Subgroups — Dic₅×#446 #12097924

Order 1280.

958 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#446	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₂×#65313	C₂	1	1	yes	maximal, characteristic
4	#10601260	C₂	1	2	yes	maximal
5	#10601259	C₂	1	1	yes	maximal, characteristic
6	Dic₅×C₄⋊C₈	C₂	1	2	yes	maximal
7	Dic₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
8	#10600603	C₂	1	2	yes	maximal
9	Dic₅×#97	C₂	1	2	yes	maximal
10	#10600550	C₂	1	1	yes	maximal, characteristic
11	C₄×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
13	C₂×Q₈×Dic₅	C₄	1	2	yes	—
14	C₅×#446	C₄	1	8	yes	complemented
15	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₅×#97	C₂²	1	2	yes	—
18	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
19	C₂×#3947	C₂²	1	2	yes	—
20	C₂×#3947	C₂²	1	2	yes	—
21	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₅	C₂²	1	2	yes	—
23	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
24	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
25	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
26	#65313	C₄	1	8	yes	complemented
27	C₄⋊C₄×Dic₅	C₄	1	4	yes	—
28	C₄⋊C₄×Dic₅	C₄	1	2	yes	—
29	C₄×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
30	C₄×C₄⋊Dic₅	C₂²	1	2	yes	—
31	C₄²×Dic₅	C₂²	1	1	yes	characteristic
32	C₄²×Dic₅	C₂²	1	2	yes	—
33	C₂×C₄⋊#168	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊#168	C₂²	1	2	yes	—
35	C₂×#3935	C₂²	1	2	yes	—
36	C₂×C₄×#168	C₂²	1	1	yes	characteristic
37	#64770	C₂²	1	2	yes	—
38	#64770	C₂²	1	2	yes	—
39	#64770	C₂²	1	2	yes	—
40	C₄×#446	—	5	1	no	maximal, Sylow
41	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₅×C₈	C₂³	1	2	yes	—
44	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
45	C₄×C₅×Q₈	C₂×C₄	1	4	yes	—
46	C₄×C₅×Q₈	C₂×C₄	1	4	yes	—
47	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
48	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₅	C₂³	1	2	yes	—
51	Q₈×Dic₅	—	2	8	no	—
52	C₂×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
53	C₂×C₄⋊Dic₅	C₂×C₄	1	4	yes	—
54	C₂×C₄×Dic₅	C₂×C₄	1	4	yes	—
55	C₂×C₄×Dic₅	C₂×C₄	1	4	yes	—
56	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
57	C₂×C₄×Dic₅	C₂×C₄	1	2	yes	—
58	C₂×C₄×Dic₅	C₂³	1	1	yes	characteristic
59	C₂×C₄×Dic₅	C₂³	1	2	yes	—
60	C₂×C₄×Dic₅	C₂³	1	1	yes	characteristic
61	C₂²×#168	C₂³	1	1	yes	characteristic
62	C₂²×#168	C₂³	1	2	yes	—
63	C₂²×#168	C₂³	1	1	yes	characteristic
64	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
65	C₅×C₄⋊C₈	C₂×C₄	1	8	yes	—
66	C₅×#97	C₂×C₄	1	8	yes	—
67	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
68	#3963	C₂×C₄	1	2	yes	—
69	#3963	C₂×C₄	1	4	yes	—
70	#3947	—	2	4	no	—
71	#3947	—	2	4	no	—
72	C₈×Dic₅	—	2	2	no	—
73	C₈×Dic₅	—	2	2	no	—
74	C₈×Dic₅	—	2	4	no	—
75	C₈×Dic₅	—	2	2	no	—
76	C₈×Dic₅	Q₈	1	4	yes	complemented
77	C₄⋊#168	C₂×C₄	1	8	yes	—
78	C₄⋊#168	C₂×C₄	1	4	yes	—
79	#3935	C₂×C₄	1	8	yes	—
80	C₄×#168	C₂×C₄	1	4	yes	—
81	C₂×#446	—	5	1	no	—
82	C₂×#446	D₅	1	1	yes	characteristic
83	C₄²×Q₈	—	5	1	no	—
84	#1771	—	5	2	no	—
85	#1771	—	5	2	no	—
86	#1770	—	5	1	no	—
87	#1770	—	5	1	no	—
88	C₄×C₄⋊C₈	—	5	2	no	—
89	C₄×C₄⋊C₈	—	5	1	no	—
90	C₄×#97	—	5	2	no	—
91	C₄²×C₈	—	5	1	no	—
92	C₂×C₅×Q₈	—	2	4	no	—
93	C₂×C₅×Q₈	C₄²	1	4	yes	—
94	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
99	C₂²×Dic₅	C₂²×C₄	1	4	yes	—
100	C₂²×Dic₅	C₂²×C₄	1	2	yes	—
101	C₂²×Dic₅	C₂²×C₄	1	2	yes	—
102	C₂×C₅×C₈	—	2	1	no	—
103	C₂×C₅×C₈	—	2	1	no	—
104	C₂×C₅×C₈	—	2	2	no	—
105	C₂×C₅×C₈	—	2	2	no	—
106	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
107	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
108	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
109	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
110	C₂×C₅×C₈	#41	1	2	yes	—
111	C₅×C₄⋊C₄	C₄²	1	8	yes	—
112	C₅×C₄⋊C₄	C₄²	1	4	yes	—
113	C₄²×C₅	C₂²×C₄	1	2	yes	—
114	C₄²×C₅	C₂²×C₄	1	4	yes	—
115	C₄²×C₅	C₂²×C₄	1	2	yes	—
116	C₄²×C₅	C₂²×C₄	1	4	yes	—
117	C₄⋊Dic₅	—	2	4	no	—
118	C₄⋊Dic₅	—	2	8	no	—
119	C₄×Dic₅	—	2	1	no	—
120	C₄×Dic₅	—	2	1	no	—
121	C₄×Dic₅	—	2	2	no	—
122	C₄×Dic₅	—	2	2	no	—
123	C₄×Dic₅	—	2	8	no	—
124	C₄×Dic₅	—	2	4	no	—
125	C₄×Dic₅	—	2	8	no	—
126	C₄×Dic₅	M₁₆	1	8	yes	—
127	C₄×Dic₅	C₂×Q₈	1	2	yes	—
128	C₄×Dic₅	#41	1	2	yes	—
129	C₂×#168	—	2	1	no	—
130	C₂×#168	—	2	1	no	—
131	C₂×#168	—	2	2	no	—
132	C₂×#168	—	2	2	no	—
133	C₂×#168	C₂²×C₄	1	2	yes	—
134	C₂×#168	C₂²×C₄	1	4	yes	—
135	C₂×#168	C₂²×C₄	1	2	yes	—
136	C₂×#168	C₂×Q₈	1	2	yes	—
137	C₂×#168	#41	1	2	yes	—
138	C₂×C₄×Q₈	—	5	2	no	—
139	C₂×C₄×Q₈	C₂×D₅	1	1	yes	characteristic
140	#446	—	5	8	no	—
141	#446	Dic₅	1	8	yes	direct product factor
142	C₂×C₄⋊C₈	—	5	1	no	—
143	C₂×C₄⋊C₈	—	5	2	no	—
144	C₂×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
145	C₂×C₄⋊C₈	C₂×D₅	1	2	yes	—
146	C₂×#97	—	5	2	no	—
147	C₂×#97	—	5	2	no	—
148	C₂×#97	—	5	2	no	—
149	C₂×#97	C₂×D₅	1	2	yes	—
150	C₂×C₄×C₈	—	5	1	no	—
151	C₂×C₄×C₈	—	5	1	no	—
152	C₂×C₄×C₈	—	5	1	no	—
153	C₂×C₄×C₈	—	5	1	no	—
154	C₂×C₄×C₈	—	5	1	no	—
155	C₂×C₄×C₈	—	5	2	no	—
156	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
157	C₄×C₄⋊C₄	—	5	2	no	—
158	C₄×C₄⋊C₄	—	5	4	no	—
159	C₄×C₄⋊C₄	—	5	1	no	—
160	C₄×C₄⋊C₄	—	5	2	no	—
161	C₄³	—	5	1	no	—
162	C₄³	—	5	2	no	—
163	#336	—	5	2	no	—
164	#336	—	5	2	no	—
165	#336	—	5	2	no	—
166	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
167	C₅×Q₈	—	2	8	no	—
168	C₂×C₄×C₅	—	2	4	no	—
169	C₂×C₄×C₅	—	2	2	no	—
170	C₂×C₄×C₅	—	2	4	no	—
171	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
172	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
173	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
174	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
175	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
176	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
182	C₂×Dic₅	—	2	2	no	—
183	C₂×Dic₅	—	2	2	no	—
184	C₂×Dic₅	—	2	4	no	—
185	C₂×Dic₅	—	2	4	no	—
186	C₂×Dic₅	—	2	4	no	—
187	C₂×Dic₅	C₄×Q₈	1	4	yes	—
188	C₂×Dic₅	C₂×M₁₆	1	4	yes	—
189	C₂×Dic₅	C₂×M₁₆	1	4	yes	—
190	C₂×Dic₅	#131	1	4	yes	—
191	C₅×C₈	—	2	2	no	—
192	C₅×C₈	—	2	4	no	—
193	C₅×C₈	C₄×Q₈	1	4	yes	complemented
194	#168	—	2	2	no	—
195	#168	—	2	4	no	—
196	#168	C₄×Q₈	1	4	yes	complemented
197	C₂²×Q₈	C₄×D₅	1	1	yes	characteristic
198	C₂²×C₈	—	5	1	no	—
199	C₂²×C₈	—	5	1	no	—
200	C₂²×C₈	—	5	2	no	—
201	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₅	1	2	yes	—
203	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
204	C₄×Q₈	—	10	8	no	—
205	C₄×Q₈	C₄×D₅	1	4	yes	—
206	C₄×Q₈	C₂×Dic₅	1	4	yes	—
207	C₂×C₄⋊C₄	—	5	4	no	—
208	C₂×C₄⋊C₄	—	5	2	no	—
209	C₂×C₄⋊C₄	C₄×D₅	1	2	yes	—
210	C₂×C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
211	C₂×C₄²	—	5	2	no	—
212	C₂×C₄²	—	5	1	no	—
213	C₂×C₄²	—	5	4	no	—
214	C₂×C₄²	—	5	2	no	—
215	C₂×C₄²	—	5	4	no	—
216	C₂×C₄²	—	5	1	no	—
217	C₂×C₄²	—	5	2	no	—
218	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
219	C₂×C₄²	C₂²×D₅	1	2	yes	—
220	C₄⋊C₈	—	5	4	no	—
221	C₄⋊C₈	—	5	8	no	—
222	C₄⋊C₈	C₂×Dic₅	1	4	yes	—
223	C₄⋊C₈	C₂×Dic₅	1	8	yes	—
224	#97	—	5	8	no	—
225	#97	—	10	4	no	—
226	#97	—	10	4	no	—
227	#97	C₂×Dic₅	1	8	yes	—
228	C₄×C₈	—	5	4	no	—
229	C₄×C₈	—	5	4	no	—
230	C₄×C₈	—	10	2	no	—
231	C₄×C₈	—	10	4	no	—
232	C₄×C₈	—	10	2	no	—
233	C₄×C₈	—	10	2	no	—
234	C₄×C₈	C₂×Dic₅	1	4	yes	—
235	#95	—	5	4	no	—
236	#95	—	5	2	no	—
237	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
238	C₂²×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
239	C₂²×C₅	C₄×#41	1	1	yes	characteristic
240	C₂²×C₅	C₂²×M₁₆	1	2	yes	—
241	C₂²×C₅	C₂×#131	1	2	yes	—
242	C₄×C₅	—	2	4	no	—
243	C₄×C₅	—	2	2	no	—
244	C₄×C₅	—	2	4	no	—
245	C₄×C₅	C₄×M₁₆	1	4	yes	—
246	C₄×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
247	C₄×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
248	C₄×C₅	C₄×#41	1	1	yes	characteristic
249	C₄×C₅	C₄×#41	1	1	yes	characteristic
250	Dic₅	—	2	4	no	—
251	Dic₅	—	4	4	no	—
252	Dic₅	#446	1	8	yes	direct product factor
253	C₂×Q₈	—	2	4	no	—
254	C₂×Q₈	C₄×Dic₅	1	4	yes	—
255	C₂²×C₄	—	5	2	no	—
256	C₂²×C₄	—	5	2	no	—
257	C₂²×C₄	—	5	4	no	—
258	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
259	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
260	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
261	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
262	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	5	2	no	—
268	C₂×C₈	—	5	2	no	—
269	C₂×C₈	—	5	2	no	—
270	C₂×C₈	—	5	2	no	—
271	C₂×C₈	—	5	4	no	—
272	C₂×C₈	—	10	1	no	—
273	C₂×C₈	—	10	2	no	—
274	C₂×C₈	—	10	1	no	—
275	C₂×C₈	—	10	2	no	—
276	C₂×C₈	Q₈×D₅	1	2	yes	—
277	C₂×C₈	#713	1	2	yes	—
278	C₂×C₈	C₂²×Dic₅	1	2	yes	—
279	C₂×C₈	C₂²×Dic₅	1	4	yes	—
280	C₂×C₈	C₂²×Dic₅	1	2	yes	—
281	C₄⋊C₄	—	10	8	no	—
282	C₄⋊C₄	—	10	4	no	—
283	C₄⋊C₄	C₄×Dic₅	1	8	yes	—
284	C₄⋊C₄	C₄×Dic₅	1	4	yes	—
285	C₄²	—	5	2	no	—
286	C₄²	—	5	2	no	—
287	C₄²	—	5	8	no	—
288	C₄²	—	10	8	no	—
289	C₄²	—	10	4	no	—
290	C₄²	—	10	8	no	—
291	C₄²	—	10	1	no	—
292	C₄²	—	10	2	no	—
293	C₄²	—	10	1	no	—
294	C₄²	—	10	2	no	—
295	C₄²	C₂×C₄×D₅	1	2	yes	—
296	C₄²	C₂×C₄×D₅	1	4	yes	—
297	C₄²	C₂²×Dic₅	1	2	yes	—
298	C₄²	C₂²×Dic₅	1	4	yes	—
299	C₂×C₅	C₄²×Q₈	1	1	yes	characteristic
300	C₂×C₅	C₂×C₄×M₁₆	1	1	yes	characteristic
301	C₂×C₅	C₄×#131	1	1	yes	characteristic
302	C₂×C₅	C₂×#446	1	2	yes	—
303	C₂×C₅	#2965	1	2	yes	—
304	C₂³	C₂²×C₄×D₅	1	1	yes	characteristic
305	Q₈	—	2	8	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	5	4	no	—
310	C₂×C₄	—	5	4	no	—
311	C₂×C₄	—	5	4	no	—
312	C₂×C₄	—	5	4	no	—
313	C₂×C₄	—	10	2	no	—
314	C₂×C₄	—	10	2	no	—
315	C₂×C₄	—	10	4	no	—
316	C₂×C₄	—	10	4	no	—
317	C₂×C₄	—	10	4	no	—
318	C₂×C₄	D₅×M₁₆	1	4	yes	—
319	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
320	C₂×C₄	C₂×C₄×Dic₅	1	4	yes	—
321	C₂×C₄	C₂×C₄×Dic₅	1	4	yes	—
322	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
323	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
324	C₂×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
325	C₂×C₄	C₂×#713	1	1	yes	characteristic
326	C₂×C₄	D₅×#41	1	1	yes	characteristic
327	C₂×C₄	D₅×#41	1	1	yes	characteristic
328	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
329	C₈	—	2	2	no	—
330	C₈	—	2	4	no	—
331	C₈	—	5	4	no	—
332	C₈	—	10	2	no	—
333	C₈	—	10	4	no	—
334	C₈	Q₈×Dic₅	1	4	yes	complemented
335	C₅	C₄×#446	1	1	yes	characteristic, complemented, Sylow
336	C₂²	C₄×Q₈×D₅	1	1	yes	characteristic
337	C₂²	C₄×#713	1	1	yes	characteristic
338	C₂²	C₂×D₅×M₁₆	1	2	yes	—
339	C₂²	D₅×#131	1	2	yes	—
340	C₂²	C₂²×C₄×Dic₅	1	1	yes	characteristic
341	C₄	—	2	4	no	—
342	C₄	—	2	2	no	—
343	C₄	—	2	4	no	—
344	C₄	—	5	8	no	—
345	C₄	—	10	4	no	—
346	C₄	—	20	4	no	—
347	C₄	M₁₆×Dic₅	1	4	yes	—
348	C₄	C₄×Q₈×D₅	1	1	yes	characteristic
349	C₄	C₄×#713	1	1	yes	characteristic
350	C₄	C₂×Q₈×Dic₅	1	1	yes	characteristic
351	C₄	#41×Dic₅	1	1	yes	characteristic
352	C₂	#10611669	1	2	yes	—
353	C₂	D₅×#446	1	2	yes	—
354	C₂	C₄×Q₈×Dic₅	1	1	yes	characteristic
355	C₂	C₂×M₁₆×Dic₅	1	1	yes	characteristic
356	C₂	Dic₅×#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅
commutator	C₂²×C₅	C₂²×C₄²
Fitting	C₂×C₅×#446	C₂
socle, prime radical	C₂³×C₅	C₂³×C₄
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	C₂×#446	D₅
hypocenter	C₅	C₄×#446
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
14. C₅×#446	C₄	semidirect
26. #65313	C₄	semidirect
76. C₈×Dic₅	Q₈	semidirect
141. #446	Dic₅	direct (N × H)
193. C₅×C₈	C₄×Q₈	semidirect
196. #168	C₄×Q₈	semidirect
252. Dic₅	#446	direct (N × H)
334. C₈	Q₈×Dic₅	semidirect
335. C₅	C₄×#446	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#446	2
C₄×Q₈×Dic₅	2
C₂×#65313	2
#10601260	2
#10601259	2
Dic₅×C₄⋊C₈	2
#10600603	2
Dic₅×#97	2
#10600550	2
C₄×C₈×Dic₅	2
C₄×#446	5

Subgroups — Dic5×#446 #12097924

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₅×#446 #12097924