Subgroups — C₅×#13035 #11935755

Order 1280.

456 conjugacy classes of subgroups.

The subgroup lattice has 332 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	C₂×C₅×#362	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2143	C₂	1	1	yes	maximal, characteristic
5	C₅×#1979	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1385	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	—	2	2	no	—
13	C₂×C₅×Q₃₂	C₄	1	2	yes	complemented
14	C₂×C₅×SD₃₂	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×#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×#359	—	2	2	no	—
27	#13035	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂×C₅×#41	—	2	1	no	—
29	C₂²×C₅×Q₈	—	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	—	2	1	no	—
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	C₂×C₄	1	2	yes	—
40	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
41	C₂×C₅×SD₁₆	—	2	1	no	—
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	—	2	1	no	—
47	C₂×C₅×M₁₆	—	4	2	no	—
48	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
49	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
50	C₅×#117	D₄	1	1	yes	characteristic
51	C₅×Q₃₂	—	2	4	no	—
52	C₅×SD₃₂	—	2	4	no	—
53	C₅×M₃₂	—	2	1	no	—
54	C₅×M₃₂	—	2	1	no	—
55	C₅×M₃₂	—	2	1	no	—
56	C₅×M₃₂	—	2	1	no	—
57	C₂×C₅×C₁₆	—	2	1	no	—
58	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
59	C₅×#108	—	2	1	no	—
60	C₅×#108	—	2	1	no	—
61	C₅×#108	—	2	1	no	—
62	C₅×#108	D₄	1	2	yes	—
63	C₅×#104	—	2	4	no	—
64	C₅×#103	—	4	2	no	—
65	C₅×#101	—	4	2	no	—
66	C₅×#97	—	2	1	no	—
67	C₄×C₅×C₈	—	2	1	no	—
68	C₂×#510	C₂×C₅	1	1	yes	characteristic
69	#2957	C₂×C₅	1	1	yes	characteristic
70	C₂×#362	C₂×C₅	1	1	yes	characteristic, complemented
71	#2143	C₂×C₅	1	1	yes	characteristic
72	#1979	C₂×C₅	1	1	yes	characteristic, complemented
73	#1385	C₂×C₅	1	1	yes	characteristic, complemented
74	#1378	C₂×C₅	1	1	yes	characteristic, complemented
75	C₅×#41	—	2	2	no	—
76	C₅×#41	—	2	2	no	—
77	C₅×#41	—	4	1	no	—
78	C₅×#41	—	4	1	no	—
79	C₂×C₅×Q₈	—	2	1	no	—
80	C₂×C₅×Q₈	—	2	2	no	—
81	C₂×C₅×Q₈	—	4	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×D₄	—	2	1	no	—
87	C₂×C₅×D₄	—	4	1	no	—
88	C₂²×C₄×C₅	—	4	1	no	—
89	C₂²×C₄×C₅	—	4	1	no	—
90	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₅×Q₁₆	—	2	2	no	—
92	C₅×Q₁₆	—	2	1	no	—
93	C₅×Q₁₆	—	2	2	no	—
94	C₅×Q₁₆	—	4	1	no	—
95	C₅×Q₁₆	—	4	1	no	—
96	C₅×Q₁₆	C₂²⋊C₄	1	2	yes	—
97	C₅×SD₁₆	—	2	2	no	—
98	C₅×SD₁₆	—	2	2	no	—
99	C₅×D₈	—	2	1	no	—
100	C₅×D₈	C₂²⋊C₄	1	2	yes	—
101	C₅×M₁₆	—	2	1	no	—
102	C₅×M₁₆	—	2	2	no	—
103	C₅×M₁₆	—	4	4	no	—
104	C₅×M₁₆	—	4	1	no	—
105	C₅×M₁₆	—	4	1	no	—
106	C₅×M₁₆	—	4	1	no	—
107	C₅×M₁₆	D₈	1	2	yes	—
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	—	4	1	no	—
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	—	2	1	no	—
119	C₅×C₂²⋊C₄	—	2	1	no	—
120	C₄²×C₅	—	2	1	no	—
121	C₅×C₁₆	—	2	2	no	—
122	C₅×C₁₆	—	4	1	no	—
123	C₂×#135	C₂²×C₅	1	1	yes	characteristic
124	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
125	#568	—	2	1	no	—
126	#510	—	2	2	no	—
127	#510	—	2	2	no	—
128	C₂×Q₃₂	C₄×C₅	1	2	yes	complemented
129	C₂×SD₃₂	C₄×C₅	1	2	yes	complemented
130	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
131	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
132	#444	—	2	2	no	—
133	#443	—	2	2	no	—
134	C₂×#108	C₂²×C₅	1	1	yes	characteristic
135	C₂×#108	C₂²×C₅	1	1	yes	characteristic
136	#421	—	2	1	no	—
137	#419	—	2	1	no	—
138	C₂×#101	—	2	1	no	—
139	#405	C₂²×C₅	1	1	yes	characteristic
140	#362	—	2	2	no	—
141	#359	—	2	2	no	—
142	C₂³×C₅	—	4	1	no	—
143	C₂³×C₅	#135	1	1	yes	characteristic
144	C₅×Q₈	—	2	2	no	—
145	C₅×Q₈	—	4	1	no	—
146	C₅×Q₈	—	4	2	no	—
147	C₅×Q₈	—	4	2	no	—
148	C₅×Q₈	—	8	1	no	—
149	C₅×Q₈	—	8	1	no	—
150	C₅×D₄	—	2	2	no	—
151	C₅×D₄	—	4	2	no	—
152	C₅×D₄	—	4	2	no	—
153	C₅×D₄	—	4	1	no	—
154	C₂×C₄×C₅	—	4	1	no	—
155	C₂×C₄×C₅	—	4	2	no	—
156	C₂×C₄×C₅	—	4	2	no	—
157	C₂×C₄×C₅	—	4	2	no	—
158	C₂×C₄×C₅	—	4	1	no	—
159	C₂×C₄×C₅	—	8	1	no	—
160	C₂×C₄×C₅	—	8	1	no	—
161	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
162	C₂×C₄×C₅	#120	1	1	yes	characteristic
163	C₂×C₄×C₅	#121	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
165	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
166	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
167	C₅×C₈	—	2	2	no	—
168	C₅×C₈	—	4	2	no	—
169	C₅×C₈	—	8	1	no	—
170	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
171	C₅×C₈	C₄×D₄	1	1	yes	characteristic
172	C₅×C₈	#121	1	1	yes	characteristic
173	C₅×C₈	#123	1	1	yes	characteristic
174	C₂×#41	—	2	1	no	—
175	C₂²×Q₈	—	2	1	no	—
176	#135	—	2	1	no	—
177	#135	—	2	1	no	—
178	#135	C₅×D₄	1	2	yes	—
179	#135	C₅×D₄	1	2	yes	—
180	C₂×Q₁₆	—	2	1	no	—
181	C₂×Q₁₆	—	2	1	no	—
182	C₂×Q₁₆	—	2	1	no	—
183	C₂×Q₁₆	—	2	1	no	—
184	C₂×Q₁₆	—	2	1	no	—
185	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	—
186	C₂×Q₁₆	C₂×C₄×C₅	1	1	yes	characteristic
187	C₂×SD₁₆	—	2	1	no	—
188	C₂×D₈	C₂×C₄×C₅	1	1	yes	characteristic
189	#131	—	2	4	no	—
190	#131	—	4	2	no	—
191	C₂×M₁₆	—	2	1	no	—
192	C₂×M₁₆	—	2	1	no	—
193	C₂×M₁₆	—	4	2	no	—
194	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
195	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
196	#117	C₅×D₄	1	1	yes	characteristic
197	Q₃₂	—	2	4	no	—
198	SD₃₂	—	2	4	no	—
199	M₃₂	—	2	1	no	—
200	M₃₂	—	2	1	no	—
201	M₃₂	—	2	1	no	—
202	M₃₂	—	2	1	no	—
203	C₂×C₁₆	—	2	1	no	—
204	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
205	#108	—	2	1	no	—
206	#108	—	2	1	no	—
207	#108	—	2	1	no	—
208	#108	C₅×D₄	1	2	yes	—
209	#104	—	2	4	no	—
210	#103	—	4	2	no	—
211	#101	—	4	2	no	—
212	#97	—	2	1	no	—
213	C₄×C₈	—	2	1	no	—
214	C₂²×C₅	—	4	1	no	—
215	C₂²×C₅	—	4	2	no	—
216	C₂²×C₅	—	16	1	no	—
217	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
218	C₂²×C₅	#449	1	1	yes	characteristic
219	C₂²×C₅	#463	1	1	yes	characteristic
220	C₄×C₅	—	4	2	no	—
221	C₄×C₅	—	8	1	no	—
222	C₄×C₅	—	8	2	no	—
223	C₄×C₅	#386	1	1	yes	characteristic
224	C₄×C₅	#442	1	1	yes	characteristic
225	C₄×C₅	#447	1	1	yes	characteristic
226	C₄×C₅	#459	1	1	yes	characteristic
227	#41	—	2	2	no	—
228	#41	—	2	2	no	—
229	#41	—	4	1	no	—
230	#41	—	4	1	no	—
231	C₂×Q₈	—	2	1	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	4	1	no	—
234	C₂×Q₈	—	4	1	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×Q₈	—	4	1	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
243	Q₁₆	—	2	2	no	—
244	Q₁₆	—	2	1	no	—
245	Q₁₆	—	2	2	no	—
246	Q₁₆	—	4	1	no	—
247	Q₁₆	—	4	1	no	—
248	Q₁₆	C₅×C₂²⋊C₄	1	2	yes	—
249	SD₁₆	—	2	2	no	—
250	SD₁₆	—	2	2	no	—
251	D₈	—	2	1	no	—
252	D₈	C₅×C₂²⋊C₄	1	2	yes	—
253	M₁₆	—	2	1	no	—
254	M₁₆	—	2	2	no	—
255	M₁₆	—	4	4	no	—
256	M₁₆	—	4	1	no	—
257	M₁₆	—	4	1	no	—
258	M₁₆	—	4	1	no	—
259	M₁₆	C₅×D₈	1	2	yes	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	4	4	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
265	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
266	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
267	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
268	C₂×C₈	C₅×#41	1	1	yes	characteristic
269	C₂×C₈	C₅×#41	1	1	yes	characteristic
270	C₄⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	2	1	no	—
272	C₄²	—	2	1	no	—
273	C₁₆	—	2	2	no	—
274	C₁₆	—	4	1	no	—
275	C₂×C₅	—	2	1	no	—
276	C₂×C₅	—	2	1	no	—
277	C₂×C₅	—	2	1	no	—
278	C₂×C₅	—	8	2	no	—
279	C₂×C₅	#1879	1	1	yes	characteristic
280	C₂³	—	4	1	no	—
281	C₂³	C₅×#135	1	1	yes	characteristic
282	Q₈	—	2	2	no	—
283	Q₈	—	4	1	no	—
284	Q₈	—	4	2	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	8	1	no	—
287	Q₈	—	8	1	no	—
288	D₄	—	2	2	no	—
289	D₄	—	4	2	no	—
290	D₄	—	4	2	no	—
291	D₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	8	1	no	—
298	C₂×C₄	—	8	1	no	—
299	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
300	C₂×C₄	C₅×#120	1	1	yes	characteristic
301	C₂×C₄	C₅×#121	1	1	yes	characteristic
302	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
303	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
304	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
305	C₈	—	2	2	no	—
306	C₈	—	4	2	no	—
307	C₈	—	8	1	no	—
308	C₈	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
309	C₈	C₄×C₅×D₄	1	1	yes	characteristic
310	C₈	C₅×#121	1	1	yes	characteristic
311	C₈	C₅×#123	1	1	yes	characteristic
312	C₅	#13035	1	1	yes	characteristic, direct product factor, Sylow
313	C₂²	—	4	1	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	16	1	no	—
316	C₂²	C₄×C₅×D₈	1	1	yes	characteristic
317	C₂²	C₅×#449	1	1	yes	characteristic
318	C₂²	C₅×#463	1	1	yes	characteristic
319	C₄	—	4	2	no	—
320	C₄	—	8	1	no	—
321	C₄	—	8	2	no	—
322	C₄	C₅×#386	1	1	yes	characteristic
323	C₄	C₅×#442	1	1	yes	characteristic
324	C₄	C₅×#447	1	1	yes	characteristic
325	C₄	C₅×#459	1	1	yes	characteristic
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	2	1	no	—
329	C₂	—	8	2	no	—
330	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
3. C₂×C₅×#362	C₂	semidirect
5. C₅×#1979	C₂	semidirect
6. C₅×#1385	C₂	semidirect
7. C₅×#1378	C₂	semidirect
13. C₂×C₅×Q₃₂	C₄	semidirect
14. C₂×C₅×SD₃₂	C₄	semidirect
27. #13035	C₅	direct (N × H)
70. C₂×#362	C₂×C₅	semidirect
72. #1979	C₂×C₅	semidirect
73. #1385	C₂×C₅	semidirect
74. #1378	C₂×C₅	semidirect
128. C₂×Q₃₂	C₄×C₅	semidirect
129. C₂×SD₃₂	C₄×C₅	semidirect
312. C₅	#13035	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
C₂×C₅×#362	2
C₅×#2143	2
C₅×#1979	2
C₅×#1385	2
C₅×#1378	2
#13035	5

Subgroups — C5×#13035 #11935755

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13035 #11935755