₂×C₅×#427	1	1	yes	characteristic
262	C₂	C₂×C₅×#430	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₄²	C₂⁴×C₅
socle	C₂²×C₅	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1572	C₂	semidirect
4. C₅×#1567	C₂	semidirect
5. C₅×#1566	C₂	semidirect
20. C₅×#335	C₂²	semidirect
21. C₅×#334	C₂²	semidirect
22. #17815	C₅	direct (N × H)
51. #1572	C₂×C₅	semidirect
52. #1567	C₂×C₅	semidirect
53. #1566	C₂×C₅	semidirect
100. #335	C₂²×C₅	semidirect
101. #334	C₂²×C₅	semidirect
240. C₅	#17815	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅×#2920	2
C₂×C₅×#431	2
C₅×#1572	2
C₅×#1567	2
C₅×#1566	2
#17815	5

Subgroups — C5×#17815 #11940535

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#17815 #11940535