Subgroups — C₂×#10602248 #12253869

Order 1280.

658 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#504	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#65432	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#65431	C₂	1	1	yes	maximal, characteristic
4	C₂²×#3932	C₂	1	1	yes	maximal, characteristic, complemented
5	#10602248	C₂	1	8	yes	maximal, direct product factor
6	C₂×#64737	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#64736	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#64726	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅×#131	C₂²	1	1	yes	characteristic
10	C₂×#4097	—	2	1	no	—
11	C₂×#4095	—	2	1	no	—
12	C₂²×Dic₂₀	C₂²	1	1	yes	characteristic
13	C₂²×D₄₀	C₂²	1	1	yes	characteristic
14	C₅×#504	—	2	2	no	—
15	C₅×#504	C₂²	1	4	yes	complemented
16	C₂×C₅×M₃₂	—	2	1	no	—
17	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
18	C₂²×C₅×C₁₆	—	2	1	no	—
19	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
20	#65432	—	2	2	no	—
21	#65432	C₂²	1	4	yes	complemented
22	#65431	—	2	2	no	—
23	#65431	C₂²	1	4	yes	—
24	C₂×#3956	—	2	1	no	—
25	C₂×#3955	—	2	1	no	—
26	C₂×#3951	C₂²	1	1	yes	characteristic
27	C₂×#3951	C₂²	1	1	yes	characteristic
28	C₂×#3932	—	2	1	no	—
29	C₂×#3932	—	2	1	no	—
30	C₂×#3932	—	2	2	no	—
31	C₂×#3932	C₂²	1	4	yes	complemented
32	#64737	C₂²	1	4	yes	complemented
33	#64736	C₂²	1	4	yes	complemented
34	#64726	C₂²	1	4	yes	complemented
35	C₂×#2223	—	5	1	no	maximal, Sylow
36	C₂×C₅×#41	D₄	1	1	yes	characteristic
37	C₂²×D₂₀	—	2	1	no	—
38	C₂²×Dic₁₀	—	2	1	no	—
39	C₅×#131	—	2	1	no	—
40	C₅×#131	—	2	1	no	—
41	C₅×#131	D₄	1	2	yes	—
42	C₅×#131	C₂³	1	2	yes	—
43	C₂×C₅×M₁₆	—	2	1	no	—
44	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
45	C₂²×C₅×C₈	—	2	1	no	—
46	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
47	#4097	—	2	4	no	—
48	#4097	—	4	2	no	—
49	#4095	—	2	4	no	—
50	#4095	—	4	2	no	—
51	C₂×#689	—	4	1	no	—
52	C₂×#688	—	4	1	no	—
53	C₂×#687	—	4	1	no	—
54	C₂×#686	—	4	1	no	—
55	C₂×#681	—	2	1	no	—
56	C₂×#681	—	2	1	no	—
57	C₂×Dic₂₀	—	2	1	no	—
58	C₂×Dic₂₀	—	2	1	no	—
59	C₂×Dic₂₀	—	2	1	no	—
60	C₂×Dic₂₀	—	2	1	no	—
61	C₂×Dic₂₀	—	2	1	no	—
62	C₂×Dic₂₀	C₂³	1	2	yes	—
63	C₂×D₄₀	—	2	1	no	—
64	C₂×D₄₀	—	2	1	no	—
65	C₂×D₄₀	—	2	1	no	—
66	C₂×D₄₀	—	2	1	no	—
67	C₂×D₄₀	—	2	1	no	—
68	C₂×D₄₀	C₂³	1	2	yes	—
69	C₅×M₃₂	—	2	1	no	—
70	C₅×M₃₂	—	2	4	no	—
71	C₅×M₃₂	C₂³	1	2	yes	—
72	C₂×C₅×C₁₆	—	2	1	no	—
73	C₂×C₅×C₁₆	—	2	4	no	—
74	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
75	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
76	C₂×C₅×C₁₆	D₄	1	2	yes	—
77	C₂×C₅×C₁₆	C₂³	1	2	yes	—
78	#3956	—	2	4	no	—
79	#3955	—	2	4	no	—
80	#3951	—	2	1	no	—
81	#3951	—	2	1	no	—
82	#3951	C₂³	1	2	yes	—
83	#3951	C₂³	1	2	yes	—
84	#3932	—	2	4	no	—
85	#3932	—	2	4	no	—
86	C₂²×SD₃₂	—	5	1	no	—
87	C₂×#504	D₅	1	1	yes	characteristic, complemented
88	C₂×#473	—	5	1	no	—
89	C₂×#472	—	5	1	no	—
90	#2223	—	5	8	no	—
91	C₂×#368	—	5	1	no	—
92	C₂×#362	—	5	1	no	—
93	C₂×#361	—	5	1	no	—
94	C₂³×D₅	—	4	1	no	—
95	C₅×#41	—	2	1	no	—
96	C₅×#41	—	2	1	no	—
97	C₅×#41	D₈	1	2	yes	—
98	C₅×#41	C₂×D₄	1	2	yes	—
99	C₂×C₅×Q₈	D₈	1	1	yes	characteristic
100	C₂×C₅×D₄	—	2	1	no	—
101	C₂×C₅×D₄	D₈	1	1	yes	characteristic
102	C₂²×C₄×C₅	—	2	1	no	—
103	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
104	C₂²×Dic₅	—	4	1	no	—
105	C₂×D₂₀	—	2	2	no	—
106	C₂×D₂₀	—	4	1	no	—
107	C₂×D₂₀	—	4	1	no	—
108	C₂×D₂₀	—	4	1	no	—
109	C₂×D₂₀	—	4	1	no	—
110	C₂×D₂₀	—	4	1	no	—
111	C₂×Dic₁₀	—	2	2	no	—
112	C₂×Dic₁₀	—	4	1	no	—
113	C₂×Dic₁₀	—	4	1	no	—
114	C₂×Dic₁₀	—	4	1	no	—
115	C₂×Dic₁₀	—	4	1	no	—
116	C₂×Dic₁₀	—	4	1	no	—
117	C₅×M₁₆	—	2	1	no	—
118	C₅×M₁₆	—	2	2	no	—
119	C₅×M₁₆	—	2	2	no	—
120	C₅×M₁₆	C₂×D₄	1	2	yes	—
121	C₂×C₅×C₈	—	2	1	no	—
122	C₂×C₅×C₈	—	2	2	no	—
123	C₂×C₅×C₈	—	2	2	no	—
124	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
125	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
126	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
127	C₂×C₅×C₈	#41	1	1	yes	characteristic
128	C₂×C₅×C₈	#41	1	1	yes	characteristic
129	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
130	#689	—	4	4	no	—
131	#688	—	4	4	no	—
132	#687	—	4	4	no	—
133	#686	—	4	4	no	—
134	#681	—	2	2	no	—
135	#681	—	2	2	no	—
136	#681	—	4	1	no	—
137	#681	—	4	1	no	—
138	C₂×#168	—	4	1	no	—
139	C₂×#168	—	4	1	no	—
140	Dic₂₀	—	2	2	no	—
141	Dic₂₀	—	2	2	no	—
142	Dic₂₀	—	4	1	no	—
143	Dic₂₀	—	4	1	no	—
144	D₄₀	—	2	2	no	—
145	D₄₀	—	2	2	no	—
146	D₄₀	—	4	1	no	—
147	D₄₀	—	4	1	no	—
148	C₅×C₁₆	—	2	2	no	—
149	C₅×C₁₆	C₂×D₄	1	2	yes	—
150	C₅×C₁₆	C₂×D₄	1	2	yes	—
151	C₂×#137	—	10	1	no	—
152	C₂×AGL(1,ℤ/8)	—	10	1	no	—
153	C₂²×Q₁₆	—	5	1	no	—
154	C₂²×D₈	—	5	1	no	—
155	C₂×#131	C₂×D₅	1	1	yes	characteristic
156	C₂×SD₃₂	—	5	4	no	—
157	C₂×SD₃₂	—	10	1	no	—
158	C₂×SD₃₂	—	10	1	no	—
159	C₂×SD₃₂	—	10	2	no	—
160	#504	—	2	2	no	—
161	#504	C₂×D₅	1	4	yes	complemented
162	C₂×M₃₂	—	2	1	no	—
163	C₂×M₃₂	C₂×D₅	1	1	yes	characteristic
164	C₂²×C₁₆	—	2	1	no	—
165	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
166	#473	—	5	4	no	—
167	#473	—	10	2	no	—
168	#472	—	5	4	no	—
169	#472	—	10	2	no	—
170	C₂×#108	—	5	1	no	—
171	C₂×#108	—	5	1	no	—
172	C₂×#101	—	10	1	no	—
173	C₂×#100	—	10	1	no	—
174	#368	—	5	4	no	—
175	#362	—	5	4	no	—
176	#361	—	5	4	no	—
177	C₂³×C₅	—	2	1	no	—
178	C₂³×C₅	#135	1	1	yes	characteristic
179	C₂²×D₅	—	4	2	no	—
180	C₂²×D₅	—	8	1	no	—
181	C₂²×D₅	—	8	1	no	—
182	C₂²×D₅	—	16	2	no	—
183	C₅×Q₈	—	2	1	no	—
184	C₅×Q₈	C₂×D₈	1	2	yes	—
185	C₅×D₄	—	2	2	no	—
186	C₅×D₄	—	2	2	no	—
187	C₅×D₄	—	2	1	no	—
188	C₅×D₄	C₂×D₈	1	2	yes	—
189	C₂×C₄×C₅	—	2	1	no	—
190	C₂×C₄×C₅	—	2	2	no	—
191	C₂×C₄×C₅	—	2	2	no	—
192	C₂×C₄×C₅	#121	1	1	yes	characteristic
193	C₂×C₄×C₅	#121	1	1	yes	characteristic
194	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
195	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
196	C₂×C₄×C₅	#135	1	1	yes	characteristic
197	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
198	C₂×Dic₅	—	4	2	no	—
199	C₂×Dic₅	—	8	1	no	—
200	C₂×Dic₅	—	8	1	no	—
201	D₂₀	—	4	2	no	—
202	D₂₀	—	4	2	no	—
203	D₂₀	—	8	1	no	—
204	D₂₀	—	8	1	no	—
205	Dic₁₀	—	4	2	no	—
206	Dic₁₀	—	4	2	no	—
207	Dic₁₀	—	8	1	no	—
208	Dic₁₀	—	8	1	no	—
209	C₅×C₈	—	2	2	no	—
210	C₅×C₈	#121	1	1	yes	characteristic
211	C₅×C₈	#121	1	1	yes	characteristic
212	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
213	C₅×C₈	C₂×#41	1	1	yes	characteristic
214	#168	—	4	2	no	—
215	#168	—	4	2	no	—
216	C₂×#41	D₂₀	1	1	yes	characteristic
217	C₂²×Q₈	—	10	1	no	—
218	C₂²×D₄	—	10	1	no	—
219	#137	—	10	4	no	—
220	#137	—	20	2	no	—
221	AGL(1,ℤ/8)	—	10	4	no	—
222	AGL(1,ℤ/8)	—	20	2	no	—
223	C₂×Q₁₆	—	5	2	no	—
224	C₂×Q₁₆	—	10	1	no	—
225	C₂×Q₁₆	—	10	1	no	—
226	C₂×Q₁₆	—	10	1	no	—
227	C₂×Q₁₆	—	10	1	no	—
228	C₂×Q₁₆	—	10	1	no	—
229	C₂×Q₁₆	—	20	1	no	—
230	C₂×SD₁₆	—	20	1	no	—
231	C₂×SD₁₆	—	20	1	no	—
232	C₂×D₈	—	5	2	no	—
233	C₂×D₈	—	10	1	no	—
234	C₂×D₈	—	10	1	no	—
235	C₂×D₈	—	10	1	no	—
236	C₂×D₈	—	10	1	no	—
237	C₂×D₈	—	10	1	no	—
238	C₂×D₈	—	20	1	no	—
239	#131	—	2	1	no	—
240	#131	—	2	1	no	—
241	#131	D₂₀	1	2	yes	—
242	#131	C₂²×D₅	1	2	yes	—
243	C₂×M₁₆	—	2	1	no	—
244	C₂×M₁₆	—	10	1	no	—
245	C₂×M₁₆	—	10	1	no	—
246	C₂×M₁₆	D₂₀	1	1	yes	characteristic
247	C₂²×C₈	—	2	1	no	—
248	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
249	SD₃₂	—	10	4	no	—
250	SD₃₂	—	10	4	no	—
251	M₃₂	—	2	1	no	—
252	M₃₂	—	2	4	no	—
253	M₃₂	C₂²×D₅	1	2	yes	—
254	C₂×C₁₆	—	2	1	no	—
255	C₂×C₁₆	—	2	4	no	—
256	C₂×C₁₆	C₂²⋊D₅	1	1	yes	characteristic
257	C₂×C₁₆	C₂²⋊D₅	1	1	yes	characteristic
258	C₂×C₁₆	C₂²⋊D₅	1	2	yes	—
259	C₂×C₁₆	C₂²×D₅	1	2	yes	—
260	#108	—	5	2	no	—
261	#108	—	5	2	no	—
262	#108	—	10	1	no	—
263	#108	—	10	1	no	—
264	#101	—	10	4	no	—
265	#100	—	10	4	no	—
266	C₂²×C₅	—	2	2	no	—
267	C₂²×C₅	—	2	2	no	—
268	C₂²×C₅	—	4	2	no	—
269	C₂²×C₅	#466	1	1	yes	characteristic
270	C₂²×C₅	#466	1	1	yes	characteristic
271	C₂²×C₅	C₂×#135	1	1	yes	characteristic
272	C₂×D₅	—	8	2	no	—
273	C₂×D₅	—	16	2	no	—
274	C₂×D₅	—	16	2	no	—
275	C₂×D₅	—	16	2	no	—
276	C₄×C₅	—	2	2	no	—
277	C₄×C₅	#466	1	1	yes	characteristic
278	C₄×C₅	#466	1	1	yes	characteristic
279	C₄×C₅	C₂×#121	1	1	yes	characteristic
280	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
281	Dic₅	—	8	2	no	—
282	C₂⁴	—	20	1	no	—
283	#41	—	2	1	no	—
284	#41	—	2	1	no	—
285	#41	D₄₀	1	2	yes	—
286	#41	C₂×D₂₀	1	2	yes	—
287	C₂×Q₈	—	10	2	no	—
288	C₂×Q₈	—	20	1	no	—
289	C₂×Q₈	—	20	1	no	—
290	C₂×Q₈	—	20	1	no	—
291	C₂×Q₈	—	20	1	no	—
292	C₂×Q₈	—	20	1	no	—
293	C₂×Q₈	D₄₀	1	1	yes	characteristic
294	C₂×D₄	—	2	1	no	—
295	C₂×D₄	—	10	2	no	—
296	C₂×D₄	—	20	1	no	—
297	C₂×D₄	—	20	1	no	—
298	C₂×D₄	—	20	1	no	—
299	C₂×D₄	—	20	1	no	—
300	C₂×D₄	—	20	1	no	—
301	C₂×D₄	D₄₀	1	1	yes	characteristic
302	C₂²×C₄	—	2	1	no	—
303	C₂²×C₄	—	20	1	no	—
304	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
305	Q₁₆	—	10	2	no	—
306	Q₁₆	—	10	2	no	—
307	Q₁₆	—	20	4	no	—
308	Q₁₆	—	20	1	no	—
309	Q₁₆	—	20	1	no	—
310	SD₁₆	—	20	4	no	—
311	SD₁₆	—	20	4	no	—
312	D₈	—	10	2	no	—
313	D₈	—	10	2	no	—
314	D₈	—	20	4	no	—
315	D₈	—	20	1	no	—
316	D₈	—	20	1	no	—
317	M₁₆	—	2	1	no	—
318	M₁₆	—	2	2	no	—
319	M₁₆	—	2	2	no	—
320	M₁₆	—	10	2	no	—
321	M₁₆	—	10	2	no	—
322	M₁₆	—	20	1	no	—
323	M₁₆	—	20	1	no	—
324	M₁₆	C₂×D₂₀	1	2	yes	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	20	1	no	—
329	C₂×C₈	—	20	1	no	—
330	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
331	C₂×C₈	#709	1	1	yes	characteristic
332	C₂×C₈	#709	1	1	yes	characteristic
333	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
334	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
335	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
336	C₁₆	—	2	2	no	—
337	C₁₆	C₂×C₂²⋊D₅	1	2	yes	—
338	C₁₆	C₂×C₂²⋊D₅	1	2	yes	—
339	C₂×C₅	—	2	1	no	—
340	C₂×C₅	—	2	1	no	—
341	C₂×C₅	—	4	2	no	—
342	C₂×C₅	#2223	1	2	yes	complemented
343	C₂×C₅	C₂×#466	1	1	yes	characteristic
344	D₅	—	16	2	no	—
345	C₂³	—	2	1	no	—
346	C₂³	—	20	2	no	—
347	C₂³	—	40	1	no	—
348	C₂³	—	40	1	no	—
349	C₂³	—	80	2	no	—
350	C₂³	#4050	1	1	yes	characteristic
351	Q₈	—	2	1	no	—
352	Q₈	—	20	2	no	—
353	Q₈	—	20	2	no	—
354	Q₈	—	40	1	no	—
355	Q₈	—	40	1	no	—
356	Q₈	C₂×D₄₀	1	2	yes	—
357	D₄	—	2	2	no	—
358	D₄	—	2	2	no	—
359	D₄	—	2	1	no	—
360	D₄	—	20	2	no	—
361	D₄	—	20	2	no	—
362	D₄	—	40	1	no	—
363	D₄	—	40	1	no	—
364	D₄	C₂×D₄₀	1	2	yes	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	20	2	no	—
369	C₂×C₄	—	40	1	no	—
370	C₂×C₄	—	40	1	no	—
371	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
372	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
373	C₂×C₄	#4050	1	1	yes	characteristic
374	C₂×C₄	#4076	1	1	yes	characteristic
375	C₂×C₄	#4076	1	1	yes	characteristic
376	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
377	C₈	—	2	2	no	—
378	C₈	—	20	2	no	—
379	C₈	—	20	2	no	—
380	C₈	#4076	1	1	yes	characteristic
381	C₈	#4076	1	1	yes	characteristic
382	C₈	C₂×#709	1	1	yes	characteristic
383	C₈	C₂²×C₂²⋊D₅	1	1	yes	characteristic
384	C₅	C₂×#2223	1	1	yes	characteristic, complemented, Sylow
385	C₂²	—	2	2	no	—
386	C₂²	—	2	2	no	—
387	C₂²	—	4	2	no	—
388	C₂²	—	40	2	no	—
389	C₂²	—	80	2	no	—
390	C₂²	—	80	2	no	—
391	C₂²	—	80	2	no	—
392	C₂²	#65404	1	1	yes	characteristic
393	C₂²	#65404	1	1	yes	characteristic
394	C₂²	C₂×#4050	1	1	yes	characteristic
395	C₄	—	2	2	no	—
396	C₄	—	40	2	no	—
397	C₄	#65404	1	1	yes	characteristic
398	C₄	#65404	1	1	yes	characteristic
399	C₄	C₂²×D₄₀	1	1	yes	characteristic
400	C₄	C₂×#4076	1	1	yes	characteristic
401	C₂	—	2	1	no	—
402	C₂	—	2	1	no	—
403	C₂	—	4	2	no	—
404	C₂	—	80	2	no	—
405	C₂	#10602248	1	2	yes	direct product factor
406	C₂	C₂×#65404	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#65404
commutator, cosocle	C₂×C₅×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂×C₅×#504	C₂
socle	C₂²×C₅	#466
prime radical	C₂×#65431	C₂
hypercenter	C₂×#504	D₅
hypocenter	C₅	C₂×#2223
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#504	C₂	semidirect
2. C₂×#65432	C₂	semidirect
4. C₂²×#3932	C₂	semidirect
5. #10602248	C₂	direct (N × H)
6. C₂×#64737	C₂	semidirect
7. C₂×#64736	C₂	semidirect
8. C₂×#64726	C₂	semidirect
15. C₅×#504	C₂²	semidirect
21. #65432	C₂²	semidirect
31. C₂×#3932	C₂²	semidirect
32. #64737	C₂²	semidirect
33. #64736	C₂²	semidirect
34. #64726	C₂²	semidirect
87. C₂×#504	D₅	semidirect
161. #504	C₂×D₅	semidirect
342. C₂×C₅	#2223	semidirect
384. C₅	C₂×#2223	semidirect
405. C₂	#10602248	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×#504	2
C₂×#65432	2
C₂×#65431	2
C₂²×#3932	2
#10602248	2
C₂×#64737	2
C₂×#64736	2
C₂×#64726	2
C₂×#2223	5

Subgroups — C2×#10602248 #12253869

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10602248 #12253869