Finite Group Databasebeta

← C2×#20288341 #28291196

Subgroups — C2×#20288341 #28291196

Order 13568.

8418 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20288341C2132yesmaximal, direct product factor
2C2×Q8×Dic106C211yesmaximal, characteristic, complemented
3C2×#16811338C213yesmaximal, complemented
4C2×#16811336C216yesmaximal, complemented
5C2×#16811295C211yesmaximal, characteristic, complemented
6C2×#16811294C213yesmaximal, complemented
7C2×#16811292C212yesmaximal, complemented
8C2×#16811173C213yesmaximal, complemented
9C2×#16811170C216yesmaximal, complemented
10C2×C4×#11719949C212yesmaximal, complemented
11C2×C4×#11719945C213yesmaximal, complemented
12C2×C4×#41×C53C211yesmaximal, characteristic, complemented
13C22×#11719949C2212yes—
14C22×#11719945C2213yes—
15C22×#11719945C2216yes—
16Q8×Dic106C22116yescomplemented
17#16811338C22148yescomplemented
18#16811336C22196yescomplemented
19#16811295C22116yescomplemented
20#16811294C22148yescomplemented
21#16811292C22132yescomplemented
22#16811173C22148yescomplemented
23#16811170C22196yescomplemented
24C2×#13133217C2213yes—
25C2×#13133216C2216yes—
26C2×#13133214C2213yes—
27C2×#13133213C2216yes—
28C2×#13133205C2216yes—
29C2×#13133203C22112yes—
30C2×#13133192C2216yes—
31C2×#13133191C2216yes—
32C2×#13133191C2216yes—
33C2×#13133190C22112yes—
34C2×#13133188C2216yes—
35C4×#11719949C22132yescomplemented
36C4×#11719945C22148yescomplemented
37C2×Q8×Dic53C2212yes—
38C2×C4×Dic106C2213yes—
39C2×C4×Dic106C2211yescharacteristic
40C2×C4×D212C2213yes—
41C2×C4×D212C2213yes—
42C2×C4×D212C2216yes—
43C2×#13133179C2216yes—
44C2×C4×C22⋊D53C2216yes—
45C2×C4×C22⋊D53C2216yes—
46C2×#13133176C2213yes—
47C2×#13133175C2216yes—
48C2×#13133174C2216yes—
49C2×C42×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—224no—
61C2×#11719945—424no—
62C2×#11719945C23124yes—
63C2×#11719945C23148yes—
64C22×Dic106C2316yes—
65C22×Dic106C2311yescharacteristic
66C22×D212C2316yes—
67C22×D212C2313yes—
68C22×D212C2316yes—
69C22×C22⋊D53C2316yes—
70C22×C22⋊D53C23112yes—
71C22×C4×D53C2316yes—
72C22×C4×D53C2316yes—
73C22×C4×D53C2316yes—
74#13133217C23124yes—
75#13133216C23148yes—
76#13133214C23124yes—
77#13133213C23148yes—
78#13133205C23148yes—
79#13133203C23196yes—
80#13133192C23148yes—
81#13133191C23148yes—
82#13133191C23148yes—
83#13133190C23196yes—
84#13133188C23148yes—
85Q8×Dic53C23116yes—
86C4×Dic106C23124yes—
87C4×Dic106C2318yes—
88C4×D212C23124yes—
89C4×D212C23124yes—
90C4×D212C23148yes—
91#13133179C23148yes—
92C4×C22⋊D53C23148yes—
93C4×C22⋊D53C23148yes—
94#13133176C23124yes—
95#13133175C23148yes—
96#13133174C23148yes—
97C2×C4⋊Dic53C2316yes—
98C2×C4⋊Dic53C2311yescharacteristic
99C2×C4⋊Dic53C2313yes—
100C2×#11719919C2312yes—
101C2×#11719919C2316yes—
102C2×C22⋊Dic53C2316yes—
103C2×#11719921C2316yes—
104C2×#11719921C2316yes—
105C2×#11719921C23112yes—
106C2×C4×Dic53C2312yes—
107C2×C4×Dic53C2316yes—
108C2×C4×Dic53C2316yes—
109C42×D53C23148yes—
110C2×#41×C53—28no—
111C2×#41×C53C2318yes—
112C22×Q8×C53C2311yescharacteristic
113C22×D4×C53C2313yes—
114C23×C4×C53C2313yes—
115C23×C4×C53C2313yes—
116C23×C4×C53C2313yes—
117C4×Q8×C53C2318yes—
118C4×Q8×C53C2318yes—
119C4×D4×C53C23124yes—
120C4×D4×C53C23124yes—
121#117×C53C23124yes—
122C2×C4⋊C4×C53C2313yes—
123C2×C4⋊C4×C53C2313yes—
124C2×C22⋊C4×C53C2316yes—
125C2×C42×C53C2311yescharacteristic
126C2×C42×C53C2313yes—
127C2×C42×C53C2313yes—
128C2×C42×C53C2313yes—
129C2×C42×C53C23124yes—
130C24×C53C2413yes—
131C23×D53C2416yes—
132#41×C53—232no—
133C2×Q8×C53#4118yescomplemented
134C2×Q8×C53C2414yes—
135C2×D4×C53—212no—
136C2×D4×C53C24112yes—
137C22×C4×C53—212no—
138C22×C4×C53—212no—
139C22×C4×C53—212no—
140C22×C4×C53C2411yescharacteristic
141C22×C4×C53C2413yes—
142C22×C4×C53C2413yes—
143C22×C4×C53C24112yes—
144C22×C4×C53C24112yes—
145C22×C4×C53C24112yes—
146C22×C4×C53C2411yescharacteristic
147C22×C4×C53C2413yes—
148C22×C4×C53C2411yescharacteristic
149C2×C22⋊D53—224no—
150C2×C22⋊D53—424no—
151C2×C22⋊D53C24124yes—
152C2×C22⋊D53C24148yes—
153C22×Dic53C2412yes—
154C22×Dic53C2416yes—
155C22×Dic53C2412yes—
156#11719949—264no—
157#11719945—296no—
158#11719945—496no—
159C2×D212—212no—
160C2×D212—224no—
161C2×D212—412no—
162C2×D212C24124yes—
163C2×D212C24112yes—
164C2×D212C24124yes—
165C2×C4×D53—224no—
166C2×C4×D53—224no—
167C2×C4×D53—224no—
168C2×C4×D53C24124yes—
169C2×C4×D53C24124yes—
170C2×C4×D53C24124yes—
171C2×Dic106—412no—
172C2×Dic106#4118yescomplemented
173C2×Dic106C24124yes—
174C2×Dic106C2414yes—
175C4⋊C4×C53C24112yes—
176C4⋊C4×C53C24112yes—
177C22⋊C4×C53C24124yes—
178C42×C53C2414yes—
179C42×C53C24112yes—
180C42×C53C24112yes—
181C42×C53C24112yes—
182C22⋊Dic53C24124yes—
183#11719921C24124yes—
184#11719921C24124yes—
185#11719921C24148yes—
186C4⋊Dic53C24124yes—
187C4⋊Dic53C2414yes—
188C4⋊Dic53C24112yes—
189#11719919C2418yes—
190#11719919C24124yes—
191C4×Dic53C2418yes—
192C4×Dic53C24124yes—
193C4×Dic53C24124yes—
194C23×C53—26no—
195C23×C53—26no—
196C23×C53—26no—
197C23×C53C2516yes—
198C23×C53C2511yescharacteristic
199C22×D53—212no—
200C22×D53—212no—
201C22×D53—212no—
202C22×D53C25112yes—
203Q8×C53C2×#41116yescomplemented
204D4×C53—224no—
205C2×C4×C53—26no—
206C2×C4×C53—22no—
207C2×C4×C53—26no—
208C2×C4×C53—224no—
209C2×C4×C53—224no—
210C2×C4×C53—224no—
211C2×C4×C53C2×#4114yes—
212C2×C4×C53C2×#41112yes—
213C2×C4×C53H(5,𝔽2)14yes—
214C2×C4×C53C2516yes—
215C2×C4×C53C2516yes—
216C2×C4×C53C2512yes—
217C2×C4×C53C2512yes—
218C2×C4×C53C2516yes—
219C2×C4×C53C2512yes—
220C22⋊D53—248no—
221C22⋊D53—448no—
222C2×Dic53—24no—
223C2×Dic53—212no—
224C2×Dic53C2×#4118yes—
225C2×Dic53C25112yes—
226C2×Dic53C2514yes—
227C2×Dic53C2514yes—
228D212—224no—
229D212—248no—
230D212—424no—
231C4×D53—248no—
232C4×D53—248no—
233C4×D53—248no—
234Dic106—424no—
235Dic106C2×#41116yescomplemented
236C2×#3483—531nomaximal, Sylow
237C22×C53—26no—
238C22×C53—26no—
239C22×C53—26no—
240C22×C53—412no—
241C22×C53C22×#4112yes—
242C22×C53C22×#4112yes—
243C22×C53C2×H(5,𝔽2)12yes—
244C22×C53C2611yescharacteristic
245C2×D53—212no—
246C2×D53—212no—
247C2×D53—212no—
248C2×D53—424no—
249C4×C53—26no—
250C4×C53—22no—
251C4×C53—26no—
252C4×C53C22×#4114yes—
253C4×C53C22×#41112yes—
254C4×C53C2×H(5,𝔽2)14yes—
255Dic53—24no—
256Dic53—212no—
257Dic53C22×#4118yes—
258#3483—5332no—
259C2×Q82—531no—
260C2×#555—533no—
261C2×#555—536no—
262C2×#550—532no—
263C2×#550—531no—
264C2×#550—533no—
265C2×#532—533no—
266C2×#532—536no—
267C2×C4×#41—533no—
268C2×C4×#41—532no—
269C2×C4×#41D5311yescharacteristic, complemented
270C2×C53—46no—
271C2×C53#348314yescomplemented
272C2×C53C23×#4111yescharacteristic
273C2×C53C23×#4111yescharacteristic
274C2×C53C22×H(5,𝔽2)11yescharacteristic
275D53—412no—
276C22×#41—532no—
277C22×#41—533no—
278C22×#41—536no—
279C22×#41C2×D5311yescharacteristic
280Q82—5316no—
281#555—5348no—
282#555—5396no—
283#550—5332no—
284#550—5316no—
285#550—5348no—
286#532—5396no—
287#532—5348no—
288C2×C4⋊Q8—536no—
289C2×C4⋊Q8—533no—
290C2×#127—536no—
291C2×#127—533no—
292C2×#124—5312no—
293C2×#124—536no—
294C2×#121—536no—
295C2×#121—536no—
296C2×#121—536no—
297C2×#121—536no—
298C2×#121—5312no—
299C4×#41—5332no—
300C4×#41—5348no—
301C4×#41C2×D53116yescomplemented
302C2×C4×Q8—532no—
303C2×C4×Q8—531no—
304C2×C4×Q8—533no—
305C2×C4×Q8C2×D5311yescharacteristic
306C2×C4×Q8C2×D5311yescharacteristic
307C2×C4×D4—536no—
308C2×C4×D4—536no—
309C2×C4×D4—536no—
310C2×C4×D4—533no—
311C2×C4×D4—536no—
312C2×C4×D4—533no—
313C2×C4×D4C2×D5313yes—
314C2×C4×D4C2×D5313yes—
315C2×#117—533no—
316C2×#117—536no—
317C2×#117—536no—
318C2×#117C2×D5313yes—
319C22×C42—536no—
320C22×C42C2×D5313yes—
321C53C2×#348311yescharacteristic, complemented, Sylow
322C2×#41—28no—
323C2×#41—5316no—
324C2×#41—5324no—
325C2×#41—5348no—
326C2×#41—10624no—
327C2×#41—10616no—
328C2×#41—21224no—
329C2×#41C22×D5318yes—
330C22×Q8—531no—
331C22×Q8—536no—
332C22×Q8C22×D5311yescharacteristic
333C22×D4—536no—
334C22×D4—5312no—
335C22×D4—533no—
336C22×D4—536no—
337C22×D4—536no—
338C22×D4C22×D5313yes—
339C23×C4—536no—
340C23×C4—536no—
341C23×C4—536no—
342C23×C4C22×D5313yes—
343C23×C4C22×D5313yes—
344C23×C4C22×D5313yes—
345C4⋊Q8—5348no—
346C4⋊Q8—5324no—
347#127—5348no—
348#127—5324no—
349#124—5396no—
350#124—5348no—
351#121—5348no—
352#121—5348no—
353#121—5348no—
354#121—5348no—
355#121—5396no—
356C4×Q8—5316no—
357C4×Q8—538no—
358C4×Q8—5324no—
359C4×Q8C22×D5318yes—
360C4×Q8C22×D5318yes—
361C4×D4—5348no—
362C4×D4—5348no—
363C4×D4—5348no—
364C4×D4—5348no—
365C4×D4—5324no—
366C4×D4—5324no—
367C4×D4C22×D53124yes—
368C4×D4C22×D53124yes—
369#117—5324no—
370#117—5348no—
371#117—5348no—
372#117C22×D53124yes—
373C2×C4⋊C4—532no—
374C2×C4⋊C4—531no—
375C2×C4⋊C4—536no—
376C2×C4⋊C4—533no—
377C2×C4⋊C4—536no—
378C2×C4⋊C4C22×D5313yes—
379C2×C4⋊C4C22×D5313yes—
380C2×C22⋊C4—536no—
381C2×C22⋊C4—536no—
382C2×C22⋊C4—5312no—
383C2×C22⋊C4—536no—
384C2×C22⋊C4C22×D5316yes—
385C2×C42—536no—
386C2×C42—536no—
387C2×C42—532no—
388C2×C42—5348no—
389C2×C42C22×D5313yes—
390C2×C42C22×D5313yes—
391C2×C42C22×D53124yes—
392C2×C42C22×D5311yescharacteristic
393C2×C42C22×D5313yes—
394C24—536no—
395C24C23×D5313yes—
396#41—232no—
397#41—10696no—
398#41—10664no—
399#41—21296no—
400C2×Q8—538no—
401C2×Q8—534no—
402C2×Q8—5324no—
403C2×Q8—21212no—
404C2×Q8#1171994518yescomplemented
405C2×Q8C23×D5314yes—
406C2×D4—212no—
407C2×D4—5324no—
408C2×D4—5348no—
409C2×D4—5312no—
410C2×D4—5324no—
411C2×D4—5324no—
412C2×D4—10624no—
413C2×D4—10612no—
414C2×D4—10624no—
415C2×D4—21212no—
416C2×D4—21224no—
417C2×D4C23×D53112yes—
418C22×C4—212no—
419C22×C4—212no—
420C22×C4—212no—
421C22×C4—532no—
422C22×C4—532no—
423C22×C4—536no—
424C22×C4—5324no—
425C22×C4—5324no—
426C22×C4—5324no—
427C22×C4—10624no—
428C22×C4—10624no—
429C22×C4—10624no—
430C22×C4C23×D5313yes—
431C22×C4C23×D5311yescharacteristic
432C22×C4C23×D5313yes—
433C22×C4C23×D53112yes—
434C22×C4C23×D53112yes—
435C22×C4C23×D53112yes—
436C22×C4C23×D5311yescharacteristic
437C22×C4C23×D5313yes—
438C22×C4C23×D5311yescharacteristic
439C4⋊C4—538no—
440C4⋊C4—534no—
441C4⋊C4—5324no—
442C4⋊C4—5312no—
443C4⋊C4—5324no—
444C4⋊C4C23×D53112yes—
445C4⋊C4C23×D53112yes—
446C22⋊C4—5324no—
447C22⋊C4—5324no—
448C22⋊C4—5348no—
449C22⋊C4—5324no—
450C22⋊C4C23×D53124yes—
451C42—538no—
452C42—5324no—
453C42—5324no—
454C42C23×D53112yes—
455C42C23×D5314yes—
456C42C23×D53112yes—
457C42C23×D53112yes—
458C23—26no—
459C23—26no—
460C23—26no—
461C23—5312no—
462C23—10612no—
463C23—10612no—
464C23—10612no—
465C23C24×D5316yes—
466C23C24×D5311yescharacteristic
467Q8—5316no—
468Q8—21224no—
469Q8C2×#11719945116yescomplemented
470D4—224no—
471D4—10648no—
472D4—10624no—
473D4—10648no—
474D4—21224no—
475D4—21248no—
476C2×C4—26no—
477C2×C4—22no—
478C2×C4—26no—
479C2×C4—224no—
480C2×C4—224no—
481C2×C4—224no—
482C2×C4—538no—
483C2×C4—534no—
484C2×C4—534no—
485C2×C4—5312no—
486C2×C4—10648no—
487C2×C4—1064no—
488C2×C4—10612no—
489C2×C4—10648no—
490C2×C4—10648no—
491C2×C4#41×D5314yes—
492C2×C4C2×#11719945112yes—
493C2×C4#1313326314yes—
494C2×C4C24×D5316yes—
495C2×C4C24×D5312yes—
496C2×C4C24×D5316yes—
497C2×C4C24×D5312yes—
498C2×C4C24×D5316yes—
499C2×C4C24×D5312yes—
500C22—26no—
501C22—26no—
502C22—26no—
503C22—412no—
504C22—10612no—
505C22—10612no—
506C22—10612no—
507C22—21224no—
508C22C2×#41×D5312yes—
509C22C22×#1171994512yes—
510C22C2×#1313326312yes—
511C22C25×D5311yescharacteristic
512C4—26no—
513C4—22no—
514C4—26no—
515C4—538no—
516C4—1064no—
517C4—10612no—
518C4C2×#41×D5314yes—
519C4C22×#11719945112yes—
520C4C2×#1313326314yes—
521C2—46no—
522C2—21212no—
523C2#2028834114yesdirect product factor
524C2C22×#41×D5311yescharacteristic
525C2C23×#1171994511yescharacteristic
526C2C22×#1313326311yescharacteristic
subgroupG/H
centerC22×C4C23×D53
commutator, cosocleC22×C53C26
FrattiniC22C25×D53
FittingC2×C4×#41×C53C2
socleC23×C53C25
hypercenterC2×C4×#41D53
hypocenterC53C2×#3483
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×C4×#41

Complements

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

NH ≅ G/Ntype
1. #20288341C2direct (N × H)
2. C2×Q8×Dic106C2semidirect
3. C2×#16811338C2semidirect
4. C2×#16811336C2semidirect
5. C2×#16811295C2semidirect
6. C2×#16811294C2semidirect
7. C2×#16811292C2semidirect
8. C2×#16811173C2semidirect
9. C2×#16811170C2semidirect
10. C2×C4×#11719949C2semidirect
11. C2×C4×#11719945C2semidirect
12. C2×C4×#41×C53C2semidirect
16. Q8×Dic106C22semidirect
17. #16811338C22semidirect
18. #16811336C22semidirect
19. #16811295C22semidirect
20. #16811294C22semidirect
21. #16811292C22semidirect
22. #16811173C22semidirect
23. #16811170C22semidirect
35. C4×#11719949C22semidirect
36. C4×#11719945C22semidirect
51. C4×#41×C53C22semidirect
133. C2×Q8×C53#41semidirect
172. C2×Dic106#41semidirect
203. Q8×C53C2×#41semidirect
235. Dic106C2×#41semidirect
269. C2×C4×#41D53semidirect
271. C2×C53#3483semidirect
301. C4×#41C2×D53semidirect
321. C53C2×#3483semidirect
404. C2×Q8#11719945semidirect
469. Q8C2×#11719945semidirect
523. C2#20288341direct (N × H)

Maximal subgroups

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

subgroupindex
#202883412
C2×Q8×Dic1062
C2×#168113382
C2×#168113362
C2×#168112952
C2×#168112942
C2×#168112922
C2×#168111732
C2×#168111702
C2×C4×#117199492
C2×C4×#117199452
C2×C4×#41×C532
C2×#348353