Subgroups — C₃×#10603815 #14285556

Order 1920.

360 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#65680	C₂	1	2	yes	maximal
2	C₃×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
3	C₃×#64883	C₂	1	2	yes	maximal, complemented
4	C₃×#64882	C₂	1	2	yes	maximal, complemented
5	#10603815	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₈×D₅	C₄	1	1	yes	characteristic
7	C₂×C₃×C₈×D₅	C₄	1	1	yes	characteristic
8	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
9	C₃×#4003	—	2	2	no	—
10	C₃×C₄⋊#170	C₂²	1	4	yes	—
11	C₃×C₄×#170	—	2	2	no	—
12	C₃×C₄×C₅×C₈	C₄	1	1	yes	characteristic
13	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
14	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
15	C₃×C₄×#168	C₄	1	1	yes	characteristic
16	C₃×#1564	—	5	1	no	maximal, Hall
17	#65680	C₂×C₃	1	2	yes	—
18	C₄×C₈×D₅	C₂×C₃	1	1	yes	characteristic
19	#64883	C₂×C₃	1	2	yes	complemented
20	#64882	C₂×C₃	1	2	yes	complemented
21	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
23	C₂×C₃×C₄×D₅	Q₈	1	1	yes	characteristic
24	C₂×C₃×#170	—	2	4	no	—
25	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
28	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
29	C₃×C₄×Dic₅	Q₈	1	1	yes	characteristic
30	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
31	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
33	C₃×C₈×D₅	—	2	2	no	—
34	C₃×C₈×D₅	C₈	1	4	yes	complemented
35	C₃×#431	—	5	2	no	—
36	C₂×C₃×C₄×C₈	—	5	1	no	—
37	C₃×#335	—	5	2	no	—
38	C₃×#334	—	5	2	no	—
39	C₂×C₈×D₅	C₃×C₄	1	1	yes	characteristic
40	C₂×C₈×D₅	C₃×C₄	1	1	yes	characteristic
41	C₄²×D₅	C₂²×C₃	1	1	yes	characteristic
42	#4003	—	2	2	no	—
43	C₄⋊#170	C₂²×C₃	1	4	yes	—
44	C₄×#170	—	2	2	no	—
45	C₄×C₅×C₈	C₃×C₄	1	1	yes	characteristic
46	C₈×Dic₅	C₂²×C₃	1	1	yes	characteristic
47	C₈×Dic₅	C₂²×C₃	1	1	yes	characteristic
48	C₄×#168	C₃×C₄	1	1	yes	characteristic
49	#1564	—	5	1	no	Sylow
50	C₂²×C₃×D₅	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
56	C₂×C₃×Dic₅	C₂×Q₈	1	1	yes	characteristic
57	C₃×C₄×D₅	—	2	2	no	—
58	C₃×C₄×D₅	—	2	2	no	—
59	C₃×C₄×D₅	C₂×C₈	1	2	yes	—
60	C₃×C₄×D₅	M₁₆	1	2	yes	—
61	C₃×#170	—	4	4	no	—
62	C₃×C₅×C₈	—	2	1	no	—
63	C₃×C₅×C₈	C₂×C₈	1	2	yes	—
64	C₃×#168	—	2	1	no	—
65	C₃×#168	C₂×C₈	1	2	yes	—
66	C₂²×C₃×C₈	—	5	1	no	—
67	C₂²×C₃×C₈	—	5	1	no	—
68	C₂×C₃×C₄²	—	5	1	no	—
69	C₃×C₄⋊C₈	—	5	4	no	—
70	C₃×C₂²⋊C₈	—	10	2	no	—
71	C₃×C₄×C₈	—	5	1	no	—
72	C₃×C₄×C₈	—	5	1	no	—
73	C₃×C₄×C₈	—	5	1	no	—
74	C₃×C₄×C₈	—	10	2	no	—
75	C₃×C₄×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
76	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
77	C₂×C₄×D₅	C₃×D₄	1	1	yes	characteristic
78	C₂×C₄×D₅	C₃×Q₈	1	1	yes	characteristic
79	C₂×#170	—	2	4	no	—
80	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
81	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
82	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
83	C₄×Dic₅	C₃×D₄	1	1	yes	characteristic
84	C₄×Dic₅	C₃×Q₈	1	1	yes	characteristic
85	C₄×Dic₅	C₂³×C₃	1	1	yes	characteristic
86	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
87	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
88	C₈×D₅	—	2	2	no	—
89	C₈×D₅	C₃×C₈	1	4	yes	complemented
90	#431	—	5	2	no	—
91	C₂×C₄×C₈	—	5	1	no	—
92	#335	—	5	2	no	—
93	#334	—	5	2	no	—
94	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₃×D₅	—	2	1	no	—
96	C₂×C₃×D₅	—	2	1	no	—
97	C₂×C₃×D₅	C₄⋊C₈	1	2	yes	—
98	C₃×C₄×C₅	—	2	1	no	—
99	C₃×C₄×C₅	#108	1	2	yes	—
100	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
101	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
102	C₃×Dic₅	—	2	1	no	—
103	C₃×Dic₅	C₄⋊C₈	1	2	yes	—
104	C₃×Dic₅	#135	1	2	yes	—
105	C₂²×C₃×C₄	—	5	1	no	—
106	C₂²×C₃×C₄	—	5	1	no	—
107	C₂²×C₃×C₄	—	5	1	no	—
108	C₂×C₃×C₈	—	5	1	no	—
109	C₂×C₃×C₈	—	5	4	no	—
110	C₂×C₃×C₈	—	5	1	no	—
111	C₂×C₃×C₈	—	10	2	no	—
112	C₂×C₃×C₈	—	10	4	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	1	no	—
116	C₃×C₄²	—	5	1	no	—
117	C₃×C₄²	—	5	1	no	—
118	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
119	C₂²×D₅	C₃×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
123	C₂×Dic₅	C₃×C₄⋊C₄	1	1	yes	characteristic
124	C₂×Dic₅	C₂×C₃×D₄	1	1	yes	characteristic
125	C₂×Dic₅	C₂×C₃×Q₈	1	1	yes	characteristic
126	C₄×D₅	—	2	2	no	—
127	C₄×D₅	—	2	2	no	—
128	C₄×D₅	C₂×C₃×C₈	1	2	yes	—
129	C₄×D₅	C₃×M₁₆	1	2	yes	—
130	#170	—	4	4	no	—
131	C₅×C₈	—	2	1	no	—
132	C₅×C₈	C₂×C₃×C₈	1	2	yes	—
133	#168	—	2	1	no	—
134	#168	C₂×C₃×C₈	1	2	yes	—
135	C₂²×C₈	—	5	1	no	—
136	C₂²×C₈	—	5	1	no	—
137	C₂×C₄²	—	5	1	no	—
138	C₄⋊C₈	—	5	4	no	—
139	C₂²⋊C₈	—	10	2	no	—
140	C₄×C₈	—	5	1	no	—
141	C₄×C₈	—	5	1	no	—
142	C₄×C₈	—	5	1	no	—
143	C₄×C₈	—	10	2	no	—
144	C₄×C₈	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
145	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
146	C₂×C₃×C₅	#427	1	1	yes	characteristic
147	C₂×C₃×C₅	C₂×#108	1	1	yes	characteristic
148	C₃×D₅	—	2	2	no	—
149	C₂³×C₃	—	5	1	no	—
150	C₂×C₃×C₄	—	5	1	no	—
151	C₂×C₃×C₄	—	5	1	no	—
152	C₂×C₃×C₄	—	5	2	no	—
153	C₂×C₃×C₄	—	5	2	no	—
154	C₂×C₃×C₄	—	5	1	no	—
155	C₂×C₃×C₄	—	10	2	no	—
156	C₂×C₃×C₄	—	10	2	no	—
157	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
158	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
159	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
160	C₃×C₈	—	2	1	no	—
161	C₃×C₈	—	5	2	no	—
162	C₃×C₈	—	10	1	no	—
163	C₃×C₈	—	20	4	no	—
164	C₃×C₈	D₅⋊C₈	1	2	yes	—
165	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
166	C₂×D₅	—	2	1	no	—
167	C₂×D₅	—	2	1	no	—
168	C₂×D₅	C₃×C₄⋊C₈	1	2	yes	—
169	C₄×C₅	—	2	1	no	—
170	C₄×C₅	C₃×#108	1	2	yes	—
171	C₄×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
172	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
173	Dic₅	—	2	1	no	—
174	Dic₅	C₃×C₄⋊C₈	1	2	yes	—
175	Dic₅	C₃×#135	1	2	yes	—
176	C₂²×C₄	—	5	1	no	—
177	C₂²×C₄	—	5	1	no	—
178	C₂²×C₄	—	5	1	no	—
179	C₂×C₈	—	5	1	no	—
180	C₂×C₈	—	5	4	no	—
181	C₂×C₈	—	5	1	no	—
182	C₂×C₈	—	10	2	no	—
183	C₂×C₈	—	10	4	no	—
184	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
185	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
186	C₄²	—	5	1	no	—
187	C₄²	—	5	1	no	—
188	C₄²	—	5	1	no	—
189	C₄²	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
190	C₃×C₅	#1564	1	1	yes	characteristic, complemented, Hall
191	C₂²×C₃	—	5	2	no	—
192	C₂²×C₃	—	10	1	no	—
193	C₂²×C₃	—	10	1	no	—
194	C₂²×C₃	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
195	C₃×C₄	—	2	1	no	—
196	C₃×C₄	—	5	2	no	—
197	C₃×C₄	—	5	2	no	—
198	C₃×C₄	—	10	1	no	—
199	C₃×C₄	#3996	1	2	yes	—
200	C₃×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
201	C₃×C₄	D₅⋊M₁₆	1	1	yes	characteristic
202	C₂×C₅	C₂×C₃×C₄⋊C₈	1	1	yes	characteristic
203	C₂×C₅	C₃×#427	1	1	yes	characteristic
204	C₂×C₅	C₂×C₃×#108	1	1	yes	characteristic
205	D₅	—	2	2	no	—
206	C₂³	—	5	1	no	—
207	C₂×C₄	—	5	1	no	—
208	C₂×C₄	—	5	1	no	—
209	C₂×C₄	—	5	2	no	—
210	C₂×C₄	—	5	2	no	—
211	C₂×C₄	—	5	1	no	—
212	C₂×C₄	—	10	2	no	—
213	C₂×C₄	—	10	2	no	—
214	C₂×C₄	C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
215	C₂×C₄	C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
216	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	5	2	no	—
219	C₈	—	10	1	no	—
220	C₈	—	20	4	no	—
221	C₈	C₃×D₅⋊C₈	1	2	yes	—
222	C₂×C₃	—	10	2	no	—
223	C₂×C₃	D₅⋊(C₄⋊C₈)	1	1	yes	characteristic
224	C₂×C₃	#65721	1	1	yes	characteristic
225	C₂×C₃	C₂×#3996	1	1	yes	characteristic
226	C₅	C₃×#1564	1	1	yes	characteristic, complemented, Sylow
227	C₂²	—	5	2	no	—
228	C₂²	—	10	1	no	—
229	C₂²	—	10	1	no	—
230	C₂²	C₂×C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
231	C₄	—	2	1	no	—
232	C₄	—	5	2	no	—
233	C₄	—	5	2	no	—
234	C₄	—	10	1	no	—
235	C₄	C₃×#3996	1	2	yes	—
236	C₄	C₂×C₃×D₅⋊C₈	1	1	yes	characteristic
237	C₄	C₃×D₅⋊M₁₆	1	1	yes	characteristic
238	C₃	#10603815	1	1	yes	characteristic, direct product factor, Sylow
239	C₂	—	10	2	no	—
240	C₂	C₃×D₅⋊(C₄⋊C₈)	1	1	yes	characteristic
241	C₂	C₃×#65721	1	1	yes	characteristic
242	C₂	C₂×C₃×#3996	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)
commutator	C₄×C₅	C₂²×C₃×C₈
Frattini	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₄×C₅×C₈	C₄
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂²×C₃×D₅	C₄⋊C₄
cosocle	C₄×Dic₅	C₂³×C₃
hypercenter	C₃×C₄×C₈	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#1564
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄² ◁ C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₃×#64883	C₂	semidirect
4. C₃×#64882	C₂	semidirect
5. #10603815	C₃	direct (N × H)
19. #64883	C₂×C₃	semidirect
20. #64882	C₂×C₃	semidirect
34. C₃×C₈×D₅	C₈	semidirect
89. C₈×D₅	C₃×C₈	semidirect
190. C₃×C₅	#1564	semidirect
226. C₅	C₃×#1564	semidirect
238. C₃	#10603815	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×#65680	2
C₃×C₄×C₈×D₅	2
C₃×#64883	2
C₃×#64882	2
#10603815	3
C₃×#1564	5

Subgroups — C3×#10603815 #14285556

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10603815 #14285556