Finite Group Databasebeta

← C2×#20288721 #28292350

Subgroups — C2×#20288721 #28292350

Order 13568.

7854 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20288721C2132yesmaximal, direct product factor
2C2×#16811352C211yesmaximal, characteristic, complemented
3C2×#16811317C216yesmaximal, complemented
4C2×#16811295C211yesmaximal, characteristic, complemented
5C2×#16811294C213yesmaximal, complemented
6C2×#16811292C212yesmaximal, complemented
7C2×#16811278C213yesmaximal, complemented
8C2×#16811224C216yesmaximal, complemented
9C2×#16811173C213yesmaximal, complemented
10C2×#16811157C213yesmaximal, complemented
11C2×#16811105C212yesmaximal, complemented
12C2×C4×#41×C53C211yesmaximal, characteristic, complemented
13C22×#11719949C2212yes—
14C22×#11719945C2216yes—
15#16811352C22116yescomplemented
16#16811317C22196yescomplemented
17#16811295C22116yescomplemented
18#16811294C22148yescomplemented
19#16811292C22132yescomplemented
20#16811278C22148yescomplemented
21#16811224C22196yescomplemented
22#16811173C22148yescomplemented
23C2×#13133217C2213yes—
24C2×#13133214C2213yes—
25C2×#13133213C2216yes—
26#16811157C22148yescomplemented
27C2×#13133211C2216yes—
28C2×#13133209C2211yescharacteristic
29C2×#13133208C2216yes—
30C2×#13133205C2213yes—
31C2×#13133200C2216yes—
32C2×#13133199C22112yes—
33C2×#13133196C2213yes—
34C2×#13133192C2216yes—
35C2×#13133191C2216yes—
36C2×#13133191C2216yes—
37C2×#13133191C2213yes—
38C2×#13133190C22112yes—
39C2×#13133188C2216yes—
40#16811105C22132yescomplemented
41C2×Q8×Dic53C2212yes—
42C2×C4×Dic106C2213yes—
43C2×C4×D212C2216yes—
44C2×C4×D212C2213yes—
45C2×C4×D212C2216yes—
46C2×#13133179C2216yes—
47C2×C4×C22⋊D53C2216yes—
48C2×#13133174C2216yes—
49C2×C4⋊C4×D53C2216yes—
50C22×#41×C53C2211yescharacteristic
51C4×#41×C53C22116yescomplemented
52C2×C4×Q8×C53C2211yescharacteristic
53C2×C4×Q8×C53C2211yescharacteristic
54C2×C4×D4×C53C2213yes—
55C2×C4×D4×C53C2213yes—
56C2×#117×C53C2213yes—
57C22×C42×C53C2213yes—
58C2×#11719949—216no—
59C2×#11719949C23116yes—
60C2×#11719945—424no—
61C2×#11719945C23148yes—
62C22×Dic106C2313yes—
63C22×D212C2316yes—
64C22×D212C2313yes—
65C22×D212C2316yes—
66C22×C22⋊D53C2316yes—
67C22×C22⋊D53C23112yes—
68C22×C4×D53C2316yes—
69C22×C4×D53C23112yes—
70#13133217C23124yes—
71#13133214C23124yes—
72#13133213C23148yes—
73#13133211C23148yes—
74#13133209C2318yes—
75#13133208C23148yes—
76#13133205C23124yes—
77#13133200C23148yes—
78#13133199C23196yes—
79#13133196C23124yes—
80#13133192C23148yes—
81#13133191C23148yes—
82#13133191C23124yes—
83#13133191C23148yes—
84#13133190C23196yes—
85#13133188C23148yes—
86Q8×Dic53C23116yes—
87C4×Dic106C23124yes—
88C4×D212C23124yes—
89C4×D212C23148yes—
90C4×D212C23148yes—
91#13133179C23148yes—
92C4×C22⋊D53C23148yes—
93#13133174C23148yes—
94C2×C4⋊Dic53C2316yes—
95C2×C4⋊Dic53C2311yescharacteristic
96C2×C4⋊Dic53C2316yes—
97C2×C4⋊Dic53C2311yescharacteristic
98C2×#11719919C2316yes—
99C2×#11719919C2312yes—
100C2×#11719919C2312yes—
101C2×#11719919C2316yes—
102C4⋊C4×D53C23148yes—
103C2×C22⋊Dic53C2316yes—
104C2×#11719921C2316yes—
105C2×#11719921C2316yes—
106C2×#11719921C2316yes—
107C2×#11719921C2316yes—
108C2×C4×Dic53C2316yes—
109C2×#41×C53—28no—
110C2×#41×C53C2318yes—
111C22×Q8×C53C2311yescharacteristic
112C22×D4×C53C2313yes—
113C23×C4×C53C2313yes—
114C23×C4×C53C2313yes—
115C23×C4×C53C2313yes—
116C4×Q8×C53C2318yes—
117C4×Q8×C53C2318yes—
118C4×D4×C53C23124yes—
119C4×D4×C53C23124yes—
120#117×C53C23124yes—
121C2×C4⋊C4×C53C2313yes—
122C2×C4⋊C4×C53C2313yes—
123C2×C22⋊C4×C53C2316yes—
124C2×C42×C53C2311yescharacteristic
125C2×C42×C53C2313yes—
126C2×C42×C53C2313yes—
127C2×C42×C53C2313yes—
128C2×C42×C53C23124yes—
129C24×C53C2413yes—
130C23×D53C2416yes—
131#41×C53—232no—
132C2×Q8×C53#4118yescomplemented
133C2×Q8×C53C2414yes—
134C2×D4×C53—212no—
135C2×D4×C53C24112yes—
136C22×C4×C53—212no—
137C22×C4×C53—212no—
138C22×C4×C53—46no—
139C22×C4×C53C2411yescharacteristic
140C22×C4×C53C2411yescharacteristic
141C22×C4×C53C2413yes—
142C22×C4×C53C2413yes—
143C22×C4×C53C24112yes—
144C22×C4×C53C24112yes—
145C22×C4×C53C24112yes—
146C22×C4×C53C2413yes—
147C22×C4×C53C2411yescharacteristic
148C2×C22⋊D53—412no—
149C2×C22⋊D53—424no—
150C2×C22⋊D53C24124yes—
151C2×C22⋊D53C24148yes—
152C22×Dic53C2412yes—
153C22×Dic53C2416yes—
154C22×Dic53C2412yes—
155#11719949—264no—
156#11719945—496no—
157C2×D212—224no—
158C2×D212—212no—
159C2×D212—412no—
160C2×D212C24124yes—
161C2×D212C24112yes—
162C2×D212C24124yes—
163C2×C4×D53—224no—
164C2×C4×D53—424no—
165C2×C4×D53C24124yes—
166C2×C4×D53C24148yes—
167C2×Dic106—212no—
168C2×Dic106C24112yes—
169C4⋊C4×C53C24112yes—
170C4⋊C4×C53C24112yes—
171C22⋊C4×C53C24124yes—
172C42×C53C2414yes—
173C42×C53C24112yes—
174C42×C53C24112yes—
175C42×C53C24112yes—
176C22⋊Dic53C24124yes—
177#11719921C24124yes—
178#11719921C24124yes—
179#11719921C24124yes—
180#11719921C24124yes—
181C4⋊Dic53C24124yes—
182C4⋊Dic53C2414yes—
183C4⋊Dic53C24124yes—
184C4⋊Dic53C2414yes—
185#11719919C24124yes—
186#11719919C2418yes—
187#11719919C2418yes—
188#11719919C24124yes—
189C4×Dic53C24124yes—
190C23×C53—26no—
191C23×C53—26no—
192C23×C53—26no—
193C23×C53C2516yes—
194C23×C53C2511yescharacteristic
195C22×D53—212no—
196C22×D53—212no—
197C22×D53—212no—
198C22×D53C25112yes—
199Q8×C53C2×#41116yescomplemented
200D4×C53—224no—
201C2×C4×C53—22no—
202C2×C4×C53—26no—
203C2×C4×C53—26no—
204C2×C4×C53—22no—
205C2×C4×C53—224no—
206C2×C4×C53—224no—
207C2×C4×C53—412no—
208C2×C4×C53C2×#41112yes—
209C2×C4×C53H(5,𝔽2)14yes—
210C2×C4×C53C2516yes—
211C2×C4×C53C2516yes—
212C2×C4×C53C2512yes—
213C2×C4×C53C2512yes—
214C2×C4×C53C2516yes—
215C2×C4×C53C2512yes—
216C22⋊D53—448no—
217C22⋊D53—424no—
218C2×Dic53—24no—
219C2×Dic53—212no—
220C2×Dic53—24no—
221C2×Dic53C25112yes—
222C2×Dic53C2514yes—
223C2×Dic53C2514yes—
224D212—248no—
225D212—224no—
226D212—424no—
227C4×D53—248no—
228C4×D53—448no—
229Dic106—224no—
230C2×#3517—531nomaximal, Sylow
231C22×C53—26no—
232C22×C53—26no—
233C22×C53—26no—
234C22×C53—412no—
235C22×C53C22×#4112yes—
236C22×C53C2×H(5,𝔽2)12yes—
237C22×C53#58512yes—
238C22×C53C2611yescharacteristic
239C2×D53—212no—
240C2×D53—212no—
241C2×D53—212no—
242C2×D53—424no—
243C4×C53—22no—
244C4×C53—26no—
245C4×C53—26no—
246C4×C53—22no—
247C4×C53C22×#41112yes—
248C4×C53C2×H(5,𝔽2)14yes—
249Dic53—24no—
250Dic53—212no—
251Dic53—24no—
252#3517—5332no—
253C2×#557—531no—
254C2×#553—536no—
255C2×#550—532no—
256C2×#550—531no—
257C2×#550—533no—
258C2×#547—533no—
259C2×#540—536no—
260C2×#532—533no—
261C2×#529—533no—
262C2×#520—532no—
263C2×C4×#41D5311yescharacteristic, complemented
264C2×C53—46no—
265C2×C53#351714yescomplemented
266C2×C53C23×#4111yescharacteristic
267C2×C53C22×H(5,𝔽2)11yescharacteristic
268C2×C53C2×#58511yescharacteristic
269D53—412no—
270C22×#41—532no—
271C22×#41—536no—
272C22×#41C2×D5311yescharacteristic
273#557—5316no—
274#553—5396no—
275#550—5332no—
276#550—5316no—
277#550—5348no—
278#547—5348no—
279#540—5396no—
280#532—5348no—
281C2×C4⋊Q8—533no—
282C2×#127—536no—
283C2×#127—533no—
284#529—5348no—
285C2×#126—536no—
286C2×#125—536no—
287C2×#125—531no—
288C2×#124—533no—
289C2×#123—536no—
290C2×#123—5312no—
291C2×C22⋊Q8—533no—
292C2×#121—536no—
293C2×#121—533no—
294C2×#121—536no—
295C2×#121—536no—
296C2×#121—536no—
297C2×#121—5312no—
298#520—5332no—
299C4×#41C2×D53116yescomplemented
300C2×C4×Q8—532no—
301C2×C4×Q8—533no—
302C2×C4×Q8C2×D5311yescharacteristic
303C2×C4×Q8C2×D5311yescharacteristic
304C2×C4×D4—536no—
305C2×C4×D4—536no—
306C2×C4×D4—536no—
307C2×C4×D4—533no—
308C2×C4×D4—536no—
309C2×C4×D4C2×D5313yes—
310C2×C4×D4C2×D5313yes—
311C2×#117—536no—
312C2×#117C2×D5313yes—
313C22×C4⋊C4—536no—
314C22×C42C2×D5313yes—
315C53C2×#351711yescharacteristic, complemented, Sylow
316C2×#41—28no—
317C2×#41—5316no—
318C2×#41—5348no—
319C2×#41—10616no—
320C2×#41—21224no—
321C2×#41C22×D5318yes—
322C22×Q8—533no—
323C22×Q8C22×D5311yescharacteristic
324C22×D4—536no—
325C22×D4—5312no—
326C22×D4—536no—
327C22×D4—533no—
328C22×D4—536no—
329C22×D4C22×D5313yes—
330C23×C4—536no—
331C23×C4—5312no—
332C23×C4C22×D5313yes—
333C23×C4C22×D5313yes—
334C23×C4C22×D5313yes—
335C4⋊Q8—5324no—
336#127—5348no—
337#127—5324no—
338#126—5348no—
339#125—5348no—
340#125—538no—
341#124—5324no—
342#123—5348no—
343#123—5396no—
344C22⋊Q8—5324no—
345#121—5348no—
346#121—5324no—
347#121—5348no—
348#121—5348no—
349#121—5348no—
350#121—5396no—
351C4×Q8—5316no—
352C4×Q8—5324no—
353C4×Q8C22×D5318yes—
354C4×Q8C22×D5318yes—
355C4×D4—5348no—
356C4×D4—5348no—
357C4×D4—5348no—
358C4×D4—5324no—
359C4×D4—5348no—
360C4×D4C22×D53124yes—
361C4×D4C22×D53124yes—
362#117—5348no—
363#117C22×D53124yes—
364C2×C4⋊C4—536no—
365C2×C4⋊C4—536no—
366C2×C4⋊C4—532no—
367C2×C4⋊C4—531no—
368C2×C4⋊C4—532no—
369C2×C4⋊C4—536no—
370C2×C4⋊C4—531no—
371C2×C4⋊C4—536no—
372C2×C4⋊C4—5348no—
373C2×C4⋊C4C22×D5313yes—
374C2×C4⋊C4C22×D5313yes—
375C2×C22⋊C4—536no—
376C2×C22⋊C4—536no—
377C2×C22⋊C4—536no—
378C2×C22⋊C4—536no—
379C2×C22⋊C4—536no—
380C2×C22⋊C4C22×D5316yes—
381C2×C42—536no—
382C2×C42C22×D5313yes—
383C2×C42C22×D5313yes—
384C2×C42C22×D5313yes—
385C2×C42C22×D53124yes—
386C2×C42C22×D5311yescharacteristic
387C24—536no—
388C24C23×D5313yes—
389#41—232no—
390#41—10664no—
391#41—21296no—
392C2×Q8—5312no—
393C2×Q8—10612no—
394C2×Q8#1171994518yescomplemented
395C2×Q8C23×D5314yes—
396C2×D4—212no—
397C2×D4—5324no—
398C2×D4—5348no—
399C2×D4—5324no—
400C2×D4—5312no—
401C2×D4—5324no—
402C2×D4—10624no—
403C2×D4—10612no—
404C2×D4—21212no—
405C2×D4—21212no—
406C2×D4—21224no—
407C2×D4C23×D53112yes—
408C22×C4—212no—
409C22×C4—212no—
410C22×C4—46no—
411C22×C4—532no—
412C22×C4—536no—
413C22×C4—532no—
414C22×C4—5324no—
415C22×C4—5348no—
416C22×C4—10624no—
417C22×C4—21224no—
418C22×C4C23×D53112yes—
419C22×C4C23×D53112yes—
420C22×C4C23×D5311yescharacteristic
421C22×C4C23×D5311yescharacteristic
422C22×C4C23×D5313yes—
423C22×C4C23×D5313yes—
424C22×C4C23×D53112yes—
425C22×C4C23×D5313yes—
426C22×C4C23×D5311yescharacteristic
427C4⋊C4—5324no—
428C4⋊C4—5324no—
429C4⋊C4—538no—
430C4⋊C4—534no—
431C4⋊C4—538no—
432C4⋊C4—5324no—
433C4⋊C4—534no—
434C4⋊C4—5324no—
435C4⋊C4C23×D53112yes—
436C4⋊C4C23×D53112yes—
437C22⋊C4—5324no—
438C22⋊C4—5324no—
439C22⋊C4—5324no—
440C22⋊C4—5324no—
441C22⋊C4—5324no—
442C22⋊C4C23×D53124yes—
443C42—5324no—
444C42C23×D5314yes—
445C42C23×D53112yes—
446C42C23×D53112yes—
447C42C23×D53112yes—
448C23—26no—
449C23—26no—
450C23—26no—
451C23—5312no—
452C23—10612no—
453C23—10612no—
454C23—10612no—
455C23C24×D5316yes—
456C23C24×D5311yescharacteristic
457Q8—10624no—
458Q8C2×#11719945116yescomplemented
459D4—224no—
460D4—10648no—
461D4—10624no—
462D4—21224no—
463D4—21224no—
464D4—21248no—
465C2×C4—22no—
466C2×C4—26no—
467C2×C4—22no—
468C2×C4—26no—
469C2×C4—224no—
470C2×C4—224no—
471C2×C4—412no—
472C2×C4—534no—
473C2×C4—5312no—
474C2×C4—534no—
475C2×C4—1064no—
476C2×C4—10612no—
477C2×C4—1064no—
478C2×C4—10648no—
479C2×C4—21248no—
480C2×C4C2×#11719945112yes—
481C2×C4#1313326314yes—
482C2×C4C24×D5312yes—
483C2×C4C24×D5312yes—
484C2×C4C24×D5316yes—
485C2×C4C24×D5316yes—
486C2×C4C24×D5316yes—
487C2×C4C24×D5312yes—
488C22—26no—
489C22—26no—
490C22—26no—
491C22—412no—
492C22—10612no—
493C22—10612no—
494C22—10612no—
495C22—21224no—
496C22C22×#1171994512yes—
497C22C2×#1313326312yes—
498C22#1681145912yes—
499C22C25×D5311yescharacteristic
500C4—22no—
501C4—26no—
502C4—22no—
503C4—26no—
504C4—1064no—
505C4—10612no—
506C4—1064no—
507C4C22×#11719945112yes—
508C4C2×#1313326314yes—
509C2—46no—
510C2—21212no—
511C2#2028872114yesdirect product factor
512C2C23×#1171994511yescharacteristic
513C2C22×#1313326311yescharacteristic
514C2C2×#1681145911yescharacteristic
subgroupG/H
centerC23C24×D53
commutator, cosocleC22×C53C26
FrattiniC22C25×D53
FittingC2×C4×#41×C53C2
socleC23×C53C25
hypercenterC2×C4×#41D53
hypocenterC53C2×#3517
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
1. #20288721C2direct (N × H)
2. C2×#16811352C2semidirect
3. C2×#16811317C2semidirect
4. C2×#16811295C2semidirect
5. C2×#16811294C2semidirect
6. C2×#16811292C2semidirect
7. C2×#16811278C2semidirect
8. C2×#16811224C2semidirect
9. C2×#16811173C2semidirect
10. C2×#16811157C2semidirect
11. C2×#16811105C2semidirect
12. C2×C4×#41×C53C2semidirect
15. #16811352C22semidirect
16. #16811317C22semidirect
17. #16811295C22semidirect
18. #16811294C22semidirect
19. #16811292C22semidirect
20. #16811278C22semidirect
21. #16811224C22semidirect
22. #16811173C22semidirect
26. #16811157C22semidirect
40. #16811105C22semidirect
51. C4×#41×C53C22semidirect
132. C2×Q8×C53#41semidirect
199. Q8×C53C2×#41semidirect
263. C2×C4×#41D53semidirect
265. C2×C53#3517semidirect
299. C4×#41C2×D53semidirect
315. C53C2×#3517semidirect
394. C2×Q8#11719945semidirect
458. Q8C2×#11719945semidirect
511. C2#20288721direct (N × H)

Maximal subgroups

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

subgroupindex
#202887212
C2×#168113522
C2×#168113172
C2×#168112952
C2×#168112942
C2×#168112922
C2×#168112782
C2×#168112242
C2×#168111732
C2×#168111572
C2×#168111052
C2×C4×#41×C532
C2×#351753