Subgroups — C₃×#10617416 #14413454

Order 1920.

952 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#66266	C₂	1	2	yes	maximal
2	C₃×D₅×#131	C₂	1	1	yes	maximal, characteristic
3	C₃×#65727	C₂	1	6	yes	maximal, complemented
4	C₃×#65721	C₂	1	3	yes	maximal, complemented
5	C₂×C₃×#3993	C₂	1	3	yes	maximal, complemented
6	#10617416	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×D₅×#41	C₂²	1	1	yes	characteristic
8	C₃×Q₈×AGL(1,𝔽₅)	—	2	2	no	—
9	C₃×D₄×AGL(1,𝔽₅)	—	2	6	no	—
10	C₃×#4130	C₂²	1	6	yes	—
11	C₂×C₃×C₄⋊AGL(1,𝔽₅)	C₂²	1	6	yes	—
12	C₃×C₅×#131	C₄	1	1	yes	characteristic, complemented
13	C₃×#4094	C₄	1	1	yes	characteristic, complemented
14	C₃×#4053	C₄	1	3	yes	complemented
15	C₃×D₅×M₁₆	C₂²	1	3	yes	—
16	C₃×#4047	C₄	1	3	yes	complemented
17	C₂×C₃×C₈×D₅	C₂²	1	3	yes	—
18	C₃×#3994	C₂²	1	6	yes	—
19	C₃×#3993	C₂²	1	3	yes	—
20	C₃×#3993	C₂²	1	6	yes	—
21	C₃×#3993	C₂²	1	1	yes	characteristic
22	C₃×#2912	—	5	1	no	maximal, Hall
23	#66266	C₂×C₃	1	2	yes	—
24	D₅×#131	C₂×C₃	1	1	yes	characteristic
25	#65727	C₂×C₃	1	6	yes	complemented
26	#65721	C₂×C₃	1	3	yes	complemented
27	C₂×#3993	C₂×C₃	1	3	yes	complemented
28	C₂²×C₃×AGL(1,𝔽₅)	—	2	6	no	—
29	C₃×C₅×#41	C₂×C₄	1	1	yes	characteristic
30	C₃×#713	Q₈	1	1	yes	characteristic
31	C₃×Q₈×D₅	D₄	1	1	yes	characteristic
32	C₃×#711	Q₈	1	3	yes	—
33	C₃×D₄×D₅	D₄	1	3	yes	—
34	C₃×#709	C₂×C₄	1	3	yes	—
35	C₂×C₃×C₄×D₅	C₂³	1	3	yes	—
36	C₃×C₂²⋊AGL(1,𝔽₅)	—	2	6	no	—
37	C₃×C₄⋊AGL(1,𝔽₅)	—	2	6	no	—
38	C₃×C₄⋊AGL(1,𝔽₅)	C₂³	1	6	yes	—
39	C₃×C₄⋊AGL(1,𝔽₅)	C₂³	1	2	yes	—
40	C₃×C₄×AGL(1,𝔽₅)	—	2	6	no	—
41	C₃×C₅×M₁₆	C₂×C₄	1	3	yes	complemented
42	C₂×C₃×C₅×C₈	C₂×C₄	1	3	yes	complemented
43	C₃×#681	C₂×C₄	1	3	yes	complemented
44	C₂×C₃×#168	C₂×C₄	1	3	yes	complemented
45	C₃×#676	C₂×C₄	1	6	yes	complemented
46	C₃×C₈×D₅	C₂×C₄	1	3	yes	complemented
47	C₃×C₈×D₅	C₂×C₄	1	3	yes	complemented
48	C₃×C₈×D₅	C₂³	1	1	yes	characteristic
49	C₃×C₈×D₅	C₂³	1	3	yes	—
50	C₂×C₃×#131	—	5	1	no	—
51	C₃×#520	—	5	2	no	—
52	C₃×#428	—	5	6	no	—
53	C₃×#427	—	5	3	no	—
54	C₂×C₃×#106	—	5	3	no	—
55	D₅×#41	C₂²×C₃	1	1	yes	characteristic
56	Q₈×AGL(1,𝔽₅)	—	2	2	no	—
57	D₄×AGL(1,𝔽₅)	—	2	6	no	—
58	#4130	C₂²×C₃	1	6	yes	—
59	C₂×C₄⋊AGL(1,𝔽₅)	C₂²×C₃	1	6	yes	—
60	C₅×#131	C₃×C₄	1	1	yes	characteristic, complemented
61	#4094	C₃×C₄	1	1	yes	characteristic, complemented
62	#4053	C₃×C₄	1	3	yes	complemented
63	D₅×M₁₆	C₂²×C₃	1	3	yes	—
64	#4047	C₃×C₄	1	3	yes	complemented
65	C₂×C₈×D₅	C₂²×C₃	1	3	yes	—
66	#3994	C₂²×C₃	1	6	yes	—
67	#3993	C₂²×C₃	1	3	yes	—
68	#3993	C₂²×C₃	1	6	yes	—
69	#3993	C₂²×C₃	1	1	yes	characteristic
70	#2912	—	5	1	no	Sylow
71	C₂²×C₃×D₅	C₂×D₄	1	3	yes	—
72	C₂×C₃×AGL(1,𝔽₅)	—	2	2	no	—
73	C₂×C₃×AGL(1,𝔽₅)	—	2	6	no	—
74	C₂×C₃×AGL(1,𝔽₅)	—	8	6	no	—
75	C₃×C₅×Q₈	C₄⋊C₄	1	1	yes	characteristic
76	C₃×C₅×D₄	C₄⋊C₄	1	3	yes	—
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	3	yes	—
78	C₃×C₂²⋊D₅	C₄⋊C₄	1	6	yes	—
79	C₂×C₃×Dic₅	C₂×Q₈	1	3	yes	—
80	C₃×D₂₀	C₄⋊C₄	1	3	yes	—
81	C₃×C₄×D₅	C₂²×C₄	1	3	yes	—
82	C₃×C₄×D₅	C₂×D₄	1	3	yes	—
83	C₃×C₄×D₅	C₂×Q₈	1	3	yes	—
84	C₃×C₄×D₅	C₂⁴	1	1	yes	characteristic
85	C₃×Dic₁₀	C₄⋊C₄	1	3	yes	—
86	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₃×C₅×C₈	C₂²×C₄	1	3	yes	—
88	C₃×#168	C₂²×C₄	1	1	yes	characteristic
89	C₃×#168	C₂²×C₄	1	3	yes	—
90	C₂×C₃×#41	—	5	1	no	—
91	C₃×#131	—	5	1	no	—
92	C₃×#131	—	5	3	no	—
93	C₃×#131	—	5	3	no	—
94	C₃×#131	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
95	C₂×C₃×M₁₆	—	5	3	no	—
96	C₂²×C₃×C₈	—	5	3	no	—
97	C₃×C₄×Q₈	—	10	2	no	—
98	C₃×C₄×D₄	—	10	6	no	—
99	C₃×#117	—	5	6	no	—
100	C₂×C₃×C₄⋊C₄	—	5	6	no	—
101	C₃×#107	—	5	6	no	—
102	C₃×#106	—	5	3	no	—
103	C₃×#106	—	5	6	no	—
104	C₃×#106	—	5	1	no	—
105	C₂²×AGL(1,𝔽₅)	—	2	6	no	—
106	C₅×#41	C₂×C₃×C₄	1	1	yes	characteristic
107	#713	C₃×Q₈	1	1	yes	characteristic
108	Q₈×D₅	C₃×D₄	1	1	yes	characteristic
109	#711	C₃×Q₈	1	3	yes	—
110	D₄×D₅	C₃×D₄	1	3	yes	—
111	#709	C₂×C₃×C₄	1	3	yes	—
112	C₂×C₄×D₅	C₂³×C₃	1	3	yes	—
113	C₂²⋊AGL(1,𝔽₅)	—	2	6	no	—
114	C₄⋊AGL(1,𝔽₅)	—	2	6	no	—
115	C₄⋊AGL(1,𝔽₅)	C₂³×C₃	1	6	yes	—
116	C₄⋊AGL(1,𝔽₅)	C₂³×C₃	1	2	yes	—
117	C₄×AGL(1,𝔽₅)	—	2	6	no	—
118	C₅×M₁₆	C₂×C₃×C₄	1	3	yes	complemented
119	C₂×C₅×C₈	C₂×C₃×C₄	1	3	yes	complemented
120	#681	C₂×C₃×C₄	1	3	yes	complemented
121	C₂×#168	C₂×C₃×C₄	1	3	yes	complemented
122	#676	C₂×C₃×C₄	1	6	yes	complemented
123	C₈×D₅	C₂×C₃×C₄	1	3	yes	complemented
124	C₈×D₅	C₂×C₃×C₄	1	3	yes	complemented
125	C₈×D₅	C₂³×C₃	1	1	yes	characteristic
126	C₈×D₅	C₂³×C₃	1	3	yes	—
127	C₂×#131	—	5	1	no	—
128	#520	—	5	2	no	—
129	#428	—	5	6	no	—
130	#427	—	5	3	no	—
131	C₂×#106	—	5	3	no	—
132	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	3	yes	—
133	C₂×C₃×D₅	—	2	6	no	—
134	C₂×C₃×D₅	C₂×C₄⋊C₄	1	3	yes	—
135	C₂×C₃×D₅	C₂²×D₄	1	1	yes	characteristic
136	C₃×AGL(1,𝔽₅)	—	4	2	no	—
137	C₃×AGL(1,𝔽₅)	—	4	6	no	—
138	C₃×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
139	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
140	C₃×Dic₅	C₂×C₄⋊C₄	1	3	yes	—
141	C₃×Dic₅	C₂²×Q₈	1	1	yes	characteristic
142	C₃×#41	—	5	1	no	—
143	C₃×#41	—	5	3	no	—
144	C₃×#41	—	5	3	no	—
145	C₃×#41	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
146	C₂×C₃×Q₈	—	5	1	no	—
147	C₂×C₃×D₄	—	5	3	no	—
148	C₂²×C₃×C₄	—	5	3	no	—
149	C₂²×C₃×C₄	—	10	6	no	—
150	C₃×M₁₆	—	5	3	no	—
151	C₃×M₁₆	—	5	6	no	—
152	C₃×M₁₆	C₂×AGL(1,𝔽₅)	1	3	yes	complemented
153	C₂×C₃×C₈	—	5	3	no	—
154	C₂×C₃×C₈	—	5	3	no	—
155	C₂×C₃×C₈	—	5	3	no	—
156	C₂×C₃×C₈	—	5	1	no	—
157	C₂×C₃×C₈	—	5	3	no	—
158	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	3	yes	complemented
159	C₃×C₄⋊C₄	—	5	6	no	—
160	C₃×C₄⋊C₄	—	5	2	no	—
161	C₃×C₄⋊C₄	—	10	6	no	—
162	C₃×C₂²⋊C₄	—	10	6	no	—
163	C₃×C₄²	—	10	6	no	—
164	C₂²×D₅	C₂×C₃×D₄	1	3	yes	—
165	C₂×AGL(1,𝔽₅)	—	2	2	no	—
166	C₂×AGL(1,𝔽₅)	—	2	6	no	—
167	C₂×AGL(1,𝔽₅)	—	8	6	no	—
168	C₅×Q₈	C₃×C₄⋊C₄	1	1	yes	characteristic
169	C₅×D₄	C₃×C₄⋊C₄	1	3	yes	—
170	C₂×C₄×C₅	C₂²×C₃×C₄	1	3	yes	—
171	C₂²⋊D₅	C₃×C₄⋊C₄	1	6	yes	—
172	C₂×Dic₅	C₂×C₃×Q₈	1	3	yes	—
173	D₂₀	C₃×C₄⋊C₄	1	3	yes	—
174	C₄×D₅	C₂²×C₃×C₄	1	3	yes	—
175	C₄×D₅	C₂×C₃×D₄	1	3	yes	—
176	C₄×D₅	C₂×C₃×Q₈	1	3	yes	—
177	C₄×D₅	C₂⁴×C₃	1	1	yes	characteristic
178	Dic₁₀	C₃×C₄⋊C₄	1	3	yes	—
179	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
180	C₅×C₈	C₂²×C₃×C₄	1	3	yes	—
181	#168	C₂²×C₃×C₄	1	1	yes	characteristic
182	#168	C₂²×C₃×C₄	1	3	yes	—
183	C₂×#41	—	5	1	no	—
184	#131	—	5	1	no	—
185	#131	—	5	3	no	—
186	#131	—	5	3	no	—
187	#131	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
188	C₂×M₁₆	—	5	3	no	—
189	C₂²×C₈	—	5	3	no	—
190	C₄×Q₈	—	10	2	no	—
191	C₄×D₄	—	10	6	no	—
192	#117	—	5	6	no	—
193	C₂×C₄⋊C₄	—	5	6	no	—
194	#107	—	5	6	no	—
195	#106	—	5	3	no	—
196	#106	—	5	6	no	—
197	#106	—	5	1	no	—
198	C₂×C₃×C₅	—	2	3	no	—
199	C₂×C₃×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
200	C₃×D₅	—	2	3	no	—
201	C₃×D₅	#577	1	2	yes	—
202	C₂³×C₃	—	5	3	no	—
203	C₃×Q₈	—	5	3	no	—
204	C₃×Q₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
205	C₃×D₄	—	5	6	no	—
206	C₃×D₄	—	5	3	no	—
207	C₃×D₄	C₄⋊AGL(1,𝔽₅)	1	3	yes	—
208	C₂×C₃×C₄	—	5	3	no	—
209	C₂×C₃×C₄	—	5	3	no	—
210	C₂×C₃×C₄	—	5	3	no	—
211	C₂×C₃×C₄	—	5	3	no	—
212	C₂×C₃×C₄	—	5	1	no	—
213	C₂×C₃×C₄	—	10	2	no	—
214	C₂×C₃×C₄	—	10	6	no	—
215	C₂×C₃×C₄	—	40	6	no	—
216	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	3	yes	—
217	C₃×C₈	—	5	1	no	—
218	C₃×C₈	—	5	3	no	—
219	C₃×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
220	C₃×C₈	C₂²×AGL(1,𝔽₅)	1	3	yes	—
221	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	3	yes	—
222	C₂×D₅	—	2	6	no	—
223	C₂×D₅	C₂×C₃×C₄⋊C₄	1	3	yes	—
224	C₂×D₅	C₂²×C₃×D₄	1	1	yes	characteristic
225	AGL(1,𝔽₅)	—	4	2	no	—
226	AGL(1,𝔽₅)	—	4	6	no	—
227	C₄×C₅	C₂×C₃×C₄⋊C₄	1	3	yes	—
228	C₄×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
229	Dic₅	C₂×C₃×C₄⋊C₄	1	3	yes	—
230	Dic₅	C₂²×C₃×Q₈	1	1	yes	characteristic
231	#41	—	5	1	no	—
232	#41	—	5	3	no	—
233	#41	—	5	3	no	—
234	#41	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
235	C₂×Q₈	—	5	1	no	—
236	C₂×D₄	—	5	3	no	—
237	C₂²×C₄	—	5	3	no	—
238	C₂²×C₄	—	10	6	no	—
239	M₁₆	—	5	3	no	—
240	M₁₆	—	5	6	no	—
241	M₁₆	C₂×C₃×AGL(1,𝔽₅)	1	3	yes	complemented
242	C₂×C₈	—	5	3	no	—
243	C₂×C₈	—	5	1	no	—
244	C₂×C₈	—	5	3	no	—
245	C₂×C₈	—	5	3	no	—
246	C₂×C₈	—	5	3	no	—
247	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	3	yes	complemented
248	C₄⋊C₄	—	5	6	no	—
249	C₄⋊C₄	—	5	2	no	—
250	C₄⋊C₄	—	10	6	no	—
251	C₂²⋊C₄	—	10	6	no	—
252	C₄²	—	10	6	no	—
253	C₃×C₅	#2912	1	1	yes	characteristic, complemented, Hall
254	C₂²×C₃	—	5	1	no	—
255	C₂²×C₃	—	5	3	no	—
256	C₂²×C₃	—	10	6	no	—
257	C₂²×C₃	C₂×C₄⋊AGL(1,𝔽₅)	1	3	yes	—
258	C₃×C₄	—	5	3	no	—
259	C₃×C₄	—	5	1	no	—
260	C₃×C₄	—	20	2	no	—
261	C₃×C₄	—	20	6	no	—
262	C₃×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	3	yes	—
263	C₃×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
264	C₂×C₅	—	2	3	no	—
265	C₂×C₅	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
266	D₅	—	2	3	no	—
267	D₅	C₃×#577	1	2	yes	—
268	C₂³	—	5	3	no	—
269	Q₈	—	5	3	no	—
270	Q₈	C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
271	D₄	—	5	6	no	—
272	D₄	—	5	3	no	—
273	D₄	C₃×C₄⋊AGL(1,𝔽₅)	1	3	yes	—
274	C₂×C₄	—	5	3	no	—
275	C₂×C₄	—	5	3	no	—
276	C₂×C₄	—	5	3	no	—
277	C₂×C₄	—	5	1	no	—
278	C₂×C₄	—	5	3	no	—
279	C₂×C₄	—	10	2	no	—
280	C₂×C₄	—	10	6	no	—
281	C₂×C₄	—	40	6	no	—
282	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	3	yes	—
283	C₈	—	5	1	no	—
284	C₈	—	5	3	no	—
285	C₈	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
286	C₈	C₂²×C₃×AGL(1,𝔽₅)	1	3	yes	—
287	C₂×C₃	—	2	3	no	—
288	C₂×C₃	—	5	2	no	—
289	C₂×C₃	—	10	3	no	—
290	C₂×C₃	C₂²×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
291	C₅	C₃×#2912	1	1	yes	characteristic, complemented, Sylow
292	C₂²	—	5	1	no	—
293	C₂²	—	5	3	no	—
294	C₂²	—	10	6	no	—
295	C₂²	C₂×C₃×C₄⋊AGL(1,𝔽₅)	1	3	yes	—
296	C₄	—	5	1	no	—
297	C₄	—	5	3	no	—
298	C₄	—	20	6	no	—
299	C₄	—	20	2	no	—
300	C₄	C₂×C₃×C₄⋊AGL(1,𝔽₅)	1	3	yes	—
301	C₄	C₂³×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
302	C₃	#10617416	1	1	yes	characteristic, direct product factor, Sylow
303	C₂	—	2	3	no	—
304	C₂	—	5	2	no	—
305	C₂	—	10	3	no	—
306	C₂	C₂²×C₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂²×C₄⋊AGL(1,𝔽₅)
commutator	C₄×C₅	C₂³×C₃×C₄
Frattini	C₄	C₂³×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×#131	C₄
socle	C₂×C₃×C₅	C₂²×C₄⋊C₄
prime radical	C₃×D₅×#41	C₂²
cosocle	C₄×D₅	C₂⁴×C₃
hypercenter	C₃×#131	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#2912
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₃×#41 ◁ C₃×#131

Complements

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

N	H ≅ G/N	type
3. C₃×#65727	C₂	semidirect
4. C₃×#65721	C₂	semidirect
5. C₂×C₃×#3993	C₂	semidirect
6. #10617416	C₃	direct (N × H)
12. C₃×C₅×#131	C₄	semidirect
13. C₃×#4094	C₄	semidirect
14. C₃×#4053	C₄	semidirect
16. C₃×#4047	C₄	semidirect
25. #65727	C₂×C₃	semidirect
26. #65721	C₂×C₃	semidirect
27. C₂×#3993	C₂×C₃	semidirect
41. C₃×C₅×M₁₆	C₂×C₄	semidirect
42. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
43. C₃×#681	C₂×C₄	semidirect
44. C₂×C₃×#168	C₂×C₄	semidirect
45. C₃×#676	C₂×C₄	semidirect
46. C₃×C₈×D₅	C₂×C₄	semidirect
47. C₃×C₈×D₅	C₂×C₄	semidirect
60. C₅×#131	C₃×C₄	semidirect
61. #4094	C₃×C₄	semidirect
62. #4053	C₃×C₄	semidirect
64. #4047	C₃×C₄	semidirect
94. C₃×#131	AGL(1,𝔽₅)	semidirect
118. C₅×M₁₆	C₂×C₃×C₄	semidirect
119. C₂×C₅×C₈	C₂×C₃×C₄	semidirect
120. #681	C₂×C₃×C₄	semidirect
121. C₂×#168	C₂×C₃×C₄	semidirect
122. #676	C₂×C₃×C₄	semidirect
123. C₈×D₅	C₂×C₃×C₄	semidirect
124. C₈×D₅	C₂×C₃×C₄	semidirect
152. C₃×M₁₆	C₂×AGL(1,𝔽₅)	semidirect
158. C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	semidirect
187. #131	C₃×AGL(1,𝔽₅)	semidirect
241. M₁₆	C₂×C₃×AGL(1,𝔽₅)	semidirect
247. C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	semidirect
253. C₃×C₅	#2912	semidirect
291. C₅	C₃×#2912	semidirect
302. C₃	#10617416	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×#66266	2
C₃×D₅×#131	2
C₃×#65727	2
C₃×#65721	2
C₂×C₃×#3993	2
#10617416	3
C₃×#2912	5

Subgroups — C3×#10617416 #14413454

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10617416 #14413454