Finite Group Databasebeta

← #27390349

Subgroups — #27390349

Order 13568.

1092 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810826C211yesmaximal, characteristic
2C2×#16810723C211yesmaximal, characteristic
3C4×#13133215C211yesmaximal, characteristic, complemented
4C4×D4×Dic53C211yesmaximal, characteristic
5#20275636C211yesmaximal, characteristic, complemented
6#20275470C211yesmaximal, characteristic, complemented
7#20275359C211yesmaximal, characteristic, complemented
8#20275355C211yesmaximal, characteristic, complemented
9#20275051C211yesmaximal, characteristic
10#20275047C211yesmaximal, characteristic
11#20274713C211yesmaximal, characteristic
12#20274569C211yesmaximal, characteristic, complemented
13#20274546C211yesmaximal, characteristic, complemented
14#20274545C211yesmaximal, characteristic, complemented
15C2×C53×#440C211yesmaximal, characteristic
16C22×#11719923C2211yescharacteristic
17C2×#13133215C2211yescharacteristic
18C2×C4×Dic106C2211yescharacteristic
19C2×D4×Dic53C2211yescharacteristic
20C2×D4×Dic53C2211yescharacteristic
21C22×C4×Dic53—21no—
22#16810826C418yescomplemented
23#16810723—24no—
24C2×#13133150C2211yescharacteristic
25C2×#13133146C2211yescharacteristic
26C2×#13133146C2211yescharacteristic
27C2×#13133142C2211yescharacteristic
28C2×#13133142C2211yescharacteristic
29C2×#13133141C2211yescharacteristic
30C2×#13133141C2211yescharacteristic
31C2×#13133140C2211yescharacteristic
32C2×#13133138C2211yescharacteristic
33C2×#13133128C2211yescharacteristic
34#16810530C2211yescharacteristic
35#16810528C2211yescharacteristic
36#16810518—21no—
37#16810518—21no—
38#16810492C2211yescharacteristic
39C4⋊C4×Dic53C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C4×#11719919—21no—
42C4×#11719919—21no—
43C4×C4⋊Dic53C2211yescharacteristic
44C22⋊C4×Dic53—21no—
45C22⋊C4×Dic53—21no—
46C4×C22⋊Dic53—21no—
47C42×Dic53C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810377C2211yescharacteristic
53#16810377C2211yescharacteristic
54C22×SD16×C53C2211yescharacteristic
55C2×C4×Q8×C53C2211yescharacteristic
56C2×C4×D4×C53C2211yescharacteristic
57C53×#440—24no—
58C2×#107×C53C2211yescharacteristic
59C2×#103×C53C2211yescharacteristic
60C2×#102×C53C2211yescharacteristic
61C2×#97×C53C2211yescharacteristic
62C22×Dic106C2311yescharacteristic
63C23×Dic53—21no—
64C23×Dic53—21no—
65C2×#11719923—22no—
66C2×#11719923—22no—
67C2×#11719923C2×C414yes—
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70#13133215—22no—
71#13133215C2×C414yescomplemented
72C4×Dic106—22no—
73C4×Dic106—22no—
74D4×Dic53—22no—
75D4×Dic53—22no—
76D4×Dic53—22no—
77D4×Dic53C2×C414yes—
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53—21no—
92C2×C4×Dic53—44no—
93C2×C4×Dic53D411yescharacteristic
94C2×C4×Dic53D411yescharacteristic
95C2×C4×Dic53D411yescharacteristic
96C2×C4×Dic53D411yescharacteristic
97C2×C4×Dic53C2311yescharacteristic
98C2×C4×Dic53C2311yescharacteristic
99#13133150—22no—
100#13133146—22no—
101#13133146C2×C414yescomplemented
102#13133142—22no—
103#13133142C2×C414yescomplemented
104#13133141—22no—
105#13133141C2×C414yescomplemented
106#13133140—22no—
107#13133138—22no—
108#13133128—22no—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112#13133124—21no—
113#13133124—21no—
114#13133124—21no—
115#13133124—21no—
116C22×Q8×C53C2311yescharacteristic
117C22×D4×C53C2311yescharacteristic
118C23×C4×C53—21no—
119C2×SD16×C53—22no—
120C2×SD16×C53—22no—
121C2×SD16×C53C2×C414yes—
122C22×C8×C53C2311yescharacteristic
123C22×C8×C53C2311yescharacteristic
124C4×Q8×C53—22no—
125C4×Q8×C53—22no—
126C4×D4×C53—22no—
127C4×D4×C53D414yes—
128C2×C4⋊C4×C53—21no—
129C2×C4⋊C4×C53C2311yescharacteristic
130C2×C4⋊C4×C53C2311yescharacteristic
131C2×C22⋊C4×C53—21no—
132C2×C42×C53—21no—
133C2×C42×C53C2311yescharacteristic
134#107×C53—22no—
135#103×C53—22no—
136#102×C53—22no—
137#97×C53—22no—
138C24×C53—21no—
139C2×Q8×C53—21no—
140C2×Q8×C53—21no—
141C2×Q8×C53—21no—
142C2×Q8×C53—21no—
143C2×Q8×C53—21no—
144C2×Q8×C53C22×C412yes—
145C2×D4×C53—21no—
146C2×D4×C53—21no—
147C2×D4×C53—21no—
148C2×D4×C53C22×C412yes—
149C2×D4×C53C2×D412yes—
150C2×D4×C53#4112yes—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—24no—
154C22×C4×C53—21no—
155C22×C4×C53—42no—
156C22×C4×C53—42no—
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53C2411yescharacteristic
160C22×Dic53—24no—
161C22×Dic53—21no—
162C22×Dic53—21no—
163C22×Dic53—21no—
164C22×Dic53—24no—
165C22×Dic53—21no—
166C22×Dic53—42no—
167C22×Dic53—42no—
168C22×Dic53—42no—
169C22×Dic53—42no—
170C22×Dic53C2×D411yescharacteristic
171C22×Dic53C2×D411yescharacteristic
172C22×Dic53C2×D411yescharacteristic
173C22×Dic53C2×D411yescharacteristic
174C2×Dic106—21no—
175C2×Dic106—21no—
176C2×Dic106—21no—
177C2×Dic106—21no—
178C2×Dic106—21no—
179C2×Dic106C22×C412yes—
180SD16×C53—28no—
181C2×C8×C53—21no—
182C2×C8×C53—21no—
183C2×C8×C53—21no—
184C2×C8×C53—21no—
185C2×C8×C53—21no—
186C2×C8×C53C22×C412yes—
187C4⋊C4×C53—21no—
188C4⋊C4×C53—21no—
189C4⋊C4×C53—21no—
190C4⋊C4×C53—42no—
191C4⋊C4×C53C2×D412yes—
192C22⋊C4×C53—24no—
193C42×C53—42no—
194C42×C53C2×D412yes—
195C42×C53#4112yes—
196C22⋊Dic53—24no—
197C22⋊Dic53—24no—
198#11719923—28no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53—21no—
203C4⋊Dic53C22×C412yes—
204C4⋊Dic53#4112yes—
205#11719919—24no—
206#11719919—24no—
207#11719919—42no—
208#11719919—42no—
209#11719919—42no—
210C4×Dic53—21no—
211C4×Dic53—21no—
212C4×Dic53—22no—
213C4×Dic53—22no—
214C4×Dic53—22no—
215C4×Dic53—41no—
216C4×Dic53—42no—
217C4×Dic53—42no—
218C4×Dic53—42no—
219C4×Dic53—42no—
220C4×Dic53C22×C412yes—
221C4×Dic53#4112yes—
222C2×#88322—21no—
223C2×#88322—21no—
224C2×#88322—21no—
225C2×#88322—21no—
226C2×#88322—21no—
227C2×#88322C22×C412yes—
228C23×C53—22no—
229C23×C53—22no—
230C23×C53—22no—
231C23×C53—41no—
232C23×C53—41no—
233C23×C53—41no—
234C23×C53—41no—
235C23×C53C22×D411yescharacteristic
236Q8×C53—22no—
237Q8×C53—22no—
238Q8×C53—41no—
239Q8×C53—41no—
240D4×C53—22no—
241D4×C53—22no—
242D4×C53—22no—
243D4×C53C4×D414yes—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—22no—
249C2×C4×C53—22no—
250C2×C4×C53—48no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
258C2×C4×C53#12112yes—
259C2×C4×C53#12112yes—
260C2×C4×C53C23×C411yescharacteristic
261C2×C4×C53C22×D411yescharacteristic
262C2×C4×C53C2×#4111yescharacteristic
263C2×C4×C53C2×#4111yescharacteristic
264C2×C4×C53C2×#4111yescharacteristic
265C2×C4×C53C2×#4111yescharacteristic
266C2×Dic53—21no—
267C2×Dic53—21no—
268C2×Dic53—21no—
269C2×Dic53—21no—
270C2×Dic53—22no—
271C2×Dic53—22no—
272C2×Dic53—22no—
273C2×Dic53—21no—
274C2×Dic53—21no—
275C2×Dic53—21no—
276C2×Dic53—48no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283C2×Dic53—48no—
284C2×Dic53—41no—
285C2×Dic53—41no—
286C2×Dic53—41no—
287C2×Dic53C4×D412yes—
288C2×Dic53C4×D412yes—
289C2×Dic53AGL(1,ℤ/8)12yes—
290C2×Dic53#13712yes—
291C2×Dic53#13712yes—
292Dic106—22no—
293Dic106—22no—
294Dic106—41no—
295Dic106—41no—
296C8×C53—22no—
297C8×C53—41no—
298#88322—22no—
299#88322—41no—
300#21504—531nomaximal, Sylow
301C22×C53—24no—
302C22×C53—42no—
303C22×C53—42no—
304C22×C53—42no—
305C22×C53—42no—
306C22×C53—42no—
307C22×C53—42no—
308C22×C53C2×C4×D411yescharacteristic
309C22×C53C2×#12111yescharacteristic
310C22×C53#52511yescharacteristic
311C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
312C22×C53C2×#13711yescharacteristic
313C22×C53C2×#13711yescharacteristic
314C22×C53#57511yescharacteristic
315C4×C53—22no—
316C4×C53—22no—
317C4×C53—42no—
318C4×C53—42no—
319C4×C53—81no—
320C4×C53C2×C4×D411yescharacteristic
321C4×C53C4×#4111yescharacteristic
322C4×C53#52911yescharacteristic
323C4×C53#53211yescharacteristic
324Dic53—22no—
325Dic53—22no—
326Dic53—42no—
327Dic53—41no—
328Dic53—42no—
329Dic53—42no—
330Dic53—81no—
331Dic53#44014yescomplemented
332C2×#461—531no—
333C2×#440—531no—
334C2×#440D5311yescharacteristic
335C4×C4⋊Q8—531no—
336C42×D4—531no—
337#1981—531no—
338#1947—531no—
339#1928—531no—
340#1928—531no—
341#1877—531no—
342#1877—531no—
343#1802—531no—
344#1767—531no—
345#1763—531no—
346#1763—531no—
347C2×C53—44no—
348C2×C53C4×#12111yescharacteristic
349C2×C53C2×#44011yescharacteristic
350C2×C53C4×#13711yescharacteristic
351C2×C53#303611yescharacteristic
352C2×C53#303911yescharacteristic
353C2×C53#311011yescharacteristic
354C2×C53#311611yescharacteristic
355C22×SD16—531no—
356C22×SD16C2×D5311yescharacteristic
357C2×C4⋊Q8—531no—
358C2×C4×Q8—531no—
359C2×C4×Q8C2×D5311yescharacteristic
360C2×C4×D4—531no—
361C2×C4×D4—531no—
362C2×C4×D4C2×D5311yescharacteristic
363C22×C42—1061no—
364#461—538no—
365#440—24no—
366#440—1064no—
367C2×#107—531no—
368C2×#107C2×D5311yescharacteristic
369C2×C4⋊C8—531no—
370C2×C4⋊C8—531no—
371C2×#103—531no—
372C2×#103—531no—
373C2×#103—531no—
374C2×#103—531no—
375C2×#103—531no—
376C2×#103C2×D5311yescharacteristic
377C2×#102—531no—
378C2×#102C2×D5311yescharacteristic
379C2×#97—531no—
380C2×#97C2×D5311yescharacteristic
381#391—531no—
382#391—531no—
383#389—1061no—
384#389—1061no—
385#384—531no—
386C4×C4⋊C4—531no—
387C4×C4⋊C4—531no—
388C4×C4⋊C4—531no—
389C4×C4⋊C4—1061no—
390C4×C4⋊C4—1061no—
391C4×C22⋊C4—1061no—
392C4×C22⋊C4—1061no—
393C4×C22⋊C4—1061no—
394C43—531no—
395#340—531no—
396#340—531no—
397#340—531no—
398#340—531no—
399#336—531no—
400#336—531no—
401C53#2150411yescharacteristic, complemented, Sylow
402C22×Q8—531no—
403C22×Q8C22×D5311yescharacteristic
404C22×D4C22×D5311yescharacteristic
405C23×C4—21no—
406C23×C4—1061no—
407C23×C4—1061no—
408C2×SD16—22no—
409C2×SD16—22no—
410C2×SD16—534no—
411C2×SD16—1062no—
412C2×SD16—1062no—
413C2×SD16C4×D5314yes—
414C22×C8—531no—
415C22×C8—531no—
416C22×C8C22×D5311yescharacteristic
417C22×C8C22×D5311yescharacteristic
418C4⋊Q8—534no—
419C4⋊Q8—1062no—
420C4×Q8—22no—
421C4×Q8—22no—
422C4×Q8—1062no—
423C4×Q8—1062no—
424C4×D4—22no—
425C4×D4—534no—
426C4×D4—1062no—
427C4×D4—1062no—
428C4×D4—1062no—
429C4×D4C22⋊D5314yes—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—531no—
432C2×C4⋊C4—531no—
433C2×C4⋊C4—531no—
434C2×C4⋊C4—1061no—
435C2×C4⋊C4—1061no—
436C2×C4⋊C4—1061no—
437C2×C4⋊C4—1061no—
438C2×C4⋊C4—1061no—
439C2×C4⋊C4C22×D5311yescharacteristic
440C2×C4⋊C4C22×D5311yescharacteristic
441C2×C22⋊C4—21no—
442C2×C22⋊C4—1061no—
443C2×C22⋊C4—1061no—
444C2×C42—21no—
445C2×C42—531no—
446C2×C42—531no—
447C2×C42—531no—
448C2×C42—531no—
449C2×C42—531no—
450C2×C42—531no—
451C2×C42—1061no—
452C2×C42—1061no—
453C2×C42—1061no—
454C2×C42—1061no—
455C2×C42—2124no—
456C2×C42C22×D5311yescharacteristic
457#107—22no—
458#107—1062no—
459C4⋊C8—534no—
460C4⋊C8—1062no—
461#103—22no—
462#103—534no—
463#103—534no—
464#103—1062no—
465#103—1062no—
466#103—1062no—
467#102—22no—
468#102—1062no—
469#97—22no—
470#97—1062no—
471#95—1061no—
472#95—1061no—
473#95—1061no—
474#95—1061no—
475#95—1061no—
476#95—1061no—
477#95—1061no—
478C24—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—532no—
485C2×Q8—1061no—
486C2×Q8—1061no—
487C2×Q8—1061no—
488C2×Q8—1061no—
489C2×Q8—1061no—
490C2×Q8C2×C4×D5312yes—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4C2×C4×D5312yes—
495C2×D4#1171994512yes—
496C2×D4C2×C22⋊D5312yes—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—24no—
500C22×C4—21no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—531no—
504C22×C4—531no—
505C22×C4—531no—
506C22×C4—531no—
507C22×C4—1064no—
508C22×C4—1064no—
509C22×C4—1061no—
510C22×C4—1061no—
511C22×C4—1061no—
512C22×C4—1061no—
513C22×C4—2122no—
514C22×C4—2122no—
515C22×C4—2122no—
516C22×C4—2122no—
517C22×C4D4×D5311yescharacteristic
518C22×C4D4×D5311yescharacteristic
519C22×C4C23×D5311yescharacteristic
520SD16—28no—
521SD16—1068no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—532no—
528C2×C8—1061no—
529C2×C8—1061no—
530C2×C8—1061no—
531C2×C8—1061no—
532C2×C8—1061no—
533C2×C8C2×C4×D5312yes—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—42no—
538C4⋊C4—532no—
539C4⋊C4—532no—
540C4⋊C4—1064no—
541C4⋊C4—1064no—
542C4⋊C4—1061no—
543C4⋊C4—1061no—
544C4⋊C4—1061no—
545C4⋊C4—1061no—
546C4⋊C4—2122no—
547C4⋊C4—2122no—
548C4⋊C4—2122no—
549C4⋊C4C2×C22⋊D5312yes—
550C22⋊C4—24no—
551C22⋊C4—1064no—
552C22⋊C4—1064no—
553C42—42no—
554C42—532no—
555C42—532no—
556C42—1062no—
557C42—1062no—
558C42—1062no—
559C42—1061no—
560C42—1061no—
561C42—2121no—
562C42—2122no—
563C42—2122no—
564C42—2122no—
565C42—2122no—
566C42#1171994712yes—
567C42C2×C22⋊D5312yes—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23C2×D4×D5311yescharacteristic
576Q8—22no—
577Q8—22no—
578Q8—41no—
579Q8—41no—
580Q8—1062no—
581Q8—1062no—
582Q8—2121no—
583Q8—2121no—
584D4—22no—
585D4—22no—
586D4—22no—
587D4C4×C22⋊D5314yes—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—48no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—532no—
603C2×C4—532no—
604C2×C4—532no—
605C2×C4—532no—
606C2×C4—532no—
607C2×C4—1061no—
608C2×C4—1061no—
609C2×C4—1061no—
610C2×C4—1061no—
611C2×C4—1062no—
612C2×C4—1062no—
613C2×C4—1061no—
614C2×C4—1062no—
615C2×C4—1061no—
616C2×C4—1061no—
617C2×C4—2128no—
618C2×C4—2128no—
619C2×C4—2121no—
620C2×C4—2121no—
621C2×C4—2121no—
622C2×C4—2121no—
623C2×C4—2121no—
624C2×C4—2121no—
625C2×C4—2121no—
626C2×C4—2121no—
627C2×C4—2121no—
628C2×C4#1313318912yes—
629C2×C4#1313318912yes—
630C2×C4C22×C4×D5311yescharacteristic
631C2×C4C22×C22⋊D5311yescharacteristic
632C2×C4#41×D5311yescharacteristic
633C2×C4#41×D5311yescharacteristic
634C2×C4C2×#1171994711yescharacteristic
635C2×C4C2×#1171994511yescharacteristic
636C8—22no—
637C8—41no—
638C8—1062no—
639C8—2121no—
640C22—24no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22C4×D4×D5311yescharacteristic
648C22C2×#1313318911yescharacteristic
649C22#1681113211yescharacteristic
650C22AGL(1,ℤ/8)×D5311yescharacteristic
651C22#137×D5311yescharacteristic
652C22#137×D5311yescharacteristic
653C22#1681141911yescharacteristic
654C4—22no—
655C4—22no—
656C4—42no—
657C4—42no—
658C4—81no—
659C4—534no—
660C4—1062no—
661C4—1062no—
662C4—4241no—
663C4—2122no—
664C4—2122no—
665C4—2121no—
666C4—2122no—
667C4C2×C4×C22⋊D5311yescharacteristic
668C4C4×#1171994711yescharacteristic
669C4#1681115311yescharacteristic
670C4#1681116911yescharacteristic
671C2—44no—
672C2C4×#1313318911yescharacteristic
673C2#440×D5311yescharacteristic
674C2#2028353711yescharacteristic
675C2#2028417611yescharacteristic
676C2#2028420011yescharacteristic
677C2#2028481211yescharacteristic
678C2#2028487311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#440C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
cosocleC22×C4×C53C24
hypercenterC2×#440D53
hypocenterC53#21504
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
3. C4×#13133215C2semidirect
5. #20275636C2semidirect
6. #20275470C2semidirect
7. #20275359C2semidirect
8. #20275355C2semidirect
12. #20274569C2semidirect
13. #20274546C2semidirect
14. #20274545C2semidirect
22. #16810826C4semidirect
71. #13133215C2×C4semidirect
101. #13133146C2×C4semidirect
103. #13133142C2×C4semidirect
105. #13133141C2×C4semidirect
331. Dic53#440semidirect
401. C53#21504semidirect

Maximal subgroups

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

subgroupindex
C2×#168108262
C2×#168107232
C4×#131332152
C4×D4×Dic532
#202756362
#202754702
#202753592
#202753552
#202750512
#202750472
#202747132
#202745692
#202745462
#202745452
C2×C53×#4402
#2150453