Finite Group Databasebeta

← #33999854

Subgroups — #33999854

Order 18688.

10698 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22624744C212yesmaximal, complemented
2#22624717C212yesmaximal, complemented
3#22624715C2112yesmaximal, complemented
4#22624713C2116yesmaximal, complemented
5#22624700C216yesmaximal, complemented
6#22624699C218yesmaximal, complemented
7C2×#17989404C211yesmaximal, characteristic, complemented
8C2×#17989401C216yesmaximal, complemented
9#578×D73C218yesmaximal, complemented
10#22624642C211yesmaximal, characteristic, complemented
11C73×#3584C211yesmaximal, characteristic, complemented
12#17989423—28no—
13#17989423C2211yescharacteristic
14#17989423C22112yescomplemented
15#17989422—212no—
16#17989422C22116yescomplemented
17C2×#14556638—28no—
18C2×#14556638C2212yescomplemented
19C2×#14556638C22112yescomplemented
202−1+4×D73—24no—
212−1+4×D73C22116yescomplemented
22#17989404C2218yescomplemented
23#17989404C2218yescomplemented
24#17989403C22148yescomplemented
25#17989402C22116yescomplemented
26#17989401C22124yescomplemented
27#17989401C22112yescomplemented
28#17989401C22124yescomplemented
29#17989401C22124yescomplemented
30#17989400C22112yescomplemented
31#17989398C22116yescomplemented
32#17989397C22148yescomplemented
33#17989394C22124yescomplemented
34#17989393C22124yescomplemented
35#17989389C2214yescomplemented
36#17989388C22112yescomplemented
37#17989385C22112yescomplemented
38#17989384C22124yescomplemented
39#17989383C22148yescomplemented
40C2×#14556624C2216yescomplemented
41C2×#14556621C22112yescomplemented
42C2×#14556620C22124yescomplemented
43#137×D73C22148yescomplemented
44C2×#14556613C2213yescomplemented
45C2×#14556610C22112yescomplemented
46#135×D73C22124yescomplemented
47C22×Dic292C2213yescomplemented
48C2×Q16×D73C22112yescomplemented
49#17989354C2211yescharacteristic, complemented
50#17989353C2216yescomplemented
51#17989352C2218yescomplemented
52C2×#14556597C2216yescomplemented
53#131×D73C2218yescomplemented
54#585×C73C2212yescomplemented
55#578×C73C2214yescomplemented
56#577×C73C2218yescomplemented
57#576×C73C2214yescomplemented
58#575×C73C2216yescomplemented
59C2×#135×C73C2216yescomplemented
60C2×#131×C73C2211yescharacteristic, complemented
61#14556639—216no—
62#14556639D412yes—
63#14556639C23124yes—
64#14556638—224no—
65#14556638—224no—
66#14556638—248no—
67#14556638—212no—
68#14556638—216no—
69#14556638—28no—
70#14556638D414yes—
71#14556638C23116yes—
72#14556638C23124yes—
73#14556638C23148yes—
74#14556637—212no—
75#14556637D414yes—
76#14556637C2318yes—
77#14556636—224no—
78#14556636—248no—
79#14556636D416yes—
80#14556636C23124yes—
81C2×#11924028—224no—
82C2×#11924028—212no—
83C2×#11924028C2316yes—
84C2×#11924028C2316yes—
85C2×#11924027—248no—
86C2×#11924027—224no—
87C2×#11924027—26no—
88C2×#11924027C23124yes—
89#41×D73—216no—
90#41×D73—224no—
91#41×D73—248no—
92#41×D73C23148yes—
93#41×D73C2318yes—
94C22×Dic146—212no—
95C22×Dic146C2316yes—
96C2×Q8×D73—216no—
97C2×Q8×D73—22no—
98C2×Q8×D73C23124yes—
99#14556624C23148yes—
100#14556623C23148yes—
101#14556622C23124yes—
102#14556621C23148yes—
103#14556621C23124yes—
104#14556620C23196yes—
105#14556620C23148yes—
106#14556618C23116yes—
107#14556617C23124yes—
108#14556616C2318yes—
109#14556615C23148yes—
110#14556614C2318yes—
111#14556613C23112yes—
112#14556613C23112yes—
113#14556612C23148yes—
114#14556611C23148yes—
115#14556610C23148yes—
116#14556610C23124yes—
117C2×Dic292C23112yes—
118C2×Dic292C2316yes—
119C2×Dic292C2313yes—
120C2×Dic292C23112yes—
121C2×#11924018C23124yes—
122Q16×D73C23148yes—
123Q16×D73C23124yes—
124C2×#11924017C2316yes—
125C2×#11924016C23124yes—
126SD16×D73C23148yes—
127C2×D584C2311yescharacteristic
128D8×D73C2318yes—
129#14556599C23124yes—
130#14556598C2318yes—
131#14556597C23112yes—
132#14556597C2314yes—
133#14556597C23112yes—
134#14556597C23112yes—
135C2×#11924012C2316yes—
136C2×#11924011C23112yes—
137M16×D73C23124yes—
138C2×C8×D73C23112yes—
1392−1+4×C73—22no—
1402−1+4×C73C2318yes—
141H(5,𝔽2)×C73—26no—
142H(5,𝔽2)×C73C2318yes—
143C2×#41×C73—28no—
144C2×#41×C73C23112yes—
145C2×#41×C73C2311yescharacteristic
146#137×C73C23124yes—
147AGL(1,ℤ/8)×C73C23124yes—
148#135×C73C23112yes—
149#135×C73C2314yes—
150#135×C73C23112yes—
151#135×C73C23112yes—
152C2×Q16×C73C2316yes—
153C2×SD16×C73C23112yes—
154C2×D8×C73C2316yes—
155#131×C73C2314yes—
156#131×C73C2314yes—
157C2×M16×C73C2313yes—
158C22×C8×C73C2313yes—
159#11924029—224no—
160#11924029C2×D418yes—
161#11924029C24116yes—
162#11924028—212no—
163#11924028—248no—
164#11924028—248no—
165#11924028—224no—
166#11924028—248no—
167#11924028—28no—
168#11924028C2×D4112yes—
169#11924028C2×D4112yes—
170#11924028C2414yes—
171#11924028C24112yes—
172#11924028C24124yes—
173#11924027—248no—
174#11924027—248no—
175#11924027—296no—
176#11924027—248no—
177#11924027—248no—
178#11924027—224no—
179#11924027C2×D4124yes—
180#11924027C24148yes—
181C2×Dic146—26no—
182C2×Dic146—212no—
183C2×Dic146—224no—
184C2×Dic146—248no—
185C2×Dic146—224no—
186C2×Dic146C2×D416yes—
187C2×Dic146C24124yes—
188C2×Dic146C2416yes—
189Q8×D73—216no—
190Q8×D73—248no—
191Q8×D73—224no—
192Q8×D73C2×D418yes—
193Q8×D73C24148yes—
194C2×D292—28no—
195C2×D292C2×D416yes—
196C2×D292C2412yes—
197C2×C22⋊D73—26no—
198C2×C22⋊D73—224no—
199C2×C22⋊D73—248no—
200C2×C22⋊D73C2×D4112yes—
201D4×D73—248no—
202D4×D73—216no—
203D4×D73—224no—
204D4×D73C2×D4124yes—
205D4×D73C24116yes—
206C22×Dic73—26no—
207C22×Dic73—224no—
208C2×C4×D73—224no—
209C2×C4×D73—26no—
210C2×C4×D73—248no—
211C2×C4×D73C24112yes—
212Dic292C2416yes—
213Dic292C24124yes—
214Dic292C2416yes—
215#11924018C24148yes—
216#11924017C24124yes—
217#11924016C24148yes—
218#11924015C24116yes—
219D584C2412yes—
220D584C2412yes—
221#11924013C24116yes—
222#11924012C24112yes—
223#11924012C2414yes—
224#11924011C24124yes—
225#11924011C2418yes—
226C2×#10596831C24112yes—
227C8×D73C2418yes—
228C8×D73C24124yes—
229#41×C73—28no—
230#41×C73—224no—
231#41×C73—212no—
232#41×C73—212no—
233#41×C73C2×D414yes—
234#41×C73C2418yes—
235#41×C73C24124yes—
236#41×C73C2414yes—
237C2×Q8×C73—28no—
238C2×Q8×C73C2×D411yescharacteristic
239C2×Q8×C73C24112yes—
240C2×D4×C73—212no—
241C2×D4×C73—224no—
242C2×D4×C73C2×D413yes—
243C2×D4×C73C24112yes—
244C22×C4×C73—212no—
245C22×C4×C73C2413yes—
246Q16×C73C2414yes—
247Q16×C73C24112yes—
248SD16×C73C2418yes—
249SD16×C73C24124yes—
250D8×C73C2414yes—
251D8×C73C24112yes—
252M16×C73C24112yes—
253C2×C8×C73C2411yescharacteristic
254C2×C8×C73C2413yes—
255C2×C8×C73C2416yes—
256C2×C8×C73C2416yes—
257C23×C73—212no—
258C23×C73C22×D413yes—
259C22×D73—22no—
260C22×D73—216no—
261C22×D73C22×D4112yes—
262Q8×C73—212no—
263Q8×C73C22×D414yes—
264Q8×C73C2518yes—
265D4×C73—28no—
266D4×C73—224no—
267D4×C73—212no—
268D4×C73C22×D4112yes—
269D4×C73C2518yes—
270C2×C4×C73—212no—
271C2×C4×C73—28no—
272C2×C4×C73—224no—
273C2×C4×C73C22×D416yes—
274C2×C4×C73C22×D413yes—
275C2×C4×C73C2511yescharacteristic
276C2×C4×C73C2516yes—
277C22⋊D73—224no—
278C22⋊D73—248no—
279C22⋊D73—216no—
280C22⋊D73C22×D418yes—
281C22⋊D73C22×D4124yes—
282C2×Dic73—26no—
283C2×Dic73—224no—
284C2×Dic73—248no—
285C2×Dic73C22×D4112yes—
286D292—216no—
287D292—22no—
288D292C22×D4112yes—
289D292C22×D414yes—
290D292C2514yes—
291C4×D73—248no—
292C4×D73—224no—
293C4×D73—216no—
294C4×D73C22×D4124yes—
295C4×D73C2518yes—
296Dic146—224no—
297Dic146—26no—
298Dic146—248no—
299Dic146C22×D4112yes—
300Dic146C22×D414yes—
301Dic146C25112yes—
302C8×C73C2511yescharacteristic
303C8×C73C2511yescharacteristic
304C8×C73C2516yes—
305#10596831C2518yes—
306C22×C73—28no—
307C22×C73—43no—
308C22×C73—812no—
309C22×C73C23×D411yescharacteristic
310C22×C73C23×D416yes—
311C2×D73—24no—
312C2×D73—412no—
313C2×D73—816no—
314C2×D73—82no—
315C2×D73C23×D418yes—
316C4×C73—28no—
317C4×C73C23×D411yescharacteristic
318C4×C73C23×D416yes—
319C4×C73C2611yescharacteristic
320Dic73—212no—
321Dic73C23×D418yes—
322#63093—731nomaximal, Sylow
323C2×C73—26no—
324C2×C73—21no—
325C2×C73—48no—
326C2×C73C24×D411yescharacteristic
327D73—28no—
328D73—44no—
3292−1+6—732no—
330#3586—732no—
331#3586—7316no—
332#3586—7312no—
333#3584—738no—
334#3584—736no—
335#3584D7311yescharacteristic, complemented
336C2×#578—738no—
337C2×#578—736no—
338C2×#578—731no—
339#3573—731no—
340C73#6309311yescharacteristic, complemented, Sylow
341#585—731no—
342#585—7316no—
343#585—7312no—
344#585—1468no—
345#585—14612no—
346#585C2×D7312yescomplemented
347C2×2−1+4—7316no—
348C2×2−1+4—7312no—
349C2×2−1+4—732no—
350C2×2−1+4—1468no—
351C2×2−1+4—1464no—
352#578—7316no—
353#578—7348no—
354#578—7324no—
355#578—7312no—
356#578—7324no—
357#578—7324no—
358#578—7312no—
359#578—738no—
360#578—738no—
361#578C2×D7314yescomplemented
362#577—7316no—
363#577—7348no—
364#577—7324no—
365#577—7324no—
366#577C2×D7318yescomplemented
367#576—734no—
368#576—7312no—
369#576C2×D7314yescomplemented
370#575—7312no—
371#575—7348no—
372#575—7324no—
373#575C2×D7316yescomplemented
374C2×#137—7312no—
375C2×#137—7348no—
376C2×#137—7324no—
377C2×#137—736no—
378C2×#135—7324no—
379C2×#135—7312no—
380C2×#135—733no—
381C2×#135C2×D7316yescomplemented
382C22×Q16—7312no—
383C22×Q16—733no—
384#568—731no—
385#568—738no—
386#568—736no—
387C2×#131—738no—
388C2×#131—736no—
389C2×#131C2×D7311yescharacteristic, complemented
3902−1+4—22no—
3912−1+4—734no—
3922−1+4—732no—
3932−1+4—7348no—
3942−1+4—7324no—
3952−1+4—7316no—
3962−1+4—7324no—
3972−1+4—14616no—
3982−1+4—14616no—
3992−1+4—14612no—
4002−1+4—14648no—
4012−1+4—14624no—
4022−1+4—14624no—
4032−1+4—1468no—
4042−1+4C22×D7318yes—
405H(5,𝔽2)—26no—
406H(5,𝔽2)—734no—
407H(5,𝔽2)—736no—
408H(5,𝔽2)—738no—
409H(5,𝔽2)—7324no—
410H(5,𝔽2)—14624no—
411H(5,𝔽2)—14648no—
412H(5,𝔽2)—14612no—
413H(5,𝔽2)C22×D7318yes—
414C2×#41—28no—
415C2×#41—738no—
416C2×#41—736no—
417C2×#41—7348no—
418C2×#41—7324no—
419C2×#41—736no—
420C2×#41—14612no—
421C2×#41—14624no—
422C2×#41—14624no—
423C2×#41—14648no—
424C2×#41—14648no—
425C2×#41—14616no—
426C2×#41—14624no—
427C2×#41—1466no—
428C2×#41C22×D73112yes—
429C2×#41C22×D7311yescharacteristic
430C22×Q8—7324no—
431C22×Q8—736no—
432C22×Q8—14612no—
433C22×Q8—14616no—
434C22×Q8—1462no—
435#137—7348no—
436#137—7324no—
437#137—7324no—
438#137—7348no—
439#137—7396no—
440#137—7348no—
441#137—7348no—
442#137C22×D73124yes—
443AGL(1,ℤ/8)—738no—
444AGL(1,ℤ/8)—7324no—
445AGL(1,ℤ/8)—7316no—
446AGL(1,ℤ/8)—7348no—
447AGL(1,ℤ/8)C22×D73124yes—
448#135—7348no—
449#135—738no—
450#135—7348no—
451#135—7324no—
452#135—7348no—
453#135—7312no—
454#135—7312no—
455#135C22×D73112yes—
456#135C22×D7314yes—
457#135C22×D73112yes—
458#135C22×D73112yes—
459C2×Q16—7324no—
460C2×Q16—7324no—
461C2×Q16—7348no—
462C2×Q16—7312no—
463C2×Q16—733no—
464C2×Q16—7312no—
465C2×Q16—736no—
466C2×Q16C22×D7316yes—
467C2×SD16—7324no—
468C2×SD16—7348no—
469C2×SD16—736no—
470C2×SD16C22×D73112yes—
471C2×D8—738no—
472C2×D8—731no—
473C2×D8C22×D7316yes—
474#131—738no—
475#131—7324no—
476#131—7312no—
477#131—734no—
478#131—7312no—
479#131—7312no—
480#131C22×D7314yes—
481#131C22×D7314yes—
482C2×M16—736no—
483C2×M16—7324no—
484C2×M16—7312no—
485C2×M16C22×D7313yes—
486C22×C8—7312no—
487C22×C8C22×D7313yes—
488#41—28no—
489#41—224no—
490#41—212no—
491#41—212no—
492#41—738no—
493#41—7324no—
494#41—7312no—
495#41—7312no—
496#41—7312no—
497#41—734no—
498#41—7316no—
499#41—7348no—
500#41—7324no—
501#41—14648no—
502#41—14648no—
503#41—14624no—
504#41—14648no—
505#41—1468no—
506#41—14648no—
507#41—14648no—
508#41—14648no—
509#41—14696no—
510#41—14624no—
511#41—14624no—
512#41—14648no—
513#41—14612no—
514#41D4×D7314yes—
515#41C23×D73124yes—
516#41C23×D7318yes—
517#41C23×D7314yes—
518C2×Q8—28no—
519C2×Q8—738no—
520C2×Q8—736no—
521C2×Q8—7348no—
522C2×Q8—7324no—
523C2×Q8—736no—
524C2×Q8—14624no—
525C2×Q8—14624no—
526C2×Q8—14648no—
527C2×Q8—14648no—
528C2×Q8—14624no—
529C2×Q8—14616no—
530C2×Q8—14612no—
531C2×Q8—1466no—
532C2×Q8D4×D7311yescharacteristic
533C2×Q8C23×D73112yes—
534C2×D4—212no—
535C2×D4—224no—
536C2×D4—7312no—
537C2×D4—7324no—
538C2×D4—7316no—
539C2×D4—736no—
540C2×D4—732no—
541C2×D4—1468no—
542C2×D4—14616no—
543C2×D4—14624no—
544C2×D4—14648no—
545C2×D4—14648no—
546C2×D4—14624no—
547C2×D4—1466no—
548C2×D4D4×D7313yes—
549C2×D4C23×D73112yes—
550C22×C4—212no—
551C22×C4—7312no—
552C22×C4—14648no—
553C22×C4—14624no—
554C22×C4—14624no—
555C22×C4—1466no—
556C22×C4—1466no—
557C22×C4C23×D7313yes—
558Q16—7348no—
559Q16—736no—
560Q16—7324no—
561Q16—736no—
562Q16C23×D7314yes—
563Q16C23×D73112yes—
564SD16—7316no—
565SD16—7348no—
566SD16—7324no—
567SD16C23×D73124yes—
568SD16C23×D7318yes—
569D8—7316no—
570D8—732no—
571D8—732no—
572D8C23×D7314yes—
573D8C23×D73112yes—
574M16—7312no—
575M16—734no—
576M16—7324no—
577M16—738no—
578M16C23×D73112yes—
579C2×C8—7312no—
580C2×C8—738no—
581C2×C8—7324no—
582C2×C8C23×D7316yes—
583C2×C8C23×D7311yescharacteristic
584C2×C8C23×D7316yes—
585C2×C8C23×D7313yes—
586C23—212no—
587C23—7312no—
588C23—14616no—
589C23—1462no—
590C23C2×D4×D7313yes—
591Q8—212no—
592Q8—734no—
593Q8—7312no—
594Q8—7312no—
595Q8—14648no—
596Q8—1466no—
597Q8—14624no—
598Q8C2×D4×D7314yes—
599Q8C24×D7318yes—
600D4—224no—
601D4—28no—
602D4—212no—
603D4—7324no—
604D4—738no—
605D4—734no—
606D4—7312no—
607D4—734no—
608D4—14616no—
609D4—14616no—
610D4—14648no—
611D4—14624no—
612D4—1462no—
613D4C2×D4×D73112yes—
614D4C24×D7318yes—
615C2×C4—212no—
616C2×C4—28no—
617C2×C4—224no—
618C2×C4—7312no—
619C2×C4—7324no—
620C2×C4—738no—
621C2×C4—14616no—
622C2×C4—14648no—
623C2×C4—14648no—
624C2×C4—14624no—
625C2×C4—1466no—
626C2×C4—14624no—
627C2×C4C2×D4×D7316yes—
628C2×C4C2×D4×D7313yes—
629C2×C4C24×D7316yes—
630C2×C4C24×D7311yescharacteristic
631C8—738no—
632C8C24×D7311yescharacteristic
633C8C24×D7316yes—
634C8C24×D7311yescharacteristic
635C22—28no—
636C22—43no—
637C22—812no—
638C22—29212no—
639C22—5842no—
640C22—58416no—
641C22—738no—
642C22—1464no—
643C22C22×D4×D7316yes—
644C22C22×D4×D7311yescharacteristic
645C4—28no—
646C4—738no—
647C4—14612no—
648C4C22×D4×D7311yescharacteristic
649C4C22×D4×D7316yes—
650C4C25×D7311yescharacteristic
651C2—21no—
652C2—26no—
653C2—48no—
654C2—2924no—
655C2—1468no—
656C2C23×D4×D7311yescharacteristic
subgroupG/H
centerC2C23×D4×D73
commutator, cosocleC4×C73C26
FrattiniC4C25×D73
FittingC73×#3584C2
socleC2×C73C24×D4
hypercenter#3584D73
hypocenterC73#63093
LCSG ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×#41 ◁ #3584

Complements

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

NH ≅ G/Ntype
1. #22624744C2semidirect
2. #22624717C2semidirect
3. #22624715C2semidirect
4. #22624713C2semidirect
5. #22624700C2semidirect
6. #22624699C2semidirect
7. C2×#17989404C2semidirect
8. C2×#17989401C2semidirect
9. #578×D73C2semidirect
10. #22624642C2semidirect
11. C73×#3584C2semidirect
14. #17989423C22semidirect
16. #17989422C22semidirect
18. C2×#14556638C22semidirect
19. C2×#14556638C22semidirect
21. 2−1+4×D73C22semidirect
22. #17989404C22semidirect
23. #17989404C22semidirect
24. #17989403C22semidirect
25. #17989402C22semidirect
26. #17989401C22semidirect
27. #17989401C22semidirect
28. #17989401C22semidirect
29. #17989401C22semidirect
30. #17989400C22semidirect
31. #17989398C22semidirect
32. #17989397C22semidirect
33. #17989394C22semidirect
34. #17989393C22semidirect
35. #17989389C22semidirect
36. #17989388C22semidirect
37. #17989385C22semidirect
38. #17989384C22semidirect
39. #17989383C22semidirect
40. C2×#14556624C22semidirect
41. C2×#14556621C22semidirect
42. C2×#14556620C22semidirect
43. #137×D73C22semidirect
44. C2×#14556613C22semidirect
45. C2×#14556610C22semidirect
46. #135×D73C22semidirect
47. C22×Dic292C22semidirect
48. C2×Q16×D73C22semidirect
49. #17989354C22semidirect
50. #17989353C22semidirect
51. #17989352C22semidirect
52. C2×#14556597C22semidirect
53. #131×D73C22semidirect
54. #585×C73C22semidirect
55. #578×C73C22semidirect
56. #577×C73C22semidirect
57. #576×C73C22semidirect
58. #575×C73C22semidirect
59. C2×#135×C73C22semidirect
60. C2×#131×C73C22semidirect
335. #3584D73semidirect
340. C73#63093semidirect
346. #585C2×D73semidirect
361. #578C2×D73semidirect
366. #577C2×D73semidirect
369. #576C2×D73semidirect
373. #575C2×D73semidirect
381. C2×#135C2×D73semidirect
389. C2×#131C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226247442
#226247172
#226247152
#226247132
#226247002
#226246992
C2×#179894042
C2×#179894012
#578×D732
#226246422
C73×#35842
#6309373