Subgroups — C₃×#10615671 #14439547

Order 1920.

1192 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	C₂×C₃×C₅×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#66164	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#4096	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#66139	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₃×#686	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×#4054	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#65465	C₂	1	2	yes	maximal, complemented
8	C₃×#65457	C₂	1	2	yes	maximal, complemented
9	C₃×#65443	C₂	1	2	yes	maximal, complemented
10	C₃×#65440	C₂	1	2	yes	maximal, complemented
11	C₃×#65411	C₂	1	1	yes	maximal, characteristic, complemented
12	#10615671	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
15	C₂×C₃×#709	C₂²	1	1	yes	characteristic
16	C₂²×C₃×D₂₀	C₂²	1	1	yes	characteristic
17	C₃×C₅×AGL(1,ℤ/8)	—	2	4	no	—
18	C₂×C₃×C₅×SD₁₆	C₂²	1	2	yes	complemented
19	C₂×C₃×C₅×D₈	C₂²	1	2	yes	complemented
20	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
21	C₃×#4099	—	2	2	no	—
22	C₃×#4096	—	2	4	no	—
23	C₃×#4093	C₂²	1	2	yes	complemented
24	C₂×C₃×#689	C₂²	1	1	yes	characteristic, complemented
25	C₂×C₃×#688	C₂²	1	1	yes	characteristic, complemented
26	C₃×#4086	C₂²	1	2	yes	complemented
27	C₃×#4086	C₂²	1	2	yes	complemented
28	C₃×#4084	—	2	2	no	—
29	C₃×#4083	—	2	2	no	—
30	C₃×#4082	C₂²	1	2	yes	complemented
31	C₂×C₃×#687	C₂²	1	1	yes	characteristic, complemented
32	C₂×C₃×#686	—	2	4	no	—
33	C₂×C₃×#686	C₂²	1	2	yes	—
34	C₂×C₃×#686	C₂²	1	2	yes	—
35	C₂×C₃×#686	C₂²	1	1	yes	characteristic, complemented
36	C₃×#4076	—	2	2	no	—
37	C₃×#4072	C₂²	1	1	yes	characteristic
38	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
39	C₃×#4054	—	2	4	no	—
40	C₂×C₃×D₄₀	C₂²	1	2	yes	complemented
41	C₂×C₃×#677	C₂²	1	2	yes	complemented
42	C₃×#3947	C₂²	1	2	yes	complemented
43	C₃×C₈×Dic₅	C₂²	1	2	yes	complemented
44	C₃×#3155	—	5	1	no	maximal, Hall
45	C₂×C₅×AGL(1,ℤ/8)	C₂×C₃	1	1	yes	characteristic, complemented
46	#66164	C₂×C₃	1	1	yes	characteristic, complemented
47	C₂×#4096	C₂×C₃	1	1	yes	characteristic, complemented
48	#66139	C₂×C₃	1	1	yes	characteristic, complemented
49	C₂²×#686	C₂×C₃	1	1	yes	characteristic, complemented
50	C₂×#4054	C₂×C₃	1	1	yes	characteristic, complemented
51	#65465	C₂×C₃	1	2	yes	complemented
52	#65457	C₂×C₃	1	2	yes	complemented
53	#65443	C₂×C₃	1	2	yes	complemented
54	#65440	C₂×C₃	1	2	yes	complemented
55	#65411	C₂×C₃	1	1	yes	characteristic, complemented
56	C₂⁴×C₃×C₅	—	2	1	no	—
57	C₂³×C₃×D₅	—	2	1	no	—
58	C₃×C₅×#41	—	2	2	no	—
59	C₃×C₅×#41	—	2	2	no	—
60	C₂×C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
61	C₂×C₃×C₅×D₄	—	2	1	no	—
62	C₂×C₃×C₅×D₄	—	2	2	no	—
63	C₂×C₃×C₅×D₄	—	2	2	no	—
64	C₂×C₃×C₅×D₄	—	2	1	no	—
65	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
66	C₂×C₃×C₅×D₄	C₂³	1	1	yes	characteristic
67	C₂²×C₃×C₄×C₅	—	2	1	no	—
68	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
69	C₂×C₃×C₂²⋊D₅	—	2	1	no	—
70	C₂×C₃×C₂²⋊D₅	—	2	4	no	—
71	C₂×C₃×C₂²⋊D₅	—	2	4	no	—
72	C₃×#709	—	2	2	no	—
73	C₃×#709	—	2	2	no	—
74	C₂×C₃×D₂₀	—	2	2	no	—
75	C₂×C₃×D₂₀	—	2	2	no	—
76	C₂×C₃×D₂₀	—	2	1	no	—
77	C₂×C₃×D₂₀	C₂³	1	2	yes	—
78	C₂×C₃×D₂₀	C₂³	1	1	yes	characteristic
79	C₂×C₃×C₄×D₅	—	2	1	no	—
80	C₂×C₃×Dic₁₀	C₂³	1	1	yes	characteristic
81	C₃×C₅×SD₁₆	—	2	4	no	—
82	C₃×C₅×D₈	—	2	4	no	—
83	C₃×C₅×M₁₆	D₄	1	4	yes	complemented
84	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
85	C₃×C₂²⋊Dic₅	—	2	4	no	—
86	C₃×C₂²⋊Dic₅	D₄	1	2	yes	—
87	C₃×#689	—	2	2	no	—
88	C₃×#688	—	2	2	no	—
89	C₃×#687	—	2	2	no	—
90	C₃×#686	—	2	4	no	—
91	C₃×#686	—	2	4	no	—
92	C₃×#686	—	2	2	no	—
93	C₃×#685	—	2	4	no	—
94	C₃×C₄⋊Dic₅	D₄	1	2	yes	—
95	C₃×C₄×Dic₅	C₂³	1	2	yes	—
96	C₂×C₃×#168	D₄	1	4	yes	complemented
97	C₂×C₃×#168	C₂³	1	1	yes	characteristic
98	C₂×C₃×#168	C₂³	1	1	yes	characteristic
99	C₃×D₄₀	—	2	4	no	—
100	C₃×#677	—	2	4	no	—
101	C₂×C₃×AGL(1,ℤ/8)	—	5	1	no	—
102	C₂×C₃×AGL(1,ℤ/8)	D₅	1	1	yes	characteristic, complemented
103	C₂×C₃×#135	—	5	1	no	—
104	C₂²×C₃×D₈	—	5	1	no	—
105	C₃×#535	—	5	1	no	—
106	C₃×#535	—	5	1	no	—
107	C₃×#496	—	5	2	no	—
108	C₃×#496	—	5	2	no	—
109	C₃×#495	—	5	2	no	—
110	C₃×#493	—	5	2	no	—
111	C₃×#405	—	5	1	no	—
112	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
113	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
114	C₂×#709	C₂²×C₃	1	1	yes	characteristic
115	C₂²×D₂₀	C₂²×C₃	1	1	yes	characteristic
116	C₅×AGL(1,ℤ/8)	—	2	4	no	—
117	C₂×C₅×SD₁₆	C₂²×C₃	1	2	yes	complemented
118	C₂×C₅×D₈	C₂²×C₃	1	2	yes	complemented
119	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
120	#4099	—	2	2	no	—
121	#4096	—	2	4	no	—
122	#4093	C₂²×C₃	1	2	yes	complemented
123	C₂×#689	C₂²×C₃	1	1	yes	characteristic, complemented
124	C₂×#688	C₂²×C₃	1	1	yes	characteristic, complemented
125	#4086	C₂²×C₃	1	2	yes	complemented
126	#4086	C₂²×C₃	1	2	yes	complemented
127	#4084	—	2	2	no	—
128	#4083	—	2	2	no	—
129	#4082	C₂²×C₃	1	2	yes	complemented
130	C₂×#687	C₂²×C₃	1	1	yes	characteristic, complemented
131	C₂×#686	—	2	4	no	—
132	C₂×#686	C₂²×C₃	1	2	yes	—
133	C₂×#686	C₂²×C₃	1	2	yes	—
134	C₂×#686	C₂²×C₃	1	1	yes	characteristic, complemented
135	#4076	—	2	2	no	—
136	#4072	C₂²×C₃	1	1	yes	characteristic
137	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
138	#4054	—	2	4	no	—
139	C₂×D₄₀	C₂²×C₃	1	2	yes	complemented
140	C₂×#677	C₂²×C₃	1	2	yes	complemented
141	#3947	C₂²×C₃	1	2	yes	complemented
142	C₈×Dic₅	C₂²×C₃	1	2	yes	complemented
143	#3155	—	5	1	no	Sylow
144	C₂³×C₃×C₅	—	2	2	no	—
145	C₂³×C₃×C₅	—	2	1	no	—
146	C₂³×C₃×C₅	—	4	2	no	—
147	C₂³×C₃×C₅	—	8	2	no	—
148	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₃×D₅	—	2	2	no	—
150	C₂²×C₃×D₅	—	2	1	no	—
151	C₂²×C₃×D₅	—	4	2	no	—
152	C₂²×C₃×D₅	—	8	2	no	—
153	C₃×C₅×Q₈	—	2	1	no	—
154	C₃×C₅×Q₈	—	2	1	no	—
155	C₃×C₅×D₄	—	2	2	no	—
156	C₃×C₅×D₄	—	2	1	no	—
157	C₃×C₅×D₄	—	2	1	no	—
158	C₃×C₅×D₄	—	2	2	no	—
159	C₃×C₅×D₄	—	4	2	no	—
160	C₃×C₅×D₄	—	4	2	no	—
161	C₂×C₃×C₄×C₅	—	2	1	no	—
162	C₂×C₃×C₄×C₅	—	4	2	no	—
163	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
164	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
165	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
166	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
167	C₃×C₂²⋊D₅	—	4	2	no	—
168	C₃×C₂²⋊D₅	—	8	4	no	—
169	C₃×C₂²⋊D₅	—	8	4	no	—
170	C₂×C₃×Dic₅	—	2	1	no	—
171	C₂×C₃×Dic₅	C₂×D₄	1	4	yes	—
172	C₃×D₂₀	—	2	2	no	—
173	C₃×D₂₀	—	2	1	no	—
174	C₃×D₂₀	—	2	2	no	—
175	C₃×D₂₀	—	2	1	no	—
176	C₃×D₂₀	—	4	2	no	—
177	C₃×C₄×D₅	—	4	2	no	—
178	C₃×Dic₁₀	—	2	1	no	—
179	C₃×Dic₁₀	—	2	1	no	—
180	C₃×C₅×C₈	C₂×D₄	1	4	yes	—
181	C₃×#168	C₂×D₄	1	2	yes	—
182	C₃×#168	C₂×D₄	1	2	yes	—
183	C₂×C₃×#41	—	5	1	no	—
184	C₂×C₃×#41	C₂×D₅	1	1	yes	characteristic
185	C₂²×C₃×D₄	—	5	1	no	—
186	C₂²×C₃×D₄	C₂×D₅	1	1	yes	characteristic
187	C₃×AGL(1,ℤ/8)	—	2	4	no	—
188	C₃×AGL(1,ℤ/8)	—	10	4	no	—
189	C₃×#135	—	10	4	no	—
190	C₂×C₃×Q₁₆	—	5	1	no	—
191	C₂×C₃×SD₁₆	—	5	1	no	—
192	C₂×C₃×SD₁₆	—	5	1	no	—
193	C₂×C₃×SD₁₆	—	5	2	no	—
194	C₂×C₃×SD₁₆	C₂×D₅	1	2	yes	complemented
195	C₂×C₃×D₈	—	5	2	no	—
196	C₂×C₃×D₈	—	5	2	no	—
197	C₂×C₃×D₈	—	5	1	no	—
198	C₂×C₃×D₈	—	5	2	no	—
199	C₂×C₃×D₈	—	10	4	no	—
200	C₂×C₃×D₈	C₂×D₅	1	2	yes	complemented
201	C₂×C₃×M₁₆	C₂×D₅	1	1	yes	characteristic
202	C₂²×C₃×C₈	—	5	1	no	—
203	C₃×#127	—	5	2	no	—
204	C₃×#127	—	5	2	no	—
205	C₃×#124	—	5	2	no	—
206	C₃×#124	—	5	2	no	—
207	C₃×#121	—	10	2	no	—
208	C₃×#121	—	10	2	no	—
209	C₃×#120	—	10	2	no	—
210	C₃×#120	—	10	2	no	—
211	C₃×#117	—	5	1	no	—
212	C₃×#97	—	5	2	no	—
213	C₃×C₄×C₈	—	5	2	no	—
214	C₂⁴×C₅	—	2	1	no	—
215	C₂³×D₅	—	2	1	no	—
216	C₅×#41	—	2	2	no	—
217	C₅×#41	—	2	2	no	—
218	C₂×C₅×Q₈	C₂³×C₃	1	1	yes	characteristic
219	C₂×C₅×D₄	—	2	1	no	—
220	C₂×C₅×D₄	—	2	2	no	—
221	C₂×C₅×D₄	—	2	2	no	—
222	C₂×C₅×D₄	—	2	1	no	—
223	C₂×C₅×D₄	C₂³×C₃	1	1	yes	characteristic
224	C₂×C₅×D₄	C₂³×C₃	1	2	yes	—
225	C₂²×C₄×C₅	—	2	1	no	—
226	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
227	C₂×C₂²⋊D₅	—	2	1	no	—
228	C₂×C₂²⋊D₅	—	2	4	no	—
229	C₂×C₂²⋊D₅	—	2	4	no	—
230	#709	—	2	2	no	—
231	#709	—	2	2	no	—
232	C₂×D₂₀	—	2	2	no	—
233	C₂×D₂₀	—	2	2	no	—
234	C₂×D₂₀	—	2	1	no	—
235	C₂×D₂₀	C₂³×C₃	1	2	yes	—
236	C₂×D₂₀	C₂³×C₃	1	1	yes	characteristic
237	C₂×C₄×D₅	—	2	1	no	—
238	C₂×Dic₁₀	C₂³×C₃	1	1	yes	characteristic
239	C₅×SD₁₆	—	2	4	no	—
240	C₅×D₈	—	2	4	no	—
241	C₅×M₁₆	C₃×D₄	1	4	yes	complemented
242	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
243	C₂²⋊Dic₅	—	2	4	no	—
244	C₂²⋊Dic₅	C₃×D₄	1	2	yes	—
245	#689	—	2	2	no	—
246	#688	—	2	2	no	—
247	#687	—	2	2	no	—
248	#686	—	2	2	no	—
249	#686	—	2	4	no	—
250	#686	—	2	4	no	—
251	#685	—	2	4	no	—
252	C₄⋊Dic₅	C₃×D₄	1	2	yes	—
253	C₄×Dic₅	C₂³×C₃	1	2	yes	—
254	C₂×#168	C₃×D₄	1	4	yes	complemented
255	C₂×#168	C₂³×C₃	1	1	yes	characteristic
256	C₂×#168	C₂³×C₃	1	1	yes	characteristic
257	D₄₀	—	2	4	no	—
258	#677	—	2	4	no	—
259	C₂×AGL(1,ℤ/8)	—	5	1	no	—
260	C₂×AGL(1,ℤ/8)	C₃×D₅	1	1	yes	characteristic, complemented
261	C₂×#135	—	5	1	no	—
262	C₂²×D₈	—	5	1	no	—
263	#535	—	5	1	no	—
264	#535	—	5	1	no	—
265	#496	—	5	2	no	—
266	#496	—	5	2	no	—
267	#495	—	5	2	no	—
268	#493	—	5	2	no	—
269	#405	—	5	1	no	—
270	C₂²×C₃×C₅	—	2	2	no	—
271	C₂²×C₃×C₅	—	4	1	no	—
272	C₂²×C₃×C₅	—	4	2	no	—
273	C₂²×C₃×C₅	—	4	2	no	—
274	C₂²×C₃×C₅	—	4	4	no	—
275	C₂²×C₃×C₅	—	8	4	no	—
276	C₂²×C₃×C₅	#127	1	2	yes	—
277	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
278	C₂×C₃×D₅	—	4	1	no	—
279	C₂×C₃×D₅	—	4	2	no	—
280	C₂×C₃×D₅	—	4	2	no	—
281	C₂×C₃×D₅	—	4	4	no	—
282	C₂×C₃×D₅	—	8	4	no	—
283	C₃×C₄×C₅	—	4	1	no	—
284	C₃×C₄×C₅	#127	1	2	yes	—
285	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
286	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
287	C₃×Dic₅	—	2	4	no	—
288	C₃×Dic₅	—	4	1	no	—
289	C₂⁴×C₃	—	2	1	no	—
290	C₂⁴×C₃	—	10	1	no	—
291	C₃×#41	—	2	2	no	—
292	C₃×#41	—	2	2	no	—
293	C₃×#41	—	10	2	no	—
294	C₃×#41	—	10	2	no	—
295	C₂×C₃×Q₈	—	5	1	no	—
296	C₂×C₃×Q₈	C₂²×D₅	1	1	yes	characteristic
297	C₂×C₃×D₄	—	2	1	no	—
298	C₂×C₃×D₄	—	2	2	no	—
299	C₂×C₃×D₄	—	2	1	no	—
300	C₂×C₃×D₄	—	2	2	no	—
301	C₂×C₃×D₄	—	5	2	no	—
302	C₂×C₃×D₄	—	5	1	no	—
303	C₂×C₃×D₄	—	10	4	no	—
304	C₂×C₃×D₄	—	10	4	no	—
305	C₂×C₃×D₄	—	10	1	no	—
306	C₂×C₃×D₄	—	10	2	no	—
307	C₂×C₃×D₄	—	10	1	no	—
308	C₂×C₃×D₄	—	10	2	no	—
309	C₂×C₃×D₄	C₂²×D₅	1	2	yes	—
310	C₂×C₃×D₄	C₂²×D₅	1	1	yes	characteristic
311	C₂²×C₃×C₄	—	2	1	no	—
312	C₂²×C₃×C₄	—	10	1	no	—
313	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
314	C₃×Q₁₆	—	10	2	no	—
315	C₃×SD₁₆	—	2	4	no	—
316	C₃×SD₁₆	—	10	2	no	—
317	C₃×SD₁₆	—	10	2	no	—
318	C₃×SD₁₆	—	10	4	no	—
319	C₃×D₈	—	2	4	no	—
320	C₃×D₈	—	10	4	no	—
321	C₃×D₈	—	10	4	no	—
322	C₃×D₈	—	10	2	no	—
323	C₃×D₈	—	10	4	no	—
324	C₃×M₁₆	C₂²⋊D₅	1	4	yes	complemented
325	C₂×C₃×C₈	—	5	4	no	—
326	C₂×C₃×C₈	—	5	1	no	—
327	C₂×C₃×C₈	—	5	1	no	—
328	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
329	C₃×C₄⋊C₄	—	5	2	no	—
330	C₃×C₂²⋊C₄	—	5	2	no	—
331	C₃×C₂²⋊C₄	—	10	4	no	—
332	C₃×C₂²⋊C₄	—	10	4	no	—
333	C₃×C₄²	—	5	2	no	—
334	C₂³×C₅	—	2	2	no	—
335	C₂³×C₅	—	2	1	no	—
336	C₂³×C₅	—	4	2	no	—
337	C₂³×C₅	—	8	2	no	—
338	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
339	C₂²×D₅	—	2	2	no	—
340	C₂²×D₅	—	2	1	no	—
341	C₂²×D₅	—	4	2	no	—
342	C₂²×D₅	—	8	2	no	—
343	C₅×Q₈	—	2	1	no	—
344	C₅×Q₈	—	2	1	no	—
345	C₅×D₄	—	2	2	no	—
346	C₅×D₄	—	2	1	no	—
347	C₅×D₄	—	2	1	no	—
348	C₅×D₄	—	2	2	no	—
349	C₅×D₄	—	4	2	no	—
350	C₅×D₄	—	4	2	no	—
351	C₂×C₄×C₅	—	2	1	no	—
352	C₂×C₄×C₅	—	4	2	no	—
353	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
354	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
355	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
356	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
357	C₂²⋊D₅	—	4	2	no	—
358	C₂²⋊D₅	—	8	4	no	—
359	C₂²⋊D₅	—	8	4	no	—
360	C₂×Dic₅	—	2	1	no	—
361	C₂×Dic₅	C₂×C₃×D₄	1	4	yes	—
362	D₂₀	—	2	2	no	—
363	D₂₀	—	2	1	no	—
364	D₂₀	—	2	2	no	—
365	D₂₀	—	2	1	no	—
366	D₂₀	—	4	2	no	—
367	C₄×D₅	—	4	2	no	—
368	Dic₁₀	—	2	1	no	—
369	Dic₁₀	—	2	1	no	—
370	C₅×C₈	C₂×C₃×D₄	1	4	yes	—
371	#168	C₂×C₃×D₄	1	2	yes	—
372	#168	C₂×C₃×D₄	1	2	yes	—
373	C₂×#41	—	5	1	no	—
374	C₂×#41	C₂×C₃×D₅	1	1	yes	characteristic
375	C₂²×D₄	—	5	1	no	—
376	C₂²×D₄	C₂×C₃×D₅	1	1	yes	characteristic
377	AGL(1,ℤ/8)	—	2	4	no	—
378	AGL(1,ℤ/8)	—	10	4	no	—
379	#135	—	10	4	no	—
380	C₂×Q₁₆	—	5	1	no	—
381	C₂×SD₁₆	—	5	1	no	—
382	C₂×SD₁₆	—	5	1	no	—
383	C₂×SD₁₆	—	5	2	no	—
384	C₂×SD₁₆	C₂×C₃×D₅	1	2	yes	complemented
385	C₂×D₈	—	5	2	no	—
386	C₂×D₈	—	5	2	no	—
387	C₂×D₈	—	5	1	no	—
388	C₂×D₈	—	5	2	no	—
389	C₂×D₈	—	10	4	no	—
390	C₂×D₈	C₂×C₃×D₅	1	2	yes	complemented
391	C₂×M₁₆	C₂×C₃×D₅	1	1	yes	characteristic
392	C₂²×C₈	—	5	1	no	—
393	#127	—	5	2	no	—
394	#127	—	5	2	no	—
395	#124	—	5	2	no	—
396	#124	—	5	2	no	—
397	#121	—	10	2	no	—
398	#121	—	10	2	no	—
399	#120	—	10	2	no	—
400	#120	—	10	2	no	—
401	#117	—	5	1	no	—
402	#97	—	5	2	no	—
403	C₄×C₈	—	5	2	no	—
404	C₂×C₃×C₅	—	2	2	no	—
405	C₂×C₃×C₅	—	8	1	no	—
406	C₂×C₃×C₅	—	8	2	no	—
407	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
408	C₂×C₃×C₅	#576	1	2	yes	—
409	C₃×D₅	—	8	1	no	—
410	C₃×D₅	—	8	2	no	—
411	C₂³×C₃	—	2	2	no	—
412	C₂³×C₃	—	2	1	no	—
413	C₂³×C₃	—	4	2	no	—
414	C₂³×C₃	—	8	2	no	—
415	C₂³×C₃	—	10	2	no	—
416	C₂³×C₃	—	10	1	no	—
417	C₂³×C₃	—	20	2	no	—
418	C₂³×C₃	—	40	2	no	—
419	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
420	C₃×Q₈	—	2	1	no	—
421	C₃×Q₈	—	2	1	no	—
422	C₃×Q₈	—	10	1	no	—
423	C₃×Q₈	—	10	1	no	—
424	C₃×D₄	—	2	2	no	—
425	C₃×D₄	—	2	1	no	—
426	C₃×D₄	—	2	1	no	—
427	C₃×D₄	—	2	2	no	—
428	C₃×D₄	—	4	2	no	—
429	C₃×D₄	—	4	2	no	—
430	C₃×D₄	—	10	2	no	—
431	C₃×D₄	—	10	1	no	—
432	C₃×D₄	—	10	1	no	—
433	C₃×D₄	—	10	2	no	—
434	C₃×D₄	—	20	2	no	—
435	C₃×D₄	—	20	2	no	—
436	C₃×D₄	—	40	4	no	—
437	C₃×D₄	—	40	4	no	—
438	C₂×C₃×C₄	—	2	1	no	—
439	C₂×C₃×C₄	—	4	2	no	—
440	C₂×C₃×C₄	—	5	4	no	—
441	C₂×C₃×C₄	—	10	1	no	—
442	C₂×C₃×C₄	—	20	2	no	—
443	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
444	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
445	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
446	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
447	C₃×C₈	—	5	2	no	—
448	C₃×C₈	—	5	2	no	—
449	C₃×C₈	C₂×C₂²⋊D₅	1	4	yes	—
450	C₂²×C₅	—	2	2	no	—
451	C₂²×C₅	—	4	2	no	—
452	C₂²×C₅	—	4	4	no	—
453	C₂²×C₅	—	4	1	no	—
454	C₂²×C₅	—	4	2	no	—
455	C₂²×C₅	—	8	4	no	—
456	C₂²×C₅	C₃×#127	1	2	yes	—
457	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
458	C₂×D₅	—	4	2	no	—
459	C₂×D₅	—	4	4	no	—
460	C₂×D₅	—	4	1	no	—
461	C₂×D₅	—	4	2	no	—
462	C₂×D₅	—	8	4	no	—
463	C₄×C₅	—	4	1	no	—
464	C₄×C₅	C₃×#127	1	2	yes	—
465	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
466	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
467	Dic₅	—	2	4	no	—
468	Dic₅	—	4	1	no	—
469	C₂⁴	—	2	1	no	—
470	C₂⁴	—	10	1	no	—
471	#41	—	2	2	no	—
472	#41	—	2	2	no	—
473	#41	—	10	2	no	—
474	#41	—	10	2	no	—
475	C₂×Q₈	—	5	1	no	—
476	C₂×Q₈	C₂²×C₃×D₅	1	1	yes	characteristic
477	C₂×D₄	—	2	1	no	—
478	C₂×D₄	—	2	2	no	—
479	C₂×D₄	—	2	1	no	—
480	C₂×D₄	—	2	2	no	—
481	C₂×D₄	—	5	2	no	—
482	C₂×D₄	—	5	1	no	—
483	C₂×D₄	—	10	4	no	—
484	C₂×D₄	—	10	4	no	—
485	C₂×D₄	—	10	1	no	—
486	C₂×D₄	—	10	2	no	—
487	C₂×D₄	—	10	1	no	—
488	C₂×D₄	—	10	2	no	—
489	C₂×D₄	C₂²×C₃×D₅	1	2	yes	—
490	C₂×D₄	C₂²×C₃×D₅	1	1	yes	characteristic
491	C₂²×C₄	—	2	1	no	—
492	C₂²×C₄	—	10	1	no	—
493	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
494	Q₁₆	—	10	2	no	—
495	SD₁₆	—	2	4	no	—
496	SD₁₆	—	10	2	no	—
497	SD₁₆	—	10	2	no	—
498	SD₁₆	—	10	4	no	—
499	D₈	—	2	4	no	—
500	D₈	—	10	4	no	—
501	D₈	—	10	4	no	—
502	D₈	—	10	2	no	—
503	D₈	—	10	4	no	—
504	M₁₆	C₃×C₂²⋊D₅	1	4	yes	complemented
505	C₂×C₈	—	5	4	no	—
506	C₂×C₈	—	5	1	no	—
507	C₂×C₈	—	5	1	no	—
508	C₂×C₈	C₂²×C₃×D₅	1	2	yes	—
509	C₄⋊C₄	—	5	2	no	—
510	C₂²⋊C₄	—	5	2	no	—
511	C₂²⋊C₄	—	10	4	no	—
512	C₂²⋊C₄	—	10	4	no	—
513	C₄²	—	5	2	no	—
514	C₃×C₅	#3155	1	1	yes	characteristic, complemented, Hall
515	C₂²×C₃	—	2	2	no	—
516	C₂²×C₃	—	4	2	no	—
517	C₂²×C₃	—	4	4	no	—
518	C₂²×C₃	—	4	1	no	—
519	C₂²×C₃	—	4	2	no	—
520	C₂²×C₃	—	8	4	no	—
521	C₂²×C₃	—	20	2	no	—
522	C₂²×C₃	—	20	4	no	—
523	C₂²×C₃	—	20	1	no	—
524	C₂²×C₃	—	20	2	no	—
525	C₂²×C₃	—	40	4	no	—
526	C₂²×C₃	#4086	1	2	yes	—
527	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
528	C₃×C₄	—	4	1	no	—
529	C₃×C₄	—	10	4	no	—
530	C₃×C₄	—	20	1	no	—
531	C₃×C₄	#4086	1	2	yes	—
532	C₃×C₄	C₂×D₄×D₅	1	1	yes	characteristic
533	C₃×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
534	C₂×C₅	—	2	2	no	—
535	C₂×C₅	—	8	2	no	—
536	C₂×C₅	—	8	1	no	—
537	C₂×C₅	C₂×C₃×#127	1	1	yes	characteristic
538	C₂×C₅	C₃×#576	1	2	yes	—
539	D₅	—	8	2	no	—
540	D₅	—	8	1	no	—
541	C₂³	—	2	2	no	—
542	C₂³	—	2	1	no	—
543	C₂³	—	4	2	no	—
544	C₂³	—	8	2	no	—
545	C₂³	—	10	2	no	—
546	C₂³	—	10	1	no	—
547	C₂³	—	20	2	no	—
548	C₂³	—	40	2	no	—
549	C₂³	C₃×D₄×D₅	1	1	yes	characteristic
550	Q₈	—	2	1	no	—
551	Q₈	—	2	1	no	—
552	Q₈	—	10	1	no	—
553	Q₈	—	10	1	no	—
554	D₄	—	2	2	no	—
555	D₄	—	2	1	no	—
556	D₄	—	2	1	no	—
557	D₄	—	2	2	no	—
558	D₄	—	4	2	no	—
559	D₄	—	4	2	no	—
560	D₄	—	10	2	no	—
561	D₄	—	10	1	no	—
562	D₄	—	10	1	no	—
563	D₄	—	10	2	no	—
564	D₄	—	20	2	no	—
565	D₄	—	20	2	no	—
566	D₄	—	40	4	no	—
567	D₄	—	40	4	no	—
568	C₂×C₄	—	2	1	no	—
569	C₂×C₄	—	4	2	no	—
570	C₂×C₄	—	5	4	no	—
571	C₂×C₄	—	10	1	no	—
572	C₂×C₄	—	20	2	no	—
573	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
574	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
575	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	2	yes	—
576	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
577	C₈	—	5	2	no	—
578	C₈	—	5	2	no	—
579	C₈	C₂×C₃×C₂²⋊D₅	1	4	yes	—
580	C₂×C₃	—	2	2	no	—
581	C₂×C₃	—	8	2	no	—
582	C₂×C₃	—	8	1	no	—
583	C₂×C₃	—	40	2	no	—
584	C₂×C₃	—	40	1	no	—
585	C₂×C₃	#66108	1	2	yes	—
586	C₂×C₃	C₂×#4086	1	1	yes	characteristic
587	C₅	C₃×#3155	1	1	yes	characteristic, complemented, Sylow
588	C₂²	—	2	2	no	—
589	C₂²	—	4	2	no	—
590	C₂²	—	4	4	no	—
591	C₂²	—	4	1	no	—
592	C₂²	—	4	2	no	—
593	C₂²	—	8	4	no	—
594	C₂²	—	20	2	no	—
595	C₂²	—	20	4	no	—
596	C₂²	—	20	1	no	—
597	C₂²	—	20	2	no	—
598	C₂²	—	40	4	no	—
599	C₂²	C₃×#4086	1	2	yes	—
600	C₂²	C₂×C₃×D₄×D₅	1	1	yes	characteristic
601	C₄	—	4	1	no	—
602	C₄	—	10	4	no	—
603	C₄	—	20	1	no	—
604	C₄	C₃×#4086	1	2	yes	—
605	C₄	C₂×C₃×D₄×D₅	1	1	yes	characteristic
606	C₄	C₂²×C₃×C₂²⋊D₅	1	1	yes	characteristic
607	C₃	#10615671	1	1	yes	characteristic, direct product factor, Sylow
608	C₂	—	2	2	no	—
609	C₂	—	8	2	no	—
610	C₂	—	8	1	no	—
611	C₂	—	40	2	no	—
612	C₂	—	40	1	no	—
613	C₂	C₃×#66108	1	2	yes	—
614	C₂	C₂×C₃×#4086	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	C₂×C₃×C₅×AGL(1,ℤ/8)	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
hypercenter	C₂×C₃×AGL(1,ℤ/8)	D₅
hypocenter	C₅	C₃×#3155
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×AGL(1,ℤ/8)	C₂	semidirect
2. C₃×#66164	C₂	semidirect
3. C₂×C₃×#4096	C₂	semidirect
4. C₃×#66139	C₂	semidirect
5. C₂²×C₃×#686	C₂	semidirect
6. C₂×C₃×#4054	C₂	semidirect
7. C₃×#65465	C₂	semidirect
8. C₃×#65457	C₂	semidirect
9. C₃×#65443	C₂	semidirect
10. C₃×#65440	C₂	semidirect
11. C₃×#65411	C₂	semidirect
12. #10615671	C₃	direct (N × H)
18. C₂×C₃×C₅×SD₁₆	C₂²	semidirect
19. C₂×C₃×C₅×D₈	C₂²	semidirect
23. C₃×#4093	C₂²	semidirect
24. C₂×C₃×#689	C₂²	semidirect
25. C₂×C₃×#688	C₂²	semidirect
26. C₃×#4086	C₂²	semidirect
27. C₃×#4086	C₂²	semidirect
30. C₃×#4082	C₂²	semidirect
31. C₂×C₃×#687	C₂²	semidirect
35. C₂×C₃×#686	C₂²	semidirect
40. C₂×C₃×D₄₀	C₂²	semidirect
41. C₂×C₃×#677	C₂²	semidirect
42. C₃×#3947	C₂²	semidirect
43. C₃×C₈×Dic₅	C₂²	semidirect
45. C₂×C₅×AGL(1,ℤ/8)	C₂×C₃	semidirect
46. #66164	C₂×C₃	semidirect
47. C₂×#4096	C₂×C₃	semidirect
48. #66139	C₂×C₃	semidirect
49. C₂²×#686	C₂×C₃	semidirect
50. C₂×#4054	C₂×C₃	semidirect
51. #65465	C₂×C₃	semidirect
52. #65457	C₂×C₃	semidirect
53. #65443	C₂×C₃	semidirect
54. #65440	C₂×C₃	semidirect
55. #65411	C₂×C₃	semidirect
83. C₃×C₅×M₁₆	D₄	semidirect
96. C₂×C₃×#168	D₄	semidirect
102. C₂×C₃×AGL(1,ℤ/8)	D₅	semidirect
117. C₂×C₅×SD₁₆	C₂²×C₃	semidirect
118. C₂×C₅×D₈	C₂²×C₃	semidirect
122. #4093	C₂²×C₃	semidirect
123. C₂×#689	C₂²×C₃	semidirect
124. C₂×#688	C₂²×C₃	semidirect
125. #4086	C₂²×C₃	semidirect
126. #4086	C₂²×C₃	semidirect
129. #4082	C₂²×C₃	semidirect
130. C₂×#687	C₂²×C₃	semidirect
134. C₂×#686	C₂²×C₃	semidirect
139. C₂×D₄₀	C₂²×C₃	semidirect
140. C₂×#677	C₂²×C₃	semidirect
141. #3947	C₂²×C₃	semidirect
142. C₈×Dic₅	C₂²×C₃	semidirect
194. C₂×C₃×SD₁₆	C₂×D₅	semidirect
200. C₂×C₃×D₈	C₂×D₅	semidirect
241. C₅×M₁₆	C₃×D₄	semidirect
254. C₂×#168	C₃×D₄	semidirect
260. C₂×AGL(1,ℤ/8)	C₃×D₅	semidirect
324. C₃×M₁₆	C₂²⋊D₅	semidirect
384. C₂×SD₁₆	C₂×C₃×D₅	semidirect
390. C₂×D₈	C₂×C₃×D₅	semidirect
504. M₁₆	C₃×C₂²⋊D₅	semidirect
514. C₃×C₅	#3155	semidirect
587. C₅	C₃×#3155	semidirect
607. C₃	#10615671	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×C₅×AGL(1,ℤ/8)	2
C₃×#66164	2
C₂×C₃×#4096	2
C₃×#66139	2
C₂²×C₃×#686	2
C₂×C₃×#4054	2
C₃×#65465	2
C₃×#65457	2
C₃×#65443	2
C₃×#65440	2
C₃×#65411	2
#10615671	3
C₃×#3155	5

Subgroups — C3×#10615671 #14439547

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10615671 #14439547