Subgroups — C₃×#10617449 #14418125

Order 1920.

916 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#66264	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#66256	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#66107	C₂	1	1	yes	maximal, characteristic
4	C₃×D₅⋊#104	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×#4008	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#65737	C₂	1	2	yes	maximal, complemented
7	C₃×#65736	C₂	1	2	yes	maximal, complemented
8	C₃×#65733	C₂	1	2	yes	maximal, complemented
9	C₃×#65732	C₂	1	2	yes	maximal, complemented
10	C₃×#65729	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×M₁₆×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic, complemented
12	#10617449	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₃×D₅×#41	C₂²	1	1	yes	characteristic
14	C₂×C₃×#711	C₂²	1	1	yes	characteristic
15	C₂×C₃×#704	—	2	2	no	—
16	C₃×#4133	C₂²	1	2	yes	complemented
17	C₃×#4131	—	2	2	no	—
18	C₃×#4131	—	2	1	no	—
19	C₃×#4131	—	2	1	no	—
20	C₃×#4131	C₂²	1	2	yes	complemented
21	C₃×#4131	C₂²	1	2	yes	complemented
22	C₃×#4131	C₂²	1	2	yes	complemented
23	C₂×C₃×C₄×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
24	C₃×D₅⋊M₁₆	—	2	1	no	—
25	C₃×D₅⋊M₁₆	C₂²	1	1	yes	characteristic
26	C₃×D₅⋊M₁₆	C₂²	1	1	yes	characteristic
27	C₂×C₃×#700	—	2	1	no	—
28	C₂×C₃×#700	C₂²	1	1	yes	characteristic
29	C₃×C₅×AGL(1,ℤ/8)	C₄	1	1	yes	characteristic, complemented
30	C₃×#4097	C₄	1	1	yes	characteristic, complemented
31	C₃×#4078	C₄	1	1	yes	characteristic, complemented
32	C₃×#4062	C₂²	1	2	yes	—
33	C₃×#4061	C₄	1	2	yes	complemented
34	C₃×#4058	C₂²	1	2	yes	—
35	C₃×#4057	C₄	1	2	yes	complemented
36	C₃×#4055	C₄	1	1	yes	characteristic, complemented
37	C₃×D₅×M₁₆	C₂²	1	1	yes	characteristic
38	C₃×#4010	C₂²	1	1	yes	characteristic, complemented
39	C₃×#4010	C₂²	1	1	yes	characteristic, complemented
40	C₃×#4008	C₂²	1	1	yes	characteristic, complemented
41	C₃×#4008	C₂²	1	2	yes	complemented
42	C₃×#4008	C₂²	1	1	yes	characteristic, complemented
43	C₃×#4008	C₂²	1	2	yes	complemented
44	C₃×#3996	C₂²	1	2	yes	complemented
45	C₃×#3995	C₂²	1	2	yes	complemented
46	C₃×#3992	C₂²	1	2	yes	complemented
47	C₃×C₈×AGL(1,𝔽₅)	C₂²	1	2	yes	complemented
48	C₃×#2956	—	5	1	no	maximal, Hall
49	#66264	C₂×C₃	1	1	yes	characteristic, complemented
50	#66256	C₂×C₃	1	1	yes	characteristic, complemented
51	#66107	C₂×C₃	1	1	yes	characteristic
52	D₅⋊#104	C₂×C₃	1	1	yes	characteristic, complemented
53	C₂×#4008	C₂×C₃	1	1	yes	characteristic, complemented
54	#65737	C₂×C₃	1	2	yes	complemented
55	#65736	C₂×C₃	1	2	yes	complemented
56	#65733	C₂×C₃	1	2	yes	complemented
57	#65732	C₂×C₃	1	2	yes	complemented
58	#65729	C₂×C₃	1	1	yes	characteristic, complemented
59	M₁₆×AGL(1,𝔽₅)	C₂×C₃	1	1	yes	characteristic, complemented
60	C₂²×C₃×AGL(1,𝔽₅)	—	2	1	no	—
61	C₃×C₅×#41	C₂×C₄	1	1	yes	characteristic
62	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
63	C₂×C₃×C₂²⋊D₅	—	2	1	no	—
64	C₂²×C₃×Dic₅	—	2	1	no	—
65	C₃×#713	C₂³	1	1	yes	characteristic
66	C₃×Q₈×D₅	C₂×C₄	1	1	yes	characteristic
67	C₃×#711	—	2	1	no	—
68	C₃×#711	—	2	1	no	—
69	C₃×#711	—	2	1	no	—
70	C₃×#711	C₂×C₄	1	2	yes	—
71	C₃×#711	C₂³	1	2	yes	—
72	C₃×#711	C₂³	1	1	yes	characteristic
73	C₃×D₄×D₅	—	2	1	no	—
74	C₃×D₄×D₅	C₂×C₄	1	1	yes	characteristic
75	C₃×#709	—	2	1	no	—
76	C₃×#709	C₂×C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×D₅	—	2	1	no	—
78	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
79	C₂×C₃×Dic₁₀	C₂×C₄	1	1	yes	characteristic
80	C₃×#704	—	2	1	no	—
81	C₃×#704	—	2	2	no	—
82	C₃×#704	—	2	2	no	—
83	C₃×#704	—	2	2	no	—
84	C₃×#704	D₄	1	2	yes	—
85	C₂×C₃×#170	—	2	1	no	—
86	C₂×C₃×#170	—	2	2	no	—
87	C₂×C₃×#170	—	2	2	no	—
88	C₂×C₃×#170	—	2	2	no	—
89	C₃×C₄×AGL(1,𝔽₅)	—	2	1	no	—
90	C₃×C₄×AGL(1,𝔽₅)	C₂³	1	1	yes	characteristic
91	C₃×C₄×AGL(1,𝔽₅)	C₂³	1	1	yes	characteristic
92	C₃×#700	—	2	1	no	—
93	C₃×#700	—	2	2	no	—
94	C₃×#700	D₄	1	1	yes	characteristic
95	C₃×#700	D₄	1	1	yes	characteristic
96	C₃×#700	C₂³	1	1	yes	characteristic
97	C₃×#700	C₂³	1	2	yes	—
98	C₃×#700	C₂³	1	1	yes	characteristic
99	C₃×D₅⋊C₈	—	2	1	no	—
100	C₃×D₅⋊C₈	—	2	2	no	—
101	C₃×D₅⋊C₈	C₂³	1	2	yes	—
102	C₃×C₅×SD₁₆	C₂×C₄	1	2	yes	complemented
103	C₃×C₅×D₈	C₂×C₄	1	2	yes	complemented
104	C₃×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
105	C₃×#689	C₂×C₄	1	2	yes	complemented
106	C₃×#687	C₂×C₄	1	2	yes	complemented
107	C₃×#687	C₂×C₄	1	2	yes	complemented
108	C₃×#686	C₂×C₄	1	2	yes	complemented
109	C₃×#681	C₂×C₄	1	1	yes	characteristic
110	C₃×Dic₂₀	C₂×C₄	1	2	yes	complemented
111	C₃×#677	C₂×C₄	1	2	yes	complemented
112	C₃×#676	C₂×C₄	1	2	yes	—
113	C₃×C₈×D₅	C₂³	1	2	yes	—
114	C₃×#575	—	5	1	no	—
115	C₃×#568	—	5	1	no	—
116	C₃×#568	—	5	1	no	—
117	C₃×#444	—	5	2	no	—
118	C₃×#444	—	5	2	no	—
119	C₃×#443	—	5	2	no	—
120	C₃×#443	—	5	2	no	—
121	C₃×#430	—	5	1	no	—
122	C₂×C₃×#104	—	5	1	no	—
123	C₂×C₃×#104	—	5	1	no	—
124	C₃×C₄×M₁₆	—	5	1	no	—
125	D₅×#41	C₂²×C₃	1	1	yes	characteristic
126	C₂×#711	C₂²×C₃	1	1	yes	characteristic
127	C₂×#704	—	2	2	no	—
128	#4133	C₂²×C₃	1	2	yes	complemented
129	#4131	—	2	2	no	—
130	#4131	—	2	1	no	—
131	#4131	—	2	1	no	—
132	#4131	C₂²×C₃	1	2	yes	complemented
133	#4131	C₂²×C₃	1	2	yes	complemented
134	#4131	C₂²×C₃	1	2	yes	complemented
135	C₂×C₄×AGL(1,𝔽₅)	C₂²×C₃	1	1	yes	characteristic
136	D₅⋊M₁₆	—	2	1	no	—
137	D₅⋊M₁₆	C₂²×C₃	1	1	yes	characteristic
138	D₅⋊M₁₆	C₂²×C₃	1	1	yes	characteristic
139	C₂×#700	—	2	1	no	—
140	C₂×#700	C₂²×C₃	1	1	yes	characteristic
141	C₅×AGL(1,ℤ/8)	C₃×C₄	1	1	yes	characteristic, complemented
142	#4097	C₃×C₄	1	1	yes	characteristic, complemented
143	#4078	C₃×C₄	1	1	yes	characteristic, complemented
144	#4062	C₂²×C₃	1	2	yes	—
145	#4061	C₃×C₄	1	2	yes	complemented
146	#4058	C₂²×C₃	1	2	yes	—
147	#4057	C₃×C₄	1	2	yes	complemented
148	#4055	C₃×C₄	1	1	yes	characteristic, complemented
149	D₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
150	#4010	C₂²×C₃	1	1	yes	characteristic, complemented
151	#4010	C₂²×C₃	1	1	yes	characteristic, complemented
152	#4008	C₂²×C₃	1	1	yes	characteristic, complemented
153	#4008	C₂²×C₃	1	1	yes	characteristic, complemented
154	#4008	C₂²×C₃	1	2	yes	complemented
155	#4008	C₂²×C₃	1	2	yes	complemented
156	#3996	C₂²×C₃	1	2	yes	complemented
157	#3995	C₂²×C₃	1	2	yes	complemented
158	#3992	C₂²×C₃	1	2	yes	complemented
159	C₈×AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	complemented
160	#2956	—	5	1	no	Sylow
161	C₂³×C₃×C₅	—	2	1	no	—
162	C₂²×C₃×D₅	—	2	1	no	—
163	C₂²×C₃×D₅	#41	1	1	yes	characteristic
164	C₂×C₃×AGL(1,𝔽₅)	—	2	1	no	—
165	C₂×C₃×AGL(1,𝔽₅)	—	2	1	no	—
166	C₂×C₃×AGL(1,𝔽₅)	—	4	2	no	—
167	C₃×C₅×Q₈	C₂²×C₄	1	1	yes	characteristic
168	C₃×C₅×D₄	—	2	1	no	—
169	C₃×C₅×D₄	—	2	1	no	—
170	C₃×C₅×D₄	C₂²×C₄	1	2	yes	—
171	C₃×C₅×D₄	C₂²×C₄	1	1	yes	characteristic
172	C₂×C₃×C₄×C₅	—	2	1	no	—
173	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
174	C₃×C₂²⋊D₅	—	2	1	no	—
175	C₃×C₂²⋊D₅	—	2	2	no	—
176	C₃×C₂²⋊D₅	—	2	1	no	—
177	C₃×C₂²⋊D₅	—	2	2	no	—
178	C₃×C₂²⋊D₅	—	2	1	no	—
179	C₂×C₃×Dic₅	—	2	1	no	—
180	C₂×C₃×Dic₅	—	2	2	no	—
181	C₂×C₃×Dic₅	—	2	1	no	—
182	C₂×C₃×Dic₅	—	2	2	no	—
183	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
184	C₃×D₂₀	—	2	1	no	—
185	C₃×D₂₀	C₂²×C₄	1	1	yes	characteristic
186	C₃×C₄×D₅	—	2	1	no	—
187	C₃×C₄×D₅	—	2	1	no	—
188	C₃×C₄×D₅	—	2	1	no	—
189	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
190	C₃×C₄×D₅	C₂×D₄	1	1	yes	characteristic
191	C₃×C₄×D₅	#41	1	1	yes	characteristic
192	C₃×C₄×D₅	C₂⁴	1	1	yes	characteristic
193	C₃×Dic₁₀	—	2	1	no	—
194	C₃×Dic₁₀	—	2	1	no	—
195	C₃×Dic₁₀	C₂²×C₄	1	2	yes	—
196	C₃×Dic₁₀	C₂²×C₄	1	1	yes	characteristic
197	C₃×#170	—	2	2	no	—
198	C₃×#170	—	2	1	no	—
199	C₃×#170	—	2	1	no	—
200	C₃×#170	C₂×D₄	1	2	yes	—
201	C₃×#170	C₂×D₄	1	2	yes	—
202	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
203	C₃×#168	C₂²×C₄	1	2	yes	—
204	C₂×C₃×#41	—	5	1	no	—
205	C₂×C₃×#41	—	5	1	no	—
206	C₃×#137	—	5	1	no	—
207	C₃×#137	—	5	2	no	—
208	C₃×#137	—	5	1	no	—
209	C₃×AGL(1,ℤ/8)	—	5	1	no	—
210	C₃×AGL(1,ℤ/8)	—	5	2	no	—
211	C₃×AGL(1,ℤ/8)	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
212	C₃×#135	—	5	2	no	—
213	C₃×#135	—	5	2	no	—
214	C₃×#131	—	5	2	no	—
215	C₃×#131	—	5	2	no	—
216	C₃×#131	—	5	2	no	—
217	C₃×#131	—	5	2	no	—
218	C₃×#131	—	10	2	no	—
219	C₃×#131	—	10	1	no	—
220	C₃×#131	—	10	1	no	—
221	C₂×C₃×M₁₆	—	5	1	no	—
222	C₂×C₃×M₁₆	—	5	1	no	—
223	C₂×C₃×M₁₆	—	5	1	no	—
224	C₂×C₃×M₁₆	—	5	1	no	—
225	C₂×C₃×M₁₆	—	10	1	no	—
226	C₂×C₃×M₁₆	—	10	1	no	—
227	C₂×C₃×M₁₆	—	10	2	no	—
228	C₂×C₃×C₄²	—	5	1	no	—
229	C₃×#108	—	5	2	no	—
230	C₃×#108	—	5	2	no	—
231	C₃×#104	—	5	1	no	—
232	C₃×#104	—	5	2	no	—
233	C₃×#104	—	5	1	no	—
234	C₃×#104	—	5	2	no	—
235	C₃×#104	—	5	1	no	—
236	C₃×#104	—	5	1	no	—
237	C₃×#97	—	5	2	no	—
238	C₃×C₄×C₈	—	5	2	no	—
239	C₂²×AGL(1,𝔽₅)	—	2	1	no	—
240	C₅×#41	C₂×C₃×C₄	1	1	yes	characteristic
241	C₂×C₅×D₄	C₂×C₃×C₄	1	1	yes	characteristic
242	C₂×C₂²⋊D₅	—	2	1	no	—
243	C₂²×Dic₅	—	2	1	no	—
244	#713	C₂³×C₃	1	1	yes	characteristic
245	Q₈×D₅	C₂×C₃×C₄	1	1	yes	characteristic
246	#711	—	2	1	no	—
247	#711	—	2	1	no	—
248	#711	—	2	1	no	—
249	#711	C₂×C₃×C₄	1	2	yes	—
250	#711	C₂³×C₃	1	2	yes	—
251	#711	C₂³×C₃	1	1	yes	characteristic
252	D₄×D₅	—	2	1	no	—
253	D₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
254	#709	—	2	1	no	—
255	#709	C₂×C₃×C₄	1	1	yes	characteristic
256	C₂×C₄×D₅	—	2	1	no	—
257	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
258	C₂×Dic₁₀	C₂×C₃×C₄	1	1	yes	characteristic
259	#704	—	2	1	no	—
260	#704	—	2	2	no	—
261	#704	—	2	2	no	—
262	#704	—	2	2	no	—
263	#704	C₃×D₄	1	2	yes	—
264	C₂×#170	—	2	1	no	—
265	C₂×#170	—	2	2	no	—
266	C₂×#170	—	2	2	no	—
267	C₂×#170	—	2	2	no	—
268	C₄×AGL(1,𝔽₅)	—	2	1	no	—
269	C₄×AGL(1,𝔽₅)	C₂³×C₃	1	1	yes	characteristic
270	C₄×AGL(1,𝔽₅)	C₂³×C₃	1	1	yes	characteristic
271	#700	—	2	1	no	—
272	#700	—	2	2	no	—
273	#700	C₃×D₄	1	1	yes	characteristic
274	#700	C₃×D₄	1	1	yes	characteristic
275	#700	C₂³×C₃	1	2	yes	—
276	#700	C₂³×C₃	1	1	yes	characteristic
277	#700	C₂³×C₃	1	1	yes	characteristic
278	D₅⋊C₈	—	2	1	no	—
279	D₅⋊C₈	—	2	2	no	—
280	D₅⋊C₈	C₂³×C₃	1	2	yes	—
281	C₅×SD₁₆	C₂×C₃×C₄	1	2	yes	complemented
282	C₅×D₈	C₂×C₃×C₄	1	2	yes	complemented
283	C₅×M₁₆	C₂×C₃×C₄	1	1	yes	characteristic
284	#689	C₂×C₃×C₄	1	2	yes	complemented
285	#687	C₂×C₃×C₄	1	2	yes	complemented
286	#687	C₂×C₃×C₄	1	2	yes	complemented
287	#686	C₂×C₃×C₄	1	2	yes	complemented
288	#681	C₂×C₃×C₄	1	1	yes	characteristic
289	Dic₂₀	C₂×C₃×C₄	1	2	yes	complemented
290	#677	C₂×C₃×C₄	1	2	yes	complemented
291	#676	C₂×C₃×C₄	1	2	yes	—
292	C₈×D₅	C₂³×C₃	1	2	yes	—
293	#575	—	5	1	no	—
294	#568	—	5	1	no	—
295	#568	—	5	1	no	—
296	#444	—	5	2	no	—
297	#444	—	5	2	no	—
298	#443	—	5	2	no	—
299	#443	—	5	2	no	—
300	#430	—	5	1	no	—
301	C₂×#104	—	5	1	no	—
302	C₂×#104	—	5	1	no	—
303	C₄×M₁₆	—	5	1	no	—
304	C₂²×C₃×C₅	—	2	2	no	—
305	C₂²×C₃×C₅	—	2	1	no	—
306	C₂²×C₃×C₅	—	8	1	no	—
307	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
308	C₂×C₃×D₅	—	2	1	no	—
309	C₂×C₃×D₅	—	4	1	no	—
310	C₂×C₃×D₅	—	8	1	no	—
311	C₂×C₃×D₅	C₄×D₄	1	1	yes	characteristic
312	C₂×C₃×D₅	C₂×#41	1	1	yes	characteristic
313	C₃×AGL(1,𝔽₅)	—	2	2	no	—
314	C₃×AGL(1,𝔽₅)	—	4	1	no	—
315	C₃×C₄×C₅	—	2	1	no	—
316	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
317	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
318	C₃×Dic₅	—	2	1	no	—
319	C₃×Dic₅	—	2	2	no	—
320	C₃×Dic₅	C₄×D₄	1	1	yes	characteristic
321	C₃×Dic₅	C₂²×D₄	1	1	yes	characteristic
322	C₃×#41	—	5	1	no	—
323	C₃×#41	—	5	2	no	—
324	C₃×#41	—	5	1	no	—
325	C₃×#41	—	5	2	no	—
326	C₃×#41	—	5	1	no	—
327	C₃×#41	—	10	1	no	—
328	C₃×#41	—	10	1	no	—
329	C₃×#41	—	10	1	no	—
330	C₃×#41	—	10	1	no	—
331	C₃×#41	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
332	C₂×C₃×Q₈	—	5	1	no	—
333	C₂×C₃×Q₈	—	5	1	no	—
334	C₂×C₃×D₄	—	5	1	no	—
335	C₂×C₃×D₄	—	10	1	no	—
336	C₂×C₃×D₄	—	10	1	no	—
337	C₂×C₃×D₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
338	C₂²×C₃×C₄	—	5	1	no	—
339	C₂²×C₃×C₄	—	10	1	no	—
340	C₂²×C₃×C₄	—	10	1	no	—
341	C₂²×C₃×C₄	—	10	1	no	—
342	C₃×Q₁₆	—	5	2	no	—
343	C₃×Q₁₆	—	5	2	no	—
344	C₃×SD₁₆	—	5	2	no	—
345	C₃×SD₁₆	—	5	2	no	—
346	C₃×SD₁₆	—	5	2	no	—
347	C₃×SD₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
348	C₃×D₈	—	5	2	no	—
349	C₃×D₈	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
350	C₃×M₁₆	—	5	1	no	—
351	C₃×M₁₆	—	5	1	no	—
352	C₃×M₁₆	—	5	2	no	—
353	C₃×M₁₆	—	5	1	no	—
354	C₃×M₁₆	—	5	2	no	—
355	C₃×M₁₆	—	5	2	no	—
356	C₃×M₁₆	—	5	1	no	—
357	C₃×M₁₆	—	5	1	no	—
358	C₃×M₁₆	—	10	2	no	—
359	C₃×M₁₆	—	10	2	no	—
360	C₃×M₁₆	—	10	2	no	—
361	C₃×M₁₆	—	10	2	no	—
362	C₃×M₁₆	—	10	1	no	—
363	C₃×M₁₆	—	10	1	no	—
364	C₃×M₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
365	C₂×C₃×C₈	—	5	2	no	—
366	C₂×C₃×C₈	—	5	2	no	—
367	C₂×C₃×C₈	—	10	2	no	—
368	C₂×C₃×C₈	—	10	2	no	—
369	C₂×C₃×C₈	—	10	2	no	—
370	C₂×C₃×C₈	—	10	2	no	—
371	C₂×C₃×C₈	—	10	1	no	—
372	C₂×C₃×C₈	—	10	1	no	—
373	C₃×C₄²	—	5	1	no	—
374	C₃×C₄²	—	5	1	no	—
375	C₃×C₄²	—	10	1	no	—
376	C₂³×C₅	—	2	1	no	—
377	C₂²×D₅	—	2	1	no	—
378	C₂²×D₅	C₃×#41	1	1	yes	characteristic
379	C₂×AGL(1,𝔽₅)	—	2	1	no	—
380	C₂×AGL(1,𝔽₅)	—	2	1	no	—
381	C₂×AGL(1,𝔽₅)	—	4	2	no	—
382	C₅×Q₈	C₂²×C₃×C₄	1	1	yes	characteristic
383	C₅×D₄	—	2	1	no	—
384	C₅×D₄	—	2	1	no	—
385	C₅×D₄	C₂²×C₃×C₄	1	2	yes	—
386	C₅×D₄	C₂²×C₃×C₄	1	1	yes	characteristic
387	C₂×C₄×C₅	—	2	1	no	—
388	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
389	C₂²⋊D₅	—	2	1	no	—
390	C₂²⋊D₅	—	2	2	no	—
391	C₂²⋊D₅	—	2	1	no	—
392	C₂²⋊D₅	—	2	2	no	—
393	C₂²⋊D₅	—	2	1	no	—
394	C₂×Dic₅	—	2	1	no	—
395	C₂×Dic₅	—	2	2	no	—
396	C₂×Dic₅	—	2	1	no	—
397	C₂×Dic₅	—	2	2	no	—
398	C₂×Dic₅	C₂×C₃×D₄	1	1	yes	characteristic
399	D₂₀	—	2	1	no	—
400	D₂₀	C₂²×C₃×C₄	1	1	yes	characteristic
401	C₄×D₅	—	2	1	no	—
402	C₄×D₅	—	2	1	no	—
403	C₄×D₅	—	2	1	no	—
404	C₄×D₅	C₂²×C₃×C₄	1	1	yes	characteristic
405	C₄×D₅	C₂×C₃×D₄	1	1	yes	characteristic
406	C₄×D₅	C₃×#41	1	1	yes	characteristic
407	C₄×D₅	C₂⁴×C₃	1	1	yes	characteristic
408	Dic₁₀	—	2	1	no	—
409	Dic₁₀	—	2	1	no	—
410	Dic₁₀	C₂²×C₃×C₄	1	2	yes	—
411	Dic₁₀	C₂²×C₃×C₄	1	1	yes	characteristic
412	#170	—	2	2	no	—
413	#170	—	2	1	no	—
414	#170	—	2	1	no	—
415	#170	C₂×C₃×D₄	1	2	yes	—
416	#170	C₂×C₃×D₄	1	2	yes	—
417	C₅×C₈	C₂²×C₃×C₄	1	2	yes	—
418	#168	C₂²×C₃×C₄	1	2	yes	—
419	C₂×#41	—	5	1	no	—
420	C₂×#41	—	5	1	no	—
421	#137	—	5	1	no	—
422	#137	—	5	2	no	—
423	#137	—	5	1	no	—
424	AGL(1,ℤ/8)	—	5	1	no	—
425	AGL(1,ℤ/8)	—	5	2	no	—
426	AGL(1,ℤ/8)	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
427	#135	—	5	2	no	—
428	#135	—	5	2	no	—
429	#131	—	5	2	no	—
430	#131	—	5	2	no	—
431	#131	—	5	2	no	—
432	#131	—	5	2	no	—
433	#131	—	10	2	no	—
434	#131	—	10	1	no	—
435	#131	—	10	1	no	—
436	C₂×M₁₆	—	5	1	no	—
437	C₂×M₁₆	—	5	1	no	—
438	C₂×M₁₆	—	5	1	no	—
439	C₂×M₁₆	—	5	1	no	—
440	C₂×M₁₆	—	10	1	no	—
441	C₂×M₁₆	—	10	1	no	—
442	C₂×M₁₆	—	10	2	no	—
443	C₂×C₄²	—	5	1	no	—
444	#108	—	5	2	no	—
445	#108	—	5	2	no	—
446	#104	—	5	1	no	—
447	#104	—	5	2	no	—
448	#104	—	5	1	no	—
449	#104	—	5	2	no	—
450	#104	—	5	1	no	—
451	#104	—	5	1	no	—
452	#97	—	5	2	no	—
453	C₄×C₈	—	5	2	no	—
454	C₂×C₃×C₅	—	2	1	no	—
455	C₂×C₃×C₅	—	4	1	no	—
456	C₂×C₃×C₅	—	4	2	no	—
457	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
458	C₃×D₅	—	2	1	no	—
459	C₃×D₅	—	2	1	no	—
460	C₃×D₅	—	4	1	no	—
461	C₂³×C₃	—	2	1	no	—
462	C₂³×C₃	—	5	1	no	—
463	C₂³×C₃	—	10	1	no	—
464	C₃×Q₈	—	5	2	no	—
465	C₃×Q₈	—	5	1	no	—
466	C₃×Q₈	—	10	1	no	—
467	C₃×Q₈	—	10	1	no	—
468	C₃×Q₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
469	C₃×D₄	—	2	1	no	—
470	C₃×D₄	—	2	1	no	—
471	C₃×D₄	—	5	1	no	—
472	C₃×D₄	—	10	1	no	—
473	C₃×D₄	—	10	1	no	—
474	C₃×D₄	—	10	1	no	—
475	C₃×D₄	—	10	2	no	—
476	C₃×D₄	—	10	2	no	—
477	C₃×D₄	—	10	1	no	—
478	C₃×D₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
479	C₃×D₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
480	C₂×C₃×C₄	—	2	1	no	—
481	C₂×C₃×C₄	—	5	1	no	—
482	C₂×C₃×C₄	—	5	1	no	—
483	C₂×C₃×C₄	—	5	1	no	—
484	C₂×C₃×C₄	—	5	1	no	—
485	C₂×C₃×C₄	—	5	1	no	—
486	C₂×C₃×C₄	—	10	1	no	—
487	C₂×C₃×C₄	—	10	1	no	—
488	C₂×C₃×C₄	—	10	1	no	—
489	C₂×C₃×C₄	—	10	2	no	—
490	C₂×C₃×C₄	—	10	2	no	—
491	C₂×C₃×C₄	—	10	1	no	—
492	C₂×C₃×C₄	—	10	1	no	—
493	C₂×C₃×C₄	—	10	1	no	—
494	C₂×C₃×C₄	—	10	1	no	—
495	C₂×C₃×C₄	—	20	2	no	—
496	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
497	C₃×C₈	—	5	2	no	—
498	C₃×C₈	—	5	2	no	—
499	C₃×C₈	—	5	2	no	—
500	C₃×C₈	—	10	2	no	—
501	C₃×C₈	—	10	1	no	—
502	C₃×C₈	—	10	1	no	—
503	C₃×C₈	C₂²×AGL(1,𝔽₅)	1	2	yes	—
504	C₂²×C₅	—	2	2	no	—
505	C₂²×C₅	—	2	1	no	—
506	C₂²×C₅	—	8	1	no	—
507	C₂²×C₅	C₃×C₄×D₄	1	1	yes	characteristic
508	C₂×D₅	—	2	1	no	—
509	C₂×D₅	—	4	1	no	—
510	C₂×D₅	—	8	1	no	—
511	C₂×D₅	C₃×C₄×D₄	1	1	yes	characteristic
512	C₂×D₅	C₂×C₃×#41	1	1	yes	characteristic
513	AGL(1,𝔽₅)	—	2	2	no	—
514	AGL(1,𝔽₅)	—	4	1	no	—
515	C₄×C₅	—	2	1	no	—
516	C₄×C₅	C₃×C₄×D₄	1	1	yes	characteristic
517	C₄×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
518	Dic₅	—	2	1	no	—
519	Dic₅	—	2	2	no	—
520	Dic₅	C₃×C₄×D₄	1	1	yes	characteristic
521	Dic₅	C₂²×C₃×D₄	1	1	yes	characteristic
522	#41	—	5	1	no	—
523	#41	—	5	2	no	—
524	#41	—	5	1	no	—
525	#41	—	5	2	no	—
526	#41	—	5	1	no	—
527	#41	—	10	1	no	—
528	#41	—	10	1	no	—
529	#41	—	10	1	no	—
530	#41	—	10	1	no	—
531	#41	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
532	C₂×Q₈	—	5	1	no	—
533	C₂×Q₈	—	5	1	no	—
534	C₂×D₄	—	5	1	no	—
535	C₂×D₄	—	10	1	no	—
536	C₂×D₄	—	10	1	no	—
537	C₂×D₄	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
538	C₂²×C₄	—	5	1	no	—
539	C₂²×C₄	—	10	1	no	—
540	C₂²×C₄	—	10	1	no	—
541	C₂²×C₄	—	10	1	no	—
542	Q₁₆	—	5	2	no	—
543	Q₁₆	—	5	2	no	—
544	SD₁₆	—	5	2	no	—
545	SD₁₆	—	5	2	no	—
546	SD₁₆	—	5	2	no	—
547	SD₁₆	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	complemented
548	D₈	—	5	2	no	—
549	D₈	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	complemented
550	M₁₆	—	5	1	no	—
551	M₁₆	—	5	1	no	—
552	M₁₆	—	5	2	no	—
553	M₁₆	—	5	1	no	—
554	M₁₆	—	5	2	no	—
555	M₁₆	—	5	2	no	—
556	M₁₆	—	5	1	no	—
557	M₁₆	—	5	1	no	—
558	M₁₆	—	10	2	no	—
559	M₁₆	—	10	2	no	—
560	M₁₆	—	10	2	no	—
561	M₁₆	—	10	2	no	—
562	M₁₆	—	10	1	no	—
563	M₁₆	—	10	1	no	—
564	M₁₆	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
565	C₂×C₈	—	5	2	no	—
566	C₂×C₈	—	5	2	no	—
567	C₂×C₈	—	10	2	no	—
568	C₂×C₈	—	10	2	no	—
569	C₂×C₈	—	10	2	no	—
570	C₂×C₈	—	10	2	no	—
571	C₂×C₈	—	10	1	no	—
572	C₂×C₈	—	10	1	no	—
573	C₄²	—	5	1	no	—
574	C₄²	—	5	1	no	—
575	C₄²	—	10	1	no	—
576	C₃×C₅	#2956	1	1	yes	characteristic, complemented, Hall
577	C₂²×C₃	—	2	1	no	—
578	C₂²×C₃	—	2	2	no	—
579	C₂²×C₃	—	5	1	no	—
580	C₂²×C₃	—	5	1	no	—
581	C₂²×C₃	—	8	1	no	—
582	C₂²×C₃	—	10	1	no	—
583	C₂²×C₃	—	20	1	no	—
584	C₂²×C₃	—	40	1	no	—
585	C₂²×C₃	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
586	C₃×C₄	—	2	1	no	—
587	C₃×C₄	—	5	1	no	—
588	C₃×C₄	—	5	1	no	—
589	C₃×C₄	—	10	2	no	—
590	C₃×C₄	—	10	1	no	—
591	C₃×C₄	—	10	2	no	—
592	C₃×C₄	—	20	1	no	—
593	C₃×C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
594	C₃×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
595	C₂×C₅	—	2	1	no	—
596	C₂×C₅	—	4	1	no	—
597	C₂×C₅	—	4	2	no	—
598	C₂×C₅	C₂×C₃×C₄×D₄	1	1	yes	characteristic
599	D₅	—	2	1	no	—
600	D₅	—	2	1	no	—
601	D₅	—	4	1	no	—
602	C₂³	—	2	1	no	—
603	C₂³	—	5	1	no	—
604	C₂³	—	10	1	no	—
605	Q₈	—	5	1	no	—
606	Q₈	—	5	2	no	—
607	Q₈	—	10	1	no	—
608	Q₈	—	10	1	no	—
609	Q₈	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
610	D₄	—	2	1	no	—
611	D₄	—	2	1	no	—
612	D₄	—	5	1	no	—
613	D₄	—	10	1	no	—
614	D₄	—	10	1	no	—
615	D₄	—	10	1	no	—
616	D₄	—	10	2	no	—
617	D₄	—	10	2	no	—
618	D₄	—	10	1	no	—
619	D₄	C₂²×C₃×AGL(1,𝔽₅)	1	2	yes	—
620	D₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
621	C₂×C₄	—	2	1	no	—
622	C₂×C₄	—	5	1	no	—
623	C₂×C₄	—	5	1	no	—
624	C₂×C₄	—	5	1	no	—
625	C₂×C₄	—	5	1	no	—
626	C₂×C₄	—	5	1	no	—
627	C₂×C₄	—	10	1	no	—
628	C₂×C₄	—	10	1	no	—
629	C₂×C₄	—	10	1	no	—
630	C₂×C₄	—	10	2	no	—
631	C₂×C₄	—	10	2	no	—
632	C₂×C₄	—	10	1	no	—
633	C₂×C₄	—	10	1	no	—
634	C₂×C₄	—	10	1	no	—
635	C₂×C₄	—	10	1	no	—
636	C₂×C₄	—	20	2	no	—
637	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
638	C₈	—	5	2	no	—
639	C₈	—	5	2	no	—
640	C₈	—	5	2	no	—
641	C₈	—	10	2	no	—
642	C₈	—	10	1	no	—
643	C₈	—	10	1	no	—
644	C₈	C₂²×C₃×AGL(1,𝔽₅)	1	2	yes	—
645	C₂×C₃	—	2	1	no	—
646	C₂×C₃	—	4	2	no	—
647	C₂×C₃	—	4	1	no	—
648	C₂×C₃	—	10	1	no	—
649	C₂×C₃	—	10	1	no	—
650	C₂×C₃	—	20	1	no	—
651	C₂×C₃	C₂×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
652	C₅	C₃×#2956	1	1	yes	characteristic, complemented, Sylow
653	C₂²	—	2	1	no	—
654	C₂²	—	2	2	no	—
655	C₂²	—	5	1	no	—
656	C₂²	—	5	1	no	—
657	C₂²	—	8	1	no	—
658	C₂²	—	10	1	no	—
659	C₂²	—	20	1	no	—
660	C₂²	—	40	1	no	—
661	C₂²	C₃×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
662	C₄	—	2	1	no	—
663	C₄	—	5	1	no	—
664	C₄	—	5	1	no	—
665	C₄	—	10	2	no	—
666	C₄	—	10	1	no	—
667	C₄	—	10	2	no	—
668	C₄	—	20	1	no	—
669	C₄	C₃×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
670	C₄	C₂³×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
671	C₃	#10617449	1	1	yes	characteristic, direct product factor, Sylow
672	C₂	—	2	1	no	—
673	C₂	—	4	2	no	—
674	C₂	—	4	1	no	—
675	C₂	—	10	1	no	—
676	C₂	—	10	1	no	—
677	C₂	—	20	1	no	—
678	C₂	C₂×C₃×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂×D₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₂³×C₃×C₄
Frattini	C₄	C₂³×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×AGL(1,ℤ/8)	C₄
socle	C₂×C₃×C₅	C₂×C₄×D₄
prime radical	C₃×#66107	C₂
cosocle	C₄×D₅	C₂⁴×C₃
hypercenter	C₃×AGL(1,ℤ/8)	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#2956
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. C₃×#66264	C₂	semidirect
2. C₃×#66256	C₂	semidirect
4. C₃×D₅⋊#104	C₂	semidirect
5. C₂×C₃×#4008	C₂	semidirect
6. C₃×#65737	C₂	semidirect
7. C₃×#65736	C₂	semidirect
8. C₃×#65733	C₂	semidirect
9. C₃×#65732	C₂	semidirect
10. C₃×#65729	C₂	semidirect
11. C₃×M₁₆×AGL(1,𝔽₅)	C₂	semidirect
12. #10617449	C₃	direct (N × H)
16. C₃×#4133	C₂²	semidirect
20. C₃×#4131	C₂²	semidirect
21. C₃×#4131	C₂²	semidirect
22. C₃×#4131	C₂²	semidirect
29. C₃×C₅×AGL(1,ℤ/8)	C₄	semidirect
30. C₃×#4097	C₄	semidirect
31. C₃×#4078	C₄	semidirect
33. C₃×#4061	C₄	semidirect
35. C₃×#4057	C₄	semidirect
36. C₃×#4055	C₄	semidirect
38. C₃×#4010	C₂²	semidirect
39. C₃×#4010	C₂²	semidirect
40. C₃×#4008	C₂²	semidirect
41. C₃×#4008	C₂²	semidirect
42. C₃×#4008	C₂²	semidirect
43. C₃×#4008	C₂²	semidirect
44. C₃×#3996	C₂²	semidirect
45. C₃×#3995	C₂²	semidirect
46. C₃×#3992	C₂²	semidirect
47. C₃×C₈×AGL(1,𝔽₅)	C₂²	semidirect
49. #66264	C₂×C₃	semidirect
50. #66256	C₂×C₃	semidirect
52. D₅⋊#104	C₂×C₃	semidirect
53. C₂×#4008	C₂×C₃	semidirect
54. #65737	C₂×C₃	semidirect
55. #65736	C₂×C₃	semidirect
56. #65733	C₂×C₃	semidirect
57. #65732	C₂×C₃	semidirect
58. #65729	C₂×C₃	semidirect
59. M₁₆×AGL(1,𝔽₅)	C₂×C₃	semidirect
102. C₃×C₅×SD₁₆	C₂×C₄	semidirect
103. C₃×C₅×D₈	C₂×C₄	semidirect
105. C₃×#689	C₂×C₄	semidirect
106. C₃×#687	C₂×C₄	semidirect
107. C₃×#687	C₂×C₄	semidirect
108. C₃×#686	C₂×C₄	semidirect
110. C₃×Dic₂₀	C₂×C₄	semidirect
111. C₃×#677	C₂×C₄	semidirect
128. #4133	C₂²×C₃	semidirect
132. #4131	C₂²×C₃	semidirect
133. #4131	C₂²×C₃	semidirect
134. #4131	C₂²×C₃	semidirect
141. C₅×AGL(1,ℤ/8)	C₃×C₄	semidirect
142. #4097	C₃×C₄	semidirect
143. #4078	C₃×C₄	semidirect
145. #4061	C₃×C₄	semidirect
147. #4057	C₃×C₄	semidirect
148. #4055	C₃×C₄	semidirect
150. #4010	C₂²×C₃	semidirect
151. #4010	C₂²×C₃	semidirect
152. #4008	C₂²×C₃	semidirect
153. #4008	C₂²×C₃	semidirect
154. #4008	C₂²×C₃	semidirect
155. #4008	C₂²×C₃	semidirect
156. #3996	C₂²×C₃	semidirect
157. #3995	C₂²×C₃	semidirect
158. #3992	C₂²×C₃	semidirect
159. C₈×AGL(1,𝔽₅)	C₂²×C₃	semidirect
211. C₃×AGL(1,ℤ/8)	AGL(1,𝔽₅)	semidirect
281. C₅×SD₁₆	C₂×C₃×C₄	semidirect
282. C₅×D₈	C₂×C₃×C₄	semidirect
284. #689	C₂×C₃×C₄	semidirect
285. #687	C₂×C₃×C₄	semidirect
286. #687	C₂×C₃×C₄	semidirect
287. #686	C₂×C₃×C₄	semidirect
289. Dic₂₀	C₂×C₃×C₄	semidirect
290. #677	C₂×C₃×C₄	semidirect
347. C₃×SD₁₆	C₂×AGL(1,𝔽₅)	semidirect
349. C₃×D₈	C₂×AGL(1,𝔽₅)	semidirect
426. AGL(1,ℤ/8)	C₃×AGL(1,𝔽₅)	semidirect
547. SD₁₆	C₂×C₃×AGL(1,𝔽₅)	semidirect
549. D₈	C₂×C₃×AGL(1,𝔽₅)	semidirect
576. C₃×C₅	#2956	semidirect
652. C₅	C₃×#2956	semidirect
671. C₃	#10617449	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×#66264	2
C₃×#66256	2
C₃×#66107	2
C₃×D₅⋊#104	2
C₂×C₃×#4008	2
C₃×#65737	2
C₃×#65736	2
C₃×#65733	2
C₃×#65732	2
C₃×#65729	2
C₃×M₁₆×AGL(1,𝔽₅)	2
#10617449	3
C₃×#2956	5

Subgroups — C3×#10617449 #14418125

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10617449 #14418125