Finite Group Databasebeta

← #27305900

Subgroups — #27305900

Order 13568.

1942 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133182C211yesmaximal, characteristic
2C2×#16810543C211yesmaximal, characteristic
3C2×C4⋊C4×Dic53C211yesmaximal, characteristic
4C2×#16810474C212yesmaximal
5#20272985C211yesmaximal, characteristic
6#20272979C212yesmaximal
7#20272978C211yesmaximal, characteristic
8#20272966C212yesmaximal
9#95×Dic53C212yesmaximal
10#95×Dic53C211yesmaximal, characteristic
11C53×#1441C211yesmaximal, characteristic
12C23×Dic106C2211yescharacteristic
13C2×#13133182—28no—
14C2×#13133182—24no—
15C22×C4⋊Dic53C2211yescharacteristic
16C22×#11719919C2212yes—
17C22×#11719919C2212yes—
18C22×#11719919C2212yes—
19C22×C4×Dic53C2211yescharacteristic
20C22×C4×Dic53C2212yes—
21C22×C4×Dic53C2211yescharacteristic
22C22×C4×Dic53C2212yes—
23C22×C4×Dic53C2211yescharacteristic
24#16810543C418yescomplemented
25C4⋊C4×Dic53—24no—
26#16810474—28no—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2211yescharacteristic
32C2×#13133124C2211yescharacteristic
33C2×#13133124C2211yescharacteristic
34C2×#13133124C2212yes—
35C22×C4⋊C4×C53C2211yescharacteristic
36C22×C4⋊C4×C53C2211yescharacteristic
37C2×#95×C53C2212yes—
38C2×#95×C53C2211yescharacteristic
39C2×#95×C53C2212yes—
40C22×Dic106—22no—
41C22×Dic106—22no—
42C22×Dic106—24no—
43C22×Dic106—24no—
44C22×Dic106C2×C414yes—
45C23×Dic53C2311yescharacteristic
46C23×Dic53C2311yescharacteristic
47C23×Dic53C2311yescharacteristic
48C23×Dic53C2312yes—
49C23×Dic53C2312yes—
50C23×Dic53C2311yescharacteristic
51#13133182—232no—
52C2×C4⋊Dic53—22no—
53C2×C4⋊Dic53—22no—
54C2×C4⋊Dic53—22no—
55C2×#11719919—24no—
56C2×#11719919—24no—
57C2×#11719919—24no—
58C2×#11719919—24no—
59C2×#11719919—24no—
60C2×#11719919—24no—
61C2×#11719919—24no—
62C2×#11719919—24no—
63C2×#11719919C2×C418yes—
64C2×C4×Dic53—24no—
65C2×C4×Dic53—24no—
66C2×C4×Dic53—24no—
67C2×C4×Dic53—24no—
68C2×C4×Dic53—22no—
69C2×C4×Dic53—24no—
70C2×C4×Dic53—22no—
71C2×C4×Dic53—24no—
72C2×C4×Dic53—22no—
73C2×C4×Dic53D418yes—
74C2×C4×Dic53D414yes—
75C2×C4×Dic53Q814yescomplemented
76C2×C4×Dic53Q818yescomplemented
77#13133124—22no—
78#13133124—24no—
79#13133124—22no—
80#13133124—24no—
81#13133124—24no—
82#13133124—24no—
83#13133124C2×C418yes—
84#13133124C2×C414yes—
85C23×C4×C53C2311yescharacteristic
86C23×C4×C53C2312yes—
87C23×C4×C53C2312yes—
88C23×C4×C53C2311yescharacteristic
89C23×C4×C53C2311yescharacteristic
90C2×C4⋊C4×C53—24no—
91C2×C4⋊C4×C53—22no—
92C2×C4⋊C4×C53—22no—
93C2×C4⋊C4×C53—22no—
94C2×C4⋊C4×C53C2×C414yes—
95#95×C53—24no—
96#95×C53—22no—
97#95×C53—24no—
98C24×C53C2411yescharacteristic
99C22×C4×C53—21no—
100C22×C4×C53—22no—
101C22×C4×C53—21no—
102C22×C4×C53—21no—
103C22×C4×C53—21no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—21no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—21no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53C22×C412yes—
122C22×C4×C53C22×C412yes—
123C22×C4×C53C22×C412yes—
124C22×C4×C53C2×D414yes—
125C22×C4×C53C2×D412yes—
126C22×C4×C53C2×Q814yes—
127C22×C4×C53C2×Q812yes—
128C22×C4×C53#4112yes—
129C22×C4×C53#4112yes—
130C22×Dic53—22no—
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—21no—
139C22×Dic53—22no—
140C22×Dic53—21no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—21no—
146C22×Dic53—22no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53C22×C414yes—
151C22×Dic53C22×C414yes—
152C22×Dic53C2×D412yes—
153C22×Dic53C2×D414yes—
154C22×Dic53C2×D414yes—
155C22×Dic53C2×D412yes—
156C22×Dic53C2×Q812yes—
157C22×Dic53C2×Q814yes—
158C22×Dic53C2×Q814yes—
159C22×Dic53C2×Q812yes—
160C22×Dic53#4112yes—
161C22×Dic53#4114yes—
162C22×Dic53#4114yes—
163C22×Dic53#4112yes—
164C2×Dic106—28no—
165C2×Dic106—216no—
166C2×Dic106—28no—
167C2×Dic106—28no—
168C2×Dic106—216no—
169C4⋊C4×C53—28no—
170C4⋊C4×C53—28no—
171C4⋊Dic53—28no—
172#11719919—216no—
173#11719919—216no—
174#11719919—216no—
175C4×Dic53—216no—
176C4×Dic53—28no—
177C4×Dic53—216no—
178C4×Dic53—28no—
179C4×Dic53—44no—
180C23×C53C23×C411yescharacteristic
181C23×C53C22×D412yes—
182C23×C53C22×D411yescharacteristic
183C23×C53C22×Q812yes—
184C23×C53C22×Q811yescharacteristic
185C23×C53C2×#4112yes—
186C23×C53C2×#4112yes—
187C23×C53C2×#4111yescharacteristic
188C23×C53C2×#4111yescharacteristic
189C23×C53C2×#4111yescharacteristic
190C23×C53C2×#4111yescharacteristic
191C2×C4×C53—24no—
192C2×C4×C53—22no—
193C2×C4×C53—22no—
194C2×C4×C53—22no—
195C2×C4×C53—22no—
196C2×C4×C53—22no—
197C2×C4×C53—24no—
198C2×C4×C53—22no—
199C2×C4×C53—22no—
200C2×C4×C53—22no—
201C2×C4×C53—24no—
202C2×C4×C53—22no—
203C2×C4×C53—24no—
204C2×C4×C53—24no—
205C2×C4×C53—24no—
206C2×C4×C53—24no—
207C2×C4×C53—24no—
208C2×C4×C53—24no—
209C2×C4×C53—41no—
210C2×C4×C53—42no—
211C2×C4×C53—41no—
212C2×C4×C53—42no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53C4×D414yes—
221C2×C4×C53C4×Q814yescomplemented
222C2×Dic53—24no—
223C2×Dic53—24no—
224C2×Dic53—24no—
225C2×Dic53—24no—
226C2×Dic53—24no—
227C2×Dic53—24no—
228C2×Dic53—24no—
229C2×Dic53—24no—
230C2×Dic53—24no—
231C2×Dic53—24no—
232C2×Dic53—24no—
233C2×Dic53—24no—
234C2×Dic53—22no—
235C2×Dic53—22no—
236C2×Dic53—22no—
237C2×Dic53—24no—
238C2×Dic53—22no—
239C2×Dic53—22no—
240C2×Dic53—24no—
241C2×Dic53—41no—
242C2×Dic53—42no—
243C2×Dic53—41no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53C4×D418yes—
250C2×Dic53C4×Q818yescomplemented
251C2×Dic53#12014yes—
252C2×Dic53C22⋊Q818yes—
253C2×Dic53C22⋊Q814yes—
254C2×Dic53C4⋊Q814yes—
255C2×Dic53C4⋊Q818yes—
256Dic106—232no—
257C4×#395—531nomaximal, Sylow
258C22×C53C2×C4×D412yes—
259C22×C53C2×C4×D411yescharacteristic
260C22×C53C2×C4×Q812yes—
261C22×C53C2×C4×Q811yescharacteristic
262C22×C53C4×#4111yescharacteristic
263C22×C53C2×#12011yescharacteristic
264C22×C53C2×C22⋊Q812yes—
265C22×C53C2×C22⋊Q811yescharacteristic
266C22×C53#52512yes—
267C22×C53#52511yescharacteristic
268C22×C53#52511yescharacteristic
269C22×C53#52512yes—
270C22×C53#52911yescharacteristic
271C22×C53#52912yes—
272C22×C53C2×C4⋊Q811yescharacteristic
273C22×C53C2×C4⋊Q812yes—
274C22×C53#53212yes—
275C22×C53#53211yescharacteristic
276C22×C53#53312yes—
277C22×C53#53312yes—
278C22×C53#53311yescharacteristic
279C22×C53#53312yes—
280C22×C53#53312yes—
281C4×C53—24no—
282C4×C53—24no—
283C4×C53—44no—
284C4×C53—44no—
285C4×C53—42no—
286Dic53—28no—
287Dic53—28no—
288Dic53—42no—
289Dic53—42no—
290Dic53—42no—
291Dic53#39518yescomplemented
292C22×C4×Q8—531no—
293C2×#395—531no—
294C2×C4×C4⋊C4—532no—
295C2×C4×C4⋊C4—531no—
296#1446—531no—
297#1444—532no—
298#1444—531no—
299#1441—532no—
300#1441D5311yescharacteristic
301C4×#95—531no—
302C4×#95—532no—
303C2×C53C4×#12011yescharacteristic
304C2×C53C4×C22⋊Q812yes—
305C2×C53C4×C22⋊Q811yescharacteristic
306C2×C53C4×C4⋊Q811yescharacteristic
307C2×C53C4×C4⋊Q812yes—
308C2×C53C2×#39511yescharacteristic
309C2×C53#238811yescharacteristic
310C2×C53#239611yescharacteristic
311C2×C53#239612yes—
312C2×C53#239912yes—
313C2×C53#239911yescharacteristic
314C23×Q8—531no—
315C2×C4×Q8—1064no—
316C2×C4×Q8—1068no—
317C22×C4⋊C4—532no—
318C22×C4⋊C4—532no—
319C22×C4⋊C4—532no—
320C22×C4⋊C4—531no—
321C22×C4⋊C4C2×D5311yescharacteristic
322C22×C4⋊C4C2×D5311yescharacteristic
323C22×C42—532no—
324C22×C42—531no—
325C22×C42—532no—
326C22×C42—531no—
327C22×C42—531no—
328#395—538no—
329C4×C4⋊C4—1068no—
330C4×C4⋊C4—1064no—
331C2×#95—532no—
332C2×#95—531no—
333C2×#95—532no—
334C2×#95—532no—
335C2×#95—532no—
336C2×#95—531no—
337C2×#95—532no—
338C2×#95—531no—
339C2×#95C2×D5312yes—
340C2×#95C2×D5311yescharacteristic
341C2×#95C2×D5312yes—
342C53C4×#39511yescharacteristic, complemented, Sylow
343C22×Q8—534no—
344C22×Q8—1064no—
345C22×Q8—1064no—
346C22×Q8—1062no—
347C22×Q8—1062no—
348C23×C4—531no—
349C23×C4—532no—
350C23×C4—532no—
351C23×C4—531no—
352C23×C4—531no—
353C23×C4—531no—
354C23×C4C22×D5311yescharacteristic
355C23×C4C22×D5312yes—
356C23×C4C22×D5312yes—
357C23×C4C22×D5311yescharacteristic
358C23×C4C22×D5311yescharacteristic
359C4×Q8—10632no—
360C2×C4⋊C4—24no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—538no—
365C2×C4⋊C4—1064no—
366C2×C4⋊C4—1064no—
367C2×C4⋊C4—1064no—
368C2×C4⋊C4—1064no—
369C2×C4⋊C4—1064no—
370C2×C4⋊C4—1064no—
371C2×C4⋊C4—1062no—
372C2×C4⋊C4—1062no—
373C2×C4⋊C4—1064no—
374C2×C4⋊C4—1064no—
375C2×C4⋊C4—1062no—
376C2×C4⋊C4C4×D5314yes—
377C2×C42—538no—
378C2×C42—534no—
379C2×C42—538no—
380C2×C42—534no—
381C2×C42—1064no—
382C2×C42—1064no—
383C2×C42—1064no—
384C2×C42—1062no—
385C2×C42—1064no—
386C2×C42—1062no—
387C2×C42—1064no—
388C2×C42—1064no—
389C2×C42—1062no—
390#95—24no—
391#95—22no—
392#95—24no—
393#95—534no—
394#95—538no—
395#95—1064no—
396#95—1064no—
397#95—1064no—
398#95—1064no—
399#95—1062no—
400#95—1062no—
401C24C23×D5311yescharacteristic
402C2×Q8—10616no—
403C2×Q8—1068no—
404C2×Q8—1068no—
405C2×Q8—10616no—
406C2×Q8—1068no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—534no—
430C22×C4—532no—
431C22×C4—534no—
432C22×C4—532no—
433C22×C4—532no—
434C22×C4—534no—
435C22×C4—534no—
436C22×C4—534no—
437C22×C4—534no—
438C22×C4—534no—
439C22×C4—534no—
440C22×C4—532no—
441C22×C4—532no—
442C22×C4—532no—
443C22×C4—1062no—
444C22×C4—1062no—
445C22×C4—1062no—
446C22×C4—1062no—
447C22×C4—1062no—
448C22×C4—1062no—
449C22×C4—1062no—
450C22×C4—1062no—
451C22×C4—1061no—
452C22×C4—1062no—
453C22×C4—1061no—
454C22×C4—1062no—
455C22×C4—1062no—
456C22×C4—1062no—
457C22×C4—1062no—
458C22×C4—1061no—
459C22×C4—1062no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4C2×C4×D5312yes—
464C22×C4C2×C4×D5312yes—
465C22×C4C2×C4×D5312yes—
466C22×C4D4×D5314yes—
467C22×C4D4×D5312yes—
468C22×C4#1171994712yes—
469C22×C4#1171994712yes—
470C22×C4Q8×D5314yes—
471C22×C4Q8×D5312yes—
472C4⋊C4—28no—
473C4⋊C4—28no—
474C4⋊C4—10616no—
475C4⋊C4—10616no—
476C4⋊C4—10616no—
477C4⋊C4—1068no—
478C42—1068no—
479C42—10616no—
480C42—1068no—
481C42—10616no—
482C42—2124no—
483C23C22×C4×D5311yescharacteristic
484C23C2×D4×D5312yes—
485C23C2×D4×D5311yescharacteristic
486C23C2×Q8×D5312yes—
487C23C2×Q8×D5311yescharacteristic
488C23#41×D5312yes—
489C23#41×D5312yes—
490C23#41×D5311yescharacteristic
491C23#41×D5311yescharacteristic
492C23C2×#1171994711yescharacteristic
493C23C2×#1171994711yescharacteristic
494Q8—10632no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—534no—
525C2×C4—538no—
526C2×C4—534no—
527C2×C4—534no—
528C2×C4—538no—
529C2×C4—538no—
530C2×C4—538no—
531C2×C4—1064no—
532C2×C4—1064no—
533C2×C4—1064no—
534C2×C4—1064no—
535C2×C4—1064no—
536C2×C4—1064no—
537C2×C4—1064no—
538C2×C4—1064no—
539C2×C4—1064no—
540C2×C4—1064no—
541C2×C4—1064no—
542C2×C4—1064no—
543C2×C4—1062no—
544C2×C4—1062no—
545C2×C4—1064no—
546C2×C4—1062no—
547C2×C4—1062no—
548C2×C4—1062no—
549C2×C4—1064no—
550C2×C4—2121no—
551C2×C4—2122no—
552C2×C4—2121no—
553C2×C4—2122no—
554C2×C4—2122no—
555C2×C4—2122no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4#1313317714yes—
559C2×C4#1313318214yescomplemented
560C22C4×D4×D5312yes—
561C22C2×#1313317711yescharacteristic
562C22C4×Q8×D5312yes—
563C22C2×#1313318211yescharacteristic
564C22#1681109111yescharacteristic
565C22#120×D5311yescharacteristic
566C22C22⋊Q8×D5312yes—
567C22C22⋊Q8×D5311yescharacteristic
568C22#1681113112yes—
569C22#1681113111yescharacteristic
570C22#1681113311yescharacteristic
571C22#1681113312yes—
572C22#1681115411yescharacteristic
573C22#1681115612yes—
574C22C4⋊Q8×D5311yescharacteristic
575C22C4⋊Q8×D5312yes—
576C22#1681116812yes—
577C22#1681117211yescharacteristic
578C22#1681117512yes—
579C22#1681117512yes—
580C22#1681117511yescharacteristic
581C22#1681117912yes—
582C22#1681117912yes—
583C4—24no—
584C4—24no—
585C4—44no—
586C4—42no—
587C4—44no—
588C4—538no—
589C4—1068no—
590C4—1068no—
591C4—2122no—
592C4—2122no—
593C4—2122no—
594C2#2027722011yescharacteristic
595C2#2027724311yescharacteristic
596C2#2027724912yes—
597C2#2027728411yescharacteristic
598C2#2027728612yes—
599C2#395×D5311yescharacteristic
600C2#2027792911yescharacteristic
601C2#2027798811yescharacteristic
602C2#2027799012yes—
603C2#2027800712yes—
604C2#2027800711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×#1441C2
socle, prime radical, cosocleC24×C53C24
hypercenter#1441D53
hypocenterC53C4×#395
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
24. #16810543C4semidirect
75. C2×C4×Dic53Q8semidirect
76. C2×C4×Dic53Q8semidirect
221. C2×C4×C53C4×Q8semidirect
250. C2×Dic53C4×Q8semidirect
291. Dic53#395semidirect
342. C53C4×#395semidirect
559. C2×C4#13133182semidirect

Maximal subgroups

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

subgroupindex
C22×#131331822
C2×#168105432
C2×C4⋊C4×Dic532
C2×#168104742
#202729852
#202729792
#202729782
#202729662
#95×Dic532
C53×#14412
C4×#39553