Subgroups — C₅×#13019 #11935739

Order 1280.

496 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×SD₃₂	C₂	1	1	yes	maximal, characteristic
2	C₅×#2948	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#361	C₂	1	1	yes	maximal, characteristic
4	C₅×#2141	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1979	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1382	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1379	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
10	C₅×#518	—	2	1	no	—
11	C₂×C₅×SD₃₂	—	2	2	no	—
12	C₂×C₅×SD₃₂	—	2	2	no	—
13	C₂×C₅×SD₃₂	C₄	1	4	yes	complemented
14	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
16	C₅×#442	—	2	2	no	—
17	C₄×C₅×D₈	—	2	2	no	—
18	C₂×C₅×#108	C₂²	1	1	yes	characteristic
19	C₅×#427	C₂²	1	1	yes	characteristic
20	C₅×#419	—	2	1	no	—
21	C₅×#418	—	2	1	no	—
22	C₂×C₅×#101	—	2	1	no	—
23	C₅×#405	C₂²	1	1	yes	characteristic
24	C₅×#361	—	2	2	no	—
25	C₅×#358	—	2	2	no	—
26	#13019	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂²×C₅×Q₈	—	2	1	no	—
28	C₂²×C₅×D₄	—	2	1	no	—
29	C₂×C₅×Q₁₆	—	2	1	no	—
30	C₂×C₅×Q₁₆	—	2	1	no	—
31	C₂×C₅×Q₁₆	—	2	1	no	—
32	C₂×C₅×Q₁₆	—	2	1	no	—
33	C₂×C₅×Q₁₆	—	2	1	no	—
34	C₂×C₅×Q₁₆	C₂×C₄	1	2	yes	complemented
35	C₂×C₅×D₈	—	2	1	no	—
36	C₂×C₅×D₈	—	2	1	no	—
37	C₂×C₅×D₈	—	2	1	no	—
38	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
39	C₂×C₅×D₈	D₄	1	2	yes	—
40	C₂×C₅×D₈	D₄	1	2	yes	—
41	C₂×C₅×M₁₆	—	2	1	no	—
42	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
43	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₄×C₅×D₄	—	2	4	no	—
45	C₄×C₅×D₄	—	4	2	no	—
46	C₅×#117	—	2	1	no	—
47	C₅×#117	D₄	1	1	yes	characteristic
48	C₂×C₅×C₂²⋊C₄	—	4	2	no	—
49	C₅×SD₃₂	—	2	8	no	—
50	C₅×M₃₂	—	2	1	no	—
51	C₅×M₃₂	—	2	1	no	—
52	C₂×C₅×C₁₆	—	2	1	no	—
53	C₂×C₅×C₁₆	—	2	1	no	—
54	C₂×C₅×C₁₆	—	2	1	no	—
55	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	complemented
56	C₅×#108	—	2	1	no	—
57	C₅×#108	—	2	1	no	—
58	C₅×#107	D₄	1	2	yes	—
59	C₅×#106	—	2	1	no	—
60	C₅×#103	—	4	2	no	—
61	C₅×#102	—	2	4	no	—
62	C₅×#101	—	4	2	no	—
63	C₅×#97	—	2	1	no	—
64	C₄×C₅×C₈	—	2	1	no	—
65	C₂²×SD₃₂	C₂×C₅	1	1	yes	characteristic
66	#2948	C₂×C₅	1	1	yes	characteristic
67	C₂×#361	C₂×C₅	1	1	yes	characteristic
68	#2141	C₂×C₅	1	1	yes	characteristic, complemented
69	#1979	C₂×C₅	1	1	yes	characteristic, complemented
70	#1382	C₂×C₅	1	1	yes	characteristic, complemented
71	#1379	C₂×C₅	1	1	yes	characteristic, complemented
72	C₂⁴×C₅	—	4	1	no	—
73	C₂×C₅×Q₈	—	2	2	no	—
74	C₂×C₅×Q₈	—	4	1	no	—
75	C₂×C₅×Q₈	—	4	1	no	—
76	C₂×C₅×Q₈	—	4	1	no	—
77	C₂×C₅×Q₈	—	4	1	no	—
78	C₂×C₅×Q₈	—	4	1	no	—
79	C₂×C₅×D₄	—	2	2	no	—
80	C₂×C₅×D₄	—	2	2	no	—
81	C₂×C₅×D₄	—	2	2	no	—
82	C₂×C₅×D₄	—	4	1	no	—
83	C₂×C₅×D₄	—	4	1	no	—
84	C₂×C₅×D₄	—	4	1	no	—
85	C₂²×C₄×C₅	—	4	4	no	—
86	C₂²×C₄×C₅	—	4	1	no	—
87	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₅×Q₁₆	—	2	2	no	—
89	C₅×Q₁₆	—	2	2	no	—
90	C₅×Q₁₆	—	4	1	no	—
91	C₅×Q₁₆	—	4	1	no	—
92	C₅×D₈	—	2	2	no	—
93	C₅×D₈	—	2	2	no	—
94	C₅×D₈	—	2	2	no	—
95	C₅×D₈	C₂²⋊C₄	1	4	yes	—
96	C₅×M₁₆	—	2	1	no	—
97	C₅×M₁₆	—	2	1	no	—
98	C₅×M₁₆	—	4	1	no	—
99	C₅×M₁₆	—	4	1	no	—
100	C₂×C₅×C₈	—	2	1	no	—
101	C₂×C₅×C₈	—	4	1	no	—
102	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
105	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
106	C₂×C₅×C₈	#41	1	1	yes	characteristic
107	C₂×C₅×C₈	#41	1	1	yes	characteristic
108	C₅×C₄⋊C₄	—	2	2	no	—
109	C₅×C₄⋊C₄	D₈	1	2	yes	—
110	C₅×C₂²⋊C₄	—	2	1	no	—
111	C₅×C₂²⋊C₄	—	4	4	no	—
112	C₅×C₂²⋊C₄	—	4	1	no	—
113	C₄²×C₅	—	2	1	no	—
114	C₄²×C₅	—	4	1	no	—
115	C₅×C₁₆	—	2	2	no	—
116	C₅×C₁₆	—	4	1	no	—
117	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
118	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
119	#518	—	2	1	no	—
120	C₂×SD₃₂	—	2	2	no	—
121	C₂×SD₃₂	—	2	2	no	—
122	C₂×SD₃₂	C₄×C₅	1	4	yes	complemented
123	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
124	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
125	#442	—	2	2	no	—
126	C₄×D₈	—	2	2	no	—
127	C₂×#108	C₂²×C₅	1	1	yes	characteristic
128	#427	C₂²×C₅	1	1	yes	characteristic
129	#419	—	2	1	no	—
130	#418	—	2	1	no	—
131	C₂×#101	—	2	1	no	—
132	#405	C₂²×C₅	1	1	yes	characteristic
133	#361	—	2	2	no	—
134	#358	—	2	2	no	—
135	C₂³×C₅	—	4	2	no	—
136	C₂³×C₅	—	4	2	no	—
137	C₂³×C₅	—	4	2	no	—
138	C₂³×C₅	—	16	2	no	—
139	C₂³×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₅×Q₈	—	4	2	no	—
141	C₅×Q₈	—	4	2	no	—
142	C₅×Q₈	—	8	1	no	—
143	C₅×Q₈	—	8	1	no	—
144	C₅×D₄	—	2	4	no	—
145	C₅×D₄	—	4	2	no	—
146	C₅×D₄	—	4	2	no	—
147	C₅×D₄	—	4	2	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	4	2	no	—
150	C₂×C₄×C₅	—	4	2	no	—
151	C₂×C₄×C₅	—	8	1	no	—
152	C₂×C₄×C₅	—	8	1	no	—
153	C₂×C₄×C₅	—	16	4	no	—
154	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
155	C₂×C₄×C₅	#120	1	1	yes	characteristic
156	C₂×C₄×C₅	#121	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
158	C₂×C₄×C₅	#135	1	1	yes	characteristic
159	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
160	C₅×C₈	—	4	1	no	—
161	C₅×C₈	—	8	1	no	—
162	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
163	C₅×C₈	C₄×D₄	1	1	yes	characteristic
164	C₅×C₈	#121	1	1	yes	characteristic
165	C₅×C₈	#123	1	1	yes	characteristic
166	C₂²×Q₈	—	2	1	no	—
167	C₂²×D₄	—	2	1	no	—
168	C₂×Q₁₆	—	2	1	no	—
169	C₂×Q₁₆	—	2	1	no	—
170	C₂×Q₁₆	—	2	1	no	—
171	C₂×Q₁₆	—	2	1	no	—
172	C₂×Q₁₆	—	2	1	no	—
173	C₂×Q₁₆	C₂×C₄×C₅	1	2	yes	complemented
174	C₂×D₈	—	2	1	no	—
175	C₂×D₈	—	2	1	no	—
176	C₂×D₈	—	2	1	no	—
177	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
178	C₂×D₈	C₅×D₄	1	2	yes	—
179	C₂×D₈	C₅×D₄	1	2	yes	—
180	C₂×M₁₆	—	2	1	no	—
181	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
182	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
183	C₄×D₄	—	2	4	no	—
184	C₄×D₄	—	4	2	no	—
185	#117	—	2	1	no	—
186	#117	C₅×D₄	1	1	yes	characteristic
187	C₂×C₂²⋊C₄	—	4	2	no	—
188	SD₃₂	—	2	8	no	—
189	M₃₂	—	2	1	no	—
190	M₃₂	—	2	1	no	—
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	—	2	1	no	—
193	C₂×C₁₆	—	2	1	no	—
194	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	complemented
195	#108	—	2	1	no	—
196	#108	—	2	1	no	—
197	#107	C₅×D₄	1	2	yes	—
198	#106	—	2	1	no	—
199	#103	—	4	2	no	—
200	#102	—	2	4	no	—
201	#101	—	4	2	no	—
202	#97	—	2	1	no	—
203	C₄×C₈	—	2	1	no	—
204	C₂²×C₅	—	2	2	no	—
205	C₂²×C₅	—	4	4	no	—
206	C₂²×C₅	—	8	4	no	—
207	C₂²×C₅	—	16	2	no	—
208	C₂²×C₅	—	16	2	no	—
209	C₂²×C₅	#442	1	1	yes	characteristic
210	C₂²×C₅	#447	1	1	yes	characteristic
211	C₂²×C₅	#463	1	1	yes	characteristic
212	C₄×C₅	—	4	2	no	—
213	C₄×C₅	—	8	2	no	—
214	C₄×C₅	—	8	2	no	—
215	C₄×C₅	#386	1	1	yes	characteristic
216	C₄×C₅	C₄×D₈	1	1	yes	characteristic
217	C₄×C₅	#449	1	1	yes	characteristic
218	C₄×C₅	#459	1	1	yes	characteristic
219	C₂⁴	—	4	1	no	—
220	C₂×Q₈	—	2	2	no	—
221	C₂×Q₈	—	4	1	no	—
222	C₂×Q₈	—	4	1	no	—
223	C₂×Q₈	—	4	1	no	—
224	C₂×Q₈	—	4	1	no	—
225	C₂×Q₈	—	4	1	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	4	1	no	—
230	C₂×D₄	—	4	1	no	—
231	C₂×D₄	—	4	1	no	—
232	C₂²×C₄	—	4	4	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
235	Q₁₆	—	2	2	no	—
236	Q₁₆	—	2	2	no	—
237	Q₁₆	—	4	1	no	—
238	Q₁₆	—	4	1	no	—
239	D₈	—	2	2	no	—
240	D₈	—	2	2	no	—
241	D₈	—	2	2	no	—
242	D₈	C₅×C₂²⋊C₄	1	4	yes	—
243	M₁₆	—	2	1	no	—
244	M₁₆	—	2	1	no	—
245	M₁₆	—	4	1	no	—
246	M₁₆	—	4	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
250	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
251	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
252	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
253	C₂×C₈	C₅×#41	1	1	yes	characteristic
254	C₂×C₈	C₅×#41	1	1	yes	characteristic
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	C₅×D₈	1	2	yes	—
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	4	4	no	—
259	C₂²⋊C₄	—	4	1	no	—
260	C₄²	—	2	1	no	—
261	C₄²	—	4	1	no	—
262	C₁₆	—	2	2	no	—
263	C₁₆	—	4	1	no	—
264	C₂×C₅	—	2	1	no	—
265	C₂×C₅	—	2	1	no	—
266	C₂×C₅	—	8	4	no	—
267	C₂×C₅	#1879	1	1	yes	characteristic
268	C₂×C₅	#2194	1	2	yes	—
269	C₂³	—	4	2	no	—
270	C₂³	—	4	2	no	—
271	C₂³	—	4	2	no	—
272	C₂³	—	16	2	no	—
273	C₂³	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
274	Q₈	—	4	2	no	—
275	Q₈	—	4	2	no	—
276	Q₈	—	8	1	no	—
277	Q₈	—	8	1	no	—
278	D₄	—	2	4	no	—
279	D₄	—	4	2	no	—
280	D₄	—	4	2	no	—
281	D₄	—	4	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	8	1	no	—
286	C₂×C₄	—	8	1	no	—
287	C₂×C₄	—	16	4	no	—
288	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
289	C₂×C₄	C₅×#120	1	1	yes	characteristic
290	C₂×C₄	C₅×#121	1	1	yes	characteristic
291	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
292	C₂×C₄	C₅×#135	1	1	yes	characteristic
293	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
294	C₈	—	4	1	no	—
295	C₈	—	8	1	no	—
296	C₈	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
297	C₈	C₄×C₅×D₄	1	1	yes	characteristic
298	C₈	C₅×#121	1	1	yes	characteristic
299	C₈	C₅×#123	1	1	yes	characteristic
300	C₅	#13019	1	1	yes	characteristic, direct product factor, Sylow
301	C₂²	—	2	2	no	—
302	C₂²	—	4	4	no	—
303	C₂²	—	8	4	no	—
304	C₂²	—	16	2	no	—
305	C₂²	—	16	2	no	—
306	C₂²	C₅×#442	1	1	yes	characteristic
307	C₂²	C₅×#447	1	1	yes	characteristic
308	C₂²	C₅×#463	1	1	yes	characteristic
309	C₄	—	4	2	no	—
310	C₄	—	8	2	no	—
311	C₄	—	8	2	no	—
312	C₄	C₅×#386	1	1	yes	characteristic
313	C₄	C₄×C₅×D₈	1	1	yes	characteristic
314	C₄	C₅×#449	1	1	yes	characteristic
315	C₄	C₅×#459	1	1	yes	characteristic
316	C₂	—	2	1	no	—
317	C₂	—	2	1	no	—
318	C₂	—	8	4	no	—
319	C₂	C₅×#1879	1	1	yes	characteristic
320	C₂	C₅×#2194	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	#447
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂²×C₅×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#405 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#2141	C₂	semidirect
5. C₅×#1979	C₂	semidirect
6. C₅×#1382	C₂	semidirect
7. C₅×#1379	C₂	semidirect
13. C₂×C₅×SD₃₂	C₄	semidirect
26. #13019	C₅	direct (N × H)
34. C₂×C₅×Q₁₆	C₂×C₄	semidirect
55. C₂×C₅×C₁₆	C₂×C₄	semidirect
68. #2141	C₂×C₅	semidirect
69. #1979	C₂×C₅	semidirect
70. #1382	C₂×C₅	semidirect
71. #1379	C₂×C₅	semidirect
122. C₂×SD₃₂	C₄×C₅	semidirect
173. C₂×Q₁₆	C₂×C₄×C₅	semidirect
194. C₂×C₁₆	C₂×C₄×C₅	semidirect
300. C₅	#13019	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×SD₃₂	2
C₅×#2948	2
C₂×C₅×#361	2
C₅×#2141	2
C₅×#1979	2
C₅×#1382	2
C₅×#1379	2
#13019	5

Subgroups — C5×#13019 #11935739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13019 #11935739