Finite Group Databasebeta

← #33239431

Subgroups — #33239431

Order 18688.

652 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22623063C211yesmaximal, characteristic
2#444×D73C211yesmaximal, characteristic, complemented
3#22618728C211yesmaximal, characteristic, complemented
4#22611832C211yesmaximal, characteristic, complemented
5#22611830C211yesmaximal, characteristic, complemented
6#22611827C211yesmaximal, characteristic, complemented
7#22611824C211yesmaximal, characteristic, complemented
8#22611822C211yesmaximal, characteristic, complemented
9#22611821C211yesmaximal, characteristic, complemented
10#22611819C211yesmaximal, characteristic, complemented
11#22611702C211yesmaximal, characteristic, complemented
12#22611677C211yesmaximal, characteristic, complemented
13#22611665C211yesmaximal, characteristic, complemented
14#347×D73C211yesmaximal, characteristic, complemented
15C73×AGL(1,ℤ/16)C211yesmaximal, characteristic, complemented
16C2×#14556610C2211yescharacteristic
17#135×D73C2211yescharacteristic
18#131×D73—21no—
19C4×#11924027—21no—
20#17989030C2211yescharacteristic
21#17989027C2211yescharacteristic
22#17989026C412yescomplemented
23#17989023C2211yescharacteristic
24#17989021C412yescomplemented
25#17989018C412yescomplemented
26#17989016C412yescomplemented
27M32×D73C2211yescharacteristic
28M32×D73C2211yescharacteristic
29#17988710C2211yescharacteristic
30#17988709C2211yescharacteristic
31#17988708C2211yescharacteristic
32#17988707—21no—
33#17988706—21no—
34#17988699C2211yescharacteristic, complemented
35#17988695C2211yescharacteristic, complemented
36#17988693C2211yescharacteristic, complemented
37#17988687—21no—
38#17988684—21no—
39#108×D73C2211yescharacteristic
40#17988622C2211yescharacteristic
41#104×D73—21no—
42#17988580—21no—
43#97×D73C2211yescharacteristic
44#17988411C2211yescharacteristic, complemented
45#17988410C2211yescharacteristic, complemented
46#17988409C2211yescharacteristic, complemented
47#17988404C2211yescharacteristic, complemented
48#17988402C2211yescharacteristic, complemented
49#17988400C2211yescharacteristic, complemented
50#17988398C2211yescharacteristic
51#17988397C2211yescharacteristic
52#17988396C2211yescharacteristic
53#17988365C2211yescharacteristic, complemented
54#17988364C2211yescharacteristic, complemented
55#17988364C2211yescharacteristic, complemented
56#509×C73C2211yescharacteristic
57#444×C73C2211yescharacteristic
58#442×C73C2211yescharacteristic, complemented
59#365×C73C2211yescharacteristic, complemented
60#361×C73C2211yescharacteristic, complemented
61#360×C73C2211yescharacteristic
62#347×C73C2211yescharacteristic, complemented
63C2×#11924027—21no—
64#41×D73—21no—
65#14556614C2×C411yescharacteristic, complemented
66#14556613C2311yescharacteristic
67#14556612—21no—
68#14556611—21no—
69#14556610—21no—
70#14556610—21no—
71#14556610C2×C412yes—
72#14556610C2×C412yes—
73#14556610C2×C411yescharacteristic, complemented
74C2×Dic292C2311yescharacteristic
75Q16×D73C2×C411yescharacteristic, complemented
76C2×#11924016—21no—
77SD16×D73—21no—
78D8×D73C2×C411yescharacteristic, complemented
79#14556599—21no—
80#14556599—41no—
81#14556598—21no—
82#14556597—21no—
83#14556597—41no—
84M16×D73—21no—
85M16×D73—41no—
86C2×C8×D73—41no—
87C2×C8×D73D411yescharacteristic
88C2×C8×D73C2311yescharacteristic
89C4×Dic146—21no—
90#14556556—21no—
91D4×Dic73—21no—
92C4×C22⋊D73—41no—
93#14556551—41no—
94#14556549—21no—
95#14556547—41no—
96C2×C4×Dic73—41no—
97C42×D73D411yescharacteristic
98Dic584C2×C412yescomplemented
99#14556537C2×C412yescomplemented
100#14556536C2×C412yescomplemented
101#14556535C2×C412yescomplemented
102#14556535C2×C412yescomplemented
103#14556532C2×C412yescomplemented
104#14556531C2311yescharacteristic
105#14556531C2311yescharacteristic
106#14556530—21no—
107#14556530C2×C412yescomplemented
108C16×D73—21no—
109C16×D73C2×C412yescomplemented
110#14556527C2311yescharacteristic
111#14556526—21no—
112#14556524—21no—
113#14556523C2311yescharacteristic
114#14556519—21no—
115#14556518—21no—
116#14556517—21no—
117#14556516—21no—
118#14556514—21no—
119#14556512—21no—
120#14556502D411yescharacteristic
121#14556501D411yescharacteristic
122#14556501C2311yescharacteristic
123#135×C73C2311yescharacteristic
124C2×D8×C73C2311yescharacteristic
125#131×C73—21no—
126C4×D4×C73—21no—
127SD32×C73C2×C412yescomplemented
128D16×C73C2×C412yescomplemented
129M32×C73C2311yescharacteristic
130M32×C73C2311yescharacteristic
131#108×C73C2311yescharacteristic
132#106×C73C2311yescharacteristic
133#104×C73—21no—
134#102×C73—21no—
135#97×C73C2311yescharacteristic
136#11924029—21no—
137#11924028—21no—
138#11924028—41no—
139#11924027—22no—
140#11924027—22no—
141#11924027—21no—
142#11924027—41no—
143#11924027—41no—
144#11924027—41no—
145C2×Dic146—21no—
146Q8×D73—21no—
147C2×C22⋊D73—41no—
148D4×D73—21no—
149D4×D73—41no—
150C22×Dic73—41no—
151C22×Dic73—41no—
152C2×C4×D73—21no—
153C2×C4×D73—41no—
154C2×C4×D73C2×D411yescharacteristic
155Dic292—21no—
156Dic292C22×C411yescharacteristic
157Dic292C22×C412yes—
158#11924018—21no—
159#11924017—21no—
160#11924016—21no—
161#11924016—22no—
162#11924016—22no—
163#11924015—21no—
164D584C22×C411yescharacteristic
165#11924013—21no—
166#11924012—21no—
167#11924012—41no—
168#11924011—41no—
169#11924011—41no—
170#11924011—41no—
171C2×#10596831—41no—
172C2×#10596831C2×D411yescharacteristic
173C2×#10596831C2×D411yescharacteristic
174C8×D73—21no—
175C8×D73—21no—
176C8×D73—41no—
177C8×D73—41no—
178C8×D73—41no—
179C8×D73C22×C411yescharacteristic
180C8×D73C22×C411yescharacteristic
181C8×D73#4111yescharacteristic
182C8×D73#4111yescharacteristic
183C4⋊Dic73—21no—
184#11924007—41no—
185#11924007—41no—
186C22⋊Dic73—41no—
187#11924005—41no—
188C4×Dic73—21no—
189C4×Dic73—41no—
190C4×Dic73—41no—
191C4×Dic73—41no—
192C4×Dic73C2×D411yescharacteristic
193#11924003—21no—
194#11924003C22×C412yes—
195#41×C73—21no—
196C2×D4×C73—21no—
197C22×C4×C73—41no—
198Q16×C73C22×C411yescharacteristic
199SD16×C73—21no—
200D8×C73—21no—
201D8×C73C22×C411yescharacteristic
202D8×C73C22×C412yes—
203M16×C73—21no—
204M16×C73—41no—
205C2×C8×C73—41no—
206C2×C8×C73C2×D411yescharacteristic
207C2×C8×C73C2411yescharacteristic
208C4⋊C4×C73—21no—
209C22⋊C4×C73—41no—
210C42×C73C2×D411yescharacteristic
211C16×C73—21no—
212C16×C73C22×C412yes—
213C23×C73—41no—
214C22×D73—41no—
215C22×D73AGL(1,ℤ/8)11yescharacteristic
216Q8×C73—21no—
217D4×C73—22no—
218D4×C73—21no—
219D4×C73—41no—
220D4×C73—41no—
221C2×C4×C73—21no—
222C2×C4×C73—41no—
223C2×C4×C73—41no—
224C2×C4×C73—82no—
225C2×C4×C73C22×D411yescharacteristic
226C22⋊D73—41no—
227C22⋊D73—41no—
228C22⋊D73—42no—
229C22⋊D73—42no—
230C22⋊D73—41no—
231C2×Dic73—21no—
232C2×Dic73—41no—
233C2×Dic73—41no—
234C2×Dic73—42no—
235C2×Dic73—42no—
236C2×Dic73—41no—
237C2×Dic73—82no—
238C2×Dic73AGL(1,ℤ/8)11yescharacteristic
239D292—21no—
240D292—41no—
241C4×D73—41no—
242C4×D73—41no—
243C4×D73—41no—
244C4×D73—81no—
245C4×D73C4×D411yescharacteristic
246C4×D73C4×D411yescharacteristic
247C4×D73AGL(1,ℤ/8)11yescharacteristic
248C4×D73AGL(1,ℤ/8)11yescharacteristic
249Dic146—22no—
250Dic146—21no—
251Dic146—41no—
252Dic146—41no—
253C8×C73—21no—
254C8×C73—41no—
255C8×C73C23×C411yescharacteristic
256C8×C73C2×#4111yescharacteristic
257#10596831—21no—
258#10596831—41no—
259#10596831C4×D411yescharacteristic
260#10596831C4×D411yescharacteristic
261C22×C73—42no—
262C22×C73—41no—
263C22×C73—81no—
264C22×C73—81no—
265C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
266C2×D73—41no—
267C2×D73—41no—
268C2×D73—82no—
269C2×D73#44211yescharacteristic
270C2×D73#44211yescharacteristic
271C4×C73—41no—
272C4×C73—41no—
273C4×C73—81no—
274C4×C73C2×C4×D411yescharacteristic
275C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
276Dic73—41no—
277Dic73—41no—
278Dic73—42no—
279Dic73—81no—
280Dic73#44211yescharacteristic
281Dic73#44211yescharacteristic
282#33468—731nomaximal, Sylow
283C2×C73—21no—
284C2×C73—81no—
285C2×C73—82no—
286C2×C73C2×#44211yescharacteristic
287D73—21no—
288D73—21no—
289D73—81no—
290#3414—731no—
291C2×#444—731no—
292#2943—731no—
293AGL(1,ℤ/16)—731no—
294AGL(1,ℤ/16)—731no—
295AGL(1,ℤ/16)—731no—
296AGL(1,ℤ/16)D7311yescharacteristic, complemented
297#2180—731no—
298#2180—731no—
299#2180—731no—
300#2180—731no—
301#2155—731no—
302#2148—731no—
303#2145—731no—
304C2×#347—731no—
305C73#3346811yescharacteristic, complemented, Sylow
306C2×#135—731no—
307C2×#135—731no—
308C2×#131—1461no—
309C4×#41—1461no—
310#510—731no—
311#510—732no—
312#510—731no—
313#509—731no—
314#509—732no—
315#509C2×D7311yescharacteristic
316#508—732no—
317#508—732no—
318C2×M32—731no—
319C2×M32—731no—
320#444—731no—
321#444—731no—
322#444—731no—
323#444—1461no—
324#444—1461no—
325#444C2×D7311yescharacteristic
326#442—731no—
327#442C2×D7311yescharacteristic, complemented
328#441—731no—
329#441—731no—
330#440—1461no—
331#440—1461no—
332C2×#108—731no—
333#427—731no—
334C2×#104—1461no—
335#416—1461no—
336C2×#97—731no—
337#365—731no—
338#365—731no—
339#365—731no—
340#365C2×D7311yescharacteristic, complemented
341#362—731no—
342#362—731no—
343#361—731no—
344#361C2×D7311yescharacteristic, complemented
345#360—731no—
346#360—731no—
347#360—731no—
348#360C2×D7311yescharacteristic
349#347—731no—
350#347—731no—
351#347—731no—
352#347C2×D7311yescharacteristic, complemented
353C2×#41—1461no—
354C2×#41—1461no—
355#135—731no—
356#135—732no—
357#135—732no—
358#135—731no—
359#135—731no—
360#135—1461no—
361#135—1461no—
362#135—1461no—
363#135—1461no—
364#135C22×D7311yescharacteristic
365C2×Q16—731no—
366C2×Q16—731no—
367C2×SD16—1461no—
368C2×SD16—1461no—
369C2×D8—731no—
370C2×D8C22×D7311yescharacteristic
371#131—21no—
372#131—2921no—
373#131—2921no—
374#131—1461no—
375#131—1461no—
376#131—1461no—
377C2×M16—2921no—
378C2×M16—1461no—
379C22×C8—2921no—
380C22×C8—731no—
381C22×C8—731no—
382C4×Q8—1461no—
383C4×Q8—1461no—
384C4×D4—21no—
385C4×D4—2921no—
386C4×D4—2921no—
387C4×D4—1461no—
388#117—2921no—
389#117—1461no—
390C2×C42—2921no—
391C2×C42—731no—
392Q32—732no—
393Q32—732no—
394SD32—732no—
395SD32—732no—
396SD32—732no—
397SD32C4×D7312yescomplemented
398D16—732no—
399D16C4×D7312yescomplemented
400M32—731no—
401M32—732no—
402M32—731no—
403M32—1461no—
404M32C22×D7311yescharacteristic
405M32C22×D7311yescharacteristic
406C2×C16—732no—
407C2×C16—1461no—
408#108—731no—
409#108—1461no—
410#108C22×D7311yescharacteristic
411#107—1461no—
412#106—731no—
413#106C22×D7311yescharacteristic
414#104—21no—
415#104—1461no—
416#104—1461no—
417#104—1461no—
418#103—1461no—
419#103—1461no—
420#102—21no—
421#102—1461no—
422#97—731no—
423#97—731no—
424#97—731no—
425#97C22×D7311yescharacteristic
426#41—21no—
427#41—2921no—
428#41—2921no—
429#41—2921no—
430#41—2921no—
431#41—1462no—
432#41—1462no—
433#41—1461no—
434#41—1461no—
435#41—1461no—
436C2×Q8—1461no—
437C2×Q8—1461no—
438C2×D4—21no—
439C2×D4—2921no—
440C2×D4—2921no—
441C2×D4—1461no—
442C22×C4—41no—
443C22×C4—2921no—
444C22×C4—2921no—
445C22×C4—2921no—
446C22×C4—731no—
447C22×C4—1461no—
448Q16—731no—
449Q16—732no—
450Q16—1461no—
451Q16—1461no—
452Q16C2×C4×D7311yescharacteristic
453SD16—21no—
454SD16—1462no—
455SD16—1462no—
456SD16—1461no—
457SD16—1461no—
458SD16—1461no—
459D8—21no—
460D8—731no—
461D8—1461no—
462D8C2×C4×D7312yes—
463D8C2×C4×D7311yescharacteristic
464M16—21no—
465M16—41no—
466M16—2921no—
467M16—2921no—
468M16—2921no—
469M16—2921no—
470M16—1461no—
471C2×C8—41no—
472C2×C8—2921no—
473C2×C8—2921no—
474C2×C8—2921no—
475C2×C8—2921no—
476C2×C8—731no—
477C2×C8—731no—
478C2×C8—731no—
479C2×C8—731no—
480C2×C8—731no—
481C2×C8—731no—
482C2×C8—1461no—
483C2×C8—1461no—
484C2×C8D4×D7311yescharacteristic
485C2×C8C23×D7311yescharacteristic
486C4⋊C4—21no—
487C4⋊C4—2921no—
488C4⋊C4—2921no—
489C4⋊C4—1461no—
490C22⋊C4—41no—
491C22⋊C4—2921no—
492C22⋊C4—2921no—
493C42—2921no—
494C42—2921no—
495C42—2921no—
496C42—731no—
497C42—1461no—
498C42D4×D7311yescharacteristic
499C16—21no—
500C16—732no—
501C16—1461no—
502C16C2×C4×D7312yes—
503C23—41no—
504C23—2921no—
505C23—731no—
506Q8—21no—
507Q8—2921no—
508Q8—2921no—
509Q8—1462no—
510Q8—1461no—
511D4—22no—
512D4—21no—
513D4—41no—
514D4—41no—
515D4—2921no—
516D4—2921no—
517D4—2922no—
518D4—2922no—
519D4—2921no—
520D4—2921no—
521D4—1461no—
522C2×C4—21no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—82no—
526C2×C4—2921no—
527C2×C4—2921no—
528C2×C4—2921no—
529C2×C4—2921no—
530C2×C4—2922no—
531C2×C4—2922no—
532C2×C4—2921no—
533C2×C4—2921no—
534C2×C4—5841no—
535C2×C4—5842no—
536C2×C4—731no—
537C2×C4—731no—
538C2×C4—731no—
539C2×C4—731no—
540C2×C4—731no—
541C2×C4—1461no—
542C2×C4C2×D4×D7311yescharacteristic
543C8—21no—
544C8—41no—
545C8—2921no—
546C8—731no—
547C8—731no—
548C8—1461no—
549C8C22×C4×D7311yescharacteristic
550C8#41×D7311yescharacteristic
551C22—41no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—2921no—
556C22—2921no—
557C22—5842no—
558C22—731no—
559C22—731no—
560C22AGL(1,ℤ/8)×D7311yescharacteristic
561C4—41no—
562C4—41no—
563C4—81no—
564C4—2921no—
565C4—2921no—
566C4—2922no—
567C4—5841no—
568C4—731no—
569C4—731no—
570C4C4×D4×D7311yescharacteristic
571C4AGL(1,ℤ/8)×D7311yescharacteristic
572C2—21no—
573C2—82no—
574C2—81no—
575C2—5841no—
576C2—1461no—
577C2—1461no—
578C2#442×D7311yescharacteristic
subgroupG/H
centerC2#442×D73
commutatorC8×C73C23×C4
FrattiniC2×C8C23×D73
FittingC73×AGL(1,ℤ/16)C2
socleC2×C73C2×#442
prime radical#22623063C2
cosocleC2×C8×C73C24
hypercenterAGL(1,ℤ/16)D73
hypocenterC73#33468
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ AGL(1,ℤ/16)

Complements

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

NH ≅ G/Ntype
2. #444×D73C2semidirect
3. #22618728C2semidirect
4. #22611832C2semidirect
5. #22611830C2semidirect
6. #22611827C2semidirect
7. #22611824C2semidirect
8. #22611822C2semidirect
9. #22611821C2semidirect
10. #22611819C2semidirect
11. #22611702C2semidirect
12. #22611677C2semidirect
13. #22611665C2semidirect
14. #347×D73C2semidirect
15. C73×AGL(1,ℤ/16)C2semidirect
22. #17989026C4semidirect
24. #17989021C4semidirect
25. #17989018C4semidirect
26. #17989016C4semidirect
34. #17988699C22semidirect
35. #17988695C22semidirect
36. #17988693C22semidirect
44. #17988411C22semidirect
45. #17988410C22semidirect
46. #17988409C22semidirect
47. #17988404C22semidirect
48. #17988402C22semidirect
49. #17988400C22semidirect
53. #17988365C22semidirect
54. #17988364C22semidirect
55. #17988364C22semidirect
58. #442×C73C22semidirect
59. #365×C73C22semidirect
60. #361×C73C22semidirect
62. #347×C73C22semidirect
65. #14556614C2×C4semidirect
73. #14556610C2×C4semidirect
75. Q16×D73C2×C4semidirect
78. D8×D73C2×C4semidirect
98. Dic584C2×C4semidirect
99. #14556537C2×C4semidirect
100. #14556536C2×C4semidirect
101. #14556535C2×C4semidirect
102. #14556535C2×C4semidirect
103. #14556532C2×C4semidirect
107. #14556530C2×C4semidirect
109. C16×D73C2×C4semidirect
127. SD32×C73C2×C4semidirect
128. D16×C73C2×C4semidirect
296. AGL(1,ℤ/16)D73semidirect
305. C73#33468semidirect
327. #442C2×D73semidirect
340. #365C2×D73semidirect
344. #361C2×D73semidirect
352. #347C2×D73semidirect
397. SD32C4×D73semidirect
399. D16C4×D73semidirect

Maximal subgroups

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

subgroupindex
#226230632
#444×D732
#226187282
#226118322
#226118302
#226118272
#226118242
#226118222
#226118212
#226118192
#226117022
#226116772
#226116652
#347×D732
C73×AGL(1,ℤ/16)2
#3346873