Finite Group Databasebeta

← #28319231

Subgroups — #28319231

Order 13568.

1426 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811646C211yesmaximal, characteristic, complemented
2C2×#16811642C211yesmaximal, characteristic, complemented
3#20290607C212yesmaximal, complemented
4#20290574C212yesmaximal, complemented
5#20290573C212yesmaximal, complemented
6#20290538C212yesmaximal
7#20290445C212yesmaximal, complemented
8#20290444C212yesmaximal, complemented
9C2×#117×D53C211yesmaximal, characteristic
10#16811646—24no—
11#16811642—24no—
12C2×#13133284C2212yescomplemented
13C2×C4⋊#88324C2212yescomplemented
14C2×#13133274C2214yes—
15C2×#13133274C2212yes—
16C2×#13133274C2212yes—
17C2×C22⋊#88324C2214yes—
18C2×#13133272C2212yescomplemented
19C2×C4×#88324C2212yescomplemented
20#16811489C2214yescomplemented
21#16811489C2214yescomplemented
22C23×C4×D53C2211yescharacteristic
23C2×#13133176C411yescharacteristic
24C2×#13133175C2211yescharacteristic
25C2×#13133175C2211yescharacteristic
26C2×#13133174C411yescharacteristic
27C2×#13133174C411yescharacteristic
28C2×#13133173C2211yescharacteristic
29C2×#13133173C2211yescharacteristic
30#117×D53—24no—
31#117×D53—24no—
32C2×C4⋊C4×D53C411yescharacteristic
33C2×C4⋊C4×D53C411yescharacteristic
34C2×C22⋊C4×D53C411yescharacteristic
35C2×C22⋊C4×D53C411yescharacteristic
36C2×C42×D53C2211yescharacteristic
37C2×C42×D53C2211yescharacteristic
38C2×#117×C53C411yescharacteristic
39C22×#88324C2314yes—
40C22×#88324C2314yes—
41#13133284—24no—
42C4⋊#88324—24no—
43#13133274—24no—
44#13133274—24no—
45#13133274—28no—
46C22⋊#88324—28no—
47#13133272—24no—
48C4×#88324—24no—
49C24×D53C2×C411yescharacteristic
50C23×Dic53C2311yescharacteristic
51C22×C4×D53—24no—
52C22×C4×D53—22no—
53C22×C4×D53—22no—
54C22×C4×D53—24no—
55C22×C4×D53C2×C411yescharacteristic
56C22×C4×D53C2×C411yescharacteristic
57C22×C4×D53C2×C411yescharacteristic
58C22×C4×D53C2×C411yescharacteristic
59C22×C4×D53C2×C411yescharacteristic
60C22×C4×D53C2×C411yescharacteristic
61C22×C4×D53C2311yescharacteristic
62C22×C4×D53C2311yescharacteristic
63#13133176—24no—
64#13133175—22no—
65#13133175—22no—
66#13133175—22no—
67#13133175—22no—
68#13133174—24no—
69#13133174—24no—
70#13133173—22no—
71#13133173—22no—
72#13133173—22no—
73#13133173—22no—
74C2×C4⋊Dic53C2×C411yescharacteristic
75C2×C4⋊Dic53C2×C411yescharacteristic
76C2×#11719919C2×C411yescharacteristic
77C2×#11719919C2×C411yescharacteristic
78C2×#11719919C2×C411yescharacteristic
79C2×#11719919C2×C411yescharacteristic
80C4⋊C4×D53—24no—
81C4⋊C4×D53—24no—
82C2×C22⋊Dic53C2×C411yescharacteristic
83C2×C22⋊Dic53C2×C411yescharacteristic
84C2×#11719921C2×C411yescharacteristic
85C2×#11719921C2×C411yescharacteristic
86C2×#11719921C2×C411yescharacteristic
87C2×#11719921C2×C411yescharacteristic
88C22⋊C4×D53—24no—
89C22⋊C4×D53—24no—
90C2×C4×Dic53C2×C411yescharacteristic
91C2×C4×Dic53C2×C411yescharacteristic
92C2×C4×Dic53C2311yescharacteristic
93C2×C4×Dic53C2311yescharacteristic
94C2×C4×Dic53C2311yescharacteristic
95C2×C4×Dic53C2311yescharacteristic
96C42×D53—22no—
97C42×D53—22no—
98C42×D53D414yes—
99C42×D53D414yes—
100C23×C4×C53C2×C411yescharacteristic
101#117×C53—24no—
102C2×C4⋊C4×C53C2×C411yescharacteristic
103C2×C4⋊C4×C53C2×C411yescharacteristic
104C2×C22⋊C4×C53C2×C411yescharacteristic
105C2×C22⋊C4×C53C2×C411yescharacteristic
106C2×C42×C53C2×C411yescharacteristic
107C2×C42×C53C2×C411yescharacteristic
108C24×C53C22×C411yescharacteristic
109C23×D53—22no—
110C23×D53—22no—
111C23×D53—22no—
112C23×D53—24no—
113C23×D53—42no—
114C23×D53C22×C411yescharacteristic
115C23×D53C22×C411yescharacteristic
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53C22×C411yescharacteristic
120C22×C4×C53C22×C411yescharacteristic
121C22×C4×C53C22×C411yescharacteristic
122C22×C4×C53C22×C411yescharacteristic
123C22×C4×C53C22×C411yescharacteristic
124C22×C4×C53C22×C411yescharacteristic
125C22×Dic53—21no—
126C22×Dic53—21no—
127C22×Dic53—22no—
128C22×Dic53C22×C411yescharacteristic
129C22×Dic53C22×C411yescharacteristic
130C22×Dic53C22×C411yescharacteristic
131C22×Dic53C22×C411yescharacteristic
132C22×Dic53C22×C411yescharacteristic
133C22×Dic53#4114yes—
134C22×Dic53C2411yescharacteristic
135C2×C4×D53—216no—
136C2×C4×D53—216no—
137C2×C4×D53—22no—
138C2×C4×D53—22no—
139C2×C4×D53—24no—
140C2×C4×D53—22no—
141C2×C4×D53—22no—
142C2×C4×D53—22no—
143C2×C4×D53—22no—
144C2×C4×D53—21no—
145C2×C4×D53—21no—
146C2×C4×D53—22no—
147C2×C4×D53—22no—
148C2×C4×D53—22no—
149C2×C4×D53—22no—
150C2×C4×D53—22no—
151C2×C4×D53—24no—
152C2×C4×D53—42no—
153C2×C4×D53—42no—
154C2×C4×D53C22⋊C414yes—
155C2×C4×D53C22⋊C414yes—
156C2×C4×D53C2×D412yes—
157C2×C4×D53C2×D412yes—
158C2×C4×D53#4114yes—
159C2×#88324—24no—
160C2×#88324—24no—
161C2×#88324—24no—
162C2×#88324—24no—
163C2×#88324—24no—
164C2×#88324—24no—
165C4⋊C4×C53—22no—
166C4⋊C4×C53—22no—
167C22⋊C4×C53—22no—
168C22⋊C4×C53—22no—
169C42×C53—22no—
170C42×C53C22⋊C414yes—
171C22⋊Dic53—22no—
172C22⋊Dic53—22no—
173#11719921—22no—
174#11719921—22no—
175#11719921—22no—
176#11719921—22no—
177C4⋊Dic53—22no—
178C4⋊Dic53—22no—
179#11719919—22no—
180#11719919—22no—
181#11719919—22no—
182#11719919—22no—
183C4×Dic53—22no—
184C4×Dic53—21no—
185C4×Dic53—21no—
186C4×Dic53—21no—
187C4×Dic53—21no—
188C4×Dic53C22⋊C414yes—
189C4×Dic53C2×D412yes—
190C4×Dic53C2×D412yes—
191C4×Dic53C2×D412yes—
192C4×Dic53C2×D412yes—
193C23×C53—21no—
194C23×C53—21no—
195C23×C53—22no—
196C23×C53—22no—
197C23×C53#11712yes—
198C23×C53C23×C411yescharacteristic
199C22×D53—21no—
200C22×D53—21no—
201C22×D53—22no—
202C22×D53—28no—
203C22×D53—21no—
204C22×D53—21no—
205C22×D53—22no—
206C22×D53—22no—
207C22×D53—48no—
208C22×D53—48no—
209C22×D53—44no—
210C22×D53—44no—
211C22×D53C2×C22⋊C412yes—
212C22×D53#11712yes—
213C22×D53#13114yes—
214C2×C4×C53—28no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53C2×C22⋊C412yes—
226C2×C4×C53C2×C22⋊C412yes—
227C2×C4×C53C2×C22⋊C412yes—
228C2×C4×C53#11712yes—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—22no—
232C2×Dic53—22no—
233C2×Dic53—24no—
234C2×Dic53—24no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—22no—
238C2×Dic53—22no—
239C2×Dic53C2×C22⋊C412yes—
240C2×Dic53C2×C22⋊C412yes—
241C2×Dic53#11712yes—
242C2×Dic53C22×D411yescharacteristic
243C2×Dic53C22×D411yescharacteristic
244C2×Dic53C2×#4112yes—
245C2×Dic53C2×#4112yes—
246C4×D53—28no—
247C4×D53—28no—
248C4×D53—28no—
249C4×D53—24no—
250C4×D53—24no—
251C4×D53—24no—
252C4×D53—24no—
253C4×D53—44no—
254C4×D53—44no—
255C4×D53—44no—
256#88324—44no—
257#88324—44no—
258#20717—531nomaximal, Sylow
259C22×C53—24no—
260C22×C53—22no—
261C22×C53—41no—
262C22×C53—41no—
263C22×C53—42no—
264C22×C53C22×C22⋊C411yescharacteristic
265C22×C53C2×#11711yescharacteristic
266C22×C53C2×#11711yescharacteristic
267C22×C53C2×#13112yes—
268C22×C53#56812yes—
269C2×D53—22no—
270C2×D53—24no—
271C2×D53—24no—
272C2×D53—22no—
273C2×D53—24no—
274C2×D53—22no—
275C2×D53—416no—
276C2×D53—41no—
277C2×D53—41no—
278C2×D53—42no—
279C2×D53#40814yes—
280C4×C53—22no—
281C4×C53—22no—
282C4×C53—22no—
283C4×C53—22no—
284C4×C53—41no—
285C4×C53—41no—
286Dic53—22no—
287Dic53—22no—
288Dic53—22no—
289Dic53—41no—
290Dic53—41no—
291Dic53#52512yes—
292Dic53#52512yes—
293C22×#117—531no—
294C2×#432—531no—
295C2×#431—531no—
296#1817—532no—
297#1798—532no—
298#1798—532no—
299#1779—532no—
300#1750—532no—
301#1750—532no—
302C2×C53—42no—
303C2×C53#228011yescharacteristic
304C2×C53C2×#40811yescharacteristic
305C2×C53#287911yescharacteristic
306C2×C53#292412yes—
307C2×C53#292412yes—
308D53—24no—
309D53—42no—
310C24×C4—531no—
311C2×#117—531no—
312C2×#117—531no—
313C2×#117—531no—
314C2×#117—531no—
315C2×#117—531no—
316C2×#117—531no—
317C2×#117—531no—
318C2×#117—1064no—
319C2×#117—1064no—
320C2×#117#615711yescharacteristic
321C22×C4⋊C4—531no—
322C22×C4⋊C4—531no—
323C22×C22⋊C4—531no—
324C22×C22⋊C4—531no—
325C22×C42—531no—
326C22×C42—531no—
327#432—1064no—
328#431—1064no—
329C2×C4⋊C8—532no—
330C2×C4⋊C8—532no—
331C2×C22⋊C8—534no—
332C2×C22⋊C8—534no—
333C2×C22⋊C8—532no—
334C2×C22⋊C8—532no—
335C2×#97—532no—
336C2×C4×C8—532no—
337#336—534no—
338#336—534no—
339C53#2071711yescharacteristic, complemented, Sylow
340C25—531no—
341C23×C4—531no—
342C23×C4—531no—
343C23×C4—531no—
344C23×C4—531no—
345C23×C4—531no—
346C23×C4—531no—
347C23×C4—531no—
348C23×C4—531no—
349C23×C4—531no—
350C23×C4—1062no—
351C23×C4—1062no—
352C23×C4—1064no—
353C23×C4—1064no—
354C23×C4C2×#615711yescharacteristic
355C22×C8—534no—
356C22×C8—534no—
357#117—24no—
358#117—1064no—
359#117—1062no—
360#117—1062no—
361#117—1062no—
362#117—1062no—
363#117—1062no—
364#117—1062no—
365#117—1062no—
366#117—1062no—
367#117—1064no—
368#117—1064no—
369C2×C4⋊C4—531no—
370C2×C4⋊C4—531no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—531no—
373C2×C4⋊C4—531no—
374C2×C4⋊C4—531no—
375C2×C4⋊C4—1064no—
376C2×C4⋊C4—1064no—
377C2×C4⋊C4C2×#615711yescharacteristic
378C2×C4⋊C4C2×#615711yescharacteristic
379C2×C22⋊C4—531no—
380C2×C22⋊C4—531no—
381C2×C22⋊C4—531no—
382C2×C22⋊C4—531no—
383C2×C22⋊C4—531no—
384C2×C22⋊C4—531no—
385C2×C22⋊C4—1064no—
386C2×C22⋊C4—1064no—
387C2×C22⋊C4C2×#615711yescharacteristic
388C2×C22⋊C4C2×#615711yescharacteristic
389C2×C42—531no—
390C2×C42—531no—
391C2×C42—531no—
392C2×C42—534no—
393C2×C42—534no—
394C2×C42—531no—
395C2×C42—531no—
396C2×C42—531no—
397C2×C42—1062no—
398C2×C42—1062no—
399C2×C42C2×#615711yescharacteristic
400C2×C42C2×#615711yescharacteristic
401C4⋊C8—1064no—
402C4⋊C8—1064no—
403C22⋊C8—1064no—
404C22⋊C8—1064no—
405C22⋊C8—1068no—
406C22⋊C8—1068no—
407#97—1064no—
408C4×C8—1064no—
409C24—531no—
410C24—531no—
411C24—1062no—
412C24—1062no—
413C24—1064no—
414C24—1062no—
415C24—2122no—
416C24C22×#615711yescharacteristic
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—531no—
421C22×C4—534no—
422C22×C4—531no—
423C22×C4—531no—
424C22×C4—531no—
425C22×C4—534no—
426C22×C4—532no—
427C22×C4—532no—
428C22×C4—534no—
429C22×C4—534no—
430C22×C4—531no—
431C22×C4—531no—
432C22×C4—1061no—
433C22×C4—1061no—
434C22×C4—1062no—
435C22×C4—1062no—
436C22×C4—1062no—
437C22×C4—1062no—
438C22×C4—1062no—
439C22×C4—1062no—
440C22×C4—1064no—
441C22×C4—1062no—
442C22×C4—10616no—
443C22×C4—10616no—
444C22×C4—1062no—
445C22×C4—1062no—
446C22×C4—1064no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—1062no—
450C22×C4—1062no—
451C22×C4—1062no—
452C22×C4—2122no—
453C22×C4—2122no—
454C22×C4C22×#615711yescharacteristic
455C22×C4C22×#615711yescharacteristic
456C22×C4C22×#615711yescharacteristic
457C22×C4C22×#615711yescharacteristic
458C22×C4C22×#615711yescharacteristic
459C22×C4C22×#615711yescharacteristic
460C2×C8—1064no—
461C2×C8—1064no—
462C2×C8—1064no—
463C2×C8—1064no—
464C2×C8—1064no—
465C2×C8—1064no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—1062no—
469C4⋊C4—1062no—
470C4⋊C4—1062no—
471C4⋊C4—1062no—
472C4⋊C4—1062no—
473C4⋊C4—1062no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—1062no—
477C22⋊C4—1062no—
478C22⋊C4—1062no—
479C22⋊C4—1062no—
480C22⋊C4—1062no—
481C22⋊C4—1062no—
482C42—22no—
483C42—532no—
484C42—532no—
485C42—532no—
486C42—532no—
487C42—534no—
488C42—1061no—
489C42—1061no—
490C42—1061no—
491C42—1061no—
492C42—1062no—
493C42C22⋊#615714yes—
494C23—21no—
495C23—21no—
496C23—22no—
497C23—22no—
498C23—532no—
499C23—534no—
500C23—532no—
501C23—1068no—
502C23—1061no—
503C23—1061no—
504C23—1062no—
505C23—1061no—
506C23—1061no—
507C23—1062no—
508C23—1062no—
509C23—2128no—
510C23—2124no—
511C23—2128no—
512C23—2124no—
513C23#1313329612yes—
514C23C23×#615711yescharacteristic
515C2×C4—28no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—531no—
527C2×C4—531no—
528C2×C4—532no—
529C2×C4—532no—
530C2×C4—532no—
531C2×C4—532no—
532C2×C4—532no—
533C2×C4—1064no—
534C2×C4—1064no—
535C2×C4—1064no—
536C2×C4—1064no—
537C2×C4—1068no—
538C2×C4—1068no—
539C2×C4—1068no—
540C2×C4—1064no—
541C2×C4—1064no—
542C2×C4—1061no—
543C2×C4—1061no—
544C2×C4—1062no—
545C2×C4—1062no—
546C2×C4—1061no—
547C2×C4—1061no—
548C2×C4—1062no—
549C2×C4—1062no—
550C2×C4—2124no—
551C2×C4—2124no—
552C2×C4—2124no—
553C2×C4C2×C22⋊#615712yes—
554C2×C4C2×C22⋊#615712yes—
555C2×C4C2×C22⋊#615712yes—
556C2×C4#1313329612yes—
557C8—2124no—
558C8—2124no—
559C22—24no—
560C22—22no—
561C22—42no—
562C22—41no—
563C22—41no—
564C22—534no—
565C22—1064no—
566C22—1064no—
567C22—1062no—
568C22—1062no—
569C22—1064no—
570C22—1062no—
571C22—21216no—
572C22—2122no—
573C22—2121no—
574C22—2121no—
575C22C22×C22⋊#615711yescharacteristic
576C22C2×#1313329611yescharacteristic
577C22C2×#1313329611yescharacteristic
578C22D53⋊#13112yes—
579C22#1681170212yes—
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—41no—
585C4—41no—
586C4—532no—
587C4—532no—
588C4—1062no—
589C4—1062no—
590C4—1062no—
591C4—2121no—
592C4—2121no—
593C2—42no—
594C2—1064no—
595C2—2122no—
596C2#2029115911yescharacteristic
597C2D53⋊#40811yescharacteristic
598C2#2029130011yescharacteristic
599C2#2029140712yes—
600C2#2029140912yes—
subgroupG/H
center, FrattiniC23C23×#6157
commutator, socleC23×C53C23×C4
FittingC2×#117×C53C4
prime radicalC24×D53C2×C4
cosocleC22×Dic53C24
hypercenterC2×#117#6157
hypocenterC53#20717
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×#16811646C2semidirect
2. C2×#16811642C2semidirect
3. #20290607C2semidirect
4. #20290574C2semidirect
5. #20290573C2semidirect
7. #20290445C2semidirect
8. #20290444C2semidirect
12. C2×#13133284C22semidirect
13. C2×C4⋊#88324C22semidirect
18. C2×#13133272C22semidirect
19. C2×C4×#88324C22semidirect
20. #16811489C22semidirect
21. #16811489C22semidirect
339. C53#20717semidirect

Maximal subgroups

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

subgroupindex
C2×#168116462
C2×#168116422
#202906072
#202905742
#202905732
#202905382
#202904452
#202904442
C2×#117×D532
#2071753