Subgroups — C₅×#13040 #11935760

Order 1280.

508 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#509	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#444	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#361	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#360	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1972	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1383	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1375	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#135	C₂²	1	1	yes	characteristic
9	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×#131	—	2	1	no	—
11	C₅×#509	—	2	2	no	—
12	C₅×#509	—	2	2	no	—
13	C₂×C₅×SD₃₂	C₄	1	2	yes	complemented
14	C₂×C₅×D₁₆	C₄	1	2	yes	complemented
15	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
17	C₅×#444	—	2	2	no	—
18	C₅×#444	—	2	2	no	—
19	C₂×C₅×#108	C₂²	1	1	yes	characteristic
20	C₂×C₅×#106	C₂²	1	1	yes	characteristic
21	C₂×C₅×#104	—	2	1	no	—
22	C₂×C₅×#102	—	2	1	no	—
23	C₂×C₅×#97	C₂²	1	1	yes	characteristic
24	C₅×#390	—	2	1	no	—
25	C₅×#361	—	2	2	no	—
26	C₅×#360	—	2	2	no	—
27	#13040	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂×C₅×#41	—	2	1	no	—
29	C₂²×C₅×D₄	—	2	1	no	—
30	C₅×#135	—	2	1	no	—
31	C₅×#135	—	2	1	no	—
32	C₅×#135	D₄	1	2	yes	—
33	C₅×#135	D₄	1	2	yes	—
34	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂×C₅×SD₁₆	—	2	1	no	—
36	C₂×C₅×D₈	—	2	1	no	—
37	C₂×C₅×D₈	—	2	1	no	—
38	C₂×C₅×D₈	—	2	1	no	—
39	C₂×C₅×D₈	—	2	1	no	—
40	C₂×C₅×D₈	—	2	1	no	—
41	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
42	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
43	C₅×#131	—	2	4	no	—
44	C₅×#131	—	4	2	no	—
45	C₂×C₅×M₁₆	—	2	1	no	—
46	C₂×C₅×M₁₆	—	4	1	no	—
47	C₂²×C₅×C₈	—	4	1	no	—
48	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
49	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
50	C₂×C₅×C₄⋊C₄	—	2	1	no	—
51	C₂×C₅×C₂²⋊C₄	—	4	2	no	—
52	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
53	C₅×SD₃₂	—	2	4	no	—
54	C₅×D₁₆	—	2	4	no	—
55	C₅×M₃₂	—	2	1	no	—
56	C₅×M₃₂	—	2	1	no	—
57	C₅×M₃₂	—	2	1	no	—
58	C₅×M₃₂	—	2	1	no	—
59	C₂×C₅×C₁₆	—	2	1	no	—
60	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
61	C₅×#108	—	2	1	no	—
62	C₅×#108	D₄	1	2	yes	—
63	C₅×#106	—	2	1	no	—
64	C₅×#106	—	2	1	no	—
65	C₅×#104	—	2	4	no	—
66	C₅×#102	—	4	2	no	—
67	C₅×#97	—	2	1	no	—
68	C₅×#97	—	2	1	no	—
69	C₂×#509	C₂×C₅	1	1	yes	characteristic
70	C₂×#444	C₂×C₅	1	1	yes	characteristic, complemented
71	C₂×#361	C₂×C₅	1	1	yes	characteristic, complemented
72	C₂×#360	C₂×C₅	1	1	yes	characteristic, complemented
73	#1972	C₂×C₅	1	1	yes	characteristic, complemented
74	#1383	C₂×C₅	1	1	yes	characteristic, complemented
75	#1375	C₂×C₅	1	1	yes	characteristic, complemented
76	C₂⁴×C₅	—	4	1	no	—
77	C₅×#41	—	2	2	no	—
78	C₅×#41	—	2	2	no	—
79	C₅×#41	—	4	1	no	—
80	C₅×#41	—	4	1	no	—
81	C₂×C₅×Q₈	—	2	1	no	—
82	C₂×C₅×D₄	—	2	1	no	—
83	C₂×C₅×D₄	—	2	2	no	—
84	C₂×C₅×D₄	—	4	1	no	—
85	C₂×C₅×D₄	—	4	1	no	—
86	C₂×C₅×D₄	—	4	1	no	—
87	C₂×C₅×D₄	—	4	1	no	—
88	C₂×C₅×D₄	—	4	1	no	—
89	C₂×C₅×D₄	—	4	1	no	—
90	C₂²×C₄×C₅	—	2	1	no	—
91	C₂²×C₄×C₅	—	4	1	no	—
92	C₂²×C₄×C₅	—	4	1	no	—
93	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
94	C₅×Q₁₆	—	2	1	no	—
95	C₅×Q₁₆	C₂²⋊C₄	1	2	yes	complemented
96	C₅×SD₁₆	—	2	2	no	—
97	C₅×SD₁₆	—	2	2	no	—
98	C₅×D₈	—	2	2	no	—
99	C₅×D₈	—	2	1	no	—
100	C₅×D₈	—	2	2	no	—
101	C₅×D₈	—	4	1	no	—
102	C₅×D₈	—	4	1	no	—
103	C₅×D₈	C₂²⋊C₄	1	2	yes	complemented
104	C₅×M₁₆	—	2	2	no	—
105	C₅×M₁₆	—	4	4	no	—
106	C₅×M₁₆	—	4	1	no	—
107	C₂×C₅×C₈	—	2	1	no	—
108	C₂×C₅×C₈	—	2	1	no	—
109	C₂×C₅×C₈	—	4	4	no	—
110	C₂×C₅×C₈	—	4	1	no	—
111	C₂×C₅×C₈	D₈	1	2	yes	—
112	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
114	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
115	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
116	C₂×C₅×C₈	#41	1	1	yes	characteristic
117	C₂×C₅×C₈	#41	1	1	yes	characteristic
118	C₅×C₄⋊C₄	—	4	1	no	—
119	C₅×C₄⋊C₄	—	4	1	no	—
120	C₅×C₂²⋊C₄	—	8	4	no	—
121	C₄²×C₅	—	2	1	no	—
122	C₄²×C₅	—	2	1	no	—
123	C₅×C₁₆	—	2	2	no	—
124	C₅×C₁₆	—	4	1	no	—
125	C₂×#135	C₂²×C₅	1	1	yes	characteristic
126	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
127	C₂×#131	—	2	1	no	—
128	#509	—	2	2	no	—
129	#509	—	2	2	no	—
130	C₂×SD₃₂	C₄×C₅	1	2	yes	complemented
131	C₂×D₁₆	C₄×C₅	1	2	yes	complemented
132	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
133	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
134	#444	—	2	2	no	—
135	#444	—	2	2	no	—
136	C₂×#108	C₂²×C₅	1	1	yes	characteristic
137	C₂×#106	C₂²×C₅	1	1	yes	characteristic
138	C₂×#104	—	2	1	no	—
139	C₂×#102	—	2	1	no	—
140	C₂×#97	C₂²×C₅	1	1	yes	characteristic
141	#390	—	2	1	no	—
142	#361	—	2	2	no	—
143	#360	—	2	2	no	—
144	C₂³×C₅	—	4	1	no	—
145	C₂³×C₅	—	4	2	no	—
146	C₂³×C₅	—	8	2	no	—
147	C₂³×C₅	—	8	2	no	—
148	C₂³×C₅	—	8	1	no	—
149	C₂³×C₅	—	8	1	no	—
150	C₂³×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₅×Q₈	—	2	2	no	—
152	C₅×Q₈	—	4	1	no	—
153	C₅×D₄	—	2	2	no	—
154	C₅×D₄	—	4	2	no	—
155	C₅×D₄	—	4	2	no	—
156	C₅×D₄	—	4	2	no	—
157	C₅×D₄	—	4	1	no	—
158	C₅×D₄	—	4	2	no	—
159	C₅×D₄	—	8	1	no	—
160	C₅×D₄	—	8	1	no	—
161	C₂×C₄×C₅	—	4	1	no	—
162	C₂×C₄×C₅	—	4	2	no	—
163	C₂×C₄×C₅	—	4	2	no	—
164	C₂×C₄×C₅	—	4	2	no	—
165	C₂×C₄×C₅	—	4	1	no	—
166	C₂×C₄×C₅	—	8	2	no	—
167	C₂×C₄×C₅	—	8	1	no	—
168	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
169	C₂×C₄×C₅	#120	1	1	yes	characteristic
170	C₂×C₄×C₅	#121	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
172	C₂×C₄×C₅	#135	1	1	yes	characteristic
173	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
174	C₅×C₈	—	2	2	no	—
175	C₅×C₈	—	4	2	no	—
176	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
177	C₅×C₈	C₄×D₄	1	1	yes	characteristic
178	C₅×C₈	#121	1	1	yes	characteristic
179	C₅×C₈	#123	1	1	yes	characteristic
180	C₂×#41	—	2	1	no	—
181	C₂²×D₄	—	2	1	no	—
182	#135	—	2	1	no	—
183	#135	—	2	1	no	—
184	#135	C₅×D₄	1	2	yes	—
185	#135	C₅×D₄	1	2	yes	—
186	C₂×Q₁₆	C₂×C₄×C₅	1	1	yes	characteristic
187	C₂×SD₁₆	—	2	1	no	—
188	C₂×D₈	—	2	1	no	—
189	C₂×D₈	—	2	1	no	—
190	C₂×D₈	—	2	1	no	—
191	C₂×D₈	—	2	1	no	—
192	C₂×D₈	—	2	1	no	—
193	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
194	C₂×D₈	C₂×C₄×C₅	1	1	yes	characteristic
195	#131	—	2	4	no	—
196	#131	—	4	2	no	—
197	C₂×M₁₆	—	2	1	no	—
198	C₂×M₁₆	—	4	1	no	—
199	C₂²×C₈	—	4	1	no	—
200	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
201	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₂²⋊C₄	—	4	2	no	—
204	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
205	SD₃₂	—	2	4	no	—
206	D₁₆	—	2	4	no	—
207	M₃₂	—	2	1	no	—
208	M₃₂	—	2	1	no	—
209	M₃₂	—	2	1	no	—
210	M₃₂	—	2	1	no	—
211	C₂×C₁₆	—	2	1	no	—
212	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
213	#108	—	2	1	no	—
214	#108	C₅×D₄	1	2	yes	—
215	#106	—	2	1	no	—
216	#106	—	2	1	no	—
217	#104	—	2	4	no	—
218	#102	—	4	2	no	—
219	#97	—	2	1	no	—
220	#97	—	2	1	no	—
221	C₂²×C₅	—	2	2	no	—
222	C₂²×C₅	—	4	2	no	—
223	C₂²×C₅	—	8	2	no	—
224	C₂²×C₅	—	8	2	no	—
225	C₂²×C₅	—	8	4	no	—
226	C₂²×C₅	—	16	2	no	—
227	C₂²×C₅	—	16	1	no	—
228	C₂²×C₅	—	16	1	no	—
229	C₂²×C₅	#442	1	1	yes	characteristic
230	C₂²×C₅	#449	1	1	yes	characteristic
231	C₂²×C₅	#459	1	1	yes	characteristic
232	C₄×C₅	—	4	2	no	—
233	C₄×C₅	—	8	1	no	—
234	C₄×C₅	—	16	1	no	—
235	C₄×C₅	#386	1	1	yes	characteristic
236	C₄×C₅	C₄×D₈	1	1	yes	characteristic
237	C₄×C₅	#447	1	1	yes	characteristic
238	C₄×C₅	#463	1	1	yes	characteristic
239	C₂⁴	—	4	1	no	—
240	#41	—	2	2	no	—
241	#41	—	2	2	no	—
242	#41	—	4	1	no	—
243	#41	—	4	1	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×D₄	—	2	1	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	4	1	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂×D₄	—	4	1	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
257	Q₁₆	—	2	1	no	—
258	Q₁₆	C₅×C₂²⋊C₄	1	2	yes	complemented
259	SD₁₆	—	2	2	no	—
260	SD₁₆	—	2	2	no	—
261	D₈	—	2	2	no	—
262	D₈	—	2	1	no	—
263	D₈	—	2	2	no	—
264	D₈	—	4	1	no	—
265	D₈	—	4	1	no	—
266	D₈	C₅×C₂²⋊C₄	1	2	yes	complemented
267	M₁₆	—	2	2	no	—
268	M₁₆	—	4	4	no	—
269	M₁₆	—	4	1	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	4	4	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	C₅×D₈	1	2	yes	—
275	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
276	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
277	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
278	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
279	C₂×C₈	C₅×#41	1	1	yes	characteristic
280	C₂×C₈	C₅×#41	1	1	yes	characteristic
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₂²⋊C₄	—	8	4	no	—
284	C₄²	—	2	1	no	—
285	C₄²	—	2	1	no	—
286	C₁₆	—	2	2	no	—
287	C₁₆	—	4	1	no	—
288	C₂×C₅	—	2	1	no	—
289	C₂×C₅	—	2	1	no	—
290	C₂×C₅	—	8	2	no	—
291	C₂×C₅	—	16	2	no	—
292	C₂×C₅	#1879	1	1	yes	characteristic
293	C₂×C₅	AGL(1,ℤ/16)	1	2	yes	—
294	C₂³	—	4	1	no	—
295	C₂³	—	4	2	no	—
296	C₂³	—	8	2	no	—
297	C₂³	—	8	2	no	—
298	C₂³	—	8	1	no	—
299	C₂³	—	8	1	no	—
300	C₂³	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
301	Q₈	—	2	2	no	—
302	Q₈	—	4	1	no	—
303	D₄	—	2	2	no	—
304	D₄	—	4	2	no	—
305	D₄	—	4	2	no	—
306	D₄	—	4	2	no	—
307	D₄	—	4	1	no	—
308	D₄	—	4	2	no	—
309	D₄	—	8	1	no	—
310	D₄	—	8	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	8	2	no	—
317	C₂×C₄	—	8	1	no	—
318	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
319	C₂×C₄	C₅×#120	1	1	yes	characteristic
320	C₂×C₄	C₅×#121	1	1	yes	characteristic
321	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
322	C₂×C₄	C₅×#135	1	1	yes	characteristic
323	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
324	C₈	—	2	2	no	—
325	C₈	—	4	2	no	—
326	C₈	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
327	C₈	C₄×C₅×D₄	1	1	yes	characteristic
328	C₈	C₅×#121	1	1	yes	characteristic
329	C₈	C₅×#123	1	1	yes	characteristic
330	C₅	#13040	1	1	yes	characteristic, direct product factor, Sylow
331	C₂²	—	2	2	no	—
332	C₂²	—	4	2	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	8	4	no	—
336	C₂²	—	16	2	no	—
337	C₂²	—	16	1	no	—
338	C₂²	—	16	1	no	—
339	C₂²	C₅×#442	1	1	yes	characteristic
340	C₂²	C₅×#449	1	1	yes	characteristic
341	C₂²	C₅×#459	1	1	yes	characteristic
342	C₄	—	4	2	no	—
343	C₄	—	8	1	no	—
344	C₄	—	16	1	no	—
345	C₄	C₅×#386	1	1	yes	characteristic
346	C₄	C₄×C₅×D₈	1	1	yes	characteristic
347	C₄	C₅×#447	1	1	yes	characteristic
348	C₄	C₅×#463	1	1	yes	characteristic
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	8	2	no	—
352	C₂	—	16	2	no	—
353	C₂	C₅×#1879	1	1	yes	characteristic
354	C₂	C₅×AGL(1,ℤ/16)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	#442
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂×C₅×#509	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#97 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#444	C₂	semidirect
3. C₂×C₅×#361	C₂	semidirect
4. C₂×C₅×#360	C₂	semidirect
5. C₅×#1972	C₂	semidirect
6. C₅×#1383	C₂	semidirect
7. C₅×#1375	C₂	semidirect
13. C₂×C₅×SD₃₂	C₄	semidirect
14. C₂×C₅×D₁₆	C₄	semidirect
27. #13040	C₅	direct (N × H)
70. C₂×#444	C₂×C₅	semidirect
71. C₂×#361	C₂×C₅	semidirect
72. C₂×#360	C₂×C₅	semidirect
73. #1972	C₂×C₅	semidirect
74. #1383	C₂×C₅	semidirect
75. #1375	C₂×C₅	semidirect
95. C₅×Q₁₆	C₂²⋊C₄	semidirect
103. C₅×D₈	C₂²⋊C₄	semidirect
130. C₂×SD₃₂	C₄×C₅	semidirect
131. C₂×D₁₆	C₄×C₅	semidirect
258. Q₁₆	C₅×C₂²⋊C₄	semidirect
266. D₈	C₅×C₂²⋊C₄	semidirect
330. C₅	#13040	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×#509	2
C₂×C₅×#444	2
C₂×C₅×#361	2
C₂×C₅×#360	2
C₅×#1972	2
C₅×#1383	2
C₅×#1375	2
#13040	5

Subgroups — C5×#13040 #11935760

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13040 #11935760