Subgroups — C₅×#13004 #11935724

Order 1280.

560 conjugacy classes of subgroups.

The subgroup lattice has 364 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×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#361	C₂	1	2	yes	maximal, complemented
4	C₅×#2112	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1387	C₂	1	2	yes	maximal, complemented
6	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
7	C₂²×C₅×M₁₆	—	2	1	no	—
8	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₅×#121	—	2	1	no	—
10	C₂×C₅×M₃₂	C₂²	1	2	yes	—
11	C₅×#466	—	2	1	no	—
12	C₅×#466	—	2	1	no	—
13	C₅×#466	C₄	1	2	yes	—
14	C₅×#466	C₄	1	2	yes	—
15	C₂×C₅×#108	—	2	4	no	—
16	C₂×C₅×#108	—	2	1	no	—
17	C₂×C₅×#108	C₂²	1	2	yes	—
18	C₂×C₅×#107	C₂²	1	1	yes	characteristic
19	C₂×C₅×#102	—	2	1	no	—
20	C₅×#361	—	2	4	no	—
21	C₅×C₂²⋊C₁₆	—	2	1	no	—
22	C₅×C₂²⋊C₁₆	—	2	1	no	—
23	#13004	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂²×C₅×D₄	—	2	1	no	—
25	C₂²×C₅×D₄	—	4	1	no	—
26	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂×C₅×D₈	—	2	2	no	—
28	C₂×C₅×D₈	—	2	1	no	—
29	C₂×C₅×D₈	—	2	2	no	—
30	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
31	C₂×C₅×M₁₆	—	2	4	no	—
32	C₂×C₅×M₁₆	—	2	2	no	—
33	C₂×C₅×M₁₆	—	4	2	no	—
34	C₂×C₅×M₁₆	—	4	1	no	—
35	C₂²×C₅×C₈	—	2	2	no	—
36	C₂²×C₅×C₈	—	2	2	no	—
37	C₂²×C₅×C₈	—	4	1	no	—
38	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
41	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
42	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
43	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₅×#121	—	2	2	no	—
45	C₅×#121	—	2	2	no	—
46	C₅×#121	—	4	1	no	—
47	C₅×#121	—	4	1	no	—
48	C₂×C₅×C₄⋊C₄	—	2	1	no	—
49	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
50	C₅×M₃₂	—	2	2	no	—
51	C₅×M₃₂	—	2	2	no	—
52	C₂×C₅×C₁₆	—	2	2	no	—
53	C₅×#108	—	2	4	no	—
54	C₅×#108	—	2	2	no	—
55	C₅×#108	D₄	1	4	yes	complemented
56	C₅×#107	—	2	1	no	—
57	C₅×#107	C₂×C₄	1	2	yes	—
58	C₅×#102	—	2	2	no	—
59	C₅×#102	—	2	2	no	—
60	C₂×#466	C₂×C₅	1	1	yes	characteristic
61	C₂²×#108	C₂×C₅	1	1	yes	characteristic, complemented
62	C₂×#361	C₂×C₅	1	2	yes	complemented
63	#2112	C₂×C₅	1	1	yes	characteristic, complemented
64	#1387	C₂×C₅	1	2	yes	complemented
65	C₂⁴×C₅	—	4	1	no	—
66	C₂⁴×C₅	SD₁₆	1	1	yes	characteristic
67	C₂×C₅×D₄	—	2	2	no	—
68	C₂×C₅×D₄	—	4	2	no	—
69	C₂×C₅×D₄	—	4	2	no	—
70	C₂×C₅×D₄	—	4	1	no	—
71	C₂×C₅×D₄	—	4	1	no	—
72	C₂×C₅×D₄	—	4	1	no	—
73	C₂×C₅×D₄	—	4	1	no	—
74	C₂×C₅×D₄	—	4	1	no	—
75	C₂×C₅×D₄	—	8	4	no	—
76	C₂²×C₄×C₅	—	2	1	no	—
77	C₂²×C₄×C₅	—	2	1	no	—
78	C₂²×C₄×C₅	—	2	1	no	—
79	C₂²×C₄×C₅	—	2	1	no	—
80	C₂²×C₄×C₅	—	4	1	no	—
81	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₅×D₈	—	2	4	no	—
88	C₅×D₈	—	4	2	no	—
89	C₅×M₁₆	—	2	4	no	—
90	C₅×M₁₆	—	4	4	no	—
91	C₅×M₁₆	—	4	2	no	—
92	C₂×C₅×C₈	—	2	1	no	—
93	C₂×C₅×C₈	—	2	1	no	—
94	C₂×C₅×C₈	—	2	2	no	—
95	C₂×C₅×C₈	—	2	2	no	—
96	C₂×C₅×C₈	—	2	2	no	—
97	C₂×C₅×C₈	—	2	2	no	—
98	C₂×C₅×C₈	—	4	4	no	—
99	C₂×C₅×C₈	—	4	2	no	—
100	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
104	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
105	C₂×C₅×C₈	#41	1	2	yes	—
106	C₅×C₄⋊C₄	—	2	2	no	—
107	C₅×C₄⋊C₄	—	4	1	no	—
108	C₅×C₂²⋊C₄	—	4	2	no	—
109	C₅×C₂²⋊C₄	—	4	2	no	—
110	C₅×C₁₆	—	4	2	no	—
111	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
112	C₂²×M₁₆	—	2	1	no	—
113	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
114	C₂×#121	—	2	1	no	—
115	C₂×M₃₂	C₂²×C₅	1	2	yes	—
116	#466	—	2	1	no	—
117	#466	—	2	1	no	—
118	#466	C₄×C₅	1	2	yes	—
119	#466	C₄×C₅	1	2	yes	—
120	C₂×#108	—	2	1	no	—
121	C₂×#108	—	2	4	no	—
122	C₂×#108	C₂²×C₅	1	2	yes	—
123	C₂×#107	C₂²×C₅	1	1	yes	characteristic
124	C₂×#102	—	2	1	no	—
125	#361	—	2	4	no	—
126	C₂²⋊C₁₆	—	2	1	no	—
127	C₂²⋊C₁₆	—	2	1	no	—
128	C₂³×C₅	—	2	2	no	—
129	C₂³×C₅	—	2	1	no	—
130	C₂³×C₅	—	2	1	no	—
131	C₂³×C₅	—	4	1	no	—
132	C₂³×C₅	—	4	2	no	—
133	C₂³×C₅	—	8	2	no	—
134	C₂³×C₅	—	8	2	no	—
135	C₂³×C₅	—	8	1	no	—
136	C₂³×C₅	—	8	1	no	—
137	C₂³×C₅	#102	1	1	yes	characteristic
138	C₂³×C₅	#102	1	1	yes	characteristic
139	C₂³×C₅	C₂×SD₁₆	1	1	yes	characteristic
140	C₅×D₄	—	4	2	no	—
141	C₅×D₄	—	4	2	no	—
142	C₅×D₄	—	8	4	no	—
143	C₅×D₄	—	8	4	no	—
144	C₅×D₄	—	8	1	no	—
145	C₅×D₄	—	8	1	no	—
146	C₂×C₄×C₅	—	2	1	no	—
147	C₂×C₄×C₅	—	2	1	no	—
148	C₂×C₄×C₅	—	2	1	no	—
149	C₂×C₄×C₅	—	2	1	no	—
150	C₂×C₄×C₅	—	2	1	no	—
151	C₂×C₄×C₅	—	2	1	no	—
152	C₂×C₄×C₅	—	2	1	no	—
153	C₂×C₄×C₅	—	2	1	no	—
154	C₂×C₄×C₅	—	2	1	no	—
155	C₂×C₄×C₅	—	2	1	no	—
156	C₂×C₄×C₅	—	4	2	no	—
157	C₂×C₄×C₅	—	8	2	no	—
158	C₂×C₄×C₅	#102	1	1	yes	characteristic
159	C₂×C₄×C₅	#102	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₂×C₄×C₅	#120	1	1	yes	characteristic
162	C₂×C₄×C₅	#123	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
164	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
165	C₂×C₄×C₅	#137	1	1	yes	characteristic
166	C₅×C₈	—	2	1	no	—
167	C₅×C₈	—	2	1	no	—
168	C₅×C₈	—	4	4	no	—
169	C₅×C₈	C₄×D₄	1	2	yes	—
170	C₅×C₈	#121	1	2	yes	—
171	C₂²×D₄	—	2	1	no	—
172	C₂²×D₄	—	4	1	no	—
173	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
174	C₂×D₈	—	2	2	no	—
175	C₂×D₈	—	2	1	no	—
176	C₂×D₈	—	2	2	no	—
177	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
178	C₂×M₁₆	—	2	4	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	4	2	no	—
181	C₂×M₁₆	—	4	1	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	4	1	no	—
185	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
186	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
187	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
188	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
189	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
190	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
191	#121	—	2	2	no	—
192	#121	—	2	2	no	—
193	#121	—	4	1	no	—
194	#121	—	4	1	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₂²⋊C₄	—	4	1	no	—
197	M₃₂	—	2	2	no	—
198	M₃₂	—	2	2	no	—
199	C₂×C₁₆	—	2	2	no	—
200	#108	—	2	4	no	—
201	#108	—	2	2	no	—
202	#108	C₅×D₄	1	4	yes	complemented
203	#107	—	2	1	no	—
204	#107	C₂×C₄×C₅	1	2	yes	—
205	#102	—	2	2	no	—
206	#102	—	2	2	no	—
207	C₂²×C₅	—	2	1	no	—
208	C₂²×C₅	—	2	1	no	—
209	C₂²×C₅	—	2	1	no	—
210	C₂²×C₅	—	2	1	no	—
211	C₂²×C₅	—	2	1	no	—
212	C₂²×C₅	—	2	1	no	—
213	C₂²×C₅	—	2	1	no	—
214	C₂²×C₅	—	4	2	no	—
215	C₂²×C₅	—	4	2	no	—
216	C₂²×C₅	—	8	2	no	—
217	C₂²×C₅	—	8	2	no	—
218	C₂²×C₅	—	8	2	no	—
219	C₂²×C₅	—	16	2	no	—
220	C₂²×C₅	—	16	2	no	—
221	C₂²×C₅	#361	1	2	yes	complemented
222	C₂²×C₅	C₂×#102	1	1	yes	characteristic
223	C₂²×C₅	#448	1	1	yes	characteristic
224	C₂²×C₅	#481	1	1	yes	characteristic
225	C₄×C₅	—	2	1	no	—
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	8	2	no	—
228	C₄×C₅	#386	1	1	yes	characteristic
229	C₄×C₅	#417	1	1	yes	characteristic
230	C₄×C₅	#447	1	1	yes	characteristic
231	C₄×C₅	#480	1	1	yes	characteristic
232	C₂⁴	—	4	1	no	—
233	C₂⁴	C₅×SD₁₆	1	1	yes	characteristic
234	C₂×D₄	—	2	2	no	—
235	C₂×D₄	—	4	2	no	—
236	C₂×D₄	—	4	2	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂×D₄	—	4	1	no	—
241	C₂×D₄	—	4	1	no	—
242	C₂×D₄	—	8	4	no	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	4	1	no	—
248	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
249	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
250	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
251	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
252	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
254	D₈	—	2	4	no	—
255	D₈	—	4	2	no	—
256	M₁₆	—	2	4	no	—
257	M₁₆	—	4	4	no	—
258	M₁₆	—	4	2	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	4	4	no	—
266	C₂×C₈	—	4	2	no	—
267	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
268	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
269	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
270	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
271	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
272	C₂×C₈	C₅×#41	1	2	yes	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₁₆	—	4	2	no	—
278	C₂×C₅	—	2	1	no	—
279	C₂×C₅	—	2	1	no	—
280	C₂×C₅	—	2	2	no	—
281	C₂×C₅	—	4	1	no	—
282	C₂×C₅	—	16	2	no	—
283	C₂×C₅	#1874	1	1	yes	characteristic
284	C₂×C₅	C₂×#361	1	1	yes	characteristic
285	C₂×C₅	#2148	1	1	yes	characteristic
286	C₂³	—	2	2	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	—	4	1	no	—
290	C₂³	—	4	2	no	—
291	C₂³	—	8	2	no	—
292	C₂³	—	8	2	no	—
293	C₂³	—	8	1	no	—
294	C₂³	—	8	1	no	—
295	C₂³	C₅×#102	1	1	yes	characteristic
296	C₂³	C₅×#102	1	1	yes	characteristic
297	C₂³	C₂×C₅×SD₁₆	1	1	yes	characteristic
298	D₄	—	4	2	no	—
299	D₄	—	4	2	no	—
300	D₄	—	8	4	no	—
301	D₄	—	8	4	no	—
302	D₄	—	8	1	no	—
303	D₄	—	8	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	8	2	no	—
316	C₂×C₄	C₅×#102	1	1	yes	characteristic
317	C₂×C₄	C₅×#102	1	1	yes	characteristic
318	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
319	C₂×C₄	C₅×#120	1	1	yes	characteristic
320	C₂×C₄	C₅×#123	1	1	yes	characteristic
321	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
322	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
323	C₂×C₄	C₅×#137	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	2	1	no	—
326	C₈	—	4	4	no	—
327	C₈	C₄×C₅×D₄	1	2	yes	—
328	C₈	C₅×#121	1	2	yes	—
329	C₅	#13004	1	1	yes	characteristic, direct product factor, Sylow
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	4	2	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	8	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	16	2	no	—
343	C₂²	—	16	2	no	—
344	C₂²	C₅×#361	1	2	yes	complemented
345	C₂²	C₂×C₅×#102	1	1	yes	characteristic
346	C₂²	C₅×#448	1	1	yes	characteristic
347	C₂²	C₅×#481	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	8	2	no	—
351	C₄	C₅×#386	1	1	yes	characteristic
352	C₄	C₅×#417	1	1	yes	characteristic
353	C₄	C₅×#447	1	1	yes	characteristic
354	C₄	C₅×#480	1	1	yes	characteristic
355	C₂	—	2	1	no	—
356	C₂	—	2	1	no	—
357	C₂	—	2	2	no	—
358	C₂	—	4	1	no	—
359	C₂	—	16	2	no	—
360	C₂	C₅×#1874	1	1	yes	characteristic
361	C₂	C₂×C₅×#361	1	1	yes	characteristic
362	C₂	C₅×#2148	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#102
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 ◁ C₂²×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×#108	C₂	semidirect
3. C₂×C₅×#361	C₂	semidirect
4. C₅×#2112	C₂	semidirect
5. C₅×#1387	C₂	semidirect
23. #13004	C₅	direct (N × H)
55. C₅×#108	D₄	semidirect
61. C₂²×#108	C₂×C₅	semidirect
62. C₂×#361	C₂×C₅	semidirect
63. #2112	C₂×C₅	semidirect
64. #1387	C₂×C₅	semidirect
202. #108	C₅×D₄	semidirect
221. C₂²×C₅	#361	semidirect
329. C₅	#13004	direct (N × H)
344. C₂²	C₅×#361	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×#108	2
C₂×C₅×#361	2
C₅×#2112	2
C₅×#1387	2
#13004	5

Subgroups — C5×#13004 #11935724

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13004 #11935724