Finite Group Databasebeta

← C2×#20273678 #27356189

Subgroups — C2×#20273678 #27356189

Order 13568.

1232 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810778C211yesmaximal, characteristic, complemented
2C2×#16810761C211yesmaximal, characteristic
3C2×#16810543C211yesmaximal, characteristic, complemented
4#20273678C218yesmaximal, direct product factor
5C2×#16810360C211yesmaximal, characteristic, complemented
6C2×#16810357C211yesmaximal, characteristic
7C2×#16810349C211yesmaximal, characteristic, complemented
8C2×C53×#539C211yesmaximal, characteristic
9C23×Dic106C2211yescharacteristic
10C22×#11719925—21no—
11C22×#11719923—21no—
12C22×#11719919—21no—
13#16810778—22no—
14#16810778C2214yescomplemented
15#16810761—22no—
16#16810761C2214yes—
17C2×#13133140—21no—
18C2×#13133140—21no—
19C2×C22⋊#88322C2211yescharacteristic
20C2×C22⋊#88322C2211yescharacteristic
21#16810543—22no—
22#16810543C2214yescomplemented
23C2×#13133124—21no—
24C2×#13133124C2211yescharacteristic
25#16810360C2214yescomplemented
26#16810357C2214yes—
27#16810349C2214yescomplemented
28C53×#539—22no—
29C53×#539—22no—
30C53×#539—22no—
31C53×#539C2214yes—
32C2×#126×C53—21no—
33C2×#125×C53—21no—
34C2×#123×C53—21no—
35C2×#123×C53—21no—
36C2×C22⋊Q8×C53—21no—
37C2×C22⋊Q8×C53C2211yescharacteristic
38C2×#121×C53C2211yescharacteristic
39C2×C4×D4×C53—21no—
40C22×C4⋊C4×C53C2211yescharacteristic
41C22×Dic106—21no—
42C22×Dic106—21no—
43C22×Dic106—21no—
44C22×Dic106—21no—
45C22×Dic106—21no—
46C22×Dic106—44no—
47C22×Dic106C2312yes—
48C23×Dic53—21no—
49C23×Dic53—21no—
50C2×#11719925—24no—
51C2×#11719925—44no—
52C2×#11719923—24no—
53C2×#11719923—44no—
54C22×#88322—21no—
55C22×#88322—21no—
56C2×#11719919—24no—
57C2×#11719919—44no—
58#13133140—24no—
59#13133140—24no—
60C22⋊#88322—21no—
61C22⋊#88322—21no—
62C22⋊#88322C2312yes—
63C22⋊#88322C2312yes—
64#13133124—21no—
65#13133124—24no—
66#13133124C2312yes—
67C22×Q8×C53D411yescharacteristic
68C22×D4×C53—21no—
69C22×D4×C53D411yescharacteristic
70C23×C4×C53—21no—
71C23×C4×C53D411yescharacteristic
72C23×C4×C53D411yescharacteristic
73C23×C4×C53C2311yescharacteristic
74#126×C53—24no—
75#126×C53—24no—
76#125×C53—24no—
77#125×C53—24no—
78#123×C53—24no—
79#123×C53—24no—
80#123×C53—24no—
81#123×C53—24no—
82C22⋊Q8×C53—24no—
83C22⋊Q8×C53—24no—
84C22⋊Q8×C53—21no—
85C22⋊Q8×C53—21no—
86C22⋊Q8×C53D412yes—
87C22⋊Q8×C53C2312yes—
88#121×C53—21no—
89#121×C53—21no—
90#121×C53D412yes—
91#121×C53C2312yes—
92C4×D4×C53—24no—
93C4×D4×C53—24no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53D411yescharacteristic
103C2×C4⋊C4×C53D411yescharacteristic
104C2×C4⋊C4×C53D412yes—
105C2×C4⋊C4×C53C2312yes—
106C2×C22⋊C4×C53—21no—
107C2×C22⋊C4×C53—21no—
108C2×C22⋊C4×C53—21no—
109C2×C22⋊C4×C53—21no—
110C2×C22⋊C4×C53—21no—
111C2×C42×C53—21no—
112C24×C53—21no—
113C24×C53C2×D411yescharacteristic
114C2×Q8×C53—21no—
115C2×Q8×C53—42no—
116C2×Q8×C53C2×D412yes—
117C2×D4×C53—22no—
118C2×D4×C53—22no—
119C2×D4×C53—21no—
120C2×D4×C53—42no—
121C2×D4×C53—42no—
122C2×D4×C53—42no—
123C2×D4×C53C2×D412yes—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—24no—
133C22×C4×C53—21no—
134C22×C4×C53—42no—
135C22×C4×C53—42no—
136C22×C4×C53—82no—
137C22×C4×C53C2×D412yes—
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D412yes—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2411yescharacteristic
145C22×Dic53—24no—
146C22×Dic53—24no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—42no—
150C22×Dic53—42no—
151C22×Dic53—41no—
152C22×Dic53—41no—
153C22×Dic53—42no—
154C2×Dic106—24no—
155C2×Dic106—24no—
156C2×Dic106—22no—
157C2×Dic106—22no—
158C2×Dic106—44no—
159C2×Dic106—44no—
160C2×Dic106—44no—
161C2×Dic106—44no—
162C2×Dic106—42no—
163C2×Dic106—42no—
164C2×Dic106—41no—
165C2×Dic106—41no—
166C4⋊C4×C53—24no—
167C4⋊C4×C53—24no—
168C4⋊C4×C53—24no—
169C4⋊C4×C53—22no—
170C4⋊C4×C53—22no—
171C4⋊C4×C53—22no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—22no—
175C4⋊C4×C53—22no—
176C4⋊C4×C53—22no—
177C4⋊C4×C53C2×D412yes—
178C4⋊C4×C53C2×D412yes—
179C22⋊C4×C53—24no—
180C22⋊C4×C53—24no—
181C22⋊C4×C53—24no—
182C22⋊C4×C53—22no—
183C22⋊C4×C53—22no—
184C22⋊C4×C53—22no—
185C22⋊C4×C53—22no—
186C42×C53—24no—
187#11719925—48no—
188#11719923—48no—
189#11719919—48no—
190C2×#88322—22no—
191C2×#88322—22no—
192C2×#88322—42no—
193C2×#88322—42no—
194C23×C53—22no—
195C23×C53—22no—
196C23×C53—22no—
197C23×C53—42no—
198C23×C53—42no—
199C23×C53—42no—
200C23×C53#12011yescharacteristic
201C23×C53#12011yescharacteristic
202C23×C53#13512yes—
203C23×C53#13712yes—
204C23×C53C22×D411yescharacteristic
205Q8×C53—42no—
206Q8×C53—81no—
207D4×C53—42no—
208D4×C53—42no—
209D4×C53—42no—
210D4×C53—44no—
211D4×C53—81no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53—81no—
233C2×C4×C53—84no—
234C2×C4×C53#12011yescharacteristic
235C2×C4×C53#12011yescharacteristic
236C2×C4×C53C22×D411yescharacteristic
237C2×C4×C53C22×D411yescharacteristic
238C2×Dic53—24no—
239C2×Dic53—24no—
240C2×Dic53—42no—
241C2×Dic53—44no—
242C2×Dic53—48no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246C2×Dic53—44no—
247Dic106—24no—
248Dic106—24no—
249Dic106—42no—
250Dic106—42no—
251Dic106—44no—
252Dic106—44no—
253#88322—42no—
254#88322—42no—
255C2×#1603—531nomaximal, Sylow
256C22×C53—24no—
257C22×C53—24no—
258C22×C53—21no—
259C22×C53—21no—
260C22×C53—21no—
261C22×C53—21no—
262C22×C53—42no—
263C22×C53—42no—
264C22×C53—42no—
265C22×C53—84no—
266C22×C53#45212yes—
267C22×C53#45212yes—
268C22×C53#45612yes—
269C22×C53B(4,𝔽2)12yes—
270C22×C53C2×#12011yescharacteristic
271C22×C53C2×#13511yescharacteristic
272C22×C53C2×#13711yescharacteristic
273C4×C53—21no—
274C4×C53—21no—
275C4×C53—21no—
276C4×C53—21no—
277C4×C53—42no—
278C4×C53—42no—
279C4×C53—42no—
280C4×C53—42no—
281C4×C53—42no—
282Dic53—44no—
283Dic53—44no—
284C2×#539D5311yescharacteristic
285C2×#451—531no—
286C2×#448—531no—
287C2×#395—531no—
288#1603—538no—
289C2×#328—531no—
290C2×#327—531no—
291C2×#324—531no—
292C2×C53—21no—
293C2×C53—21no—
294C2×C53—22no—
295C2×C53—82no—
296C2×C53#160314yescomplemented
297C2×C53C2×#45211yescharacteristic
298C2×C53C2×#45611yescharacteristic
299C2×C53C2×B(4,𝔽2)11yescharacteristic
300C23×Q8—531no—
301C22×Q16—1061no—
302C22×SD16—1061no—
303#539—22no—
304#539—22no—
305#539—22no—
306#539C2×D5314yes—
307C2×#126—21no—
308C2×#125—21no—
309C2×#123—21no—
310C2×#123—21no—
311C2×C22⋊Q8—21no—
312C2×C22⋊Q8C2×D5311yescharacteristic
313C2×#121C2×D5311yescharacteristic
314C2×C4×D4—21no—
315C22×C4⋊C4—1061no—
316C22×C4⋊C4C2×D5311yescharacteristic
317#451—534no—
318#451—1062no—
319#448—534no—
320#448—1062no—
321C2×#103—1061no—
322C2×#103—1061no—
323C2×C22⋊C8—531no—
324C2×C22⋊C8—531no—
325#395—534no—
326#395—1062no—
327C2×#95—531no—
328C2×#95—1061no—
329#328—534no—
330#327—534no—
331#324—534no—
332C53C2×#160311yescharacteristic, complemented, Sylow
333C22×Q8—532no—
334C22×Q8—1061no—
335C22×Q8—1061no—
336C22×Q8—1061no—
337C22×Q8—1061no—
338C22×Q8—1061no—
339C22×Q8—2124no—
340C22×Q8C22⋊D5311yescharacteristic
341C22×D4—21no—
342C22×D4C22⋊D5311yescharacteristic
343C23×C4—21no—
344C23×C4—1061no—
345C23×C4—1061no—
346C23×C4C22⋊D5311yescharacteristic
347C23×C4C22⋊D5311yescharacteristic
348C23×C4C22×D5311yescharacteristic
349C2×Q16—1064no—
350C2×Q16—2124no—
351C2×SD16—1064no—
352C2×SD16—2124no—
353C22×C8—1061no—
354C22×C8—1061no—
355#126—24no—
356#126—24no—
357#125—24no—
358#125—24no—
359#123—24no—
360#123—24no—
361#123—24no—
362#123—24no—
363C22⋊Q8—24no—
364C22⋊Q8—24no—
365C22⋊Q8—21no—
366C22⋊Q8—21no—
367C22⋊Q8C22⋊D5312yes—
368C22⋊Q8C22×D5312yes—
369#121—21no—
370#121—21no—
371#121C22⋊D5312yes—
372#121C22×D5312yes—
373C4×D4—24no—
374C4×D4—24no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—1064no—
384C2×C4⋊C4—2124no—
385C2×C4⋊C4C22⋊D5312yes—
386C2×C4⋊C4C22⋊D5311yescharacteristic
387C2×C4⋊C4C22⋊D5311yescharacteristic
388C2×C4⋊C4C22×D5312yes—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C42—21no—
395#103—1064no—
396#103—1064no—
397C22⋊C8—532no—
398C22⋊C8—532no—
399C22⋊C8—1061no—
400C22⋊C8—1061no—
401#95—532no—
402#95—1061no—
403#95—1064no—
404C24—21no—
405C24C2×C22⋊D5311yescharacteristic
406C2×Q8—21no—
407C2×Q8—42no—
408C2×Q8—1064no—
409C2×Q8—1064no—
410C2×Q8—1062no—
411C2×Q8—1062no—
412C2×Q8—2122no—
413C2×Q8—2122no—
414C2×Q8—2124no—
415C2×Q8—2124no—
416C2×Q8—2124no—
417C2×Q8—2124no—
418C2×Q8—2121no—
419C2×Q8—2121no—
420C2×Q8C2×C22⋊D5312yes—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—21no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4C2×C22⋊D5312yes—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—82no—
441C22×C4—1064no—
442C22×C4—1064no—
443C22×C4—1062no—
444C22×C4—1062no—
445C22×C4—2122no—
446C22×C4—2122no—
447C22×C4—2121no—
448C22×C4—2121no—
449C22×C4—2122no—
450C22×C4C2×C22⋊D5312yes—
451C22×C4C2×C22⋊D5311yescharacteristic
452C22×C4C2×C22⋊D5311yescharacteristic
453C22×C4C2×C22⋊D5312yes—
454C22×C4C2×C22⋊D5311yescharacteristic
455C22×C4C2×C22⋊D5311yescharacteristic
456C22×C4C2×C22⋊D5311yescharacteristic
457C22×C4C23×D5311yescharacteristic
458Q16—2128no—
459SD16—2128no—
460C2×C8—1062no—
461C2×C8—1062no—
462C2×C8—2122no—
463C2×C8—2122no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—2128no—
476C4⋊C4C2×C22⋊D5312yes—
477C4⋊C4C2×C22⋊D5312yes—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C42—24no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23#1313318611yescharacteristic
493C23#1313318611yescharacteristic
494C23#1313323712yes—
495C23#1313324712yes—
496C23C22×C22⋊D5311yescharacteristic
497Q8—42no—
498Q8—81no—
499Q8—1064no—
500Q8—1064no—
501Q8—2122no—
502Q8—2122no—
503Q8—2124no—
504Q8—2124no—
505D4—42no—
506D4—44no—
507D4—42no—
508D4—42no—
509D4—81no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—81no—
531C2×C4—84no—
532C2×C4—1064no—
533C2×C4—1064no—
534C2×C4—2122no—
535C2×C4—2128no—
536C2×C4—2122no—
537C2×C4—2124no—
538C2×C4—2124no—
539C2×C4—2122no—
540C2×C4—2122no—
541C2×C4#1313318611yescharacteristic
542C2×C4#1313318611yescharacteristic
543C2×C4C22×C22⋊D5311yescharacteristic
544C2×C4C22×C22⋊D5311yescharacteristic
545C8—2122no—
546C8—2122no—
547C22—24no—
548C22—24no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—84no—
557C22#1681078512yes—
558C22#1681078512yes—
559C22#1681080612yes—
560C22#1681081012yes—
561C22C2×#1313318611yescharacteristic
562C22C2×#1313323711yescharacteristic
563C22C2×#1313324711yescharacteristic
564C4—21no—
565C4—21no—
566C4—21no—
567C4—21no—
568C4—42no—
569C4—42no—
570C4—42no—
571C4—42no—
572C4—42no—
573C4—2124no—
574C4—2124no—
575C2—21no—
576C2—22no—
577C2—21no—
578C2—82no—
579C2#2027367814yesdirect product factor
580C2C2×#1681078511yescharacteristic
581C2C2×#1681080611yescharacteristic
582C2C2×#1681081011yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#539C2
socleC23×C53#120
prime radicalC2×#121×C53C22
hypercenterC2×#539D53
hypocenterC53C2×#1603
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#539

Complements

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

NH ≅ G/Ntype
1. C2×#16810778C2semidirect
3. C2×#16810543C2semidirect
4. #20273678C2direct (N × H)
5. C2×#16810360C2semidirect
7. C2×#16810349C2semidirect
14. #16810778C22semidirect
22. #16810543C22semidirect
25. #16810360C22semidirect
27. #16810349C22semidirect
296. C2×C53#1603semidirect
332. C53C2×#1603semidirect
579. C2#20273678direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168107782
C2×#168107612
C2×#168105432
#202736782
C2×#168103602
C2×#168103572
C2×#168103492
C2×C53×#5392
C2×#160353