Subgroups — C₂×#10598863 #12064898

Order 1280.

966 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#64974	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#676	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#322	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#64674	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#64674	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#64674	C₂	1	1	yes	maximal, characteristic, complemented
8	#10598863	C₂	1	8	yes	maximal, direct product factor
9	C₂²×#676	C₄	1	2	yes	—
10	C₂²×C₈×D₅	C₄	1	2	yes	—
11	C₂×C₄²×D₅	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₂×#3952	C₄	1	2	yes	—
16	C₂×#3947	C₄	1	2	yes	—
17	C₂×#3946	C₄	1	2	yes	—
18	C₂×C₈×Dic₅	C₄	1	2	yes	—
19	C₂×C₄×#168	C₂²	1	1	yes	characteristic
20	C₂×C₄×#168	C₂²	1	1	yes	characteristic
21	C₂×C₄×#168	C₂²	1	1	yes	characteristic
22	C₄×#676	C₄	1	4	yes	—
23	C₄×#676	C₂²	1	4	yes	—
24	#64974	C₄	1	4	yes	—
25	#64974	C₂²	1	4	yes	—
26	C₄×C₈×D₅	C₄	1	4	yes	—
27	C₄×C₈×D₅	C₂²	1	4	yes	—
28	C₅×#322	C₂²	1	4	yes	complemented
29	#64674	C₂²	1	4	yes	complemented
30	#64674	C₂²	1	4	yes	complemented
31	#64674	C₂²	1	4	yes	complemented
32	C₂×#1451	—	5	1	no	maximal, Sylow
33	C₂²×C₄×D₅	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×D₅	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×D₅	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
37	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
38	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
43	C₂²×#168	C₂×C₄	1	2	yes	—
44	C₂²×#168	C₂×C₄	1	2	yes	—
45	C₂²×#168	C₂×C₄	1	2	yes	—
46	C₂×#676	—	2	8	no	—
47	C₂×#676	C₂×C₄	1	8	yes	—
48	C₂×C₈×D₅	—	2	8	no	—
49	C₂×C₈×D₅	C₂×C₄	1	8	yes	—
50	C₄²×D₅	C₂×C₄	1	2	yes	—
51	C₄²×D₅	C₂×C₄	1	2	yes	—
52	C₄²×D₅	C₂×C₄	1	2	yes	—
53	C₄²×D₅	C₂³	1	2	yes	—
54	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
55	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
56	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
57	C₄×C₅×C₈	C₂³	1	2	yes	—
58	C₄×C₅×C₈	C₂³	1	2	yes	—
59	C₄×C₅×C₈	C₂³	1	2	yes	—
60	#3952	C₂×C₄	1	8	yes	—
61	#3947	C₂×C₄	1	8	yes	—
62	#3946	C₂×C₄	1	8	yes	—
63	C₈×Dic₅	C₂×C₄	1	8	yes	—
64	C₄×#168	C₂×C₄	1	2	yes	—
65	C₄×#168	C₂×C₄	1	2	yes	—
66	C₄×#168	C₂×C₄	1	2	yes	—
67	C₄×#168	C₂³	1	2	yes	—
68	C₄×#168	C₂³	1	2	yes	—
69	C₄×#168	C₂³	1	2	yes	—
70	C₂×#431	—	5	1	no	—
71	C₂×C₄×M₁₆	—	5	1	no	—
72	C₂²×C₄×C₈	—	5	1	no	—
73	#1451	—	5	8	no	—
74	C₂×#322	—	5	1	no	—
75	C₂×#322	—	5	1	no	—
76	C₂×#322	—	5	1	no	—
77	C₂×#322	D₅	1	1	yes	characteristic, complemented
78	C₂³×D₅	C₄²	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₂²×Dic₅	C₄²	1	1	yes	characteristic
83	C₂²×Dic₅	C₄²	1	1	yes	characteristic
84	C₂²×Dic₅	C₄²	1	1	yes	characteristic
85	C₂×C₄×D₅	—	2	4	no	—
86	C₂×C₄×D₅	—	2	4	no	—
87	C₂×C₄×D₅	C₄²	1	2	yes	—
88	C₂×C₄×D₅	C₄²	1	2	yes	—
89	C₂×C₄×D₅	C₄²	1	2	yes	—
90	C₂×C₄×D₅	M₁₆	1	8	yes	—
91	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
92	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
93	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
94	C₂×C₅×C₈	—	2	4	no	—
95	C₂×C₅×C₈	—	2	4	no	—
96	C₂×C₅×C₈	—	2	4	no	—
97	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
98	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
99	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
100	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₅	C₂⁴	1	1	yes	characteristic
104	C₄×Dic₅	C₄²	1	2	yes	—
105	C₄×Dic₅	C₄²	1	2	yes	—
106	C₄×Dic₅	C₄²	1	2	yes	—
107	C₄×Dic₅	C₂²×C₄	1	2	yes	—
108	C₄×Dic₅	C₂²×C₄	1	2	yes	—
109	C₄×Dic₅	C₂²×C₄	1	2	yes	—
110	C₂×#168	—	2	4	no	—
111	C₂×#168	—	2	4	no	—
112	C₂×#168	M₁₆	1	8	yes	complemented
113	C₂×#168	C₂²×C₄	1	4	yes	—
114	C₂×#168	C₂²×C₄	1	4	yes	—
115	C₂×#168	C₂²×C₄	1	4	yes	—
116	#676	—	2	16	no	—
117	C₈×D₅	—	2	16	no	—
118	C₂²×M₁₆	—	5	2	no	—
119	C₂³×C₈	—	5	2	no	—
120	C₂²×C₄²	—	5	1	no	—
121	#431	—	5	4	no	—
122	#431	—	5	4	no	—
123	C₂×C₄⋊C₈	—	5	2	no	—
124	C₂×C₂²⋊C₈	—	5	2	no	—
125	C₄×M₁₆	—	5	4	no	—
126	C₄×M₁₆	—	5	4	no	—
127	C₂×#97	—	5	2	no	—
128	C₂×C₄×C₈	—	5	2	no	—
129	C₂×C₄×C₈	—	5	1	no	—
130	C₂×C₄×C₈	—	5	1	no	—
131	C₂×C₄×C₈	—	5	1	no	—
132	C₂×C₄×C₈	—	5	4	no	—
133	C₂×C₄×C₈	—	5	4	no	—
134	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
135	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
136	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
137	#322	—	5	4	no	—
138	#322	—	5	4	no	—
139	#322	—	5	4	no	—
140	#322	C₂×D₅	1	4	yes	complemented
141	C₂³×C₅	C₂×C₄²	1	1	yes	characteristic
142	C₂²×D₅	—	2	2	no	—
143	C₂²×D₅	—	2	2	no	—
144	C₂²×D₅	#97	1	4	yes	—
145	C₂²×D₅	C₂×C₄²	1	2	yes	—
146	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
150	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
151	C₂×C₄×C₅	#131	1	4	yes	—
152	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
155	C₂×Dic₅	—	2	2	no	—
156	C₂×Dic₅	—	2	2	no	—
157	C₂×Dic₅	#97	1	4	yes	—
158	C₂×Dic₅	C₂×C₄²	1	2	yes	—
159	C₂×Dic₅	C₂×C₄²	1	2	yes	—
160	C₂×Dic₅	C₂×C₄²	1	2	yes	—
161	C₄×D₅	—	2	4	no	—
162	C₄×D₅	—	2	4	no	—
163	C₄×D₅	—	2	4	no	—
164	C₄×D₅	—	2	4	no	—
165	C₄×D₅	#97	1	8	yes	—
166	C₄×D₅	C₂×M₁₆	1	8	yes	—
167	C₅×C₈	—	2	4	no	—
168	C₅×C₈	—	2	4	no	—
169	C₅×C₈	—	2	4	no	—
170	#168	—	2	4	no	—
171	#168	—	2	4	no	—
172	#168	C₂×M₁₆	1	8	yes	complemented
173	C₂³×C₄	—	5	1	no	—
174	C₂³×C₄	—	5	1	no	—
175	C₂³×C₄	—	5	1	no	—
176	C₂×M₁₆	—	5	8	no	—
177	C₂×M₁₆	—	10	8	no	—
178	C₂²×C₈	—	5	2	no	—
179	C₂²×C₈	—	5	2	no	—
180	C₂²×C₈	—	5	2	no	—
181	C₂²×C₈	—	5	8	no	—
182	C₂²×C₈	—	10	8	no	—
183	C₂²×C₈	C₄×D₅	1	2	yes	—
184	C₂²×C₈	C₄×D₅	1	2	yes	—
185	C₂²×C₈	C₄×D₅	1	2	yes	—
186	C₂×C₄²	—	5	1	no	—
187	C₂×C₄²	—	5	1	no	—
188	C₂×C₄²	—	5	1	no	—
189	C₂×C₄²	—	5	2	no	—
190	C₂×C₄²	—	5	2	no	—
191	C₂×C₄²	—	5	2	no	—
192	C₂×C₄²	—	5	2	no	—
193	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
194	C₄⋊C₈	—	5	8	no	—
195	C₂²⋊C₈	—	5	8	no	—
196	#97	—	5	8	no	—
197	C₄×C₈	—	5	8	no	—
198	C₄×C₈	—	5	2	no	—
199	C₄×C₈	—	5	2	no	—
200	C₄×C₈	—	5	2	no	—
201	C₄×C₈	—	5	2	no	—
202	C₄×C₈	—	5	2	no	—
203	C₄×C₈	—	5	2	no	—
204	C₄×C₈	C₄×D₅	1	2	yes	—
205	C₄×C₈	C₄×D₅	1	2	yes	—
206	C₄×C₈	C₄×D₅	1	2	yes	—
207	C₄×C₈	C₂²×D₅	1	2	yes	—
208	C₄×C₈	C₂²×D₅	1	2	yes	—
209	C₄×C₈	C₂²×D₅	1	2	yes	—
210	C₂²×C₅	C₂×#97	1	2	yes	—
211	C₂²×C₅	C₄×M₁₆	1	2	yes	—
212	C₂²×C₅	#405	1	2	yes	—
213	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
214	C₂×D₅	—	2	2	no	—
215	C₂×D₅	—	2	2	no	—
216	C₂×D₅	—	2	8	no	—
217	C₂×D₅	C₂×#97	1	4	yes	—
218	C₄×C₅	C₂×#97	1	2	yes	—
219	C₄×C₅	C₄×M₁₆	1	2	yes	—
220	C₄×C₅	#405	1	2	yes	—
221	C₄×C₅	C₂²×M₁₆	1	2	yes	—
222	C₄×C₅	C₂²×M₁₆	1	2	yes	—
223	C₄×C₅	C₂×#131	1	2	yes	—
224	Dic₅	—	2	2	no	—
225	Dic₅	—	2	2	no	—
226	Dic₅	C₂×#97	1	4	yes	—
227	C₂⁴	—	5	1	no	—
228	C₂²×C₄	—	5	1	no	—
229	C₂²×C₄	—	5	2	no	—
230	C₂²×C₄	—	5	2	no	—
231	C₂²×C₄	—	5	8	no	—
232	C₂²×C₄	—	5	1	no	—
233	C₂²×C₄	—	5	1	no	—
234	C₂²×C₄	—	5	2	no	—
235	C₂²×C₄	—	5	2	no	—
236	C₂²×C₄	—	5	2	no	—
237	C₂²×C₄	—	5	2	no	—
238	C₂²×C₄	—	10	4	no	—
239	C₂²×C₄	—	10	4	no	—
240	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
243	M₁₆	—	10	16	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	5	4	no	—
248	C₂×C₈	—	5	8	no	—
249	C₂×C₈	—	5	4	no	—
250	C₂×C₈	—	5	4	no	—
251	C₂×C₈	—	10	16	no	—
252	C₂×C₈	—	10	4	no	—
253	C₂×C₈	—	10	4	no	—
254	C₂×C₈	C₂×C₄×D₅	1	4	yes	—
255	C₂×C₈	C₂×C₄×D₅	1	4	yes	—
256	C₂×C₈	C₂×C₄×D₅	1	4	yes	—
257	C₄²	—	5	2	no	—
258	C₄²	—	5	2	no	—
259	C₄²	—	5	2	no	—
260	C₄²	—	5	2	no	—
261	C₄²	—	5	2	no	—
262	C₄²	—	5	2	no	—
263	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
264	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
265	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
266	C₄²	C₂³×D₅	1	1	yes	characteristic
267	C₂×C₅	#1451	1	4	yes	complemented
268	C₂×C₅	C₂²×#97	1	1	yes	characteristic
269	C₂×C₅	C₂×C₄×M₁₆	1	1	yes	characteristic
270	C₂×C₅	C₂×#405	1	1	yes	characteristic
271	D₅	—	2	4	no	—
272	C₂³	—	5	2	no	—
273	C₂³	—	5	4	no	—
274	C₂³	—	10	2	no	—
275	C₂³	—	10	2	no	—
276	C₂³	C₄²×D₅	1	1	yes	characteristic
277	C₂×C₄	—	5	2	no	—
278	C₂×C₄	—	5	4	no	—
279	C₂×C₄	—	5	8	no	—
280	C₂×C₄	—	5	8	no	—
281	C₂×C₄	—	5	2	no	—
282	C₂×C₄	—	5	2	no	—
283	C₂×C₄	—	10	2	no	—
284	C₂×C₄	—	10	2	no	—
285	C₂×C₄	—	10	4	no	—
286	C₂×C₄	—	10	4	no	—
287	C₂×C₄	—	10	4	no	—
288	C₂×C₄	—	10	4	no	—
289	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
290	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
291	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
292	C₂×C₄	#4047	1	4	yes	—
293	C₂×C₄	D₅×M₁₆	1	4	yes	—
294	C₂×C₄	D₅×M₁₆	1	4	yes	—
295	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
296	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
297	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
298	C₈	—	2	4	no	—
299	C₈	—	2	4	no	—
300	C₈	—	2	4	no	—
301	C₈	—	5	8	no	—
302	C₈	—	10	4	no	—
303	C₈	—	10	4	no	—
304	C₅	C₂×#1451	1	1	yes	characteristic, complemented, Sylow
305	C₂²	—	5	4	no	—
306	C₂²	—	10	2	no	—
307	C₂²	—	10	2	no	—
308	C₂²	—	10	8	no	—
309	C₂²	#64978	1	2	yes	—
310	C₂²	D₅×#97	1	2	yes	—
311	C₂²	#64996	1	2	yes	—
312	C₂²	C₂×C₄²×D₅	1	1	yes	characteristic
313	C₄	—	5	4	no	—
314	C₄	—	10	2	no	—
315	C₄	—	10	2	no	—
316	C₄	#64978	1	2	yes	—
317	C₄	D₅×#97	1	2	yes	—
318	C₄	#64996	1	2	yes	—
319	C₄	C₂×#4047	1	2	yes	—
320	C₄	C₂×D₅×M₁₆	1	2	yes	—
321	C₄	C₂×D₅×M₁₆	1	2	yes	—
322	C₂	—	10	4	no	—
323	C₂	#10598863	1	4	yes	direct product factor
324	C₂	C₂×#64978	1	1	yes	characteristic
325	C₂	C₂×D₅×#97	1	1	yes	characteristic
326	C₂	C₂×#64996	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²×D₅
commutator	C₂²×C₅	C₂²×C₄²
Frattini	C₄²	C₂³×D₅
Fitting	C₂×C₅×#322	C₂
socle	C₂³×C₅	C₂×C₄²
prime radical	C₂³×D₅	C₄²
cosocle	C₄²×C₅	C₂⁴
hypercenter	C₂×#322	D₅
hypocenter	C₅	C₂×#1451
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
4. C₂×C₅×#322	C₂	semidirect
5. C₂×#64674	C₂	semidirect
6. C₂×#64674	C₂	semidirect
7. C₂×#64674	C₂	semidirect
8. #10598863	C₂	direct (N × H)
28. C₅×#322	C₂²	semidirect
29. #64674	C₂²	semidirect
30. #64674	C₂²	semidirect
31. #64674	C₂²	semidirect
77. C₂×#322	D₅	semidirect
112. C₂×#168	M₁₆	semidirect
140. #322	C₂×D₅	semidirect
172. #168	C₂×M₁₆	semidirect
267. C₂×C₅	#1451	semidirect
304. C₅	C₂×#1451	semidirect
323. C₂	#10598863	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×#64974	2
C₂×C₄×#676	2
C₂×C₄×C₈×D₅	2
C₂×C₅×#322	2
C₂×#64674	2
#10598863	2
C₂×#1451	5

Subgroups — C2×#10598863 #12064898

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10598863 #12064898