Finite Group Databasebeta

← C2×#20272833 #27299031

Subgroups — C2×#20272833 #27299031

Order 13568.

1202 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133134C211yesmaximal, characteristic
2C22×#13133132C211yesmaximal, characteristic
3#20272833C214yesmaximal, direct product factor
4C2×C53×#386C211yesmaximal, characteristic
5C22×#11719917—21no—
6C22×C22⋊Dic53—21no—
7C2×#13133134—22no—
8C2×#13133134—21no—
9C2×#13133134C414yescomplemented
10C2×#13133134C2212yes—
11C2×#13133132—21no—
12C2×#13133132—22no—
13C2×#13133132C414yescomplemented
14C2×#13133132C2212yes—
15C23×D4×C53C2211yescharacteristic
16C24×C4×C53C411yescharacteristic, complemented
17C22×C22⋊C4×C53—21no—
18C22×C22⋊C4×C53C411yescharacteristic, complemented
19C53×#386—21no—
20C53×#386—21no—
21C53×#386C412yescomplemented
22C53×#386C2212yes—
23C2×#95×C53—21no—
24C23×Dic53—41no—
25C2×#11719917—24no—
26C2×#11719917—22no—
27C2×#11719917—42no—
28C2×#11719917—41no—
29C22×#88322—41no—
30C2×C22⋊Dic53—24no—
31C2×C22⋊Dic53—22no—
32C2×C22⋊Dic53—42no—
33C2×C22⋊Dic53—41no—
34#13133134—24no—
35#13133134—22no—
36#13133134C2×C414yescomplemented
37#13133132—24no—
38#13133132—22no—
39#13133132C2×C414yescomplemented
40C25×C53C2×C411yescharacteristic
41C25×C53D411yescharacteristic
42C22×D4×C53—41no—
43C22×D4×C53—42no—
44C22×D4×C53—41no—
45C22×D4×C53C2×C411yescharacteristic
46C22×D4×C53C2×C412yes—
47C22×D4×C53C2×C412yes—
48C22×D4×C53D411yescharacteristic
49C22×D4×C53D412yes—
50C22×D4×C53D411yescharacteristic
51C22×D4×C53Q812yes—
52C22×D4×C53C2311yescharacteristic
53C23×C4×C53—21no—
54C23×C4×C53—21no—
55C23×C4×C53—21no—
56C23×C4×C53—44no—
57C23×C4×C53—41no—
58C23×C4×C53—41no—
59C23×C4×C53C2×C412yescomplemented
60C23×C4×C53D411yescharacteristic
61C2×C22⋊C4×C53—22no—
62C2×C22⋊C4×C53—21no—
63C2×C22⋊C4×C53—22no—
64C2×C22⋊C4×C53—44no—
65C2×C22⋊C4×C53—42no—
66C2×C22⋊C4×C53C2×C412yescomplemented
67#95×C53—22no—
68#95×C53—22no—
69C24×C53—22no—
70C24×C53—21no—
71C24×C53—21no—
72C24×C53—21no—
73C24×C53—22no—
74C24×C53—21no—
75C24×C53—42no—
76C24×C53—42no—
77C24×C53—42no—
78C24×C53—42no—
79C24×C53C4212yes—
80C24×C53C22⋊C411yescharacteristic
81C24×C53C22⋊C412yes—
82C24×C53C22⋊C411yescharacteristic
83C24×C53C22⋊C411yescharacteristic
84C24×C53C4⋊C412yes—
85C24×C53C4⋊C412yes—
86C24×C53C22×C411yescharacteristic
87C24×C53C2×D411yescharacteristic
88C2×D4×C53—42no—
89C2×D4×C53—42no—
90C2×D4×C53—42no—
91C2×D4×C53—42no—
92C2×D4×C53—41no—
93C2×D4×C53—41no—
94C2×D4×C53—42no—
95C2×D4×C53—42no—
96C2×D4×C53—41no—
97C2×D4×C53—41no—
98C2×D4×C53—41no—
99C2×D4×C53—41no—
100C2×D4×C53—42no—
101C2×D4×C53—42no—
102C2×D4×C53—41no—
103C2×D4×C53—41no—
104C2×D4×C53C4212yes—
105C2×D4×C53C22⋊C412yes—
106C2×D4×C53C22⋊C412yes—
107C2×D4×C53C4⋊C412yes—
108C2×D4×C53C4⋊C412yes—
109C2×D4×C53C4⋊C412yes—
110C2×D4×C53C22×C411yescharacteristic
111C2×D4×C53C22×C411yescharacteristic
112C2×D4×C53C2×D411yescharacteristic
113C2×D4×C53C2×Q811yescharacteristic
114C22×C4×C53—21no—
115C22×C4×C53—22no—
116C22×C4×C53—21no—
117C22×C4×C53—24no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—44no—
121C22×C4×C53—44no—
122C22×C4×C53—42no—
123C22×C4×C53—42no—
124C22×C4×C53—48no—
125C22×C4×C53—44no—
126C22×C4×C53—41no—
127C22×C4×C53—41no—
128C22×C4×C53—42no—
129C22×C4×C53—44no—
130C22×C4×C53—42no—
131C22×C4×C53—42no—
132C22×C4×C53—42no—
133C22×C4×C53C22⋊C411yescharacteristic
134C22×C4×C53C22⋊C412yes—
135C22×C4×C53C4⋊C412yes—
136C22×C4×C53C2×D411yescharacteristic
137C22×Dic53—44no—
138C22×Dic53—42no—
139C22×Dic53—82no—
140C22×Dic53—82no—
141C22⋊C4×C53—44no—
142C22⋊C4×C53—44no—
143C22⋊C4×C53—42no—
144C22⋊C4×C53—42no—
145C22⋊Dic53—24no—
146C22⋊Dic53—44no—
147C22⋊Dic53—42no—
148#11719917—24no—
149#11719917—44no—
150#11719917—42no—
151C2×#88322—44no—
152C2×#88322—42no—
153C23×C53—22no—
154C23×C53—22no—
155C23×C53—22no—
156C23×C53—22no—
157C23×C53—22no—
158C23×C53—22no—
159C23×C53—22no—
160C23×C53—22no—
161C23×C53—21no—
162C23×C53—21no—
163C23×C53—21no—
164C23×C53—21no—
165C23×C53—21no—
166C23×C53—21no—
167C23×C53—22no—
168C23×C53—22no—
169C23×C53—21no—
170C23×C53—21no—
171C23×C53—41no—
172C23×C53—41no—
173C23×C53—44no—
174C23×C53—41no—
175C23×C53—44no—
176C23×C53—41no—
177C23×C53—42no—
178C23×C53—42no—
179C23×C53—48no—
180C23×C53—44no—
181C23×C53—41no—
182C23×C53—41no—
183C23×C53—44no—
184C23×C53—41no—
185C23×C53—41no—
186C23×C53—42no—
187C23×C53—42no—
188C23×C53—42no—
189C23×C53—42no—
190C23×C53—81no—
191C23×C53—81no—
192C23×C53—84no—
193C23×C53#9512yes—
194C23×C53#9512yes—
195C23×C53#9512yes—
196C23×C53#9911yescharacteristic
197C23×C53#9912yes—
198C23×C53#9911yescharacteristic
199C23×C53C2×C4211yescharacteristic
200C23×C53C2×C22⋊C411yescharacteristic
201C23×C53C2×C22⋊C411yescharacteristic
202C23×C53C2×C4⋊C411yescharacteristic
203C23×C53C2×C4⋊C411yescharacteristic
204D4×C53—42no—
205D4×C53—42no—
206D4×C53—42no—
207D4×C53—42no—
208D4×C53—42no—
209D4×C53—42no—
210D4×C53—41no—
211D4×C53—41no—
212D4×C53—41no—
213D4×C53—41no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—21no—
222C2×C4×C53—24no—
223C2×C4×C53—42no—
224C2×C4×C53—44no—
225C2×C4×C53—48no—
226C2×C4×C53—44no—
227C2×C4×C53—44no—
228C2×C4×C53—42no—
229C2×C4×C53—44no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×C4×C53—84no—
234C2×C4×C53#9512yes—
235C2×C4×C53C2×C22⋊C411yescharacteristic
236C2×C4×C53C2×C4⋊C411yescharacteristic
237C2×Dic53—44no—
238C2×Dic53—82no—
239C2×Dic53—82no—
240C2×Dic53—88no—
241#88322—44no—
242C2×#1390—531nomaximal, Sylow
243C22×C53—21no—
244C22×C53—21no—
245C22×C53—21no—
246C22×C53—21no—
247C22×C53—24no—
248C22×C53—24no—
249C22×C53—21no—
250C22×C53—22no—
251C22×C53—22no—
252C22×C53—21no—
253C22×C53—21no—
254C22×C53—21no—
255C22×C53—42no—
256C22×C53—42no—
257C22×C53—41no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53—41no—
261C22×C53—42no—
262C22×C53—42no—
263C22×C53—44no—
264C22×C53—44no—
265C22×C53—41no—
266C22×C53—41no—
267C22×C53—41no—
268C22×C53—41no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53—41no—
272C22×C53—41no—
273C22×C53—44no—
274C22×C53—44no—
275C22×C53—44no—
276C22×C53—44no—
277C22×C53—81no—
278C22×C53—81no—
279C22×C53—82no—
280C22×C53—82no—
281C22×C53#34212yes—
282C22×C53#34212yes—
283C22×C53AΣL(1,𝔽16)12yes—
284C22×C53#35212yes—
285C22×C53C2×#9511yescharacteristic
286C22×C53C2×#9911yescharacteristic
287C22×C53C2×#9911yescharacteristic
288C4×C53—21no—
289C4×C53—22no—
290C4×C53—21no—
291C4×C53—44no—
292C4×C53—82no—
293Dic53—84no—
294C22×#100—531no—
295C22×#99—531no—
296C2×#386D5311yescharacteristic
297#1390—534no—
298C2×C53—21no—
299C2×C53—21no—
300C2×C53—22no—
301C2×C53—41no—
302C2×C53—42no—
303C2×C53—41no—
304C2×C53—41no—
305C2×C53—42no—
306C2×C53—41no—
307C2×C53#139014yescomplemented
308C2×C53C2×#34211yescharacteristic
309C2×C53C2×AΣL(1,𝔽16)11yescharacteristic
310C2×C53C2×#35211yescharacteristic
311C23×D4C2×D5311yescharacteristic
312C24×C4Dic5311yescharacteristic, complemented
313C22×M16—1061no—
314C22×C22⋊C4—21no—
315C22×C22⋊C4—1061no—
316C22×C22⋊C4Dic5311yescharacteristic, complemented
317C2×#100—534no—
318C2×#100—532no—
319C2×#100—1062no—
320C2×#100—1061no—
321C2×#99—534no—
322C2×#99—532no—
323C2×#99—1062no—
324C2×#99—1061no—
325#386—21no—
326#386—21no—
327#386Dic5312yescomplemented
328#386C2×D5312yes—
329C2×#95—21no—
330C53C2×#139011yescharacteristic, complemented, Sylow
331C25C2×Dic5311yescharacteristic
332C25C22⋊D5311yescharacteristic
333C22×D4—42no—
334C22×D4—41no—
335C22×D4—41no—
336C22×D4Dic10612yes—
337C22×D4C4×D5312yes—
338C22×D4C4×D5312yes—
339C22×D4D21212yes—
340C22×D4C2×Dic5311yescharacteristic
341C22×D4C22⋊D5311yescharacteristic
342C22×D4C22⋊D5311yescharacteristic
343C22×D4C22×D5311yescharacteristic
344C23×C4—21no—
345C23×C4—21no—
346C23×C4—21no—
347C23×C4—44no—
348C23×C4—41no—
349C23×C4—41no—
350C23×C4—2121no—
351C23×C4C2×Dic5312yescomplemented
352C23×C4C22⋊D5311yescharacteristic
353C2×M16—1062no—
354C2×M16—1064no—
355C2×M16—2121no—
356C2×M16—2122no—
357C22×C8—2121no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—44no—
362C2×C22⋊C4—42no—
363C2×C22⋊C4—1062no—
364C2×C22⋊C4—1064no—
365C2×C22⋊C4—2122no—
366C2×C22⋊C4—2121no—
367C2×C22⋊C4C2×Dic5312yescomplemented
368#100—534no—
369#100—1062no—
370#100—1064no—
371#99—534no—
372#99—1062no—
373#99—1064no—
374#95—22no—
375#95—22no—
376C24—22no—
377C24—21no—
378C24—21no—
379C24—21no—
380C24—21no—
381C24—22no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24C4×Dic5312yes—
387C24#1171991912yes—
388C24C4⋊Dic5312yes—
389C24#1171992112yes—
390C24C22⋊Dic5311yescharacteristic
391C24C22⋊Dic5311yescharacteristic
392C24C22⋊Dic5311yescharacteristic
393C24C22×Dic5311yescharacteristic
394C24C2×C22⋊D5311yescharacteristic
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4C4×Dic5312yes—
412C2×D4#1171991912yes—
413C2×D4#1171991912yes—
414C2×D4C4⋊Dic5312yes—
415C2×D4#1171992112yes—
416C2×D4#1171992112yes—
417C2×D4C2×Dic10611yescharacteristic
418C2×D4C2×C4×D5311yescharacteristic
419C2×D4C2×C4×D5311yescharacteristic
420C2×D4C2×D21211yescharacteristic
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—48no—
432C22×C4—44no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—4242no—
441C22×C4—4242no—
442C22×C4—2122no—
443C22×C4—2124no—
444C22×C4#1171991912yes—
445C22×C4#1171992112yes—
446C22×C4C22⋊Dic5311yescharacteristic
447C22×C4C2×C22⋊D5311yescharacteristic
448M16—1064no—
449M16—2122no—
450M16—2124no—
451C2×C8—2122no—
452C2×C8—2124no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—1064no—
458C22⋊C4—2122no—
459C22⋊C4—2124no—
460C23—22no—
461C23—22no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—21no—
475C23—21no—
476C23—22no—
477C23—22no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—44no—
482C23—44no—
483C23—41no—
484C23—42no—
485C23—42no—
486C23—48no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—44no—
491C23—44no—
492C23—41no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—81no—
498C23—81no—
499C23—84no—
500C23#1313312412yes—
501C23#1313312412yes—
502C23#1313312412yes—
503C23#1313313211yescharacteristic
504C23#1313313212yes—
505C23#1313313211yescharacteristic
506C23C2×C4×Dic5311yescharacteristic
507C23C2×#1171992111yescharacteristic
508C23C2×C22⋊Dic5311yescharacteristic
509C23C2×#1171991911yescharacteristic
510C23C2×C4⋊Dic5311yescharacteristic
511D4—41no—
512D4—41no—
513D4—41no—
514D4—41no—
515D4—42no—
516D4—42no—
517D4—42no—
518D4—42no—
519D4—42no—
520D4—42no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—48no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—84no—
541C2×C4—4242no—
542C2×C4—4242no—
543C2×C4—4248no—
544C2×C4—2124no—
545C2×C4#1313312412yes—
546C2×C4C2×#1171992111yescharacteristic
547C2×C4C2×#1171991911yescharacteristic
548C8—2124no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—24no—
554C22—24no—
555C22—21no—
556C22—22no—
557C22—22no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—42no—
562C22—42no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—42no—
568C22—42no—
569C22—44no—
570C22—44no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—81no—
584C22—81no—
585C22—82no—
586C22—82no—
587C22#1681039012yes—
588C22#1681039012yes—
589C22#1681041012yes—
590C22#1681041312yes—
591C22C2×#1313312411yescharacteristic
592C22C2×#1313313211yescharacteristic
593C22C2×#1313313211yescharacteristic
594C4—21no—
595C4—22no—
596C4—21no—
597C4—44no—
598C4—82no—
599C4—4244no—
600C2—21no—
601C2—21no—
602C2—22no—
603C2—41no—
604C2—42no—
605C2—41no—
606C2—41no—
607C2—42no—
608C2—41no—
609C2#2027283314yesdirect product factor
610C2C2×#1681039011yescharacteristic
611C2C2×#1681041011yescharacteristic
612C2C2×#1681041311yescharacteristic
subgroupG/H
centerC23#13133132
commutatorC23×C53C2×C42
FrattiniC22×D4C22×D53
FittingC2×C53×#386C2
socleC23×C53#99
prime radicalC23×D4×C53C22
cosocleC22×D4×C53C23
hypercenterC2×#386D53
hypocenterC53C2×#1390
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24 ◁ C23×D4 ◁ C2×#386

Complements

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

NH ≅ G/Ntype
3. #20272833C2direct (N × H)
9. C2×#13133134C4semidirect
13. C2×#13133132C4semidirect
16. C24×C4×C53C4semidirect
18. C22×C22⋊C4×C53C4semidirect
21. C53×#386C4semidirect
36. #13133134C2×C4semidirect
39. #13133132C2×C4semidirect
59. C23×C4×C53C2×C4semidirect
66. C2×C22⋊C4×C53C2×C4semidirect
307. C2×C53#1390semidirect
312. C24×C4Dic53semidirect
316. C22×C22⋊C4Dic53semidirect
327. #386Dic53semidirect
330. C53C2×#1390semidirect
351. C23×C4C2×Dic53semidirect
367. C2×C22⋊C4C2×Dic53semidirect
609. C2#20272833direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131331342
C22×#131331322
#202728332
C2×C53×#3862
C2×#139053