Finite Group Databasebeta

← #27401820

Subgroups — #27401820

Order 13568.

782 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810665C211yesmaximal, characteristic
2#20277598C211yesmaximal, characteristic
3#20277590C211yesmaximal, characteristic
4#20275453C211yesmaximal, characteristic, complemented
5#20275448C211yesmaximal, characteristic, complemented
6#20275308C211yesmaximal, characteristic, complemented
7#20275301C211yesmaximal, characteristic, complemented
8#20275294C211yesmaximal, characteristic, complemented
9#20275287C211yesmaximal, characteristic, complemented
10#20274811C211yesmaximal, characteristic
11#20274802C211yesmaximal, characteristic
12#20274748C211yesmaximal, characteristic
13#20274612C211yesmaximal, characteristic, complemented
14C4×#13133149C211yesmaximal, characteristic, complemented
15C53×#1753C211yesmaximal, characteristic
16C2×#13133176C2211yescharacteristic
17C22×C4⋊Dic53C2211yescharacteristic
18#16810665C418yescomplemented
19C2×#13133151C2211yescharacteristic
20C2×#13133151C2211yescharacteristic
21C2×#13133151C2211yescharacteristic
22C2×#13133149C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133149C2211yescharacteristic
25#16810521C2211yescharacteristic
26#16810521C2211yescharacteristic
27#16810521C2211yescharacteristic
28#16810521C2211yescharacteristic
29#16810511—21no—
30#16810511—21no—
31#16810506—21no—
32#16810506—21no—
33#16810487—21no—
34#16810483C2211yescharacteristic
35#16810483C2211yescharacteristic
36#16810481—21no—
37C4×C4⋊Dic53C2211yescharacteristic
38C4×C4⋊Dic53C2211yescharacteristic
39C4×C4⋊Dic53C2211yescharacteristic
40C4×C4⋊Dic53C2211yescharacteristic
41C22⋊C4×Dic53—21no—
42#16810468—21no—
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52#16810386C2211yescharacteristic
53C22×M16×C53C2211yescharacteristic
54C2×C22⋊C8×C53C2211yescharacteristic
55C2×#97×C53C2211yescharacteristic
56C2×C4×C8×C53C2211yescharacteristic
57C4×C22⋊C4×C53C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C23×Dic53—21no—
61#13133176—22no—
62#13133176C2×C414yes—
63C2×C4⋊Dic53—21no—
64C2×C4⋊Dic53—22no—
65C2×C4⋊Dic53C2×C414yes—
66C2×C4⋊Dic53C2311yescharacteristic
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86#13133151—22no—
87#13133151C2×C414yes—
88#13133151C2×C414yes—
89#13133149—22no—
90#13133149C2×C414yes—
91#13133149C2×C414yes—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102C23×C4×C53C2311yescharacteristic
103C2×M16×C53—22no—
104C2×M16×C53—22no—
105C2×M16×C53C2×C414yes—
106C22×C8×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C2×C22⋊C4×C53D411yescharacteristic
111C2×C22⋊C4×C53D411yescharacteristic
112C2×C42×C53C2311yescharacteristic
113C2×C42×C53C2311yescharacteristic
114C22⋊C8×C53—22no—
115#97×C53—22no—
116C4×C8×C53—22no—
117#95×C53D411yescharacteristic
118#95×C53D411yescharacteristic
119C24×C53C2×D411yescharacteristic
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53C22×C412yes—
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53#4112yes—
131C22×C4×C53C2411yescharacteristic
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53—24no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—81no—
142C22×Dic53—81no—
143M16×C53—28no—
144C2×C8×C53—21no—
145C2×C8×C53—21no—
146C2×C8×C53—21no—
147C2×C8×C53—21no—
148C2×C8×C53—21no—
149C2×C8×C53—21no—
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53C22×C412yes—
153C2×C8×C53C22×C412yes—
154C2×C8×C53#4112yes—
155C2×C8×C53#4112yes—
156C22⋊C4×C53—22no—
157C22⋊C4×C53—41no—
158C42×C53—21no—
159C42×C53—21no—
160C42×C53—21no—
161C42×C53—21no—
162C22⋊Dic53—24no—
163C22⋊Dic53—42no—
164C22⋊Dic53—81no—
165C4⋊Dic53—24no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53C22×C412yes—
179C4⋊Dic53C22×C412yes—
180C4⋊Dic53C22×C412yes—
181C4⋊Dic53C22×C412yes—
182#11719919—81no—
183#11719919—81no—
184#11719919—81no—
185#11719919—81no—
186C4×Dic53—24no—
187C4×Dic53—42no—
188C4×Dic53—42no—
189C4×Dic53—81no—
190C4×Dic53—81no—
191C23×C53—21no—
192C23×C53—21no—
193C23×C53—21no—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53C4×D412yes—
198C23×C53C22×D411yescharacteristic
199C2×C4×C53—21no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—22no—
204C2×C4×C53—22no—
205C2×C4×C53—22no—
206C2×C4×C53—22no—
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—21no—
216C2×C4×C53C4×D412yes—
217C2×C4×C53#13712yes—
218C2×C4×C53C23×C411yescharacteristic
219C2×C4×C53C2×#4111yescharacteristic
220C2×C4×C53C2×#4111yescharacteristic
221C2×C4×C53C2×#4111yescharacteristic
222C2×C4×C532−1+411yescharacteristic
223C2×C4×C532−1+411yescharacteristic
224C2×Dic53—22no—
225C2×Dic53—22no—
226C2×Dic53—22no—
227C2×Dic53—22no—
228C2×Dic53—41no—
229C2×Dic53—41no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—84no—
249C8×C53—22no—
250C8×C53—41no—
251C8×C53—41no—
252C8×C53#11714yes—
253#22621—531nomaximal, Sylow
254C22×C53—22no—
255C22×C53—22no—
256C22×C53—22no—
257C22×C53—41no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53—41no—
261C22×C53C2×C4×D411yescharacteristic
262C22×C53#54711yescharacteristic
263C22×C53#54711yescharacteristic
264C22×C53C2×#13711yescharacteristic
265C22×C53#57511yescharacteristic
266C22×C53#57611yescharacteristic
267C22×C53#57711yescharacteristic
268C4×C53—22no—
269C4×C53—22no—
270C4×C53—41no—
271C4×C53—41no—
272C4×C53—41no—
273C4×C53C2×#11711yescharacteristic
274C4×C53#51911yescharacteristic
275C4×C53#55211yescharacteristic
276C4×C53#55211yescharacteristic
277Dic53—42no—
278Dic53—42no—
279Dic53—42no—
280Dic53—42no—
281Dic53—81no—
282Dic53—81no—
283Dic53—81no—
284Dic53—81no—
285C2×#428—531no—
286#2346—531no—
287#2345—531no—
288#1943—531no—
289#1942—531no—
290#1921—531no—
291#1920—531no—
292#1919—531no—
293#1918—531no—
294#1825—531no—
295#1823—531no—
296#1809—531no—
297#1776—531no—
298C4×#106—531no—
299#1753D5311yescharacteristic
300C2×C53—42no—
301C2×C53#234411yescharacteristic
302C2×C53C4×#13711yescharacteristic
303C2×C53#294611yescharacteristic
304C2×C53#334411yescharacteristic
305C2×C53#334711yescharacteristic
306C2×C53#335111yescharacteristic
307C2×C53#335711yescharacteristic
308C22×M16C2×D5311yescharacteristic
309C2×#117—531no—
310C22×C4⋊C4—531no—
311#428—538no—
312C2×#107—531no—
313C2×#107—531no—
314C2×#107—531no—
315C2×#106—531no—
316C2×#106—531no—
317C2×#106—531no—
318C2×C22⋊C8C2×D5311yescharacteristic
319C2×#97C2×D5311yescharacteristic
320C2×C4×C8C2×D5311yescharacteristic
321#389—531no—
322#389—531no—
323#389—531no—
324#389—531no—
325#388—1061no—
326#388—1061no—
327#387—1061no—
328#387—1061no—
329#382—1061no—
330#380—531no—
331#380—531no—
332#380—1061no—
333C4×C4⋊C4—531no—
334C4×C4⋊C4—531no—
335C4×C4⋊C4—531no—
336C4×C4⋊C4—531no—
337C4×C22⋊C4—1061no—
338C4×C22⋊C4C2×D5311yescharacteristic
339#376—1061no—
340#340—531no—
341#340—531no—
342#340—531no—
343#340—531no—
344#340—531no—
345#340—531no—
346#340—531no—
347#340—531no—
348#340—531no—
349#340—531no—
350#336C2×D5311yescharacteristic
351#336C2×D5311yescharacteristic
352C53#2262111yescharacteristic, complemented, Sylow
353C23×C4—1061no—
354C23×C4C22×D5311yescharacteristic
355C2×M16—22no—
356C2×M16—22no—
357C2×M16C4×D5314yes—
358C22×C8C22×D5311yescharacteristic
359C22×C8C22×D5311yescharacteristic
360C22×C8C22×D5311yescharacteristic
361C22×C8C22×D5311yescharacteristic
362#117—534no—
363#117—1062no—
364C2×C4⋊C4—534no—
365C2×C4⋊C4—531no—
366C2×C4⋊C4—531no—
367C2×C4⋊C4—531no—
368C2×C4⋊C4—531no—
369C2×C4⋊C4—531no—
370C2×C4⋊C4—531no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—531no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4—1062no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4—1061no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4—1061no—
379C2×C22⋊C4—1061no—
380C2×C22⋊C4—1061no—
381C2×C22⋊C4—1061no—
382C2×C22⋊C4D21211yescharacteristic
383C2×C22⋊C4D21211yescharacteristic
384C2×C42—1061no—
385C2×C42—1061no—
386C2×C42—1061no—
387C2×C42—1061no—
388C2×C42—1061no—
389C2×C42C22×D5311yescharacteristic
390C2×C42C22×D5311yescharacteristic
391#107—534no—
392#107—534no—
393#107—1062no—
394#106—534no—
395#106—534no—
396#106—1062no—
397C22⋊C8—22no—
398#97—22no—
399C4×C8—22no—
400#95—1061no—
401#95—1061no—
402#95—1061no—
403#95—1061no—
404#95—1061no—
405#95—1061no—
406#95—1061no—
407#95—1061no—
408#95—1061no—
409#95—1061no—
410#95D21211yescharacteristic
411#95D21211yescharacteristic
412C24C2×D21211yescharacteristic
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—1064no—
418C22×C4—1061no—
419C22×C4—1061no—
420C22×C4—1061no—
421C22×C4—1061no—
422C22×C4—1061no—
423C22×C4—1061no—
424C22×C4—1061no—
425C22×C4—1061no—
426C22×C4—4241no—
427C22×C4—4241no—
428C22×C4C2×C4×D5312yes—
429C22×C4C2×D21211yescharacteristic
430C22×C4C2×D21211yescharacteristic
431C22×C4C2×D21211yescharacteristic
432C22×C4C2×D21211yescharacteristic
433C22×C4C2×D21211yescharacteristic
434C22×C4#1171994512yes—
435C22×C4C23×D5311yescharacteristic
436M16—28no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8C2×C4×D5312yes—
446C2×C8C2×C4×D5312yes—
447C2×C8#1171994712yes—
448C2×C8#1171994712yes—
449C4⋊C4—532no—
450C4⋊C4—532no—
451C4⋊C4—532no—
452C4⋊C4—532no—
453C4⋊C4—1064no—
454C4⋊C4—1061no—
455C4⋊C4—1061no—
456C4⋊C4—1061no—
457C4⋊C4—1061no—
458C4⋊C4—1061no—
459C4⋊C4—1061no—
460C4⋊C4—1061no—
461C4⋊C4—1061no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—4241no—
467C4⋊C4—4241no—
468C4⋊C4—4241no—
469C4⋊C4—4241no—
470C22⋊C4—22no—
471C22⋊C4—41no—
472C22⋊C4—1064no—
473C22⋊C4—4241no—
474C22⋊C4—2122no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—1064no—
480C42—4241no—
481C42—4241no—
482C42—2122no—
483C42—2122no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23C4×D21212yes—
491C23C22×D21211yescharacteristic
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—1062no—
510C2×C4—1062no—
511C2×C4—1062no—
512C2×C4—1062no—
513C2×C4—4244no—
514C2×C4—2121no—
515C2×C4—2121no—
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—2121no—
526C2×C4—2121no—
527C2×C4—2121no—
528C2×C4—2121no—
529C2×C4—2121no—
530C2×C4—2121no—
531C2×C4—2121no—
532C2×C4—2121no—
533C2×C4—2121no—
534C2×C4C4×D21212yes—
535C2×C4#1313325012yes—
536C2×C4C22×C4×D5311yescharacteristic
537C2×C4C2×#1171994711yescharacteristic
538C2×C4C2×#1171994711yescharacteristic
539C2×C4C2×#1171994511yescharacteristic
540C2×C4#1313326411yescharacteristic
541C2×C4#1313326411yescharacteristic
542C8—22no—
543C8—41no—
544C8—41no—
545C8#1313317314yes—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22C2×C4×D21211yescharacteristic
554C22#1681127711yescharacteristic
555C22#1681127711yescharacteristic
556C22C2×#1313325011yescharacteristic
557C22#1681142311yescharacteristic
558C22#1681142811yescharacteristic
559C22#1681143511yescharacteristic
560C4—22no—
561C4—22no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—4241no—
566C4—4241no—
567C4—4241no—
568C4—4241no—
569C4—2122no—
570C4—2122no—
571C4—2122no—
572C4—2122no—
573C4C2×#1313317311yescharacteristic
574C4#1681110011yescharacteristic
575C4#1681131311yescharacteristic
576C4#1681131311yescharacteristic
577C2—42no—
578C2#2027758211yescharacteristic
579C2C4×#1313325011yescharacteristic
580C2#2028354911yescharacteristic
581C2#2028717211yescharacteristic
582C2#2028721711yescharacteristic
583C2#2028727711yescharacteristic
584C2#2028734211yescharacteristic
subgroupG/H
centerC23C22×D212
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1753C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
cosocleC22×C4×C53C24
hypercenter#1753D53
hypocenterC53#22621
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
4. #20275453C2semidirect
5. #20275448C2semidirect
6. #20275308C2semidirect
7. #20275301C2semidirect
8. #20275294C2semidirect
9. #20275287C2semidirect
13. #20274612C2semidirect
14. C4×#13133149C2semidirect
18. #16810665C4semidirect
352. C53#22621semidirect

Maximal subgroups

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

subgroupindex
C2×#168106652
#202775982
#202775902
#202754532
#202754482
#202753082
#202753012
#202752942
#202752872
#202748112
#202748022
#202747482
#202746122
C4×#131331492
C53×#17532
#2262153