Finite Group Databasebeta

← #27444697

Subgroups — #27444697

Order 13568.

874 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20278166C211yesmaximal, characteristic
2D4×C4⋊Dic53C211yesmaximal, characteristic
3#20276274C211yesmaximal, characteristic, complemented
4#20276148C211yesmaximal, characteristic, complemented
5#20276125C211yesmaximal, characteristic, complemented
6#20275912C211yesmaximal, characteristic, complemented
7#20275350C211yesmaximal, characteristic, complemented
8#20275301C211yesmaximal, characteristic, complemented
9#20275138C211yesmaximal, characteristic, complemented
10#20275020C211yesmaximal, characteristic, complemented
11#20275020C211yesmaximal, characteristic, complemented
12#20274910C211yesmaximal, characteristic, complemented
13#20274802C211yesmaximal, characteristic, complemented
14C2×#16810386C211yesmaximal, characteristic, complemented
15C53×#1869C211yesmaximal, characteristic
16C2×#13133196C2211yescharacteristic
17C2×#13133196C2211yescharacteristic
18C2×D4×Dic53—21no—
19C22×C4⋊Dic53C2211yescharacteristic
20C22×C4⋊Dic53C2211yescharacteristic
21C22×C4⋊Dic53C2211yescharacteristic
22C2×#13133149C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133142C2211yescharacteristic
25C2×#13133142C2211yescharacteristic
26C2×#13133142C2211yescharacteristic
27C2×#13133142C2211yescharacteristic
28#16810556—21no—
29#16810546—21no—
30#16810529C2211yescharacteristic
31#16810522—21no—
32#16810521C2211yescharacteristic
33#16810503—21no—
34#16810498—21no—
35#16810498—21no—
36#16810493C2211yescharacteristic
37#16810483C2211yescharacteristic
38#16810483C2211yescharacteristic
39#16810483C2211yescharacteristic
40#16810481—21no—
41C4×C4⋊Dic53C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52C23×C8×C53C2211yescharacteristic
53C2×C4×D4×C53C2211yescharacteristic
54C2×C22⋊C8×C53C2211yescharacteristic
55C2×C22⋊C8×C53C2211yescharacteristic
56C2×C22⋊C8×C53C2211yescharacteristic
57C53×#336C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×Dic106C2311yescharacteristic
60C23×Dic53—21no—
61C23×Dic53—21no—
62C23×Dic53—21no—
63#13133196—22no—
64#13133196—22no—
65#13133196—22no—
66#13133196—41no—
67D4×Dic53—44no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—22no—
74C2×C4⋊Dic53—22no—
75C2×C4⋊Dic53—22no—
76C2×C4⋊Dic53—41no—
77C2×C4⋊Dic53—41no—
78C2×C4⋊Dic53—41no—
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×C4⋊Dic53C2311yescharacteristic
84C2×C4⋊Dic53C2311yescharacteristic
85C2×C4⋊Dic53C2311yescharacteristic
86C2×#11719919—21no—
87C2×#11719919—21no—
88C2×#11719919—21no—
89C2×C22⋊Dic53—21no—
90C2×C22⋊Dic53—21no—
91C2×C22⋊Dic53—21no—
92C2×C22⋊Dic53—21no—
93C2×C22⋊Dic53—21no—
94C2×C4×Dic53—21no—
95#13133149—22no—
96#13133149—41no—
97#13133142—22no—
98#13133142—41no—
99#13133142—41no—
100#13133142—41no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108C22×D4×C53D411yescharacteristic
109C23×C4×C53C2311yescharacteristic
110C23×C4×C53C2311yescharacteristic
111C22×C8×C53—22no—
112C22×C8×C53—41no—
113C22×C8×C53—41no—
114C22×C8×C53C2311yescharacteristic
115C22×C8×C53C2311yescharacteristic
116C22×C8×C53C2311yescharacteristic
117C22×C8×C53C2311yescharacteristic
118C4×D4×C53—22no—
119C4×D4×C53—22no—
120C2×C4⋊C4×C53D411yescharacteristic
121C2×C22⋊C4×C53D411yescharacteristic
122C2×C22⋊C4×C53D411yescharacteristic
123C2×C42×C53C2311yescharacteristic
124C22⋊C8×C53—22no—
125C22⋊C8×C53—41no—
126C22⋊C8×C53—41no—
127C24×C53C2×D411yescharacteristic
128C24×C53C2×D411yescharacteristic
129C2×D4×C53—22no—
130C2×D4×C53—22no—
131C2×D4×C53SD1614yes—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53#4112yes—
147C22×C4×C53#4112yes—
148C22×C4×C53C2411yescharacteristic
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—42no—
158C22×Dic53—42no—
159C22×Dic53—42no—
160C22×Dic53—42no—
161C22×Dic53—42no—
162C22×Dic53—42no—
163C22×Dic53—81no—
164C22×Dic53—81no—
165C22×Dic53—81no—
166C2×Dic106—21no—
167C2×Dic106—21no—
168C2×Dic106—21no—
169C2×Dic106—21no—
170C2×Dic106—21no—
171C2×Dic106—21no—
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×C53—21no—
183C2×C8×C53—21no—
184C2×C8×C53—44no—
185C4⋊C4×C53—22no—
186C22⋊C4×C53—22no—
187C22⋊C4×C53—22no—
188C42×C53—21no—
189C42×C53—21no—
190C22⋊Dic53—42no—
191C22⋊Dic53—42no—
192C22⋊Dic53—42no—
193C22⋊Dic53—42no—
194C22⋊Dic53—81no—
195C4⋊Dic53—21no—
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—42no—
208C4⋊Dic53—42no—
209C4⋊Dic53—42no—
210C4⋊Dic53—42no—
211C4⋊Dic53—81no—
212C4⋊Dic53#4112yes—
213C4⋊Dic53#4112yes—
214#11719919—42no—
215#11719919—42no—
216#11719919—81no—
217C4×Dic53—42no—
218C23×C53—21no—
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#12312yes—
229C23×C53C2×SD1612yes—
230C23×C53C2×SD1612yes—
231C23×C53#13712yes—
232C23×C53C22×D411yescharacteristic
233D4×C53—28no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
240C2×C4×C53—22no—
241C2×C4×C53—22no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53#12312yes—
253C2×C4×C53C2×SD1612yes—
254C2×C4×C53C2×#4111yescharacteristic
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—22no—
261C2×Dic53—22no—
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—
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—84no—
285C2×Dic53—84no—
286C2×Dic53—84no—
287Dic106—41no—
288Dic106—41no—
289Dic106—41no—
290Dic106—41no—
291C8×C53—41no—
292C8×C53—41no—
293C8×C53—41no—
294C8×C53—41no—
295#26200—531nomaximal, Sylow
296C22×C53—22no—
297C22×C53—22no—
298C22×C53—22no—
299C22×C53—22no—
300C22×C53—22no—
301C22×C53—22no—
302C22×C53—22no—
303C22×C53—22no—
304C22×C53—22no—
305C22×C53—41no—
306C22×C53—41no—
307C22×C53—41no—
308C22×C53—41no—
309C22×C53#48314yes—
310C22×C53C2×#12311yescharacteristic
311C22×C53#54611yescharacteristic
312C22×C53#54611yescharacteristic
313C22×C53C22×SD1611yescharacteristic
314C22×C53C2×#13711yescharacteristic
315C22×C53#57611yescharacteristic
316C22×C53#57811yescharacteristic
317C4×C53—22no—
318C4×C53—22no—
319C4×C53—41no—
320C4×C53—41no—
321C4×C53#52911yescharacteristic
322C4×C53#53811yescharacteristic
323C4×C53#55311yescharacteristic
324C4×C53#55311yescharacteristic
325Dic53—42no—
326Dic53—42no—
327Dic53—81no—
328Dic53—81no—
329Dic53—81no—
330Dic53—81no—
331Dic53—81no—
332Dic53—81no—
333#2417—531no—
334D4×C4⋊C4—531no—
335#2099—531no—
336#2077—531no—
337#2072—531no—
338#2035—531no—
339#1927—531no—
340#1920—531no—
341#1892—531no—
342#1872—531no—
343#1872—531no—
344#1869D5311yescharacteristic
345#1853—531no—
346#1823—531no—
347C2×#340—531no—
348C2×C53—24no—
349C2×C53—42no—
350C2×C53#244411yescharacteristic
351C2×C53C2×#48311yescharacteristic
352C2×C53#309811yescharacteristic
353C2×C53#329511yescharacteristic
354C2×C53#329811yescharacteristic
355C2×C53#331211yescharacteristic
356C2×C53#331411yescharacteristic
357C23×C8C2×D5311yescharacteristic
358C2×C22⋊Q8—531no—
359C2×C22⋊Q8—531no—
360C2×C4×D4—1061no—
361C2×C4×D4C2×D5311yescharacteristic
362C22×C4⋊C4—531no—
363C22×C4⋊C4—531no—
364C22×C4⋊C4—531no—
365C2×#106—531no—
366C2×#106—531no—
367C2×#103—531no—
368C2×#103—531no—
369C2×#103—531no—
370C2×#103—531no—
371C2×C22⋊C8C2×D5311yescharacteristic
372C2×C22⋊C8C2×D5311yescharacteristic
373C2×C22⋊C8C2×D5311yescharacteristic
374#398—1061no—
375#396—1061no—
376#391—531no—
377#389—531no—
378#389—1061no—
379#386—1061no—
380#385—1061no—
381#385—1061no—
382#384—531no—
383#380—531no—
384#380—531no—
385#380—531no—
386#380—1061no—
387C4×C4⋊C4—531no—
388#340—531no—
389#340—531no—
390#340—531no—
391#340—531no—
392#340—531no—
393#340—531no—
394#340—531no—
395#340—531no—
396#340—531no—
397#340—531no—
398#336C2×D5311yescharacteristic
399#336C2×D5311yescharacteristic
400C53#2620011yescharacteristic, complemented, Sylow
401C22×Q8—531no—
402C22×D4D21211yescharacteristic
403C23×C4—1061no—
404C23×C4—1061no—
405C23×C4—1061no—
406C23×C4C22×D5311yescharacteristic
407C23×C4C22×D5311yescharacteristic
408C22×C8—22no—
409C22×C8—41no—
410C22×C8—41no—
411C22×C8C22×D5311yescharacteristic
412C22×C8C22×D5311yescharacteristic
413C22×C8C22×D5311yescharacteristic
414C22×C8C22×D5311yescharacteristic
415C22⋊Q8—1062no—
416C22⋊Q8—1062no—
417C22⋊Q8—1062no—
418C22⋊Q8—2121no—
419C4×D4—22no—
420C4×D4—22no—
421C4×D4—2124no—
422C2×C4⋊C4—531no—
423C2×C4⋊C4—531no—
424C2×C4⋊C4—531no—
425C2×C4⋊C4—531no—
426C2×C4⋊C4—531no—
427C2×C4⋊C4—531no—
428C2×C4⋊C4—531no—
429C2×C4⋊C4—1062no—
430C2×C4⋊C4—1062no—
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—1061no—
439C2×C4⋊C4—1061no—
440C2×C4⋊C4—2121no—
441C2×C4⋊C4—2121no—
442C2×C4⋊C4—2121no—
443C2×C4⋊C4D21211yescharacteristic
444C2×C22⋊C4—1061no—
445C2×C22⋊C4—1061no—
446C2×C22⋊C4—1061no—
447C2×C22⋊C4—1061no—
448C2×C22⋊C4—1061no—
449C2×C22⋊C4D21211yescharacteristic
450C2×C22⋊C4D21211yescharacteristic
451C2×C42—1061no—
452C2×C42C22×D5311yescharacteristic
453#106—1062no—
454#106—2121no—
455#103—1062no—
456#103—2121no—
457#103—2121no—
458#103—2121no—
459C22⋊C8—22no—
460C22⋊C8—41no—
461C22⋊C8—41no—
462#95—1061no—
463#95—1061no—
464#95—1061no—
465#95—1061no—
466#95—1061no—
467#95—1061no—
468#95—1061no—
469C24C2×D21211yescharacteristic
470C24C2×D21211yescharacteristic
471C2×Q8—1061no—
472C2×Q8—1061no—
473C2×Q8—1061no—
474C2×Q8—1061no—
475C2×Q8—1061no—
476C2×Q8—1061no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4#1171991314yes—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—1061no—
491C22×C4—1061no—
492C22×C4—1061no—
493C22×C4—1061no—
494C22×C4—1061no—
495C22×C4—1061no—
496C22×C4—1061no—
497C22×C4—1061no—
498C22×C4—4241no—
499C22×C4—4241no—
500C22×C4—4241no—
501C22×C4—2122no—
502C22×C4—2122no—
503C22×C4—2122no—
504C22×C4—2122no—
505C22×C4—2122no—
506C22×C4—2122no—
507C22×C4C2×D21211yescharacteristic
508C22×C4C2×D21211yescharacteristic
509C22×C4C2×D21211yescharacteristic
510C22×C4C2×D21211yescharacteristic
511C22×C4#1171994712yes—
512C22×C4#1171994712yes—
513C22×C4C23×D5311yescharacteristic
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×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—44no—
527C4⋊C4—22no—
528C4⋊C4—532no—
529C4⋊C4—532no—
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—1061no—
541C4⋊C4—1061no—
542C4⋊C4—4241no—
543C4⋊C4—4241no—
544C4⋊C4—2122no—
545C4⋊C4—2122no—
546C4⋊C4—2122no—
547C4⋊C4—2122no—
548C4⋊C4—2122no—
549C4⋊C4—2122no—
550C22⋊C4—22no—
551C22⋊C4—22no—
552C22⋊C4—4241no—
553C22⋊C4—2122no—
554C22⋊C4—2122no—
555C22⋊C4—2122no—
556C22⋊C4—2122no—
557C42—21no—
558C42—21no—
559C42—2122no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23#1313320112yes—
571C23C2×#1171991312yes—
572C23C2×#1171991312yes—
573C23#1313325012yes—
574C23C22×D21211yescharacteristic
575Q8—2121no—
576Q8—2121no—
577Q8—2121no—
578Q8—2121no—
579D4—28no—
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—4244no—
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—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—2121no—
624C2×C4—2121no—
625C2×C4#1313320112yes—
626C2×C4C2×#1171991312yes—
627C2×C4#41×D5311yescharacteristic
628C2×C4#41×D5311yescharacteristic
629C2×C4C2×#1171994711yescharacteristic
630C2×C4C2×#1171994711yescharacteristic
631C2×C4#1313326211yescharacteristic
632C2×C4#1313326211yescharacteristic
633C8—41no—
634C8—41no—
635C8—41no—
636C8—41no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22#1681095414yes—
651C22C2×#1313320111yescharacteristic
652C22#1681126811yescharacteristic
653C22#1681126811yescharacteristic
654C22C22×#1171991311yescharacteristic
655C22C2×#1313325011yescharacteristic
656C22#1681142811yescharacteristic
657C22#1681144011yescharacteristic
658C4—22no—
659C4—22no—
660C4—41no—
661C4—41no—
662C4—4241no—
663C4—4241no—
664C4—4241no—
665C4—4241no—
666C4—4241no—
667C4—4241no—
668C4—2122no—
669C4—2122no—
670C4#1681115011yescharacteristic
671C4#1681120811yescharacteristic
672C4#1681131611yescharacteristic
673C4#1681131611yescharacteristic
674C2—24no—
675C2—42no—
676C2#2027846311yescharacteristic
677C2C2×#1681095411yescharacteristic
678C2#2028469511yescharacteristic
679C2#2028654011yescharacteristic
680C2#2028658111yescharacteristic
681C2#2028676811yescharacteristic
682C2#2028679811yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1869C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenter#1869D53
hypocenterC53#26200
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1869

Complements

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

NH ≅ G/Ntype
3. #20276274C2semidirect
4. #20276148C2semidirect
5. #20276125C2semidirect
6. #20275912C2semidirect
7. #20275350C2semidirect
8. #20275301C2semidirect
9. #20275138C2semidirect
10. #20275020C2semidirect
11. #20275020C2semidirect
12. #20274910C2semidirect
13. #20274802C2semidirect
14. C2×#16810386C2semidirect
400. C53#26200semidirect

Maximal subgroups

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

subgroupindex
#202781662
D4×C4⋊Dic532
#202762742
#202761482
#202761252
#202759122
#202753502
#202753012
#202751382
#202750202
#202749102
#202748022
C2×#168103862
C53×#18692
#2620053