Subgroups — C₂×C₃×C₅×C₂²⋊C₁₆ #14275145

Order 1920.

544 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₅×C₁₆	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×C₂²⋊C₁₆	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅×C₂²⋊C₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂³×C₃×C₄×C₅	C₄	1	1	yes	characteristic
6	C₂²×C₃×C₅×C₈	—	2	4	no	—
7	C₂²×C₃×C₅×C₈	C₄	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	C₄	1	2	yes	—
9	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×C₁₆	—	2	4	no	—
12	C₂×C₃×C₅×C₁₆	C₂²	1	4	yes	complemented
13	C₂×C₃×C₂²⋊C₁₆	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂³×C₅×C₈	C₂×C₃	1	1	yes	characteristic
15	C₂²×C₅×C₁₆	C₂×C₃	1	2	yes	complemented
16	C₅×C₂²⋊C₁₆	C₂×C₃	1	4	yes	direct product factor
17	C₂⁴×C₃×C₅	C₈	1	1	yes	characteristic
18	C₂²×C₃×C₄×C₅	—	2	2	no	—
19	C₂²×C₃×C₄×C₅	—	2	2	no	—
20	C₂²×C₃×C₄×C₅	C₈	1	1	yes	characteristic
21	C₂²×C₃×C₄×C₅	C₈	1	2	yes	—
22	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₅×C₈	—	2	2	no	—
25	C₂×C₃×C₅×C₈	—	2	4	no	—
26	C₂×C₃×C₅×C₈	—	2	4	no	—
27	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₅×C₈	D₄	1	4	yes	—
30	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
31	C₃×C₅×C₁₆	—	2	4	no	—
32	C₂³×C₃×C₈	C₂×C₅	1	1	yes	characteristic
33	C₂²×C₃×C₁₆	C₂×C₅	1	2	yes	complemented
34	C₃×C₂²⋊C₁₆	C₂×C₅	1	4	yes	direct product factor
35	C₂³×C₄×C₅	C₃×C₄	1	1	yes	characteristic
36	C₂²×C₅×C₈	—	2	4	no	—
37	C₂²×C₅×C₈	C₃×C₄	1	1	yes	characteristic
38	C₂²×C₅×C₈	C₃×C₄	1	2	yes	—
39	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
40	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
41	C₂×C₅×C₁₆	—	2	4	no	—
42	C₂×C₅×C₁₆	C₂²×C₃	1	4	yes	complemented
43	C₂×C₂²⋊C₁₆	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
44	C₂³×C₃×C₅	—	2	2	no	—
45	C₂³×C₃×C₅	—	2	2	no	—
46	C₂³×C₃×C₅	C₁₆	1	4	yes	complemented
47	C₂³×C₃×C₅	C₂×C₈	1	2	yes	—
48	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	—	2	2	no	—
50	C₂×C₃×C₄×C₅	—	2	4	no	—
51	C₂×C₃×C₄×C₅	—	2	2	no	—
52	C₂×C₃×C₄×C₅	—	2	2	no	—
53	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
54	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
55	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
57	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₃×C₅×C₈	—	2	2	no	—
59	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
60	C₃×C₅×C₈	C₂×D₄	1	2	yes	—
61	C₂³×C₃×C₄	C₄×C₅	1	1	yes	characteristic
62	C₂²×C₃×C₈	—	2	4	no	—
63	C₂²×C₃×C₈	C₄×C₅	1	1	yes	characteristic
64	C₂²×C₃×C₈	C₄×C₅	1	2	yes	—
65	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
66	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
67	C₂×C₃×C₁₆	—	2	4	no	—
68	C₂×C₃×C₁₆	C₂²×C₅	1	4	yes	complemented
69	C₂⁴×C₅	C₃×C₈	1	1	yes	characteristic
70	C₂²×C₄×C₅	—	2	2	no	—
71	C₂²×C₄×C₅	—	2	2	no	—
72	C₂²×C₄×C₅	C₃×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₃×C₈	1	2	yes	—
74	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
75	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
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	C₂×C₃×C₄	1	2	yes	—
80	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
81	C₂×C₅×C₈	C₃×D₄	1	4	yes	—
82	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
83	C₅×C₁₆	—	2	4	no	—
84	C₂³×C₈	C₂×C₃×C₅	1	1	yes	characteristic
85	C₂²×C₁₆	C₂×C₃×C₅	1	2	yes	complemented
86	C₂²⋊C₁₆	C₂×C₃×C₅	1	4	yes	direct product factor
87	C₂²×C₃×C₅	—	2	8	no	—
88	C₂²×C₃×C₅	—	2	2	no	—
89	C₂²×C₃×C₅	—	2	2	no	—
90	C₂²×C₃×C₅	C₂²⋊C₈	1	2	yes	—
91	C₂²×C₃×C₅	C₂×C₁₆	1	2	yes	—
92	C₂²×C₃×C₅	C₂×C₁₆	1	4	yes	complemented
93	C₂²×C₃×C₅	M₃₂	1	2	yes	—
94	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
95	C₃×C₄×C₅	—	2	2	no	—
96	C₃×C₄×C₅	C₂²⋊C₈	1	2	yes	—
97	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
99	C₂⁴×C₃	C₅×C₈	1	1	yes	characteristic
100	C₂²×C₃×C₄	—	2	2	no	—
101	C₂²×C₃×C₄	—	2	2	no	—
102	C₂²×C₃×C₄	C₅×C₈	1	1	yes	characteristic
103	C₂²×C₃×C₄	C₅×C₈	1	2	yes	—
104	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
105	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
106	C₂×C₃×C₈	—	2	2	no	—
107	C₂×C₃×C₈	—	2	4	no	—
108	C₂×C₃×C₈	—	2	4	no	—
109	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
110	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
111	C₂×C₃×C₈	C₅×D₄	1	4	yes	—
112	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
113	C₃×C₁₆	—	2	4	no	—
114	C₂³×C₅	—	2	2	no	—
115	C₂³×C₅	—	2	2	no	—
116	C₂³×C₅	C₃×C₁₆	1	4	yes	complemented
117	C₂³×C₅	C₂×C₃×C₈	1	2	yes	—
118	C₂³×C₅	C₂×C₃×C₈	1	1	yes	characteristic
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	—	2	4	no	—
121	C₂×C₄×C₅	—	2	2	no	—
122	C₂×C₄×C₅	—	2	2	no	—
123	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
124	C₂×C₄×C₅	C₂×C₃×C₈	1	2	yes	—
125	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
126	C₂×C₄×C₅	C₃×M₁₆	1	2	yes	—
127	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
128	C₅×C₈	—	2	2	no	—
129	C₅×C₈	C₃×C₂²⋊C₄	1	2	yes	—
130	C₅×C₈	C₂×C₃×D₄	1	2	yes	—
131	C₂³×C₄	C₃×C₄×C₅	1	1	yes	characteristic
132	C₂²×C₈	—	2	4	no	—
133	C₂²×C₈	C₃×C₄×C₅	1	1	yes	characteristic
134	C₂²×C₈	C₃×C₄×C₅	1	2	yes	—
135	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
136	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
137	C₂×C₁₆	—	2	4	no	—
138	C₂×C₁₆	C₂²×C₃×C₅	1	4	yes	complemented
139	C₂×C₃×C₅	—	2	4	no	—
140	C₂×C₃×C₅	C₂²⋊C₁₆	1	4	yes	direct product factor
141	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
142	C₂×C₃×C₅	C₂²×C₁₆	1	1	yes	characteristic
143	C₂×C₃×C₅	C₂×M₃₂	1	1	yes	characteristic
144	C₂³×C₃	—	2	2	no	—
145	C₂³×C₃	—	2	2	no	—
146	C₂³×C₃	C₅×C₁₆	1	4	yes	complemented
147	C₂³×C₃	C₂×C₅×C₈	1	2	yes	—
148	C₂³×C₃	C₂×C₅×C₈	1	1	yes	characteristic
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	4	no	—
153	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
154	C₂×C₃×C₄	C₂×C₅×C₈	1	2	yes	—
155	C₂×C₃×C₄	C₂×C₅×C₈	1	1	yes	characteristic
156	C₂×C₃×C₄	C₅×M₁₆	1	2	yes	—
157	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
158	C₃×C₈	—	2	2	no	—
159	C₃×C₈	C₅×C₂²⋊C₄	1	2	yes	—
160	C₃×C₈	C₂×C₅×D₄	1	2	yes	—
161	C₂²×C₅	—	2	2	no	—
162	C₂²×C₅	—	2	2	no	—
163	C₂²×C₅	—	2	8	no	—
164	C₂²×C₅	C₃×C₂²⋊C₈	1	2	yes	—
165	C₂²×C₅	C₂×C₃×C₁₆	1	2	yes	—
166	C₂²×C₅	C₂×C₃×C₁₆	1	4	yes	complemented
167	C₂²×C₅	C₃×M₃₂	1	2	yes	—
168	C₂²×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
169	C₄×C₅	—	2	2	no	—
170	C₄×C₅	C₃×C₂²⋊C₈	1	2	yes	—
171	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
172	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
173	C₂⁴	C₃×C₅×C₈	1	1	yes	characteristic
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	C₃×C₅×C₈	1	1	yes	characteristic
177	C₂²×C₄	C₃×C₅×C₈	1	2	yes	—
178	C₂²×C₄	C₂×C₃×C₄×C₅	1	2	yes	—
179	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
184	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
185	C₂×C₈	C₃×C₅×D₄	1	4	yes	—
186	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
187	C₁₆	—	2	4	no	—
188	C₃×C₅	C₂×C₂²⋊C₁₆	1	1	yes	characteristic, direct product factor, Hall
189	C₂²×C₃	—	2	2	no	—
190	C₂²×C₃	—	2	8	no	—
191	C₂²×C₃	—	2	2	no	—
192	C₂²×C₃	C₅×C₂²⋊C₈	1	2	yes	—
193	C₂²×C₃	C₂×C₅×C₁₆	1	2	yes	—
194	C₂²×C₃	C₂×C₅×C₁₆	1	4	yes	complemented
195	C₂²×C₃	C₅×M₃₂	1	2	yes	—
196	C₂²×C₃	C₂²×C₅×C₈	1	1	yes	characteristic
197	C₃×C₄	—	2	2	no	—
198	C₃×C₄	C₅×C₂²⋊C₈	1	2	yes	—
199	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
200	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
201	C₂×C₅	—	2	4	no	—
202	C₂×C₅	C₃×C₂²⋊C₁₆	1	4	yes	direct product factor
203	C₂×C₅	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
204	C₂×C₅	C₂²×C₃×C₁₆	1	1	yes	characteristic
205	C₂×C₅	C₂×C₃×M₃₂	1	1	yes	characteristic
206	C₂³	—	2	2	no	—
207	C₂³	—	2	2	no	—
208	C₂³	C₃×C₅×C₁₆	1	4	yes	complemented
209	C₂³	C₂×C₃×C₅×C₈	1	2	yes	—
210	C₂³	C₂×C₃×C₅×C₈	1	1	yes	characteristic
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
216	C₂×C₄	C₂×C₃×C₅×C₈	1	2	yes	—
217	C₂×C₄	C₂×C₃×C₅×C₈	1	1	yes	characteristic
218	C₂×C₄	C₃×C₅×M₁₆	1	2	yes	—
219	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	C₃×C₅×C₂²⋊C₄	1	2	yes	—
222	C₈	C₂×C₃×C₅×D₄	1	2	yes	—
223	C₂×C₃	—	2	4	no	—
224	C₂×C₃	C₅×C₂²⋊C₁₆	1	4	yes	direct product factor
225	C₂×C₃	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
226	C₂×C₃	C₂²×C₅×C₁₆	1	1	yes	characteristic
227	C₂×C₃	C₂×C₅×M₃₂	1	1	yes	characteristic
228	C₅	C₂×C₃×C₂²⋊C₁₆	1	1	yes	characteristic, direct product factor, Sylow
229	C₂²	—	2	2	no	—
230	C₂²	—	2	8	no	—
231	C₂²	—	2	2	no	—
232	C₂²	C₃×C₅×C₂²⋊C₈	1	2	yes	—
233	C₂²	C₂×C₃×C₅×C₁₆	1	2	yes	—
234	C₂²	C₂×C₃×C₅×C₁₆	1	4	yes	complemented
235	C₂²	C₃×C₅×M₃₂	1	2	yes	—
236	C₂²	C₂²×C₃×C₅×C₈	1	1	yes	characteristic
237	C₄	—	2	2	no	—
238	C₄	C₃×C₅×C₂²⋊C₈	1	2	yes	—
239	C₄	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
240	C₄	C₂×C₃×C₅×M₁₆	1	1	yes	characteristic
241	C₃	C₂×C₅×C₂²⋊C₁₆	1	1	yes	characteristic, direct product factor, Sylow
242	C₂	—	2	4	no	—
243	C₂	C₃×C₅×C₂²⋊C₁₆	1	4	yes	direct product factor
244	C₂	C₂×C₃×C₅×C₂²⋊C₈	1	1	yes	characteristic
245	C₂	C₂²×C₃×C₅×C₁₆	1	1	yes	characteristic
246	C₂	C₂×C₃×C₅×M₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₅×C₈	C₂²
commutator	C₂	C₂²×C₃×C₅×C₁₆
Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
socle	C₂³×C₃×C₅	C₂×C₈
prime radical	C₂⁴×C₃×C₅	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×C₅×C₁₆	C₂	semidirect
3. C₃×C₅×C₂²⋊C₁₆	C₂	direct (N × H)
4. C₂×C₅×C₂²⋊C₁₆	C₃	direct (N × H)
12. C₂×C₃×C₅×C₁₆	C₂²	semidirect
13. C₂×C₃×C₂²⋊C₁₆	C₅	direct (N × H)
15. C₂²×C₅×C₁₆	C₂×C₃	semidirect
16. C₅×C₂²⋊C₁₆	C₂×C₃	direct (N × H)
33. C₂²×C₃×C₁₆	C₂×C₅	semidirect
34. C₃×C₂²⋊C₁₆	C₂×C₅	direct (N × H)
42. C₂×C₅×C₁₆	C₂²×C₃	semidirect
43. C₂×C₂²⋊C₁₆	C₃×C₅	direct (N × H)
46. C₂³×C₃×C₅	C₁₆	semidirect
68. C₂×C₃×C₁₆	C₂²×C₅	semidirect
85. C₂²×C₁₆	C₂×C₃×C₅	semidirect
86. C₂²⋊C₁₆	C₂×C₃×C₅	direct (N × H)
92. C₂²×C₃×C₅	C₂×C₁₆	semidirect
116. C₂³×C₅	C₃×C₁₆	semidirect
138. C₂×C₁₆	C₂²×C₃×C₅	semidirect
140. C₂×C₃×C₅	C₂²⋊C₁₆	direct (N × H)
146. C₂³×C₃	C₅×C₁₆	semidirect
166. C₂²×C₅	C₂×C₃×C₁₆	semidirect
188. C₃×C₅	C₂×C₂²⋊C₁₆	direct (N × H)
194. C₂²×C₃	C₂×C₅×C₁₆	semidirect
202. C₂×C₅	C₃×C₂²⋊C₁₆	direct (N × H)
208. C₂³	C₃×C₅×C₁₆	semidirect
224. C₂×C₃	C₅×C₂²⋊C₁₆	direct (N × H)
228. C₅	C₂×C₃×C₂²⋊C₁₆	direct (N × H)
234. C₂²	C₂×C₃×C₅×C₁₆	semidirect
241. C₃	C₂×C₅×C₂²⋊C₁₆	direct (N × H)
243. C₂	C₃×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
C₂³×C₃×C₅×C₈	2
C₂²×C₃×C₅×C₁₆	2
C₃×C₅×C₂²⋊C₁₆	2
C₂×C₅×C₂²⋊C₁₆	3
C₂×C₃×C₂²⋊C₁₆	5

Subgroups — C2×C3×C5×C22⋊C16 #14275145

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×C₂²⋊C₁₆ #14275145