Finite Group Databasebeta

← #27384126

Subgroups — #27384126

Order 13568.

1402 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11719947C211yesmaximal, characteristic
2C2×#16810675C211yesmaximal, characteristic, complemented
3#20277120C211yesmaximal, characteristic, complemented
4#20274808C212yesmaximal, complemented
5#20274721C211yesmaximal, characteristic, complemented
6#20274717C211yesmaximal, characteristic, complemented
7#20274715C211yesmaximal, characteristic, complemented
8#20274652C212yesmaximal
9#20274534C212yesmaximal, complemented
10#102×Dic53C212yesmaximal, complemented
11C53×#1802C211yesmaximal, characteristic, complemented
12C22×#11719947C2211yescharacteristic
13C4×#11719947—24no—
14C4×#11719947C418yescomplemented
15C2×C4×Dic106C2211yescharacteristic
16C2×#13133182C2211yescharacteristic
17C2×D4×Dic53C2211yescharacteristic
18C2×C4×C22⋊D53—21no—
19C2×#13133177—21no—
20C2×#13133175—21no—
21C2×#13133175C2211yescharacteristic
22C2×#13133173—21no—
23C22×C4×Dic53—21no—
24C2×C42×D53C2211yescharacteristic
25#16810675—24no—
26C2×C4⋊#88322C2211yescharacteristic
27C2×#13133142C2212yes—
28C2×#13133140C2212yes—
29C2×#13133138C2212yes—
30C2×#13133129C2212yes—
31C2×C8×Dic53C2212yes—
32#16810493C2211yescharacteristic
33#16810488—21no—
34#16810480—21no—
35#16810480C2212yes—
36C4⋊C4×Dic53—21no—
37C4⋊C4×Dic53C2212yes—
38#16810386C2212yes—
39#16810385C2212yes—
40#16810385C2212yes—
41C2×C4×D4×C53C2211yescharacteristic
42C2×C4⋊C8×C53C2211yescharacteristic
43C2×#102×C53C2212yes—
44C53×#384C2211yescharacteristic
45C53×#340C2212yes—
46C2×#11719947—22no—
47C2×#11719947—22no—
48C2×#11719947—22no—
49C2×#11719947—22no—
50C2×#11719947—22no—
51C2×#11719947C2×C414yes—
52C22×Dic106C2311yescharacteristic
53C22×C22⋊D53—21no—
54C23×Dic53—21no—
55C23×Dic53—21no—
56C22×C4×D53D411yescharacteristic
57C22×C4×D53D411yescharacteristic
58C22×C4×D53C2311yescharacteristic
59C22×#88322C2312yes—
60C4×Dic106—22no—
61C4×Dic106C2×C414yes—
62#13133182—22no—
63#13133182C2×C414yes—
64D4×Dic53—22no—
65D4×Dic53C2×C414yes—
66C4×C22⋊D53—28no—
67#13133177—28no—
68#13133175—22no—
69#13133175—44no—
70#13133175C2×C414yes—
71#13133173—28no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53C2312yes—
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×C22⋊Dic53—21no—
79C2×#11719921—21no—
80C2×#11719921—22no—
81C2×C4×Dic53—28no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—22no—
86C2×C4×Dic53D411yescharacteristic
87C2×C4×Dic53D411yescharacteristic
88C2×C4×Dic53C2311yescharacteristic
89C42×D53C2×C414yes—
90C42×D53D414yes—
91C4⋊#88322—22no—
92#13133142—24no—
93#13133140—24no—
94#13133138—24no—
95#13133129—24no—
96C8×Dic53—24no—
97#13133124—22no—
98#13133124—22no—
99C22×D4×C53C2311yescharacteristic
100C23×C4×C53—21no—
101C22×C8×C53C2312yes—
102C4×D4×C53—22no—
103C4×D4×C53C2×C414yes—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53C2312yes—
107C2×C4⋊C4×C53C2311yescharacteristic
108C2×C22⋊C4×C53—21no—
109C2×C42×C53C2311yescharacteristic
110C4⋊C8×C53—22no—
111#102×C53—24no—
112C24×C53—21no—
113C23×D53C2×D411yescharacteristic
114C2×D4×C53—21no—
115C2×D4×C53—21no—
116C2×D4×C53—21no—
117C2×D4×C53C22×C412yes—
118C2×D4×C53#4112yes—
119C2×D4×C53#4112yes—
120C22×C4×C53—21no—
121C22×C4×C53—24no—
122C22×C4×C53—22no—
123C22×C4×C53—42no—
124C22×C4×C53—42no—
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2411yescharacteristic
128C2×C22⋊D53—24no—
129C2×C22⋊D53—24no—
130C2×C22⋊D53—24no—
131C22×Dic53—24no—
132C22×Dic53—24no—
133C22×Dic53—21no—
134C22×Dic53—24no—
135C22×Dic53—21no—
136C22×Dic53—24no—
137C22×Dic53—22no—
138C22×Dic53—42no—
139C22×Dic53—42no—
140C22×Dic53C2×D411yescharacteristic
141C22×Dic53C2×D411yescharacteristic
142C22×Dic53C2×D411yescharacteristic
143#11719947—28no—
144#11719947—28no—
145#11719947—28no—
146#11719947—28no—
147C2×C4×D53—22no—
148C2×C4×D53—22no—
149C2×C4×D53—21no—
150C2×C4×D53—21no—
151C2×C4×D53—21no—
152C2×C4×D53—21no—
153C2×C4×D53—22no—
154C2×C4×D53—22no—
155C2×C4×D53C22⋊C414yes—
156C2×C4×D53C22⋊C414yes—
157C2×C4×D53C22×C412yes—
158C2×C4×D53C2×D412yes—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106—21no—
162C2×Dic106C22×C412yes—
163C2×Dic106#4112yes—
164C2×Dic106#4112yes—
165C2×C8×C53—22no—
166C2×C8×C53—22no—
167C2×C8×C53—22no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—22no—
170C4⋊C4×C53—22no—
171C4⋊C4×C53—42no—
172C4⋊C4×C53—42no—
173C4⋊C4×C53C22×C412yes—
174C22⋊C4×C53—24no—
175C42×C53C22×C412yes—
176C42×C53C2×D412yes—
177C22⋊Dic53—24no—
178#11719921—24no—
179#11719921—44no—
180C4⋊Dic53—21no—
181C4⋊Dic53—22no—
182C4⋊Dic53—22no—
183C4⋊Dic53—42no—
184C4⋊Dic53—42no—
185C4⋊Dic53C22×C412yes—
186#11719919—24no—
187#11719919—24no—
188C4×Dic53—24no—
189C4×Dic53—24no—
190C4×Dic53—24no—
191C4×Dic53—44no—
192C4×Dic53C22⋊C414yes—
193C4×Dic53C22⋊C414yes—
194C4×Dic53C22×C412yes—
195C4×Dic53C2×D412yes—
196C2×#88322—22no—
197C2×#88322—22no—
198C2×#88322—22no—
199C23×C53—22no—
200C23×C53—22no—
201C23×C53—22no—
202C23×C53—41no—
203C23×C53—41no—
204C23×C53—41no—
205C23×C53—41no—
206C23×C53C22×D411yescharacteristic
207C22×D53—21no—
208C22×D53—21no—
209C22×D53—21no—
210C22×D53—21no—
211C22×D53—21no—
212C22×D53—21no—
213C22×D53C2×C22⋊C412yes—
214D4×C53—22no—
215D4×C53—22no—
216D4×C53—22no—
217D4×C53#11714yescomplemented
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—22no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—48no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53C2×C22⋊C412yes—
228C2×C4×C53C2×C22⋊C412yes—
229C2×C4×C53#13512yes—
230C2×C4×C53#13512yes—
231C2×C4×C53C23×C411yescharacteristic
232C2×C4×C53C22×D411yescharacteristic
233C2×C4×C53C2×#4111yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C22⋊D53—216no—
238C2×Dic53—216no—
239C2×Dic53—22no—
240C2×Dic53—22no—
241C2×Dic53—22no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—22no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—48no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53C2×C22⋊C412yes—
254C2×Dic53C2×C22⋊C412yes—
255C2×Dic53C2×C22⋊C412yes—
256C2×Dic53AGL(1,ℤ/8)12yes—
257C2×Dic53#13712yes—
258C4×D53—22no—
259C4×D53—22no—
260C4×D53—22no—
261C4×D53—22no—
262C4×D53—28no—
263C4×D53—28no—
264Dic106—22no—
265Dic106—22no—
266Dic106—22no—
267Dic106#11714yescomplemented
268C8×C53—42no—
269#88322—42no—
270#20737—531nomaximal, Sylow
271C22×C53—24no—
272C22×C53—42no—
273C22×C53—42no—
274C22×C53—42no—
275C22×C53—42no—
276C22×C53—42no—
277C22×C53—42no—
278C22×C53C22×C22⋊C411yescharacteristic
279C22×C53#52511yescharacteristic
280C22×C53#52511yescharacteristic
281C22×C53C2×#13511yescharacteristic
282C22×C53C2×#13511yescharacteristic
283C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C53C2×#13711yescharacteristic
285C2×D53—22no—
286C2×D53—22no—
287C2×D53—22no—
288C2×D53—41no—
289C2×D53—41no—
290C2×D53—41no—
291C2×D53—41no—
292C4×C53—22no—
293C4×C53—42no—
294C4×C53—82no—
295C4×C53#41614yes—
296C4×C53C2×#11711yescharacteristic
297C4×C53C2×#11711yescharacteristic
298C4×C53#52511yescharacteristic
299C4×C53#52511yescharacteristic
300Dic53—22no—
301Dic53—24no—
302Dic53—22no—
303Dic53—42no—
304Dic53—82no—
305Dic53#41714yes—
306C2×C4×#41—531no—
307C2×#431—531no—
308#2285—531no—
309#1825—532no—
310#1803—531no—
311#1803—531no—
312#1802—531no—
313#1802D5311yescharacteristic, complemented
314#1788—532no—
315C4×#103—532no—
316C4×#102—532no—
317C2×C53—44no—
318C2×C53#228011yescharacteristic
319C2×C53C2×#41611yescharacteristic
320C2×C53C2×#41711yescharacteristic
321C2×C53#310611yescharacteristic
322C2×C53#310611yescharacteristic
323C2×C53#310711yescharacteristic
324C2×C53#310711yescharacteristic
325D53—42no—
326C22×#41—531no—
327C4×#41—538no—
328C4×#41—1064no—
329C2×C4×Q8—531no—
330C2×C4×Q8—531no—
331C2×C4×D4—531no—
332C2×C4×D4—1061no—
333C2×C4×D4—1061no—
334C2×C4×D4C2×D5311yescharacteristic
335C2×#117—531no—
336C2×#117—1061no—
337C2×#117—1061no—
338C22×C42—531no—
339C22×C42—1061no—
340#431—1064no—
341C2×C4⋊C8—531no—
342C2×C4⋊C8C2×D5311yescharacteristic
343C2×#103—532no—
344C2×#103—532no—
345C2×#102—532no—
346C2×#102C2×D5312yes—
347C2×C22⋊C8—532no—
348C2×C4×C8—532no—
349#384—531no—
350#384C2×D5311yescharacteristic
351#382—1061no—
352#380—532no—
353#380—1061no—
354C4×C4⋊C4—532no—
355C4×C4⋊C4—1061no—
356#340—532no—
357#340—532no—
358#340—532no—
359#340C2×D5312yes—
360C53#2073711yescharacteristic, complemented, Sylow
361C2×#41—534no—
362C2×#41—1062no—
363C2×#41—1062no—
364C2×#41—1062no—
365C2×#41—1062no—
366C2×#41—1062no—
367C22×Q8—531no—
368C22×D4—1061no—
369C22×D4C22×D5311yescharacteristic
370C23×C4—21no—
371C23×C4—531no—
372C23×C4—531no—
373C23×C4—531no—
374C23×C4—1061no—
375C23×C4—1061no—
376C22×C8—532no—
377C22×C8C22×D5312yes—
378C4×Q8—534no—
379C4×Q8—534no—
380C4×Q8—1062no—
381C4×Q8—1062no—
382C4×D4—22no—
383C4×D4—534no—
384C4×D4—1062no—
385C4×D4—1068no—
386C4×D4—1068no—
387C4×D4C4×D5314yes—
388#117—534no—
389#117—1068no—
390#117—1062no—
391#117—2124no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—532no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4C22×D5312yes—
401C2×C4⋊C4C22×D5311yescharacteristic
402C2×C22⋊C4—21no—
403C2×C22⋊C4—1061no—
404C2×C22⋊C4—1061no—
405C2×C22⋊C4—1062no—
406C2×C42—531no—
407C2×C42—531no—
408C2×C42—531no—
409C2×C42—534no—
410C2×C42—534no—
411C2×C42—1068no—
412C2×C42—1061no—
413C2×C42—1061no—
414C2×C42—1061no—
415C2×C42—1062no—
416C2×C42C22×D5311yescharacteristic
417C4⋊C8—22no—
418C4⋊C8—1062no—
419#103—1064no—
420#103—1064no—
421#102—24no—
422#102—1064no—
423C22⋊C8—1064no—
424C4×C8—1064no—
425#95—1062no—
426#95—1062no—
427C24—21no—
428C24—531no—
429#41—1068no—
430#41—1068no—
431#41—1068no—
432#41—1068no—
433C2×Q8—532no—
434C2×Q8—532no—
435C2×Q8—532no—
436C2×Q8—1061no—
437C2×Q8—1061no—
438C2×Q8—1061no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—1064no—
443C2×D4—1064no—
444C2×D4—1064no—
445C2×D4C2×C4×D5312yes—
446C2×D4#1171994712yes—
447C2×D4#1171994912yes—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—531no—
454C22×C4—531no—
455C22×C4—531no—
456C22×C4—534no—
457C22×C4—534no—
458C22×C4—532no—
459C22×C4—532no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—1064no—
463C22×C4—1064no—
464C22×C4—1064no—
465C22×C4—1064no—
466C22×C4—1061no—
467C22×C4—1062no—
468C22×C4—1062no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4—1061no—
473C22×C4—1061no—
474C22×C4—1062no—
475C22×C4—2122no—
476C22×C4—2122no—
477C22×C4D4×D5311yescharacteristic
478C22×C4D4×D5311yescharacteristic
479C22×C4C23×D5311yescharacteristic
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—1062no—
484C2×C8—1062no—
485C2×C8—1062no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—532no—
492C4⋊C4—1064no—
493C4⋊C4—1064no—
494C4⋊C4—1061no—
495C4⋊C4—1062no—
496C4⋊C4—1062no—
497C4⋊C4—2122no—
498C4⋊C4—2122no—
499C4⋊C4C2×C4×D5312yes—
500C22⋊C4—24no—
501C22⋊C4—1064no—
502C22⋊C4—1064no—
503C22⋊C4—2124no—
504C42—534no—
505C42—534no—
506C42—532no—
507C42—532no—
508C42—1064no—
509C42—1064no—
510C42—1064no—
511C42—2124no—
512C42C2×C4×D5312yes—
513C42D4×D5312yes—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—532no—
522C23—1061no—
523C23—1061no—
524C23—1061no—
525C23—1061no—
526C23—1061no—
527C23—1061no—
528C23C2×D4×D5311yescharacteristic
529Q8—534no—
530Q8—1062no—
531Q8—1062no—
532Q8—1062no—
533D4—22no—
534D4—22no—
535D4—22no—
536D4—10616no—
537D4#1313317514yescomplemented
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—48no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—532no—
548C2×C4—532no—
549C2×C4—532no—
550C2×C4—532no—
551C2×C4—532no—
552C2×C4—1068no—
553C2×C4—10616no—
554C2×C4—1062no—
555C2×C4—1061no—
556C2×C4—1062no—
557C2×C4—1061no—
558C2×C4—1062no—
559C2×C4—1061no—
560C2×C4—1061no—
561C2×C4—1062no—
562C2×C4—1062no—
563C2×C4—1068no—
564C2×C4—1062no—
565C2×C4—1062no—
566C2×C4—1062no—
567C2×C4—2128no—
568C2×C4—2122no—
569C2×C4—2122no—
570C2×C4—2122no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4C22⋊C4×D5312yes—
574C2×C4C22⋊C4×D5312yes—
575C2×C4#1313323612yes—
576C2×C4#1313323812yes—
577C2×C4C22×C4×D5311yescharacteristic
578C2×C4C2×D4×D5311yescharacteristic
579C2×C4#41×D5311yescharacteristic
580C2×C4#41×D5311yescharacteristic
581C2×C4C2×#1171994711yescharacteristic
582C2×C4C2×#1171994911yescharacteristic
583C8—42no—
584C8—2122no—
585C22—24no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—1062no—
593C22—1062no—
594C22—1062no—
595C22—2121no—
596C22—2121no—
597C22—2121no—
598C22—2121no—
599C22C2×C22⋊C4×D5311yescharacteristic
600C22#1681113311yescharacteristic
601C22#1681113611yescharacteristic
602C22C2×#1313323611yescharacteristic
603C22C2×#1313323811yescharacteristic
604C22AGL(1,ℤ/8)×D5311yescharacteristic
605C22#137×D5311yescharacteristic
606C4—22no—
607C4—42no—
608C4—82no—
609C4—534no—
610C4—1062no—
611C4—1062no—
612C4—1064no—
613C4—4242no—
614C4—2122no—
615C4#1681061614yes—
616C4#117×D5311yescharacteristic
617C4C2×#1313317511yescharacteristic
618C4#1681113311yescharacteristic
619C4#1681113611yescharacteristic
620C2—44no—
621C2—2122no—
622C2#2027709711yescharacteristic
623C2C2×#1681061611yescharacteristic
624C2#417×D5311yescharacteristic
625C2#2028477211yescharacteristic
626C2#2028477711yescharacteristic
627C2#2028478311yescharacteristic
628C2#2028478811yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1802C2
socleC23×C53C22×D4
prime radicalC22×#11719947C22
cosocleC22×C4×C53C24
hypercenter#1802D53
hypocenterC53#20737
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. C2×#16810675C2semidirect
3. #20277120C2semidirect
4. #20274808C2semidirect
5. #20274721C2semidirect
6. #20274717C2semidirect
7. #20274715C2semidirect
9. #20274534C2semidirect
10. #102×Dic53C2semidirect
11. C53×#1802C2semidirect
14. C4×#11719947C4semidirect
217. D4×C53#117semidirect
267. Dic106#117semidirect
313. #1802D53semidirect
360. C53#20737semidirect
537. D4#13133175semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117199472
C2×#168106752
#202771202
#202748082
#202747212
#202747172
#202747152
#202746522
#202745342
#102×Dic532
C53×#18022
#2073753