Subgroups — Dic₅×#830 #14295024

Order 1920.

872 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×Dic₅	C₂	1	1	yes	maximal, characteristic
2	C₂×#91114	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#830	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₅×C₃⋊C₈	C₂	1	2	yes	maximal, complemented
5	#11745966	C₂	1	2	yes	maximal, complemented
6	Dic₅×C₂²⋊C₈	—	3	1	no	maximal, Hall
7	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₂³×C₃×Dic₅	C₄	1	2	yes	—
9	C₂²×C₃⋊#168	C₂²	1	2	yes	—
10	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
11	C₂×C₃×C₄×Dic₅	—	2	4	no	—
12	C₂×C₃×C₄×Dic₅	C₄	1	2	yes	—
13	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
15	#91114	C₄	1	4	yes	complemented
16	C₅×#830	C₄	1	4	yes	complemented
17	Dic₅×C₃⋊C₈	—	2	4	no	—
18	C₄×#830	—	5	1	no	maximal, Hall
19	C₂²×C₄×Dic₅	D₃	1	1	yes	characteristic
20	C₂×C₅×C₂²⋊C₈	—	3	1	no	—
21	C₂×C₂²⋊#168	—	3	1	no	—
22	C₂×C₈×Dic₅	—	3	2	no	—
23	#64770	—	3	2	no	—
24	C₂⁴×C₃×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	—	2	2	no	—
26	C₂²×C₃×C₄×C₅	—	2	2	no	—
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
29	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
30	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂²×C₃×Dic₅	—	2	4	no	—
32	C₂²×C₃×Dic₅	—	2	4	no	—
33	C₂²×C₃×Dic₅	C₈	1	8	yes	complemented
34	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	—
35	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	—
36	C₂×C₃⋊#168	—	2	2	no	—
37	C₂×C₃⋊#168	—	2	2	no	—
38	C₂×C₃⋊#168	C₂×C₄	1	4	yes	complemented
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	complemented
42	C₃×C₄×Dic₅	—	2	2	no	—
43	C₃×C₄×Dic₅	—	2	4	no	—
44	C₃×C₄×Dic₅	D₄	1	4	yes	—
45	C₂²×C₃×C₄²	—	5	1	no	—
46	C₂×#830	—	5	1	no	—
47	C₂×#830	D₅	1	1	yes	characteristic
48	C₂×C₄×C₃⋊C₈	—	5	2	no	—
49	#4524	—	5	2	no	—
50	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
51	C₂³×Dic₅	Dic₃	1	2	yes	—
52	C₂²×C₅×C₈	—	3	2	no	—
53	C₂×C₄×Dic₅	—	2	4	no	—
54	C₂×C₄×Dic₅	Dic₃	1	2	yes	—
55	C₂×C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
56	C₂×C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
57	C₂²×#168	—	3	2	no	—
58	C₅×C₂²⋊C₈	—	3	4	no	—
59	C₂²⋊#168	—	3	4	no	—
60	C₈×Dic₅	—	6	4	no	—
61	C₄×C₂²⋊C₈	—	15	1	no	Sylow
62	C₂³×C₃×C₅	—	2	1	no	—
63	C₂³×C₃×C₅	—	2	1	no	—
64	C₂³×C₃×C₅	—	2	2	no	—
65	C₂³×C₃×C₅	C₄²	1	2	yes	—
66	C₂³×C₃×C₅	C₂×C₈	1	4	yes	—
67	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	—	2	1	no	—
69	C₂×C₃×C₄×C₅	—	2	1	no	—
70	C₂×C₃×C₄×C₅	—	2	4	no	—
71	C₂×C₃×C₄×C₅	—	2	4	no	—
72	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
73	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
76	C₂×C₃×C₄×C₅	#41	1	2	yes	—
77	C₂×C₃×Dic₅	—	2	2	no	—
78	C₂×C₃×Dic₅	—	2	2	no	—
79	C₂×C₃×Dic₅	—	2	16	no	—
80	C₂×C₃×Dic₅	C₂²⋊C₄	1	4	yes	—
81	C₂×C₃×Dic₅	C₂×C₈	1	4	yes	—
82	C₂×C₃×Dic₅	C₂×C₈	1	4	yes	—
83	C₂×C₃×Dic₅	M₁₆	1	4	yes	—
84	C₃⋊#168	—	2	4	no	—
85	C₅×C₃⋊C₈	—	2	4	no	—
86	C₂³×C₃×C₄	—	5	2	no	—
87	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
88	C₂×C₃×C₄²	—	5	1	no	—
89	C₂×C₃×C₄²	—	5	2	no	—
90	C₂×C₃×C₄²	—	5	1	no	—
91	C₂×C₃×C₄²	—	10	4	no	—
92	C₂²×C₃⋊C₈	—	5	2	no	—
93	C₂²×C₃⋊C₈	C₂×D₅	1	2	yes	—
94	#830	—	5	4	no	—
95	#830	Dic₅	1	4	yes	direct product factor
96	C₄×C₃⋊C₈	—	10	4	no	—
97	C₂⁴×C₅	C₂×Dic₃	1	1	yes	characteristic
98	C₂²×C₄×C₅	—	2	2	no	—
99	C₂²×C₄×C₅	—	2	2	no	—
100	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
101	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
102	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
103	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
104	C₂²×Dic₅	—	2	4	no	—
105	C₂²×Dic₅	—	2	4	no	—
106	C₂²×Dic₅	C₃⋊C₈	1	8	yes	complemented
107	C₂²×Dic₅	C₂×Dic₃	1	2	yes	—
108	C₂²×Dic₅	C₂×Dic₃	1	2	yes	—
109	C₂×C₅×C₈	—	3	4	no	—
110	C₂×C₅×C₈	—	6	2	no	—
111	C₂×C₅×C₈	—	6	2	no	—
112	C₄×Dic₅	—	2	2	no	—
113	C₄×Dic₅	—	2	4	no	—
114	C₄×Dic₅	#67	1	4	yes	—
115	C₂×#168	—	3	4	no	—
116	C₂×#168	—	6	2	no	—
117	C₂×#168	—	6	2	no	—
118	C₂²×C₄²	—	5	1	no	—
119	C₂×C₂²⋊C₈	—	3	1	no	—
120	C₂×C₂²⋊C₈	—	15	1	no	—
121	C₂×C₄×C₈	—	15	2	no	—
122	#336	—	15	2	no	—
123	C₂²×C₃×C₅	—	2	2	no	—
124	C₂²×C₃×C₅	—	2	2	no	—
125	C₂²×C₃×C₅	—	2	4	no	—
126	C₂²×C₃×C₅	—	2	4	no	—
127	C₂²×C₃×C₅	C₄×C₈	1	4	yes	complemented
128	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
129	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₃×C₅	#117	1	1	yes	characteristic
131	C₂²×C₃×C₅	C₂²×C₈	1	2	yes	—
132	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
133	C₃×C₄×C₅	—	2	2	no	—
134	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
135	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
136	C₃×Dic₅	—	2	4	no	—
137	C₃×Dic₅	C₂²⋊C₈	1	8	yes	complemented
138	C₂⁴×C₃	C₄×D₅	1	1	yes	characteristic
139	C₂²×C₃×C₄	—	2	2	no	—
140	C₂²×C₃×C₄	—	2	2	no	—
141	C₂²×C₃×C₄	—	5	2	no	—
142	C₂²×C₃×C₄	—	5	8	no	—
143	C₂²×C₃×C₄	—	5	2	no	—
144	C₂²×C₃×C₄	—	10	4	no	—
145	C₂²×C₃×C₄	—	10	4	no	—
146	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
147	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
148	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
149	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
150	C₃×C₄²	—	5	4	no	—
151	C₃×C₄²	—	10	4	no	—
152	C₃×C₄²	—	10	2	no	—
153	C₂×C₃⋊C₈	—	2	2	no	—
154	C₂×C₃⋊C₈	—	2	2	no	—
155	C₂×C₃⋊C₈	—	5	4	no	—
156	C₂×C₃⋊C₈	—	10	2	no	—
157	C₂×C₃⋊C₈	—	10	2	no	—
158	C₂×C₃⋊C₈	C₂×Dic₅	1	4	yes	complemented
159	C₂³×C₅	—	2	2	no	—
160	C₂³×C₅	—	2	1	no	—
161	C₂³×C₅	—	2	1	no	—
162	C₂³×C₅	C₂×C₃⋊C₈	1	4	yes	—
163	C₂³×C₅	C₄×Dic₃	1	2	yes	—
164	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
165	C₂×C₄×C₅	—	2	1	no	—
166	C₂×C₄×C₅	—	2	1	no	—
167	C₂×C₄×C₅	—	2	4	no	—
168	C₂×C₄×C₅	—	2	4	no	—
169	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
170	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
172	C₂×C₄×C₅	#235	1	2	yes	—
173	C₂×C₄×C₅	C₂×#67	1	2	yes	—
174	C₂×Dic₅	—	2	2	no	—
175	C₂×Dic₅	—	2	2	no	—
176	C₂×Dic₅	—	2	16	no	—
177	C₂×Dic₅	C₂×C₃⋊C₈	1	4	yes	—
178	C₂×Dic₅	C₂×C₃⋊C₈	1	4	yes	—
179	C₂×Dic₅	#208	1	4	yes	—
180	C₂×Dic₅	#217	1	4	yes	—
181	C₅×C₈	—	6	4	no	—
182	#168	—	6	4	no	—
183	C₂³×C₄	—	5	2	no	—
184	C₂³×C₄	D₃×D₅	1	1	yes	characteristic
185	C₂²×C₈	—	3	2	no	—
186	C₂²×C₈	—	15	2	no	—
187	C₂×C₄²	—	5	1	no	—
188	C₂×C₄²	—	5	2	no	—
189	C₂×C₄²	—	5	1	no	—
190	C₂×C₄²	—	10	4	no	—
191	C₂²⋊C₈	—	3	4	no	—
192	C₂²⋊C₈	—	15	4	no	—
193	C₄×C₈	—	30	4	no	—
194	C₂×C₃×C₅	—	2	4	no	—
195	C₂×C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
196	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
197	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
198	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	2	yes	—
199	C₂×C₃×C₅	#431	1	2	yes	—
200	C₂³×C₃	—	2	1	no	—
201	C₂³×C₃	—	2	1	no	—
202	C₂³×C₃	—	2	2	no	—
203	C₂³×C₃	C₈×D₅	1	4	yes	—
204	C₂³×C₃	C₄×Dic₅	1	2	yes	—
205	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
206	C₂×C₃×C₄	—	2	1	no	—
207	C₂×C₃×C₄	—	2	1	no	—
208	C₂×C₃×C₄	—	2	4	no	—
209	C₂×C₃×C₄	—	2	4	no	—
210	C₂×C₃×C₄	—	5	4	no	—
211	C₂×C₃×C₄	—	5	4	no	—
212	C₂×C₃×C₄	—	5	4	no	—
213	C₂×C₃×C₄	—	5	4	no	—
214	C₂×C₃×C₄	—	10	16	no	—
215	C₂×C₃×C₄	—	10	2	no	—
216	C₂×C₃×C₄	—	10	2	no	—
217	C₂×C₃×C₄	C₄×Dic₅	1	2	yes	—
218	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
219	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
220	C₂×C₃×C₄	#711	1	2	yes	—
221	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
222	C₃⋊C₈	—	2	4	no	—
223	C₃⋊C₈	—	10	4	no	—
224	C₂²×C₅	—	2	2	no	—
225	C₂²×C₅	—	2	4	no	—
226	C₂²×C₅	—	2	4	no	—
227	C₂²×C₅	—	2	2	no	—
228	C₂²×C₅	C₄×C₃⋊C₈	1	4	yes	complemented
229	C₂²×C₅	C₂²×C₃⋊C₈	1	2	yes	—
230	C₂²×C₅	C₂×#208	1	2	yes	—
231	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
232	C₂²×C₅	#926	1	1	yes	characteristic
233	C₂²×C₅	C₂×#217	1	1	yes	characteristic
234	C₄×C₅	—	2	2	no	—
235	C₄×C₅	C₄×#67	1	2	yes	—
236	C₄×C₅	C₄×#67	1	2	yes	—
237	Dic₅	—	2	4	no	—
238	Dic₅	#830	1	8	yes	direct product factor
239	C₂⁴	D₅×Dic₃	1	1	yes	characteristic
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	5	2	no	—
243	C₂²×C₄	—	5	2	no	—
244	C₂²×C₄	—	5	8	no	—
245	C₂²×C₄	—	10	4	no	—
246	C₂²×C₄	—	10	4	no	—
247	C₂²×C₄	D₅×Dic₃	1	1	yes	characteristic
248	C₂²×C₄	D₃×Dic₅	1	2	yes	—
249	C₂²×C₄	#1200	1	2	yes	—
250	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
251	C₂×C₈	—	3	4	no	—
252	C₂×C₈	—	6	2	no	—
253	C₂×C₈	—	6	2	no	—
254	C₂×C₈	—	15	4	no	—
255	C₂×C₈	—	30	2	no	—
256	C₂×C₈	—	30	2	no	—
257	C₄²	—	5	4	no	—
258	C₄²	—	10	4	no	—
259	C₄²	—	10	2	no	—
260	C₃×C₅	C₄×C₂²⋊C₈	1	1	yes	characteristic, complemented, Hall
261	C₂²×C₃	—	2	2	no	—
262	C₂²×C₃	—	2	2	no	—
263	C₂²×C₃	—	2	4	no	—
264	C₂²×C₃	—	2	4	no	—
265	C₂²×C₃	C₈×Dic₅	1	4	yes	complemented
266	C₂²×C₃	#4023	1	1	yes	characteristic
267	C₂²×C₃	D₅×C₂²⋊C₄	1	1	yes	characteristic
268	C₂²×C₃	C₂×C₈×D₅	1	2	yes	—
269	C₂²×C₃	D₅×M₁₆	1	2	yes	—
270	C₂²×C₃	C₂×C₄×Dic₅	1	1	yes	characteristic
271	C₃×C₄	—	2	2	no	—
272	C₃×C₄	—	5	8	no	—
273	C₃×C₄	—	10	4	no	—
274	C₃×C₄	#4027	1	2	yes	—
275	C₃×C₄	D₄×Dic₅	1	2	yes	—
276	C₂×C₅	—	2	4	no	—
277	C₂×C₅	C₂×C₄×C₃⋊C₈	1	1	yes	characteristic
278	C₂×C₅	C₄×#208	1	1	yes	characteristic
279	C₂×C₅	#4925	1	2	yes	—
280	C₂×C₅	C₂×#830	1	2	yes	—
281	C₂×C₅	C₄×#217	1	1	yes	characteristic
282	C₂³	—	2	1	no	—
283	C₂³	—	2	2	no	—
284	C₂³	—	2	1	no	—
285	C₂³	D₅×C₃⋊C₈	1	4	yes	—
286	C₂³	Dic₃×Dic₅	1	2	yes	—
287	C₂³	C₂×D₅×Dic₃	1	1	yes	characteristic
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	5	4	no	—
293	C₂×C₄	—	5	4	no	—
294	C₂×C₄	—	5	4	no	—
295	C₂×C₄	—	5	4	no	—
296	C₂×C₄	—	10	16	no	—
297	C₂×C₄	—	10	2	no	—
298	C₂×C₄	—	10	2	no	—
299	C₂×C₄	Dic₃×Dic₅	1	2	yes	—
300	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
301	C₂×C₄	#6782	1	2	yes	—
302	C₂×C₄	C₂×#1200	1	1	yes	characteristic
303	C₂×C₄	D₅×#67	1	2	yes	—
304	C₈	—	6	4	no	—
305	C₈	—	30	4	no	—
306	C₂×C₃	—	2	4	no	—
307	C₂×C₃	#65007	1	2	yes	—
308	C₂×C₃	D₅×C₂²⋊C₈	1	2	yes	—
309	C₂×C₃	C₂²⋊C₄×Dic₅	1	1	yes	characteristic
310	C₂×C₃	C₂×C₈×Dic₅	1	1	yes	characteristic
311	C₂×C₃	M₁₆×Dic₅	1	1	yes	characteristic
312	C₅	C₄×#830	1	1	yes	characteristic, complemented, Sylow
313	C₂²	—	2	2	no	—
314	C₂²	—	2	4	no	—
315	C₂²	—	2	4	no	—
316	C₂²	—	2	2	no	—
317	C₂²	Dic₅×C₃⋊C₈	1	4	yes	complemented
318	C₂²	C₂×D₅×C₃⋊C₈	1	2	yes	—
319	C₂²	D₅×#208	1	2	yes	—
320	C₂²	C₂×Dic₃×Dic₅	1	1	yes	characteristic
321	C₂²	#91528	1	1	yes	characteristic
322	C₂²	D₅×#217	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	5	8	no	—
325	C₄	—	10	4	no	—
326	C₄	Dic₅×#67	1	2	yes	—
327	C₄	#91552	1	2	yes	—
328	C₃	Dic₅×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
329	C₂	—	2	4	no	—
330	C₂	C₂×Dic₅×C₃⋊C₈	1	1	yes	characteristic
331	C₂	#11748316	1	2	yes	—
332	C₂	Dic₅×#208	1	1	yes	characteristic
333	C₂	D₅×#830	1	2	yes	—
334	C₂	Dic₅×#217	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×D₃×D₅
commutator	C₂×C₃×C₅	C₂×C₄×C₈
Fitting	C₂³×C₃×C₄×C₅	C₂²
socle	C₂³×C₃×C₅	C₂²×C₄
prime radical	C₂⁴×C₃×C₅	C₂×C₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂³×C₄	D₃×D₅
hypocenter	C₃×C₅	C₄×C₂²⋊C₈
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
4. C₂×Dic₅×C₃⋊C₈	C₂	semidirect
5. #11745966	C₂	semidirect
15. #91114	C₄	semidirect
16. C₅×#830	C₄	semidirect
33. C₂²×C₃×Dic₅	C₈	semidirect
38. C₂×C₃⋊#168	C₂×C₄	semidirect
41. C₂×C₅×C₃⋊C₈	C₂×C₄	semidirect
95. #830	Dic₅	direct (N × H)
106. C₂²×Dic₅	C₃⋊C₈	semidirect
127. C₂²×C₃×C₅	C₄×C₈	semidirect
137. C₃×Dic₅	C₂²⋊C₈	semidirect
158. C₂×C₃⋊C₈	C₂×Dic₅	semidirect
228. C₂²×C₅	C₄×C₃⋊C₈	semidirect
238. Dic₅	#830	direct (N × H)
260. C₃×C₅	C₄×C₂²⋊C₈	semidirect
265. C₂²×C₃	C₈×Dic₅	semidirect
312. C₅	C₄×#830	semidirect
317. C₂²	Dic₅×C₃⋊C₈	semidirect
328. C₃	Dic₅×C₂²⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₄×Dic₅	2
C₂×#91114	2
C₂×C₅×#830	2
C₂×Dic₅×C₃⋊C₈	2
#11745966	2
Dic₅×C₂²⋊C₈	3
C₄×#830	5

Subgroups — Dic5×#830 #14295024

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₅×#830 #14295024