Subgroups — C₃×#10612672 #14458171

Order 1920.

952 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#66108	C₂	1	2	yes	maximal, complemented
2	C₃×#66091	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#66086	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#472	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#65487	C₂	1	2	yes	maximal, complemented
6	C₃×#65440	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#65433	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#65186	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#65184	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#65038	C₂	1	2	yes	maximal, complemented
11	C₃×#65036	C₂	1	2	yes	maximal, complemented
12	#10612672	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₃×#4149	C₂²	1	2	yes	complemented
14	C₃×#4144	C₂²	1	2	yes	complemented
15	C₃×C₅×AGL(1,ℤ/8)	C₂²	1	2	yes	complemented
16	C₂×C₃×C₅×D₈	C₂²	1	1	yes	characteristic, complemented
17	C₃×C₅×#131	C₂²	1	1	yes	characteristic, complemented
18	C₃×#4096	C₂²	1	1	yes	characteristic
19	C₃×#4086	C₂²	1	2	yes	complemented
20	C₃×#4078	C₂²	1	2	yes	complemented
21	C₂×C₃×#686	C₂²	1	2	yes	complemented
22	C₃×#4062	—	2	2	no	—
23	C₃×#4060	—	2	2	no	—
24	C₃×#4058	—	2	1	no	—
25	C₃×#4057	—	2	2	no	—
26	C₃×#4057	—	2	2	no	—
27	C₃×D₅×D₈	—	2	2	no	—
28	C₃×D₅×D₈	—	2	1	no	—
29	C₃×#4054	—	2	2	no	—
30	C₃×#4054	C₂²	1	2	yes	complemented
31	C₃×#4054	C₂²	1	2	yes	complemented
32	C₃×#4053	C₂²	1	2	yes	complemented
33	C₃×#4050	—	2	1	no	—
34	C₃×#4050	C₂²	1	1	yes	characteristic
35	C₂×C₃×D₄₀	—	2	1	no	—
36	C₂×C₃×D₄₀	C₂²	1	1	yes	characteristic, complemented
37	C₃×#4047	C₂²	1	1	yes	characteristic, complemented
38	C₃×C₅×#108	C₂²	1	1	yes	characteristic, complemented
39	C₃×C₅×#100	C₂²	1	2	yes	complemented
40	C₃×#3969	C₂²	1	1	yes	characteristic, complemented
41	C₃×#3957	C₂²	1	1	yes	characteristic, complemented
42	C₃×#3957	C₂²	1	2	yes	complemented
43	C₃×#3955	C₂²	1	2	yes	complemented
44	C₃×#3954	C₂²	1	1	yes	characteristic, complemented
45	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic, complemented
46	C₃×#3292	—	5	1	no	maximal, Hall
47	#66108	C₂×C₃	1	2	yes	complemented
48	#66091	C₂×C₃	1	1	yes	characteristic, complemented
49	#66086	C₂×C₃	1	1	yes	characteristic, complemented
50	C₅×#472	C₂×C₃	1	1	yes	characteristic, complemented
51	#65487	C₂×C₃	1	2	yes	complemented
52	#65440	C₂×C₃	1	1	yes	characteristic, complemented
53	#65433	C₂×C₃	1	1	yes	characteristic, complemented
54	#65186	C₂×C₃	1	1	yes	characteristic, complemented
55	#65184	C₂×C₃	1	1	yes	characteristic, complemented
56	#65038	C₂×C₃	1	2	yes	complemented
57	#65036	C₂×C₃	1	2	yes	complemented
58	C₃×C₅×#41	C₂³	1	1	yes	characteristic
59	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
60	C₂×C₃×C₂²⋊D₅	—	4	2	no	—
61	C₂×C₃×C₂²⋊D₅	—	4	2	no	—
62	C₃×#713	—	2	2	no	—
63	C₃×#711	—	2	2	no	—
64	C₃×#711	—	2	2	no	—
65	C₃×D₄×D₅	—	2	2	no	—
66	C₃×D₄×D₅	—	2	2	no	—
67	C₃×D₄×D₅	—	2	2	no	—
68	C₃×D₄×D₅	—	4	2	no	—
69	C₃×#709	—	2	2	no	—
70	C₃×#709	C₂³	1	2	yes	—
71	C₃×#709	C₂³	1	1	yes	characteristic
72	C₂×C₃×D₂₀	—	2	2	no	—
73	C₂×C₃×D₂₀	C₂³	1	2	yes	—
74	C₃×C₅×SD₁₆	—	2	2	no	—
75	C₃×C₅×D₈	—	2	2	no	—
76	C₃×C₅×D₈	—	2	1	no	—
77	C₃×C₅×D₈	—	2	1	no	—
78	C₃×C₅×M₁₆	—	2	1	no	—
79	C₃×C₅×M₁₆	C₂³	1	1	yes	characteristic
80	C₃×C₅×M₁₆	C₂³	1	2	yes	—
81	C₂×C₃×C₅×C₈	—	2	1	no	—
82	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
83	C₃×#689	D₄	1	1	yes	characteristic, complemented
84	C₃×#688	D₄	1	1	yes	characteristic, complemented
85	C₃×#687	—	2	2	no	—
86	C₃×#687	D₄	1	1	yes	characteristic, complemented
87	C₃×#686	—	2	2	no	—
88	C₃×#686	—	2	2	no	—
89	C₃×#686	—	2	2	no	—
90	C₃×#686	D₄	1	1	yes	characteristic, complemented
91	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
92	C₃×#681	C₂³	1	1	yes	characteristic
93	C₂×C₃×#168	C₂³	1	1	yes	characteristic
94	C₃×Dic₂₀	D₄	1	1	yes	characteristic, complemented
95	C₃×D₄₀	—	2	2	no	—
96	C₃×D₄₀	—	2	2	no	—
97	C₃×D₄₀	—	2	1	no	—
98	C₃×D₄₀	—	2	1	no	—
99	C₃×D₄₀	D₄	1	1	yes	characteristic, complemented
100	C₃×#677	—	2	2	no	—
101	C₃×#677	—	2	2	no	—
102	C₃×#677	D₄	1	2	yes	complemented
103	C₃×#676	—	2	1	no	—
104	C₃×#676	—	2	2	no	—
105	C₃×C₈×D₅	—	2	1	no	—
106	C₃×C₈×D₅	—	2	2	no	—
107	C₃×#576	—	5	2	no	—
108	C₃×#576	—	5	1	no	—
109	C₃×#576	—	5	1	no	—
110	C₃×#493	—	5	1	no	—
111	C₃×#472	—	5	1	no	—
112	C₃×#472	—	5	2	no	—
113	C₃×#472	D₅	1	1	yes	characteristic, complemented
114	C₃×#453	—	5	2	no	—
115	C₃×#453	—	5	2	no	—
116	C₃×#443	—	5	1	no	—
117	C₃×#443	—	5	1	no	—
118	#4149	C₂²×C₃	1	2	yes	complemented
119	#4144	C₂²×C₃	1	2	yes	complemented
120	C₅×AGL(1,ℤ/8)	C₂²×C₃	1	2	yes	complemented
121	C₂×C₅×D₈	C₂²×C₃	1	1	yes	characteristic, complemented
122	C₅×#131	C₂²×C₃	1	1	yes	characteristic, complemented
123	#4096	C₂²×C₃	1	1	yes	characteristic
124	#4086	C₂²×C₃	1	2	yes	complemented
125	#4078	C₂²×C₃	1	2	yes	complemented
126	C₂×#686	C₂²×C₃	1	2	yes	complemented
127	#4062	—	2	2	no	—
128	#4060	—	2	2	no	—
129	#4058	—	2	1	no	—
130	#4057	—	2	2	no	—
131	#4057	—	2	2	no	—
132	D₅×D₈	—	2	2	no	—
133	D₅×D₈	—	2	1	no	—
134	#4054	—	2	2	no	—
135	#4054	C₂²×C₃	1	2	yes	complemented
136	#4054	C₂²×C₃	1	2	yes	complemented
137	#4053	C₂²×C₃	1	2	yes	complemented
138	#4050	—	2	1	no	—
139	#4050	C₂²×C₃	1	1	yes	characteristic
140	C₂×D₄₀	—	2	1	no	—
141	C₂×D₄₀	C₂²×C₃	1	1	yes	characteristic, complemented
142	#4047	C₂²×C₃	1	1	yes	characteristic, complemented
143	C₅×#108	C₂²×C₃	1	1	yes	characteristic, complemented
144	C₅×#100	C₂²×C₃	1	2	yes	complemented
145	#3969	C₂²×C₃	1	1	yes	characteristic, complemented
146	#3957	C₂²×C₃	1	1	yes	characteristic, complemented
147	#3957	C₂²×C₃	1	2	yes	complemented
148	#3955	C₂²×C₃	1	2	yes	complemented
149	#3954	C₂²×C₃	1	1	yes	characteristic, complemented
150	C₈×Dic₅	C₂²×C₃	1	1	yes	characteristic, complemented
151	#3292	—	5	1	no	Sylow
152	C₂³×C₃×C₅	—	2	2	no	—
153	C₂²×C₃×D₅	—	2	2	no	—
154	C₂²×C₃×D₅	—	4	2	no	—
155	C₂²×C₃×D₅	—	4	2	no	—
156	C₃×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
157	C₃×C₅×D₄	—	2	1	no	—
158	C₃×C₅×D₄	—	2	2	no	—
159	C₃×C₅×D₄	—	2	2	no	—
160	C₃×C₅×D₄	C₂×D₄	1	1	yes	characteristic
161	C₂×C₃×C₄×C₅	—	2	1	no	—
162	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
163	C₃×C₂²⋊D₅	—	2	2	no	—
164	C₃×C₂²⋊D₅	—	2	1	no	—
165	C₃×C₂²⋊D₅	—	4	2	no	—
166	C₃×C₂²⋊D₅	—	4	2	no	—
167	C₃×C₂²⋊D₅	—	4	2	no	—
168	C₃×C₂²⋊D₅	—	8	4	no	—
169	C₂×C₃×Dic₅	—	2	1	no	—
170	C₂×C₃×Dic₅	—	4	2	no	—
171	C₃×D₂₀	—	2	2	no	—
172	C₃×D₂₀	—	2	2	no	—
173	C₃×D₂₀	—	4	2	no	—
174	C₃×D₂₀	C₂×D₄	1	2	yes	—
175	C₃×D₂₀	C₂×D₄	1	1	yes	characteristic
176	C₃×C₄×D₅	—	2	1	no	—
177	C₃×C₄×D₅	—	2	2	no	—
178	C₃×C₄×D₅	—	4	2	no	—
179	C₃×Dic₁₀	C₂×D₄	1	1	yes	characteristic
180	C₃×Dic₁₀	C₂×D₄	1	2	yes	—
181	C₃×C₅×C₈	—	2	2	no	—
182	C₃×C₅×C₈	—	2	1	no	—
183	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
184	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
185	C₃×#168	—	2	1	no	—
186	C₃×#168	C₂×D₄	1	1	yes	characteristic
187	C₃×#168	C₂×D₄	1	1	yes	characteristic
188	C₃×H(5,𝔽₂)	—	5	2	no	—
189	C₃×H(5,𝔽₂)	—	5	2	no	—
190	C₃×AGL(1,ℤ/8)	—	5	2	no	—
191	C₃×AGL(1,ℤ/8)	—	5	2	no	—
192	C₃×AGL(1,ℤ/8)	—	5	2	no	—
193	C₃×AGL(1,ℤ/8)	—	10	2	no	—
194	C₃×AGL(1,ℤ/8)	—	10	2	no	—
195	C₃×AGL(1,ℤ/8)	—	10	2	no	—
196	C₃×AGL(1,ℤ/8)	—	10	2	no	—
197	C₃×AGL(1,ℤ/8)	C₂×D₅	1	2	yes	complemented
198	C₃×#135	—	5	1	no	—
199	C₃×#135	—	5	1	no	—
200	C₃×#135	—	10	1	no	—
201	C₃×#135	—	10	1	no	—
202	C₃×#135	—	10	2	no	—
203	C₂×C₃×D₈	—	5	2	no	—
204	C₂×C₃×D₈	—	5	1	no	—
205	C₂×C₃×D₈	—	10	1	no	—
206	C₂×C₃×D₈	—	10	2	no	—
207	C₂×C₃×D₈	—	10	1	no	—
208	C₂×C₃×D₈	C₂×D₅	1	1	yes	characteristic, complemented
209	C₃×#131	—	5	2	no	—
210	C₃×#131	—	5	1	no	—
211	C₃×#131	C₂×D₅	1	1	yes	characteristic, complemented
212	C₃×#127	—	5	2	no	—
213	C₃×#108	—	5	1	no	—
214	C₃×#108	C₂×D₅	1	1	yes	characteristic, complemented
215	C₃×#104	—	5	1	no	—
216	C₃×#104	—	5	2	no	—
217	C₃×#104	—	5	1	no	—
218	C₃×#100	—	5	2	no	—
219	C₃×#100	C₂×D₅	1	2	yes	complemented
220	C₃×C₄×C₈	—	5	1	no	—
221	C₅×#41	C₂³×C₃	1	1	yes	characteristic
222	C₂×C₅×D₄	C₂³×C₃	1	2	yes	—
223	C₂×C₂²⋊D₅	—	4	2	no	—
224	C₂×C₂²⋊D₅	—	4	2	no	—
225	#713	—	2	2	no	—
226	#711	—	2	2	no	—
227	#711	—	2	2	no	—
228	D₄×D₅	—	2	2	no	—
229	D₄×D₅	—	2	2	no	—
230	D₄×D₅	—	2	2	no	—
231	D₄×D₅	—	4	2	no	—
232	#709	—	2	2	no	—
233	#709	C₂³×C₃	1	1	yes	characteristic
234	#709	C₂³×C₃	1	2	yes	—
235	C₂×D₂₀	—	2	2	no	—
236	C₂×D₂₀	C₂³×C₃	1	2	yes	—
237	C₅×SD₁₆	—	2	2	no	—
238	C₅×D₈	—	2	2	no	—
239	C₅×D₈	—	2	1	no	—
240	C₅×D₈	—	2	1	no	—
241	C₅×M₁₆	—	2	1	no	—
242	C₅×M₁₆	C₂³×C₃	1	1	yes	characteristic
243	C₅×M₁₆	C₂³×C₃	1	2	yes	—
244	C₂×C₅×C₈	—	2	1	no	—
245	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
246	#689	C₃×D₄	1	1	yes	characteristic, complemented
247	#688	C₃×D₄	1	1	yes	characteristic, complemented
248	#687	—	2	2	no	—
249	#687	C₃×D₄	1	1	yes	characteristic, complemented
250	#686	—	2	2	no	—
251	#686	—	2	2	no	—
252	#686	—	2	2	no	—
253	#686	C₃×D₄	1	1	yes	characteristic, complemented
254	C₄×Dic₅	C₂³×C₃	1	1	yes	characteristic
255	#681	C₂³×C₃	1	1	yes	characteristic
256	C₂×#168	C₂³×C₃	1	1	yes	characteristic
257	Dic₂₀	C₃×D₄	1	1	yes	characteristic, complemented
258	D₄₀	—	2	2	no	—
259	D₄₀	—	2	2	no	—
260	D₄₀	—	2	1	no	—
261	D₄₀	—	2	1	no	—
262	D₄₀	C₃×D₄	1	1	yes	characteristic, complemented
263	#677	—	2	2	no	—
264	#677	—	2	2	no	—
265	#677	C₃×D₄	1	2	yes	complemented
266	#676	—	2	1	no	—
267	#676	—	2	2	no	—
268	C₈×D₅	—	2	1	no	—
269	C₈×D₅	—	2	2	no	—
270	#576	—	5	2	no	—
271	#576	—	5	1	no	—
272	#576	—	5	1	no	—
273	#493	—	5	1	no	—
274	#472	—	5	1	no	—
275	#472	—	5	2	no	—
276	#472	C₃×D₅	1	1	yes	characteristic, complemented
277	#453	—	5	2	no	—
278	#453	—	5	2	no	—
279	#443	—	5	1	no	—
280	#443	—	5	1	no	—
281	C₂²×C₃×C₅	—	2	1	no	—
282	C₂²×C₃×C₅	—	4	2	no	—
283	C₂²×C₃×C₅	—	8	2	no	—
284	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
285	C₂×C₃×D₅	—	2	2	no	—
286	C₂×C₃×D₅	—	2	1	no	—
287	C₂×C₃×D₅	—	4	2	no	—
288	C₂×C₃×D₅	—	8	2	no	—
289	C₂×C₃×D₅	—	16	2	no	—
290	C₂×C₃×D₅	—	16	2	no	—
291	C₃×C₄×C₅	—	2	1	no	—
292	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
293	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
294	C₃×Dic₅	—	2	2	no	—
295	C₃×Dic₅	—	2	2	no	—
296	C₃×Dic₅	—	2	1	no	—
297	C₃×#41	—	5	2	no	—
298	C₃×#41	—	5	1	no	—
299	C₃×#41	—	10	2	no	—
300	C₃×#41	—	10	2	no	—
301	C₃×#41	—	10	2	no	—
302	C₃×#41	—	10	2	no	—
303	C₃×#41	C₂²×D₅	1	1	yes	characteristic
304	C₂×C₃×D₄	—	5	2	no	—
305	C₂×C₃×D₄	—	10	2	no	—
306	C₂×C₃×D₄	—	10	2	no	—
307	C₂×C₃×D₄	—	10	2	no	—
308	C₂×C₃×D₄	—	10	2	no	—
309	C₂×C₃×D₄	—	20	2	no	—
310	C₂×C₃×D₄	—	20	2	no	—
311	C₂×C₃×D₄	—	20	2	no	—
312	C₂×C₃×D₄	C₂²×D₅	1	2	yes	—
313	C₃×Q₁₆	—	5	1	no	—
314	C₃×Q₁₆	—	5	1	no	—
315	C₃×SD₁₆	—	2	2	no	—
316	C₃×SD₁₆	—	5	1	no	—
317	C₃×SD₁₆	—	5	1	no	—
318	C₃×SD₁₆	—	5	2	no	—
319	C₃×SD₁₆	—	10	2	no	—
320	C₃×SD₁₆	—	10	2	no	—
321	C₃×SD₁₆	—	10	2	no	—
322	C₃×D₈	—	2	2	no	—
323	C₃×D₈	—	2	1	no	—
324	C₃×D₈	—	2	1	no	—
325	C₃×D₈	—	5	1	no	—
326	C₃×D₈	—	5	1	no	—
327	C₃×D₈	—	10	2	no	—
328	C₃×D₈	—	10	2	no	—
329	C₃×D₈	—	10	2	no	—
330	C₃×D₈	—	10	2	no	—
331	C₃×D₈	—	10	2	no	—
332	C₃×D₈	—	10	1	no	—
333	C₃×D₈	—	10	1	no	—
334	C₃×M₁₆	—	2	1	no	—
335	C₃×M₁₆	—	5	1	no	—
336	C₃×M₁₆	—	10	2	no	—
337	C₃×M₁₆	—	10	1	no	—
338	C₃×M₁₆	C₂²×D₅	1	2	yes	—
339	C₃×M₁₆	C₂²×D₅	1	1	yes	characteristic
340	C₂×C₃×C₈	—	2	1	no	—
341	C₂×C₃×C₈	—	5	1	no	—
342	C₂×C₃×C₈	—	10	2	no	—
343	C₂×C₃×C₈	—	10	1	no	—
344	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
345	C₃×C₄²	—	5	1	no	—
346	C₂³×C₅	—	2	2	no	—
347	C₂²×D₅	—	2	2	no	—
348	C₂²×D₅	—	4	2	no	—
349	C₂²×D₅	—	4	2	no	—
350	C₅×Q₈	C₂×C₃×D₄	1	1	yes	characteristic
351	C₅×D₄	—	2	1	no	—
352	C₅×D₄	—	2	2	no	—
353	C₅×D₄	—	2	2	no	—
354	C₅×D₄	C₂×C₃×D₄	1	1	yes	characteristic
355	C₂×C₄×C₅	—	2	1	no	—
356	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
357	C₂²⋊D₅	—	2	2	no	—
358	C₂²⋊D₅	—	2	1	no	—
359	C₂²⋊D₅	—	4	2	no	—
360	C₂²⋊D₅	—	4	2	no	—
361	C₂²⋊D₅	—	4	2	no	—
362	C₂²⋊D₅	—	8	4	no	—
363	C₂×Dic₅	—	2	1	no	—
364	C₂×Dic₅	—	4	2	no	—
365	D₂₀	—	2	2	no	—
366	D₂₀	—	2	2	no	—
367	D₂₀	—	4	2	no	—
368	D₂₀	C₂×C₃×D₄	1	2	yes	—
369	D₂₀	C₂×C₃×D₄	1	1	yes	characteristic
370	C₄×D₅	—	2	1	no	—
371	C₄×D₅	—	2	2	no	—
372	C₄×D₅	—	4	2	no	—
373	Dic₁₀	C₂×C₃×D₄	1	1	yes	characteristic
374	Dic₁₀	C₂×C₃×D₄	1	2	yes	—
375	C₅×C₈	—	2	2	no	—
376	C₅×C₈	—	2	1	no	—
377	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
378	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
379	#168	—	2	1	no	—
380	#168	C₂×C₃×D₄	1	1	yes	characteristic
381	#168	C₂×C₃×D₄	1	1	yes	characteristic
382	H(5,𝔽₂)	—	5	2	no	—
383	H(5,𝔽₂)	—	5	2	no	—
384	AGL(1,ℤ/8)	—	5	2	no	—
385	AGL(1,ℤ/8)	—	5	2	no	—
386	AGL(1,ℤ/8)	—	5	2	no	—
387	AGL(1,ℤ/8)	—	10	2	no	—
388	AGL(1,ℤ/8)	—	10	2	no	—
389	AGL(1,ℤ/8)	—	10	2	no	—
390	AGL(1,ℤ/8)	—	10	2	no	—
391	AGL(1,ℤ/8)	C₂×C₃×D₅	1	2	yes	complemented
392	#135	—	5	1	no	—
393	#135	—	5	1	no	—
394	#135	—	10	1	no	—
395	#135	—	10	1	no	—
396	#135	—	10	2	no	—
397	C₂×D₈	—	5	2	no	—
398	C₂×D₈	—	5	1	no	—
399	C₂×D₈	—	10	1	no	—
400	C₂×D₈	—	10	2	no	—
401	C₂×D₈	—	10	1	no	—
402	C₂×D₈	C₂×C₃×D₅	1	1	yes	characteristic, complemented
403	#131	—	5	2	no	—
404	#131	—	5	1	no	—
405	#131	C₂×C₃×D₅	1	1	yes	characteristic, complemented
406	#127	—	5	2	no	—
407	#108	—	5	1	no	—
408	#108	C₂×C₃×D₅	1	1	yes	characteristic, complemented
409	#104	—	5	1	no	—
410	#104	—	5	2	no	—
411	#104	—	5	1	no	—
412	#100	—	5	2	no	—
413	#100	C₂×C₃×D₅	1	2	yes	complemented
414	C₄×C₈	—	5	1	no	—
415	C₂×C₃×C₅	—	2	1	no	—
416	C₂×C₃×C₅	—	4	1	no	—
417	C₂×C₃×C₅	—	8	2	no	—
418	C₂×C₃×C₅	D₄²	1	1	yes	characteristic
419	C₃×D₅	—	4	2	no	—
420	C₃×D₅	—	4	1	no	—
421	C₃×D₅	—	8	2	no	—
422	C₂³×C₃	—	2	2	no	—
423	C₂³×C₃	—	10	2	no	—
424	C₂³×C₃	—	20	2	no	—
425	C₂³×C₃	—	20	2	no	—
426	C₃×Q₈	—	5	1	no	—
427	C₃×Q₈	—	5	2	no	—
428	C₃×Q₈	D₄×D₅	1	1	yes	characteristic
429	C₃×D₄	—	2	1	no	—
430	C₃×D₄	—	2	2	no	—
431	C₃×D₄	—	2	2	no	—
432	C₃×D₄	—	5	1	no	—
433	C₃×D₄	—	5	2	no	—
434	C₃×D₄	—	10	2	no	—
435	C₃×D₄	—	10	1	no	—
436	C₃×D₄	—	10	2	no	—
437	C₃×D₄	—	10	2	no	—
438	C₃×D₄	—	20	2	no	—
439	C₃×D₄	—	20	2	no	—
440	C₃×D₄	—	20	2	no	—
441	C₃×D₄	—	20	2	no	—
442	C₃×D₄	—	40	4	no	—
443	C₃×D₄	D₄×D₅	1	1	yes	characteristic
444	C₂×C₃×C₄	—	2	1	no	—
445	C₂×C₃×C₄	—	10	1	no	—
446	C₂×C₃×C₄	—	10	2	no	—
447	C₂×C₃×C₄	—	10	1	no	—
448	C₂×C₃×C₄	—	20	2	no	—
449	C₂×C₃×C₄	—	20	2	no	—
450	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
451	C₃×C₈	—	2	1	no	—
452	C₃×C₈	—	2	2	no	—
453	C₃×C₈	—	5	1	no	—
454	C₃×C₈	—	5	1	no	—
455	C₃×C₈	—	10	1	no	—
456	C₃×C₈	D₄×D₅	1	1	yes	characteristic
457	C₃×C₈	D₄×D₅	1	1	yes	characteristic
458	C₂²×C₅	—	2	1	no	—
459	C₂²×C₅	—	4	2	no	—
460	C₂²×C₅	—	8	2	no	—
461	C₂²×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
462	C₂×D₅	—	2	2	no	—
463	C₂×D₅	—	2	1	no	—
464	C₂×D₅	—	4	2	no	—
465	C₂×D₅	—	8	2	no	—
466	C₂×D₅	—	16	2	no	—
467	C₂×D₅	—	16	2	no	—
468	C₄×C₅	—	2	1	no	—
469	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
470	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
471	Dic₅	—	2	2	no	—
472	Dic₅	—	2	2	no	—
473	Dic₅	—	2	1	no	—
474	#41	—	5	2	no	—
475	#41	—	5	1	no	—
476	#41	—	10	2	no	—
477	#41	—	10	2	no	—
478	#41	—	10	2	no	—
479	#41	—	10	2	no	—
480	#41	C₂²×C₃×D₅	1	1	yes	characteristic
481	C₂×D₄	—	5	2	no	—
482	C₂×D₄	—	10	2	no	—
483	C₂×D₄	—	10	2	no	—
484	C₂×D₄	—	10	2	no	—
485	C₂×D₄	—	10	2	no	—
486	C₂×D₄	—	20	2	no	—
487	C₂×D₄	—	20	2	no	—
488	C₂×D₄	—	20	2	no	—
489	C₂×D₄	C₂²×C₃×D₅	1	2	yes	—
490	Q₁₆	—	5	1	no	—
491	Q₁₆	—	5	1	no	—
492	SD₁₆	—	2	2	no	—
493	SD₁₆	—	5	1	no	—
494	SD₁₆	—	5	1	no	—
495	SD₁₆	—	5	2	no	—
496	SD₁₆	—	10	2	no	—
497	SD₁₆	—	10	2	no	—
498	SD₁₆	—	10	2	no	—
499	D₈	—	2	2	no	—
500	D₈	—	2	1	no	—
501	D₈	—	2	1	no	—
502	D₈	—	5	1	no	—
503	D₈	—	5	1	no	—
504	D₈	—	10	2	no	—
505	D₈	—	10	2	no	—
506	D₈	—	10	2	no	—
507	D₈	—	10	2	no	—
508	D₈	—	10	2	no	—
509	D₈	—	10	1	no	—
510	D₈	—	10	1	no	—
511	M₁₆	—	2	1	no	—
512	M₁₆	—	5	1	no	—
513	M₁₆	—	10	2	no	—
514	M₁₆	—	10	1	no	—
515	M₁₆	C₂²×C₃×D₅	1	2	yes	—
516	M₁₆	C₂²×C₃×D₅	1	1	yes	characteristic
517	C₂×C₈	—	2	1	no	—
518	C₂×C₈	—	5	1	no	—
519	C₂×C₈	—	10	2	no	—
520	C₂×C₈	—	10	1	no	—
521	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
522	C₄²	—	5	1	no	—
523	C₃×C₅	#3292	1	1	yes	characteristic, complemented, Hall
524	C₂²×C₃	—	2	1	no	—
525	C₂²×C₃	—	4	2	no	—
526	C₂²×C₃	—	8	2	no	—
527	C₂²×C₃	—	10	2	no	—
528	C₂²×C₃	—	10	1	no	—
529	C₂²×C₃	—	20	2	no	—
530	C₂²×C₃	—	40	2	no	—
531	C₂²×C₃	—	80	2	no	—
532	C₂²×C₃	—	80	2	no	—
533	C₂²×C₃	#4149	1	1	yes	characteristic
534	C₃×C₄	—	2	1	no	—
535	C₃×C₄	—	10	2	no	—
536	C₃×C₄	—	10	2	no	—
537	C₃×C₄	—	10	1	no	—
538	C₃×C₄	C₂×D₄×D₅	1	1	yes	characteristic
539	C₃×C₄	C₂×D₄×D₅	1	1	yes	characteristic
540	C₂×C₅	—	2	1	no	—
541	C₂×C₅	—	4	1	no	—
542	C₂×C₅	—	8	2	no	—
543	C₂×C₅	C₃×D₄²	1	1	yes	characteristic
544	D₅	—	4	2	no	—
545	D₅	—	4	1	no	—
546	D₅	—	8	2	no	—
547	C₂³	—	2	2	no	—
548	C₂³	—	10	2	no	—
549	C₂³	—	20	2	no	—
550	C₂³	—	20	2	no	—
551	Q₈	—	5	1	no	—
552	Q₈	—	5	2	no	—
553	Q₈	C₃×D₄×D₅	1	1	yes	characteristic
554	D₄	—	2	1	no	—
555	D₄	—	2	2	no	—
556	D₄	—	2	2	no	—
557	D₄	—	5	1	no	—
558	D₄	—	5	2	no	—
559	D₄	—	10	2	no	—
560	D₄	—	10	1	no	—
561	D₄	—	10	2	no	—
562	D₄	—	10	2	no	—
563	D₄	—	20	2	no	—
564	D₄	—	20	2	no	—
565	D₄	—	20	2	no	—
566	D₄	—	20	2	no	—
567	D₄	—	40	4	no	—
568	D₄	C₃×D₄×D₅	1	1	yes	characteristic
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	10	1	no	—
571	C₂×C₄	—	10	2	no	—
572	C₂×C₄	—	10	1	no	—
573	C₂×C₄	—	20	2	no	—
574	C₂×C₄	—	20	2	no	—
575	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
576	C₈	—	2	1	no	—
577	C₈	—	2	2	no	—
578	C₈	—	5	1	no	—
579	C₈	—	5	1	no	—
580	C₈	—	10	1	no	—
581	C₈	C₃×D₄×D₅	1	1	yes	characteristic
582	C₈	C₃×D₄×D₅	1	1	yes	characteristic
583	C₂×C₃	—	2	1	no	—
584	C₂×C₃	—	4	1	no	—
585	C₂×C₃	—	8	2	no	—
586	C₂×C₃	—	20	2	no	—
587	C₂×C₃	—	20	1	no	—
588	C₂×C₃	—	40	2	no	—
589	C₂×C₃	#65943	1	1	yes	characteristic
590	C₅	C₃×#3292	1	1	yes	characteristic, complemented, Sylow
591	C₂²	—	2	1	no	—
592	C₂²	—	4	2	no	—
593	C₂²	—	8	2	no	—
594	C₂²	—	10	2	no	—
595	C₂²	—	10	1	no	—
596	C₂²	—	20	2	no	—
597	C₂²	—	40	2	no	—
598	C₂²	—	80	2	no	—
599	C₂²	—	80	2	no	—
600	C₂²	C₃×#4149	1	1	yes	characteristic
601	C₄	—	2	1	no	—
602	C₄	—	10	2	no	—
603	C₄	—	10	2	no	—
604	C₄	—	10	1	no	—
605	C₄	C₂×C₃×D₄×D₅	1	1	yes	characteristic
606	C₄	C₂×C₃×D₄×D₅	1	1	yes	characteristic
607	C₃	#10612672	1	1	yes	characteristic, direct product factor, Sylow
608	C₂	—	2	1	no	—
609	C₂	—	4	1	no	—
610	C₂	—	8	2	no	—
611	C₂	—	20	2	no	—
612	C₂	—	20	1	no	—
613	C₂	—	40	2	no	—
614	C₂	C₃×#65943	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	#65943
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₃×C₅×#472	C₂
socle	C₂×C₃×C₅	D₄²
hypercenter	C₃×#472	D₅
hypocenter	C₅	C₃×#3292
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#472

Complements

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

N	H ≅ G/N	type
1. C₃×#66108	C₂	semidirect
2. C₃×#66091	C₂	semidirect
3. C₃×#66086	C₂	semidirect
4. C₃×C₅×#472	C₂	semidirect
5. C₃×#65487	C₂	semidirect
6. C₃×#65440	C₂	semidirect
7. C₃×#65433	C₂	semidirect
8. C₃×#65186	C₂	semidirect
9. C₃×#65184	C₂	semidirect
10. C₃×#65038	C₂	semidirect
11. C₃×#65036	C₂	semidirect
12. #10612672	C₃	direct (N × H)
13. C₃×#4149	C₂²	semidirect
14. C₃×#4144	C₂²	semidirect
15. C₃×C₅×AGL(1,ℤ/8)	C₂²	semidirect
16. C₂×C₃×C₅×D₈	C₂²	semidirect
17. C₃×C₅×#131	C₂²	semidirect
19. C₃×#4086	C₂²	semidirect
20. C₃×#4078	C₂²	semidirect
21. C₂×C₃×#686	C₂²	semidirect
30. C₃×#4054	C₂²	semidirect
31. C₃×#4054	C₂²	semidirect
32. C₃×#4053	C₂²	semidirect
36. C₂×C₃×D₄₀	C₂²	semidirect
37. C₃×#4047	C₂²	semidirect
38. C₃×C₅×#108	C₂²	semidirect
39. C₃×C₅×#100	C₂²	semidirect
40. C₃×#3969	C₂²	semidirect
41. C₃×#3957	C₂²	semidirect
42. C₃×#3957	C₂²	semidirect
43. C₃×#3955	C₂²	semidirect
44. C₃×#3954	C₂²	semidirect
45. C₃×C₈×Dic₅	C₂²	semidirect
47. #66108	C₂×C₃	semidirect
48. #66091	C₂×C₃	semidirect
49. #66086	C₂×C₃	semidirect
50. C₅×#472	C₂×C₃	semidirect
51. #65487	C₂×C₃	semidirect
52. #65440	C₂×C₃	semidirect
53. #65433	C₂×C₃	semidirect
54. #65186	C₂×C₃	semidirect
55. #65184	C₂×C₃	semidirect
56. #65038	C₂×C₃	semidirect
57. #65036	C₂×C₃	semidirect
83. C₃×#689	D₄	semidirect
84. C₃×#688	D₄	semidirect
86. C₃×#687	D₄	semidirect
90. C₃×#686	D₄	semidirect
94. C₃×Dic₂₀	D₄	semidirect
99. C₃×D₄₀	D₄	semidirect
102. C₃×#677	D₄	semidirect
113. C₃×#472	D₅	semidirect
118. #4149	C₂²×C₃	semidirect
119. #4144	C₂²×C₃	semidirect
120. C₅×AGL(1,ℤ/8)	C₂²×C₃	semidirect
121. C₂×C₅×D₈	C₂²×C₃	semidirect
122. C₅×#131	C₂²×C₃	semidirect
124. #4086	C₂²×C₃	semidirect
125. #4078	C₂²×C₃	semidirect
126. C₂×#686	C₂²×C₃	semidirect
135. #4054	C₂²×C₃	semidirect
136. #4054	C₂²×C₃	semidirect
137. #4053	C₂²×C₃	semidirect
141. C₂×D₄₀	C₂²×C₃	semidirect
142. #4047	C₂²×C₃	semidirect
143. C₅×#108	C₂²×C₃	semidirect
144. C₅×#100	C₂²×C₃	semidirect
145. #3969	C₂²×C₃	semidirect
146. #3957	C₂²×C₃	semidirect
147. #3957	C₂²×C₃	semidirect
148. #3955	C₂²×C₃	semidirect
149. #3954	C₂²×C₃	semidirect
150. C₈×Dic₅	C₂²×C₃	semidirect
197. C₃×AGL(1,ℤ/8)	C₂×D₅	semidirect
208. C₂×C₃×D₈	C₂×D₅	semidirect
211. C₃×#131	C₂×D₅	semidirect
214. C₃×#108	C₂×D₅	semidirect
219. C₃×#100	C₂×D₅	semidirect
246. #689	C₃×D₄	semidirect
247. #688	C₃×D₄	semidirect
249. #687	C₃×D₄	semidirect
253. #686	C₃×D₄	semidirect
257. Dic₂₀	C₃×D₄	semidirect
262. D₄₀	C₃×D₄	semidirect
265. #677	C₃×D₄	semidirect
276. #472	C₃×D₅	semidirect
391. AGL(1,ℤ/8)	C₂×C₃×D₅	semidirect
402. C₂×D₈	C₂×C₃×D₅	semidirect
405. #131	C₂×C₃×D₅	semidirect
408. #108	C₂×C₃×D₅	semidirect
413. #100	C₂×C₃×D₅	semidirect
523. C₃×C₅	#3292	semidirect
590. C₅	C₃×#3292	semidirect
607. C₃	#10612672	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×#66108	2
C₃×#66091	2
C₃×#66086	2
C₃×C₅×#472	2
C₃×#65487	2
C₃×#65440	2
C₃×#65433	2
C₃×#65186	2
C₃×#65184	2
C₃×#65038	2
C₃×#65036	2
#10612672	3
C₃×#3292	5

Subgroups — C3×#10612672 #14458171

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10612672 #14458171