Finite Group Databasebeta

← #27401431

Subgroups — #27401431

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×#16810633C211yesmaximal, characteristic
2#20277598C211yesmaximal, characteristic
3C22⋊C4×Dic106C211yesmaximal, characteristic
4#20275405C211yesmaximal, characteristic, complemented
5#20275391C211yesmaximal, characteristic, complemented
6#20275357C211yesmaximal, characteristic, complemented
7#20275350C211yesmaximal, characteristic, complemented
8#20274986C211yesmaximal, characteristic, complemented
9#20274972C211yesmaximal, characteristic, complemented
10#20274811C211yesmaximal, characteristic
11#20274748C211yesmaximal, characteristic
12#20274651C211yesmaximal, characteristic
13#20274546C211yesmaximal, characteristic, complemented
14C4×#13133142C211yesmaximal, characteristic, complemented
15C53×#1753C211yesmaximal, characteristic
16C23×Dic106C2211yescharacteristic
17C2×C4×Dic106C2211yescharacteristic
18C2×C4×Dic106C2211yescharacteristic
19C2×#13133176C2211yescharacteristic
20C2×#13133151C2211yescharacteristic
21C2×#13133149C2211yescharacteristic
22#16810633C418yescomplemented
23C2×#13133142C2211yescharacteristic
24C2×#13133142C2211yescharacteristic
25C2×#13133142C2211yescharacteristic
26C2×#13133142C2211yescharacteristic
27C2×#13133142C2211yescharacteristic
28C2×#13133142C2211yescharacteristic
29C2×#13133142C2211yescharacteristic
30C2×#13133142C2211yescharacteristic
31#16810529C2211yescharacteristic
32#16810529C2211yescharacteristic
33#16810527—21no—
34#16810527—21no—
35#16810521C2211yescharacteristic
36#16810521C2211yescharacteristic
37#16810511—21no—
38#16810511—21no—
39#16810487—21no—
40#16810483C2211yescharacteristic
41#16810483C2211yescharacteristic
42#16810481—21no—
43C4×C4⋊Dic53C2211yescharacteristic
44C4×C4⋊Dic53C2211yescharacteristic
45C22⋊C4×Dic53—21no—
46#16810468—21no—
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52#16810386C2211yescharacteristic
53C22×M16×C53C2211yescharacteristic
54C2×C22⋊C8×C53C2211yescharacteristic
55C2×#97×C53C2211yescharacteristic
56C2×C4×C8×C53C2211yescharacteristic
57C4×C22⋊C4×C53C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C22×Dic106—22no—
61C22×Dic106—22no—
62C22×Dic106—21no—
63C22×Dic106—22no—
64C22×Dic106—41no—
65C22×Dic106—41no—
66C22×Dic106C2×C414yes—
67C22×Dic106C2311yescharacteristic
68C22×Dic106C2311yescharacteristic
69C23×Dic53—21no—
70C4×Dic106—22no—
71C4×Dic106—22no—
72C4×Dic106—41no—
73C4×Dic106—41no—
74#13133176—22no—
75#13133176C2×C414yes—
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53—21no—
88C2×C22⋊Dic53—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53—21no—
94#13133151—41no—
95#13133149—41no—
96#13133142—22no—
97#13133142—22no—
98#13133142—41no—
99#13133142—41no—
100#13133142C2×C414yes—
101#13133142C2×C414yes—
102#13133142C2×C414yes—
103#13133142C2×C414yes—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112#13133124—21no—
113#13133124—21no—
114C23×C4×C53C2311yescharacteristic
115C2×M16×C53—41no—
116C2×M16×C53—41no—
117C2×M16×C53C2×C414yes—
118C22×C8×C53C2311yescharacteristic
119C22×C8×C53C2311yescharacteristic
120C22×C8×C53C2311yescharacteristic
121C22×C8×C53C2311yescharacteristic
122C2×C22⋊C4×C53D411yescharacteristic
123C2×C22⋊C4×C53D411yescharacteristic
124C2×C42×C53C2311yescharacteristic
125C2×C42×C53C2311yescharacteristic
126C22⋊C8×C53—22no—
127#97×C53—41no—
128C4×C8×C53—41no—
129#95×C53D411yescharacteristic
130#95×C53D411yescharacteristic
131C24×C53C2×D411yescharacteristic
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53C22×C412yes—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53#4112yes—
143C22×C4×C53C2411yescharacteristic
144C22×Dic53—21no—
145C22×Dic53—24no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—42no—
154C22×Dic53—42no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106—21no—
162C2×Dic106—28no—
163C2×Dic106—24no—
164C2×Dic106—21no—
165C2×Dic106—44no—
166C2×Dic106—44no—
167C2×Dic106—44no—
168C2×Dic106—42no—
169C2×Dic106—42no—
170C2×Dic106C22×C412yes—
171C2×Dic106C22×C412yes—
172C2×Dic106#4112yes—
173C2×Dic106#4112yes—
174M16×C53—44no—
175C2×C8×C53—21no—
176C2×C8×C53—21no—
177C2×C8×C53—21no—
178C2×C8×C53—21no—
179C2×C8×C53—21no—
180C2×C8×C53—21no—
181C2×C8×C53—21no—
182C2×C8×C53—21no—
183C2×C8×C53—21no—
184C2×C8×C53—21no—
185C2×C8×C53C22×C412yes—
186C2×C8×C53C22×C412yes—
187C22⋊C4×C53—22no—
188C22⋊C4×C53—41no—
189C42×C53—21no—
190C42×C53—21no—
191C42×C53—21no—
192C42×C53—21no—
193C22⋊Dic53—24no—
194C22⋊Dic53—42no—
195C22⋊Dic53—81no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53—21no—
203C4⋊Dic53—21no—
204C4⋊Dic53—21no—
205C4⋊Dic53—21no—
206C4⋊Dic53C22×C412yes—
207C4⋊Dic53C22×C412yes—
208#11719919—42no—
209#11719919—42no—
210#11719919—81no—
211#11719919—81no—
212C4×Dic53—24no—
213C4×Dic53—42no—
214C4×Dic53—42no—
215C4×Dic53—42no—
216C4×Dic53—42no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53—21no—
222C23×C53—21no—
223C23×C53C4×D412yes—
224C23×C53C22×D411yescharacteristic
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53C4×D412yes—
243C2×C4×C53#13712yes—
244C2×C4×C53C23×C411yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53H(5,𝔽2)11yescharacteristic
249C2×C4×C53H(5,𝔽2)11yescharacteristic
250C2×Dic53—22no—
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—22no—
254C2×Dic53—22no—
255C2×Dic53—22no—
256C2×Dic53—48no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275Dic106—22no—
276Dic106—22no—
277Dic106—22no—
278Dic106—22no—
279Dic106—22no—
280Dic106—48no—
281Dic106—41no—
282Dic106—41no—
283Dic106#11714yes—
284C8×C53—22no—
285C8×C53—22no—
286C8×C53—41no—
287C8×C53—41no—
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—22no—
305C4×C53—41no—
306C4×C53—41no—
307C4×C53—41no—
308C4×C53C2×#11711yescharacteristic
309C4×C53#51811yescharacteristic
310C4×C53#55111yescharacteristic
311C4×C53#55111yescharacteristic
312Dic53—24no—
313Dic53—42no—
314Dic53—42no—
315Dic53—42no—
316Dic53—81no—
317Dic53—81no—
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#1787—531no—
332#1763—531no—
333C4×#103—531no—
334#1753D5311yescharacteristic
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×Q8—531no—
344C22×M16C2×D5311yescharacteristic
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×#103—531no—
356C2×#103—531no—
357C2×#103—531no—
358C2×#103—531no—
359C2×C22⋊C8C2×D5311yescharacteristic
360C2×#97C2×D5311yescharacteristic
361C2×C4×C8C2×D5311yescharacteristic
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—531no—
373#380—1061no—
374C4×C4⋊C4—531no—
375C4×C4⋊C4—531no—
376C4×C22⋊C4—1061no—
377C4×C22⋊C4C2×D5311yescharacteristic
378#376—1061no—
379#340—531no—
380#340—531no—
381#340—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385#336C2×D5311yescharacteristic
386#336C2×D5311yescharacteristic
387C53#2258511yescharacteristic, complemented, Sylow
388C22×Q8—534no—
389C22×Q8—531no—
390C22×Q8—531no—
391C22×Q8—1062no—
392C22×Q8—1062no—
393C22×Q8—1061no—
394C22×Q8—1062no—
395C22×Q8—2121no—
396C22×Q8—2121no—
397C23×C4—1061no—
398C23×C4C22×D5311yescharacteristic
399C2×M16—41no—
400C2×M16—41no—
401C2×M16C4×D5314yes—
402C22×C8C22×D5311yescharacteristic
403C22×C8C22×D5311yescharacteristic
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—531no—
417C2×C4⋊C4—531no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4—1061no—
420C2×C4⋊C4—1061no—
421C2×C4⋊C4—1061no—
422C2×C22⋊C4—1061no—
423C2×C22⋊C4—1061no—
424C2×C22⋊C4—1061no—
425C2×C22⋊C4D21211yescharacteristic
426C2×C22⋊C4D21211yescharacteristic
427C2×C42—1061no—
428C2×C42—1061no—
429C2×C42—1061no—
430C2×C42—1061no—
431C2×C42—1061no—
432C2×C42C22×D5311yescharacteristic
433C2×C42C22×D5311yescharacteristic
434#107—2121no—
435#106—2121no—
436#103—534no—
437#103—534no—
438#103—534no—
439#103—534no—
440#103—1062no—
441#103—1062no—
442#103—2121no—
443#103—2121no—
444C22⋊C8—22no—
445#97—41no—
446C4×C8—41no—
447#95—1061no—
448#95—1061no—
449#95—1061no—
450#95—1061no—
451#95—1061no—
452#95—1061no—
453#95—1061no—
454#95—1061no—
455#95—1061no—
456#95—1061no—
457#95D21211yescharacteristic
458#95D21211yescharacteristic
459C24C2×D21211yescharacteristic
460C2×Q8—532no—
461C2×Q8—532no—
462C2×Q8—532no—
463C2×Q8—532no—
464C2×Q8—1068no—
465C2×Q8—1064no—
466C2×Q8—1061no—
467C2×Q8—1061no—
468C2×Q8—1061no—
469C2×Q8—1061no—
470C2×Q8—1061no—
471C2×Q8—1061no—
472C2×Q8—1061no—
473C2×Q8—1061no—
474C2×Q8—2122no—
475C2×Q8—2122no—
476C2×Q8—2124no—
477C2×Q8—2124no—
478C2×Q8—2124no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—1061no—
484C22×C4—1064no—
485C22×C4—1061no—
486C22×C4—1061no—
487C22×C4—1061no—
488C22×C4—1061no—
489C22×C4—1061no—
490C22×C4—1061no—
491C22×C4—1061no—
492C22×C4—2122no—
493C22×C4—2122no—
494C22×C4C2×C4×D5312yes—
495C22×C4C2×D21211yescharacteristic
496C22×C4C2×D21211yescharacteristic
497C22×C4C2×D21211yescharacteristic
498C22×C4C2×D21211yescharacteristic
499C22×C4C2×D21211yescharacteristic
500C22×C4#1171994512yes—
501C22×C4C23×D5311yescharacteristic
502M16—44no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8C2×C4×D5312yes—
514C2×C8C2×C4×D5312yes—
515C4⋊C4—532no—
516C4⋊C4—532no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—1061no—
520C4⋊C4—1061no—
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—22no—
532C22⋊C4—41no—
533C22⋊C4—1064no—
534C22⋊C4—4241no—
535C22⋊C4—2122no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—1064no—
541C42—2122no—
542C42—2122no—
543C42—2122no—
544C42—2122no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23C4×D21212yes—
552C23C22×D21211yescharacteristic
553Q8—534no—
554Q8—1062no—
555Q8—1062no—
556Q8—1062no—
557Q8—1062no—
558Q8—1062no—
559Q8—2128no—
560Q8—2121no—
561Q8—2121no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—1062no—
580C2×C4—1062no—
581C2×C4—1062no—
582C2×C4—1062no—
583C2×C4—1062no—
584C2×C4—1062no—
585C2×C4—2128no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
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×C4C4×D21212yes—
605C2×C4#1313325012yes—
606C2×C4C22×C4×D5311yescharacteristic
607C2×C4#41×D5311yescharacteristic
608C2×C4#41×D5311yescharacteristic
609C2×C4C2×#1171994511yescharacteristic
610C2×C4#1313326211yescharacteristic
611C2×C4#1313326211yescharacteristic
612C8—22no—
613C8—22no—
614C8—41no—
615C8—41no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22C2×C4×D21211yescharacteristic
624C22#1681126811yescharacteristic
625C22#1681126811yescharacteristic
626C22C2×#1313325011yescharacteristic
627C22#1681142311yescharacteristic
628C22#1681143511yescharacteristic
629C22#1681144011yescharacteristic
630C4—22no—
631C4—22no—
632C4—41no—
633C4—41no—
634C4—41no—
635C4—1064no—
636C4—4241no—
637C4—4241no—
638C4—4241no—
639C4—4241no—
640C4—2122no—
641C4—2122no—
642C4—2122no—
643C4#117×D5311yescharacteristic
644C4#1681109611yescharacteristic
645C4#1681130211yescharacteristic
646C4#1681130211yescharacteristic
647C2—42no—
648C2#2027756711yescharacteristic
649C2C4×#1313325011yescharacteristic
650C2#2028356011yescharacteristic
651C2#2028676811yescharacteristic
652C2#2028690311yescharacteristic
653C2#2028698511yescharacteristic
654C2#2028701011yescharacteristic
subgroupG/H
centerC23C22×D212
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1753C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
cosocleC22×C4×C53C24
hypercenter#1753D53
hypocenterC53#22585
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
4. #20275405C2semidirect
5. #20275391C2semidirect
6. #20275357C2semidirect
7. #20275350C2semidirect
8. #20274986C2semidirect
9. #20274972C2semidirect
13. #20274546C2semidirect
14. C4×#13133142C2semidirect
22. #16810633C4semidirect
387. C53#22585semidirect

Maximal subgroups

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

subgroupindex
C2×#168106332
#202775982
C22⋊C4×Dic1062
#202754052
#202753912
#202753572
#202753502
#202749862
#202749722
#202748112
#202747482
#202746512
#202745462
C4×#131331422
C53×#17532
#2258553