Finite Group Databasebeta

← C4×C47×#550 #26073858

Subgroups — C4×C47×#550 #26073858

Order 12032.

3420 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C47×#550C211yesmaximal, characteristic
2C2×C4×#41×C47C212yesmaximal, complemented
3C47×#2366C213yesmaximal, complemented
4C47×#2363C216yesmaximal, complemented
5C47×#2354C213yesmaximal, complemented
6Q8×C4⋊C4×C47C211yesmaximal, characteristic, complemented
7C47×#2341C212yesmaximal, complemented
8C4×#127×C47C213yesmaximal, complemented
9C4×#121×C47C216yesmaximal, complemented
10C42×Q8×C47C211yesmaximal, characteristic, complemented
11C42×D4×C47C213yesmaximal, complemented
12C22×#41×C47C2212yes—
13C47×#550C4116yesdirect product factor
14C2×#127×C47C2213yes—
15C2×#121×C47C2216yes—
16C4×#41×C47—216no—
17C2×C4×Q8×C47C2211yescharacteristic
18C2×C4×Q8×C47C2212yes—
19C2×C4×Q8×C47C2211yescharacteristic
20C2×C4×Q8×C47C2211yescharacteristic
21C2×C4×D4×C47C2213yes—
22C2×C4×D4×C47C2213yes—
23C2×C4×D4×C47C2216yes—
24C2×C4×D4×C47C2216yes—
25C2×C4×D4×C47C2216yes—
26C2×C4×D4×C47C2213yes—
27C2×#117×C47C2216yes—
28C22×C42×C47C2216yes—
29C47×#390C2216yes—
30C47×#390C2216yes—
31C47×#390C2216yes—
32C47×#390C2216yes—
33C47×#389C2216yes—
34C47×#388C22112yes—
35C47×#386C22112yes—
36C47×#384C2213yes—
37C47×#380C2216yes—
38C4×C4⋊C4×C47C2211yescharacteristic
39C4×C4⋊C4×C47C2213yes—
40C4×C4⋊C4×C47C2213yes—
41C4×C4⋊C4×C47C2213yes—
42C4×C4⋊C4×C47C2213yes—
43C4×C4⋊C4×C47C2213yes—
44C4×C22⋊C4×C47C2216yes—
45C4×C22⋊C4×C47C2216yes—
46C4×C22⋊C4×C47C2216yes—
47C43×C47C2213yes—
48C2×#41×C47—28no—
49C2×#41×C47—28no—
50C2×#41×C47C2×C4116yescomplemented
51C22×Q8×C47C2311yescharacteristic
52C22×D4×C47C2313yes—
53C22×D4×C47C2316yes—
54C22×D4×C47C2316yes—
55C23×C4×C47C2316yes—
56C23×C4×C47C2316yes—
57C23×C4×C47C2316yes—
58#127×C47C2×C4124yescomplemented
59#121×C47C2×C4148yescomplemented
60C4×Q8×C47—28no—
61C4×Q8×C47—24no—
62C4×Q8×C47C2×C418yescomplemented
63C4×Q8×C47D418yescomplemented
64C4×D4×C47—224no—
65C4×D4×C47—212no—
66C4×D4×C47—224no—
67C4×D4×C47—212no—
68C4×D4×C47—412no—
69C4×D4×C47C2×C4124yescomplemented
70#117×C47—224no—
71C2×C4⋊C4×C47C2316yes—
72C2×C4⋊C4×C47C2313yes—
73C2×C4⋊C4×C47C2313yes—
74C2×C4⋊C4×C47C2316yes—
75C2×C4⋊C4×C47C2311yescharacteristic
76C2×C4⋊C4×C47C2313yes—
77C2×C4⋊C4×C47C2313yes—
78C2×C4⋊C4×C47C2311yescharacteristic
79C2×C22⋊C4×C47C23112yes—
80C2×C22⋊C4×C47C2316yes—
81C2×C22⋊C4×C47C2316yes—
82C2×C22⋊C4×C47C23112yes—
83C2×C22⋊C4×C47C2316yes—
84C2×C42×C47—224no—
85C2×C42×C47C2316yes—
86C2×C42×C47C2316yes—
87C2×C42×C47C2313yes—
88C2×C42×C47C2313yes—
89C2×C42×C47C2312yes—
90C2×C42×C47C2313yes—
91C2×C42×C47C2313yes—
92C2×C42×C47C2313yes—
93C2×C42×C47C2313yes—
94C2×C42×C47C2311yescharacteristic
95#95×C47C2312yes—
96#95×C47C2316yes—
97#95×C47C2316yes—
98#95×C47C2316yes—
99C24×C47C2416yes—
100#41×C47—264no—
101C2×Q8×C47C22×C414yes—
102C2×Q8×C47C2×D414yes—
103C2×Q8×C47#4114yes—
104C2×D4×C47—212no—
105C2×D4×C47—26no—
106C2×D4×C47—26no—
107C2×D4×C47—212no—
108C2×D4×C47—46no—
109C2×D4×C47—46no—
110C2×D4×C47C22×C4112yes—
111C2×D4×C47C22×C4124yes—
112C2×D4×C47C22×C4124yes—
113C22×C4×C47—212no—
114C22×C4×C47—212no—
115C22×C4×C47—212no—
116C22×C4×C47—212no—
117C22×C4×C47—212no—
118C22×C4×C47—212no—
119C22×C4×C47—212no—
120C22×C4×C47—46no—
121C22×C4×C47C22×C4124yes—
122C22×C4×C47C2413yes—
123C22×C4×C47C2416yes—
124C22×C4×C47C2412yes—
125C22×C4×C47C2413yes—
126C22×C4×C47C2413yes—
127C22×C4×C47C2411yescharacteristic
128C22×C4×C47C2412yes—
129C22×C4×C47C2413yes—
130C22×C4×C47C2411yescharacteristic
131C22×C4×C47C2411yescharacteristic
132C4⋊C4×C47—212no—
133C4⋊C4×C47—26no—
134C4⋊C4×C47—26no—
135C4⋊C4×C47—46no—
136C4⋊C4×C47C22×C414yes—
137C4⋊C4×C47C22×C4112yes—
138C4⋊C4×C47C2×D4112yes—
139C4⋊C4×C47#4114yescomplemented
140C22⋊C4×C47—224no—
141C22⋊C4×C47—212no—
142C22⋊C4×C47—212no—
143C22⋊C4×C47—412no—
144C22⋊C4×C47C22×C4124yes—
145C42×C47—24no—
146C42×C47—212no—
147C42×C47—212no—
148C42×C47—26no—
149C42×C47—26no—
150C42×C47—26no—
151C42×C47—43no—
152C42×C47C22×C4112yes—
153C42×C47C2×D4112yes—
154C42×C47#4114yescomplemented
155C23×C47—26no—
156C23×C47—26no—
157C23×C47—26no—
158C23×C47—26no—
159C23×C47—26no—
160C23×C47—26no—
161C23×C47C23×C4112yes—
162C23×C47C2511yescharacteristic
163Q8×C47C4×D4116yescomplemented
164D4×C47—248no—
165D4×C47—224no—
166D4×C47—424no—
167C2×C4×C47—22no—
168C2×C4×C47—22no—
169C2×C4×C47—21no—
170C2×C4×C47—23no—
171C2×C4×C47—22no—
172C2×C4×C47—23no—
173C2×C4×C47—21no—
174C2×C4×C47—23no—
175C2×C4×C47—23no—
176C2×C4×C47—22no—
177C2×C4×C47—248no—
178C2×C4×C47—26no—
179C2×C4×C47—23no—
180C2×C4×C47—23no—
181C2×C4×C47—26no—
182C2×C4×C47—23no—
183C2×C4×C47—26no—
184C2×C4×C47—424no—
185C2×C4×C47—424no—
186C2×C4×C47C23×C416yes—
187C2×C4×C47C23×C414yes—
188C2×C4×C47C23×C416yes—
189C2×C4×C47C23×C412yes—
190C2×C4×C47C22×D416yes—
191C2×C4×C47C22×D416yes—
192C2×C4×C47C22×D412yes—
193C2×C4×C47C2×#4112yes—
194C2×C4×C47C2×#4114yes—
195C2×C4×C47C2×#4116yes—
196C2×C4×C47C2×#4112yes—
197C2×C4×C47C2×#4112yes—
198C2×C4×C47C2×#4112yes—
199C2×C4×C47H(5,𝔽2)12yes—
200C4×#550C4711yesmaximal, characteristic, direct product factor, Sylow
201C22×C47—212no—
202C22×C47—212no—
203C22×C47—212no—
204C22×C47—46no—
205C22×C47—46no—
206C22×C47—46no—
207C22×C47—46no—
208C22×C47C24×C411yescharacteristic
209C22×C47C23×D411yescharacteristic
210C22×C47C22×#4111yescharacteristic
211C22×C47C22×#4111yescharacteristic
212C22×C47C22×#4111yescharacteristic
213C22×C47C2×H(5,𝔽2)11yescharacteristic
214C22×C47#58511yescharacteristic
215C4×C47—24no—
216C4×C47—26no—
217C4×C47—24no—
218C4×C47—26no—
219C4×C47—22no—
220C4×C47—46no—
221C4×C47—43no—
222C4×C47C2×C4×D4112yes—
223C4×C47C4×#4114yescomplemented
224C4×C47#55014yesdirect product factor
225C2×#550C2×C4711yescharacteristic
226C2×C4×#41C2×C4712yescomplemented
227#2366C2×C4713yescomplemented
228#2363C2×C4716yescomplemented
229#2354C2×C4713yescomplemented
230Q8×C4⋊C4C2×C4711yescharacteristic, complemented
231#2341C2×C4712yescomplemented
232C4×#127C2×C4713yescomplemented
233C4×#121C2×C4716yescomplemented
234C42×Q8C2×C4711yescharacteristic, complemented
235C42×D4C2×C4713yescomplemented
236C2×C47—412no—
237C2×C47C22×C4×D411yescharacteristic
238C2×C47C2×C4×#4111yescharacteristic
239C2×C47C4×H(5,𝔽2)11yescharacteristic
240C2×C47C2×#55011yescharacteristic
241C2×C47D4×#4111yescharacteristic
242C2×C47#348011yescharacteristic
243C2×C47#348311yescharacteristic
244C22×#41C22×C4712yes—
245#550C4×C47116yesdirect product factor
246C2×#127C22×C4713yes—
247C2×#121C22×C4716yes—
248C4×#41—216no—
249C2×C4×Q8C22×C4711yescharacteristic
250C2×C4×Q8C22×C4712yes—
251C2×C4×Q8C22×C4711yescharacteristic
252C2×C4×Q8C22×C4711yescharacteristic
253C2×C4×D4C22×C4713yes—
254C2×C4×D4C22×C4716yes—
255C2×C4×D4C22×C4716yes—
256C2×C4×D4C22×C4713yes—
257C2×C4×D4C22×C4716yes—
258C2×C4×D4C22×C4713yes—
259C2×#117C22×C4716yes—
260C22×C42C22×C4716yes—
261#390C22×C4716yes—
262#390C22×C4716yes—
263#390C22×C4716yes—
264#390C22×C4716yes—
265#389C22×C4716yes—
266#388C22×C47112yes—
267#386C22×C47112yes—
268#384C22×C4713yes—
269#380C22×C4716yes—
270C4×C4⋊C4C22×C4713yes—
271C4×C4⋊C4C22×C4713yes—
272C4×C4⋊C4C22×C4713yes—
273C4×C4⋊C4C22×C4713yes—
274C4×C4⋊C4C22×C4711yescharacteristic
275C4×C4⋊C4C22×C4713yes—
276C4×C22⋊C4C22×C4716yes—
277C4×C22⋊C4C22×C4716yes—
278C4×C22⋊C4C22×C4716yes—
279C43C22×C4713yes—
280C47C4×#55011yescharacteristic, direct product factor, Sylow
281C2×#41—28no—
282C2×#41—28no—
283C2×#41C2×C4×C47116yescomplemented
284C22×Q8C23×C4711yescharacteristic
285C22×D4C23×C4713yes—
286C22×D4C23×C4716yes—
287C22×D4C23×C4716yes—
288C23×C4C23×C4716yes—
289C23×C4C23×C4716yes—
290C23×C4C23×C4716yes—
291#127C2×C4×C47124yescomplemented
292#121C2×C4×C47148yescomplemented
293C4×Q8—28no—
294C4×Q8—24no—
295C4×Q8C2×C4×C4718yescomplemented
296C4×Q8D4×C4718yescomplemented
297C4×D4—224no—
298C4×D4—212no—
299C4×D4—224no—
300C4×D4—212no—
301C4×D4—412no—
302C4×D4C2×C4×C47124yescomplemented
303#117—224no—
304C2×C4⋊C4C23×C4716yes—
305C2×C4⋊C4C23×C4713yes—
306C2×C4⋊C4C23×C4713yes—
307C2×C4⋊C4C23×C4716yes—
308C2×C4⋊C4C23×C4711yescharacteristic
309C2×C4⋊C4C23×C4713yes—
310C2×C4⋊C4C23×C4713yes—
311C2×C4⋊C4C23×C4711yescharacteristic
312C2×C22⋊C4C23×C47112yes—
313C2×C22⋊C4C23×C4716yes—
314C2×C22⋊C4C23×C4716yes—
315C2×C22⋊C4C23×C47112yes—
316C2×C22⋊C4C23×C4716yes—
317C2×C42—224no—
318C2×C42C23×C4716yes—
319C2×C42C23×C4713yes—
320C2×C42C23×C4716yes—
321C2×C42C23×C4713yes—
322C2×C42C23×C4713yes—
323C2×C42C23×C4712yes—
324C2×C42C23×C4713yes—
325C2×C42C23×C4713yes—
326C2×C42C23×C4713yes—
327C2×C42C23×C4711yescharacteristic
328#95C23×C4712yes—
329#95C23×C4716yes—
330#95C23×C4716yes—
331#95C23×C4716yes—
332C24C24×C4716yes—
333#41—264no—
334C2×Q8C22×C4×C4714yes—
335C2×Q8C2×D4×C4714yes—
336C2×Q8#41×C4714yes—
337C2×D4—212no—
338C2×D4—26no—
339C2×D4—26no—
340C2×D4—212no—
341C2×D4—46no—
342C2×D4—46no—
343C2×D4C22×C4×C47112yes—
344C2×D4C22×C4×C47124yes—
345C2×D4C22×C4×C47124yes—
346C22×C4—212no—
347C22×C4—212no—
348C22×C4—212no—
349C22×C4—212no—
350C22×C4—212no—
351C22×C4—212no—
352C22×C4—212no—
353C22×C4—46no—
354C22×C4C22×C4×C47124yes—
355C22×C4C24×C4713yes—
356C22×C4C24×C4716yes—
357C22×C4C24×C4712yes—
358C22×C4C24×C4713yes—
359C22×C4C24×C4713yes—
360C22×C4C24×C4712yes—
361C22×C4C24×C4711yescharacteristic
362C22×C4C24×C4713yes—
363C22×C4C24×C4711yescharacteristic
364C22×C4C24×C4711yescharacteristic
365C4⋊C4—212no—
366C4⋊C4—26no—
367C4⋊C4—26no—
368C4⋊C4—46no—
369C4⋊C4C22×C4×C4714yes—
370C4⋊C4C22×C4×C47112yes—
371C4⋊C4C2×D4×C47112yes—
372C4⋊C4#41×C4714yescomplemented
373C22⋊C4—224no—
374C22⋊C4—212no—
375C22⋊C4—212no—
376C22⋊C4—412no—
377C22⋊C4C22×C4×C47124yes—
378C42—24no—
379C42—212no—
380C42—212no—
381C42—26no—
382C42—26no—
383C42—26no—
384C42—43no—
385C42C22×C4×C47112yes—
386C42C2×D4×C47112yes—
387C42#41×C4714yescomplemented
388C23—26no—
389C23—26no—
390C23—26no—
391C23—26no—
392C23—26no—
393C23—26no—
394C23C23×C4×C47112yes—
395C23C25×C4711yescharacteristic
396Q8C4×D4×C47116yescomplemented
397D4—248no—
398D4—224no—
399D4—424no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—23no—
404C2×C4—22no—
405C2×C4—23no—
406C2×C4—21no—
407C2×C4—23no—
408C2×C4—23no—
409C2×C4—22no—
410C2×C4—248no—
411C2×C4—26no—
412C2×C4—23no—
413C2×C4—23no—
414C2×C4—26no—
415C2×C4—23no—
416C2×C4—26no—
417C2×C4—424no—
418C2×C4—424no—
419C2×C4C23×C4×C4716yes—
420C2×C4C23×C4×C4714yes—
421C2×C4C23×C4×C4716yes—
422C2×C4C23×C4×C4712yes—
423C2×C4C22×D4×C4716yes—
424C2×C4C22×D4×C4716yes—
425C2×C4C22×D4×C4712yes—
426C2×C4C2×#41×C4712yes—
427C2×C4C2×#41×C4714yes—
428C2×C4C2×#41×C4716yes—
429C2×C4C2×#41×C4712yes—
430C2×C4C2×#41×C4712yes—
431C2×C4C2×#41×C4712yes—
432C2×C4H(5,𝔽2)×C4712yes—
433C22—212no—
434C22—212no—
435C22—212no—
436C22—46no—
437C22—46no—
438C22—46no—
439C22—46no—
440C22C24×C4×C4711yescharacteristic
441C22C23×D4×C4711yescharacteristic
442C22C22×#41×C4711yescharacteristic
443C22C22×#41×C4711yescharacteristic
444C22C22×#41×C4711yescharacteristic
445C22C2×H(5,𝔽2)×C4711yescharacteristic
446C22C47×#58511yescharacteristic
447C4—24no—
448C4—26no—
449C4—24no—
450C4—26no—
451C4—22no—
452C4—46no—
453C4—43no—
454C4C2×C4×D4×C47112yes—
455C4C4×#41×C4714yescomplemented
456C4C47×#55014yesdirect product factor
457C2—412no—
458C2C22×C4×D4×C4711yescharacteristic
459C2C2×C4×#41×C4711yescharacteristic
460C2C4×H(5,𝔽2)×C4711yescharacteristic
461C2C2×C47×#55011yescharacteristic
462C2D4×#41×C4711yescharacteristic
463C2C47×#348011yescharacteristic
464C2C47×#348311yescharacteristic
subgroupG/H
centerC22×C4×C47C24
commutatorC22C24×C4×C47
Frattini, cosocleC23C25×C47
socleC23×C47C25
prime radicalC2×C47×#550C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4×C47 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C4×#41×C47C2semidirect
3. C47×#2366C2semidirect
4. C47×#2363C2semidirect
5. C47×#2354C2semidirect
6. Q8×C4⋊C4×C47C2semidirect
7. C47×#2341C2semidirect
8. C4×#127×C47C2semidirect
9. C4×#121×C47C2semidirect
10. C42×Q8×C47C2semidirect
11. C42×D4×C47C2semidirect
13. C47×#550C4direct (N × H)
50. C2×#41×C47C2×C4semidirect
58. #127×C47C2×C4semidirect
59. #121×C47C2×C4semidirect
62. C4×Q8×C47C2×C4semidirect
63. C4×Q8×C47D4semidirect
69. C4×D4×C47C2×C4semidirect
139. C4⋊C4×C47#41semidirect
154. C42×C47#41semidirect
163. Q8×C47C4×D4semidirect
200. C4×#550C47direct (N × H)
223. C4×C47C4×#41semidirect
224. C4×C47#550direct (N × H)
226. C2×C4×#41C2×C47semidirect
227. #2366C2×C47semidirect
228. #2363C2×C47semidirect
229. #2354C2×C47semidirect
230. Q8×C4⋊C4C2×C47semidirect
231. #2341C2×C47semidirect
232. C4×#127C2×C47semidirect
233. C4×#121C2×C47semidirect
234. C42×Q8C2×C47semidirect
235. C42×D4C2×C47semidirect
245. #550C4×C47direct (N × H)
280. C47C4×#550direct (N × H)
283. C2×#41C2×C4×C47semidirect
291. #127C2×C4×C47semidirect
292. #121C2×C4×C47semidirect
295. C4×Q8C2×C4×C47semidirect
296. C4×Q8D4×C47semidirect
302. C4×D4C2×C4×C47semidirect
372. C4⋊C4#41×C47semidirect
387. C42#41×C47semidirect
396. Q8C4×D4×C47semidirect
455. C4C4×#41×C47semidirect
456. C4C47×#550direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C47×#5502
C2×C4×#41×C472
C47×#23662
C47×#23632
C47×#23542
Q8×C4⋊C4×C472
C47×#23412
C4×#127×C472
C4×#121×C472
C42×Q8×C472
C42×D4×C472
C4×#55047