Finite Group Databasebeta

← C2×#20273729 #27356334

Subgroups — C2×#20273729 #27356334

Order 13568.

1156 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810951C211yesmaximal, characteristic, complemented
2C2×#16810934C211yesmaximal, characteristic, complemented
3C2×#16810548C211yesmaximal, characteristic, complemented
4#20273729C218yesmaximal, direct product factor
5C2×#16810360C211yesmaximal, characteristic, complemented
6C2×#16810357C211yesmaximal, characteristic, complemented
7C2×#16810347C211yesmaximal, characteristic, complemented
8C2×C53×#538C211yesmaximal, characteristic
9C22×C4⋊Dic53C2211yescharacteristic
10C22×C22⋊Dic53—21no—
11#16810951—22no—
12#16810951C2214yescomplemented
13#16810934—22no—
14#16810934C2214yescomplemented
15C2×#13133150—21no—
16C2×#13133148—21no—
17C2×#13133141—21no—
18C2×#13133138—21no—
19C2×C22⋊#88322C2211yescharacteristic
20C2×C22⋊#88322C2211yescharacteristic
21#16810548—22no—
22#16810548C2214yescomplemented
23C2×#13133124—21no—
24C2×#13133124C2211yescharacteristic
25#16810360C2214yescomplemented
26#16810357C2214yescomplemented
27#16810347C2214yescomplemented
28C53×#538—22no—
29C53×#538—22no—
30C53×#538—22no—
31C53×#538C2214yes—
32C2×#126×C53—21no—
33C2×#124×C53—21no—
34C2×#123×C53—21no—
35C2×C22⋊Q8×C53C2211yescharacteristic
36C2×#121×C53—21no—
37C2×#121×C53C2211yescharacteristic
38C2×#120×C53—21no—
39C2×C4×D4×C53—21no—
40C22×C22⋊C4×C53C2211yescharacteristic
41C23×Dic53—21no—
42C23×Dic53—21no—
43C22×#88322—21no—
44C22×#88322—21no—
45C2×C4⋊Dic53—21no—
46C2×C4⋊Dic53—21no—
47C2×C4⋊Dic53—21no—
48C2×C4⋊Dic53—21no—
49C2×C4⋊Dic53—21no—
50C2×C4⋊Dic53C2312yes—
51C2×C22⋊Dic53—24no—
52C2×C22⋊Dic53—44no—
53#13133150—24no—
54#13133148—24no—
55#13133141—24no—
56#13133138—24no—
57C22⋊#88322—21no—
58C22⋊#88322—21no—
59C22⋊#88322C2312yes—
60C22⋊#88322C2312yes—
61#13133124—21no—
62#13133124—24no—
63#13133124C2312yes—
64C25×C53D411yescharacteristic
65C22×Q8×C53D411yescharacteristic
66C22×D4×C53—21no—
67C22×D4×C53—21no—
68C22×D4×C53—21no—
69C22×D4×C53D411yescharacteristic
70C23×C4×C53—21no—
71C23×C4×C53D411yescharacteristic
72C23×C4×C53C2311yescharacteristic
73#126×C53—24no—
74#126×C53—24no—
75#124×C53—24no—
76#124×C53—24no—
77#123×C53—24no—
78#123×C53—24no—
79C22⋊Q8×C53—21no—
80C22⋊Q8×C53—21no—
81C22⋊Q8×C53D412yes—
82C22⋊Q8×C53C2312yes—
83#121×C53—24no—
84#121×C53—24no—
85#121×C53—21no—
86#121×C53—21no—
87#121×C53D412yes—
88#121×C53C2312yes—
89#120×C53—24no—
90#120×C53—24no—
91C4×D4×C53—24no—
92C4×D4×C53—24no—
93C2×C4⋊C4×C53—21no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53D411yescharacteristic
96C2×C4⋊C4×C53D411yescharacteristic
97C2×C22⋊C4×C53—21no—
98C2×C22⋊C4×C53—21no—
99C2×C22⋊C4×C53—21no—
100C2×C22⋊C4×C53—21no—
101C2×C22⋊C4×C53—21no—
102C2×C22⋊C4×C53—21no—
103C2×C22⋊C4×C53—21no—
104C2×C22⋊C4×C53—21no—
105C2×C22⋊C4×C53—21no—
106C2×C22⋊C4×C53D412yes—
107C2×C22⋊C4×C53C2312yes—
108C2×C42×C53—21no—
109C24×C53—21no—
110C24×C53—21no—
111C24×C53—22no—
112C24×C53—21no—
113C24×C53—22no—
114C24×C53—42no—
115C24×C53—42no—
116C24×C53C2×D411yescharacteristic
117C24×C53C2×D412yes—
118C2×Q8×C53—21no—
119C2×Q8×C53—42no—
120C2×Q8×C53C2×D412yes—
121C2×D4×C53—22no—
122C2×D4×C53—22no—
123C2×D4×C53—24no—
124C2×D4×C53—24no—
125C2×D4×C53—21no—
126C2×D4×C53—42no—
127C2×D4×C53—42no—
128C2×D4×C53—42no—
129C2×D4×C53—82no—
130C2×D4×C53—82no—
131C2×D4×C53C2×D412yes—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—24no—
137C22×C4×C53—21no—
138C22×C4×C53—42no—
139C22×C4×C53—42no—
140C22×C4×C53—82no—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D412yes—
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2411yescharacteristic
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—42no—
151C22×Dic53—42no—
152C22×Dic53—41no—
153C22×Dic53—41no—
154C22×Dic53—42no—
155C22×Dic53—42no—
156C22×Dic53—42no—
157C4⋊C4×C53—24no—
158C4⋊C4×C53—22no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—22no—
162C4⋊C4×C53C2×D412yes—
163C4⋊C4×C53C2×D412yes—
164C22⋊C4×C53—24no—
165C22⋊C4×C53—24no—
166C22⋊C4×C53—24no—
167C22⋊C4×C53—22no—
168C22⋊C4×C53—22no—
169C22⋊C4×C53—22no—
170C22⋊C4×C53—22no—
171C22⋊C4×C53—22no—
172C22⋊C4×C53—22no—
173C22⋊C4×C53—22no—
174C22⋊C4×C53—22no—
175C42×C53—24no—
176C22⋊Dic53—48no—
177C4⋊Dic53—22no—
178C4⋊Dic53—22no—
179C4⋊Dic53—41no—
180C4⋊Dic53—41no—
181C2×#88322—22no—
182C2×#88322—22no—
183C2×#88322—42no—
184C2×#88322—42no—
185C23×C53—22no—
186C23×C53—22no—
187C23×C53—22no—
188C23×C53—22no—
189C23×C53—22no—
190C23×C53—22no—
191C23×C53—42no—
192C23×C53—42no—
193C23×C53—48no—
194C23×C53—42no—
195C23×C53—42no—
196C23×C53—42no—
197C23×C53—42no—
198C23×C53—42no—
199C23×C53—42no—
200C23×C53—42no—
201C23×C53—81no—
202C23×C53—81no—
203C23×C53—84no—
204C23×C53#12011yescharacteristic
205C23×C53#12011yescharacteristic
206C23×C53#13512yes—
207C23×C53AGL(1,ℤ/8)12yes—
208C23×C53C22×D411yescharacteristic
209Q8×C53—42no—
210Q8×C53—81no—
211D4×C53—42no—
212D4×C53—42no—
213D4×C53—42no—
214D4×C53—44no—
215D4×C53—81no—
216D4×C53—84no—
217D4×C53—84no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×C4×C53—81no—
234C2×C4×C53—84no—
235C2×C4×C53#12011yescharacteristic
236C2×C4×C53#12011yescharacteristic
237C2×C4×C53C22×D411yescharacteristic
238C2×C4×C53C22×D411yescharacteristic
239C2×Dic53—42no—
240C2×Dic53—42no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—84no—
246C2×Dic53—82no—
247C2×Dic53—82no—
248#88322—42no—
249#88322—42no—
250C2×#1612—531nomaximal, Sylow
251C22×C53—24no—
252C22×C53—24no—
253C22×C53—21no—
254C22×C53—21no—
255C22×C53—21no—
256C22×C53—21no—
257C22×C53—44no—
258C22×C53—42no—
259C22×C53—42no—
260C22×C53—42no—
261C22×C53—42no—
262C22×C53—42no—
263C22×C53—42no—
264C22×C53—82no—
265C22×C53—84no—
266C22×C53—84no—
267C22×C53—84no—
268C22×C53#44912yes—
269C22×C53#44912yes—
270C22×C53#45512yes—
271C22×C53#45812yes—
272C22×C53C2×#12011yescharacteristic
273C22×C53C2×#13511yescharacteristic
274C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
275C4×C53—21no—
276C4×C53—21no—
277C4×C53—21no—
278C4×C53—21no—
279C4×C53—42no—
280C4×C53—42no—
281C4×C53—42no—
282C4×C53—42no—
283Dic53—82no—
284Dic53—82no—
285C2×#538D5311yescharacteristic
286C2×#483—531no—
287C2×#480—531no—
288C2×#397—531no—
289#1612—538no—
290C2×#328—531no—
291C2×#327—531no—
292C2×#323—531no—
293C2×C53—21no—
294C2×C53—21no—
295C2×C53—22no—
296C2×C53—82no—
297C2×C53—82no—
298C2×C53#161214yescomplemented
299C2×C53C2×#44911yescharacteristic
300C2×C53C2×#45511yescharacteristic
301C2×C53C2×#45811yescharacteristic
302#538—22no—
303#538—22no—
304#538—22no—
305#538C2×D5314yes—
306C2×#126—21no—
307C2×#124—21no—
308C2×#123—21no—
309C2×C22⋊Q8C2×D5311yescharacteristic
310C2×#121—21no—
311C2×#121C2×D5311yescharacteristic
312C2×#120—21no—
313C2×C4×D4—21no—
314C22×C4⋊C4—531no—
315C22×C22⋊C4—1061no—
316C22×C22⋊C4C2×D5311yescharacteristic
317#483—534no—
318#483—1062no—
319#480—534no—
320#480—1062no—
321C2×#107—1061no—
322C2×#106—1061no—
323C2×#103—1061no—
324C2×#102—1061no—
325C2×C22⋊C8—531no—
326C2×C22⋊C8—531no—
327#397—534no—
328#397—1062no—
329C2×#95—531no—
330C2×#95—1061no—
331#328—534no—
332#327—534no—
333#323—534no—
334C53C2×#161211yescharacteristic, complemented, Sylow
335C25C22⋊D5311yescharacteristic
336C22×Q8C22⋊D5311yescharacteristic
337C22×D4—21no—
338C22×D4—21no—
339C22×D4—21no—
340C22×D4C22⋊D5311yescharacteristic
341C23×C4—21no—
342C23×C4—1061no—
343C23×C4—1061no—
344C23×C4C22⋊D5311yescharacteristic
345C23×C4C22×D5311yescharacteristic
346C22×C8—1061no—
347C22×C8—1061no—
348#126—24no—
349#126—24no—
350#124—24no—
351#124—24no—
352#123—24no—
353#123—24no—
354C22⋊Q8—21no—
355C22⋊Q8—21no—
356C22⋊Q8C22⋊D5312yes—
357C22⋊Q8C22×D5312yes—
358#121—24no—
359#121—24no—
360#121—21no—
361#121—21no—
362#121C22⋊D5312yes—
363#121C22×D5312yes—
364#120—24no—
365#120—24no—
366C4×D4—24no—
367C4×D4—24no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—532no—
371C2×C4⋊C4—1061no—
372C2×C4⋊C4—1061no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4C22⋊D5311yescharacteristic
377C2×C4⋊C4C22⋊D5311yescharacteristic
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—1064no—
388C2×C22⋊C4—2124no—
389C2×C22⋊C4C22⋊D5312yes—
390C2×C22⋊C4C22×D5312yes—
391C2×C42—21no—
392#107—1064no—
393#106—1064no—
394#103—1064no—
395#102—1064no—
396C22⋊C8—532no—
397C22⋊C8—532no—
398C22⋊C8—1061no—
399C22⋊C8—1061no—
400#95—532no—
401#95—1061no—
402#95—1064no—
403C24—21no—
404C24—21no—
405C24—22no—
406C24—21no—
407C24—22no—
408C24—42no—
409C24—42no—
410C24C2×C22⋊D5311yescharacteristic
411C24C2×C22⋊D5312yes—
412C2×Q8—21no—
413C2×Q8—42no—
414C2×Q8C2×C22⋊D5312yes—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—21no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—82no—
424C2×D4—82no—
425C2×D4C2×C22⋊D5312yes—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—82no—
435C22×C4—1062no—
436C22×C4—1062no—
437C22×C4—2122no—
438C22×C4—2122no—
439C22×C4—2121no—
440C22×C4—2121no—
441C22×C4—2122no—
442C22×C4—2122no—
443C22×C4—2122no—
444C22×C4C2×C22⋊D5311yescharacteristic
445C22×C4C2×C22⋊D5311yescharacteristic
446C22×C4C2×C22⋊D5312yes—
447C22×C4C2×C22⋊D5311yescharacteristic
448C22×C4C2×C22⋊D5311yescharacteristic
449C22×C4C2×C22⋊D5311yescharacteristic
450C22×C4C23×D5311yescharacteristic
451C2×C8—1062no—
452C2×C8—1062no—
453C2×C8—2122no—
454C2×C8—2122no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—1062no—
461C4⋊C4—1062no—
462C4⋊C4—2121no—
463C4⋊C4—2121no—
464C4⋊C4C2×C22⋊D5312yes—
465C4⋊C4C2×C22⋊D5312yes—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—2128no—
478C42—24no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—42no—
486C23—42no—
487C23—48no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—81no—
496C23—81no—
497C23—84no—
498C23#1313318611yescharacteristic
499C23#1313318611yescharacteristic
500C23#1313323712yes—
501C23#1313324212yes—
502C23C22×C22⋊D5311yescharacteristic
503Q8—42no—
504Q8—81no—
505D4—42no—
506D4—44no—
507D4—42no—
508D4—42no—
509D4—81no—
510D4—84no—
511D4—84no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—81no—
528C2×C4—84no—
529C2×C4—4244no—
530C2×C4—4242no—
531C2×C4—4242no—
532C2×C4—2122no—
533C2×C4—2122no—
534C2×C4—2122no—
535C2×C4—2122no—
536C2×C4—2122no—
537C2×C4—2122no—
538C2×C4#1313318611yescharacteristic
539C2×C4#1313318611yescharacteristic
540C2×C4C22×C22⋊D5311yescharacteristic
541C2×C4C22×C22⋊D5311yescharacteristic
542C8—2122no—
543C8—2122no—
544C22—24no—
545C22—24no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—44no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—82no—
558C22—84no—
559C22—84no—
560C22—84no—
561C22#1681076812yes—
562C22#1681076812yes—
563C22#1681080112yes—
564C22#1681081312yes—
565C22C2×#1313318611yescharacteristic
566C22C2×#1313323711yescharacteristic
567C22C2×#1313324211yescharacteristic
568C4—21no—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—42no—
576C4—4242no—
577C4—4242no—
578C2—21no—
579C2—22no—
580C2—21no—
581C2—82no—
582C2—82no—
583C2#2027372914yesdirect product factor
584C2C2×#1681076811yescharacteristic
585C2C2×#1681080111yescharacteristic
586C2C2×#1681081311yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#538C2
socleC23×C53#120
hypercenterC2×#538D53
hypocenterC53C2×#1612
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#538

Complements

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

NH ≅ G/Ntype
1. C2×#16810951C2semidirect
2. C2×#16810934C2semidirect
3. C2×#16810548C2semidirect
4. #20273729C2direct (N × H)
5. C2×#16810360C2semidirect
6. C2×#16810357C2semidirect
7. C2×#16810347C2semidirect
12. #16810951C22semidirect
14. #16810934C22semidirect
22. #16810548C22semidirect
25. #16810360C22semidirect
26. #16810357C22semidirect
27. #16810347C22semidirect
298. C2×C53#1612semidirect
334. C53C2×#1612semidirect
583. C2#20273729direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168109512
C2×#168109342
C2×#168105482
#202737292
C2×#168103602
C2×#168103572
C2×#168103472
C2×C53×#5382
C2×#161253