Subgroups — C₅×#12567 #11935287

Order 1280.

586 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#427	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#427	C₂	1	1	yes	maximal, characteristic
4	C₅×#1386	C₂	1	4	yes	maximal, complemented
5	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
6	C₂×C₅×#117	—	2	1	no	—
7	C₂×C₅×#117	—	2	1	no	—
8	C₂×C₅×M₃₂	—	2	1	no	—
9	C₂×C₅×M₃₂	—	2	2	no	—
10	C₂×C₅×M₃₂	C₄	1	4	yes	complemented
11	C₂×C₅×M₃₂	C₂²	1	2	yes	—
12	C₂²×C₅×C₁₆	—	2	1	no	—
13	C₅×#427	—	2	1	no	—
14	C₅×#427	—	2	2	no	—
15	C₅×#427	—	2	1	no	—
16	C₅×#427	C₄	1	2	yes	complemented
17	C₅×#427	C₄	1	4	yes	complemented
18	C₅×#427	C₂²	1	2	yes	—
19	C₂×C₅×#107	—	2	1	no	—
20	C₂×C₅×#107	C₄	1	2	yes	complemented
21	C₂×C₅×#106	—	2	1	no	—
22	C₂×C₅×#106	C₂²	1	2	yes	—
23	#12567	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	D₄	1	2	yes	—
33	C₂²×C₅×C₈	Q₈	1	2	yes	—
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₅×#117	—	2	4	no	—
36	C₅×#117	—	2	2	no	—
37	C₅×#117	—	2	2	no	—
38	C₅×#117	—	4	1	no	—
39	C₅×#117	—	4	2	no	—
40	C₅×#117	—	4	1	no	—
41	C₂×C₅×C₄⋊C₄	—	2	2	no	—
42	C₂×C₅×C₄⋊C₄	—	2	2	no	—
43	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
44	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
45	C₂×C₄²×C₅	—	4	1	no	—
46	C₂×C₄²×C₅	—	4	1	no	—
47	C₅×M₃₂	—	2	4	no	—
48	C₅×M₃₂	—	2	2	no	—
49	C₅×M₃₂	C₂×C₄	1	4	yes	complemented
50	C₂×C₅×C₁₆	—	2	4	no	—
51	C₂×C₅×C₁₆	—	2	2	no	—
52	C₅×#107	—	2	4	no	—
53	C₅×#107	—	2	2	no	—
54	C₅×#107	C₂×C₄	1	4	yes	—
55	C₅×#106	—	2	2	no	—
56	C₅×#106	—	2	2	no	—
57	C₅×#106	—	2	2	no	—
58	C₅×#106	C₂×C₄	1	4	yes	complemented
59	C₂²×M₃₂	C₂×C₅	1	1	yes	characteristic
60	C₂×#427	C₂×C₅	1	1	yes	characteristic
61	C₂×#427	C₂×C₅	1	1	yes	characteristic
62	#1386	C₂×C₅	1	4	yes	complemented
63	C₂⁴×C₅	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₅	—	4	2	no	—
65	C₂²×C₄×C₅	—	4	2	no	—
66	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
71	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
72	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
73	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
74	C₂²×C₄×C₅	Q₁₆	1	2	yes	—
75	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₅×C₈	C₄²	1	2	yes	—
77	C₂×C₅×C₈	C₄²	1	2	yes	—
78	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
81	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
82	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
83	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
84	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
85	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
86	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
87	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
88	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
94	C₅×C₄⋊C₄	—	2	4	no	—
95	C₅×C₄⋊C₄	—	2	4	no	—
96	C₅×C₄⋊C₄	—	4	2	no	—
97	C₅×C₄⋊C₄	—	4	2	no	—
98	C₅×C₂²⋊C₄	—	4	2	no	—
99	C₅×C₂²⋊C₄	—	4	4	no	—
100	C₅×C₂²⋊C₄	—	4	2	no	—
101	C₄²×C₅	—	4	4	no	—
102	C₄²×C₅	—	4	2	no	—
103	C₄²×C₅	—	4	2	no	—
104	C₅×C₁₆	—	2	4	no	—
105	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
106	C₂×#117	—	2	1	no	—
107	C₂×#117	—	2	1	no	—
108	C₂×M₃₂	—	2	2	no	—
109	C₂×M₃₂	—	2	1	no	—
110	C₂×M₃₂	C₄×C₅	1	4	yes	complemented
111	C₂×M₃₂	C₂²×C₅	1	2	yes	—
112	C₂²×C₁₆	—	2	1	no	—
113	#427	—	2	1	no	—
114	#427	—	2	1	no	—
115	#427	—	2	2	no	—
116	#427	C₄×C₅	1	2	yes	complemented
117	#427	C₄×C₅	1	4	yes	complemented
118	#427	C₂²×C₅	1	2	yes	—
119	C₂×#107	—	2	1	no	—
120	C₂×#107	C₄×C₅	1	2	yes	complemented
121	C₂×#106	—	2	1	no	—
122	C₂×#106	C₂²×C₅	1	2	yes	—
123	C₂³×C₅	—	4	2	no	—
124	C₂³×C₅	#102	1	1	yes	characteristic
125	C₂³×C₅	#102	1	1	yes	characteristic
126	C₂³×C₅	#102	1	2	yes	—
127	C₂³×C₅	#107	1	2	yes	—
128	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₅	—	4	4	no	—
130	C₂×C₄×C₅	—	4	4	no	—
131	C₂×C₄×C₅	—	8	4	no	—
132	C₂×C₄×C₅	—	8	4	no	—
133	C₂×C₄×C₅	#95	1	2	yes	—
134	C₂×C₄×C₅	#95	1	2	yes	—
135	C₂×C₄×C₅	#102	1	2	yes	—
136	C₂×C₄×C₅	#102	1	1	yes	characteristic
137	C₂×C₄×C₅	#102	1	1	yes	characteristic
138	C₂×C₄×C₅	#103	1	2	yes	—
139	C₂×C₄×C₅	#103	1	2	yes	—
140	C₂×C₄×C₅	#103	1	2	yes	—
141	C₂×C₄×C₅	#103	1	2	yes	—
142	C₂×C₄×C₅	#106	1	2	yes	—
143	C₂×C₄×C₅	#106	1	2	yes	—
144	C₂×C₄×C₅	#107	1	2	yes	—
145	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
151	C₅×C₈	#95	1	2	yes	—
152	C₅×C₈	#95	1	2	yes	—
153	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
154	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
155	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
158	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
159	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
160	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
161	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
162	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
163	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
164	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
165	C₂²×C₈	C₅×D₄	1	2	yes	—
166	C₂²×C₈	C₅×Q₈	1	2	yes	—
167	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
168	#117	—	2	2	no	—
169	#117	—	2	4	no	—
170	#117	—	2	2	no	—
171	#117	—	4	1	no	—
172	#117	—	4	1	no	—
173	#117	—	4	2	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₂²⋊C₄	—	4	1	no	—
177	C₂×C₂²⋊C₄	—	4	1	no	—
178	C₂×C₄²	—	4	1	no	—
179	C₂×C₄²	—	4	1	no	—
180	M₃₂	—	2	4	no	—
181	M₃₂	—	2	2	no	—
182	M₃₂	C₂×C₄×C₅	1	4	yes	complemented
183	C₂×C₁₆	—	2	4	no	—
184	C₂×C₁₆	—	2	2	no	—
185	#107	—	2	4	no	—
186	#107	—	2	2	no	—
187	#107	C₂×C₄×C₅	1	4	yes	—
188	#106	—	2	2	no	—
189	#106	—	2	2	no	—
190	#106	—	2	2	no	—
191	#106	C₂×C₄×C₅	1	4	yes	complemented
192	C₂²×C₅	—	2	2	no	—
193	C₂²×C₅	—	2	2	no	—
194	C₂²×C₅	—	2	2	no	—
195	C₂²×C₅	—	4	2	no	—
196	C₂²×C₅	—	4	2	no	—
197	C₂²×C₅	#340	1	2	yes	—
198	C₂²×C₅	#340	1	2	yes	—
199	C₂²×C₅	C₂×#102	1	1	yes	characteristic
200	C₂²×C₅	C₂×#102	1	1	yes	characteristic
201	C₂²×C₅	C₂×#107	1	1	yes	characteristic
202	C₄×C₅	—	8	4	no	—
203	C₄×C₅	—	8	4	no	—
204	C₄×C₅	#340	1	2	yes	—
205	C₄×C₅	#340	1	2	yes	—
206	C₄×C₅	C₂×#95	1	1	yes	characteristic
207	C₄×C₅	C₂×#103	1	1	yes	characteristic
208	C₄×C₅	C₂×#103	1	1	yes	characteristic
209	C₄×C₅	C₂×#106	1	1	yes	characteristic
210	C₂⁴	C₅×D₈	1	1	yes	characteristic
211	C₂²×C₄	—	4	2	no	—
212	C₂²×C₄	—	4	2	no	—
213	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
214	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
215	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
216	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
217	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
218	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
219	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
220	C₂²×C₄	C₅×SD₁₆	1	2	yes	—
221	C₂²×C₄	C₅×Q₁₆	1	2	yes	—
222	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
223	C₂×C₈	C₄²×C₅	1	2	yes	—
224	C₂×C₈	C₄²×C₅	1	2	yes	—
225	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
226	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
227	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
228	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
229	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
230	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
231	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
232	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
233	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
234	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
235	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
236	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
237	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
238	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
239	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
240	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₂²⋊C₄	—	4	2	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₂²⋊C₄	—	4	4	no	—
248	C₄²	—	4	2	no	—
249	C₄²	—	4	4	no	—
250	C₄²	—	4	2	no	—
251	C₁₆	—	2	4	no	—
252	C₂×C₅	—	2	1	no	—
253	C₂×C₅	—	2	2	no	—
254	C₂×C₅	—	2	2	no	—
255	C₂×C₅	—	2	1	no	—
256	C₂×C₅	C₂×#340	1	1	yes	characteristic
257	C₂×C₅	#2145	1	2	yes	—
258	C₂³	—	4	2	no	—
259	C₂³	C₅×#102	1	1	yes	characteristic
260	C₂³	C₅×#102	1	1	yes	characteristic
261	C₂³	C₅×#102	1	2	yes	—
262	C₂³	C₅×#107	1	2	yes	—
263	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	8	4	no	—
267	C₂×C₄	—	8	4	no	—
268	C₂×C₄	C₅×#95	1	2	yes	—
269	C₂×C₄	C₅×#95	1	2	yes	—
270	C₂×C₄	C₅×#102	1	2	yes	—
271	C₂×C₄	C₅×#102	1	1	yes	characteristic
272	C₂×C₄	C₅×#102	1	1	yes	characteristic
273	C₂×C₄	C₅×#103	1	2	yes	—
274	C₂×C₄	C₅×#103	1	2	yes	—
275	C₂×C₄	C₅×#103	1	2	yes	—
276	C₂×C₄	C₅×#103	1	2	yes	—
277	C₂×C₄	C₅×#106	1	2	yes	—
278	C₂×C₄	C₅×#106	1	2	yes	—
279	C₂×C₄	C₅×#107	1	2	yes	—
280	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
281	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
282	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
283	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
284	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
285	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
286	C₈	C₅×#95	1	2	yes	—
287	C₈	C₅×#95	1	2	yes	—
288	C₈	C₂×C₄²×C₅	1	1	yes	characteristic
289	C₈	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
290	C₈	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
291	C₈	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
292	C₅	#12567	1	1	yes	characteristic, direct product factor, Sylow
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	4	2	no	—
297	C₂²	—	4	2	no	—
298	C₂²	C₅×#340	1	2	yes	—
299	C₂²	C₅×#340	1	2	yes	—
300	C₂²	C₂×C₅×#102	1	1	yes	characteristic
301	C₂²	C₂×C₅×#102	1	1	yes	characteristic
302	C₂²	C₂×C₅×#107	1	1	yes	characteristic
303	C₄	—	8	4	no	—
304	C₄	—	8	4	no	—
305	C₄	C₅×#340	1	2	yes	—
306	C₄	C₅×#340	1	2	yes	—
307	C₄	C₂×C₅×#95	1	1	yes	characteristic
308	C₄	C₂×C₅×#103	1	1	yes	characteristic
309	C₄	C₂×C₅×#103	1	1	yes	characteristic
310	C₄	C₂×C₅×#106	1	1	yes	characteristic
311	C₂	—	2	1	no	—
312	C₂	—	2	2	no	—
313	C₂	—	2	2	no	—
314	C₂	—	2	1	no	—
315	C₂	C₂×C₅×#340	1	1	yes	characteristic
316	C₂	C₅×#2145	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	#102
commutator	C₈	C₂×C₄²×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×#102
prime radical	C₂⁴×C₅	D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1386	C₂	semidirect
10. C₂×C₅×M₃₂	C₄	semidirect
16. C₅×#427	C₄	semidirect
17. C₅×#427	C₄	semidirect
20. C₂×C₅×#107	C₄	semidirect
23. #12567	C₅	direct (N × H)
49. C₅×M₃₂	C₂×C₄	semidirect
58. C₅×#106	C₂×C₄	semidirect
62. #1386	C₂×C₅	semidirect
110. C₂×M₃₂	C₄×C₅	semidirect
116. #427	C₄×C₅	semidirect
117. #427	C₄×C₅	semidirect
120. C₂×#107	C₄×C₅	semidirect
182. M₃₂	C₂×C₄×C₅	semidirect
191. #106	C₂×C₄×C₅	semidirect
292. C₅	#12567	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×M₃₂	2
C₂×C₅×#427	2
C₅×#1386	2
#12567	5

Subgroups — C5×#12567 #11935287

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12567 #11935287