Finite Group Databasebeta

← C2×#20272839 #27299058

Subgroups — C2×#20272839 #27299058

Order 13568.

1154 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×#13133132C211yesmaximal, characteristic
2C2×#16810526C211yesmaximal, characteristic
3#20272839C214yesmaximal, direct product factor
4C22×#99×C53C211yesmaximal, characteristic
5C22×C4⋊Dic53C411yescharacteristic, complemented
6C22×C22⋊Dic53—21no—
7C22×C22⋊Dic53—21no—
8C22×C22⋊Dic53—21no—
9C22×C4×Dic53C411yescharacteristic, complemented
10C2×#13133132—21no—
11C2×#13133132—22no—
12C2×#13133132C414yescomplemented
13C2×#13133132C2212yes—
14#16810526—21no—
15#16810526—21no—
16#16810526C412yescomplemented
17#16810526C2212yes—
18C23×D4×C53C2211yescharacteristic
19C22×C22⋊C4×C53—21no—
20C2×#99×C53—21no—
21C2×#99×C53—22no—
22C2×#99×C53C414yescomplemented
23C2×#99×C53C2212yes—
24C23×Dic53—21no—
25C23×Dic53—21no—
26C23×Dic53—41no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53—41no—
29C2×C4⋊Dic53—41no—
30C2×C4⋊Dic53C2×C412yescomplemented
31C2×C22⋊Dic53—22no—
32C2×C22⋊Dic53—22no—
33C2×C22⋊Dic53—22no—
34C2×C22⋊Dic53—22no—
35C2×C22⋊Dic53—24no—
36C2×C22⋊Dic53—22no—
37C2×C22⋊Dic53—44no—
38C2×C22⋊Dic53—44no—
39C2×C22⋊Dic53—42no—
40C2×C22⋊Dic53—41no—
41C2×C4×Dic53—21no—
42C2×C4×Dic53—42no—
43C2×C4×Dic53C2×C412yescomplemented
44#13133132—24no—
45#13133132—22no—
46#13133132C2×C414yescomplemented
47C25×C53D411yescharacteristic
48C25×C53D411yescharacteristic
49C22×D4×C53—42no—
50C22×D4×C53—41no—
51C22×D4×C53—41no—
52C22×D4×C53C2×C412yes—
53C22×D4×C53C2×C412yes—
54C22×D4×C53C2×C411yescharacteristic
55C22×D4×C53D411yescharacteristic
56C22×D4×C53D411yescharacteristic
57C22×D4×C53D412yes—
58C22×D4×C53Q812yes—
59C22×D4×C53C2311yescharacteristic
60C23×C4×C53—41no—
61C23×C4×C53C2×C411yescharacteristic
62C2×C22⋊C4×C53—24no—
63C2×C22⋊C4×C53—22no—
64C2×C22⋊C4×C53—42no—
65C2×C22⋊C4×C53—41no—
66#99×C53—24no—
67#99×C53—22no—
68#99×C53C2×C414yescomplemented
69C24×C53—21no—
70C24×C53—21no—
71C24×C53—22no—
72C24×C53—21no—
73C24×C53—22no—
74C24×C53—21no—
75C24×C53—42no—
76C24×C53—42no—
77C24×C53—42no—
78C24×C53—42no—
79C24×C53C22⋊C411yescharacteristic
80C24×C53C22⋊C412yes—
81C24×C53C22⋊C411yescharacteristic
82C24×C53C22⋊C411yescharacteristic
83C24×C53C22⋊C412yes—
84C24×C53C4⋊C412yes—
85C24×C53C4⋊C412yes—
86C24×C53C2×D411yescharacteristic
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—22no—
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—44no—
118C22×C4×C53—42no—
119C22×C4×C53—82no—
120C22×C4×C53—82no—
121C22×C4×C53C4212yes—
122C22×C4×C53C22⋊C411yescharacteristic
123C22×C4×C53C4⋊C412yes—
124C22×C4×C53C22×C411yescharacteristic
125C22×Dic53—24no—
126C22×Dic53—22no—
127C22×Dic53—22no—
128C22×Dic53—42no—
129C22×Dic53—42no—
130C22×Dic53—44no—
131C22×Dic53—42no—
132C22×Dic53—42no—
133C22×Dic53—42no—
134C22×Dic53—41no—
135C22×Dic53—41no—
136C22×Dic53—82no—
137C22×Dic53—82no—
138C22⋊C4×C53—24no—
139C22⋊C4×C53—44no—
140C22⋊C4×C53—42no—
141C22⋊Dic53—24no—
142C22⋊Dic53—44no—
143C22⋊Dic53—44no—
144C22⋊Dic53—44no—
145C22⋊Dic53—44no—
146C22⋊Dic53—44no—
147C22⋊Dic53—42no—
148C4⋊Dic53—42no—
149C4⋊Dic53—42no—
150C4×Dic53—42no—
151C4×Dic53—42no—
152C23×C53—22no—
153C23×C53—22no—
154C23×C53—22no—
155C23×C53—22no—
156C23×C53—22no—
157C23×C53—22no—
158C23×C53—21no—
159C23×C53—21no—
160C23×C53—21no—
161C23×C53—21no—
162C23×C53—22no—
163C23×C53—22no—
164C23×C53—22no—
165C23×C53—22no—
166C23×C53—21no—
167C23×C53—21no—
168C23×C53—21no—
169C23×C53—21no—
170C23×C53—41no—
171C23×C53—41no—
172C23×C53—44no—
173C23×C53—44no—
174C23×C53—41no—
175C23×C53—41no—
176C23×C53—44no—
177C23×C53—44no—
178C23×C53—41no—
179C23×C53—41no—
180C23×C53—41no—
181C23×C53—41no—
182C23×C53—42no—
183C23×C53—42no—
184C23×C53—42no—
185C23×C53—42no—
186C23×C53—81no—
187C23×C53—81no—
188C23×C53—81no—
189C23×C53—81no—
190C23×C53—84no—
191C23×C53—84no—
192C23×C53#9512yes—
193C23×C53#9512yes—
194C23×C53#9512yes—
195C23×C53#9911yescharacteristic
196C23×C53#9912yes—
197C23×C53#9911yescharacteristic
198C23×C53C2×C22⋊C411yescharacteristic
199C23×C53C2×C22⋊C411yescharacteristic
200C23×C53C2×C22⋊C411yescharacteristic
201C23×C53C2×C4⋊C411yescharacteristic
202C23×C53C2×C4⋊C411yescharacteristic
203D4×C53—42no—
204D4×C53—42no—
205D4×C53—42no—
206D4×C53—42no—
207D4×C53—42no—
208D4×C53—42no—
209D4×C53—41no—
210D4×C53—41no—
211D4×C53—41no—
212D4×C53—41no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—44no—
222C2×C4×C53—82no—
223C2×C4×C53—82no—
224C2×C4×C53—88no—
225C2×C4×C53#9512yes—
226C2×C4×C53C2×C4211yescharacteristic
227C2×C4×C53C2×C4⋊C411yescharacteristic
228C2×Dic53—24no—
229C2×Dic53—42no—
230C2×Dic53—48no—
231C2×Dic53—42no—
232C2×Dic53—42no—
233C2×Dic53—42no—
234C2×Dic53—44no—
235C2×Dic53—42no—
236C2×Dic53—82no—
237C2×Dic53—82no—
238C2×Dic53—82no—
239C2×Dic53—88no—
240C2×Dic53—82no—
241C2×#1393—531nomaximal, Sylow
242C22×C53—21no—
243C22×C53—21no—
244C22×C53—21no—
245C22×C53—21no—
246C22×C53—24no—
247C22×C53—24no—
248C22×C53—21no—
249C22×C53—21no—
250C22×C53—22no—
251C22×C53—22no—
252C22×C53—21no—
253C22×C53—21no—
254C22×C53—42no—
255C22×C53—42no—
256C22×C53—41no—
257C22×C53—41no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53—41no—
261C22×C53—41no—
262C22×C53—41no—
263C22×C53—41no—
264C22×C53—41no—
265C22×C53—41no—
266C22×C53—41no—
267C22×C53—41no—
268C22×C53—44no—
269C22×C53—44no—
270C22×C53—44no—
271C22×C53—44no—
272C22×C53—81no—
273C22×C53—81no—
274C22×C53—81no—
275C22×C53—81no—
276C22×C53—82no—
277C22×C53—82no—
278C22×C53—82no—
279C22×C53—82no—
280C22×C53#34212yes—
281C22×C53#34212yes—
282C22×C53#35312yes—
283C22×C53#35412yes—
284C22×C53C2×#9511yescharacteristic
285C22×C53C2×#9911yescharacteristic
286C22×C53C2×#9911yescharacteristic
287C4×C53—22no—
288C4×C53—21no—
289C4×C53—21no—
290C4×C53—84no—
291Dic53—44no—
292Dic53—84no—
293Dic53—82no—
294C22×#99—531no—
295C22×#99D5311yescharacteristic
296C2×#390—531no—
297#1393—534no—
298C2×C53—21no—
299C2×C53—21no—
300C2×C53—22no—
301C2×C53—41no—
302C2×C53—41no—
303C2×C53—42no—
304C2×C53—42no—
305C2×C53—41no—
306C2×C53—41no—
307C2×C53#139314yescomplemented
308C2×C53C2×#34211yescharacteristic
309C2×C53C2×#35311yescharacteristic
310C2×C53C2×#35411yescharacteristic
311C23×D4C2×D5311yescharacteristic
312C22×C4⋊C4—531no—
313C22×C22⋊C4—21no—
314C22×C22⋊C4—1061no—
315C22×C22⋊C4—1061no—
316C22×C22⋊C4—1061no—
317C22×C42—531no—
318C2×#99—22no—
319C2×#99—21no—
320C2×#99—534no—
321C2×#99—532no—
322C2×#99—1061no—
323C2×#99—1062no—
324C2×#99Dic5314yescomplemented
325C2×#99C2×D5312yes—
326#390—532no—
327#390—532no—
328#390—1061no—
329#390—1061no—
330C53C2×#139311yescharacteristic, complemented, Sylow
331C25D21211yescharacteristic
332C25C22⋊D5311yescharacteristic
333C22×D4—42no—
334C22×D4—41no—
335C22×D4—41no—
336C22×D4Dic10612yes—
337C22×D4C4×D5311yescharacteristic
338C22×D4C4×D5312yes—
339C22×D4D21211yescharacteristic
340C22×D4C2×Dic5312yes—
341C22×D4C22⋊D5312yes—
342C22×D4C22⋊D5311yescharacteristic
343C22×D4C22×D5311yescharacteristic
344C23×C4—41no—
345C23×C4—1061no—
346C23×C4—1061no—
347C23×C4—2121no—
348C23×C4C4×D5311yescharacteristic
349C2×C4⋊C4—532no—
350C2×C4⋊C4—1061no—
351C2×C4⋊C4—2121no—
352C2×C4⋊C4—2121no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—42no—
356C2×C22⋊C4—41no—
357C2×C22⋊C4—1062no—
358C2×C22⋊C4—1064no—
359C2×C22⋊C4—1062no—
360C2×C22⋊C4—1062no—
361C2×C22⋊C4—1062no—
362C2×C22⋊C4—1062no—
363C2×C22⋊C4—2124no—
364C2×C22⋊C4—2124no—
365C2×C22⋊C4—2122no—
366C2×C22⋊C4—2121no—
367C2×C42—532no—
368C2×C42—1061no—
369C2×C42—2122no—
370#99—22no—
371#99—24no—
372#99—534no—
373#99—1062no—
374#99—1064no—
375#99C2×Dic5314yescomplemented
376C24—21no—
377C24—21no—
378C24—21no—
379C24—22no—
380C24—21no—
381C24—22no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24#1171991912yes—
387C24C4⋊Dic5312yes—
388C24#1171992111yescharacteristic
389C24#1171992111yescharacteristic
390C24#1171992112yes—
391C24#1171992111yescharacteristic
392C24C22⋊Dic5312yes—
393C24C2×D21211yescharacteristic
394C24C2×C22⋊D5311yescharacteristic
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4C4×Dic5312yes—
412C2×D4#1171991912yes—
413C2×D4#1171991912yes—
414C2×D4C4⋊Dic5312yes—
415C2×D4#1171992112yes—
416C2×D4C22⋊Dic5312yes—
417C2×D4C2×Dic10611yescharacteristic
418C2×D4C2×C4×D5311yescharacteristic
419C2×D4C22×Dic5311yescharacteristic
420C2×D4C2×C22⋊D5311yescharacteristic
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—42no—
425C22×C4—44no—
426C22×C4—82no—
427C22×C4—82no—
428C22×C4—1064no—
429C22×C4—1062no—
430C22×C4—1062no—
431C22×C4—4242no—
432C22×C4—4242no—
433C22×C4—2122no—
434C22×C4—2124no—
435C22×C4—2122no—
436C22×C4—2122no—
437C22×C4—2121no—
438C22×C4—2121no—
439C22×C4—2122no—
440C22×C4—2122no—
441C22×C4C4×Dic5312yes—
442C22×C4#1171991912yes—
443C22×C4#1171992111yescharacteristic
444C22×C4C2×C4×D5311yescharacteristic
445C4⋊C4—2122no—
446C4⋊C4—2122no—
447C22⋊C4—24no—
448C22⋊C4—42no—
449C22⋊C4—44no—
450C22⋊C4—1064no—
451C22⋊C4—2124no—
452C22⋊C4—2124no—
453C22⋊C4—2124no—
454C22⋊C4—2124no—
455C22⋊C4—2122no—
456C22⋊C4—2124no—
457C42—2122no—
458C42—2122no—
459C23—22no—
460C23—22no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—44no—
484C23—44no—
485C23—41no—
486C23—41no—
487C23—44no—
488C23—44no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—81no—
494C23—81no—
495C23—81no—
496C23—81no—
497C23—84no—
498C23—84no—
499C23#1313312412yes—
500C23#1313312412yes—
501C23#1313312412yes—
502C23#1313313111yescharacteristic
503C23#1313313112yes—
504C23#1313313111yescharacteristic
505C23C2×#1171992111yescharacteristic
506C23C2×#1171992111yescharacteristic
507C23C2×C22⋊Dic5311yescharacteristic
508C23C2×#1171991911yescharacteristic
509C23C2×C4⋊Dic5311yescharacteristic
510D4—41no—
511D4—41no—
512D4—41no—
513D4—41no—
514D4—42no—
515D4—42no—
516D4—42no—
517D4—42no—
518D4—42no—
519D4—42no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—44no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—88no—
532C2×C4—1064no—
533C2×C4—4242no—
534C2×C4—4242no—
535C2×C4—4242no—
536C2×C4—4248no—
537C2×C4—4242no—
538C2×C4—2124no—
539C2×C4—2122no—
540C2×C4—2122no—
541C2×C4—2122no—
542C2×C4—2122no—
543C2×C4—2128no—
544C2×C4—2122no—
545C2×C4#1313312412yes—
546C2×C4C2×C4×Dic5311yescharacteristic
547C2×C4C2×#1171991911yescharacteristic
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—24no—
553C22—24no—
554C22—21no—
555C22—21no—
556C22—22no—
557C22—22no—
558C22—21no—
559C22—21no—
560C22—42no—
561C22—42no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22#1681038912yes—
587C22#1681038912yes—
588C22#1681041512yes—
589C22#1681041712yes—
590C22C2×#1313312411yescharacteristic
591C22C2×#1313313111yescharacteristic
592C22C2×#1313313111yescharacteristic
593C4—22no—
594C4—21no—
595C4—21no—
596C4—84no—
597C4—4244no—
598C4—4242no—
599C4—2124no—
600C2—21no—
601C2—21no—
602C2—22no—
603C2—41no—
604C2—41no—
605C2—42no—
606C2—42no—
607C2—41no—
608C2—41no—
609C2#2027283914yesdirect product factor
610C2C2×#1681038911yescharacteristic
611C2C2×#1681041511yescharacteristic
612C2C2×#1681041711yescharacteristic
subgroupG/H
centerC23#13133131
commutatorC2×C4×C53C2×C42
FrattiniC22×D4C22×D53
FittingC22×#99×C53C2
socleC23×C53#99
prime radicalC23×D4×C53C22
cosocleC22×D4×C53C23
hypercenterC22×#99D53
hypocenterC53C2×#1393
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24 ◁ C23×D4 ◁ C22×#99

Complements

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

NH ≅ G/Ntype
3. #20272839C2direct (N × H)
5. C22×C4⋊Dic53C4semidirect
9. C22×C4×Dic53C4semidirect
12. C2×#13133132C4semidirect
16. #16810526C4semidirect
22. C2×#99×C53C4semidirect
30. C2×C4⋊Dic53C2×C4semidirect
43. C2×C4×Dic53C2×C4semidirect
46. #13133132C2×C4semidirect
68. #99×C53C2×C4semidirect
307. C2×C53#1393semidirect
324. C2×#99Dic53semidirect
330. C53C2×#1393semidirect
375. #99C2×Dic53semidirect
609. C2#20272839direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131331322
C2×#168105262
#202728392
C22×#99×C532
C2×#139353