Finite Group Databasebeta

← #33249293

Subgroups — #33249293

Order 18688.

1898 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989142C211yesmaximal, characteristic
2#22613092C214yesmaximal
3#22613029C212yesmaximal, complemented
4#22613026C212yesmaximal, complemented
5#22612895C214yesmaximal, complemented
6#22612820C212yesmaximal
7#22612815C212yesmaximal
8#22612683C214yesmaximal
9#22612680C211yesmaximal, characteristic, complemented
10#22612679C211yesmaximal, characteristic, complemented
11#22612677C211yesmaximal, characteristic, complemented
12#22612580C214yesmaximal, complemented
13#22612324C211yesmaximal, characteristic
14#22612323C211yesmaximal, characteristic
15C73×#2331C211yesmaximal, characteristic, complemented
16#17989142C4116yescomplemented
17C2×#14556591C2211yescharacteristic
18C2×#14556582C2212yes—
19C2×#14556571C2212yes—
20C2×#14556567C2212yes—
21C2×Q8×Dic73C2212yes—
22C2×#14556556C2212yes—
23C2×#14556549C2212yes—
24C2×C42×D73C2211yescharacteristic
25#17988494C2214yes—
26#17988491C2214yes—
27#17988491C2214yes—
28#17988486C2212yes—
29#17988485C2212yes—
30#17988484C2214yes—
31#17988483C2212yes—
32#17988482C2214yes—
33#17988479C2214yes—
34#17988478C2214yes—
35#17988477C2214yes—
36#17988476C2214yes—
37#17988473C2214yes—
38#17988472C2214yes—
39#17988472C2214yes—
40#17988471C2214yes—
41#17988470C2214yes—
42#17988470C2214yes—
43#17988469C2214yes—
44#17988454C2214yes—
45#17988454C2212yes—
46#17988454C2212yes—
47#17988453C2211yescharacteristic
48#17988452C2214yes—
49#17988451C2212yes—
50#17988449C2212yes—
51#17988449C2212yes—
52#17988446C2212yes—
53#17988444C2212yes—
54C4⋊C4×Dic73C2212yes—
55C4⋊C4×Dic73C2212yes—
56C4⋊C4×Dic73C2214yes—
57C4⋊C4×Dic73C2212yes—
58C4×#11924007C2211yescharacteristic
59C4×#11924007C2211yescharacteristic
60C4×#11924007C2214yes—
61C4×C4⋊Dic73C2212yes—
62C4×C4⋊Dic73C2214yes—
63C4×#11924005C2211yescharacteristic
64C4×#11924005C2211yescharacteristic
65#17988434C2212yes—
66#17988433C2211yescharacteristic
67#17988432C2214yes—
68#17988432C2212yes—
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—
85#14556556C2×C4116yes—
86#14556549C2×C4116yes—
87C2×C4⋊Dic73C2312yes—
88C2×C4⋊Dic73C2314yes—
89C2×C4⋊Dic73C2312yes—
90C2×C4⋊Dic73C2312yes—
91C2×C4⋊Dic73C2314yes—
92C2×#11924007C2312yes—
93C2×#11924007C2312yes—
94C2×#11924007C2314yes—
95C2×#11924007C2314yes—
96C2×#11924007C2314yes—
97C2×#11924005C2314yes—
98C2×#11924005C2312yes—
99C2×#11924005C2312yes—
100C2×#11924005C2314yes—
101C2×C4×Dic73C2311yescharacteristic
102C2×C4×Dic73C2311yescharacteristic
103C2×C4×Dic73C2314yes—
104C2×C4×Dic73C2314yes—
105C2×C4×Dic73C2314yes—
106C2×C4×Dic73C2314yes—
107C2×C4×Dic73C2311yescharacteristic
108C2×C4×Dic73C2314yes—
109C2×C4×Dic73C2311yescharacteristic
110C2×C4×Dic73C2311yescharacteristic
111C42×D73C2×C418yes—
112#14556498C2312yes—
113#14556498C2314yes—
114#14556498C2312yes—
115#14556498C2314yes—
116#14556498C2312yes—
117#14556498C2312yes—
118#14556498C2312yes—
119#14556498C2312yes—
120#14556498C2312yes—
121#14556498C2314yes—
122#14556498C2314yes—
123#14556498C2312yes—
124#14556498C2314yes—
125#14556498C2314yes—
126#14556498C2314yes—
127#14556498C2314yes—
128C22×Q8×C73C2312yes—
129C4⋊Q8×C73C2×C418yes—
130C2×C4⋊C4×C73C2312yes—
131C2×C4⋊C4×C73C2314yes—
132C2×C4⋊C4×C73C2312yes—
133C2×C4⋊C4×C73C2312yes—
134C2×C42×C73C2311yescharacteristic
135C2×C42×C73C2311yescharacteristic
136C2×C42×C73C2314yes—
137C2×C42×C73C2311yescharacteristic
138#95×C73C2314yes—
139#95×C73C2312yes—
140#95×C73C2312yes—
141#95×C73C2312yes—
142#95×C73C2314yes—
143#95×C73C2312yes—
144C23×D73C2411yescharacteristic
145C2×Dic146—44no—
146C2×Dic146C22×C418yes—
147C22×Dic73C2412yes—
148C22×Dic73C2412yes—
149C22×Dic73C2414yes—
150C22×Dic73C2411yescharacteristic
151C22×Dic73C2411yescharacteristic
152C22×Dic73C2414yes—
153C22×Dic73C2411yescharacteristic
154C2×C4×D73—22no—
155C2×C4×D73—22no—
156C2×C4×D73—42no—
157C2×C4×D73—42no—
158C2×C4×D73C22×C414yes—
159C2×C4×D73C22×C414yes—
160C2×C4×D73C22×C414yes—
161C4⋊Dic73—24no—
162C4⋊Dic73—44no—
163C4⋊Dic73C22×C418yes—
164C4⋊Dic73C22×C418yes—
165C4⋊Dic73C22×C4116yes—
166#11924007—44no—
167#11924007—44no—
168#11924007—42no—
169#11924007—42no—
170#11924007C22×C4116yes—
171#11924005—42no—
172#11924005—44no—
173#11924005—42no—
174#11924005C22×C4116yes—
175C4×Dic73—22no—
176C4×Dic73—22no—
177C4×Dic73—44no—
178C4×Dic73—44no—
179C4×Dic73—44no—
180C4×Dic73—44no—
181C4×Dic73C22×C414yes—
182C4×Dic73C22×C414yes—
183C4×Dic73C22×C414yes—
184C4×Dic73C22×C4116yes—
185C2×Q8×C73—44no—
186C2×Q8×C73C22×C418yes—
187C22×C4×C73C2412yes—
188C22×C4×C73C2412yes—
189C22×C4×C73C2414yes—
190C22×C4×C73C2411yescharacteristic
191C22×C4×C73C2411yescharacteristic
192C22×C4×C73C2414yes—
193C22×C4×C73C2411yescharacteristic
194C4⋊C4×C73—24no—
195C4⋊C4×C73—44no—
196C4⋊C4×C73C22×C418yes—
197C4⋊C4×C73C22×C418yes—
198C42×C73—22no—
199C42×C73—22no—
200C42×C73—44no—
201C42×C73C22×C414yes—
202C23×C73C2511yescharacteristic
203C22×D73—21no—
204C22×D73—21no—
205C22×D73—22no—
206C22×D73—22no—
207C22×D73C23×C412yes—
208Q8×C73—48no—
209C2×C4×C73—24no—
210C2×C4×C73—24no—
211C2×C4×C73—22no—
212C2×C4×C73—22no—
213C2×C4×C73—22no—
214C2×C4×C73—22no—
215C2×C4×C73—24no—
216C2×C4×C73—22no—
217C2×C4×C73—24no—
218C2×C4×C73—22no—
219C2×C4×C73—22no—
220C2×C4×C73—24no—
221C2×C4×C73—22no—
222C2×C4×C73C23×C412yes—
223C2×C4×C73C23×C412yes—
224C2×C4×C73C23×C418yes—
225C2×C4×C73C23×C412yes—
226C2×C4×C732−1+412yes—
227C2×C4×C732−1+412yes—
228C2×Dic73—21no—
229C2×Dic73—21no—
230C2×Dic73—22no—
231C2×Dic73—24no—
232C2×Dic73—24no—
233C2×Dic73—22no—
234C2×Dic73—22no—
235C2×Dic73—22no—
236C2×Dic73—24no—
237C2×Dic73—22no—
238C2×Dic73—22no—
239C2×Dic73—24no—
240C2×Dic73—24no—
241C2×Dic73—22no—
242C2×Dic73—22no—
243C2×Dic73C23×C412yes—
244C2×Dic73C23×C412yes—
245C2×Dic73C23×C412yes—
246C2×Dic73C23×C418yes—
247C4×D73—28no—
248C4×D73—44no—
249C4×D73—44no—
250Dic146—48no—
251C22×C73C24×C411yescharacteristic
252C22×C73C2×2−1+411yescharacteristic
253C22×C73C2×2−1+411yescharacteristic
254C22×C73#58512yes—
255C22×C73#58512yes—
256C2×D73—22no—
257C2×D73—22no—
258C2×D73—22no—
259C2×D73—42no—
260C2×D73—42no—
261C4×C73—22no—
262C4×C73—22no—
263C4×C73—28no—
264C4×C73—42no—
265C4×C73—42no—
266C4×C73—44no—
267C4×C73#51914yes—
268Dic73—22no—
269Dic73—28no—
270Dic73—22no—
271Dic73—22no—
272Dic73—42no—
273Dic73—44no—
274Dic73—42no—
275#34403—731nomaximal, Sylow
276C2×C73C2×#51911yescharacteristic
277C2×C73#342811yescharacteristic
278C2×C73#342811yescharacteristic
279C2×C73#356512yes—
280C2×C73#356512yes—
281D73—42no—
282C2×#533—731no—
283#2378—734no—
284#2370—732no—
285#2370—732no—
286#2356—734no—
287#2347—732no—
288#2346—732no—
289#2332—734no—
290#2331—731no—
291#2331—731no—
292#2331—731no—
293#2331D7311yescharacteristic, complemented
294#2320—734no—
295#2283—731no—
296#2283—731no—
297C73#3440311yescharacteristic, complemented, Sylow
298#533—7316no—
299C2×C4⋊Q8—731no—
300C2×C4⋊Q8C2×D7311yescharacteristic
301C2×#125—732no—
302C2×C22⋊Q8—732no—
303C2×C22⋊Q8—732no—
304C2×C4×Q8—732no—
305C2×C4×Q8—732no—
306C2×#117—732no—
307C22×C42—731no—
308#391—734no—
309#391—734no—
310#391—734no—
311#391C2×D7314yes—
312#389—734no—
313#389—732no—
314#389—732no—
315#389—732no—
316#389—734no—
317#389C2×D7312yes—
318#388—734no—
319#388—734no—
320#388—734no—
321#388—734no—
322#387—734no—
323#387—734no—
324#387—734no—
325#387—734no—
326#387—734no—
327#387—734no—
328#387—734no—
329#387C2×D7314yes—
330#383—732no—
331#383—734no—
332#383—732no—
333#382—734no—
334#382—731no—
335#382—732no—
336#382C2×D7311yescharacteristic
337#381—732no—
338#381—732no—
339#380—732no—
340#380—732no—
341C4×C4⋊C4—732no—
342C4×C4⋊C4—732no—
343C4×C4⋊C4—734no—
344C4×C4⋊C4—732no—
345C4×C4⋊C4—734no—
346C4×C4⋊C4—732no—
347C4×C4⋊C4—734no—
348C4×C4⋊C4—731no—
349C4×C4⋊C4—731no—
350C4×C4⋊C4C2×D7312yes—
351C4×C22⋊C4—731no—
352C4×C22⋊C4—731no—
353C4×C22⋊C4—732no—
354#376—731no—
355#376—734no—
356#376—732no—
357#376C2×D7311yescharacteristic
358C22×Q8—732no—
359C22×Q8C22×D7312yes—
360C23×C4—731no—
361C23×C4—731no—
362C23×C4—731no—
363C4⋊Q8—738no—
364C4⋊Q8C4×D7318yes—
365#125—7316no—
366C22⋊Q8—7316no—
367C22⋊Q8—7316no—
368C4×Q8—7316no—
369C4×Q8—7316no—
370#117—7316no—
371C2×C4⋊C4—732no—
372C2×C4⋊C4—732no—
373C2×C4⋊C4—734no—
374C2×C4⋊C4—734no—
375C2×C4⋊C4—734no—
376C2×C4⋊C4—732no—
377C2×C4⋊C4—734no—
378C2×C4⋊C4—734no—
379C2×C4⋊C4—732no—
380C2×C4⋊C4—732no—
381C2×C4⋊C4C22×D7312yes—
382C2×C4⋊C4C22×D7314yes—
383C2×C4⋊C4C22×D7312yes—
384C2×C4⋊C4C22×D7312yes—
385C2×C22⋊C4—732no—
386C2×C22⋊C4—734no—
387C2×C22⋊C4—732no—
388C2×C22⋊C4—734no—
389C2×C42—734no—
390C2×C42—734no—
391C2×C42—734no—
392C2×C42—734no—
393C2×C42—734no—
394C2×C42—731no—
395C2×C42—731no—
396C2×C42—731no—
397C2×C42—738no—
398C2×C42—731no—
399C2×C42—731no—
400C2×C42C22×D7311yescharacteristic
401C2×C42C22×D7311yescharacteristic
402C2×C42C22×D7314yes—
403C2×C42C22×D7311yescharacteristic
404#95—732no—
405#95—734no—
406#95—732no—
407#95—732no—
408#95—732no—
409#95—734no—
410#95—732no—
411#95—732no—
412#95—732no—
413#95—732no—
414#95—734no—
415#95—734no—
416#95—734no—
417#95—734no—
418#95—734no—
419#95—734no—
420#95C22×D7312yes—
421#95C22×D7314yes—
422#95C22×D7312yes—
423#95C22×D7312yes—
424#95C22×D7312yes—
425#95C22×D7314yes—
426C24—731no—
427C2×Q8—44no—
428C2×Q8—2924no—
429C2×Q8—738no—
430C2×Q8C2×C4×D7318yes—
431C22×C4—2922no—
432C22×C4—2922no—
433C22×C4—734no—
434C22×C4—734no—
435C22×C4—731no—
436C22×C4—731no—
437C22×C4—734no—
438C22×C4—731no—
439C22×C4—734no—
440C22×C4—732no—
441C22×C4—732no—
442C22×C4—734no—
443C22×C4—1462no—
444C22×C4—1462no—
445C22×C4C23×D7312yes—
446C22×C4C23×D7314yes—
447C22×C4C23×D7312yes—
448C22×C4C23×D7311yescharacteristic
449C22×C4C23×D7311yescharacteristic
450C22×C4C23×D7314yes—
451C22×C4C23×D7311yescharacteristic
452C4⋊C4—24no—
453C4⋊C4—44no—
454C4⋊C4—2922no—
455C4⋊C4—2922no—
456C4⋊C4—2924no—
457C4⋊C4—2924no—
458C4⋊C4—2924no—
459C4⋊C4—7316no—
460C4⋊C4—738no—
461C4⋊C4—7316no—
462C4⋊C4—738no—
463C4⋊C4—1464no—
464C4⋊C4C2×C4×D7318yes—
465C4⋊C4C2×C4×D7318yes—
466C22⋊C4—2924no—
467C22⋊C4—2922no—
468C22⋊C4—2922no—
469C22⋊C4—7316no—
470C42—22no—
471C42—22no—
472C42—44no—
473C42—2924no—
474C42—2924no—
475C42—2924no—
476C42—2924no—
477C42—734no—
478C42—734no—
479C42—734no—
480C42—7316no—
481C42—1462no—
482C42—1462no—
483C42C2×C4×D7314yes—
484C23—732no—
485C23—1461no—
486C23—1461no—
487C23—1462no—
488C23—1462no—
489C23C24×D7311yescharacteristic
490Q8—48no—
491Q8—2928no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—2924no—
506C2×C4—2924no—
507C2×C4—732no—
508C2×C4—732no—
509C2×C4—732no—
510C2×C4—738no—
511C2×C4—1462no—
512C2×C4—1462no—
513C2×C4—1464no—
514C2×C4—1462no—
515C2×C4—1464no—
516C2×C4—1462no—
517C2×C4—1462no—
518C2×C4—1464no—
519C2×C4—1462no—
520C2×C4—1461no—
521C2×C4—1461no—
522C2×C4—1462no—
523C2×C4—1462no—
524C2×C4—1464no—
525C2×C4—1464no—
526C2×C4—1468no—
527C2×C4C22×C4×D7312yes—
528C2×C4C22×C4×D7312yes—
529C2×C4C22×C4×D7318yes—
530C2×C4C22×C4×D7312yes—
531C2×C4#1455663812yes—
532C2×C4#1455663912yes—
533C22—2922no—
534C22—2922no—
535C22—1462no—
536C22—1462no—
537C22—1462no—
538C22C23×C4×D7311yescharacteristic
539C22C2×#1455663811yescharacteristic
540C22C2×#1455663911yescharacteristic
541C22#1798942212yes—
542C22#1798942412yes—
543C4—22no—
544C4—28no—
545C4—22no—
546C4—42no—
547C4—42no—
548C4—44no—
549C4—2922no—
550C4—2924no—
551C4—2922no—
552C4—1462no—
553C4—1462no—
554C4—1462no—
555C4—1468no—
556C4#1798906314yes—
557C2—2922no—
558C2C2×#1798906311yescharacteristic
559C2#2262312911yescharacteristic
560C2#2262313311yescharacteristic
561C2#2262456412yes—
562C2#2262458212yes—
subgroupG/H
center, FrattiniC23C24×D73
commutatorC22×C73C24×C4
FittingC73×#2331C2
socle, cosocleC23×C73C25
prime radicalC23×D73C24
hypercenter#2331D73
hypocenterC73#34403
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ #2331

Complements

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

NH ≅ G/Ntype
3. #22613029C2semidirect
4. #22613026C2semidirect
5. #22612895C2semidirect
9. #22612680C2semidirect
10. #22612679C2semidirect
11. #22612677C2semidirect
12. #22612580C2semidirect
15. C73×#2331C2semidirect
16. #17989142C4semidirect
293. #2331D73semidirect
297. C73#34403semidirect

Maximal subgroups

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

subgroupindex
C2×#179891422
#226130922
#226130292
#226130262
#226128952
#226128202
#226128152
#226126832
#226126802
#226126792
#226126772
#226125802
#226123242
#226123232
C73×#23312
#3440373