Subgroups — C₃×#10599899 #14312821

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₈×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3930	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₂²⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64727	C₂	1	2	yes	maximal, complemented
6	#10599899	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₄×D₅	C₄	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₃×#168	C₄	1	1	yes	characteristic
10	C₂×C₃×C₈×D₅	—	2	4	no	—
11	C₂×C₃×C₈×D₅	C₄	1	1	yes	characteristic
12	C₂×C₃×C₈×D₅	C₄	1	1	yes	characteristic
13	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	complemented
16	C₂×C₃×#672	C₂²	1	2	yes	complemented
17	C₃×#3930	—	2	4	no	—
18	C₃×#2112	—	5	1	no	maximal, Hall
19	C₂²×C₈×D₅	C₂×C₃	1	1	yes	characteristic
20	C₂×#3930	C₂×C₃	1	2	yes	complemented
21	C₅×C₂²⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂²⋊#672	C₂×C₃	1	1	yes	characteristic, complemented
23	#64727	C₂×C₃	1	2	yes	complemented
24	C₂³×C₃×D₅	C₈	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×Dic₅	C₈	1	1	yes	characteristic
27	C₂×C₃×C₄×D₅	—	2	2	no	—
28	C₂×C₃×C₄×D₅	—	2	2	no	—
29	C₂×C₃×C₄×D₅	C₈	1	1	yes	characteristic
30	C₂×C₃×C₄×D₅	C₈	1	1	yes	characteristic
31	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	—	2	2	no	—
34	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₂×C₃×#168	—	2	2	no	—
37	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
39	C₃×C₈×D₅	—	2	2	no	—
40	C₃×C₈×D₅	—	2	2	no	—
41	C₃×C₈×D₅	—	2	2	no	—
42	C₃×C₈×D₅	D₄	1	4	yes	—
43	C₃×C₅×C₁₆	—	2	2	no	—
44	C₃×#672	—	2	2	no	—
45	C₂³×C₃×C₈	—	5	1	no	—
46	C₂×C₃×M₃₂	—	5	2	no	—
47	C₃×C₂²⋊C₁₆	—	5	1	no	—
48	C₃×C₂²⋊C₁₆	—	5	2	no	—
49	C₃×C₂²⋊C₁₆	D₅	1	1	yes	characteristic, complemented
50	C₂²×C₄×D₅	C₃×C₄	1	1	yes	characteristic
51	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
52	C₂²×#168	C₃×C₄	1	1	yes	characteristic
53	C₂×C₈×D₅	—	2	4	no	—
54	C₂×C₈×D₅	C₃×C₄	1	1	yes	characteristic
55	C₂×C₈×D₅	C₃×C₄	1	1	yes	characteristic
56	C₂×C₈×D₅	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₈×D₅	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	complemented
59	C₂×#672	C₂²×C₃	1	2	yes	complemented
60	#3930	—	2	4	no	—
61	#2112	—	5	1	no	Sylow
62	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₃×D₅	—	2	2	no	—
64	C₂²×C₃×D₅	—	2	2	no	—
65	C₂²×C₃×D₅	—	2	2	no	—
66	C₂²×C₃×D₅	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₃×D₅	C₂×C₈	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	—	2	1	no	—
69	C₂×C₃×C₄×C₅	—	2	1	no	—
70	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₃×Dic₅	—	2	2	no	—
73	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
74	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
75	C₃×C₄×D₅	—	2	2	no	—
76	C₃×C₄×D₅	—	2	1	no	—
77	C₃×C₄×D₅	—	2	1	no	—
78	C₃×C₄×D₅	—	2	2	no	—
79	C₃×C₄×D₅	C₂²⋊C₄	1	2	yes	—
80	C₃×C₄×D₅	M₁₆	1	2	yes	—
81	C₃×C₅×C₈	—	2	1	no	—
82	C₃×C₅×C₈	C₂×D₄	1	2	yes	—
83	C₃×#168	—	2	1	no	—
84	C₃×#168	C₂²⋊C₄	1	2	yes	—
85	C₂³×C₃×C₄	—	5	1	no	—
86	C₂²×C₃×C₈	—	5	1	no	—
87	C₂²×C₃×C₈	—	5	1	no	—
88	C₂²×C₃×C₈	—	5	1	no	—
89	C₂²×C₃×C₈	—	5	1	no	—
90	C₂²×C₃×C₈	—	5	1	no	—
91	C₂²×C₃×C₈	—	10	4	no	—
92	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
93	C₃×M₃₂	—	10	4	no	—
94	C₂×C₃×C₁₆	—	5	2	no	—
95	C₂×C₃×C₁₆	C₂×D₅	1	2	yes	complemented
96	C₂³×D₅	C₃×C₈	1	1	yes	characteristic
97	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
98	C₂²×Dic₅	C₃×C₈	1	1	yes	characteristic
99	C₂×C₄×D₅	—	2	2	no	—
100	C₂×C₄×D₅	—	2	2	no	—
101	C₂×C₄×D₅	C₃×C₈	1	1	yes	characteristic
102	C₂×C₄×D₅	C₃×C₈	1	1	yes	characteristic
103	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
104	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
105	C₂×C₅×C₈	—	2	2	no	—
106	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
107	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
108	C₂×#168	—	2	2	no	—
109	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
110	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
111	C₈×D₅	—	2	2	no	—
112	C₈×D₅	—	2	2	no	—
113	C₈×D₅	—	2	2	no	—
114	C₈×D₅	C₃×D₄	1	4	yes	—
115	C₅×C₁₆	—	2	2	no	—
116	#672	—	2	2	no	—
117	C₂³×C₈	—	5	1	no	—
118	C₂×M₃₂	—	5	2	no	—
119	C₂²⋊C₁₆	—	5	1	no	—
120	C₂²⋊C₁₆	—	5	2	no	—
121	C₂²⋊C₁₆	C₃×D₅	1	1	yes	characteristic, complemented
122	C₂²×C₃×C₅	—	2	1	no	—
123	C₂²×C₃×C₅	—	2	1	no	—
124	C₂²×C₃×C₅	M₃₂	1	2	yes	complemented
125	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₃×D₅	—	2	8	no	—
127	C₂×C₃×D₅	—	2	1	no	—
128	C₂×C₃×D₅	—	2	1	no	—
129	C₂×C₃×D₅	—	2	1	no	—
130	C₂×C₃×D₅	—	2	1	no	—
131	C₂×C₃×D₅	C₂²⋊C₈	1	2	yes	—
132	C₂×C₃×D₅	M₃₂	1	2	yes	complemented
133	C₃×C₄×C₅	—	2	1	no	—
134	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
136	C₃×Dic₅	—	2	1	no	—
137	C₃×Dic₅	C₂²⋊C₈	1	2	yes	—
138	C₂⁴×C₃	—	5	1	no	—
139	C₂²×C₃×C₄	—	5	1	no	—
140	C₂²×C₃×C₄	—	5	1	no	—
141	C₂²×C₃×C₄	—	5	1	no	—
142	C₂²×C₃×C₄	—	5	1	no	—
143	C₂²×C₃×C₄	—	5	1	no	—
144	C₂²×C₃×C₄	—	10	2	no	—
145	C₂²×C₃×C₄	—	10	2	no	—
146	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
147	C₂×C₃×C₈	—	2	2	no	—
148	C₂×C₃×C₈	—	5	1	no	—
149	C₂×C₃×C₈	—	5	4	no	—
150	C₂×C₃×C₈	—	5	1	no	—
151	C₂×C₃×C₈	—	10	2	no	—
152	C₂×C₃×C₈	—	10	2	no	—
153	C₂×C₃×C₈	—	10	2	no	—
154	C₂×C₃×C₈	—	10	2	no	—
155	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
156	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
157	C₃×C₁₆	—	2	2	no	—
158	C₃×C₁₆	—	10	2	no	—
159	C₂³×C₅	C₂×C₃×C₈	1	1	yes	characteristic
160	C₂²×D₅	—	2	2	no	—
161	C₂²×D₅	—	2	2	no	—
162	C₂²×D₅	—	2	2	no	—
163	C₂²×D₅	C₂×C₃×C₈	1	1	yes	characteristic
164	C₂²×D₅	C₂×C₃×C₈	1	1	yes	characteristic
165	C₂×C₄×C₅	—	2	1	no	—
166	C₂×C₄×C₅	—	2	1	no	—
167	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
169	C₂×Dic₅	—	2	2	no	—
170	C₂×Dic₅	C₂×C₃×C₈	1	1	yes	characteristic
171	C₂×Dic₅	C₂×C₃×C₈	1	1	yes	characteristic
172	C₄×D₅	—	2	2	no	—
173	C₄×D₅	—	2	1	no	—
174	C₄×D₅	—	2	1	no	—
175	C₄×D₅	—	2	2	no	—
176	C₄×D₅	C₃×C₂²⋊C₄	1	2	yes	—
177	C₄×D₅	C₃×M₁₆	1	2	yes	—
178	C₅×C₈	—	2	1	no	—
179	C₅×C₈	C₂×C₃×D₄	1	2	yes	—
180	#168	—	2	1	no	—
181	#168	C₃×C₂²⋊C₄	1	2	yes	—
182	C₂³×C₄	—	5	1	no	—
183	C₂²×C₈	—	5	1	no	—
184	C₂²×C₈	—	5	1	no	—
185	C₂²×C₈	—	5	1	no	—
186	C₂²×C₈	—	5	1	no	—
187	C₂²×C₈	—	5	1	no	—
188	C₂²×C₈	—	10	4	no	—
189	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
190	M₃₂	—	10	4	no	—
191	C₂×C₁₆	—	5	2	no	—
192	C₂×C₁₆	C₂×C₃×D₅	1	2	yes	complemented
193	C₂×C₃×C₅	—	2	2	no	—
194	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
195	C₂×C₃×C₅	C₂×M₃₂	1	1	yes	characteristic
196	C₂×C₃×C₅	C₂×M₃₂	1	1	yes	characteristic
197	C₃×D₅	—	2	2	no	—
198	C₃×D₅	—	2	2	no	—
199	C₂³×C₃	—	5	1	no	—
200	C₂³×C₃	—	5	1	no	—
201	C₂³×C₃	—	10	2	no	—
202	C₂³×C₃	—	10	2	no	—
203	C₂³×C₃	—	10	2	no	—
204	C₂³×C₃	C₈×D₅	1	1	yes	characteristic
205	C₂×C₃×C₄	—	2	1	no	—
206	C₂×C₃×C₄	—	2	1	no	—
207	C₂×C₃×C₄	—	5	1	no	—
208	C₂×C₃×C₄	—	5	2	no	—
209	C₂×C₃×C₄	—	5	2	no	—
210	C₂×C₃×C₄	—	5	1	no	—
211	C₂×C₃×C₄	—	10	2	no	—
212	C₂×C₃×C₄	—	10	2	no	—
213	C₂×C₃×C₄	—	10	2	no	—
214	C₂×C₃×C₄	—	10	1	no	—
215	C₂×C₃×C₄	—	10	1	no	—
216	C₂×C₃×C₄	C₈×D₅	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
218	C₃×C₈	—	2	1	no	—
219	C₃×C₈	—	5	2	no	—
220	C₃×C₈	—	10	1	no	—
221	C₃×C₈	D₄×D₅	1	2	yes	—
222	C₂²×C₅	—	2	1	no	—
223	C₂²×C₅	—	2	1	no	—
224	C₂²×C₅	C₃×M₃₂	1	2	yes	complemented
225	C₂²×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
226	C₂×D₅	—	2	8	no	—
227	C₂×D₅	—	2	1	no	—
228	C₂×D₅	—	2	1	no	—
229	C₂×D₅	—	2	1	no	—
230	C₂×D₅	—	2	1	no	—
231	C₂×D₅	C₃×C₂²⋊C₈	1	2	yes	—
232	C₂×D₅	C₃×M₃₂	1	2	yes	complemented
233	C₄×C₅	—	2	1	no	—
234	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
235	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
236	Dic₅	—	2	1	no	—
237	Dic₅	C₃×C₂²⋊C₈	1	2	yes	—
238	C₂⁴	—	5	1	no	—
239	C₂²×C₄	—	5	1	no	—
240	C₂²×C₄	—	5	1	no	—
241	C₂²×C₄	—	5	1	no	—
242	C₂²×C₄	—	5	1	no	—
243	C₂²×C₄	—	5	1	no	—
244	C₂²×C₄	—	10	2	no	—
245	C₂²×C₄	—	10	2	no	—
246	C₂²×C₄	C₃×C₄×D₅	1	1	yes	characteristic
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	5	1	no	—
249	C₂×C₈	—	5	4	no	—
250	C₂×C₈	—	5	1	no	—
251	C₂×C₈	—	10	2	no	—
252	C₂×C₈	—	10	2	no	—
253	C₂×C₈	—	10	2	no	—
254	C₂×C₈	—	10	2	no	—
255	C₂×C₈	C₃×C₄×D₅	1	1	yes	characteristic
256	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
257	C₁₆	—	2	2	no	—
258	C₁₆	—	10	2	no	—
259	C₃×C₅	#2112	1	1	yes	characteristic, complemented, Hall
260	C₂²×C₃	—	2	1	no	—
261	C₂²×C₃	—	2	1	no	—
262	C₂²×C₃	—	5	2	no	—
263	C₂²×C₃	—	5	2	no	—
264	C₂²×C₃	—	10	8	no	—
265	C₂²×C₃	—	10	1	no	—
266	C₂²×C₃	—	10	1	no	—
267	C₂²×C₃	—	10	1	no	—
268	C₂²×C₃	—	10	1	no	—
269	C₂²×C₃	#3930	1	2	yes	complemented
270	C₂²×C₃	C₂×C₈×D₅	1	1	yes	characteristic
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	—	5	2	no	—
273	C₃×C₄	—	10	1	no	—
274	C₃×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
275	C₃×C₄	D₅×M₁₆	1	1	yes	characteristic
276	C₂×C₅	—	2	2	no	—
277	C₂×C₅	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
278	C₂×C₅	C₂×C₃×M₃₂	1	1	yes	characteristic
279	C₂×C₅	C₂×C₃×M₃₂	1	1	yes	characteristic
280	D₅	—	2	2	no	—
281	D₅	—	2	2	no	—
282	C₂³	—	5	1	no	—
283	C₂³	—	5	1	no	—
284	C₂³	—	10	2	no	—
285	C₂³	—	10	2	no	—
286	C₂³	—	10	2	no	—
287	C₂³	C₃×C₈×D₅	1	1	yes	characteristic
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	5	1	no	—
291	C₂×C₄	—	5	2	no	—
292	C₂×C₄	—	5	2	no	—
293	C₂×C₄	—	5	1	no	—
294	C₂×C₄	—	10	2	no	—
295	C₂×C₄	—	10	2	no	—
296	C₂×C₄	—	10	2	no	—
297	C₂×C₄	—	10	1	no	—
298	C₂×C₄	—	10	1	no	—
299	C₂×C₄	C₃×C₈×D₅	1	1	yes	characteristic
300	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	5	2	no	—
303	C₈	—	10	1	no	—
304	C₈	C₃×D₄×D₅	1	2	yes	—
305	C₂×C₃	—	2	2	no	—
306	C₂×C₃	—	10	2	no	—
307	C₂×C₃	—	10	2	no	—
308	C₂×C₃	D₅×C₂²⋊C₈	1	1	yes	characteristic
309	C₂×C₃	C₂×#3930	1	1	yes	characteristic
310	C₂×C₃	D₅×M₃₂	1	1	yes	characteristic
311	C₅	C₃×#2112	1	1	yes	characteristic, complemented, Sylow
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	5	2	no	—
315	C₂²	—	5	2	no	—
316	C₂²	—	10	8	no	—
317	C₂²	—	10	1	no	—
318	C₂²	—	10	1	no	—
319	C₂²	—	10	1	no	—
320	C₂²	—	10	1	no	—
321	C₂²	C₃×#3930	1	2	yes	complemented
322	C₂²	C₂×C₃×C₈×D₅	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	5	2	no	—
325	C₄	—	10	1	no	—
326	C₄	C₃×D₅×C₂²⋊C₄	1	1	yes	characteristic
327	C₄	C₃×D₅×M₁₆	1	1	yes	characteristic
328	C₃	#10599899	1	1	yes	characteristic, direct product factor, Sylow
329	C₂	—	2	2	no	—
330	C₂	—	10	2	no	—
331	C₂	—	10	2	no	—
332	C₂	C₃×D₅×C₂²⋊C₈	1	1	yes	characteristic
333	C₂	C₂×C₃×#3930	1	1	yes	characteristic
334	C₂	C₃×D₅×M₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	C₂²×D₅
commutator	C₂²×C₅	C₂²×C₃×C₈
Frattini	C₂×C₈	C₂²×C₃×D₅
Fitting	C₃×C₅×C₂²⋊C₁₆	C₂
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂³×C₃×D₅	C₈
cosocle	C₂×C₅×C₈	C₂³×C₃
hypercenter	C₃×C₂²⋊C₁₆	D₅
hypocenter	C₅	C₃×#2112
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₃×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#3930	C₂	semidirect
3. C₃×C₅×C₂²⋊C₁₆	C₂	semidirect
4. C₃×C₂²⋊#672	C₂	semidirect
5. C₃×#64727	C₂	semidirect
6. #10599899	C₃	direct (N × H)
15. C₂×C₃×C₅×C₁₆	C₂²	semidirect
16. C₂×C₃×#672	C₂²	semidirect
20. C₂×#3930	C₂×C₃	semidirect
21. C₅×C₂²⋊C₁₆	C₂×C₃	semidirect
22. C₂²⋊#672	C₂×C₃	semidirect
23. #64727	C₂×C₃	semidirect
49. C₃×C₂²⋊C₁₆	D₅	semidirect
58. C₂×C₅×C₁₆	C₂²×C₃	semidirect
59. C₂×#672	C₂²×C₃	semidirect
95. C₂×C₃×C₁₆	C₂×D₅	semidirect
121. C₂²⋊C₁₆	C₃×D₅	semidirect
124. C₂²×C₃×C₅	M₃₂	semidirect
132. C₂×C₃×D₅	M₃₂	semidirect
192. C₂×C₁₆	C₂×C₃×D₅	semidirect
224. C₂²×C₅	C₃×M₃₂	semidirect
232. C₂×D₅	C₃×M₃₂	semidirect
259. C₃×C₅	#2112	semidirect
269. C₂²×C₃	#3930	semidirect
311. C₅	C₃×#2112	semidirect
321. C₂²	C₃×#3930	semidirect
328. C₃	#10599899	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₈×D₅	2
C₂×C₃×#3930	2
C₃×C₅×C₂²⋊C₁₆	2
C₃×C₂²⋊#672	2
C₃×#64727	2
#10599899	3
C₃×#2112	5

Subgroups — C3×#10599899 #14312821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10599899 #14312821