Subgroups — C₅×Dic₃×C₄⋊C₈ #14296097

Order 1920.

632 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₈×Dic₃	C₂	1	2	yes	maximal
3	C₄²×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	C₅×#4550	C₂	1	2	yes	maximal
6	C₄×C₅×C₄⋊C₈	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₅×Dic₃	C₄	1	2	yes	—
10	C₂×C₄×C₅×Dic₃	C₄	1	2	yes	—
11	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
13	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
14	C₃×C₅×C₄⋊C₈	C₄	1	4	yes	complemented
15	C₅×C₈×Dic₃	—	2	4	no	—
16	C₅×C₄⋊(C₃⋊C₈)	C₄	1	4	yes	complemented
17	Dic₃×C₄⋊C₈	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×C₄⋊C₈	—	3	1	no	—
19	C₂×C₅×C₄⋊C₈	D₃	1	1	yes	characteristic
20	C₂×C₄×C₅×C₈	—	3	2	no	—
21	C₄³×C₅	—	3	1	no	—
22	C₅×#336	—	3	2	no	—
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂²×C₅×Dic₃	C₂×C₄	1	2	yes	—
27	C₂²×C₅×Dic₃	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₅×C₈	—	2	2	no	—
29	C₂×C₃×C₅×C₈	—	2	2	no	—
30	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	—
31	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
32	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
33	C₄×C₅×Dic₃	—	2	1	no	—
34	C₄×C₅×Dic₃	—	2	1	no	—
35	C₄×C₅×Dic₃	—	2	4	no	—
36	C₄×C₅×Dic₃	C₈	1	8	yes	complemented
37	C₄×C₅×Dic₃	D₄	1	2	yes	—
38	C₄×C₅×Dic₃	Q₈	1	2	yes	—
39	C₂×C₅×C₃⋊C₈	—	2	2	no	—
40	C₂×C₅×C₃⋊C₈	—	2	2	no	—
41	C₂×C₅×C₃⋊C₈	C₂×C₄	1	4	yes	—
42	C₂×C₃×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
43	C₂×C₈×Dic₃	C₂×C₅	1	2	yes	—
44	C₄²×Dic₃	C₂×C₅	1	1	yes	characteristic
45	C₂×C₄⋊(C₃⋊C₈)	C₂×C₅	1	1	yes	characteristic
46	#4550	C₂×C₅	1	2	yes	—
47	C₂²×C₅×C₈	—	3	2	no	—
48	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
49	C₂×C₄²×C₅	—	3	2	no	—
50	C₂×C₄²×C₅	—	3	1	no	—
51	C₂×C₄²×C₅	—	3	2	no	—
52	C₂×C₄²×C₅	—	3	1	no	—
53	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
54	C₅×C₄⋊C₈	—	3	4	no	—
55	C₅×C₄⋊C₈	Dic₃	1	4	yes	direct product factor
56	C₄×C₅×C₈	—	6	4	no	—
57	C₄×C₄⋊C₈	—	3	1	no	Sylow
58	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	—	2	2	no	—
60	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
61	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
62	C₂×C₃×C₄×C₅	C₂×C₈	1	4	yes	—
63	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
69	C₂×C₅×Dic₃	—	2	2	no	—
70	C₂×C₅×Dic₃	—	2	2	no	—
71	C₂×C₅×Dic₃	C₄⋊C₄	1	4	yes	—
72	C₂×C₅×Dic₃	C₂×C₈	1	4	yes	—
73	C₂×C₅×Dic₃	C₂×C₈	1	4	yes	—
74	C₂×C₅×Dic₃	M₁₆	1	4	yes	—
75	C₃×C₅×C₈	—	2	4	no	—
76	C₅×C₃⋊C₈	—	2	4	no	—
77	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
78	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
79	C₂×C₄×Dic₃	C₄×C₅	1	2	yes	—
80	C₂×C₄×Dic₃	C₄×C₅	1	2	yes	—
81	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
82	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
83	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
84	C₃×C₄⋊C₈	C₄×C₅	1	4	yes	complemented
85	C₈×Dic₃	—	2	4	no	—
86	C₄⋊(C₃⋊C₈)	C₄×C₅	1	4	yes	complemented
87	C₂²×C₄×C₅	—	3	2	no	—
88	C₂²×C₄×C₅	—	3	2	no	—
89	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
90	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
91	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
92	C₂×C₅×C₈	—	2	2	no	—
93	C₂×C₅×C₈	—	2	2	no	—
94	C₂×C₅×C₈	—	3	4	no	—
95	C₂×C₅×C₈	—	6	2	no	—
96	C₂×C₅×C₈	—	6	2	no	—
97	C₂×C₅×C₈	C₂×Dic₃	1	4	yes	—
98	C₄²×C₅	—	3	8	no	—
99	C₄²×C₅	—	3	2	no	—
100	C₄²×C₅	—	3	2	no	—
101	C₄²×C₅	—	6	4	no	—
102	C₄²×C₅	—	6	1	no	—
103	C₄²×C₅	—	6	1	no	—
104	C₄²×C₅	C₄×D₃	1	2	yes	—
105	C₄²×C₅	C₂×Dic₃	1	2	yes	—
106	C₂×C₄⋊C₈	—	3	1	no	—
107	C₂×C₄⋊C₈	C₅×D₃	1	1	yes	characteristic
108	C₂×C₄×C₈	—	3	2	no	—
109	C₄³	—	3	1	no	—
110	#336	—	3	2	no	—
111	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
112	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₃×C₅	#117	1	1	yes	characteristic
114	C₂²×C₃×C₅	C₂²×C₈	1	2	yes	—
115	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
116	C₃×C₄×C₅	—	2	2	no	—
117	C₃×C₄×C₅	C₄×C₈	1	4	yes	complemented
118	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
119	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
120	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
121	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
122	C₅×Dic₃	—	2	4	no	—
123	C₅×Dic₃	C₄⋊C₈	1	8	yes	direct product factor
124	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
125	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
126	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
127	C₂²×Dic₃	C₂×C₄×C₅	1	2	yes	—
128	C₂²×Dic₃	C₂×C₄×C₅	1	2	yes	—
129	C₂×C₃×C₈	—	2	2	no	—
130	C₂×C₃×C₈	—	2	2	no	—
131	C₂×C₃×C₈	C₂×C₄×C₅	1	4	yes	—
132	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
133	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
134	C₄×Dic₃	—	2	1	no	—
135	C₄×Dic₃	—	2	1	no	—
136	C₄×Dic₃	—	2	4	no	—
137	C₄×Dic₃	C₅×C₈	1	8	yes	complemented
138	C₄×Dic₃	C₅×D₄	1	2	yes	—
139	C₄×Dic₃	C₅×Q₈	1	2	yes	—
140	C₂×C₃⋊C₈	—	2	2	no	—
141	C₂×C₃⋊C₈	—	2	2	no	—
142	C₂×C₃⋊C₈	C₂×C₄×C₅	1	4	yes	—
143	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
144	C₂×C₄×C₅	—	2	2	no	—
145	C₂×C₄×C₅	—	3	4	no	—
146	C₂×C₄×C₅	—	3	4	no	—
147	C₂×C₄×C₅	—	3	4	no	—
148	C₂×C₄×C₅	—	3	4	no	—
149	C₂×C₄×C₅	—	6	2	no	—
150	C₂×C₄×C₅	—	6	2	no	—
151	C₂×C₄×C₅	D₃×C₈	1	4	yes	—
152	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
153	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
154	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
155	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
156	C₂×C₄×C₅	#237	1	1	yes	characteristic
157	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
158	C₂×C₄×C₅	#239	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
160	C₅×C₈	—	2	4	no	—
161	C₅×C₈	—	6	4	no	—
162	C₂²×C₈	—	3	2	no	—
163	C₂²×C₈	C₂×C₅×D₃	1	2	yes	—
164	C₂×C₄²	—	3	2	no	—
165	C₂×C₄²	—	3	1	no	—
166	C₂×C₄²	—	3	2	no	—
167	C₂×C₄²	—	3	1	no	—
168	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
169	C₄⋊C₈	—	3	4	no	—
170	C₄⋊C₈	C₅×Dic₃	1	4	yes	direct product factor
171	C₄×C₈	—	6	4	no	—
172	C₂×C₃×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
174	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
175	C₂×C₃×C₅	C₂×C₄⋊C₈	1	2	yes	—
176	C₂×C₃×C₅	#431	1	2	yes	—
177	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	C₄²×C₅	1	2	yes	—
180	C₂×C₃×C₄	C₄²×C₅	1	2	yes	—
181	C₂×C₃×C₄	C₂×C₅×C₈	1	4	yes	—
182	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
186	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
187	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
188	C₂×Dic₃	—	2	2	no	—
189	C₂×Dic₃	—	2	2	no	—
190	C₂×Dic₃	C₅×C₄⋊C₄	1	4	yes	—
191	C₂×Dic₃	C₂×C₅×C₈	1	4	yes	—
192	C₂×Dic₃	C₂×C₅×C₈	1	4	yes	—
193	C₂×Dic₃	C₅×M₁₆	1	4	yes	—
194	C₃×C₈	—	2	4	no	—
195	C₃⋊C₈	—	2	4	no	—
196	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
197	C₂²×C₅	#892	1	1	yes	characteristic
198	C₂²×C₅	C₂×D₃×C₈	1	2	yes	—
199	C₂²×C₅	D₃×M₁₆	1	2	yes	—
200	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
201	C₄×C₅	—	2	2	no	—
202	C₄×C₅	—	3	8	no	—
203	C₄×C₅	—	6	4	no	—
204	C₄×C₅	C₈×Dic₃	1	4	yes	complemented
205	C₄×C₅	#887	1	1	yes	characteristic
206	C₄×C₅	#893	1	1	yes	characteristic
207	C₄×C₅	D₄×Dic₃	1	1	yes	characteristic
208	C₄×C₅	Q₈×Dic₃	1	1	yes	characteristic
209	C₂²×C₄	—	3	2	no	—
210	C₂²×C₄	—	3	2	no	—
211	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
212	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
213	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	3	4	no	—
217	C₂×C₈	—	6	2	no	—
218	C₂×C₈	—	6	2	no	—
219	C₂×C₈	C₂×C₅×Dic₃	1	4	yes	—
220	C₄²	—	3	8	no	—
221	C₄²	—	3	2	no	—
222	C₄²	—	3	2	no	—
223	C₄²	—	6	4	no	—
224	C₄²	—	6	1	no	—
225	C₄²	—	6	1	no	—
226	C₄²	C₄×C₅×D₃	1	2	yes	—
227	C₄²	C₂×C₅×Dic₃	1	2	yes	—
228	C₃×C₅	C₄×C₄⋊C₈	1	1	yes	characteristic, complemented, Hall
229	C₂²×C₃	C₂×C₄²×C₅	1	1	yes	characteristic
230	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
231	C₂²×C₃	C₅×#117	1	1	yes	characteristic
232	C₂²×C₃	C₂²×C₅×C₈	1	2	yes	—
233	C₂²×C₃	C₂×C₅×M₁₆	1	2	yes	—
234	C₃×C₄	—	2	2	no	—
235	C₃×C₄	C₄×C₅×C₈	1	4	yes	complemented
236	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
237	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
238	C₃×C₄	C₄×C₅×Q₈	1	1	yes	characteristic
239	C₃×C₄	C₄×C₅×Q₈	1	1	yes	characteristic
240	Dic₃	—	2	4	no	—
241	Dic₃	C₅×C₄⋊C₈	1	8	yes	direct product factor
242	C₂×C₅	D₃×C₄⋊C₈	1	2	yes	—
243	C₂×C₅	#4833	1	2	yes	—
244	C₂×C₅	Dic₃×C₄⋊C₄	1	1	yes	characteristic
245	C₂×C₅	C₂×C₈×Dic₃	1	1	yes	characteristic
246	C₂×C₅	Dic₃×M₁₆	1	1	yes	characteristic
247	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	3	4	no	—
250	C₂×C₄	—	3	4	no	—
251	C₂×C₄	—	3	4	no	—
252	C₂×C₄	—	3	4	no	—
253	C₂×C₄	—	6	2	no	—
254	C₂×C₄	—	6	2	no	—
255	C₂×C₄	C₅×D₃×C₈	1	4	yes	—
256	C₂×C₄	C₄×C₅×Dic₃	1	2	yes	—
257	C₂×C₄	C₄×C₅×Dic₃	1	2	yes	—
258	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
259	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
260	C₂×C₄	C₅×#237	1	1	yes	characteristic
261	C₂×C₄	C₅×D₃×Q₈	1	1	yes	characteristic
262	C₂×C₄	C₅×#239	1	1	yes	characteristic
263	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
264	C₈	—	2	4	no	—
265	C₈	—	6	4	no	—
266	C₂×C₃	C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
267	C₂×C₃	C₂×C₄×C₅×C₈	1	1	yes	characteristic
268	C₂×C₃	C₄×C₅×M₁₆	1	1	yes	characteristic
269	C₂×C₃	C₂×C₅×C₄⋊C₈	1	2	yes	—
270	C₂×C₃	C₅×#431	1	2	yes	—
271	C₅	Dic₃×C₄⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
272	C₂²	C₅×D₃×C₄⋊C₄	1	1	yes	characteristic
273	C₂²	C₅×#892	1	1	yes	characteristic
274	C₂²	C₂×C₅×D₃×C₈	1	2	yes	—
275	C₂²	C₅×D₃×M₁₆	1	2	yes	—
276	C₂²	C₂×C₄×C₅×Dic₃	1	1	yes	characteristic
277	C₄	—	2	2	no	—
278	C₄	—	3	8	no	—
279	C₄	—	6	4	no	—
280	C₄	C₅×C₈×Dic₃	1	4	yes	complemented
281	C₄	C₅×#887	1	1	yes	characteristic
282	C₄	C₅×#893	1	1	yes	characteristic
283	C₄	C₅×D₄×Dic₃	1	1	yes	characteristic
284	C₄	C₅×Q₈×Dic₃	1	1	yes	characteristic
285	C₃	C₄×C₅×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
286	C₂	C₅×D₃×C₄⋊C₈	1	2	yes	—
287	C₂	C₅×#4833	1	2	yes	—
288	C₂	C₅×Dic₃×C₄⋊C₄	1	1	yes	characteristic
289	C₂	C₂×C₅×C₈×Dic₃	1	1	yes	characteristic
290	C₂	C₅×Dic₃×M₁₆	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
14. C₃×C₅×C₄⋊C₈	C₄	semidirect
16. C₅×C₄⋊(C₃⋊C₈)	C₄	semidirect
17. Dic₃×C₄⋊C₈	C₅	direct (N × H)
36. C₄×C₅×Dic₃	C₈	semidirect
55. C₅×C₄⋊C₈	Dic₃	direct (N × H)
84. C₃×C₄⋊C₈	C₄×C₅	semidirect
86. C₄⋊(C₃⋊C₈)	C₄×C₅	semidirect
117. C₃×C₄×C₅	C₄×C₈	semidirect
123. C₅×Dic₃	C₄⋊C₈	direct (N × H)
137. C₄×Dic₃	C₅×C₈	semidirect
170. C₄⋊C₈	C₅×Dic₃	direct (N × H)
204. C₄×C₅	C₈×Dic₃	semidirect
228. C₃×C₅	C₄×C₄⋊C₈	semidirect
235. C₃×C₄	C₄×C₅×C₈	semidirect
241. Dic₃	C₅×C₄⋊C₈	direct (N × H)
271. C₅	Dic₃×C₄⋊C₈	direct (N × H)
280. C₄	C₅×C₈×Dic₃	semidirect
285. C₃	C₄×C₅×C₄⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×C₄⋊C₈	2
C₂×C₅×C₈×Dic₃	2
C₄²×C₅×Dic₃	2
C₂×C₅×C₄⋊(C₃⋊C₈)	2
C₅×#4550	2
C₄×C₅×C₄⋊C₈	3
Dic₃×C₄⋊C₈	5

Subgroups — C5×Dic3×C4⋊C8 #14296097

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×Dic₃×C₄⋊C₈ #14296097