Subgroups — C₃×#10604737 #14280712

Order 1920.

428 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×#704	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	#10604737	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×#704	C₄	1	4	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×#4002	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×#1455	—	5	1	no	maximal, Hall
19	C₄×#704	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	C₃×#704	—	2	8	no	—
27	C₂×C₃×#170	C₂×C₄	1	4	yes	—
28	C₂×C₃×#170	C₂×C₄	1	4	yes	—
29	C₃×C₅×M₁₆	—	2	2	no	—
30	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
33	C₃×C₄×Dic₅	C₂×C₄	1	2	yes	—
34	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
35	C₃×#681	—	2	2	no	—
36	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
38	C₃×C₄×M₁₆	—	5	2	no	—
39	C₃×C₄×M₁₆	—	5	1	no	—
40	C₃×#322	—	5	1	no	—
41	C₃×#322	—	5	2	no	—
42	C₃×C₈²	—	5	1	no	—
43	C₂×#704	C₃×C₄	1	4	yes	—
44	C₂×C₅×M₁₆	C₃×C₄	1	1	yes	characteristic
45	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
46	C₂×#681	C₃×C₄	1	1	yes	characteristic
47	#4002	C₃×C₄	1	4	yes	—
48	C₄×#170	C₂²×C₃	1	2	yes	—
49	C₄×#170	C₂²×C₃	1	2	yes	—
50	#3947	C₃×C₄	1	1	yes	characteristic
51	#3947	C₃×C₄	1	1	yes	characteristic
52	C₈×Dic₅	C₂²×C₃	1	1	yes	characteristic
53	C₈×Dic₅	C₂²×C₃	1	1	yes	characteristic
54	#1455	—	5	1	no	Sylow
55	C₂³×C₃×C₅	C₄²	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	—	2	1	no	—
57	C₂×C₃×C₄×C₅	—	2	1	no	—
58	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₃×Dic₅	—	2	4	no	—
61	C₂×C₃×Dic₅	C₄²	1	2	yes	—
62	C₂×C₃×Dic₅	C₂²×C₄	1	2	yes	—
63	C₃×#170	—	2	4	no	—
64	C₃×#170	M₁₆	1	8	yes	complemented
65	C₃×C₅×C₈	—	2	1	no	—
66	C₃×C₅×C₈	M₁₆	1	2	yes	complemented
67	C₃×#168	—	2	1	no	—
68	C₃×#168	M₁₆	1	2	yes	complemented
69	C₂×C₃×M₁₆	—	5	4	no	—
70	C₂×C₃×M₁₆	—	5	1	no	—
71	C₂×C₃×M₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic
72	C₂×C₃×C₄²	—	5	1	no	—
73	C₃×#97	—	5	4	no	—
74	C₃×#97	—	5	1	no	—
75	C₃×#97	—	5	1	no	—
76	C₃×C₄×C₈	—	5	2	no	—
77	C₃×C₄×C₈	—	5	2	no	—
78	C₃×C₄×C₈	—	5	1	no	—
79	C₃×C₄×C₈	—	5	1	no	—
80	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
81	C₂²×Dic₅	C₂×C₃×C₄	1	2	yes	—
82	#704	—	2	8	no	—
83	C₂×#170	C₂×C₃×C₄	1	4	yes	—
84	C₂×#170	C₂×C₃×C₄	1	4	yes	—
85	C₅×M₁₆	—	2	2	no	—
86	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
87	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
88	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
89	C₄×Dic₅	C₂×C₃×C₄	1	2	yes	—
90	C₄×Dic₅	C₂³×C₃	1	1	yes	characteristic
91	#681	—	2	2	no	—
92	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
93	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
94	C₄×M₁₆	—	5	2	no	—
95	C₄×M₁₆	—	5	1	no	—
96	#322	—	5	1	no	—
97	#322	—	5	2	no	—
98	C₈²	—	5	1	no	—
99	C₂²×C₃×C₅	—	2	2	no	—
100	C₂²×C₃×C₅	—	2	1	no	—
101	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
102	C₃×C₄×C₅	—	2	1	no	—
103	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
104	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
105	C₃×Dic₅	—	2	2	no	—
106	C₃×Dic₅	C₂×M₁₆	1	4	yes	—
107	C₂²×C₃×C₄	—	5	2	no	—
108	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
109	C₃×M₁₆	—	2	2	no	—
110	C₃×M₁₆	—	10	2	no	—
111	C₃×M₁₆	—	10	8	no	—
112	C₂×C₃×C₈	—	5	4	no	—
113	C₂×C₃×C₈	—	5	1	no	—
114	C₂×C₃×C₈	—	5	4	no	—
115	C₂×C₃×C₈	—	5	1	no	—
116	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
117	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
118	C₃×C₄²	—	5	2	no	—
119	C₃×C₄²	—	5	1	no	—
120	C₃×C₄²	—	5	1	no	—
121	C₂³×C₅	C₃×C₄²	1	1	yes	characteristic
122	C₂×C₄×C₅	—	2	1	no	—
123	C₂×C₄×C₅	—	2	1	no	—
124	C₂×C₄×C₅	C₃×C₄²	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
126	C₂×Dic₅	—	2	4	no	—
127	C₂×Dic₅	C₃×C₄²	1	2	yes	—
128	C₂×Dic₅	C₂²×C₃×C₄	1	2	yes	—
129	#170	—	2	4	no	—
130	#170	C₃×M₁₆	1	8	yes	complemented
131	C₅×C₈	—	2	1	no	—
132	C₅×C₈	C₃×M₁₆	1	2	yes	complemented
133	#168	—	2	1	no	—
134	#168	C₃×M₁₆	1	2	yes	complemented
135	C₂×M₁₆	—	5	4	no	—
136	C₂×M₁₆	—	5	1	no	—
137	C₂×M₁₆	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
138	C₂×C₄²	—	5	1	no	—
139	#97	—	5	4	no	—
140	#97	—	5	1	no	—
141	#97	—	5	1	no	—
142	C₄×C₈	—	5	2	no	—
143	C₄×C₈	—	5	2	no	—
144	C₄×C₈	—	5	1	no	—
145	C₄×C₈	—	5	1	no	—
146	C₂×C₃×C₅	—	4	1	no	—
147	C₂×C₃×C₅	C₄×M₁₆	1	2	yes	—
148	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
149	C₂³×C₃	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
150	C₂×C₃×C₄	—	2	1	no	—
151	C₂×C₃×C₄	—	2	1	no	—
152	C₂×C₃×C₄	—	5	2	no	—
153	C₂×C₃×C₄	—	5	2	no	—
154	C₂×C₃×C₄	—	10	4	no	—
155	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
157	C₃×C₈	—	2	1	no	—
158	C₃×C₈	—	5	8	no	—
159	C₃×C₈	—	5	2	no	—
160	C₃×C₈	—	10	4	no	—
161	C₃×C₈	—	10	1	no	—
162	C₃×C₈	#704	1	2	yes	complemented
163	C₂²×C₅	—	2	2	no	—
164	C₂²×C₅	—	2	1	no	—
165	C₂²×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
166	C₄×C₅	—	2	1	no	—
167	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
168	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
169	Dic₅	—	2	2	no	—
170	Dic₅	C₂×C₃×M₁₆	1	4	yes	—
171	C₂²×C₄	—	5	2	no	—
172	C₂²×C₄	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
173	M₁₆	—	2	2	no	—
174	M₁₆	—	10	2	no	—
175	M₁₆	—	10	8	no	—
176	C₂×C₈	—	5	4	no	—
177	C₂×C₈	—	5	1	no	—
178	C₂×C₈	—	5	4	no	—
179	C₂×C₈	—	5	1	no	—
180	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
181	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
182	C₄²	—	5	2	no	—
183	C₄²	—	5	1	no	—
184	C₄²	—	5	1	no	—
185	C₃×C₅	#1455	1	1	yes	characteristic, complemented, Hall
186	C₂²×C₃	—	2	2	no	—
187	C₂²×C₃	—	2	1	no	—
188	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
189	C₃×C₄	—	2	1	no	—
190	C₃×C₄	—	5	4	no	—
191	C₃×C₄	—	10	2	no	—
192	C₃×C₄	D₅⋊M₁₆	1	1	yes	characteristic
193	C₃×C₄	C₂×#704	1	1	yes	characteristic
194	C₂×C₅	—	4	1	no	—
195	C₂×C₅	C₃×C₄×M₁₆	1	2	yes	—
196	C₂×C₅	C₃×C₄×M₁₆	1	1	yes	characteristic
197	C₂³	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	5	2	no	—
201	C₂×C₄	—	5	2	no	—
202	C₂×C₄	—	10	4	no	—
203	C₂×C₄	C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
204	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
205	C₈	—	2	1	no	—
206	C₈	—	5	8	no	—
207	C₈	—	5	2	no	—
208	C₈	—	10	4	no	—
209	C₈	—	10	1	no	—
210	C₈	C₃×#704	1	2	yes	complemented
211	C₂×C₃	—	4	1	no	—
212	C₂×C₃	M₁₆×AGL(1,𝔽₅)	1	2	yes	—
213	C₂×C₃	C₄×#704	1	1	yes	characteristic
214	C₅	C₃×#1455	1	1	yes	characteristic, complemented, Sylow
215	C₂²	—	2	2	no	—
216	C₂²	—	2	1	no	—
217	C₂²	C₂×C₃×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	5	4	no	—
220	C₄	—	10	2	no	—
221	C₄	C₃×D₅⋊M₁₆	1	1	yes	characteristic
222	C₄	C₂×C₃×#704	1	1	yes	characteristic
223	C₃	#10604737	1	1	yes	characteristic, direct product factor, Sylow
224	C₂	—	4	1	no	—
225	C₂	C₃×M₁₆×AGL(1,𝔽₅)	1	2	yes	—
226	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×#1455
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. #10604737	C₃	direct (N × H)
21. #170⋊C₈	C₂×C₃	semidirect
22. #64879	C₂×C₃	semidirect
23. C₈×#170	C₂×C₃	semidirect
64. C₃×#170	M₁₆	semidirect
66. C₃×C₅×C₈	M₁₆	semidirect
68. C₃×#168	M₁₆	semidirect
130. #170	C₃×M₁₆	semidirect
132. C₅×C₈	C₃×M₁₆	semidirect
134. #168	C₃×M₁₆	semidirect
162. C₃×C₈	#704	semidirect
185. C₃×C₅	#1455	semidirect
210. C₈	C₃×#704	semidirect
214. C₅	C₃×#1455	semidirect
223. C₃	#10604737	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×#704	2
C₃×M₁₆×Dic₅	2
C₃×#170⋊C₈	2
C₃×#64879	2
C₃×C₈×#170	2
#10604737	3
C₃×#1455	5

Subgroups — C3×#10604737 #14280712

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10604737 #14280712