Subgroups — C₄×C₅×C₂²⋊C₁₆ #11935318

Order 1280.

614 conjugacy classes of subgroups.

The subgroup lattice has 256 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	C₂×C₅×C₂²⋊C₁₆	C₂	1	2	yes	maximal
3	C₂×C₄×C₅×C₁₆	C₂	1	2	yes	maximal, complemented
4	C₅×#1374	C₂	1	2	yes	maximal, complemented
5	C₂³×C₅×C₈	C₄	1	1	yes	characteristic
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₅	C₄	1	1	yes	characteristic
8	C₂²×C₅×C₁₆	C₂²	1	4	yes	—
9	C₂×C₄×C₅×C₈	—	2	4	no	—
10	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
11	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₅×C₂²⋊C₁₆	C₄	1	8	yes	direct product factor
15	C₄×C₅×C₁₆	—	2	4	no	—
16	C₄×C₂²⋊C₁₆	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂³×C₄×C₅	C₈	1	2	yes	—
18	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₅×C₈	—	2	2	no	—
20	C₂²×C₅×C₈	—	2	2	no	—
21	C₂²×C₅×C₈	—	2	2	no	—
22	C₂²×C₅×C₈	—	2	2	no	—
23	C₂²×C₅×C₈	C₈	1	4	yes	—
24	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
26	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₅	—	2	2	no	—
30	C₂×C₄²×C₅	—	2	2	no	—
31	C₂×C₄²×C₅	C₈	1	2	yes	—
32	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₂×C₅×C₁₆	—	2	4	no	—
35	C₂×C₅×C₁₆	—	2	4	no	—
36	C₂×C₅×C₁₆	C₂×C₄	1	8	yes	complemented
37	C₄×C₅×C₈	—	2	2	no	—
38	C₄×C₅×C₈	—	2	2	no	—
39	C₄×C₅×C₈	—	2	2	no	—
40	C₄×C₅×C₈	D₄	1	4	yes	—
41	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic
42	C₂×C₂²⋊C₁₆	C₂×C₅	1	2	yes	—
43	C₂×C₄×C₁₆	C₂×C₅	1	2	yes	complemented
44	#1374	C₂×C₅	1	2	yes	complemented
45	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₅	—	2	4	no	—
47	C₂²×C₄×C₅	—	2	4	no	—
48	C₂²×C₄×C₅	—	2	1	no	—
49	C₂²×C₄×C₅	—	2	1	no	—
50	C₂²×C₄×C₅	—	2	1	no	—
51	C₂²×C₄×C₅	—	2	1	no	—
52	C₂²×C₄×C₅	C₁₆	1	8	yes	complemented
53	C₂²×C₄×C₅	C₄²	1	2	yes	—
54	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₅×C₈	—	2	1	no	—
60	C₂×C₅×C₈	—	2	1	no	—
61	C₂×C₅×C₈	—	2	1	no	—
62	C₂×C₅×C₈	—	2	1	no	—
63	C₂×C₅×C₈	—	2	8	no	—
64	C₂×C₅×C₈	—	2	8	no	—
65	C₂×C₅×C₈	C₄²	1	2	yes	—
66	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
67	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
68	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
69	C₂×C₅×C₈	M₁₆	1	2	yes	—
70	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
71	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
72	C₂×C₅×C₈	#41	1	2	yes	—
73	C₄²×C₅	—	2	4	no	—
74	C₄²×C₅	—	2	1	no	—
75	C₄²×C₅	—	2	1	no	—
76	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₅	M₁₆	1	2	yes	—
78	C₅×C₁₆	—	2	8	no	—
79	C₂³×C₈	C₄×C₅	1	1	yes	characteristic
80	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
81	C₂²×C₄²	C₄×C₅	1	1	yes	characteristic
82	C₂²×C₁₆	C₂²×C₅	1	4	yes	—
83	C₂×C₄×C₈	—	2	4	no	—
84	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
85	C₂×C₄×C₈	C₄×C₅	1	1	yes	characteristic
86	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
87	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
88	C₂²⋊C₁₆	C₄×C₅	1	8	yes	direct product factor
89	C₄×C₁₆	—	2	4	no	—
90	C₂³×C₅	—	2	1	no	—
91	C₂³×C₅	—	2	1	no	—
92	C₂³×C₅	—	2	2	no	—
93	C₂³×C₅	C₄×C₈	1	2	yes	—
94	C₂³×C₅	C₂×C₁₆	1	4	yes	—
95	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
96	C₂×C₄×C₅	—	2	2	no	—
97	C₂×C₄×C₅	—	2	1	no	—
98	C₂×C₄×C₅	—	2	1	no	—
99	C₂×C₄×C₅	—	2	2	no	—
100	C₂×C₄×C₅	—	2	16	no	—
101	C₂×C₄×C₅	—	2	2	no	—
102	C₂×C₄×C₅	—	2	2	no	—
103	C₂×C₄×C₅	—	2	2	no	—
104	C₂×C₄×C₅	—	2	2	no	—
105	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
106	C₂×C₄×C₅	C₂²⋊C₈	1	4	yes	—
107	C₂×C₄×C₅	C₂×C₁₆	1	4	yes	—
108	C₂×C₄×C₅	C₂×C₁₆	1	4	yes	—
109	C₂×C₄×C₅	M₃₂	1	4	yes	—
110	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₅	#117	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
116	C₅×C₈	—	2	2	no	—
117	C₅×C₈	—	2	2	no	—
118	C₅×C₈	C₂²⋊C₈	1	4	yes	—
119	C₅×C₈	C₄×D₄	1	4	yes	—
120	C₂³×C₄	C₅×C₈	1	2	yes	—
121	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
122	C₂²×C₈	—	2	2	no	—
123	C₂²×C₈	—	2	2	no	—
124	C₂²×C₈	—	2	2	no	—
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	C₅×C₈	1	4	yes	—
127	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
128	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
129	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
130	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
131	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
132	C₂×C₄²	—	2	2	no	—
133	C₂×C₄²	—	2	2	no	—
134	C₂×C₄²	C₅×C₈	1	2	yes	—
135	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
136	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
137	C₂×C₁₆	—	2	4	no	—
138	C₂×C₁₆	—	2	4	no	—
139	C₂×C₁₆	C₂×C₄×C₅	1	8	yes	complemented
140	C₄×C₈	—	2	2	no	—
141	C₄×C₈	—	2	2	no	—
142	C₄×C₈	—	2	2	no	—
143	C₄×C₈	C₅×D₄	1	4	yes	—
144	C₂²×C₅	—	2	2	no	—
145	C₂²×C₅	—	2	4	no	—
146	C₂²×C₅	—	2	4	no	—
147	C₂²×C₅	—	2	2	no	—
148	C₂²×C₅	C₄×C₁₆	1	4	yes	complemented
149	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
150	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
151	C₂²×C₅	#431	1	1	yes	characteristic
152	C₂²×C₅	C₂²×C₁₆	1	2	yes	—
153	C₂²×C₅	C₂×M₃₂	1	2	yes	—
154	C₄×C₅	—	2	4	no	—
155	C₄×C₅	—	2	2	no	—
156	C₄×C₅	C₂²⋊C₁₆	1	8	yes	direct product factor
157	C₄×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
158	C₄×C₅	C₄×M₁₆	1	1	yes	characteristic
159	C₄×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
160	C₄×C₅	#431	1	1	yes	characteristic
161	C₂⁴	C₂×C₅×C₈	1	1	yes	characteristic
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	C₅×C₁₆	1	8	yes	complemented
169	C₂²×C₄	C₄²×C₅	1	2	yes	—
170	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
171	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
173	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
174	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	8	no	—
180	C₂×C₈	—	2	8	no	—
181	C₂×C₈	C₄²×C₅	1	2	yes	—
182	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
183	C₂×C₈	C₂×C₅×C₈	1	2	yes	—
184	C₂×C₈	C₂×C₅×C₈	1	2	yes	—
185	C₂×C₈	C₅×M₁₆	1	2	yes	—
186	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
187	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
188	C₂×C₈	C₅×#41	1	2	yes	—
189	C₄²	—	2	1	no	—
190	C₄²	—	2	1	no	—
191	C₄²	—	2	4	no	—
192	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
193	C₄²	C₅×M₁₆	1	2	yes	—
194	C₁₆	—	2	8	no	—
195	C₂×C₅	—	2	4	no	—
196	C₂×C₅	C₄×C₂²⋊C₈	1	1	yes	characteristic
197	C₂×C₅	C₂×C₄×C₁₆	1	1	yes	characteristic
198	C₂×C₅	C₄×M₃₂	1	1	yes	characteristic
199	C₂×C₅	C₂×C₂²⋊C₁₆	1	2	yes	—
200	C₂×C₅	#2162	1	2	yes	—
201	C₂³	—	2	1	no	—
202	C₂³	—	2	1	no	—
203	C₂³	—	2	2	no	—
204	C₂³	C₄×C₅×C₈	1	2	yes	—
205	C₂³	C₂×C₅×C₁₆	1	4	yes	—
206	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	16	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	C₄×C₅×C₈	1	2	yes	—
217	C₂×C₄	C₅×C₂²⋊C₈	1	4	yes	—
218	C₂×C₄	C₂×C₅×C₁₆	1	4	yes	—
219	C₂×C₄	C₂×C₅×C₁₆	1	4	yes	—
220	C₂×C₄	C₅×M₃₂	1	4	yes	—
221	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
222	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
223	C₂×C₄	C₅×#117	1	1	yes	characteristic
224	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
225	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
226	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
227	C₈	—	2	2	no	—
228	C₈	—	2	2	no	—
229	C₈	C₅×C₂²⋊C₈	1	4	yes	—
230	C₈	C₄×C₅×D₄	1	4	yes	—
231	C₅	C₄×C₂²⋊C₁₆	1	1	yes	characteristic, direct product factor, Sylow
232	C₂²	—	2	2	no	—
233	C₂²	—	2	4	no	—
234	C₂²	—	2	4	no	—
235	C₂²	—	2	2	no	—
236	C₂²	C₄×C₅×C₁₆	1	4	yes	complemented
237	C₂²	C₂×C₄×C₅×C₈	1	1	yes	characteristic
238	C₂²	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
239	C₂²	C₅×#431	1	1	yes	characteristic
240	C₂²	C₂²×C₅×C₁₆	1	2	yes	—
241	C₂²	C₂×C₅×M₃₂	1	2	yes	—
242	C₄	—	2	4	no	—
243	C₄	—	2	2	no	—
244	C₄	C₅×C₂²⋊C₁₆	1	8	yes	direct product factor
245	C₄	C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic
246	C₄	C₄×C₅×M₁₆	1	1	yes	characteristic
247	C₄	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
248	C₄	C₅×#431	1	1	yes	characteristic
249	C₂	—	2	4	no	—
250	C₂	C₄×C₅×C₂²⋊C₈	1	1	yes	characteristic
251	C₂	C₂×C₄×C₅×C₁₆	1	1	yes	characteristic
252	C₂	C₄×C₅×M₃₂	1	1	yes	characteristic
253	C₂	C₂×C₅×C₂²⋊C₁₆	1	2	yes	—
254	C₂	C₅×#2162	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅×C₈	C₂²
commutator	C₂	C₂×C₄×C₅×C₁₆
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂⁴×C₅	C₂×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×C₅×C₁₆	C₂	semidirect
4. C₅×#1374	C₂	semidirect
14. C₅×C₂²⋊C₁₆	C₄	direct (N × H)
16. C₄×C₂²⋊C₁₆	C₅	direct (N × H)
36. C₂×C₅×C₁₆	C₂×C₄	semidirect
43. C₂×C₄×C₁₆	C₂×C₅	semidirect
44. #1374	C₂×C₅	semidirect
52. C₂²×C₄×C₅	C₁₆	semidirect
88. C₂²⋊C₁₆	C₄×C₅	direct (N × H)
139. C₂×C₁₆	C₂×C₄×C₅	semidirect
148. C₂²×C₅	C₄×C₁₆	semidirect
156. C₄×C₅	C₂²⋊C₁₆	direct (N × H)
168. C₂²×C₄	C₅×C₁₆	semidirect
231. C₅	C₄×C₂²⋊C₁₆	direct (N × H)
236. C₂²	C₄×C₅×C₁₆	semidirect
244. C₄	C₅×C₂²⋊C₁₆	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
C₂×C₅×C₂²⋊C₁₆	2
C₂×C₄×C₅×C₁₆	2
C₅×#1374	2
C₄×C₂²⋊C₁₆	5

Subgroups — C4×C5×C22⋊C16 #11935318

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×C₂²⋊C₁₆ #11935318