Subgroups — C₅×#13031 #11935751

Order 1280.

456 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#510	C₂	1	1	yes	maximal, characteristic
2	C₅×#2957	C₂	1	1	yes	maximal, characteristic
3	C₅×#2148	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#362	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1979	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1388	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1378	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#135	C₂²	1	1	yes	characteristic
9	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
10	C₅×#568	—	2	1	no	—
11	C₅×#510	—	2	2	no	—
12	C₅×#510	C₄	1	4	yes	complemented
13	C₂×C₅×Q₃₂	—	2	1	no	—
14	C₂×C₅×SD₃₂	—	2	1	no	—
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×#443	—	2	2	no	—
19	C₂×C₅×#108	C₂²	1	1	yes	characteristic
20	C₂×C₅×#108	C₂²	1	1	yes	characteristic
21	C₅×#421	—	2	1	no	—
22	C₅×#419	—	2	1	no	—
23	C₂×C₅×#101	—	2	1	no	—
24	C₅×#405	C₂²	1	1	yes	characteristic
25	C₅×#362	—	2	2	no	—
26	C₅×#362	—	2	1	no	—
27	C₅×#361	—	2	1	no	—
28	#13031	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₂×C₅×#41	—	2	1	no	—
30	C₂²×C₅×Q₈	—	2	1	no	—
31	C₅×#135	—	2	1	no	—
32	C₅×#135	—	2	1	no	—
33	C₅×#135	C₂×C₄	1	2	yes	—
34	C₅×#135	D₄	1	2	yes	—
35	C₂×C₅×Q₁₆	—	2	1	no	—
36	C₂×C₅×Q₁₆	—	2	1	no	—
37	C₂×C₅×Q₁₆	—	2	1	no	—
38	C₂×C₅×Q₁₆	—	2	1	no	—
39	C₂×C₅×Q₁₆	—	2	1	no	—
40	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	—
41	C₂×C₅×Q₁₆	D₄	1	1	yes	characteristic
42	C₂×C₅×SD₁₆	—	2	1	no	—
43	C₂×C₅×D₈	D₄	1	1	yes	characteristic
44	C₅×#131	—	2	4	no	—
45	C₅×#131	—	4	2	no	—
46	C₂×C₅×M₁₆	—	2	1	no	—
47	C₂×C₅×M₁₆	—	2	1	no	—
48	C₂×C₅×M₁₆	—	4	2	no	—
49	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
50	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
51	C₅×#117	D₄	1	1	yes	characteristic
52	C₅×Q₃₂	—	2	4	no	—
53	C₅×SD₃₂	—	2	4	no	—
54	C₅×M₃₂	—	2	1	no	—
55	C₅×M₃₂	—	2	1	no	—
56	C₅×M₃₂	—	2	1	no	—
57	C₅×M₃₂	C₂×C₄	1	2	yes	—
58	C₂×C₅×C₁₆	—	2	1	no	—
59	C₂×C₅×C₁₆	—	2	1	no	—
60	C₅×#108	—	2	1	no	—
61	C₅×#108	—	2	1	no	—
62	C₅×#108	—	2	1	no	—
63	C₅×#108	D₄	1	2	yes	—
64	C₅×#104	—	2	4	no	—
65	C₅×#103	—	4	2	no	—
66	C₅×#101	—	4	2	no	—
67	C₅×#97	—	2	1	no	—
68	C₄×C₅×C₈	—	2	1	no	—
69	C₂×#510	C₂×C₅	1	1	yes	characteristic
70	#2957	C₂×C₅	1	1	yes	characteristic
71	#2148	C₂×C₅	1	1	yes	characteristic
72	C₂×#362	C₂×C₅	1	1	yes	characteristic, complemented
73	#1979	C₂×C₅	1	1	yes	characteristic, complemented
74	#1388	C₂×C₅	1	1	yes	characteristic, complemented
75	#1378	C₂×C₅	1	1	yes	characteristic, complemented
76	C₅×#41	—	2	2	no	—
77	C₅×#41	—	2	2	no	—
78	C₅×#41	—	4	1	no	—
79	C₅×#41	—	4	1	no	—
80	C₂×C₅×Q₈	—	2	2	no	—
81	C₂×C₅×Q₈	—	2	1	no	—
82	C₂×C₅×Q₈	—	4	1	no	—
83	C₂×C₅×Q₈	—	4	1	no	—
84	C₂×C₅×Q₈	—	4	1	no	—
85	C₂×C₅×Q₈	—	4	1	no	—
86	C₂×C₅×Q₈	—	4	1	no	—
87	C₂×C₅×D₄	—	2	1	no	—
88	C₂×C₅×D₄	—	4	1	no	—
89	C₂²×C₄×C₅	—	4	1	no	—
90	C₂²×C₄×C₅	—	4	1	no	—
91	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	C₅×Q₁₆	—	2	2	no	—
93	C₅×Q₁₆	—	2	2	no	—
94	C₅×Q₁₆	—	2	1	no	—
95	C₅×Q₁₆	—	4	1	no	—
96	C₅×Q₁₆	—	4	1	no	—
97	C₅×Q₁₆	C₂²⋊C₄	1	2	yes	—
98	C₅×SD₁₆	—	2	2	no	—
99	C₅×SD₁₆	—	2	2	no	—
100	C₅×D₈	—	2	1	no	—
101	C₅×D₈	C₂²⋊C₄	1	2	yes	—
102	C₅×M₁₆	—	2	1	no	—
103	C₅×M₁₆	—	2	2	no	—
104	C₅×M₁₆	—	4	4	no	—
105	C₅×M₁₆	—	4	1	no	—
106	C₅×M₁₆	—	4	1	no	—
107	C₅×M₁₆	—	4	1	no	—
108	C₅×M₁₆	D₈	1	2	yes	—
109	C₂×C₅×C₈	—	2	1	no	—
110	C₂×C₅×C₈	—	4	4	no	—
111	C₂×C₅×C₈	—	4	1	no	—
112	C₂×C₅×C₈	—	4	1	no	—
113	C₂×C₅×C₈	C₂²×C₄	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	C₂×D₄	1	1	yes	characteristic
117	C₂×C₅×C₈	#41	1	1	yes	characteristic
118	C₂×C₅×C₈	#41	1	1	yes	characteristic
119	C₅×C₄⋊C₄	—	2	1	no	—
120	C₅×C₂²⋊C₄	—	2	1	no	—
121	C₄²×C₅	—	2	1	no	—
122	C₅×C₁₆	—	2	2	no	—
123	C₅×C₁₆	—	4	1	no	—
124	C₂×#135	C₂²×C₅	1	1	yes	characteristic
125	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
126	#568	—	2	1	no	—
127	#510	—	2	2	no	—
128	#510	C₄×C₅	1	4	yes	complemented
129	C₂×Q₃₂	—	2	1	no	—
130	C₂×SD₃₂	—	2	1	no	—
131	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
132	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
133	#444	—	2	2	no	—
134	#443	—	2	2	no	—
135	C₂×#108	C₂²×C₅	1	1	yes	characteristic
136	C₂×#108	C₂²×C₅	1	1	yes	characteristic
137	#421	—	2	1	no	—
138	#419	—	2	1	no	—
139	C₂×#101	—	2	1	no	—
140	#405	C₂²×C₅	1	1	yes	characteristic
141	#362	—	2	1	no	—
142	#362	—	2	2	no	—
143	#361	—	2	1	no	—
144	C₂³×C₅	—	4	1	no	—
145	C₂³×C₅	#135	1	1	yes	characteristic
146	C₅×Q₈	—	2	2	no	—
147	C₅×Q₈	—	4	1	no	—
148	C₅×Q₈	—	4	2	no	—
149	C₅×Q₈	—	4	2	no	—
150	C₅×Q₈	—	8	1	no	—
151	C₅×Q₈	—	8	1	no	—
152	C₅×D₄	—	2	2	no	—
153	C₅×D₄	—	4	2	no	—
154	C₅×D₄	—	4	2	no	—
155	C₅×D₄	—	4	1	no	—
156	C₂×C₄×C₅	—	4	1	no	—
157	C₂×C₄×C₅	—	4	2	no	—
158	C₂×C₄×C₅	—	4	2	no	—
159	C₂×C₄×C₅	—	4	2	no	—
160	C₂×C₄×C₅	—	4	1	no	—
161	C₂×C₄×C₅	—	8	1	no	—
162	C₂×C₄×C₅	—	8	1	no	—
163	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
164	C₂×C₄×C₅	#120	1	1	yes	characteristic
165	C₂×C₄×C₅	#121	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
167	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
168	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
169	C₅×C₈	—	2	2	no	—
170	C₅×C₈	—	4	2	no	—
171	C₅×C₈	—	8	1	no	—
172	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
173	C₅×C₈	C₄×D₄	1	1	yes	characteristic
174	C₅×C₈	#121	1	1	yes	characteristic
175	C₅×C₈	#123	1	1	yes	characteristic
176	C₂×#41	—	2	1	no	—
177	C₂²×Q₈	—	2	1	no	—
178	#135	—	2	1	no	—
179	#135	—	2	1	no	—
180	#135	C₂×C₄×C₅	1	2	yes	—
181	#135	C₅×D₄	1	2	yes	—
182	C₂×Q₁₆	—	2	1	no	—
183	C₂×Q₁₆	—	2	1	no	—
184	C₂×Q₁₆	—	2	1	no	—
185	C₂×Q₁₆	—	2	1	no	—
186	C₂×Q₁₆	—	2	1	no	—
187	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
188	C₂×Q₁₆	C₅×D₄	1	1	yes	characteristic
189	C₂×SD₁₆	—	2	1	no	—
190	C₂×D₈	C₅×D₄	1	1	yes	characteristic
191	#131	—	2	4	no	—
192	#131	—	4	2	no	—
193	C₂×M₁₆	—	2	1	no	—
194	C₂×M₁₆	—	2	1	no	—
195	C₂×M₁₆	—	4	2	no	—
196	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
197	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
198	#117	C₅×D₄	1	1	yes	characteristic
199	Q₃₂	—	2	4	no	—
200	SD₃₂	—	2	4	no	—
201	M₃₂	—	2	1	no	—
202	M₃₂	—	2	1	no	—
203	M₃₂	—	2	1	no	—
204	M₃₂	C₂×C₄×C₅	1	2	yes	—
205	C₂×C₁₆	—	2	1	no	—
206	C₂×C₁₆	—	2	1	no	—
207	#108	—	2	1	no	—
208	#108	—	2	1	no	—
209	#108	—	2	1	no	—
210	#108	C₅×D₄	1	2	yes	—
211	#104	—	2	4	no	—
212	#103	—	4	2	no	—
213	#101	—	4	2	no	—
214	#97	—	2	1	no	—
215	C₄×C₈	—	2	1	no	—
216	C₂²×C₅	—	4	1	no	—
217	C₂²×C₅	—	4	2	no	—
218	C₂²×C₅	—	16	1	no	—
219	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
220	C₂²×C₅	#449	1	1	yes	characteristic
221	C₂²×C₅	#463	1	1	yes	characteristic
222	C₄×C₅	—	4	2	no	—
223	C₄×C₅	—	8	1	no	—
224	C₄×C₅	—	8	2	no	—
225	C₄×C₅	#386	1	1	yes	characteristic
226	C₄×C₅	#442	1	1	yes	characteristic
227	C₄×C₅	#447	1	1	yes	characteristic
228	C₄×C₅	#459	1	1	yes	characteristic
229	#41	—	2	2	no	—
230	#41	—	2	2	no	—
231	#41	—	4	1	no	—
232	#41	—	4	1	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	2	1	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×Q₈	—	4	1	no	—
238	C₂×Q₈	—	4	1	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	4	1	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
245	Q₁₆	—	2	2	no	—
246	Q₁₆	—	2	2	no	—
247	Q₁₆	—	2	1	no	—
248	Q₁₆	—	4	1	no	—
249	Q₁₆	—	4	1	no	—
250	Q₁₆	C₅×C₂²⋊C₄	1	2	yes	—
251	SD₁₆	—	2	2	no	—
252	SD₁₆	—	2	2	no	—
253	D₈	—	2	1	no	—
254	D₈	C₅×C₂²⋊C₄	1	2	yes	—
255	M₁₆	—	2	1	no	—
256	M₁₆	—	2	2	no	—
257	M₁₆	—	4	4	no	—
258	M₁₆	—	4	1	no	—
259	M₁₆	—	4	1	no	—
260	M₁₆	—	4	1	no	—
261	M₁₆	C₅×D₈	1	2	yes	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	4	4	no	—
264	C₂×C₈	—	4	1	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
267	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
268	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
269	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
270	C₂×C₈	C₅×#41	1	1	yes	characteristic
271	C₂×C₈	C₅×#41	1	1	yes	characteristic
272	C₄⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	2	1	no	—
274	C₄²	—	2	1	no	—
275	C₁₆	—	2	2	no	—
276	C₁₆	—	4	1	no	—
277	C₂×C₅	—	2	1	no	—
278	C₂×C₅	—	2	1	no	—
279	C₂×C₅	—	2	1	no	—
280	C₂×C₅	—	8	2	no	—
281	C₂×C₅	#1879	1	1	yes	characteristic
282	C₂³	—	4	1	no	—
283	C₂³	C₅×#135	1	1	yes	characteristic
284	Q₈	—	2	2	no	—
285	Q₈	—	4	1	no	—
286	Q₈	—	4	2	no	—
287	Q₈	—	4	2	no	—
288	Q₈	—	8	1	no	—
289	Q₈	—	8	1	no	—
290	D₄	—	2	2	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	8	1	no	—
300	C₂×C₄	—	8	1	no	—
301	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
302	C₂×C₄	C₅×#120	1	1	yes	characteristic
303	C₂×C₄	C₅×#121	1	1	yes	characteristic
304	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
305	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
306	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
307	C₈	—	2	2	no	—
308	C₈	—	4	2	no	—
309	C₈	—	8	1	no	—
310	C₈	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
311	C₈	C₄×C₅×D₄	1	1	yes	characteristic
312	C₈	C₅×#121	1	1	yes	characteristic
313	C₈	C₅×#123	1	1	yes	characteristic
314	C₅	#13031	1	1	yes	characteristic, direct product factor, Sylow
315	C₂²	—	4	1	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	16	1	no	—
318	C₂²	C₄×C₅×D₈	1	1	yes	characteristic
319	C₂²	C₅×#449	1	1	yes	characteristic
320	C₂²	C₅×#463	1	1	yes	characteristic
321	C₄	—	4	2	no	—
322	C₄	—	8	1	no	—
323	C₄	—	8	2	no	—
324	C₄	C₅×#386	1	1	yes	characteristic
325	C₄	C₅×#442	1	1	yes	characteristic
326	C₄	C₅×#447	1	1	yes	characteristic
327	C₄	C₅×#459	1	1	yes	characteristic
328	C₂	—	2	1	no	—
329	C₂	—	2	1	no	—
330	C₂	—	2	1	no	—
331	C₂	—	8	2	no	—
332	C₂	C₅×#1879	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅	#1879
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂×C₅×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#405 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₅×#362	C₂	semidirect
5. C₅×#1979	C₂	semidirect
6. C₅×#1388	C₂	semidirect
7. C₅×#1378	C₂	semidirect
12. C₅×#510	C₄	semidirect
28. #13031	C₅	direct (N × H)
72. C₂×#362	C₂×C₅	semidirect
73. #1979	C₂×C₅	semidirect
74. #1388	C₂×C₅	semidirect
75. #1378	C₂×C₅	semidirect
128. #510	C₄×C₅	semidirect
314. C₅	#13031	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×#510	2
C₅×#2957	2
C₅×#2148	2
C₂×C₅×#362	2
C₅×#1979	2
C₅×#1388	2
C₅×#1378	2
#13031	5

Subgroups — C5×#13031 #11935751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13031 #11935751