Subgroups — C₅×#8730 #11931450

Order 1280.

522 conjugacy classes of subgroups.

The subgroup lattice has 274 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×#1458	C₂	1	2	yes	maximal
3	C₅×#1276	C₂	1	2	yes	maximal, complemented
4	C₅×#1274	C₂	1	2	yes	maximal, complemented
5	C₂×C₅×#117	C₄	1	1	yes	characteristic
6	C₂×C₅×#117	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₅	C₄	1	1	yes	characteristic
8	C₅×#431	C₂²	1	4	yes	—
9	C₄×C₅×C₄⋊C₄	—	2	2	no	—
10	C₄×C₅×C₂²⋊C₄	—	2	2	no	—
11	C₅×#376	C₄	1	1	yes	characteristic
12	C₅×#376	C₂²	1	1	yes	characteristic
13	C₄³×C₅	C₄	1	1	yes	characteristic
14	C₄³×C₅	C₂²	1	1	yes	characteristic
15	C₅×#324	—	2	2	no	—
16	C₅×#323	—	2	2	no	—
17	#8730	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₄×C₅	—	2	1	no	—
19	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₅×#117	—	2	2	no	—
21	C₅×#117	—	2	2	no	—
22	C₅×#117	—	2	2	no	—
23	C₅×#117	—	2	2	no	—
24	C₂×C₅×C₄⋊C₄	—	2	1	no	—
25	C₂×C₅×C₄⋊C₄	D₄	1	2	yes	—
26	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
27	C₂×C₅×C₂²⋊C₄	D₄	1	2	yes	—
28	C₂×C₄²×C₅	—	2	1	no	—
29	C₂×C₄²×C₅	—	2	2	no	—
30	C₂×C₄²×C₅	—	2	2	no	—
31	C₂×C₄²×C₅	—	2	2	no	—
32	C₂×C₄²×C₅	C₈	1	4	yes	complemented
33	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
39	C₅×C₄⋊C₈	—	2	4	no	—
40	C₅×C₂²⋊C₈	—	2	4	no	—
41	C₄×C₅×C₈	—	2	4	no	—
42	C₅×#95	—	2	2	no	—
43	C₅×#95	—	2	2	no	—
44	C₄×#117	C₂×C₅	1	1	yes	characteristic
45	#1458	C₂×C₅	1	2	yes	—
46	#1276	C₂×C₅	1	2	yes	complemented
47	#1274	C₂×C₅	1	2	yes	complemented
48	C₂⁴×C₅	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅	—	2	1	no	—
50	C₂²×C₄×C₅	—	2	1	no	—
51	C₂²×C₄×C₅	—	2	1	no	—
52	C₂²×C₄×C₅	—	2	1	no	—
53	C₂²×C₄×C₅	—	2	1	no	—
54	C₂²×C₄×C₅	—	2	4	no	—
55	C₂²×C₄×C₅	—	2	1	no	—
56	C₂²×C₄×C₅	—	2	1	no	—
57	C₂²×C₄×C₅	—	2	1	no	—
58	C₂²×C₄×C₅	—	4	4	no	—
59	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₅	M₁₆	1	2	yes	—
63	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
65	C₂×C₅×C₈	—	4	4	no	—
66	C₅×C₄⋊C₄	—	2	4	no	—
67	C₅×C₄⋊C₄	—	2	4	no	—
68	C₅×C₂²⋊C₄	—	2	4	no	—
69	C₅×C₂²⋊C₄	—	2	4	no	—
70	C₄²×C₅	—	2	4	no	—
71	C₄²×C₅	—	2	1	no	—
72	C₄²×C₅	—	2	1	no	—
73	C₄²×C₅	—	2	1	no	—
74	C₄²×C₅	—	2	1	no	—
75	C₄²×C₅	—	2	1	no	—
76	C₄²×C₅	—	2	1	no	—
77	C₄²×C₅	—	4	4	no	—
78	C₄²×C₅	—	4	4	no	—
79	C₄²×C₅	C₂×C₈	1	2	yes	—
80	C₄²×C₅	C₂×C₈	1	2	yes	—
81	C₄²×C₅	M₁₆	1	2	yes	—
82	C₄²×C₅	M₁₆	1	2	yes	—
83	C₄²×C₅	#41	1	2	yes	—
84	C₄²×C₅	#41	1	2	yes	—
85	C₂×#117	C₄×C₅	1	1	yes	characteristic
86	C₂×#117	C₂²×C₅	1	1	yes	characteristic
87	C₂²×C₄²	C₄×C₅	1	1	yes	characteristic
88	#431	C₂²×C₅	1	4	yes	—
89	C₄×C₄⋊C₄	—	2	2	no	—
90	C₄×C₂²⋊C₄	—	2	2	no	—
91	#376	C₄×C₅	1	1	yes	characteristic
92	#376	C₂²×C₅	1	1	yes	characteristic
93	C₄³	C₄×C₅	1	1	yes	characteristic
94	C₄³	C₂²×C₅	1	1	yes	characteristic
95	#324	—	2	2	no	—
96	#323	—	2	2	no	—
97	C₂³×C₅	—	2	1	no	—
98	C₂³×C₅	—	2	1	no	—
99	C₂³×C₅	—	4	2	no	—
100	C₂³×C₅	C₂²⋊C₈	1	2	yes	—
101	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₅	—	2	4	no	—
103	C₂×C₄×C₅	—	2	1	no	—
104	C₂×C₄×C₅	—	2	2	no	—
105	C₂×C₄×C₅	—	2	1	no	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	—	2	2	no	—
109	C₂×C₄×C₅	—	2	2	no	—
110	C₂×C₄×C₅	—	2	2	no	—
111	C₂×C₄×C₅	—	2	2	no	—
112	C₂×C₄×C₅	—	2	2	no	—
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	4	2	no	—
115	C₂×C₄×C₅	—	4	8	no	—
116	C₂×C₄×C₅	—	4	2	no	—
117	C₂×C₄×C₅	—	4	2	no	—
118	C₂×C₄×C₅	—	4	2	no	—
119	C₂×C₄×C₅	—	4	2	no	—
120	C₂×C₄×C₅	C₂²⋊C₈	1	2	yes	—
121	C₂×C₄×C₅	#117	1	1	yes	characteristic
122	C₂×C₄×C₅	#117	1	1	yes	characteristic
123	C₂×C₄×C₅	#123	1	4	yes	—
124	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
126	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
128	C₅×C₈	—	8	4	no	—
129	C₂³×C₄	—	2	1	no	—
130	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
131	#117	—	2	2	no	—
132	#117	—	2	2	no	—
133	#117	—	2	2	no	—
134	#117	—	2	2	no	—
135	C₂×C₄⋊C₄	—	2	1	no	—
136	C₂×C₄⋊C₄	C₅×D₄	1	2	yes	—
137	C₂×C₂²⋊C₄	—	2	1	no	—
138	C₂×C₂²⋊C₄	C₅×D₄	1	2	yes	—
139	C₂×C₄²	—	2	1	no	—
140	C₂×C₄²	—	2	2	no	—
141	C₂×C₄²	—	2	2	no	—
142	C₂×C₄²	—	2	2	no	—
143	C₂×C₄²	C₅×C₈	1	4	yes	complemented
144	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
145	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
146	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
147	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
148	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
149	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
150	C₄⋊C₈	—	2	4	no	—
151	C₂²⋊C₈	—	2	4	no	—
152	C₄×C₈	—	2	4	no	—
153	#95	—	2	2	no	—
154	#95	—	2	2	no	—
155	C₂²×C₅	—	2	2	no	—
156	C₂²×C₅	—	2	2	no	—
157	C₂²×C₅	—	4	2	no	—
158	C₂²×C₅	—	4	2	no	—
159	C₂²×C₅	—	4	2	no	—
160	C₂²×C₅	#381	1	1	yes	characteristic
161	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
162	C₂²×C₅	#407	1	1	yes	characteristic
163	C₄×C₅	—	2	4	no	—
164	C₄×C₅	—	2	2	no	—
165	C₄×C₅	—	2	4	no	—
166	C₄×C₅	—	4	2	no	—
167	C₄×C₅	—	4	2	no	—
168	C₄×C₅	—	4	2	no	—
169	C₄×C₅	—	4	2	no	—
170	C₄×C₅	#431	1	1	yes	characteristic
171	C₄×C₅	#431	1	1	yes	characteristic
172	C₄×C₅	#432	1	1	yes	characteristic
173	C₄×C₅	#432	1	1	yes	characteristic
174	C₂⁴	C₅×C₂²⋊C₄	1	1	yes	characteristic
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	4	4	no	—
185	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
186	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
187	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
188	C₂²×C₄	C₅×M₁₆	1	2	yes	—
189	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
191	C₂×C₈	—	4	4	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	2	4	no	—
194	C₂²⋊C₄	—	2	4	no	—
195	C₂²⋊C₄	—	2	4	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	2	1	no	—
198	C₄²	—	2	1	no	—
199	C₄²	—	2	1	no	—
200	C₄²	—	2	1	no	—
201	C₄²	—	2	1	no	—
202	C₄²	—	2	4	no	—
203	C₄²	—	4	4	no	—
204	C₄²	—	4	4	no	—
205	C₄²	C₂×C₅×C₈	1	2	yes	—
206	C₄²	C₂×C₅×C₈	1	2	yes	—
207	C₄²	C₅×M₁₆	1	2	yes	—
208	C₄²	C₅×M₁₆	1	2	yes	—
209	C₄²	C₅×#41	1	2	yes	—
210	C₄²	C₅×#41	1	2	yes	—
211	C₂×C₅	—	2	1	no	—
212	C₂×C₅	—	2	1	no	—
213	C₂×C₅	—	4	2	no	—
214	C₂×C₅	#1817	1	1	yes	characteristic
215	C₂×C₅	#1827	1	2	yes	—
216	C₂³	—	2	1	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	4	2	no	—
219	C₂³	C₅×C₂²⋊C₈	1	2	yes	—
220	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	8	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	C₅×C₂²⋊C₈	1	2	yes	—
240	C₂×C₄	C₅×#117	1	1	yes	characteristic
241	C₂×C₄	C₅×#117	1	1	yes	characteristic
242	C₂×C₄	C₅×#123	1	4	yes	—
243	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
244	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
245	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
246	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
247	C₈	—	8	4	no	—
248	C₅	#8730	1	1	yes	characteristic, direct product factor, Sylow
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	4	2	no	—
254	C₂²	C₅×#381	1	1	yes	characteristic
255	C₂²	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
256	C₂²	C₅×#407	1	1	yes	characteristic
257	C₄	—	2	4	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	4	no	—
260	C₄	—	4	2	no	—
261	C₄	—	4	2	no	—
262	C₄	—	4	2	no	—
263	C₄	—	4	2	no	—
264	C₄	C₅×#431	1	1	yes	characteristic
265	C₄	C₅×#431	1	1	yes	characteristic
266	C₄	C₅×#432	1	1	yes	characteristic
267	C₄	C₅×#432	1	1	yes	characteristic
268	C₂	—	2	1	no	—
269	C₂	—	2	1	no	—
270	C₂	—	4	2	no	—
271	C₂	C₅×#1817	1	1	yes	characteristic
272	C₂	C₅×#1827	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	#117
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#381
prime radical	C₂⁴×C₅	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₅×#376 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1276	C₂	semidirect
4. C₅×#1274	C₂	semidirect
17. #8730	C₅	direct (N × H)
32. C₂×C₄²×C₅	C₈	semidirect
46. #1276	C₂×C₅	semidirect
47. #1274	C₂×C₅	semidirect
143. C₂×C₄²	C₅×C₈	semidirect
248. C₅	#8730	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×#1458	2
C₅×#1276	2
C₅×#1274	2
#8730	5

Subgroups — C5×#8730 #11931450

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8730 #11931450