Subgroups — C₃×#10604743 #14312867

Order 1920.

456 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×#4053	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3990	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×D₅⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₂²⋊#674	C₂	1	2	yes	maximal, complemented
5	C₃×#64887	C₂	1	2	yes	maximal, complemented
6	#10604743	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×#709	C₄	1	1	yes	characteristic
8	C₂×C₃×C₅×M₁₆	C₄	1	1	yes	characteristic
9	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
10	C₃×#4053	—	2	2	no	—
11	C₃×#4053	—	2	2	no	—
12	C₂×C₃×#676	C₄	1	1	yes	characteristic
13	C₂×C₃×#676	C₄	1	1	yes	characteristic
14	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
16	C₂×C₃×#674	C₂²	1	2	yes	—
17	C₂×C₃×#674	C₂²	1	2	yes	—
18	C₃×#3990	—	2	2	no	—
19	C₃×D₅⋊C₁₆	—	2	2	no	—
20	C₃×#2113	—	5	1	no	maximal, Hall
21	C₂×#4053	C₂×C₃	1	1	yes	characteristic
22	C₂×#3990	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂×D₅⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂²⋊#674	C₂×C₃	1	2	yes	complemented
25	#64887	C₂×C₃	1	2	yes	complemented
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₂²⋊D₅	C₈	1	1	yes	characteristic
28	C₂×C₃×C₂²⋊D₅	C₈	1	1	yes	characteristic
29	C₃×#709	—	2	2	no	—
30	C₃×#709	—	2	2	no	—
31	C₂×C₃×D₂₀	C₈	1	1	yes	characteristic
32	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×Dic₁₀	C₈	1	1	yes	characteristic
35	C₃×C₅×M₁₆	—	2	2	no	—
36	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×#168	—	2	1	no	—
39	C₂×C₃×#168	—	2	1	no	—
40	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×#168	C₂³	1	1	yes	characteristic
42	C₃×#676	—	2	2	no	—
43	C₃×#676	—	2	2	no	—
44	C₃×C₈×D₅	—	2	1	no	—
45	C₃×C₈×D₅	—	2	1	no	—
46	C₃×C₈×D₅	D₄	1	2	yes	—
47	C₃×C₈×D₅	D₄	1	2	yes	—
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	C₂²×C₃×C₁₆	—	5	1	no	—
53	C₃×C₂²⋊C₁₆	—	5	2	no	—
54	C₃×C₂²⋊C₁₆	—	5	2	no	—
55	C₂×#709	C₃×C₄	1	1	yes	characteristic
56	C₂×C₅×M₁₆	C₃×C₄	1	1	yes	characteristic
57	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
58	#4053	—	2	2	no	—
59	#4053	—	2	2	no	—
60	C₂×#676	C₃×C₄	1	1	yes	characteristic
61	C₂×#676	C₃×C₄	1	1	yes	characteristic
62	C₂×C₈×D₅	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₈×D₅	C₂²×C₃	1	1	yes	characteristic
64	C₂×#674	C₂²×C₃	1	2	yes	—
65	C₂×#674	C₂²×C₃	1	2	yes	—
66	#3990	—	2	2	no	—
67	D₅⋊C₁₆	—	2	2	no	—
68	#2113	—	5	1	no	Sylow
69	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₃×D₅	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₃×D₅	C₂×C₈	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	—	2	1	no	—
73	C₂×C₃×C₄×C₅	—	2	1	no	—
74	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₃×C₂²⋊D₅	—	2	2	no	—
77	C₃×C₂²⋊D₅	—	2	2	no	—
78	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
79	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
80	C₃×D₂₀	—	2	2	no	—
81	C₃×C₄×D₅	—	2	1	no	—
82	C₃×C₄×D₅	—	2	1	no	—
83	C₃×C₄×D₅	C₂²⋊C₄	1	2	yes	—
84	C₃×C₄×D₅	M₁₆	1	2	yes	—
85	C₃×Dic₁₀	—	2	2	no	—
86	C₃×C₅×C₈	—	2	1	no	—
87	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
88	C₃×#168	—	2	1	no	—
89	C₃×#168	C₂×D₄	1	1	yes	characteristic
90	C₃×#168	C₂×D₄	1	1	yes	characteristic
91	C₂×C₃×#41	—	5	1	no	—
92	C₃×#131	—	10	2	no	—
93	C₃×#131	—	10	2	no	—
94	C₂×C₃×M₁₆	—	5	1	no	—
95	C₂×C₃×M₁₆	—	5	1	no	—
96	C₂×C₃×M₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic
97	C₂²×C₃×C₈	—	5	1	no	—
98	C₂²×C₃×C₈	—	5	1	no	—
99	C₂²×C₃×C₈	—	5	1	no	—
100	C₃×M₃₂	—	10	2	no	—
101	C₂×C₃×C₁₆	—	5	2	no	—
102	C₂×C₃×C₁₆	—	5	2	no	—
103	C₂×C₃×C₁₆	—	10	2	no	—
104	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
105	C₂×C₂²⋊D₅	C₃×C₈	1	1	yes	characteristic
106	C₂×C₂²⋊D₅	C₃×C₈	1	1	yes	characteristic
107	#709	—	2	2	no	—
108	#709	—	2	2	no	—
109	C₂×D₂₀	C₃×C₈	1	1	yes	characteristic
110	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
111	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
112	C₂×Dic₁₀	C₃×C₈	1	1	yes	characteristic
113	C₅×M₁₆	—	2	2	no	—
114	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
115	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
116	C₂×#168	—	2	1	no	—
117	C₂×#168	—	2	1	no	—
118	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
119	C₂×#168	C₂³×C₃	1	1	yes	characteristic
120	#676	—	2	2	no	—
121	#676	—	2	2	no	—
122	C₈×D₅	—	2	1	no	—
123	C₈×D₅	—	2	1	no	—
124	C₈×D₅	C₃×D₄	1	2	yes	—
125	C₈×D₅	C₃×D₄	1	2	yes	—
126	#674	—	2	2	no	—
127	#674	—	2	2	no	—
128	C₂×#131	—	5	1	no	—
129	C₂×M₃₂	—	5	1	no	—
130	C₂²×C₁₆	—	5	1	no	—
131	C₂²⋊C₁₆	—	5	2	no	—
132	C₂²⋊C₁₆	—	5	2	no	—
133	C₂²×C₃×C₅	—	2	1	no	—
134	C₂²×C₃×C₅	—	2	2	no	—
135	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₃×D₅	—	2	2	no	—
137	C₂×C₃×D₅	—	2	1	no	—
138	C₂×C₃×D₅	—	2	2	no	—
139	C₂×C₃×D₅	C₂²⋊C₈	1	2	yes	—
140	C₃×C₄×C₅	—	2	1	no	—
141	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
143	C₃×Dic₅	—	2	1	no	—
144	C₃×Dic₅	C₂²⋊C₈	1	2	yes	—
145	C₃×#41	—	10	2	no	—
146	C₃×#41	—	10	2	no	—
147	C₂×C₃×Q₈	—	5	1	no	—
148	C₂×C₃×D₄	—	5	1	no	—
149	C₂×C₃×D₄	—	5	1	no	—
150	C₂×C₃×D₄	—	5	1	no	—
151	C₂²×C₃×C₄	—	5	1	no	—
152	C₂²×C₃×C₄	—	5	1	no	—
153	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
154	C₃×M₁₆	—	2	2	no	—
155	C₃×M₁₆	—	10	2	no	—
156	C₃×M₁₆	—	10	2	no	—
157	C₂×C₃×C₈	—	5	1	no	—
158	C₂×C₃×C₈	—	5	2	no	—
159	C₂×C₃×C₈	—	5	2	no	—
160	C₂×C₃×C₈	—	5	1	no	—
161	C₂×C₃×C₈	—	10	1	no	—
162	C₂×C₃×C₈	—	10	1	no	—
163	C₂×C₃×C₈	—	10	1	no	—
164	C₂×C₃×C₈	—	10	1	no	—
165	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
166	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
167	C₃×C₁₆	—	10	2	no	—
168	C₃×C₁₆	—	10	2	no	—
169	C₂³×C₅	C₂×C₃×C₈	1	1	yes	characteristic
170	C₂²×D₅	C₂×C₃×C₈	1	1	yes	characteristic
171	C₂²×D₅	C₂×C₃×C₈	1	1	yes	characteristic
172	C₂×C₄×C₅	—	2	1	no	—
173	C₂×C₄×C₅	—	2	1	no	—
174	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
176	C₂²⋊D₅	—	2	2	no	—
177	C₂²⋊D₅	—	2	2	no	—
178	C₂×Dic₅	C₂×C₃×C₈	1	1	yes	characteristic
179	C₂×Dic₅	C₂×C₃×C₈	1	1	yes	characteristic
180	D₂₀	—	2	2	no	—
181	C₄×D₅	—	2	1	no	—
182	C₄×D₅	—	2	1	no	—
183	C₄×D₅	C₃×C₂²⋊C₄	1	2	yes	—
184	C₄×D₅	C₃×M₁₆	1	2	yes	—
185	Dic₁₀	—	2	2	no	—
186	C₅×C₈	—	2	1	no	—
187	C₅×C₈	C₃×C₂²⋊C₄	1	2	yes	—
188	#168	—	2	1	no	—
189	#168	C₂×C₃×D₄	1	1	yes	characteristic
190	#168	C₂×C₃×D₄	1	1	yes	characteristic
191	C₂×#41	—	5	1	no	—
192	#131	—	10	2	no	—
193	#131	—	10	2	no	—
194	C₂×M₁₆	—	5	1	no	—
195	C₂×M₁₆	—	5	1	no	—
196	C₂×M₁₆	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
197	C₂²×C₈	—	5	1	no	—
198	C₂²×C₈	—	5	1	no	—
199	C₂²×C₈	—	5	1	no	—
200	M₃₂	—	10	2	no	—
201	C₂×C₁₆	—	5	2	no	—
202	C₂×C₁₆	—	5	2	no	—
203	C₂×C₁₆	—	10	2	no	—
204	C₂×C₃×C₅	—	4	1	no	—
205	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
206	C₂×C₃×C₅	#504	1	2	yes	—
207	C₃×D₅	—	2	2	no	—
208	C₃×D₅	—	4	1	no	—
209	C₂³×C₃	—	5	1	no	—
210	C₂³×C₃	—	5	1	no	—
211	C₂³×C₃	D₅⋊C₈	1	1	yes	characteristic
212	C₃×Q₈	—	10	2	no	—
213	C₃×D₄	—	10	2	no	—
214	C₃×D₄	—	10	2	no	—
215	C₃×D₄	—	10	2	no	—
216	C₂×C₃×C₄	—	2	1	no	—
217	C₂×C₃×C₄	—	2	1	no	—
218	C₂×C₃×C₄	—	5	1	no	—
219	C₂×C₃×C₄	—	5	2	no	—
220	C₂×C₃×C₄	—	5	2	no	—
221	C₂×C₃×C₄	—	5	1	no	—
222	C₂×C₃×C₄	—	10	1	no	—
223	C₂×C₃×C₄	—	10	1	no	—
224	C₂×C₃×C₄	D₅⋊C₈	1	1	yes	characteristic
225	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
226	C₃×C₈	—	2	1	no	—
227	C₃×C₈	—	5	1	no	—
228	C₃×C₈	—	5	1	no	—
229	C₃×C₈	—	10	1	no	—
230	C₃×C₈	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
231	C₂²×C₅	—	2	2	no	—
232	C₂²×C₅	—	2	1	no	—
233	C₂²×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
234	C₂×D₅	—	2	2	no	—
235	C₂×D₅	—	2	2	no	—
236	C₂×D₅	—	2	1	no	—
237	C₂×D₅	C₃×C₂²⋊C₈	1	2	yes	—
238	C₄×C₅	—	2	1	no	—
239	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
240	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
241	Dic₅	—	2	1	no	—
242	Dic₅	C₃×C₂²⋊C₈	1	2	yes	—
243	#41	—	10	2	no	—
244	#41	—	10	2	no	—
245	C₂×Q₈	—	5	1	no	—
246	C₂×D₄	—	5	1	no	—
247	C₂×D₄	—	5	1	no	—
248	C₂×D₄	—	5	1	no	—
249	C₂²×C₄	—	5	1	no	—
250	C₂²×C₄	—	5	1	no	—
251	C₂²×C₄	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
252	M₁₆	—	2	2	no	—
253	M₁₆	—	10	2	no	—
254	M₁₆	—	10	2	no	—
255	C₂×C₈	—	5	1	no	—
256	C₂×C₈	—	5	2	no	—
257	C₂×C₈	—	5	2	no	—
258	C₂×C₈	—	5	1	no	—
259	C₂×C₈	—	10	1	no	—
260	C₂×C₈	—	10	1	no	—
261	C₂×C₈	—	10	1	no	—
262	C₂×C₈	—	10	1	no	—
263	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
264	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
265	C₁₆	—	10	2	no	—
266	C₁₆	—	10	2	no	—
267	C₃×C₅	#2113	1	1	yes	characteristic, complemented, Hall
268	C₂²×C₃	—	2	2	no	—
269	C₂²×C₃	—	2	1	no	—
270	C₂²×C₃	—	5	2	no	—
271	C₂²×C₃	—	10	2	no	—
272	C₂²×C₃	—	10	2	no	—
273	C₂²×C₃	—	10	1	no	—
274	C₂²×C₃	C₂×D₅⋊C₈	1	1	yes	characteristic
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	5	2	no	—
277	C₃×C₄	—	10	1	no	—
278	C₃×C₄	D₅⋊M₁₆	1	1	yes	characteristic
279	C₃×C₄	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
280	C₂×C₅	—	4	1	no	—
281	C₂×C₅	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
282	C₂×C₅	C₃×#504	1	2	yes	—
283	D₅	—	2	2	no	—
284	D₅	—	4	1	no	—
285	C₂³	—	5	1	no	—
286	C₂³	—	5	1	no	—
287	C₂³	C₃×D₅⋊C₈	1	1	yes	characteristic
288	Q₈	—	10	2	no	—
289	D₄	—	10	2	no	—
290	D₄	—	10	2	no	—
291	D₄	—	10	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	5	1	no	—
295	C₂×C₄	—	5	2	no	—
296	C₂×C₄	—	5	2	no	—
297	C₂×C₄	—	5	1	no	—
298	C₂×C₄	—	10	1	no	—
299	C₂×C₄	—	10	1	no	—
300	C₂×C₄	C₃×D₅⋊C₈	1	1	yes	characteristic
301	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	5	1	no	—
304	C₈	—	5	1	no	—
305	C₈	—	10	1	no	—
306	C₈	C₃×C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
307	C₂×C₃	—	4	1	no	—
308	C₂×C₃	—	10	2	no	—
309	C₂×C₃	—	20	1	no	—
310	C₂×C₃	#65725	1	2	yes	—
311	C₂×C₃	D₅⋊(C₂²⋊C₈)	1	1	yes	characteristic
312	C₅	C₃×#2113	1	1	yes	characteristic, complemented, Sylow
313	C₂²	—	2	2	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	5	2	no	—
316	C₂²	—	10	2	no	—
317	C₂²	—	10	2	no	—
318	C₂²	—	10	1	no	—
319	C₂²	C₂×C₃×D₅⋊C₈	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	5	2	no	—
322	C₄	—	10	1	no	—
323	C₄	C₃×D₅⋊M₁₆	1	1	yes	characteristic
324	C₄	C₂×C₃×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
325	C₃	#10604743	1	1	yes	characteristic, direct product factor, Sylow
326	C₂	—	4	1	no	—
327	C₂	—	10	2	no	—
328	C₂	—	20	1	no	—
329	C₂	C₃×#65725	1	2	yes	—
330	C₂	C₃×D₅⋊(C₂²⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)
commutator	C₂²×C₅	C₂²×C₃×C₈
Frattini	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting	C₂×C₃×C₅×M₁₆	C₄
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂×C₃×#709	C₄
cosocle	C₂×#168	C₂³×C₃
hypercenter	C₂×C₃×M₁₆	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#2113
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#3990	C₂	semidirect
3. C₂×C₃×D₅⋊C₁₆	C₂	semidirect
4. C₃×C₂²⋊#674	C₂	semidirect
5. C₃×#64887	C₂	semidirect
6. #10604743	C₃	direct (N × H)
22. C₂×#3990	C₂×C₃	semidirect
23. C₂×D₅⋊C₁₆	C₂×C₃	semidirect
24. C₂²⋊#674	C₂×C₃	semidirect
25. #64887	C₂×C₃	semidirect
267. C₃×C₅	#2113	semidirect
312. C₅	C₃×#2113	semidirect
325. C₃	#10604743	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×#4053	2
C₂×C₃×#3990	2
C₂×C₃×D₅⋊C₁₆	2
C₃×C₂²⋊#674	2
C₃×#64887	2
#10604743	3
C₃×#2113	5

Subgroups — C3×#10604743 #14312867

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10604743 #14312867