Finite Group Databasebeta

← #27299566

Subgroups — #27299566

Order 13568.

1228 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×#16810413C212yesmaximal, complemented
2C2×#16810410C212yesmaximal, complemented
3#20274636C211yesmaximal, characteristic, complemented
4#20274630C211yesmaximal, characteristic, complemented
5C2×C53×#534C211yesmaximal, characteristic
6C2×#13133134—21no—
7C2×#13133134C2212yes—
8C2×#13133132—21no—
9C2×#13133132C2212yes—
10#16810587—21no—
11#16810479—21no—
12#16810413—24no—
13#16810410—24no—
14C23×D4×C53C411yescharacteristic, complemented
15C23×D4×C53C2211yescharacteristic
16C53×#534—24no—
17C53×#534—24no—
18C2×#123×C53—21no—
19C2×#123×C53C411yescharacteristic, complemented
20C2×#123×C53C2211yescharacteristic
21C2×#121×C53—21no—
22C2×#121×C53—21no—
23C2×#120×C53—21no—
24C2×#120×C53C411yescharacteristic, complemented
25C2×#120×C53C2211yescharacteristic
26C22×C22⋊C4×C53C411yescharacteristic, complemented
27C2×#11719917—22no—
28C2×C22⋊Dic53—22no—
29C2×C22⋊Dic53—42no—
30#13133134—22no—
31#13133134—22no—
32#13133134—42no—
33#13133132—22no—
34#13133132—22no—
35#13133132—42no—
36C22⋊#88322—42no—
37C25×C53C2×C411yescharacteristic
38C25×C53C2×C411yescharacteristic
39C25×C53D411yescharacteristic
40C22×D4×C53—21no—
41C22×D4×C53—21no—
42C22×D4×C53—22no—
43C22×D4×C53—21no—
44C22×D4×C53—21no—
45C22×D4×C53—22no—
46C22×D4×C53—22no—
47C22×D4×C53—22no—
48C22×D4×C53—42no—
49C22×D4×C53—42no—
50C22×D4×C53—42no—
51C22×D4×C53—42no—
52C22×D4×C53C2×C411yescharacteristic
53C22×D4×C53D411yescharacteristic
54C22×D4×C53D411yescharacteristic
55C22×D4×C53C2311yescharacteristic
56C23×C4×C53—21no—
57C23×C4×C53C2×C411yescharacteristic
58C23×C4×C53D411yescharacteristic
59#123×C53—28no—
60#123×C53—22no—
61#123×C53—22no—
62#123×C53—22no—
63#123×C53D414yes—
64#121×C53—28no—
65#121×C53—28no—
66#120×C53—28no—
67#120×C53—22no—
68#120×C53—22no—
69#120×C53—22no—
70#120×C53D414yes—
71C2×C4⋊C4×C53—21no—
72C2×C4⋊C4×C53—21no—
73C2×C22⋊C4×C53—22no—
74C2×C22⋊C4×C53—22no—
75C2×C22⋊C4×C53—21no—
76C2×C22⋊C4×C53—21no—
77C2×C22⋊C4×C53—21no—
78C2×C22⋊C4×C53—21no—
79C2×C22⋊C4×C53—21no—
80C2×C22⋊C4×C53C2×C411yescharacteristic
81C2×C22⋊C4×C53C2×C411yescharacteristic
82C24×C53—22no—
83C24×C53—22no—
84C24×C53—21no—
85C24×C53—21no—
86C24×C53—21no—
87C24×C53—22no—
88C24×C53—21no—
89C24×C53—22no—
90C24×C53—41no—
91C24×C53—41no—
92C24×C53—41no—
93C24×C53—82no—
94C24×C53—82no—
95C24×C53—82no—
96C24×C53C22⋊C411yescharacteristic
97C24×C53C22⋊C411yescharacteristic
98C24×C53C22⋊C411yescharacteristic
99C24×C53C22⋊C412yescomplemented
100C24×C53C22×C411yescharacteristic
101C24×C53C2×D411yescharacteristic
102C2×D4×C53—24no—
103C2×D4×C53—24no—
104C2×D4×C53—22no—
105C2×D4×C53—22no—
106C2×D4×C53—21no—
107C2×D4×C53—21no—
108C2×D4×C53—22no—
109C2×D4×C53—24no—
110C2×D4×C53—24no—
111C2×D4×C53—22no—
112C2×D4×C53—41no—
113C2×D4×C53—41no—
114C2×D4×C53—41no—
115C2×D4×C53—41no—
116C2×D4×C53—41no—
117C2×D4×C53—41no—
118C2×D4×C53—48no—
119C2×D4×C53—48no—
120C2×D4×C53—48no—
121C2×D4×C53—42no—
122C2×D4×C53—42no—
123C2×D4×C53—48no—
124C2×D4×C53—42no—
125C2×D4×C53—42no—
126C2×D4×C53—82no—
127C2×D4×C53—82no—
128C2×D4×C53C2×D412yes—
129C2×D4×C53C2×D412yes—
130C22×C4×C53—22no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—24no—
136C22×C4×C53—22no—
137C22×C4×C53—41no—
138C22×C4×C53—41no—
139C22×C4×C53—41no—
140C22×C4×C53—82no—
141C22×C4×C53C22⋊C411yescharacteristic
142C22×C4×C53C22⋊C412yescomplemented
143C22×C4×C53C2×D411yescharacteristic
144C22×Dic53—42no—
145C4⋊C4×C53—24no—
146C4⋊C4×C53—24no—
147C22⋊C4×C53—24no—
148C22⋊C4×C53—24no—
149C22⋊C4×C53—24no—
150C22⋊C4×C53—21no—
151C22⋊C4×C53—21no—
152C22⋊C4×C53—24no—
153C22⋊C4×C53—24no—
154C22⋊C4×C53C22⋊C412yescomplemented
155C22⋊C4×C53C22⋊C412yescomplemented
156C22⋊Dic53—42no—
157C22⋊Dic53—42no—
158C22⋊Dic53—84no—
159#11719917—42no—
160#11719917—42no—
161C2×#88322—42no—
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—21no—
171C23×C53—21no—
172C23×C53—21no—
173C23×C53—22no—
174C23×C53—21no—
175C23×C53—21no—
176C23×C53—42no—
177C23×C53—44no—
178C23×C53—42no—
179C23×C53—42no—
180C23×C53—42no—
181C23×C53—42no—
182C23×C53—42no—
183C23×C53—42no—
184C23×C53—42no—
185C23×C53—44no—
186C23×C53—42no—
187C23×C53—41no—
188C23×C53—41no—
189C23×C53—82no—
190C23×C53—82no—
191C23×C53—82no—
192C23×C53—82no—
193C23×C53—84no—
194C23×C53—82no—
195C23×C53—82no—
196C23×C53—82no—
197C23×C53—84no—
198C23×C53—84no—
199C23×C53—84no—
200C23×C53—84no—
201C23×C53#9912yes—
202C23×C53#9912yes—
203C23×C53C2×C22⋊C411yescharacteristic
204C23×C53C2×C22⋊C411yescharacteristic
205C23×C53C2×C22⋊C411yescharacteristic
206C23×C53#12012yes—
207D4×C53—42no—
208D4×C53—42no—
209D4×C53—44no—
210D4×C53—44no—
211D4×C53—42no—
212D4×C53—42no—
213D4×C53—48no—
214D4×C53—48no—
215D4×C53—84no—
216D4×C53—84no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—21no—
222C2×C4×C53—22no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—44no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—42no—
232C2×C4×C53—84no—
233C2×C4×C53#12012yes—
234C2×Dic53—82no—
235C2×Dic53—84no—
236#88322—82no—
237#12690—531nomaximal, Sylow
238C22×C53—22no—
239C22×C53—22no—
240C22×C53—22no—
241C22×C53—22no—
242C22×C53—22no—
243C22×C53—21no—
244C22×C53—21no—
245C22×C53—22no—
246C22×C53—42no—
247C22×C53—41no—
248C22×C53—41no—
249C22×C53—42no—
250C22×C53—42no—
251C22×C53—42no—
252C22×C53—42no—
253C22×C53—42no—
254C22×C53—42no—
255C22×C53—42no—
256C22×C53—42no—
257C22×C53—42no—
258C22×C53—42no—
259C22×C53—42no—
260C22×C53—42no—
261C22×C53—41no—
262C22×C53—42no—
263C22×C53—81no—
264C22×C53—81no—
265C22×C53—82no—
266C22×C53—84no—
267C22×C53—84no—
268C22×C53—88no—
269C22×C53—84no—
270C22×C53—84no—
271C22×C53—82no—
272C22×C53#37911yescharacteristic
273C22×C53C2×#9911yescharacteristic
274C22×C53C2×#9911yescharacteristic
275C4×C53—22no—
276C4×C53—42no—
277C4×C53—42no—
278C4×C53—41no—
279C4×C53—42no—
280Dic53—162no—
281C2×#534D5311yescharacteristic
282C2×#352—532no—
283C2×AΣL(1,𝔽16)—532no—
284#1783—531no—
285#1781—531no—
286C2×C53—21no—
287C2×C53—21no—
288C2×C53—42no—
289C2×C53—42no—
290C2×C53—42no—
291C2×C53—41no—
292C2×C53—41no—
293C2×C53—81no—
294C2×C53—82no—
295C2×C53#178111yescharacteristic
296C2×C53#212112yes—
297C23×D4Dic5311yescharacteristic, complemented
298C23×D4C2×D5311yescharacteristic
299#534—24no—
300#534—24no—
301C2×#123—21no—
302C2×#123Dic5311yescharacteristic, complemented
303C2×#123C2×D5311yescharacteristic
304C2×#121—21no—
305C2×#121—21no—
306C2×#120—21no—
307C2×#120Dic5311yescharacteristic, complemented
308C2×#120C2×D5311yescharacteristic
309C22×C22⋊C4Dic5311yescharacteristic, complemented
310C2×#100—532no—
311C2×#100—1061no—
312C2×#99—532no—
313C2×#99—1061no—
314#407—1061no—
315#379—1061no—
316#352—1064no—
317AΣL(1,𝔽16)—1064no—
318C53#1269011yescharacteristic, complemented, Sylow
319C25C2×Dic5311yescharacteristic
320C25C2×Dic5311yescharacteristic
321C25C22⋊D5311yescharacteristic
322C22×D4—21no—
323C22×D4—21no—
324C22×D4—22no—
325C22×D4—21no—
326C22×D4—21no—
327C22×D4—22no—
328C22×D4—22no—
329C22×D4—22no—
330C22×D4—42no—
331C22×D4—42no—
332C22×D4—42no—
333C22×D4—42no—
334C22×D4C2×Dic5311yescharacteristic
335C22×D4C22⋊D5311yescharacteristic
336C22×D4C22⋊D5311yescharacteristic
337C22×D4C22×D5311yescharacteristic
338C23×C4—21no—
339C23×C4C2×Dic5311yescharacteristic
340C23×C4C22⋊D5311yescharacteristic
341C2×M16—1062no—
342#123—28no—
343#123—22no—
344#123—22no—
345#123—22no—
346#123C22⋊D5314yes—
347#121—28no—
348#121—28no—
349#120—28no—
350#120—22no—
351#120—22no—
352#120—22no—
353#120C22⋊D5314yes—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—1062no—
364C2×C22⋊C4—2122no—
365C2×C22⋊C4C2×Dic5311yescharacteristic
366C2×C22⋊C4C2×Dic5311yescharacteristic
367#100—1062no—
368#100—1062no—
369#100—2122no—
370#99—1062no—
371#99—1062no—
372#99—2122no—
373C22⋊C8—2122no—
374C24—22no—
375C24—22no—
376C24—21no—
377C24—21no—
378C24—21no—
379C24—22no—
380C24—21no—
381C24—22no—
382C24—41no—
383C24—41no—
384C24—41no—
385C24—82no—
386C24—82no—
387C24—82no—
388C24C22⋊Dic5311yescharacteristic
389C24C22⋊Dic5311yescharacteristic
390C24C22⋊Dic5311yescharacteristic
391C24C22⋊Dic5312yescomplemented
392C24C22×Dic5311yescharacteristic
393C24C2×C22⋊D5311yescharacteristic
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—22no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—48no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—48no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—82no—
419C2×D4—82no—
420C2×D4C2×C22⋊D5312yes—
421C2×D4C2×C22⋊D5312yes—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—82no—
433C22×C4—2122no—
434C22×C4C22⋊Dic5311yescharacteristic
435C22×C4C22⋊Dic5312yescomplemented
436C22×C4C2×C22⋊D5311yescharacteristic
437M16—2122no—
438M16—2122no—
439C2×C8—2122no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—21no—
446C22⋊C4—21no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—4244no—
450C22⋊C4—2122no—
451C22⋊C4—2122no—
452C22⋊C4C22⋊Dic5312yescomplemented
453C22⋊C4C22⋊Dic5312yescomplemented
454C23—24no—
455C23—24no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—22no—
467C23—21no—
468C23—42no—
469C23—44no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—44no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23—84no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23—84no—
490C23—84no—
491C23—84no—
492C23—84no—
493C23#1313313212yes—
494C23#1313313212yes—
495C23C2×C22⋊Dic5311yescharacteristic
496C23C2×C22⋊Dic5311yescharacteristic
497C23C2×C22⋊Dic5311yescharacteristic
498C23#1313318612yes—
499D4—42no—
500D4—42no—
501D4—44no—
502D4—44no—
503D4—42no—
504D4—42no—
505D4—48no—
506D4—48no—
507D4—84no—
508D4—84no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—84no—
525C2×C4—4242no—
526C2×C4—4244no—
527C2×C4#1313318612yes—
528C8—4242no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—21no—
534C22—21no—
535C22—22no—
536C22—22no—
537C22—42no—
538C22—41no—
539C22—41no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—41no—
553C22—42no—
554C22—81no—
555C22—81no—
556C22—82no—
557C22—84no—
558C22—84no—
559C22—88no—
560C22—84no—
561C22—84no—
562C22—82no—
563C22#1681047911yescharacteristic
564C22C2×#1313313211yescharacteristic
565C22C2×#1313313211yescharacteristic
566C4—22no—
567C4—42no—
568C4—42no—
569C4—41no—
570C4—42no—
571C4—8482no—
572C2—21no—
573C2—21no—
574C2—42no—
575C2—42no—
576C2—41no—
577C2—41no—
578C2—42no—
579C2—81no—
580C2—82no—
581C2#2027463011yescharacteristic
582C2#2027636112yes—
subgroupG/H
centerC22C2×#13133132
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
Fitting, prime radicalC2×C53×#534C2
socleC22×C53C2×#99
cosocleC22×D4×C53C23
hypercenterC2×#534D53
hypocenterC53#12690
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ C2×#534

Complements

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

NH ≅ G/Ntype
1. C2×#16810413C2semidirect
2. C2×#16810410C2semidirect
3. #20274636C2semidirect
4. #20274630C2semidirect
14. C23×D4×C53C4semidirect
19. C2×#123×C53C4semidirect
24. C2×#120×C53C4semidirect
26. C22×C22⋊C4×C53C4semidirect
99. C24×C53C22⋊C4semidirect
142. C22×C4×C53C22⋊C4semidirect
154. C22⋊C4×C53C22⋊C4semidirect
155. C22⋊C4×C53C22⋊C4semidirect
297. C23×D4Dic53semidirect
302. C2×#123Dic53semidirect
307. C2×#120Dic53semidirect
309. C22×C22⋊C4Dic53semidirect
318. C53#12690semidirect
391. C24C22⋊Dic53semidirect
435. C22×C4C22⋊Dic53semidirect
452. C22⋊C4C22⋊Dic53semidirect
453. C22⋊C4C22⋊Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×#168104132
C2×#168104102
#202746362
#202746302
C2×C53×#5342
#1269053