Subgroups — C₅×#9102 #11931822

Order 1280.

546 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2982	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#2326	C₂	1	1	yes	maximal, characteristic
3	C₅×#1478	C₂	1	2	yes	maximal
4	C₅×#1461	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1274	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#124	C₄	1	1	yes	characteristic
7	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
9	C₅×C₈×Q₈	—	2	4	no	—
10	C₅×#431	—	2	2	no	—
11	C₅×#431	C₂²	1	2	yes	—
12	C₅×#431	C₂²	1	2	yes	—
13	C₅×#391	C₄	1	1	yes	characteristic
14	C₅×#390	C₄	1	1	yes	characteristic
15	C₅×#388	—	2	2	no	—
16	C₅×#385	—	2	2	no	—
17	C₅×#384	C₄	1	1	yes	characteristic
18	C₅×#376	C₂²	1	1	yes	characteristic
19	C₅×D₄⋊C₈	—	2	4	no	—
20	C₅×#324	—	2	2	no	—
21	#9102	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×D₄	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₅	—	2	1	no	—
25	C₅×#124	—	2	2	no	—
26	C₅×#124	C₈	1	4	yes	complemented
27	C₄×C₅×Q₈	—	2	2	no	—
28	C₄×C₅×Q₈	—	2	2	no	—
29	C₄×C₅×D₄	—	2	2	no	—
30	C₄×C₅×D₄	—	2	2	no	—
31	C₂×C₅×C₄⋊C₄	—	2	2	no	—
32	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
33	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
34	C₂×C₅×C₄⋊C₄	D₄	1	2	yes	—
35	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
36	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
37	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
38	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅	D₄	1	2	yes	—
41	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₅×C₄⋊C₈	—	2	4	no	—
43	C₅×C₄⋊C₈	—	2	2	no	—
44	C₅×C₄⋊C₈	—	2	2	no	—
45	C₅×C₂²⋊C₈	—	2	2	no	—
46	C₅×C₂²⋊C₈	—	2	2	no	—
47	C₄×C₅×C₈	—	2	4	no	—
48	C₄×C₅×C₈	—	2	2	no	—
49	C₄×C₅×C₈	D₄	1	4	yes	—
50	C₅×#95	—	2	2	no	—
51	C₅×#95	—	2	1	no	—
52	C₅×#95	—	2	1	no	—
53	#2982	C₂×C₅	1	1	yes	characteristic, complemented
54	#2326	C₂×C₅	1	1	yes	characteristic
55	#1478	C₂×C₅	1	2	yes	—
56	#1461	C₂×C₅	1	1	yes	characteristic, complemented
57	#1274	C₂×C₅	1	2	yes	complemented
58	C₂⁴×C₅	—	2	1	no	—
59	C₂×C₅×Q₈	—	2	2	no	—
60	C₂×C₅×Q₈	—	2	2	no	—
61	C₂×C₅×Q₈	C₂×C₈	1	2	yes	—
62	C₂×C₅×Q₈	M₁₆	1	2	yes	—
63	C₂×C₅×D₄	—	2	2	no	—
64	C₂×C₅×D₄	—	2	1	no	—
65	C₂×C₅×D₄	—	2	2	no	—
66	C₂×C₅×D₄	C₂×C₈	1	2	yes	—
67	C₂²×C₄×C₅	—	2	1	no	—
68	C₂²×C₄×C₅	—	2	1	no	—
69	C₂²×C₄×C₅	—	2	1	no	—
70	C₂²×C₄×C₅	—	4	2	no	—
71	C₂²×C₄×C₅	—	8	2	no	—
72	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
73	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₂×C₅×C₈	—	2	4	no	—
78	C₂×C₅×C₈	—	4	2	no	—
79	C₅×C₄⋊C₄	—	2	2	no	—
80	C₅×C₄⋊C₄	—	2	4	no	—
81	C₅×C₄⋊C₄	—	2	2	no	—
82	C₅×C₄⋊C₄	—	8	2	no	—
83	C₅×C₂²⋊C₄	—	2	8	no	—
84	C₅×C₂²⋊C₄	—	4	1	no	—
85	C₅×C₂²⋊C₄	—	4	1	no	—
86	C₅×C₂²⋊C₄	—	8	2	no	—
87	C₄²×C₅	—	2	2	no	—
88	C₄²×C₅	—	2	2	no	—
89	C₄²×C₅	—	2	1	no	—
90	C₄²×C₅	—	2	2	no	—
91	C₄²×C₅	C₂×C₈	1	2	yes	—
92	C₄²×C₅	C₂×D₄	1	2	yes	—
93	C₄²×C₅	#41	1	2	yes	—
94	C₂×#124	C₄×C₅	1	1	yes	characteristic
95	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
96	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
97	C₈×Q₈	—	2	4	no	—
98	#431	—	2	2	no	—
99	#431	C₂²×C₅	1	2	yes	—
100	#431	C₂²×C₅	1	2	yes	—
101	#391	C₄×C₅	1	1	yes	characteristic
102	#390	C₄×C₅	1	1	yes	characteristic
103	#388	—	2	2	no	—
104	#385	—	2	2	no	—
105	#384	C₄×C₅	1	1	yes	characteristic
106	#376	C₂²×C₅	1	1	yes	characteristic
107	D₄⋊C₈	—	2	4	no	—
108	#324	—	2	2	no	—
109	C₂³×C₅	—	2	2	no	—
110	C₂³×C₅	—	4	2	no	—
111	C₂³×C₅	—	4	2	no	—
112	C₂³×C₅	—	4	1	no	—
113	C₂³×C₅	—	4	1	no	—
114	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₅×Q₈	—	2	4	no	—
116	C₅×Q₈	—	2	4	no	—
117	C₅×D₄	—	2	4	no	—
118	C₅×D₄	—	4	2	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	—	2	2	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	4	1	no	—
124	C₂×C₄×C₅	—	4	1	no	—
125	C₂×C₄×C₅	—	4	1	no	—
126	C₂×C₄×C₅	—	4	1	no	—
127	C₂×C₄×C₅	—	4	2	no	—
128	C₂×C₄×C₅	—	4	1	no	—
129	C₂×C₄×C₅	—	4	2	no	—
130	C₂×C₄×C₅	—	4	1	no	—
131	C₂×C₄×C₅	—	4	1	no	—
132	C₂×C₄×C₅	—	4	1	no	—
133	C₂×C₄×C₅	—	8	4	no	—
134	C₂×C₄×C₅	C₂²⋊C₈	1	4	yes	—
135	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
136	C₂×C₄×C₅	#120	1	1	yes	characteristic
137	C₂×C₄×C₅	#121	1	2	yes	—
138	C₂×C₄×C₅	#123	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
141	C₂×C₄×C₅	#131	1	2	yes	—
142	C₅×C₈	—	4	4	no	—
143	C₅×C₈	—	8	2	no	—
144	C₂²×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
145	C₂²×D₄	C₂×C₄×C₅	1	1	yes	characteristic
146	C₂³×C₄	—	2	1	no	—
147	#124	—	2	2	no	—
148	#124	C₅×C₈	1	4	yes	complemented
149	C₄×Q₈	—	2	2	no	—
150	C₄×Q₈	—	2	2	no	—
151	C₄×D₄	—	2	2	no	—
152	C₄×D₄	—	2	2	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
155	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
156	C₂×C₄⋊C₄	C₅×D₄	1	2	yes	—
157	C₂×C₂²⋊C₄	—	2	2	no	—
158	C₂×C₂²⋊C₄	—	2	1	no	—
159	C₂×C₂²⋊C₄	—	2	2	no	—
160	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
161	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
162	C₂×C₄²	C₅×D₄	1	2	yes	—
163	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
164	C₄⋊C₈	—	2	4	no	—
165	C₄⋊C₈	—	2	2	no	—
166	C₄⋊C₈	—	2	2	no	—
167	C₂²⋊C₈	—	2	2	no	—
168	C₂²⋊C₈	—	2	2	no	—
169	C₄×C₈	—	2	4	no	—
170	C₄×C₈	—	2	2	no	—
171	C₄×C₈	C₅×D₄	1	4	yes	—
172	#95	—	2	2	no	—
173	#95	—	2	1	no	—
174	#95	—	2	1	no	—
175	C₂²×C₅	—	2	1	no	—
176	C₂²×C₅	—	2	1	no	—
177	C₂²×C₅	—	4	4	no	—
178	C₂²×C₅	—	4	2	no	—
179	C₂²×C₅	—	8	2	no	—
180	C₂²×C₅	—	8	1	no	—
181	C₂²×C₅	—	8	1	no	—
182	C₂²×C₅	#386	1	1	yes	characteristic
183	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
184	C₂²×C₅	#408	1	1	yes	characteristic
185	C₄×C₅	—	2	4	no	—
186	C₄×C₅	—	2	2	no	—
187	C₄×C₅	—	2	1	no	—
188	C₄×C₅	—	2	1	no	—
189	C₄×C₅	—	4	2	no	—
190	C₄×C₅	—	4	2	no	—
191	C₄×C₅	—	8	1	no	—
192	C₄×C₅	—	8	1	no	—
193	C₄×C₅	C₈×D₄	1	2	yes	—
194	C₄×C₅	#435	1	2	yes	—
195	C₂⁴	—	2	1	no	—
196	C₂×Q₈	—	2	2	no	—
197	C₂×Q₈	—	2	2	no	—
198	C₂×Q₈	C₂×C₅×C₈	1	2	yes	—
199	C₂×Q₈	C₅×M₁₆	1	2	yes	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	C₂×C₅×C₈	1	2	yes	—
204	C₂²×C₄	—	2	1	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	—	8	2	no	—
209	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
210	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
211	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	4	2	no	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	8	2	no	—
220	C₂²⋊C₄	—	2	8	no	—
221	C₂²⋊C₄	—	4	1	no	—
222	C₂²⋊C₄	—	4	1	no	—
223	C₂²⋊C₄	—	8	2	no	—
224	C₄²	—	2	2	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	2	1	no	—
227	C₄²	—	2	2	no	—
228	C₄²	C₂×C₅×C₈	1	2	yes	—
229	C₄²	C₂×C₅×D₄	1	2	yes	—
230	C₄²	C₅×#41	1	2	yes	—
231	C₂×C₅	—	2	1	no	—
232	C₂×C₅	—	2	1	no	—
233	C₂×C₅	—	8	2	no	—
234	C₂×C₅	#1869	1	1	yes	characteristic
235	C₂×C₅	#1910	1	1	yes	characteristic
236	C₂×C₅	#1912	1	1	yes	characteristic
237	C₂³	—	2	2	no	—
238	C₂³	—	4	2	no	—
239	C₂³	—	4	2	no	—
240	C₂³	—	4	1	no	—
241	C₂³	—	4	1	no	—
242	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
243	Q₈	—	2	4	no	—
244	Q₈	—	2	4	no	—
245	D₄	—	2	4	no	—
246	D₄	—	4	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	8	4	no	—
262	C₂×C₄	C₅×C₂²⋊C₈	1	4	yes	—
263	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
264	C₂×C₄	C₅×#120	1	1	yes	characteristic
265	C₂×C₄	C₅×#121	1	2	yes	—
266	C₂×C₄	C₅×#123	1	1	yes	characteristic
267	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
268	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
269	C₂×C₄	C₅×#131	1	2	yes	—
270	C₈	—	4	4	no	—
271	C₈	—	8	2	no	—
272	C₅	#9102	1	1	yes	characteristic, direct product factor, Sylow
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	4	4	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	8	2	no	—
278	C₂²	—	8	1	no	—
279	C₂²	—	8	1	no	—
280	C₂²	C₅×#386	1	1	yes	characteristic
281	C₂²	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
282	C₂²	C₅×#408	1	1	yes	characteristic
283	C₄	—	2	4	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	2	no	—
289	C₄	—	8	1	no	—
290	C₄	—	8	1	no	—
291	C₄	C₅×C₈×D₄	1	2	yes	—
292	C₄	C₅×#435	1	2	yes	—
293	C₂	—	2	1	no	—
294	C₂	—	2	1	no	—
295	C₂	—	8	2	no	—
296	C₂	C₅×#1869	1	1	yes	characteristic
297	C₂	C₅×#1910	1	1	yes	characteristic
298	C₂	C₅×#1912	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	#386
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂²×C₅×D₄	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#2982	C₂	semidirect
4. C₅×#1461	C₂	semidirect
5. C₅×#1274	C₂	semidirect
21. #9102	C₅	direct (N × H)
26. C₅×#124	C₈	semidirect
53. #2982	C₂×C₅	semidirect
56. #1461	C₂×C₅	semidirect
57. #1274	C₂×C₅	semidirect
148. #124	C₅×C₈	semidirect
272. C₅	#9102	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×#2982	2
C₅×#2326	2
C₅×#1478	2
C₅×#1461	2
C₅×#1274	2
#9102	5

Subgroups — C5×#9102 #11931822

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9102 #11931822