Finite Group Databasebeta

← C2×#20285904 #28256196

Subgroups — C2×#20285904 #28256196

Order 13568.

2050 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811168C211yesmaximal, characteristic
2C22×#13133208C211yesmaximal, characteristic, complemented
3#20285904C2116yesmaximal, direct product factor
4C2×#16811030C212yesmaximal, complemented
5C2×#16810986C212yesmaximal, complemented
6C2×#16810941C211yesmaximal, characteristic, complemented
7C2×#16810937C211yesmaximal, characteristic, complemented
8C2×#16810934C211yesmaximal, characteristic, complemented
9C2×#16810922C212yesmaximal, complemented
10C2×#16810921C212yesmaximal, complemented
11C2×#16810677C211yesmaximal, characteristic, complemented
12C2×C53×#481C211yesmaximal, characteristic, complemented
13C22×#11719947—21no—
14#16811168—24no—
15#16811168C2218yes—
16C2×#13133216C2211yescharacteristic
17C2×#13133213C2211yescharacteristic
18C2×#13133208—21no—
19C2×#13133208—21no—
20C2×#13133208—24no—
21C2×#13133208C2212yes—
22C2×#13133208C2212yes—
23C2×#13133208C2218yescomplemented
24C2×#13133203—21no—
25C2×#13133191C2211yescharacteristic
26C2×#13133189—21no—
27C2×D4×Dic53C2211yescharacteristic
28C2×#13133179C2211yescharacteristic
29C2×#13133177—21no—
30C22×C4⋊Dic53—21no—
31C22×C4⋊Dic53C2211yescharacteristic
32C22×#11719919—21no—
33C22×C4×Dic53C2211yescharacteristic
34#16811030C22116yescomplemented
35#16810986C22116yescomplemented
36#16810941C2218yescomplemented
37#16810937C2218yescomplemented
38#16810934C2218yescomplemented
39#16810922C22116yescomplemented
40#16810921C22116yescomplemented
41#16810677C2218yescomplemented
42C2×#13133151C2212yes—
43C2×#13133150C2212yes—
44C2×#13133148C2212yes—
45C2×#13133146C2212yes—
46C2×#13133139C2212yes—
47C2×#13133138C2212yes—
48C2×#13133137C2212yes—
49C2×C22⋊#88322C2211yescharacteristic
50C2×#13133127C2212yes—
51C2×#121×C53C2211yescharacteristic
52C22×C4⋊C4×C53C2211yescharacteristic
53C53×#481C2218yescomplemented
54C2×#106×C53C2212yes—
55C2×#102×C53C2212yes—
56C2×C22⋊C8×C53C2211yescharacteristic
57C2×#11719947—28no—
58C2×#11719947—48no—
59C22×Dic106—21no—
60C22×D212C2311yescharacteristic
61C22×C22⋊D53—21no—
62C22×C22⋊D53—21no—
63C22×C22⋊D53—21no—
64C23×Dic53—21no—
65C23×Dic53—21no—
66C23×Dic53D411yescharacteristic
67C23×Dic53D411yescharacteristic
68C22×C4×D53—21no—
69C22×#88322C2312yes—
70#13133216—22no—
71#13133216C2314yes—
72#13133213—22no—
73#13133213C2314yes—
74#13133208—24no—
75#13133208—24no—
76#13133208—28no—
77#13133208—28no—
78#13133208C2318yes—
79#13133208C2318yes—
80#13133203—28no—
81#13133191—22no—
82#13133191C2314yes—
83#13133189—28no—
84D4×Dic53—22no—
85D4×Dic53C2314yes—
86#13133179—22no—
87#13133179C2314yes—
88#13133177—28no—
89C2×C4⋊Dic53—21no—
90C2×C4⋊Dic53—22no—
91C2×C4⋊Dic53—22no—
92C2×C4⋊Dic53—22no—
93C2×C4⋊Dic53—28no—
94C2×C4⋊Dic53C2312yes—
95C2×C4⋊Dic53C2314yes—
96C2×C4⋊Dic53C2311yescharacteristic
97C2×#11719919—22no—
98C2×#11719919—22no—
99C2×#11719919—28no—
100C2×#11719919—21no—
101C2×C22⋊Dic53—21no—
102C2×#11719921—21no—
103C2×#11719921—21no—
104C2×C4×Dic53D411yescharacteristic
105C2×C4×Dic53D411yescharacteristic
106C2×C4×Dic53D414yes—
107C2×C4×Dic53C2311yescharacteristic
108C2×C4×Dic53C2311yescharacteristic
109C2×C4×Dic53C2314yes—
110#13133151C2318yes—
111#13133150C2318yes—
112#13133148C2318yes—
113#13133146C2318yes—
114#13133139C2318yes—
115#13133138C2318yes—
116#13133137C2318yes—
117C22⋊#88322C2314yes—
118#13133127C2318yes—
119C22×D4×C53—21no—
120C22×D4×C53C2311yescharacteristic
121C23×C4×C53—21no—
122C23×C4×C53C2311yescharacteristic
123C22×C8×C53C2312yes—
124#121×C53—22no—
125#121×C53C2314yes—
126C2×C4⋊C4×C53—21no—
127C2×C4⋊C4×C53—22no—
128C2×C4⋊C4×C53C2312yes—
129C2×C4⋊C4×C53C2314yes—
130C2×C4⋊C4×C53C2311yescharacteristic
131C2×C22⋊C4×C53—21no—
132#106×C53C2318yes—
133#102×C53C2318yes—
134C22⋊C8×C53C2314yes—
135C24×C53—21no—
136C24×C53C2×D411yescharacteristic
137C23×D53—21no—
138C2×D4×C53—24no—
139C2×D4×C53—21no—
140C2×D4×C53—24no—
141C2×D4×C53—44no—
142C2×D4×C53C2412yes—
143C22×C4×C53—24no—
144C22×C4×C53—21no—
145C22×C4×C53—24no—
146C22×C4×C53—22no—
147C22×C4×C53—44no—
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D412yes—
150C22×C4×C53C2412yes—
151C22×C4×C53C2411yescharacteristic
152C2×C22⋊D53—24no—
153C2×C22⋊D53—24no—
154C2×C22⋊D53—24no—
155C2×C22⋊D53—44no—
156C2×C22⋊D53—44no—
157C2×C22⋊D53—44no—
158C22×Dic53—24no—
159C22×Dic53—21no—
160C22×Dic53—24no—
161C22×Dic53—22no—
162C22×Dic53—24no—
163C22×Dic53—42no—
164C22×Dic53—44no—
165C22×Dic53—44no—
166C22×Dic53C2×D411yescharacteristic
167C22×Dic53C2×D411yescharacteristic
168C22×Dic53C2×D411yescharacteristic
169C22×Dic53C2×D411yescharacteristic
170C22×Dic53C2×D414yes—
171C22×Dic53C2×D414yes—
172#11719947—432no—
173C2×D212—24no—
174C2×D212—21no—
175C2×D212C2412yes—
176C2×C4×D53—24no—
177C2×C4×D53—44no—
178C2×Dic106—24no—
179C2×Dic106—44no—
180C2×C8×C53—24no—
181C2×C8×C53C2414yes—
182C4⋊C4×C53—24no—
183C4⋊C4×C53—22no—
184C4⋊C4×C53—21no—
185C4⋊C4×C53C2414yes—
186C4⋊C4×C53C2412yes—
187C22⋊C4×C53—24no—
188C22⋊Dic53—24no—
189#11719921—24no—
190#11719921—24no—
191C4⋊Dic53—24no—
192C4⋊Dic53—22no—
193C4⋊Dic53—21no—
194C4⋊Dic53—28no—
195C4⋊Dic53—28no—
196C4⋊Dic53C2414yes—
197C4⋊Dic53C2412yes—
198#11719919—28no—
199#11719919—28no—
200#11719919—24no—
201C4×Dic53C2×D414yes—
202C4×Dic53C2×D412yes—
203C4×Dic53C2×D414yes—
204C4×Dic53C2×D412yes—
205C4×Dic53C2412yes—
206C4×Dic53C2412yes—
207C2×#88322—24no—
208C2×#88322C2414yes—
209C23×C53—22no—
210C23×C53—22no—
211C23×C53—22no—
212C23×C53—42no—
213C23×C53—42no—
214C23×C53—42no—
215C23×C53AGL(1,ℤ/8)14yes—
216C23×C53C22×D412yes—
217C23×C53C22×D411yescharacteristic
218C22×D53—22no—
219C22×D53—42no—
220C22×D53—42no—
221C22×D53—42no—
222D4×C53—24no—
223D4×C53—48no—
224D4×C53—42no—
225C2×C4×C53—22no—
226C2×C4×C53—24no—
227C2×C4×C53—24no—
228C2×C4×C53—22no—
229C2×C4×C53—24no—
230C2×C4×C53—48no—
231C2×C4×C53—42no—
232C2×C4×C53—44no—
233C2×C4×C53C22×D412yes—
234C2×C4×C53C22×D411yescharacteristic
235C2×C4×C532−1+414yes—
236C2×C4×C53C2511yescharacteristic
237C22⋊D53—48no—
238C22⋊D53—48no—
239C22⋊D53—48no—
240C2×Dic53—22no—
241C2×Dic53—22no—
242C2×Dic53—28no—
243C2×Dic53—22no—
244C2×Dic53—22no—
245C2×Dic53—22no—
246C2×Dic53—24no—
247C2×Dic53—44no—
248C2×Dic53—48no—
249C2×Dic53—42no—
250C2×Dic53—48no—
251C2×Dic53—44no—
252C2×Dic53C22×D412yes—
253C2×Dic53C22×D412yes—
254C2×Dic53C22×D412yes—
255C2×Dic53C22×D412yes—
256D212—24no—
257D212—42no—
258C4×D53—48no—
259Dic106—48no—
260C8×C53—24no—
261#88322—24no—
262C2×#3234—531nomaximal, Sylow
263C22×C53—22no—
264C22×C53—28no—
265C22×C53—22no—
266C22×C53—42no—
267C22×C53—42no—
268C22×C53—42no—
269C22×C53—84no—
270C22×C53#53612yes—
271C22×C53C2×AGL(1,ℤ/8)14yes—
272C22×C53C2×AGL(1,ℤ/8)12yes—
273C22×C53#57512yes—
274C22×C53C23×D411yescharacteristic
275C2×D53—42no—
276C2×D53—42no—
277C2×D53—42no—
278C2×D53—84no—
279C4×C53—22no—
280C4×C53—42no—
281C4×C53—44no—
282C4×C53#53612yes—
283C4×C53C2×2−1+412yes—
284Dic53—22no—
285Dic53—22no—
286Dic53—22no—
287Dic53—22no—
288Dic53—42no—
289Dic53—44no—
290C2×#532—531no—
291C22×#125—531no—
292#3234—5316no—
293C2×#501—532no—
294C2×#490—532no—
295C2×#481—531no—
296C2×#481D5311yescharacteristic, complemented
297C2×#480—531no—
298C2×#480—531no—
299C2×#478—532no—
300C2×#478—532no—
301C2×#432—531no—
302C2×C53—24no—
303C2×C53—82no—
304C2×C53#323414yescomplemented
305C2×C53C2×#53611yescharacteristic
306C2×C53C22×AGL(1,ℤ/8)11yescharacteristic
307C2×C53C2×#57511yescharacteristic
308D53—82no—
309C22×#41—1061no—
310#532—538no—
311#532—1064no—
312C2×C4⋊Q8—531no—
313C2×#127—531no—
314C2×#125—532no—
315C2×#125—532no—
316C2×#125—538no—
317C2×#125—1061no—
318C2×#125—1061no—
319C2×#125—1064no—
320C2×#124—1061no—
321C2×#121—531no—
322C2×#121—1061no—
323C2×#121C2×D5311yescharacteristic
324C2×C4×D4—531no—
325C2×C4×D4—531no—
326C2×C4×D4—1061no—
327C22×C4⋊C4—531no—
328C22×C4⋊C4—1061no—
329C22×C4⋊C4—1061no—
330C22×C4⋊C4C2×D5311yescharacteristic
331C22×C42—531no—
332#501—5316no—
333#490—5316no—
334#481—538no—
335#481C2×D5318yescomplemented
336#480—538no—
337#480—538no—
338#478—5316no—
339#478—5316no—
340#432—538no—
341C2×#107—532no—
342C2×#107—532no—
343C2×#106—532no—
344C2×#106C2×D5312yes—
345C2×C4⋊C8—532no—
346C2×#102—532no—
347C2×#102—532no—
348C2×#102—532no—
349C2×#102C2×D5312yes—
350C2×C22⋊C8—531no—
351C2×C22⋊C8C2×D5311yescharacteristic
352C2×#97—532no—
353C53C2×#323411yescharacteristic, complemented, Sylow
354C2×#41—1068no—
355C2×#41—2128no—
356C22×Q8—1061no—
357C22×D4—21no—
358C22×D4—531no—
359C22×D4—1061no—
360C22×D4—1061no—
361C22×D4—1061no—
362C22×D4C22×D5311yescharacteristic
363C23×C4—21no—
364C23×C4—531no—
365C23×C4—531no—
366C23×C4—1061no—
367C23×C4—1061no—
368C23×C4—1061no—
369C23×C4C22×D5311yescharacteristic
370C22×C8—532no—
371C22×C8C22×D5312yes—
372C4⋊Q8—534no—
373C4⋊Q8—1062no—
374#127—534no—
375#127—1062no—
376#125—538no—
377#125—538no—
378#125—1064no—
379#125—1068no—
380#125—1068no—
381#125—1064no—
382#124—1068no—
383#121—22no—
384#121—534no—
385#121—1068no—
386#121—1062no—
387#121C22×D5314yes—
388C4×D4—534no—
389C4×D4—534no—
390C4×D4—1068no—
391C4×D4—1062no—
392C4×D4—1062no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—532no—
396C2×C4⋊C4—534no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—1068no—
399C2×C4⋊C4—1062no—
400C2×C4⋊C4—1062no—
401C2×C4⋊C4—1068no—
402C2×C4⋊C4—1062no—
403C2×C4⋊C4—1062no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—1062no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4C22×D5312yes—
408C2×C4⋊C4C22×D5314yes—
409C2×C4⋊C4C22×D5311yescharacteristic
410C2×C22⋊C4—21no—
411C2×C22⋊C4—1061no—
412C2×C22⋊C4—1061no—
413C2×C22⋊C4—1061no—
414C2×C42—531no—
415C2×C42—531no—
416C2×C42—534no—
417C2×C42—531no—
418C2×C42—531no—
419C2×C42—534no—
420#107—538no—
421#107—538no—
422#106—538no—
423#106C22×D5318yes—
424C4⋊C8—538no—
425#102—538no—
426#102—538no—
427#102—538no—
428#102C22×D5318yes—
429C22⋊C8—534no—
430C22⋊C8C22×D5314yes—
431#97—538no—
432C24—21no—
433C24—1061no—
434C24D4×D5311yescharacteristic
435#41—21232no—
436C2×Q8—1064no—
437C2×Q8—2124no—
438C2×D4—24no—
439C2×D4—21no—
440C2×D4—24no—
441C2×D4—44no—
442C2×D4—532no—
443C2×D4—1064no—
444C2×D4—1064no—
445C2×D4—1064no—
446C2×D4—1061no—
447C2×D4—1064no—
448C2×D4—2124no—
449C2×D4—2124no—
450C2×D4—2124no—
451C2×D4C23×D5312yes—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—44no—
457C22×C4—534no—
458C22×C4—531no—
459C22×C4—531no—
460C22×C4—531no—
461C22×C4—531no—
462C22×C4—534no—
463C22×C4—1064no—
464C22×C4—1064no—
465C22×C4—1064no—
466C22×C4—1064no—
467C22×C4—1061no—
468C22×C4—1062no—
469C22×C4—2124no—
470C22×C4—2122no—
471C22×C4—2124no—
472C22×C4—2124no—
473C22×C4D4×D5312yes—
474C22×C4D4×D5311yescharacteristic
475C22×C4C23×D5312yes—
476C22×C4C23×D5311yescharacteristic
477C2×C8—24no—
478C2×C8—534no—
479C2×C8—1064no—
480C2×C8C23×D5314yes—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—534no—
485C4⋊C4—532no—
486C4⋊C4—1068no—
487C4⋊C4—1068no—
488C4⋊C4—1068no—
489C4⋊C4—1068no—
490C4⋊C4—1064no—
491C4⋊C4—1064no—
492C4⋊C4—1062no—
493C4⋊C4—1061no—
494C4⋊C4C23×D5314yes—
495C4⋊C4C23×D5312yes—
496C22⋊C4—24no—
497C22⋊C4—1064no—
498C22⋊C4—1064no—
499C22⋊C4—1064no—
500C42—532no—
501C42—534no—
502C42—534no—
503C42—532no—
504C42—532no—
505C42—532no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—1062no—
513C23—2122no—
514C23—2122no—
515C23—2122no—
516C23#1313324414yes—
517C23C2×D4×D5312yes—
518C23C2×D4×D5311yescharacteristic
519Q8—2128no—
520D4—24no—
521D4—48no—
522D4—42no—
523D4—1064no—
524D4—2128no—
525D4—2128no—
526D4—2128no—
527D4—2122no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—48no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—532no—
537C2×C4—532no—
538C2×C4—532no—
539C2×C4—532no—
540C2×C4—1062no—
541C2×C4—1062no—
542C2×C4—1068no—
543C2×C4—1062no—
544C2×C4—1062no—
545C2×C4—1062no—
546C2×C4—1064no—
547C2×C4—2128no—
548C2×C4—2124no—
549C2×C4—2128no—
550C2×C4—2128no—
551C2×C4—2124no—
552C2×C4—2122no—
553C2×C4C2×D4×D5312yes—
554C2×C4C2×D4×D5311yescharacteristic
555C2×C4#1313326414yes—
556C2×C4C24×D5311yescharacteristic
557C8—24no—
558C8—1064no—
559C22—22no—
560C22—22no—
561C22—28no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—84no—
566C22—4244no—
567C22—2122no—
568C22—2122no—
569C22—2122no—
570C22#1681119512yes—
571C22C2×#1313324414yes—
572C22C2×#1313324412yes—
573C22#1681141912yes—
574C22C22×D4×D5311yescharacteristic
575C4—22no—
576C4—42no—
577C4—44no—
578C4—1062no—
579C4—1062no—
580C4—1062no—
581C4—1062no—
582C4—2124no—
583C4—2122no—
584C4#1681119512yes—
585C4C2×#1313326412yes—
586C2—24no—
587C2—82no—
588C2—4242no—
589C2#2028590414yesdirect product factor
590C2C2×#1681119511yescharacteristic
591C2C22×#1313324411yescharacteristic
592C2C2×#1681141911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC2×C4×C53C25
FrattiniC2×C4C24×D53
FittingC2×C53×#481C2
socleC23×C53C22×D4
prime radicalC2×#16811168C2
hypercenterC2×#481D53
hypocenterC53C2×#3234
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#481

Complements

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

NH ≅ G/Ntype
2. C22×#13133208C2semidirect
3. #20285904C2direct (N × H)
4. C2×#16811030C2semidirect
5. C2×#16810986C2semidirect
6. C2×#16810941C2semidirect
7. C2×#16810937C2semidirect
8. C2×#16810934C2semidirect
9. C2×#16810922C2semidirect
10. C2×#16810921C2semidirect
11. C2×#16810677C2semidirect
12. C2×C53×#481C2semidirect
23. C2×#13133208C22semidirect
34. #16811030C22semidirect
35. #16810986C22semidirect
36. #16810941C22semidirect
37. #16810937C22semidirect
38. #16810934C22semidirect
39. #16810922C22semidirect
40. #16810921C22semidirect
41. #16810677C22semidirect
53. C53×#481C22semidirect
296. C2×#481D53semidirect
304. C2×C53#3234semidirect
335. #481C2×D53semidirect
353. C53C2×#3234semidirect
589. C2#20285904direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111682
C22×#131332082
#202859042
C2×#168110302
C2×#168109862
C2×#168109412
C2×#168109372
C2×#168109342
C2×#168109222
C2×#168109212
C2×#168106772
C2×C53×#4812
C2×#323453