Finite Group Databasebeta

← C4×C41×#560 #23810552

Subgroups — C4×C41×#560 #23810552

Order 10496.

2568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C41×#560C211yesmaximal, characteristic
2C41×#2383C212yesmaximal, complemented
3C41×#2380C211yesmaximal, characteristic, complemented
4C41×#2377C216yesmaximal, complemented
5C41×#2376C213yesmaximal, complemented
6C41×#2375C213yesmaximal, complemented
7C4×#127×C41C211yesmaximal, characteristic, complemented
8C4×#126×C41C212yesmaximal, complemented
9C4×#123×C41C213yesmaximal, complemented
10C4×#121×C41C216yesmaximal, complemented
11C4×#120×C41C213yesmaximal, complemented
12C24×C4×C41C2211yescharacteristic
13C41×#560C4116yesdirect product factor
14C2×#127×C41C2211yescharacteristic
15C2×#126×C41C2212yes—
16C2×#123×C41C2213yes—
17C2×#121×C41C2216yes—
18C2×#120×C41C2213yes—
19C2×C4×D4×C41C2216yes—
20C2×C4×D4×C41C2213yes—
21C2×C4×D4×C41C2213yescomplemented
22C22×C42×C41C2213yescomplemented
23C41×#390C2216yes—
24C41×#390C2213yes—
25C41×#390C2213yescomplemented
26C41×#389C2216yescomplemented
27C41×#388C2216yes—
28C41×#388C2216yes—
29C41×#388C2216yes—
30C41×#388C2216yescomplemented
31C41×#387C2216yescomplemented
32C41×#387C2216yescomplemented
33C41×#386C2213yes—
34C41×#386C2216yes—
35C41×#386C2216yes—
36C41×#386C2213yes—
37C41×#386C2216yescomplemented
38C41×#385C2213yes—
39C41×#385C2216yes—
40C41×#385C2213yescomplemented
41C41×#384C2211yescharacteristic, complemented
42C41×#383C2212yescomplemented
43C41×#381C2213yescomplemented
44C41×#380C2216yescomplemented
45C41×#379C2213yes—
46C4×C4⋊C4×C41C2216yescomplemented
47C4×C22⋊C4×C41C2216yes—
48C4×C22⋊C4×C41C2213yes—
49C4×C22⋊C4×C41C2213yescomplemented
50C43×C41C2211yescharacteristic, complemented
51C25×C41C2311yescharacteristic
52C22×D4×C41C2316yes—
53C22×D4×C41C2313yes—
54C22×D4×C41C2313yes—
55C23×C4×C41—46no—
56C23×C4×C41C2313yes—
57C23×C4×C41C2312yes—
58C23×C4×C41C2311yescharacteristic
59C23×C4×C41C2313yes—
60C23×C4×C41C2311yescharacteristic
61C23×C4×C41C2312yes—
62C23×C4×C41C2313yes—
63#127×C41C2×C418yescomplemented
64#126×C41C2×C4116yescomplemented
65#123×C41C2×C4124yescomplemented
66#121×C41C2×C4148yescomplemented
67#120×C41C2×C4124yescomplemented
68C4×D4×C41—412no—
69C4×D4×C41—46no—
70C4×D4×C41—46no—
71C2×C4⋊C4×C41C2316yes—
72C2×C4⋊C4×C41C2313yes—
73C2×C4⋊C4×C41C2313yes—
74C2×C4⋊C4×C41C2316yes—
75C2×C4⋊C4×C41C2316yes—
76C2×C22⋊C4×C41C2316yes—
77C2×C22⋊C4×C41C2313yes—
78C2×C22⋊C4×C41C2316yes—
79C2×C22⋊C4×C41C2313yes—
80C2×C22⋊C4×C41C2316yes—
81C2×C22⋊C4×C41C2313yes—
82C2×C22⋊C4×C41C2316yes—
83C2×C22⋊C4×C41C2313yes—
84C2×C22⋊C4×C41C2316yes—
85C2×C22⋊C4×C41C2313yes—
86C2×C22⋊C4×C41C2313yes—
87C2×C42×C41—46no—
88C2×C42×C41C2316yes—
89C2×C42×C41C2313yes—
90C2×C42×C41C2316yes—
91C2×C42×C41C2313yes—
92C2×C42×C41C2311yescharacteristic
93#95×C41C2316yes—
94#95×C41C2316yes—
95#95×C41C2316yes—
96#95×C41C2316yes—
97#95×C41C2316yes—
98#95×C41C2316yes—
99C24×C41—43no—
100C24×C41—43no—
101C24×C41C22×C414yes—
102C24×C41C2412yes—
103C24×C41C2411yescharacteristic
104C24×C41C2413yes—
105C2×D4×C41—46no—
106C2×D4×C41—43no—
107C2×D4×C41—43no—
108C2×D4×C41—43no—
109C2×D4×C41—43no—
110C2×D4×C41—46no—
111C2×D4×C41C22×C4124yes—
112C2×D4×C41C22×C4112yes—
113C2×D4×C41C22×C4112yescomplemented
114C22×C4×C41—26no—
115C22×C4×C41—212no—
116C22×C4×C41—212no—
117C22×C4×C41—26no—
118C22×C4×C41—43no—
119C22×C4×C41—43no—
120C22×C4×C41—43no—
121C22×C4×C41—46no—
122C22×C4×C41—43no—
123C22×C4×C41—46no—
124C22×C4×C41—416no—
125C22×C4×C41C22×C4112yescomplemented
126C22×C4×C41C2416yes—
127C22×C4×C41C2412yes—
128C22×C4×C41C2411yescharacteristic
129C22×C4×C41C2413yes—
130C22×C4×C41C2413yes—
131C22×C4×C41C2413yes—
132C22×C4×C41C2416yes—
133C22×C4×C41C2411yescharacteristic
134C4⋊C4×C41—46no—
135C4⋊C4×C41—46no—
136C4⋊C4×C41—43no—
137C4⋊C4×C41—43no—
138C4⋊C4×C41C22×C4124yescomplemented
139C22⋊C4×C41—46no—
140C22⋊C4×C41—43no—
141C22⋊C4×C41—43no—
142C22⋊C4×C41—46no—
143C22⋊C4×C41—46no—
144C22⋊C4×C41—43no—
145C22⋊C4×C41—46no—
146C22⋊C4×C41—43no—
147C22⋊C4×C41C22×C4124yes—
148C22⋊C4×C41C22×C4112yes—
149C22⋊C4×C41C22×C4112yescomplemented
150C42×C41—212no—
151C42×C41—26no—
152C42×C41—46no—
153C42×C41—43no—
154C42×C41C22×C414yescomplemented
155C23×C41—26no—
156C23×C41—23no—
157C23×C41—26no—
158C23×C41—23no—
159C23×C41—23no—
160C23×C41—26no—
161C23×C41—23no—
162C23×C41—26no—
163C23×C41—412no—
164C23×C41—44no—
165C23×C41—412no—
166C23×C41C23×C414yes—
167C23×C41C23×C412yes—
168C23×C41C23×C416yes—
169C23×C41C2511yescharacteristic
170D4×C41—424no—
171D4×C41—412no—
172D4×C41—412no—
173C2×C4×C41—26no—
174C2×C4×C41—23no—
175C2×C4×C41—23no—
176C2×C4×C41—26no—
177C2×C4×C41—26no—
178C2×C4×C41—23no—
179C2×C4×C41—23no—
180C2×C4×C41—26no—
181C2×C4×C41—26no—
182C2×C4×C41—23no—
183C2×C4×C41—26no—
184C2×C4×C41—26no—
185C2×C4×C41—26no—
186C2×C4×C41—412no—
187C2×C4×C41—412no—
188C2×C4×C41—44no—
189C2×C4×C41—48no—
190C2×C4×C41—412no—
191C2×C4×C41—48no—
192C2×C4×C41—44no—
193C2×C4×C41C23×C416yes—
194C2×C4×C41C23×C4112yes—
195C2×C4×C41H(5,𝔽2)16yes—
196C4×#560C4111yesmaximal, characteristic, direct product factor, Sylow
197C22×C41—26no—
198C22×C41—212no—
199C22×C41—212no—
200C22×C41—26no—
201C22×C41—416no—
202C22×C41—42no—
203C22×C41—41no—
204C22×C41—43no—
205C22×C41—43no—
206C22×C41—46no—
207C22×C41—46no—
208C22×C41—43no—
209C22×C41C24×C411yescharacteristic
210C22×C41C2×H(5,𝔽2)13yes—
211C22×C41#58513yes—
212C4×C41—212no—
213C4×C41—26no—
214C4×C41—43no—
215C4×C41—42no—
216C4×C41—41no—
217C4×C41—43no—
218C4×C41—46no—
219C4×C41#56014yesdirect product factor
220C2×#560C2×C4111yescharacteristic
221#2383C2×C4112yescomplemented
222#2380C2×C4111yescharacteristic, complemented
223#2377C2×C4116yescomplemented
224#2376C2×C4113yescomplemented
225#2375C2×C4113yescomplemented
226C4×#127C2×C4111yescharacteristic, complemented
227C4×#126C2×C4112yescomplemented
228C4×#123C2×C4113yescomplemented
229C4×#121C2×C4116yescomplemented
230C4×#120C2×C4113yescomplemented
231C2×C41—46no—
232C2×C41—44no—
233C2×C41—42no—
234C2×C41C4×H(5,𝔽2)13yes—
235C2×C41C2×#56011yescharacteristic
236C2×C41#352913yes—
237C24×C4C22×C4111yescharacteristic
238#560C4×C41116yesdirect product factor
239C2×#127C22×C4111yescharacteristic
240C2×#126C22×C4112yes—
241C2×#123C22×C4113yes—
242C2×#121C22×C4116yes—
243C2×#120C22×C4113yes—
244C2×C4×D4C22×C4113yes—
245C2×C4×D4C22×C4116yes—
246C2×C4×D4C22×C4113yescomplemented
247C22×C42C22×C4113yescomplemented
248#390C22×C4113yes—
249#390C22×C4116yes—
250#390C22×C4113yescomplemented
251#389C22×C4116yescomplemented
252#388C22×C4116yes—
253#388C22×C4116yes—
254#388C22×C4116yes—
255#388C22×C4116yescomplemented
256#387C22×C4116yescomplemented
257#387C22×C4116yescomplemented
258#386C22×C4116yes—
259#386C22×C4113yes—
260#386C22×C4113yes—
261#386C22×C4116yes—
262#386C22×C4116yescomplemented
263#385C22×C4113yes—
264#385C22×C4116yes—
265#385C22×C4113yescomplemented
266#384C22×C4111yescharacteristic, complemented
267#383C22×C4112yescomplemented
268#381C22×C4113yescomplemented
269#380C22×C4116yescomplemented
270#379C22×C4113yes—
271C4×C4⋊C4C22×C4116yescomplemented
272C4×C22⋊C4C22×C4116yes—
273C4×C22⋊C4C22×C4113yes—
274C4×C22⋊C4C22×C4113yescomplemented
275C43C22×C4111yescharacteristic, complemented
276C41C4×#56011yescharacteristic, direct product factor, Sylow
277C25C23×C4111yescharacteristic
278C22×D4C23×C4116yes—
279C22×D4C23×C4113yes—
280C22×D4C23×C4113yes—
281C23×C4—46no—
282C23×C4C23×C4113yes—
283C23×C4C23×C4111yescharacteristic
284C23×C4C23×C4112yes—
285C23×C4C23×C4111yescharacteristic
286C23×C4C23×C4112yes—
287C23×C4C23×C4113yes—
288C23×C4C23×C4113yes—
289#127C2×C4×C4118yescomplemented
290#126C2×C4×C41116yescomplemented
291#123C2×C4×C41124yescomplemented
292#121C2×C4×C41148yescomplemented
293#120C2×C4×C41124yescomplemented
294C4×D4—412no—
295C4×D4—46no—
296C4×D4—46no—
297C2×C4⋊C4C23×C4116yes—
298C2×C4⋊C4C23×C4113yes—
299C2×C4⋊C4C23×C4113yes—
300C2×C4⋊C4C23×C4116yes—
301C2×C4⋊C4C23×C4116yes—
302C2×C22⋊C4C23×C4116yes—
303C2×C22⋊C4C23×C4113yes—
304C2×C22⋊C4C23×C4116yes—
305C2×C22⋊C4C23×C4113yes—
306C2×C22⋊C4C23×C4116yes—
307C2×C22⋊C4C23×C4113yes—
308C2×C22⋊C4C23×C4116yes—
309C2×C22⋊C4C23×C4113yes—
310C2×C22⋊C4C23×C4116yes—
311C2×C22⋊C4C23×C4113yes—
312C2×C22⋊C4C23×C4113yes—
313C2×C42—46no—
314C2×C42C23×C4113yes—
315C2×C42C23×C4116yes—
316C2×C42C23×C4113yes—
317C2×C42C23×C4116yes—
318C2×C42C23×C4111yescharacteristic
319#95C23×C4116yes—
320#95C23×C4116yes—
321#95C23×C4116yes—
322#95C23×C4116yes—
323#95C23×C4116yes—
324#95C23×C4116yes—
325C24—43no—
326C24—43no—
327C24C22×C4×C4114yes—
328C24C24×C4112yes—
329C24C24×C4111yescharacteristic
330C24C24×C4113yes—
331C2×D4—46no—
332C2×D4—43no—
333C2×D4—43no—
334C2×D4—43no—
335C2×D4—43no—
336C2×D4—46no—
337C2×D4C22×C4×C41124yes—
338C2×D4C22×C4×C41112yes—
339C2×D4C22×C4×C41112yescomplemented
340C22×C4—26no—
341C22×C4—212no—
342C22×C4—212no—
343C22×C4—26no—
344C22×C4—43no—
345C22×C4—43no—
346C22×C4—43no—
347C22×C4—46no—
348C22×C4—43no—
349C22×C4—46no—
350C22×C4—416no—
351C22×C4C22×C4×C41112yescomplemented
352C22×C4C24×C4116yes—
353C22×C4C24×C4112yes—
354C22×C4C24×C4111yescharacteristic
355C22×C4C24×C4113yes—
356C22×C4C24×C4113yes—
357C22×C4C24×C4113yes—
358C22×C4C24×C4116yes—
359C22×C4C24×C4111yescharacteristic
360C4⋊C4—46no—
361C4⋊C4—43no—
362C4⋊C4—43no—
363C4⋊C4—46no—
364C4⋊C4C22×C4×C41124yescomplemented
365C22⋊C4—46no—
366C22⋊C4—46no—
367C22⋊C4—43no—
368C22⋊C4—46no—
369C22⋊C4—43no—
370C22⋊C4—46no—
371C22⋊C4—43no—
372C22⋊C4—43no—
373C22⋊C4C22×C4×C41124yes—
374C22⋊C4C22×C4×C41112yes—
375C22⋊C4C22×C4×C41112yescomplemented
376C42—212no—
377C42—26no—
378C42—46no—
379C42—43no—
380C42C22×C4×C4114yescomplemented
381C23—26no—
382C23—23no—
383C23—26no—
384C23—23no—
385C23—23no—
386C23—26no—
387C23—23no—
388C23—26no—
389C23—412no—
390C23—44no—
391C23—412no—
392C23C23×C4×C4114yes—
393C23C23×C4×C4112yes—
394C23C23×C4×C4116yes—
395C23C25×C4111yescharacteristic
396D4—424no—
397D4—412no—
398D4—412no—
399C2×C4—26no—
400C2×C4—23no—
401C2×C4—23no—
402C2×C4—26no—
403C2×C4—26no—
404C2×C4—23no—
405C2×C4—23no—
406C2×C4—26no—
407C2×C4—26no—
408C2×C4—23no—
409C2×C4—26no—
410C2×C4—26no—
411C2×C4—26no—
412C2×C4—412no—
413C2×C4—44no—
414C2×C4—48no—
415C2×C4—412no—
416C2×C4—48no—
417C2×C4—44no—
418C2×C4—412no—
419C2×C4C23×C4×C4116yes—
420C2×C4C23×C4×C41112yes—
421C2×C4H(5,𝔽2)×C4116yes—
422C22—26no—
423C22—212no—
424C22—212no—
425C22—26no—
426C22—416no—
427C22—42no—
428C22—41no—
429C22—43no—
430C22—43no—
431C22—46no—
432C22—43no—
433C22—46no—
434C22C24×C4×C4111yescharacteristic
435C22C2×H(5,𝔽2)×C4113yes—
436C22C41×#58513yes—
437C4—212no—
438C4—26no—
439C4—42no—
440C4—41no—
441C4—43no—
442C4—46no—
443C4—43no—
444C4C41×#56014yesdirect product factor
445C2—44no—
446C2—42no—
447C2—46no—
448C2C4×H(5,𝔽2)×C4113yes—
449C2C2×C41×#56011yescharacteristic
450C2C41×#352913yes—
subgroupG/H
centerC22×C4×C41C24
commutatorC22C24×C4×C41
Frattini, cosocleC23C25×C41
socleC23×C41C25
prime radicalC2×C41×#560C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C41×#2383C2semidirect
3. C41×#2380C2semidirect
4. C41×#2377C2semidirect
5. C41×#2376C2semidirect
6. C41×#2375C2semidirect
7. C4×#127×C41C2semidirect
8. C4×#126×C41C2semidirect
9. C4×#123×C41C2semidirect
10. C4×#121×C41C2semidirect
11. C4×#120×C41C2semidirect
13. C41×#560C4direct (N × H)
21. C2×C4×D4×C41C22semidirect
22. C22×C42×C41C22semidirect
25. C41×#390C22semidirect
26. C41×#389C22semidirect
30. C41×#388C22semidirect
31. C41×#387C22semidirect
32. C41×#387C22semidirect
37. C41×#386C22semidirect
40. C41×#385C22semidirect
41. C41×#384C22semidirect
42. C41×#383C22semidirect
43. C41×#381C22semidirect
44. C41×#380C22semidirect
46. C4×C4⋊C4×C41C22semidirect
49. C4×C22⋊C4×C41C22semidirect
50. C43×C41C22semidirect
63. #127×C41C2×C4semidirect
64. #126×C41C2×C4semidirect
65. #123×C41C2×C4semidirect
66. #121×C41C2×C4semidirect
67. #120×C41C2×C4semidirect
113. C2×D4×C41C22×C4semidirect
125. C22×C4×C41C22×C4semidirect
138. C4⋊C4×C41C22×C4semidirect
149. C22⋊C4×C41C22×C4semidirect
154. C42×C41C22×C4semidirect
196. C4×#560C41direct (N × H)
219. C4×C41#560direct (N × H)
221. #2383C2×C41semidirect
222. #2380C2×C41semidirect
223. #2377C2×C41semidirect
224. #2376C2×C41semidirect
225. #2375C2×C41semidirect
226. C4×#127C2×C41semidirect
227. C4×#126C2×C41semidirect
228. C4×#123C2×C41semidirect
229. C4×#121C2×C41semidirect
230. C4×#120C2×C41semidirect
238. #560C4×C41direct (N × H)
246. C2×C4×D4C22×C41semidirect
247. C22×C42C22×C41semidirect
250. #390C22×C41semidirect
251. #389C22×C41semidirect
255. #388C22×C41semidirect
256. #387C22×C41semidirect
257. #387C22×C41semidirect
262. #386C22×C41semidirect
265. #385C22×C41semidirect
266. #384C22×C41semidirect
267. #383C22×C41semidirect
268. #381C22×C41semidirect
269. #380C22×C41semidirect
271. C4×C4⋊C4C22×C41semidirect
274. C4×C22⋊C4C22×C41semidirect
275. C43C22×C41semidirect
276. C41C4×#560direct (N × H)
289. #127C2×C4×C41semidirect
290. #126C2×C4×C41semidirect
291. #123C2×C4×C41semidirect
292. #121C2×C4×C41semidirect
293. #120C2×C4×C41semidirect
339. C2×D4C22×C4×C41semidirect
351. C22×C4C22×C4×C41semidirect
364. C4⋊C4C22×C4×C41semidirect
375. C22⋊C4C22×C4×C41semidirect
380. C42C22×C4×C41semidirect
444. C4C41×#560direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C41×#5602
C41×#23832
C41×#23802
C41×#23772
C41×#23762
C41×#23752
C4×#127×C412
C4×#126×C412
C4×#123×C412
C4×#121×C412
C4×#120×C412
C4×#56041