Subgroups — C₃×#10600373 #14305802

Order 1920.

872 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4020	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#384	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3941	C₂	1	4	yes	maximal, complemented
4	C₂×C₃×C₄×#168	C₂	1	1	yes	maximal, characteristic, complemented
5	#10600373	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×D₂₀	—	2	2	no	—
7	C₂²×C₃×D₂₀	C₂²	1	2	yes	—
8	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
9	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂²×C₃×#168	C₂²	1	2	yes	—
12	C₃×#4020	—	2	2	no	—
13	C₃×#4020	C₄	1	4	yes	complemented
14	C₃×#3941	—	2	8	no	—
15	C₃×C₄×#168	—	2	2	no	—
16	C₃×#1957	—	5	1	no	maximal, Hall
17	C₂×#4020	C₂×C₃	1	1	yes	characteristic
18	C₅×#384	C₂×C₃	1	1	yes	characteristic, complemented
19	C₂×#3941	C₂×C₃	1	4	yes	complemented
20	C₂×C₄×#168	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂³×C₃×D₅	—	2	2	no	—
22	C₂²×C₃×C₄×C₅	—	2	2	no	—
23	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
24	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂×C₃×D₂₀	—	2	4	no	—
26	C₂×C₃×D₂₀	—	2	4	no	—
27	C₂×C₃×D₂₀	—	2	8	no	—
28	C₂×C₃×D₂₀	—	2	2	no	—
29	C₂×C₃×D₂₀	—	4	4	no	—
30	C₂×C₃×D₂₀	—	4	4	no	—
31	C₂×C₃×D₂₀	C₂×C₄	1	4	yes	—
32	C₃×C₅×C₄⋊C₄	—	2	4	no	—
33	C₃×C₅×C₄⋊C₄	—	4	4	no	—
34	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
35	C₃×C₄²×C₅	D₄	1	2	yes	—
36	C₂×C₃×#168	—	2	2	no	—
37	C₂×C₃×#168	—	2	2	no	—
38	C₂×C₃×#168	D₄	1	4	yes	complemented
39	C₂×C₃×#127	—	5	1	no	—
40	C₂×C₃×#102	—	5	4	no	—
41	C₂×C₃×C₄×C₈	—	5	1	no	—
42	C₃×#384	D₅	1	1	yes	characteristic, complemented
43	C₂²×D₂₀	—	2	2	no	—
44	C₂²×D₂₀	C₂²×C₃	1	2	yes	—
45	C₂×C₅×C₄⋊C₄	—	2	2	no	—
46	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
47	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
48	C₂²×#168	C₂²×C₃	1	2	yes	—
49	#4020	—	2	2	no	—
50	#4020	C₃×C₄	1	4	yes	complemented
51	#3941	—	2	8	no	—
52	C₄×#168	—	2	2	no	—
53	#1957	—	5	1	no	Sylow
54	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₃×D₅	—	2	4	no	—
56	C₂²×C₃×D₅	—	4	2	no	—
57	C₂²×C₃×D₅	—	4	2	no	—
58	C₂²×C₃×D₅	—	4	4	no	—
59	C₂²×C₃×D₅	—	4	4	no	—
60	C₂×C₃×C₄×C₅	—	4	2	no	—
61	C₂×C₃×C₄×C₅	—	4	4	no	—
62	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
63	C₂×C₃×C₄×C₅	D₈	1	4	yes	—
64	C₂×C₃×C₄×C₅	SD₁₆	1	4	yes	—
65	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
67	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	#41	1	2	yes	—
69	C₃×D₂₀	—	2	8	no	—
70	C₃×D₂₀	—	4	16	no	—
71	C₃×D₂₀	—	4	4	no	—
72	C₃×#168	—	2	4	no	—
73	C₂²×C₃×D₄	—	5	2	no	—
74	C₂²×C₃×D₄	—	10	2	no	—
75	C₂²×C₃×C₈	—	5	2	no	—
76	C₃×#127	—	5	4	no	—
77	C₃×#127	—	10	2	no	—
78	C₂×C₃×C₄⋊C₄	—	2	2	no	—
79	C₂×C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
80	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
81	C₃×#102	—	10	8	no	—
82	C₃×C₄×C₈	—	10	2	no	—
83	C₂³×D₅	—	2	2	no	—
84	C₂²×C₄×C₅	—	2	2	no	—
85	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
86	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
87	C₂×D₂₀	—	2	4	no	—
88	C₂×D₂₀	—	2	4	no	—
89	C₂×D₂₀	—	2	8	no	—
90	C₂×D₂₀	—	2	2	no	—
91	C₂×D₂₀	—	4	4	no	—
92	C₂×D₂₀	—	4	4	no	—
93	C₂×D₂₀	C₂×C₃×C₄	1	4	yes	—
94	C₅×C₄⋊C₄	—	2	4	no	—
95	C₅×C₄⋊C₄	—	4	4	no	—
96	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
97	C₄²×C₅	C₃×D₄	1	2	yes	—
98	C₂×#168	—	2	2	no	—
99	C₂×#168	—	2	2	no	—
100	C₂×#168	C₃×D₄	1	4	yes	complemented
101	C₂×#127	—	5	1	no	—
102	C₂×#102	—	5	4	no	—
103	C₂×C₄×C₈	—	5	1	no	—
104	#384	C₃×D₅	1	1	yes	characteristic, complemented
105	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂²×C₃×C₅	#124	1	1	yes	characteristic
107	C₂²×C₃×C₅	#127	1	1	yes	characteristic
108	C₂²×C₃×C₅	C₂×D₈	1	2	yes	—
109	C₂²×C₃×C₅	C₂×SD₁₆	1	2	yes	—
110	C₂×C₃×D₅	—	4	4	no	—
111	C₂×C₃×D₅	—	4	8	no	—
112	C₂×C₃×D₅	—	8	4	no	—
113	C₂×C₃×D₅	—	8	2	no	—
114	C₂×C₃×D₅	—	8	2	no	—
115	C₃×C₄×C₅	—	8	2	no	—
116	C₃×C₄×C₅	#102	1	8	yes	complemented
117	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
118	C₃×C₄×C₅	#121	1	2	yes	—
119	C₂⁴×C₃	—	10	2	no	—
120	C₂×C₃×D₄	—	5	4	no	—
121	C₂×C₃×D₄	—	10	8	no	—
122	C₂×C₃×D₄	—	10	2	no	—
123	C₂×C₃×D₄	—	10	4	no	—
124	C₂×C₃×D₄	—	10	4	no	—
125	C₂×C₃×D₄	—	20	4	no	—
126	C₂×C₃×D₄	—	20	4	no	—
127	C₂²×C₃×C₄	—	2	2	no	—
128	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
129	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
130	C₂×C₃×C₈	—	5	4	no	—
131	C₂×C₃×C₈	—	10	2	no	—
132	C₂×C₃×C₈	—	10	2	no	—
133	C₃×C₄⋊C₄	—	2	4	no	—
134	C₃×C₄⋊C₄	—	4	4	no	—
135	C₃×C₄²	C₄×D₅	1	2	yes	—
136	C₃×C₄²	D₂₀	1	2	yes	—
137	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
138	C₂²×D₅	—	2	4	no	—
139	C₂²×D₅	—	4	2	no	—
140	C₂²×D₅	—	4	2	no	—
141	C₂²×D₅	—	4	4	no	—
142	C₂²×D₅	—	4	4	no	—
143	C₂×C₄×C₅	—	4	2	no	—
144	C₂×C₄×C₅	—	4	4	no	—
145	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	4	yes	—
146	C₂×C₄×C₅	C₃×D₈	1	4	yes	—
147	C₂×C₄×C₅	C₃×SD₁₆	1	4	yes	—
148	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
151	C₂×C₄×C₅	C₃×#41	1	2	yes	—
152	D₂₀	—	2	8	no	—
153	D₂₀	—	4	16	no	—
154	D₂₀	—	4	4	no	—
155	#168	—	2	4	no	—
156	C₂²×D₄	—	5	2	no	—
157	C₂²×D₄	—	10	2	no	—
158	C₂²×C₈	—	5	2	no	—
159	#127	—	5	4	no	—
160	#127	—	10	2	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
163	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
164	#102	—	10	8	no	—
165	C₄×C₈	—	10	2	no	—
166	C₂×C₃×C₅	#390	1	1	yes	characteristic
167	C₂×C₃×C₅	C₂×#102	1	2	yes	—
168	C₂×C₃×C₅	#486	1	2	yes	—
169	C₂×C₃×C₅	#492	1	1	yes	characteristic
170	C₂×C₃×C₅	#493	1	1	yes	characteristic
171	C₃×D₅	—	8	4	no	—
172	C₂³×C₃	—	10	4	no	—
173	C₂³×C₃	—	20	4	no	—
174	C₂³×C₃	—	20	2	no	—
175	C₂³×C₃	—	20	4	no	—
176	C₂³×C₃	—	20	2	no	—
177	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
178	C₃×D₄	—	10	8	no	—
179	C₃×D₄	—	20	16	no	—
180	C₃×D₄	—	20	4	no	—
181	C₂×C₃×C₄	—	4	4	no	—
182	C₂×C₃×C₄	—	4	2	no	—
183	C₂×C₃×C₄	#685	1	4	yes	—
184	C₂×C₃×C₄	#686	1	4	yes	—
185	C₂×C₃×C₄	#688	1	4	yes	—
186	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
188	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
189	C₂×C₃×C₄	#713	1	2	yes	—
190	C₃×C₈	—	10	4	no	—
191	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
192	C₂²×C₅	C₃×#124	1	1	yes	characteristic
193	C₂²×C₅	C₃×#127	1	1	yes	characteristic
194	C₂²×C₅	C₂×C₃×D₈	1	2	yes	—
195	C₂²×C₅	C₂×C₃×SD₁₆	1	2	yes	—
196	C₂×D₅	—	4	4	no	—
197	C₂×D₅	—	4	8	no	—
198	C₂×D₅	—	8	4	no	—
199	C₂×D₅	—	8	2	no	—
200	C₂×D₅	—	8	2	no	—
201	C₄×C₅	—	8	2	no	—
202	C₄×C₅	C₃×#102	1	8	yes	complemented
203	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
204	C₄×C₅	C₃×#121	1	2	yes	—
205	C₂⁴	—	10	2	no	—
206	C₂×D₄	—	5	4	no	—
207	C₂×D₄	—	10	8	no	—
208	C₂×D₄	—	10	2	no	—
209	C₂×D₄	—	10	4	no	—
210	C₂×D₄	—	10	4	no	—
211	C₂×D₄	—	20	4	no	—
212	C₂×D₄	—	20	4	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	C₃×C₂²⋊D₅	1	2	yes	—
215	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
216	C₂×C₈	—	5	4	no	—
217	C₂×C₈	—	10	2	no	—
218	C₂×C₈	—	10	2	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	4	4	no	—
221	C₄²	C₃×C₄×D₅	1	2	yes	—
222	C₄²	C₃×D₂₀	1	2	yes	—
223	C₃×C₅	#1957	1	1	yes	characteristic, complemented, Hall
224	C₂²×C₃	—	20	8	no	—
225	C₂²×C₃	—	20	4	no	—
226	C₂²×C₃	—	40	2	no	—
227	C₂²×C₃	—	40	2	no	—
228	C₂²×C₃	—	40	4	no	—
229	C₂²×C₃	C₂×#685	1	1	yes	characteristic
230	C₂²×C₃	C₂×#686	1	2	yes	—
231	C₂²×C₃	#4086	1	1	yes	characteristic
232	C₂²×C₃	C₂×#688	1	2	yes	—
233	C₂²×C₃	#4093	1	1	yes	characteristic
234	C₃×C₄	—	8	2	no	—
235	C₃×C₄	#3941	1	8	yes	complemented
236	C₃×C₄	#4039	1	2	yes	—
237	C₃×C₄	#4041	1	2	yes	—
238	C₂×C₅	C₃×#390	1	1	yes	characteristic
239	C₂×C₅	C₂×C₃×#102	1	2	yes	—
240	C₂×C₅	C₃×#486	1	2	yes	—
241	C₂×C₅	C₃×#492	1	1	yes	characteristic
242	C₂×C₅	C₃×#493	1	1	yes	characteristic
243	D₅	—	8	4	no	—
244	C₂³	—	10	4	no	—
245	C₂³	—	20	4	no	—
246	C₂³	—	20	2	no	—
247	C₂³	—	20	4	no	—
248	C₂³	—	20	2	no	—
249	C₂³	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
250	D₄	—	10	8	no	—
251	D₄	—	20	16	no	—
252	D₄	—	20	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	C₃×#685	1	4	yes	—
256	C₂×C₄	C₃×#686	1	4	yes	—
257	C₂×C₄	C₃×#688	1	4	yes	—
258	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
259	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
260	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
261	C₂×C₄	C₃×#713	1	2	yes	—
262	C₈	—	10	4	no	—
263	C₂×C₃	—	40	4	no	—
264	C₂×C₃	C₂×#3941	1	2	yes	—
265	C₂×C₃	#65277	1	1	yes	characteristic
266	C₂×C₃	#65362	1	1	yes	characteristic
267	C₂×C₃	#65374	1	1	yes	characteristic
268	C₂×C₃	#65377	1	2	yes	—
269	C₅	C₃×#1957	1	1	yes	characteristic, complemented, Sylow
270	C₂²	—	20	8	no	—
271	C₂²	—	20	4	no	—
272	C₂²	—	40	2	no	—
273	C₂²	—	40	2	no	—
274	C₂²	—	40	4	no	—
275	C₂²	C₂×C₃×#685	1	1	yes	characteristic
276	C₂²	C₂×C₃×#686	1	2	yes	—
277	C₂²	C₃×#4086	1	1	yes	characteristic
278	C₂²	C₂×C₃×#688	1	2	yes	—
279	C₂²	C₃×#4093	1	1	yes	characteristic
280	C₄	—	8	2	no	—
281	C₄	C₃×#3941	1	8	yes	complemented
282	C₄	C₃×#4039	1	2	yes	—
283	C₄	C₃×#4041	1	2	yes	—
284	C₃	#10600373	1	1	yes	characteristic, direct product factor, Sylow
285	C₂	—	40	4	no	—
286	C₂	C₂×C₃×#3941	1	2	yes	—
287	C₂	C₃×#65277	1	1	yes	characteristic
288	C₂	C₃×#65362	1	1	yes	characteristic
289	C₂	C₃×#65374	1	1	yes	characteristic
290	C₂	C₃×#65377	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂×C₂²⋊D₅
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₃×C₅×#384	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂×C₃×#4020	C₂
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₃×#384	D₅
hypocenter	C₅	C₃×#1957
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₃×#384

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#384	C₂	semidirect
3. C₂×C₃×#3941	C₂	semidirect
4. C₂×C₃×C₄×#168	C₂	semidirect
5. #10600373	C₃	direct (N × H)
13. C₃×#4020	C₄	semidirect
18. C₅×#384	C₂×C₃	semidirect
19. C₂×#3941	C₂×C₃	semidirect
20. C₂×C₄×#168	C₂×C₃	semidirect
38. C₂×C₃×#168	D₄	semidirect
42. C₃×#384	D₅	semidirect
50. #4020	C₃×C₄	semidirect
100. C₂×#168	C₃×D₄	semidirect
104. #384	C₃×D₅	semidirect
116. C₃×C₄×C₅	#102	semidirect
202. C₄×C₅	C₃×#102	semidirect
223. C₃×C₅	#1957	semidirect
235. C₃×C₄	#3941	semidirect
269. C₅	C₃×#1957	semidirect
281. C₄	C₃×#3941	semidirect
284. C₃	#10600373	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×#4020	2
C₃×C₅×#384	2
C₂×C₃×#3941	2
C₂×C₃×C₄×#168	2
#10600373	3
C₃×#1957	5

Subgroups — C3×#10600373 #14305802

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10600373 #14305802