Subgroups — C₂×D₅⋊(C₄⋊C₁₆) #13043370

Order 1280.

862 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₅⋊C₁₆	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
3	C₂×#64888	C₂	1	2	yes	maximal, complemented
4	C₂×C₄⋊#674	C₂	1	2	yes	maximal, complemented
5	D₅⋊(C₄⋊C₁₆)	C₂	1	8	yes	maximal, direct product factor
6	C₂²×C₈×D₅	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₅	C₄	1	1	yes	characteristic
9	C₂²×#674	C₂²	1	4	yes	—
10	C₂×D₅⋊C₁₆	—	2	8	no	—
11	C₂×D₅⋊C₁₆	C₂²	1	8	yes	—
12	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₅	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
15	C₂×C₄×#168	C₂²	1	1	yes	characteristic
16	C₄×C₈×D₅	C₄	1	4	yes	—
17	C₄×C₈×D₅	C₂²	1	4	yes	—
18	#64888	C₂²	1	8	yes	complemented
19	C₄⋊#674	C₂²	1	8	yes	complemented
20	C₂²×C₄⋊C₁₆	—	5	1	no	maximal, Sylow
21	C₂²×C₄×D₅	C₈	1	1	yes	characteristic
22	C₂²×C₄×D₅	C₈	1	1	yes	characteristic
23	C₂²×C₄×D₅	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₅	C₈	1	1	yes	characteristic
28	C₂×C₄×Dic₅	C₈	1	1	yes	characteristic
29	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
30	C₂²×#168	C₂×C₄	1	1	yes	characteristic
31	C₂²×#168	C₂³	1	1	yes	characteristic
32	C₂×C₈×D₅	—	2	4	no	—
33	C₂×C₈×D₅	C₂×C₄	1	4	yes	—
34	C₂×C₈×D₅	C₂×C₄	1	2	yes	—
35	C₂×C₈×D₅	D₄	1	4	yes	—
36	C₂×C₈×D₅	Q₈	1	4	yes	—
37	C₂×C₈×D₅	C₂³	1	2	yes	—
38	C₄²×D₅	C₈	1	4	yes	—
39	C₄²×D₅	C₂×C₄	1	4	yes	—
40	C₂×#674	—	2	8	no	—
41	C₂×#674	C₂³	1	8	yes	—
42	D₅⋊C₁₆	—	2	16	no	—
43	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
44	C₈×Dic₅	C₂×C₄	1	4	yes	—
45	C₈×Dic₅	C₂×C₄	1	2	yes	—
46	C₈×Dic₅	C₂³	1	2	yes	—
47	C₄×#168	C₂×C₄	1	2	yes	—
48	C₄×#168	C₂³	1	2	yes	—
49	C₂³×C₁₆	—	5	2	no	—
50	C₂²×C₄×C₈	—	5	1	no	—
51	C₂×C₄⋊C₁₆	—	5	8	no	—
52	C₂×C₄⋊C₁₆	—	5	2	no	—
53	C₂×C₄⋊C₁₆	—	5	2	no	—
54	C₂³×D₅	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
59	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
60	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
61	C₂×C₄×D₅	—	2	4	no	—
62	C₂×C₄×D₅	C₁₆	1	8	yes	complemented
63	C₂×C₄×D₅	C₄⋊C₄	1	4	yes	—
64	C₂×C₄×D₅	C₂×C₈	1	4	yes	—
65	C₂×C₄×D₅	C₂×C₈	1	2	yes	—
66	C₂×C₄×D₅	C₂×C₈	1	4	yes	—
67	C₂×C₄×D₅	M₁₆	1	4	yes	—
68	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
69	C₂×C₅×C₈	—	2	2	no	—
70	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
71	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
72	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
73	C₄²×C₅	C₂×C₈	1	2	yes	—
74	C₄²×C₅	C₂²×C₄	1	2	yes	—
75	C₄×Dic₅	C₂×C₈	1	2	yes	—
76	C₄×Dic₅	C₂×C₈	1	4	yes	—
77	C₄×Dic₅	C₂×C₈	1	4	yes	—
78	C₄×Dic₅	C₂²×C₄	1	2	yes	—
79	C₂×#168	—	2	2	no	—
80	C₂×#168	C₂²×C₄	1	2	yes	—
81	C₂×#168	C₂²×C₄	1	1	yes	characteristic
82	C₂×#168	C₂×D₄	1	2	yes	—
83	C₂×#168	C₂×Q₈	1	2	yes	—
84	C₂×#168	C₂⁴	1	1	yes	characteristic
85	C₈×D₅	—	2	8	no	—
86	C₈×D₅	C₄⋊C₄	1	8	yes	—
87	C₈×D₅	C₂×D₄	1	4	yes	—
88	C₈×D₅	C₂×Q₈	1	4	yes	—
89	#674	—	2	8	no	—
90	C₂³×C₈	—	5	1	no	—
91	C₂³×C₈	—	5	1	no	—
92	C₂²×C₄²	—	5	1	no	—
93	C₂²×C₁₆	—	5	8	no	—
94	C₂²×C₁₆	—	5	4	no	—
95	C₂²×C₁₆	—	10	8	no	—
96	C₂×C₄×C₈	—	5	1	no	—
97	C₂×C₄×C₈	—	5	1	no	—
98	C₂×C₄×C₈	—	5	1	no	—
99	C₂×C₄×C₈	—	5	4	no	—
100	C₂×C₄×C₈	—	5	4	no	—
101	C₂×C₄×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
102	C₄⋊C₁₆	—	5	8	no	—
103	C₄⋊C₁₆	—	5	8	no	—
104	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
105	C₂²×D₅	C₄⋊C₈	1	4	yes	—
106	C₂²×D₅	C₂×C₁₆	1	4	yes	—
107	C₂²×D₅	M₃₂	1	4	yes	—
108	C₂²×D₅	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₅	—	2	2	no	—
110	C₂×C₄×C₅	C₂×C₁₆	1	4	yes	complemented
111	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
113	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
115	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
116	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
117	C₂×Dic₅	—	2	2	no	—
118	C₂×Dic₅	C₄⋊C₈	1	4	yes	—
119	C₂×Dic₅	C₂×C₁₆	1	4	yes	complemented
120	C₂×Dic₅	C₂²×C₈	1	2	yes	—
121	C₂×Dic₅	C₂²×C₈	1	2	yes	—
122	C₂×Dic₅	C₂²×C₈	1	2	yes	—
123	C₄×D₅	—	2	8	no	—
124	C₄×D₅	C₄⋊C₈	1	8	yes	—
125	C₄×D₅	C₂×C₁₆	1	16	yes	complemented
126	C₄×D₅	C₂×C₄⋊C₄	1	4	yes	—
127	C₄×D₅	C₂×M₁₆	1	4	yes	—
128	C₅×C₈	—	2	2	no	—
129	C₅×C₈	C₂×C₄⋊C₄	1	4	yes	—
130	#168	—	2	2	no	—
131	#168	C₂×C₄⋊C₄	1	2	yes	—
132	#168	C₂²×D₄	1	1	yes	characteristic
133	#168	C₂²×Q₈	1	1	yes	characteristic
134	C₂³×C₄	—	5	1	no	—
135	C₂³×C₄	—	5	1	no	—
136	C₂³×C₄	—	5	1	no	—
137	C₂²×C₈	—	5	1	no	—
138	C₂²×C₈	—	5	4	no	—
139	C₂²×C₈	—	5	4	no	—
140	C₂²×C₈	—	5	2	no	—
141	C₂²×C₈	—	5	2	no	—
142	C₂²×C₈	—	5	1	no	—
143	C₂²×C₈	—	5	4	no	—
144	C₂²×C₈	—	10	4	no	—
145	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
146	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
147	C₂×C₄²	—	5	1	no	—
148	C₂×C₄²	—	5	1	no	—
149	C₂×C₄²	—	5	1	no	—
150	C₂×C₄²	—	5	4	no	—
151	C₂×C₄²	—	5	4	no	—
152	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
153	C₂×C₁₆	—	5	8	no	—
154	C₂×C₁₆	—	10	16	no	—
155	C₂×C₁₆	—	10	8	no	—
156	C₄×C₈	—	5	2	no	—
157	C₄×C₈	—	5	4	no	—
158	C₄×C₈	—	5	2	no	—
159	C₄×C₈	—	5	2	no	—
160	C₄×C₈	—	5	2	no	—
161	C₄×C₈	C₂×AGL(1,𝔽₅)	1	4	yes	—
162	C₂²×C₅	C₂×C₄⋊C₈	1	2	yes	—
163	C₂²×C₅	C₂²×C₁₆	1	2	yes	—
164	C₂²×C₅	C₂×M₃₂	1	2	yes	—
165	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
166	C₂×D₅	C₄⋊C₁₆	1	16	yes	complemented
167	C₂×D₅	C₂×C₄⋊C₈	1	4	yes	—
168	C₂×D₅	C₂²×C₁₆	1	4	yes	—
169	C₂×D₅	C₂×M₃₂	1	4	yes	—
170	C₄×C₅	—	2	2	no	—
171	C₄×C₅	C₂×C₄⋊C₈	1	2	yes	—
172	C₄×C₅	C₂²×C₁₆	1	4	yes	complemented
173	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
174	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
175	Dic₅	—	2	2	no	—
176	Dic₅	C₂×C₄⋊C₈	1	4	yes	—
177	Dic₅	C₂²×C₁₆	1	4	yes	complemented
178	C₂⁴	—	5	1	no	—
179	C₂²×C₄	—	5	1	no	—
180	C₂²×C₄	—	5	1	no	—
181	C₂²×C₄	—	5	2	no	—
182	C₂²×C₄	—	5	4	no	—
183	C₂²×C₄	—	5	4	no	—
184	C₂²×C₄	—	5	4	no	—
185	C₂²×C₄	—	5	2	no	—
186	C₂²×C₄	—	5	8	no	—
187	C₂²×C₄	—	5	1	no	—
188	C₂²×C₄	—	5	4	no	—
189	C₂²×C₄	—	10	4	no	—
190	C₂²×C₄	D₅⋊C₈	1	1	yes	characteristic
191	C₂²×C₄	D₅⋊C₈	1	1	yes	characteristic
192	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	5	2	no	—
195	C₂×C₈	—	5	2	no	—
196	C₂×C₈	—	5	1	no	—
197	C₂×C₈	—	5	1	no	—
198	C₂×C₈	—	5	4	no	—
199	C₂×C₈	—	5	8	no	—
200	C₂×C₈	—	5	4	no	—
201	C₂×C₈	—	5	2	no	—
202	C₂×C₈	—	10	8	no	—
203	C₂×C₈	—	10	2	no	—
204	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	4	yes	—
205	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	2	yes	—
206	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	2	yes	—
207	C₄²	—	5	4	no	—
208	C₄²	—	5	2	no	—
209	C₄²	—	5	4	no	—
210	C₄²	—	5	2	no	—
211	C₄²	D₅⋊C₈	1	2	yes	—
212	C₄²	C₂²×AGL(1,𝔽₅)	1	2	yes	—
213	C₁₆	—	10	8	no	—
214	C₂×C₅	C₂×C₄⋊C₁₆	1	4	yes	complemented
215	C₂×C₅	C₂²×C₄⋊C₈	1	1	yes	characteristic
216	C₂×C₅	C₂³×C₁₆	1	1	yes	characteristic
217	C₂×C₅	C₂²×M₃₂	1	1	yes	characteristic
218	D₅	C₂×C₄⋊C₁₆	1	8	yes	complemented
219	C₂³	—	5	4	no	—
220	C₂³	—	5	4	no	—
221	C₂³	—	5	4	no	—
222	C₂³	—	5	2	no	—
223	C₂³	C₂×D₅⋊C₈	1	1	yes	characteristic
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	5	4	no	—
226	C₂×C₄	—	5	4	no	—
227	C₂×C₄	—	5	2	no	—
228	C₂×C₄	—	5	2	no	—
229	C₂×C₄	—	5	8	no	—
230	C₂×C₄	—	5	16	no	—
231	C₂×C₄	—	5	4	no	—
232	C₂×C₄	—	5	4	no	—
233	C₂×C₄	—	5	2	no	—
234	C₂×C₄	—	10	8	no	—
235	C₂×C₄	—	10	2	no	—
236	C₂×C₄	D₅⋊C₁₆	1	4	yes	complemented
237	C₂×C₄	C₂×D₅⋊C₈	1	2	yes	—
238	C₂×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
239	C₂×C₄	C₂×D₅⋊C₈	1	2	yes	—
240	C₂×C₄	D₅⋊M₁₆	1	2	yes	—
241	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	2	yes	—
242	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	5	1	no	—
245	C₈	—	5	2	no	—
246	C₈	—	5	1	no	—
247	C₈	—	10	2	no	—
248	C₈	C₂×C₄⋊AGL(1,𝔽₅)	1	4	yes	—
249	C₅	C₂²×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
250	C₂²	—	5	4	no	—
251	C₂²	—	5	4	no	—
252	C₂²	—	5	16	no	—
253	C₂²	—	5	4	no	—
254	C₂²	D₅⋊(C₄⋊C₈)	1	2	yes	—
255	C₂²	C₂×D₅⋊C₁₆	1	2	yes	—
256	C₂²	D₅⋊M₃₂	1	2	yes	—
257	C₂²	C₂²×D₅⋊C₈	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	5	4	no	—
260	C₄	—	5	4	no	—
261	C₄	—	10	2	no	—
262	C₄	D₅⋊(C₄⋊C₈)	1	2	yes	—
263	C₄	C₂×D₅⋊C₁₆	1	4	yes	complemented
264	C₄	C₂×D₅⋊M₁₆	1	1	yes	characteristic
265	C₄	C₂²×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
266	C₂	—	5	8	no	—
267	C₂	D₅⋊(C₄⋊C₁₆)	1	4	yes	direct product factor
268	C₂	C₂×D₅⋊(C₄⋊C₈)	1	1	yes	characteristic
269	C₂	C₂²×D₅⋊C₁₆	1	1	yes	characteristic
270	C₂	C₂×D₅⋊M₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂³×C₁₆
Frattini	C₂×C₄	C₂³×AGL(1,𝔽₅)
Fitting	C₂×C₄×C₅×C₈	C₄
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂³×D₅	C₂×C₈
cosocle	C₂×#168	C₂⁴
hypercenter	C₂×C₄×C₈	AGL(1,𝔽₅)
hypocenter	C₅	C₂²×C₄⋊C₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#64888	C₂	semidirect
4. C₂×C₄⋊#674	C₂	semidirect
5. D₅⋊(C₄⋊C₁₆)	C₂	direct (N × H)
18. #64888	C₂²	semidirect
19. C₄⋊#674	C₂²	semidirect
62. C₂×C₄×D₅	C₁₆	semidirect
110. C₂×C₄×C₅	C₂×C₁₆	semidirect
119. C₂×Dic₅	C₂×C₁₆	semidirect
125. C₄×D₅	C₂×C₁₆	semidirect
166. C₂×D₅	C₄⋊C₁₆	semidirect
172. C₄×C₅	C₂²×C₁₆	semidirect
177. Dic₅	C₂²×C₁₆	semidirect
214. C₂×C₅	C₂×C₄⋊C₁₆	semidirect
218. D₅	C₂×C₄⋊C₁₆	semidirect
236. C₂×C₄	D₅⋊C₁₆	semidirect
249. C₅	C₂²×C₄⋊C₁₆	semidirect
263. C₄	C₂×D₅⋊C₁₆	semidirect
267. C₂	D₅⋊(C₄⋊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×D₅⋊C₁₆	2
C₂×C₄×C₈×D₅	2
C₂×#64888	2
C₂×C₄⋊#674	2
D₅⋊(C₄⋊C₁₆)	2
C₂²×C₄⋊C₁₆	5

Subgroups — C2×D5⋊(C4⋊C16) #13043370

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₅⋊(C₄⋊C₁₆) #13043370