Finite Group Databasebeta

← #27442954

Subgroups — #27442954

Order 13568.

818 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811033C211yesmaximal, characteristic, complemented
2C2×#16810666C211yesmaximal, characteristic
3#20278417C211yesmaximal, characteristic
4#20276192C211yesmaximal, characteristic, complemented
5#20276186C211yesmaximal, characteristic, complemented
6#20276050C211yesmaximal, characteristic, complemented
7#20276048C211yesmaximal, characteristic, complemented
8#20275450C211yesmaximal, characteristic, complemented
9#20275445C211yesmaximal, characteristic, complemented
10#20274914C211yesmaximal, characteristic
11#20274907C211yesmaximal, characteristic
12#20274871C211yesmaximal, characteristic, complemented
13#20274746C211yesmaximal, characteristic
14#20274499C211yesmaximal, characteristic
15C53×#2441C211yesmaximal, characteristic
16C22×#11719917C2211yescharacteristic
17C2×#13133217C2211yescharacteristic
18C2×#13133194—21no—
19C22×C4⋊Dic53C2211yescharacteristic
20#16811033—24no—
21#16810666—24no—
22C2×#13133150C2211yescharacteristic
23C2×#13133150C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133150C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×#13133148C2211yescharacteristic
28C2×#13133148C2211yescharacteristic
29C2×#13133148C2211yescharacteristic
30C2×#13133148C2211yescharacteristic
31C2×#13133148C2211yescharacteristic
32C2×C22⋊#88322C2211yescharacteristic
33C2×#13133128C2211yescharacteristic
34C2×C4×#88322C2211yescharacteristic
35#16810555—21no—
36#16810544—21no—
37#16810521C2211yescharacteristic
38#16810521C2211yescharacteristic
39#16810498—21no—
40#16810493C2211yescharacteristic
41#16810483C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×#125×C53C2211yescharacteristic
51C2×#117×C53C2211yescharacteristic
52C53×#399—21no—
53C53×#396—21no—
54C53×#389—21no—
55C53×#389C2211yescharacteristic
56C53×#389C2211yescharacteristic
57C53×#387—21no—
58C53×#384C2211yescharacteristic
59C53×#380C2211yescharacteristic
60C4×C22⋊C4×C53C2211yescharacteristic
61C22×Dic106—21no—
62C23×Dic53—21no—
63C2×#11719917—22no—
64C2×#11719917—22no—
65C2×#11719917Q814yescomplemented
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70#13133217—22no—
71#13133217—41no—
72#13133194—44no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53—22no—
78C2×C4⋊Dic53—41no—
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×C22⋊Dic53—21no—
86#13133150—22no—
87#13133150—22no—
88#13133150—22no—
89#13133150D414yes—
90#13133148—22no—
91#13133148—22no—
92#13133148—22no—
93#13133148—22no—
94#13133148—41no—
95#13133148—41no—
96C22⋊#88322Q814yescomplemented
97#13133128—22no—
98C4×#88322—22no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102C23×C4×C53C2311yescharacteristic
103#125×C53—22no—
104#125×C53—41no—
105#117×C53—22no—
106#117×C53—41no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53—21no—
110C2×C4⋊C4×C53—21no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C22⋊C4×C53—21no—
118C2×C22⋊C4×C53—21no—
119C2×C22⋊C4×C53D411yescharacteristic
120C2×C22⋊C4×C53D411yescharacteristic
121C2×C42×C53—21no—
122C2×C42×C53C2311yescharacteristic
123C2×C42×C53C2311yescharacteristic
124#95×C53—21no—
125#95×C53—21no—
126#95×C53—21no—
127#95×C53D411yescharacteristic
128#95×C53D411yescharacteristic
129C24×C53C2×D411yescharacteristic
130C22×C4×C53—21no—
131C22×C4×C53—21no—
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×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×Q812yes—
144C22×C4×C53C2×Q812yes—
145C22×C4×C53C2411yescharacteristic
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—42no—
151C22×Dic53—42no—
152C22×Dic53—81no—
153C2×Dic106—42no—
154C2×Dic106—42no—
155C2×Dic106—81no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—42no—
164C4⋊C4×C53—42no—
165C4⋊C4×C53—42no—
166C4⋊C4×C53—42no—
167C4⋊C4×C53—81no—
168C4⋊C4×C53—81no—
169C4⋊C4×C53C2×D412yes—
170C22⋊C4×C53—22no—
171C22⋊C4×C53—22no—
172C22⋊C4×C53—42no—
173C22⋊C4×C53—81no—
174C42×C53—21no—
175C42×C53—21no—
176C42×C53—21no—
177C42×C53—21no—
178C42×C53—42no—
179C22⋊Dic53—42no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—42no—
188C4⋊Dic53—42no—
189C4⋊Dic53—42no—
190C4⋊Dic53—81no—
191C4⋊Dic53C2×D412yes—
192#11719919—42no—
193#11719919—42no—
194#11719917—28no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322C2×D412yes—
202C2×#88322C2×Q812yes—
203C2×#88322C2×Q812yes—
204C2×#88322C2×Q812yes—
205C2×#88322C2×Q812yes—
206C2×#88322#4112yes—
207C23×C53—21no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53C4⋊Q812yes—
214C23×C53C22×D411yescharacteristic
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—21no—
226C2×C4×C53—22no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53C4⋊Q812yes—
244C2×C4×C53AGL(1,ℤ/8)12yes—
245C2×C4×C53#13712yes—
246C2×C4×C53C22×D411yescharacteristic
247C2×C4×C53C22×Q811yescharacteristic
248C2×C4×C53C22×Q811yescharacteristic
249C2×C4×C53C2×#4111yescharacteristic
250C2×C4×C53H(5,𝔽2)11yescharacteristic
251C2×C4×C532−1+411yescharacteristic
252C2×Dic53—22no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—84no—
265Dic106—84no—
266#88322—22no—
267#88322—22no—
268#88322—22no—
269#88322C22⋊Q814yescomplemented
270#26078—531nomaximal, Sylow
271C22×C53—22no—
272C22×C53—22no—
273C22×C53—22no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53—41no—
278C22×C53C2×C4⋊Q811yescharacteristic
279C22×C53D4211yescharacteristic
280C22×C53#54711yescharacteristic
281C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C53C2×#13711yescharacteristic
283C22×C53#57711yescharacteristic
284C22×C53#57711yescharacteristic
285C4×C53—22no—
286C4×C53—22no—
287C4×C53—22no—
288C4×C53—41no—
289C4×C53—42no—
290C4×C53—41no—
291C4×C53—81no—
292C4×C53—81no—
293C4×C53—81no—
294C4×C53C2×C22⋊Q811yescharacteristic
295C4×C53#54411yescharacteristic
296C4×C53D4×Q811yescharacteristic
297C4×C53#55411yescharacteristic
298Dic53—42no—
299Dic53—81no—
300Dic53—81no—
301Dic53—81no—
302C2×#501—531no—
303C2×#428—531no—
304#2441D5311yescharacteristic
305#2440—531no—
306#2084—531no—
307#2083—531no—
308#2059—531no—
309#2058—531no—
310#1943—531no—
311#1942—531no—
312#1854—531no—
313#1853—531no—
314#1844—531no—
315#1809—531no—
316#1753—531no—
317C2×C53—42no—
318C2×C53#244011yescharacteristic
319C2×C53#316311yescharacteristic
320C2×C53#316511yescharacteristic
321C2×C53#327211yescharacteristic
322C2×C53#327811yescharacteristic
323C2×C53#334411yescharacteristic
324C2×C53#334511yescharacteristic
325C22×M16—531no—
326C2×C4⋊Q8—531no—
327C2×#125C2×D5311yescharacteristic
328C2×C22⋊Q8—1061no—
329C2×#117C2×D5311yescharacteristic
330C22×C4⋊C4—531no—
331#501—1064no—
332#428—1064no—
333C2×#107—531no—
334C2×#107—531no—
335C2×#107—531no—
336C2×#107—531no—
337C2×#106—531no—
338C2×#106—531no—
339C2×#106—531no—
340C2×#106—531no—
341C2×#106—531no—
342C2×#106—531no—
343C2×C22⋊C8—531no—
344C2×#97—531no—
345C2×C4×C8—531no—
346#399—21no—
347#398—1061no—
348#396—21no—
349#395—1061no—
350#389—21no—
351#389—531no—
352#389—531no—
353#389C2×D5311yescharacteristic
354#389C2×D5311yescharacteristic
355#387—21no—
356#385—1061no—
357#384—531no—
358#384C2×D5311yescharacteristic
359#380—531no—
360#380C2×D5311yescharacteristic
361C4×C22⋊C4C2×D5311yescharacteristic
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#336—531no—
369#336—531no—
370C53#2607811yescharacteristic, complemented, Sylow
371C22×Q8—1061no—
372C23×C4—1061no—
373C23×C4C22×D5311yescharacteristic
374C2×M16—534no—
375C2×M16—1062no—
376C2×M16—1062no—
377C22×C8—531no—
378C22×C8—531no—
379C22×C8—531no—
380C22×C8—531no—
381C4⋊Q8—1062no—
382C4⋊Q8—2121no—
383#125—22no—
384#125—41no—
385C22⋊Q8—2124no—
386#117—22no—
387#117—41no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—531no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—531no—
398C2×C4⋊C4—1062no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—1061no—
402C2×C4⋊C4—1061no—
403C2×C4⋊C4—1061no—
404C2×C4⋊C4—1061no—
405C2×C4⋊C4—2121no—
406C2×C4⋊C4C22×D5311yescharacteristic
407C2×C4⋊C4C22×D5311yescharacteristic
408C2×C4⋊C4C22×D5311yescharacteristic
409C2×C4⋊C4C22×D5311yescharacteristic
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—1061no—
413C2×C22⋊C4C22⋊D5311yescharacteristic
414C2×C22⋊C4C22⋊D5311yescharacteristic
415C2×C42—21no—
416C2×C42C22×D5311yescharacteristic
417C2×C42C22×D5311yescharacteristic
418#107—534no—
419#107—1062no—
420#107—1062no—
421#107—1062no—
422#106—1062no—
423#106—1062no—
424#106—1062no—
425#106—1062no—
426#106—2121no—
427#106—2121no—
428C22⋊C8—534no—
429#97—1062no—
430C4×C8—1062no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437#95C22⋊D5311yescharacteristic
438#95C22⋊D5311yescharacteristic
439C24C2×C22⋊D5311yescharacteristic
440C2×Q8—4241no—
441C2×Q8—2122no—
442C2×Q8—2122no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—1061no—
452C22×C4—1061no—
453C22×C4—1061no—
454C22×C4—1061no—
455C22×C4—4241no—
456C22×C4—2122no—
457C22×C4—2122no—
458C22×C4Q8×D5312yes—
459C22×C4Q8×D5312yes—
460C22×C4C2×C22⋊D5311yescharacteristic
461C22×C4C2×C22⋊D5311yescharacteristic
462C22×C4C2×C22⋊D5311yescharacteristic
463C22×C4C2×C22⋊D5311yescharacteristic
464C22×C4C2×C22⋊D5311yescharacteristic
465C22×C4C23×D5311yescharacteristic
466M16—1068no—
467C2×C8—532no—
468C2×C8—532no—
469C2×C8—532no—
470C2×C8—532no—
471C2×C8—532no—
472C2×C8—532no—
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8—1061no—
478C2×C8—1061no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—81no—
491C4⋊C4—81no—
492C4⋊C4—532no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—4241no—
501C4⋊C4—2122no—
502C4⋊C4—2122no—
503C4⋊C4—2122no—
504C4⋊C4—2122no—
505C4⋊C4—2122no—
506C4⋊C4D4×D5312yes—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—42no—
510C22⋊C4—81no—
511C22⋊C4—2122no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—42no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23#1313321512yes—
524C23C22×C22⋊D5311yescharacteristic
525Q8—4244no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—1062no—
555C2×C4—4244no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4#1313321512yes—
568C2×C4#1313324212yes—
569C2×C4#1313324712yes—
570C2×C4C2×D4×D5311yescharacteristic
571C2×C4C2×Q8×D5311yescharacteristic
572C2×C4C2×Q8×D5311yescharacteristic
573C2×C4#41×D5311yescharacteristic
574C2×C4#1313326211yescharacteristic
575C2×C4#1313326411yescharacteristic
576C8—534no—
577C8—1062no—
578C8—1062no—
579C8—1062no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22C2×#1313321511yescharacteristic
588C22D4×C22⋊D5311yescharacteristic
589C22#1681127211yescharacteristic
590C22C2×#1313324211yescharacteristic
591C22C2×#1313324711yescharacteristic
592C22#1681143111yescharacteristic
593C22#1681143411yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—41no—
598C4—42no—
599C4—41no—
600C4—81no—
601C4—81no—
602C4—81no—
603C4—4241no—
604C4—4241no—
605C4—4241no—
606C4—2122no—
607C4C22⋊Q8×D5311yescharacteristic
608C4#1681125211yescharacteristic
609C4#1681128611yescharacteristic
610C4#1681132711yescharacteristic
611C2—42no—
612C2#2027841411yescharacteristic
613C2#2028524111yescharacteristic
614C2#2028526211yescharacteristic
615C2#2028631511yescharacteristic
616C2#2028638111yescharacteristic
617C2#2028717011yescharacteristic
618C2#2028717911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2441C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#2441D53
hypocenterC53#26078
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2441

Complements

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

NH ≅ G/Ntype
1. C2×#16811033C2semidirect
4. #20276192C2semidirect
5. #20276186C2semidirect
6. #20276050C2semidirect
7. #20276048C2semidirect
8. #20275450C2semidirect
9. #20275445C2semidirect
12. #20274871C2semidirect
65. C2×#11719917Q8semidirect
96. C22⋊#88322Q8semidirect
269. #88322C22⋊Q8semidirect
370. C53#26078semidirect

Maximal subgroups

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

subgroupindex
C2×#168110332
C2×#168106662
#202784172
#202761922
#202761862
#202760502
#202760482
#202754502
#202754452
#202749142
#202749072
#202748712
#202747462
#202744992
C53×#24412
#2607853