Finite Group Databasebeta

← #33239455

Subgroups — #33239455

Order 18688.

668 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
1#509×D73C211yesmaximal, characteristic
2#22618845C211yesmaximal, characteristic, complemented
3#22618780C211yesmaximal, characteristic, complemented
4#22611832C211yesmaximal, characteristic, complemented
5#22611831C211yesmaximal, characteristic, complemented
6#22611830C211yesmaximal, characteristic, complemented
7#22611829C211yesmaximal, characteristic, complemented
8#22611828C211yesmaximal, characteristic, complemented
9#22611827C211yesmaximal, characteristic, complemented
10#22611826C211yesmaximal, characteristic, complemented
11#22611702C211yesmaximal, characteristic, complemented
12#361×D73C211yesmaximal, characteristic, complemented
13#22611665C211yesmaximal, characteristic, complemented
14#22611534C211yesmaximal, characteristic, complemented
15C73×AGL(1,ℤ/16)C211yesmaximal, characteristic, complemented
16#135×D73C2211yescharacteristic
17C2×D8×D73C2211yescharacteristic
18#17989352—21no—
19#17989059—21no—
20#17989025C2211yescharacteristic
21#17989024C412yescomplemented
22#17989023C2211yescharacteristic
23#17989022C2211yescharacteristic
24#17989021C412yescomplemented
25SD32×D73C412yescomplemented
26D16×D73C412yescomplemented
27M32×D73C2211yescharacteristic
28M32×D73C2211yescharacteristic
29#17988710C2211yescharacteristic, complemented
30#17988709C2211yescharacteristic, complemented
31#17988708C2211yescharacteristic, complemented
32#17988702—21no—
33#17988701—21no—
34#17988700C2211yescharacteristic, complemented
35#17988699C2211yescharacteristic, complemented
36#17988698C2211yescharacteristic, complemented
37C4×#11924013—21no—
38#17988667—21no—
39#108×D73C2211yescharacteristic
40#17988622C2211yescharacteristic
41#17988602—21no—
42#17988590—21no—
43#17988544C2211yescharacteristic
44#17988411C2211yescharacteristic, complemented
45#17988410C2211yescharacteristic, complemented
46#17988409C2211yescharacteristic, complemented
47#17988401C2211yescharacteristic, complemented
48#17988400C2211yescharacteristic, complemented
49#17988399C2211yescharacteristic, complemented
50#17988398C2211yescharacteristic, complemented
51#17988397C2211yescharacteristic, complemented
52#17988396C2211yescharacteristic, complemented
53#17988365C2211yescharacteristic, complemented
54#17988364C2211yescharacteristic, complemented
55#17988364C2211yescharacteristic, complemented
56#509×C73C2211yescharacteristic
57#444×C73C2211yescharacteristic, complemented
58#442×C73C2211yescharacteristic, complemented
59#365×C73C2211yescharacteristic, complemented
60#361×C73C2211yescharacteristic, complemented
61#360×C73C2211yescharacteristic, complemented
62#347×C73C2211yescharacteristic, complemented
63#41×D73—21no—
64C2×D4×D73—21no—
65#14556614C2×C411yescharacteristic, complemented
66#14556613C2311yescharacteristic
67#14556612—21no—
68#14556611—21no—
69#14556610C2×C411yescharacteristic, complemented
70Q16×D73C2×C411yescharacteristic, complemented
71SD16×D73—21no—
72C2×D584C2311yescharacteristic
73C2×#11924013—21no—
74D8×D73—21no—
75D8×D73—21no—
76D8×D73C2×C412yes—
77D8×D73C2×C412yes—
78D8×D73C2×C411yescharacteristic, complemented
79#14556599—21no—
80#14556599—41no—
81#14556598—21no—
82#14556597—21no—
83#14556597—41no—
84C2×#11924011—41no—
85C2×#11924011D411yescharacteristic
86M16×D73—21no—
87M16×D73—41no—
88C2×C8×D73C2311yescharacteristic
89C4×D292—21no—
90D4×Dic73—21no—
91#14556553—21no—
92C4×C22⋊D73—41no—
93#14556551—41no—
94#14556549—21no—
95#14556548D411yescharacteristic
96C2×#11924005—41no—
97C22⋊C4×D73—41no—
98#14556536C2×C412yescomplemented
99#14556535C2×C412yescomplemented
100#14556534C2×C412yescomplemented
101D1168C2×C412yescomplemented
102#14556532C2×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#14556513—21no—
118#14556512—21no—
119#14556511—21no—
120#14556501C2311yescharacteristic
121C4×#10596831D411yescharacteristic
122C8×Dic73D411yescharacteristic
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
136C23×D73—41no—
137#11924029—21no—
138#11924028—21no—
139#11924028—41no—
140#11924027—21no—
141#11924027—41no—
142Q8×D73—21no—
143C2×D292—21no—
144C2×C22⋊D73—41no—
145D4×D73—22no—
146D4×D73—22no—
147D4×D73—21no—
148D4×D73—41no—
149D4×D73—41no—
150D4×D73—41no—
151C22×Dic73—41no—
152C2×C4×D73—41no—
153C2×C4×D73—41no—
154C2×C4×D73—41no—
155C2×C4×D73C2×D411yescharacteristic
156Dic292C22×C411yescharacteristic
157#11924018—21no—
158#11924017—21no—
159#11924016—21no—
160#11924015—21no—
161D584—21no—
162D584C22×C411yescharacteristic
163D584C22×C412yes—
164#11924013—22no—
165#11924013—22no—
166#11924013—21no—
167#11924012—21no—
168#11924012—41no—
169#11924011—21no—
170#11924011—21no—
171#11924011—41no—
172#11924011—41no—
173#11924011—41no—
174C2×#10596831—41no—
175C2×#10596831C2×D411yescharacteristic
176C2×#10596831C2×D411yescharacteristic
177C8×D73—41no—
178C8×D73—41no—
179C8×D73—41no—
180C8×D73C22×C411yescharacteristic
181C8×D73C22×C411yescharacteristic
182C8×D73#4111yescharacteristic
183C8×D73#4111yescharacteristic
184C4⋊Dic73—21no—
185#11924007—21no—
186C22⋊Dic73—41no—
187#11924005—21no—
188#11924005—41no—
189#11924005—41no—
190#11924005—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×D73—41no—
216C22×D73—42no—
217C22×D73—42no—
218C22×D73—161no—
219C22×D73—161no—
220C22×D73AGL(1,ℤ/8)11yescharacteristic
221Q8×C73—21no—
222D4×C73—22no—
223D4×C73—21no—
224D4×C73—41no—
225D4×C73—41no—
226C2×C4×C73—21no—
227C2×C4×C73—41no—
228C2×C4×C73—41no—
229C2×C4×C73—161no—
230C2×C4×C73C22×D411yescharacteristic
231C22⋊D73—41no—
232C22⋊D73—41no—
233C22⋊D73—42no—
234C22⋊D73—42no—
235C22⋊D73—41no—
236C2×Dic73—21no—
237C2×Dic73—41no—
238C2×Dic73—41no—
239C2×Dic73—161no—
240C2×Dic73AGL(1,ℤ/8)11yescharacteristic
241D292—22no—
242D292—21no—
243D292—41no—
244D292—41no—
245C4×D73—41no—
246C4×D73—41no—
247C4×D73—41no—
248C4×D73—161no—
249C4×D73—161no—
250C4×D73C4×D411yescharacteristic
251C4×D73C4×D411yescharacteristic
252C4×D73AGL(1,ℤ/8)11yescharacteristic
253C4×D73AGL(1,ℤ/8)11yescharacteristic
254Dic146—21no—
255Dic146—41no—
256C8×C73—21no—
257C8×C73—41no—
258C8×C73C23×C411yescharacteristic
259C8×C73C2×#4111yescharacteristic
260#10596831—21no—
261#10596831—41no—
262#10596831C4×D411yescharacteristic
263#10596831C4×D411yescharacteristic
264C22×C73—41no—
265C22×C73—42no—
266C22×C73—81no—
267C22×C73—81no—
268C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
269C2×D73—41no—
270C2×D73—41no—
271C2×D73—42no—
272C2×D73—81no—
273C2×D73—81no—
274C2×D73—161no—
275C2×D73—162no—
276C2×D73—162no—
277C2×D73#44211yescharacteristic
278C2×D73#44211yescharacteristic
279C4×C73—41no—
280C4×C73—41no—
281C4×C73—81no—
282C4×C73C2×C4×D411yescharacteristic
283C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
284Dic73—41no—
285Dic73—41no—
286Dic73—81no—
287Dic73#44211yescharacteristic
288Dic73#44211yescharacteristic
289#33471—731nomaximal, Sylow
290C2×C73—21no—
291C2×C73—82no—
292C2×C73—81no—
293C2×C73C2×#44211yescharacteristic
294D73—21no—
295D73—21no—
296D73—82no—
297D73—81no—
298C2×#509—731no—
299#2957—731no—
300#2948—731no—
301AGL(1,ℤ/16)—731no—
302AGL(1,ℤ/16)—731no—
303AGL(1,ℤ/16)—731no—
304AGL(1,ℤ/16)—731no—
305AGL(1,ℤ/16)—731no—
306AGL(1,ℤ/16)—731no—
307AGL(1,ℤ/16)—731no—
308AGL(1,ℤ/16)D7311yescharacteristic, complemented
309#2155—731no—
310C2×#361—731no—
311#2145—731no—
312#2110—731no—
313C73#3347111yescharacteristic, complemented, Sylow
314C2×#135—731no—
315C22×D8—731no—
316#568—1461no—
317#518—1461no—
318#509—731no—
319#509—731no—
320#509—732no—
321#509—732no—
322#509—731no—
323#509C2×D7311yescharacteristic
324C2×SD32—732no—
325C2×D16—732no—
326C2×M32—731no—
327C2×M32—731no—
328#444—731no—
329#444—731no—
330#444—731no—
331#444C2×D7311yescharacteristic, complemented
332#443—1461no—
333#443—1461no—
334#442—731no—
335#442—731no—
336#442—731no—
337#442C2×D7311yescharacteristic, complemented
338C4×D8—1461no—
339C4×D8—1461no—
340C2×#108—731no—
341#427—731no—
342#421—1461no—
343#418—1461no—
344#405—731no—
345#365—731no—
346#365—731no—
347#365—731no—
348#365C2×D7311yescharacteristic, complemented
349#361—731no—
350#361—731no—
351#361—731no—
352#361C2×D7311yescharacteristic, complemented
353#360—731no—
354#360—731no—
355#360—731no—
356#360C2×D7311yescharacteristic, complemented
357#347—731no—
358#347—731no—
359#347—731no—
360#347C2×D7311yescharacteristic, complemented
361C2×#41—1461no—
362C22×D4—1461no—
363#135—731no—
364#135—731no—
365#135—731no—
366#135—1461no—
367#135—1461no—
368#135C22×D7311yescharacteristic
369C2×Q16—731no—
370C2×SD16—1461no—
371C2×D8—732no—
372C2×D8—732no—
373C2×D8—731no—
374C2×D8—731no—
375C2×D8—1461no—
376C2×D8—1461no—
377C2×D8—1461no—
378C2×D8C22×D7311yescharacteristic
379#131—21no—
380#131—2921no—
381#131—2921no—
382#131—1461no—
383#131—1461no—
384#131—1461no—
385C2×M16—2921no—
386C2×M16—2921no—
387C2×M16—731no—
388C2×M16—1461no—
389C22×C8—731no—
390C4×D4—21no—
391C4×D4—2921no—
392C4×D4—2921no—
393C4×D4—1461no—
394C4×D4—1461no—
395C4×D4—1461no—
396#117—731no—
397#117—1461no—
398C2×C22⋊C4—2921no—
399C2×C22⋊C4—2921no—
400SD32—732no—
401SD32—732no—
402SD32—732no—
403SD32C4×D7312yescomplemented
404D16—732no—
405D16—732no—
406D16—732no—
407D16C4×D7312yescomplemented
408M32—731no—
409M32—732no—
410M32—731no—
411M32—1461no—
412M32C22×D7311yescharacteristic
413M32C22×D7311yescharacteristic
414C2×C16—732no—
415C2×C16—1461no—
416#108—731no—
417#108—1461no—
418#108C22×D7311yescharacteristic
419#107—1461no—
420#106—731no—
421#106C22×D7311yescharacteristic
422#104—21no—
423#104—1461no—
424#104—1461no—
425#104—1461no—
426#102—21no—
427#102—1461no—
428#102—1461no—
429#102—1461no—
430#97—731no—
431#97C22×D7311yescharacteristic
432C4×C8—731no—
433C4×C8—731no—
434C24—2921no—
435#41—21no—
436#41—2921no—
437#41—2921no—
438#41—1461no—
439#41—1461no—
440#41—1461no—
441C2×Q8—1461no—
442C2×D4—21no—
443C2×D4—2921no—
444C2×D4—2921no—
445C2×D4—2921no—
446C2×D4—2921no—
447C2×D4—1462no—
448C2×D4—1462no—
449C2×D4—1461no—
450C2×D4—1461no—
451C22×C4—41no—
452C22×C4—2921no—
453C22×C4—2921no—
454C22×C4—2921no—
455C22×C4—2921no—
456C22×C4—731no—
457Q16—731no—
458Q16—1461no—
459Q16C2×C4×D7311yescharacteristic
460SD16—21no—
461SD16—1461no—
462SD16—1461no—
463SD16—1461no—
464D8—21no—
465D8—731no—
466D8—732no—
467D8—1462no—
468D8—1462no—
469D8—1461no—
470D8—1461no—
471D8C2×C4×D7311yescharacteristic
472D8C2×C4×D7312yes—
473M16—21no—
474M16—41no—
475M16—2921no—
476M16—2921no—
477M16—2921no—
478M16—2921no—
479M16—1461no—
480M16—1461no—
481M16—1461no—
482C2×C8—41no—
483C2×C8—2921no—
484C2×C8—2921no—
485C2×C8—2921no—
486C2×C8—2921no—
487C2×C8—731no—
488C2×C8—731no—
489C2×C8—731no—
490C2×C8—731no—
491C2×C8—731no—
492C2×C8—731no—
493C2×C8D4×D7311yescharacteristic
494C2×C8C23×D7311yescharacteristic
495C4⋊C4—21no—
496C4⋊C4—1461no—
497C4⋊C4—1461no—
498C22⋊C4—41no—
499C22⋊C4—2921no—
500C22⋊C4—2921no—
501C22⋊C4—2921no—
502C22⋊C4—2921no—
503C22⋊C4—1461no—
504C42—2921no—
505C42—731no—
506C42D4×D7311yescharacteristic
507C16—21no—
508C16—732no—
509C16—1461no—
510C16C2×C4×D7312yes—
511C23—41no—
512C23—2921no—
513C23—2922no—
514C23—2922no—
515C23—2921no—
516C23—731no—
517C23—11681no—
518C23—11681no—
519Q8—21no—
520Q8—2921no—
521Q8—1461no—
522D4—22no—
523D4—21no—
524D4—41no—
525D4—41no—
526D4—2921no—
527D4—2921no—
528D4—2921no—
529D4—2922no—
530D4—2922no—
531D4—2921no—
532D4—2921no—
533D4—1462no—
534D4—1461no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—161no—
539C2×C4—2921no—
540C2×C4—2921no—
541C2×C4—2921no—
542C2×C4—2921no—
543C2×C4—2921no—
544C2×C4—731no—
545C2×C4—731no—
546C2×C4—731no—
547C2×C4—731no—
548C2×C4—731no—
549C2×C4—11681no—
550C2×C4—11681no—
551C2×C4—11681no—
552C2×C4—1461no—
553C2×C4C2×D4×D7311yescharacteristic
554C8—21no—
555C8—41no—
556C8—2921no—
557C8—731no—
558C8—731no—
559C8—1461no—
560C8C22×C4×D7311yescharacteristic
561C8#41×D7311yescharacteristic
562C22—41no—
563C22—42no—
564C22—81no—
565C22—81no—
566C22—2921no—
567C22—2921no—
568C22—2922no—
569C22—5841no—
570C22—5841no—
571C22—731no—
572C22—731no—
573C22—11682no—
574C22—11682no—
575C22—11681no—
576C22AGL(1,ℤ/8)×D7311yescharacteristic
577C4—41no—
578C4—41no—
579C4—81no—
580C4—2921no—
581C4—2921no—
582C4—5841no—
583C4—731no—
584C4—731no—
585C4C4×D4×D7311yescharacteristic
586C4AGL(1,ℤ/8)×D7311yescharacteristic
587C2—21no—
588C2—82no—
589C2—81no—
590C2—5842no—
591C2—5841no—
592C2—1461no—
593C2—1461no—
594C2#442×D7311yescharacteristic
subgroupG/H
centerC2#442×D73
commutatorC8×C73C23×C4
FrattiniC2×C8C23×D73
FittingC73×AGL(1,ℤ/16)C2
socleC2×C73C2×#442
prime radical#509×D73C2
cosocleC2×C8×C73C24
hypercenterAGL(1,ℤ/16)D73
hypocenterC73#33471
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. #22618845C2semidirect
3. #22618780C2semidirect
4. #22611832C2semidirect
5. #22611831C2semidirect
6. #22611830C2semidirect
7. #22611829C2semidirect
8. #22611828C2semidirect
9. #22611827C2semidirect
10. #22611826C2semidirect
11. #22611702C2semidirect
12. #361×D73C2semidirect
13. #22611665C2semidirect
14. #22611534C2semidirect
15. C73×AGL(1,ℤ/16)C2semidirect
21. #17989024C4semidirect
24. #17989021C4semidirect
25. SD32×D73C4semidirect
26. D16×D73C4semidirect
29. #17988710C22semidirect
30. #17988709C22semidirect
31. #17988708C22semidirect
34. #17988700C22semidirect
35. #17988699C22semidirect
36. #17988698C22semidirect
44. #17988411C22semidirect
45. #17988410C22semidirect
46. #17988409C22semidirect
47. #17988401C22semidirect
48. #17988400C22semidirect
49. #17988399C22semidirect
50. #17988398C22semidirect
51. #17988397C22semidirect
52. #17988396C22semidirect
53. #17988365C22semidirect
54. #17988364C22semidirect
55. #17988364C22semidirect
57. #444×C73C22semidirect
58. #442×C73C22semidirect
59. #365×C73C22semidirect
60. #361×C73C22semidirect
61. #360×C73C22semidirect
62. #347×C73C22semidirect
65. #14556614C2×C4semidirect
69. #14556610C2×C4semidirect
70. Q16×D73C2×C4semidirect
78. D8×D73C2×C4semidirect
98. #14556536C2×C4semidirect
99. #14556535C2×C4semidirect
100. #14556534C2×C4semidirect
101. D1168C2×C4semidirect
102. #14556532C2×C4semidirect
103. #14556532C2×C4semidirect
107. #14556530C2×C4semidirect
109. C16×D73C2×C4semidirect
127. SD32×C73C2×C4semidirect
128. D16×C73C2×C4semidirect
308. AGL(1,ℤ/16)D73semidirect
313. C73#33471semidirect
331. #444C2×D73semidirect
337. #442C2×D73semidirect
348. #365C2×D73semidirect
352. #361C2×D73semidirect
356. #360C2×D73semidirect
360. #347C2×D73semidirect
403. SD32C4×D73semidirect
407. D16C4×D73semidirect

Maximal subgroups

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

subgroupindex
#509×D732
#226188452
#226187802
#226118322
#226118312
#226118302
#226118292
#226118282
#226118272
#226118262
#226117022
#361×D732
#226116652
#226115342
C73×AGL(1,ℤ/16)2
#3347173