Subgroups — C₃×#10610617 #14424658

Order 1920.

1528 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4149	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#4096	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#4054	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65844	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#417	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#65418	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×#3941	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#65100	C₂	1	2	yes	maximal, complemented
9	C₃×#65099	C₂	1	2	yes	maximal, complemented
10	C₃×#65065	C₂	1	2	yes	maximal, complemented
11	C₃×#65062	C₂	1	2	yes	maximal, complemented
12	#10610617	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
14	C₃×#4149	—	2	4	no	—
15	C₃×#4149	—	2	4	no	—
16	C₂×C₃×#713	C₂²	1	2	yes	—
17	C₂×C₃×D₄×D₅	—	2	2	no	—
18	C₂×C₃×D₄×D₅	C₂²	1	2	yes	—
19	C₂×C₃×#709	—	2	1	no	—
20	C₂×C₃×#709	C₂²	1	1	yes	characteristic
21	C₂×C₃×#709	C₂²	1	1	yes	characteristic, complemented
22	C₂²×C₃×D₂₀	—	2	1	no	—
23	C₂²×C₃×D₂₀	C₂²	1	1	yes	characteristic
24	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic, complemented
25	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
26	C₃×#4096	—	2	4	no	—
27	C₂×C₃×#689	C₂²	1	1	yes	characteristic, complemented
28	C₂×C₃×#688	C₂²	1	1	yes	characteristic, complemented
29	C₂×C₃×#687	C₂²	1	1	yes	characteristic, complemented
30	C₂×C₃×#686	C₂²	1	1	yes	characteristic, complemented
31	C₃×#4076	—	2	2	no	—
32	C₂²×C₃×#168	C₂²	1	1	yes	characteristic, complemented
33	C₃×#4054	—	2	4	no	—
34	C₂×C₃×D₄₀	C₂²	1	2	yes	complemented
35	C₂×C₃×#677	C₂²	1	2	yes	complemented
36	C₃×#4043	—	2	1	no	—
37	C₃×#4043	C₂²	1	2	yes	complemented
38	C₃×#4041	—	2	1	no	—
39	C₃×#4041	C₂²	1	2	yes	complemented
40	C₃×C₅×#103	C₂²	1	2	yes	complemented
41	C₃×C₅×#102	C₂²	1	2	yes	complemented
42	C₃×#3952	C₂²	1	2	yes	complemented
43	C₃×#3952	C₂²	1	2	yes	complemented
44	C₃×#3941	C₂²	1	2	yes	complemented
45	C₃×#3941	C₂²	1	2	yes	complemented
46	C₃×#3012	—	5	1	no	maximal, Hall
47	C₂×#4149	C₂×C₃	1	1	yes	characteristic, complemented
48	C₂×#4096	C₂×C₃	1	1	yes	characteristic, complemented
49	C₂×#4054	C₂×C₃	1	1	yes	characteristic, complemented
50	#65844	C₂×C₃	1	1	yes	characteristic, complemented
51	C₅×#417	C₂×C₃	1	1	yes	characteristic, complemented
52	#65418	C₂×C₃	1	1	yes	characteristic, complemented
53	C₂×#3941	C₂×C₃	1	1	yes	characteristic, complemented
54	#65100	C₂×C₃	1	2	yes	complemented
55	#65099	C₂×C₃	1	2	yes	complemented
56	#65065	C₂×C₃	1	2	yes	complemented
57	#65062	C₂×C₃	1	2	yes	complemented
58	C₂³×C₃×D₅	—	2	1	no	—
59	C₂³×C₃×D₅	—	2	2	no	—
60	C₃×C₅×#41	—	2	2	no	—
61	C₃×C₅×#41	D₄	1	4	yes	complemented
62	C₂×C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
63	C₂×C₃×C₅×D₄	—	2	1	no	—
64	C₂×C₃×C₅×D₄	C₂³	1	1	yes	characteristic
65	C₂²×C₃×C₄×C₅	—	2	1	no	—
66	C₂²×C₃×C₄×C₅	—	2	1	no	—
67	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
68	C₂×C₃×C₂²⋊D₅	—	2	1	no	—
69	C₂×C₃×C₂²⋊D₅	—	2	2	no	—
70	C₂×C₃×C₂²⋊D₅	D₄	1	2	yes	—
71	C₃×#713	—	2	4	no	—
72	C₃×#713	—	2	4	no	—
73	C₃×D₄×D₅	—	2	16	no	—
74	C₃×D₄×D₅	—	2	4	no	—
75	C₃×D₄×D₅	—	2	4	no	—
76	C₃×#709	—	2	8	no	—
77	C₃×#709	—	2	2	no	—
78	C₃×#709	—	2	2	no	—
79	C₃×#709	—	2	2	no	—
80	C₃×#709	—	2	2	no	—
81	C₂×C₃×D₂₀	—	2	8	no	—
82	C₂×C₃×D₂₀	—	2	2	no	—
83	C₂×C₃×D₂₀	—	2	1	no	—
84	C₂×C₃×D₂₀	—	2	2	no	—
85	C₂×C₃×D₂₀	—	2	2	no	—
86	C₂×C₃×D₂₀	D₄	1	1	yes	characteristic
87	C₂×C₃×D₂₀	D₄	1	4	yes	complemented
88	C₂×C₃×D₂₀	C₂³	1	1	yes	characteristic
89	C₂×C₃×D₂₀	C₂³	1	2	yes	—
90	C₂×C₃×C₄×D₅	—	2	2	no	—
91	C₂×C₃×C₄×D₅	—	2	1	no	—
92	C₂×C₃×C₄×D₅	C₂³	1	2	yes	—
93	C₂×C₃×Dic₁₀	D₄	1	1	yes	characteristic
94	C₂×C₃×Dic₁₀	C₂³	1	1	yes	characteristic
95	C₃×C₅×M₁₆	—	2	2	no	—
96	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
97	C₃×C₅×C₄⋊C₄	—	2	1	no	—
98	C₃×C₅×C₄⋊C₄	C₂³	1	2	yes	—
99	C₃×#689	—	2	2	no	—
100	C₃×#688	—	2	2	no	—
101	C₃×#687	—	2	2	no	—
102	C₃×#686	—	2	2	no	—
103	C₃×#685	—	2	2	no	—
104	C₃×#685	—	2	2	no	—
105	C₃×C₄⋊Dic₅	—	2	2	no	—
106	C₂×C₃×#168	—	2	2	no	—
107	C₂×C₃×#168	C₂³	1	1	yes	characteristic
108	C₂×C₃×#168	C₂³	1	1	yes	characteristic
109	C₃×D₄₀	—	2	4	no	—
110	C₃×#677	—	2	4	no	—
111	C₂×C₃×H(5,𝔽₂)	—	5	1	no	—
112	C₂×C₃×AGL(1,ℤ/8)	—	5	1	no	—
113	C₂×C₃×#135	—	5	1	no	—
114	C₃×#537	—	5	1	no	—
115	C₃×#452	—	5	2	no	—
116	C₃×#450	—	5	2	no	—
117	C₃×#449	—	5	2	no	—
118	C₃×#447	—	5	2	no	—
119	C₃×#417	D₅	1	1	yes	characteristic, complemented
120	C₂×C₃×#102	—	5	1	no	—
121	C₃×#408	—	5	1	no	—
122	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
123	#4149	—	2	4	no	—
124	#4149	—	2	4	no	—
125	C₂×#713	C₂²×C₃	1	2	yes	—
126	C₂×D₄×D₅	—	2	2	no	—
127	C₂×D₄×D₅	C₂²×C₃	1	2	yes	—
128	C₂×#709	—	2	1	no	—
129	C₂×#709	C₂²×C₃	1	1	yes	characteristic
130	C₂×#709	C₂²×C₃	1	1	yes	characteristic, complemented
131	C₂²×D₂₀	—	2	1	no	—
132	C₂²×D₂₀	C₂²×C₃	1	1	yes	characteristic
133	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic, complemented
134	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic, complemented
135	#4096	—	2	4	no	—
136	C₂×#689	C₂²×C₃	1	1	yes	characteristic, complemented
137	C₂×#688	C₂²×C₃	1	1	yes	characteristic, complemented
138	C₂×#687	C₂²×C₃	1	1	yes	characteristic, complemented
139	C₂×#686	C₂²×C₃	1	1	yes	characteristic, complemented
140	#4076	—	2	2	no	—
141	C₂²×#168	C₂²×C₃	1	1	yes	characteristic, complemented
142	#4054	—	2	4	no	—
143	C₂×D₄₀	C₂²×C₃	1	2	yes	complemented
144	C₂×#677	C₂²×C₃	1	2	yes	complemented
145	#4043	—	2	1	no	—
146	#4043	C₂²×C₃	1	2	yes	complemented
147	#4041	—	2	1	no	—
148	#4041	C₂²×C₃	1	2	yes	complemented
149	C₅×#103	C₂²×C₃	1	2	yes	complemented
150	C₅×#102	C₂²×C₃	1	2	yes	complemented
151	#3952	C₂²×C₃	1	2	yes	complemented
152	#3952	C₂²×C₃	1	2	yes	complemented
153	#3941	C₂²×C₃	1	2	yes	complemented
154	#3941	C₂²×C₃	1	2	yes	complemented
155	#3012	—	5	1	no	Sylow
156	C₂³×C₃×C₅	—	2	1	no	—
157	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₃×D₅	—	2	2	no	—
159	C₂²×C₃×D₅	—	2	8	no	—
160	C₂²×C₃×D₅	—	2	4	no	—
161	C₂²×C₃×D₅	—	2	1	no	—
162	C₂²×C₃×D₅	—	8	2	no	—
163	C₂²×C₃×D₅	—	8	2	no	—
164	C₂²×C₃×D₅	—	8	1	no	—
165	C₂²×C₃×D₅	—	8	1	no	—
166	C₂²×C₃×D₅	C₂×D₄	1	2	yes	—
167	C₃×C₅×Q₈	—	2	1	no	—
168	C₃×C₅×Q₈	C₂×D₄	1	2	yes	—
169	C₃×C₅×D₄	—	2	4	no	—
170	C₃×C₅×D₄	—	2	1	no	—
171	C₃×C₅×D₄	C₂×D₄	1	2	yes	—
172	C₂×C₃×C₄×C₅	—	2	1	no	—
173	C₂×C₃×C₄×C₅	—	2	4	no	—
174	C₂×C₃×C₄×C₅	—	2	2	no	—
175	C₂×C₃×C₄×C₅	—	8	1	no	—
176	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
177	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
178	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
179	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
180	C₃×C₂²⋊D₅	—	2	8	no	—
181	C₃×C₂²⋊D₅	—	2	4	no	—
182	C₃×C₂²⋊D₅	—	4	2	no	—
183	C₂×C₃×Dic₅	—	2	1	no	—
184	C₂×C₃×Dic₅	C₂×D₄	1	2	yes	—
185	C₃×D₂₀	—	2	8	no	—
186	C₃×D₂₀	—	2	2	no	—
187	C₃×D₂₀	—	2	4	no	—
188	C₃×D₂₀	—	2	2	no	—
189	C₃×D₂₀	—	2	1	no	—
190	C₃×D₂₀	—	2	1	no	—
191	C₃×D₂₀	—	8	2	no	—
192	C₃×D₂₀	C₂×D₄	1	4	yes	—
193	C₃×C₄×D₅	—	2	8	no	—
194	C₃×C₄×D₅	—	2	2	no	—
195	C₃×C₄×D₅	—	2	2	no	—
196	C₃×C₄×D₅	—	4	2	no	—
197	C₃×Dic₁₀	—	2	2	no	—
198	C₃×Dic₁₀	—	2	1	no	—
199	C₃×Dic₁₀	—	2	1	no	—
200	C₃×C₅×C₈	—	2	2	no	—
201	C₃×#168	—	2	1	no	—
202	C₃×#168	—	2	1	no	—
203	C₃×H(5,𝔽₂)	—	10	4	no	—
204	C₃×H(5,𝔽₂)	—	10	4	no	—
205	C₂×C₃×#41	—	5	1	no	—
206	C₂×C₃×#41	—	5	2	no	—
207	C₂×C₃×#41	—	5	1	no	—
208	C₂×C₃×#41	—	10	1	no	—
209	C₂×C₃×#41	C₂×D₅	1	1	yes	characteristic
210	C₂²×C₃×D₄	—	5	2	no	—
211	C₂²×C₃×D₄	—	5	1	no	—
212	C₂²×C₃×D₄	—	10	1	no	—
213	C₂²×C₃×D₄	—	10	2	no	—
214	C₃×AGL(1,ℤ/8)	—	10	4	no	—
215	C₃×#135	—	10	4	no	—
216	C₂×C₃×Q₁₆	—	5	1	no	—
217	C₂×C₃×SD₁₆	—	5	1	no	—
218	C₂×C₃×SD₁₆	—	5	1	no	—
219	C₂×C₃×SD₁₆	—	5	2	no	—
220	C₂×C₃×D₈	—	5	1	no	—
221	C₂×C₃×D₈	—	5	2	no	—
222	C₂×C₃×M₁₆	C₂×D₅	1	1	yes	characteristic, complemented
223	C₂²×C₃×C₈	—	5	1	no	—
224	C₃×C₂²⋊Q₈	—	5	2	no	—
225	C₃×C₂²⋊Q₈	—	10	1	no	—
226	C₃×#121	—	5	2	no	—
227	C₃×#121	—	10	1	no	—
228	C₃×#121	—	10	2	no	—
229	C₂×C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic, complemented
230	C₃×#103	C₂×D₅	1	2	yes	complemented
231	C₃×#102	—	5	2	no	—
232	C₃×#102	—	5	2	no	—
233	C₃×#102	C₂×D₅	1	2	yes	complemented
234	C₃×C₂²⋊C₈	—	5	2	no	—
235	C₃×C₂²⋊C₈	—	5	2	no	—
236	C₂³×D₅	—	2	1	no	—
237	C₂³×D₅	—	2	2	no	—
238	C₅×#41	—	2	2	no	—
239	C₅×#41	C₃×D₄	1	4	yes	complemented
240	C₂×C₅×Q₈	C₂³×C₃	1	1	yes	characteristic
241	C₂×C₅×D₄	—	2	1	no	—
242	C₂×C₅×D₄	C₂³×C₃	1	1	yes	characteristic
243	C₂²×C₄×C₅	—	2	1	no	—
244	C₂²×C₄×C₅	—	2	1	no	—
245	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
246	C₂×C₂²⋊D₅	—	2	1	no	—
247	C₂×C₂²⋊D₅	—	2	2	no	—
248	C₂×C₂²⋊D₅	C₃×D₄	1	2	yes	—
249	#713	—	2	4	no	—
250	#713	—	2	4	no	—
251	D₄×D₅	—	2	16	no	—
252	D₄×D₅	—	2	4	no	—
253	D₄×D₅	—	2	4	no	—
254	#709	—	2	8	no	—
255	#709	—	2	2	no	—
256	#709	—	2	2	no	—
257	#709	—	2	2	no	—
258	#709	—	2	2	no	—
259	C₂×D₂₀	—	2	8	no	—
260	C₂×D₂₀	—	2	2	no	—
261	C₂×D₂₀	—	2	1	no	—
262	C₂×D₂₀	—	2	2	no	—
263	C₂×D₂₀	—	2	2	no	—
264	C₂×D₂₀	C₃×D₄	1	1	yes	characteristic
265	C₂×D₂₀	C₃×D₄	1	4	yes	complemented
266	C₂×D₂₀	C₂³×C₃	1	1	yes	characteristic
267	C₂×D₂₀	C₂³×C₃	1	2	yes	—
268	C₂×C₄×D₅	—	2	2	no	—
269	C₂×C₄×D₅	—	2	1	no	—
270	C₂×C₄×D₅	C₂³×C₃	1	2	yes	—
271	C₂×Dic₁₀	C₃×D₄	1	1	yes	characteristic
272	C₂×Dic₁₀	C₂³×C₃	1	1	yes	characteristic
273	C₅×M₁₆	—	2	2	no	—
274	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
275	C₅×C₄⋊C₄	—	2	1	no	—
276	C₅×C₄⋊C₄	C₂³×C₃	1	2	yes	—
277	#689	—	2	2	no	—
278	#688	—	2	2	no	—
279	#687	—	2	2	no	—
280	#686	—	2	2	no	—
281	#685	—	2	2	no	—
282	#685	—	2	2	no	—
283	C₄⋊Dic₅	—	2	2	no	—
284	C₂×#168	—	2	2	no	—
285	C₂×#168	C₂³×C₃	1	1	yes	characteristic
286	C₂×#168	C₂³×C₃	1	1	yes	characteristic
287	D₄₀	—	2	4	no	—
288	#677	—	2	4	no	—
289	C₂×H(5,𝔽₂)	—	5	1	no	—
290	C₂×AGL(1,ℤ/8)	—	5	1	no	—
291	C₂×#135	—	5	1	no	—
292	#537	—	5	1	no	—
293	#452	—	5	2	no	—
294	#450	—	5	2	no	—
295	#449	—	5	2	no	—
296	#447	—	5	2	no	—
297	#417	C₃×D₅	1	1	yes	characteristic, complemented
298	C₂×#102	—	5	1	no	—
299	#408	—	5	1	no	—
300	C₂²×C₃×C₅	—	2	1	no	—
301	C₂²×C₃×C₅	—	2	1	no	—
302	C₂²×C₃×C₅	—	2	2	no	—
303	C₂²×C₃×C₅	—	4	1	no	—
304	C₂²×C₃×C₅	—	4	1	no	—
305	C₂²×C₃×C₅	#120	1	2	yes	—
306	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
307	C₂×C₃×D₅	—	2	2	no	—
308	C₂×C₃×D₅	—	2	4	no	—
309	C₂×C₃×D₅	—	2	2	no	—
310	C₂×C₃×D₅	—	2	2	no	—
311	C₂×C₃×D₅	—	4	2	no	—
312	C₂×C₃×D₅	—	4	2	no	—
313	C₂×C₃×D₅	—	4	1	no	—
314	C₂×C₃×D₅	—	4	1	no	—
315	C₂×C₃×D₅	—	4	1	no	—
316	C₂×C₃×D₅	—	8	4	no	—
317	C₂×C₃×D₅	—	8	8	no	—
318	C₃×C₄×C₅	—	2	2	no	—
319	C₃×C₄×C₅	—	4	2	no	—
320	C₃×C₄×C₅	#120	1	2	yes	—
321	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
322	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
323	C₃×Dic₅	—	2	2	no	—
324	C₃×Dic₅	—	4	1	no	—
325	C₂⁴×C₃	—	10	2	no	—
326	C₂⁴×C₃	—	10	1	no	—
327	C₃×#41	—	2	2	no	—
328	C₃×#41	—	10	8	no	—
329	C₃×#41	—	10	4	no	—
330	C₃×#41	—	10	4	no	—
331	C₃×#41	—	10	2	no	—
332	C₃×#41	—	10	2	no	—
333	C₃×#41	—	10	2	no	—
334	C₃×#41	—	10	2	no	—
335	C₃×#41	D₂₀	1	4	yes	complemented
336	C₂×C₃×Q₈	—	5	1	no	—
337	C₂×C₃×Q₈	—	5	1	no	—
338	C₂×C₃×Q₈	C₂²×D₅	1	1	yes	characteristic
339	C₂×C₃×D₄	—	2	1	no	—
340	C₂×C₃×D₄	—	5	2	no	—
341	C₂×C₃×D₄	—	5	4	no	—
342	C₂×C₃×D₄	—	5	1	no	—
343	C₂×C₃×D₄	—	5	2	no	—
344	C₂×C₃×D₄	—	5	1	no	—
345	C₂×C₃×D₄	—	10	2	no	—
346	C₂×C₃×D₄	—	10	8	no	—
347	C₂×C₃×D₄	—	10	2	no	—
348	C₂×C₃×D₄	—	10	2	no	—
349	C₂×C₃×D₄	—	10	4	no	—
350	C₂×C₃×D₄	—	10	16	no	—
351	C₂×C₃×D₄	—	10	4	no	—
352	C₂×C₃×D₄	—	10	1	no	—
353	C₂×C₃×D₄	—	10	2	no	—
354	C₂×C₃×D₄	—	10	1	no	—
355	C₂×C₃×D₄	C₂²×D₅	1	1	yes	characteristic
356	C₂²×C₃×C₄	—	2	1	no	—
357	C₂²×C₃×C₄	—	2	1	no	—
358	C₂²×C₃×C₄	—	5	2	no	—
359	C₂²×C₃×C₄	—	10	2	no	—
360	C₂²×C₃×C₄	—	10	1	no	—
361	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
362	C₃×Q₁₆	—	10	2	no	—
363	C₃×SD₁₆	—	10	4	no	—
364	C₃×SD₁₆	—	10	2	no	—
365	C₃×SD₁₆	—	10	2	no	—
366	C₃×D₈	—	10	4	no	—
367	C₃×D₈	—	10	2	no	—
368	C₃×M₁₆	—	2	2	no	—
369	C₂×C₃×C₈	—	5	1	no	—
370	C₂×C₃×C₈	—	5	1	no	—
371	C₂×C₃×C₈	—	10	2	no	—
372	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
373	C₃×C₄⋊C₄	—	2	1	no	—
374	C₃×C₄⋊C₄	—	10	2	no	—
375	C₃×C₄⋊C₄	C₂²×D₅	1	2	yes	—
376	C₃×C₂²⋊C₄	—	10	2	no	—
377	C₃×C₂²⋊C₄	—	10	2	no	—
378	C₂³×C₅	—	2	1	no	—
379	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
380	C₂²×D₅	—	2	2	no	—
381	C₂²×D₅	—	2	8	no	—
382	C₂²×D₅	—	2	4	no	—
383	C₂²×D₅	—	2	1	no	—
384	C₂²×D₅	—	8	2	no	—
385	C₂²×D₅	—	8	2	no	—
386	C₂²×D₅	—	8	1	no	—
387	C₂²×D₅	—	8	1	no	—
388	C₂²×D₅	C₂×C₃×D₄	1	2	yes	—
389	C₅×Q₈	—	2	1	no	—
390	C₅×Q₈	C₂×C₃×D₄	1	2	yes	—
391	C₅×D₄	—	2	4	no	—
392	C₅×D₄	—	2	1	no	—
393	C₅×D₄	C₂×C₃×D₄	1	2	yes	—
394	C₂×C₄×C₅	—	2	1	no	—
395	C₂×C₄×C₅	—	2	4	no	—
396	C₂×C₄×C₅	—	2	2	no	—
397	C₂×C₄×C₅	—	8	1	no	—
398	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
399	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
400	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
401	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
402	C₂²⋊D₅	—	2	8	no	—
403	C₂²⋊D₅	—	2	4	no	—
404	C₂²⋊D₅	—	4	2	no	—
405	C₂×Dic₅	—	2	1	no	—
406	C₂×Dic₅	C₂×C₃×D₄	1	2	yes	—
407	D₂₀	—	2	8	no	—
408	D₂₀	—	2	2	no	—
409	D₂₀	—	2	4	no	—
410	D₂₀	—	2	2	no	—
411	D₂₀	—	2	1	no	—
412	D₂₀	—	2	1	no	—
413	D₂₀	—	8	2	no	—
414	D₂₀	C₂×C₃×D₄	1	4	yes	—
415	C₄×D₅	—	2	8	no	—
416	C₄×D₅	—	2	2	no	—
417	C₄×D₅	—	2	2	no	—
418	C₄×D₅	—	4	2	no	—
419	Dic₁₀	—	2	2	no	—
420	Dic₁₀	—	2	1	no	—
421	Dic₁₀	—	2	1	no	—
422	C₅×C₈	—	2	2	no	—
423	#168	—	2	1	no	—
424	#168	—	2	1	no	—
425	H(5,𝔽₂)	—	10	4	no	—
426	H(5,𝔽₂)	—	10	4	no	—
427	C₂×#41	—	5	1	no	—
428	C₂×#41	—	5	2	no	—
429	C₂×#41	—	5	1	no	—
430	C₂×#41	—	10	1	no	—
431	C₂×#41	C₂×C₃×D₅	1	1	yes	characteristic
432	C₂²×D₄	—	5	2	no	—
433	C₂²×D₄	—	5	1	no	—
434	C₂²×D₄	—	10	1	no	—
435	C₂²×D₄	—	10	2	no	—
436	AGL(1,ℤ/8)	—	10	4	no	—
437	#135	—	10	4	no	—
438	C₂×Q₁₆	—	5	1	no	—
439	C₂×SD₁₆	—	5	1	no	—
440	C₂×SD₁₆	—	5	1	no	—
441	C₂×SD₁₆	—	5	2	no	—
442	C₂×D₈	—	5	1	no	—
443	C₂×D₈	—	5	2	no	—
444	C₂×M₁₆	C₂×C₃×D₅	1	1	yes	characteristic, complemented
445	C₂²×C₈	—	5	1	no	—
446	C₂²⋊Q₈	—	5	2	no	—
447	C₂²⋊Q₈	—	10	1	no	—
448	#121	—	5	2	no	—
449	#121	—	10	1	no	—
450	#121	—	10	2	no	—
451	C₂×C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic, complemented
452	#103	C₂×C₃×D₅	1	2	yes	complemented
453	#102	—	5	2	no	—
454	#102	—	5	2	no	—
455	#102	C₂×C₃×D₅	1	2	yes	complemented
456	C₂²⋊C₈	—	5	2	no	—
457	C₂²⋊C₈	—	5	2	no	—
458	C₂×C₃×C₅	—	2	2	no	—
459	C₂×C₃×C₅	—	4	2	no	—
460	C₂×C₃×C₅	C₂×#120	1	1	yes	characteristic
461	C₂×C₃×C₅	#576	1	1	yes	characteristic
462	C₂×C₃×C₅	#577	1	1	yes	characteristic
463	C₃×D₅	—	4	2	no	—
464	C₃×D₅	—	4	4	no	—
465	C₃×D₅	—	8	1	no	—
466	C₂³×C₃	—	2	1	no	—
467	C₂³×C₃	—	5	2	no	—
468	C₂³×C₃	—	10	8	no	—
469	C₂³×C₃	—	10	2	no	—
470	C₂³×C₃	—	10	4	no	—
471	C₂³×C₃	—	10	1	no	—
472	C₂³×C₃	—	40	1	no	—
473	C₂³×C₃	—	40	1	no	—
474	C₂³×C₃	—	40	2	no	—
475	C₂³×C₃	—	40	2	no	—
476	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
477	C₃×Q₈	—	2	1	no	—
478	C₃×Q₈	—	10	2	no	—
479	C₃×Q₈	—	10	1	no	—
480	C₃×Q₈	—	10	1	no	—
481	C₃×Q₈	C₂×D₂₀	1	2	yes	—
482	C₃×D₄	—	2	4	no	—
483	C₃×D₄	—	2	1	no	—
484	C₃×D₄	—	5	4	no	—
485	C₃×D₄	—	10	8	no	—
486	C₃×D₄	—	10	8	no	—
487	C₃×D₄	—	10	4	no	—
488	C₃×D₄	—	10	2	no	—
489	C₃×D₄	—	10	1	no	—
490	C₃×D₄	—	10	1	no	—
491	C₃×D₄	—	10	4	no	—
492	C₃×D₄	—	10	2	no	—
493	C₃×D₄	—	20	2	no	—
494	C₃×D₄	—	40	2	no	—
495	C₃×D₄	C₂×D₂₀	1	2	yes	—
496	C₂×C₃×C₄	—	2	1	no	—
497	C₂×C₃×C₄	—	2	4	no	—
498	C₂×C₃×C₄	—	2	2	no	—
499	C₂×C₃×C₄	—	5	2	no	—
500	C₂×C₃×C₄	—	8	1	no	—
501	C₂×C₃×C₄	—	10	8	no	—
502	C₂×C₃×C₄	—	10	2	no	—
503	C₂×C₃×C₄	—	10	2	no	—
504	C₂×C₃×C₄	—	10	1	no	—
505	C₂×C₃×C₄	—	20	2	no	—
506	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
507	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
508	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
509	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
510	C₃×C₈	—	2	2	no	—
511	C₃×C₈	—	10	1	no	—
512	C₃×C₈	—	10	1	no	—
513	C₂²×C₅	—	2	1	no	—
514	C₂²×C₅	—	2	1	no	—
515	C₂²×C₅	—	2	2	no	—
516	C₂²×C₅	—	4	1	no	—
517	C₂²×C₅	—	4	1	no	—
518	C₂²×C₅	C₃×#120	1	2	yes	—
519	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
520	C₂×D₅	—	2	2	no	—
521	C₂×D₅	—	2	2	no	—
522	C₂×D₅	—	2	2	no	—
523	C₂×D₅	—	2	4	no	—
524	C₂×D₅	—	4	2	no	—
525	C₂×D₅	—	4	2	no	—
526	C₂×D₅	—	4	1	no	—
527	C₂×D₅	—	4	1	no	—
528	C₂×D₅	—	4	1	no	—
529	C₂×D₅	—	8	4	no	—
530	C₂×D₅	—	8	8	no	—
531	C₄×C₅	—	2	2	no	—
532	C₄×C₅	—	4	2	no	—
533	C₄×C₅	C₃×#120	1	2	yes	—
534	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
535	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
536	Dic₅	—	2	2	no	—
537	Dic₅	—	4	1	no	—
538	C₂⁴	—	10	2	no	—
539	C₂⁴	—	10	1	no	—
540	#41	—	2	2	no	—
541	#41	—	10	8	no	—
542	#41	—	10	4	no	—
543	#41	—	10	4	no	—
544	#41	—	10	2	no	—
545	#41	—	10	2	no	—
546	#41	—	10	2	no	—
547	#41	—	10	2	no	—
548	#41	C₃×D₂₀	1	4	yes	complemented
549	C₂×Q₈	—	5	1	no	—
550	C₂×Q₈	—	5	1	no	—
551	C₂×Q₈	C₂²×C₃×D₅	1	1	yes	characteristic
552	C₂×D₄	—	2	1	no	—
553	C₂×D₄	—	5	2	no	—
554	C₂×D₄	—	5	4	no	—
555	C₂×D₄	—	5	1	no	—
556	C₂×D₄	—	5	2	no	—
557	C₂×D₄	—	5	1	no	—
558	C₂×D₄	—	10	2	no	—
559	C₂×D₄	—	10	8	no	—
560	C₂×D₄	—	10	2	no	—
561	C₂×D₄	—	10	2	no	—
562	C₂×D₄	—	10	4	no	—
563	C₂×D₄	—	10	16	no	—
564	C₂×D₄	—	10	4	no	—
565	C₂×D₄	—	10	1	no	—
566	C₂×D₄	—	10	2	no	—
567	C₂×D₄	—	10	1	no	—
568	C₂×D₄	C₂²×C₃×D₅	1	1	yes	characteristic
569	C₂²×C₄	—	2	1	no	—
570	C₂²×C₄	—	2	1	no	—
571	C₂²×C₄	—	5	2	no	—
572	C₂²×C₄	—	10	2	no	—
573	C₂²×C₄	—	10	1	no	—
574	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
575	Q₁₆	—	10	2	no	—
576	SD₁₆	—	10	4	no	—
577	SD₁₆	—	10	2	no	—
578	SD₁₆	—	10	2	no	—
579	D₈	—	10	4	no	—
580	D₈	—	10	2	no	—
581	M₁₆	—	2	2	no	—
582	C₂×C₈	—	5	1	no	—
583	C₂×C₈	—	5	1	no	—
584	C₂×C₈	—	10	2	no	—
585	C₂×C₈	C₂²×C₃×D₅	1	2	yes	—
586	C₄⋊C₄	—	2	1	no	—
587	C₄⋊C₄	—	10	2	no	—
588	C₄⋊C₄	C₂²×C₃×D₅	1	2	yes	—
589	C₂²⋊C₄	—	10	2	no	—
590	C₂²⋊C₄	—	10	2	no	—
591	C₃×C₅	#3012	1	1	yes	characteristic, complemented, Hall
592	C₂²×C₃	—	2	1	no	—
593	C₂²×C₃	—	2	1	no	—
594	C₂²×C₃	—	2	2	no	—
595	C₂²×C₃	—	4	1	no	—
596	C₂²×C₃	—	4	1	no	—
597	C₂²×C₃	—	10	4	no	—
598	C₂²×C₃	—	10	2	no	—
599	C₂²×C₃	—	10	2	no	—
600	C₂²×C₃	—	10	2	no	—
601	C₂²×C₃	—	20	2	no	—
602	C₂²×C₃	—	20	2	no	—
603	C₂²×C₃	—	20	1	no	—
604	C₂²×C₃	—	20	1	no	—
605	C₂²×C₃	—	20	1	no	—
606	C₂²×C₃	—	40	4	no	—
607	C₂²×C₃	—	40	8	no	—
608	C₂²×C₃	#4028	1	2	yes	—
609	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
610	C₃×C₄	—	2	2	no	—
611	C₃×C₄	—	4	2	no	—
612	C₃×C₄	—	10	2	no	—
613	C₃×C₄	—	20	1	no	—
614	C₃×C₄	#4028	1	2	yes	—
615	C₃×C₄	C₂²×D₂₀	1	1	yes	characteristic
616	C₃×C₄	C₂×D₄×D₅	1	1	yes	characteristic
617	C₂×C₅	—	2	2	no	—
618	C₂×C₅	—	4	2	no	—
619	C₂×C₅	C₂×C₃×#120	1	1	yes	characteristic
620	C₂×C₅	C₃×#576	1	1	yes	characteristic
621	C₂×C₅	C₃×#577	1	1	yes	characteristic
622	D₅	—	4	2	no	—
623	D₅	—	4	4	no	—
624	D₅	—	8	1	no	—
625	C₂³	—	2	1	no	—
626	C₂³	—	5	2	no	—
627	C₂³	—	10	8	no	—
628	C₂³	—	10	2	no	—
629	C₂³	—	10	4	no	—
630	C₂³	—	10	1	no	—
631	C₂³	—	40	1	no	—
632	C₂³	—	40	1	no	—
633	C₂³	—	40	2	no	—
634	C₂³	—	40	2	no	—
635	C₂³	C₃×D₄×D₅	1	1	yes	characteristic
636	Q₈	—	2	1	no	—
637	Q₈	—	10	2	no	—
638	Q₈	—	10	1	no	—
639	Q₈	—	10	1	no	—
640	Q₈	C₂×C₃×D₂₀	1	2	yes	—
641	D₄	—	2	4	no	—
642	D₄	—	2	1	no	—
643	D₄	—	5	4	no	—
644	D₄	—	10	8	no	—
645	D₄	—	10	8	no	—
646	D₄	—	10	4	no	—
647	D₄	—	10	2	no	—
648	D₄	—	10	1	no	—
649	D₄	—	10	1	no	—
650	D₄	—	10	4	no	—
651	D₄	—	10	2	no	—
652	D₄	—	20	2	no	—
653	D₄	—	40	2	no	—
654	D₄	C₂×C₃×D₂₀	1	2	yes	—
655	C₂×C₄	—	2	1	no	—
656	C₂×C₄	—	2	4	no	—
657	C₂×C₄	—	2	2	no	—
658	C₂×C₄	—	5	2	no	—
659	C₂×C₄	—	8	1	no	—
660	C₂×C₄	—	10	8	no	—
661	C₂×C₄	—	10	2	no	—
662	C₂×C₄	—	10	2	no	—
663	C₂×C₄	—	10	1	no	—
664	C₂×C₄	—	20	2	no	—
665	C₂×C₄	C₂×C₃×D₂₀	1	2	yes	—
666	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
667	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
668	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
669	C₈	—	2	2	no	—
670	C₈	—	10	1	no	—
671	C₈	—	10	1	no	—
672	C₂×C₃	—	2	2	no	—
673	C₂×C₃	—	4	2	no	—
674	C₂×C₃	—	20	2	no	—
675	C₂×C₃	—	20	4	no	—
676	C₂×C₃	—	40	1	no	—
677	C₂×C₃	C₂×#4028	1	1	yes	characteristic
678	C₂×C₃	#66108	1	1	yes	characteristic
679	C₂×C₃	#66112	1	1	yes	characteristic
680	C₅	C₃×#3012	1	1	yes	characteristic, complemented, Sylow
681	C₂²	—	2	1	no	—
682	C₂²	—	2	1	no	—
683	C₂²	—	2	2	no	—
684	C₂²	—	4	1	no	—
685	C₂²	—	4	1	no	—
686	C₂²	—	10	2	no	—
687	C₂²	—	10	2	no	—
688	C₂²	—	10	4	no	—
689	C₂²	—	10	2	no	—
690	C₂²	—	20	2	no	—
691	C₂²	—	20	2	no	—
692	C₂²	—	20	1	no	—
693	C₂²	—	20	1	no	—
694	C₂²	—	20	1	no	—
695	C₂²	—	40	4	no	—
696	C₂²	—	40	8	no	—
697	C₂²	C₃×#4028	1	2	yes	—
698	C₂²	C₂×C₃×D₄×D₅	1	1	yes	characteristic
699	C₄	—	2	2	no	—
700	C₄	—	4	2	no	—
701	C₄	—	10	2	no	—
702	C₄	—	20	1	no	—
703	C₄	C₃×#4028	1	2	yes	—
704	C₄	C₂²×C₃×D₂₀	1	1	yes	characteristic
705	C₄	C₂×C₃×D₄×D₅	1	1	yes	characteristic
706	C₃	#10610617	1	1	yes	characteristic, direct product factor, Sylow
707	C₂	—	2	2	no	—
708	C₂	—	4	2	no	—
709	C₂	—	20	2	no	—
710	C₂	—	20	4	no	—
711	C₂	—	40	1	no	—
712	C₂	C₂×C₃×#4028	1	1	yes	characteristic
713	C₂	C₃×#66108	1	1	yes	characteristic
714	C₂	C₃×#66112	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₃×C₅×#417	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
hypercenter	C₃×#417	D₅
hypocenter	C₅	C₃×#3012
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×#417

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#4149	C₂	semidirect
2. C₂×C₃×#4096	C₂	semidirect
3. C₂×C₃×#4054	C₂	semidirect
4. C₃×#65844	C₂	semidirect
5. C₃×C₅×#417	C₂	semidirect
6. C₃×#65418	C₂	semidirect
7. C₂×C₃×#3941	C₂	semidirect
8. C₃×#65100	C₂	semidirect
9. C₃×#65099	C₂	semidirect
10. C₃×#65065	C₂	semidirect
11. C₃×#65062	C₂	semidirect
12. #10610617	C₃	direct (N × H)
21. C₂×C₃×#709	C₂²	semidirect
24. C₂×C₃×C₅×M₁₆	C₂²	semidirect
25. C₂×C₃×C₅×C₄⋊C₄	C₂²	semidirect
27. C₂×C₃×#689	C₂²	semidirect
28. C₂×C₃×#688	C₂²	semidirect
29. C₂×C₃×#687	C₂²	semidirect
30. C₂×C₃×#686	C₂²	semidirect
32. C₂²×C₃×#168	C₂²	semidirect
34. C₂×C₃×D₄₀	C₂²	semidirect
35. C₂×C₃×#677	C₂²	semidirect
37. C₃×#4043	C₂²	semidirect
39. C₃×#4041	C₂²	semidirect
40. C₃×C₅×#103	C₂²	semidirect
41. C₃×C₅×#102	C₂²	semidirect
42. C₃×#3952	C₂²	semidirect
43. C₃×#3952	C₂²	semidirect
44. C₃×#3941	C₂²	semidirect
45. C₃×#3941	C₂²	semidirect
47. C₂×#4149	C₂×C₃	semidirect
48. C₂×#4096	C₂×C₃	semidirect
49. C₂×#4054	C₂×C₃	semidirect
50. #65844	C₂×C₃	semidirect
51. C₅×#417	C₂×C₃	semidirect
52. #65418	C₂×C₃	semidirect
53. C₂×#3941	C₂×C₃	semidirect
54. #65100	C₂×C₃	semidirect
55. #65099	C₂×C₃	semidirect
56. #65065	C₂×C₃	semidirect
57. #65062	C₂×C₃	semidirect
61. C₃×C₅×#41	D₄	semidirect
87. C₂×C₃×D₂₀	D₄	semidirect
119. C₃×#417	D₅	semidirect
130. C₂×#709	C₂²×C₃	semidirect
133. C₂×C₅×M₁₆	C₂²×C₃	semidirect
134. C₂×C₅×C₄⋊C₄	C₂²×C₃	semidirect
136. C₂×#689	C₂²×C₃	semidirect
137. C₂×#688	C₂²×C₃	semidirect
138. C₂×#687	C₂²×C₃	semidirect
139. C₂×#686	C₂²×C₃	semidirect
141. C₂²×#168	C₂²×C₃	semidirect
143. C₂×D₄₀	C₂²×C₃	semidirect
144. C₂×#677	C₂²×C₃	semidirect
146. #4043	C₂²×C₃	semidirect
148. #4041	C₂²×C₃	semidirect
149. C₅×#103	C₂²×C₃	semidirect
150. C₅×#102	C₂²×C₃	semidirect
151. #3952	C₂²×C₃	semidirect
152. #3952	C₂²×C₃	semidirect
153. #3941	C₂²×C₃	semidirect
154. #3941	C₂²×C₃	semidirect
222. C₂×C₃×M₁₆	C₂×D₅	semidirect
229. C₂×C₃×C₄⋊C₄	C₂×D₅	semidirect
230. C₃×#103	C₂×D₅	semidirect
233. C₃×#102	C₂×D₅	semidirect
239. C₅×#41	C₃×D₄	semidirect
265. C₂×D₂₀	C₃×D₄	semidirect
297. #417	C₃×D₅	semidirect
335. C₃×#41	D₂₀	semidirect
444. C₂×M₁₆	C₂×C₃×D₅	semidirect
451. C₂×C₄⋊C₄	C₂×C₃×D₅	semidirect
452. #103	C₂×C₃×D₅	semidirect
455. #102	C₂×C₃×D₅	semidirect
548. #41	C₃×D₂₀	semidirect
591. C₃×C₅	#3012	semidirect
680. C₅	C₃×#3012	semidirect
706. C₃	#10610617	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×#4149	2
C₂×C₃×#4096	2
C₂×C₃×#4054	2
C₃×#65844	2
C₃×C₅×#417	2
C₃×#65418	2
C₂×C₃×#3941	2
C₃×#65100	2
C₃×#65099	2
C₃×#65065	2
C₃×#65062	2
#10610617	3
C₃×#3012	5

Subgroups — C3×#10610617 #14424658

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10610617 #14424658