Subgroups — C₅×#13036 #11935756

Order 1280.

476 conjugacy classes of subgroups.

The subgroup lattice has 342 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	C₅×#2957	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#361	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2143	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1978	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×D₈	C₂²	1	1	yes	characteristic
10	C₅×#568	—	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×#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×#418	—	2	1	no	—
23	C₂×C₅×#100	—	2	1	no	—
24	C₅×#405	C₂²	1	1	yes	characteristic
25	C₅×#361	—	2	2	no	—
26	C₅×#359	—	2	2	no	—
27	#13036	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	—	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×SD₃₂	—	2	4	no	—
52	C₅×D₁₆	—	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×#102	—	4	2	no	—
65	C₅×#100	—	4	2	no	—
66	C₅×#97	—	2	1	no	—
67	C₄×C₅×C₈	—	2	1	no	—
68	C₂×#509	C₂×C₅	1	1	yes	characteristic
69	#2957	C₂×C₅	1	1	yes	characteristic, complemented
70	C₂×#361	C₂×C₅	1	1	yes	characteristic, complemented
71	#2143	C₂×C₅	1	1	yes	characteristic, complemented
72	#1978	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	C₂⁴×C₅	—	4	1	no	—
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	1	no	—
81	C₂×C₅×D₄	—	2	1	no	—
82	C₂×C₅×D₄	—	2	2	no	—
83	C₂×C₅×D₄	—	4	1	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	C₂²×C₄×C₅	—	4	1	no	—
90	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₅×Q₁₆	—	2	1	no	—
92	C₅×Q₁₆	C₂²⋊C₄	1	2	yes	—
93	C₅×SD₁₆	—	2	2	no	—
94	C₅×SD₁₆	—	2	2	no	—
95	C₅×D₈	—	2	2	no	—
96	C₅×D₈	—	2	1	no	—
97	C₅×D₈	—	2	2	no	—
98	C₅×D₈	—	4	1	no	—
99	C₅×D₈	—	4	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×D₈	C₂²×C₅	1	1	yes	characteristic
125	#568	—	2	1	no	—
126	#509	—	2	2	no	—
127	#509	—	2	2	no	—
128	C₂×SD₃₂	C₄×C₅	1	2	yes	complemented
129	C₂×D₁₆	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	#418	—	2	1	no	—
138	C₂×#100	—	2	1	no	—
139	#405	C₂²×C₅	1	1	yes	characteristic
140	#361	—	2	2	no	—
141	#359	—	2	2	no	—
142	C₂³×C₅	—	4	1	no	—
143	C₂³×C₅	—	4	2	no	—
144	C₂³×C₅	—	8	1	no	—
145	C₂³×C₅	—	8	1	no	—
146	C₂³×C₅	—	16	2	no	—
147	C₂³×C₅	#135	1	1	yes	characteristic
148	C₅×Q₈	—	2	2	no	—
149	C₅×Q₈	—	4	1	no	—
150	C₅×D₄	—	2	2	no	—
151	C₅×D₄	—	4	2	no	—
152	C₅×D₄	—	4	2	no	—
153	C₅×D₄	—	4	2	no	—
154	C₅×D₄	—	4	1	no	—
155	C₅×D₄	—	4	2	no	—
156	C₅×D₄	—	8	1	no	—
157	C₅×D₄	—	8	1	no	—
158	C₂×C₄×C₅	—	4	1	no	—
159	C₂×C₄×C₅	—	4	2	no	—
160	C₂×C₄×C₅	—	4	2	no	—
161	C₂×C₄×C₅	—	4	1	no	—
162	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
163	C₂×C₄×C₅	#120	1	1	yes	characteristic
164	C₂×C₄×C₅	#121	1	1	yes	characteristic
165	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
166	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
167	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
168	C₅×C₈	—	2	2	no	—
169	C₅×C₈	—	4	2	no	—
170	C₅×C₈	—	8	1	no	—
171	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
172	C₅×C₈	C₄×D₄	1	1	yes	characteristic
173	C₅×C₈	#121	1	1	yes	characteristic
174	C₅×C₈	#123	1	1	yes	characteristic
175	C₂×#41	—	2	1	no	—
176	C₂²×D₄	—	2	1	no	—
177	#135	—	2	1	no	—
178	#135	—	2	1	no	—
179	#135	C₅×D₄	1	2	yes	—
180	#135	C₅×D₄	1	2	yes	—
181	C₂×Q₁₆	C₂×C₄×C₅	1	1	yes	characteristic
182	C₂×SD₁₆	—	2	1	no	—
183	C₂×D₈	—	2	1	no	—
184	C₂×D₈	—	2	1	no	—
185	C₂×D₈	—	2	1	no	—
186	C₂×D₈	—	2	1	no	—
187	C₂×D₈	—	2	1	no	—
188	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
189	C₂×D₈	C₂×C₄×C₅	1	1	yes	characteristic
190	#131	—	2	4	no	—
191	#131	—	4	2	no	—
192	C₂×M₁₆	—	2	1	no	—
193	C₂×M₁₆	—	2	1	no	—
194	C₂×M₁₆	—	4	2	no	—
195	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
196	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
197	#117	C₅×D₄	1	1	yes	characteristic
198	SD₃₂	—	2	4	no	—
199	D₁₆	—	2	4	no	—
200	M₃₂	—	2	1	no	—
201	M₃₂	—	2	1	no	—
202	M₃₂	—	2	1	no	—
203	M₃₂	—	2	1	no	—
204	C₂×C₁₆	—	2	1	no	—
205	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
206	#108	—	2	1	no	—
207	#108	—	2	1	no	—
208	#108	—	2	1	no	—
209	#108	C₅×D₄	1	2	yes	—
210	#104	—	2	4	no	—
211	#102	—	4	2	no	—
212	#100	—	4	2	no	—
213	#97	—	2	1	no	—
214	C₄×C₈	—	2	1	no	—
215	C₂²×C₅	—	4	1	no	—
216	C₂²×C₅	—	4	2	no	—
217	C₂²×C₅	—	8	2	no	—
218	C₂²×C₅	—	16	1	no	—
219	C₂²×C₅	—	16	2	no	—
220	C₂²×C₅	—	16	2	no	—
221	C₂²×C₅	—	16	2	no	—
222	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
223	C₂²×C₅	#449	1	1	yes	characteristic
224	C₂²×C₅	#463	1	1	yes	characteristic
225	C₄×C₅	—	4	2	no	—
226	C₄×C₅	—	8	1	no	—
227	C₄×C₅	#386	1	1	yes	characteristic
228	C₄×C₅	#442	1	1	yes	characteristic
229	C₄×C₅	#447	1	1	yes	characteristic
230	C₄×C₅	#459	1	1	yes	characteristic
231	C₂⁴	—	4	1	no	—
232	#41	—	2	2	no	—
233	#41	—	2	2	no	—
234	#41	—	4	1	no	—
235	#41	—	4	1	no	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×D₄	—	2	1	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂×D₄	—	4	1	no	—
241	C₂×D₄	—	4	1	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
247	Q₁₆	—	2	1	no	—
248	Q₁₆	C₅×C₂²⋊C₄	1	2	yes	—
249	SD₁₆	—	2	2	no	—
250	SD₁₆	—	2	2	no	—
251	D₈	—	2	2	no	—
252	D₈	—	2	1	no	—
253	D₈	—	2	2	no	—
254	D₈	—	4	1	no	—
255	D₈	—	4	1	no	—
256	D₈	C₅×C₂²⋊C₄	1	2	yes	—
257	M₁₆	—	2	1	no	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	4	4	no	—
260	M₁₆	—	4	1	no	—
261	M₁₆	—	4	1	no	—
262	M₁₆	—	4	1	no	—
263	M₁₆	C₅×D₈	1	2	yes	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	4	4	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
269	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
270	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
271	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
272	C₂×C₈	C₅×#41	1	1	yes	characteristic
273	C₂×C₈	C₅×#41	1	1	yes	characteristic
274	C₄⋊C₄	—	2	1	no	—
275	C₂²⋊C₄	—	2	1	no	—
276	C₄²	—	2	1	no	—
277	C₁₆	—	2	2	no	—
278	C₁₆	—	4	1	no	—
279	C₂×C₅	—	2	1	no	—
280	C₂×C₅	—	2	1	no	—
281	C₂×C₅	—	2	1	no	—
282	C₂×C₅	—	8	2	no	—
283	C₂×C₅	—	16	2	no	—
284	C₂×C₅	#1879	1	1	yes	characteristic
285	C₂³	—	4	1	no	—
286	C₂³	—	4	2	no	—
287	C₂³	—	8	1	no	—
288	C₂³	—	8	1	no	—
289	C₂³	—	16	2	no	—
290	C₂³	C₅×#135	1	1	yes	characteristic
291	Q₈	—	2	2	no	—
292	Q₈	—	4	1	no	—
293	D₄	—	2	2	no	—
294	D₄	—	4	2	no	—
295	D₄	—	4	2	no	—
296	D₄	—	4	2	no	—
297	D₄	—	4	1	no	—
298	D₄	—	4	2	no	—
299	D₄	—	8	1	no	—
300	D₄	—	8	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
306	C₂×C₄	C₅×#120	1	1	yes	characteristic
307	C₂×C₄	C₅×#121	1	1	yes	characteristic
308	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
309	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
310	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	4	2	no	—
313	C₈	—	8	1	no	—
314	C₈	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
315	C₈	C₄×C₅×D₄	1	1	yes	characteristic
316	C₈	C₅×#121	1	1	yes	characteristic
317	C₈	C₅×#123	1	1	yes	characteristic
318	C₅	#13036	1	1	yes	characteristic, direct product factor, Sylow
319	C₂²	—	4	1	no	—
320	C₂²	—	4	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	16	1	no	—
323	C₂²	—	16	2	no	—
324	C₂²	—	16	2	no	—
325	C₂²	—	16	2	no	—
326	C₂²	C₄×C₅×D₈	1	1	yes	characteristic
327	C₂²	C₅×#449	1	1	yes	characteristic
328	C₂²	C₅×#463	1	1	yes	characteristic
329	C₄	—	4	2	no	—
330	C₄	—	8	1	no	—
331	C₄	C₅×#386	1	1	yes	characteristic
332	C₄	C₅×#442	1	1	yes	characteristic
333	C₄	C₅×#447	1	1	yes	characteristic
334	C₄	C₅×#459	1	1	yes	characteristic
335	C₂	—	2	1	no	—
336	C₂	—	2	1	no	—
337	C₂	—	2	1	no	—
338	C₂	—	8	2	no	—
339	C₂	—	16	2	no	—
340	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×#509	C₂
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
2. C₅×#2957	C₂	semidirect
3. C₂×C₅×#361	C₂	semidirect
4. C₅×#2143	C₂	semidirect
5. C₅×#1978	C₂	semidirect
6. C₅×#1385	C₂	semidirect
7. C₅×#1378	C₂	semidirect
13. C₂×C₅×SD₃₂	C₄	semidirect
14. C₂×C₅×D₁₆	C₄	semidirect
27. #13036	C₅	direct (N × H)
69. #2957	C₂×C₅	semidirect
70. C₂×#361	C₂×C₅	semidirect
71. #2143	C₂×C₅	semidirect
72. #1978	C₂×C₅	semidirect
73. #1385	C₂×C₅	semidirect
74. #1378	C₂×C₅	semidirect
128. C₂×SD₃₂	C₄×C₅	semidirect
129. C₂×D₁₆	C₄×C₅	semidirect
318. C₅	#13036	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
C₅×#2957	2
C₂×C₅×#361	2
C₅×#2143	2
C₅×#1978	2
C₅×#1385	2
C₅×#1378	2
#13036	5

Subgroups — C5×#13036 #11935756

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13036 #11935756