Subgroups — C₃×#10601643 #14291491

Order 1920.

344 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×C₅×C₈×D₄	C₂	1	1	yes	maximal, characteristic
2	C₃×#65296	C₂	1	2	yes	maximal
3	C₃×#64988	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64702	C₂	1	2	yes	maximal, complemented
5	C₃×#64678	C₂	1	1	yes	maximal, characteristic, complemented
6	#10601643	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×C₈	—	2	1	no	—
8	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
9	C₂×C₃×Dic₂₀	—	2	1	no	—
10	C₂×C₃×Dic₂₀	—	2	1	no	—
11	C₃×#4015	C₂²	1	2	yes	—
12	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
14	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₃×#3964	—	2	2	no	—
16	C₃×#3939	—	2	2	no	—
17	C₃×C₄⋊#168	C₂²	1	2	yes	—
18	C₃×#1669	—	5	1	no	maximal, Hall
19	C₅×C₈×D₄	C₂×C₃	1	1	yes	characteristic
20	#65296	C₂×C₃	1	2	yes	—
21	#64988	C₂×C₃	1	1	yes	characteristic, complemented
22	#64702	C₂×C₃	1	2	yes	complemented
23	#64678	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×C₃×C₅×D₄	D₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	—	2	1	no	—
26	C₂×C₃×Dic₁₀	—	2	2	no	—
27	C₂×C₃×C₅×C₈	—	2	1	no	—
28	C₂×C₃×C₅×C₈	—	4	2	no	—
29	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
30	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
31	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
32	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
33	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
34	C₃×C₄⋊Dic₅	—	2	2	no	—
35	C₃×C₄⋊Dic₅	—	2	2	no	—
36	C₂×C₃×#168	—	2	2	no	—
37	C₃×Dic₂₀	—	4	2	no	—
38	C₃×Dic₂₀	—	4	2	no	—
39	C₃×#494	—	5	1	no	—
40	C₃×#474	—	5	2	no	—
41	C₃×C₈×D₄	D₅	1	1	yes	characteristic
42	C₃×#335	—	5	1	no	—
43	C₃×#332	—	5	2	no	—
44	C₂²×C₅×C₈	—	2	1	no	—
45	C₄×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
46	C₂×Dic₂₀	—	2	1	no	—
47	C₂×Dic₂₀	—	2	1	no	—
48	#4015	C₂²×C₃	1	2	yes	—
49	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
50	C₅×C₂²⋊C₈	—	2	1	no	—
51	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
52	#3964	—	2	2	no	—
53	#3939	—	2	2	no	—
54	C₄⋊#168	C₂²×C₃	1	2	yes	—
55	#1669	—	5	1	no	Sylow
56	C₂³×C₃×C₅	—	2	1	no	—
57	C₃×C₅×D₄	—	2	1	no	—
58	C₃×C₅×D₄	Q₁₆	1	2	yes	complemented
59	C₂×C₃×C₄×C₅	—	2	1	no	—
60	C₂×C₃×C₄×C₅	—	4	1	no	—
61	C₂×C₃×C₄×C₅	—	4	1	no	—
62	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
65	C₂×C₃×Dic₅	—	4	2	no	—
66	C₃×Dic₁₀	—	4	2	no	—
67	C₃×Dic₁₀	—	4	2	no	—
68	C₃×C₅×C₈	—	2	1	no	—
69	C₃×C₅×C₈	—	4	1	no	—
70	C₃×C₅×C₈	D₈	1	2	yes	—
71	C₃×#168	—	4	2	no	—
72	C₂×C₃×Q₁₆	—	10	1	no	—
73	C₂×C₃×Q₁₆	—	10	1	no	—
74	C₂²×C₃×C₈	—	2	1	no	—
75	C₃×C₄⋊Q₈	—	5	2	no	—
76	C₃×C₄×D₄	C₂×D₅	1	1	yes	characteristic
77	C₃×#107	—	10	2	no	—
78	C₃×C₄⋊C₈	—	5	2	no	—
79	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
80	C₃×#102	—	10	2	no	—
81	C₃×C₂²⋊C₈	—	2	1	no	—
82	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
83	C₂×C₅×D₄	C₃×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	—	2	1	no	—
85	C₂×Dic₁₀	—	2	2	no	—
86	C₂×C₅×C₈	—	2	1	no	—
87	C₂×C₅×C₈	—	4	2	no	—
88	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
89	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
90	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
91	C₅×C₂²⋊C₄	—	2	1	no	—
92	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
93	C₄⋊Dic₅	—	2	2	no	—
94	C₄⋊Dic₅	—	2	2	no	—
95	C₂×#168	—	2	2	no	—
96	Dic₂₀	—	4	2	no	—
97	Dic₂₀	—	4	2	no	—
98	#494	—	5	1	no	—
99	#474	—	5	2	no	—
100	C₈×D₄	C₃×D₅	1	1	yes	characteristic
101	#335	—	5	1	no	—
102	#332	—	5	2	no	—
103	C₂²×C₃×C₅	—	2	2	no	—
104	C₂²×C₃×C₅	—	4	1	no	—
105	C₂²×C₃×C₅	—	4	1	no	—
106	C₂²×C₃×C₅	#121	1	1	yes	characteristic
107	C₃×C₄×C₅	—	2	1	no	—
108	C₃×C₄×C₅	—	4	1	no	—
109	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
110	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
111	C₃×C₄×C₅	#135	1	1	yes	characteristic
112	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₃×Dic₅	—	8	2	no	—
114	C₂×C₃×Q₈	—	10	2	no	—
115	C₂×C₃×D₄	D₂₀	1	1	yes	characteristic
116	C₂²×C₃×C₄	—	2	1	no	—
117	C₃×Q₁₆	—	20	2	no	—
118	C₃×Q₁₆	—	20	2	no	—
119	C₂×C₃×C₈	—	2	1	no	—
120	C₂×C₃×C₈	—	4	2	no	—
121	C₂×C₃×C₈	—	10	2	no	—
122	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
123	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
124	C₃×C₄⋊C₄	—	10	2	no	—
125	C₃×C₄⋊C₄	—	10	2	no	—
126	C₃×C₄⋊C₄	D₂₀	1	1	yes	characteristic
127	C₃×C₂²⋊C₄	—	2	1	no	—
128	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
129	C₂³×C₅	—	2	1	no	—
130	C₅×D₄	—	2	1	no	—
131	C₅×D₄	C₃×Q₁₆	1	2	yes	complemented
132	C₂×C₄×C₅	—	2	1	no	—
133	C₂×C₄×C₅	—	4	1	no	—
134	C₂×C₄×C₅	—	4	1	no	—
135	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
138	C₂×Dic₅	—	4	2	no	—
139	Dic₁₀	—	4	2	no	—
140	Dic₁₀	—	4	2	no	—
141	C₅×C₈	—	2	1	no	—
142	C₅×C₈	—	4	1	no	—
143	C₅×C₈	C₃×D₈	1	2	yes	—
144	#168	—	4	2	no	—
145	C₂×Q₁₆	—	10	1	no	—
146	C₂×Q₁₆	—	10	1	no	—
147	C₂²×C₈	—	2	1	no	—
148	C₄⋊Q₈	—	5	2	no	—
149	C₄×D₄	C₂×C₃×D₅	1	1	yes	characteristic
150	#107	—	10	2	no	—
151	C₄⋊C₈	—	5	2	no	—
152	C₄⋊C₈	C₂×C₃×D₅	1	1	yes	characteristic
153	#102	—	10	2	no	—
154	C₂²⋊C₈	—	2	1	no	—
155	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
156	C₂×C₃×C₅	—	4	2	no	—
157	C₂×C₃×C₅	#459	1	1	yes	characteristic
158	C₂×C₃×C₅	#467	1	1	yes	characteristic
159	C₂×C₃×C₅	#473	1	1	yes	characteristic
160	C₂³×C₃	—	2	1	no	—
161	C₃×Q₈	—	20	2	no	—
162	C₃×Q₈	—	20	2	no	—
163	C₃×D₄	—	2	1	no	—
164	C₃×D₄	Dic₂₀	1	2	yes	complemented
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	4	1	no	—
167	C₂×C₃×C₄	—	4	1	no	—
168	C₂×C₃×C₄	—	20	2	no	—
169	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
170	C₂×C₃×C₄	#709	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
172	C₃×C₈	—	2	1	no	—
173	C₃×C₈	—	4	1	no	—
174	C₃×C₈	—	20	2	no	—
175	C₃×C₈	#686	1	2	yes	—
176	C₂²×C₅	—	2	2	no	—
177	C₂²×C₅	—	4	1	no	—
178	C₂²×C₅	—	4	1	no	—
179	C₂²×C₅	C₃×#121	1	1	yes	characteristic
180	C₄×C₅	—	2	1	no	—
181	C₄×C₅	—	4	1	no	—
182	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
183	C₄×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
184	C₄×C₅	C₃×#135	1	1	yes	characteristic
185	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
186	Dic₅	—	8	2	no	—
187	C₂×Q₈	—	10	2	no	—
188	C₂×D₄	C₃×D₂₀	1	1	yes	characteristic
189	C₂²×C₄	—	2	1	no	—
190	Q₁₆	—	20	2	no	—
191	Q₁₆	—	20	2	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	10	2	no	—
195	C₂×C₈	C₃×C₂²⋊D₅	1	1	yes	characteristic
196	C₂×C₈	C₃×C₂²⋊D₅	1	1	yes	characteristic
197	C₄⋊C₄	—	10	2	no	—
198	C₄⋊C₄	—	10	2	no	—
199	C₄⋊C₄	C₃×D₂₀	1	1	yes	characteristic
200	C₂²⋊C₄	—	2	1	no	—
201	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
202	C₃×C₅	#1669	1	1	yes	characteristic, complemented, Hall
203	C₂²×C₃	—	2	2	no	—
204	C₂²×C₃	—	4	1	no	—
205	C₂²×C₃	—	4	1	no	—
206	C₂²×C₃	#4076	1	1	yes	characteristic
207	C₃×C₄	—	2	1	no	—
208	C₃×C₄	—	4	1	no	—
209	C₃×C₄	—	40	2	no	—
210	C₃×C₄	#4050	1	1	yes	characteristic
211	C₃×C₄	C₂×Dic₂₀	1	1	yes	characteristic
212	C₃×C₄	C₂×#686	1	1	yes	characteristic
213	C₃×C₄	#4095	1	1	yes	characteristic
214	C₂×C₅	—	4	2	no	—
215	C₂×C₅	C₃×#459	1	1	yes	characteristic
216	C₂×C₅	C₃×#467	1	1	yes	characteristic
217	C₂×C₅	C₃×#473	1	1	yes	characteristic
218	C₂³	—	2	1	no	—
219	Q₈	—	20	2	no	—
220	Q₈	—	20	2	no	—
221	D₄	—	2	1	no	—
222	D₄	C₃×Dic₂₀	1	2	yes	complemented
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	20	2	no	—
227	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
228	C₂×C₄	C₃×#709	1	1	yes	characteristic
229	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
230	C₈	—	2	1	no	—
231	C₈	—	4	1	no	—
232	C₈	—	20	2	no	—
233	C₈	C₃×#686	1	2	yes	—
234	C₂×C₃	—	4	2	no	—
235	C₂×C₃	#65304	1	1	yes	characteristic
236	C₂×C₃	#65405	1	1	yes	characteristic
237	C₂×C₃	#65432	1	1	yes	characteristic
238	C₅	C₃×#1669	1	1	yes	characteristic, complemented, Sylow
239	C₂²	—	2	2	no	—
240	C₂²	—	4	1	no	—
241	C₂²	—	4	1	no	—
242	C₂²	C₃×#4076	1	1	yes	characteristic
243	C₄	—	2	1	no	—
244	C₄	—	4	1	no	—
245	C₄	—	40	2	no	—
246	C₄	C₃×#4050	1	1	yes	characteristic
247	C₄	C₂×C₃×Dic₂₀	1	1	yes	characteristic
248	C₄	C₂×C₃×#686	1	1	yes	characteristic
249	C₄	C₃×#4095	1	1	yes	characteristic
250	C₃	#10601643	1	1	yes	characteristic, direct product factor, Sylow
251	C₂	—	4	2	no	—
252	C₂	C₃×#65304	1	1	yes	characteristic
253	C₂	C₃×#65405	1	1	yes	characteristic
254	C₂	C₃×#65432	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	#4076
commutator, cosocle	C₄²×C₅	C₂³×C₃
Frattini	C₄²	C₂²×C₃×D₅
Fitting	C₃×C₅×C₈×D₄	C₂
socle	C₂²×C₃×C₅	#121
prime radical	C₂×C₃×C₅×D₄	D₄
hypercenter	C₃×C₈×D₄	D₅
hypocenter	C₅	C₃×#1669
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₈×D₄

Complements

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

N	H ≅ G/N	type
3. C₃×#64988	C₂	semidirect
4. C₃×#64702	C₂	semidirect
5. C₃×#64678	C₂	semidirect
6. #10601643	C₃	direct (N × H)
21. #64988	C₂×C₃	semidirect
22. #64702	C₂×C₃	semidirect
23. #64678	C₂×C₃	semidirect
58. C₃×C₅×D₄	Q₁₆	semidirect
131. C₅×D₄	C₃×Q₁₆	semidirect
164. C₃×D₄	Dic₂₀	semidirect
202. C₃×C₅	#1669	semidirect
222. D₄	C₃×Dic₂₀	semidirect
238. C₅	C₃×#1669	semidirect
250. C₃	#10601643	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×C₅×C₈×D₄	2
C₃×#65296	2
C₃×#64988	2
C₃×#64702	2
C₃×#64678	2
#10601643	3
C₃×#1669	5

Subgroups — C3×#10601643 #14291491

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10601643 #14291491