Subgroups — C₃×M₁₆×#170 #14280372

Order 1920.

448 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#65748	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×#170	C₂	1	1	yes	maximal, characteristic
3	C₃×M₁₆×Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₃×#64879	C₂	1	2	yes	maximal, complemented
5	C₃×C₈×#170	C₂	1	2	yes	maximal, complemented
6	M₁₆×#170	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×#170	C₄	1	2	yes	—
8	C₂×C₃×C₅×M₁₆	C₄	1	1	yes	characteristic
9	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
10	C₂×C₃×#681	C₄	1	1	yes	characteristic
11	C₃×C₂²⋊#170	C₄	1	2	yes	—
12	C₃×C₄⋊#170	C₄	1	2	yes	—
13	C₃×C₄×#170	C₄	1	2	yes	—
14	C₃×C₄×#170	C₂²	1	1	yes	characteristic
15	C₃×C₄×#170	C₂²	1	1	yes	characteristic
16	C₃×C₄×#170	C₂²	1	2	yes	—
17	C₃×#3947	C₄	1	2	yes	—
18	C₃×C₈×Dic₅	C₂²	1	2	yes	—
19	C₃×C₈×M₁₆	—	5	1	no	maximal, Hall
20	#65748	C₂×C₃	1	1	yes	characteristic
21	C₂×C₄×#170	C₂×C₃	1	1	yes	characteristic
22	M₁₆×Dic₅	C₂×C₃	1	1	yes	characteristic
23	#64879	C₂×C₃	1	2	yes	complemented
24	C₈×#170	C₂×C₃	1	2	yes	complemented
25	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×#170	—	2	4	no	—
29	C₂×C₃×#170	C₂×C₄	1	2	yes	—
30	C₂×C₃×#170	C₂×C₄	1	4	yes	—
31	C₂×C₃×#170	C₂×C₄	1	2	yes	—
32	C₃×C₅×M₁₆	C₈	1	4	yes	complemented
33	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
35	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
36	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
38	C₃×#681	C₈	1	4	yes	complemented
39	C₂×C₃×#168	C₂×C₄	1	2	yes	—
40	C₃×#431	—	5	1	no	—
41	C₃×C₄×M₁₆	—	5	1	no	—
42	C₂×C₃×C₄×C₈	—	5	1	no	—
43	C₃×#322	—	5	2	no	—
44	C₃×C₈²	—	5	2	no	—
45	C₂²×#170	C₃×C₄	1	2	yes	—
46	C₂×C₅×M₁₆	C₃×C₄	1	1	yes	characteristic
47	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
48	C₂×#681	C₃×C₄	1	1	yes	characteristic
49	C₂²⋊#170	C₃×C₄	1	2	yes	—
50	C₄⋊#170	C₃×C₄	1	2	yes	—
51	C₄×#170	C₃×C₄	1	2	yes	—
52	C₄×#170	C₂²×C₃	1	1	yes	characteristic
53	C₄×#170	C₂²×C₃	1	2	yes	—
54	C₄×#170	C₂²×C₃	1	1	yes	characteristic
55	#3947	C₃×C₄	1	2	yes	—
56	C₈×Dic₅	C₂²×C₃	1	2	yes	—
57	C₈×M₁₆	—	5	1	no	Sylow
58	C₂³×C₃×C₅	C₄²	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
61	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
62	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₃×Dic₅	—	2	2	no	—
64	C₂×C₃×Dic₅	—	2	2	no	—
65	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
66	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
67	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
69	C₃×#170	—	2	2	no	—
70	C₃×#170	—	2	4	no	—
71	C₃×#170	M₁₆	1	4	yes	direct product factor
72	C₃×C₅×C₈	C₂×C₈	1	4	yes	complemented
73	C₃×#168	C₂×C₈	1	4	yes	complemented
74	C₂×C₃×M₁₆	—	5	1	no	—
75	C₂×C₃×M₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic
76	C₂²×C₃×C₈	—	5	2	no	—
77	C₂×C₃×C₄²	—	5	1	no	—
78	C₃×C₄⋊C₈	—	5	2	no	—
79	C₃×C₂²⋊C₈	—	5	2	no	—
80	C₃×#97	—	5	2	no	—
81	C₃×C₄×C₈	—	5	2	no	—
82	C₃×C₄×C₈	—	5	1	no	—
83	C₃×C₄×C₈	—	5	1	no	—
84	C₃×C₄×C₈	—	5	2	no	—
85	C₃×C₄×C₈	—	5	2	no	—
86	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
87	C₂²×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
88	C₂²×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
89	C₂×#170	—	2	4	no	—
90	C₂×#170	C₂×C₃×C₄	1	2	yes	—
91	C₂×#170	C₂×C₃×C₄	1	4	yes	—
92	C₂×#170	C₂×C₃×C₄	1	2	yes	—
93	C₅×M₁₆	C₃×C₈	1	4	yes	complemented
94	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
95	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
96	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
97	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
98	C₄×Dic₅	C₂³×C₃	1	1	yes	characteristic
99	#681	C₃×C₈	1	4	yes	complemented
100	C₂×#168	C₂×C₃×C₄	1	2	yes	—
101	#431	—	5	1	no	—
102	C₄×M₁₆	—	5	1	no	—
103	C₂×C₄×C₈	—	5	1	no	—
104	#322	—	5	2	no	—
105	C₈²	—	5	2	no	—
106	C₂²×C₃×C₅	—	2	1	no	—
107	C₂²×C₃×C₅	—	2	1	no	—
108	C₂²×C₃×C₅	C₄×C₈	1	2	yes	—
109	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
110	C₃×C₄×C₅	C₄×C₈	1	2	yes	—
111	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
112	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
113	C₃×Dic₅	—	2	1	no	—
114	C₃×Dic₅	—	2	1	no	—
115	C₃×Dic₅	C₂×M₁₆	1	2	yes	—
116	C₃×Dic₅	#131	1	2	yes	—
117	C₂²×C₃×C₄	—	5	1	no	—
118	C₂²×C₃×C₄	—	5	1	no	—
119	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
120	C₃×M₁₆	—	5	4	no	—
121	C₃×M₁₆	#170	1	4	yes	direct product factor
122	C₂×C₃×C₈	—	5	2	no	—
123	C₂×C₃×C₈	—	5	2	no	—
124	C₂×C₃×C₈	—	5	2	no	—
125	C₂×C₃×C₈	—	5	4	no	—
126	C₂×C₃×C₈	—	10	4	no	—
127	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
128	C₃×C₄²	—	5	1	no	—
129	C₃×C₄²	—	5	1	no	—
130	C₃×C₄²	—	5	1	no	—
131	C₃×C₄²	—	5	1	no	—
132	C₂³×C₅	C₃×C₄²	1	1	yes	characteristic
133	C₂×C₄×C₅	C₃×C₄²	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×C₃×C₈	1	2	yes	—
135	C₂×C₄×C₅	C₂×C₃×C₈	1	2	yes	—
136	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
137	C₂×Dic₅	—	2	2	no	—
138	C₂×Dic₅	—	2	2	no	—
139	C₂×Dic₅	C₃×C₄²	1	1	yes	characteristic
140	C₂×Dic₅	C₃×C₄²	1	1	yes	characteristic
141	C₂×Dic₅	C₂²×C₃×C₄	1	1	yes	characteristic
142	C₂×Dic₅	C₂²×C₃×C₄	1	1	yes	characteristic
143	#170	—	2	2	no	—
144	#170	—	2	4	no	—
145	#170	C₃×M₁₆	1	4	yes	direct product factor
146	C₅×C₈	C₂×C₃×C₈	1	4	yes	complemented
147	#168	C₂×C₃×C₈	1	4	yes	complemented
148	C₂×M₁₆	—	5	1	no	—
149	C₂×M₁₆	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
150	C₂²×C₈	—	5	2	no	—
151	C₂×C₄²	—	5	1	no	—
152	C₄⋊C₈	—	5	2	no	—
153	C₂²⋊C₈	—	5	2	no	—
154	#97	—	5	2	no	—
155	C₄×C₈	—	5	1	no	—
156	C₄×C₈	—	5	2	no	—
157	C₄×C₈	—	5	1	no	—
158	C₄×C₈	—	5	2	no	—
159	C₄×C₈	—	5	2	no	—
160	C₂×C₃×C₅	—	2	2	no	—
161	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
162	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
163	C₂×C₃×C₅	#405	1	1	yes	characteristic
164	C₂³×C₃	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
165	C₂×C₃×C₄	—	5	1	no	—
166	C₂×C₃×C₄	—	5	1	no	—
167	C₂×C₃×C₄	—	5	1	no	—
168	C₂×C₃×C₄	—	5	1	no	—
169	C₂×C₃×C₄	—	10	2	no	—
170	C₂×C₃×C₄	—	10	2	no	—
171	C₂×C₃×C₄	D₅⋊C₈	1	2	yes	—
172	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×#170	1	2	yes	—
174	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
175	C₃×C₈	—	5	4	no	—
176	C₃×C₈	—	5	4	no	—
177	C₃×C₈	—	10	4	no	—
178	C₃×C₈	—	10	2	no	—
179	C₃×C₈	C₂×#170	1	4	yes	complemented
180	C₂²×C₅	—	2	1	no	—
181	C₂²×C₅	—	2	1	no	—
182	C₂²×C₅	C₃×C₄×C₈	1	2	yes	—
183	C₂²×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
184	C₄×C₅	C₃×C₄×C₈	1	2	yes	—
185	C₄×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
186	C₄×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
187	Dic₅	—	2	1	no	—
188	Dic₅	—	2	1	no	—
189	Dic₅	C₂×C₃×M₁₆	1	2	yes	—
190	Dic₅	C₃×#131	1	2	yes	—
191	C₂²×C₄	—	5	1	no	—
192	C₂²×C₄	—	5	1	no	—
193	C₂²×C₄	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
194	M₁₆	—	5	4	no	—
195	M₁₆	C₃×#170	1	4	yes	direct product factor
196	C₂×C₈	—	5	2	no	—
197	C₂×C₈	—	5	2	no	—
198	C₂×C₈	—	5	2	no	—
199	C₂×C₈	—	5	4	no	—
200	C₂×C₈	—	10	4	no	—
201	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
202	C₄²	—	5	1	no	—
203	C₄²	—	5	1	no	—
204	C₄²	—	5	1	no	—
205	C₄²	—	5	1	no	—
206	C₃×C₅	C₈×M₁₆	1	1	yes	characteristic, complemented, Hall
207	C₂²×C₃	—	2	1	no	—
208	C₂²×C₃	—	2	1	no	—
209	C₂²×C₃	C₄×#170	1	2	yes	—
210	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
211	C₃×C₄	—	5	2	no	—
212	C₃×C₄	—	5	2	no	—
213	C₃×C₄	—	10	1	no	—
214	C₃×C₄	—	10	1	no	—
215	C₃×C₄	C₄×#170	1	2	yes	—
216	C₃×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
217	C₃×C₄	C₂²×#170	1	1	yes	characteristic
218	C₂×C₅	—	2	2	no	—
219	C₂×C₅	C₂×C₃×C₄×C₈	1	1	yes	characteristic
220	C₂×C₅	C₃×C₄×M₁₆	1	1	yes	characteristic
221	C₂×C₅	C₃×#405	1	1	yes	characteristic
222	C₂³	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
223	C₂×C₄	—	5	1	no	—
224	C₂×C₄	—	5	1	no	—
225	C₂×C₄	—	5	1	no	—
226	C₂×C₄	—	5	1	no	—
227	C₂×C₄	—	10	2	no	—
228	C₂×C₄	—	10	2	no	—
229	C₂×C₄	C₃×D₅⋊C₈	1	2	yes	—
230	C₂×C₄	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
231	C₂×C₄	C₂×C₃×#170	1	2	yes	—
232	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
233	C₈	—	5	4	no	—
234	C₈	—	5	4	no	—
235	C₈	—	10	2	no	—
236	C₈	—	10	4	no	—
237	C₈	C₂×C₃×#170	1	4	yes	complemented
238	C₂×C₃	—	2	2	no	—
239	C₂×C₃	M₁₆×AGL(1,𝔽₅)	1	1	yes	characteristic
240	C₂×C₃	#65728	1	1	yes	characteristic
241	C₂×C₃	C₂×C₄×#170	1	1	yes	characteristic
242	C₅	C₃×C₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	C₃×C₄×#170	1	2	yes	—
246	C₂²	C₂×C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
247	C₄	—	5	2	no	—
248	C₄	—	5	2	no	—
249	C₄	—	10	1	no	—
250	C₄	—	10	1	no	—
251	C₄	C₃×C₄×#170	1	2	yes	—
252	C₄	C₂×C₃×D₅⋊C₈	1	1	yes	characteristic
253	C₄	C₂²×C₃×#170	1	1	yes	characteristic
254	C₃	M₁₆×#170	1	1	yes	characteristic, direct product factor, Sylow
255	C₂	—	2	2	no	—
256	C₂	C₃×M₁₆×AGL(1,𝔽₅)	1	1	yes	characteristic
257	C₂	C₃×#65728	1	1	yes	characteristic
258	C₂	C₂×C₃×C₄×#170	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₃×C₄×C₈
Frattini	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₂×C₃×C₅×M₁₆	C₄
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂³×C₃×C₅	C₄²
cosocle	C₄×Dic₅	C₂³×C₃
hypercenter	C₂×C₃×M₁₆	AGL(1,𝔽₅)
hypocenter	C₅	C₃×C₈×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×M₁₆

Complements

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

N	H ≅ G/N	type
4. C₃×#64879	C₂	semidirect
5. C₃×C₈×#170	C₂	semidirect
6. M₁₆×#170	C₃	direct (N × H)
23. #64879	C₂×C₃	semidirect
24. C₈×#170	C₂×C₃	semidirect
32. C₃×C₅×M₁₆	C₈	semidirect
38. C₃×#681	C₈	semidirect
71. C₃×#170	M₁₆	direct (N × H)
72. C₃×C₅×C₈	C₂×C₈	semidirect
73. C₃×#168	C₂×C₈	semidirect
93. C₅×M₁₆	C₃×C₈	semidirect
99. #681	C₃×C₈	semidirect
121. C₃×M₁₆	#170	direct (N × H)
145. #170	C₃×M₁₆	direct (N × H)
146. C₅×C₈	C₂×C₃×C₈	semidirect
147. #168	C₂×C₃×C₈	semidirect
179. C₃×C₈	C₂×#170	semidirect
195. M₁₆	C₃×#170	direct (N × H)
206. C₃×C₅	C₈×M₁₆	semidirect
237. C₈	C₂×C₃×#170	semidirect
242. C₅	C₃×C₈×M₁₆	semidirect
254. C₃	M₁₆×#170	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#65748	2
C₂×C₃×C₄×#170	2
C₃×M₁₆×Dic₅	2
C₃×#64879	2
C₃×C₈×#170	2
M₁₆×#170	3
C₃×C₈×M₁₆	5

Subgroups — C3×M16×#170 #14280372

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×M₁₆×#170 #14280372