Subgroups — C₃×#10604736 #14280616

Order 1920.

380 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#65754	C₂	1	2	yes	maximal
2	C₃×M₁₆×Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₃×#170⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₃×#64879	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₈×#170	C₂	1	1	yes	maximal, characteristic, complemented
6	#10604736	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×M₁₆	C₄	1	1	yes	characteristic
8	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
9	C₂×C₃×#681	C₄	1	1	yes	characteristic
10	C₃×C₂²⋊#170	C₄	1	4	yes	—
11	C₃×#4004	C₄	1	4	yes	—
12	C₃×C₄×#170	C₂²	1	2	yes	—
13	C₃×C₄×#170	C₂²	1	2	yes	—
14	C₃×#3947	C₄	1	1	yes	characteristic
15	C₃×#3947	C₄	1	1	yes	characteristic
16	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
17	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
18	C₃×#1453	—	5	1	no	maximal, Hall
19	#65754	C₂×C₃	1	2	yes	—
20	M₁₆×Dic₅	C₂×C₃	1	1	yes	characteristic
21	#170⋊C₈	C₂×C₃	1	2	yes	complemented
22	#64879	C₂×C₃	1	1	yes	characteristic, complemented
23	C₈×#170	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	—
26	C₂×C₃×#170	C₂×C₄	1	4	yes	—
27	C₂×C₃×#170	C₂×C₄	1	4	yes	—
28	C₃×C₅×M₁₆	—	2	2	no	—
29	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
31	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
32	C₃×C₄×Dic₅	C₂×C₄	1	2	yes	—
33	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
34	C₃×#681	—	2	2	no	—
35	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
37	C₃×#431	—	5	2	no	—
38	C₃×C₄×M₁₆	—	5	1	no	—
39	C₃×#322	—	5	1	no	—
40	C₃×#322	—	5	2	no	—
41	C₃×C₈²	—	5	1	no	—
42	C₂×C₅×M₁₆	C₃×C₄	1	1	yes	characteristic
43	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
44	C₂×#681	C₃×C₄	1	1	yes	characteristic
45	C₂²⋊#170	C₃×C₄	1	4	yes	—
46	#4004	C₃×C₄	1	4	yes	—
47	C₄×#170	C₂²×C₃	1	2	yes	—
48	C₄×#170	C₂²×C₃	1	2	yes	—
49	#3947	C₃×C₄	1	1	yes	characteristic
50	#3947	C₃×C₄	1	1	yes	characteristic
51	C₈×Dic₅	C₂²×C₃	1	1	yes	characteristic
52	C₈×Dic₅	C₂²×C₃	1	1	yes	characteristic
53	#1453	—	5	1	no	Sylow
54	C₂³×C₃×C₅	C₄²	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	—	2	1	no	—
56	C₂×C₃×C₄×C₅	—	2	1	no	—
57	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₃×Dic₅	—	2	4	no	—
60	C₂×C₃×Dic₅	C₄²	1	2	yes	—
61	C₂×C₃×Dic₅	C₂²×C₄	1	2	yes	—
62	C₃×#170	—	2	4	no	—
63	C₃×#170	—	2	4	no	—
64	C₃×C₅×C₈	—	2	1	no	—
65	C₃×C₅×C₈	M₁₆	1	2	yes	—
66	C₃×#168	—	2	1	no	—
67	C₃×#168	M₁₆	1	2	yes	—
68	C₂×C₃×M₁₆	—	5	1	no	—
69	C₂×C₃×M₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic
70	C₂×C₃×C₄²	—	5	1	no	—
71	C₃×C₄⋊C₈	—	5	4	no	—
72	C₃×C₂²⋊C₈	—	5	4	no	—
73	C₃×#97	—	5	1	no	—
74	C₃×#97	—	5	1	no	—
75	C₃×C₄×C₈	—	5	2	no	—
76	C₃×C₄×C₈	—	5	2	no	—
77	C₃×C₄×C₈	—	5	1	no	—
78	C₃×C₄×C₈	—	5	1	no	—
79	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
80	C₂²×Dic₅	C₂×C₃×C₄	1	2	yes	—
81	C₂×#170	C₂×C₃×C₄	1	4	yes	—
82	C₂×#170	C₂×C₃×C₄	1	4	yes	—
83	C₅×M₁₆	—	2	2	no	—
84	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
85	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
86	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
87	C₄×Dic₅	C₂×C₃×C₄	1	2	yes	—
88	C₄×Dic₅	C₂³×C₃	1	1	yes	characteristic
89	#681	—	2	2	no	—
90	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
91	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
92	#431	—	5	2	no	—
93	C₄×M₁₆	—	5	1	no	—
94	#322	—	5	1	no	—
95	#322	—	5	2	no	—
96	C₈²	—	5	1	no	—
97	C₂²×C₃×C₅	—	2	1	no	—
98	C₂²×C₃×C₅	—	2	2	no	—
99	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
100	C₃×C₄×C₅	—	2	1	no	—
101	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
102	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
103	C₃×Dic₅	—	2	2	no	—
104	C₃×Dic₅	#131	1	4	yes	—
105	C₂²×C₃×C₄	—	5	2	no	—
106	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
107	C₃×M₁₆	—	2	2	no	—
108	C₃×M₁₆	—	10	2	no	—
109	C₂×C₃×C₈	—	5	1	no	—
110	C₂×C₃×C₈	—	5	4	no	—
111	C₂×C₃×C₈	—	5	4	no	—
112	C₂×C₃×C₈	—	5	1	no	—
113	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
114	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
115	C₃×C₄²	—	5	2	no	—
116	C₃×C₄²	—	5	1	no	—
117	C₃×C₄²	—	5	1	no	—
118	C₂³×C₅	C₃×C₄²	1	1	yes	characteristic
119	C₂×C₄×C₅	—	2	1	no	—
120	C₂×C₄×C₅	—	2	1	no	—
121	C₂×C₄×C₅	C₃×C₄²	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
123	C₂×Dic₅	—	2	4	no	—
124	C₂×Dic₅	C₃×C₄²	1	2	yes	—
125	C₂×Dic₅	C₂²×C₃×C₄	1	2	yes	—
126	#170	—	2	4	no	—
127	#170	—	2	4	no	—
128	C₅×C₈	—	2	1	no	—
129	C₅×C₈	C₃×M₁₆	1	2	yes	—
130	#168	—	2	1	no	—
131	#168	C₃×M₁₆	1	2	yes	—
132	C₂×M₁₆	—	5	1	no	—
133	C₂×M₁₆	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
134	C₂×C₄²	—	5	1	no	—
135	C₄⋊C₈	—	5	4	no	—
136	C₂²⋊C₈	—	5	4	no	—
137	#97	—	5	1	no	—
138	#97	—	5	1	no	—
139	C₄×C₈	—	5	2	no	—
140	C₄×C₈	—	5	2	no	—
141	C₄×C₈	—	5	1	no	—
142	C₄×C₈	—	5	1	no	—
143	C₂×C₃×C₅	—	4	1	no	—
144	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
145	C₂×C₃×C₅	#405	1	2	yes	—
146	C₂³×C₃	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	—	2	1	no	—
149	C₂×C₃×C₄	—	5	2	no	—
150	C₂×C₃×C₄	—	5	2	no	—
151	C₂×C₃×C₄	—	10	4	no	—
152	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
154	C₃×C₈	—	2	1	no	—
155	C₃×C₈	—	5	2	no	—
156	C₃×C₈	—	10	4	no	—
157	C₃×C₈	—	10	4	no	—
158	C₃×C₈	—	10	1	no	—
159	C₃×C₈	#704	1	2	yes	—
160	C₂²×C₅	—	2	1	no	—
161	C₂²×C₅	—	2	2	no	—
162	C₂²×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
163	C₄×C₅	—	2	1	no	—
164	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
165	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
166	Dic₅	—	2	2	no	—
167	Dic₅	C₃×#131	1	4	yes	—
168	C₂²×C₄	—	5	2	no	—
169	C₂²×C₄	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
170	M₁₆	—	2	2	no	—
171	M₁₆	—	10	2	no	—
172	C₂×C₈	—	5	1	no	—
173	C₂×C₈	—	5	4	no	—
174	C₂×C₈	—	5	4	no	—
175	C₂×C₈	—	5	1	no	—
176	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
177	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
178	C₄²	—	5	2	no	—
179	C₄²	—	5	1	no	—
180	C₄²	—	5	1	no	—
181	C₃×C₅	#1453	1	1	yes	characteristic, complemented, Hall
182	C₂²×C₃	—	2	2	no	—
183	C₂²×C₃	—	2	1	no	—
184	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
185	C₃×C₄	—	2	1	no	—
186	C₃×C₄	—	5	4	no	—
187	C₃×C₄	—	10	2	no	—
188	C₃×C₄	D₅⋊M₁₆	1	1	yes	characteristic
189	C₃×C₄	C₂×#704	1	1	yes	characteristic
190	C₂×C₅	—	4	1	no	—
191	C₂×C₅	C₃×C₄×M₁₆	1	1	yes	characteristic
192	C₂×C₅	C₃×#405	1	2	yes	—
193	C₂³	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	5	2	no	—
197	C₂×C₄	—	5	2	no	—
198	C₂×C₄	—	10	4	no	—
199	C₂×C₄	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
200	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
201	C₈	—	2	1	no	—
202	C₈	—	5	2	no	—
203	C₈	—	10	4	no	—
204	C₈	—	10	4	no	—
205	C₈	—	10	1	no	—
206	C₈	C₃×#704	1	2	yes	—
207	C₂×C₃	—	4	1	no	—
208	C₂×C₃	#65728	1	2	yes	—
209	C₂×C₃	C₄×#704	1	1	yes	characteristic
210	C₅	C₃×#1453	1	1	yes	characteristic, complemented, Sylow
211	C₂²	—	2	2	no	—
212	C₂²	—	2	1	no	—
213	C₂²	C₂×C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	5	4	no	—
216	C₄	—	10	2	no	—
217	C₄	C₃×D₅⋊M₁₆	1	1	yes	characteristic
218	C₄	C₂×C₃×#704	1	1	yes	characteristic
219	C₃	#10604736	1	1	yes	characteristic, direct product factor, Sylow
220	C₂	—	4	1	no	—
221	C₂	C₃×#65728	1	2	yes	—
222	C₂	C₃×C₄×#704	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)
commutator	C₂²×C₅	C₂×C₃×C₄²
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×#1453
LCS	G ▷ C₂²×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
3. C₃×#170⋊C₈	C₂	semidirect
4. C₃×#64879	C₂	semidirect
5. C₃×C₈×#170	C₂	semidirect
6. #10604736	C₃	direct (N × H)
21. #170⋊C₈	C₂×C₃	semidirect
22. #64879	C₂×C₃	semidirect
23. C₈×#170	C₂×C₃	semidirect
181. C₃×C₅	#1453	semidirect
210. C₅	C₃×#1453	semidirect
219. C₃	#10604736	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×#65754	2
C₃×M₁₆×Dic₅	2
C₃×#170⋊C₈	2
C₃×#64879	2
C₃×C₈×#170	2
#10604736	3
C₃×#1453	5

Subgroups — C3×#10604736 #14280616

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10604736 #14280616