Finite Group Databasebeta

← C4×C23×#560 #19181977

Subgroups — C4×C23×#560 #19181977

Order 5888.

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×C23×#560C211yesmaximal, characteristic
2C23×#2383C212yesmaximal, complemented
3C23×#2380C211yesmaximal, characteristic, complemented
4C23×#2377C216yesmaximal, complemented
5C23×#2376C213yesmaximal, complemented
6C23×#2375C213yesmaximal, complemented
7C4×C23×#127C211yesmaximal, characteristic, complemented
8C4×C23×#126C212yesmaximal, complemented
9C4×C23×#123C213yesmaximal, complemented
10C4×C23×#121C216yesmaximal, complemented
11C4×C23×#120C213yesmaximal, complemented
12C24×C4×C23C2211yescharacteristic
13C23×#560C4116yesdirect product factor
14C2×C23×#127C2211yescharacteristic
15C2×C23×#126C2212yes—
16C2×C23×#123C2213yes—
17C2×C23×#121C2216yes—
18C2×C23×#120C2213yes—
19C2×C4×D4×C23C2216yes—
20C2×C4×D4×C23C2213yes—
21C2×C4×D4×C23C2213yescomplemented
22C22×C42×C23C2213yescomplemented
23C23×#390C2216yes—
24C23×#390C2213yes—
25C23×#390C2213yescomplemented
26C23×#389C2216yescomplemented
27C23×#388C2216yes—
28C23×#388C2216yes—
29C23×#388C2216yes—
30C23×#388C2216yescomplemented
31C23×#387C2216yescomplemented
32C23×#387C2216yescomplemented
33C23×#386C2213yes—
34C23×#386C2216yes—
35C23×#386C2216yes—
36C23×#386C2213yes—
37C23×#386C2216yescomplemented
38C23×#385C2213yes—
39C23×#385C2216yes—
40C23×#385C2213yescomplemented
41C23×#384C2211yescharacteristic, complemented
42C23×#383C2212yescomplemented
43C23×#381C2213yescomplemented
44C23×#380C2216yescomplemented
45C23×#379C2213yes—
46C4×C4⋊C4×C23C2216yescomplemented
47C4×C22⋊C4×C23C2216yes—
48C4×C22⋊C4×C23C2213yes—
49C4×C22⋊C4×C23C2213yescomplemented
50C43×C23C2211yescharacteristic, complemented
51C25×C23C2311yescharacteristic
52C22×D4×C23C2316yes—
53C22×D4×C23C2313yes—
54C22×D4×C23C2313yes—
55C23×C4×C23—46no—
56C23×C4×C23C2312yes—
57C23×C4×C23C2311yescharacteristic
58C23×C4×C23C2313yes—
59C23×C4×C23C2311yescharacteristic
60C23×C4×C23C2312yes—
61C23×C4×C23C2313yes—
62C23×C4×C23C2313yes—
63C23×#127C2×C418yescomplemented
64C23×#126C2×C4116yescomplemented
65C23×#123C2×C4124yescomplemented
66C23×#121C2×C4148yescomplemented
67C23×#120C2×C4124yescomplemented
68C4×D4×C23—412no—
69C4×D4×C23—46no—
70C4×D4×C23—46no—
71C2×C4⋊C4×C23C2316yes—
72C2×C4⋊C4×C23C2316yes—
73C2×C4⋊C4×C23C2313yes—
74C2×C4⋊C4×C23C2313yes—
75C2×C4⋊C4×C23C2316yes—
76C2×C22⋊C4×C23C2313yes—
77C2×C22⋊C4×C23C2316yes—
78C2×C22⋊C4×C23C2313yes—
79C2×C22⋊C4×C23C2316yes—
80C2×C22⋊C4×C23C2313yes—
81C2×C22⋊C4×C23C2316yes—
82C2×C22⋊C4×C23C2316yes—
83C2×C22⋊C4×C23C2313yes—
84C2×C22⋊C4×C23C2316yes—
85C2×C22⋊C4×C23C2313yes—
86C2×C22⋊C4×C23C2313yes—
87C2×C42×C23—46no—
88C2×C42×C23C2313yes—
89C2×C42×C23C2316yes—
90C2×C42×C23C2313yes—
91C2×C42×C23C2316yes—
92C2×C42×C23C2311yescharacteristic
93C23×#95C2316yes—
94C23×#95C2316yes—
95C23×#95C2316yes—
96C23×#95C2316yes—
97C23×#95C2316yes—
98C23×#95C2316yes—
99C24×C23—43no—
100C24×C23—43no—
101C24×C23C22×C414yes—
102C24×C23C2411yescharacteristic
103C24×C23C2412yes—
104C24×C23C2413yes—
105C2×D4×C23—46no—
106C2×D4×C23—43no—
107C2×D4×C23—43no—
108C2×D4×C23—43no—
109C2×D4×C23—43no—
110C2×D4×C23—46no—
111C2×D4×C23C22×C4124yes—
112C2×D4×C23C22×C4112yes—
113C2×D4×C23C22×C4112yescomplemented
114C22×C4×C23—26no—
115C22×C4×C23—212no—
116C22×C4×C23—212no—
117C22×C4×C23—26no—
118C22×C4×C23—43no—
119C22×C4×C23—43no—
120C22×C4×C23—43no—
121C22×C4×C23—46no—
122C22×C4×C23—43no—
123C22×C4×C23—46no—
124C22×C4×C23—416no—
125C22×C4×C23C22×C4112yescomplemented
126C22×C4×C23C2413yes—
127C22×C4×C23C2413yes—
128C22×C4×C23C2416yes—
129C22×C4×C23C2412yes—
130C22×C4×C23C2411yescharacteristic
131C22×C4×C23C2413yes—
132C22×C4×C23C2416yes—
133C22×C4×C23C2411yescharacteristic
134C4⋊C4×C23—46no—
135C4⋊C4×C23—46no—
136C4⋊C4×C23—43no—
137C4⋊C4×C23—43no—
138C4⋊C4×C23C22×C4124yescomplemented
139C22⋊C4×C23—46no—
140C22⋊C4×C23—43no—
141C22⋊C4×C23—43no—
142C22⋊C4×C23—46no—
143C22⋊C4×C23—46no—
144C22⋊C4×C23—43no—
145C22⋊C4×C23—46no—
146C22⋊C4×C23—43no—
147C22⋊C4×C23C22×C4124yes—
148C22⋊C4×C23C22×C4112yes—
149C22⋊C4×C23C22×C4112yescomplemented
150C42×C23—212no—
151C42×C23—26no—
152C42×C23—46no—
153C42×C23—43no—
154C42×C23C22×C414yescomplemented
155C4×#560C2311yesmaximal, characteristic, direct product factor, Sylow
156C23×C23—26no—
157C23×C23—23no—
158C23×C23—26no—
159C23×C23—23no—
160C23×C23—23no—
161C23×C23—26no—
162C23×C23—23no—
163C23×C23—26no—
164C23×C23—412no—
165C23×C23—44no—
166C23×C23—412no—
167C23×C23C23×C414yes—
168C23×C23C23×C412yes—
169C23×C23C23×C416yes—
170C23×C23C2511yescharacteristic
171D4×C23—424no—
172D4×C23—412no—
173D4×C23—412no—
174C2×C4×C23—26no—
175C2×C4×C23—23no—
176C2×C4×C23—23no—
177C2×C4×C23—26no—
178C2×C4×C23—26no—
179C2×C4×C23—23no—
180C2×C4×C23—23no—
181C2×C4×C23—26no—
182C2×C4×C23—26no—
183C2×C4×C23—23no—
184C2×C4×C23—26no—
185C2×C4×C23—26no—
186C2×C4×C23—26no—
187C2×C4×C23—412no—
188C2×C4×C23—412no—
189C2×C4×C23—44no—
190C2×C4×C23—48no—
191C2×C4×C23—412no—
192C2×C4×C23—48no—
193C2×C4×C23—44no—
194C2×C4×C23C23×C416yes—
195C2×C4×C23C23×C4112yes—
196C2×C4×C23H(5,𝔽2)16yes—
197C2×#560C2×C2311yescharacteristic
198#2383C2×C2312yescomplemented
199#2380C2×C2311yescharacteristic, complemented
200#2377C2×C2316yescomplemented
201#2376C2×C2313yescomplemented
202#2375C2×C2313yescomplemented
203C4×#127C2×C2311yescharacteristic, complemented
204C4×#126C2×C2312yescomplemented
205C4×#123C2×C2313yescomplemented
206C4×#121C2×C2316yescomplemented
207C4×#120C2×C2313yescomplemented
208C22×C23—26no—
209C22×C23—212no—
210C22×C23—212no—
211C22×C23—26no—
212C22×C23—416no—
213C22×C23—42no—
214C22×C23—41no—
215C22×C23—43no—
216C22×C23—43no—
217C22×C23—46no—
218C22×C23—46no—
219C22×C23—43no—
220C22×C23C24×C411yescharacteristic
221C22×C23C2×H(5,𝔽2)13yes—
222C22×C23#58513yes—
223C4×C23—212no—
224C4×C23—26no—
225C4×C23—43no—
226C4×C23—42no—
227C4×C23—41no—
228C4×C23—43no—
229C4×C23—46no—
230C4×C23#56014yesdirect product factor
231C24×C4C22×C2311yescharacteristic
232#560C4×C23116yesdirect product factor
233C2×#127C22×C2311yescharacteristic
234C2×#126C22×C2312yes—
235C2×#123C22×C2313yes—
236C2×#121C22×C2316yes—
237C2×#120C22×C2313yes—
238C2×C4×D4C22×C2313yes—
239C2×C4×D4C22×C2316yes—
240C2×C4×D4C22×C2313yescomplemented
241C22×C42C22×C2313yescomplemented
242#390C22×C2313yes—
243#390C22×C2316yes—
244#390C22×C2313yescomplemented
245#389C22×C2316yescomplemented
246#388C22×C2316yes—
247#388C22×C2316yes—
248#388C22×C2316yes—
249#388C22×C2316yescomplemented
250#387C22×C2316yescomplemented
251#387C22×C2316yescomplemented
252#386C22×C2316yes—
253#386C22×C2313yes—
254#386C22×C2313yes—
255#386C22×C2316yes—
256#386C22×C2316yescomplemented
257#385C22×C2313yes—
258#385C22×C2316yes—
259#385C22×C2313yescomplemented
260#384C22×C2311yescharacteristic, complemented
261#383C22×C2312yescomplemented
262#381C22×C2313yescomplemented
263#380C22×C2316yescomplemented
264#379C22×C2313yes—
265C4×C4⋊C4C22×C2316yescomplemented
266C4×C22⋊C4C22×C2316yes—
267C4×C22⋊C4C22×C2313yes—
268C4×C22⋊C4C22×C2313yescomplemented
269C43C22×C2311yescharacteristic, complemented
270C2×C23—46no—
271C2×C23—44no—
272C2×C23—42no—
273C2×C23C4×H(5,𝔽2)13yes—
274C2×C23C2×#56011yescharacteristic
275C2×C23#352913yes—
276C25C23×C2311yescharacteristic
277C22×D4C23×C2316yes—
278C22×D4C23×C2313yes—
279C22×D4C23×C2313yes—
280C23×C4—46no—
281C23×C4C23×C2313yes—
282C23×C4C23×C2311yescharacteristic
283C23×C4C23×C2312yes—
284C23×C4C23×C2311yescharacteristic
285C23×C4C23×C2312yes—
286C23×C4C23×C2313yes—
287C23×C4C23×C2313yes—
288#127C2×C4×C2318yescomplemented
289#126C2×C4×C23116yescomplemented
290#123C2×C4×C23124yescomplemented
291#121C2×C4×C23148yescomplemented
292#120C2×C4×C23124yescomplemented
293C4×D4—412no—
294C4×D4—46no—
295C4×D4—46no—
296C2×C4⋊C4C23×C2316yes—
297C2×C4⋊C4C23×C2313yes—
298C2×C4⋊C4C23×C2313yes—
299C2×C4⋊C4C23×C2316yes—
300C2×C4⋊C4C23×C2316yes—
301C2×C22⋊C4C23×C2316yes—
302C2×C22⋊C4C23×C2313yes—
303C2×C22⋊C4C23×C2316yes—
304C2×C22⋊C4C23×C2313yes—
305C2×C22⋊C4C23×C2316yes—
306C2×C22⋊C4C23×C2313yes—
307C2×C22⋊C4C23×C2316yes—
308C2×C22⋊C4C23×C2313yes—
309C2×C22⋊C4C23×C2316yes—
310C2×C22⋊C4C23×C2313yes—
311C2×C22⋊C4C23×C2313yes—
312C2×C42—46no—
313C2×C42C23×C2313yes—
314C2×C42C23×C2316yes—
315C2×C42C23×C2313yes—
316C2×C42C23×C2316yes—
317C2×C42C23×C2311yescharacteristic
318#95C23×C2316yes—
319#95C23×C2316yes—
320#95C23×C2316yes—
321#95C23×C2316yes—
322#95C23×C2316yes—
323#95C23×C2316yes—
324C23C4×#56011yescharacteristic, direct product factor, Sylow
325C24—43no—
326C24—43no—
327C24C22×C4×C2314yes—
328C24C24×C2312yes—
329C24C24×C2311yescharacteristic
330C24C24×C2313yes—
331C2×D4—46no—
332C2×D4—43no—
333C2×D4—43no—
334C2×D4—43no—
335C2×D4—43no—
336C2×D4—46no—
337C2×D4C22×C4×C23124yes—
338C2×D4C22×C4×C23112yes—
339C2×D4C22×C4×C23112yescomplemented
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×C23112yescomplemented
352C22×C4C24×C2316yes—
353C22×C4C24×C2312yes—
354C22×C4C24×C2311yescharacteristic
355C22×C4C24×C2313yes—
356C22×C4C24×C2313yes—
357C22×C4C24×C2313yes—
358C22×C4C24×C2316yes—
359C22×C4C24×C2311yescharacteristic
360C4⋊C4—46no—
361C4⋊C4—43no—
362C4⋊C4—43no—
363C4⋊C4—46no—
364C4⋊C4C22×C4×C23124yescomplemented
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×C23124yes—
374C22⋊C4C22×C4×C23112yes—
375C22⋊C4C22×C4×C23112yescomplemented
376C42—212no—
377C42—26no—
378C42—46no—
379C42—43no—
380C42C22×C4×C2314yescomplemented
381C23—26no—
382C23—23no—
383C23—26no—
384C23—23no—
385C23—23no—
386C23—26no—
387C23—23no—
388C23—26no—
389C23—412no—
390C23—44no—
391C23—412no—
392C23C23×C4×C2314yes—
393C23C23×C4×C2312yes—
394C23C23×C4×C2316yes—
395C23C25×C2311yescharacteristic
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×C2316yes—
420C2×C4C23×C4×C23112yes—
421C2×C4C23×H(5,𝔽2)16yes—
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×C2311yescharacteristic
435C22C2×C23×H(5,𝔽2)13yes—
436C22C23×#58513yes—
437C4—212no—
438C4—26no—
439C4—42no—
440C4—41no—
441C4—43no—
442C4—46no—
443C4—43no—
444C4C23×#56014yesdirect product factor
445C2—44no—
446C2—42no—
447C2—46no—
448C2C4×C23×H(5,𝔽2)13yes—
449C2C2×C23×#56011yescharacteristic
450C2C23×#352913yes—
subgroupG/H
centerC22×C4×C23C24
commutatorC22C24×C4×C23
Frattini, cosocleC23C25×C23
socleC23×C23C25
prime radicalC2×C23×#560C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2383C2semidirect
3. C23×#2380C2semidirect
4. C23×#2377C2semidirect
5. C23×#2376C2semidirect
6. C23×#2375C2semidirect
7. C4×C23×#127C2semidirect
8. C4×C23×#126C2semidirect
9. C4×C23×#123C2semidirect
10. C4×C23×#121C2semidirect
11. C4×C23×#120C2semidirect
13. C23×#560C4direct (N × H)
21. C2×C4×D4×C23C22semidirect
22. C22×C42×C23C22semidirect
25. C23×#390C22semidirect
26. C23×#389C22semidirect
30. C23×#388C22semidirect
31. C23×#387C22semidirect
32. C23×#387C22semidirect
37. C23×#386C22semidirect
40. C23×#385C22semidirect
41. C23×#384C22semidirect
42. C23×#383C22semidirect
43. C23×#381C22semidirect
44. C23×#380C22semidirect
46. C4×C4⋊C4×C23C22semidirect
49. C4×C22⋊C4×C23C22semidirect
50. C43×C23C22semidirect
63. C23×#127C2×C4semidirect
64. C23×#126C2×C4semidirect
65. C23×#123C2×C4semidirect
66. C23×#121C2×C4semidirect
67. C23×#120C2×C4semidirect
113. C2×D4×C23C22×C4semidirect
125. C22×C4×C23C22×C4semidirect
138. C4⋊C4×C23C22×C4semidirect
149. C22⋊C4×C23C22×C4semidirect
154. C42×C23C22×C4semidirect
155. C4×#560C23direct (N × H)
198. #2383C2×C23semidirect
199. #2380C2×C23semidirect
200. #2377C2×C23semidirect
201. #2376C2×C23semidirect
202. #2375C2×C23semidirect
203. C4×#127C2×C23semidirect
204. C4×#126C2×C23semidirect
205. C4×#123C2×C23semidirect
206. C4×#121C2×C23semidirect
207. C4×#120C2×C23semidirect
230. C4×C23#560direct (N × H)
232. #560C4×C23direct (N × H)
240. C2×C4×D4C22×C23semidirect
241. C22×C42C22×C23semidirect
244. #390C22×C23semidirect
245. #389C22×C23semidirect
249. #388C22×C23semidirect
250. #387C22×C23semidirect
251. #387C22×C23semidirect
256. #386C22×C23semidirect
259. #385C22×C23semidirect
260. #384C22×C23semidirect
261. #383C22×C23semidirect
262. #381C22×C23semidirect
263. #380C22×C23semidirect
265. C4×C4⋊C4C22×C23semidirect
268. C4×C22⋊C4C22×C23semidirect
269. C43C22×C23semidirect
288. #127C2×C4×C23semidirect
289. #126C2×C4×C23semidirect
290. #123C2×C4×C23semidirect
291. #121C2×C4×C23semidirect
292. #120C2×C4×C23semidirect
324. C23C4×#560direct (N × H)
339. C2×D4C22×C4×C23semidirect
351. C22×C4C22×C4×C23semidirect
364. C4⋊C4C22×C4×C23semidirect
375. C22⋊C4C22×C4×C23semidirect
380. C42C22×C4×C23semidirect
444. C4C23×#560direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5602
C23×#23832
C23×#23802
C23×#23772
C23×#23762
C23×#23752
C4×C23×#1272
C4×C23×#1262
C4×C23×#1232
C4×C23×#1212
C4×C23×#1202
C4×#56023