Subgroups — C₃×#10600318 #14297012

Order 1920.

1184 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×Dic₁₀	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#380	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₂²⋊#168	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×#3942	C₂	1	4	yes	maximal, complemented
5	#10600318	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₂³×C₃×Dic₅	—	2	1	no	—
8	C₂²×C₃×Dic₁₀	—	2	1	no	—
9	C₂²×C₃×Dic₁₀	—	2	2	no	—
10	C₂²×C₃×Dic₁₀	—	2	8	no	—
11	C₂²×C₃×Dic₁₀	C₄	1	4	yes	complemented
12	C₂²×C₃×Dic₁₀	C₂²	1	2	yes	—
13	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₃×C₅×C₂²⋊C₄	—	2	1	no	—
15	C₂²×C₃×#168	C₂²	1	2	yes	—
16	C₃×C₅×#95	—	2	1	no	—
17	C₃×C₂²⋊#168	—	2	2	no	—
18	C₃×#3942	—	2	8	no	—
19	C₃×#1787	—	5	1	no	maximal, Hall
20	C₂³×Dic₁₀	C₂×C₃	1	1	yes	characteristic
21	C₅×#380	C₂×C₃	1	1	yes	characteristic
22	C₂×C₂²⋊#168	C₂×C₃	1	1	yes	characteristic
23	C₂×#3942	C₂×C₃	1	4	yes	complemented
24	C₂⁴×C₃×C₅	D₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	—	2	4	no	—
26	C₂²×C₃×C₄×C₅	—	2	2	no	—
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
30	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂²×C₃×Dic₅	—	2	8	no	—
32	C₂²×C₃×Dic₅	—	2	4	no	—
33	C₂²×C₃×Dic₅	—	2	2	no	—
34	C₂×C₃×Dic₁₀	—	2	8	no	—
35	C₂×C₃×Dic₁₀	—	2	16	no	—
36	C₂×C₃×Dic₁₀	—	2	4	no	—
37	C₂×C₃×Dic₁₀	—	2	4	no	—
38	C₂×C₃×Dic₁₀	—	2	2	no	—
39	C₂×C₃×Dic₁₀	—	2	16	no	—
40	C₂×C₃×Dic₁₀	C₂×C₄	1	4	yes	—
41	C₂×C₃×Dic₁₀	D₄	1	8	yes	—
42	C₃×C₅×C₄⋊C₄	—	2	4	no	—
43	C₃×C₅×C₂²⋊C₄	—	4	2	no	—
44	C₂×C₃×#168	—	2	4	no	—
45	C₂×C₃×#168	—	2	2	no	—
46	C₂³×C₃×Q₈	—	5	1	no	—
47	C₂×C₃×#103	—	5	4	no	—
48	C₂×C₃×C₂²⋊C₈	—	5	1	no	—
49	C₃×#380	D₅	1	1	yes	characteristic
50	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
51	C₂³×Dic₅	—	2	1	no	—
52	C₂²×Dic₁₀	—	2	2	no	—
53	C₂²×Dic₁₀	—	2	1	no	—
54	C₂²×Dic₁₀	—	2	8	no	—
55	C₂²×Dic₁₀	C₃×C₄	1	4	yes	complemented
56	C₂²×Dic₁₀	C₂²×C₃	1	2	yes	—
57	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
58	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
59	C₂²×#168	C₂²×C₃	1	2	yes	—
60	C₅×#95	—	2	1	no	—
61	C₂²⋊#168	—	2	2	no	—
62	#3942	—	2	8	no	—
63	#1787	—	5	1	no	Sylow
64	C₂³×C₃×C₅	—	2	2	no	—
65	C₂³×C₃×C₅	—	2	2	no	—
66	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
67	C₂³×C₃×C₅	SD₁₆	1	2	yes	—
68	C₂³×C₃×C₅	Q₁₆	1	2	yes	—
69	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	—	2	2	no	—
71	C₂×C₃×C₄×C₅	—	2	4	no	—
72	C₂×C₃×C₄×C₅	—	2	4	no	—
73	C₂×C₃×C₄×C₅	—	4	2	no	—
74	C₂×C₃×C₄×C₅	—	4	2	no	—
75	C₂×C₃×C₄×C₅	—	4	2	no	—
76	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
79	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×Dic₅	—	2	16	no	—
81	C₂×C₃×Dic₅	—	2	4	no	—
82	C₂×C₃×Dic₅	—	2	8	no	—
83	C₃×Dic₁₀	—	2	16	no	—
84	C₃×Dic₁₀	—	2	4	no	—
85	C₃×Dic₁₀	—	2	8	no	—
86	C₃×Dic₁₀	C₂²⋊C₄	1	8	yes	complemented
87	C₃×#168	—	4	2	no	—
88	C₂²×C₃×Q₈	—	5	4	no	—
89	C₂²×C₃×Q₈	—	5	2	no	—
90	C₂²×C₃×Q₈	—	10	8	no	—
91	C₂²×C₃×Q₈	—	10	2	no	—
92	C₂²×C₃×Q₈	—	10	1	no	—
93	C₂³×C₃×C₄	—	10	1	no	—
94	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
95	C₂²×C₃×C₈	—	5	2	no	—
96	C₂×C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
97	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
98	C₃×#103	—	10	8	no	—
99	C₃×C₂²⋊C₈	—	10	2	no	—
100	C₃×#95	—	2	1	no	—
101	C₂⁴×C₅	C₃×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₅	—	2	4	no	—
103	C₂²×C₄×C₅	—	2	2	no	—
104	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
105	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
106	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
108	C₂²×Dic₅	—	2	8	no	—
109	C₂²×Dic₅	—	2	4	no	—
110	C₂²×Dic₅	—	2	2	no	—
111	C₂×Dic₁₀	—	2	8	no	—
112	C₂×Dic₁₀	—	2	16	no	—
113	C₂×Dic₁₀	—	2	4	no	—
114	C₂×Dic₁₀	—	2	2	no	—
115	C₂×Dic₁₀	—	2	4	no	—
116	C₂×Dic₁₀	—	2	16	no	—
117	C₂×Dic₁₀	C₂×C₃×C₄	1	4	yes	—
118	C₂×Dic₁₀	C₃×D₄	1	8	yes	—
119	C₅×C₄⋊C₄	—	2	4	no	—
120	C₅×C₂²⋊C₄	—	4	2	no	—
121	C₂×#168	—	2	4	no	—
122	C₂×#168	—	2	2	no	—
123	C₂³×Q₈	—	5	1	no	—
124	C₂×#103	—	5	4	no	—
125	C₂×C₂²⋊C₈	—	5	1	no	—
126	#380	C₃×D₅	1	1	yes	characteristic
127	C₂²×C₃×C₅	—	2	2	no	—
128	C₂²×C₃×C₅	—	2	2	no	—
129	C₂²×C₃×C₅	—	2	4	no	—
130	C₂²×C₃×C₅	—	2	4	no	—
131	C₂²×C₃×C₅	#103	1	4	yes	complemented
132	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₃×C₅	#120	1	2	yes	—
134	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
135	C₂²×C₃×C₅	C₂×Q₁₆	1	1	yes	characteristic
136	C₂²×C₃×C₅	#137	1	2	yes	—
137	C₃×C₄×C₅	—	2	2	no	—
138	C₃×C₄×C₅	—	8	2	no	—
139	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
140	C₃×C₄×C₅	#120	1	2	yes	—
141	C₃×Dic₅	—	2	8	no	—
142	C₂⁴×C₃	C₂²⋊D₅	1	1	yes	characteristic
143	C₂×C₃×Q₈	—	5	4	no	—
144	C₂×C₃×Q₈	—	5	8	no	—
145	C₂×C₃×Q₈	—	10	16	no	—
146	C₂×C₃×Q₈	—	10	8	no	—
147	C₂×C₃×Q₈	—	10	16	no	—
148	C₂×C₃×Q₈	—	10	4	no	—
149	C₂×C₃×Q₈	—	10	2	no	—
150	C₂×C₃×Q₈	—	10	4	no	—
151	C₂²×C₃×C₄	—	2	4	no	—
152	C₂²×C₃×C₄	—	2	2	no	—
153	C₂²×C₃×C₄	—	10	8	no	—
154	C₂²×C₃×C₄	—	10	4	no	—
155	C₂²×C₃×C₄	—	10	2	no	—
156	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
157	C₂²×C₃×C₄	D₂₀	1	2	yes	—
158	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
159	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
160	C₂×C₃×C₈	—	10	2	no	—
161	C₂×C₃×C₈	—	10	4	no	—
162	C₃×C₄⋊C₄	—	2	4	no	—
163	C₃×C₂²⋊C₄	—	4	2	no	—
164	C₂³×C₅	—	2	2	no	—
165	C₂³×C₅	—	2	2	no	—
166	C₂³×C₅	C₃×C₂²⋊C₄	1	2	yes	—
167	C₂³×C₅	C₃×SD₁₆	1	2	yes	—
168	C₂³×C₅	C₃×Q₁₆	1	2	yes	—
169	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
170	C₂×C₄×C₅	—	2	2	no	—
171	C₂×C₄×C₅	—	2	4	no	—
172	C₂×C₄×C₅	—	2	4	no	—
173	C₂×C₄×C₅	—	4	2	no	—
174	C₂×C₄×C₅	—	4	2	no	—
175	C₂×C₄×C₅	—	4	2	no	—
176	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
177	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
180	C₂×Dic₅	—	2	16	no	—
181	C₂×Dic₅	—	2	4	no	—
182	C₂×Dic₅	—	2	8	no	—
183	Dic₁₀	—	2	16	no	—
184	Dic₁₀	—	2	4	no	—
185	Dic₁₀	—	2	8	no	—
186	Dic₁₀	C₃×C₂²⋊C₄	1	8	yes	complemented
187	#168	—	4	2	no	—
188	C₂²×Q₈	—	5	4	no	—
189	C₂²×Q₈	—	5	2	no	—
190	C₂²×Q₈	—	10	8	no	—
191	C₂²×Q₈	—	10	2	no	—
192	C₂²×Q₈	—	10	1	no	—
193	C₂³×C₄	—	10	1	no	—
194	C₂³×C₄	C₂×C₃×D₅	1	1	yes	characteristic
195	C₂²×C₈	—	5	2	no	—
196	C₂×C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
197	C₂×C₂²⋊C₄	—	2	1	no	—
198	#103	—	10	8	no	—
199	C₂²⋊C₈	—	10	2	no	—
200	#95	—	2	1	no	—
201	C₂×C₃×C₅	—	2	4	no	—
202	C₂×C₃×C₅	#379	1	1	yes	characteristic
203	C₂×C₃×C₅	C₂×#103	1	1	yes	characteristic
204	C₂×C₃×C₅	#419	1	1	yes	characteristic
205	C₂×C₃×C₅	#448	1	2	yes	—
206	C₂×C₃×C₅	#451	1	2	yes	—
207	C₂³×C₃	—	2	2	no	—
208	C₂³×C₃	—	2	2	no	—
209	C₂³×C₃	#685	1	2	yes	—
210	C₂³×C₃	#687	1	2	yes	—
211	C₂³×C₃	#689	1	2	yes	—
212	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
213	C₃×Q₈	—	5	8	no	—
214	C₃×Q₈	—	10	16	no	—
215	C₃×Q₈	—	10	4	no	—
216	C₃×Q₈	—	10	8	no	—
217	C₂×C₃×C₄	—	2	2	no	—
218	C₂×C₃×C₄	—	2	4	no	—
219	C₂×C₃×C₄	—	2	4	no	—
220	C₂×C₃×C₄	—	4	2	no	—
221	C₂×C₃×C₄	—	4	2	no	—
222	C₂×C₃×C₄	—	4	2	no	—
223	C₂×C₃×C₄	—	10	16	no	—
224	C₂×C₃×C₄	—	10	4	no	—
225	C₂×C₃×C₄	—	10	8	no	—
226	C₂×C₃×C₄	#685	1	2	yes	—
227	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
229	C₂×C₃×C₄	D₄×D₅	1	4	yes	—
230	C₃×C₈	—	20	2	no	—
231	C₂²×C₅	—	2	2	no	—
232	C₂²×C₅	—	2	2	no	—
233	C₂²×C₅	—	2	4	no	—
234	C₂²×C₅	—	2	4	no	—
235	C₂²×C₅	C₃×#103	1	4	yes	complemented
236	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
237	C₂²×C₅	C₃×#120	1	2	yes	—
238	C₂²×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
239	C₂²×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
240	C₂²×C₅	C₃×#137	1	2	yes	—
241	C₄×C₅	—	2	2	no	—
242	C₄×C₅	—	8	2	no	—
243	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	2	yes	—
244	C₄×C₅	C₃×#120	1	2	yes	—
245	Dic₅	—	2	8	no	—
246	C₂⁴	C₃×C₂²⋊D₅	1	1	yes	characteristic
247	C₂×Q₈	—	5	4	no	—
248	C₂×Q₈	—	5	8	no	—
249	C₂×Q₈	—	10	16	no	—
250	C₂×Q₈	—	10	8	no	—
251	C₂×Q₈	—	10	16	no	—
252	C₂×Q₈	—	10	4	no	—
253	C₂×Q₈	—	10	2	no	—
254	C₂×Q₈	—	10	4	no	—
255	C₂²×C₄	—	2	4	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	10	8	no	—
258	C₂²×C₄	—	10	4	no	—
259	C₂²×C₄	—	10	2	no	—
260	C₂²×C₄	C₃×C₄×D₅	1	2	yes	—
261	C₂²×C₄	C₃×D₂₀	1	2	yes	—
262	C₂²×C₄	C₃×C₂²⋊D₅	1	1	yes	characteristic
263	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
264	C₂×C₈	—	10	2	no	—
265	C₂×C₈	—	10	4	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₃×C₅	#1787	1	1	yes	characteristic, complemented, Hall
269	C₂²×C₃	—	2	2	no	—
270	C₂²×C₃	—	2	4	no	—
271	C₂²×C₃	—	2	4	no	—
272	C₂²×C₃	—	2	2	no	—
273	C₂²×C₃	#3942	1	4	yes	complemented
274	C₂²×C₃	C₂×#685	1	1	yes	characteristic
275	C₂²×C₃	C₂×#687	1	1	yes	characteristic
276	C₂²×C₃	#4083	1	2	yes	—
277	C₂²×C₃	C₂×#689	1	1	yes	characteristic
278	C₂²×C₃	#4097	1	2	yes	—
279	C₃×C₄	—	2	2	no	—
280	C₃×C₄	—	8	2	no	—
281	C₃×C₄	—	10	8	no	—
282	C₃×C₄	D₅×C₂²⋊C₄	1	2	yes	—
283	C₃×C₄	#4028	1	2	yes	—
284	C₂×C₅	—	2	4	no	—
285	C₂×C₅	C₃×#379	1	1	yes	characteristic
286	C₂×C₅	C₂×C₃×#103	1	1	yes	characteristic
287	C₂×C₅	C₃×#419	1	1	yes	characteristic
288	C₂×C₅	C₃×#448	1	2	yes	—
289	C₂×C₅	C₃×#451	1	2	yes	—
290	C₂³	—	2	2	no	—
291	C₂³	—	2	2	no	—
292	C₂³	C₃×#685	1	2	yes	—
293	C₂³	C₃×#687	1	2	yes	—
294	C₂³	C₃×#689	1	2	yes	—
295	C₂³	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
296	Q₈	—	5	8	no	—
297	Q₈	—	10	16	no	—
298	Q₈	—	10	4	no	—
299	Q₈	—	10	8	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	10	16	no	—
307	C₂×C₄	—	10	8	no	—
308	C₂×C₄	—	10	4	no	—
309	C₂×C₄	C₃×#685	1	2	yes	—
310	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
311	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
312	C₂×C₄	C₃×D₄×D₅	1	4	yes	—
313	C₈	—	20	2	no	—
314	C₂×C₃	—	2	4	no	—
315	C₂×C₃	#65244	1	1	yes	characteristic
316	C₂×C₃	C₂×#3942	1	1	yes	characteristic
317	C₂×C₃	#65293	1	1	yes	characteristic
318	C₂×C₃	#65329	1	2	yes	—
319	C₂×C₃	#65339	1	2	yes	—
320	C₅	C₃×#1787	1	1	yes	characteristic, complemented, Sylow
321	C₂²	—	2	2	no	—
322	C₂²	—	2	4	no	—
323	C₂²	—	2	4	no	—
324	C₂²	—	2	2	no	—
325	C₂²	C₃×#3942	1	4	yes	complemented
326	C₂²	C₂×C₃×#685	1	1	yes	characteristic
327	C₂²	C₂×C₃×#687	1	1	yes	characteristic
328	C₂²	C₃×#4083	1	2	yes	—
329	C₂²	C₂×C₃×#689	1	1	yes	characteristic
330	C₂²	C₃×#4097	1	2	yes	—
331	C₄	—	2	2	no	—
332	C₄	—	8	2	no	—
333	C₄	—	10	8	no	—
334	C₄	C₃×D₅×C₂²⋊C₄	1	2	yes	—
335	C₄	C₃×#4028	1	2	yes	—
336	C₃	#10600318	1	1	yes	characteristic, direct product factor, Sylow
337	C₂	—	2	4	no	—
338	C₂	C₃×#65244	1	1	yes	characteristic
339	C₂	C₂×C₃×#3942	1	1	yes	characteristic
340	C₂	C₃×#65293	1	1	yes	characteristic
341	C₂	C₃×#65329	1	2	yes	—
342	C₂	C₃×#65339	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×#380	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂⁴×C₃×C₅	D₄
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₃×#380	D₅
hypocenter	C₅	C₃×#1787
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂³×C₃×C₄ ◁ C₃×#380

Complements

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

N	H ≅ G/N	type
4. C₂×C₃×#3942	C₂	semidirect
5. #10600318	C₃	direct (N × H)
11. C₂²×C₃×Dic₁₀	C₄	semidirect
23. C₂×#3942	C₂×C₃	semidirect
55. C₂²×Dic₁₀	C₃×C₄	semidirect
86. C₃×Dic₁₀	C₂²⋊C₄	semidirect
131. C₂²×C₃×C₅	#103	semidirect
186. Dic₁₀	C₃×C₂²⋊C₄	semidirect
235. C₂²×C₅	C₃×#103	semidirect
268. C₃×C₅	#1787	semidirect
273. C₂²×C₃	#3942	semidirect
320. C₅	C₃×#1787	semidirect
325. C₂²	C₃×#3942	semidirect
336. C₃	#10600318	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×Dic₁₀	2
C₃×C₅×#380	2
C₂×C₃×C₂²⋊#168	2
C₂×C₃×#3942	2
#10600318	3
C₃×#1787	5

Subgroups — C3×#10600318 #14297012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10600318 #14297012