Finite Group Databasebeta

← #27406397

Subgroups — #27406397

Order 13568.

958 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810842C211yesmaximal, characteristic
2C2×#16810715C211yesmaximal, characteristic, complemented
3C2×#16810714C211yesmaximal, characteristic
4#20277790C211yesmaximal, characteristic
5#20277632C211yesmaximal, characteristic, complemented
6#20275481C211yesmaximal, characteristic, complemented
7#20275395C211yesmaximal, characteristic
8#20275388C211yesmaximal, characteristic, complemented
9#20275067C211yesmaximal, characteristic
10#20275053C211yesmaximal, characteristic, complemented
11#20274981C211yesmaximal, characteristic, complemented
12#20274960C211yesmaximal, characteristic
13C8×C4⋊Dic53C211yesmaximal, characteristic, complemented
14C4×#13133142C211yesmaximal, characteristic, complemented
15C53×#1958C211yesmaximal, characteristic
16C22×#11719925C2211yescharacteristic
17C22×#11719923C2211yescharacteristic
18C2×Q8×Dic53C2211yescharacteristic
19C2×C4×Dic106C2211yescharacteristic
20C2×#13133182C2211yescharacteristic
21C2×#13133182C2211yescharacteristic
22C2×D4×Dic53C2211yescharacteristic
23#16810842C418yescomplemented
24#16810715—24no—
25#16810714—24no—
26C2×#13133150C2211yescharacteristic
27C2×#13133148C2211yescharacteristic
28C2×C4⋊#88322C2211yescharacteristic
29C2×#13133142C2211yescharacteristic
30C2×#13133142C2211yescharacteristic
31C2×#13133141C2211yescharacteristic
32C2×#13133138C2211yescharacteristic
33C2×C8×Dic53C2211yescharacteristic
34C2×C8×Dic53C2211yescharacteristic
35#16810530C2211yescharacteristic
36#16810526C2211yescharacteristic
37#16810520—21no—
38#16810520—21no—
39#16810519C2211yescharacteristic
40#16810519C2211yescharacteristic
41#16810516—21no—
42#16810516—21no—
43#16810498—21no—
44#16810487—21no—
45C4×C4⋊Dic53C2211yescharacteristic
46C4×C4⋊Dic53C2211yescharacteristic
47#16810474—21no—
48C22⋊C4×Dic53—21no—
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53#16810377C2211yescharacteristic
54#16810377C2211yescharacteristic
55C2×#124×C53C2211yescharacteristic
56C2×#103×C53C2211yescharacteristic
57C2×#103×C53C2211yescharacteristic
58C2×#102×C53C2211yescharacteristic
59C2×#102×C53C2211yescharacteristic
60C2×C4×C8×C53C2211yescharacteristic
61C53×#382C2211yescharacteristic
62C22×Dic106C2311yescharacteristic
63C23×Dic53—21no—
64C2×#11719925—22no—
65C2×#11719925—22no—
66C2×#11719925C2×C414yescomplemented
67C2×#11719923—22no—
68C2×#11719923—22no—
69C2×#11719923C2×C414yes—
70C22×#88322C2311yescharacteristic
71C22×#88322C2311yescharacteristic
72Q8×Dic53—22no—
73Q8×Dic53—41no—
74C4×Dic106—22no—
75C4×Dic106C2×C414yescomplemented
76#13133182—22no—
77#13133182—22no—
78#13133182—41no—
79#13133182—41no—
80D4×Dic53—22no—
81D4×Dic53—41no—
82C2×C4⋊Dic53—21no—
83C2×C4⋊Dic53D411yescharacteristic
84C2×C4⋊Dic53D411yescharacteristic
85C2×C4⋊Dic53C2311yescharacteristic
86C2×C4⋊Dic53C2311yescharacteristic
87C2×C4⋊Dic53C2311yescharacteristic
88C2×#11719919—21no—
89C2×#11719919—21no—
90C2×#11719919—21no—
91C2×#11719919—21no—
92C2×#11719919—21no—
93C2×C22⋊Dic53—21no—
94C2×C22⋊Dic53—21no—
95C2×C22⋊Dic53—21no—
96C2×C4×Dic53—21no—
97C2×C4×Dic53—21no—
98C2×C4×Dic53—21no—
99C2×C4×Dic53—21no—
100C2×C4×Dic53—21no—
101C2×C4×Dic53C2311yescharacteristic
102C2×C4×Dic53C2311yescharacteristic
103#13133150—22no—
104#13133148—22no—
105C4⋊#88322C2×C414yescomplemented
106#13133142—22no—
107#13133142D414yes—
108#13133141C2×C414yescomplemented
109#13133138C2×C414yes—
110C8×Dic53—22no—
111C8×Dic53—22no—
112#13133124—21no—
113#13133124—21no—
114#13133124—21no—
115#13133124—21no—
116#13133124—21no—
117#13133124D411yescharacteristic
118#13133124D411yescharacteristic
119C22×Q8×C53C2311yescharacteristic
120C22×D4×C53C2311yescharacteristic
121C22×C8×C53C2311yescharacteristic
122C22×C8×C53C2311yescharacteristic
123#124×C53—22no—
124#124×C53C2×C414yes—
125C2×C4⋊C4×C53C2311yescharacteristic
126C2×C4⋊C4×C53C2311yescharacteristic
127C2×C22⋊C4×C53—21no—
128C2×C22⋊C4×C53—21no—
129C2×C42×C53C2311yescharacteristic
130#103×C53—22no—
131#103×C53—41no—
132#102×C53—22no—
133#102×C53—41no—
134C4×C8×C53—22no—
135#95×C53—21no—
136#95×C53—21no—
137C24×C53—21no—
138C2×Q8×C53—21no—
139C2×Q8×C53—21no—
140C2×Q8×C53—21no—
141C2×Q8×C53—21no—
142C2×Q8×C53—21no—
143C2×Q8×C53C22×C412yes—
144C2×D4×C53—21no—
145C2×D4×C53—21no—
146C2×D4×C53—21no—
147C2×D4×C53—21no—
148C2×D4×C53—21no—
149C2×D4×C53C22×C412yes—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2411yescharacteristic
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—42no—
161C22×Dic53—42no—
162C22×Dic53—81no—
163C22×Dic53C2×D411yescharacteristic
164C22×Dic53C2×D411yescharacteristic
165C22×Dic53C2×D411yescharacteristic
166C22×Dic53C2×D411yescharacteristic
167C2×Dic106—21no—
168C2×Dic106—21no—
169C2×Dic106—21no—
170C2×Dic106C22×C412yes—
171C2×Dic106C2×D412yes—
172C2×Dic106#4112yes—
173C2×C8×C53—21no—
174C2×C8×C53—21no—
175C2×C8×C53—21no—
176C2×C8×C53—21no—
177C2×C8×C53—21no—
178C2×C8×C53C2×D412yes—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—21no—
183C22⋊C4×C53—24no—
184C22⋊C4×C53—24no—
185C42×C53C22×C412yes—
186C42×C53#4112yes—
187C22⋊Dic53—42no—
188C22⋊Dic53—42no—
189C22⋊Dic53—81no—
190#11719925—28no—
191#11719923—28no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—22no—
195C4⋊Dic53—22no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—42no—
199C4⋊Dic53C22×C412yes—
200C4⋊Dic53C2×D412yes—
201#11719919—24no—
202#11719919—42no—
203#11719919—42no—
204#11719919—42no—
205#11719919—42no—
206C4×Dic53—24no—
207C4×Dic53—21no—
208C4×Dic53—21no—
209C4×Dic53—21no—
210C4×Dic53—21no—
211C4×Dic53—42no—
212C4×Dic53—42no—
213C4×Dic53—42no—
214C4×Dic53—81no—
215C2×#88322—21no—
216C2×#88322—21no—
217C2×#88322—21no—
218C2×#88322—21no—
219C2×#88322C22×C412yes—
220C2×#88322C22×C412yes—
221C23×C53—22no—
222C23×C53—41no—
223C23×C53—41no—
224C23×C53—41no—
225C23×C53—41no—
226C23×C53—41no—
227C23×C53—41no—
228C23×C53C22×D411yescharacteristic
229Q8×C53—22no—
230Q8×C53—22no—
231Q8×C53—41no—
232Q8×C53—41no—
233D4×C53—22no—
234D4×C53—22no—
235D4×C53—41no—
236D4×C53—41no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—22no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53C4×D412yes—
251C2×C4×C53C4×D412yes—
252C2×C4×C53C23×C411yescharacteristic
253C2×C4×C53C22×D411yescharacteristic
254C2×C4×C53C2×#4111yescharacteristic
255C2×C4×C53C2×#4111yescharacteristic
256C2×C4×C53H(5,𝔽2)11yescharacteristic
257C2×C4×C532−1+411yescharacteristic
258C2×Dic53—22no—
259C2×Dic53—22no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—22no—
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—22no—
266C2×Dic53—22no—
267C2×Dic53—21no—
268C2×Dic53—21no—
269C2×Dic53—21no—
270C2×Dic53—21no—
271C2×Dic53—21no—
272C2×Dic53—21no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—84no—
281C2×Dic53#13512yes—
282C2×Dic53#13512yes—
283Dic106—22no—
284Dic106—22no—
285Dic106—22no—
286Dic106C4×D414yes—
287C8×C53—22no—
288C8×C53—41no—
289#88322—22no—
290#88322—22no—
291#23000—531nomaximal, Sylow
292C22×C53—42no—
293C22×C53—42no—
294C22×C53—42no—
295C22×C53—81no—
296C22×C53—81no—
297C22×C53—81no—
298C22×C53—81no—
299C22×C53C2×C4×D411yescharacteristic
300C22×C53D4211yescharacteristic
301C22×C53#54811yescharacteristic
302C22×C53C2×#13511yescharacteristic
303C22×C53C2×#13511yescharacteristic
304C22×C53#57711yescharacteristic
305C22×C53#57811yescharacteristic
306C4×C53—22no—
307C4×C53—22no—
308C4×C53—42no—
309C4×C53—81no—
310C4×C53—81no—
311C4×C53C2×C4×D411yescharacteristic
312C4×C53#52011yescharacteristic
313C4×C53#54711yescharacteristic
314C4×C53#55311yescharacteristic
315Dic53—24no—
316Dic53—22no—
317Dic53—22no—
318Dic53—42no—
319Dic53—41no—
320Dic53—42no—
321Dic53—41no—
322Dic53—81no—
323C2×#464—531no—
324C2×C4×Q16—531no—
325C2×C4×SD16—531no—
326#2367—531no—
327#2351—531no—
328#1958D5311yescharacteristic
329#1949—531no—
330#1934—531no—
331#1933—531no—
332#1880—531no—
333#1878—531no—
334#1865—531no—
335#1862—531no—
336C8×C4⋊C4—531no—
337C4×#103—531no—
338C2×C53—82no—
339C2×C53#236111yescharacteristic
340C2×C53C4×#13511yescharacteristic
341C2×C53#294711yescharacteristic
342C2×C53#328211yescharacteristic
343C2×C53#328511yescharacteristic
344C2×C53#336411yescharacteristic
345C2×C53#336511yescharacteristic
346C22×Q16—531no—
347C22×SD16—531no—
348C2×#124C2×D5311yescharacteristic
349C2×C4×Q8—531no—
350C2×C4×Q8—531no—
351C2×C4×Q8—531no—
352C2×C4×Q8—531no—
353C2×C4×D4—531no—
354#464—538no—
355C4×Q16—1064no—
356C4×SD16—1064no—
357C2×#107—531no—
358C2×#106—531no—
359C2×C4⋊C8—531no—
360C2×#103—531no—
361C2×#103—531no—
362C2×#103—531no—
363C2×#103C2×D5311yescharacteristic
364C2×#103C2×D5311yescharacteristic
365C2×#102—531no—
366C2×#102C2×D5311yescharacteristic
367C2×#102C2×D5311yescharacteristic
368C2×C4×C8—531no—
369C2×C4×C8—531no—
370C2×C4×C8C2×D5311yescharacteristic
371#391—531no—
372#390—531no—
373#389—531no—
374#389—531no—
375#389—1061no—
376#389—1061no—
377#388—1061no—
378#388—1061no—
379#385—1061no—
380#382—1061no—
381#382C2×D5311yescharacteristic
382C4×C4⋊C4—531no—
383C4×C4⋊C4—531no—
384C4×C4⋊C4—1061no—
385C4×C22⋊C4—1061no—
386#340—531no—
387#340—531no—
388#340—531no—
389#340—531no—
390#336—531no—
391#336—531no—
392C53#2300011yescharacteristic, complemented, Sylow
393C22×Q8—531no—
394C22×Q8C22×D5311yescharacteristic
395C22×D4C22×D5311yescharacteristic
396C23×C4—1061no—
397C2×Q16—534no—
398C2×Q16—1062no—
399C2×Q16—1062no—
400C2×SD16—534no—
401C2×SD16—1062no—
402C2×SD16—1062no—
403C22×C8—531no—
404C22×C8—531no—
405C22×C8C22×D5311yescharacteristic
406C22×C8C22×D5311yescharacteristic
407#124—22no—
408#124C4×D5314yes—
409C4×Q8—534no—
410C4×Q8—1062no—
411C4×Q8—1062no—
412C4×Q8—1062no—
413C4×Q8—1062no—
414C4×Q8—2121no—
415C4×Q8—2121no—
416C4×Q8—2121no—
417C4×D4—1062no—
418C4×D4—2121no—
419C2×C4⋊C4—531no—
420C2×C4⋊C4—531no—
421C2×C4⋊C4—531no—
422C2×C4⋊C4—531no—
423C2×C4⋊C4—531no—
424C2×C4⋊C4—1061no—
425C2×C4⋊C4—1061no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4—1061no—
428C2×C4⋊C4—1061no—
429C2×C4⋊C4—1061no—
430C2×C4⋊C4C22×D5311yescharacteristic
431C2×C4⋊C4C22×D5311yescharacteristic
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—1061no—
435C2×C22⋊C4—1061no—
436C2×C22⋊C4—1061no—
437C2×C42—531no—
438C2×C42—531no—
439C2×C42—1061no—
440C2×C42—1061no—
441C2×C42—1061no—
442C2×C42—1061no—
443C2×C42—1061no—
444C2×C42C22×D5311yescharacteristic
445#107—1062no—
446#106—1062no—
447C4⋊C8—534no—
448#103—22no—
449#103—41no—
450#103—534no—
451#103—534no—
452#103—1062no—
453#102—22no—
454#102—41no—
455#102—534no—
456C4×C8—22no—
457C4×C8—1062no—
458C4×C8—1062no—
459#95—21no—
460#95—21no—
461#95—531no—
462#95—531no—
463#95—1061no—
464#95—1061no—
465#95—1061no—
466#95—1061no—
467#95—1061no—
468C24—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—532no—
475C2×Q8—532no—
476C2×Q8—532no—
477C2×Q8—1061no—
478C2×Q8—1061no—
479C2×Q8—1061no—
480C2×Q8C2×C4×D5312yes—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4C2×C4×D5312yes—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—531no—
491C22×C4—531no—
492C22×C4—531no—
493C22×C4—531no—
494C22×C4—1061no—
495C22×C4—1061no—
496C22×C4—1061no—
497C22×C4—1061no—
498C22×C4—4241no—
499C22×C4—2122no—
500C22×C4—2122no—
501C22×C4D4×D5311yescharacteristic
502C22×C4D4×D5311yescharacteristic
503C22×C4C23×D5311yescharacteristic
504Q16—1068no—
505SD16—1068no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—532no—
512C2×C8—532no—
513C2×C8—1061no—
514C2×C8—1061no—
515C2×C8—1061no—
516C2×C8—1061no—
517C2×C8D4×D5312yes—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—532no—
523C4⋊C4—532no—
524C4⋊C4—1062no—
525C4⋊C4—1062no—
526C4⋊C4—1064no—
527C4⋊C4—1061no—
528C4⋊C4—1061no—
529C4⋊C4—1061no—
530C4⋊C4—1061no—
531C4⋊C4—2122no—
532C4⋊C4—2122no—
533C4⋊C4—2122no—
534C4⋊C4—2122no—
535C4⋊C4—2122no—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—4241no—
539C22⋊C4—2122no—
540C22⋊C4—2122no—
541C42—1064no—
542C42—1061no—
543C42—1061no—
544C42—1061no—
545C42—1061no—
546C42—4241no—
547C42—2122no—
548C42—2122no—
549C42—2122no—
550C42C2×C4×D5312yes—
551C42#1171994712yes—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23C2×D4×D5311yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—41no—
563Q8—41no—
564Q8—534no—
565Q8—1062no—
566Q8—1062no—
567Q8—1062no—
568D4—22no—
569D4—22no—
570D4—41no—
571D4—41no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—532no—
586C2×C4—532no—
587C2×C4—1062no—
588C2×C4—1062no—
589C2×C4—1061no—
590C2×C4—1061no—
591C2×C4—1061no—
592C2×C4—1061no—
593C2×C4—1062no—
594C2×C4—1062no—
595C2×C4—1061no—
596C2×C4—1061no—
597C2×C4—1061no—
598C2×C4—1061no—
599C2×C4—1061no—
600C2×C4—1061no—
601C2×C4—1062no—
602C2×C4—4244no—
603C2×C4—2121no—
604C2×C4—2121no—
605C2×C4—2121no—
606C2×C4—2121no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4#1313317712yes—
611C2×C4#1313317712yes—
612C2×C4C22×C4×D5311yescharacteristic
613C2×C4C2×D4×D5311yescharacteristic
614C2×C4#41×D5311yescharacteristic
615C2×C4C2×#1171994711yescharacteristic
616C2×C4#1313326211yescharacteristic
617C2×C4#1313326411yescharacteristic
618C8—22no—
619C8—41no—
620C8—1062no—
621C8—1062no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22C2×#1313317711yescharacteristic
630C22#1681125811yescharacteristic
631C22#1681128111yescharacteristic
632C22#135×D5311yescharacteristic
633C22#135×D5311yescharacteristic
634C22#1681143011yescharacteristic
635C22#1681143711yescharacteristic
636C4—22no—
637C4—22no—
638C4—42no—
639C4—81no—
640C4—81no—
641C4—1062no—
642C4—1064no—
643C4—1062no—
644C4—4241no—
645C4—2122no—
646C4—2121no—
647C4—2122no—
648C4—2121no—
649C4C4×D4×D5311yescharacteristic
650C4#1681110211yescharacteristic
651C4#1681127311yescharacteristic
652C4#1681132011yescharacteristic
653C2—82no—
654C2#2027773611yescharacteristic
655C2#2028348311yescharacteristic
656C2#2028356111yescharacteristic
657C2#2028642111yescharacteristic
658C2#2028645511yescharacteristic
659C2#2028739711yescharacteristic
660C2#2028740811yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1958C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
cosocleC22×C4×C53C24
hypercenter#1958D53
hypocenterC53#23000
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1958

Complements

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

NH ≅ G/Ntype
2. C2×#16810715C2semidirect
5. #20277632C2semidirect
6. #20275481C2semidirect
8. #20275388C2semidirect
10. #20275053C2semidirect
11. #20274981C2semidirect
13. C8×C4⋊Dic53C2semidirect
14. C4×#13133142C2semidirect
23. #16810842C4semidirect
66. C2×#11719925C2×C4semidirect
75. C4×Dic106C2×C4semidirect
105. C4⋊#88322C2×C4semidirect
108. #13133141C2×C4semidirect
392. C53#23000semidirect

Maximal subgroups

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

subgroupindex
C2×#168108422
C2×#168107152
C2×#168107142
#202777902
#202776322
#202754812
#202753952
#202753882
#202750672
#202750532
#202749812
#202749602
C8×C4⋊Dic532
C4×#131331422
C53×#19582
#2300053