Finite Group Databasebeta

← #27401430

Subgroups — #27401430

Order 13568.

934 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810632C211yesmaximal, characteristic
2#20277602C211yesmaximal, characteristic
3Q8×C22⋊Dic53C211yesmaximal, characteristic
4#20275402C211yesmaximal, characteristic, complemented
5#20275388C211yesmaximal, characteristic, complemented
6#20275359C211yesmaximal, characteristic, complemented
7#20275352C211yesmaximal, characteristic, complemented
8#20274981C211yesmaximal, characteristic, complemented
9#20274967C211yesmaximal, characteristic, complemented
10#20274811C211yesmaximal, characteristic
11#20274746C211yesmaximal, characteristic
12#20274547C211yesmaximal, characteristic, complemented
13#103×Dic53C211yesmaximal, characteristic, complemented
14#20274501C211yesmaximal, characteristic
15C53×#1787C211yesmaximal, characteristic
16C22×#11719917C2211yescharacteristic
17C2×Q8×Dic53C2211yescharacteristic
18C2×Q8×Dic53C2211yescharacteristic
19C2×#13133176C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22#16810632C418yescomplemented
23C2×#13133141C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133127C2211yescharacteristic
28C2×C8×Dic53C2211yescharacteristic
29#16810530—21no—
30#16810530—21no—
31#16810530C2211yescharacteristic
32#16810530C2211yescharacteristic
33#16810519C2211yescharacteristic
34#16810519C2211yescharacteristic
35#16810517—21no—
36#16810517—21no—
37#16810487—21no—
38#16810483—21no—
39#16810483C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
42C4×C22⋊Dic53—21no—
43C4×C22⋊Dic53C2211yescharacteristic
44#16810468—21no—
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C23×Q8×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C2×#103×C53C2211yescharacteristic
57C2×#103×C53C2211yescharacteristic
58C2×C22⋊C8×C53C2211yescharacteristic
59C53×#380C2211yescharacteristic
60C2×#11719917—41no—
61C2×#11719917—41no—
62C2×#11719917C2×C414yes—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65Q8×Dic53—22no—
66Q8×Dic53—22no—
67Q8×Dic53—41no—
68Q8×Dic53—41no—
69#13133176—22no—
70#13133176C2×C414yes—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53D411yescharacteristic
82C2×C22⋊Dic53D411yescharacteristic
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53C2311yescharacteristic
89C2×C4×Dic53C2311yescharacteristic
90#13133150—41no—
91#13133148—41no—
92#13133141C2×C414yes—
93#13133141C2×C414yes—
94#13133141C2×C414yes—
95#13133141C2×C414yes—
96#13133127—41no—
97C8×Dic53—41no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124D411yescharacteristic
108#13133124D411yescharacteristic
109C22×Q8×C53—21no—
110C22×Q8×C53—22no—
111C22×Q8×C53—22no—
112C22×Q8×C53—22no—
113C22×Q8×C53—41no—
114C22×Q8×C53—41no—
115C22×Q8×C53C2×C414yes—
116C22×Q8×C53C2311yescharacteristic
117C22×Q8×C53C2311yescharacteristic
118C23×C4×C53—21no—
119C23×C4×C53C2311yescharacteristic
120C22×C8×C53C2311yescharacteristic
121C22×C8×C53C2311yescharacteristic
122C2×C4⋊C4×C53C2311yescharacteristic
123C2×C4⋊C4×C53C2311yescharacteristic
124C2×C22⋊C4×C53—21no—
125#103×C53—22no—
126#103×C53—22no—
127#103×C53—41no—
128#103×C53—41no—
129C22⋊C8×C53—22no—
130#95×C53—21no—
131C24×C53C2×D411yescharacteristic
132C2×Q8×C53—24no—
133C2×Q8×C53—21no—
134C2×Q8×C53—21no—
135C2×Q8×C53—21no—
136C2×Q8×C53—21no—
137C2×Q8×C53—28no—
138C2×Q8×C53—21no—
139C2×Q8×C53—21no—
140C2×Q8×C53—21no—
141C2×Q8×C53—21no—
142C2×Q8×C53—42no—
143C2×Q8×C53—42no—
144C2×Q8×C53—44no—
145C2×Q8×C53—44no—
146C2×Q8×C53—44no—
147C2×Q8×C53C22×C412yes—
148C2×Q8×C53C22×C412yes—
149C2×Q8×C53#4112yes—
150C2×Q8×C53#4112yes—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53—21no—
155C22×C4×C53—21no—
156C22×C4×C53—24no—
157C22×C4×C53—21no—
158C22×C4×C53—21no—
159C22×C4×C53—21no—
160C22×C4×C53—42no—
161C22×C4×C53—42no—
162C22×C4×C53C22×C412yes—
163C22×C4×C53C2×D411yescharacteristic
164C22×C4×C53#4112yes—
165C22×C4×C53C2411yescharacteristic
166C22×Dic53—21no—
167C22×Dic53—21no—
168C22×Dic53—21no—
169C22×Dic53—21no—
170C22×Dic53C2×D411yescharacteristic
171C22×Dic53C2×D411yescharacteristic
172C22×Dic53C2×D411yescharacteristic
173C22×Dic53C2×D411yescharacteristic
174C2×C8×C53—21no—
175C2×C8×C53—21no—
176C2×C8×C53—21no—
177C2×C8×C53—21no—
178C2×C8×C53—21no—
179C2×C8×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53—21no—
184C22⋊C4×C53—81no—
185C22⋊Dic53—24no—
186C22⋊Dic53—22no—
187C22⋊Dic53—41no—
188C22⋊Dic53—42no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—42no—
196C4⋊Dic53—42no—
197C4⋊Dic53C22×C412yes—
198C4⋊Dic53C22×C412yes—
199#11719919—81no—
200#11719919—81no—
201C4×Dic53—24no—
202C4×Dic53—21no—
203C4×Dic53—21no—
204C4×Dic53—21no—
205C4×Dic53—21no—
206C4×Dic53—42no—
207C4×Dic53—42no—
208C4×Dic53—42no—
209C4×Dic53—42no—
210#11719917—44no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—21no—
215C2×#88322C22×C412yes—
216C2×#88322C22×C412yes—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53—21no—
222C23×C53—21no—
223C23×C53C4×D412yes—
224C23×C53C22×D411yescharacteristic
225Q8×C53—22no—
226Q8×C53—22no—
227Q8×C53—22no—
228Q8×C53—22no—
229Q8×C53—22no—
230Q8×C53—48no—
231Q8×C53—41no—
232Q8×C53—41no—
233Q8×C53#11714yes—
234C2×C4×C53—22no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
240C2×C4×C53—22no—
241C2×C4×C53—22no—
242C2×C4×C53—22no—
243C2×C4×C53—22no—
244C2×C4×C53—48no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53C4×D412yes—
254C2×C4×C53C23×C411yescharacteristic
255C2×C4×C53C2×#4111yescharacteristic
256C2×C4×C53C2×#4111yescharacteristic
257C2×C4×C53C2×#4111yescharacteristic
258C2×C4×C53H(5,𝔽2)11yescharacteristic
259C2×C4×C53H(5,𝔽2)11yescharacteristic
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—22no—
263C2×Dic53—22no—
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—21no—
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—41no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283C2×Dic53#13712yes—
284C8×C53—41no—
285C8×C53—41no—
286#88322—22no—
287#88322—22no—
288#22585—531nomaximal, Sylow
289C22×C53—22no—
290C22×C53—22no—
291C22×C53—22no—
292C22×C53—41no—
293C22×C53—41no—
294C22×C53—41no—
295C22×C53—41no—
296C22×C53C2×C4×D411yescharacteristic
297C22×C53#54611yescharacteristic
298C22×C53#54611yescharacteristic
299C22×C53C2×#13711yescharacteristic
300C22×C53#57511yescharacteristic
301C22×C53#57711yescharacteristic
302C22×C53#57811yescharacteristic
303C4×C53—22no—
304C4×C53—24no—
305C4×C53—42no—
306C4×C53—81no—
307C4×C53—81no—
308C4×C53C2×#11711yescharacteristic
309C4×C53#51811yescharacteristic
310C4×C53#55111yescharacteristic
311C4×C53#55111yescharacteristic
312Dic53—22no—
313Dic53—41no—
314Dic53—42no—
315Dic53—42no—
316Dic53—41no—
317Dic53—41no—
318Dic53—81no—
319Dic53—81no—
320C2×#419—531no—
321#2346—531no—
322Q8×C22⋊C4—531no—
323#1935—531no—
324#1933—531no—
325#1928—531no—
326#1927—531no—
327#1865—531no—
328#1863—531no—
329#1825—531no—
330#1809—531no—
331#1787D5311yescharacteristic
332#1763—531no—
333C4×#103—531no—
334#1753—531no—
335C2×C53—42no—
336C2×C53#234211yescharacteristic
337C2×C53C4×#13711yescharacteristic
338C2×C53#294711yescharacteristic
339C2×C53#331211yescharacteristic
340C2×C53#332111yescharacteristic
341C2×C53#332711yescharacteristic
342C2×C53#332911yescharacteristic
343C23×Q8C2×D5311yescharacteristic
344C22×M16—531no—
345C2×C4×Q8—531no—
346C2×C4×Q8—531no—
347C2×#117—531no—
348C2×#107—531no—
349C2×#106—531no—
350#419—538no—
351C2×#103—531no—
352C2×#103—531no—
353C2×#103—531no—
354C2×#103—531no—
355C2×#103C2×D5311yescharacteristic
356C2×#103C2×D5311yescharacteristic
357C2×#103C2×D5311yescharacteristic
358C2×#103C2×D5311yescharacteristic
359C2×C22⋊C8C2×D5311yescharacteristic
360C2×#97—531no—
361C2×C4×C8—531no—
362#391—531no—
363#391—531no—
364#391—1061no—
365#391—1061no—
366#389—531no—
367#389—531no—
368#388—1061no—
369#388—1061no—
370#382—1061no—
371#380—531no—
372#380—1061no—
373#380C2×D5311yescharacteristic
374C4×C4⋊C4—531no—
375C4×C4⋊C4—531no—
376C4×C22⋊C4—531no—
377C4×C22⋊C4—1061no—
378#376—1061no—
379#340—531no—
380#340—531no—
381#340—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385#336—531no—
386#336—531no—
387C53#2258511yescharacteristic, complemented, Sylow
388C22×Q8—21no—
389C22×Q8—22no—
390C22×Q8—22no—
391C22×Q8—22no—
392C22×Q8—41no—
393C22×Q8—41no—
394C22×Q8C4×D5314yes—
395C22×Q8C22×D5311yescharacteristic
396C22×Q8C22×D5311yescharacteristic
397C23×C4—21no—
398C23×C4C22×D5311yescharacteristic
399C2×M16—534no—
400C2×M16—2121no—
401C2×M16—2121no—
402C22×C8—531no—
403C22×C8—531no—
404C22×C8C22×D5311yescharacteristic
405C22×C8C22×D5311yescharacteristic
406C4×Q8—1062no—
407C4×Q8—1062no—
408C4×Q8—2121no—
409C4×Q8—2121no—
410#117—534no—
411#117—1062no—
412C2×C4⋊C4—531no—
413C2×C4⋊C4—531no—
414C2×C4⋊C4—531no—
415C2×C4⋊C4—531no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4—1061no—
420C2×C4⋊C4C22×D5311yescharacteristic
421C2×C4⋊C4C22×D5311yescharacteristic
422C2×C22⋊C4—21no—
423C2×C22⋊C4—531no—
424C2×C22⋊C4—531no—
425C2×C22⋊C4—1061no—
426C2×C22⋊C4—1061no—
427C2×C42—531no—
428C2×C42—531no—
429C2×C42—1061no—
430C2×C42—1061no—
431C2×C42—1061no—
432C2×C42—1061no—
433C2×C42—1061no—
434#107—2121no—
435#106—2121no—
436#103—22no—
437#103—22no—
438#103—41no—
439#103—41no—
440#103—534no—
441#103—534no—
442#103—534no—
443#103—534no—
444C22⋊C8—22no—
445#97—2121no—
446C4×C8—2121no—
447#95—21no—
448#95—531no—
449#95—531no—
450#95—1061no—
451#95—1061no—
452#95—1061no—
453#95—1061no—
454#95—1061no—
455#95—1061no—
456#95—1061no—
457#95—1061no—
458#95—1061no—
459C24D4×D5311yescharacteristic
460C2×Q8—24no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—28no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—42no—
471C2×Q8—42no—
472C2×Q8—44no—
473C2×Q8—44no—
474C2×Q8—44no—
475C2×Q8C2×C4×D5312yes—
476C2×Q8C2×C4×D5312yes—
477C2×Q8#1171994712yes—
478C2×Q8#1171994712yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—24no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—42no—
489C22×C4—42no—
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×C4C2×C4×D5312yes—
499C22×C4D4×D5311yescharacteristic
500C22×C4#1171994712yes—
501C22×C4C23×D5311yescharacteristic
502M16—2124no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—532no—
510C2×C8—532no—
511C2×C8—1061no—
512C2×C8—1061no—
513C2×C8—1061no—
514C2×C8—1061no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—532no—
520C4⋊C4—532no—
521C4⋊C4—1061no—
522C4⋊C4—1061no—
523C4⋊C4—1061no—
524C4⋊C4—1061no—
525C4⋊C4—1061no—
526C4⋊C4—1061no—
527C4⋊C4—4241no—
528C4⋊C4—4241no—
529C4⋊C4—2122no—
530C4⋊C4—2122no—
531C22⋊C4—81no—
532C22⋊C4—1062no—
533C22⋊C4—1064no—
534C22⋊C4—2121no—
535C22⋊C4—2122no—
536C42—1064no—
537C42—1061no—
538C42—1061no—
539C42—1061no—
540C42—1061no—
541C42—2122no—
542C42—2122no—
543C42—2122no—
544C42—2122no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23#1313317712yes—
552C23C2×D4×D5311yescharacteristic
553Q8—22no—
554Q8—22no—
555Q8—22no—
556Q8—22no—
557Q8—22no—
558Q8—48no—
559Q8—41no—
560Q8—41no—
561Q8#1313317314yes—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—48no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—532no—
582C2×C4—1061no—
583C2×C4—1061no—
584C2×C4—1062no—
585C2×C4—1062no—
586C2×C4—1061no—
587C2×C4—1061no—
588C2×C4—1061no—
589C2×C4—1061no—
590C2×C4—1061no—
591C2×C4—1061no—
592C2×C4—1061no—
593C2×C4—1061no—
594C2×C4—1061no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4—2121no—
604C2×C4—2121no—
605C2×C4#1313317712yes—
606C2×C4C22×C4×D5311yescharacteristic
607C2×C4C2×#1171994711yescharacteristic
608C2×C4C2×#1171994711yescharacteristic
609C2×C4C2×#1171994711yescharacteristic
610C2×C4#1313326211yescharacteristic
611C2×C4#1313326211yescharacteristic
612C8—41no—
613C8—41no—
614C8—1062no—
615C8—1062no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22C2×#1313317711yescharacteristic
624C22#1681126011yescharacteristic
625C22#1681126011yescharacteristic
626C22#137×D5311yescharacteristic
627C22#1681142211yescharacteristic
628C22#1681142911yescharacteristic
629C22#1681143611yescharacteristic
630C4—22no—
631C4—24no—
632C4—42no—
633C4—81no—
634C4—81no—
635C4—1062no—
636C4—4241no—
637C4—4241no—
638C4—2121no—
639C4—2121no—
640C4—2122no—
641C4—2122no—
642C4—2121no—
643C4C2×#1313317311yescharacteristic
644C4#1681109611yescharacteristic
645C4#1681130411yescharacteristic
646C4#1681130411yescharacteristic
647C2—42no—
648C2#2027756611yescharacteristic
649C2#2028353011yescharacteristic
650C2#2028355211yescharacteristic
651C2#2028675911yescharacteristic
652C2#2028689411yescharacteristic
653C2#2028698711yescharacteristic
654C2#2028700211yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1787C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
cosocleC22×C4×C53C24
hypercenter#1787D53
hypocenterC53#22585
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
4. #20275402C2semidirect
5. #20275388C2semidirect
6. #20275359C2semidirect
7. #20275352C2semidirect
8. #20274981C2semidirect
9. #20274967C2semidirect
12. #20274547C2semidirect
13. #103×Dic53C2semidirect
22. #16810632C4semidirect
387. C53#22585semidirect

Maximal subgroups

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

subgroupindex
C2×#168106322
#202776022
Q8×C22⋊Dic532
#202754022
#202753882
#202753592
#202753522
#202749812
#202749672
#202748112
#202747462
#202745472
#103×Dic532
#202745012
C53×#17872
#2258553