Subgroups — C₅×#12815 #11935535

Order 1280.

422 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	C₄×C₅×#135	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#359	C₂	1	2	yes	maximal
4	C₅×#1848	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1382	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×#135	C₂²	1	1	yes	characteristic
7	C₄×C₅×#41	—	2	1	no	—
8	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
9	C₄×C₅×Q₁₆	C₄	1	1	yes	characteristic
10	C₄×C₅×Q₁₆	C₄	1	1	yes	characteristic
11	C₄×C₅×SD₁₆	—	2	1	no	—
12	C₄×C₅×SD₁₆	—	2	1	no	—
13	C₄×C₅×D₈	C₄	1	1	yes	characteristic
14	C₄×C₅×D₈	C₄	1	1	yes	characteristic
15	C₂×C₅×#108	—	2	1	no	—
16	C₂×C₅×#108	C₂²	1	2	yes	—
17	C₅×#427	C₂²	1	1	yes	characteristic
18	C₅×#423	—	2	1	no	—
19	C₅×#416	—	2	1	no	—
20	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₅×C₄⋊C₁₆	—	2	1	no	—
22	C₅×C₄⋊C₁₆	—	2	1	no	—
23	C₅×#359	—	2	4	no	—
24	#12815	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂×C₅×#41	—	2	1	no	—
26	C₅×#135	—	2	1	no	—
27	C₅×#135	—	2	1	no	—
28	C₅×#135	D₄	1	2	yes	—
29	C₅×#135	Q₈	1	2	yes	—
30	C₂×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂×C₅×SD₁₆	—	2	1	no	—
32	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₅×M₁₆	—	2	2	no	—
34	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₄×C₅×Q₈	—	2	1	no	—
37	C₄×C₅×Q₈	—	2	1	no	—
38	C₄×C₅×D₄	—	2	1	no	—
39	C₄×C₅×D₄	—	2	1	no	—
40	C₄×C₅×D₄	—	4	1	no	—
41	C₄×C₅×D₄	—	4	1	no	—
42	C₅×#117	—	2	1	no	—
43	C₅×#117	—	4	1	no	—
44	C₂×C₄²×C₅	—	4	1	no	—
45	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
46	C₂×C₅×C₁₆	—	2	2	no	—
47	C₂×C₅×C₁₆	—	2	2	no	—
48	C₂×C₅×C₁₆	—	2	2	no	—
49	C₅×#108	—	2	2	no	—
50	C₅×#108	—	2	2	no	—
51	C₅×#108	—	4	2	no	—
52	C₅×#107	C₂×C₄	1	1	yes	characteristic
53	C₅×#107	C₂×C₄	1	1	yes	characteristic
54	C₅×#106	—	2	1	no	—
55	C₅×C₄⋊C₈	—	4	1	no	—
56	C₅×C₄⋊C₈	—	4	1	no	—
57	C₅×#103	—	2	1	no	—
58	C₅×#103	—	2	1	no	—
59	C₅×#102	—	2	1	no	—
60	C₅×#102	—	2	1	no	—
61	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
62	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
63	C₄×C₅×C₈	D₄	1	1	yes	characteristic
64	C₄×C₅×C₈	D₄	1	1	yes	characteristic
65	C₄×#135	C₂×C₅	1	1	yes	characteristic
66	C₂×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
67	C₂×#359	C₂×C₅	1	2	yes	—
68	#1848	C₂×C₅	1	1	yes	characteristic, complemented
69	#1382	C₂×C₅	1	2	yes	complemented
70	C₅×#41	—	2	2	no	—
71	C₅×#41	—	2	2	no	—
72	C₅×#41	—	4	1	no	—
73	C₅×#41	—	4	1	no	—
74	C₂×C₅×Q₈	—	2	1	no	—
75	C₂×C₅×D₄	—	2	1	no	—
76	C₂×C₅×D₄	—	4	1	no	—
77	C₂²×C₄×C₅	—	4	1	no	—
78	C₂²×C₄×C₅	—	4	1	no	—
79	C₂²×C₄×C₅	—	4	1	no	—
80	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
81	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₅×Q₁₆	—	2	1	no	—
84	C₅×Q₁₆	C₄⋊C₄	1	2	yes	—
85	C₅×SD₁₆	—	2	2	no	—
86	C₅×SD₁₆	—	2	2	no	—
87	C₅×D₈	—	2	1	no	—
88	C₅×D₈	C₄⋊C₄	1	2	yes	—
89	C₅×M₁₆	—	4	2	no	—
90	C₅×M₁₆	—	4	2	no	—
91	C₂×C₅×C₈	—	2	1	no	—
92	C₂×C₅×C₈	—	2	1	no	—
93	C₂×C₅×C₈	—	4	2	no	—
94	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
100	C₂×C₅×C₈	#41	1	1	yes	characteristic
101	C₂×C₅×C₈	#41	1	1	yes	characteristic
102	C₅×C₄⋊C₄	—	2	1	no	—
103	C₅×C₄⋊C₄	—	2	1	no	—
104	C₅×C₄⋊C₄	—	4	1	no	—
105	C₅×C₄⋊C₄	—	4	1	no	—
106	C₅×C₂²⋊C₄	—	4	1	no	—
107	C₅×C₂²⋊C₄	—	4	1	no	—
108	C₅×C₂²⋊C₄	—	4	1	no	—
109	C₄²×C₅	—	4	1	no	—
110	C₄²×C₅	—	4	1	no	—
111	C₄²×C₅	—	4	1	no	—
112	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
113	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
114	C₄²×C₅	D₈	1	1	yes	characteristic
115	C₄²×C₅	D₈	1	1	yes	characteristic
116	C₅×C₁₆	—	4	2	no	—
117	C₂×#135	C₂²×C₅	1	1	yes	characteristic
118	C₄×#41	—	2	1	no	—
119	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
120	C₄×Q₁₆	C₄×C₅	1	1	yes	characteristic
121	C₄×Q₁₆	C₄×C₅	1	1	yes	characteristic
122	C₄×SD₁₆	—	2	1	no	—
123	C₄×SD₁₆	—	2	1	no	—
124	C₄×D₈	C₄×C₅	1	1	yes	characteristic
125	C₄×D₈	C₄×C₅	1	1	yes	characteristic
126	C₂×#108	—	2	1	no	—
127	C₂×#108	C₂²×C₅	1	2	yes	—
128	#427	C₂²×C₅	1	1	yes	characteristic
129	#423	—	2	1	no	—
130	#416	—	2	1	no	—
131	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
132	C₄⋊C₁₆	—	2	1	no	—
133	C₄⋊C₁₆	—	2	1	no	—
134	#359	—	2	4	no	—
135	C₂³×C₅	—	4	1	no	—
136	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
137	C₅×Q₈	—	2	2	no	—
138	C₅×Q₈	—	4	1	no	—
139	C₅×D₄	—	2	2	no	—
140	C₅×D₄	—	4	2	no	—
141	C₅×D₄	—	4	2	no	—
142	C₅×D₄	—	4	1	no	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	—	4	1	no	—
145	C₂×C₄×C₅	—	4	1	no	—
146	C₂×C₄×C₅	—	4	2	no	—
147	C₂×C₄×C₅	—	4	2	no	—
148	C₂×C₄×C₅	—	4	1	no	—
149	C₂×C₄×C₅	—	4	1	no	—
150	C₂×C₄×C₅	—	8	2	no	—
151	C₂×C₄×C₅	—	8	2	no	—
152	C₂×C₄×C₅	#102	1	1	yes	characteristic
153	C₂×C₄×C₅	#102	1	1	yes	characteristic
154	C₂×C₄×C₅	#102	1	1	yes	characteristic
155	C₂×C₄×C₅	#102	1	1	yes	characteristic
156	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
157	C₂×C₄×C₅	#121	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
160	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₂×C₄×C₅	#137	1	1	yes	characteristic
162	C₅×C₈	—	2	1	no	—
163	C₅×C₈	—	2	1	no	—
164	C₅×C₈	—	8	2	no	—
165	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
166	C₅×C₈	#117	1	1	yes	characteristic
167	C₅×C₈	#121	1	1	yes	characteristic
168	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
169	C₂×#41	—	2	1	no	—
170	#135	—	2	1	no	—
171	#135	—	2	1	no	—
172	#135	C₅×D₄	1	2	yes	—
173	#135	C₅×Q₈	1	2	yes	—
174	C₂×Q₁₆	C₂×C₄×C₅	1	1	yes	characteristic
175	C₂×SD₁₆	—	2	1	no	—
176	C₂×D₈	C₂×C₄×C₅	1	1	yes	characteristic
177	C₂×M₁₆	—	2	2	no	—
178	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
179	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
180	C₄×Q₈	—	2	1	no	—
181	C₄×Q₈	—	2	1	no	—
182	C₄×D₄	—	2	1	no	—
183	C₄×D₄	—	2	1	no	—
184	C₄×D₄	—	4	1	no	—
185	C₄×D₄	—	4	1	no	—
186	#117	—	2	1	no	—
187	#117	—	4	1	no	—
188	C₂×C₄²	—	4	1	no	—
189	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
190	C₂×C₁₆	—	2	2	no	—
191	C₂×C₁₆	—	2	2	no	—
192	C₂×C₁₆	—	2	2	no	—
193	#108	—	2	2	no	—
194	#108	—	2	2	no	—
195	#108	—	4	2	no	—
196	#107	C₂×C₄×C₅	1	1	yes	characteristic
197	#107	C₂×C₄×C₅	1	1	yes	characteristic
198	#106	—	2	1	no	—
199	C₄⋊C₈	—	4	1	no	—
200	C₄⋊C₈	—	4	1	no	—
201	#103	—	2	1	no	—
202	#103	—	2	1	no	—
203	#102	—	2	1	no	—
204	#102	—	2	1	no	—
205	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
206	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
207	C₄×C₈	C₅×D₄	1	1	yes	characteristic
208	C₄×C₈	C₅×D₄	1	1	yes	characteristic
209	C₂²×C₅	—	2	1	no	—
210	C₂²×C₅	—	2	1	no	—
211	C₂²×C₅	—	4	2	no	—
212	C₂²×C₅	—	8	1	no	—
213	C₂²×C₅	—	8	1	no	—
214	C₂²×C₅	C₂×#102	1	1	yes	characteristic
215	C₂²×C₅	#461	1	1	yes	characteristic
216	C₂²×C₅	#476	1	1	yes	characteristic
217	C₄×C₅	—	2	1	no	—
218	C₄×C₅	—	2	1	no	—
219	C₄×C₅	—	2	1	no	—
220	C₄×C₅	—	4	2	no	—
221	C₄×C₅	—	8	1	no	—
222	C₄×C₅	—	8	1	no	—
223	C₄×C₅	#359	1	2	yes	complemented
224	C₄×C₅	#380	1	1	yes	characteristic
225	C₄×C₅	#417	1	1	yes	characteristic
226	C₄×C₅	#459	1	1	yes	characteristic
227	C₄×C₅	#474	1	1	yes	characteristic
228	#41	—	2	2	no	—
229	#41	—	2	2	no	—
230	#41	—	4	1	no	—
231	#41	—	4	1	no	—
232	C₂×Q₈	—	2	1	no	—
233	C₂×D₄	—	2	1	no	—
234	C₂×D₄	—	4	1	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
239	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
241	Q₁₆	—	2	1	no	—
242	Q₁₆	C₅×C₄⋊C₄	1	2	yes	—
243	SD₁₆	—	2	2	no	—
244	SD₁₆	—	2	2	no	—
245	D₈	—	2	1	no	—
246	D₈	C₅×C₄⋊C₄	1	2	yes	—
247	M₁₆	—	4	2	no	—
248	M₁₆	—	4	2	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
253	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
254	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
255	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
256	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
257	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
258	C₂×C₈	C₅×#41	1	1	yes	characteristic
259	C₂×C₈	C₅×#41	1	1	yes	characteristic
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₂²⋊C₄	—	4	1	no	—
265	C₂²⋊C₄	—	4	1	no	—
266	C₂²⋊C₄	—	4	1	no	—
267	C₄²	—	4	1	no	—
268	C₄²	—	4	1	no	—
269	C₄²	—	4	1	no	—
270	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
271	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
272	C₄²	C₅×D₈	1	1	yes	characteristic
273	C₄²	C₅×D₈	1	1	yes	characteristic
274	C₁₆	—	4	2	no	—
275	C₂×C₅	—	2	1	no	—
276	C₂×C₅	—	2	1	no	—
277	C₂×C₅	—	8	2	no	—
278	C₂×C₅	#1802	1	1	yes	characteristic
279	C₂×C₅	C₂×#359	1	1	yes	characteristic
280	C₂×C₅	#2143	1	1	yes	characteristic
281	C₂³	—	4	1	no	—
282	C₂³	C₂×C₅×SD₁₆	1	1	yes	characteristic
283	Q₈	—	2	2	no	—
284	Q₈	—	4	1	no	—
285	D₄	—	2	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	4	1	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	8	2	no	—
297	C₂×C₄	—	8	2	no	—
298	C₂×C₄	C₅×#102	1	1	yes	characteristic
299	C₂×C₄	C₅×#102	1	1	yes	characteristic
300	C₂×C₄	C₅×#102	1	1	yes	characteristic
301	C₂×C₄	C₅×#102	1	1	yes	characteristic
302	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
303	C₂×C₄	C₅×#121	1	1	yes	characteristic
304	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
305	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
306	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
307	C₂×C₄	C₅×#137	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	8	2	no	—
311	C₈	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
312	C₈	C₅×#117	1	1	yes	characteristic
313	C₈	C₅×#121	1	1	yes	characteristic
314	C₈	C₅×C₂²⋊Q₈	1	1	yes	characteristic
315	C₅	#12815	1	1	yes	characteristic, direct product factor, Sylow
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	2	no	—
319	C₂²	—	8	1	no	—
320	C₂²	—	8	1	no	—
321	C₂²	C₂×C₅×#102	1	1	yes	characteristic
322	C₂²	C₅×#461	1	1	yes	characteristic
323	C₂²	C₅×#476	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	4	2	no	—
328	C₄	—	8	1	no	—
329	C₄	—	8	1	no	—
330	C₄	C₅×#359	1	2	yes	complemented
331	C₄	C₅×#380	1	1	yes	characteristic
332	C₄	C₅×#417	1	1	yes	characteristic
333	C₄	C₅×#459	1	1	yes	characteristic
334	C₄	C₅×#474	1	1	yes	characteristic
335	C₂	—	2	1	no	—
336	C₂	—	2	1	no	—
337	C₂	—	8	2	no	—
338	C₂	C₅×#1802	1	1	yes	characteristic
339	C₂	C₂×C₅×#359	1	1	yes	characteristic
340	C₂	C₅×#2143	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×#102
prime radical	C₂×C₅×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×C₄⋊C₁₆	C₂	semidirect
4. C₅×#1848	C₂	semidirect
5. C₅×#1382	C₂	semidirect
24. #12815	C₅	direct (N × H)
66. C₂×C₄⋊C₁₆	C₂×C₅	semidirect
68. #1848	C₂×C₅	semidirect
69. #1382	C₂×C₅	semidirect
223. C₄×C₅	#359	semidirect
315. C₅	#12815	direct (N × H)
330. C₄	C₅×#359	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×C₅×#135	2
C₂×C₅×C₄⋊C₁₆	2
C₂×C₅×#359	2
C₅×#1848	2
C₅×#1382	2
#12815	5

Subgroups — C5×#12815 #11935535

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12815 #11935535