Subgroups — C₃×#10613505 #14458099

Order 1920.

920 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×#66096	C₂	1	4	yes	maximal, complemented
2	C₃×C₅×#492	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65466	C₂	1	4	yes	maximal, complemented
4	C₃×#65384	C₂	1	2	yes	maximal, complemented
5	C₃×#65366	C₂	1	2	yes	maximal, complemented
6	C₃×#64989	C₂	1	2	yes	maximal, complemented
7	#10613505	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₃×#4147	C₂²	1	2	yes	—
9	C₃×#4144	C₂²	1	2	yes	complemented
10	C₂×C₃×C₅×SD₁₆	C₂²	1	4	yes	complemented
11	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
12	C₃×C₅×#127	C₂²	1	1	yes	characteristic
13	C₃×#4088	C₂²	1	4	yes	complemented
14	C₃×#4078	C₂²	1	4	yes	—
15	C₃×#4062	—	2	4	no	—
16	C₃×#4061	—	2	4	no	—
17	C₃×#4060	—	2	4	no	—
18	C₃×D₅×SD₁₆	—	2	4	no	—
19	C₃×#4050	C₂²	1	2	yes	—
20	C₃×#4047	C₂²	1	4	yes	complemented
21	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic, complemented
22	C₃×#3968	C₂²	1	2	yes	complemented
23	C₃×#3965	C₂²	1	2	yes	complemented
24	C₃×#3951	C₂²	1	2	yes	complemented
25	C₃×#3937	C₂²	1	4	yes	complemented
26	C₃×#3291	—	5	1	no	maximal, Hall
27	#66096	C₂×C₃	1	4	yes	complemented
28	C₅×#492	C₂×C₃	1	1	yes	characteristic, complemented
29	#65466	C₂×C₃	1	4	yes	complemented
30	#65384	C₂×C₃	1	2	yes	complemented
31	#65366	C₂×C₃	1	2	yes	complemented
32	#64989	C₂×C₃	1	2	yes	complemented
33	C₂×C₃×C₅×Q₈	C₂³	1	2	yes	—
34	C₂×C₃×C₅×D₄	—	4	1	no	—
35	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
36	C₂×C₃×C₂²⋊D₅	—	4	2	no	—
37	C₃×#713	—	2	4	no	—
38	C₃×Q₈×D₅	—	2	4	no	—
39	C₃×#711	—	2	4	no	—
40	C₃×D₄×D₅	—	2	4	no	—
41	C₃×#709	—	4	2	no	—
42	C₃×#709	C₂³	1	4	yes	—
43	C₃×C₅×SD₁₆	—	2	4	no	—
44	C₃×C₅×SD₁₆	—	2	4	no	—
45	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
46	C₃×C₅×C₄⋊C₄	—	4	1	no	—
47	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
48	C₃×#689	—	2	4	no	—
49	C₃×#688	—	2	4	no	—
50	C₃×#687	—	2	4	no	—
51	C₃×#686	—	2	4	no	—
52	C₃×#681	C₂³	1	4	yes	—
53	C₃×Dic₂₀	D₄	1	2	yes	complemented
54	C₃×D₄₀	D₄	1	2	yes	complemented
55	C₃×#677	D₄	1	4	yes	complemented
56	C₃×#676	—	2	4	no	—
57	C₃×C₈×D₅	—	2	4	no	—
58	C₃×#577	—	5	4	no	—
59	C₃×#492	D₅	1	1	yes	characteristic, complemented
60	C₃×#471	—	5	4	no	—
61	C₃×#456	—	5	2	no	—
62	C₃×#453	—	5	2	no	—
63	C₃×#443	—	5	2	no	—
64	#4147	C₂²×C₃	1	2	yes	—
65	#4144	C₂²×C₃	1	2	yes	complemented
66	C₂×C₅×SD₁₆	C₂²×C₃	1	4	yes	complemented
67	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
68	C₅×#127	C₂²×C₃	1	1	yes	characteristic
69	#4088	C₂²×C₃	1	4	yes	complemented
70	#4078	C₂²×C₃	1	4	yes	—
71	#4062	—	2	4	no	—
72	#4061	—	2	4	no	—
73	#4060	—	2	4	no	—
74	D₅×SD₁₆	—	2	4	no	—
75	#4050	C₂²×C₃	1	2	yes	—
76	#4047	C₂²×C₃	1	4	yes	complemented
77	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic, complemented
78	#3968	C₂²×C₃	1	2	yes	complemented
79	#3965	C₂²×C₃	1	2	yes	complemented
80	#3951	C₂²×C₃	1	2	yes	complemented
81	#3937	C₂²×C₃	1	4	yes	complemented
82	#3291	—	5	1	no	Sylow
83	C₂³×C₃×C₅	—	2	2	no	—
84	C₂²×C₃×D₅	—	4	2	no	—
85	C₃×C₅×Q₈	—	2	4	no	—
86	C₃×C₅×D₄	—	2	4	no	—
87	C₃×C₅×D₄	—	8	2	no	—
88	C₂×C₃×C₄×C₅	—	2	1	no	—
89	C₂×C₃×C₄×C₅	—	2	2	no	—
90	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
91	C₃×C₂²⋊D₅	—	2	4	no	—
92	C₃×C₂²⋊D₅	—	4	4	no	—
93	C₂×C₃×Dic₅	—	4	2	no	—
94	C₃×D₂₀	—	4	2	no	—
95	C₃×D₂₀	C₂×D₄	1	4	yes	—
96	C₃×C₄×D₅	—	2	4	no	—
97	C₃×C₄×D₅	—	4	4	no	—
98	C₃×Dic₁₀	—	4	2	no	—
99	C₃×Dic₁₀	C₂×D₄	1	4	yes	—
100	C₃×C₅×C₈	C₂×D₄	1	2	yes	—
101	C₃×C₅×C₈	C₂×D₄	1	2	yes	—
102	C₃×#168	—	2	4	no	—
103	C₃×2₋¹⁺⁴	—	5	2	no	—
104	C₃×H(5,𝔽₂)	—	5	2	no	—
105	C₃×#137	—	5	4	no	—
106	C₃×#137	—	10	4	no	—
107	C₃×AGL(1,ℤ/8)	—	5	4	no	—
108	C₃×AGL(1,ℤ/8)	—	10	4	no	—
109	C₃×#135	—	5	2	no	—
110	C₃×#135	—	10	4	no	—
111	C₂×C₃×SD₁₆	—	10	4	no	—
112	C₂×C₃×SD₁₆	C₂×D₅	1	4	yes	complemented
113	C₃×#131	—	5	4	no	—
114	C₃×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic, complemented
115	C₃×#127	C₂×D₅	1	1	yes	characteristic
116	C₃×#108	—	5	2	no	—
117	C₃×#104	—	5	4	no	—
118	C₃×#101	—	5	2	no	—
119	C₃×#100	—	5	2	no	—
120	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic, complemented
121	C₂×C₅×Q₈	C₂³×C₃	1	2	yes	—
122	C₂×C₅×D₄	—	4	1	no	—
123	C₂×C₅×D₄	C₂³×C₃	1	2	yes	—
124	C₂×C₂²⋊D₅	—	4	2	no	—
125	#713	—	2	4	no	—
126	Q₈×D₅	—	2	4	no	—
127	#711	—	2	4	no	—
128	D₄×D₅	—	2	4	no	—
129	#709	—	4	2	no	—
130	#709	C₂³×C₃	1	4	yes	—
131	C₅×SD₁₆	—	2	4	no	—
132	C₅×SD₁₆	—	2	4	no	—
133	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
134	C₅×C₄⋊C₄	—	4	1	no	—
135	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
136	#689	—	2	4	no	—
137	#688	—	2	4	no	—
138	#687	—	2	4	no	—
139	#686	—	2	4	no	—
140	#681	C₂³×C₃	1	4	yes	—
141	Dic₂₀	C₃×D₄	1	2	yes	complemented
142	D₄₀	C₃×D₄	1	2	yes	complemented
143	#677	C₃×D₄	1	4	yes	complemented
144	#676	—	2	4	no	—
145	C₈×D₅	—	2	4	no	—
146	#577	—	5	4	no	—
147	#492	C₃×D₅	1	1	yes	characteristic, complemented
148	#471	—	5	4	no	—
149	#456	—	5	2	no	—
150	#453	—	5	2	no	—
151	#443	—	5	2	no	—
152	C₂²×C₃×C₅	—	4	2	no	—
153	C₂²×C₃×C₅	—	8	2	no	—
154	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂×C₃×D₅	—	2	4	no	—
156	C₂×C₃×D₅	—	16	2	no	—
157	C₃×C₄×C₅	—	2	2	no	—
158	C₃×C₄×C₅	—	4	2	no	—
159	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
160	C₃×Dic₅	—	2	4	no	—
161	C₃×#41	—	5	4	no	—
162	C₃×#41	—	10	4	no	—
163	C₃×#41	—	10	4	no	—
164	C₃×#41	—	20	2	no	—
165	C₂×C₃×Q₈	—	10	4	no	—
166	C₂×C₃×Q₈	C₂²×D₅	1	2	yes	—
167	C₂×C₃×D₄	—	4	1	no	—
168	C₂×C₃×D₄	—	10	4	no	—
169	C₂×C₃×D₄	—	20	2	no	—
170	C₂×C₃×D₄	C₂²×D₅	1	2	yes	—
171	C₃×Q₁₆	—	5	2	no	—
172	C₃×Q₁₆	—	10	4	no	—
173	C₃×SD₁₆	—	2	4	no	—
174	C₃×SD₁₆	—	2	4	no	—
175	C₃×SD₁₆	—	5	4	no	—
176	C₃×SD₁₆	—	10	4	no	—
177	C₃×SD₁₆	—	10	4	no	—
178	C₃×D₈	—	5	2	no	—
179	C₃×D₈	—	10	4	no	—
180	C₃×M₁₆	—	5	4	no	—
181	C₃×M₁₆	—	10	4	no	—
182	C₂×C₃×C₈	—	10	4	no	—
183	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
184	C₃×C₄⋊C₄	—	4	1	no	—
185	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
186	C₂³×C₅	—	2	2	no	—
187	C₂²×D₅	—	4	2	no	—
188	C₅×Q₈	—	2	4	no	—
189	C₅×D₄	—	2	4	no	—
190	C₅×D₄	—	8	2	no	—
191	C₂×C₄×C₅	—	2	1	no	—
192	C₂×C₄×C₅	—	2	2	no	—
193	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
194	C₂²⋊D₅	—	2	4	no	—
195	C₂²⋊D₅	—	4	4	no	—
196	C₂×Dic₅	—	4	2	no	—
197	D₂₀	—	4	2	no	—
198	D₂₀	C₂×C₃×D₄	1	4	yes	—
199	C₄×D₅	—	2	4	no	—
200	C₄×D₅	—	4	4	no	—
201	Dic₁₀	—	4	2	no	—
202	Dic₁₀	C₂×C₃×D₄	1	4	yes	—
203	C₅×C₈	C₂×C₃×D₄	1	2	yes	—
204	C₅×C₈	C₂×C₃×D₄	1	2	yes	—
205	#168	—	2	4	no	—
206	2₋¹⁺⁴	—	5	2	no	—
207	H(5,𝔽₂)	—	5	2	no	—
208	#137	—	5	4	no	—
209	#137	—	10	4	no	—
210	AGL(1,ℤ/8)	—	5	4	no	—
211	AGL(1,ℤ/8)	—	10	4	no	—
212	#135	—	5	2	no	—
213	#135	—	10	4	no	—
214	C₂×SD₁₆	—	10	4	no	—
215	C₂×SD₁₆	C₂×C₃×D₅	1	4	yes	complemented
216	#131	—	5	4	no	—
217	C₄⋊Q₈	C₂×C₃×D₅	1	1	yes	characteristic, complemented
218	#127	C₂×C₃×D₅	1	1	yes	characteristic
219	#108	—	5	2	no	—
220	#104	—	5	4	no	—
221	#101	—	5	2	no	—
222	#100	—	5	2	no	—
223	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic, complemented
224	C₂×C₃×C₅	—	2	1	no	—
225	C₂×C₃×C₅	—	8	2	no	—
226	C₂×C₃×C₅	D₄²	1	1	yes	characteristic
227	C₃×D₅	—	4	4	no	—
228	C₂³×C₃	—	2	2	no	—
229	C₂³×C₃	—	20	2	no	—
230	C₃×Q₈	—	2	4	no	—
231	C₃×Q₈	—	5	4	no	—
232	C₃×Q₈	—	20	2	no	—
233	C₃×D₄	—	2	4	no	—
234	C₃×D₄	—	5	4	no	—
235	C₃×D₄	—	8	2	no	—
236	C₃×D₄	—	10	4	no	—
237	C₃×D₄	—	20	2	no	—
238	C₃×D₄	—	20	4	no	—
239	C₂×C₃×C₄	—	2	1	no	—
240	C₂×C₃×C₄	—	2	2	no	—
241	C₂×C₃×C₄	—	10	4	no	—
242	C₂×C₃×C₄	—	20	4	no	—
243	C₂×C₃×C₄	—	20	2	no	—
244	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
245	C₃×C₈	—	10	4	no	—
246	C₃×C₈	D₄×D₅	1	2	yes	—
247	C₃×C₈	D₄×D₅	1	2	yes	—
248	C₂²×C₅	—	4	2	no	—
249	C₂²×C₅	—	8	2	no	—
250	C₂²×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
251	C₂×D₅	—	2	4	no	—
252	C₂×D₅	—	16	2	no	—
253	C₄×C₅	—	2	2	no	—
254	C₄×C₅	—	4	2	no	—
255	C₄×C₅	C₂²×C₃×D₄	1	2	yes	—
256	Dic₅	—	2	4	no	—
257	#41	—	5	4	no	—
258	#41	—	10	4	no	—
259	#41	—	10	4	no	—
260	#41	—	20	2	no	—
261	C₂×Q₈	—	10	4	no	—
262	C₂×Q₈	C₂²×C₃×D₅	1	2	yes	—
263	C₂×D₄	—	4	1	no	—
264	C₂×D₄	—	10	4	no	—
265	C₂×D₄	—	20	2	no	—
266	C₂×D₄	C₂²×C₃×D₅	1	2	yes	—
267	Q₁₆	—	5	2	no	—
268	Q₁₆	—	10	4	no	—
269	SD₁₆	—	2	4	no	—
270	SD₁₆	—	2	4	no	—
271	SD₁₆	—	5	4	no	—
272	SD₁₆	—	10	4	no	—
273	SD₁₆	—	10	4	no	—
274	D₈	—	5	2	no	—
275	D₈	—	10	4	no	—
276	M₁₆	—	5	4	no	—
277	M₁₆	—	10	4	no	—
278	C₂×C₈	—	10	4	no	—
279	C₂×C₈	C₂²×C₃×D₅	1	2	yes	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
282	C₃×C₅	#3291	1	1	yes	characteristic, complemented, Hall
283	C₂²×C₃	—	4	2	no	—
284	C₂²×C₃	—	8	2	no	—
285	C₂²×C₃	—	10	4	no	—
286	C₂²×C₃	—	80	2	no	—
287	C₂²×C₃	#4144	1	1	yes	characteristic
288	C₃×C₄	—	2	2	no	—
289	C₃×C₄	—	4	2	no	—
290	C₃×C₄	—	10	4	no	—
291	C₃×C₄	C₂×D₄×D₅	1	2	yes	—
292	C₂×C₅	—	2	1	no	—
293	C₂×C₅	—	8	2	no	—
294	C₂×C₅	C₃×D₄²	1	1	yes	characteristic
295	D₅	—	4	4	no	—
296	C₂³	—	2	2	no	—
297	C₂³	—	20	2	no	—
298	Q₈	—	2	4	no	—
299	Q₈	—	5	4	no	—
300	Q₈	—	20	2	no	—
301	D₄	—	2	4	no	—
302	D₄	—	5	4	no	—
303	D₄	—	8	2	no	—
304	D₄	—	10	4	no	—
305	D₄	—	20	2	no	—
306	D₄	—	20	4	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	10	4	no	—
310	C₂×C₄	—	20	4	no	—
311	C₂×C₄	—	20	2	no	—
312	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
313	C₈	—	10	4	no	—
314	C₈	C₃×D₄×D₅	1	2	yes	—
315	C₈	C₃×D₄×D₅	1	2	yes	—
316	C₂×C₃	—	2	1	no	—
317	C₂×C₃	—	8	2	no	—
318	C₂×C₃	—	20	4	no	—
319	C₂×C₃	#66051	1	1	yes	characteristic
320	C₅	C₃×#3291	1	1	yes	characteristic, complemented, Sylow
321	C₂²	—	4	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	10	4	no	—
324	C₂²	—	80	2	no	—
325	C₂²	C₃×#4144	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	4	2	no	—
328	C₄	—	10	4	no	—
329	C₄	C₂×C₃×D₄×D₅	1	2	yes	—
330	C₃	#10613505	1	1	yes	characteristic, direct product factor, Sylow
331	C₂	—	2	1	no	—
332	C₂	—	8	2	no	—
333	C₂	—	20	4	no	—
334	C₂	C₃×#66051	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	#66051
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₃×C₅×#492	C₂
socle	C₂×C₃×C₅	D₄²
hypercenter	C₃×#492	D₅
hypocenter	C₅	C₃×#3291
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#492

Complements

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

N	H ≅ G/N	type
1. C₃×#66096	C₂	semidirect
2. C₃×C₅×#492	C₂	semidirect
3. C₃×#65466	C₂	semidirect
4. C₃×#65384	C₂	semidirect
5. C₃×#65366	C₂	semidirect
6. C₃×#64989	C₂	semidirect
7. #10613505	C₃	direct (N × H)
9. C₃×#4144	C₂²	semidirect
10. C₂×C₃×C₅×SD₁₆	C₂²	semidirect
11. C₃×C₅×C₄⋊Q₈	C₂²	semidirect
13. C₃×#4088	C₂²	semidirect
20. C₃×#4047	C₂²	semidirect
21. C₃×C₄×C₅×C₈	C₂²	semidirect
22. C₃×#3968	C₂²	semidirect
23. C₃×#3965	C₂²	semidirect
24. C₃×#3951	C₂²	semidirect
25. C₃×#3937	C₂²	semidirect
27. #66096	C₂×C₃	semidirect
28. C₅×#492	C₂×C₃	semidirect
29. #65466	C₂×C₃	semidirect
30. #65384	C₂×C₃	semidirect
31. #65366	C₂×C₃	semidirect
32. #64989	C₂×C₃	semidirect
53. C₃×Dic₂₀	D₄	semidirect
54. C₃×D₄₀	D₄	semidirect
55. C₃×#677	D₄	semidirect
59. C₃×#492	D₅	semidirect
65. #4144	C₂²×C₃	semidirect
66. C₂×C₅×SD₁₆	C₂²×C₃	semidirect
67. C₅×C₄⋊Q₈	C₂²×C₃	semidirect
69. #4088	C₂²×C₃	semidirect
76. #4047	C₂²×C₃	semidirect
77. C₄×C₅×C₈	C₂²×C₃	semidirect
78. #3968	C₂²×C₃	semidirect
79. #3965	C₂²×C₃	semidirect
80. #3951	C₂²×C₃	semidirect
81. #3937	C₂²×C₃	semidirect
112. C₂×C₃×SD₁₆	C₂×D₅	semidirect
114. C₃×C₄⋊Q₈	C₂×D₅	semidirect
120. C₃×C₄×C₈	C₂×D₅	semidirect
141. Dic₂₀	C₃×D₄	semidirect
142. D₄₀	C₃×D₄	semidirect
143. #677	C₃×D₄	semidirect
147. #492	C₃×D₅	semidirect
215. C₂×SD₁₆	C₂×C₃×D₅	semidirect
217. C₄⋊Q₈	C₂×C₃×D₅	semidirect
223. C₄×C₈	C₂×C₃×D₅	semidirect
282. C₃×C₅	#3291	semidirect
320. C₅	C₃×#3291	semidirect
330. C₃	#10613505	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×#66096	2
C₃×C₅×#492	2
C₃×#65466	2
C₃×#65384	2
C₃×#65366	2
C₃×#64989	2
#10613505	3
C₃×#3291	5

Subgroups — C3×#10613505 #14458099

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10613505 #14458099