Subgroups — C₃×#10605009 #14313011

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×#4094	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3998	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3998	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64932	C₂	1	4	yes	maximal, complemented
5	#10605009	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₅×#41	C₄	1	1	yes	characteristic
7	C₃×#4094	—	2	1	no	—
8	C₃×#4094	—	2	1	no	—
9	C₃×#4094	C₄	1	2	yes	—
10	C₃×#4094	C₂²	1	2	yes	—
11	C₂×C₃×#681	—	2	1	no	—
12	C₂×C₃×#681	C₄	1	1	yes	characteristic
13	C₂²×C₃×#168	—	2	1	no	—
14	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
15	C₃×#3998	—	2	1	no	—
16	C₃×#3998	—	2	1	no	—
17	C₃×#3998	—	2	1	no	—
18	C₃×#3998	C₂²	1	2	yes	complemented
19	C₂×C₃×#674	—	2	1	no	—
20	C₂×C₃×#674	C₂²	1	2	yes	complemented
21	C₃×#2117	—	5	1	no	maximal, Hall
22	C₂×#4094	C₂×C₃	1	1	yes	characteristic
23	C₂×#3998	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×#3998	C₂×C₃	1	1	yes	characteristic, complemented
25	#64932	C₂×C₃	1	4	yes	complemented
26	C₃×C₅×#41	—	2	1	no	—
27	C₃×C₅×#41	—	2	1	no	—
28	C₃×C₅×#41	C₈	1	2	yes	—
29	C₃×C₅×#41	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₅×Q₈	C₈	1	1	yes	characteristic
31	C₂×C₃×C₅×D₄	—	2	1	no	—
32	C₂×C₃×C₅×D₄	C₈	1	1	yes	characteristic
33	C₂²×C₃×C₄×C₅	—	2	1	no	—
34	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
35	C₃×#681	—	2	1	no	—
36	C₃×#681	—	2	2	no	—
37	C₃×#681	—	2	2	no	—
38	C₃×#681	C₂×C₄	1	2	yes	—
39	C₂×C₃×#168	—	2	1	no	—
40	C₂×C₃×#168	—	2	2	no	—
41	C₂×C₃×#168	—	2	2	no	—
42	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃×#168	D₄	1	1	yes	characteristic
44	C₂×C₃×#168	D₄	1	1	yes	characteristic
45	C₂×C₃×#168	D₄	1	1	yes	characteristic
46	C₂×C₃×#168	D₄	1	1	yes	characteristic
47	C₂×C₃×#168	C₂³	1	1	yes	characteristic
48	C₃×#674	—	2	2	no	—
49	C₃×#674	—	2	2	no	—
50	C₂×C₃×#131	—	5	1	no	—
51	C₂×C₃×M₃₂	—	5	1	no	—
52	C₂×C₃×M₃₂	—	5	1	no	—
53	C₃×#350	—	5	4	no	—
54	C₂×C₅×#41	C₃×C₄	1	1	yes	characteristic
55	#4094	—	2	1	no	—
56	#4094	—	2	1	no	—
57	#4094	C₃×C₄	1	2	yes	—
58	#4094	C₂²×C₃	1	2	yes	—
59	C₂×#681	—	2	1	no	—
60	C₂×#681	C₃×C₄	1	1	yes	characteristic
61	C₂²×#168	—	2	1	no	—
62	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
63	#3998	—	2	1	no	—
64	#3998	—	2	1	no	—
65	#3998	—	2	1	no	—
66	#3998	C₂²×C₃	1	2	yes	complemented
67	C₂×#674	—	2	1	no	—
68	C₂×#674	C₂²×C₃	1	2	yes	complemented
69	#2117	—	5	1	no	Sylow
70	C₂³×C₃×C₅	—	2	1	no	—
71	C₂³×C₃×C₅	M₁₆	1	1	yes	characteristic
72	C₃×C₅×Q₈	—	2	1	no	—
73	C₃×C₅×Q₈	C₂×C₈	1	2	yes	—
74	C₃×C₅×D₄	—	2	2	no	—
75	C₃×C₅×D₄	—	2	2	no	—
76	C₃×C₅×D₄	—	2	1	no	—
77	C₃×C₅×D₄	C₂×C₈	1	2	yes	—
78	C₂×C₃×C₄×C₅	—	2	1	no	—
79	C₂×C₃×C₄×C₅	—	2	2	no	—
80	C₂×C₃×C₄×C₅	—	2	2	no	—
81	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	M₁₆	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₃×#168	—	2	2	no	—
88	C₃×#168	C₂²⋊C₄	1	1	yes	characteristic
89	C₃×#168	C₂²⋊C₄	1	1	yes	characteristic
90	C₃×#168	C₂×D₄	1	1	yes	characteristic
91	C₃×#168	C₂×D₄	1	1	yes	characteristic
92	C₂×C₃×#41	AGL(1,𝔽₅)	1	1	yes	characteristic
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	—	5	1	no	—
100	C₂²×C₃×C₈	—	10	1	no	—
101	C₃×M₃₂	—	5	2	no	—
102	C₃×M₃₂	—	10	1	no	—
103	C₃×M₃₂	—	10	1	no	—
104	C₃×M₃₂	—	10	1	no	—
105	C₂×C₃×C₁₆	—	5	2	no	—
106	C₂×C₃×C₁₆	—	10	1	no	—
107	C₅×#41	—	2	1	no	—
108	C₅×#41	—	2	1	no	—
109	C₅×#41	C₃×C₈	1	2	yes	—
110	C₅×#41	C₂×C₃×C₄	1	2	yes	—
111	C₂×C₅×Q₈	C₃×C₈	1	1	yes	characteristic
112	C₂×C₅×D₄	—	2	1	no	—
113	C₂×C₅×D₄	C₃×C₈	1	1	yes	characteristic
114	C₂²×C₄×C₅	—	2	1	no	—
115	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
116	#681	—	2	1	no	—
117	#681	—	2	2	no	—
118	#681	—	2	2	no	—
119	#681	C₂×C₃×C₄	1	2	yes	—
120	C₂×#168	—	2	1	no	—
121	C₂×#168	—	2	2	no	—
122	C₂×#168	—	2	2	no	—
123	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
124	C₂×#168	C₃×D₄	1	1	yes	characteristic
125	C₂×#168	C₃×D₄	1	1	yes	characteristic
126	C₂×#168	C₃×D₄	1	1	yes	characteristic
127	C₂×#168	C₃×D₄	1	1	yes	characteristic
128	C₂×#168	C₂³×C₃	1	1	yes	characteristic
129	#674	—	2	2	no	—
130	#674	—	2	2	no	—
131	C₂×#131	—	5	1	no	—
132	C₂×M₃₂	—	5	1	no	—
133	C₂×M₃₂	—	5	1	no	—
134	#350	—	5	4	no	—
135	C₂²×C₃×C₅	—	2	2	no	—
136	C₂²×C₃×C₅	—	4	1	no	—
137	C₂²×C₃×C₅	—	4	2	no	—
138	C₂²×C₃×C₅	C₂²⋊C₈	1	1	yes	characteristic
139	C₂²×C₃×C₅	C₂²⋊C₈	1	1	yes	characteristic
140	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
141	C₃×C₄×C₅	—	2	2	no	—
142	C₃×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
143	C₃×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
144	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
146	C₃×#41	—	2	1	no	—
147	C₃×#41	—	2	1	no	—
148	C₃×#41	#170	1	2	yes	—
149	C₃×#41	C₂×AGL(1,𝔽₅)	1	2	yes	—
150	C₂×C₃×Q₈	#170	1	1	yes	characteristic
151	C₂×C₃×D₄	—	2	1	no	—
152	C₂×C₃×D₄	#170	1	1	yes	characteristic
153	C₂²×C₃×C₄	—	2	1	no	—
154	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
155	C₃×M₁₆	—	5	2	no	—
156	C₃×M₁₆	—	10	2	no	—
157	C₃×M₁₆	—	10	2	no	—
158	C₃×M₁₆	—	10	1	no	—
159	C₂×C₃×C₈	—	5	1	no	—
160	C₂×C₃×C₈	—	5	1	no	—
161	C₂×C₃×C₈	—	5	1	no	—
162	C₂×C₃×C₈	—	5	1	no	—
163	C₂×C₃×C₈	—	5	1	no	—
164	C₂×C₃×C₈	—	5	1	no	—
165	C₂×C₃×C₈	—	10	2	no	—
166	C₂×C₃×C₈	—	10	2	no	—
167	C₂×C₃×C₈	—	10	1	no	—
168	C₃×C₁₆	—	10	2	no	—
169	C₃×C₁₆	—	10	2	no	—
170	C₂³×C₅	—	2	1	no	—
171	C₂³×C₅	C₃×M₁₆	1	1	yes	characteristic
172	C₅×Q₈	—	2	1	no	—
173	C₅×Q₈	C₂×C₃×C₈	1	2	yes	—
174	C₅×D₄	—	2	2	no	—
175	C₅×D₄	—	2	2	no	—
176	C₅×D₄	—	2	1	no	—
177	C₅×D₄	C₂×C₃×C₈	1	2	yes	—
178	C₂×C₄×C₅	—	2	1	no	—
179	C₂×C₄×C₅	—	2	2	no	—
180	C₂×C₄×C₅	—	2	2	no	—
181	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
182	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
183	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
185	C₂×C₄×C₅	C₃×M₁₆	1	1	yes	characteristic
186	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
187	#168	—	2	2	no	—
188	#168	C₃×C₂²⋊C₄	1	1	yes	characteristic
189	#168	C₃×C₂²⋊C₄	1	1	yes	characteristic
190	#168	C₂×C₃×D₄	1	1	yes	characteristic
191	#168	C₂×C₃×D₄	1	1	yes	characteristic
192	C₂×#41	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
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	—	5	1	no	—
200	C₂²×C₈	—	10	1	no	—
201	M₃₂	—	5	2	no	—
202	M₃₂	—	10	1	no	—
203	M₃₂	—	10	1	no	—
204	M₃₂	—	10	1	no	—
205	C₂×C₁₆	—	5	2	no	—
206	C₂×C₁₆	—	10	1	no	—
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	—	4	2	no	—
211	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
212	C₂³×C₃	—	2	1	no	—
213	C₂³×C₃	#704	1	1	yes	characteristic
214	C₃×Q₈	—	2	1	no	—
215	C₃×Q₈	C₂×#170	1	2	yes	—
216	C₃×D₄	—	2	2	no	—
217	C₃×D₄	—	2	2	no	—
218	C₃×D₄	—	2	1	no	—
219	C₃×D₄	C₂×#170	1	2	yes	—
220	C₂×C₃×C₄	—	2	1	no	—
221	C₂×C₃×C₄	—	2	2	no	—
222	C₂×C₃×C₄	—	2	2	no	—
223	C₂×C₃×C₄	C₂×#170	1	1	yes	characteristic
224	C₂×C₃×C₄	C₂×#170	1	1	yes	characteristic
225	C₂×C₃×C₄	#704	1	1	yes	characteristic
226	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
227	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
229	C₃×C₈	—	5	1	no	—
230	C₃×C₈	—	5	1	no	—
231	C₃×C₈	—	5	1	no	—
232	C₃×C₈	—	5	1	no	—
233	C₃×C₈	—	10	2	no	—
234	C₂²×C₅	—	2	2	no	—
235	C₂²×C₅	—	4	1	no	—
236	C₂²×C₅	—	4	2	no	—
237	C₂²×C₅	C₃×C₂²⋊C₈	1	1	yes	characteristic
238	C₂²×C₅	C₃×C₂²⋊C₈	1	1	yes	characteristic
239	C₂²×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
240	C₄×C₅	—	2	2	no	—
241	C₄×C₅	C₃×C₂²⋊C₈	1	1	yes	characteristic
242	C₄×C₅	C₃×C₂²⋊C₈	1	1	yes	characteristic
243	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
244	C₄×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
245	#41	—	2	1	no	—
246	#41	—	2	1	no	—
247	#41	C₃×#170	1	2	yes	—
248	#41	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	—
249	C₂×Q₈	C₃×#170	1	1	yes	characteristic
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	C₃×#170	1	1	yes	characteristic
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
254	M₁₆	—	5	2	no	—
255	M₁₆	—	10	2	no	—
256	M₁₆	—	10	2	no	—
257	M₁₆	—	10	1	no	—
258	C₂×C₈	—	5	1	no	—
259	C₂×C₈	—	5	1	no	—
260	C₂×C₈	—	5	1	no	—
261	C₂×C₈	—	5	1	no	—
262	C₂×C₈	—	5	1	no	—
263	C₂×C₈	—	5	1	no	—
264	C₂×C₈	—	10	2	no	—
265	C₂×C₈	—	10	2	no	—
266	C₂×C₈	—	10	1	no	—
267	C₁₆	—	10	2	no	—
268	C₁₆	—	10	2	no	—
269	C₃×C₅	#2117	1	1	yes	characteristic, complemented, Hall
270	C₂²×C₃	—	2	2	no	—
271	C₂²×C₃	—	4	1	no	—
272	C₂²×C₃	—	4	2	no	—
273	C₂²×C₃	C₂²⋊#170	1	1	yes	characteristic
274	C₂²×C₃	C₂²⋊#170	1	1	yes	characteristic
275	C₂²×C₃	C₂×#704	1	1	yes	characteristic
276	C₃×C₄	—	2	2	no	—
277	C₃×C₄	C₂²⋊#170	1	1	yes	characteristic
278	C₃×C₄	C₂²⋊#170	1	1	yes	characteristic
279	C₃×C₄	C₂²×#170	1	1	yes	characteristic
280	C₃×C₄	C₂×C₂²⋊AGL(1,𝔽₅)	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	—	4	2	no	—
285	C₂×C₅	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
286	C₂³	—	2	1	no	—
287	C₂³	C₃×#704	1	1	yes	characteristic
288	Q₈	—	2	1	no	—
289	Q₈	C₂×C₃×#170	1	2	yes	—
290	D₄	—	2	2	no	—
291	D₄	—	2	2	no	—
292	D₄	—	2	1	no	—
293	D₄	C₂×C₃×#170	1	2	yes	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	C₂×C₃×#170	1	1	yes	characteristic
298	C₂×C₄	C₂×C₃×#170	1	1	yes	characteristic
299	C₂×C₄	C₃×#704	1	1	yes	characteristic
300	C₂×C₄	C₃×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
301	C₂×C₄	C₃×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
302	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
303	C₈	—	5	1	no	—
304	C₈	—	5	1	no	—
305	C₈	—	5	1	no	—
306	C₈	—	5	1	no	—
307	C₈	—	10	2	no	—
308	C₂×C₃	—	2	1	no	—
309	C₂×C₃	—	2	1	no	—
310	C₂×C₃	—	2	1	no	—
311	C₂×C₃	—	4	2	no	—
312	C₂×C₃	C₂×C₂²⋊#170	1	1	yes	characteristic
313	C₅	C₃×#2117	1	1	yes	characteristic, complemented, Sylow
314	C₂²	—	2	2	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	2	no	—
317	C₂²	C₃×C₂²⋊#170	1	1	yes	characteristic
318	C₂²	C₃×C₂²⋊#170	1	1	yes	characteristic
319	C₂²	C₂×C₃×#704	1	1	yes	characteristic
320	C₄	—	2	2	no	—
321	C₄	C₃×C₂²⋊#170	1	1	yes	characteristic
322	C₄	C₃×C₂²⋊#170	1	1	yes	characteristic
323	C₄	C₂²×C₃×#170	1	1	yes	characteristic
324	C₄	C₂×C₃×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
325	C₃	#10605009	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	—	4	2	no	—
330	C₂	C₂×C₃×C₂²⋊#170	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	C₂×C₂²⋊AGL(1,𝔽₅)
commutator	C₄×C₅	C₂²×C₃×C₈
Frattini	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting, prime radical	C₂×C₃×C₅×#41	C₄
socle	C₂×C₃×C₅	C₂×C₂²⋊C₈
cosocle	C₂×#168	C₂³×C₃
hypercenter	C₂×C₃×#41	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#2117
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#41

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#3998	C₂	semidirect
3. C₂×C₃×#3998	C₂	semidirect
4. C₃×#64932	C₂	semidirect
5. #10605009	C₃	direct (N × H)
18. C₃×#3998	C₂²	semidirect
20. C₂×C₃×#674	C₂²	semidirect
23. C₂×#3998	C₂×C₃	semidirect
24. C₂×#3998	C₂×C₃	semidirect
25. #64932	C₂×C₃	semidirect
66. #3998	C₂²×C₃	semidirect
68. C₂×#674	C₂²×C₃	semidirect
269. C₃×C₅	#2117	semidirect
313. C₅	C₃×#2117	semidirect
325. C₃	#10605009	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×#4094	2
C₂×C₃×#3998	2
C₃×#64932	2
#10605009	3
C₃×#2117	5

Subgroups — C3×#10605009 #14313011

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10605009 #14313011