Subgroups — C₅×#12937 #11935657

Order 1280.

490 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#367	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#357	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#357	C₂	1	1	yes	maximal, characteristic
5	C₅×#1942	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1380	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₅×D₄	—	2	1	no	—
10	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
12	C₄×C₅×D₈	—	2	2	no	—
13	C₄×C₅×D₈	—	2	2	no	—
14	C₂×C₅×#107	C₂²	1	1	yes	characteristic
15	C₂×C₅×#107	C₂²	1	1	yes	characteristic
16	C₂×C₅×#107	C₂²	1	1	yes	characteristic
17	C₂×C₅×#102	—	2	1	no	—
18	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₅×#389	—	2	1	no	—
20	C₅×#367	—	2	2	no	—
21	C₅×#357	—	2	2	no	—
22	C₅×#357	C₄	1	4	yes	complemented
23	C₅×#340	—	2	1	no	—
24	#12937	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂²×C₅×D₄	—	2	1	no	—
26	C₂³×C₄×C₅	—	4	1	no	—
27	C₂×C₅×D₈	—	2	1	no	—
28	C₂×C₅×D₈	—	2	1	no	—
29	C₂×C₅×D₈	—	2	1	no	—
30	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
31	C₂×C₅×D₈	D₄	1	2	yes	—
32	C₂×C₅×D₈	Q₈	1	2	yes	—
33	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₄×C₅×D₄	—	2	4	no	—
36	C₄×C₅×D₄	—	4	2	no	—
37	C₂×C₅×C₄⋊C₄	—	2	1	no	—
38	C₂×C₅×C₄⋊C₄	—	2	1	no	—
39	C₂×C₅×C₄⋊C₄	—	2	1	no	—
40	C₂×C₅×C₄⋊C₄	—	4	1	no	—
41	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
42	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
43	C₂×C₅×C₁₆	—	2	1	no	—
44	C₂×C₅×C₁₆	—	2	1	no	—
45	C₂×C₅×C₁₆	—	2	1	no	—
46	C₂×C₅×C₁₆	—	2	1	no	—
47	C₂×C₅×C₁₆	—	2	1	no	—
48	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	complemented
49	C₅×#107	—	2	1	no	—
50	C₅×#107	—	2	1	no	—
51	C₅×#107	—	2	1	no	—
52	C₅×#107	—	2	1	no	—
53	C₅×#107	C₂×C₄	1	2	yes	complemented
54	C₅×#107	D₄	1	2	yes	—
55	C₅×#102	—	2	4	no	—
56	C₄×C₅×C₈	—	2	1	no	—
57	C₄×C₅×C₈	—	2	1	no	—
58	C₅×#95	—	4	1	no	—
59	C₂×C₄×D₈	C₂×C₅	1	1	yes	characteristic
60	C₂×#367	C₂×C₅	1	1	yes	characteristic, complemented
61	C₂×#357	C₂×C₅	1	1	yes	characteristic
62	C₂×#357	C₂×C₅	1	1	yes	characteristic
63	#1942	C₂×C₅	1	1	yes	characteristic, complemented
64	#1380	C₂×C₅	1	1	yes	characteristic, complemented
65	#1374	C₂×C₅	1	1	yes	characteristic, complemented
66	C₂⁴×C₅	—	4	1	no	—
67	C₂×C₅×D₄	—	2	2	no	—
68	C₂×C₅×D₄	—	2	2	no	—
69	C₂×C₅×D₄	—	2	2	no	—
70	C₂×C₅×D₄	—	4	1	no	—
71	C₂×C₅×D₄	—	4	1	no	—
72	C₂×C₅×D₄	—	4	1	no	—
73	C₂²×C₄×C₅	—	4	4	no	—
74	C₂²×C₄×C₅	—	4	1	no	—
75	C₂²×C₄×C₅	—	4	1	no	—
76	C₂²×C₄×C₅	—	4	1	no	—
77	C₂²×C₄×C₅	—	8	2	no	—
78	C₂²×C₄×C₅	—	8	2	no	—
79	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₅×D₈	—	2	2	no	—
83	C₅×D₈	—	2	2	no	—
84	C₅×D₈	—	2	2	no	—
85	C₅×D₈	C₄⋊C₄	1	4	yes	complemented
86	C₂×C₅×C₈	—	2	1	no	—
87	C₂×C₅×C₈	—	2	1	no	—
88	C₂×C₅×C₈	—	2	1	no	—
89	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
91	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
93	C₂×C₅×C₈	#41	1	1	yes	characteristic
94	C₂×C₅×C₈	#41	1	1	yes	characteristic
95	C₅×C₄⋊C₄	—	2	2	no	—
96	C₅×C₄⋊C₄	—	2	2	no	—
97	C₅×C₄⋊C₄	—	4	1	no	—
98	C₅×C₄⋊C₄	—	4	1	no	—
99	C₅×C₄⋊C₄	—	4	1	no	—
100	C₅×C₄⋊C₄	—	4	1	no	—
101	C₅×C₄⋊C₄	—	8	2	no	—
102	C₅×C₂²⋊C₄	—	4	4	no	—
103	C₄²×C₅	—	2	1	no	—
104	C₄²×C₅	—	2	1	no	—
105	C₅×C₁₆	—	2	2	no	—
106	C₅×C₁₆	—	4	1	no	—
107	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
108	C₂×C₄×D₄	—	2	1	no	—
109	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
110	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
111	C₄×D₈	—	2	2	no	—
112	C₄×D₈	—	2	2	no	—
113	C₂×#107	C₂²×C₅	1	1	yes	characteristic
114	C₂×#107	C₂²×C₅	1	1	yes	characteristic
115	C₂×#107	C₂²×C₅	1	1	yes	characteristic
116	C₂×#102	—	2	1	no	—
117	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
118	#389	—	2	1	no	—
119	#367	—	2	2	no	—
120	#357	—	2	2	no	—
121	#357	C₄×C₅	1	4	yes	complemented
122	#340	—	2	1	no	—
123	C₂³×C₅	—	4	2	no	—
124	C₂³×C₅	—	4	2	no	—
125	C₂³×C₅	—	4	2	no	—
126	C₂³×C₅	—	8	1	no	—
127	C₂³×C₅	—	8	1	no	—
128	C₂³×C₅	—	8	1	no	—
129	C₂³×C₅	—	8	1	no	—
130	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
131	C₅×D₄	—	2	4	no	—
132	C₅×D₄	—	4	2	no	—
133	C₅×D₄	—	4	2	no	—
134	C₅×D₄	—	4	2	no	—
135	C₂×C₄×C₅	—	2	1	no	—
136	C₂×C₄×C₅	—	2	1	no	—
137	C₂×C₄×C₅	—	2	1	no	—
138	C₂×C₄×C₅	—	2	1	no	—
139	C₂×C₄×C₅	—	2	1	no	—
140	C₂×C₄×C₅	—	4	2	no	—
141	C₂×C₄×C₅	—	4	2	no	—
142	C₂×C₄×C₅	—	8	8	no	—
143	C₂×C₄×C₅	—	8	1	no	—
144	C₂×C₄×C₅	—	8	1	no	—
145	C₂×C₄×C₅	—	8	1	no	—
146	C₂×C₄×C₅	—	8	1	no	—
147	C₂×C₄×C₅	—	8	1	no	—
148	C₂×C₄×C₅	—	8	1	no	—
149	C₂×C₄×C₅	—	8	1	no	—
150	C₂×C₄×C₅	SD₃₂	1	2	yes	—
151	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₅	#120	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
154	C₂×C₄×C₅	#135	1	1	yes	characteristic
155	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
156	C₂×C₄×C₅	#137	1	1	yes	characteristic
157	C₅×C₈	—	4	1	no	—
158	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₅×C₈	C₄×D₄	1	1	yes	characteristic
160	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
161	C₅×C₈	#123	1	1	yes	characteristic
162	C₂²×D₄	—	2	1	no	—
163	C₂³×C₄	—	4	1	no	—
164	C₂×D₈	—	2	1	no	—
165	C₂×D₈	—	2	1	no	—
166	C₂×D₈	—	2	1	no	—
167	C₂×D₈	C₂×C₄×C₅	1	2	yes	—
168	C₂×D₈	C₅×D₄	1	2	yes	—
169	C₂×D₈	C₅×Q₈	1	2	yes	—
170	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
171	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
172	C₄×D₄	—	2	4	no	—
173	C₄×D₄	—	4	2	no	—
174	C₂×C₄⋊C₄	—	2	1	no	—
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	—	2	1	no	—
177	C₂×C₄⋊C₄	—	4	1	no	—
178	C₂×C₂²⋊C₄	—	4	1	no	—
179	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
180	C₂×C₁₆	—	2	1	no	—
181	C₂×C₁₆	—	2	1	no	—
182	C₂×C₁₆	—	2	1	no	—
183	C₂×C₁₆	—	2	1	no	—
184	C₂×C₁₆	—	2	1	no	—
185	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	complemented
186	#107	—	2	1	no	—
187	#107	—	2	1	no	—
188	#107	—	2	1	no	—
189	#107	—	2	1	no	—
190	#107	C₂×C₄×C₅	1	2	yes	complemented
191	#107	C₅×D₄	1	2	yes	—
192	#102	—	2	4	no	—
193	C₄×C₈	—	2	1	no	—
194	C₄×C₈	—	2	1	no	—
195	#95	—	4	1	no	—
196	C₂²×C₅	—	4	4	no	—
197	C₂²×C₅	—	8	2	no	—
198	C₂²×C₅	—	8	2	no	—
199	C₂²×C₅	—	8	2	no	—
200	C₂²×C₅	—	8	2	no	—
201	C₂²×C₅	—	8	2	no	—
202	C₂²×C₅	—	8	2	no	—
203	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
204	C₂²×C₅	#447	1	1	yes	characteristic
205	C₂²×C₅	#474	1	1	yes	characteristic
206	C₂²×C₅	C₂×SD₃₂	1	1	yes	characteristic
207	C₂²×C₅	#508	1	1	yes	characteristic
208	C₂²×C₅	#509	1	1	yes	characteristic
209	C₂²×C₅	#509	1	1	yes	characteristic
210	C₄×C₅	—	2	2	no	—
211	C₄×C₅	—	4	1	no	—
212	C₄×C₅	—	8	2	no	—
213	C₄×C₅	—	8	2	no	—
214	C₄×C₅	—	16	1	no	—
215	C₄×C₅	#385	1	1	yes	characteristic
216	C₄×C₅	#440	1	1	yes	characteristic
217	C₄×C₅	#452	1	1	yes	characteristic
218	C₄×C₅	#478	1	1	yes	characteristic
219	C₂⁴	—	4	1	no	—
220	C₂×D₄	—	2	2	no	—
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	—	2	2	no	—
223	C₂×D₄	—	4	1	no	—
224	C₂×D₄	—	4	1	no	—
225	C₂×D₄	—	4	1	no	—
226	C₂²×C₄	—	4	4	no	—
227	C₂²×C₄	—	4	1	no	—
228	C₂²×C₄	—	4	1	no	—
229	C₂²×C₄	—	4	1	no	—
230	C₂²×C₄	—	8	2	no	—
231	C₂²×C₄	—	8	2	no	—
232	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
233	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
235	D₈	—	2	2	no	—
236	D₈	—	2	2	no	—
237	D₈	—	2	2	no	—
238	D₈	C₅×C₄⋊C₄	1	4	yes	complemented
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
243	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
244	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
245	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
246	C₂×C₈	C₅×#41	1	1	yes	characteristic
247	C₂×C₈	C₅×#41	1	1	yes	characteristic
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	4	1	no	—
251	C₄⋊C₄	—	4	1	no	—
252	C₄⋊C₄	—	4	1	no	—
253	C₄⋊C₄	—	4	1	no	—
254	C₄⋊C₄	—	8	2	no	—
255	C₂²⋊C₄	—	4	4	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	1	no	—
258	C₁₆	—	2	2	no	—
259	C₁₆	—	4	1	no	—
260	C₂×C₅	—	8	4	no	—
261	C₂×C₅	#1862	1	1	yes	characteristic
262	C₂×C₅	C₄×SD₃₂	1	1	yes	characteristic
263	C₂×C₅	#2177	1	1	yes	characteristic
264	C₂×C₅	#2186	1	1	yes	characteristic
265	C₂×C₅	#2187	1	1	yes	characteristic
266	C₂×C₅	#2226	1	1	yes	characteristic
267	C₂×C₅	#2228	1	1	yes	characteristic
268	C₂³	—	4	2	no	—
269	C₂³	—	4	2	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	C₂×C₅×D₈	1	1	yes	characteristic
276	D₄	—	2	4	no	—
277	D₄	—	4	2	no	—
278	D₄	—	4	2	no	—
279	D₄	—	4	2	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	8	8	no	—
288	C₂×C₄	—	8	1	no	—
289	C₂×C₄	—	8	1	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	—	8	1	no	—
293	C₂×C₄	—	8	1	no	—
294	C₂×C₄	—	8	1	no	—
295	C₂×C₄	C₅×SD₃₂	1	2	yes	—
296	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
297	C₂×C₄	C₅×#120	1	1	yes	characteristic
298	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
299	C₂×C₄	C₅×#135	1	1	yes	characteristic
300	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
301	C₂×C₄	C₅×#137	1	1	yes	characteristic
302	C₈	—	4	1	no	—
303	C₈	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
304	C₈	C₄×C₅×D₄	1	1	yes	characteristic
305	C₈	C₅×C₂²⋊Q₈	1	1	yes	characteristic
306	C₈	C₅×#123	1	1	yes	characteristic
307	C₅	#12937	1	1	yes	characteristic, direct product factor, Sylow
308	C₂²	—	4	4	no	—
309	C₂²	—	8	2	no	—
310	C₂²	—	8	2	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	C₄×C₅×D₈	1	1	yes	characteristic
316	C₂²	C₅×#447	1	1	yes	characteristic
317	C₂²	C₅×#474	1	1	yes	characteristic
318	C₂²	C₂×C₅×SD₃₂	1	1	yes	characteristic
319	C₂²	C₅×#508	1	1	yes	characteristic
320	C₂²	C₅×#509	1	1	yes	characteristic
321	C₂²	C₅×#509	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	4	1	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	—	16	1	no	—
327	C₄	C₅×#385	1	1	yes	characteristic
328	C₄	C₅×#440	1	1	yes	characteristic
329	C₄	C₅×#452	1	1	yes	characteristic
330	C₄	C₅×#478	1	1	yes	characteristic
331	C₂	—	8	4	no	—
332	C₂	C₅×#1862	1	1	yes	characteristic
333	C₂	C₄×C₅×SD₃₂	1	1	yes	characteristic
334	C₂	C₅×#2177	1	1	yes	characteristic
335	C₂	C₅×#2186	1	1	yes	characteristic
336	C₂	C₅×#2187	1	1	yes	characteristic
337	C₂	C₅×#2226	1	1	yes	characteristic
338	C₂	C₅×#2228	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	C₂²×C₅×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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×#367	C₂	semidirect
5. C₅×#1942	C₂	semidirect
6. C₅×#1380	C₂	semidirect
7. C₅×#1374	C₂	semidirect
22. C₅×#357	C₄	semidirect
24. #12937	C₅	direct (N × H)
48. C₂×C₅×C₁₆	C₂×C₄	semidirect
53. C₅×#107	C₂×C₄	semidirect
60. C₂×#367	C₂×C₅	semidirect
63. #1942	C₂×C₅	semidirect
64. #1380	C₂×C₅	semidirect
65. #1374	C₂×C₅	semidirect
85. C₅×D₈	C₄⋊C₄	semidirect
121. #357	C₄×C₅	semidirect
185. C₂×C₁₆	C₂×C₄×C₅	semidirect
190. #107	C₂×C₄×C₅	semidirect
238. D₈	C₅×C₄⋊C₄	semidirect
307. C₅	#12937	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×C₄×C₅×D₈	2
C₂×C₅×#367	2
C₂×C₅×#357	2
C₅×#1942	2
C₅×#1380	2
C₅×#1374	2
#12937	5

Subgroups — C5×#12937 #11935657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12937 #11935657