Subgroups — C₅×C₈×#239 #14418893

Order 1920.

912 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#239	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×C₈×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×Q₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4834	C₂	1	3	yes	maximal, complemented
5	C₅×#4833	C₂	1	3	yes	maximal, complemented
6	C₅×C₈×D₁₂	C₂	1	3	yes	maximal, complemented
7	C₄×C₅×D₃×C₈	C₂	1	3	yes	maximal, complemented
8	C₅×C₈×#41	—	3	1	no	maximal, Hall
9	C₂×C₅×#239	C₄	1	1	yes	characteristic
10	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₅×Q₈×Dic₃	C₄	1	1	yes	characteristic
12	C₂×C₅×D₃×C₈	C₂²	1	3	yes	—
13	C₂×C₅×D₃×C₈	C₂²	1	3	yes	—
14	C₅×#893	C₄	1	3	yes	—
15	C₅×#892	C₄	1	3	yes	—
16	C₄×C₅×D₁₂	C₂²	1	3	yes	—
17	C₄²×C₅×D₃	C₂²	1	3	yes	—
18	C₃×C₅×C₄⋊C₈	C₂²	1	3	yes	—
19	C₃×C₄×C₅×C₈	C₂²	1	3	yes	—
20	C₅×(C₂×D₃)⋊C₈	C₂²	1	6	yes	—
21	C₅×C₈×Dic₃	C₂²	1	1	yes	characteristic
22	C₅×C₈×Dic₃	C₂²	1	3	yes	—
23	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	3	yes	—
24	C₄×C₅×C₃⋊C₈	C₂²	1	3	yes	—
25	C₈×#239	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
26	C₄×C₅×#41	—	3	1	no	—
27	C₅×C₈×Q₈	—	3	1	no	—
28	C₅×C₈×Q₈	D₃	1	1	yes	characteristic, complemented
29	C₅×C₈×D₄	—	3	3	no	—
30	C₅×C₈×D₄	—	3	3	no	—
31	C₅×#431	—	3	3	no	—
32	C₂×C₄×C₅×C₈	—	3	3	no	—
33	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
34	C₅×#239	C₈	1	8	yes	direct product factor
35	C₂×C₅×D₁₂	C₂×C₄	1	3	yes	—
36	C₂×C₄×C₅×D₃	C₂×C₄	1	3	yes	—
37	C₂×C₄×C₅×D₃	C₂×C₄	1	3	yes	—
38	C₂×C₄×C₅×D₃	C₂³	1	3	yes	—
39	C₂×C₃×C₅×C₈	C₂³	1	3	yes	—
40	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₃×C₅×C₄⋊C₄	C₂×C₄	1	3	yes	—
42	C₃×C₄²×C₅	C₂³	1	3	yes	—
43	C₅×(C₂×D₃)⋊C₄	C₂×C₄	1	6	yes	—
44	C₅×C₄⋊Dic₃	C₂×C₄	1	3	yes	—
45	C₄×C₅×Dic₃	C₂×C₄	1	3	yes	—
46	C₄×C₅×Dic₃	C₂×C₄	1	3	yes	—
47	C₄×C₅×Dic₃	C₂³	1	1	yes	characteristic
48	C₂×C₅×C₃⋊C₈	C₂³	1	3	yes	—
49	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
50	C₅×D₃×C₈	—	2	6	no	—
51	C₅×D₃×C₈	—	2	6	no	—
52	C₄×#239	C₂×C₅	1	1	yes	characteristic
53	C₃×C₈×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
54	Q₈×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
55	#4834	C₂×C₅	1	3	yes	complemented
56	#4833	C₂×C₅	1	3	yes	complemented
57	C₈×D₁₂	C₂×C₅	1	3	yes	complemented
58	C₄×D₃×C₈	C₂×C₅	1	3	yes	complemented
59	C₂×C₅×#41	—	3	1	no	—
60	C₂²×C₅×C₈	—	3	3	no	—
61	C₂²×C₅×C₈	—	3	3	no	—
62	C₄×C₅×Q₈	—	3	1	no	—
63	C₄×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
64	C₄×C₅×D₄	—	3	3	no	—
65	C₄×C₅×D₄	—	3	3	no	—
66	C₅×#117	—	3	3	no	—
67	C₂×C₄²×C₅	—	3	3	no	—
68	C₅×C₄⋊C₈	—	3	3	no	—
69	C₅×C₄⋊C₈	C₂×D₃	1	3	yes	—
70	C₅×C₂²⋊C₈	—	3	6	no	—
71	C₄×C₅×C₈	—	3	3	no	—
72	C₄×C₅×C₈	—	3	1	no	—
73	C₄×C₅×C₈	—	3	3	no	—
74	C₄×C₅×C₈	C₂×D₃	1	3	yes	—
75	C₈×#41	—	3	1	no	Sylow
76	C₂²×C₅×D₃	C₂²×C₄	1	3	yes	—
77	C₃×C₅×Q₈	C₂×C₈	1	4	yes	complemented
78	C₂×C₃×C₄×C₅	C₂²×C₄	1	3	yes	—
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	3	yes	—
80	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
81	C₂×C₅×Dic₃	C₂²×C₄	1	3	yes	—
82	C₂×C₅×Dic₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₅×Dic₃	C₂²×C₄	1	1	yes	characteristic
84	C₅×D₁₂	C₂×C₈	1	12	yes	complemented
85	C₄×C₅×D₃	—	2	3	no	—
86	C₄×C₅×D₃	—	2	3	no	—
87	C₄×C₅×D₃	—	2	6	no	—
88	C₄×C₅×D₃	C₂×C₈	1	12	yes	complemented
89	C₃×C₅×C₈	—	2	3	no	—
90	C₃×C₅×C₈	#41	1	2	yes	complemented
91	C₅×C₃⋊C₈	—	2	3	no	—
92	C₅×C₃⋊C₈	#41	1	2	yes	complemented
93	C₂×#239	C₄×C₅	1	1	yes	characteristic
94	C₃×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
95	Q₈×Dic₃	C₄×C₅	1	1	yes	characteristic
96	C₂×D₃×C₈	C₂²×C₅	1	3	yes	—
97	C₂×D₃×C₈	C₂²×C₅	1	3	yes	—
98	#893	C₄×C₅	1	3	yes	—
99	#892	C₄×C₅	1	3	yes	—
100	C₄×D₁₂	C₂²×C₅	1	3	yes	—
101	C₄²×D₃	C₂²×C₅	1	3	yes	—
102	C₃×C₄⋊C₈	C₂²×C₅	1	3	yes	—
103	C₃×C₄×C₈	C₂²×C₅	1	3	yes	—
104	(C₂×D₃)⋊C₈	C₂²×C₅	1	6	yes	—
105	C₈×Dic₃	C₂²×C₅	1	1	yes	characteristic
106	C₈×Dic₃	C₂²×C₅	1	3	yes	—
107	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	3	yes	—
108	C₄×C₃⋊C₈	C₂²×C₅	1	3	yes	—
109	C₅×#41	—	3	8	no	—
110	C₂×C₅×Q₈	C₄×D₃	1	1	yes	characteristic
111	C₂×C₅×D₄	—	3	3	no	—
112	C₂²×C₄×C₅	—	3	3	no	—
113	C₂²×C₄×C₅	—	3	3	no	—
114	C₂²×C₄×C₅	—	3	3	no	—
115	C₂×C₅×C₈	—	3	1	no	—
116	C₂×C₅×C₈	—	3	3	no	—
117	C₂×C₅×C₈	—	6	6	no	—
118	C₂×C₅×C₈	—	6	6	no	—
119	C₂×C₅×C₈	C₂²×D₃	1	3	yes	—
120	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
121	C₅×C₄⋊C₄	—	3	3	no	—
122	C₅×C₄⋊C₄	C₄×D₃	1	3	yes	—
123	C₅×C₂²⋊C₄	—	3	6	no	—
124	C₄²×C₅	—	3	3	no	—
125	C₄²×C₅	—	3	3	no	—
126	C₄²×C₅	—	3	1	no	—
127	C₄²×C₅	C₂²×D₃	1	3	yes	—
128	C₄×#41	—	3	1	no	—
129	C₈×Q₈	—	3	1	no	—
130	C₈×Q₈	C₅×D₃	1	1	yes	characteristic, complemented
131	C₈×D₄	—	3	3	no	—
132	C₈×D₄	—	3	3	no	—
133	#431	—	3	3	no	—
134	C₂×C₄×C₈	—	3	3	no	—
135	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₅×D₃	—	2	3	no	—
137	C₂×C₅×D₃	—	2	3	no	—
138	C₂×C₅×D₃	C₂²×C₈	1	6	yes	—
139	C₃×C₄×C₅	—	2	3	no	—
140	C₃×C₄×C₅	C₂²×C₈	1	6	yes	—
141	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
142	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
143	C₅×Dic₃	—	2	3	no	—
144	C₅×Dic₃	C₂²×C₈	1	2	yes	—
145	C₅×Dic₃	#131	1	2	yes	—
146	C₂×C₃×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
147	#239	C₅×C₈	1	8	yes	direct product factor
148	C₂×D₁₂	C₂×C₄×C₅	1	3	yes	—
149	C₂×C₄×D₃	C₂×C₄×C₅	1	3	yes	—
150	C₂×C₄×D₃	C₂×C₄×C₅	1	3	yes	—
151	C₂×C₄×D₃	C₂³×C₅	1	3	yes	—
152	C₂×C₃×C₈	C₂³×C₅	1	3	yes	—
153	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
154	C₃×C₄⋊C₄	C₂×C₄×C₅	1	3	yes	—
155	C₃×C₄²	C₂³×C₅	1	3	yes	—
156	(C₂×D₃)⋊C₄	C₂×C₄×C₅	1	6	yes	—
157	C₄⋊Dic₃	C₂×C₄×C₅	1	3	yes	—
158	C₄×Dic₃	C₂×C₄×C₅	1	3	yes	—
159	C₄×Dic₃	C₂×C₄×C₅	1	3	yes	—
160	C₄×Dic₃	C₂³×C₅	1	1	yes	characteristic
161	C₂×C₃⋊C₈	C₂³×C₅	1	3	yes	—
162	C₂×C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
163	D₃×C₈	—	2	6	no	—
164	D₃×C₈	—	2	6	no	—
165	C₂³×C₅	—	3	3	no	—
166	C₅×Q₈	D₃×C₈	1	4	yes	complemented
167	C₅×D₄	—	3	12	no	—
168	C₂×C₄×C₅	—	3	1	no	—
169	C₂×C₄×C₅	—	3	1	no	—
170	C₂×C₄×C₅	—	3	12	no	—
171	C₂×C₄×C₅	—	3	3	no	—
172	C₂×C₄×C₅	—	6	6	no	—
173	C₂×C₄×C₅	—	6	3	no	—
174	C₂×C₄×C₅	—	6	3	no	—
175	C₂×C₄×C₅	C₂×C₄×D₃	1	3	yes	—
176	C₂×C₄×C₅	C₂×C₄×D₃	1	3	yes	—
177	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
178	C₅×C₈	—	2	3	no	—
179	C₅×C₈	—	3	2	no	—
180	C₅×C₈	—	6	3	no	—
181	C₅×C₈	#239	1	2	yes	direct product factor
182	C₂×#41	—	3	1	no	—
183	C₂²×C₈	—	3	3	no	—
184	C₂²×C₈	—	3	3	no	—
185	C₄×Q₈	—	3	1	no	—
186	C₄×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
187	C₄×D₄	—	3	3	no	—
188	C₄×D₄	—	3	3	no	—
189	#117	—	3	3	no	—
190	C₂×C₄²	—	3	3	no	—
191	C₄⋊C₈	—	3	3	no	—
192	C₄⋊C₈	C₂×C₅×D₃	1	3	yes	—
193	C₂²⋊C₈	—	3	6	no	—
194	C₄×C₈	—	3	1	no	—
195	C₄×C₈	—	3	3	no	—
196	C₄×C₈	—	3	3	no	—
197	C₄×C₈	C₂×C₅×D₃	1	3	yes	—
198	C₂×C₃×C₅	C₄×#41	1	1	yes	characteristic
199	C₂×C₃×C₅	C₂³×C₈	1	1	yes	characteristic
200	C₂×C₃×C₅	C₂×#131	1	1	yes	characteristic
201	C₅×D₃	—	2	6	no	—
202	C₂²×D₃	C₂²×C₄×C₅	1	3	yes	—
203	C₃×Q₈	C₂×C₅×C₈	1	4	yes	complemented
204	C₂×C₃×C₄	C₂²×C₄×C₅	1	3	yes	—
205	C₂×C₃×C₄	C₂²×C₄×C₅	1	3	yes	—
206	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
207	C₂×Dic₃	C₂²×C₄×C₅	1	3	yes	—
208	C₂×Dic₃	C₂²×C₄×C₅	1	1	yes	characteristic
209	C₂×Dic₃	C₂²×C₄×C₅	1	1	yes	characteristic
210	D₁₂	C₂×C₅×C₈	1	12	yes	complemented
211	C₄×D₃	—	2	3	no	—
212	C₄×D₃	—	2	3	no	—
213	C₄×D₃	—	2	6	no	—
214	C₄×D₃	C₂×C₅×C₈	1	12	yes	complemented
215	C₃×C₈	—	2	3	no	—
216	C₃×C₈	C₅×#41	1	2	yes	complemented
217	C₃⋊C₈	—	2	3	no	—
218	C₃⋊C₈	C₅×#41	1	2	yes	complemented
219	C₂²×C₅	—	3	6	no	—
220	C₂²×C₅	—	6	3	no	—
221	C₂²×C₅	—	6	3	no	—
222	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
223	C₄×C₅	—	2	3	no	—
224	C₄×C₅	—	3	2	no	—
225	C₄×C₅	—	3	2	no	—
226	C₄×C₅	—	6	3	no	—
227	C₄×C₅	C₂×D₃×C₈	1	6	yes	—
228	C₄×C₅	C₂×#239	1	1	yes	characteristic
229	C₄×C₅	D₃×#41	1	1	yes	characteristic
230	#41	—	3	8	no	—
231	C₂×Q₈	C₄×C₅×D₃	1	1	yes	characteristic
232	C₂×D₄	—	3	3	no	—
233	C₂²×C₄	—	3	3	no	—
234	C₂²×C₄	—	3	3	no	—
235	C₂²×C₄	—	3	3	no	—
236	C₂×C₈	—	3	1	no	—
237	C₂×C₈	—	3	3	no	—
238	C₂×C₈	—	6	6	no	—
239	C₂×C₈	—	6	6	no	—
240	C₂×C₈	C₂²×C₅×D₃	1	3	yes	—
241	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
242	C₄⋊C₄	—	3	3	no	—
243	C₄⋊C₄	C₄×C₅×D₃	1	3	yes	—
244	C₂²⋊C₄	—	3	6	no	—
245	C₄²	—	3	3	no	—
246	C₄²	—	3	3	no	—
247	C₄²	—	3	1	no	—
248	C₄²	C₂²×C₅×D₃	1	3	yes	—
249	C₃×C₅	C₈×#41	1	1	yes	characteristic, complemented, Hall
250	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
251	C₂×D₃	—	2	3	no	—
252	C₂×D₃	—	2	3	no	—
253	C₂×D₃	C₂²×C₅×C₈	1	6	yes	—
254	C₃×C₄	—	2	3	no	—
255	C₃×C₄	C₂²×C₅×C₈	1	6	yes	—
256	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
257	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
258	Dic₃	—	2	3	no	—
259	Dic₃	C₂²×C₅×C₈	1	2	yes	—
260	Dic₃	C₅×#131	1	2	yes	—
261	C₂×C₅	—	6	6	no	—
262	C₂×C₅	C₄×#239	1	1	yes	characteristic
263	C₂×C₅	C₂²×D₃×C₈	1	1	yes	characteristic
264	C₂×C₅	D₃×#131	1	1	yes	characteristic
265	C₂³	—	3	3	no	—
266	Q₈	C₅×D₃×C₈	1	4	yes	complemented
267	D₄	—	3	12	no	—
268	C₂×C₄	—	3	1	no	—
269	C₂×C₄	—	3	1	no	—
270	C₂×C₄	—	3	12	no	—
271	C₂×C₄	—	3	3	no	—
272	C₂×C₄	—	6	6	no	—
273	C₂×C₄	—	6	3	no	—
274	C₂×C₄	—	6	3	no	—
275	C₂×C₄	C₂×C₄×C₅×D₃	1	3	yes	—
276	C₂×C₄	C₂×C₄×C₅×D₃	1	3	yes	—
277	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
278	C₈	—	2	3	no	—
279	C₈	—	3	2	no	—
280	C₈	—	6	3	no	—
281	C₈	C₅×#239	1	2	yes	direct product factor
282	C₂×C₃	C₄×C₅×#41	1	1	yes	characteristic
283	C₂×C₃	C₂³×C₅×C₈	1	1	yes	characteristic
284	C₂×C₃	C₂×C₅×#131	1	1	yes	characteristic
285	D₃	—	2	6	no	—
286	C₅	C₈×#239	1	1	yes	characteristic, direct product factor, Sylow
287	C₂²	—	3	6	no	—
288	C₂²	—	6	3	no	—
289	C₂²	—	6	3	no	—
290	C₂²	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
291	C₄	—	2	3	no	—
292	C₄	—	3	2	no	—
293	C₄	—	3	2	no	—
294	C₄	—	6	3	no	—
295	C₄	C₂×C₅×D₃×C₈	1	6	yes	—
296	C₄	C₂×C₅×#239	1	1	yes	characteristic
297	C₄	C₅×D₃×#41	1	1	yes	characteristic
298	C₃	C₅×C₈×#41	1	1	yes	characteristic, complemented, Sylow
299	C₂	—	6	6	no	—
300	C₂	C₄×C₅×#239	1	1	yes	characteristic
301	C₂	C₂²×C₅×D₃×C₈	1	1	yes	characteristic
302	C₂	C₅×D₃×#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅×C₈	C₂²×D₃
commutator	C₂×C₃	C₂³×C₅×C₈
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×C₈×Q₈	C₂
socle	C₂²×C₃×C₅	C₂³×C₄
prime radical	C₂×C₅×#239	C₄
cosocle	C₂×C₃×C₄	C₂⁴×C₅
hypercenter	C₅×C₈×Q₈	D₃
hypocenter	C₃	C₅×C₈×#41
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅×C₈ ◁ C₅×C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×C₈×Q₈	C₂	semidirect
3. C₅×Q₈×C₃⋊C₈	C₂	semidirect
4. C₅×#4834	C₂	semidirect
5. C₅×#4833	C₂	semidirect
6. C₅×C₈×D₁₂	C₂	semidirect
7. C₄×C₅×D₃×C₈	C₂	semidirect
25. C₈×#239	C₅	direct (N × H)
28. C₅×C₈×Q₈	D₃	semidirect
34. C₅×#239	C₈	direct (N × H)
53. C₃×C₈×Q₈	C₂×C₅	semidirect
54. Q₈×C₃⋊C₈	C₂×C₅	semidirect
55. #4834	C₂×C₅	semidirect
56. #4833	C₂×C₅	semidirect
57. C₈×D₁₂	C₂×C₅	semidirect
58. C₄×D₃×C₈	C₂×C₅	semidirect
77. C₃×C₅×Q₈	C₂×C₈	semidirect
84. C₅×D₁₂	C₂×C₈	semidirect
88. C₄×C₅×D₃	C₂×C₈	semidirect
90. C₃×C₅×C₈	#41	semidirect
92. C₅×C₃⋊C₈	#41	semidirect
130. C₈×Q₈	C₅×D₃	semidirect
147. #239	C₅×C₈	direct (N × H)
166. C₅×Q₈	D₃×C₈	semidirect
181. C₅×C₈	#239	direct (N × H)
203. C₃×Q₈	C₂×C₅×C₈	semidirect
210. D₁₂	C₂×C₅×C₈	semidirect
214. C₄×D₃	C₂×C₅×C₈	semidirect
216. C₃×C₈	C₅×#41	semidirect
218. C₃⋊C₈	C₅×#41	semidirect
249. C₃×C₅	C₈×#41	semidirect
266. Q₈	C₅×D₃×C₈	semidirect
281. C₈	C₅×#239	direct (N × H)
286. C₅	C₈×#239	direct (N × H)
298. C₃	C₅×C₈×#41	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×C₅×#239	2
C₃×C₅×C₈×Q₈	2
C₅×Q₈×C₃⋊C₈	2
C₅×#4834	2
C₅×#4833	2
C₅×C₈×D₁₂	2
C₄×C₅×D₃×C₈	2
C₅×C₈×#41	3
C₈×#239	5

Subgroups — C5×C8×#239 #14418893

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₈×#239 #14418893