Finite Group Databasebeta

← #33248885

Subgroups — #33248885

Order 18688.

1998 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989142C211yesmaximal, characteristic
2#22613092C214yesmaximal
3#22612960C212yesmaximal
4#22612957C212yesmaximal
5#22612887C214yesmaximal, complemented
6#22612884C212yesmaximal, complemented
7#22612845C212yesmaximal, complemented
8#22612731C214yesmaximal, complemented
9#22612680C211yesmaximal, characteristic, complemented
10#22612650C211yesmaximal, characteristic, complemented
11#22612648C211yesmaximal, characteristic, complemented
12#22612537C214yesmaximal
13#22612337C211yesmaximal, characteristic
14#22612323C211yesmaximal, characteristic
15C73×#2331C211yesmaximal, characteristic, complemented
16#17989142C4116yescomplemented
17C2×#14556591C2211yescharacteristic
18C2×#14556582C2212yes—
19C2×#14556571C2212yes—
20C2×#14556567C2212yes—
21C2×Q8×Dic73C2212yes—
22C2×C4×Dic146C2212yes—
23C2×#14556556C2212yes—
24C2×#14556556C2212yes—
25C2×#14556549C2212yes—
26C2×#14556549C2212yes—
27C2×C42×D73C2211yescharacteristic
28#17988494C2214yes—
29#17988493C2212yes—
30#17988492C2212yes—
31#17988486C2212yes—
32#17988484C2214yes—
33#17988482C2214yes—
34#17988479C2214yes—
35#17988478C2214yes—
36#17988477C2214yes—
37#17988476C2214yes—
38#17988473C2214yes—
39#17988472C2214yes—
40#17988472C2214yes—
41#17988472C2214yes—
42#17988471C2214yes—
43#17988470C2214yes—
44#17988470C2214yes—
45#17988470C2214yes—
46#17988469C2214yes—
47#17988456C2214yes—
48#17988456C2212yes—
49#17988454C2212yes—
50#17988453C2211yescharacteristic
51#17988451C2214yes—
52#17988451C2212yes—
53#17988449C2212yes—
54#17988446C2212yes—
55#17988446C2212yes—
56#17988446C2212yes—
57C4⋊C4×Dic73C2212yes—
58C4⋊C4×Dic73C2214yes—
59C4×#11924007C2214yes—
60C4×#11924007C2211yescharacteristic
61C4×#11924007C2211yescharacteristic
62C4×C4⋊Dic73C2214yes—
63#17988438C2214yes—
64#17988438C2212yes—
65#17988438C2212yes—
66C4×#11924005C2211yescharacteristic
67C4×#11924005C2211yescharacteristic
68#17988433C2211yescharacteristic
69C2×C4⋊Q8×C73C2211yescharacteristic
70#391×C73C2214yes—
71#389×C73C2212yes—
72#387×C73C2214yes—
73#382×C73C2211yescharacteristic
74C4×C4⋊C4×C73C2212yes—
75#376×C73C2211yescharacteristic
76C22×Dic146C2312yes—
77C22×C4×D73C2311yescharacteristic
78C22×C4×D73C2311yescharacteristic
79C22×C4×D73C2311yescharacteristic
80#14556591C2×C418yes—
81#14556582C2×C4116yes—
82#14556571C2×C4116yes—
83#14556567C2×C4116yes—
84Q8×Dic73C2×C4116yes—
85C4×Dic146—44no—
86#14556556—44no—
87#14556556C2×C4116yes—
88#14556549—44no—
89#14556549C2×C4116yes—
90C2×C4⋊Dic73C2312yes—
91C2×C4⋊Dic73C2314yes—
92C2×C4⋊Dic73C2312yes—
93C2×C4⋊Dic73C2312yes—
94C2×C4⋊Dic73C2314yes—
95C2×#11924007C2312yes—
96C2×#11924007C2314yes—
97C2×#11924007C2312yes—
98C2×#11924007C2314yes—
99C2×#11924007C2314yes—
100C2×#11924007C2314yes—
101C2×#11924007C2314yes—
102C2×#11924005C2312yes—
103C2×#11924005C2312yes—
104C2×#11924005C2314yes—
105C2×#11924005C2314yes—
106C2×C4×Dic73C2311yescharacteristic
107C2×C4×Dic73C2311yescharacteristic
108C2×C4×Dic73C2314yes—
109C2×C4×Dic73C2314yes—
110C2×C4×Dic73C2314yes—
111C2×C4×Dic73C2314yes—
112C2×C4×Dic73C2311yescharacteristic
113C2×C4×Dic73C2311yescharacteristic
114C2×C4×Dic73C2314yes—
115C2×C4×Dic73C2311yescharacteristic
116C42×D73C2×C418yes—
117#14556498C2312yes—
118#14556498C2312yes—
119#14556498C2312yes—
120#14556498C2312yes—
121#14556498C2312yes—
122#14556498C2312yes—
123#14556498C2314yes—
124#14556498C2314yes—
125#14556498C2312yes—
126#14556498C2312yes—
127#14556498C2314yes—
128#14556498C2314yes—
129#14556498C2314yes—
130#14556498C2314yes—
131C22×Q8×C73C2312yes—
132C4⋊Q8×C73C2×C418yes—
133C2×C4⋊C4×C73C2312yes—
134C2×C4⋊C4×C73C2314yes—
135C2×C4⋊C4×C73C2312yes—
136C2×C4⋊C4×C73C2312yes—
137C2×C42×C73C2311yescharacteristic
138C2×C42×C73C2311yescharacteristic
139C2×C42×C73C2314yes—
140C2×C42×C73C2311yescharacteristic
141#95×C73C2314yes—
142#95×C73C2312yes—
143#95×C73C2312yes—
144#95×C73C2312yes—
145#95×C73C2314yes—
146#95×C73C2312yes—
147C23×D73C2411yescharacteristic
148C2×Dic146—24no—
149C2×Dic146—24no—
150C2×Dic146C22×C418yes—
151C22×Dic73C2412yes—
152C22×Dic73C2412yes—
153C22×Dic73C2414yes—
154C22×Dic73C2411yescharacteristic
155C22×Dic73C2414yes—
156C22×Dic73C2411yescharacteristic
157C22×Dic73C2411yescharacteristic
158C2×C4×D73—22no—
159C2×C4×D73—22no—
160C2×C4×D73—24no—
161C2×C4×D73—42no—
162C2×C4×D73C22×C414yes—
163C2×C4×D73C22×C414yes—
164C2×C4×D73C22×C414yes—
165C4⋊Dic73—24no—
166C4⋊Dic73—44no—
167C4⋊Dic73C22×C418yes—
168C4⋊Dic73C22×C418yes—
169C4⋊Dic73C22×C4116yes—
170#11924007—24no—
171#11924007—28no—
172#11924007—24no—
173#11924007—44no—
174#11924007—44no—
175#11924007—44no—
176#11924007C22×C4116yes—
177#11924005—42no—
178#11924005—42no—
179#11924005—44no—
180#11924005C22×C4116yes—
181C4×Dic73—22no—
182C4×Dic73—22no—
183C4×Dic73—44no—
184C4×Dic73—44no—
185C4×Dic73—44no—
186C4×Dic73—44no—
187C4×Dic73C22×C414yes—
188C4×Dic73C22×C414yes—
189C4×Dic73C22×C414yes—
190C4×Dic73C22×C4116yes—
191C2×Q8×C73—44no—
192C2×Q8×C73C22×C418yes—
193C22×C4×C73C2412yes—
194C22×C4×C73C2412yes—
195C22×C4×C73C2414yes—
196C22×C4×C73C2411yescharacteristic
197C22×C4×C73C2411yescharacteristic
198C22×C4×C73C2414yes—
199C22×C4×C73C2411yescharacteristic
200C4⋊C4×C73—24no—
201C4⋊C4×C73—44no—
202C4⋊C4×C73C22×C418yes—
203C4⋊C4×C73C22×C418yes—
204C42×C73—22no—
205C42×C73—22no—
206C42×C73—44no—
207C42×C73C22×C414yes—
208C23×C73C2511yescharacteristic
209C22×D73—21no—
210C22×D73—22no—
211C22×D73—21no—
212C22×D73—22no—
213C22×D73C23×C412yes—
214Q8×C73—48no—
215C2×C4×C73—22no—
216C2×C4×C73—24no—
217C2×C4×C73—24no—
218C2×C4×C73—22no—
219C2×C4×C73—22no—
220C2×C4×C73—24no—
221C2×C4×C73—22no—
222C2×C4×C73—22no—
223C2×C4×C73—24no—
224C2×C4×C73—22no—
225C2×C4×C73—22no—
226C2×C4×C73—24no—
227C2×C4×C73—22no—
228C2×C4×C73C23×C412yes—
229C2×C4×C73C23×C412yes—
230C2×C4×C73C23×C418yes—
231C2×C4×C73C23×C412yes—
232C2×C4×C73H(5,𝔽2)12yes—
233C2×C4×C732−1+412yes—
234C2×Dic73—21no—
235C2×Dic73—24no—
236C2×Dic73—21no—
237C2×Dic73—24no—
238C2×Dic73—22no—
239C2×Dic73—22no—
240C2×Dic73—24no—
241C2×Dic73—22no—
242C2×Dic73—22no—
243C2×Dic73—24no—
244C2×Dic73—22no—
245C2×Dic73—22no—
246C2×Dic73—24no—
247C2×Dic73—22no—
248C2×Dic73C23×C412yes—
249C2×Dic73C23×C412yes—
250C2×Dic73C23×C418yes—
251C2×Dic73C23×C412yes—
252C2×Dic732−1+414yes—
253C4×D73—28no—
254C4×D73—28no—
255C4×D73—44no—
256Dic146—216no—
257C22×C73C24×C411yescharacteristic
258C22×C73C2×H(5,𝔽2)11yescharacteristic
259C22×C73C2×2−1+412yes—
260C22×C73C2×2−1+411yescharacteristic
261C22×C73#58512yes—
262C2×D73—22no—
263C2×D73—22no—
264C2×D73—22no—
265C2×D73—42no—
266C2×D73—42no—
267C4×C73—22no—
268C4×C73—22no—
269C4×C73—28no—
270C4×C73—42no—
271C4×C73—44no—
272C4×C73—42no—
273C4×C73#52014yes—
274Dic73—28no—
275Dic73—22no—
276Dic73—22no—
277Dic73—42no—
278Dic73—44no—
279Dic73—42no—
280Dic73#51914yes—
281#34373—731nomaximal, Sylow
282C2×C73C2×#51911yescharacteristic
283C2×C73C2×#52011yescharacteristic
284C2×C73#342811yescharacteristic
285C2×C73#355412yes—
286C2×C73#355512yes—
287D73—42no—
288C2×#533—731no—
289#2378—734no—
290#2362—732no—
291#2362—732no—
292#2355—734no—
293#2355—732no—
294Q8×C4⋊C4—732no—
295#2336—734no—
296#2331—731no—
297#2331D7311yescharacteristic, complemented
298#2327—731no—
299#2327—731no—
300#2314—734no—
301#2285—731no—
302#2283—731no—
303C73#3437311yescharacteristic, complemented, Sylow
304#533—7316no—
305C2×C4⋊Q8—731no—
306C2×C4⋊Q8C2×D7311yescharacteristic
307C2×#125—732no—
308C2×C22⋊Q8—732no—
309C2×C22⋊Q8—732no—
310C2×C4×Q8—732no—
311C2×C4×Q8—732no—
312C2×C4×Q8—732no—
313C2×C4×Q8—732no—
314C2×#117—732no—
315C2×#117—732no—
316C22×C42—731no—
317#391—734no—
318#391—732no—
319#391—732no—
320#391C2×D7314yes—
321#389—732no—
322#389—734no—
323#389—734no—
324#389C2×D7312yes—
325#388—734no—
326#388—734no—
327#388—734no—
328#388—734no—
329#387—734no—
330#387—734no—
331#387—734no—
332#387—734no—
333#387—734no—
334#387—734no—
335#387—734no—
336#387—734no—
337#387—734no—
338#387C2×D7314yes—
339#384—734no—
340#384—732no—
341#383—732no—
342#382—731no—
343#382—734no—
344#382—732no—
345#382C2×D7311yescharacteristic
346#381—732no—
347#380—732no—
348#380—732no—
349#380—732no—
350C4×C4⋊C4—732no—
351C4×C4⋊C4—734no—
352C4×C4⋊C4—734no—
353C4×C4⋊C4—734no—
354C4×C4⋊C4—731no—
355C4×C4⋊C4—731no—
356C4×C4⋊C4—734no—
357C4×C4⋊C4—732no—
358C4×C4⋊C4—732no—
359C4×C4⋊C4C2×D7312yes—
360C4×C22⋊C4—731no—
361C4×C22⋊C4—731no—
362#376—731no—
363#376C2×D7311yescharacteristic
364C22×Q8—732no—
365C22×Q8C22×D7312yes—
366C23×C4—731no—
367C23×C4—731no—
368C23×C4—731no—
369C4⋊Q8—738no—
370C4⋊Q8C4×D7318yes—
371#125—7316no—
372C22⋊Q8—7316no—
373C22⋊Q8—7316no—
374C4×Q8—2924no—
375C4×Q8—2924no—
376C4×Q8—7316no—
377C4×Q8—7316no—
378#117—2924no—
379#117—7316no—
380C2×C4⋊C4—734no—
381C2×C4⋊C4—734no—
382C2×C4⋊C4—734no—
383C2×C4⋊C4—734no—
384C2×C4⋊C4—732no—
385C2×C4⋊C4—734no—
386C2×C4⋊C4—734no—
387C2×C4⋊C4—732no—
388C2×C4⋊C4—732no—
389C2×C4⋊C4—732no—
390C2×C4⋊C4—734no—
391C2×C4⋊C4—732no—
392C2×C4⋊C4C22×D7312yes—
393C2×C4⋊C4C22×D7314yes—
394C2×C4⋊C4C22×D7312yes—
395C2×C4⋊C4C22×D7312yes—
396C2×C22⋊C4—734no—
397C2×C22⋊C4—732no—
398C2×C22⋊C4—732no—
399C2×C22⋊C4—734no—
400C2×C42—734no—
401C2×C42—734no—
402C2×C42—734no—
403C2×C42—734no—
404C2×C42—734no—
405C2×C42—731no—
406C2×C42—731no—
407C2×C42—731no—
408C2×C42—738no—
409C2×C42—731no—
410C2×C42—731no—
411C2×C42C22×D7311yescharacteristic
412C2×C42C22×D7311yescharacteristic
413C2×C42C22×D7314yes—
414C2×C42C22×D7311yescharacteristic
415#95—732no—
416#95—732no—
417#95—732no—
418#95—732no—
419#95—732no—
420#95—732no—
421#95—732no—
422#95—732no—
423#95—734no—
424#95—734no—
425#95—734no—
426#95—734no—
427#95—734no—
428#95—734no—
429#95C22×D7312yes—
430#95C22×D7312yes—
431#95C22×D7312yes—
432#95C22×D7312yes—
433#95C22×D7314yes—
434#95C22×D7314yes—
435C24—731no—
436C2×Q8—44no—
437C2×Q8—738no—
438C2×Q8—1464no—
439C2×Q8—1464no—
440C2×Q8C2×C4×D7318yes—
441C22×C4—2922no—
442C22×C4—734no—
443C22×C4—731no—
444C22×C4—731no—
445C22×C4—734no—
446C22×C4—731no—
447C22×C4—734no—
448C22×C4—734no—
449C22×C4—732no—
450C22×C4—732no—
451C22×C4—734no—
452C22×C4—1464no—
453C22×C4—1462no—
454C22×C4—1462no—
455C22×C4C23×D7312yes—
456C22×C4C23×D7312yes—
457C22×C4C23×D7314yes—
458C22×C4C23×D7311yescharacteristic
459C22×C4C23×D7311yescharacteristic
460C22×C4C23×D7314yes—
461C22×C4C23×D7311yescharacteristic
462C4⋊C4—24no—
463C4⋊C4—44no—
464C4⋊C4—2924no—
465C4⋊C4—2924no—
466C4⋊C4—2924no—
467C4⋊C4—2924no—
468C4⋊C4—7316no—
469C4⋊C4—738no—
470C4⋊C4—7316no—
471C4⋊C4—738no—
472C4⋊C4—1464no—
473C4⋊C4—1468no—
474C4⋊C4—1464no—
475C4⋊C4—1464no—
476C4⋊C4C2×C4×D7318yes—
477C4⋊C4C2×C4×D7318yes—
478C22⋊C4—2922no—
479C22⋊C4—2922no—
480C22⋊C4—2924no—
481C22⋊C4—7316no—
482C42—22no—
483C42—22no—
484C42—44no—
485C42—2924no—
486C42—2924no—
487C42—2924no—
488C42—2924no—
489C42—734no—
490C42—734no—
491C42—734no—
492C42—7316no—
493C42—1462no—
494C42—1462no—
495C42C2×C4×D7314yes—
496C23—732no—
497C23—1461no—
498C23—1462no—
499C23—1461no—
500C23—1462no—
501C23C24×D7311yescharacteristic
502Q8—48no—
503Q8—14616no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—2924no—
518C2×C4—732no—
519C2×C4—734no—
520C2×C4—732no—
521C2×C4—732no—
522C2×C4—738no—
523C2×C4—1468no—
524C2×C4—1462no—
525C2×C4—1464no—
526C2×C4—1462no—
527C2×C4—1464no—
528C2×C4—1462no—
529C2×C4—1462no—
530C2×C4—1462no—
531C2×C4—1464no—
532C2×C4—1462no—
533C2×C4—1461no—
534C2×C4—1464no—
535C2×C4—1461no—
536C2×C4—1464no—
537C2×C4—1462no—
538C2×C4—1468no—
539C2×C4C22×C4×D7312yes—
540C2×C4C22×C4×D7312yes—
541C2×C4C22×C4×D7318yes—
542C2×C4C22×C4×D7312yes—
543C2×C4#1455663712yes—
544C2×C4#1455663912yes—
545C22—2922no—
546C22—2922no—
547C22—1462no—
548C22—1462no—
549C22—1462no—
550C22C23×C4×D7311yescharacteristic
551C22C2×#1455663711yescharacteristic
552C222−1+4×D7312yes—
553C22C2×#1455663911yescharacteristic
554C22#1798942212yes—
555C4—22no—
556C4—22no—
557C4—28no—
558C4—42no—
559C4—44no—
560C4—42no—
561C4—2922no—
562C4—2924no—
563C4—2922no—
564C4—734no—
565C4—1462no—
566C4—1468no—
567C4—1462no—
568C4#1798906914yes—
569C2—2922no—
570C2#519×D7311yescharacteristic
571C2C2×#1798906911yescharacteristic
572C2#2262312911yescharacteristic
573C2#2262439912yes—
574C2#2262441612yes—
subgroupG/H
center, FrattiniC23C24×D73
commutatorC22×C73C24×C4
FittingC73×#2331C2
socle, cosocleC23×C73C25
prime radicalC23×D73C24
hypercenter#2331D73
hypocenterC73#34373
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ #2331

Complements

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

NH ≅ G/Ntype
5. #22612887C2semidirect
6. #22612884C2semidirect
7. #22612845C2semidirect
8. #22612731C2semidirect
9. #22612680C2semidirect
10. #22612650C2semidirect
11. #22612648C2semidirect
15. C73×#2331C2semidirect
16. #17989142C4semidirect
297. #2331D73semidirect
303. C73#34373semidirect

Maximal subgroups

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

subgroupindex
C2×#179891422
#226130922
#226129602
#226129572
#226128872
#226128842
#226128452
#226127312
#226126802
#226126502
#226126482
#226125372
#226123372
#226123232
C73×#23312
#3437373