Subgroups — C₂×#10613647 #12968904

Order 1280.

1942 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	C₂×#66022	C₂	1	2	yes	maximal
2	C₂×C₅×#499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#65353	C₂	1	2	yes	maximal, complemented
4	#10613647	C₂	1	16	yes	maximal, direct product factor
5	C₂×#65169	C₂	1	2	yes	maximal
6	C₂×#65163	C₂	1	2	yes	maximal, complemented
7	C₂×#65149	C₂	1	2	yes	maximal
8	C₂×#65144	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
10	C₂×#64972	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅×#125	C₂²	1	2	yes	—
12	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
13	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
14	#66022	—	2	8	no	—
15	#66022	C₂²	1	16	yes	—
16	C₂×#4042	—	2	4	no	—
17	C₂×#4038	C₂²	1	4	yes	—
18	C₂×#4034	C₂²	1	4	yes	—
19	C₂×#4033	—	2	4	no	—
20	C₂×C₄²×D₅	C₂²	1	1	yes	characteristic
21	C₂×#4016	C₂²	1	2	yes	—
22	C₅×#499	C₂²	1	8	yes	complemented
23	C₂×C₅×#107	C₂²	1	2	yes	—
24	C₂×C₅×#106	C₂²	1	2	yes	—
25	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
26	C₂×#3950	C₂²	1	2	yes	—
27	C₂×#3949	C₂²	1	2	yes	—
28	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
29	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
30	#65353	C₂²	1	16	yes	complemented
31	C₂×#3940	C₂²	1	4	yes	—
32	C₂×#3939	C₂²	1	4	yes	—
33	C₂×C₄×#168	C₂²	1	1	yes	characteristic
34	#65169	C₂²	1	16	yes	—
35	#65163	C₂²	1	16	yes	complemented
36	#65149	C₂²	1	16	yes	—
37	#65144	C₂²	1	16	yes	complemented
38	C₄×C₈×D₅	C₂²	1	8	yes	—
39	#64972	C₂²	1	8	yes	complemented
40	C₂×#3160	—	5	1	no	maximal, Sylow
41	C₂²×C₄×D₅	D₄	1	2	yes	—
42	C₂²×C₄×D₅	C₂³	1	1	yes	characteristic
43	C₂²×Dic₁₀	—	2	4	no	—
44	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
45	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
46	C₅×#125	—	2	4	no	—
47	C₅×#125	C₂³	1	8	yes	—
48	C₂×C₅×C₄⋊C₄	—	2	4	no	—
49	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
50	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
51	C₂×#685	—	2	4	no	—
52	C₂×C₄⋊Dic₅	C₂³	1	4	yes	—
53	C₂×#683	—	2	4	no	—
54	C₂×#683	—	2	4	no	—
55	C₂×C₄×Dic₅	—	2	4	no	—
56	C₂×C₄×Dic₅	D₄	1	2	yes	—
57	C₂×C₄×Dic₅	C₂³	1	1	yes	characteristic
58	C₂²×#168	C₂³	1	1	yes	characteristic
59	C₂²×#168	C₂³	1	1	yes	characteristic
60	C₂×C₈×D₅	Q₈	1	8	yes	complemented
61	C₂×C₈×D₅	Q₈	1	8	yes	complemented
62	C₂×C₈×D₅	C₂³	1	4	yes	—
63	C₂×C₈×D₅	C₂³	1	4	yes	—
64	#4042	—	2	32	no	—
65	#4038	—	2	8	no	—
66	#4038	C₂³	1	16	yes	—
67	#4034	—	2	8	no	—
68	#4034	C₂³	1	16	yes	—
69	#4033	—	2	32	no	—
70	C₄²×D₅	D₄	1	4	yes	—
71	C₄²×D₅	C₂³	1	4	yes	—
72	#4016	—	2	4	no	—
73	#4016	C₂³	1	8	yes	—
74	C₅×#107	C₂³	1	8	yes	—
75	C₅×#106	C₂³	1	8	yes	—
76	C₄×C₅×C₈	C₂³	1	4	yes	—
77	#3950	C₂³	1	8	yes	—
78	#3949	C₂³	1	8	yes	—
79	C₈×Dic₅	C₂³	1	4	yes	—
80	C₈×Dic₅	C₂³	1	4	yes	—
81	#3940	C₂³	1	16	yes	—
82	#3939	C₂³	1	16	yes	—
83	C₄×#168	C₂³	1	4	yes	—
84	C₂×#533	—	5	2	no	—
85	#3160	—	5	16	no	—
86	C₂×#500	—	5	2	no	—
87	C₂×#499	—	5	1	no	—
88	C₂×#499	—	5	2	no	—
89	C₂×#499	D₅	1	1	yes	characteristic, complemented
90	C₂×#498	—	5	2	no	—
91	C₂×#427	—	5	2	no	—
92	C₂×#427	—	5	2	no	—
93	C₂²×C₄×C₈	—	5	1	no	—
94	C₂³×D₅	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₅	—	2	4	no	—
96	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
98	C₂²×Dic₅	—	2	4	no	—
99	C₂²×Dic₅	C₂×D₄	1	1	yes	characteristic
100	C₂²×Dic₅	C₂×D₄	1	2	yes	—
101	C₂×C₄×D₅	—	2	8	no	—
102	C₂×C₄×D₅	C₂×D₄	1	8	yes	—
103	C₂×C₄×D₅	C₂×D₄	1	2	yes	—
104	C₂×C₄×D₅	C₂×Q₈	1	4	yes	—
105	C₂×C₄×D₅	C₂×Q₈	1	4	yes	—
106	C₂×C₄×D₅	C₂⁴	1	2	yes	—
107	C₂×Dic₁₀	—	2	16	no	—
108	C₂×Dic₁₀	—	4	16	no	—
109	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
110	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
111	C₂×C₅×C₈	C₂⁴	1	2	yes	—
112	C₂×C₅×C₈	C₂⁴	1	2	yes	—
113	C₅×C₄⋊C₄	—	2	16	no	—
114	C₅×C₄⋊C₄	—	2	4	no	—
115	C₅×C₄⋊C₄	C₂⁴	1	8	yes	—
116	C₄²×C₅	C₂×D₄	1	2	yes	—
117	C₄²×C₅	C₂⁴	1	2	yes	—
118	#685	—	2	16	no	—
119	C₄⋊Dic₅	—	2	4	no	—
120	C₄⋊Dic₅	C₂⁴	1	8	yes	—
121	#683	—	2	16	no	—
122	#683	—	2	16	no	—
123	C₄×Dic₅	—	2	16	no	—
124	C₄×Dic₅	C₂×D₄	1	2	yes	—
125	C₄×Dic₅	C₂×D₄	1	8	yes	—
126	C₄×Dic₅	C₂⁴	1	2	yes	—
127	C₂×#168	C₂×Q₈	1	4	yes	—
128	C₂×#168	C₂×Q₈	1	4	yes	—
129	C₂×#168	C₂⁴	1	2	yes	—
130	C₂×#168	C₂⁴	1	2	yes	—
131	C₈×D₅	C₂×Q₈	1	16	yes	complemented
132	C₈×D₅	C₂×Q₈	1	16	yes	complemented
133	C₂³×C₈	—	5	1	no	—
134	C₂³×C₈	—	5	1	no	—
135	#533	—	5	16	no	—
136	#533	—	10	8	no	—
137	C₂×C₄⋊Q₈	—	5	4	no	—
138	C₂×#125	—	5	2	no	—
139	C₂×#125	C₂×D₅	1	2	yes	—
140	C₂×C₂²⋊Q₈	—	10	4	no	—
141	C₂×C₄×Q₈	—	10	4	no	—
142	C₂×#117	—	5	4	no	—
143	C₂²×C₄²	—	5	1	no	—
144	#500	—	5	16	no	—
145	#499	—	5	16	no	—
146	#499	—	5	8	no	—
147	#499	C₂×D₅	1	8	yes	complemented
148	#498	—	5	16	no	—
149	#427	—	5	16	no	—
150	#427	—	5	16	no	—
151	C₂×#107	—	5	2	no	—
152	C₂×#107	—	5	4	no	—
153	C₂×#107	C₂×D₅	1	2	yes	—
154	C₂×#106	—	5	2	no	—
155	C₂×#106	—	5	4	no	—
156	C₂×#106	C₂×D₅	1	2	yes	—
157	C₂×C₄×C₈	—	5	1	no	—
158	C₂×C₄×C₈	—	5	1	no	—
159	C₂×C₄×C₈	—	5	1	no	—
160	C₂×C₄×C₈	—	5	8	no	—
161	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
162	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
163	C₂²×D₅	—	2	4	no	—
164	C₂²×D₅	C₄⋊Q₈	1	4	yes	—
165	C₂²×D₅	C₂²×D₄	1	2	yes	—
166	C₂×C₄×C₅	—	2	4	no	—
167	C₂×C₄×C₅	—	2	8	no	—
168	C₂×C₄×C₅	—	4	8	no	—
169	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
170	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
172	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
173	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
174	C₂×Dic₅	—	2	8	no	—
175	C₂×Dic₅	—	4	8	no	—
176	C₂×Dic₅	C₄⋊Q₈	1	4	yes	—
177	C₂×Dic₅	#135	1	8	yes	—
178	C₂×Dic₅	C₂²×D₄	1	2	yes	—
179	C₂×Dic₅	C₂²×D₄	1	4	yes	—
180	C₄×D₅	—	2	16	no	—
181	C₄×D₅	C₄⋊Q₈	1	8	yes	—
182	C₄×D₅	C₂²×Q₈	1	4	yes	—
183	C₄×D₅	C₂²×Q₈	1	4	yes	—
184	Dic₁₀	—	4	32	no	—
185	C₅×C₈	C₂²×Q₈	1	4	yes	—
186	C₅×C₈	C₂²×Q₈	1	4	yes	—
187	#168	C₂²×Q₈	1	4	yes	—
188	#168	C₂²×Q₈	1	4	yes	—
189	C₂²×Q₈	—	10	4	no	—
190	C₂³×C₄	—	5	1	no	—
191	C₂³×C₄	—	5	2	no	—
192	C₂²×C₈	—	5	1	no	—
193	C₂²×C₈	—	5	1	no	—
194	C₂²×C₈	—	5	8	no	—
195	C₂²×C₈	—	5	8	no	—
196	C₂²×C₈	—	5	4	no	—
197	C₂²×C₈	—	5	4	no	—
198	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
199	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
200	C₄⋊Q₈	—	5	16	no	—
201	C₄⋊Q₈	—	10	8	no	—
202	#125	—	2	4	no	—
203	#125	—	5	8	no	—
204	#125	—	10	4	no	—
205	#125	C₂²×D₅	1	8	yes	—
206	C₂²⋊Q₈	—	10	32	no	—
207	C₄×Q₈	—	10	32	no	—
208	#117	—	5	16	no	—
209	#117	—	10	8	no	—
210	C₂×C₄⋊C₄	—	2	4	no	—
211	C₂×C₄⋊C₄	—	5	4	no	—
212	C₂×C₄⋊C₄	—	10	4	no	—
213	C₂×C₄⋊C₄	—	10	4	no	—
214	C₂×C₄⋊C₄	C₂²×D₅	1	4	yes	—
215	C₂×C₂²⋊C₄	—	10	4	no	—
216	C₂×C₄²	—	5	2	no	—
217	C₂×C₄²	—	5	1	no	—
218	C₂×C₄²	—	5	4	no	—
219	C₂×C₄²	—	5	4	no	—
220	C₂×C₄²	—	10	4	no	—
221	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
222	#107	—	5	8	no	—
223	#107	—	5	16	no	—
224	#107	C₂²×D₅	1	8	yes	—
225	#106	—	5	8	no	—
226	#106	—	5	16	no	—
227	#106	C₂²×D₅	1	8	yes	—
228	C₄×C₈	—	5	4	no	—
229	C₄×C₈	—	5	4	no	—
230	C₄×C₈	—	5	4	no	—
231	C₄×C₈	C₂²×D₅	1	4	yes	—
232	C₂²×C₅	C₂×C₄⋊Q₈	1	2	yes	—
233	C₂²×C₅	C₂×#135	1	4	yes	—
234	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
235	C₂×D₅	—	2	8	no	—
236	C₂×D₅	—	2	4	no	—
237	C₂×D₅	C₂×C₄⋊Q₈	1	4	yes	—
238	C₄×C₅	—	2	4	no	—
239	C₄×C₅	—	4	8	no	—
240	C₄×C₅	C₂×C₄⋊Q₈	1	2	yes	—
241	C₄×C₅	C₂³×Q₈	1	1	yes	characteristic
242	C₄×C₅	C₂³×Q₈	1	1	yes	characteristic
243	Dic₅	—	4	8	no	—
244	Dic₅	C₂×C₄⋊Q₈	1	4	yes	—
245	Dic₅	C₂×#135	1	8	yes	—
246	C₂⁴	—	5	1	no	—
247	C₂×Q₈	—	10	16	no	—
248	C₂×Q₈	—	20	16	no	—
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	5	1	no	—
251	C₂²×C₄	—	5	2	no	—
252	C₂²×C₄	—	5	2	no	—
253	C₂²×C₄	—	5	4	no	—
254	C₂²×C₄	—	5	4	no	—
255	C₂²×C₄	—	5	2	no	—
256	C₂²×C₄	—	5	8	no	—
257	C₂²×C₄	—	10	8	no	—
258	C₂²×C₄	—	10	4	no	—
259	C₂²×C₄	D₄×D₅	1	2	yes	—
260	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
261	C₂×C₈	—	5	4	no	—
262	C₂×C₈	—	5	4	no	—
263	C₂×C₈	—	5	2	no	—
264	C₂×C₈	—	5	2	no	—
265	C₂×C₈	—	5	16	no	—
266	C₂×C₈	—	5	16	no	—
267	C₂×C₈	Q₈×D₅	1	4	yes	—
268	C₂×C₈	Q₈×D₅	1	4	yes	—
269	C₂×C₈	C₂³×D₅	1	2	yes	—
270	C₂×C₈	C₂³×D₅	1	2	yes	—
271	C₄⋊C₄	—	2	16	no	—
272	C₄⋊C₄	—	2	4	no	—
273	C₄⋊C₄	—	5	8	no	—
274	C₄⋊C₄	—	10	16	no	—
275	C₄⋊C₄	—	10	4	no	—
276	C₄⋊C₄	—	10	16	no	—
277	C₄⋊C₄	C₂³×D₅	1	8	yes	—
278	C₂²⋊C₄	—	10	16	no	—
279	C₄²	—	5	8	no	—
280	C₄²	—	5	2	no	—
281	C₄²	—	5	2	no	—
282	C₄²	—	10	16	no	—
283	C₄²	D₄×D₅	1	2	yes	—
284	C₄²	C₂³×D₅	1	2	yes	—
285	C₂×C₅	#3160	1	4	yes	complemented
286	C₂×C₅	C₂²×C₄⋊Q₈	1	1	yes	characteristic
287	C₂×C₅	C₂²×#135	1	2	yes	—
288	D₅	—	2	4	no	—
289	C₂³	—	5	4	no	—
290	C₂³	—	5	2	no	—
291	C₂³	—	10	4	no	—
292	C₂³	C₂×D₄×D₅	1	1	yes	characteristic
293	Q₈	—	20	32	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	8	no	—
296	C₂×C₄	—	4	8	no	—
297	C₂×C₄	—	5	4	no	—
298	C₂×C₄	—	5	8	no	—
299	C₂×C₄	—	5	2	no	—
300	C₂×C₄	—	5	4	no	—
301	C₂×C₄	—	5	8	no	—
302	C₂×C₄	—	5	4	no	—
303	C₂×C₄	—	5	4	no	—
304	C₂×C₄	—	10	16	no	—
305	C₂×C₄	—	10	8	no	—
306	C₂×C₄	—	20	8	no	—
307	C₂×C₄	C₂×D₄×D₅	1	4	yes	—
308	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
309	C₂×C₄	C₂×Q₈×D₅	1	2	yes	—
310	C₂×C₄	C₂×Q₈×D₅	1	2	yes	—
311	C₂×C₄	C₂⁴×D₅	1	1	yes	characteristic
312	C₈	—	5	4	no	—
313	C₈	—	5	4	no	—
314	C₈	C₂×Q₈×D₅	1	4	yes	—
315	C₈	C₂×Q₈×D₅	1	4	yes	—
316	C₅	C₂×#3160	1	1	yes	characteristic, complemented, Sylow
317	C₂²	—	5	4	no	—
318	C₂²	—	10	8	no	—
319	C₂²	—	10	4	no	—
320	C₂²	D₅×C₄⋊Q₈	1	2	yes	—
321	C₂²	D₅×#135	1	4	yes	—
322	C₂²	C₂²×D₄×D₅	1	1	yes	characteristic
323	C₄	—	2	4	no	—
324	C₄	—	4	8	no	—
325	C₄	—	5	8	no	—
326	C₄	—	5	4	no	—
327	C₄	—	20	8	no	—
328	C₄	D₅×C₄⋊Q₈	1	2	yes	—
329	C₄	C₂²×Q₈×D₅	1	1	yes	characteristic
330	C₄	C₂²×Q₈×D₅	1	1	yes	characteristic
331	C₂	—	10	4	no	—
332	C₂	#10613647	1	4	yes	direct product factor
333	C₂	C₂×D₅×C₄⋊Q₈	1	1	yes	characteristic
334	C₂	C₂×D₅×#135	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×#499	C₂
socle	C₂³×C₅	C₂²×D₄
prime radical	C₂³×D₅	C₂×D₄
hypercenter	C₂×#499	D₅
hypocenter	C₅	C₂×#3160
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#499	C₂	semidirect
3. C₂×#65353	C₂	semidirect
4. #10613647	C₂	direct (N × H)
6. C₂×#65163	C₂	semidirect
8. C₂×#65144	C₂	semidirect
10. C₂×#64972	C₂	semidirect
22. C₅×#499	C₂²	semidirect
30. #65353	C₂²	semidirect
35. #65163	C₂²	semidirect
37. #65144	C₂²	semidirect
39. #64972	C₂²	semidirect
60. C₂×C₈×D₅	Q₈	semidirect
61. C₂×C₈×D₅	Q₈	semidirect
89. C₂×#499	D₅	semidirect
131. C₈×D₅	C₂×Q₈	semidirect
132. C₈×D₅	C₂×Q₈	semidirect
147. #499	C₂×D₅	semidirect
285. C₂×C₅	#3160	semidirect
316. C₅	C₂×#3160	semidirect
332. C₂	#10613647	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×#66022	2
C₂×C₅×#499	2
C₂×#65353	2
#10613647	2
C₂×#65169	2
C₂×#65163	2
C₂×#65149	2
C₂×#65144	2
C₂×C₄×C₈×D₅	2
C₂×#64972	2
C₂×#3160	5

Subgroups — C2×#10613647 #12968904

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10613647 #12968904