Subgroups — C₅×D₄×D₁₆ #11956565

Order 1280.

812 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×C₅×D₁₆	C₂	1	2	yes	maximal, complemented
2	C₅×D₄×D₈	C₂	1	2	yes	maximal, complemented
3	C₅×#2246	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2215	C₂	1	2	yes	maximal, complemented
5	C₅×#2206	C₂	1	2	yes	maximal, complemented
6	C₅×#2184	C₂	1	4	yes	maximal, complemented
7	C₄×C₅×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×D₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₅×D₈	C₂²	1	4	yes	complemented
10	C₅×D₄²	—	2	2	no	—
11	C₂×C₅×D₁₆	—	2	8	no	—
12	C₂×C₅×D₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₅×D₁₆	C₂²	1	4	yes	complemented
14	C₂×C₅×D₁₆	C₂²	1	2	yes	complemented
15	C₂×C₅×D₁₆	C₂²	1	2	yes	complemented
16	C₂²×C₅×C₁₆	C₂²	1	2	yes	complemented
17	C₅×#493	C₂²	1	2	yes	complemented
18	C₅×#466	C₂²	1	4	yes	complemented
19	C₅×#459	—	2	2	no	—
20	C₅×#447	—	2	4	no	—
21	C₄×C₅×D₈	C₂²	1	2	yes	complemented
22	C₅×C₈×D₄	C₂²	1	1	yes	characteristic
23	C₅×#366	C₂²	1	1	yes	characteristic, complemented
24	C₅×C₄⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
25	C₅×#357	C₂²	1	2	yes	complemented
26	C₅×#357	C₂²	1	4	yes	complemented
27	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	complemented
28	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic, complemented
29	D₄×D₁₆	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₅×D₄	—	2	4	no	—
31	C₂²×C₅×D₄	—	4	2	no	—
32	C₂×C₅×D₈	—	2	8	no	—
33	C₂×C₅×D₈	—	2	4	no	—
34	C₂×C₅×D₈	—	2	2	no	—
35	C₂×C₅×D₈	—	4	4	no	—
36	C₂×C₅×D₈	C₂³	1	4	yes	—
37	C₂×C₅×D₈	C₂³	1	2	yes	—
38	C₂²×C₅×C₈	C₂³	1	2	yes	—
39	C₅×#127	—	2	2	no	—
40	C₅×#121	—	2	4	no	—
41	C₅×#121	—	4	2	no	—
42	C₅×#120	—	4	4	no	—
43	C₄×C₅×D₄	—	2	2	no	—
44	C₄×C₅×D₄	D₄	1	1	yes	characteristic
45	C₅×D₁₆	—	2	8	no	—
46	C₅×D₁₆	—	2	4	no	—
47	C₅×D₁₆	—	4	2	no	—
48	C₅×D₁₆	D₄	1	4	yes	direct product factor
49	C₂×C₅×C₁₆	—	2	4	no	—
50	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
51	C₂×C₅×C₁₆	C₂³	1	2	yes	—
52	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
53	C₅×#107	C₂³	1	2	yes	—
54	C₅×C₄⋊C₈	D₄	1	1	yes	characteristic
55	C₅×#102	—	2	2	no	—
56	C₅×#102	—	2	4	no	—
57	C₅×C₂²⋊C₈	D₄	1	2	yes	—
58	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
59	C₂²×D₁₆	C₂×C₅	1	2	yes	complemented
60	D₄×D₈	C₂×C₅	1	2	yes	complemented
61	#2246	C₂×C₅	1	1	yes	characteristic, complemented
62	#2215	C₂×C₅	1	2	yes	complemented
63	#2206	C₂×C₅	1	2	yes	complemented
64	#2184	C₂×C₅	1	4	yes	complemented
65	C₄×D₁₆	C₂×C₅	1	1	yes	characteristic, complemented
66	D₄×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
67	C₂⁴×C₅	—	4	4	no	—
68	C₂×C₅×D₄	—	2	2	no	—
69	C₂×C₅×D₄	—	2	4	no	—
70	C₂×C₅×D₄	—	4	8	no	—
71	C₂×C₅×D₄	—	4	4	no	—
72	C₂×C₅×D₄	—	4	4	no	—
73	C₂×C₅×D₄	—	4	2	no	—
74	C₂×C₅×D₄	—	4	2	no	—
75	C₂×C₅×D₄	—	8	4	no	—
76	C₂×C₅×D₄	—	8	8	no	—
77	C₂×C₅×D₄	D₈	1	1	yes	characteristic
78	C₂²×C₄×C₅	—	4	2	no	—
79	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
80	C₅×D₈	—	2	4	no	—
81	C₅×D₈	—	2	2	no	—
82	C₅×D₈	—	2	4	no	—
83	C₅×D₈	—	4	8	no	—
84	C₅×D₈	—	8	2	no	—
85	C₅×D₈	C₂×D₄	1	4	yes	complemented
86	C₂×C₅×C₈	—	2	2	no	—
87	C₂×C₅×C₈	—	2	2	no	—
88	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
89	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
91	C₅×C₄⋊C₄	—	2	2	no	—
92	C₅×C₄⋊C₄	D₈	1	1	yes	characteristic
93	C₅×C₂²⋊C₄	—	4	2	no	—
94	C₅×C₂²⋊C₄	—	4	4	no	—
95	C₅×C₂²⋊C₄	D₈	1	2	yes	—
96	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
97	C₅×C₁₆	—	2	2	no	—
98	C₅×C₁₆	—	2	1	no	—
99	C₅×C₁₆	C₂×D₄	1	2	yes	complemented
100	C₂²×D₈	C₂²×C₅	1	4	yes	complemented
101	D₄²	—	2	2	no	—
102	C₂×D₁₆	—	2	8	no	—
103	C₂×D₁₆	C₂²×C₅	1	1	yes	characteristic
104	C₂×D₁₆	C₂²×C₅	1	4	yes	complemented
105	C₂×D₁₆	C₂²×C₅	1	2	yes	complemented
106	C₂×D₁₆	C₂²×C₅	1	2	yes	complemented
107	C₂²×C₁₆	C₂²×C₅	1	2	yes	complemented
108	#493	C₂²×C₅	1	2	yes	complemented
109	#466	C₂²×C₅	1	4	yes	complemented
110	#459	—	2	2	no	—
111	#447	—	2	4	no	—
112	C₄×D₈	C₂²×C₅	1	2	yes	complemented
113	C₈×D₄	C₂²×C₅	1	1	yes	characteristic
114	#366	C₂²×C₅	1	1	yes	characteristic, complemented
115	C₄⋊C₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
116	#357	C₂²×C₅	1	2	yes	complemented
117	#357	C₂²×C₅	1	4	yes	complemented
118	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	complemented
119	C₄×C₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
120	C₂³×C₅	—	4	2	no	—
121	C₂³×C₅	—	4	4	no	—
122	C₂³×C₅	—	8	8	no	—
123	C₂³×C₅	—	8	8	no	—
124	C₂³×C₅	—	16	4	no	—
125	C₂³×C₅	C₂×D₈	1	2	yes	—
126	C₅×D₄	—	2	4	no	—
127	C₅×D₄	—	4	4	no	—
128	C₅×D₄	—	4	2	no	—
129	C₅×D₄	—	4	4	no	—
130	C₅×D₄	—	8	8	no	—
131	C₅×D₄	—	8	8	no	—
132	C₅×D₄	—	8	4	no	—
133	C₅×D₄	—	16	2	no	—
134	C₅×D₄	D₁₆	1	4	yes	direct product factor
135	C₂×C₄×C₅	—	2	2	no	—
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	—	4	2	no	—
138	C₂×C₄×C₅	—	8	4	no	—
139	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
141	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
143	C₅×C₈	—	2	2	no	—
144	C₅×C₈	—	2	1	no	—
145	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
146	C₅×C₈	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂²×D₄	—	2	4	no	—
148	C₂²×D₄	—	4	2	no	—
149	C₂×D₈	—	2	8	no	—
150	C₂×D₈	—	2	4	no	—
151	C₂×D₈	—	2	2	no	—
152	C₂×D₈	—	4	4	no	—
153	C₂×D₈	C₂³×C₅	1	4	yes	—
154	C₂×D₈	C₂³×C₅	1	2	yes	—
155	C₂²×C₈	C₂³×C₅	1	2	yes	—
156	#127	—	2	2	no	—
157	#121	—	2	4	no	—
158	#121	—	4	2	no	—
159	#120	—	4	4	no	—
160	C₄×D₄	—	2	2	no	—
161	C₄×D₄	C₅×D₄	1	1	yes	characteristic
162	D₁₆	—	2	8	no	—
163	D₁₆	—	2	4	no	—
164	D₁₆	—	4	2	no	—
165	D₁₆	C₅×D₄	1	4	yes	direct product factor
166	C₂×C₁₆	—	2	4	no	—
167	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
168	C₂×C₁₆	C₂³×C₅	1	2	yes	—
169	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
170	#107	C₂³×C₅	1	2	yes	—
171	C₄⋊C₈	C₅×D₄	1	1	yes	characteristic
172	#102	—	2	2	no	—
173	#102	—	2	4	no	—
174	C₂²⋊C₈	C₅×D₄	1	2	yes	—
175	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
176	C₂²×C₅	—	2	2	no	—
177	C₂²×C₅	—	2	2	no	—
178	C₂²×C₅	—	4	4	no	—
179	C₂²×C₅	—	8	2	no	—
180	C₂²×C₅	—	8	2	no	—
181	C₂²×C₅	—	8	4	no	—
182	C₂²×C₅	—	8	4	no	—
183	C₂²×C₅	—	8	4	no	—
184	C₂²×C₅	—	16	16	no	—
185	C₂²×C₅	C₂×D₁₆	1	4	yes	complemented
186	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
187	C₄×C₅	—	2	1	no	—
188	C₄×C₅	—	2	2	no	—
189	C₄×C₅	—	8	2	no	—
190	C₄×C₅	C₂×D₁₆	1	2	yes	complemented
191	C₄×C₅	D₄²	1	1	yes	characteristic
192	C₄×C₅	#576	1	1	yes	characteristic
193	C₂⁴	—	4	4	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	2	4	no	—
196	C₂×D₄	—	4	8	no	—
197	C₂×D₄	—	4	4	no	—
198	C₂×D₄	—	4	4	no	—
199	C₂×D₄	—	4	2	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂×D₄	—	8	4	no	—
202	C₂×D₄	—	8	8	no	—
203	C₂×D₄	C₅×D₈	1	1	yes	characteristic
204	C₂²×C₄	—	4	2	no	—
205	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
206	D₈	—	2	4	no	—
207	D₈	—	2	2	no	—
208	D₈	—	2	4	no	—
209	D₈	—	4	8	no	—
210	D₈	—	8	2	no	—
211	D₈	C₂×C₅×D₄	1	4	yes	complemented
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
215	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
216	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	C₅×D₈	1	1	yes	characteristic
219	C₂²⋊C₄	—	4	2	no	—
220	C₂²⋊C₄	—	4	4	no	—
221	C₂²⋊C₄	C₅×D₈	1	2	yes	—
222	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
223	C₁₆	—	2	2	no	—
224	C₁₆	—	2	1	no	—
225	C₁₆	C₂×C₅×D₄	1	2	yes	complemented
226	C₂×C₅	—	2	4	no	—
227	C₂×C₅	—	8	4	no	—
228	C₂×C₅	—	16	4	no	—
229	C₂×C₅	D₄×D₈	1	1	yes	characteristic
230	C₂×C₅	C₂²×D₁₆	1	1	yes	characteristic
231	C₂×C₅	#3415	1	1	yes	characteristic
232	C₂³	—	4	2	no	—
233	C₂³	—	4	4	no	—
234	C₂³	—	8	8	no	—
235	C₂³	—	8	8	no	—
236	C₂³	—	16	4	no	—
237	C₂³	C₂×C₅×D₈	1	2	yes	—
238	D₄	—	2	4	no	—
239	D₄	—	4	4	no	—
240	D₄	—	4	2	no	—
241	D₄	—	4	4	no	—
242	D₄	—	8	8	no	—
243	D₄	—	8	8	no	—
244	D₄	—	8	4	no	—
245	D₄	—	16	2	no	—
246	D₄	C₅×D₁₆	1	4	yes	direct product factor
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	8	4	no	—
251	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
252	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
253	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
254	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	2	1	no	—
257	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
258	C₈	C₅×H(5,𝔽₂)	1	1	yes	characteristic
259	C₅	D₄×D₁₆	1	1	yes	characteristic, direct product factor, Sylow
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	4	4	no	—
263	C₂²	—	8	2	no	—
264	C₂²	—	8	2	no	—
265	C₂²	—	8	4	no	—
266	C₂²	—	8	4	no	—
267	C₂²	—	8	4	no	—
268	C₂²	—	16	16	no	—
269	C₂²	C₂×C₅×D₁₆	1	4	yes	complemented
270	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	1	no	—
273	C₄	—	8	2	no	—
274	C₄	C₂×C₅×D₁₆	1	2	yes	complemented
275	C₄	C₅×D₄²	1	1	yes	characteristic
276	C₄	C₅×#576	1	1	yes	characteristic
277	C₂	—	2	4	no	—
278	C₂	—	8	4	no	—
279	C₂	—	16	4	no	—
280	C₂	C₅×D₄×D₈	1	1	yes	characteristic
281	C₂	C₂²×C₅×D₁₆	1	1	yes	characteristic
282	C₂	C₅×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄×C₅×D₄ ◁ C₅×C₈×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×D₁₆	C₂	semidirect
2. C₅×D₄×D₈	C₂	semidirect
3. C₅×#2246	C₂	semidirect
4. C₅×#2215	C₂	semidirect
5. C₅×#2206	C₂	semidirect
6. C₅×#2184	C₂	semidirect
7. C₄×C₅×D₁₆	C₂	semidirect
8. C₅×D₄×C₁₆	C₂	semidirect
9. C₂²×C₅×D₈	C₂²	semidirect
13. C₂×C₅×D₁₆	C₂²	semidirect
14. C₂×C₅×D₁₆	C₂²	semidirect
15. C₂×C₅×D₁₆	C₂²	semidirect
16. C₂²×C₅×C₁₆	C₂²	semidirect
17. C₅×#493	C₂²	semidirect
18. C₅×#466	C₂²	semidirect
21. C₄×C₅×D₈	C₂²	semidirect
23. C₅×#366	C₂²	semidirect
24. C₅×C₄⋊C₁₆	C₂²	semidirect
25. C₅×#357	C₂²	semidirect
26. C₅×#357	C₂²	semidirect
27. C₅×C₂²⋊C₁₆	C₂²	semidirect
28. C₄×C₅×C₁₆	C₂²	semidirect
29. D₄×D₁₆	C₅	direct (N × H)
48. C₅×D₁₆	D₄	direct (N × H)
59. C₂²×D₁₆	C₂×C₅	semidirect
60. D₄×D₈	C₂×C₅	semidirect
61. #2246	C₂×C₅	semidirect
62. #2215	C₂×C₅	semidirect
63. #2206	C₂×C₅	semidirect
64. #2184	C₂×C₅	semidirect
65. C₄×D₁₆	C₂×C₅	semidirect
66. D₄×C₁₆	C₂×C₅	semidirect
85. C₅×D₈	C₂×D₄	semidirect
99. C₅×C₁₆	C₂×D₄	semidirect
100. C₂²×D₈	C₂²×C₅	semidirect
104. C₂×D₁₆	C₂²×C₅	semidirect
105. C₂×D₁₆	C₂²×C₅	semidirect
106. C₂×D₁₆	C₂²×C₅	semidirect
107. C₂²×C₁₆	C₂²×C₅	semidirect
108. #493	C₂²×C₅	semidirect
109. #466	C₂²×C₅	semidirect
112. C₄×D₈	C₂²×C₅	semidirect
114. #366	C₂²×C₅	semidirect
115. C₄⋊C₁₆	C₂²×C₅	semidirect
116. #357	C₂²×C₅	semidirect
117. #357	C₂²×C₅	semidirect
118. C₂²⋊C₁₆	C₂²×C₅	semidirect
119. C₄×C₁₆	C₂²×C₅	semidirect
134. C₅×D₄	D₁₆	direct (N × H)
165. D₁₆	C₅×D₄	direct (N × H)
185. C₂²×C₅	C₂×D₁₆	semidirect
190. C₄×C₅	C₂×D₁₆	semidirect
211. D₈	C₂×C₅×D₄	semidirect
225. C₁₆	C₂×C₅×D₄	semidirect
246. D₄	C₅×D₁₆	direct (N × H)
259. C₅	D₄×D₁₆	direct (N × H)
269. C₂²	C₂×C₅×D₁₆	semidirect
274. C₄	C₂×C₅×D₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×D₁₆	2
C₅×D₄×D₈	2
C₅×#2246	2
C₅×#2215	2
C₅×#2206	2
C₅×#2184	2
C₄×C₅×D₁₆	2
C₅×D₄×C₁₆	2
D₄×D₁₆	5

Subgroups — C5×D4×D16 #11956565

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×D₄×D₁₆ #11956565