Subgroups — C₂×#10601542 #12068356

Order 1280.

770 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×#65213	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊#168	C₂	1	1	yes	maximal, characteristic
4	#10601542	C₂	1	8	yes	maximal, direct product factor
5	C₂×#64677	C₂	1	4	yes	maximal, complemented
6	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₂²×#681	—	2	1	no	—
8	C₂³×#168	—	2	1	no	—
9	C₅×#431	C₄	1	4	yes	—
10	C₅×#431	C₂²	1	4	yes	—
11	C₂×C₅×C₄⋊C₈	C₄	1	2	yes	—
12	C₂×C₅×C₂²⋊C₈	C₄	1	2	yes	—
13	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
14	#65213	—	2	2	no	—
15	#65213	C₂²	1	4	yes	—
16	C₂×C₄⋊#168	—	2	1	no	—
17	C₂×C₄⋊#168	—	2	2	no	—
18	C₂×C₄⋊#168	—	2	1	no	—
19	C₂×C₄⋊#168	C₂²	1	4	yes	—
20	C₂×C₄⋊#168	C₂²	1	2	yes	—
21	C₂×C₄⋊#168	C₂²	1	2	yes	—
22	#64677	C₂²	1	16	yes	complemented
23	C₂×#1570	—	5	1	no	maximal, Sylow
24	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₅	Q₈	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₅	D₄	1	2	yes	—
32	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₅	Q₈	1	2	yes	—
34	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₅	C₂³	1	2	yes	—
36	C₂×#681	—	2	4	no	—
37	C₂×#681	—	4	4	no	—
38	C₂²×#168	—	2	4	no	—
39	C₂²×#168	—	2	2	no	—
40	C₂²×#168	—	2	2	no	—
41	C₂²×#168	—	4	4	no	—
42	C₅×C₄⋊C₈	C₂×C₄	1	8	yes	—
43	C₅×C₂²⋊C₈	C₂×C₄	1	8	yes	—
44	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
45	C₄×C₅×C₈	C₂³	1	4	yes	—
46	C₄⋊#168	—	2	4	no	—
47	C₄⋊#168	—	2	2	no	—
48	C₄⋊#168	—	2	4	no	—
49	C₄⋊#168	—	2	2	no	—
50	C₄⋊#168	C₂³	1	4	yes	—
51	C₄⋊#168	C₂³	1	4	yes	—
52	C₂×#431	D₅	1	1	yes	characteristic
53	C₂×#423	—	5	1	no	—
54	C₂²×C₄⋊C₈	—	5	1	no	—
55	#1570	—	5	8	no	—
56	C₂×#334	—	5	4	no	—
57	C₂⁴×C₅	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅	—	2	4	no	—
59	C₂²×C₄×C₅	—	4	2	no	—
60	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
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	SD₁₆	1	4	yes	—
67	C₂²×C₄×C₅	SD₁₆	1	4	yes	—
68	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
74	C₂×C₅×C₈	—	2	8	no	—
75	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
76	C₄²×C₅	C₄⋊C₄	1	2	yes	—
77	C₄²×C₅	C₄⋊C₄	1	2	yes	—
78	C₄²×C₅	C₄⋊C₄	1	2	yes	—
79	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₅	C₂²×C₄	1	2	yes	—
81	C₄²×C₅	C₂×D₄	1	2	yes	—
82	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₅	C₂×Q₈	1	2	yes	—
84	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
85	C₄²×C₅	C₂⁴	1	1	yes	characteristic
86	#681	—	4	8	no	—
87	C₂×#168	—	2	4	no	—
88	C₂×#168	—	2	4	no	—
89	C₂×#168	—	4	8	no	—
90	C₂×#168	—	4	4	no	—
91	C₂×#168	—	4	4	no	—
92	C₂²×M₁₆	—	10	1	no	—
93	C₂³×C₈	—	10	1	no	—
94	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
95	#431	Dic₅	1	4	yes	—
96	#431	C₂×D₅	1	4	yes	—
97	#423	—	5	4	no	—
98	#423	—	10	2	no	—
99	C₂×C₄⋊C₈	—	5	2	no	—
100	C₂×C₄⋊C₈	—	5	2	no	—
101	C₂×C₄⋊C₈	—	5	4	no	—
102	C₂×C₄⋊C₈	—	10	2	no	—
103	C₂×C₄⋊C₈	—	10	1	no	—
104	C₂×C₄⋊C₈	—	10	1	no	—
105	C₂×C₄⋊C₈	Dic₅	1	2	yes	—
106	C₂×C₂²⋊C₈	Dic₅	1	2	yes	—
107	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
108	#334	—	5	16	no	—
109	C₂³×C₅	—	2	2	no	—
110	C₂³×C₅	—	2	2	no	—
111	C₂³×C₅	#106	1	4	yes	—
112	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
113	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	2	2	no	—
117	C₂×C₄×C₅	—	2	4	no	—
118	C₂×C₄×C₅	—	2	4	no	—
119	C₂×C₄×C₅	—	4	2	no	—
120	C₂×C₄×C₅	—	4	2	no	—
121	C₂×C₄×C₅	#106	1	8	yes	—
122	C₂×C₄×C₅	#106	1	4	yes	—
123	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₅	#131	1	4	yes	—
130	C₂×C₄×C₅	C₂×SD₁₆	1	4	yes	—
131	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
132	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
133	C₂×C₄×C₅	C₂×SD₁₆	1	4	yes	—
134	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
137	C₅×C₈	—	2	8	no	—
138	#168	—	4	4	no	—
139	#168	—	4	4	no	—
140	C₂³×C₄	Dic₁₀	1	1	yes	characteristic
141	C₂³×C₄	D₂₀	1	1	yes	characteristic
142	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
143	C₂×M₁₆	—	10	4	no	—
144	C₂×M₁₆	—	20	4	no	—
145	C₂²×C₈	—	10	4	no	—
146	C₂²×C₈	—	10	2	no	—
147	C₂²×C₈	—	10	2	no	—
148	C₂²×C₈	—	20	4	no	—
149	C₂²×C₈	C₂×Dic₅	1	4	yes	—
150	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
151	C₂×C₄²	Dic₁₀	1	2	yes	—
152	C₂×C₄²	D₂₀	1	2	yes	—
153	C₂×C₄²	D₂₀	1	1	yes	characteristic
154	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
155	C₂×C₄²	C₂×Dic₅	1	2	yes	—
156	C₂×C₄²	C₂²×D₅	1	2	yes	—
157	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
158	C₄⋊C₈	—	5	4	no	—
159	C₄⋊C₈	—	5	4	no	—
160	C₄⋊C₈	—	10	2	no	—
161	C₄⋊C₈	—	10	4	no	—
162	C₄⋊C₈	—	10	2	no	—
163	C₄⋊C₈	—	10	4	no	—
164	C₄⋊C₈	C₂×Dic₅	1	8	yes	—
165	C₂²⋊C₈	C₂×Dic₅	1	8	yes	—
166	C₄×C₈	C₂×Dic₅	1	4	yes	—
167	C₄×C₈	C₂²×D₅	1	4	yes	—
168	C₂²×C₅	—	2	8	no	—
169	C₂²×C₅	—	2	2	no	—
170	C₂²×C₅	—	2	2	no	—
171	C₂²×C₅	#424	1	2	yes	—
172	C₂²×C₅	C₂×#106	1	4	yes	—
173	C₂²×C₅	C₂×#106	1	2	yes	—
174	C₂²×C₅	#430	1	2	yes	—
175	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
176	C₄×C₅	—	2	2	no	—
177	C₄×C₅	—	2	2	no	—
178	C₄×C₅	—	2	1	no	—
179	C₄×C₅	—	2	1	no	—
180	C₄×C₅	#424	1	2	yes	—
181	C₄×C₅	C₂×#106	1	2	yes	—
182	C₄×C₅	C₂×#106	1	4	yes	—
183	C₄×C₅	C₂×#131	1	2	yes	—
184	C₄×C₅	C₂²×SD₁₆	1	1	yes	characteristic
185	C₄×C₅	C₂²×SD₁₆	1	1	yes	characteristic
186	C₂⁴	C₄⋊Dic₅	1	1	yes	characteristic
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	#677	1	4	yes	—
190	C₂²×C₄	#677	1	4	yes	—
191	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
192	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
193	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
194	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
195	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
196	C₂²×C₄	C₄⋊Dic₅	1	2	yes	—
197	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
198	C₂²×C₄	C₂×Dic₁₀	1	2	yes	—
199	C₂²×C₄	C₂×D₂₀	1	2	yes	—
200	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
201	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
202	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
203	M₁₆	—	20	8	no	—
204	C₂×C₈	—	2	8	no	—
205	C₂×C₈	—	10	4	no	—
206	C₂×C₈	—	10	4	no	—
207	C₂×C₈	—	20	8	no	—
208	C₂×C₈	—	20	4	no	—
209	C₂×C₈	—	20	4	no	—
210	C₂×C₈	C₂²×Dic₅	1	8	yes	—
211	C₄²	C₄⋊Dic₅	1	2	yes	—
212	C₄²	C₄⋊Dic₅	1	2	yes	—
213	C₄²	C₄⋊Dic₅	1	2	yes	—
214	C₄²	C₂×Dic₁₀	1	2	yes	—
215	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
216	C₄²	C₂×D₂₀	1	2	yes	—
217	C₄²	C₂×D₂₀	1	1	yes	characteristic
218	C₄²	C₂²×Dic₅	1	1	yes	characteristic
219	C₄²	C₂²×Dic₅	1	2	yes	—
220	C₄²	C₂³×D₅	1	1	yes	characteristic
221	C₂×C₅	—	2	4	no	—
222	C₂×C₅	#1570	1	4	yes	complemented
223	C₂×C₅	C₂×#424	1	1	yes	characteristic
224	C₂×C₅	C₂²×#106	1	1	yes	characteristic
225	C₂×C₅	C₂×#430	1	1	yes	characteristic
226	C₂³	—	2	2	no	—
227	C₂³	—	2	2	no	—
228	C₂³	#3949	1	4	yes	—
229	C₂³	C₂×C₄⋊Dic₅	1	2	yes	—
230	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	#3949	1	8	yes	—
239	C₂×C₄	#3949	1	4	yes	—
240	C₂×C₄	C₂×#677	1	4	yes	—
241	C₂×C₄	C₂×#677	1	2	yes	—
242	C₂×C₄	C₂×#677	1	2	yes	—
243	C₂×C₄	C₂×#677	1	4	yes	—
244	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
245	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
246	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
247	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
249	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
250	C₂×C₄	#4094	1	4	yes	—
251	C₂×C₄	C₂²×Dic₁₀	1	1	yes	characteristic
252	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
253	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
254	C₈	—	2	8	no	—
255	C₈	—	20	4	no	—
256	C₈	—	20	4	no	—
257	C₅	C₂×#1570	1	1	yes	characteristic, complemented, Sylow
258	C₂²	—	2	8	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	#65288	1	2	yes	—
262	C₂²	C₂×#3949	1	4	yes	—
263	C₂²	C₂×#3949	1	2	yes	—
264	C₂²	#65414	1	2	yes	—
265	C₂²	C₂²×C₄⋊Dic₅	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	#65288	1	2	yes	—
271	C₄	C₂×#3949	1	4	yes	—
272	C₄	C₂×#3949	1	2	yes	—
273	C₄	C₂²×#677	1	1	yes	characteristic
274	C₄	C₂²×#677	1	1	yes	characteristic
275	C₄	C₂×#4094	1	2	yes	—
276	C₂	—	2	4	no	—
277	C₂	#10601542	1	4	yes	direct product factor
278	C₂	C₂×#65288	1	1	yes	characteristic
279	C₂	C₂²×#3949	1	1	yes	characteristic
280	C₂	C₂×#65414	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₂×C₅×#431	C₂
socle	C₂³×C₅	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅	C₄⋊C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	C₂×#431	D₅
hypocenter	C₅	C₂×#1570
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
4. #10601542	C₂	direct (N × H)
5. C₂×#64677	C₂	semidirect
22. #64677	C₂²	semidirect
222. C₂×C₅	#1570	semidirect
257. C₅	C₂×#1570	semidirect
277. C₂	#10601542	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×C₅×#431	2
C₂×#65213	2
C₂²×C₄⋊#168	2
#10601542	2
C₂×#64677	2
C₂×#1570	5

Subgroups — C2×#10601542 #12068356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10601542 #12068356