Finite Group Databasebeta

← #27444022

Subgroups — #27444022

Order 13568.

854 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810965C211yesmaximal, characteristic, complemented
2C2×#16810665C211yesmaximal, characteristic, complemented
3#20278166C211yesmaximal, characteristic
4#20277664C211yesmaximal, characteristic
5#20276257C211yesmaximal, characteristic, complemented
6#20276193C211yesmaximal, characteristic, complemented
7#20276148C211yesmaximal, characteristic, complemented
8#20275912C211yesmaximal, characteristic, complemented
9#20275357C211yesmaximal, characteristic, complemented
10#20275308C211yesmaximal, characteristic, complemented
11#20275148C211yesmaximal, characteristic, complemented
12#20275025C211yesmaximal, characteristic, complemented
13#20274916C211yesmaximal, characteristic, complemented
14#20274910C211yesmaximal, characteristic, complemented
15C53×#1870C211yesmaximal, characteristic
16C2×#13133196C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19C2×#13133176C2211yescharacteristic
20C22×C4⋊Dic53C2211yescharacteristic
21#16810965—24no—
22#16810665—24no—
23C2×#13133151C2211yescharacteristic
24C2×#13133151C2211yescharacteristic
25C2×#13133149C2211yescharacteristic
26C2×#13133149C2211yescharacteristic
27C2×#13133149C2211yescharacteristic
28C2×#13133149C2211yescharacteristic
29C2×#13133142C2211yescharacteristic
30C2×#13133142C2211yescharacteristic
31C2×#13133142C2211yescharacteristic
32C2×#13133142C2211yescharacteristic
33#16810556—21no—
34#16810546—21no—
35#16810529C2211yescharacteristic
36#16810526—21no—
37#16810521C2211yescharacteristic
38#16810516—21no—
39#16810516—21no—
40#16810503—21no—
41#16810493C2211yescharacteristic
42#16810483C2211yescharacteristic
43#16810483C2211yescharacteristic
44#16810483C2211yescharacteristic
45#16810481—21no—
46C4⋊C4×Dic53—21no—
47C4×C4⋊Dic53C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52#16810386C2211yescharacteristic
53#16810386C2211yescharacteristic
54C22×M16×C53C2211yescharacteristic
55C2×C4×D4×C53C2211yescharacteristic
56C2×C22⋊C8×C53C2211yescharacteristic
57C2×C22⋊C8×C53C2211yescharacteristic
58C2×C22⋊C8×C53C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C53×#336C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62C23×Dic53—21no—
63#13133196—22no—
64#13133196—22no—
65#13133196—22no—
66#13133196—41no—
67#13133176—22no—
68#13133176—22no—
69#13133176—41no—
70#13133176—41no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—22no—
75C2×C4⋊Dic53—41no—
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53—21no—
88C2×C22⋊Dic53—21no—
89C2×C22⋊Dic53—21no—
90C2×C22⋊Dic53—21no—
91C2×C22⋊Dic53—21no—
92C2×C22⋊Dic53—21no—
93C2×C4×Dic53—21no—
94C2×C4×Dic53—21no—
95C2×C4×Dic53—21no—
96#13133151—22no—
97#13133151D414yescomplemented
98#13133149—22no—
99#13133149—22no—
100#13133149—22no—
101#13133149—41no—
102#13133142—22no—
103#13133142—22no—
104#13133142—22no—
105#13133142—41no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112#13133124—21no—
113C22×D4×C53D411yescharacteristic
114C23×C4×C53C2311yescharacteristic
115C23×C4×C53C2311yescharacteristic
116C2×M16×C53—22no—
117C2×M16×C53—22no—
118C2×M16×C53—22no—
119C22×C8×C53C2311yescharacteristic
120C22×C8×C53C2311yescharacteristic
121C22×C8×C53C2311yescharacteristic
122C22×C8×C53C2311yescharacteristic
123C4×D4×C53—22no—
124C4×D4×C53—22no—
125C2×C4⋊C4×C53D411yescharacteristic
126C2×C22⋊C4×C53D411yescharacteristic
127C2×C22⋊C4×C53D411yescharacteristic
128C2×C42×C53C2311yescharacteristic
129C22⋊C8×C53—22no—
130C22⋊C8×C53—22no—
131C22⋊C8×C53—22no—
132C24×C53C2×D411yescharacteristic
133C24×C53C2×D411yescharacteristic
134C2×D4×C53—22no—
135C2×D4×C53—22no—
136C2×D4×C53—41no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53#4112yes—
152C22×C4×C53#4112yes—
153C22×C4×C53C2411yescharacteristic
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53—21no—
162C22×Dic53—42no—
163C22×Dic53—42no—
164C22×Dic53—81no—
165C2×Dic106—21no—
166C2×Dic106—21no—
167C2×Dic106—21no—
168C2×Dic106—21no—
169C2×Dic106—21no—
170C2×Dic106—21no—
171M16×C53—28no—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53—21no—
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×C53C2×D412yes—
183C2×C8×C53#4112yes—
184C4⋊C4×C53—22no—
185C22⋊C4×C53—22no—
186C22⋊C4×C53—22no—
187C42×C53—21no—
188C42×C53—21no—
189C22⋊Dic53—42no—
190C22⋊Dic53—42no—
191C22⋊Dic53—42no—
192C22⋊Dic53—42no—
193C22⋊Dic53—81no—
194C22⋊Dic53—81no—
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⋊Dic53—21no—
207C4⋊Dic53—21no—
208C4⋊Dic53—42no—
209C4⋊Dic53—42no—
210C4⋊Dic53—81no—
211C4⋊Dic53C2×D412yes—
212C4⋊Dic53C2×D412yes—
213#11719919—42no—
214#11719919—42no—
215#11719919—81no—
216C4×Dic53—42no—
217C4×Dic53—42no—
218C4×Dic53—42no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53—21no—
222C23×C53—21no—
223C23×C53—21no—
224C23×C53—21no—
225C23×C53—21no—
226C23×C53—21no—
227C23×C53—21no—
228C23×C53—21no—
229C23×C53—21no—
230C23×C53—21no—
231C23×C53—21no—
232C23×C53#12312yes—
233C23×C53C22×D411yescharacteristic
234D4×C53—44no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—22no—
240C2×C4×C53—22no—
241C2×C4×C53—22no—
242C2×C4×C53—22no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—21no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53#12312yes—
254C2×C4×C53#13712yes—
255C2×C4×C53C22×D411yescharacteristic
256C2×C4×C53C2×#4111yescharacteristic
257C2×C4×C53C2×#4111yescharacteristic
258C2×C4×C53C2×#4111yescharacteristic
259C2×C4×C53H(5,𝔽2)11yescharacteristic
260C2×C4×C532−1+411yescharacteristic
261C2×Dic53—22no—
262C2×Dic53—22no—
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—
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—41no—
284C2×Dic53—41no—
285C2×Dic53—84no—
286Dic106—41no—
287Dic106—41no—
288Dic106—41no—
289Dic106—41no—
290C8×C53—22no—
291C8×C53—22no—
292C8×C53—41no—
293C8×C53—41no—
294#26155—531nomaximal, Sylow
295C22×C53—22no—
296C22×C53—22no—
297C22×C53—22no—
298C22×C53—41no—
299C22×C53—41no—
300C22×C53—41no—
301C22×C53—41no—
302C22×C53—41no—
303C22×C53—41no—
304C22×C53—41no—
305C22×C53—41no—
306C22×C53—41no—
307C22×C53—41no—
308C22×C53—41no—
309C22×C53C2×#12311yescharacteristic
310C22×C53D4211yescharacteristic
311C22×C53#54711yescharacteristic
312C22×C53C2×#13711yescharacteristic
313C22×C53#57511yescharacteristic
314C22×C53#57711yescharacteristic
315C22×C53#57711yescharacteristic
316C4×C53—22no—
317C4×C53—22no—
318C4×C53—41no—
319C4×C53—41no—
320C4×C53#52511yescharacteristic
321C4×C53#53911yescharacteristic
322C4×C53#54811yescharacteristic
323C4×C53#55311yescharacteristic
324Dic53—42no—
325Dic53—42no—
326Dic53—81no—
327Dic53—81no—
328Dic53—81no—
329Dic53—81no—
330Dic53—81no—
331Dic53—81no—
332C2×#485—531no—
333C2×#428—531no—
334#2417—531no—
335#2354—531no—
336#2096—531no—
337#2084—531no—
338#2077—531no—
339#2035—531no—
340#1928—531no—
341#1921—531no—
342#1894—531no—
343#1873—531no—
344#1870D5311yescharacteristic
345#1854—531no—
346#1853—531no—
347C2×C53—42no—
348C2×C53—81no—
349C2×C53#244311yescharacteristic
350C2×C53#309511yescharacteristic
351C2×C53#309911yescharacteristic
352C2×C53#326711yescharacteristic
353C2×C53#327811yescharacteristic
354C2×C53#334811yescharacteristic
355C2×C53#335611yescharacteristic
356C22×M16C2×D5311yescharacteristic
357C2×C22⋊Q8—531no—
358C2×C22⋊Q8—531no—
359C2×C4×D4C2×D5311yescharacteristic
360C2×#117—531no—
361C2×#117—531no—
362C22×C4⋊C4—531no—
363#485—1064no—
364#428—1064no—
365C2×#107—531no—
366C2×#107—531no—
367C2×#106—531no—
368C2×#106—531no—
369C2×#106—531no—
370C2×#106—531no—
371C2×#103—531no—
372C2×#103—531no—
373C2×#103—531no—
374C2×#103—531no—
375C2×C22⋊C8C2×D5311yescharacteristic
376C2×C22⋊C8C2×D5311yescharacteristic
377C2×C22⋊C8C2×D5311yescharacteristic
378#398—1061no—
379#396—1061no—
380#391—531no—
381#390—1061no—
382#389—531no—
383#388—1061no—
384#388—1061no—
385#386—1061no—
386#384—531no—
387#380—531no—
388#380—531no—
389#380—531no—
390#380—1061no—
391C4×C4⋊C4—531no—
392C4×C4⋊C4—1061no—
393#340—531no—
394#340—531no—
395#340—531no—
396#340—531no—
397#340—531no—
398#340—531no—
399#336C2×D5311yescharacteristic
400#336C2×D5311yescharacteristic
401C53#2615511yescharacteristic, complemented, Sylow
402C22×Q8—531no—
403C22×D4D21211yescharacteristic
404C23×C4—1061no—
405C23×C4C22×D5311yescharacteristic
406C23×C4C22×D5311yescharacteristic
407C2×M16—22no—
408C2×M16—22no—
409C2×M16—22no—
410C22×C8C22×D5311yescharacteristic
411C22×C8C22×D5311yescharacteristic
412C22×C8C22×D5311yescharacteristic
413C22×C8C22×D5311yescharacteristic
414C22⋊Q8—1062no—
415C22⋊Q8—1062no—
416C22⋊Q8—1062no—
417C22⋊Q8—2121no—
418C4×D4—22no—
419C4×D4—22no—
420#117—1062no—
421#117—1062no—
422#117—2121no—
423#117—2121no—
424C2×C4⋊C4—531no—
425C2×C4⋊C4—531no—
426C2×C4⋊C4—531no—
427C2×C4⋊C4—531no—
428C2×C4⋊C4—531no—
429C2×C4⋊C4—531no—
430C2×C4⋊C4—531no—
431C2×C4⋊C4—1062no—
432C2×C4⋊C4—1061no—
433C2×C4⋊C4—1061no—
434C2×C4⋊C4—1061no—
435C2×C4⋊C4—1061no—
436C2×C4⋊C4—1061no—
437C2×C4⋊C4—1061no—
438C2×C4⋊C4—2121no—
439C2×C4⋊C4D21211yescharacteristic
440C2×C22⋊C4—1061no—
441C2×C22⋊C4—1061no—
442C2×C22⋊C4—1061no—
443C2×C22⋊C4—1061no—
444C2×C22⋊C4—1061no—
445C2×C22⋊C4—1061no—
446C2×C22⋊C4—1061no—
447C2×C22⋊C4D21211yescharacteristic
448C2×C22⋊C4D21211yescharacteristic
449C2×C42—1061no—
450C2×C42—1061no—
451C2×C42—1061no—
452C2×C42C22×D5311yescharacteristic
453#107—534no—
454#107—1062no—
455#106—1062no—
456#106—1062no—
457#106—1062no—
458#106—2121no—
459#103—1062no—
460#103—1062no—
461#103—1062no—
462#103—2121no—
463C22⋊C8—22no—
464C22⋊C8—22no—
465C22⋊C8—22no—
466#95—1061no—
467#95—1061no—
468#95—1061no—
469#95—1061no—
470#95—1061no—
471#95—1061no—
472#95—1061no—
473C24C2×D21211yescharacteristic
474C24C2×D21211yescharacteristic
475C2×Q8—1061no—
476C2×Q8—1061no—
477C2×Q8—1061no—
478C2×Q8—1061no—
479C2×Q8—1061no—
480C2×Q8—1061no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—41no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—1061no—
495C22×C4—1061no—
496C22×C4—1061no—
497C22×C4—1061no—
498C22×C4—1061no—
499C22×C4—1061no—
500C22×C4—1061no—
501C22×C4—1061no—
502C22×C4—4241no—
503C22×C4—2122no—
504C22×C4—2122no—
505C22×C4C2×D21211yescharacteristic
506C22×C4C2×D21211yescharacteristic
507C22×C4C2×D21211yescharacteristic
508C22×C4C2×D21211yescharacteristic
509C22×C4#1171994712yes—
510C22×C4#1171994712yes—
511C22×C4C23×D5311yescharacteristic
512M16—28no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8D4×D5312yes—
524C2×C8#1171994712yes—
525C4⋊C4—22no—
526C4⋊C4—532no—
527C4⋊C4—532no—
528C4⋊C4—1061no—
529C4⋊C4—1061no—
530C4⋊C4—1061no—
531C4⋊C4—1061no—
532C4⋊C4—1061no—
533C4⋊C4—1061no—
534C4⋊C4—1061no—
535C4⋊C4—1061no—
536C4⋊C4—1061no—
537C4⋊C4—1061no—
538C4⋊C4—1061no—
539C4⋊C4—1061no—
540C4⋊C4—4241no—
541C4⋊C4—4241no—
542C4⋊C4—2122no—
543C4⋊C4—2122no—
544C4⋊C4—2122no—
545C4⋊C4—2122no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—4241no—
549C22⋊C4—4241no—
550C22⋊C4—4241no—
551C22⋊C4—2122no—
552C22⋊C4—2122no—
553C22⋊C4—2122no—
554C22⋊C4—2122no—
555C42—21no—
556C42—21no—
557C42—2122no—
558C42—2122no—
559C42—2122no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23#1313320112yes—
574C23C22×D21211yescharacteristic
575Q8—2121no—
576Q8—2121no—
577Q8—2121no—
578Q8—2121no—
579D4—44no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—1062no—
599C2×C4—1062no—
600C2×C4—4244no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4—2121no—
604C2×C4—2121no—
605C2×C4—2121no—
606C2×C4—2121no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4—2121no—
611C2×C4—2121no—
612C2×C4—2121no—
613C2×C4—2121no—
614C2×C4—2121no—
615C2×C4—2121no—
616C2×C4—2121no—
617C2×C4—2121no—
618C2×C4—2121no—
619C2×C4—2121no—
620C2×C4—2121no—
621C2×C4—2121no—
622C2×C4—2121no—
623C2×C4#1313320112yes—
624C2×C4#1313325012yes—
625C2×C4C2×D4×D5311yescharacteristic
626C2×C4C2×#1171994711yescharacteristic
627C2×C4C2×#1171994711yescharacteristic
628C2×C4C2×#1171994711yescharacteristic
629C2×C4#1313326311yescharacteristic
630C2×C4#1313326411yescharacteristic
631C8—22no—
632C8—22no—
633C8—41no—
634C8—41no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22C2×#1313320111yescharacteristic
650C22D4×D21211yescharacteristic
651C22#1681127711yescharacteristic
652C22C2×#1313325011yescharacteristic
653C22#1681142311yescharacteristic
654C22#1681143511yescharacteristic
655C22#1681143511yescharacteristic
656C4—22no—
657C4—22no—
658C4—41no—
659C4—41no—
660C4—4241no—
661C4—4241no—
662C4—4241no—
663C4—4241no—
664C4—4241no—
665C4—4241no—
666C4—2122no—
667C4—2122no—
668C4#1681112911yescharacteristic
669C4#1681121711yescharacteristic
670C4#1681127911yescharacteristic
671C4#1681132511yescharacteristic
672C2—42no—
673C2—81no—
674C2#2027844811yescharacteristic
675C2#2028466411yescharacteristic
676C2#2028470611yescharacteristic
677C2#2028624511yescharacteristic
678C2#2028638311yescharacteristic
679C2#2028723211yescharacteristic
680C2#2028733111yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1870C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenter#1870D53
hypocenterC53#26155
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1870

Complements

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

NH ≅ G/Ntype
1. C2×#16810965C2semidirect
2. C2×#16810665C2semidirect
5. #20276257C2semidirect
6. #20276193C2semidirect
7. #20276148C2semidirect
8. #20275912C2semidirect
9. #20275357C2semidirect
10. #20275308C2semidirect
11. #20275148C2semidirect
12. #20275025C2semidirect
13. #20274916C2semidirect
14. #20274910C2semidirect
97. #13133151D4semidirect
401. C53#26155semidirect

Maximal subgroups

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

subgroupindex
C2×#168109652
C2×#168106652
#202781662
#202776642
#202762572
#202761932
#202761482
#202759122
#202753572
#202753082
#202751482
#202750252
#202749162
#202749102
C53×#18702
#2615553