Subgroups — C₅×#33504 #11956224

Order 1280.

536 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₈×#41	C₂	1	1	yes	maximal, characteristic
2	C₅×#2183	C₂	1	2	yes	maximal, complemented
3	C₅×#2168	C₂	1	4	yes	maximal, complemented
4	C₅×D₄×C₁₆	C₂	1	2	yes	maximal, complemented
5	C₅×#2163	C₂	1	2	yes	maximal, complemented
6	C₅×#2162	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#2150	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×C₄⋊C₁₆	C₂	1	2	yes	maximal, complemented
9	C₄×C₅×#41	C₄	1	1	yes	characteristic
10	C₂×C₅×M₃₂	C₂²	1	2	yes	—
11	C₂²×C₅×C₁₆	C₂²	1	4	yes	—
12	C₅×C₈×Q₈	C₄	1	1	yes	characteristic
13	C₅×C₈×Q₈	C₂²	1	1	yes	characteristic
14	C₅×C₈×D₄	C₄	1	2	yes	—
15	C₅×C₈×D₄	C₄	1	1	yes	characteristic
16	C₅×C₈×D₄	C₂²	1	1	yes	characteristic
17	C₅×C₈×D₄	C₂²	1	2	yes	—
18	C₅×#431	C₄	1	2	yes	—
19	C₅×#431	C₄	1	1	yes	characteristic
20	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
21	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₅×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
23	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
24	C₅×C₄⋊C₁₆	C₂²	1	4	yes	—
25	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
26	C₅×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
27	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₅×C₂²⋊C₁₆	C₂²	1	4	yes	—
29	C₅×#346	C₂²	1	4	yes	—
30	C₄×C₅×C₁₆	C₂²	1	2	yes	—
31	#33504	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂×C₅×#41	C₈	1	1	yes	characteristic
33	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₂³	1	2	yes	—
36	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
37	C₄×C₅×Q₈	C₈	1	1	yes	characteristic
38	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
39	C₄×C₅×D₄	C₈	1	2	yes	—
40	C₄×C₅×D₄	C₈	1	1	yes	characteristic
41	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
42	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
43	C₅×#117	C₈	1	1	yes	characteristic
44	C₅×#117	C₈	1	2	yes	—
45	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
47	C₅×M₃₂	—	4	2	no	—
48	C₂×C₅×C₁₆	—	4	4	no	—
49	C₂×C₅×C₁₆	C₂³	1	2	yes	—
50	C₂×C₅×C₁₆	C₂³	1	2	yes	—
51	C₂×C₅×C₁₆	C₂³	1	4	yes	—
52	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
53	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₅×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
55	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
56	C₅×C₂²⋊C₈	C₂×C₄	1	2	yes	—
57	C₅×C₂²⋊C₈	C₂×C₄	1	2	yes	—
58	C₅×C₂²⋊C₈	C₂×C₄	1	2	yes	—
59	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
60	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
61	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
62	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
63	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
64	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
65	C₄×C₅×C₈	C₂³	1	2	yes	—
66	C₈×#41	C₂×C₅	1	1	yes	characteristic
67	#2183	C₂×C₅	1	2	yes	complemented
68	#2168	C₂×C₅	1	4	yes	complemented
69	D₄×C₁₆	C₂×C₅	1	2	yes	complemented
70	#2163	C₂×C₅	1	2	yes	complemented
71	#2162	C₂×C₅	1	1	yes	characteristic, complemented
72	#2150	C₂×C₅	1	1	yes	characteristic, complemented
73	C₂×C₄⋊C₁₆	C₂×C₅	1	2	yes	complemented
74	C₅×#41	—	2	4	no	—
75	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
76	C₂×C₅×D₄	C₂×C₈	1	2	yes	—
77	C₂×C₅×D₄	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
79	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
82	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
84	C₂×C₅×C₈	—	2	2	no	—
85	C₂×C₅×C₈	—	2	1	no	—
86	C₂×C₅×C₈	—	2	2	no	—
87	C₂×C₅×C₈	—	2	1	no	—
88	C₂×C₅×C₈	—	4	2	no	—
89	C₂×C₅×C₈	—	4	1	no	—
90	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
93	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
94	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
96	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
97	C₅×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
98	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
99	C₅×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
100	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
101	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
102	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
103	C₄²×C₅	C₂×C₈	1	2	yes	—
104	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
105	C₄²×C₅	C₂×C₈	1	2	yes	—
106	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
107	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₅	C₂²×C₄	1	2	yes	—
109	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
110	C₅×C₁₆	—	2	2	no	—
111	C₅×C₁₆	—	2	4	no	—
112	C₅×C₁₆	—	2	2	no	—
113	C₄×#41	C₄×C₅	1	1	yes	characteristic
114	C₂×M₃₂	C₂²×C₅	1	2	yes	—
115	C₂²×C₁₆	C₂²×C₅	1	4	yes	—
116	C₈×Q₈	C₄×C₅	1	1	yes	characteristic
117	C₈×Q₈	C₂²×C₅	1	1	yes	characteristic
118	C₈×D₄	C₄×C₅	1	2	yes	—
119	C₈×D₄	C₄×C₅	1	1	yes	characteristic
120	C₈×D₄	C₂²×C₅	1	1	yes	characteristic
121	C₈×D₄	C₂²×C₅	1	2	yes	—
122	#431	C₄×C₅	1	1	yes	characteristic
123	#431	C₄×C₅	1	2	yes	—
124	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
125	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
126	C₄⋊C₁₆	C₂²×C₅	1	1	yes	characteristic
127	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
128	C₄⋊C₁₆	C₂²×C₅	1	4	yes	—
129	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
130	C₄⋊C₁₆	C₂²×C₅	1	1	yes	characteristic
131	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
132	C₂²⋊C₁₆	C₂²×C₅	1	4	yes	—
133	#346	C₂²×C₅	1	4	yes	—
134	C₄×C₁₆	C₂²×C₅	1	2	yes	—
135	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
136	C₂³×C₅	C₂²×C₈	1	2	yes	—
137	C₅×Q₈	—	2	2	no	—
138	C₅×D₄	—	2	4	no	—
139	C₅×D₄	M₃₂	1	4	yes	complemented
140	C₂×C₄×C₅	—	2	4	no	—
141	C₂×C₄×C₅	—	2	2	no	—
142	C₂×C₄×C₅	—	2	1	no	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	—	2	1	no	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	2	4	no	—
147	C₂×C₄×C₅	—	2	2	no	—
148	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
150	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
152	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
154	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
155	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
156	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
157	C₅×C₈	—	2	2	no	—
158	C₅×C₈	—	2	1	no	—
159	C₅×C₈	—	2	1	no	—
160	C₅×C₈	—	2	1	no	—
161	C₅×C₈	—	2	2	no	—
162	C₅×C₈	H(5,𝔽₂)	1	1	yes	characteristic
163	C₅×C₈	2₋¹⁺⁴	1	1	yes	characteristic
164	C₂×#41	C₅×C₈	1	1	yes	characteristic
165	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
166	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
167	C₂²×C₈	C₂³×C₅	1	2	yes	—
168	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
169	C₄×Q₈	C₅×C₈	1	1	yes	characteristic
170	C₄×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
171	C₄×D₄	C₅×C₈	1	2	yes	—
172	C₄×D₄	C₅×C₈	1	1	yes	characteristic
173	C₄×D₄	C₂×C₄×C₅	1	2	yes	—
174	C₄×D₄	C₂×C₄×C₅	1	1	yes	characteristic
175	#117	C₅×C₈	1	1	yes	characteristic
176	#117	C₅×C₈	1	2	yes	—
177	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
178	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
179	M₃₂	—	4	2	no	—
180	C₂×C₁₆	—	4	4	no	—
181	C₂×C₁₆	C₂³×C₅	1	2	yes	—
182	C₂×C₁₆	C₂³×C₅	1	2	yes	—
183	C₂×C₁₆	C₂³×C₅	1	4	yes	—
184	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
185	C₄⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
186	C₄⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
187	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
188	C₂²⋊C₈	C₂×C₄×C₅	1	2	yes	—
189	C₂²⋊C₈	C₂×C₄×C₅	1	2	yes	—
190	C₂²⋊C₈	C₂×C₄×C₅	1	2	yes	—
191	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
192	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
193	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
194	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
195	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
196	C₄×C₈	C₂³×C₅	1	2	yes	—
197	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
198	C₂²×C₅	—	2	2	no	—
199	C₂²×C₅	—	2	2	no	—
200	C₂²×C₅	—	2	1	no	—
201	C₂²×C₅	—	2	1	no	—
202	C₂²×C₅	—	2	1	no	—
203	C₂²×C₅	C₂×M₃₂	1	4	yes	—
204	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
205	C₄×C₅	—	2	1	no	—
206	C₄×C₅	—	2	2	no	—
207	C₄×C₅	—	2	2	no	—
208	C₄×C₅	—	2	1	no	—
209	C₄×C₅	—	2	2	no	—
210	C₄×C₅	—	2	1	no	—
211	C₄×C₅	C₂×M₃₂	1	2	yes	—
212	C₄×C₅	#504	1	2	yes	—
213	C₄×C₅	#520	1	1	yes	characteristic
214	C₄×C₅	#568	1	1	yes	characteristic
215	#41	—	2	4	no	—
216	C₂×Q₈	C₂×C₅×C₈	1	1	yes	characteristic
217	C₂×D₄	C₂×C₅×C₈	1	2	yes	—
218	C₂×D₄	C₂×C₅×C₈	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
220	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
223	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
224	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
232	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
233	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
234	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
235	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
236	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
237	C₄⋊C₄	C₂×C₅×C₈	1	2	yes	—
238	C₄⋊C₄	C₂×C₅×C₈	1	1	yes	characteristic
239	C₄⋊C₄	C₂×C₅×C₈	1	2	yes	—
240	C₄⋊C₄	C₂×C₅×C₈	1	1	yes	characteristic
241	C₂²⋊C₄	C₂×C₅×C₈	1	2	yes	—
242	C₂²⋊C₄	C₂×C₅×C₈	1	2	yes	—
243	C₂²⋊C₄	C₂×C₅×C₈	1	2	yes	—
244	C₄²	C₂×C₅×C₈	1	2	yes	—
245	C₄²	C₂×C₅×C₈	1	1	yes	characteristic
246	C₄²	C₂×C₅×C₈	1	2	yes	—
247	C₄²	C₂×C₅×C₈	1	1	yes	characteristic
248	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
249	C₄²	C₂²×C₄×C₅	1	2	yes	—
250	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
251	C₁₆	—	2	2	no	—
252	C₁₆	—	2	4	no	—
253	C₁₆	—	2	2	no	—
254	C₂×C₅	—	2	4	no	—
255	C₂×C₅	—	4	1	no	—
256	C₂×C₅	#2988	1	1	yes	characteristic
257	C₂×C₅	C₂²×M₃₂	1	1	yes	characteristic
258	C₂×C₅	C₂×#504	1	1	yes	characteristic
259	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
260	C₂³	C₂²×C₅×C₈	1	2	yes	—
261	Q₈	—	2	2	no	—
262	D₄	—	2	4	no	—
263	D₄	C₅×M₃₂	1	4	yes	complemented
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
273	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
274	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
275	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
276	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
277	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
278	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
279	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
280	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
281	C₈	—	2	2	no	—
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	—	2	1	no	—
285	C₈	—	2	2	no	—
286	C₈	C₅×H(5,𝔽₂)	1	1	yes	characteristic
287	C₈	C₅×2₋¹⁺⁴	1	1	yes	characteristic
288	C₅	#33504	1	1	yes	characteristic, direct product factor, Sylow
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	C₂×C₅×M₃₂	1	4	yes	—
295	C₂²	C₂³×C₅×C₈	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	1	no	—
302	C₄	C₂×C₅×M₃₂	1	2	yes	—
303	C₄	C₅×#504	1	2	yes	—
304	C₄	C₅×#520	1	1	yes	characteristic
305	C₄	C₅×#568	1	1	yes	characteristic
306	C₂	—	2	4	no	—
307	C₂	—	4	1	no	—
308	C₂	C₅×#2988	1	1	yes	characteristic
309	C₂	C₂²×C₅×M₃₂	1	1	yes	characteristic
310	C₂	C₂×C₅×#504	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅×C₈	C₂⁴
commutator	C₂²	C₂³×C₅×C₈
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂²×C₅	C₂³×C₈
prime radical	C₂×C₅×#41	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2183	C₂	semidirect
3. C₅×#2168	C₂	semidirect
4. C₅×D₄×C₁₆	C₂	semidirect
5. C₅×#2163	C₂	semidirect
6. C₅×#2162	C₂	semidirect
7. C₅×#2150	C₂	semidirect
8. C₂×C₅×C₄⋊C₁₆	C₂	semidirect
31. #33504	C₅	direct (N × H)
67. #2183	C₂×C₅	semidirect
68. #2168	C₂×C₅	semidirect
69. D₄×C₁₆	C₂×C₅	semidirect
70. #2163	C₂×C₅	semidirect
71. #2162	C₂×C₅	semidirect
72. #2150	C₂×C₅	semidirect
73. C₂×C₄⋊C₁₆	C₂×C₅	semidirect
139. C₅×D₄	M₃₂	semidirect
263. D₄	C₅×M₃₂	semidirect
288. C₅	#33504	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×C₈×#41	2
C₅×#2183	2
C₅×#2168	2
C₅×D₄×C₁₆	2
C₅×#2163	2
C₅×#2162	2
C₅×#2150	2
C₂×C₅×C₄⋊C₁₆	2
#33504	5

Subgroups — C5×#33504 #11956224

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33504 #11956224