Finite Group Databasebeta

← #27452178

Subgroups — #27452178

Order 13568.

826 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279080C211yesmaximal, characteristic
2#20277724C211yesmaximal, characteristic, complemented
3#20276028C211yesmaximal, characteristic, complemented
4#20275908C211yesmaximal, characteristic, complemented
5#20275854C211yesmaximal, characteristic, complemented
6#20275665C211yesmaximal, characteristic, complemented
7#20275400C211yesmaximal, characteristic, complemented
8#20275345C211yesmaximal, characteristic, complemented
9#20275318C211yesmaximal, characteristic, complemented
10#20275303C211yesmaximal, characteristic, complemented
11#20275072C211yesmaximal, characteristic, complemented
12#20275033C211yesmaximal, characteristic
13#20274806C211yesmaximal, characteristic, complemented
14#103×Dic53C211yesmaximal, characteristic, complemented
15C53×#2000C211yesmaximal, characteristic
16C2×#13133216C2211yescharacteristic
17C2×#13133204—21no—
18C2×Q8×Dic53C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×#13133146C2211yescharacteristic
22C2×#13133141C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×#13133138C2211yescharacteristic
26C2×C22⋊#88322C2211yescharacteristic
27C2×C8×Dic53C2211yescharacteristic
28#16810537—21no—
29#16810535—21no—
30#16810530C2211yescharacteristic
31#16810526C2211yescharacteristic
32#16810526C2211yescharacteristic
33#16810522—21no—
34#16810522C2211yescharacteristic
35#16810517—21no—
36#16810509—21no—
37#16810498—21no—
38#16810483C2211yescharacteristic
39C4⋊C4×Dic53C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C22⋊C4×Dic53—21no—
42C4×C22⋊Dic53C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C22×SD16×C53C2211yescharacteristic
53C2×C22⋊Q8×C53C2211yescharacteristic
54C2×#121×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C2×C22⋊C8×C53C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×Dic106—21no—
60C23×Dic53—21no—
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63#13133216—22no—
64#13133216—41no—
65#13133204—44no—
66Q8×Dic53—22no—
67Q8×Dic53—41no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—22no—
72C2×C4⋊Dic53—41no—
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×#11719919—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53D411yescharacteristic
84C2×C22⋊Dic53D411yescharacteristic
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53C2311yescharacteristic
89C2×C4×Dic53C2311yescharacteristic
90#13133148—22no—
91#13133146—41no—
92#13133141—22no—
93#13133141—41no—
94#13133138—22no—
95#13133138—41no—
96C22⋊#88322—41no—
97C8×Dic53—41no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124D411yescharacteristic
105#13133124D411yescharacteristic
106C22×Q8×C53C2311yescharacteristic
107C22×D4×C53—21no—
108C22×D4×C53C2311yescharacteristic
109C23×C4×C53C2311yescharacteristic
110C2×SD16×C53—22no—
111C2×SD16×C53—41no—
112C2×SD16×C53—41no—
113C22×C8×C53C2311yescharacteristic
114C22×C8×C53C2311yescharacteristic
115C22⋊Q8×C53—22no—
116C22⋊Q8×C53—22no—
117#121×C53—22no—
118#121×C53—41no—
119C2×C4⋊C4×C53—21no—
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C4⋊C4×C53C2311yescharacteristic
122C2×C22⋊C4×C53—21no—
123C2×C22⋊C4×C53—21no—
124#103×C53—22no—
125#102×C53—22no—
126C22⋊C8×C53—41no—
127C24×C53—21no—
128C24×C53C2×D411yescharacteristic
129C2×Q8×C53—21no—
130C2×Q8×C53—21no—
131C2×Q8×C53—21no—
132C2×Q8×C53—21no—
133C2×Q8×C53—21no—
134C2×Q8×C53#4112yes—
135C2×D4×C53—21no—
136C2×D4×C53—21no—
137C2×D4×C53—21no—
138C2×D4×C53—21no—
139C2×D4×C53—21no—
140C2×D4×C53—21no—
141C2×D4×C53—42no—
142C2×D4×C53—42no—
143C2×D4×C53—81no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53C2411yescharacteristic
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—42no—
160C22×Dic53—42no—
161C22×Dic53—81no—
162C22×Dic53C2×D411yescharacteristic
163C22×Dic53C2×D411yescharacteristic
164C22×Dic53C2×D411yescharacteristic
165C22×Dic53C2×D411yescharacteristic
166C2×Dic106—42no—
167C2×Dic106—42no—
168C2×Dic106—81no—
169SD16×C53—44no—
170C2×C8×C53—21no—
171C2×C8×C53—21no—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53—21no—
175C2×C8×C53—21no—
176C4⋊C4×C53—24no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53#4112yes—
180C4⋊C4×C53#4112yes—
181C22⋊C4×C53—24no—
182C22⋊C4×C53—42no—
183C22⋊Dic53—41no—
184C22⋊Dic53—42no—
185C22⋊Dic53—42no—
186C22⋊Dic53—42no—
187C22⋊Dic53—42no—
188C22⋊Dic53—81no—
189C22⋊Dic53SD1614yescomplemented
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—42no—
198C4⋊Dic53—42no—
199C4⋊Dic53—81no—
200C4⋊Dic53#4112yes—
201#11719919—42no—
202C4×Dic53—21no—
203C4×Dic53—21no—
204C4×Dic53—21no—
205C4×Dic53—21no—
206C4×Dic53—42no—
207C4×Dic53—42no—
208C4×Dic53—81no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53—41no—
222C23×C53—41no—
223C23×C53—41no—
224C23×C53—41no—
225C23×C53—41no—
226C23×C53—41no—
227C23×C53—41no—
228C23×C53C2×SD1612yes—
229C23×C53C22×D411yescharacteristic
230Q8×C53—22no—
231Q8×C53—22no—
232Q8×C53—41no—
233Q8×C53—41no—
234D4×C53—41no—
235D4×C53—41no—
236D4×C53—41no—
237D4×C53—41no—
238D4×C53—84no—
239C2×C4×C53—21no—
240C2×C4×C53—22no—
241C2×C4×C53—22no—
242C2×C4×C53—22no—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
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×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53C2×#4111yescharacteristic
256C2×C4×C53C2×#4111yescharacteristic
257C2×C4×C53C2×#4111yescharacteristic
258C2×C4×C53C2×#4111yescharacteristic
259C2×C4×C53H(5,𝔽2)11yescharacteristic
260C2×C4×C53H(5,𝔽2)11yescharacteristic
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—22no—
266C2×Dic53—21no—
267C2×Dic53—21no—
268C2×Dic53—21no—
269C2×Dic53—21no—
270C2×Dic53—21no—
271C2×Dic53—21no—
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—84no—
284C2×Dic53C2×SD1612yes—
285C2×Dic53C2×SD1612yes—
286Dic106—84no—
287C8×C53—41no—
288C8×C53—41no—
289#88322—41no—
290#88322—41no—
291#26724—531nomaximal, Sylow
292C22×C53—22no—
293C22×C53—22no—
294C22×C53—22no—
295C22×C53—41no—
296C22×C53—41no—
297C22×C53—41no—
298C22×C53—41no—
299C22×C53—81no—
300C22×C53—81no—
301C22×C53—81no—
302C22×C53—81no—
303C22×C53—81no—
304C22×C53—81no—
305C22×C53—81no—
306C22×C53#52511yescharacteristic
307C22×C53#54611yescharacteristic
308C22×C53#54611yescharacteristic
309C22×C53C22×SD1611yescharacteristic
310C22×C53#57511yescharacteristic
311C22×C53#57611yescharacteristic
312C22×C53#57811yescharacteristic
313C4×C53—42no—
314C4×C53—41no—
315C4×C53—42no—
316C4×C53—42no—
317C4×C53#52911yescharacteristic
318C4×C53#53811yescharacteristic
319C4×C53#55511yescharacteristic
320C4×C53#55911yescharacteristic
321Dic53—22no—
322Dic53—22no—
323Dic53—41no—
324Dic53—42no—
325Dic53—41no—
326Dic53—81no—
327Dic53—81no—
328Dic53—81no—
329#2491—531no—
330#2360—531no—
331#2055—531no—
332#2035—531no—
333#2024—531no—
334#2000D5311yescharacteristic
335#1988—531no—
336#1934—531no—
337#1926—531no—
338#1923—531no—
339#1920—531no—
340#1880—531no—
341#1874—531no—
342#1824—531no—
343C4×#103—531no—
344C2×C53—42no—
345C2×C53—161no—
346C2×C53#245911yescharacteristic
347C2×C53#310311yescharacteristic
348C2×C53#312411yescharacteristic
349C2×C53#329511yescharacteristic
350C2×C53#329811yescharacteristic
351C2×C53#331911yescharacteristic
352C2×C53#332711yescharacteristic
353C22×SD16C2×D5311yescharacteristic
354C2×C4⋊Q8—531no—
355C2×#124—1061no—
356C2×C22⋊Q8C2×D5311yescharacteristic
357C2×#121C2×D5311yescharacteristic
358C2×C4×Q8—531no—
359C22×C4⋊C4—531no—
360C2×#106—531no—
361C2×C4⋊C8—531no—
362C2×#103—531no—
363C2×#103—531no—
364C2×#103C2×D5311yescharacteristic
365C2×#102—531no—
366C2×#102—531no—
367C2×#102C2×D5311yescharacteristic
368C2×C22⋊C8—531no—
369C2×C22⋊C8C2×D5311yescharacteristic
370C2×C4×C8—531no—
371#394—1061no—
372#393—1061no—
373#391—531no—
374#390—531no—
375#390—531no—
376#389—531no—
377#389—1061no—
378#388—1061no—
379#387—1061no—
380#385—1061no—
381#380—531no—
382C4×C4⋊C4—531no—
383C4×C4⋊C4—531no—
384C4×C22⋊C4—531no—
385C4×C22⋊C4—1061no—
386#340—531no—
387#340—531no—
388#340—531no—
389#340—531no—
390#340—531no—
391#340—531no—
392#340—531no—
393#340C2×D5311yescharacteristic
394#336—531no—
395#336—531no—
396C53#2672411yescharacteristic, complemented, Sylow
397C22×Q8—1061no—
398C22×Q8C22×D5311yescharacteristic
399C22×D4—21no—
400C22×D4C22×D5311yescharacteristic
401C23×C4—1061no—
402C23×C4C22×D5311yescharacteristic
403C2×SD16—22no—
404C2×SD16—41no—
405C2×SD16—41no—
406C22×C8—531no—
407C22×C8—531no—
408C22×C8C22×D5311yescharacteristic
409C22×C8C22×D5311yescharacteristic
410C4⋊Q8—1062no—
411C4⋊Q8—2121no—
412#124—2124no—
413C22⋊Q8—22no—
414C22⋊Q8—22no—
415#121—22no—
416#121—41no—
417C4×Q8—1062no—
418C4×Q8—2121no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—531no—
421C2×C4⋊C4—531no—
422C2×C4⋊C4—531no—
423C2×C4⋊C4—531no—
424C2×C4⋊C4—1062no—
425C2×C4⋊C4—1061no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4—1061no—
428C2×C4⋊C4—1061no—
429C2×C4⋊C4—2121no—
430C2×C4⋊C4C22×D5311yescharacteristic
431C2×C4⋊C4C22×D5311yescharacteristic
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—531no—
435C2×C22⋊C4—531no—
436C2×C22⋊C4—1061no—
437C2×C22⋊C4—1061no—
438C2×C22⋊C4—1061no—
439C2×C22⋊C4—1061no—
440C2×C22⋊C4—1061no—
441C2×C42—531no—
442C2×C42—531no—
443C2×C42—1061no—
444C2×C42—1061no—
445C2×C42—1061no—
446#106—1062no—
447C4⋊C8—2121no—
448#103—22no—
449#103—1062no—
450#103—2121no—
451#102—22no—
452#102—1062no—
453#102—2121no—
454C22⋊C8—41no—
455C22⋊C8—2121no—
456C4×C8—2121no—
457#95—531no—
458#95—531no—
459#95—1061no—
460#95—1061no—
461#95—1061no—
462#95—1061no—
463#95—1061no—
464#95—1061no—
465C24—21no—
466C24D4×D5311yescharacteristic
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—4241no—
473C2×Q8—2122no—
474C2×Q8—2122no—
475C2×Q8#1171994712yes—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—42no—
483C2×D4—42no—
484C2×D4—81no—
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—531no—
495C22×C4—531no—
496C22×C4—531no—
497C22×C4—531no—
498C22×C4—1061no—
499C22×C4—1061no—
500C22×C4—1061no—
501C22×C4—1061no—
502C22×C4—4241no—
503C22×C4—2122no—
504C22×C4—2122no—
505C22×C4D4×D5311yescharacteristic
506C22×C4C23×D5311yescharacteristic
507SD16—44no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—1061no—
515C2×C8—1061no—
516C2×C8—1061no—
517C2×C8—1061no—
518C2×C8—1061no—
519C2×C8—1061no—
520C4⋊C4—24no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—532no—
524C4⋊C4—1061no—
525C4⋊C4—1061no—
526C4⋊C4—1061no—
527C4⋊C4—1061no—
528C4⋊C4—1061no—
529C4⋊C4—1061no—
530C4⋊C4—1061no—
531C4⋊C4—4241no—
532C4⋊C4—2122no—
533C4⋊C4—2122no—
534C4⋊C4—2122no—
535C4⋊C4#1171994712yes—
536C4⋊C4#1171994712yes—
537C22⋊C4—24no—
538C22⋊C4—42no—
539C22⋊C4—534no—
540C22⋊C4—4241no—
541C22⋊C4—2122no—
542C22⋊C4—2122no—
543C22⋊C4—2121no—
544C22⋊C4—2122no—
545C22⋊C4—2122no—
546C42—1061no—
547C42—1061no—
548C42—1061no—
549C42—1061no—
550C42—4241no—
551C42—2122no—
552C42—2122no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23SD16×D5312yes—
567C23C2×D4×D5311yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—41no—
571Q8—41no—
572Q8—4244no—
573D4—41no—
574D4—41no—
575D4—41no—
576D4—41no—
577D4—84no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—532no—
595C2×C4—532no—
596C2×C4—1061no—
597C2×C4—1061no—
598C2×C4—1061no—
599C2×C4—1061no—
600C2×C4—1061no—
601C2×C4—1061no—
602C2×C4—1061no—
603C2×C4—1061no—
604C2×C4—1061no—
605C2×C4—1061no—
606C2×C4—1062no—
607C2×C4—4244no—
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#41×D5311yescharacteristic
620C2×C4C2×#1171994711yescharacteristic
621C2×C4C2×#1171994711yescharacteristic
622C2×C4C2×#1171994711yescharacteristic
623C2×C4#1313326211yescharacteristic
624C2×C4#1313326211yescharacteristic
625C8—41no—
626C8—41no—
627C8—2121no—
628C8—2121no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22#1681112911yescharacteristic
644C22#1681126011yescharacteristic
645C22#1681126211yescharacteristic
646C22C2×SD16×D5311yescharacteristic
647C22#1681142211yescharacteristic
648C22#1681142711yescharacteristic
649C22#1681143711yescharacteristic
650C4—42no—
651C4—42no—
652C4—41no—
653C4—42no—
654C4—1062no—
655C4—1062no—
656C4—4241no—
657C4—4241no—
658C4—4241no—
659C4—2121no—
660C4—2121no—
661C4—2122no—
662C4#1681115411yescharacteristic
663C4#1681120811yescharacteristic
664C4#1681133711yescharacteristic
665C4#1681135911yescharacteristic
666C2—42no—
667C2—161no—
668C2#2027864711yescharacteristic
669C2#2028474111yescharacteristic
670C2#2028497011yescharacteristic
671C2#2028653211yescharacteristic
672C2#2028657511yescharacteristic
673C2#2028686611yescharacteristic
674C2#2028697611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2000C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#2000D53
hypocenterC53#26724
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2000

Complements

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

NH ≅ G/Ntype
2. #20277724C2semidirect
3. #20276028C2semidirect
4. #20275908C2semidirect
5. #20275854C2semidirect
6. #20275665C2semidirect
7. #20275400C2semidirect
8. #20275345C2semidirect
9. #20275318C2semidirect
10. #20275303C2semidirect
11. #20275072C2semidirect
13. #20274806C2semidirect
14. #103×Dic53C2semidirect
189. C22⋊Dic53SD16semidirect
396. C53#26724semidirect

Maximal subgroups

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

subgroupindex
#202790802
#202777242
#202760282
#202759082
#202758542
#202756652
#202754002
#202753452
#202753182
#202753032
#202750722
#202750332
#202748062
#103×Dic532
C53×#20002
#2672453