Subgroups — C₃×#10602823 #14303038

Order 1920.

680 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	C₃×C₅×#568	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#66164	C₂	1	1	yes	maximal, characteristic
3	C₃×#65526	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#3955	C₂	1	2	yes	maximal, complemented
5	C₃×#64779	C₂	1	2	yes	maximal, complemented
6	#10602823	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₂²×C₃×D₂₀	C₂²	1	1	yes	characteristic
9	C₃×C₅×#131	—	2	4	no	—
10	C₂×C₃×C₅×M₁₆	—	2	2	no	—
11	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
12	C₃×#4093	—	2	1	no	—
13	C₃×#4086	—	2	1	no	—
14	C₃×#4083	C₄	1	4	yes	—
15	C₃×#4076	—	2	2	no	—
16	C₃×#4072	C₂²	1	1	yes	characteristic
17	C₂×C₃×#681	C₂²	1	2	yes	—
18	C₃×#3968	—	2	1	no	—
19	C₃×#3965	—	2	1	no	—
20	C₃×#3955	—	2	4	no	—
21	C₃×#1914	—	5	1	no	maximal, Hall
22	C₅×#568	C₂×C₃	1	1	yes	characteristic, complemented
23	#66164	C₂×C₃	1	1	yes	characteristic
24	#65526	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂×#3955	C₂×C₃	1	2	yes	complemented
26	#64779	C₂×C₃	1	2	yes	complemented
27	C₂³×C₃×D₅	C₂×C₄	1	2	yes	—
28	C₃×C₅×#41	—	2	2	no	—
29	C₃×C₅×#41	—	2	2	no	—
30	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
31	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
32	C₂×C₃×C₅×D₄	D₄	1	1	yes	characteristic
33	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
34	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂×C₃×C₂²⋊D₅	—	2	4	no	—
36	C₂×C₃×D₂₀	—	2	1	no	—
37	C₂×C₃×D₂₀	—	2	2	no	—
38	C₂×C₃×D₂₀	—	2	1	no	—
39	C₂×C₃×D₂₀	—	2	2	no	—
40	C₃×C₅×M₁₆	—	2	4	no	—
41	C₃×C₅×M₁₆	—	2	2	no	—
42	C₃×C₅×M₁₆	D₄	1	4	yes	complemented
43	C₂×C₃×C₅×C₈	—	2	4	no	—
44	C₂×C₃×C₅×C₈	—	2	2	no	—
45	C₃×C₂²⋊Dic₅	C₂×C₄	1	2	yes	—
46	C₃×#685	—	2	4	no	—
47	C₃×C₄⋊Dic₅	—	2	1	no	—
48	C₃×C₄×Dic₅	—	2	1	no	—
49	C₃×#681	—	2	2	no	—
50	C₃×#681	—	2	2	no	—
51	C₂×C₃×#168	—	2	2	no	—
52	C₃×#568	D₅	1	1	yes	characteristic, complemented
53	C₃×#535	—	5	1	no	—
54	C₃×#413	—	5	1	no	—
55	C₂×C₃×#100	—	5	2	no	—
56	C₃×#344	—	5	2	no	—
57	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
58	C₂²×D₂₀	C₂²×C₃	1	1	yes	characteristic
59	C₅×#131	—	2	4	no	—
60	C₂×C₅×M₁₆	—	2	2	no	—
61	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
62	#4093	—	2	1	no	—
63	#4086	—	2	1	no	—
64	#4083	C₃×C₄	1	4	yes	—
65	#4076	—	2	2	no	—
66	#4072	C₂²×C₃	1	1	yes	characteristic
67	C₂×#681	C₂²×C₃	1	2	yes	—
68	#3968	—	2	1	no	—
69	#3965	—	2	1	no	—
70	#3955	—	2	4	no	—
71	#1914	—	5	1	no	Sylow
72	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
73	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₃×D₅	—	2	2	no	—
75	C₂²×C₃×D₅	—	2	4	no	—
76	C₂²×C₃×D₅	—	8	2	no	—
77	C₃×C₅×Q₈	—	2	2	no	—
78	C₃×C₅×D₄	—	2	4	no	—
79	C₃×C₅×D₄	—	2	2	no	—
80	C₂×C₃×C₄×C₅	—	2	2	no	—
81	C₂×C₃×C₄×C₅	—	2	2	no	—
82	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
83	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
84	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	#41	1	2	yes	—
87	C₃×C₂²⋊D₅	—	8	4	no	—
88	C₂×C₃×Dic₅	—	2	2	no	—
89	C₃×D₂₀	—	4	1	no	—
90	C₃×D₂₀	—	4	2	no	—
91	C₃×D₂₀	—	4	1	no	—
92	C₃×C₅×C₈	—	2	4	no	—
93	C₃×#168	—	4	2	no	—
94	C₂×C₃×#41	C₂×D₅	1	1	yes	characteristic
95	C₂²×C₃×D₄	—	5	1	no	—
96	C₃×#131	—	2	4	no	—
97	C₂×C₃×M₁₆	—	2	2	no	—
98	C₂×C₃×M₁₆	—	5	2	no	—
99	C₂×C₃×M₁₆	C₂×D₅	1	2	yes	—
100	C₃×#127	—	10	1	no	—
101	C₃×#124	—	10	1	no	—
102	C₃×#121	—	10	2	no	—
103	C₃×#120	—	5	4	no	—
104	C₃×#117	—	5	1	no	—
105	C₃×#101	—	10	1	no	—
106	C₃×#100	—	10	1	no	—
107	C₃×#100	—	10	4	no	—
108	C₂³×D₅	C₂×C₃×C₄	1	2	yes	—
109	C₅×#41	—	2	2	no	—
110	C₅×#41	—	2	2	no	—
111	C₂×C₅×Q₈	C₃×D₄	1	1	yes	characteristic
112	C₂×C₅×D₄	C₂×C₃×C₄	1	2	yes	—
113	C₂×C₅×D₄	C₃×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
115	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
116	C₂×C₂²⋊D₅	—	2	4	no	—
117	C₂×D₂₀	—	2	2	no	—
118	C₂×D₂₀	—	2	1	no	—
119	C₂×D₂₀	—	2	1	no	—
120	C₂×D₂₀	—	2	2	no	—
121	C₅×M₁₆	—	2	4	no	—
122	C₅×M₁₆	—	2	2	no	—
123	C₅×M₁₆	C₃×D₄	1	4	yes	complemented
124	C₂×C₅×C₈	—	2	4	no	—
125	C₂×C₅×C₈	—	2	2	no	—
126	C₂²⋊Dic₅	C₂×C₃×C₄	1	2	yes	—
127	#685	—	2	4	no	—
128	C₄⋊Dic₅	—	2	1	no	—
129	C₄×Dic₅	—	2	1	no	—
130	#681	—	2	2	no	—
131	#681	—	2	2	no	—
132	C₂×#168	—	2	2	no	—
133	#568	C₃×D₅	1	1	yes	characteristic, complemented
134	#535	—	5	1	no	—
135	#413	—	5	1	no	—
136	C₂×#100	—	5	2	no	—
137	#344	—	5	2	no	—
138	C₂²×C₃×C₅	—	2	2	no	—
139	C₂²×C₃×C₅	—	4	1	no	—
140	C₂²×C₃×C₅	—	4	2	no	—
141	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
143	C₂×C₃×D₅	—	4	4	no	—
144	C₂×C₃×D₅	—	4	2	no	—
145	C₂×C₃×D₅	—	8	4	no	—
146	C₃×C₄×C₅	—	2	2	no	—
147	C₃×C₄×C₅	#120	1	1	yes	characteristic
148	C₃×C₄×C₅	#121	1	2	yes	—
149	C₃×C₄×C₅	#123	1	1	yes	characteristic
150	C₃×Dic₅	—	4	2	no	—
151	C₂⁴×C₃	—	5	2	no	—
152	C₃×#41	—	2	2	no	—
153	C₃×#41	—	2	2	no	—
154	C₂×C₃×Q₈	D₂₀	1	1	yes	characteristic
155	C₂×C₃×D₄	—	10	4	no	—
156	C₂×C₃×D₄	—	10	1	no	—
157	C₂×C₃×D₄	—	10	2	no	—
158	C₂×C₃×D₄	—	10	2	no	—
159	C₂×C₃×D₄	—	10	1	no	—
160	C₂×C₃×D₄	C₄×D₅	1	2	yes	—
161	C₂×C₃×D₄	D₂₀	1	1	yes	characteristic
162	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
163	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
164	C₃×M₁₆	—	2	4	no	—
165	C₃×M₁₆	—	2	2	no	—
166	C₃×M₁₆	—	10	2	no	—
167	C₃×M₁₆	—	10	2	no	—
168	C₃×M₁₆	C₂²⋊D₅	1	4	yes	complemented
169	C₂×C₃×C₈	—	2	4	no	—
170	C₂×C₃×C₈	—	2	2	no	—
171	C₂×C₃×C₈	—	10	2	no	—
172	C₃×C₄⋊C₄	—	10	1	no	—
173	C₃×C₂²⋊C₄	—	5	2	no	—
174	C₃×C₂²⋊C₄	—	10	4	no	—
175	C₃×C₄²	—	10	1	no	—
176	C₂³×C₅	C₃×C₂²⋊C₄	1	2	yes	—
177	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
178	C₂²×D₅	—	2	2	no	—
179	C₂²×D₅	—	2	4	no	—
180	C₂²×D₅	—	8	2	no	—
181	C₅×Q₈	—	2	2	no	—
182	C₅×D₄	—	2	4	no	—
183	C₅×D₄	—	2	2	no	—
184	C₂×C₄×C₅	—	2	2	no	—
185	C₂×C₄×C₅	—	2	2	no	—
186	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
187	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
188	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
189	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
190	C₂×C₄×C₅	C₃×#41	1	2	yes	—
191	C₂²⋊D₅	—	8	4	no	—
192	C₂×Dic₅	—	2	2	no	—
193	D₂₀	—	4	1	no	—
194	D₂₀	—	4	2	no	—
195	D₂₀	—	4	1	no	—
196	C₅×C₈	—	2	4	no	—
197	#168	—	4	2	no	—
198	C₂×#41	C₂×C₃×D₅	1	1	yes	characteristic
199	C₂²×D₄	—	5	1	no	—
200	#131	—	2	4	no	—
201	C₂×M₁₆	—	2	2	no	—
202	C₂×M₁₆	—	5	2	no	—
203	C₂×M₁₆	C₂×C₃×D₅	1	2	yes	—
204	#127	—	10	1	no	—
205	#124	—	10	1	no	—
206	#121	—	10	2	no	—
207	#120	—	5	4	no	—
208	#117	—	5	1	no	—
209	#101	—	10	1	no	—
210	#100	—	10	1	no	—
211	#100	—	10	4	no	—
212	C₂×C₃×C₅	—	2	1	no	—
213	C₂×C₃×C₅	—	2	2	no	—
214	C₂×C₃×C₅	—	4	2	no	—
215	C₂×C₃×C₅	#386	1	1	yes	characteristic
216	C₃×D₅	—	8	2	no	—
217	C₂³×C₃	—	10	2	no	—
218	C₂³×C₃	—	10	4	no	—
219	C₂³×C₃	—	40	2	no	—
220	C₂³×C₃	#685	1	2	yes	—
221	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
222	C₃×Q₈	—	2	2	no	—
223	C₃×D₄	—	2	4	no	—
224	C₃×D₄	—	2	2	no	—
225	C₃×D₄	—	20	1	no	—
226	C₃×D₄	—	20	1	no	—
227	C₃×D₄	—	20	2	no	—
228	C₃×D₄	—	40	4	no	—
229	C₂×C₃×C₄	—	2	2	no	—
230	C₂×C₃×C₄	—	2	2	no	—
231	C₂×C₃×C₄	—	10	2	no	—
232	C₂×C₃×C₄	#685	1	2	yes	—
233	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
234	C₂×C₃×C₄	#709	1	2	yes	—
235	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
236	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
237	C₃×C₈	—	2	4	no	—
238	C₃×C₈	—	20	2	no	—
239	C₂²×C₅	—	2	2	no	—
240	C₂²×C₅	—	4	1	no	—
241	C₂²×C₅	—	4	2	no	—
242	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
243	C₂²×C₅	C₃×C₄×D₄	1	2	yes	—
244	C₂×D₅	—	4	4	no	—
245	C₂×D₅	—	4	2	no	—
246	C₂×D₅	—	8	4	no	—
247	C₄×C₅	—	2	2	no	—
248	C₄×C₅	C₃×#120	1	1	yes	characteristic
249	C₄×C₅	C₃×#121	1	2	yes	—
250	C₄×C₅	C₃×#123	1	1	yes	characteristic
251	Dic₅	—	4	2	no	—
252	C₂⁴	—	5	2	no	—
253	#41	—	2	2	no	—
254	#41	—	2	2	no	—
255	C₂×Q₈	C₃×D₂₀	1	1	yes	characteristic
256	C₂×D₄	—	10	4	no	—
257	C₂×D₄	—	10	1	no	—
258	C₂×D₄	—	10	2	no	—
259	C₂×D₄	—	10	2	no	—
260	C₂×D₄	—	10	1	no	—
261	C₂×D₄	C₃×C₄×D₅	1	2	yes	—
262	C₂×D₄	C₃×D₂₀	1	1	yes	characteristic
263	C₂²×C₄	C₃×C₂²⋊D₅	1	2	yes	—
264	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
265	M₁₆	—	2	4	no	—
266	M₁₆	—	2	2	no	—
267	M₁₆	—	10	2	no	—
268	M₁₆	—	10	2	no	—
269	M₁₆	C₃×C₂²⋊D₅	1	4	yes	complemented
270	C₂×C₈	—	2	4	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	10	2	no	—
273	C₄⋊C₄	—	10	1	no	—
274	C₂²⋊C₄	—	5	2	no	—
275	C₂²⋊C₄	—	10	4	no	—
276	C₄²	—	10	1	no	—
277	C₃×C₅	#1914	1	1	yes	characteristic, complemented, Hall
278	C₂²×C₃	—	2	2	no	—
279	C₂²×C₃	—	4	1	no	—
280	C₂²×C₃	—	4	2	no	—
281	C₂²×C₃	—	20	4	no	—
282	C₂²×C₃	—	20	2	no	—
283	C₂²×C₃	—	40	4	no	—
284	C₂²×C₃	C₂×#685	1	1	yes	characteristic
285	C₂²×C₃	C₄×C₂²⋊D₅	1	2	yes	—
286	C₃×C₄	—	2	2	no	—
287	C₃×C₄	—	20	2	no	—
288	C₃×C₄	#4075	1	1	yes	characteristic
289	C₃×C₄	#4076	1	2	yes	—
290	C₃×C₄	#4099	1	1	yes	characteristic
291	C₂×C₅	—	2	1	no	—
292	C₂×C₅	—	2	2	no	—
293	C₂×C₅	—	4	2	no	—
294	C₂×C₅	C₃×#386	1	1	yes	characteristic
295	D₅	—	8	2	no	—
296	C₂³	—	10	2	no	—
297	C₂³	—	10	4	no	—
298	C₂³	—	40	2	no	—
299	C₂³	C₃×#685	1	2	yes	—
300	C₂³	C₂×C₃×C₄×D₅	1	1	yes	characteristic
301	Q₈	—	2	2	no	—
302	D₄	—	2	4	no	—
303	D₄	—	2	2	no	—
304	D₄	—	20	1	no	—
305	D₄	—	20	1	no	—
306	D₄	—	20	2	no	—
307	D₄	—	40	4	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	10	2	no	—
311	C₂×C₄	C₃×#685	1	2	yes	—
312	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
313	C₂×C₄	C₃×#709	1	2	yes	—
314	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
315	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	2	yes	—
316	C₈	—	2	4	no	—
317	C₈	—	20	2	no	—
318	C₂×C₃	—	2	1	no	—
319	C₂×C₃	—	2	2	no	—
320	C₂×C₃	—	4	2	no	—
321	C₂×C₃	—	40	2	no	—
322	C₂×C₃	#65502	1	1	yes	characteristic
323	C₅	C₃×#1914	1	1	yes	characteristic, complemented, Sylow
324	C₂²	—	2	2	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	4	2	no	—
327	C₂²	—	20	4	no	—
328	C₂²	—	20	2	no	—
329	C₂²	—	40	4	no	—
330	C₂²	C₂×C₃×#685	1	1	yes	characteristic
331	C₂²	C₃×C₄×C₂²⋊D₅	1	2	yes	—
332	C₄	—	2	2	no	—
333	C₄	—	20	2	no	—
334	C₄	C₃×#4075	1	1	yes	characteristic
335	C₄	C₃×#4076	1	2	yes	—
336	C₄	C₃×#4099	1	1	yes	characteristic
337	C₃	#10602823	1	1	yes	characteristic, direct product factor, Sylow
338	C₂	—	2	1	no	—
339	C₂	—	2	2	no	—
340	C₂	—	4	2	no	—
341	C₂	—	40	2	no	—
342	C₂	C₃×#65502	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	#65502
commutator	C₂³×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₃×C₅×#568	C₂
socle	C₂×C₃×C₅	#386
prime radical	C₃×#66164	C₂
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₃×#568	D₅
hypocenter	C₅	C₃×#1914
LCS	G ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×#568

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#568	C₂	semidirect
3. C₃×#65526	C₂	semidirect
4. C₂×C₃×#3955	C₂	semidirect
5. C₃×#64779	C₂	semidirect
6. #10602823	C₃	direct (N × H)
22. C₅×#568	C₂×C₃	semidirect
24. #65526	C₂×C₃	semidirect
25. C₂×#3955	C₂×C₃	semidirect
26. #64779	C₂×C₃	semidirect
42. C₃×C₅×M₁₆	D₄	semidirect
52. C₃×#568	D₅	semidirect
123. C₅×M₁₆	C₃×D₄	semidirect
133. #568	C₃×D₅	semidirect
168. C₃×M₁₆	C₂²⋊D₅	semidirect
269. M₁₆	C₃×C₂²⋊D₅	semidirect
277. C₃×C₅	#1914	semidirect
323. C₅	C₃×#1914	semidirect
337. C₃	#10602823	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×C₅×#568	2
C₃×#66164	2
C₃×#65526	2
C₂×C₃×#3955	2
C₃×#64779	2
#10602823	3
C₃×#1914	5

Subgroups — C3×#10602823 #14303038

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10602823 #14303038