Subgroups — C₂×#10601539 #12068280

Order 1280.

834 conjugacy classes of subgroups.

The subgroup lattice has 324 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	C₂²×C₄⋊#168	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#681	C₂	1	1	yes	maximal, characteristic
4	#10601539	C₂	1	8	yes	maximal, direct product factor
5	C₂×#64699	C₂	1	2	yes	maximal, complemented
6	C₂×#64700	C₂	1	2	yes	maximal, complemented
7	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂²×#681	C₄	1	1	yes	characteristic
9	C₂²×#681	C₄	1	1	yes	characteristic
10	C₂³×#168	—	2	1	no	—
11	C₅×#431	—	2	2	no	—
12	C₅×#431	C₂²	1	4	yes	—
13	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₅×C₂²⋊C₈	—	2	1	no	—
16	C₂×C₄×C₅×C₈	—	2	1	no	—
17	C₂×C₄⋊#168	—	2	1	no	—
18	C₂×C₄⋊#168	—	2	2	no	—
19	C₂×C₄⋊#168	C₂²	1	4	yes	—
20	C₂×C₄⋊#168	C₂²	1	2	yes	—
21	C₄×#681	C₄	1	4	yes	—
22	C₄×#681	C₂²	1	4	yes	—
23	C₂×#3935	C₄	1	2	yes	—
24	C₂×C₄×#168	C₂²	1	2	yes	—
25	#64700	C₂²	1	8	yes	complemented
26	#64699	C₂²	1	8	yes	complemented
27	C₂×#1565	—	5	1	no	maximal, Sylow
28	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₅	Q₈	1	1	yes	characteristic
31	C₂²×C₅×C₈	—	2	1	no	—
32	C₂²×C₅×C₈	—	2	1	no	—
33	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	D₄	1	2	yes	—
37	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₅	Q₈	1	2	yes	—
39	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₅	C₂³	1	2	yes	—
41	C₂×#681	—	2	4	no	—
42	C₂×#681	C₈	1	8	yes	complemented
43	C₂×#681	C₂×C₄	1	4	yes	—
44	C₂×#681	C₂×C₄	1	4	yes	—
45	C₂²×#168	—	2	4	no	—
46	C₂²×#168	—	2	2	no	—
47	C₂²×#168	—	4	4	no	—
48	C₂²×#168	C₂×C₄	1	2	yes	—
49	C₂²×#168	C₂×C₄	1	2	yes	—
50	C₅×C₄⋊C₈	—	2	1	no	—
51	C₅×C₄⋊C₈	—	2	1	no	—
52	C₅×C₄⋊C₈	C₂³	1	2	yes	—
53	C₅×C₄⋊C₈	C₂³	1	2	yes	—
54	C₅×C₂²⋊C₈	—	2	4	no	—
55	C₄×C₅×C₈	—	2	4	no	—
56	C₄⋊#168	—	2	4	no	—
57	C₄⋊#168	—	2	2	no	—
58	C₄⋊#168	C₂³	1	4	yes	—
59	#3935	C₂×C₄	1	8	yes	—
60	C₄×#168	C₂×C₄	1	4	yes	—
61	C₄×#168	C₂³	1	4	yes	—
62	C₂×#431	D₅	1	1	yes	characteristic
63	C₂²×C₄⋊C₈	—	5	1	no	—
64	C₂×C₄×M₁₆	—	5	1	no	—
65	#1565	—	5	8	no	—
66	C₂×#335	—	5	2	no	—
67	C₂×#334	—	5	2	no	—
68	C₂⁴×C₅	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	—	2	2	no	—
70	C₂²×C₄×C₅	—	2	2	no	—
71	C₂²×C₄×C₅	—	4	2	no	—
72	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
75	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
76	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
77	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
79	C₂²×C₄×C₅	M₁₆	1	4	yes	—
80	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
86	C₂×C₅×C₈	—	2	2	no	—
87	C₂×C₅×C₈	—	2	2	no	—
88	C₂×C₅×C₈	—	4	2	no	—
89	C₂×C₅×C₈	—	4	2	no	—
90	C₄²×C₅	C₄⋊C₄	1	2	yes	—
91	C₄²×C₅	C₄⋊C₄	1	2	yes	—
92	C₄²×C₅	C₄⋊C₄	1	2	yes	—
93	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₅	C₂²×C₄	1	2	yes	—
95	C₄²×C₅	C₂×D₄	1	2	yes	—
96	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₅	C₂×Q₈	1	2	yes	—
98	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
99	C₄²×C₅	C₂⁴	1	1	yes	characteristic
100	#681	—	2	8	no	—
101	#681	C₂×C₈	1	16	yes	complemented
102	C₂×#168	—	2	4	no	—
103	C₂×#168	—	2	4	no	—
104	C₂×#168	—	4	8	no	—
105	C₂×#168	—	4	4	no	—
106	C₂×#168	C₂×C₈	1	8	yes	complemented
107	C₂×#168	C₂²×C₄	1	4	yes	—
108	C₂×#168	C₂²×C₄	1	4	yes	—
109	C₂²×M₁₆	—	5	1	no	—
110	C₂²×M₁₆	—	5	1	no	—
111	C₂³×C₈	—	10	1	no	—
112	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
113	#431	—	2	2	no	—
114	#431	C₂×D₅	1	4	yes	—
115	C₂×C₄⋊C₈	—	5	2	no	—
116	C₂×C₄⋊C₈	—	5	4	no	—
117	C₂×C₄⋊C₈	—	10	2	no	—
118	C₂×C₄⋊C₈	—	10	1	no	—
119	C₂×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
120	C₂×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
121	C₂×C₂²⋊C₈	—	2	1	no	—
122	C₄×M₁₆	—	5	4	no	—
123	C₄×M₁₆	—	5	4	no	—
124	C₂×#97	—	5	2	no	—
125	C₂×C₄×C₈	—	2	1	no	—
126	C₂×C₄×C₈	—	5	2	no	—
127	#335	—	5	8	no	—
128	#334	—	5	8	no	—
129	C₂³×C₅	—	2	2	no	—
130	C₂³×C₅	—	2	2	no	—
131	C₂³×C₅	C₄⋊C₈	1	4	yes	—
132	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
133	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₅	—	2	2	no	—
135	C₂×C₄×C₅	—	2	2	no	—
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	—	2	4	no	—
140	C₂×C₄×C₅	—	4	2	no	—
141	C₂×C₄×C₅	—	4	2	no	—
142	C₂×C₄×C₅	C₄⋊C₈	1	8	yes	—
143	C₂×C₄×C₅	C₄⋊C₈	1	4	yes	—
144	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
146	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
147	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
148	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
151	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
152	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
153	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
154	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
155	C₂×C₄×C₅	#137	1	2	yes	—
156	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
159	C₅×C₈	—	4	2	no	—
160	C₅×C₈	—	4	2	no	—
161	#168	—	2	4	no	—
162	#168	—	4	4	no	—
163	#168	C₂²×C₈	1	8	yes	complemented
164	C₂³×C₄	Dic₁₀	1	1	yes	characteristic
165	C₂³×C₄	C₄×D₅	1	1	yes	characteristic
166	C₂³×C₄	C₂²⋊D₅	1	1	yes	characteristic
167	C₂×M₁₆	—	5	8	no	—
168	C₂×M₁₆	—	5	4	no	—
169	C₂×M₁₆	—	5	4	no	—
170	C₂×M₁₆	—	10	4	no	—
171	C₂²×C₈	—	2	1	no	—
172	C₂²×C₈	—	2	1	no	—
173	C₂²×C₈	—	5	2	no	—
174	C₂²×C₈	—	5	2	no	—
175	C₂²×C₈	—	10	4	no	—
176	C₂²×C₈	—	10	2	no	—
177	C₂²×C₈	—	20	4	no	—
178	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
179	C₂×C₄²	Dic₁₀	1	2	yes	—
180	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
181	C₂×C₄²	C₄×D₅	1	2	yes	—
182	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
183	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
184	C₂×C₄²	C₂²×D₅	1	2	yes	—
185	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
186	C₄⋊C₈	—	2	1	no	—
187	C₄⋊C₈	—	2	1	no	—
188	C₄⋊C₈	—	5	4	no	—
189	C₄⋊C₈	—	10	4	no	—
190	C₄⋊C₈	—	10	2	no	—
191	C₄⋊C₈	C₂²×D₅	1	2	yes	—
192	C₄⋊C₈	C₂²×D₅	1	2	yes	—
193	C₂²⋊C₈	—	2	4	no	—
194	#97	—	5	8	no	—
195	C₄×C₈	—	2	4	no	—
196	C₄×C₈	—	5	4	no	—
197	C₄×C₈	—	5	4	no	—
198	C₂²×C₅	—	2	8	no	—
199	C₂²×C₅	—	2	2	no	—
200	C₂²×C₅	—	2	2	no	—
201	C₂²×C₅	C₂×C₄⋊C₈	1	4	yes	—
202	C₂²×C₅	C₂×C₄⋊C₈	1	2	yes	—
203	C₂²×C₅	#428	1	2	yes	—
204	C₂²×C₅	#430	1	2	yes	—
205	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
206	C₄×C₅	—	2	2	no	—
207	C₄×C₅	—	2	2	no	—
208	C₄×C₅	—	2	1	no	—
209	C₄×C₅	—	2	1	no	—
210	C₄×C₅	C₂×C₄⋊C₈	1	2	yes	—
211	C₄×C₅	C₂×C₄⋊C₈	1	4	yes	—
212	C₄×C₅	#428	1	2	yes	—
213	C₄×C₅	C₂³×C₈	1	1	yes	characteristic
214	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
215	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
216	C₄×C₅	C₂×#137	1	1	yes	characteristic
217	C₂⁴	#683	1	1	yes	characteristic
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	C₈×D₅	1	4	yes	—
222	C₂²×C₄	#676	1	4	yes	—
223	C₂²×C₄	#683	1	1	yes	characteristic
224	C₂²×C₄	#683	1	2	yes	—
225	C₂²×C₄	#683	1	1	yes	characteristic
226	C₂²×C₄	#683	1	2	yes	—
227	C₂²×C₄	#683	1	2	yes	—
228	C₂²×C₄	#683	1	1	yes	characteristic
229	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
230	C₂²×C₄	C₂×Dic₁₀	1	2	yes	—
231	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
233	C₂²×C₄	C₂×C₂²⋊D₅	1	2	yes	—
234	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
235	M₁₆	—	5	16	no	—
236	M₁₆	—	10	8	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	5	4	no	—
242	C₂×C₈	—	5	8	no	—
243	C₂×C₈	—	5	4	no	—
244	C₂×C₈	—	10	4	no	—
245	C₂×C₈	—	10	4	no	—
246	C₂×C₈	—	20	8	no	—
247	C₂×C₈	—	20	4	no	—
248	C₄²	#683	1	2	yes	—
249	C₄²	#683	1	2	yes	—
250	C₄²	#683	1	2	yes	—
251	C₄²	C₂×Dic₁₀	1	2	yes	—
252	C₄²	C₂×Dic₁₀	1	1	yes	characteristic
253	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
254	C₄²	C₂×C₄×D₅	1	2	yes	—
255	C₄²	C₂×C₂²⋊D₅	1	2	yes	—
256	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
257	C₄²	C₂³×D₅	1	1	yes	characteristic
258	C₂×C₅	—	2	4	no	—
259	C₂×C₅	#1565	1	4	yes	complemented
260	C₂×C₅	C₂²×C₄⋊C₈	1	1	yes	characteristic
261	C₂×C₅	C₂×#428	1	1	yes	characteristic
262	C₂×C₅	C₂×#430	1	1	yes	characteristic
263	C₂³	—	2	2	no	—
264	C₂³	—	2	2	no	—
265	C₂³	#3946	1	4	yes	—
266	C₂³	C₂×#683	1	2	yes	—
267	C₂³	C₂×#683	1	1	yes	characteristic
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	#3946	1	8	yes	—
277	C₂×C₄	#3946	1	4	yes	—
278	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
279	C₂×C₄	C₂×C₈×D₅	1	4	yes	—
280	C₂×C₄	C₂×#676	1	2	yes	—
281	C₂×C₄	C₂×#676	1	4	yes	—
282	C₂×C₄	C₂×#683	1	2	yes	—
283	C₂×C₄	C₂×#683	1	1	yes	characteristic
284	C₂×C₄	C₂×#683	1	2	yes	—
285	C₂×C₄	C₂×#683	1	1	yes	characteristic
286	C₂×C₄	C₂×#683	1	2	yes	—
287	C₂×C₄	C₂×#683	1	1	yes	characteristic
288	C₂×C₄	#4095	1	2	yes	—
289	C₂×C₄	#4097	1	2	yes	—
290	C₂×C₄	C₂²×Dic₁₀	1	1	yes	characteristic
291	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
293	C₈	—	4	2	no	—
294	C₈	—	4	2	no	—
295	C₈	—	5	8	no	—
296	C₈	—	10	4	no	—
297	C₈	—	20	4	no	—
298	C₅	C₂×#1565	1	1	yes	characteristic, complemented, Sylow
299	C₂²	—	2	8	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	2	no	—
302	C₂²	#65284	1	2	yes	—
303	C₂²	C₂×#3946	1	4	yes	—
304	C₂²	C₂×#3946	1	2	yes	—
305	C₂²	#65413	1	2	yes	—
306	C₂²	C₂²×#683	1	1	yes	characteristic
307	C₄	—	2	2	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	2	no	—
310	C₄	—	2	1	no	—
311	C₄	#65284	1	2	yes	—
312	C₄	C₂×#3946	1	2	yes	—
313	C₄	C₂×#3946	1	4	yes	—
314	C₄	C₂²×C₈×D₅	1	1	yes	characteristic
315	C₄	C₂²×#676	1	1	yes	characteristic
316	C₄	C₂×#4095	1	1	yes	characteristic
317	C₄	C₂×#4097	1	1	yes	characteristic
318	C₂	—	2	4	no	—
319	C₂	#10601539	1	4	yes	direct product factor
320	C₂	C₂×#65284	1	1	yes	characteristic
321	C₂	C₂²×#3946	1	1	yes	characteristic
322	C₂	C₂×#65413	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅
commutator	C₄×C₅	C₂³×C₈
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×#1565
LCS	G ▷ 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. #10601539	C₂	direct (N × H)
5. C₂×#64699	C₂	semidirect
6. C₂×#64700	C₂	semidirect
25. #64700	C₂²	semidirect
26. #64699	C₂²	semidirect
42. C₂×#681	C₈	semidirect
101. #681	C₂×C₈	semidirect
106. C₂×#168	C₂×C₈	semidirect
163. #168	C₂²×C₈	semidirect
259. C₂×C₅	#1565	semidirect
298. C₅	C₂×#1565	semidirect
319. C₂	#10601539	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
C₂²×C₄⋊#168	2
C₂×C₄×#681	2
#10601539	2
C₂×#64699	2
C₂×#64700	2
C₂×#1565	5

Subgroups — C2×#10601539 #12068280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10601539 #12068280