Subgroups — C₅×#33418 #11956138

Order 1280.

682 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#2163	C₂	1	4	yes	maximal, complemented
3	C₅×#2162	C₂	1	4	yes	maximal, complemented
4	C₅×#2150	C₂	1	2	yes	maximal, complemented
5	C₅×#2112	C₂	1	2	yes	maximal
6	C₄×C₅×M₃₂	C₂	1	2	yes	maximal, complemented
7	C₂³×C₅×C₈	C₄	1	1	yes	characteristic
8	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₅	C₄	1	1	yes	characteristic
10	C₂×C₅×M₃₂	C₂²	1	4	yes	—
11	C₂×C₄×C₅×C₈	C₄	1	2	yes	—
12	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
13	C₂×C₄×C₅×C₈	C₄	1	2	yes	—
14	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
17	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
19	C₅×C₄⋊C₁₆	C₂²	1	8	yes	complemented
20	C₅×C₂²⋊C₁₆	C₂²	1	8	yes	—
21	C₅×#346	C₂²	1	4	yes	complemented
22	C₄×C₅×C₁₆	C₂²	1	4	yes	complemented
23	#33418	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂³×C₄×C₅	C₈	1	2	yes	—
25	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅×C₈	—	2	4	no	—
27	C₂²×C₅×C₈	—	2	4	no	—
28	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
32	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
33	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	C₂³	1	2	yes	—
35	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₅	C₈	1	2	yes	—
37	C₂×C₄²×C₅	C₈	1	2	yes	—
38	C₂×C₄²×C₅	C₈	1	2	yes	—
39	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
43	C₅×M₃₂	—	2	8	no	—
44	C₂×C₅×C₁₆	C₂³	1	8	yes	—
45	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
46	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
47	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
48	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
49	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
50	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
51	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
52	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
53	C₄×C₅×C₈	C₂³	1	2	yes	—
54	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic
56	#2163	C₂×C₅	1	4	yes	complemented
57	#2162	C₂×C₅	1	4	yes	complemented
58	#2150	C₂×C₅	1	2	yes	complemented
59	#2112	C₂×C₅	1	2	yes	—
60	C₄×M₃₂	C₂×C₅	1	2	yes	complemented
61	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₅	—	2	8	no	—
63	C₂²×C₄×C₅	—	2	2	no	—
64	C₂²×C₄×C₅	—	2	2	no	—
65	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
67	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
68	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
69	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
70	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
71	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
72	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
73	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₅×C₈	—	2	2	no	—
76	C₂×C₅×C₈	—	2	2	no	—
77	C₂×C₅×C₈	—	2	4	no	—
78	C₂×C₅×C₈	—	2	4	no	—
79	C₂×C₅×C₈	—	2	4	no	—
80	C₂×C₅×C₈	—	2	4	no	—
81	C₂×C₅×C₈	M₁₆	1	4	yes	—
82	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
83	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
84	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₅×C₈	#41	1	4	yes	—
88	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
89	C₄²×C₅	C₂×C₈	1	2	yes	—
90	C₄²×C₅	C₂×C₈	1	2	yes	—
91	C₄²×C₅	C₂×C₈	1	2	yes	—
92	C₄²×C₅	C₂×C₈	1	2	yes	—
93	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
94	C₄²×C₅	C₂×C₈	1	2	yes	—
95	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
96	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₅	C₂²×C₄	1	2	yes	—
98	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
99	C₅×C₁₆	—	2	8	no	—
100	C₂³×C₈	C₄×C₅	1	1	yes	characteristic
101	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
102	C₂²×C₄²	C₄×C₅	1	1	yes	characteristic
103	C₂×M₃₂	C₂²×C₅	1	4	yes	—
104	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
105	C₂×C₄×C₈	C₄×C₅	1	2	yes	—
106	C₂×C₄×C₈	C₄×C₅	1	2	yes	—
107	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
110	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
111	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
112	C₄⋊C₁₆	C₂²×C₅	1	8	yes	complemented
113	C₂²⋊C₁₆	C₂²×C₅	1	8	yes	—
114	#346	C₂²×C₅	1	4	yes	complemented
115	C₄×C₁₆	C₂²×C₅	1	4	yes	complemented
116	C₂³×C₅	—	2	2	no	—
117	C₂³×C₅	—	2	4	no	—
118	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
119	C₂³×C₅	C₂²×C₈	1	2	yes	—
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	—	2	4	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	2	4	no	—
125	C₂×C₄×C₅	—	2	4	no	—
126	C₂×C₄×C₅	—	2	4	no	—
127	C₂×C₄×C₅	—	2	4	no	—
128	C₂×C₄×C₅	—	2	4	no	—
129	C₂×C₄×C₅	M₃₂	1	8	yes	complemented
130	C₂×C₄×C₅	#117	1	2	yes	—
131	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
132	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
133	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
134	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
136	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
137	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
138	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
139	C₅×C₈	—	2	2	no	—
140	C₅×C₈	—	2	2	no	—
141	C₅×C₈	#117	1	2	yes	—
142	C₅×C₈	C₂×M₁₆	1	2	yes	—
143	C₅×C₈	C₂×M₁₆	1	2	yes	—
144	C₅×C₈	C₂×#41	1	2	yes	—
145	C₂³×C₄	C₅×C₈	1	2	yes	—
146	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
147	C₂²×C₈	—	2	4	no	—
148	C₂²×C₈	—	2	4	no	—
149	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
150	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
151	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
152	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
153	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
154	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
155	C₂²×C₈	C₂³×C₅	1	2	yes	—
156	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
157	C₂×C₄²	C₅×C₈	1	2	yes	—
158	C₂×C₄²	C₅×C₈	1	2	yes	—
159	C₂×C₄²	C₅×C₈	1	2	yes	—
160	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
161	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
162	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
163	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
164	M₃₂	—	2	8	no	—
165	C₂×C₁₆	C₂³×C₅	1	8	yes	—
166	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
167	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
168	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
169	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
170	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
171	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
172	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
173	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
174	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
175	C₄×C₈	C₂³×C₅	1	2	yes	—
176	C₂²×C₅	—	2	8	no	—
177	C₂²×C₅	—	2	2	no	—
178	C₂²×C₅	—	2	2	no	—
179	C₂²×C₅	—	2	2	no	—
180	C₂²×C₅	#431	1	2	yes	—
181	C₂²×C₅	C₂×M₃₂	1	4	yes	—
182	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
183	C₄×C₅	—	2	2	no	—
184	C₄×C₅	—	2	2	no	—
185	C₄×C₅	—	2	2	no	—
186	C₄×C₅	#431	1	2	yes	—
187	C₄×C₅	C₂×M₃₂	1	4	yes	—
188	C₄×C₅	C₂×M₃₂	1	4	yes	—
189	C₄×C₅	C₂×#117	1	1	yes	characteristic
190	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
191	C₂⁴	C₂×C₅×C₈	1	1	yes	characteristic
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	8	no	—
195	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
197	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
198	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
199	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
200	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
201	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
202	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
203	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
204	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	C₅×M₁₆	1	4	yes	—
212	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
213	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
214	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
215	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
216	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
217	C₂×C₈	C₅×#41	1	4	yes	—
218	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
219	C₄²	C₂×C₅×C₈	1	2	yes	—
220	C₄²	C₂×C₅×C₈	1	2	yes	—
221	C₄²	C₂×C₅×C₈	1	2	yes	—
222	C₄²	C₂×C₅×C₈	1	2	yes	—
223	C₄²	C₂×C₅×C₈	1	1	yes	characteristic
224	C₄²	C₂×C₅×C₈	1	2	yes	—
225	C₄²	C₂×C₅×C₈	1	1	yes	characteristic
226	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
227	C₄²	C₂²×C₄×C₅	1	2	yes	—
228	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
229	C₁₆	—	2	8	no	—
230	C₂×C₅	—	2	2	no	—
231	C₂×C₅	—	2	4	no	—
232	C₂×C₅	C₂×#431	1	1	yes	characteristic
233	C₂×C₅	C₂²×M₃₂	1	2	yes	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	4	no	—
236	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
237	C₂³	C₂²×C₅×C₈	1	2	yes	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	C₅×M₃₂	1	8	yes	complemented
248	C₂×C₄	C₅×#117	1	2	yes	—
249	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
250	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
251	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
252	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
253	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
254	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
255	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
256	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	2	2	no	—
259	C₈	C₅×#117	1	2	yes	—
260	C₈	C₂×C₅×M₁₆	1	2	yes	—
261	C₈	C₂×C₅×M₁₆	1	2	yes	—
262	C₈	C₂×C₅×#41	1	2	yes	—
263	C₅	#33418	1	1	yes	characteristic, direct product factor, Sylow
264	C₂²	—	2	8	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	C₅×#431	1	2	yes	—
269	C₂²	C₂×C₅×M₃₂	1	4	yes	—
270	C₂²	C₂³×C₅×C₈	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	C₅×#431	1	2	yes	—
275	C₄	C₂×C₅×M₃₂	1	4	yes	—
276	C₄	C₂×C₅×M₃₂	1	4	yes	—
277	C₄	C₂×C₅×#117	1	1	yes	characteristic
278	C₄	C₂²×C₅×M₁₆	1	1	yes	characteristic
279	C₂	—	2	2	no	—
280	C₂	—	2	4	no	—
281	C₂	C₂×C₅×#431	1	1	yes	characteristic
282	C₂	C₂²×C₅×M₃₂	1	2	yes	—

	subgroup	G/H
center	C₄×C₅×C₈	C₂³
commutator	C₂²	C₂³×C₅×C₈
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂²×C₅	C₂³×C₈
prime radical	C₂⁴×C₅	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2163	C₂	semidirect
3. C₅×#2162	C₂	semidirect
4. C₅×#2150	C₂	semidirect
6. C₄×C₅×M₃₂	C₂	semidirect
19. C₅×C₄⋊C₁₆	C₂²	semidirect
21. C₅×#346	C₂²	semidirect
22. C₄×C₅×C₁₆	C₂²	semidirect
23. #33418	C₅	direct (N × H)
56. #2163	C₂×C₅	semidirect
57. #2162	C₂×C₅	semidirect
58. #2150	C₂×C₅	semidirect
60. C₄×M₃₂	C₂×C₅	semidirect
112. C₄⋊C₁₆	C₂²×C₅	semidirect
114. #346	C₂²×C₅	semidirect
115. C₄×C₁₆	C₂²×C₅	semidirect
129. C₂×C₄×C₅	M₃₂	semidirect
247. C₂×C₄	C₅×M₃₂	semidirect
263. C₅	#33418	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₅×C₈	2
C₅×#2163	2
C₅×#2162	2
C₅×#2150	2
C₅×#2112	2
C₄×C₅×M₃₂	2
#33418	5

Subgroups — C5×#33418 #11956138

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33418 #11956138