Finite Group Databasebeta

← #33084567

Subgroups — #33084567

Order 18688.

1014 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988706C211yesmaximal, characteristic
2C2×#17988702C211yesmaximal, characteristic
3#22618727C211yesmaximal, characteristic
4#22618358C211yesmaximal, characteristic, complemented
5#22618357C211yesmaximal, characteristic, complemented
6#22609836C211yesmaximal, characteristic, complemented
7#22609814C211yesmaximal, characteristic, complemented
8#22609811C211yesmaximal, characteristic, complemented
9#22609775C211yesmaximal, characteristic, complemented
10#22609771C211yesmaximal, characteristic, complemented
11#22609767C211yesmaximal, characteristic, complemented
12#104×Dic73C211yesmaximal, characteristic, complemented
13#22609735C211yesmaximal, characteristic, complemented
14#97×Dic73C211yesmaximal, characteristic, complemented
15C73×#1765C211yesmaximal, characteristic, complemented
16C2×#14556611C2211yescharacteristic
17C2×#14556599C2211yescharacteristic
18C2×#14556598C2211yescharacteristic
19C2×#14556597C2211yescharacteristic
20#41×Dic73C2211yescharacteristic
21#17989055C2211yescharacteristic
22#17988706C418yescomplemented
23#17988702C418yescomplemented
24#17988696C412yescomplemented
25#17988691C412yescomplemented
26#17988690C412yescomplemented
27#17988684C412yescomplemented
28C2×#14556526C2211yescharacteristic
29C2×#14556526C2211yescharacteristic
30#17988623C2211yescharacteristic
31C2×#14556519C2211yescharacteristic
32C2×#14556518C2211yescharacteristic
33C2×#14556517C2211yescharacteristic
34#17988584C2211yescharacteristic
35#17988545—21no—
36#17988545C2211yescharacteristic
37#17988544—21no—
38#17988543C2211yescharacteristic
39#17988542—21no—
40#17988542C2211yescharacteristic
41#17988541—21no—
42M16×Dic73—21no—
43M16×Dic73C2211yescharacteristic
44M16×Dic73C2211yescharacteristic
45C4×#11924011—21no—
46C2×#14556502C2211yescharacteristic
47C2×#14556501—21no—
48C2×#14556501C2211yescharacteristic
49C2×#14556501C2211yescharacteristic
50#97×D73—21no—
51C2×C8×Dic73C2211yescharacteristic
52C2×C8×Dic73C2211yescharacteristic
53C42×Dic73C2211yescharacteristic
54#17988352C2211yescharacteristic
55#17988348C2211yescharacteristic
56#17988347C2211yescharacteristic
57#17988346C2211yescharacteristic
58C2×#131×C73C2211yescharacteristic
59C2×#104×C73C2211yescharacteristic
60#416×C73C2211yescharacteristic
61#405×C73C2211yescharacteristic
62C2×#97×C73C2211yescharacteristic
63#341×C73C2211yescharacteristic
64#339×C73C2211yescharacteristic
65C2×#11924028C2311yescharacteristic
66#14556611C2×C414yes—
67#14556611C2×C414yes—
68C2×#11924018C2×C411yescharacteristic
69C2×#11924016C2×C411yescharacteristic
70C2×#11924015C2×C411yescharacteristic
71C2×#11924013C2×C411yescharacteristic
72#14556599—22no—
73#14556599C2×C414yescomplemented
74#14556598—22no—
75#14556598C2×C414yescomplemented
76#14556597—22no—
77#14556597C2×C414yescomplemented
78C2×#11924012—21no—
79C2×#11924012C2311yescharacteristic
80C2×#11924012C2311yescharacteristic
81C2×#11924011—21no—
82C2×#11924011—21no—
83C22×#10596831—21no—
84C22×#10596831C2311yescharacteristic
85C22×#10596831C2311yescharacteristic
86C2×C8×D73—21no—
87C2×C8×D73—21no—
88Q8×Dic73C2×C412yes—
89#14556556C2×C412yes—
90D4×Dic73—22no—
91D4×Dic73C2×C412yes—
92#14556553C2×C412yes—
93#14556551—22no—
94#14556550—21no—
95#14556550C2311yescharacteristic
96#14556548—21no—
97C2×C4×Dic73—21no—
98C2×C4×Dic73—22no—
99C2×C4×Dic73C2311yescharacteristic
100C2×C4×Dic73C2311yescharacteristic
101C42×D73—21no—
102#14556526C2×C414yescomplemented
103#14556526C2×C414yescomplemented
104#14556524C2×C412yes—
105#14556522C2×C412yes—
106#14556519C2×C414yescomplemented
107#14556518C2×C414yescomplemented
108#14556517C2×C414yescomplemented
109#14556516C2×C412yes—
110#14556513C2×C412yes—
111#14556502—21no—
112#14556502C2×C414yescomplemented
113#14556501—21no—
114#14556501—21no—
115#14556501—21no—
116#14556501—21no—
117#14556501C2×C412yes—
118#14556501C2×C412yes—
119#14556501D411yescharacteristic
120#14556501D411yescharacteristic
121#14556501D411yescharacteristic
122#14556501D411yescharacteristic
123C4×#10596831—21no—
124C8×Dic73—21no—
125C8×Dic73—22no—
126C8×Dic73—21no—
127C8×Dic73—21no—
128C8×Dic73—21no—
129C8×Dic73C2×C414yescomplemented
130C2×#41×C73C2311yescharacteristic
131#131×C73—22no—
132#131×C73C2×C414yescomplemented
133C2×M16×C73—21no—
134C2×M16×C73C2311yescharacteristic
135C2×M16×C73C2311yescharacteristic
136C22×C8×C73—21no—
137C22×C8×C73C2311yescharacteristic
138C22×C8×C73C2311yescharacteristic
139#117×C73C2311yescharacteristic
140C2×C42×C73C2311yescharacteristic
141#104×C73C2×C414yescomplemented
142#103×C73C2×C412yes—
143#102×C73C2×C412yes—
144#97×C73—21no—
145#97×C73—22no—
146C4×C8×C73—21no—
147#11924028—21no—
148#11924028—21no—
149#11924028C22×C412yes—
150#11924028C22×C412yes—
151C2×Dic146C22×C411yescharacteristic
152C2×D292C22×C411yescharacteristic
153C2×C22⋊D73—21no—
154C22×Dic73—21no—
155C22×Dic73—22no—
156C22×Dic73#4112yes—
157C2×C4×D73—21no—
158C2×C4×D73—22no—
159#11924018C4214yescomplemented
160#11924016C4214yescomplemented
161#11924015C4214yescomplemented
162#11924013C4214yescomplemented
163#11924012—21no—
164#11924012—21no—
165#11924012—24no—
166#11924012C22×C412yes—
167#11924012C22×C412yes—
168#11924011—24no—
169#11924011—24no—
170C2×#10596831—21no—
171C2×#10596831—21no—
172C2×#10596831—24no—
173C2×#10596831C22×C412yes—
174C2×#10596831C22×C411yescharacteristic
175C2×#10596831C22×C411yescharacteristic
176C2×#10596831C22×C412yes—
177C2×#10596831C22×C412yes—
178C2×#10596831C2×D411yescharacteristic
179C2×#10596831C2×D411yescharacteristic
180C2×#10596831#4112yes—
181C8×D73—24no—
182C8×D73—24no—
183C4⋊Dic73—22no—
184C4⋊Dic73C22×C412yes—
185#11924007—22no—
186C22⋊Dic73—21no—
187C22⋊Dic73—22no—
188#11924005—22no—
189C4×Dic73—21no—
190C4×Dic73—21no—
191C4×Dic73—22no—
192C4×Dic73—22no—
193C4×Dic73—44no—
194C4×Dic73C22×C412yes—
195C4×Dic73C22×C412yes—
196C4×Dic73C2×D411yescharacteristic
197C4×Dic73C2×D411yescharacteristic
198#41×C73—21no—
199#41×C73—21no—
200#41×C73C22×C412yes—
201#41×C73C22×C412yes—
202C2×Q8×C73C22×C411yescharacteristic
203C2×D4×C73—21no—
204C2×D4×C73C22×C411yescharacteristic
205C22×C4×C73—21no—
206C22×C4×C73—21no—
207C22×C4×C73C2411yescharacteristic
208M16×C73—21no—
209M16×C73—21no—
210M16×C73—24no—
211M16×C73C22×C412yes—
212M16×C73C22×C412yes—
213C2×C8×C73—21no—
214C2×C8×C73—21no—
215C2×C8×C73—21no—
216C2×C8×C73—21no—
217C2×C8×C73—24no—
218C2×C8×C73C22×C412yes—
219C2×C8×C73C22×C412yes—
220C2×C8×C73C2×D411yescharacteristic
221C2×C8×C73C2×D411yescharacteristic
222C2×C8×C73#4112yes—
223C4⋊C4×C73C22×C412yes—
224C22⋊C4×C73—21no—
225C42×C73—21no—
226C42×C73—21no—
227C42×C73C22×C412yes—
228C23×C73—21no—
229C23×C73C2×#4111yescharacteristic
230C22×D73—21no—
231Q8×C73—21no—
232Q8×C73C2×C4212yes—
233D4×C73—22no—
234D4×C73—22no—
235D4×C73—21no—
236D4×C73C2×C4212yes—
237C2×C4×C73—21no—
238C2×C4×C73—22no—
239C2×C4×C73—22no—
240C2×C4×C73—22no—
241C2×C4×C73—22no—
242C2×C4×C73—42no—
243C2×C4×C73C23×C411yescharacteristic
244C2×C4×C73C23×C411yescharacteristic
245C2×C4×C73C23×C411yescharacteristic
246C2×C4×C73C22×D411yescharacteristic
247C2×C4×C73C2×#4111yescharacteristic
248C2×C4×C73C2×#4111yescharacteristic
249C22⋊D73—22no—
250C22⋊D73—22no—
251C2×Dic73—24no—
252C2×Dic73—21no—
253C2×Dic73—22no—
254C2×Dic73—22no—
255C2×Dic73—42no—
256C2×Dic73—82no—
257C2×Dic73C4×D412yes—
258C2×Dic73C4×D412yes—
259D292—21no—
260D292C2×C4212yes—
261C4×D73—22no—
262C4×D73—22no—
263C4×D73—82no—
264Dic146—21no—
265Dic146C2×C4212yes—
266C8×C73—22no—
267C8×C73—22no—
268C8×C73—22no—
269C8×C73C4×D412yes—
270C8×C73C4×D412yes—
271#10596831—22no—
272#10596831—22no—
273#10596831C2×C4212yes—
274#10596831C2×C4212yes—
275#10596831C4×D412yes—
276#10596831C4×D412yes—
277C22×C73—22no—
278C22×C73—22no—
279C22×C73—42no—
280C22×C73C2×C4×D411yescharacteristic
281C22×C73C4×#4111yescharacteristic
282C22×C73C4×#4111yescharacteristic
283C2×D73—22no—
284C2×D73—42no—
285C4×C73—22no—
286C4×C73—42no—
287C4×C73—42no—
288C4×C73C22×C4211yescharacteristic
289C4×C73C2×C4×D411yescharacteristic
290C4×C73C2×C4×D411yescharacteristic
291C4×C73C4×#4111yescharacteristic
292Dic73—24no—
293Dic73—22no—
294Dic73—22no—
295Dic73—22no—
296Dic73—42no—
297#20512—731nomaximal, Sylow
298C2×C73—21no—
299C2×C73—21no—
300C2×C73—42no—
301C2×C73C42×D411yescharacteristic
302C2×C73#295612yes—
303D73—42no—
304C2×#444—731no—
305C2×#443—731no—
306#2943—731no—
307#2875—731no—
308#2875—731no—
309#1778—731no—
310#1773—731no—
311#1772—731no—
312#1765—731no—
313#1765D7311yescharacteristic, complemented
314#1764—731no—
315#1764—731no—
316C4×#104—731no—
317C4×#104—731no—
318C4×#97—731no—
319C73#2051211yescharacteristic, complemented, Sylow
320C2×#135—731no—
321C2×#131—731no—
322C2×#131—731no—
323C2×#131—731no—
324C2×#131C2×D7311yescharacteristic
325C4×#41—731no—
326C4×#41—731no—
327#444—738no—
328#443—738no—
329#442—732no—
330#441—732no—
331#440—732no—
332#440—732no—
333C2×#108—731no—
334C2×#108—731no—
335#427—731no—
336C2×#104—731no—
337C2×#104—731no—
338C2×#104—731no—
339C2×#104C2×D7311yescharacteristic
340#416—731no—
341#416C2×D7311yescharacteristic
342#405—731no—
343#405—731no—
344#405—731no—
345#405—1461no—
346#405—1461no—
347#405—1461no—
348#405—1461no—
349#405C2×D7311yescharacteristic
350C4×M16—731no—
351C4×M16—731no—
352C4×M16—1461no—
353C4×M16—1461no—
354C2×#97—731no—
355C2×#97—731no—
356C2×#97—731no—
357C2×#97—1461no—
358C2×#97—1461no—
359C2×#97C2×D7311yescharacteristic
360C2×C4×C8—731no—
361C2×C4×C8—731no—
362C43—731no—
363#341—731no—
364#341C2×D7311yescharacteristic
365#339—731no—
366#339—731no—
367#339—731no—
368#339C2×D7311yescharacteristic
369C2×#41—731no—
370C2×#41C22×D7311yescharacteristic
371#135—734no—
372#135—734no—
373C2×Q16—731no—
374C2×SD16—731no—
375C2×SD16—731no—
376C2×D8—731no—
377#131—22no—
378#131—734no—
379#131—734no—
380#131—734no—
381#131—1462no—
382#131—1462no—
383#131—1462no—
384#131C4×D7314yescomplemented
385C2×M16—21no—
386C2×M16—731no—
387C2×M16—731no—
388C2×M16—1461no—
389C2×M16—1461no—
390C2×M16—1461no—
391C2×M16C22×D7311yescharacteristic
392C2×M16C22×D7311yescharacteristic
393C22×C8—21no—
394C22×C8—731no—
395C22×C8—731no—
396C22×C8—1461no—
397C22×C8—1461no—
398C22×C8—1461no—
399C22×C8C22×D7311yescharacteristic
400C22×C8C22×D7311yescharacteristic
401C4×Q8—732no—
402C4×Q8—732no—
403C4×D4—732no—
404C4×D4—732no—
405C4×D4—1462no—
406C4×D4—1462no—
407#117—731no—
408#117—1461no—
409#117—1461no—
410#117C22×D7311yescharacteristic
411C2×C42—731no—
412C2×C42—731no—
413C2×C42—1461no—
414C2×C42—1462no—
415C2×C42—1461no—
416C2×C42C22×D7311yescharacteristic
417#108—734no—
418#108—734no—
419#107—732no—
420#106—732no—
421#104—734no—
422#104—734no—
423#104—734no—
424#104C4×D7314yescomplemented
425#103—732no—
426#103C4×D7312yes—
427#102—732no—
428#102C4×D7312yes—
429#97—21no—
430#97—22no—
431#97—732no—
432#97—732no—
433#97—731no—
434#97—731no—
435#97—731no—
436#97—731no—
437#97—734no—
438#97—1461no—
439#97—1461no—
440#97—1461no—
441#97—1461no—
442#97—1461no—
443C4×C8—21no—
444C4×C8—734no—
445C4×C8—1461no—
446C4×C8—1461no—
447C4×C8—1461no—
448C4×C8—1462no—
449C4×C8—1461no—
450C4×C8—1461no—
451#41—21no—
452#41—21no—
453#41—732no—
454#41—732no—
455#41—1461no—
456#41—1461no—
457#41C2×C4×D7312yes—
458#41C2×C4×D7312yes—
459C2×Q8—731no—
460C2×Q8C2×C4×D7311yescharacteristic
461C2×D4—21no—
462C2×D4—731no—
463C2×D4—1461no—
464C2×D4C2×C4×D7311yescharacteristic
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—732no—
468C22×C4—1462no—
469C22×C4—1462no—
470C22×C4—1461no—
471C22×C4—1461no—
472C22×C4C23×D7311yescharacteristic
473Q16—734no—
474SD16—734no—
475SD16—734no—
476D8—734no—
477M16—21no—
478M16—21no—
479M16—24no—
480M16—732no—
481M16—732no—
482M16—1464no—
483M16—1464no—
484M16—1461no—
485M16—1461no—
486M16—1464no—
487M16C2×C4×D7312yes—
488M16C2×C4×D7312yes—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—24no—
494C2×C8—732no—
495C2×C8—731no—
496C2×C8—731no—
497C2×C8—732no—
498C2×C8—732no—
499C2×C8—731no—
500C2×C8—731no—
501C2×C8—732no—
502C2×C8—1464no—
503C2×C8—1464no—
504C2×C8—1461no—
505C2×C8—1461no—
506C2×C8—1464no—
507C2×C8C2×C4×D7312yes—
508C2×C8C2×C4×D7312yes—
509C2×C8D4×D7311yescharacteristic
510C2×C8D4×D7311yescharacteristic
511C2×C8#1192402712yes—
512C4⋊C4—732no—
513C4⋊C4—1462no—
514C4⋊C4—1462no—
515C4⋊C4C2×C4×D7312yes—
516C22⋊C4—21no—
517C22⋊C4—1462no—
518C22⋊C4—1462no—
519C22⋊C4—1461no—
520C42—21no—
521C42—21no—
522C42—2924no—
523C42—732no—
524C42—731no—
525C42—731no—
526C42—732no—
527C42—1462no—
528C42—1462no—
529C42—1461no—
530C42—1461no—
531C42C2×C4×D7312yes—
532C23—21no—
533C23—1461no—
534C23#41×D7311yescharacteristic
535Q8—21no—
536Q8—732no—
537Q8—1461no—
538Q8C42×D7312yes—
539D4—22no—
540D4—22no—
541D4—21no—
542D4—732no—
543D4—1462no—
544D4—1462no—
545D4—1461no—
546D4C42×D7312yes—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—42no—
553C2×C4—2922no—
554C2×C4—5842no—
555C2×C4—5842no—
556C2×C4—732no—
557C2×C4—732no—
558C2×C4—1462no—
559C2×C4—1464no—
560C2×C4—1461no—
561C2×C4—1462no—
562C2×C4—1462no—
563C2×C4—1462no—
564C2×C4C22×C4×D7311yescharacteristic
565C2×C4C22×C4×D7311yescharacteristic
566C2×C4C22×C4×D7311yescharacteristic
567C2×C4C2×D4×D7311yescharacteristic
568C2×C4#41×D7311yescharacteristic
569C2×C4C2×#1192402711yescharacteristic
570C8—22no—
571C8—22no—
572C8—22no—
573C8—732no—
574C8—732no—
575C8—732no—
576C8—732no—
577C8—1462no—
578C8—1462no—
579C8#1455655112yes—
580C8#1455655112yes—
581C22—22no—
582C22—22no—
583C22—42no—
584C22—2922no—
585C22—1462no—
586C22C4×D4×D7311yescharacteristic
587C22C4×#1192402711yescharacteristic
588C22#1798905511yescharacteristic
589C4—22no—
590C4—42no—
591C4—42no—
592C4—2922no—
593C4—1464no—
594C4—1462no—
595C4—1462no—
596C4—1462no—
597C4C2×C42×D7311yescharacteristic
598C4C4×D4×D7311yescharacteristic
599C4C2×#1455655111yescharacteristic
600C4C4×#1192402711yescharacteristic
601C2—21no—
602C2—21no—
603C2—42no—
604C2—2922no—
605C2C4×#1455655111yescharacteristic
606C2#2261883912yes—
subgroupG/H
centerC2×C4C2×D4×D73
commutatorC4×C73C22×C42
FrattiniC22×C4C23×D73
FittingC73×#1765C2
socleC22×C73C2×C4×D4
prime radicalC2×#14556611C22
cosocleC22×C4×C73C24
hypercenter#1765D73
hypocenterC73#20512
LCSG ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1765

Complements

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

NH ≅ G/Ntype
4. #22618358C2semidirect
5. #22618357C2semidirect
6. #22609836C2semidirect
7. #22609814C2semidirect
8. #22609811C2semidirect
9. #22609775C2semidirect
10. #22609771C2semidirect
11. #22609767C2semidirect
12. #104×Dic73C2semidirect
13. #22609735C2semidirect
14. #97×Dic73C2semidirect
15. C73×#1765C2semidirect
22. #17988706C4semidirect
23. #17988702C4semidirect
24. #17988696C4semidirect
25. #17988691C4semidirect
26. #17988690C4semidirect
27. #17988684C4semidirect
73. #14556599C2×C4semidirect
75. #14556598C2×C4semidirect
77. #14556597C2×C4semidirect
102. #14556526C2×C4semidirect
103. #14556526C2×C4semidirect
106. #14556519C2×C4semidirect
107. #14556518C2×C4semidirect
108. #14556517C2×C4semidirect
112. #14556502C2×C4semidirect
129. C8×Dic73C2×C4semidirect
132. #131×C73C2×C4semidirect
141. #104×C73C2×C4semidirect
159. #11924018C42semidirect
160. #11924016C42semidirect
161. #11924015C42semidirect
162. #11924013C42semidirect
313. #1765D73semidirect
319. C73#20512semidirect
384. #131C4×D73semidirect
424. #104C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179887062
C2×#179887022
#226187272
#226183582
#226183572
#226098362
#226098142
#226098112
#226097752
#226097712
#226097672
#104×Dic732
#226097352
#97×Dic732
C73×#17652
#2051273