Subgroups — C₂×#10600522 #12068274

Order 1280.

830 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#65214	C₂	1	2	yes	maximal
3	#10600522	C₂	1	8	yes	maximal, direct product factor
4	C₂×#64678	C₂	1	2	yes	maximal, complemented
5	C₂×#64677	C₂	1	2	yes	maximal, complemented
6	C₂³×C₅×C₈	C₄	1	1	yes	characteristic
7	C₂³×C₅×C₈	C₄	1	1	yes	characteristic
8	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₅×C₈	C₄	1	1	yes	characteristic
11	C₂×C₄×C₅×C₈	C₄	1	4	yes	—
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	4	yes	—
15	C₂×C₂²⋊#168	—	2	2	no	—
16	#65214	—	2	4	no	—
17	#65214	C₂²	1	8	yes	—
18	C₂×C₄⋊#168	C₂²	1	4	yes	—
19	C₂×C₄×#168	—	2	2	no	—
20	#64678	C₂²	1	8	yes	complemented
21	#64677	C₂²	1	8	yes	complemented
22	C₂×#1564	—	5	1	no	maximal, Sylow
23	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₅	Q₈	1	1	yes	characteristic
26	C₂²×C₅×C₈	—	2	4	no	—
27	C₂²×C₅×C₈	C₈	1	8	yes	complemented
28	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
31	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	D₄	1	2	yes	—
37	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
39	C₂×C₄²×C₅	Q₈	1	2	yes	—
40	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₅	C₂³	1	2	yes	—
42	C₂²×#168	—	2	4	no	—
43	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
44	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
45	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
46	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
47	C₄×C₅×C₈	C₂³	1	2	yes	—
48	C₄×C₅×C₈	C₂³	1	2	yes	—
49	C₂²⋊#168	—	2	8	no	—
50	C₄⋊#168	—	2	4	no	—
51	C₄⋊#168	C₂³	1	8	yes	—
52	C₄×#168	—	2	8	no	—
53	C₂×#431	—	5	2	no	—
54	C₂²×C₄×C₈	D₅	1	1	yes	characteristic
55	#1564	—	5	8	no	—
56	C₂×#335	—	5	2	no	—
57	C₂×#334	—	5	2	no	—
58	C₂⁴×C₅	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	—	2	4	no	—
60	C₂²×C₄×C₅	—	2	4	no	—
61	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
68	C₂²×C₄×C₅	M₁₆	1	4	yes	—
69	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
75	C₂×C₅×C₈	—	2	8	no	—
76	C₂×C₅×C₈	—	2	2	no	—
77	C₂×C₅×C₈	—	2	2	no	—
78	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
79	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
80	C₂×C₅×C₈	C₂×C₈	1	16	yes	complemented
81	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
82	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
83	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
84	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
85	C₄²×C₅	C₄⋊C₄	1	2	yes	—
86	C₄²×C₅	C₄⋊C₄	1	2	yes	—
87	C₄²×C₅	C₄⋊C₄	1	2	yes	—
88	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₅	C₂²×C₄	1	2	yes	—
90	C₄²×C₅	C₂×D₄	1	2	yes	—
91	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₅	C₂×Q₈	1	2	yes	—
93	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
94	C₄²×C₅	C₂⁴	1	1	yes	characteristic
95	C₂×#168	—	2	8	no	—
96	C₂×#168	—	4	8	no	—
97	C₂³×C₈	Dic₅	1	1	yes	characteristic
98	C₂³×C₈	Dic₅	1	1	yes	characteristic
99	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
100	#431	—	5	8	no	—
101	#431	—	10	4	no	—
102	C₂×C₄⋊C₈	—	5	4	no	—
103	C₂×C₂²⋊C₈	—	10	2	no	—
104	C₂×C₄×C₈	—	10	2	no	—
105	C₂×C₄×C₈	Dic₅	1	1	yes	characteristic
106	C₂×C₄×C₈	Dic₅	1	1	yes	characteristic
107	C₂×C₄×C₈	Dic₅	1	4	yes	—
108	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
110	C₂×C₄×C₈	C₂×D₅	1	4	yes	—
111	#335	—	5	8	no	—
112	#334	—	5	8	no	—
113	C₂³×C₅	—	2	2	no	—
114	C₂³×C₅	—	2	2	no	—
115	C₂³×C₅	C₄⋊C₈	1	4	yes	—
116	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
117	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₅	—	2	2	no	—
119	C₂×C₄×C₅	—	2	2	no	—
120	C₂×C₄×C₅	—	2	4	no	—
121	C₂×C₄×C₅	—	2	4	no	—
122	C₂×C₄×C₅	—	2	4	no	—
123	C₂×C₄×C₅	—	2	4	no	—
124	C₂×C₄×C₅	C₄⋊C₈	1	8	yes	—
125	C₂×C₄×C₅	C₄⋊C₈	1	4	yes	—
126	C₂×C₄×C₅	#108	1	4	yes	—
127	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
133	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
134	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
135	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
136	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
137	C₂×C₄×C₅	#135	1	4	yes	—
138	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
141	C₅×C₈	—	2	2	no	—
142	C₅×C₈	—	2	2	no	—
143	C₅×C₈	C₂²×C₈	1	4	yes	—
144	C₅×C₈	C₂²×C₈	1	4	yes	—
145	#168	—	4	8	no	—
146	C₂³×C₄	Dic₁₀	1	1	yes	characteristic
147	C₂³×C₄	D₂₀	1	1	yes	characteristic
148	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
149	C₂²×C₈	—	2	4	no	—
150	C₂²×C₈	—	10	4	no	—
151	C₂²×C₈	#168	1	8	yes	complemented
152	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
153	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
154	C₂²×C₈	C₂×Dic₅	1	4	yes	—
155	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
156	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
157	C₂²×C₈	C₂×Dic₅	1	4	yes	—
158	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
159	C₂×C₄²	Dic₁₀	1	2	yes	—
160	C₂×C₄²	D₂₀	1	2	yes	—
161	C₂×C₄²	D₂₀	1	1	yes	characteristic
162	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
163	C₂×C₄²	C₂×Dic₅	1	2	yes	—
164	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
165	C₂×C₄²	C₂²×D₅	1	2	yes	—
166	C₄⋊C₈	—	5	8	no	—
167	C₄⋊C₈	—	10	4	no	—
168	C₂²⋊C₈	—	10	8	no	—
169	C₄×C₈	—	10	8	no	—
170	C₄×C₈	C₂×Dic₅	1	2	yes	—
171	C₄×C₈	C₂×Dic₅	1	2	yes	—
172	C₄×C₈	C₂×Dic₅	1	4	yes	—
173	C₄×C₈	C₂×Dic₅	1	4	yes	—
174	C₄×C₈	C₂²×D₅	1	2	yes	—
175	C₄×C₈	C₂²×D₅	1	2	yes	—
176	C₂²×C₅	—	2	8	no	—
177	C₂²×C₅	—	2	2	no	—
178	C₂²×C₅	—	2	2	no	—
179	C₂²×C₅	C₂×C₄⋊C₈	1	4	yes	—
180	C₂²×C₅	C₂×C₄⋊C₈	1	2	yes	—
181	C₂²×C₅	#427	1	2	yes	—
182	C₂²×C₅	C₂×#108	1	2	yes	—
183	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
184	C₄×C₅	—	2	2	no	—
185	C₄×C₅	—	2	1	no	—
186	C₄×C₅	—	2	1	no	—
187	C₄×C₅	C₂×C₄⋊C₈	1	2	yes	—
188	C₄×C₅	C₂×C₄⋊C₈	1	4	yes	—
189	C₄×C₅	#427	1	2	yes	—
190	C₄×C₅	C₂×#108	1	4	yes	—
191	C₄×C₅	C₂³×C₈	1	1	yes	characteristic
192	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
193	C₄×C₅	C₂×#135	1	2	yes	—
194	C₂⁴	C₄⋊Dic₅	1	1	yes	characteristic
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	C₂×#168	1	4	yes	—
198	C₂²×C₄	#681	1	4	yes	—
199	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
200	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
201	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
202	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
203	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
204	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
205	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
206	C₂²×C₄	C₂×Dic₁₀	1	2	yes	—
207	C₂²×C₄	C₂×D₂₀	1	2	yes	—
208	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
209	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
210	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
211	C₂×C₈	—	2	8	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	10	8	no	—
215	C₂×C₈	—	20	8	no	—
216	C₂×C₈	C₂×#168	1	4	yes	—
217	C₂×C₈	C₂×#168	1	4	yes	—
218	C₂×C₈	C₂×#168	1	16	yes	complemented
219	C₂×C₈	C₂²×Dic₅	1	2	yes	—
220	C₂×C₈	C₂²×Dic₅	1	2	yes	—
221	C₂×C₈	C₂²×Dic₅	1	2	yes	—
222	C₂×C₈	C₂²×Dic₅	1	2	yes	—
223	C₄²	C₄⋊Dic₅	1	2	yes	—
224	C₄²	C₄⋊Dic₅	1	2	yes	—
225	C₄²	C₄⋊Dic₅	1	2	yes	—
226	C₄²	C₂×Dic₁₀	1	2	yes	—
227	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
228	C₄²	C₂×D₂₀	1	2	yes	—
229	C₄²	C₂×D₂₀	1	1	yes	characteristic
230	C₄²	C₂²×Dic₅	1	2	yes	—
231	C₄²	C₂²×Dic₅	1	1	yes	characteristic
232	C₄²	C₂³×D₅	1	1	yes	characteristic
233	C₂×C₅	—	2	4	no	—
234	C₂×C₅	#1564	1	4	yes	complemented
235	C₂×C₅	C₂²×C₄⋊C₈	1	1	yes	characteristic
236	C₂×C₅	C₂×#427	1	1	yes	characteristic
237	C₂×C₅	C₂²×#108	1	1	yes	characteristic
238	C₂³	—	2	2	no	—
239	C₂³	—	2	2	no	—
240	C₂³	C₄⋊#168	1	4	yes	—
241	C₂³	C₂×C₄⋊Dic₅	1	2	yes	—
242	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	C₄⋊#168	1	8	yes	—
250	C₂×C₄	C₄⋊#168	1	4	yes	—
251	C₂×C₄	#3951	1	4	yes	—
252	C₂×C₄	#4050	1	4	yes	—
253	C₂×C₄	C₂²×#168	1	2	yes	—
254	C₂×C₄	C₂²×#168	1	4	yes	—
255	C₂×C₄	C₂×#681	1	4	yes	—
256	C₂×C₄	C₂×#681	1	2	yes	—
257	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
258	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
259	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
260	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
261	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
262	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
263	C₂×C₄	C₂²×Dic₁₀	1	1	yes	characteristic
264	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
265	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	—	2	2	no	—
268	C₈	—	20	8	no	—
269	C₈	C₂²×#168	1	4	yes	—
270	C₈	C₂²×#168	1	4	yes	—
271	C₅	C₂×#1564	1	1	yes	characteristic, complemented, Sylow
272	C₂²	—	2	8	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	C₂×C₄⋊#168	1	4	yes	—
276	C₂²	C₂×C₄⋊#168	1	2	yes	—
277	C₂²	#65395	1	2	yes	—
278	C₂²	C₂×#3951	1	2	yes	—
279	C₂²	C₂²×C₄⋊Dic₅	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	1	no	—
283	C₄	C₂×C₄⋊#168	1	2	yes	—
284	C₄	C₂×C₄⋊#168	1	4	yes	—
285	C₄	#65395	1	2	yes	—
286	C₄	C₂×#3951	1	4	yes	—
287	C₄	C₂×#4050	1	2	yes	—
288	C₄	C₂³×#168	1	1	yes	characteristic
289	C₄	C₂²×#681	1	1	yes	characteristic
290	C₂	—	2	4	no	—
291	C₂	#10600522	1	4	yes	direct product factor
292	C₂	C₂²×C₄⋊#168	1	1	yes	characteristic
293	C₂	C₂×#65395	1	1	yes	characteristic
294	C₂	C₂²×#3951	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	D₂₀
commutator	C₄×C₅	C₂³×C₈
Frattini	C₄²	C₂³×D₅
Fitting	C₂²×C₄×C₅×C₈	C₂
socle	C₂³×C₅	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅	C₄⋊C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	C₂²×C₄×C₈	D₅
hypocenter	C₅	C₂×#1564
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #10600522	C₂	direct (N × H)
4. C₂×#64678	C₂	semidirect
5. C₂×#64677	C₂	semidirect
20. #64678	C₂²	semidirect
21. #64677	C₂²	semidirect
27. C₂²×C₅×C₈	C₈	semidirect
80. C₂×C₅×C₈	C₂×C₈	semidirect
151. C₂²×C₈	#168	semidirect
218. C₂×C₈	C₂×#168	semidirect
234. C₂×C₅	#1564	semidirect
271. C₅	C₂×#1564	semidirect
291. C₂	#10600522	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₅×C₈	2
C₂×#65214	2
#10600522	2
C₂×#64678	2
C₂×#64677	2
C₂×#1564	5

Subgroups — C2×#10600522 #12068274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10600522 #12068274