Subgroups — C₃×#10604009 #14280382

Order 1920.

448 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	C₃×C₄×#700	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×D₅⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×#65122	C₂	1	1	yes	maximal, characteristic
4	C₃×#170⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₃×C₈×#170	C₂	1	2	yes	maximal, complemented
6	#10604009	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×#700	C₄	1	2	yes	—
8	C₂×C₃×D₅⋊C₈	C₄	1	2	yes	—
9	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
10	C₃×#4002	C₄	1	2	yes	—
11	C₃×C₄×#170	C₄	1	2	yes	—
12	C₃×C₄×#170	C₂²	1	1	yes	characteristic
13	C₃×C₄×#170	C₂²	1	1	yes	characteristic
14	C₃×C₄×#170	C₂²	1	2	yes	—
15	C₃×C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
16	C₃×#3952	C₄	1	2	yes	—
17	C₃×C₈×Dic₅	C₂²	1	2	yes	—
18	C₃×C₄⋊#168	C₄	1	1	yes	characteristic
19	C₃×C₈×M₁₆	—	5	1	no	maximal, Hall
20	C₄×#700	C₂×C₃	1	1	yes	characteristic
21	C₄×D₅⋊C₈	C₂×C₃	1	1	yes	characteristic
22	#65122	C₂×C₃	1	1	yes	characteristic
23	#170⋊C₈	C₂×C₃	1	2	yes	complemented
24	C₈×#170	C₂×C₃	1	2	yes	complemented
25	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×#170	C₂×C₄	1	2	yes	—
29	C₂×C₃×#170	C₂×C₄	1	4	yes	—
30	C₂×C₃×#170	C₂×C₄	1	2	yes	—
31	C₃×#700	C₈	1	8	yes	complemented
32	C₃×D₅⋊C₈	—	2	4	no	—
33	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₃×C₄²×C₅	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	C₂×C₃×#168	C₂×C₄	1	2	yes	—
39	C₃×#431	—	5	1	no	—
40	C₃×C₄×M₁₆	—	5	1	no	—
41	C₂×C₃×C₄×C₈	—	5	1	no	—
42	C₃×#322	—	5	2	no	—
43	C₃×C₈²	—	5	2	no	—
44	C₂×#700	C₃×C₄	1	2	yes	—
45	C₂×D₅⋊C₈	C₃×C₄	1	2	yes	—
46	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
47	#4002	C₃×C₄	1	2	yes	—
48	C₄×#170	C₃×C₄	1	2	yes	—
49	C₄×#170	C₂²×C₃	1	1	yes	characteristic
50	C₄×#170	C₂²×C₃	1	2	yes	—
51	C₄×#170	C₂²×C₃	1	1	yes	characteristic
52	C₅×C₄⋊C₈	C₃×C₄	1	1	yes	characteristic
53	#3952	C₃×C₄	1	2	yes	—
54	C₈×Dic₅	C₂²×C₃	1	2	yes	—
55	C₄⋊#168	C₃×C₄	1	1	yes	characteristic
56	C₈×M₁₆	—	5	1	no	Sylow
57	C₂²×C₃×D₅	C₄²	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
62	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
64	C₃×C₄×D₅	—	2	1	no	—
65	C₃×C₄×D₅	—	2	1	no	—
66	C₃×C₄×D₅	—	2	2	no	—
67	C₃×C₄×D₅	C₂×C₈	1	4	yes	—
68	C₃×#170	—	2	2	no	—
69	C₃×#170	C₂×C₈	1	8	yes	complemented
70	C₃×#170	M₁₆	1	4	yes	complemented
71	C₃×C₅×C₈	—	2	2	no	—
72	C₃×#168	—	2	2	no	—
73	C₂×C₃×M₁₆	—	5	2	no	—
74	C₂²×C₃×C₈	—	5	2	no	—
75	C₂×C₃×C₄²	—	5	1	no	—
76	C₃×C₄⋊C₈	—	5	1	no	—
77	C₃×C₄⋊C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
78	C₃×C₂²⋊C₈	—	5	2	no	—
79	C₃×#97	—	5	2	no	—
80	C₃×C₄×C₈	—	5	2	no	—
81	C₃×C₄×C₈	—	5	1	no	—
82	C₃×C₄×C₈	—	5	1	no	—
83	C₃×C₄×C₈	—	5	2	no	—
84	C₃×C₄×C₈	—	5	2	no	—
85	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
86	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
87	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
88	C₂×#170	C₂×C₃×C₄	1	2	yes	—
89	C₂×#170	C₂×C₃×C₄	1	4	yes	—
90	C₂×#170	C₂×C₃×C₄	1	2	yes	—
91	#700	C₃×C₈	1	8	yes	complemented
92	D₅⋊C₈	—	2	4	no	—
93	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
94	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
95	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
96	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
97	C₄×Dic₅	C₂³×C₃	1	1	yes	characteristic
98	C₂×#168	C₂×C₃×C₄	1	2	yes	—
99	#431	—	5	1	no	—
100	C₄×M₁₆	—	5	1	no	—
101	C₂×C₄×C₈	—	5	1	no	—
102	#322	—	5	2	no	—
103	C₈²	—	5	2	no	—
104	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₃×D₅	—	2	1	no	—
106	C₂×C₃×D₅	—	2	1	no	—
107	C₂×C₃×D₅	C₄×C₈	1	2	yes	—
108	C₃×C₄×C₅	—	2	1	no	—
109	C₃×C₄×C₅	C₄×C₈	1	2	yes	—
110	C₃×C₄×C₅	#131	1	1	yes	characteristic
111	C₃×C₄×C₅	#131	1	1	yes	characteristic
112	C₃×Dic₅	—	2	1	no	—
113	C₃×Dic₅	C₂²×C₈	1	2	yes	—
114	C₃×Dic₅	C₂×M₁₆	1	2	yes	—
115	C₂²×C₃×C₄	—	5	1	no	—
116	C₂²×C₃×C₄	—	5	1	no	—
117	C₂²×C₃×C₄	—	5	1	no	—
118	C₃×M₁₆	—	5	8	no	—
119	C₂×C₃×C₈	—	5	2	no	—
120	C₂×C₃×C₈	—	5	4	no	—
121	C₂×C₃×C₈	—	5	2	no	—
122	C₂×C₃×C₈	—	5	2	no	—
123	C₂×C₃×C₈	—	10	4	no	—
124	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
125	C₃×C₄²	—	5	1	no	—
126	C₃×C₄²	—	5	1	no	—
127	C₃×C₄²	—	5	1	no	—
128	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
129	C₂²×D₅	C₃×C₄²	1	1	yes	characteristic
130	C₂×C₄×C₅	C₃×C₄²	1	1	yes	characteristic
131	C₂×C₄×C₅	C₃×C₄²	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
133	C₂×Dic₅	C₃×C₄²	1	1	yes	characteristic
134	C₂×Dic₅	C₂²×C₃×C₄	1	1	yes	characteristic
135	C₂×Dic₅	C₂²×C₃×C₄	1	1	yes	characteristic
136	C₄×D₅	—	2	1	no	—
137	C₄×D₅	—	2	1	no	—
138	C₄×D₅	—	2	2	no	—
139	C₄×D₅	C₂×C₃×C₈	1	4	yes	—
140	#170	—	2	2	no	—
141	#170	C₂×C₃×C₈	1	8	yes	complemented
142	#170	C₃×M₁₆	1	4	yes	complemented
143	C₅×C₈	—	2	2	no	—
144	#168	—	2	2	no	—
145	C₂×M₁₆	—	5	2	no	—
146	C₂²×C₈	—	5	2	no	—
147	C₂×C₄²	—	5	1	no	—
148	C₄⋊C₈	—	5	1	no	—
149	C₄⋊C₈	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
150	C₂²⋊C₈	—	5	2	no	—
151	#97	—	5	2	no	—
152	C₄×C₈	—	5	1	no	—
153	C₄×C₈	—	5	2	no	—
154	C₄×C₈	—	5	1	no	—
155	C₄×C₈	—	5	2	no	—
156	C₄×C₈	—	5	2	no	—
157	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
158	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
159	C₂×C₃×C₅	#405	1	1	yes	characteristic
160	C₃×D₅	—	2	2	no	—
161	C₂³×C₃	—	5	1	no	—
162	C₂×C₃×C₄	—	5	1	no	—
163	C₂×C₃×C₄	—	5	1	no	—
164	C₂×C₃×C₄	—	5	4	no	—
165	C₂×C₃×C₄	—	5	1	no	—
166	C₂×C₃×C₄	—	10	2	no	—
167	C₂×C₃×C₄	—	10	1	no	—
168	C₂×C₃×C₄	—	10	1	no	—
169	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
170	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
172	C₃×C₈	—	2	2	no	—
173	C₃×C₈	—	5	8	no	—
174	C₃×C₈	—	5	4	no	—
175	C₃×C₈	—	10	2	no	—
176	C₃×C₈	—	10	2	no	—
177	C₂²×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
178	C₂×D₅	—	2	1	no	—
179	C₂×D₅	—	2	1	no	—
180	C₂×D₅	C₃×C₄×C₈	1	2	yes	—
181	C₄×C₅	—	2	1	no	—
182	C₄×C₅	C₃×C₄×C₈	1	2	yes	—
183	C₄×C₅	C₃×#131	1	1	yes	characteristic
184	C₄×C₅	C₃×#131	1	1	yes	characteristic
185	Dic₅	—	2	1	no	—
186	Dic₅	C₂²×C₃×C₈	1	2	yes	—
187	Dic₅	C₂×C₃×M₁₆	1	2	yes	—
188	C₂²×C₄	—	5	1	no	—
189	C₂²×C₄	—	5	1	no	—
190	C₂²×C₄	—	5	1	no	—
191	M₁₆	—	5	8	no	—
192	C₂×C₈	—	5	2	no	—
193	C₂×C₈	—	5	4	no	—
194	C₂×C₈	—	5	2	no	—
195	C₂×C₈	—	5	2	no	—
196	C₂×C₈	—	10	4	no	—
197	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
198	C₄²	—	5	1	no	—
199	C₄²	—	5	1	no	—
200	C₄²	—	5	1	no	—
201	C₄²	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
202	C₃×C₅	C₈×M₁₆	1	1	yes	characteristic, complemented, Hall
203	C₂²×C₃	—	5	2	no	—
204	C₂²×C₃	—	10	1	no	—
205	C₂²×C₃	—	10	1	no	—
206	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
207	C₃×C₄	—	2	1	no	—
208	C₃×C₄	—	5	2	no	—
209	C₃×C₄	—	5	2	no	—
210	C₃×C₄	—	10	1	no	—
211	C₃×C₄	C₈×AGL(1,𝔽₅)	1	2	yes	—
212	C₃×C₄	#4131	1	1	yes	characteristic
213	C₃×C₄	#4133	1	1	yes	characteristic
214	C₂×C₅	C₂×C₃×C₄×C₈	1	1	yes	characteristic
215	C₂×C₅	C₃×C₄×M₁₆	1	1	yes	characteristic
216	C₂×C₅	C₃×#405	1	1	yes	characteristic
217	D₅	—	2	2	no	—
218	C₂³	—	5	1	no	—
219	C₂×C₄	—	5	1	no	—
220	C₂×C₄	—	5	1	no	—
221	C₂×C₄	—	5	4	no	—
222	C₂×C₄	—	5	1	no	—
223	C₂×C₄	—	10	2	no	—
224	C₂×C₄	—	10	1	no	—
225	C₂×C₄	—	10	1	no	—
226	C₂×C₄	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
227	C₂×C₄	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
228	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
229	C₈	—	2	2	no	—
230	C₈	—	5	4	no	—
231	C₈	—	5	8	no	—
232	C₈	—	10	2	no	—
233	C₈	—	10	2	no	—
234	C₂×C₃	—	10	2	no	—
235	C₂×C₃	#65701	1	1	yes	characteristic
236	C₂×C₃	C₂×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
237	C₂×C₃	M₁₆×AGL(1,𝔽₅)	1	1	yes	characteristic
238	C₅	C₃×C₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
239	C₂²	—	5	2	no	—
240	C₂²	—	10	1	no	—
241	C₂²	—	10	1	no	—
242	C₂²	C₂×C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
243	C₄	—	2	1	no	—
244	C₄	—	5	2	no	—
245	C₄	—	5	2	no	—
246	C₄	—	10	1	no	—
247	C₄	C₃×C₈×AGL(1,𝔽₅)	1	2	yes	—
248	C₄	C₃×#4131	1	1	yes	characteristic
249	C₄	C₃×#4133	1	1	yes	characteristic
250	C₃	#10604009	1	1	yes	characteristic, direct product factor, Sylow
251	C₂	—	10	2	no	—
252	C₂	C₃×#65701	1	1	yes	characteristic
253	C₂	C₂×C₃×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
254	C₂	C₃×M₁₆×AGL(1,𝔽₅)	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	C₃×C₅×C₄⋊C₈	C₄
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂²×C₃×D₅	C₄²
cosocle	C₄×Dic₅	C₂³×C₃
hypercenter	C₃×C₄⋊C₈	AGL(1,𝔽₅)
hypocenter	C₅	C₃×C₈×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
4. C₃×#170⋊C₈	C₂	semidirect
5. C₃×C₈×#170	C₂	semidirect
6. #10604009	C₃	direct (N × H)
23. #170⋊C₈	C₂×C₃	semidirect
24. C₈×#170	C₂×C₃	semidirect
31. C₃×#700	C₈	semidirect
69. C₃×#170	C₂×C₈	semidirect
70. C₃×#170	M₁₆	semidirect
91. #700	C₃×C₈	semidirect
141. #170	C₂×C₃×C₈	semidirect
142. #170	C₃×M₁₆	semidirect
202. C₃×C₅	C₈×M₁₆	semidirect
238. C₅	C₃×C₈×M₁₆	semidirect
250. C₃	#10604009	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×C₄×#700	2
C₃×C₄×D₅⋊C₈	2
C₃×#65122	2
C₃×#170⋊C₈	2
C₃×C₈×#170	2
#10604009	3
C₃×C₈×M₁₆	5

Subgroups — C3×#10604009 #14280382

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10604009 #14280382