Subgroups — C₂²×M₁₆×Dic₅ #12962686

Order 1280.

3950 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄×Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₂³×#681	C₂	1	1	yes	maximal, characteristic
4	C₂×M₁₆×Dic₅	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#3947	C₂	1	2	yes	maximal, complemented
6	C₂²×C₈×Dic₅	C₂	1	2	yes	maximal, complemented
7	C₂⁴×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₂⁴×Dic₅	C₄	1	2	yes	—
9	C₂²×C₅×M₁₆	C₄	1	16	yes	complemented
10	C₂²×C₅×M₁₆	C₂²	1	12	yes	—
11	C₂³×C₅×C₈	C₂²	1	2	yes	—
12	C₂²×C₄×Dic₅	C₄	1	12	yes	—
13	C₂²×C₄×Dic₅	C₄	1	2	yes	—
14	C₂²×C₄×Dic₅	C₂²	1	1	yes	characteristic
15	C₂²×C₄×Dic₅	C₂²	1	1	yes	characteristic
16	C₂²×C₄×Dic₅	C₂²	1	12	yes	—
17	C₂²×#681	C₄	1	16	yes	complemented
18	C₂²×#681	C₂²	1	12	yes	—
19	C₂³×#168	C₂²	1	2	yes	—
20	M₁₆×Dic₅	C₂²	1	64	yes	direct product factor
21	C₂×#3947	C₂²	1	24	yes	complemented
22	C₂×C₈×Dic₅	C₂²	1	24	yes	complemented
23	C₂²×C₄×M₁₆	—	5	1	no	maximal, Sylow
24	C₂⁵×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₅	C₂×C₄	1	6	yes	—
26	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₅	C₂×C₄	1	8	yes	—
28	C₂³×C₄×C₅	C₂×C₄	1	8	yes	—
29	C₂³×C₄×C₅	C₂³	1	6	yes	—
30	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂³×Dic₅	—	2	16	no	—
32	C₂³×Dic₅	C₂×C₄	1	2	yes	—
33	C₂³×Dic₅	C₂×C₄	1	24	yes	—
34	C₂³×Dic₅	C₂×C₄	1	2	yes	—
35	C₂×C₅×M₁₆	C₂×C₄	1	96	yes	complemented
36	C₂×C₅×M₁₆	C₂³	1	16	yes	—
37	C₂²×C₅×C₈	C₂×C₄	1	16	yes	complemented
38	C₂²×C₅×C₈	C₂³	1	12	yes	—
39	C₂×C₄×Dic₅	C₂×C₄	1	6	yes	—
40	C₂×C₄×Dic₅	C₂×C₄	1	48	yes	—
41	C₂×C₄×Dic₅	C₂×C₄	1	24	yes	—
42	C₂×C₄×Dic₅	C₂×C₄	1	6	yes	—
43	C₂×C₄×Dic₅	C₂³	1	6	yes	—
44	C₂×C₄×Dic₅	C₂³	1	6	yes	—
45	C₂×C₄×Dic₅	C₂³	1	16	yes	—
46	C₂×#681	C₂×C₄	1	96	yes	complemented
47	C₂×#681	C₂³	1	16	yes	—
48	C₂²×#168	C₂×C₄	1	16	yes	complemented
49	C₂²×#168	C₂³	1	12	yes	—
50	#3947	C₂³	1	32	yes	complemented
51	C₈×Dic₅	C₂³	1	32	yes	complemented
52	C₂³×M₁₆	—	5	1	no	—
53	C₂³×M₁₆	D₅	1	1	yes	characteristic
54	C₂³×C₄²	—	5	1	no	—
55	C₂×C₄×M₁₆	—	5	24	no	—
56	C₂²×#97	—	5	2	no	—
57	C₂²×C₄×C₈	—	5	2	no	—
58	C₂⁴×C₅	—	2	8	no	—
59	C₂⁴×C₅	C₄²	1	8	yes	—
60	C₂⁴×C₅	C₂²×C₄	1	6	yes	—
61	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₄²	1	48	yes	—
63	C₂²×C₄×C₅	C₄²	1	8	yes	—
64	C₂²×C₄×C₅	C₂²×C₄	1	12	yes	—
65	C₂²×C₄×C₅	C₂²×C₄	1	6	yes	—
66	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
67	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₅	C₂²×C₄	1	24	yes	—
69	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₅	C₂²×C₄	1	24	yes	—
71	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
72	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
73	C₂²×Dic₅	—	2	8	no	—
74	C₂²×Dic₅	—	2	96	no	—
75	C₂²×Dic₅	M₁₆	1	16	yes	direct product factor
76	C₂²×Dic₅	C₂²×C₄	1	12	yes	—
77	C₂²×Dic₅	C₂²×C₄	1	32	yes	—
78	C₂²×Dic₅	C₂²×C₄	1	12	yes	—
79	C₅×M₁₆	C₂²×C₄	1	64	yes	complemented
80	C₂×C₅×C₈	C₂²×C₄	1	48	yes	complemented
81	C₂×C₅×C₈	C₂⁴	1	8	yes	—
82	C₄×Dic₅	C₂²×C₄	1	12	yes	—
83	C₄×Dic₅	C₂²×C₄	1	32	yes	—
84	C₄×Dic₅	C₂²×C₄	1	12	yes	—
85	C₄×Dic₅	C₂⁴	1	4	yes	—
86	C₄×Dic₅	C₂⁴	1	4	yes	—
87	#681	C₂²×C₄	1	64	yes	complemented
88	C₂×#168	C₂²×C₄	1	48	yes	complemented
89	C₂×#168	C₂⁴	1	8	yes	—
90	C₂⁴×C₄	—	5	2	no	—
91	C₂⁴×C₄	C₂×D₅	1	1	yes	characteristic
92	C₂²×M₁₆	—	5	16	no	—
93	C₂²×M₁₆	—	5	12	no	—
94	C₂²×M₁₆	Dic₅	1	16	yes	direct product factor
95	C₂²×M₁₆	C₂×D₅	1	12	yes	—
96	C₂³×C₈	—	5	2	no	—
97	C₂³×C₈	C₂×D₅	1	2	yes	—
98	C₂²×C₄²	—	5	2	no	—
99	C₂²×C₄²	—	5	12	no	—
100	C₂²×C₄²	—	5	12	no	—
101	C₂²×C₄²	—	5	1	no	—
102	C₂²×C₄²	—	5	1	no	—
103	C₄×M₁₆	—	5	64	no	—
104	C₂×#97	—	5	24	no	—
105	C₂×C₄×C₈	—	5	24	no	—
106	C₂³×C₅	—	2	24	no	—
107	C₂³×C₅	—	2	32	no	—
108	C₂³×C₅	C₂×C₄²	1	24	yes	—
109	C₂³×C₅	C₂×C₄²	1	4	yes	—
110	C₂³×C₅	C₂×M₁₆	1	8	yes	—
111	C₂³×C₅	C₂³×C₄	1	4	yes	—
112	C₂³×C₅	C₂³×C₄	1	3	yes	—
113	C₂×C₄×C₅	C₂×C₄²	1	48	yes	—
114	C₂×C₄×C₅	C₂×C₄²	1	12	yes	—
115	C₂×C₄×C₅	C₂×C₄²	1	24	yes	—
116	C₂×C₄×C₅	C₂³×C₄	1	6	yes	—
117	C₂×C₄×C₅	C₂³×C₄	1	8	yes	—
118	C₂×C₄×C₅	C₂³×C₄	1	3	yes	—
119	C₂×C₄×C₅	C₂³×C₄	1	4	yes	—
120	C₂×C₄×C₅	C₂³×C₄	1	6	yes	—
121	C₂×C₄×C₅	C₂³×C₄	1	8	yes	—
122	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
123	C₂×Dic₅	—	2	24	no	—
124	C₂×Dic₅	—	2	64	no	—
125	C₂×Dic₅	C₂×M₁₆	1	48	yes	direct product factor
126	C₂×Dic₅	C₂³×C₄	1	8	yes	—
127	C₂×Dic₅	C₂³×C₄	1	8	yes	—
128	C₅×C₈	C₂³×C₄	1	16	yes	complemented
129	#168	C₂³×C₄	1	16	yes	complemented
130	C₂⁵	C₄×D₅	1	1	yes	characteristic
131	C₂³×C₄	—	5	2	no	—
132	C₂³×C₄	—	5	24	no	—
133	C₂³×C₄	—	5	2	no	—
134	C₂³×C₄	—	10	16	no	—
135	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
136	C₂³×C₄	C₄×D₅	1	6	yes	—
137	C₂³×C₄	C₄×D₅	1	8	yes	—
138	C₂³×C₄	C₂×Dic₅	1	8	yes	—
139	C₂³×C₄	C₂²×D₅	1	6	yes	—
140	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
141	C₂×M₁₆	—	5	96	no	—
142	C₂×M₁₆	—	5	16	no	—
143	C₂×M₁₆	C₂×Dic₅	1	96	yes	direct product factor
144	C₂×M₁₆	C₂²×D₅	1	16	yes	—
145	C₂²×C₈	—	5	16	no	—
146	C₂²×C₈	—	5	12	no	—
147	C₂²×C₈	C₂×Dic₅	1	16	yes	complemented
148	C₂²×C₈	C₂²×D₅	1	12	yes	—
149	C₂×C₄²	—	5	6	no	—
150	C₂×C₄²	—	5	24	no	—
151	C₂×C₄²	—	5	48	no	—
152	C₂×C₄²	—	5	6	no	—
153	C₂×C₄²	—	5	6	no	—
154	C₂×C₄²	—	5	6	no	—
155	C₂×C₄²	—	5	16	no	—
156	#97	—	5	32	no	—
157	C₄×C₈	—	5	32	no	—
158	C₂²×C₅	—	2	48	no	—
159	C₂²×C₅	—	2	8	no	—
160	C₂²×C₅	C₄×M₁₆	1	16	yes	complemented
161	C₂²×C₅	C₂²×C₄²	1	8	yes	—
162	C₂²×C₅	C₂²×C₄²	1	6	yes	—
163	C₂²×C₅	C₂²×M₁₆	1	12	yes	—
164	C₂²×C₅	C₂⁴×C₄	1	1	yes	characteristic
165	C₄×C₅	C₂²×C₄²	1	6	yes	—
166	C₄×C₅	C₂²×C₄²	1	8	yes	—
167	C₄×C₅	C₂⁴×C₄	1	1	yes	characteristic
168	C₄×C₅	C₂⁴×C₄	1	1	yes	characteristic
169	Dic₅	—	2	8	no	—
170	Dic₅	C₂²×M₁₆	1	16	yes	direct product factor
171	C₂⁴	—	2	8	no	—
172	C₂⁴	C₄×Dic₅	1	8	yes	—
173	C₂⁴	C₂×C₄×D₅	1	6	yes	—
174	C₂⁴	C₂×C₄×D₅	1	1	yes	characteristic
175	C₂²×C₄	—	5	12	no	—
176	C₂²×C₄	—	5	16	no	—
177	C₂²×C₄	—	5	32	no	—
178	C₂²×C₄	—	5	12	no	—
179	C₂²×C₄	—	10	96	no	—
180	C₂²×C₄	—	10	8	no	—
181	C₂²×C₄	C₄×Dic₅	1	48	yes	—
182	C₂²×C₄	C₄×Dic₅	1	8	yes	—
183	C₂²×C₄	C₂×C₄×D₅	1	4	yes	—
184	C₂²×C₄	C₂×C₄×D₅	1	24	yes	—
185	C₂²×C₄	C₂×C₄×D₅	1	12	yes	—
186	C₂²×C₄	C₂×C₄×D₅	1	6	yes	—
187	C₂²×C₄	C₂×C₄×D₅	1	3	yes	—
188	C₂²×C₄	C₂²×Dic₅	1	4	yes	—
189	C₂²×C₄	C₂²×Dic₅	1	24	yes	—
190	C₂²×C₄	C₂³×D₅	1	3	yes	—
191	C₂²×C₄	C₂³×D₅	1	4	yes	—
192	M₁₆	—	5	64	no	—
193	M₁₆	C₂²×Dic₅	1	64	yes	direct product factor
194	C₂×C₈	—	5	48	no	—
195	C₂×C₈	—	5	8	no	—
196	C₂×C₈	C₂²×Dic₅	1	48	yes	complemented
197	C₂×C₈	C₂³×D₅	1	8	yes	—
198	C₄²	—	5	12	no	—
199	C₄²	—	5	32	no	—
200	C₄²	—	5	12	no	—
201	C₄²	—	5	4	no	—
202	C₄²	—	5	4	no	—
203	C₂×C₅	—	2	8	no	—
204	C₂×C₅	C₂×C₄×M₁₆	1	12	yes	complemented
205	C₂×C₅	C₂³×C₄²	1	1	yes	characteristic
206	C₂×C₅	C₂³×M₁₆	1	2	yes	—
207	C₂³	—	2	24	no	—
208	C₂³	—	2	32	no	—
209	C₂³	D₅×M₁₆	1	8	yes	—
210	C₂³	C₂×C₄×Dic₅	1	24	yes	—
211	C₂³	C₂×C₄×Dic₅	1	4	yes	—
212	C₂³	C₂²×C₄×D₅	1	4	yes	—
213	C₂³	C₂²×C₄×D₅	1	3	yes	—
214	C₂×C₄	—	5	48	no	—
215	C₂×C₄	—	5	8	no	—
216	C₂×C₄	—	5	8	no	—
217	C₂×C₄	—	10	64	no	—
218	C₂×C₄	—	10	24	no	—
219	C₂×C₄	C₂×C₄×Dic₅	1	48	yes	—
220	C₂×C₄	C₂×C₄×Dic₅	1	12	yes	—
221	C₂×C₄	C₂×C₄×Dic₅	1	24	yes	—
222	C₂×C₄	C₂²×C₄×D₅	1	4	yes	—
223	C₂×C₄	C₂²×C₄×D₅	1	3	yes	—
224	C₂×C₄	C₂²×C₄×D₅	1	6	yes	—
225	C₂×C₄	C₂²×C₄×D₅	1	8	yes	—
226	C₂×C₄	C₂³×Dic₅	1	6	yes	—
227	C₂×C₄	C₂³×Dic₅	1	8	yes	—
228	C₂×C₄	C₂⁴×D₅	1	1	yes	characteristic
229	C₈	—	5	16	no	—
230	C₈	C₂³×Dic₅	1	16	yes	complemented
231	C₅	C₂²×C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
232	C₂²	—	2	8	no	—
233	C₂²	—	2	48	no	—
234	C₂²	M₁₆×Dic₅	1	16	yes	direct product factor
235	C₂²	C₂×D₅×M₁₆	1	12	yes	—
236	C₂²	C₂²×C₄×Dic₅	1	8	yes	—
237	C₂²	C₂²×C₄×Dic₅	1	6	yes	—
238	C₂²	C₂³×C₄×D₅	1	1	yes	characteristic
239	C₄	—	5	16	no	—
240	C₄	—	10	8	no	—
241	C₄	C₂²×C₄×Dic₅	1	6	yes	—
242	C₄	C₂²×C₄×Dic₅	1	8	yes	—
243	C₄	C₂³×C₄×D₅	1	1	yes	characteristic
244	C₄	C₂⁴×Dic₅	1	1	yes	characteristic
245	C₂	—	2	8	no	—
246	C₂	C₂×M₁₆×Dic₅	1	12	yes	direct product factor
247	C₂	C₂²×D₅×M₁₆	1	2	yes	—
248	C₂	C₂³×C₄×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅
commutator	C₂×C₅	C₂³×C₄²
Frattini	C₂×C₄	C₂⁴×D₅
Fitting	C₂³×C₅×M₁₆	C₂
socle	C₂⁴×C₅	C₂²×C₄
prime radical	C₂⁵×C₅	C₂×C₄
cosocle	C₂×C₄×C₅	C₂⁵
hypercenter	C₂³×M₁₆	D₅
hypocenter	C₅	C₂²×C₄×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
4. C₂×M₁₆×Dic₅	C₂	direct (N × H)
5. C₂²×#3947	C₂	semidirect
6. C₂²×C₈×Dic₅	C₂	semidirect
9. C₂²×C₅×M₁₆	C₄	semidirect
17. C₂²×#681	C₄	semidirect
20. M₁₆×Dic₅	C₂²	direct (N × H)
21. C₂×#3947	C₂²	semidirect
22. C₂×C₈×Dic₅	C₂²	semidirect
35. C₂×C₅×M₁₆	C₂×C₄	semidirect
37. C₂²×C₅×C₈	C₂×C₄	semidirect
46. C₂×#681	C₂×C₄	semidirect
48. C₂²×#168	C₂×C₄	semidirect
50. #3947	C₂³	semidirect
51. C₈×Dic₅	C₂³	semidirect
75. C₂²×Dic₅	M₁₆	direct (N × H)
79. C₅×M₁₆	C₂²×C₄	semidirect
80. C₂×C₅×C₈	C₂²×C₄	semidirect
87. #681	C₂²×C₄	semidirect
88. C₂×#168	C₂²×C₄	semidirect
94. C₂²×M₁₆	Dic₅	direct (N × H)
125. C₂×Dic₅	C₂×M₁₆	direct (N × H)
128. C₅×C₈	C₂³×C₄	semidirect
129. #168	C₂³×C₄	semidirect
143. C₂×M₁₆	C₂×Dic₅	direct (N × H)
147. C₂²×C₈	C₂×Dic₅	semidirect
160. C₂²×C₅	C₄×M₁₆	semidirect
170. Dic₅	C₂²×M₁₆	direct (N × H)
193. M₁₆	C₂²×Dic₅	direct (N × H)
196. C₂×C₈	C₂²×Dic₅	semidirect
204. C₂×C₅	C₂×C₄×M₁₆	semidirect
230. C₈	C₂³×Dic₅	semidirect
231. C₅	C₂²×C₄×M₁₆	semidirect
234. C₂²	M₁₆×Dic₅	direct (N × H)
246. C₂	C₂×M₁₆×Dic₅	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₅×M₁₆	2
C₂³×C₄×Dic₅	2
C₂³×#681	2
C₂×M₁₆×Dic₅	2
C₂²×#3947	2
C₂²×C₈×Dic₅	2
C₂²×C₄×M₁₆	5

Subgroups — C22×M16×Dic5 #12962686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×M₁₆×Dic₅ #12962686