Subgroups — C₃×#10615548 #14412906

Order 1920.

1476 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	C₃×#66283	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#568	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#41×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65428	C₂	1	6	yes	maximal, complemented
5	C₂×C₃×#3957	C₂	1	6	yes	maximal, complemented
6	#10615548	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₃×#4150	C₄	1	4	yes	complemented
9	C₃×#4149	C₄	1	4	yes	complemented
10	C₃×D₅×#41	—	2	4	no	—
11	C₃×#4147	—	2	1	no	—
12	C₃×#4144	—	2	3	no	—
13	C₂×C₃×#709	C₂²	1	6	yes	—
14	C₃×C₅×#131	—	2	4	no	—
15	C₂×C₃×C₅×M₁₆	C₂²	1	6	yes	complemented
16	C₃×Q₈×Dic₅	—	2	1	no	—
17	C₃×D₄×Dic₅	—	2	3	no	—
18	C₃×#4072	C₂²	1	3	yes	complemented
19	C₂×C₃×C₄×Dic₅	C₂²	1	3	yes	complemented
20	C₃×#3957	C₂²	1	4	yes	complemented
21	C₃×#3957	C₂²	1	12	yes	complemented
22	C₃×#2901	—	5	1	no	maximal, Hall
23	#66283	C₂×C₃	1	1	yes	characteristic
24	C₅×#568	C₂×C₃	1	1	yes	characteristic, complemented
25	#41×Dic₅	C₂×C₃	1	1	yes	characteristic, complemented
26	#65428	C₂×C₃	1	6	yes	complemented
27	C₂×#3957	C₂×C₃	1	6	yes	complemented
28	C₃×C₅×#41	C₂×C₄	1	4	yes	—
29	C₃×C₅×#41	D₄	1	4	yes	—
30	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
31	C₂×C₃×C₅×D₄	D₄	1	3	yes	—
32	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
33	C₂×C₃×C₂²⋊D₅	—	2	6	no	—
34	C₂²×C₃×Dic₅	—	2	3	no	—
35	C₃×#713	—	2	4	no	—
36	C₃×Q₈×D₅	—	2	4	no	—
37	C₃×#711	—	2	12	no	—
38	C₃×D₄×D₅	—	2	12	no	—
39	C₃×#709	—	2	6	no	—
40	C₃×#709	—	2	6	no	—
41	C₃×#709	C₂×C₄	1	12	yes	complemented
42	C₃×#709	C₂³	1	4	yes	—
43	C₂×C₃×D₂₀	C₂×C₄	1	6	yes	complemented
44	C₂×C₃×C₄×D₅	—	2	6	no	—
45	C₂×C₃×Dic₁₀	C₂×C₄	1	6	yes	complemented
46	C₃×C₅×M₁₆	—	2	6	no	—
47	C₃×C₅×M₁₆	C₂³	1	4	yes	—
48	C₂×C₃×C₅×C₈	—	2	6	no	—
49	C₃×C₂²⋊Dic₅	—	2	3	no	—
50	C₃×C₄⋊Dic₅	—	2	3	no	—
51	C₃×C₄×Dic₅	—	2	3	no	—
52	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
53	C₃×C₄×Dic₅	C₂³	1	3	yes	—
54	C₃×#585	—	5	1	no	—
55	C₃×#568	D₅	1	1	yes	characteristic, complemented
56	C₃×C₄×#41	—	5	1	no	—
57	C₃×#421	—	5	6	no	—
58	C₂×C₃×#104	—	5	6	no	—
59	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
60	#4150	C₃×C₄	1	4	yes	complemented
61	#4149	C₃×C₄	1	4	yes	complemented
62	D₅×#41	—	2	4	no	—
63	#4147	—	2	1	no	—
64	#4144	—	2	3	no	—
65	C₂×#709	C₂²×C₃	1	6	yes	—
66	C₅×#131	—	2	4	no	—
67	C₂×C₅×M₁₆	C₂²×C₃	1	6	yes	complemented
68	Q₈×Dic₅	—	2	1	no	—
69	D₄×Dic₅	—	2	3	no	—
70	#4072	C₂²×C₃	1	3	yes	complemented
71	C₂×C₄×Dic₅	C₂²×C₃	1	3	yes	complemented
72	#3957	C₂²×C₃	1	4	yes	complemented
73	#3957	C₂²×C₃	1	12	yes	complemented
74	#2901	—	5	1	no	Sylow
75	C₂³×C₃×C₅	C₂×D₄	1	3	yes	—
76	C₂²×C₃×D₅	—	2	6	no	—
77	C₃×C₅×Q₈	C₂²⋊C₄	1	4	yes	—
78	C₃×C₅×D₄	C₂²⋊C₄	1	12	yes	—
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
80	C₂×C₃×C₄×C₅	C₂×D₄	1	3	yes	—
81	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
82	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
83	C₃×C₂²⋊D₅	—	2	12	no	—
84	C₃×C₂²⋊D₅	—	2	4	no	—
85	C₂×C₃×Dic₅	—	2	6	no	—
86	C₂×C₃×Dic₅	—	2	1	no	—
87	C₂×C₃×Dic₅	—	2	3	no	—
88	C₂×C₃×Dic₅	—	4	6	no	—
89	C₃×D₂₀	—	2	6	no	—
90	C₃×D₂₀	C₂²×C₄	1	4	yes	—
91	C₃×C₄×D₅	—	2	12	no	—
92	C₃×C₄×D₅	—	2	4	no	—
93	C₃×Dic₁₀	—	2	6	no	—
94	C₃×Dic₁₀	C₂²×C₄	1	4	yes	—
95	C₃×C₅×C₈	—	2	4	no	—
96	C₃×2₋¹⁺⁴	—	5	4	no	—
97	C₃×2₋¹⁺⁴	—	10	1	no	—
98	C₃×H(5,𝔽₂)	—	5	4	no	—
99	C₃×H(5,𝔽₂)	—	10	3	no	—
100	C₂×C₃×#41	—	5	6	no	—
101	C₂×C₃×#41	—	10	4	no	—
102	C₂×C₃×#41	C₂×D₅	1	1	yes	characteristic
103	C₃×#131	—	2	4	no	—
104	C₂×C₃×M₁₆	C₂×D₅	1	6	yes	complemented
105	C₃×C₄×Q₈	—	10	1	no	—
106	C₃×C₄×D₄	—	10	3	no	—
107	C₃×#117	—	5	3	no	—
108	C₂×C₃×C₄²	—	5	3	no	—
109	C₃×#104	—	5	4	no	—
110	C₃×#104	—	5	12	no	—
111	C₅×#41	C₂×C₃×C₄	1	4	yes	—
112	C₅×#41	C₃×D₄	1	4	yes	—
113	C₂×C₅×Q₈	C₃×D₄	1	1	yes	characteristic
114	C₂×C₅×D₄	C₃×D₄	1	3	yes	—
115	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
116	C₂×C₂²⋊D₅	—	2	6	no	—
117	C₂²×Dic₅	—	2	3	no	—
118	#713	—	2	4	no	—
119	Q₈×D₅	—	2	4	no	—
120	#711	—	2	12	no	—
121	D₄×D₅	—	2	12	no	—
122	#709	—	2	6	no	—
123	#709	—	2	6	no	—
124	#709	C₂×C₃×C₄	1	12	yes	complemented
125	#709	C₂³×C₃	1	4	yes	—
126	C₂×D₂₀	C₂×C₃×C₄	1	6	yes	complemented
127	C₂×C₄×D₅	—	2	6	no	—
128	C₂×Dic₁₀	C₂×C₃×C₄	1	6	yes	complemented
129	C₅×M₁₆	—	2	6	no	—
130	C₅×M₁₆	C₂³×C₃	1	4	yes	—
131	C₂×C₅×C₈	—	2	6	no	—
132	C₂²⋊Dic₅	—	2	3	no	—
133	C₄⋊Dic₅	—	2	3	no	—
134	C₄×Dic₅	—	2	3	no	—
135	C₄×Dic₅	C₂³×C₃	1	1	yes	characteristic
136	C₄×Dic₅	C₂³×C₃	1	3	yes	—
137	#585	—	5	1	no	—
138	#568	C₃×D₅	1	1	yes	characteristic, complemented
139	C₄×#41	—	5	1	no	—
140	#421	—	5	6	no	—
141	C₂×#104	—	5	6	no	—
142	C₂²×C₃×C₅	—	4	3	no	—
143	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	6	yes	—
144	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₃×D₅	—	2	4	no	—
146	C₂×C₃×D₅	—	8	6	no	—
147	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	6	yes	—
148	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
149	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
150	C₃×Dic₅	—	2	2	no	—
151	C₃×Dic₅	—	2	4	no	—
152	C₃×Dic₅	—	4	3	no	—
153	C₃×#41	—	5	12	no	—
154	C₃×#41	—	5	4	no	—
155	C₃×#41	—	10	4	no	—
156	C₃×#41	—	10	12	no	—
157	C₃×#41	—	10	6	no	—
158	C₃×#41	—	10	6	no	—
159	C₃×#41	C₄×D₅	1	4	yes	—
160	C₃×#41	C₂²⋊D₅	1	4	yes	—
161	C₂×C₃×Q₈	—	5	6	no	—
162	C₂×C₃×Q₈	—	10	4	no	—
163	C₂×C₃×Q₈	D₂₀	1	1	yes	characteristic
164	C₂×C₃×D₄	—	5	6	no	—
165	C₂×C₃×D₄	—	10	6	no	—
166	C₂×C₃×D₄	—	10	12	no	—
167	C₂×C₃×D₄	D₂₀	1	3	yes	—
168	C₂²×C₃×C₄	—	10	6	no	—
169	C₂²×C₃×C₄	—	10	3	no	—
170	C₂²×C₃×C₄	C₂²×D₅	1	3	yes	—
171	C₃×M₁₆	—	2	6	no	—
172	C₃×M₁₆	C₂²×D₅	1	4	yes	—
173	C₂×C₃×C₈	—	2	6	no	—
174	C₃×C₄⋊C₄	—	10	3	no	—
175	C₃×C₂²⋊C₄	—	10	3	no	—
176	C₃×C₄²	—	5	1	no	—
177	C₃×C₄²	—	5	3	no	—
178	C₃×C₄²	—	10	3	no	—
179	C₂³×C₅	C₂×C₃×D₄	1	3	yes	—
180	C₂²×D₅	—	2	6	no	—
181	C₅×Q₈	C₃×C₂²⋊C₄	1	4	yes	—
182	C₅×D₄	C₃×C₂²⋊C₄	1	12	yes	—
183	C₂×C₄×C₅	C₂²×C₃×C₄	1	6	yes	—
184	C₂×C₄×C₅	C₂×C₃×D₄	1	3	yes	—
185	C₂×C₄×C₅	C₂×C₃×D₄	1	6	yes	—
186	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
187	C₂²⋊D₅	—	2	12	no	—
188	C₂²⋊D₅	—	2	4	no	—
189	C₂×Dic₅	—	2	6	no	—
190	C₂×Dic₅	—	2	1	no	—
191	C₂×Dic₅	—	2	3	no	—
192	C₂×Dic₅	—	4	6	no	—
193	D₂₀	—	2	6	no	—
194	D₂₀	C₂²×C₃×C₄	1	4	yes	—
195	C₄×D₅	—	2	12	no	—
196	C₄×D₅	—	2	4	no	—
197	Dic₁₀	—	2	6	no	—
198	Dic₁₀	C₂²×C₃×C₄	1	4	yes	—
199	C₅×C₈	—	2	4	no	—
200	2₋¹⁺⁴	—	5	4	no	—
201	2₋¹⁺⁴	—	10	1	no	—
202	H(5,𝔽₂)	—	5	4	no	—
203	H(5,𝔽₂)	—	10	3	no	—
204	C₂×#41	—	5	6	no	—
205	C₂×#41	—	10	4	no	—
206	C₂×#41	C₂×C₃×D₅	1	1	yes	characteristic
207	#131	—	2	4	no	—
208	C₂×M₁₆	C₂×C₃×D₅	1	6	yes	complemented
209	C₄×Q₈	—	10	1	no	—
210	C₄×D₄	—	10	3	no	—
211	#117	—	5	3	no	—
212	C₂×C₄²	—	5	3	no	—
213	#104	—	5	4	no	—
214	#104	—	5	12	no	—
215	C₂×C₃×C₅	—	2	1	no	—
216	C₂×C₃×C₅	—	2	6	no	—
217	C₂×C₃×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
218	C₃×D₅	—	4	4	no	—
219	C₂³×C₃	—	10	6	no	—
220	C₂³×C₃	C₂×D₂₀	1	3	yes	—
221	C₃×Q₈	—	5	4	no	—
222	C₃×Q₈	—	10	6	no	—
223	C₃×Q₈	#685	1	4	yes	—
224	C₃×D₄	—	5	4	no	—
225	C₃×D₄	—	10	4	no	—
226	C₃×D₄	—	10	12	no	—
227	C₃×D₄	—	10	6	no	—
228	C₃×D₄	#685	1	12	yes	—
229	C₂×C₃×C₄	—	10	1	no	—
230	C₂×C₃×C₄	—	10	6	no	—
231	C₂×C₃×C₄	—	10	4	no	—
232	C₂×C₃×C₄	—	10	12	no	—
233	C₂×C₃×C₄	—	10	3	no	—
234	C₂×C₃×C₄	—	20	6	no	—
235	C₂×C₃×C₄	C₂×C₄×D₅	1	6	yes	—
236	C₂×C₃×C₄	C₂×D₂₀	1	3	yes	—
237	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	6	yes	—
238	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
239	C₃×C₈	—	2	4	no	—
240	C₂²×C₅	—	4	3	no	—
241	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	6	yes	—
242	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
243	C₂×D₅	—	2	4	no	—
244	C₂×D₅	—	8	6	no	—
245	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	6	yes	—
246	C₄×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
247	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
248	Dic₅	—	2	2	no	—
249	Dic₅	—	2	4	no	—
250	Dic₅	—	4	3	no	—
251	#41	—	5	12	no	—
252	#41	—	5	4	no	—
253	#41	—	10	4	no	—
254	#41	—	10	12	no	—
255	#41	—	10	6	no	—
256	#41	—	10	6	no	—
257	#41	C₃×C₄×D₅	1	4	yes	—
258	#41	C₃×C₂²⋊D₅	1	4	yes	—
259	C₂×Q₈	—	5	6	no	—
260	C₂×Q₈	—	10	4	no	—
261	C₂×Q₈	C₃×D₂₀	1	1	yes	characteristic
262	C₂×D₄	—	5	6	no	—
263	C₂×D₄	—	10	6	no	—
264	C₂×D₄	—	10	12	no	—
265	C₂×D₄	C₃×D₂₀	1	3	yes	—
266	C₂²×C₄	—	10	6	no	—
267	C₂²×C₄	—	10	3	no	—
268	C₂²×C₄	C₂²×C₃×D₅	1	3	yes	—
269	M₁₆	—	2	6	no	—
270	M₁₆	C₂²×C₃×D₅	1	4	yes	—
271	C₂×C₈	—	2	6	no	—
272	C₄⋊C₄	—	10	3	no	—
273	C₂²⋊C₄	—	10	3	no	—
274	C₄²	—	5	1	no	—
275	C₄²	—	5	3	no	—
276	C₄²	—	10	3	no	—
277	C₃×C₅	#2901	1	1	yes	characteristic, complemented, Hall
278	C₂²×C₃	—	4	3	no	—
279	C₂²×C₃	—	10	4	no	—
280	C₂²×C₃	—	40	6	no	—
281	C₂²×C₃	C₂×#685	1	6	yes	—
282	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
283	C₃×C₄	—	10	2	no	—
284	C₃×C₄	—	10	4	no	—
285	C₃×C₄	—	20	3	no	—
286	C₃×C₄	C₂×#685	1	6	yes	—
287	C₃×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
288	C₃×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
289	C₂×C₅	—	2	1	no	—
290	C₂×C₅	—	2	6	no	—
291	C₂×C₅	C₂²×C₃×C₂²⋊C₄	1	1	yes	characteristic
292	D₅	—	4	4	no	—
293	C₂³	—	10	6	no	—
294	C₂³	C₂×C₃×D₂₀	1	3	yes	—
295	Q₈	—	5	4	no	—
296	Q₈	—	10	6	no	—
297	Q₈	C₃×#685	1	4	yes	—
298	D₄	—	5	4	no	—
299	D₄	—	10	4	no	—
300	D₄	—	10	12	no	—
301	D₄	—	10	6	no	—
302	D₄	C₃×#685	1	12	yes	—
303	C₂×C₄	—	10	1	no	—
304	C₂×C₄	—	10	6	no	—
305	C₂×C₄	—	10	4	no	—
306	C₂×C₄	—	10	12	no	—
307	C₂×C₄	—	10	3	no	—
308	C₂×C₄	—	20	6	no	—
309	C₂×C₄	C₂×C₃×C₄×D₅	1	6	yes	—
310	C₂×C₄	C₂×C₃×D₂₀	1	3	yes	—
311	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	6	yes	—
312	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
313	C₈	—	2	4	no	—
314	C₂×C₃	—	2	1	no	—
315	C₂×C₃	—	2	6	no	—
316	C₂×C₃	—	20	4	no	—
317	C₂×C₃	C₂²×#685	1	1	yes	characteristic
318	C₅	C₃×#2901	1	1	yes	characteristic, complemented, Sylow
319	C₂²	—	4	3	no	—
320	C₂²	—	10	4	no	—
321	C₂²	—	40	6	no	—
322	C₂²	C₂×C₃×#685	1	6	yes	—
323	C₂²	C₂²×C₃×D₂₀	1	1	yes	characteristic
324	C₄	—	10	2	no	—
325	C₄	—	10	4	no	—
326	C₄	—	20	3	no	—
327	C₄	C₂×C₃×#685	1	6	yes	—
328	C₄	C₂²×C₃×C₄×D₅	1	1	yes	characteristic
329	C₄	C₂²×C₃×C₂²⋊D₅	1	1	yes	characteristic
330	C₃	#10615548	1	1	yes	characteristic, direct product factor, Sylow
331	C₂	—	2	1	no	—
332	C₂	—	2	6	no	—
333	C₂	—	20	4	no	—
334	C₂	C₂²×C₃×#685	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	C₂²×C₂²⋊D₅
commutator	C₄×C₅	C₂³×C₃×C₄
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₃×C₅×#568	C₂
socle	C₂×C₃×C₅	C₂²×C₂²⋊C₄
prime radical	C₃×#66283	C₂
cosocle	C₂×C₄×C₅	C₂⁴×C₃
hypercenter	C₃×#568	D₅
hypocenter	C₅	C₃×#2901
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×#41 ◁ C₃×#568

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#568	C₂	semidirect
3. C₃×#41×Dic₅	C₂	semidirect
4. C₃×#65428	C₂	semidirect
5. C₂×C₃×#3957	C₂	semidirect
6. #10615548	C₃	direct (N × H)
8. C₃×#4150	C₄	semidirect
9. C₃×#4149	C₄	semidirect
15. C₂×C₃×C₅×M₁₆	C₂²	semidirect
18. C₃×#4072	C₂²	semidirect
19. C₂×C₃×C₄×Dic₅	C₂²	semidirect
20. C₃×#3957	C₂²	semidirect
21. C₃×#3957	C₂²	semidirect
24. C₅×#568	C₂×C₃	semidirect
25. #41×Dic₅	C₂×C₃	semidirect
26. #65428	C₂×C₃	semidirect
27. C₂×#3957	C₂×C₃	semidirect
41. C₃×#709	C₂×C₄	semidirect
43. C₂×C₃×D₂₀	C₂×C₄	semidirect
45. C₂×C₃×Dic₁₀	C₂×C₄	semidirect
55. C₃×#568	D₅	semidirect
60. #4150	C₃×C₄	semidirect
61. #4149	C₃×C₄	semidirect
67. C₂×C₅×M₁₆	C₂²×C₃	semidirect
70. #4072	C₂²×C₃	semidirect
71. C₂×C₄×Dic₅	C₂²×C₃	semidirect
72. #3957	C₂²×C₃	semidirect
73. #3957	C₂²×C₃	semidirect
104. C₂×C₃×M₁₆	C₂×D₅	semidirect
124. #709	C₂×C₃×C₄	semidirect
126. C₂×D₂₀	C₂×C₃×C₄	semidirect
128. C₂×Dic₁₀	C₂×C₃×C₄	semidirect
138. #568	C₃×D₅	semidirect
208. C₂×M₁₆	C₂×C₃×D₅	semidirect
277. C₃×C₅	#2901	semidirect
318. C₅	C₃×#2901	semidirect
330. C₃	#10615548	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×#66283	2
C₃×C₅×#568	2
C₃×#41×Dic₅	2
C₃×#65428	2
C₂×C₃×#3957	2
#10615548	3
C₃×#2901	5

Subgroups — C3×#10615548 #14412906

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10615548 #14412906