Finite Group Databasebeta

← #33953664

Subgroups — #33953664

Order 18688.

2054 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22623757C211yesmaximal, characteristic, complemented
2#22619793C213yesmaximal, complemented
3#22619782C213yesmaximal, complemented
4#22619762C213yesmaximal, complemented
5#22619762C213yesmaximal, complemented
6#22619753C213yesmaximal, complemented
7#22619728C213yesmaximal, complemented
8#22619719C213yesmaximal, complemented
9C2×#17988887C213yesmaximal, complemented
10#22618529C211yesmaximal, characteristic, complemented
11#22618524C211yesmaximal, characteristic, complemented
12#22618499C211yesmaximal, characteristic, complemented
13#22618457C211yesmaximal, characteristic
14#22618457C211yesmaximal, characteristic
15C4×H(5,𝔽2)×C73C211yesmaximal, characteristic
16C2×#14556598C2211yescharacteristic
17C2×#14556598C2211yescharacteristic
18#17989313—21no—
19#17989313—21no—
20#17989296—23no—
21#17989296—23no—
22#17989215C2213yescomplemented
23#17989215C2213yescomplemented
24#17989144C2213yescomplemented
25C2×#14556583C2213yescomplemented
26#17989070C2211yescharacteristic
27#17989070C2211yescharacteristic
28#17988894C2211yescharacteristic, complemented
29#17988894C2213yescomplemented
30#17988894C2216yescomplemented
31#17988887C2213yescomplemented
32#17988887C2216yescomplemented
33#17988887C2216yescomplemented
34#17988887C2216yescomplemented
35#17988887C2213yescomplemented
36#17988887C2216yescomplemented
37#17988882C2213yescomplemented
38#17988877C2213yescomplemented
39#17988877C2216yescomplemented
40#17988872C2213yescomplemented
41#17988867C2213yescomplemented
42#17988867C2216yescomplemented
43#17988630C2213yescomplemented
44#17988630C2213yescomplemented
45#17988627C2213yescomplemented
46#17988627C2213yescomplemented
47#17988623C2213yescomplemented
48#17988623C2213yescomplemented
49C2×#14556524C2213yescomplemented
50C2×#14556522C2213yescomplemented
51#17988614C2213yescomplemented
52#17988614C2213yescomplemented
53#17988611C2213yescomplemented
54C2×C4⋊#10596831C2213yescomplemented
55#17988592C2213yes—
56#17988592C2213yes—
57#17988588C2213yes—
58#17988588C2213yes—
59#17988583C2213yes—
60#17988583C2213yes—
61C2×#14556512C2213yes—
62C2×#14556512C2213yes—
63C2×H(5,𝔽2)×C73C2211yescharacteristic
64#520×C73—21no—
65#520×C73C2213yes—
66#520×C73C2211yescharacteristic
67#518×C73—23no—
68#518×C73C2213yes—
69C4×#41×C73—21no—
70C4×#41×C73C2213yes—
71C4×#41×C73C2211yescharacteristic
72C2×C4×D4×C73—23no—
73C2×C4×D4×C73C2213yes—
74#14556598—24no—
75#14556598—24no—
76C2×#11924012C2313yes—
77C2×#11924012C2313yes—
78C22×#10596831C2313yes—
79C22×#10596831C2313yes—
80#14556591C2313yes—
81#14556591C2313yes—
82#14556590—23no—
83#14556590—23no—
84#14556583—23no—
85#14556583—23no—
86#14556583C2311yescharacteristic
87#14556583C2313yes—
88#14556583C2316yes—
89#14556582—23no—
90#14556582—23no—
91#14556570—23no—
92#14556570—23no—
93#14556568—26no—
94#14556568—26no—
95Q8×Dic73—21no—
96Q8×Dic73—21no—
97C4×Dic146—23no—
98C4×Dic146—23no—
99D4×Dic73—23no—
100D4×Dic73—23no—
101#14556550C2313yes—
102#14556550C2313yes—
103C2×C4⋊Dic73C2313yes—
104C2×C4⋊Dic73C2313yes—
105C2×#11924007—23no—
106C2×#11924007—23no—
107#14556524C2311yescharacteristic
108#14556524C2313yes—
109#14556524C2316yes—
110#14556524C2313yes—
111#14556524C2313yes—
112#14556522C2313yes—
113#14556522C2313yes—
114#14556522C2316yes—
115#14556522C2311yescharacteristic
116#14556522C2313yes—
117C4⋊#10596831C2311yescharacteristic
118C4⋊#10596831C2313yes—
119C4⋊#10596831C2313yes—
120C4⋊#10596831C2313yes—
121C4⋊#10596831C2316yes—
122#14556515C2311yescharacteristic
123#14556515C2313yes—
124#14556515C2311yescharacteristic
125#14556515C2313yes—
126#14556512C2313yes—
127#14556512C2313yes—
128#14556512C2313yes—
129#14556512C2316yes—
130#14556512C2313yes—
131#14556512C2316yes—
132H(5,𝔽2)×C73—24no—
133H(5,𝔽2)×C73Q818yescomplemented
134C2×#41×C73—21no—
135C2×#41×C73C2313yes—
136C2×#41×C73C2311yescharacteristic
137C22×D4×C73—23no—
138C22×D4×C73C2313yes—
139C23×C4×C73—23no—
140C4×Q8×C73—21no—
141C4×Q8×C73D411yescharacteristic
142C4×Q8×C73D411yescharacteristic
143C4×Q8×C73C2313yes—
144C4×Q8×C73C2311yescharacteristic
145C4×D4×C73—23no—
146C4×D4×C73—23no—
147C4×D4×C73—23no—
148C4×D4×C73—23no—
149C4×D4×C73—23no—
150C4×D4×C73—26no—
151C4×D4×C73—23no—
152C4×D4×C73—23no—
153C4×D4×C73D413yes—
154C4×D4×C73D413yes—
155C4×D4×C73C2316yes—
156C4×D4×C73C2313yes—
157C4×D4×C73C2313yes—
158#117×C73—23no—
159#117×C73—23no—
160#117×C73C2313yes—
161#117×C73C2313yes—
162C2×C4⋊C4×C73—23no—
163C2×C4⋊C4×C73C2313yes—
164C2×C22⋊C4×C73—23no—
165C2×C22⋊C4×C73—26no—
166C2×C42×C73—23no—
167C2×C42×C73C2313yes—
168C2×Dic146—23no—
169C2×Dic146—23no—
170C22×Dic73—23no—
171C22×Dic73—23no—
172#11924012—26no—
173#11924012—26no—
174C2×#10596831—26no—
175C2×#10596831—26no—
176C2×#10596831C2411yescharacteristic
177C2×#10596831C2413yes—
178C2×#10596831C2413yes—
179C2×#10596831C2411yescharacteristic
180C4⋊Dic73—23no—
181C4⋊Dic73—23no—
182C4⋊Dic73—23no—
183C4⋊Dic73—23no—
184C4⋊Dic73C2413yes—
185C4⋊Dic73C2411yescharacteristic
186C4⋊Dic73C2413yes—
187C4⋊Dic73C2411yescharacteristic
188#11924007—23no—
189#11924007—23no—
190#11924007—26no—
191#11924007—26no—
192#11924007—21no—
193#11924007—21no—
194C22⋊Dic73—23no—
195C22⋊Dic73—23no—
196C4×Dic73—23no—
197C4×Dic73—23no—
198C24×C73—23no—
199#41×C73—28no—
200#41×C73—26no—
201#41×C73C2×Q8112yes—
202#41×C73C2×Q814yes—
203#41×C73#4114yes—
204C2×Q8×C73C2×D411yescharacteristic
205C2×Q8×C73C2411yescharacteristic
206C2×D4×C73—23no—
207C2×D4×C73—224no—
208C2×D4×C73—23no—
209C2×D4×C73—26no—
210C2×D4×C73C2×D413yes—
211C2×D4×C73C2×Q8112yes—
212C2×D4×C73C2413yes—
213C22×C4×C73—23no—
214C22×C4×C73—23no—
215C22×C4×C73—23no—
216C22×C4×C73—23no—
217C22×C4×C73—26no—
218C22×C4×C73—86no—
219C22×C4×C73C2×D413yes—
220C22×C4×C73C2×D413yes—
221C22×C4×C73C2413yes—
222C4⋊C4×C73—23no—
223C4⋊C4×C73—23no—
224C4⋊C4×C73—21no—
225C4⋊C4×C73C2×D413yes—
226C4⋊C4×C73C2×D413yes—
227C4⋊C4×C73C2413yes—
228C4⋊C4×C73C2411yescharacteristic
229C22⋊C4×C73—23no—
230C22⋊C4×C73—23no—
231C22⋊C4×C73—26no—
232C22⋊C4×C73—23no—
233C22⋊C4×C73C2×D416yes—
234C42×C73—21no—
235C42×C73—23no—
236C42×C73—23no—
237C42×C73C2×D413yes—
238C42×C73C2×D413yes—
239C42×C73C2411yescharacteristic
240C42×C73C2413yes—
241C23×C73—23no—
242C23×C73—212no—
243C23×C73—83no—
244C23×C73—83no—
245C23×C73C22×D413yes—
246Q8×C73—21no—
247Q8×C73C22⋊Q814yes—
248Q8×C73C22×Q812yes—
249D4×C73—212no—
250D4×C73—212no—
251D4×C73—23no—
252D4×C73C22⋊Q8112yes—
253D4×C73C22×Q816yes—
254C2×C4×C73—21no—
255C2×C4×C73—212no—
256C2×C4×C73—21no—
257C2×C4×C73—23no—
258C2×C4×C73—43no—
259C2×C4×C73—43no—
260C2×C4×C73—46no—
261C2×C4×C73—83no—
262C2×C4×C73—86no—
263C2×C4×C73—83no—
264C2×C4×C73C22×D411yescharacteristic
265C2×C4×C73C22×D413yes—
266C2×C4×C73C22×D411yescharacteristic
267C2×C4×C73C22×D413yes—
268C2×C4×C73C22×D413yes—
269C2×C4×C73C22×Q816yes—
270C2×C4×C73C2×#4116yes—
271C2×C4×C73C2511yescharacteristic
272C2×Dic73—21no—
273C2×Dic73—21no—
274C2×Dic73—23no—
275C2×Dic73—23no—
276C2×Dic73—83no—
277C2×Dic73—83no—
278Dic146—83no—
279Dic146—83no—
280#10596831—21no—
281#10596831—21no—
282#10596831—23no—
283#10596831—23no—
284C22×C73—23no—
285C22×C73—23no—
286C22×C73—26no—
287C22×C73—43no—
288C22×C73—43no—
289C22×C73—812no—
290C22×C73C2×C22⋊Q816yes—
291C22×C73C23×D411yescharacteristic
292C4×C73—23no—
293C4×C73—21no—
294C4×C73—23no—
295C4×C73—21no—
296C4×C73—22no—
297C4×C73—41no—
298C4×C73—43no—
299C4×C73C2×C22⋊Q816yes—
300C4×C73C23×Q811yescharacteristic
301C4×C73C22×#4111yescharacteristic
302Dic73—41no—
303Dic73—43no—
304Dic73—41no—
305Dic73—43no—
306#60496—731nomaximal, Sylow
307C2×C73—26no—
308C2×C73—46no—
309C2×C73C22×C22⋊Q811yescharacteristic
310C2×C73#358411yescharacteristic
311C2×C73#358511yescharacteristic
312#3504—731no—
313C4×H(5,𝔽2)D7311yescharacteristic
314#3084—733no—
315#3083—733no—
316#3081—733no—
317#3081—733no—
318#3080—733no—
319#3077—733no—
320#3076—733no—
321C2×#478—733no—
322#2913—731no—
323#2912—731no—
324#2906—731no—
325#2897—731no—
326#2897—731no—
327C73#6049611yescharacteristic, complemented, Sylow
328C2×H(5,𝔽2)C2×D7311yescharacteristic
329C2×#131—731no—
330C2×#131—731no—
331#557—1461no—
332#557—1461no—
333#554—1463no—
334#554—1463no—
335#544—733no—
336#544—733no—
337#533—733no—
338C2×#125—733no—
339#520—21no—
340#520—731no—
341#520—731no—
342#520C2×D7313yes—
343#520C2×D7311yescharacteristic
344#518—23no—
345#518C2×D7313yes—
346C4×#41—21no—
347C4×#41C2×D7313yes—
348C4×#41C2×D7311yescharacteristic
349C2×C4×D4—23no—
350C2×C4×D4C2×D7313yes—
351#479—731no—
352#479—733no—
353#479—736no—
354#478—733no—
355#478—736no—
356#478—736no—
357#478—736no—
358#478—733no—
359#478—736no—
360#477—733no—
361#476—733no—
362#476—736no—
363#475—733no—
364#474—733no—
365#474—736no—
366#428—733no—
367#428—733no—
368#428—733no—
369#428—733no—
370#427—733no—
371#427—733no—
372C2×#107—733no—
373C2×#106—733no—
374#424—733no—
375#424—733no—
376#423—733no—
377C2×C4⋊C8—733no—
378#418—733no—
379#418—733no—
380#417—733no—
381#417—733no—
382#416—733no—
383#416—733no—
384C2×#102—733no—
385C2×#102—733no—
386H(5,𝔽2)—24no—
387H(5,𝔽2)Dic14618yescomplemented
388C2×#41—21no—
389C2×#41C22×D7313yes—
390C2×#41C22×D7311yescharacteristic
391C22×D4—23no—
392C22×D4C22×D7313yes—
393C23×C4—23no—
394#131—1464no—
395#131—1464no—
396C2×M16—733no—
397C2×M16—733no—
398C22×C8—733no—
399C22×C8—733no—
400C4⋊Q8—733no—
401C4⋊Q8—733no—
402C4⋊Q8—1463no—
403C4⋊Q8—1463no—
404#125—733no—
405#125—736no—
406#125—731no—
407#125—1463no—
408#125—1463no—
409#125—1463no—
410#125—1463no—
411C22⋊Q8—1463no—
412C22⋊Q8—1463no—
413C22⋊Q8—1466no—
414C22⋊Q8—1466no—
415C4×Q8—21no—
416C4×Q8—1461no—
417C4×Q8—1461no—
418C4×Q8—1463no—
419C4×Q8—1463no—
420C4×Q8C22⋊D7311yescharacteristic
421C4×Q8C22⋊D7311yescharacteristic
422C4×Q8C22×D7311yescharacteristic
423C4×Q8C22×D7313yes—
424C4×D4—23no—
425C4×D4—23no—
426C4×D4—23no—
427C4×D4—23no—
428C4×D4—23no—
429C4×D4—26no—
430C4×D4—23no—
431C4×D4—23no—
432C4×D4—1463no—
433C4×D4—1463no—
434C4×D4C22⋊D7313yes—
435C4×D4C22⋊D7313yes—
436C4×D4C22×D7313yes—
437C4×D4C22×D7316yes—
438C4×D4C22×D7313yes—
439#117—23no—
440#117—23no—
441#117—733no—
442#117—733no—
443#117C22×D7313yes—
444#117C22×D7313yes—
445C2×C4⋊C4—23no—
446C2×C4⋊C4—733no—
447C2×C4⋊C4—733no—
448C2×C4⋊C4—1463no—
449C2×C4⋊C4—1463no—
450C2×C4⋊C4C22×D7313yes—
451C2×C22⋊C4—23no—
452C2×C22⋊C4—26no—
453C2×C42—23no—
454C2×C42C22×D7313yes—
455#107—733no—
456#107—736no—
457#107—731no—
458#107—733no—
459#107—733no—
460#106—733no—
461#106—736no—
462#106—731no—
463#106—733no—
464#106—733no—
465C4⋊C8—731no—
466C4⋊C8—733no—
467C4⋊C8—733no—
468C4⋊C8—733no—
469C4⋊C8—736no—
470#103—731no—
471#103—731no—
472#103—733no—
473#103—733no—
474#102—733no—
475#102—733no—
476#102—736no—
477#102—736no—
478#102—733no—
479#102—733no—
480C24—23no—
481#41—28no—
482#41—26no—
483#41C2×Dic146112yes—
484#41C2×Dic14614yes—
485#41#1192402814yes—
486C2×Q8—1463no—
487C2×Q8—1463no—
488C2×Q8C2×C22⋊D7311yescharacteristic
489C2×Q8C23×D7311yescharacteristic
490C2×D4—23no—
491C2×D4—224no—
492C2×D4—23no—
493C2×D4—26no—
494C2×D4C2×C22⋊D7313yes—
495C2×D4C2×Dic146112yes—
496C2×D4C23×D7313yes—
497C22×C4—23no—
498C22×C4—23no—
499C22×C4—23no—
500C22×C4—23no—
501C22×C4—26no—
502C22×C4—86no—
503C22×C4—1463no—
504C22×C4—1463no—
505C22×C4C2×C22⋊D7313yes—
506C22×C4C2×C22⋊D7313yes—
507C22×C4C23×D7313yes—
508M16—1466no—
509M16—1466no—
510C2×C8—731no—
511C2×C8—733no—
512C2×C8—733no—
513C2×C8—731no—
514C2×C8—1466no—
515C2×C8—1466no—
516C4⋊C4—23no—
517C4⋊C4—23no—
518C4⋊C4—21no—
519C4⋊C4—733no—
520C4⋊C4—731no—
521C4⋊C4—733no—
522C4⋊C4—731no—
523C4⋊C4—1463no—
524C4⋊C4—1463no—
525C4⋊C4—1466no—
526C4⋊C4—1466no—
527C4⋊C4—1461no—
528C4⋊C4—1461no—
529C4⋊C4—1463no—
530C4⋊C4—1463no—
531C4⋊C4—1463no—
532C4⋊C4—1463no—
533C4⋊C4C2×C22⋊D7313yes—
534C4⋊C4C2×C22⋊D7313yes—
535C4⋊C4C23×D7313yes—
536C4⋊C4C23×D7311yescharacteristic
537C22⋊C4—23no—
538C22⋊C4—23no—
539C22⋊C4—26no—
540C22⋊C4—23no—
541C22⋊C4—1463no—
542C22⋊C4—1463no—
543C22⋊C4C2×C22⋊D7316yes—
544C42—21no—
545C42—23no—
546C42—23no—
547C42—1463no—
548C42—1463no—
549C42C2×C22⋊D7313yes—
550C42C2×C22⋊D7313yes—
551C42C23×D7311yescharacteristic
552C42C23×D7313yes—
553C23—23no—
554C23—212no—
555C23—83no—
556C23—83no—
557C23C22×C22⋊D7313yes—
558Q8—21no—
559Q8—5843no—
560Q8—5843no—
561Q8#1455657014yes—
562Q8C22×Dic14612yes—
563D4—212no—
564D4—23no—
565D4—212no—
566D4#14556570112yes—
567D4C22×Dic14616yes—
568C2×C4—21no—
569C2×C4—212no—
570C2×C4—21no—
571C2×C4—23no—
572C2×C4—43no—
573C2×C4—43no—
574C2×C4—46no—
575C2×C4—86no—
576C2×C4—83no—
577C2×C4—83no—
578C2×C4—5843no—
579C2×C4—5843no—
580C2×C4—1461no—
581C2×C4—1461no—
582C2×C4—1463no—
583C2×C4—1463no—
584C2×C4C22×C22⋊D7311yescharacteristic
585C2×C4C22×C22⋊D7313yes—
586C2×C4C22×C22⋊D7311yescharacteristic
587C2×C4C22×C22⋊D7313yes—
588C2×C4C22×C22⋊D7313yes—
589C2×C4C22×Dic14616yes—
590C2×C4C2×#1192402816yes—
591C2×C4C24×D7311yescharacteristic
592C8—1461no—
593C8—1461no—
594C8—1463no—
595C8—1463no—
596C22—23no—
597C22—23no—
598C22—26no—
599C22—43no—
600C22—43no—
601C22—812no—
602C22C2×#1455657016yes—
603C22C23×C22⋊D7311yescharacteristic
604C4—22no—
605C4—21no—
606C4—23no—
607C4—21no—
608C4—23no—
609C4—41no—
610C4—43no—
611C4—2923no—
612C4—2921no—
613C4—2923no—
614C4—2921no—
615C4C2×#1455657016yes—
616C4C23×Dic14611yescharacteristic
617C4C22×#1192402811yescharacteristic
618C2—26no—
619C2—46no—
620C2C22×#1455657011yescharacteristic
621C2#2262470211yescharacteristic
622C2#2262471111yescharacteristic
subgroupG/H
centerC22C23×C22⋊D73
commutator, cosocleC2×C4×C73C25
FrattiniC2×C4C24×D73
FittingC4×H(5,𝔽2)×C73C2
socleC22×C73C23×D4
prime radicalC2×H(5,𝔽2)×C73C22
hypercenterC4×H(5,𝔽2)D73
hypocenterC73#60496
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C4×#41 ◁ C4×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #22623757C2semidirect
2. #22619793C2semidirect
3. #22619782C2semidirect
4. #22619762C2semidirect
5. #22619762C2semidirect
6. #22619753C2semidirect
7. #22619728C2semidirect
8. #22619719C2semidirect
9. C2×#17988887C2semidirect
10. #22618529C2semidirect
11. #22618524C2semidirect
12. #22618499C2semidirect
22. #17989215C22semidirect
23. #17989215C22semidirect
24. #17989144C22semidirect
25. C2×#14556583C22semidirect
28. #17988894C22semidirect
29. #17988894C22semidirect
30. #17988894C22semidirect
31. #17988887C22semidirect
32. #17988887C22semidirect
33. #17988887C22semidirect
34. #17988887C22semidirect
35. #17988887C22semidirect
36. #17988887C22semidirect
37. #17988882C22semidirect
38. #17988877C22semidirect
39. #17988877C22semidirect
40. #17988872C22semidirect
41. #17988867C22semidirect
42. #17988867C22semidirect
43. #17988630C22semidirect
44. #17988630C22semidirect
45. #17988627C22semidirect
46. #17988627C22semidirect
47. #17988623C22semidirect
48. #17988623C22semidirect
49. C2×#14556524C22semidirect
50. C2×#14556522C22semidirect
51. #17988614C22semidirect
52. #17988614C22semidirect
53. #17988611C22semidirect
54. C2×C4⋊#10596831C22semidirect
133. H(5,𝔽2)×C73Q8semidirect
327. C73#60496semidirect
387. H(5,𝔽2)Dic146semidirect

Maximal subgroups

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

subgroupindex
#226237572
#226197932
#226197822
#226197622
#226197532
#226197282
#226197192
C2×#179888872
#226185292
#226185242
#226184992
#226184572
C4×H(5,𝔽2)×C732
#6049673