Finite Group Databasebeta

← #27291260

Subgroups — #27291260

Order 13568.

878 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810651C211yesmaximal, characteristic, complemented
2C2×#16810639C211yesmaximal, characteristic
3C2×#16810639C211yesmaximal, characteristic
4C2×#16810591C211yesmaximal, characteristic
5C2×#16810380C211yesmaximal, characteristic, complemented
6#20274413C211yesmaximal, characteristic, complemented
7C53×#1738C211yesmaximal, characteristic, complemented
8C22×#11719945C2211yescharacteristic
9C23×#88322C2211yescharacteristic
10C2×#13133176C2211yescharacteristic
11C2×#13133176C2211yescharacteristic
12#16810651—22no—
13#16810651—22no—
14C2×#13133146—22no—
15#16810639—22no—
16#16810639—22no—
17#16810639—22no—
18#16810639C414yescomplemented
19C2×#13133143—22no—
20C2×#13133143—22no—
21#16810591—22no—
22#16810591—22no—
23C2×#13133129—22no—
24#16810380—22no—
25#16810377—22no—
26C22×M16×C53C2211yescharacteristic
27C22×M16×C53C2211yescharacteristic
28C22×M16×C53C2211yescharacteristic
29C53×#341—22no—
30C2×#11719945—21no—
31C2×#11719945—21no—
32C2×#11719945—21no—
33C2×#11719945—21no—
34C2×#11719945—21no—
35C2×#11719945—21no—
36C2×#11719945—21no—
37C2×#11719945—21no—
38C2×#11719945—21no—
39C2×#11719945—21no—
40C2×#11719945C2×C412yes—
41C2×#11719945D412yes—
42C22×Dic106Q811yescharacteristic
43C22×D212Q811yescharacteristic
44C22×C22⋊D53—21no—
45C22×C4×D53—21no—
46C22×#88322—21no—
47C22×#88322—21no—
48C22×#88322—21no—
49C22×#88322—21no—
50C22×#88322—21no—
51C22×#88322—21no—
52C22×#88322D412yes—
53#13133176—21no—
54#13133176—21no—
55#13133176—21no—
56#13133176—21no—
57#13133176—21no—
58#13133176—21no—
59#13133176C2×C412yes—
60#13133176D412yes—
61C2×C4⋊Dic53—21no—
62C2×C4⋊Dic53—21no—
63C2×C22⋊Dic53—21no—
64C2×C22⋊Dic53—21no—
65C2×C4×Dic53D412yes—
66C2×C4×Dic53D412yes—
67#13133146—28no—
68#13133143—28no—
69#13133143—28no—
70#13133129—28no—
71C23×C4×C53C2311yescharacteristic
72C2×M16×C53—21no—
73C2×M16×C53—21no—
74C2×M16×C53—21no—
75C2×M16×C53—21no—
76C2×M16×C53—21no—
77C2×M16×C53—21no—
78C2×M16×C53—21no—
79C2×M16×C53—21no—
80C2×M16×C53—21no—
81C2×M16×C53—21no—
82C2×M16×C53—21no—
83C2×M16×C53—21no—
84C2×M16×C53—21no—
85C2×M16×C53—21no—
86C2×M16×C53—21no—
87C2×M16×C53—21no—
88C2×M16×C53C2×C412yes—
89C2×M16×C53D412yes—
90C22×C8×C53—21no—
91C22×C8×C53—21no—
92C22×C8×C53D412yes—
93C24×C53#4111yescharacteristic
94C23×D53—21no—
95C22×C4×C53C22×C411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×Q811yescharacteristic
104C22×C4×C53#4111yescharacteristic
105C22×C4×C53#4111yescharacteristic
106C22×C4×C53#4111yescharacteristic
107C22×C4×C53#4111yescharacteristic
108C22×C4×C53#4111yescharacteristic
109C2×C22⋊D53—22no—
110C2×C22⋊D53—22no—
111C2×C22⋊D53—41no—
112C2×C22⋊D53—41no—
113C2×C22⋊D53—41no—
114C2×C22⋊D53—41no—
115C22×Dic53—21no—
116C22×Dic53—22no—
117C22×Dic53—22no—
118#11719945—22no—
119#11719945—22no—
120#11719945—22no—
121#11719945—22no—
122#11719945—22no—
123#11719945—22no—
124#11719945—22no—
125#11719945—22no—
126#11719945—41no—
127#11719945—41no—
128#11719945—41no—
129#11719945—41no—
130#11719945—41no—
131#11719945—41no—
132#11719945—41no—
133#11719945—41no—
134C2×D212—21no—
135C2×D212—21no—
136C2×D212—21no—
137C2×D212—21no—
138C2×D212—21no—
139C2×D212C4⋊C412yes—
140C2×C4×D53—22no—
141C2×C4×D53—22no—
142C2×C4×D53—41no—
143C2×C4×D53—41no—
144C2×C4×D53—41no—
145C2×C4×D53—41no—
146C2×Dic106—21no—
147C2×Dic106—21no—
148C2×Dic106—21no—
149C2×Dic106—21no—
150C2×Dic106—21no—
151C2×Dic106C4⋊C412yes—
152M16×C53—22no—
153M16×C53—22no—
154M16×C53—22no—
155M16×C53—22no—
156M16×C53—41no—
157M16×C53—41no—
158M16×C53—41no—
159M16×C53—41no—
160M16×C53—41no—
161M16×C53—41no—
162M16×C53—41no—
163M16×C53—41no—
164C2×C8×C53—22no—
165C2×C8×C53—22no—
166C2×C8×C53—22no—
167C2×C8×C53—22no—
168C2×C8×C53—22no—
169C2×C8×C53—41no—
170C2×C8×C53—41no—
171C2×C8×C53—41no—
172C2×C8×C53—41no—
173C22⋊Dic53—22no—
174C22⋊Dic53—22no—
175C22⋊Dic53—41no—
176C22⋊Dic53—41no—
177C4⋊Dic53—22no—
178C4⋊Dic53—22no—
179C4⋊Dic53—41no—
180C4⋊Dic53—41no—
181C4×Dic53—22no—
182C4×Dic53—22no—
183C4×Dic53—22no—
184C4×Dic53C22⋊C414yes—
185C2×#88322—22no—
186C2×#88322—22no—
187C2×#88322—22no—
188C2×#88322—41no—
189C2×#88322—41no—
190C2×#88322—41no—
191C2×#88322—41no—
192C23×C53—42no—
193C23×C53C4×D411yescharacteristic
194C23×C53#12111yescharacteristic
195C23×C53C22⋊Q811yescharacteristic
196C23×C53#12311yescharacteristic
197C23×C53#12311yescharacteristic
198C23×C53#12311yescharacteristic
199C23×C53#12611yescharacteristic
200C22×D53—22no—
201C22×D53—44no—
202C22×D53—41no—
203C22×D53—41no—
204C2×C4×C53C2×C22⋊C411yescharacteristic
205C2×C4×C53C2×C4⋊C411yescharacteristic
206C2×C4×C53#11711yescharacteristic
207C2×C4×C53C4×D411yescharacteristic
208C2×C4×C53C4×D411yescharacteristic
209C2×C4×C53C4×D411yescharacteristic
210C2×C4×C53#12011yescharacteristic
211C2×C4×C53#12011yescharacteristic
212C2×C4×C53#12011yescharacteristic
213C2×C4×C53#12111yescharacteristic
214C2×C4×C53#12111yescharacteristic
215C2×C4×C53#12111yescharacteristic
216C2×C4×C53#12111yescharacteristic
217C2×C4×C53#12111yescharacteristic
218C2×C4×C53#12111yescharacteristic
219C2×C4×C53#12111yescharacteristic
220C2×C4×C53C22⋊Q811yescharacteristic
221C2×C4×C53C22⋊Q811yescharacteristic
222C2×C4×C53C22⋊Q811yescharacteristic
223C2×C4×C53C22⋊Q811yescharacteristic
224C2×C4×C53C22⋊Q811yescharacteristic
225C2×C4×C53#12311yescharacteristic
226C2×C4×C53#12311yescharacteristic
227C2×C4×C53#12411yescharacteristic
228C2×C4×C53#12411yescharacteristic
229C2×C4×C53#12411yescharacteristic
230C2×C4×C53#12611yescharacteristic
231C2×C4×C53#12711yescharacteristic
232C22⋊D53—42no—
233C22⋊D53—42no—
234C22⋊D53—42no—
235C22⋊D53—42no—
236C2×Dic53—22no—
237C2×Dic53—24no—
238C2×Dic53—44no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—44no—
242C2×Dic53—42no—
243D212—22no—
244D212—22no—
245D212—41no—
246D212—41no—
247C4×D53—42no—
248C4×D53—42no—
249C4×D53—42no—
250C4×D53—42no—
251Dic106—22no—
252Dic106—22no—
253Dic106—41no—
254Dic106—41no—
255C8×C53—42no—
256C8×C53—42no—
257C8×C53—42no—
258#88322—42no—
259#11084—531nomaximal, Sylow
260C22×C53—22no—
261C22×C53—22no—
262C22×C53—22no—
263C22×C53—41no—
264C22×C53—41no—
265C22×C53—41no—
266C22×C53—41no—
267C22×C53#38511yescharacteristic
268C22×C53#38611yescharacteristic
269C22×C53#38811yescharacteristic
270C22×C53#39411yescharacteristic
271C22×C53#39611yescharacteristic
272C22×C53#39711yescharacteristic
273C22×C53#39911yescharacteristic
274C2×D53—44no—
275C2×D53—42no—
276C2×D53—82no—
277C2×D53—82no—
278C4×C53#38011yescharacteristic
279C4×C53#38511yescharacteristic
280C4×C53#38811yescharacteristic
281C4×C53#39011yescharacteristic
282C4×C53#39211yescharacteristic
283C4×C53#39311yescharacteristic
284C4×C53#39411yescharacteristic
285C4×C53#39611yescharacteristic
286Dic53—44no—
287Dic53—42no—
288Dic53—82no—
289C2×#424—531no—
290C2×#421—531no—
291C2×#421—531no—
292C2×#408—531no—
293#1738D5311yescharacteristic, complemented
294C2×H(3,ℤ/4)—531no—
295#1729—531no—
296C2×C53—21no—
297C2×C53—21no—
298C2×C53—21no—
299C2×C53—21no—
300C2×C53—21no—
301C2×C53—21no—
302C2×C53#143911yescharacteristic
303C2×C53#209812yes—
304D53—82no—
305C22×#41—531no—
306C22×M16C2×D5311yescharacteristic
307C22×M16C2×D5311yescharacteristic
308C22×M16C2×D5311yescharacteristic
309C23×C8—531no—
310C2×#117—531no—
311C2×#117—531no—
312#424—1062no—
313#424—1062no—
314C2×C4⋊C8—1062no—
315#421—534no—
316#421—1062no—
317#421—1062no—
318#421—1062no—
319C2×#104—1062no—
320C2×#104—1062no—
321#408—1062no—
322#408—1062no—
323C2×C22⋊C8—1062no—
324#341—22no—
325H(3,ℤ/4)—1062no—
326#336—1062no—
327C53#1108411yescharacteristic, complemented, Sylow
328C2×#41—532no—
329C2×#41—532no—
330C2×#41—1061no—
331C2×#41—1061no—
332C2×#41—1061no—
333C2×#41—1061no—
334C2×#41—1061no—
335C2×#41—1061no—
336C2×#41—1061no—
337C2×#41—1061no—
338C2×#41—1061no—
339C2×#41—1061no—
340C22×Q8—531no—
341C22×D4—531no—
342C22×D4—1061no—
343C23×C4—1061no—
344C23×C4C22×D5311yescharacteristic
345C2×M16—21no—
346C2×M16—21no—
347C2×M16—21no—
348C2×M16—21no—
349C2×M16—21no—
350C2×M16—21no—
351C2×M16—21no—
352C2×M16—21no—
353C2×M16—21no—
354C2×M16—21no—
355C2×M16—21no—
356C2×M16—21no—
357C2×M16—21no—
358C2×M16—21no—
359C2×M16—21no—
360C2×M16—21no—
361C2×M16C4×D5312yes—
362C2×M16D21212yes—
363C22×C8—21no—
364C22×C8—21no—
365C22×C8—532no—
366C22×C8—1061no—
367C22×C8—1061no—
368C22×C8—1061no—
369C22×C8—1061no—
370C22×C8—1061no—
371C22×C8—1061no—
372C22×C8C22⋊D5312yes—
373#117—532no—
374#117—532no—
375#117—1061no—
376#117—1061no—
377#117—1061no—
378#117—1061no—
379#117—1061no—
380#117—1061no—
381C2×C4⋊C4—1061no—
382C2×C4⋊C4—1061no—
383C2×C22⋊C4—1061no—
384C2×C22⋊C4—1061no—
385C2×C42—532no—
386C2×C42—532no—
387C4⋊C8—1068no—
388#104—1068no—
389#104—1068no—
390C22⋊C8—1068no—
391C24—1061no—
392C24#1171994911yescharacteristic
393#41—1062no—
394#41—1062no—
395#41—1062no—
396#41—1062no—
397#41—1062no—
398#41—1062no—
399#41—1062no—
400#41—1062no—
401#41—2121no—
402#41—2121no—
403#41—2121no—
404#41—2121no—
405#41—2121no—
406#41—2121no—
407#41—2121no—
408#41—2121no—
409C2×Q8—532no—
410C2×Q8—1061no—
411C2×Q8—1061no—
412C2×Q8—1061no—
413C2×Q8—1061no—
414C2×Q8—1061no—
415C2×D4—532no—
416C2×D4—1062no—
417C2×D4—1062no—
418C2×D4—1061no—
419C2×D4—1061no—
420C2×D4—1061no—
421C2×D4—1061no—
422C2×D4—1061no—
423C2×D4—2121no—
424C2×D4—2121no—
425C2×D4—2121no—
426C2×D4—2121no—
427C22×C4—1061no—
428C22×C4—1062no—
429C22×C4—1062no—
430C22×C4—1062no—
431C22×C4—1062no—
432C22×C4—2121no—
433C22×C4—2121no—
434C22×C4—2121no—
435C22×C4—2121no—
436C22×C4C2×C4×D5311yescharacteristic
437C22×C4C2×D21211yescharacteristic
438C22×C4#1171994511yescharacteristic
439C22×C4#1171994511yescharacteristic
440C22×C4#1171994511yescharacteristic
441C22×C4#1171994511yescharacteristic
442C22×C4D4×D5311yescharacteristic
443C22×C4D4×D5311yescharacteristic
444C22×C4D4×D5311yescharacteristic
445C22×C4D4×D5311yescharacteristic
446C22×C4D4×D5311yescharacteristic
447C22×C4#1171994711yescharacteristic
448C22×C4Q8×D5311yescharacteristic
449C22×C4C2×C22⋊D5311yescharacteristic
450M16—22no—
451M16—22no—
452M16—22no—
453M16—22no—
454M16—41no—
455M16—41no—
456M16—41no—
457M16—41no—
458M16—41no—
459M16—41no—
460M16—41no—
461M16—41no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—1062no—
472C2×C8—1062no—
473C2×C8—1062no—
474C2×C8—2121no—
475C2×C8—2121no—
476C2×C8—2121no—
477C2×C8—2121no—
478C4⋊C4—1062no—
479C4⋊C4—1062no—
480C4⋊C4—2121no—
481C4⋊C4—2121no—
482C22⋊C4—1062no—
483C22⋊C4—1062no—
484C22⋊C4—2121no—
485C22⋊C4—2121no—
486C42—534no—
487C42—1062no—
488C42—1062no—
489C42—1062no—
490C23—42no—
491C23—1062no—
492C23—2124no—
493C23—2121no—
494C23—2121no—
495C23#1313317911yescharacteristic
496C23#1313319211yescharacteristic
497C23#1313319311yescharacteristic
498C23#1313320211yescharacteristic
499C23#1313320211yescharacteristic
500C23#1313320211yescharacteristic
501C23#1313321211yescharacteristic
502Q8—1062no—
503Q8—1062no—
504Q8—2121no—
505Q8—2121no—
506D4—1062no—
507D4—1062no—
508D4—2122no—
509D4—2122no—
510D4—2122no—
511D4—2122no—
512D4—2121no—
513D4—2121no—
514C2×C4—1064no—
515C2×C4—1062no—
516C2×C4—2124no—
517C2×C4—2122no—
518C2×C4—2124no—
519C2×C4—2121no—
520C2×C4—2121no—
521C2×C4—2122no—
522C2×C4—2122no—
523C2×C4—2122no—
524C2×C4—2122no—
525C2×C4C22⋊C4×D5311yescharacteristic
526C2×C4C4⋊C4×D5311yescharacteristic
527C2×C4#1313317411yescharacteristic
528C2×C4#1313317711yescharacteristic
529C2×C4C4×C22⋊D5311yescharacteristic
530C2×C4C4×D21211yescharacteristic
531C2×C4#1313318511yescharacteristic
532C2×C4#1313318711yescharacteristic
533C2×C4#1313318711yescharacteristic
534C2×C4#1313318911yescharacteristic
535C2×C4#1313319011yescharacteristic
536C2×C4#1313319011yescharacteristic
537C2×C4#1313319011yescharacteristic
538C2×C4#1313319011yescharacteristic
539C2×C4#1313319011yescharacteristic
540C2×C4#1313319111yescharacteristic
541C2×C4#1313319511yescharacteristic
542C2×C4#1313319511yescharacteristic
543C2×C4#1313319511yescharacteristic
544C2×C4#1313319511yescharacteristic
545C2×C4#1313319711yescharacteristic
546C2×C4#1313319911yescharacteristic
547C2×C4#1313320011yescharacteristic
548C2×C4#1313320311yescharacteristic
549C2×C4#1313320311yescharacteristic
550C2×C4#1313320511yescharacteristic
551C2×C4#1313321111yescharacteristic
552C2×C4#1313321311yescharacteristic
553C8—42no—
554C8—42no—
555C8—42no—
556C8—2122no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—4242no—
565C22—4242no—
566C22—2124no—
567C22—2122no—
568C22#1681049411yescharacteristic
569C22#1681050111yescharacteristic
570C22#1681051211yescharacteristic
571C22#1681054111yescharacteristic
572C22#1681054511yescharacteristic
573C22#1681055211yescharacteristic
574C22#1681055811yescharacteristic
575C4—4242no—
576C4—2124no—
577C4—2122no—
578C4#1681048211yescharacteristic
579C4#1681049711yescharacteristic
580C4#1681051411yescharacteristic
581C4#1681052311yescharacteristic
582C4#1681053211yescharacteristic
583C4#1681053611yescharacteristic
584C4#1681053811yescharacteristic
585C4#1681054711yescharacteristic
586C2—21no—
587C2—21no—
588C2—21no—
589C2—21no—
590C2—21no—
591C2—21no—
592C2—4242no—
593C2#2027295911yescharacteristic
594C2#2027626712yes—
subgroupG/H
centerC2×C4#13133213
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1738C2
socleC22×C53#399
prime radicalC22×#11719945C22
cosocleC23×C4×C53C23
hypercenter#1738D53
hypocenterC53#11084
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1738

Complements

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

NH ≅ G/Ntype
1. C2×#16810651C2semidirect
5. C2×#16810380C2semidirect
6. #20274413C2semidirect
7. C53×#1738C2semidirect
18. #16810639C4semidirect
293. #1738D53semidirect
327. C53#11084semidirect

Maximal subgroups

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

subgroupindex
C2×#168106512
C2×#168106392
C2×#168105912
C2×#168103802
#202744132
C53×#17382
#1108453