Finite Group Databasebeta

← #33246431

Subgroups — #33246431

Order 18688.

2310 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989143C211yesmaximal, characteristic
2#22612939C214yesmaximal
3#22612933C214yesmaximal
4#22612885C214yesmaximal, complemented
5#22612845C214yesmaximal, complemented
6#22612673C211yesmaximal, characteristic, complemented
7#22612672C211yesmaximal, characteristic, complemented
8#22612670C211yesmaximal, characteristic, complemented
9#22612553C212yesmaximal
10#22612502C212yesmaximal, complemented
11C4×#14556590C212yesmaximal, complemented
12#384×D73C211yesmaximal, characteristic
13#22612285C212yesmaximal
14#22612270C211yesmaximal, characteristic
15C73×#2330C211yesmaximal, characteristic, complemented
16#17989143C4116yescomplemented
17C2×#14556590C2212yes—
18C2×#14556583C2211yescharacteristic
19C2×#14556569C2214yes—
20C2×C4×Dic146C2214yes—
21C2×#14556556C2214yes—
22C2×#14556556C2214yes—
23C2×#14556549C2212yes—
24C2×#14556548C2211yescharacteristic
25C2×#14556548C2211yescharacteristic
26C2×C4⋊C4×D73C2212yes—
27C2×C4⋊C4×D73C2214yes—
28C2×C42×D73C2211yescharacteristic
29#17988491C2214yes—
30#17988491C2214yes—
31#17988486C2212yes—
32#17988486C2214yes—
33#17988485C2212yes—
34#17988484C2214yes—
35#17988484C2214yes—
36#17988483C2212yes—
37#17988482C2214yes—
38#17988482C2214yes—
39#17988474C2214yes—
40#17988473C2214yes—
41#17988468C2214yes—
42#17988461C2214yes—
43#17988458C2214yes—
44#17988457C2211yescharacteristic
45#17988456C2212yes—
46#17988456C2214yes—
47#17988456C2214yes—
48#17988451C2214yes—
49C4⋊C4×Dic73C2212yes—
50C4⋊C4×Dic73C2212yes—
51C4×#11924007C2214yes—
52C4×#11924007C2214yes—
53C4×C4⋊Dic73C2212yes—
54#17988438C2214yes—
55#17988438C2212yes—
56#17988438C2212yes—
57#17988438C2214yes—
58C4×#11924005C2214yes—
59#17988434C2212yes—
60#17988434C2212yes—
61#17988434C2214yes—
62#17988433C2211yescharacteristic
63C42×Dic73C2211yescharacteristic
64C42×Dic73C2211yescharacteristic
65C2×#125×C73C2211yescharacteristic
66#389×C73C2212yes—
67#389×C73C2214yes—
68#387×C73C2214yes—
69#384×C73C2211yescharacteristic
70C4×C4⋊C4×C73C2212yes—
71#376×C73C2211yescharacteristic
72C22×Dic146C2314yes—
73C22×C4×D73C2312yes—
74C22×C4×D73C2314yes—
75C22×C4×D73C2311yescharacteristic
76#14556590C2×C4116yescomplemented
77#14556583C2×C418yescomplemented
78#14556569C2×C4132yes—
79C4×Dic146—48no—
80#14556556—48no—
81#14556556C2×C4132yescomplemented
82#14556549C2×C4116yes—
83#14556548—24no—
84#14556548Q818yescomplemented
85C2×C4⋊Dic73C2312yes—
86C2×C4⋊Dic73C2314yes—
87C2×C4⋊Dic73C2314yes—
88C2×C4⋊Dic73C2312yes—
89C2×#11924007C2311yescharacteristic
90C2×#11924007C2311yescharacteristic
91C2×#11924007C2314yes—
92C2×#11924007C2314yes—
93C2×#11924007C2314yes—
94C2×#11924007C2314yes—
95C2×#11924007C2314yes—
96C2×#11924007C2314yes—
97C2×#11924007C2314yes—
98C2×#11924007C2312yes—
99C4⋊C4×D73—28no—
100C4⋊C4×D73—48no—
101C2×#11924005C2311yescharacteristic
102C2×#11924005C2311yescharacteristic
103C2×#11924005C2314yes—
104C2×#11924005C2312yes—
105C2×C4×Dic73C2311yescharacteristic
106C2×C4×Dic73C2312yes—
107C2×C4×Dic73C2312yes—
108C2×C4×Dic73C2312yes—
109C2×C4×Dic73C2312yes—
110C2×C4×Dic73C2314yes—
111C2×C4×Dic73C2314yes—
112C2×C4×Dic73C2314yes—
113C2×C4×Dic73C2311yescharacteristic
114C2×C4×Dic73C2312yes—
115C2×C4×Dic73C2314yes—
116C2×C4×Dic73C2311yescharacteristic
117C42×D73C2×C418yes—
118#14556498C2312yes—
119#14556498C2312yes—
120#14556498C2314yes—
121#14556498C2314yes—
122#14556498C2314yes—
123#14556498C2314yes—
124#14556498C2314yes—
125#14556498C2314yes—
126#125×C73C2×C418yescomplemented
127C2×C4⋊C4×C73C2312yes—
128C2×C4⋊C4×C73C2314yes—
129C2×C4⋊C4×C73C2314yes—
130C2×C4⋊C4×C73C2312yes—
131C2×C4⋊C4×C73C2314yes—
132C2×C42×C73C2311yescharacteristic
133C2×C42×C73C2314yes—
134C2×C42×C73C2311yescharacteristic
135C2×C42×C73C2311yescharacteristic
136#95×C73C2314yes—
137#95×C73C2314yes—
138#95×C73C2312yes—
139#95×C73C2312yes—
140C23×D73C2411yescharacteristic
141C2×Dic146—28no—
142C2×Dic146—28no—
143C2×Dic146C22×C4116yes—
144C22×Dic73C2411yescharacteristic
145C22×Dic73C2414yes—
146C22×Dic73C2411yescharacteristic
147C22×Dic73C2414yes—
148C22×Dic73C2412yes—
149C22×Dic73C2411yescharacteristic
150C22×Dic73C2412yes—
151C2×C4×D73—24no—
152C2×C4×D73—24no—
153C2×C4×D73—28no—
154C2×C4×D73—44no—
155C2×C4×D73—44no—
156C2×C4×D73C22×C418yes—
157C2×C4×D73C22×C414yes—
158C2×C4×D73C2×Q814yes—
159C2×C4×D73#4114yes—
160C4⋊Dic73—24no—
161C4⋊Dic73—44no—
162C4⋊Dic73—44no—
163C4⋊Dic73C22×C418yes—
164#11924007—28no—
165#11924007—22no—
166#11924007—22no—
167#11924007—28no—
168#11924007—28no—
169#11924007—44no—
170#11924007—44no—
171#11924007C22×C4116yes—
172#11924007C22×C4116yes—
173#11924007C2×Q818yes—
174#11924005—22no—
175#11924005—22no—
176#11924005C22×C4116yes—
177#11924005C2×Q818yes—
178C4×Dic73—22no—
179C4×Dic73—24no—
180C4×Dic73—24no—
181C4×Dic73—24no—
182C4×Dic73—24no—
183C4×Dic73—44no—
184C4×Dic73—44no—
185C4×Dic73—44no—
186C4×Dic73C22×C418yes—
187C4×Dic73C22×C414yes—
188C4×Dic73C22×C4116yes—
189C4×Dic73C2×Q814yes—
190C22×C4×C73C2411yescharacteristic
191C22×C4×C73C2414yes—
192C22×C4×C73C2411yescharacteristic
193C22×C4×C73C2414yes—
194C22×C4×C73C2412yes—
195C22×C4×C73C2412yes—
196C22×C4×C73C2411yescharacteristic
197C4⋊C4×C73—24no—
198C4⋊C4×C73—44no—
199C4⋊C4×C73—44no—
200C4⋊C4×C73C22×C418yes—
201C4⋊C4×C73C22×C4116yes—
202C42×C73—22no—
203C42×C73—44no—
204C42×C73C22×C414yes—
205C42×C73C2×Q814yes—
206C23×C73C2511yescharacteristic
207C22×D73—22no—
208C22×D73—22no—
209C22×D73C23×C412yes—
210C22×D73C22×Q812yes—
211C22×D73C2×#4112yes—
212C2×C4×C73—22no—
213C2×C4×C73—24no—
214C2×C4×C73—22no—
215C2×C4×C73—22no—
216C2×C4×C73—22no—
217C2×C4×C73—24no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—24no—
221C2×C4×C73—24no—
222C2×C4×C73—22no—
223C2×C4×C73—24no—
224C2×C4×C73—22no—
225C2×C4×C73C23×C414yes—
226C2×C4×C73C23×C418yes—
227C2×C4×C73C23×C412yes—
228C2×C4×C73C22×Q814yes—
229C2×C4×C73C22×Q812yes—
230C2×C4×C73C2×#4112yes—
231C2×Dic73—22no—
232C2×Dic73—24no—
233C2×Dic73—22no—
234C2×Dic73—24no—
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—24no—
238C2×Dic73—22no—
239C2×Dic73—24no—
240C2×Dic73—24no—
241C2×Dic73—22no—
242C2×Dic73C23×C418yes—
243C2×Dic73C23×C412yes—
244C2×Dic73C23×C414yes—
245C2×Dic73C22×Q814yes—
246C2×Dic73C22×Q812yes—
247C2×Dic73C2×#4112yes—
248C2×Dic73H(5,𝔽2)14yes—
249C2×Dic732−1+414yes—
250C4×D73—216no—
251C4×D73—88no—
252C4×D73C4×Q8116yescomplemented
253Dic146—232no—
254C22×C73C24×C411yescharacteristic
255C22×C73C23×Q811yescharacteristic
256C22×C73C22×#4111yescharacteristic
257C22×C73C2×H(5,𝔽2)12yes—
258C22×C73C2×2−1+412yes—
259C2×D73—24no—
260C2×D73—24no—
261C2×D73—22no—
262C2×D73—22no—
263C2×D73C2×C4×Q814yes—
264C4×C73—28no—
265C4×C73—24no—
266C4×C73—24no—
267C4×C73—44no—
268C4×C73—41no—
269C4×C73—41no—
270C4×C73C2×C4×Q814yes—
271Dic73—28no—
272Dic73—24no—
273Dic73—44no—
274Dic73—41no—
275Dic73—41no—
276Dic73C2×C4×Q814yes—
277Dic73#52018yes—
278#34214—731nomaximal, Sylow
279C2×C73C22×C4×Q811yescharacteristic
280C2×C73C2×#52012yes—
281C2×C73#350312yes—
282C2×C73#350811yescharacteristic
283C2×C73#350911yescharacteristic
284D73—24no—
285C2×#533—731no—
286#2360—734no—
287#2360—734no—
288#2355—734no—
289Q8×C4⋊C4—734no—
290#2330—731no—
291#2330—731no—
292#2330—731no—
293#2330D7311yescharacteristic, complemented
294#2317—732no—
295C4×C4⋊Q8—732no—
296C4×C4⋊Q8—732no—
297C2×#384—731no—
298#2274—732no—
299C4×#117—731no—
300C73#3421411yescharacteristic, complemented, Sylow
301#533—7316no—
302C2×C4⋊Q8—732no—
303C2×#125—731no—
304C2×#125C2×D7311yescharacteristic
305C2×C22⋊Q8—734no—
306C2×C4×Q8—734no—
307C2×C4×Q8—734no—
308C2×C4×Q8—734no—
309C2×#117—732no—
310C2×#117—731no—
311C2×#117—731no—
312C22×C4⋊C4—734no—
313C22×C4⋊C4—732no—
314C22×C42—731no—
315#391—734no—
316#391—734no—
317#389—732no—
318#389—732no—
319#389—734no—
320#389—734no—
321#389—734no—
322#389—732no—
323#389—734no—
324#389—734no—
325#389C2×D7312yes—
326#389C2×D7314yes—
327#387—734no—
328#387—734no—
329#387—734no—
330#387C2×D7314yes—
331#385—734no—
332#385—734no—
333#384—732no—
334#384—734no—
335#384—734no—
336#384—731no—
337#384C2×D7311yescharacteristic
338#382—734no—
339C4×C4⋊C4—732no—
340C4×C4⋊C4—732no—
341C4×C4⋊C4—732no—
342C4×C4⋊C4—734no—
343C4×C4⋊C4—734no—
344C4×C4⋊C4—734no—
345C4×C4⋊C4—732no—
346C4×C4⋊C4—734no—
347C4×C4⋊C4—732no—
348C4×C4⋊C4C2×D7312yes—
349C4×C22⋊C4—734no—
350C4×C22⋊C4—732no—
351C4×C22⋊C4—732no—
352C4×C22⋊C4—734no—
353#376—731no—
354#376C2×D7311yescharacteristic
355C43—731no—
356C43—731no—
357C22×Q8—734no—
358C23×C4—734no—
359C23×C4—731no—
360C23×C4—732no—
361C4⋊Q8—7316no—
362#125—738no—
363#125C4×D7318yescomplemented
364C22⋊Q8—7332no—
365C4×Q8—2928no—
366C4×Q8—2928no—
367C4×Q8—7332no—
368#117—7316no—
369#117—738no—
370#117—1464no—
371C2×C4⋊C4—2928no—
372C2×C4⋊C4—734no—
373C2×C4⋊C4—734no—
374C2×C4⋊C4—734no—
375C2×C4⋊C4—734no—
376C2×C4⋊C4—732no—
377C2×C4⋊C4—732no—
378C2×C4⋊C4—734no—
379C2×C4⋊C4—734no—
380C2×C4⋊C4—734no—
381C2×C4⋊C4—734no—
382C2×C4⋊C4—734no—
383C2×C4⋊C4—732no—
384C2×C4⋊C4—731no—
385C2×C4⋊C4—731no—
386C2×C4⋊C4—1468no—
387C2×C4⋊C4C22×D7312yes—
388C2×C4⋊C4C22×D7314yes—
389C2×C4⋊C4C22×D7314yes—
390C2×C4⋊C4C22×D7312yes—
391C2×C4⋊C4C22×D7314yes—
392C2×C22⋊C4—734no—
393C2×C22⋊C4—732no—
394C2×C22⋊C4—731no—
395C2×C22⋊C4—731no—
396C2×C42—734no—
397C2×C42—734no—
398C2×C42—734no—
399C2×C42—734no—
400C2×C42—731no—
401C2×C42—732no—
402C2×C42—731no—
403C2×C42—738no—
404C2×C42—731no—
405C2×C42—732no—
406C2×C42—732no—
407C2×C42—732no—
408C2×C42—732no—
409C2×C42C22×D7311yescharacteristic
410C2×C42C22×D7314yes—
411C2×C42C22×D7311yescharacteristic
412C2×C42C22×D7311yescharacteristic
413#95—732no—
414#95—732no—
415#95—734no—
416#95—734no—
417#95—734no—
418#95—734no—
419#95—734no—
420#95—734no—
421#95C22×D7312yes—
422#95C22×D7312yes—
423#95C22×D7314yes—
424#95C22×D7314yes—
425C24—731no—
426C2×Q8—7316no—
427C2×Q8—1468no—
428C2×Q8—1468no—
429C22×C4—2924no—
430C22×C4—2924no—
431C22×C4—731no—
432C22×C4—732no—
433C22×C4—734no—
434C22×C4—734no—
435C22×C4—734no—
436C22×C4—732no—
437C22×C4—734no—
438C22×C4—738no—
439C22×C4—731no—
440C22×C4—734no—
441C22×C4—731no—
442C22×C4—1468no—
443C22×C4—1464no—
444C22×C4—1464no—
445C22×C4C23×D7311yescharacteristic
446C22×C4C23×D7314yes—
447C22×C4C23×D7311yescharacteristic
448C22×C4C23×D7314yes—
449C22×C4C23×D7312yes—
450C22×C4C23×D7312yes—
451C22×C4C23×D7311yescharacteristic
452C4⋊C4—24no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—2924no—
456C4⋊C4—2924no—
457C4⋊C4—2924no—
458C4⋊C4—2924no—
459C4⋊C4—7316no—
460C4⋊C4—7316no—
461C4⋊C4—738no—
462C4⋊C4—738no—
463C4⋊C4—1468no—
464C4⋊C4—1468no—
465C4⋊C4—1468no—
466C4⋊C4—1462no—
467C4⋊C4—1462no—
468C4⋊C4—1464no—
469C4⋊C4C2×C4×D7318yes—
470C4⋊C4C2×C4×D73116yes—
471C22⋊C4—7316no—
472C22⋊C4—738no—
473C22⋊C4—1462no—
474C22⋊C4—1462no—
475C42—22no—
476C42—44no—
477C42—2924no—
478C42—2924no—
479C42—2924no—
480C42—738no—
481C42—734no—
482C42—7316no—
483C42—734no—
484C42—1464no—
485C42—1464no—
486C42—1464no—
487C42—1464no—
488C42—1462no—
489C42C2×C4×D7314yes—
490C42Q8×D7314yes—
491C23—732no—
492C23—732no—
493C23—732no—
494C23—1462no—
495C23—1462no—
496C23C24×D7311yescharacteristic
497Q8—14632no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—5848no—
512C2×C4—732no—
513C2×C4—734no—
514C2×C4—732no—
515C2×C4—732no—
516C2×C4—734no—
517C2×C4—734no—
518C2×C4—7316no—
519C2×C4—738no—
520C2×C4—734no—
521C2×C4—14616no—
522C2×C4—1461no—
523C2×C4—1461no—
524C2×C4—1464no—
525C2×C4—1464no—
526C2×C4—1464no—
527C2×C4—1462no—
528C2×C4—1462no—
529C2×C4—1462no—
530C2×C4—1464no—
531C2×C4—1462no—
532C2×C4—1464no—
533C2×C4C22×C4×D7314yes—
534C2×C4C22×C4×D7318yes—
535C2×C4C22×C4×D7312yes—
536C2×C4C2×Q8×D7314yes—
537C2×C4C2×Q8×D7312yes—
538C2×C4#41×D7312yes—
539C22—734no—
540C22—1464no—
541C22—1462no—
542C22—1462no—
543C22—1464no—
544C22C23×C4×D7311yescharacteristic
545C22C22×Q8×D7311yescharacteristic
546C22C2×#41×D7311yescharacteristic
547C22H(5,𝔽2)×D7312yes—
548C222−1+4×D7312yes—
549C4—24no—
550C4—24no—
551C4—28no—
552C4—44no—
553C4—41no—
554C4—41no—
555C4—2924no—
556C4—2921no—
557C4—2921no—
558C4—734no—
559C4—738no—
560C4—1464no—
561C4—1468no—
562C4C4×Q8×D7314yes—
563C2—1464no—
564C2C2×C4×Q8×D7311yescharacteristic
565C2#520×D7312yes—
566C2#2262374612yes—
567C2#2262380311yescharacteristic
568C2#2262381311yescharacteristic
subgroupG/H
center, FrattiniC23C24×D73
commutatorC22×C73C24×C4
FittingC73×#2330C2
socle, cosocleC23×C73C25
prime radicalC23×D73C24
hypercenter#2330D73
hypocenterC73#34214
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ #2330

Complements

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

NH ≅ G/Ntype
4. #22612885C2semidirect
5. #22612845C2semidirect
6. #22612673C2semidirect
7. #22612672C2semidirect
8. #22612670C2semidirect
10. #22612502C2semidirect
11. C4×#14556590C2semidirect
15. C73×#2330C2semidirect
16. #17989143C4semidirect
76. #14556590C2×C4semidirect
77. #14556583C2×C4semidirect
81. #14556556C2×C4semidirect
84. #14556548Q8semidirect
126. #125×C73C2×C4semidirect
252. C4×D73C4×Q8semidirect
293. #2330D73semidirect
300. C73#34214semidirect
363. #125C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179891432
#226129392
#226129332
#226128852
#226128452
#226126732
#226126722
#226126702
#226125532
#226125022
C4×#145565902
#384×D732
#226122852
#226122702
C73×#23302
#3421473