Subgroups — C₂×D₅×#498 #12968886

Order 1280.

2398 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅×C₄⋊Q₈	C₂	1	2	yes	maximal
2	C₂×C₅×#498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#65368	C₂	1	2	yes	maximal, complemented
4	D₅×#498	C₂	1	16	yes	maximal, direct product factor
5	C₂×D₅×#106	C₂	1	4	yes	maximal
6	C₂×#65144	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
8	C₂×#64969	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₈×D₅	—	2	2	no	—
10	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×#4090	—	2	2	no	—
12	C₂²×C₈×D₅	C₂²	1	2	yes	—
13	D₅×C₄⋊Q₈	—	2	8	no	—
14	D₅×C₄⋊Q₈	C₂²	1	16	yes	—
15	C₂×D₅×C₄⋊C₄	—	2	2	no	—
16	C₂×D₅×C₄⋊C₄	C₂²	1	4	yes	—
17	C₂×#4034	—	2	2	no	—
18	C₂×#4034	C₂²	1	4	yes	—
19	C₂×C₄²×D₅	C₂²	1	1	yes	characteristic
20	C₂×#4015	C₂²	1	2	yes	—
21	C₅×#498	C₂²	1	8	yes	complemented
22	C₂×C₅×#106	C₂²	1	4	yes	—
23	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
24	C₂×#3949	C₂²	1	4	yes	—
25	C₂×C₈×Dic₅	C₂²	1	2	yes	—
26	#65368	C₂²	1	16	yes	complemented
27	C₂×#3940	C₂²	1	8	yes	—
28	C₂×C₄×#168	C₂²	1	1	yes	characteristic
29	D₅×#106	C₂²	1	32	yes	—
30	#65144	C₂²	1	32	yes	complemented
31	C₄×C₈×D₅	C₂²	1	8	yes	—
32	#64969	C₂²	1	8	yes	complemented
33	C₂²×#498	—	5	1	no	maximal, Sylow
34	C₂²×C₅×Q₈	—	2	2	no	—
35	C₂×Q₈×D₅	—	2	16	no	—
36	C₂×Q₈×D₅	—	4	16	no	—
37	C₂²×C₄×D₅	—	2	4	no	—
38	C₂²×C₄×D₅	D₄	1	2	yes	—
39	C₂²×C₄×D₅	C₂³	1	1	yes	characteristic
40	C₂²×Dic₁₀	—	2	2	no	—
41	C₂²×Dic₁₀	—	2	4	no	—
42	C₂²×C₅×C₈	C₂³	1	2	yes	—
43	C₅×C₄⋊Q₈	—	2	4	no	—
44	C₅×C₄⋊Q₈	C₂³	1	8	yes	—
45	C₂×C₅×C₄⋊C₄	—	2	2	no	—
46	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
47	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
48	#4090	—	2	16	no	—
49	C₂×C₄⋊Dic₅	—	2	2	no	—
50	C₂×C₄⋊Dic₅	C₂³	1	4	yes	—
51	C₂×#683	—	2	4	no	—
52	C₂×#683	—	2	4	no	—
53	C₂×C₄×Dic₅	D₄	1	2	yes	—
54	C₂×C₄×Dic₅	C₂³	1	1	yes	characteristic
55	C₂²×#168	C₂³	1	2	yes	—
56	C₂×C₈×D₅	Q₈	1	16	yes	complemented
57	C₂×C₈×D₅	C₂³	1	8	yes	—
58	D₅×C₄⋊C₄	—	2	8	no	—
59	D₅×C₄⋊C₄	—	2	16	no	—
60	D₅×C₄⋊C₄	C₂³	1	16	yes	—
61	#4034	—	2	8	no	—
62	#4034	—	2	16	no	—
63	#4034	C₂³	1	16	yes	—
64	C₄²×D₅	D₄	1	4	yes	—
65	C₄²×D₅	C₂³	1	4	yes	—
66	#4015	—	2	4	no	—
67	#4015	C₂³	1	8	yes	—
68	C₅×#106	C₂³	1	16	yes	—
69	C₄×C₅×C₈	C₂³	1	4	yes	—
70	#3949	C₂³	1	16	yes	—
71	C₈×Dic₅	C₂³	1	8	yes	—
72	#3940	C₂³	1	32	yes	—
73	C₄×#168	C₂³	1	4	yes	—
74	C₂²×C₄⋊Q₈	—	5	2	no	—
75	C₂×#498	—	5	16	no	—
76	C₂×#498	—	5	4	no	—
77	C₂×#498	—	5	1	no	—
78	C₂×#498	—	5	2	no	—
79	C₂×#498	D₅	1	1	yes	characteristic, direct product factor
80	C₂²×#106	—	5	4	no	—
81	C₂²×C₄×C₈	—	5	1	no	—
82	C₂³×D₅	C₂×D₄	1	1	yes	characteristic
83	C₂×C₅×Q₈	—	2	8	no	—
84	C₂×C₅×Q₈	—	4	8	no	—
85	C₂²×C₄×C₅	—	2	4	no	—
86	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
88	C₂²×Dic₅	—	2	4	no	—
89	C₂²×Dic₅	C₂×D₄	1	1	yes	characteristic
90	C₂²×Dic₅	C₂×D₄	1	2	yes	—
91	Q₈×D₅	—	4	64	no	—
92	C₂×C₄×D₅	—	2	16	no	—
93	C₂×C₄×D₅	—	4	16	no	—
94	C₂×C₄×D₅	SD₁₆	1	16	yes	—
95	C₂×C₄×D₅	C₂×D₄	1	2	yes	—
96	C₂×C₄×D₅	C₂×D₄	1	8	yes	—
97	C₂×C₄×D₅	C₂×Q₈	1	8	yes	—
98	C₂×C₄×D₅	C₂⁴	1	2	yes	—
99	C₂×Dic₁₀	—	2	8	no	—
100	C₂×Dic₁₀	—	2	16	no	—
101	C₂×Dic₁₀	—	4	8	no	—
102	C₂×Dic₁₀	—	4	16	no	—
103	C₂×C₅×C₈	C₂×Q₈	1	8	yes	complemented
104	C₂×C₅×C₈	C₂⁴	1	4	yes	—
105	C₅×C₄⋊C₄	—	2	8	no	—
106	C₅×C₄⋊C₄	—	2	4	no	—
107	C₅×C₄⋊C₄	C₂⁴	1	8	yes	—
108	C₄²×C₅	C₂×D₄	1	2	yes	—
109	C₄²×C₅	C₂⁴	1	2	yes	—
110	C₄⋊Dic₅	—	2	4	no	—
111	C₄⋊Dic₅	—	2	8	no	—
112	C₄⋊Dic₅	C₂⁴	1	8	yes	—
113	#683	—	2	16	no	—
114	#683	—	2	16	no	—
115	C₄×Dic₅	C₂×D₄	1	2	yes	—
116	C₄×Dic₅	C₂×D₄	1	8	yes	—
117	C₄×Dic₅	C₂⁴	1	2	yes	—
118	C₂×#168	C₂×Q₈	1	8	yes	complemented
119	C₂×#168	C₂⁴	1	4	yes	—
120	C₈×D₅	C₂×Q₈	1	32	yes	complemented
121	C₂³×Q₈	—	10	2	no	—
122	C₂³×C₈	—	5	2	no	—
123	C₂×C₄⋊Q₈	—	5	16	no	—
124	C₂×C₄⋊Q₈	—	5	4	no	—
125	C₂×C₄⋊Q₈	—	5	2	no	—
126	C₂×C₄⋊Q₈	—	10	2	no	—
127	C₂×C₄⋊Q₈	—	10	8	no	—
128	C₂×C₄⋊Q₈	—	10	2	no	—
129	C₂×C₄⋊Q₈	C₂×D₅	1	2	yes	—
130	C₂²×C₄⋊C₄	—	5	4	no	—
131	C₂²×C₄⋊C₄	—	10	2	no	—
132	C₂²×C₄²	—	5	1	no	—
133	#498	—	5	16	no	—
134	#498	—	5	32	no	—
135	#498	—	5	8	no	—
136	#498	C₂×D₅	1	8	yes	direct product factor
137	C₂×#106	—	5	4	no	—
138	C₂×#106	—	5	8	no	—
139	C₂×#106	—	5	32	no	—
140	C₂×#106	C₂×D₅	1	4	yes	—
141	C₂×C₄×C₈	—	5	2	no	—
142	C₂×C₄×C₈	—	5	1	no	—
143	C₂×C₄×C₈	—	5	8	no	—
144	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
145	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
146	C₂²×D₅	C₄⋊Q₈	1	4	yes	—
147	C₂²×D₅	C₂×SD₁₆	1	8	yes	—
148	C₂²×D₅	C₂²×D₄	1	2	yes	—
149	C₅×Q₈	—	4	16	no	—
150	C₂×C₄×C₅	—	2	8	no	—
151	C₂×C₄×C₅	—	4	8	no	—
152	C₂×C₄×C₅	C₂×SD₁₆	1	8	yes	—
153	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
155	C₂×C₄×C₅	C₂²×Q₈	1	4	yes	—
156	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
157	C₂×Dic₅	—	2	8	no	—
158	C₂×Dic₅	—	4	8	no	—
159	C₂×Dic₅	C₄⋊Q₈	1	4	yes	—
160	C₂×Dic₅	C₂×SD₁₆	1	8	yes	—
161	C₂×Dic₅	C₂²×D₄	1	2	yes	—
162	C₂×Dic₅	C₂²×D₄	1	4	yes	—
163	C₄×D₅	—	4	32	no	—
164	C₄×D₅	C₄⋊Q₈	1	8	yes	—
165	C₄×D₅	C₂×SD₁₆	1	32	yes	—
166	C₄×D₅	C₂²×Q₈	1	8	yes	—
167	Dic₁₀	—	4	16	no	—
168	Dic₁₀	—	4	32	no	—
169	C₅×C₈	C₂²×Q₈	1	8	yes	complemented
170	#168	C₂²×Q₈	1	8	yes	complemented
171	C₂²×Q₈	—	2	2	no	—
172	C₂²×Q₈	—	10	4	no	—
173	C₂²×Q₈	—	10	16	no	—
174	C₂²×Q₈	—	10	2	no	—
175	C₂²×Q₈	—	20	16	no	—
176	C₂³×C₄	—	5	2	no	—
177	C₂³×C₄	—	5	1	no	—
178	C₂³×C₄	—	10	4	no	—
179	C₂²×C₈	—	5	2	no	—
180	C₂²×C₈	—	5	16	no	—
181	C₂²×C₈	—	5	8	no	—
182	C₂²×C₈	C₂²×D₅	1	2	yes	—
183	C₄⋊Q₈	—	2	4	no	—
184	C₄⋊Q₈	—	5	16	no	—
185	C₄⋊Q₈	—	5	8	no	—
186	C₄⋊Q₈	—	10	16	no	—
187	C₄⋊Q₈	—	10	8	no	—
188	C₄⋊Q₈	—	10	16	no	—
189	C₄⋊Q₈	—	10	4	no	—
190	C₄⋊Q₈	C₂²×D₅	1	8	yes	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	5	4	no	—
193	C₂×C₄⋊C₄	—	5	16	no	—
194	C₂×C₄⋊C₄	—	10	8	no	—
195	C₂×C₄⋊C₄	—	10	16	no	—
196	C₂×C₄⋊C₄	—	10	4	no	—
197	C₂×C₄⋊C₄	—	10	4	no	—
198	C₂×C₄⋊C₄	—	10	2	no	—
199	C₂×C₄⋊C₄	C₂²×D₅	1	4	yes	—
200	C₂×C₄²	—	5	2	no	—
201	C₂×C₄²	—	5	1	no	—
202	C₂×C₄²	—	5	4	no	—
203	C₂×C₄²	—	5	4	no	—
204	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
205	#106	—	5	16	no	—
206	#106	—	5	32	no	—
207	#106	C₂²×D₅	1	16	yes	—
208	C₄×C₈	—	5	8	no	—
209	C₄×C₈	—	5	4	no	—
210	C₄×C₈	C₂²×D₅	1	4	yes	—
211	C₂²×C₅	C₂×C₄⋊Q₈	1	2	yes	—
212	C₂²×C₅	C₂²×SD₁₆	1	4	yes	—
213	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
214	C₂×D₅	#498	1	16	yes	direct product factor
215	C₂×D₅	C₂×C₄⋊Q₈	1	4	yes	—
216	C₂×D₅	C₂²×SD₁₆	1	8	yes	—
217	C₄×C₅	—	4	8	no	—
218	C₄×C₅	C₂×C₄⋊Q₈	1	2	yes	—
219	C₄×C₅	C₂²×SD₁₆	1	8	yes	—
220	C₄×C₅	C₂³×Q₈	1	2	yes	—
221	Dic₅	—	4	8	no	—
222	Dic₅	C₂×C₄⋊Q₈	1	4	yes	—
223	Dic₅	C₂²×SD₁₆	1	8	yes	—
224	C₂⁴	—	5	1	no	—
225	C₂×Q₈	—	2	8	no	—
226	C₂×Q₈	—	4	8	no	—
227	C₂×Q₈	—	10	16	no	—
228	C₂×Q₈	—	10	8	no	—
229	C₂×Q₈	—	20	16	no	—
230	C₂×Q₈	—	20	64	no	—
231	C₂×Q₈	—	20	8	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	5	1	no	—
234	C₂²×C₄	—	5	2	no	—
235	C₂²×C₄	—	5	2	no	—
236	C₂²×C₄	—	5	8	no	—
237	C₂²×C₄	—	5	8	no	—
238	C₂²×C₄	—	5	2	no	—
239	C₂²×C₄	—	5	16	no	—
240	C₂²×C₄	—	10	16	no	—
241	C₂²×C₄	—	10	4	no	—
242	C₂²×C₄	—	20	16	no	—
243	C₂²×C₄	D₄×D₅	1	2	yes	—
244	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
245	C₂×C₈	—	5	8	no	—
246	C₂×C₈	—	5	4	no	—
247	C₂×C₈	—	5	32	no	—
248	C₂×C₈	Q₈×D₅	1	8	yes	complemented
249	C₂×C₈	C₂³×D₅	1	4	yes	—
250	C₄⋊C₄	—	2	8	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	5	8	no	—
253	C₄⋊C₄	—	10	16	no	—
254	C₄⋊C₄	—	10	16	no	—
255	C₄⋊C₄	—	10	4	no	—
256	C₄⋊C₄	—	10	8	no	—
257	C₄⋊C₄	C₂³×D₅	1	8	yes	—
258	C₄²	—	5	8	no	—
259	C₄²	—	5	2	no	—
260	C₄²	—	5	2	no	—
261	C₄²	D₄×D₅	1	2	yes	—
262	C₄²	C₂³×D₅	1	2	yes	—
263	C₂×C₅	C₂×#498	1	4	yes	complemented
264	C₂×C₅	C₂²×C₄⋊Q₈	1	1	yes	characteristic
265	C₂×C₅	C₂³×SD₁₆	1	2	yes	—
266	D₅	C₂×#498	1	8	yes	direct product factor
267	C₂³	—	5	4	no	—
268	C₂³	—	5	8	no	—
269	C₂³	—	5	2	no	—
270	C₂³	C₂×D₄×D₅	1	1	yes	characteristic
271	Q₈	—	4	16	no	—
272	Q₈	—	20	32	no	—
273	Q₈	—	20	16	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	4	8	no	—
276	C₂×C₄	—	5	4	no	—
277	C₂×C₄	—	5	8	no	—
278	C₂×C₄	—	5	2	no	—
279	C₂×C₄	—	5	4	no	—
280	C₂×C₄	—	5	8	no	—
281	C₂×C₄	—	5	32	no	—
282	C₂×C₄	—	5	8	no	—
283	C₂×C₄	—	10	8	no	—
284	C₂×C₄	—	20	32	no	—
285	C₂×C₄	—	20	8	no	—
286	C₂×C₄	D₅×SD₁₆	1	8	yes	—
287	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
288	C₂×C₄	C₂×D₄×D₅	1	4	yes	—
289	C₂×C₄	C₂×Q₈×D₅	1	4	yes	—
290	C₂×C₄	C₂⁴×D₅	1	1	yes	characteristic
291	C₈	—	5	8	no	—
292	C₈	C₂×Q₈×D₅	1	8	yes	complemented
293	C₅	C₂²×#498	1	1	yes	characteristic, complemented, Sylow
294	C₂²	—	5	16	no	—
295	C₂²	—	5	8	no	—
296	C₂²	—	5	4	no	—
297	C₂²	D₅×C₄⋊Q₈	1	2	yes	—
298	C₂²	C₂×D₅×SD₁₆	1	4	yes	—
299	C₂²	C₂²×D₄×D₅	1	1	yes	characteristic
300	C₄	—	4	8	no	—
301	C₄	—	5	8	no	—
302	C₄	—	5	4	no	—
303	C₄	—	20	8	no	—
304	C₄	D₅×C₄⋊Q₈	1	2	yes	—
305	C₄	C₂×D₅×SD₁₆	1	8	yes	—
306	C₄	C₂²×Q₈×D₅	1	2	yes	—
307	C₂	—	5	8	no	—
308	C₂	D₅×#498	1	4	yes	direct product factor
309	C₂	C₂×D₅×C₄⋊Q₈	1	1	yes	characteristic
310	C₂	C₂²×D₅×SD₁₆	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅
Fitting	C₂×C₅×#498	C₂
socle	C₂³×C₅	C₂²×D₄
prime radical	C₂³×D₅	C₂×D₄
hypercenter	C₂×#498	D₅
hypocenter	C₅	C₂²×#498
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#498	C₂	semidirect
3. C₂×#65368	C₂	semidirect
4. D₅×#498	C₂	direct (N × H)
6. C₂×#65144	C₂	semidirect
8. C₂×#64969	C₂	semidirect
21. C₅×#498	C₂²	semidirect
26. #65368	C₂²	semidirect
30. #65144	C₂²	semidirect
32. #64969	C₂²	semidirect
56. C₂×C₈×D₅	Q₈	semidirect
79. C₂×#498	D₅	direct (N × H)
103. C₂×C₅×C₈	C₂×Q₈	semidirect
118. C₂×#168	C₂×Q₈	semidirect
120. C₈×D₅	C₂×Q₈	semidirect
136. #498	C₂×D₅	direct (N × H)
169. C₅×C₈	C₂²×Q₈	semidirect
170. #168	C₂²×Q₈	semidirect
214. C₂×D₅	#498	direct (N × H)
248. C₂×C₈	Q₈×D₅	semidirect
263. C₂×C₅	C₂×#498	semidirect
266. D₅	C₂×#498	direct (N × H)
292. C₈	C₂×Q₈×D₅	semidirect
293. C₅	C₂²×#498	semidirect
308. C₂	D₅×#498	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×D₅×C₄⋊Q₈	2
C₂×C₅×#498	2
C₂×#65368	2
D₅×#498	2
C₂×D₅×#106	2
C₂×#65144	2
C₂×C₄×C₈×D₅	2
C₂×#64969	2
C₂²×#498	5

Subgroups — C2×D5×#498 #12968886

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₅×#498 #12968886