Subgroups — C₃×#10602230 #14313003

Order 1920.

440 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4047	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3944	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64728	C₂	1	4	yes	maximal, complemented
5	#10602230	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×#709	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×#681	C₄	1	1	yes	characteristic
9	C₃×#4047	—	2	1	no	—
10	C₃×#4047	—	2	1	no	—
11	C₃×#4047	C₄	1	2	yes	—
12	C₃×#4047	C₂²	1	2	yes	—
13	C₂×C₃×#676	—	2	1	no	—
14	C₂×C₃×C₈×D₅	—	2	1	no	—
15	C₃×C₅×M₃₂	—	2	1	no	—
16	C₃×C₅×M₃₂	—	2	1	no	—
17	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	complemented
18	C₃×#3944	—	2	1	no	—
19	C₃×#3944	C₂²	1	2	yes	complemented
20	C₂×C₃×#672	—	2	1	no	—
21	C₃×#2117	—	5	1	no	maximal, Hall
22	C₂×#4047	C₂×C₃	1	1	yes	characteristic
23	C₂×C₅×M₃₂	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×#3944	C₂×C₃	1	1	yes	characteristic, complemented
25	#64728	C₂×C₃	1	4	yes	complemented
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₂²⋊D₅	—	2	1	no	—
28	C₃×#709	—	2	1	no	—
29	C₃×#709	—	2	1	no	—
30	C₃×#709	C₈	1	2	yes	—
31	C₃×#709	C₂×C₄	1	2	yes	—
32	C₂×C₃×D₂₀	C₈	1	1	yes	characteristic
33	C₂×C₃×C₄×D₅	—	2	1	no	—
34	C₂×C₃×Dic₁₀	C₈	1	1	yes	characteristic
35	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
37	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
38	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
39	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
40	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₃×#681	—	2	1	no	—
42	C₃×#681	C₂×C₄	1	2	yes	—
43	C₂×C₃×#168	—	2	1	no	—
44	C₃×#676	—	2	2	no	—
45	C₃×#676	—	2	2	no	—
46	C₃×C₈×D₅	—	2	2	no	—
47	C₃×C₈×D₅	—	2	2	no	—
48	C₃×C₅×C₁₆	—	2	2	no	—
49	C₃×#672	—	2	2	no	—
50	C₂×C₃×#131	—	5	1	no	—
51	C₂×C₃×M₃₂	—	5	1	no	—
52	C₂×C₃×M₃₂	D₅	1	1	yes	characteristic, complemented
53	C₃×#350	—	5	4	no	—
54	C₂×#709	C₃×C₄	1	1	yes	characteristic
55	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
56	C₂×#681	C₃×C₄	1	1	yes	characteristic
57	#4047	—	2	1	no	—
58	#4047	—	2	1	no	—
59	#4047	C₃×C₄	1	2	yes	—
60	#4047	C₂²×C₃	1	2	yes	—
61	C₂×#676	—	2	1	no	—
62	C₂×C₈×D₅	—	2	1	no	—
63	C₅×M₃₂	—	2	1	no	—
64	C₅×M₃₂	—	2	1	no	—
65	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	complemented
66	#3944	—	2	1	no	—
67	#3944	C₂²×C₃	1	2	yes	complemented
68	C₂×#672	—	2	1	no	—
69	#2117	—	5	1	no	Sylow
70	C₂³×C₃×C₅	M₁₆	1	1	yes	characteristic
71	C₂²×C₃×D₅	—	2	1	no	—
72	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	M₁₆	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₃×C₂²⋊D₅	—	2	2	no	—
79	C₃×C₂²⋊D₅	—	2	2	no	—
80	C₂×C₃×Dic₅	—	2	1	no	—
81	C₃×D₂₀	—	2	1	no	—
82	C₃×D₂₀	C₂×C₈	1	2	yes	—
83	C₃×C₄×D₅	—	2	2	no	—
84	C₃×C₄×D₅	—	2	2	no	—
85	C₃×Dic₁₀	—	2	1	no	—
86	C₃×Dic₁₀	C₂×C₈	1	2	yes	—
87	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
88	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
89	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
90	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
91	C₃×#168	—	2	2	no	—
92	C₂×C₃×#41	—	5	1	no	—
93	C₃×#131	—	5	2	no	—
94	C₃×#131	—	5	2	no	—
95	C₃×#131	—	10	1	no	—
96	C₃×#131	—	10	1	no	—
97	C₂×C₃×M₁₆	—	5	1	no	—
98	C₂×C₃×M₁₆	—	10	1	no	—
99	C₂²×C₃×C₈	—	10	1	no	—
100	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
101	C₃×M₃₂	—	2	1	no	—
102	C₃×M₃₂	—	2	1	no	—
103	C₃×M₃₂	—	5	2	no	—
104	C₃×M₃₂	—	10	1	no	—
105	C₂×C₃×C₁₆	—	10	1	no	—
106	C₂×C₃×C₁₆	C₂×D₅	1	2	yes	complemented
107	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
108	C₂×C₂²⋊D₅	—	2	1	no	—
109	#709	—	2	1	no	—
110	#709	—	2	1	no	—
111	#709	C₃×C₈	1	2	yes	—
112	#709	C₂×C₃×C₄	1	2	yes	—
113	C₂×D₂₀	C₃×C₈	1	1	yes	characteristic
114	C₂×C₄×D₅	—	2	1	no	—
115	C₂×Dic₁₀	C₃×C₈	1	1	yes	characteristic
116	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
117	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
118	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
119	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
120	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
121	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
122	#681	—	2	1	no	—
123	#681	C₂×C₃×C₄	1	2	yes	—
124	C₂×#168	—	2	1	no	—
125	#676	—	2	2	no	—
126	#676	—	2	2	no	—
127	C₈×D₅	—	2	2	no	—
128	C₈×D₅	—	2	2	no	—
129	C₅×C₁₆	—	2	2	no	—
130	#672	—	2	2	no	—
131	C₂×#131	—	5	1	no	—
132	C₂×M₃₂	—	5	1	no	—
133	C₂×M₃₂	C₃×D₅	1	1	yes	characteristic, complemented
134	#350	—	5	4	no	—
135	C₂²×C₃×C₅	—	4	1	no	—
136	C₂²×C₃×C₅	C₂²⋊C₈	1	1	yes	characteristic
137	C₂²×C₃×C₅	C₂²⋊C₈	1	1	yes	characteristic
138	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
139	C₂×C₃×D₅	—	2	2	no	—
140	C₂×C₃×D₅	—	4	2	no	—
141	C₃×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
142	C₃×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
143	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
145	C₃×Dic₅	—	2	2	no	—
146	C₃×#41	—	5	2	no	—
147	C₃×#41	—	5	2	no	—
148	C₃×#41	—	10	1	no	—
149	C₃×#41	—	10	1	no	—
150	C₂×C₃×Q₈	—	5	1	no	—
151	C₂×C₃×D₄	—	5	1	no	—
152	C₂×C₃×D₄	—	10	1	no	—
153	C₂²×C₃×C₄	—	10	1	no	—
154	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
155	C₃×M₁₆	—	5	2	no	—
156	C₃×M₁₆	—	10	1	no	—
157	C₃×M₁₆	—	10	2	no	—
158	C₃×M₁₆	—	10	2	no	—
159	C₂×C₃×C₈	—	10	1	no	—
160	C₂×C₃×C₈	—	10	2	no	—
161	C₂×C₃×C₈	—	10	2	no	—
162	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
163	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
164	C₂×C₃×C₈	D₂₀	1	1	yes	characteristic
165	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
166	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
167	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
168	C₃×C₁₆	—	2	2	no	—
169	C₃×C₁₆	—	10	2	no	—
170	C₂³×C₅	C₃×M₁₆	1	1	yes	characteristic
171	C₂²×D₅	—	2	1	no	—
172	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
173	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
176	C₂×C₄×C₅	C₃×M₁₆	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
178	C₂²⋊D₅	—	2	2	no	—
179	C₂²⋊D₅	—	2	2	no	—
180	C₂×Dic₅	—	2	1	no	—
181	D₂₀	—	2	1	no	—
182	D₂₀	C₂×C₃×C₈	1	2	yes	—
183	C₄×D₅	—	2	2	no	—
184	C₄×D₅	—	2	2	no	—
185	Dic₁₀	—	2	1	no	—
186	Dic₁₀	C₂×C₃×C₈	1	2	yes	—
187	C₅×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
188	C₅×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
189	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
190	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
191	#168	—	2	2	no	—
192	C₂×#41	—	5	1	no	—
193	#131	—	5	2	no	—
194	#131	—	5	2	no	—
195	#131	—	10	1	no	—
196	#131	—	10	1	no	—
197	C₂×M₁₆	—	5	1	no	—
198	C₂×M₁₆	—	10	1	no	—
199	C₂²×C₈	—	10	1	no	—
200	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
201	M₃₂	—	2	1	no	—
202	M₃₂	—	2	1	no	—
203	M₃₂	—	5	2	no	—
204	M₃₂	—	10	1	no	—
205	C₂×C₁₆	—	10	1	no	—
206	C₂×C₁₆	C₂×C₃×D₅	1	2	yes	complemented
207	C₂×C₃×C₅	—	2	1	no	—
208	C₂×C₃×C₅	—	2	1	no	—
209	C₂×C₃×C₅	—	2	1	no	—
210	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
211	C₃×D₅	—	4	2	no	—
212	C₂³×C₃	—	10	1	no	—
213	C₂³×C₃	#676	1	1	yes	characteristic
214	C₃×Q₈	—	5	2	no	—
215	C₃×Q₈	—	10	1	no	—
216	C₃×D₄	—	5	2	no	—
217	C₃×D₄	—	10	2	no	—
218	C₃×D₄	—	10	2	no	—
219	C₃×D₄	—	10	1	no	—
220	C₂×C₃×C₄	—	10	1	no	—
221	C₂×C₃×C₄	—	10	2	no	—
222	C₂×C₃×C₄	—	10	2	no	—
223	C₂×C₃×C₄	C₈×D₅	1	1	yes	characteristic
224	C₂×C₃×C₄	C₈×D₅	1	1	yes	characteristic
225	C₂×C₃×C₄	#676	1	1	yes	characteristic
226	C₂×C₃×C₄	#685	1	1	yes	characteristic
227	C₂×C₃×C₄	#685	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
229	C₃×C₈	—	10	2	no	—
230	C₃×C₈	#685	1	1	yes	characteristic
231	C₃×C₈	#685	1	1	yes	characteristic
232	C₃×C₈	C₂×D₂₀	1	1	yes	characteristic
233	C₃×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
234	C₂²×C₅	—	4	1	no	—
235	C₂²×C₅	C₃×C₂²⋊C₈	1	1	yes	characteristic
236	C₂²×C₅	C₃×C₂²⋊C₈	1	1	yes	characteristic
237	C₂²×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
238	C₂×D₅	—	2	2	no	—
239	C₂×D₅	—	4	2	no	—
240	C₄×C₅	C₃×C₂²⋊C₈	1	1	yes	characteristic
241	C₄×C₅	C₃×C₂²⋊C₈	1	1	yes	characteristic
242	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
243	C₄×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
244	Dic₅	—	2	2	no	—
245	#41	—	5	2	no	—
246	#41	—	5	2	no	—
247	#41	—	10	1	no	—
248	#41	—	10	1	no	—
249	C₂×Q₈	—	5	1	no	—
250	C₂×D₄	—	5	1	no	—
251	C₂×D₄	—	10	1	no	—
252	C₂²×C₄	—	10	1	no	—
253	C₂²×C₄	C₃×C₄×D₅	1	1	yes	characteristic
254	M₁₆	—	5	2	no	—
255	M₁₆	—	10	1	no	—
256	M₁₆	—	10	2	no	—
257	M₁₆	—	10	2	no	—
258	C₂×C₈	—	10	1	no	—
259	C₂×C₈	—	10	2	no	—
260	C₂×C₈	—	10	2	no	—
261	C₂×C₈	C₃×C₄×D₅	1	1	yes	characteristic
262	C₂×C₈	C₃×D₂₀	1	1	yes	characteristic
263	C₂×C₈	C₃×D₂₀	1	1	yes	characteristic
264	C₂×C₈	C₃×C₂²⋊D₅	1	1	yes	characteristic
265	C₂×C₈	C₃×C₂²⋊D₅	1	1	yes	characteristic
266	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
267	C₁₆	—	2	2	no	—
268	C₁₆	—	10	2	no	—
269	C₃×C₅	#2117	1	1	yes	characteristic, complemented, Hall
270	C₂²×C₃	—	4	1	no	—
271	C₂²×C₃	—	10	2	no	—
272	C₂²×C₃	—	20	2	no	—
273	C₂²×C₃	#3952	1	1	yes	characteristic
274	C₂²×C₃	#3952	1	1	yes	characteristic
275	C₂²×C₃	C₂×#676	1	1	yes	characteristic
276	C₃×C₄	—	10	2	no	—
277	C₃×C₄	#3952	1	1	yes	characteristic
278	C₃×C₄	#3952	1	1	yes	characteristic
279	C₃×C₄	C₂×C₈×D₅	1	1	yes	characteristic
280	C₃×C₄	C₂×#685	1	1	yes	characteristic
281	C₂×C₅	—	2	1	no	—
282	C₂×C₅	—	2	1	no	—
283	C₂×C₅	—	2	1	no	—
284	C₂×C₅	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
285	D₅	—	4	2	no	—
286	C₂³	—	10	1	no	—
287	C₂³	C₃×#676	1	1	yes	characteristic
288	Q₈	—	5	2	no	—
289	Q₈	—	10	1	no	—
290	D₄	—	5	2	no	—
291	D₄	—	10	2	no	—
292	D₄	—	10	2	no	—
293	D₄	—	10	1	no	—
294	C₂×C₄	—	10	1	no	—
295	C₂×C₄	—	10	2	no	—
296	C₂×C₄	—	10	2	no	—
297	C₂×C₄	C₃×C₈×D₅	1	1	yes	characteristic
298	C₂×C₄	C₃×C₈×D₅	1	1	yes	characteristic
299	C₂×C₄	C₃×#676	1	1	yes	characteristic
300	C₂×C₄	C₃×#685	1	1	yes	characteristic
301	C₂×C₄	C₃×#685	1	1	yes	characteristic
302	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
303	C₈	—	10	2	no	—
304	C₈	C₃×#685	1	1	yes	characteristic
305	C₈	C₃×#685	1	1	yes	characteristic
306	C₈	C₂×C₃×D₂₀	1	1	yes	characteristic
307	C₈	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
308	C₂×C₃	—	2	1	no	—
309	C₂×C₃	—	2	1	no	—
310	C₂×C₃	—	2	1	no	—
311	C₂×C₃	—	20	2	no	—
312	C₂×C₃	C₂×#3952	1	1	yes	characteristic
313	C₅	C₃×#2117	1	1	yes	characteristic, complemented, Sylow
314	C₂²	—	4	1	no	—
315	C₂²	—	10	2	no	—
316	C₂²	—	20	2	no	—
317	C₂²	C₃×#3952	1	1	yes	characteristic
318	C₂²	C₃×#3952	1	1	yes	characteristic
319	C₂²	C₂×C₃×#676	1	1	yes	characteristic
320	C₄	—	10	2	no	—
321	C₄	C₃×#3952	1	1	yes	characteristic
322	C₄	C₃×#3952	1	1	yes	characteristic
323	C₄	C₂×C₃×C₈×D₅	1	1	yes	characteristic
324	C₄	C₂×C₃×#685	1	1	yes	characteristic
325	C₃	#10602230	1	1	yes	characteristic, direct product factor, Sylow
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	2	1	no	—
329	C₂	—	20	2	no	—
330	C₂	C₂×C₃×#3952	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₈	C₂×D₂₀
commutator	C₄×C₅	C₂²×C₃×C₈
Frattini	C₂×C₈	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×M₃₂	C₂
socle	C₂×C₃×C₅	C₂×C₂²⋊C₈
prime radical	C₂×C₃×#709	C₄
cosocle	C₂×C₅×C₈	C₂³×C₃
hypercenter	C₂×C₃×M₃₂	D₅
hypocenter	C₅	C₃×#2117
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₈ ◁ C₂²×C₃×C₈ ◁ C₂×C₃×M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×M₃₂	C₂	semidirect
3. C₂×C₃×#3944	C₂	semidirect
4. C₃×#64728	C₂	semidirect
5. #10602230	C₃	direct (N × H)
17. C₂×C₃×C₅×C₁₆	C₂²	semidirect
19. C₃×#3944	C₂²	semidirect
23. C₂×C₅×M₃₂	C₂×C₃	semidirect
24. C₂×#3944	C₂×C₃	semidirect
25. #64728	C₂×C₃	semidirect
52. C₂×C₃×M₃₂	D₅	semidirect
65. C₂×C₅×C₁₆	C₂²×C₃	semidirect
67. #3944	C₂²×C₃	semidirect
106. C₂×C₃×C₁₆	C₂×D₅	semidirect
133. C₂×M₃₂	C₃×D₅	semidirect
206. C₂×C₁₆	C₂×C₃×D₅	semidirect
269. C₃×C₅	#2117	semidirect
313. C₅	C₃×#2117	semidirect
325. C₃	#10602230	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×#4047	2
C₂×C₃×C₅×M₃₂	2
C₂×C₃×#3944	2
C₃×#64728	2
#10602230	3
C₃×#2117	5

Subgroups — C3×#10602230 #14313003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10602230 #14313003