Finite Group Databasebeta

← #27442141

Subgroups — #27442141

Order 13568.

704 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277590C211yesmaximal, characteristic
2#20276277C211yesmaximal, characteristic, complemented
3#20276216C211yesmaximal, characteristic, complemented
4#20276206C211yesmaximal, characteristic, complemented
5#20276050C211yesmaximal, characteristic, complemented
6#20275673C211yesmaximal, characteristic, complemented
7#20275471C211yesmaximal, characteristic, complemented
8#20275297C211yesmaximal, characteristic, complemented
9#20275290C211yesmaximal, characteristic, complemented
10#20274914C211yesmaximal, characteristic
11#20274900C211yesmaximal, characteristic
12#20274799C211yesmaximal, characteristic
13#20274746C211yesmaximal, characteristic
14#20274436C211yesmaximal, characteristic
15C53×#2441C211yesmaximal, characteristic
16C22×#11719917C2211yescharacteristic
17C22×C4⋊Dic53C2211yescharacteristic
18C2×#13133150C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×C4⋊#88322C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×C22⋊#88322C2211yescharacteristic
25#16810521C2211yescharacteristic
26#16810521C2211yescharacteristic
27#16810506—21no—
28#16810506—21no—
29#16810483C2211yescharacteristic
30#16810481—21no—
31C4×C4⋊Dic53C2211yescharacteristic
32C4×C4⋊Dic53C2211yescharacteristic
33C22⋊C4×Dic53—21no—
34#16810385C2211yescharacteristic
35#16810385C2211yescharacteristic
36#16810385C2211yescharacteristic
37#16810385C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#125×C53C2211yescharacteristic
49C2×#117×C53C2211yescharacteristic
50C53×#399—21no—
51C53×#396—21no—
52C53×#389—21no—
53C53×#389C2211yescharacteristic
54C53×#389C2211yescharacteristic
55C53×#387—21no—
56C53×#384C2211yescharacteristic
57C53×#380C2211yescharacteristic
58C4×C22⋊C4×C53C2211yescharacteristic
59C23×Dic53—21no—
60C2×#11719917—22no—
61C2×#11719917—22no—
62C2×#11719917—41no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—22no—
69C2×C4⋊Dic53Q814yes—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×C22⋊Dic53—21no—
77C2×C4×Dic53—21no—
78C2×C4×Dic53—21no—
79#13133150—41no—
80#13133150—41no—
81#13133148—22no—
82#13133148—22no—
83C4⋊#88322—22no—
84C4⋊#88322—41no—
85C22⋊#88322—22no—
86#13133124—21no—
87#13133124—21no—
88#13133124—21no—
89#13133124—21no—
90#13133124—21no—
91C23×C4×C53C2311yescharacteristic
92#125×C53—22no—
93#125×C53—41no—
94#117×C53—22no—
95#117×C53—41no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53C2311yescharacteristic
103C2×C4⋊C4×C53C2311yescharacteristic
104C2×C4⋊C4×C53C2311yescharacteristic
105C2×C4⋊C4×C53C2311yescharacteristic
106C2×C22⋊C4×C53—21no—
107C2×C22⋊C4×C53—21no—
108C2×C22⋊C4×C53D411yescharacteristic
109C2×C22⋊C4×C53D411yescharacteristic
110C2×C42×C53—21no—
111C2×C42×C53C2311yescharacteristic
112C2×C42×C53C2311yescharacteristic
113#95×C53—21no—
114#95×C53—21no—
115#95×C53—21no—
116#95×C53D411yescharacteristic
117#95×C53D411yescharacteristic
118C24×C53C2×D411yescharacteristic
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×Q812yes—
133C22×C4×C53#4112yes—
134C22×C4×C53C2411yescharacteristic
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—24no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—81no—
141C22×Dic53—81no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—42no—
150C4⋊C4×C53—42no—
151C4⋊C4×C53—42no—
152C4⋊C4×C53—42no—
153C4⋊C4×C53—81no—
154C4⋊C4×C53—81no—
155C4⋊C4×C53#4112yes—
156C22⋊C4×C53—22no—
157C22⋊C4×C53—41no—
158C22⋊C4×C53—42no—
159C22⋊C4×C53—81no—
160C42×C53—21no—
161C42×C53—21no—
162C42×C53—21no—
163C42×C53—21no—
164C42×C53—42no—
165C22⋊Dic53—42no—
166C4⋊Dic53—24no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53C2×Q812yes—
174C4⋊Dic53C2×Q812yes—
175#11719919—81no—
176#11719919—81no—
177C4×Dic53—81no—
178C4×Dic53—81no—
179#11719917—44no—
180C2×#88322—21no—
181C2×#88322—21no—
182C2×#88322—21no—
183C2×#88322—21no—
184C2×#88322—21no—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322#4112yes—
192C23×C53—21no—
193C23×C53—21no—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53C22⋊Q812yes—
199C23×C53C22×D411yescharacteristic
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—22no—
205C2×C4×C53—22no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
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×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53C22⋊Q812yes—
230C2×C4×C53#13712yes—
231C2×C4×C53C22×Q811yescharacteristic
232C2×C4×C53C2×#4111yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53H(5,𝔽2)11yescharacteristic
236C2×C4×C532−1+411yescharacteristic
237C2×Dic53—22no—
238C2×Dic53—22no—
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—41no—
249C2×Dic53—84no—
250#88322—22no—
251#88322—41no—
252#88322—41no—
253#88322—41no—
254#26018—531nomaximal, Sylow
255C22×C53—22no—
256C22×C53—22no—
257C22×C53—22no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53—41no—
261C22×C53—41no—
262C22×C53C2×C22⋊Q811yescharacteristic
263C22×C53#54611yescharacteristic
264C22×C53#54711yescharacteristic
265C22×C53C2×#13711yescharacteristic
266C22×C53#57511yescharacteristic
267C22×C53#57611yescharacteristic
268C22×C53#57711yescharacteristic
269C4×C53—22no—
270C4×C53—22no—
271C4×C53—41no—
272C4×C53—42no—
273C4×C53—41no—
274C4×C53—41no—
275C4×C53—81no—
276C4×C53—81no—
277C4×C53—81no—
278C4×C53#53311yescharacteristic
279C4×C53#54411yescharacteristic
280C4×C53#55111yescharacteristic
281C4×C53#55211yescharacteristic
282Dic53—42no—
283Dic53—42no—
284Dic53—81no—
285Dic53—81no—
286#2441D5311yescharacteristic
287#2345—531no—
288#2100—531no—
289#2088—531no—
290#2086—531no—
291#2059—531no—
292#1989—531no—
293#1947—531no—
294#1920—531no—
295#1919—531no—
296#1854—531no—
297#1851—531no—
298#1823—531no—
299#1809—531no—
300#1736—531no—
301C2×C53—42no—
302C2×C53#243811yescharacteristic
303C2×C53#307811yescharacteristic
304C2×C53#308111yescharacteristic
305C2×C53#330911yescharacteristic
306C2×C53#333111yescharacteristic
307C2×C53#334611yescharacteristic
308C2×C53#335111yescharacteristic
309C22×M16—531no—
310C2×#125C2×D5311yescharacteristic
311C2×#117C2×D5311yescharacteristic
312C22×C4⋊C4—531no—
313C2×#107—531no—
314C2×#107—531no—
315C2×#106—531no—
316C2×#106—531no—
317C2×C4⋊C8—531no—
318C2×C4⋊C8—531no—
319C2×C22⋊C8—531no—
320#399—21no—
321#396—21no—
322#389—21no—
323#389—531no—
324#389—531no—
325#389C2×D5311yescharacteristic
326#389C2×D5311yescharacteristic
327#387—21no—
328#387—1061no—
329#387—1061no—
330#384C2×D5311yescharacteristic
331#380—531no—
332#380—1061no—
333#380C2×D5311yescharacteristic
334C4×C4⋊C4—531no—
335C4×C4⋊C4—531no—
336C4×C22⋊C4—1061no—
337C4×C22⋊C4C2×D5311yescharacteristic
338#340—531no—
339#340—531no—
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#336—531no—
351#336—531no—
352C53#2601811yescharacteristic, complemented, Sylow
353C23×C4—1061no—
354C23×C4C22×D5311yescharacteristic
355C2×M16—1062no—
356C2×M16—1062no—
357C2×M16—2121no—
358C22×C8—531no—
359C22×C8—531no—
360C22×C8—531no—
361C22×C8—531no—
362#125—22no—
363#125—41no—
364#117—22no—
365#117—41no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—534no—
373C2×C4⋊C4—531no—
374C2×C4⋊C4—531no—
375C2×C4⋊C4—531no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—1062no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4—1061no—
380C2×C4⋊C4—1061no—
381C2×C4⋊C4C22×D5311yescharacteristic
382C2×C4⋊C4C22×D5311yescharacteristic
383C2×C4⋊C4C22×D5311yescharacteristic
384C2×C4⋊C4C22×D5311yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—1061no—
388C2×C22⋊C4C22⋊D5311yescharacteristic
389C2×C22⋊C4C22⋊D5311yescharacteristic
390C2×C42—21no—
391C2×C42—1061no—
392C2×C42—1061no—
393C2×C42C22×D5311yescharacteristic
394C2×C42C22×D5311yescharacteristic
395#107—2121no—
396#107—2121no—
397#106—1062no—
398#106—1062no—
399C4⋊C8—1062no—
400C4⋊C8—2121no—
401C22⋊C8—1062no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—1061no—
406#95—1061no—
407#95—1061no—
408#95—1061no—
409#95—1061no—
410#95C22⋊D5311yescharacteristic
411#95C22⋊D5311yescharacteristic
412C24C2×C22⋊D5311yescharacteristic
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—1064no—
422C22×C4—1061no—
423C22×C4—1061no—
424C22×C4—1061no—
425C22×C4—1061no—
426C22×C4—4241no—
427C22×C4—4241no—
428C22×C4Q8×D5312yes—
429C22×C4#1171994912yes—
430C22×C4C2×C22⋊D5311yescharacteristic
431C22×C4C2×C22⋊D5311yescharacteristic
432C22×C4C2×C22⋊D5311yescharacteristic
433C22×C4C2×C22⋊D5311yescharacteristic
434C22×C4C2×C22⋊D5311yescharacteristic
435C22×C4C23×D5311yescharacteristic
436M16—2124no—
437C2×C8—532no—
438C2×C8—1061no—
439C2×C8—1061no—
440C2×C8—1061no—
441C2×C8—1061no—
442C2×C8—1061no—
443C2×C8—1061no—
444C2×C8—1061no—
445C2×C8—1061no—
446C2×C8—1061no—
447C2×C8—1061no—
448C2×C8—1061no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—81no—
461C4⋊C4—81no—
462C4⋊C4—532no—
463C4⋊C4—532no—
464C4⋊C4—1064no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—4241no—
472C4⋊C4—4241no—
473C4⋊C4#1171994712yes—
474C22⋊C4—22no—
475C22⋊C4—41no—
476C22⋊C4—42no—
477C22⋊C4—81no—
478C22⋊C4—2122no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—42no—
484C42—4241no—
485C42—4241no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23#1313319312yes—
493C23C22×C22⋊D5311yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—1062no—
524C2×C4—1062no—
525C2×C4—4244no—
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×C4—2121no—
535C2×C4—2121no—
536C2×C4#1313319312yes—
537C2×C4#1313324712yes—
538C2×C4C2×Q8×D5311yescharacteristic
539C2×C4#41×D5311yescharacteristic
540C2×C4C2×#1171994711yescharacteristic
541C2×C4C2×#1171994911yescharacteristic
542C2×C4#1313326211yescharacteristic
543C2×C4#1313326411yescharacteristic
544C8—1062no—
545C8—2121no—
546C8—2121no—
547C8—2121no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×#1313319311yescharacteristic
556C22#1681126711yescharacteristic
557C22#1681127211yescharacteristic
558C22C2×#1313324711yescharacteristic
559C22#1681142111yescharacteristic
560C22#1681142611yescharacteristic
561C22#1681143111yescharacteristic
562C4—22no—
563C4—22no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—42no—
568C4—81no—
569C4—81no—
570C4—81no—
571C4—4241no—
572C4—4241no—
573C4—2122no—
574C4—2122no—
575C4#1681117511yescharacteristic
576C4#1681125211yescharacteristic
577C4#1681130011yescharacteristic
578C4#1681131511yescharacteristic
579C2—42no—
580C2#2027838711yescharacteristic
581C2#2028451611yescharacteristic
582C2#2028454011yescharacteristic
583C2#2028672111yescharacteristic
584C2#2028703111yescharacteristic
585C2#2028720011yescharacteristic
586C2#2028727511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2441C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2441D53
hypocenterC53#26018
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2441

Complements

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

NH ≅ G/Ntype
2. #20276277C2semidirect
3. #20276216C2semidirect
4. #20276206C2semidirect
5. #20276050C2semidirect
6. #20275673C2semidirect
7. #20275471C2semidirect
8. #20275297C2semidirect
9. #20275290C2semidirect
352. C53#26018semidirect

Maximal subgroups

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

subgroupindex
#202775902
#202762772
#202762162
#202762062
#202760502
#202756732
#202754712
#202752972
#202752902
#202749142
#202749002
#202747992
#202747462
#202744362
C53×#24412
#2601853