Finite Group Databasebeta

← #27279463

Subgroups — #27279463

Order 13568.

1136 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810502C211yesmaximal, characteristic
2#20274625C211yesmaximal, characteristic, complemented
3C2×#16810346C212yesmaximal, complemented
4#20272557C212yesmaximal, complemented
5C22×C22⋊C8×C53C211yesmaximal, characteristic, complemented
6C23×C22⋊D53C411yescharacteristic
7C22×C22⋊Dic53C411yescharacteristic
8C22×#11719921C411yescharacteristic
9C22×#11719921C2211yescharacteristic
10C2×C22⋊#88322—21no—
11C2×C22⋊#88322—21no—
12C2×C22⋊#88322C2212yes—
13#16810502—21no—
14#16810502—21no—
15#16810502—21no—
16#16810502—21no—
17C2×#13133124C411yescharacteristic
18C2×#13133124C2211yescharacteristic
19#16810346—24no—
20C24×C4×C53C2211yescharacteristic
21C23×C8×C53—21no—
22C2×C22⋊C8×C53—21no—
23C2×C22⋊C8×C53—24no—
24C2×C22⋊C8×C53C2212yes—
25C24×D53C2×C411yescharacteristic
26C22×C22⋊D53—21no—
27C22×C22⋊D53—21no—
28C22×C22⋊D53—22no—
29C22×C22⋊D53—42no—
30C22×C22⋊D53—42no—
31C22×C22⋊D53C814yescomplemented
32C23×Dic53C2×C411yescharacteristic
33C23×Dic53C2×C411yescharacteristic
34C23×Dic53C2×C411yescharacteristic
35C22×#88322—22no—
36C2×C22⋊Dic53—21no—
37C2×C22⋊Dic53—21no—
38C2×C22⋊Dic53—42no—
39C2×#11719921—21no—
40C2×#11719921—21no—
41C2×#11719921—21no—
42C2×#11719921—21no—
43C2×#11719921—22no—
44C2×#11719921—22no—
45C2×#11719921—42no—
46C22⋊#88322—22no—
47C22⋊#88322—22no—
48C22⋊#88322—42no—
49C22⋊#88322—42no—
50#13133124—21no—
51#13133124—21no—
52#13133124—21no—
53#13133124—21no—
54C25×C53C2×C411yescharacteristic
55C23×C4×C53—22no—
56C23×C4×C53—22no—
57C23×C4×C53—42no—
58C23×C4×C53—42no—
59C23×C4×C53C2×C411yescharacteristic
60C23×C4×C53D411yescharacteristic
61C23×C4×C53D411yescharacteristic
62C23×C4×C53D411yescharacteristic
63C23×C4×C53D411yescharacteristic
64C23×C4×C53C2311yescharacteristic
65C22×C8×C53—24no—
66C22×C8×C53—22no—
67C22×C8×C53—42no—
68C22×C8×C53—42no—
69C22⋊C8×C53—24no—
70C22⋊C8×C53—22no—
71C22⋊C8×C53D414yescomplemented
72C24×C53—22no—
73C24×C53—22no—
74C24×C53—22no—
75C24×C53—22no—
76C24×C53—41no—
77C24×C53—41no—
78C24×C53C22⋊C411yescharacteristic
79C24×C53C22⋊C411yescharacteristic
80C24×C53C2×C812yes—
81C24×C53M1612yes—
82C24×C53C22×C411yescharacteristic
83C23×D53—21no—
84C23×D53—21no—
85C23×D53—22no—
86C23×D53—22no—
87C23×D53—41no—
88C23×D53—41no—
89C23×D53—41no—
90C23×D53—41no—
91C23×D53C2×C812yescomplemented
92C22×C4×C53—22no—
93C22×C4×C53—22no—
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53—22no—
97C22×C4×C53—22no—
98C22×C4×C53—22no—
99C22×C4×C53—22no—
100C22×C4×C53—21no—
101C22×C4×C53—22no—
102C22×C4×C53—22no—
103C22×C4×C53—21no—
104C22×C4×C53—41no—
105C22×C4×C53—41no—
106C22×C4×C53—41no—
107C22×C4×C53—41no—
108C22×C4×C53—41no—
109C22×C4×C53—41no—
110C22×C4×C53—41no—
111C22×C4×C53—48no—
112C22×C4×C53—48no—
113C22×C4×C53C22⋊C411yescharacteristic
114C22×C4×C53C22⋊C411yescharacteristic
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×D412yes—
118C22×C4×C53#4112yes—
119C2×C22⋊D53—24no—
120C2×C22⋊D53—24no—
121C2×C22⋊D53—48no—
122C2×C22⋊D53—42no—
123C2×C22⋊D53—42no—
124C2×C22⋊D53—42no—
125C2×C22⋊D53—42no—
126C2×C22⋊D53—48no—
127C22×Dic53—21no—
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53—21no—
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—41no—
134C22×Dic53—41no—
135C22×Dic53—41no—
136C22×Dic53—41no—
137C22×Dic53—41no—
138C22×Dic53—41no—
139C22×Dic53C2×C812yescomplemented
140C2×C8×C53—24no—
141C2×C8×C53—48no—
142C2×C8×C53—42no—
143C2×C8×C53—42no—
144C22⋊Dic53—42no—
145C22⋊Dic53—42no—
146#11719921—42no—
147#11719921—42no—
148#11719921—41no—
149#11719921—42no—
150#11719921—41no—
151C2×#88322—42no—
152C2×#88322—42no—
153C2×#88322—42no—
154C23×C53—21no—
155C23×C53—21no—
156C23×C53—21no—
157C23×C53—21no—
158C23×C53—21no—
159C23×C53—21no—
160C23×C53—21no—
161C23×C53—21no—
162C23×C53—24no—
163C23×C53—24no—
164C23×C53—21no—
165C23×C53—21no—
166C23×C53—21no—
167C23×C53—21no—
168C23×C53—21no—
169C23×C53—21no—
170C23×C53—24no—
171C23×C53—24no—
172C23×C53—42no—
173C23×C53—42no—
174C23×C53—44no—
175C23×C53—44no—
176C23×C53—44no—
177C23×C53—44no—
178C23×C53C22⋊C812yes—
179C23×C53C22⋊C812yes—
180C23×C53#9912yes—
181C23×C53#10012yes—
182C23×C53C2×C22⋊C411yescharacteristic
183C23×C53C4×D412yes—
184C23×C53C22×C811yescharacteristic
185C23×C53C2×M1611yescharacteristic
186C23×C53#13112yes—
187C22×D53—22no—
188C22×D53—24no—
189C22×D53—42no—
190C22×D53—42no—
191C22×D53—41no—
192C22×D53—41no—
193C22×D53—41no—
194C22×D53—41no—
195C22×D53—41no—
196C22×D53—41no—
197C22×D53—42no—
198C22×D53—42no—
199C22×D53—42no—
200C22×D53—82no—
201C22×D53—82no—
202C22×D53—82no—
203C22×D53—82no—
204C2×C4×C53—24no—
205C2×C4×C53—24no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×C4×C53—44no—
218C2×C4×C53—44no—
219C2×C4×C53—44no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53#9912yes—
223C2×C4×C53#12011yescharacteristic
224C2×C4×C53#12112yes—
225C2×C4×C53#12311yescharacteristic
226C22⋊D53—48no—
227C22⋊D53—48no—
228C2×Dic53—22no—
229C2×Dic53—42no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53—42no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—82no—
240C2×Dic53—82no—
241C8×C53—44no—
242#88322—82no—
243#9079—531nomaximal, Sylow
244C22×C53—22no—
245C22×C53—22no—
246C22×C53—22no—
247C22×C53—22no—
248C22×C53—22no—
249C22×C53—21no—
250C22×C53—21no—
251C22×C53—21no—
252C22×C53—21no—
253C22×C53—22no—
254C22×C53—22no—
255C22×C53—22no—
256C22×C53—22no—
257C22×C53—22no—
258C22×C53—22no—
259C22×C53—22no—
260C22×C53—22no—
261C22×C53—48no—
262C22×C53—48no—
263C22×C53—41no—
264C22×C53—41no—
265C22×C53—41no—
266C22×C53—41no—
267C22×C53#32314yescomplemented
268C22×C53#38611yescharacteristic
269C22×C53C2×C22⋊C811yescharacteristic
270C22×C53#40811yescharacteristic
271C22×C53C2×#9911yescharacteristic
272C22×C53C2×#9911yescharacteristic
273C22×C53C2×#10011yescharacteristic
274C22×C53#41311yescharacteristic
275C22×C53C8×D412yes—
276C22×C53#43512yes—
277C2×D53—42no—
278C2×D53—42no—
279C2×D53—42no—
280C2×D53—84no—
281C2×D53—81no—
282C2×D53—81no—
283C2×D53—81no—
284C2×D53—81no—
285C2×D53—84no—
286C4×C53—22no—
287C4×C53—21no—
288C4×C53—22no—
289C4×C53—21no—
290C4×C53—41no—
291Dic53—42no—
292Dic53—81no—
293Dic53—81no—
294C22×C22⋊C8D5311yescharacteristic, complemented
295C2×#386—531no—
296#1779—531no—
297C2×#323—532no—
298#1280—532no—
299C2×C53—22no—
300C2×C53—22no—
301C2×C53—24no—
302C2×C53—42no—
303C2×C53C2×#32311yescharacteristic
304C2×C53#145911yescharacteristic
305C2×C53#146811yescharacteristic
306C2×C53#147111yescharacteristic
307C2×C53#186911yescharacteristic
308C2×C53#189811yescharacteristic
309C2×C53#190011yescharacteristic
310D53—82no—
311C23×D4—531no—
312C24×C4C2×D5311yescharacteristic
313C23×C8—21no—
314C22×C22⋊C4—531no—
315C22×C22⋊C4—531no—
316C22×C22⋊C4—531no—
317C2×C22⋊C8—21no—
318C2×C22⋊C8—24no—
319C2×C22⋊C8—532no—
320C2×C22⋊C8—1061no—
321C2×C22⋊C8—1061no—
322C2×C22⋊C8C2×D5312yes—
323#386—1061no—
324#386—1061no—
325#386—1061no—
326#386—1061no—
327C2×#95—531no—
328C2×#95—531no—
329#323—1064no—
330C53#907911yescharacteristic, complemented, Sylow
331C25—531no—
332C25C4×D5311yescharacteristic
333C22×D4—534no—
334C22×D4—1062no—
335C22×D4—1061no—
336C22×D4—1061no—
337C22×D4—2122no—
338C22×D4—2122no—
339C23×C4—22no—
340C23×C4—22no—
341C23×C4—42no—
342C23×C4—42no—
343C23×C4—531no—
344C23×C4—531no—
345C23×C4—531no—
346C23×C4C4×D5311yescharacteristic
347C23×C4D21211yescharacteristic
348C23×C4D21211yescharacteristic
349C23×C4C22⋊D5311yescharacteristic
350C23×C4C22⋊D5311yescharacteristic
351C23×C4C22×D5311yescharacteristic
352C22×C8—24no—
353C22×C8—22no—
354C22×C8—42no—
355C22×C8—42no—
356C22×C8—1062no—
357C2×C22⋊C4—1061no—
358C2×C22⋊C4—1061no—
359C2×C22⋊C4—1061no—
360C2×C22⋊C4—1062no—
361C2×C22⋊C4—1062no—
362C2×C22⋊C4—1061no—
363C2×C22⋊C4—1061no—
364C2×C22⋊C4—1061no—
365C2×C22⋊C4—2122no—
366C2×C22⋊C4—2122no—
367C22⋊C8—24no—
368C22⋊C8—22no—
369C22⋊C8—1062no—
370C22⋊C8—1062no—
371C22⋊C8—2122no—
372C22⋊C8—2122no—
373C22⋊C8C22⋊D5314yescomplemented
374#95—1061no—
375#95—1061no—
376#95—1061no—
377#95—1061no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—41no—
383C24—41no—
384C24—532no—
385C24—1062no—
386C24—1062no—
387C24—1061no—
388C24—1061no—
389C24—2121no—
390C24—2121no—
391C24—2121no—
392C24—2121no—
393C24C8×D5312yes—
394C24#1171991212yes—
395C24#1171992111yescharacteristic
396C24#1171992111yescharacteristic
397C24C2×C4×D5311yescharacteristic
398C2×D4—1064no—
399C2×D4—1064no—
400C2×D4—2128no—
401C2×D4—2122no—
402C2×D4—2122no—
403C2×D4—2122no—
404C2×D4—2122no—
405C2×D4—2128no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—532no—
428C22×C4—1061no—
429C22×C4—1061no—
430C22×C4—1062no—
431C22×C4—1062no—
432C22×C4—1061no—
433C22×C4—1061no—
434C22×C4—2121no—
435C22×C4—2121no—
436C22×C4—2121no—
437C22×C4—2121no—
438C22×C4—2121no—
439C22×C4—2121no—
440C22×C4#1171992111yescharacteristic
441C22×C4#1171992111yescharacteristic
442C22×C4C2×D21211yescharacteristic
443C22×C4#1171994512yes—
444C22×C4C2×C22⋊D5311yescharacteristic
445C22×C4C2×C22⋊D5312yes—
446C2×C8—24no—
447C2×C8—48no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—2122no—
451C2×C8—2122no—
452C2×C8—2122no—
453C22⋊C4—2122no—
454C22⋊C4—2122no—
455C22⋊C4—2122no—
456C22⋊C4—2122no—
457C22⋊C4—2121no—
458C22⋊C4—2122no—
459C22⋊C4—2121no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—24no—
474C23—24no—
475C23—21no—
476C23—24no—
477C23—24no—
478C23—42no—
479C23—42no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—1064no—
485C23—1062no—
486C23—4242no—
487C23—4242no—
488C23—4242no—
489C23—4242no—
490C23—2122no—
491C23—2122no—
492C23—2122no—
493C23—2122no—
494C23—2121no—
495C23—2121no—
496C23—2121no—
497C23—2121no—
498C23—2121no—
499C23—2121no—
500C23—2122no—
501C23#1313312912yes—
502C23#1313312912yes—
503C23#1313313112yes—
504C23#1313313312yes—
505C23C2×#1171992111yescharacteristic
506C23C4×C22⋊D5312yes—
507C23C2×C8×D5311yescharacteristic
508C23C2×#1171991211yescharacteristic
509C23#1313322312yes—
510D4—2128no—
511D4—2128no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—1062no—
531C2×C4—4242no—
532C2×C4—4242no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4—2121no—
536C2×C4—2121no—
537C2×C4—2121no—
538C2×C4—2121no—
539C2×C4—2122no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2122no—
543C2×C4#1313313112yes—
544C2×C4#1313318611yescharacteristic
545C2×C4#1313319112yes—
546C2×C4#1313320011yescharacteristic
547C8—44no—
548C8—4242no—
549C22—22no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—48no—
567C22—48no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—4244no—
573C22—4244no—
574C22—4241no—
575C22—4241no—
576C22—4241no—
577C22—4241no—
578C22—2122no—
579C22—2122no—
580C22—2122no—
581C22#1681034614yescomplemented
582C22#1681050211yescharacteristic
583C22C2×#1313312911yescharacteristic
584C22#1681058911yescharacteristic
585C22C2×#1313313111yescharacteristic
586C22C2×#1313313111yescharacteristic
587C22C2×#1313313311yescharacteristic
588C22#1681060511yescharacteristic
589C22C8×C22⋊D5312yes—
590C22#1681069412yes—
591C4—21no—
592C4—22no—
593C4—22no—
594C4—21no—
595C4—41no—
596C4—4241no—
597C4—4241no—
598C4—2122no—
599C2—22no—
600C2—22no—
601C2—24no—
602C2—42no—
603C2—4242no—
604C2C2×#1681034611yescharacteristic
605C2#2027303111yescharacteristic
606C2#2027306111yescharacteristic
607C2#2027307411yescharacteristic
608C2#2027500711yescharacteristic
609C2#2027516811yescharacteristic
610C2#2027517811yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC23×C53C22×C8
FrattiniC23×C4C22×D53
FittingC22×C22⋊C8×C53C2
socleC23×C53C2×C22⋊C4
prime radicalC23×C22⋊D53C4
cosocleC23×C4×C53C23
hypercenterC22×C22⋊C8D53
hypocenterC53#9079
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. #20274625C2semidirect
3. C2×#16810346C2semidirect
4. #20272557C2semidirect
5. C22×C22⋊C8×C53C2semidirect
31. C22×C22⋊D53C8semidirect
71. C22⋊C8×C53D4semidirect
91. C23×D53C2×C8semidirect
139. C22×Dic53C2×C8semidirect
267. C22×C53#323semidirect
294. C22×C22⋊C8D53semidirect
330. C53#9079semidirect
373. C22⋊C8C22⋊D53semidirect
581. C22#16810346semidirect

Maximal subgroups

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

subgroupindex
C2×#168105022
#202746252
C2×#168103462
#202725572
C22×C22⋊C8×C532
#907953