Subgroups — C₂×#10600518 #12064899

Order 1280.

966 conjugacy classes of subgroups.

The subgroup lattice has 292 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	1	yes	maximal, characteristic
3	C₂×C₄×#681	C₂	1	1	yes	maximal, characteristic
4	#10600518	C₂	1	8	yes	maximal, direct product factor
5	C₂×#64675	C₂	1	2	yes	maximal, complemented
6	C₂×#64674	C₂	1	2	yes	maximal, complemented
7	C₂³×C₅×C₈	C₄	1	2	yes	—
8	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂²×#681	C₄	1	2	yes	—
10	C₂×C₄×C₅×C₈	C₄	1	4	yes	—
11	C₂×C₄×C₅×C₈	C₄	1	2	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	C₄	1	2	yes	—
16	#65214	C₄	1	4	yes	—
17	#65214	C₂²	1	4	yes	—
18	C₂×C₄⋊#168	C₄	1	2	yes	—
19	C₄×#681	C₄	1	4	yes	—
20	C₄×#681	C₂²	1	4	yes	—
21	C₂×#3935	C₄	1	2	yes	—
22	C₂×C₄×#168	C₂²	1	2	yes	—
23	C₂×C₄×#168	C₂²	1	2	yes	—
24	#64675	C₂²	1	8	yes	complemented
25	#64674	C₂²	1	8	yes	complemented
26	C₂×#1451	—	5	1	no	maximal, Sylow
27	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₅×C₈	—	2	8	no	—
31	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
32	C₂²×C₅×C₈	C₂×C₄	1	8	yes	—
33	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅	C₂³	1	2	yes	—
41	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₂×#681	—	2	8	no	—
43	C₂×#681	C₂×C₄	1	8	yes	—
44	C₂²×#168	C₂×C₄	1	4	yes	—
45	C₂²×#168	C₂×C₄	1	4	yes	—
46	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
47	C₄×C₅×C₈	C₂×C₄	1	8	yes	—
48	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
49	C₄×C₅×C₈	C₂³	1	2	yes	—
50	C₄×C₅×C₈	C₂³	1	2	yes	—
51	C₂²⋊#168	C₂×C₄	1	8	yes	—
52	C₄⋊#168	C₂×C₄	1	8	yes	—
53	#3935	C₂×C₄	1	8	yes	—
54	C₄×#168	C₂×C₄	1	4	yes	—
55	C₄×#168	C₂×C₄	1	4	yes	—
56	C₄×#168	C₂³	1	4	yes	—
57	C₄×#168	C₂³	1	4	yes	—
58	C₂×#431	—	5	1	no	—
59	C₂×C₄×M₁₆	—	5	1	no	—
60	C₂²×C₄×C₈	D₅	1	1	yes	characteristic
61	#1451	—	5	8	no	—
62	C₂×#322	—	5	2	no	—
63	C₂×#322	—	5	2	no	—
64	C₂⁴×C₅	C₄²	1	1	yes	characteristic
65	C₂²×C₄×C₅	—	2	4	no	—
66	C₂²×C₄×C₅	—	2	4	no	—
67	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₄²	1	2	yes	—
69	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₄²	1	2	yes	—
71	C₂²×C₄×C₅	C₄²	1	2	yes	—
72	C₂²×C₄×C₅	C₄²	1	1	yes	characteristic
73	C₂²×C₄×C₅	M₁₆	1	8	yes	—
74	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₅×C₈	—	2	4	no	—
81	C₂×C₅×C₈	—	2	16	no	—
82	C₂×C₅×C₈	M₁₆	1	8	yes	complemented
83	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
84	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
85	C₄²×C₅	C₄²	1	2	yes	—
86	C₄²×C₅	C₄²	1	2	yes	—
87	C₄²×C₅	C₄²	1	2	yes	—
88	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₅	C₂²×C₄	1	2	yes	—
91	C₄²×C₅	C₂²×C₄	1	2	yes	—
92	C₄²×C₅	C₂²×C₄	1	2	yes	—
93	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₅	C₂⁴	1	1	yes	characteristic
95	#681	—	2	16	no	—
96	C₂×#168	—	2	8	no	—
97	C₂×#168	—	2	8	no	—
98	C₂×#168	C₂²×C₄	1	8	yes	—
99	C₂×#168	C₂²×C₄	1	8	yes	—
100	C₂²×M₁₆	—	5	2	no	—
101	C₂³×C₈	Dic₅	1	2	yes	—
102	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
103	#431	—	5	4	no	—
104	#431	—	5	4	no	—
105	C₂×C₄⋊C₈	—	5	2	no	—
106	C₂×C₂²⋊C₈	—	5	2	no	—
107	C₄×M₁₆	—	5	4	no	—
108	C₄×M₁₆	—	5	4	no	—
109	C₂×#97	—	5	2	no	—
110	C₂×C₄×C₈	—	5	2	no	—
111	C₂×C₄×C₈	—	5	2	no	—
112	C₂×C₄×C₈	Dic₅	1	4	yes	—
113	C₂×C₄×C₈	Dic₅	1	2	yes	—
114	C₂×C₄×C₈	C₂×D₅	1	4	yes	—
115	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
116	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
117	#322	—	5	8	no	—
118	#322	—	5	8	no	—
119	C₂³×C₅	—	2	2	no	—
120	C₂³×C₅	—	2	2	no	—
121	C₂³×C₅	#97	1	4	yes	—
122	C₂³×C₅	C₂×C₄²	1	2	yes	—
123	C₂³×C₅	C₂×C₄²	1	1	yes	characteristic
124	C₂×C₄×C₅	—	2	2	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	2	4	no	—
127	C₂×C₄×C₅	—	2	4	no	—
128	C₂×C₄×C₅	—	2	4	no	—
129	C₂×C₄×C₅	—	2	4	no	—
130	C₂×C₄×C₅	#97	1	8	yes	—
131	C₂×C₄×C₅	#97	1	4	yes	—
132	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
133	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
134	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
136	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
139	C₂×C₄×C₅	C₂×M₁₆	1	8	yes	—
140	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
141	C₂×C₄×C₅	#131	1	4	yes	—
142	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
145	C₅×C₈	—	2	4	no	—
146	C₅×C₈	C₂×M₁₆	1	8	yes	complemented
147	#168	—	2	8	no	—
148	#168	—	2	8	no	—
149	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
150	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
151	C₂³×C₄	C₂×Dic₅	1	1	yes	characteristic
152	C₂×M₁₆	—	5	8	no	—
153	C₂×M₁₆	—	10	8	no	—
154	C₂²×C₈	—	2	8	no	—
155	C₂²×C₈	—	5	4	no	—
156	C₂²×C₈	—	5	4	no	—
157	C₂²×C₈	C₂×Dic₅	1	2	yes	—
158	C₂²×C₈	C₂×Dic₅	1	8	yes	—
159	C₂²×C₈	C₂×Dic₅	1	2	yes	—
160	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
161	C₂×C₄²	C₄×D₅	1	2	yes	—
162	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
163	C₂×C₄²	C₄×D₅	1	2	yes	—
164	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
165	C₂×C₄²	C₂×Dic₅	1	2	yes	—
166	C₂×C₄²	C₂²×D₅	1	2	yes	—
167	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
168	C₄⋊C₈	—	5	8	no	—
169	C₂²⋊C₈	—	5	8	no	—
170	#97	—	5	8	no	—
171	C₄×C₈	—	5	4	no	—
172	C₄×C₈	—	5	4	no	—
173	C₄×C₈	—	5	4	no	—
174	C₄×C₈	—	5	4	no	—
175	C₄×C₈	C₂×Dic₅	1	8	yes	—
176	C₄×C₈	C₂×Dic₅	1	2	yes	—
177	C₄×C₈	C₂×Dic₅	1	2	yes	—
178	C₄×C₈	C₂²×D₅	1	2	yes	—
179	C₄×C₈	C₂²×D₅	1	2	yes	—
180	C₂²×C₅	—	2	2	no	—
181	C₂²×C₅	—	2	2	no	—
182	C₂²×C₅	—	2	8	no	—
183	C₂²×C₅	C₂×#97	1	4	yes	—
184	C₂²×C₅	C₂×#97	1	2	yes	—
185	C₂²×C₅	C₄×M₁₆	1	2	yes	—
186	C₂²×C₅	#405	1	2	yes	—
187	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
188	C₄×C₅	—	2	2	no	—
189	C₄×C₅	—	2	2	no	—
190	C₄×C₅	C₂×#97	1	2	yes	—
191	C₄×C₅	C₂×#97	1	4	yes	—
192	C₄×C₅	C₄×M₁₆	1	2	yes	—
193	C₄×C₅	#405	1	2	yes	—
194	C₄×C₅	C₂²×M₁₆	1	2	yes	—
195	C₄×C₅	C₂²×M₁₆	1	2	yes	—
196	C₄×C₅	C₂×#131	1	2	yes	—
197	C₂⁴	C₄×Dic₅	1	1	yes	characteristic
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	#676	1	8	yes	—
201	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
202	C₂²×C₄	C₄×Dic₅	1	2	yes	—
203	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
204	C₂²×C₄	C₄×Dic₅	1	2	yes	—
205	C₂²×C₄	C₄×Dic₅	1	2	yes	—
206	C₂²×C₄	C₄×Dic₅	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
208	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
209	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
211	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
212	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
213	M₁₆	—	10	16	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	16	no	—
216	C₂×C₈	—	5	8	no	—
217	C₂×C₈	—	5	8	no	—
218	C₂×C₈	—	10	8	no	—
219	C₂×C₈	—	10	8	no	—
220	C₂×C₈	#681	1	8	yes	complemented
221	C₂×C₈	C₂²×Dic₅	1	4	yes	—
222	C₂×C₈	C₂²×Dic₅	1	4	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×C₄×D₅	1	1	yes	characteristic
227	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
228	C₄²	C₂×C₄×D₅	1	2	yes	—
229	C₄²	C₂×C₄×D₅	1	2	yes	—
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	#1451	1	4	yes	complemented
235	C₂×C₅	C₂²×#97	1	1	yes	characteristic
236	C₂×C₅	C₂×C₄×M₁₆	1	1	yes	characteristic
237	C₂×C₅	C₂×#405	1	1	yes	characteristic
238	C₂³	—	2	2	no	—
239	C₂³	—	2	2	no	—
240	C₂³	#3947	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	#3947	1	8	yes	—
250	C₂×C₄	#3947	1	4	yes	—
251	C₂×C₄	C₂×#676	1	4	yes	—
252	C₂×C₄	C₂×#676	1	8	yes	—
253	C₂×C₄	#4047	1	4	yes	—
254	C₂×C₄	C₂×#681	1	4	yes	—
255	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
256	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
257	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
258	C₂×C₄	C₂×C₄×Dic₅	1	2	yes	—
259	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
260	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
261	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
262	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
263	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
264	C₈	—	2	4	no	—
265	C₈	—	10	8	no	—
266	C₈	—	10	8	no	—
267	C₈	C₂×#681	1	8	yes	complemented
268	C₅	C₂×#1451	1	1	yes	characteristic, complemented, Sylow
269	C₂²	—	2	2	no	—
270	C₂²	—	2	8	no	—
271	C₂²	—	2	2	no	—
272	C₂²	C₄×#681	1	2	yes	—
273	C₂²	C₂×#3947	1	4	yes	—
274	C₂²	C₂×#3947	1	2	yes	—
275	C₂²	#65390	1	2	yes	—
276	C₂²	C₂²×C₄×Dic₅	1	1	yes	characteristic
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	C₄×#681	1	2	yes	—
280	C₄	C₂×#3947	1	2	yes	—
281	C₄	C₂×#3947	1	4	yes	—
282	C₄	#65390	1	2	yes	—
283	C₄	C₂²×#676	1	2	yes	—
284	C₄	C₂×#4047	1	2	yes	—
285	C₄	C₂²×#681	1	2	yes	—
286	C₂	—	2	4	no	—
287	C₂	#10600518	1	4	yes	direct product factor
288	C₂	C₂×C₄×#681	1	1	yes	characteristic
289	C₂	C₂²×#3947	1	1	yes	characteristic
290	C₂	C₂×#65390	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²×D₅
commutator	C₂²×C₅	C₂²×C₄²
Frattini	C₄²	C₂³×D₅
Fitting	C₂²×C₄×C₅×C₈	C₂
socle	C₂³×C₅	C₂×C₄²
prime radical	C₂⁴×C₅	C₄²
cosocle	C₄²×C₅	C₂⁴
hypercenter	C₂²×C₄×C₈	D₅
hypocenter	C₅	C₂×#1451
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #10600518	C₂	direct (N × H)
5. C₂×#64675	C₂	semidirect
6. C₂×#64674	C₂	semidirect
24. #64675	C₂²	semidirect
25. #64674	C₂²	semidirect
82. C₂×C₅×C₈	M₁₆	semidirect
146. C₅×C₈	C₂×M₁₆	semidirect
220. C₂×C₈	#681	semidirect
234. C₂×C₅	#1451	semidirect
267. C₈	C₂×#681	semidirect
268. C₅	C₂×#1451	semidirect
287. C₂	#10600518	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
C₂×C₄×#681	2
#10600518	2
C₂×#64675	2
C₂×#64674	2
C₂×#1451	5

Subgroups — C2×#10600518 #12064899

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10600518 #12064899