Subgroups — C₅×#12991 #11935711

Order 1280.

626 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#466	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×#107	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#357	C₂	1	2	yes	maximal, complemented
4	C₂×C₅×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1380	C₂	1	2	yes	maximal, complemented
6	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₅×#121	—	2	1	no	—
9	C₂²×C₅×C₄⋊C₄	—	2	1	no	—
10	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
11	C₅×#466	—	2	2	no	—
12	C₅×#466	C₄	1	4	yes	complemented
13	C₂×C₅×#107	—	2	4	no	—
14	C₂×C₅×#107	—	2	1	no	—
15	C₂×C₅×#107	C₂²	1	1	yes	characteristic
16	C₂×C₅×#107	C₂²	1	2	yes	—
17	C₂×C₅×#102	—	2	1	no	—
18	C₅×#357	—	2	4	no	—
19	C₅×C₂²⋊C₁₆	—	2	2	no	—
20	#12991	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×D₄	—	2	1	no	—
22	C₂²×C₅×D₄	—	4	1	no	—
23	C₂³×C₄×C₅	—	4	1	no	—
24	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
25	C₂×C₅×D₈	—	2	2	no	—
26	C₂×C₅×D₈	—	2	1	no	—
27	C₂×C₅×D₈	—	2	2	no	—
28	C₂×C₅×D₈	C₂×C₄	1	2	yes	complemented
29	C₂²×C₅×C₈	—	2	2	no	—
30	C₂²×C₅×C₈	—	2	2	no	—
31	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
32	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
33	C₂²×C₅×C₈	D₄	1	2	yes	—
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₅×#121	—	2	4	no	—
36	C₅×#121	—	4	2	no	—
37	C₂×C₅×C₄⋊C₄	—	2	4	no	—
38	C₂×C₅×C₄⋊C₄	—	2	1	no	—
39	C₂×C₅×C₄⋊C₄	—	2	2	no	—
40	C₂×C₅×C₄⋊C₄	—	4	2	no	—
41	C₂×C₅×C₄⋊C₄	—	4	1	no	—
42	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
43	C₂×C₅×C₁₆	—	2	2	no	—
44	C₂×C₅×C₁₆	—	2	2	no	—
45	C₂×C₅×C₁₆	—	2	2	no	—
46	C₅×#107	—	2	4	no	—
47	C₅×#107	—	2	2	no	—
48	C₅×#107	—	2	1	no	—
49	C₅×#107	C₂×C₄	1	2	yes	complemented
50	C₅×#107	D₄	1	4	yes	complemented
51	C₅×#102	—	2	4	no	—
52	C₂×#466	C₂×C₅	1	1	yes	characteristic
53	C₂²×#107	C₂×C₅	1	1	yes	characteristic, complemented
54	C₂×#357	C₂×C₅	1	2	yes	complemented
55	C₂×C₂²⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
56	#1380	C₂×C₅	1	2	yes	complemented
57	C₂⁴×C₅	—	4	1	no	—
58	C₂⁴×C₅	D₈	1	1	yes	characteristic
59	C₂×C₅×D₄	—	2	2	no	—
60	C₂×C₅×D₄	—	4	4	no	—
61	C₂×C₅×D₄	—	4	1	no	—
62	C₂×C₅×D₄	—	4	1	no	—
63	C₂×C₅×D₄	—	4	1	no	—
64	C₂×C₅×D₄	—	4	1	no	—
65	C₂×C₅×D₄	—	4	1	no	—
66	C₂×C₅×D₄	—	8	2	no	—
67	C₂×C₅×D₄	—	8	2	no	—
68	C₂²×C₄×C₅	—	2	1	no	—
69	C₂²×C₄×C₅	—	2	1	no	—
70	C₂²×C₄×C₅	—	2	1	no	—
71	C₂²×C₄×C₅	—	2	1	no	—
72	C₂²×C₄×C₅	—	4	4	no	—
73	C₂²×C₄×C₅	—	4	1	no	—
74	C₂²×C₄×C₅	—	4	2	no	—
75	C₂²×C₄×C₅	—	8	2	no	—
76	C₂²×C₄×C₅	—	8	2	no	—
77	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
78	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
79	C₂²×C₄×C₅	SD₁₆	1	2	yes	—
80	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₅×D₈	—	2	4	no	—
82	C₅×D₈	—	4	2	no	—
83	C₂×C₅×C₈	—	2	1	no	—
84	C₂×C₅×C₈	—	2	1	no	—
85	C₂×C₅×C₈	—	2	4	no	—
86	C₂×C₅×C₈	—	2	4	no	—
87	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
88	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
90	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
91	C₂×C₅×C₈	#41	1	2	yes	—
92	C₅×C₄⋊C₄	—	2	2	no	—
93	C₅×C₄⋊C₄	—	2	4	no	—
94	C₅×C₄⋊C₄	—	4	4	no	—
95	C₅×C₄⋊C₄	—	4	1	no	—
96	C₅×C₄⋊C₄	—	4	2	no	—
97	C₅×C₂²⋊C₄	—	4	4	no	—
98	C₅×C₁₆	—	4	2	no	—
99	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
100	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
101	C₂×#121	—	2	1	no	—
102	C₂²×C₄⋊C₄	—	2	1	no	—
103	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
104	#466	—	2	2	no	—
105	#466	C₄×C₅	1	4	yes	complemented
106	C₂×#107	—	2	1	no	—
107	C₂×#107	—	2	4	no	—
108	C₂×#107	C₂²×C₅	1	1	yes	characteristic
109	C₂×#107	C₂²×C₅	1	2	yes	—
110	C₂×#102	—	2	1	no	—
111	#357	—	2	4	no	—
112	C₂²⋊C₁₆	—	2	2	no	—
113	C₂³×C₅	—	2	1	no	—
114	C₂³×C₅	—	2	1	no	—
115	C₂³×C₅	—	2	1	no	—
116	C₂³×C₅	—	2	1	no	—
117	C₂³×C₅	—	4	2	no	—
118	C₂³×C₅	—	8	1	no	—
119	C₂³×C₅	—	8	1	no	—
120	C₂³×C₅	—	8	1	no	—
121	C₂³×C₅	—	8	1	no	—
122	C₂³×C₅	—	8	1	no	—
123	C₂³×C₅	—	8	1	no	—
124	C₂³×C₅	#102	1	2	yes	—
125	C₂³×C₅	D₁₆	1	2	yes	—
126	C₂³×C₅	SD₃₂	1	2	yes	—
127	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
128	C₅×D₄	—	4	2	no	—
129	C₅×D₄	—	4	2	no	—
130	C₅×D₄	—	8	8	no	—
131	C₅×D₄	—	8	1	no	—
132	C₅×D₄	—	8	1	no	—
133	C₂×C₄×C₅	—	2	1	no	—
134	C₂×C₄×C₅	—	2	1	no	—
135	C₂×C₄×C₅	—	2	2	no	—
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	—	4	2	no	—
140	C₂×C₄×C₅	—	4	4	no	—
141	C₂×C₄×C₅	—	8	8	no	—
142	C₂×C₄×C₅	—	8	2	no	—
143	C₂×C₄×C₅	—	8	1	no	—
144	C₂×C₄×C₅	—	8	1	no	—
145	C₂×C₄×C₅	—	8	2	no	—
146	C₂×C₄×C₅	#102	1	2	yes	—
147	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₅	#120	1	1	yes	characteristic
149	C₂×C₄×C₅	#123	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
151	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂×C₄×C₅	#137	1	1	yes	characteristic
153	C₅×C₈	—	2	2	no	—
154	C₅×C₈	C₄×D₄	1	2	yes	—
155	C₅×C₈	#121	1	2	yes	—
156	C₂²×D₄	—	2	1	no	—
157	C₂²×D₄	—	4	1	no	—
158	C₂³×C₄	—	4	1	no	—
159	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
160	C₂×D₈	—	2	2	no	—
161	C₂×D₈	—	2	1	no	—
162	C₂×D₈	—	2	2	no	—
163	C₂×D₈	C₂×C₄×C₅	1	2	yes	complemented
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
167	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
168	C₂²×C₈	C₅×D₄	1	2	yes	—
169	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
170	#121	—	2	4	no	—
171	#121	—	4	2	no	—
172	C₂×C₄⋊C₄	—	2	4	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	—	2	1	no	—
175	C₂×C₄⋊C₄	—	4	2	no	—
176	C₂×C₄⋊C₄	—	4	1	no	—
177	C₂×C₂²⋊C₄	—	4	1	no	—
178	C₂×C₁₆	—	2	2	no	—
179	C₂×C₁₆	—	2	2	no	—
180	C₂×C₁₆	—	2	2	no	—
181	#107	—	2	4	no	—
182	#107	—	2	2	no	—
183	#107	—	2	1	no	—
184	#107	C₂×C₄×C₅	1	2	yes	complemented
185	#107	C₅×D₄	1	4	yes	complemented
186	#102	—	2	4	no	—
187	C₂²×C₅	—	2	2	no	—
188	C₂²×C₅	—	2	2	no	—
189	C₂²×C₅	—	2	2	no	—
190	C₂²×C₅	—	2	2	no	—
191	C₂²×C₅	—	2	2	no	—
192	C₂²×C₅	—	2	2	no	—
193	C₂²×C₅	—	8	2	no	—
194	C₂²×C₅	—	8	2	no	—
195	C₂²×C₅	—	8	2	no	—
196	C₂²×C₅	—	16	1	no	—
197	C₂²×C₅	—	16	1	no	—
198	C₂²×C₅	—	16	1	no	—
199	C₂²×C₅	—	16	1	no	—
200	C₂²×C₅	#357	1	4	yes	complemented
201	C₂²×C₅	C₂×#102	1	1	yes	characteristic
202	C₂²×C₅	#447	1	1	yes	characteristic
203	C₂²×C₅	#480	1	1	yes	characteristic
204	C₂²×C₅	C₂×D₁₆	1	1	yes	characteristic
205	C₂²×C₅	C₂×SD₃₂	1	1	yes	characteristic
206	C₂²×C₅	#509	1	1	yes	characteristic
207	C₂²×C₅	#510	1	1	yes	characteristic
208	C₄×C₅	—	2	2	no	—
209	C₄×C₅	—	8	2	no	—
210	C₄×C₅	—	8	4	no	—
211	C₄×C₅	#386	1	1	yes	characteristic
212	C₄×C₅	#417	1	1	yes	characteristic
213	C₄×C₅	#448	1	1	yes	characteristic
214	C₄×C₅	#481	1	1	yes	characteristic
215	C₂⁴	—	4	1	no	—
216	C₂⁴	C₅×D₈	1	1	yes	characteristic
217	C₂×D₄	—	2	2	no	—
218	C₂×D₄	—	4	4	no	—
219	C₂×D₄	—	4	1	no	—
220	C₂×D₄	—	4	1	no	—
221	C₂×D₄	—	4	1	no	—
222	C₂×D₄	—	4	1	no	—
223	C₂×D₄	—	4	1	no	—
224	C₂×D₄	—	8	2	no	—
225	C₂×D₄	—	8	2	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	4	4	no	—
231	C₂²×C₄	—	4	1	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	—	8	2	no	—
234	C₂²×C₄	—	8	2	no	—
235	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
236	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
237	C₂²×C₄	C₅×SD₁₆	1	2	yes	—
238	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
239	D₈	—	2	4	no	—
240	D₈	—	4	2	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
246	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
247	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
248	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
249	C₂×C₈	C₅×#41	1	2	yes	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	4	4	no	—
253	C₄⋊C₄	—	4	1	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₂²⋊C₄	—	4	4	no	—
256	C₁₆	—	4	2	no	—
257	C₂×C₅	—	2	4	no	—
258	C₂×C₅	—	16	2	no	—
259	C₂×C₅	#1874	1	1	yes	characteristic
260	C₂×C₅	C₂×#357	1	1	yes	characteristic
261	C₂×C₅	#2139	1	1	yes	characteristic
262	C₂×C₅	#2184	1	1	yes	characteristic
263	C₂×C₅	#2185	1	1	yes	characteristic
264	C₂×C₅	#2232	1	1	yes	characteristic
265	C₂×C₅	#2233	1	1	yes	characteristic
266	C₂³	—	2	1	no	—
267	C₂³	—	2	1	no	—
268	C₂³	—	2	1	no	—
269	C₂³	—	2	1	no	—
270	C₂³	—	4	2	no	—
271	C₂³	—	8	1	no	—
272	C₂³	—	8	1	no	—
273	C₂³	—	8	1	no	—
274	C₂³	—	8	1	no	—
275	C₂³	—	8	1	no	—
276	C₂³	—	8	1	no	—
277	C₂³	C₅×#102	1	2	yes	—
278	C₂³	C₅×D₁₆	1	2	yes	—
279	C₂³	C₅×SD₃₂	1	2	yes	—
280	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	8	8	no	—
284	D₄	—	8	1	no	—
285	D₄	—	8	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	8	8	no	—
295	C₂×C₄	—	8	2	no	—
296	C₂×C₄	—	8	1	no	—
297	C₂×C₄	—	8	1	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	C₅×#102	1	2	yes	—
300	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
301	C₂×C₄	C₅×#120	1	1	yes	characteristic
302	C₂×C₄	C₅×#123	1	1	yes	characteristic
303	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
304	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
305	C₂×C₄	C₅×#137	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	C₄×C₅×D₄	1	2	yes	—
308	C₈	C₅×#121	1	2	yes	—
309	C₅	#12991	1	1	yes	characteristic, direct product factor, Sylow
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	—	16	1	no	—
320	C₂²	—	16	1	no	—
321	C₂²	—	16	1	no	—
322	C₂²	—	16	1	no	—
323	C₂²	C₅×#357	1	4	yes	complemented
324	C₂²	C₂×C₅×#102	1	1	yes	characteristic
325	C₂²	C₅×#447	1	1	yes	characteristic
326	C₂²	C₅×#480	1	1	yes	characteristic
327	C₂²	C₂×C₅×D₁₆	1	1	yes	characteristic
328	C₂²	C₂×C₅×SD₃₂	1	1	yes	characteristic
329	C₂²	C₅×#509	1	1	yes	characteristic
330	C₂²	C₅×#510	1	1	yes	characteristic
331	C₄	—	2	2	no	—
332	C₄	—	8	2	no	—
333	C₄	—	8	4	no	—
334	C₄	C₅×#386	1	1	yes	characteristic
335	C₄	C₅×#417	1	1	yes	characteristic
336	C₄	C₅×#448	1	1	yes	characteristic
337	C₄	C₅×#481	1	1	yes	characteristic
338	C₂	—	2	4	no	—
339	C₂	—	16	2	no	—
340	C₂	C₅×#1874	1	1	yes	characteristic
341	C₂	C₂×C₅×#357	1	1	yes	characteristic
342	C₂	C₅×#2139	1	1	yes	characteristic
343	C₂	C₅×#2184	1	1	yes	characteristic
344	C₂	C₅×#2185	1	1	yes	characteristic
345	C₂	C₅×#2232	1	1	yes	characteristic
346	C₂	C₅×#2233	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂×C₅×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×#107	C₂	semidirect
3. C₂×C₅×#357	C₂	semidirect
4. C₂×C₅×C₂²⋊C₁₆	C₂	semidirect
5. C₅×#1380	C₂	semidirect
12. C₅×#466	C₄	semidirect
20. #12991	C₅	direct (N × H)
28. C₂×C₅×D₈	C₂×C₄	semidirect
49. C₅×#107	C₂×C₄	semidirect
50. C₅×#107	D₄	semidirect
53. C₂²×#107	C₂×C₅	semidirect
54. C₂×#357	C₂×C₅	semidirect
55. C₂×C₂²⋊C₁₆	C₂×C₅	semidirect
56. #1380	C₂×C₅	semidirect
105. #466	C₄×C₅	semidirect
163. C₂×D₈	C₂×C₄×C₅	semidirect
184. #107	C₂×C₄×C₅	semidirect
185. #107	C₅×D₄	semidirect
200. C₂²×C₅	#357	semidirect
309. C₅	#12991	direct (N × H)
323. C₂²	C₅×#357	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#466	2
C₂²×C₅×#107	2
C₂×C₅×#357	2
C₂×C₅×C₂²⋊C₁₆	2
C₅×#1380	2
#12991	5

Subgroups — C5×#12991 #11935711

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12991 #11935711