Finite Group Databasebeta

← #27279655

Subgroups — #27279655

Order 13568.

1062 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊#88322C211yesmaximal, characteristic, complemented
2#20282997C211yesmaximal, characteristic
3C2×#16810356C212yesmaximal, complemented
4#20272594C212yesmaximal, complemented
5C53×C22⋊#95C211yesmaximal, characteristic, complemented
6C23×#88322—21no—
7C2×#13133191C411yescharacteristic, complemented
8C2×#13133191C411yescharacteristic, complemented
9C2×#13133191C2211yescharacteristic
10C2×#13133186—21no—
11C2×C22⋊#88322—24no—
12C2×C22⋊#88322—21no—
13C2×C22⋊#88322C2212yes—
14#16810502—21no—
15#16810502—21no—
16#16810502—21no—
17#16810483C411yescharacteristic, complemented
18#16810483C411yescharacteristic, complemented
19#16810483C2211yescharacteristic
20#16810356—24no—
21C24×C4×C53C2211yescharacteristic
22C22×C22⋊C4×C53—21no—
23C2×#95×C53—21no—
24C2×#95×C53C2212yes—
25C22×D212C2×C411yescharacteristic
26C22×C22⋊D53—21no—
27C22×C22⋊D53—21no—
28C22×C22⋊D53—21no—
29C22×#88322—24no—
30C22×#88322—22no—
31C22×#88322—42no—
32C22×#88322—42no—
33#13133191—22no—
34#13133191—22no—
35#13133191—42no—
36#13133191—42no—
37#13133186—42no—
38#13133186—42no—
39C2×C4⋊Dic53C2×C411yescharacteristic
40C2×C4⋊Dic53C2×C411yescharacteristic
41C2×C4⋊Dic53C2×C411yescharacteristic
42C2×C22⋊Dic53—21no—
43C2×C22⋊Dic53—21no—
44C2×C22⋊Dic53—21no—
45C2×#11719921—21no—
46C2×#11719921—21no—
47C2×#11719921—21no—
48C22⋊#88322—24no—
49C22⋊#88322—22no—
50C22⋊#88322D414yescomplemented
51#13133124—21no—
52#13133124—21no—
53#13133124—21no—
54C25×C53D411yescharacteristic
55C23×C4×C53—22no—
56C23×C4×C53—22no—
57C23×C4×C53—22no—
58C23×C4×C53—42no—
59C23×C4×C53—42no—
60C23×C4×C53C2×C411yescharacteristic
61C23×C4×C53C2×C411yescharacteristic
62C23×C4×C53D411yescharacteristic
63C23×C4×C53D411yescharacteristic
64C23×C4×C53D411yescharacteristic
65C23×C4×C53C2311yescharacteristic
66C2×C22⋊C4×C53—24no—
67C2×C22⋊C4×C53—42no—
68C2×C22⋊C4×C53—42no—
69#95×C53—22no—
70#95×C53—22no—
71#95×C53—42no—
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×C53D812yes—
81C24×C53SD1612yes—
82C24×C53C2×D411yescharacteristic
83C23×D53—21no—
84C22×C4×C53—22no—
85C22×C4×C53—22no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—22no—
89C22×C4×C53—22no—
90C22×C4×C53—22no—
91C22×C4×C53—22no—
92C22×C4×C53—22no—
93C22×C4×C53—22no—
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53—24no—
97C22×C4×C53—41no—
98C22×C4×C53—41no—
99C22×C4×C53—41no—
100C22×C4×C53—41no—
101C22×C4×C53—41no—
102C22×C4×C53—41no—
103C22×C4×C53—41no—
104C22×C4×C53—48no—
105C22×C4×C53—48no—
106C22×C4×C53—42no—
107C22×C4×C53—42no—
108C22×C4×C53—42no—
109C22×C4×C53—42no—
110C22×C4×C53—42no—
111C22×C4×C53—42no—
112C22×C4×C53C22⋊C411yescharacteristic
113C22×C4×C53C22⋊C411yescharacteristic
114C22×C4×C53C22×C411yescharacteristic
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53C2×D412yes—
117C22×C4×C53#4112yes—
118C2×C22⋊D53—42no—
119C2×C22⋊D53—42no—
120C2×C22⋊D53—42no—
121C2×C22⋊D53—42no—
122C2×C22⋊D53—42no—
123C2×C22⋊D53—42no—
124C2×C22⋊D53—42no—
125C2×C22⋊D53—41no—
126C2×C22⋊D53—41no—
127C2×C22⋊D53—42no—
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53—21no—
131C2×D212—22no—
132C2×D212—41no—
133C2×D212—41no—
134C22⋊C4×C53—48no—
135C22⋊Dic53—42no—
136C22⋊Dic53—42no—
137C22⋊Dic53—41no—
138C22⋊Dic53—41no—
139#11719921—42no—
140#11719921—42no—
141#11719921—41no—
142#11719921—41no—
143C4⋊Dic53—22no—
144C4⋊Dic53—41no—
145C4⋊Dic53—41no—
146C2×#88322—24no—
147C2×#88322—48no—
148C2×#88322—42no—
149C2×#88322—42no—
150C23×C53—21no—
151C23×C53—21no—
152C23×C53—21no—
153C23×C53—21no—
154C23×C53—21no—
155C23×C53—21no—
156C23×C53—21no—
157C23×C53—21no—
158C23×C53—24no—
159C23×C53—24no—
160C23×C53—21no—
161C23×C53—21no—
162C23×C53—21no—
163C23×C53—21no—
164C23×C53—21no—
165C23×C53—21no—
166C23×C53—24no—
167C23×C53—24no—
168C23×C53—42no—
169C23×C53—42no—
170C23×C53—44no—
171C23×C53—44no—
172C23×C53—44no—
173C23×C53—44no—
174C23×C53#9912yes—
175C23×C53#10212yes—
176C23×C53#10212yes—
177C23×C53#10412yes—
178C23×C53C2×C22⋊C411yescharacteristic
179C23×C53#12112yes—
180C23×C53C2×D811yescharacteristic
181C23×C53C2×SD1611yescharacteristic
182C23×C53#13512yes—
183C22×D53—41no—
184C22×D53—41no—
185C22×D53—41no—
186C22×D53—41no—
187C22×D53—41no—
188C22×D53—41no—
189C22×D53—41no—
190C2×C4×C53—24no—
191C2×C4×C53—24no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—42no—
202C2×C4×C53—42no—
203C2×C4×C53—44no—
204C2×C4×C53—44no—
205C2×C4×C53—44no—
206C2×C4×C53—42no—
207C2×C4×C53—42no—
208C2×C4×C53—44no—
209C2×C4×C53—44no—
210C2×C4×C53—44no—
211C2×C4×C53—82no—
212C2×C4×C53—82no—
213C2×C4×C53—82no—
214C2×C4×C53—82no—
215C2×C4×C53#9912yes—
216C2×C4×C53C4×D412yes—
217C2×C4×C53#12011yescharacteristic
218C2×C4×C53#12311yescharacteristic
219C22⋊D53—84no—
220C22⋊D53—84no—
221C22⋊D53—82no—
222C22⋊D53—82no—
223C2×Dic53—41no—
224C2×Dic53—41no—
225C2×Dic53—41no—
226C2×Dic53—41no—
227C2×Dic53—41no—
228C2×Dic53—41no—
229C2×Dic53—41no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232D212—82no—
233#88322—44no—
234#9113—531nomaximal, Sylow
235C22×C53—22no—
236C22×C53—22no—
237C22×C53—22no—
238C22×C53—22no—
239C22×C53—22no—
240C22×C53—21no—
241C22×C53—21no—
242C22×C53—22no—
243C22×C53—22no—
244C22×C53—22no—
245C22×C53—22no—
246C22×C53—21no—
247C22×C53—22no—
248C22×C53—21no—
249C22×C53—22no—
250C22×C53—22no—
251C22×C53—22no—
252C22×C53—48no—
253C22×C53—48no—
254C22×C53—41no—
255C22×C53—41no—
256C22×C53—41no—
257C22×C53—41no—
258C22×C53#32714yescomplemented
259C22×C53#38611yescharacteristic
260C22×C53C2×#9911yescharacteristic
261C22×C53C2×#9911yescharacteristic
262C22×C53C2×#10211yescharacteristic
263C22×C53#41611yescharacteristic
264C22×C53C2×#10411yescharacteristic
265C22×C53#42111yescharacteristic
266C22×C53#46512yes—
267C22×C53#46612yes—
268C2×D53—81no—
269C2×D53—81no—
270C2×D53—81no—
271C2×D53—81no—
272C2×D53—81no—
273C2×D53—81no—
274C2×D53—81no—
275C4×C53—21no—
276C4×C53—22no—
277C4×C53—22no—
278C4×C53—21no—
279C4×C53—41no—
280C4×C53—84no—
281Dic53—81no—
282Dic53—81no—
283Dic53—81no—
284C22×C22⋊C8—531no—
285#2847—531no—
286C2×#327—532no—
287C22⋊#95D5311yescharacteristic, complemented
288#1294—532no—
289C2×C53—22no—
290C2×C53—24no—
291C2×C53—22no—
292C2×C53—42no—
293C2×C53C2×#32711yescharacteristic
294C2×C53#150111yescharacteristic
295C2×C53#151611yescharacteristic
296C2×C53#151911yescharacteristic
297C2×C53#187111yescharacteristic
298C2×C53#189311yescharacteristic
299C2×C53#189811yescharacteristic
300D53—161no—
301C24×C4C2×D5311yescharacteristic
302C23×C8—1061no—
303C2×#121—531no—
304C2×#121—531no—
305C2×#121—531no—
306C2×#120—1061no—
307C22×C22⋊C4—21no—
308C2×C22⋊C8—532no—
309C2×C22⋊C8—1064no—
310C2×C22⋊C8—1061no—
311#386—1061no—
312#386—1061no—
313#386—1061no—
314#380—531no—
315#380—531no—
316#380—531no—
317C2×#95—21no—
318C2×#95C2×D5312yes—
319#327—1064no—
320C53#911311yescharacteristic, complemented, Sylow
321C25C22⋊D5311yescharacteristic
322C22×D4—531no—
323C22×D4—1061no—
324C22×D4—1061no—
325C22×D4—1061no—
326C23×C4—22no—
327C23×C4—22no—
328C23×C4—22no—
329C23×C4—42no—
330C23×C4—42no—
331C23×C4C4×D5311yescharacteristic
332C23×C4C4×D5311yescharacteristic
333C23×C4D21211yescharacteristic
334C23×C4D21211yescharacteristic
335C23×C4C22⋊D5311yescharacteristic
336C23×C4C22×D5311yescharacteristic
337C22×C8—1064no—
338C22×C8—1062no—
339C22×C8—2122no—
340C22×C8—2122no—
341#121—1062no—
342#121—1062no—
343#121—2122no—
344#121—2122no—
345#120—2122no—
346#120—2122no—
347C2×C4⋊C4—531no—
348C2×C4⋊C4—531no—
349C2×C4⋊C4—531no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—42no—
352C2×C22⋊C4—42no—
353C2×C22⋊C4—1061no—
354C2×C22⋊C4—1061no—
355C2×C22⋊C4—1061no—
356C2×C22⋊C4—1061no—
357C2×C22⋊C4—1061no—
358C2×C22⋊C4—1061no—
359C22⋊C8—534no—
360C22⋊C8—1064no—
361C22⋊C8—1062no—
362#95—22no—
363#95—22no—
364#95—42no—
365#95—1061no—
366#95—1061no—
367#95—1061no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—41no—
373C24—41no—
374C24—1061no—
375C24#1171992111yescharacteristic
376C24#1171992111yescharacteristic
377C24#1171992212yes—
378C24#1171992412yes—
379C24C2×C22⋊D5311yescharacteristic
380C2×D4—1062no—
381C2×D4—2121no—
382C2×D4—2121no—
383C2×D4—2122no—
384C2×D4—2122no—
385C2×D4—2122no—
386C2×D4—2122no—
387C2×D4—2122no—
388C2×D4—2122no—
389C2×D4—2122no—
390C2×D4—2121no—
391C2×D4—2121no—
392C2×D4—2122no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—1061no—
422C22×C4—1061no—
423C22×C4—1061no—
424C22×C4#1171992111yescharacteristic
425C22×C4#1171992111yescharacteristic
426C22×C4C2×C4×D5311yescharacteristic
427C22×C4C2×D21211yescharacteristic
428C22×C4D4×D5312yes—
429C22×C4#1171994712yes—
430C2×C8—1064no—
431C2×C8—2128no—
432C2×C8—2122no—
433C2×C8—2122no—
434C4⋊C4—1062no—
435C4⋊C4—2121no—
436C4⋊C4—2121no—
437C22⋊C4—48no—
438C22⋊C4—2122no—
439C22⋊C4—2122no—
440C22⋊C4—2122no—
441C22⋊C4—2121no—
442C22⋊C4—2121no—
443C22⋊C4—2122no—
444C22⋊C4—2121no—
445C22⋊C4—2121no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—2121no—
471C23—2121no—
472C23—2121no—
473C23—2121no—
474C23—2121no—
475C23—2121no—
476C23—2121no—
477C23#1313313112yes—
478C23#1313313712yes—
479C23#1313313712yes—
480C23#1313314512yes—
481C23C2×#1171992111yescharacteristic
482C23#1313318912yes—
483C23C2×#1171992211yescharacteristic
484C23C2×#1171992411yescharacteristic
485C23#1313323712yes—
486D4—4242no—
487D4—4244no—
488D4—4242no—
489D4—4242no—
490D4—4244no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—2121no—
517C2×C4—2121no—
518C2×C4—2121no—
519C2×C4—2121no—
520C2×C4—2121no—
521C2×C4—2121no—
522C2×C4—2121no—
523C2×C4—2121no—
524C2×C4—2121no—
525C2×C4#1313313112yes—
526C2×C4#1313317712yes—
527C2×C4#1313318711yescharacteristic
528C2×C4#1313320111yescharacteristic
529C8—2124no—
530C22—21no—
531C22—21no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—21no—
539C22—22no—
540C22—21no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—48no—
548C22—48no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—4241no—
554C22—4241no—
555C22—4241no—
556C22—4241no—
557C22—4241no—
558C22—4241no—
559C22—4241no—
560C22#1681035614yescomplemented
561C22#1681050011yescharacteristic
562C22C2×#1313313111yescharacteristic
563C22C2×#1313313111yescharacteristic
564C22C2×#1313313711yescharacteristic
565C22#1681061711yescharacteristic
566C22C2×#1313314511yescharacteristic
567C22#1681064111yescharacteristic
568C22#1681085412yes—
569C22#1681085712yes—
570C4—21no—
571C4—22no—
572C4—21no—
573C4—22no—
574C4—41no—
575C4—84no—
576C4—4241no—
577C4—4241no—
578C4—4241no—
579C2—22no—
580C2—22no—
581C2—24no—
582C2—42no—
583C2—8481no—
584C2C2×#1681035611yescharacteristic
585C2#2027320411yescharacteristic
586C2#2027326911yescharacteristic
587C2#2027328211yescharacteristic
588C2#2027501511yescharacteristic
589C2#2027514111yescharacteristic
590C2#2027516611yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×C22⋊#95C2
socleC23×C53C2×C22⋊C4
prime radical#20282997C2
cosocleC23×C4×C53C23
hypercenterC22⋊#95D53
hypocenterC53#9113
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊#88322C2semidirect
3. C2×#16810356C2semidirect
4. #20272594C2semidirect
5. C53×C22⋊#95C2semidirect
7. C2×#13133191C4semidirect
8. C2×#13133191C4semidirect
17. #16810483C4semidirect
18. #16810483C4semidirect
50. C22⋊#88322D4semidirect
258. C22×C53#327semidirect
287. C22⋊#95D53semidirect
320. C53#9113semidirect
560. C22#16810356semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊#883222
#202829972
C2×#168103562
#202725942
C53×C22⋊#952
#911353