Finite Group Databasebeta

← #33246438

Subgroups — #33246438

Order 18688.

2110 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989143C211yesmaximal, characteristic
2#22613015C214yesmaximal
3#22613008C214yesmaximal
4#22612885C214yesmaximal, complemented
5#22612884C214yesmaximal, complemented
6#22612678C212yesmaximal, complemented
7#22612675C212yesmaximal, complemented
8#22612553C212yesmaximal
9#125×Dic73C211yesmaximal, characteristic, complemented
10C4×#14556583C211yesmaximal, characteristic, complemented
11#22612486C211yesmaximal, characteristic, complemented
12#382×D73C211yesmaximal, characteristic
13#22612285C212yesmaximal
14C4×#14556548C211yesmaximal, characteristic
15C4×#125×C73C211yesmaximal, characteristic, complemented
16#17989143C4116yescomplemented
17C2×#14556590C2212yes—
18C2×#14556583C2211yescharacteristic
19C2×#14556569C2214yes—
20C2×#14556556C2214yes—
21C2×#14556549C2212yes—
22C2×#14556548C2211yescharacteristic
23C2×#14556548C2211yescharacteristic
24C2×C4⋊C4×D73C2212yes—
25C2×C42×D73C2211yescharacteristic
26#17988493C2214yes—
27#17988492C2214yes—
28#17988491C2214yes—
29#17988491C2214yes—
30#17988486C2212yes—
31#17988485C2212yes—
32#17988483C2212yes—
33#17988474C2214yes—
34#17988473C2214yes—
35#17988471C2214yes—
36#17988471C2214yes—
37#17988469C2214yes—
38#17988469C2214yes—
39#17988468C2214yes—
40#17988461C2214yes—
41#17988458C2214yes—
42#17988456C2214yes—
43#17988453C2211yescharacteristic
44#17988452C2212yes—
45#17988451C2214yes—
46#17988451C2214yes—
47C4⋊C4×Dic73C2212yes—
48C4⋊C4×Dic73C2214yes—
49C4⋊C4×Dic73C2212yes—
50C4×#11924007C2214yes—
51C4×#11924007C2212yes—
52C4×#11924007C2212yes—
53C4×#11924007C2214yes—
54C4×C4⋊Dic73C2212yes—
55#17988438C2214yes—
56#17988438C2214yes—
57C4×#11924005C2212yes—
58C4×#11924005C2212yes—
59C4×#11924005C2214yes—
60#17988434C2214yes—
61#17988433C2211yescharacteristic
62#17988433C2211yescharacteristic
63#95×D73C2214yes—
64C42×Dic73C2211yescharacteristic
65C2×#125×C73C2211yescharacteristic
66#389×C73C2212yes—
67#387×C73C2214yes—
68#382×C73C2211yescharacteristic
69C4×C4⋊C4×C73C2214yes—
70C4×C4⋊C4×C73C2212yes—
71C43×C73C2211yescharacteristic
72C22×Dic146C2314yes—
73C22×C4×D73C2314yes—
74C22×C4×D73C2312yes—
75C22×C4×D73C2311yescharacteristic
76#14556590C2×C4116yescomplemented
77#14556583C2×C418yescomplemented
78#14556569C2×C4132yes—
79#14556556C2×C4132yescomplemented
80#14556549C2×C4116yes—
81#14556548—24no—
82#14556548Q818yescomplemented
83C2×C4⋊Dic73C2312yes—
84C2×C4⋊Dic73C2314yes—
85C2×C4⋊Dic73C2312yes—
86C2×#11924007C2314yes—
87C2×#11924007C2312yes—
88C2×#11924007C2314yes—
89C2×#11924007C2314yes—
90C2×#11924007C2314yes—
91C2×#11924007C2314yes—
92C2×#11924007C2311yescharacteristic
93C2×#11924007C2311yescharacteristic
94C4⋊C4×D73—28no—
95C2×#11924005C2312yes—
96C2×#11924005C2314yes—
97C2×#11924005C2311yescharacteristic
98C2×#11924005C2311yescharacteristic
99C2×C4×Dic73C2311yescharacteristic
100C2×C4×Dic73C2312yes—
101C2×C4×Dic73C2312yes—
102C2×C4×Dic73C2314yes—
103C2×C4×Dic73C2314yes—
104C2×C4×Dic73C2314yes—
105C2×C4×Dic73C2311yescharacteristic
106C2×C4×Dic73C2314yes—
107C2×C4×Dic73C2312yes—
108C2×C4×Dic73C2311yescharacteristic
109C42×D73C2×C418yes—
110#14556498C2314yes—
111#14556498C2314yes—
112#14556498C2314yes—
113#14556498C2312yes—
114#14556498C2312yes—
115#14556498C2312yes—
116#14556498C2312yes—
117#14556498C2314yes—
118#14556498C2314yes—
119#14556498C2314yes—
120#14556498C2314yes—
121#14556498C2314yes—
122#14556498C2314yes—
123#125×C73C2×C418yescomplemented
124C2×C4⋊C4×C73C2312yes—
125C2×C4⋊C4×C73C2314yes—
126C2×C4⋊C4×C73C2314yes—
127C2×C4⋊C4×C73C2312yes—
128C2×C42×C73C2311yescharacteristic
129C2×C42×C73C2314yes—
130C2×C42×C73C2312yes—
131C2×C42×C73C2312yes—
132C2×C42×C73C2311yescharacteristic
133C2×C42×C73C2311yescharacteristic
134#95×C73C2314yes—
135#95×C73C2314yes—
136#95×C73C2314yes—
137C23×D73C2411yescharacteristic
138C2×Dic146—44no—
139C2×Dic146—44no—
140C2×Dic146C22×C4116yes—
141C22×Dic73C2414yes—
142C22×Dic73C2412yes—
143C22×Dic73C2412yes—
144C22×Dic73C2414yes—
145C22×Dic73C2411yescharacteristic
146C22×Dic73C2411yescharacteristic
147C22×Dic73C2411yescharacteristic
148C2×C4×D73—28no—
149C2×C4×D73—42no—
150C2×C4×D73—42no—
151C2×C4×D73—44no—
152C2×C4×D73—44no—
153C2×C4×D73C22×C418yes—
154C2×C4×D73C22×C414yes—
155C2×C4×D73C2×Q814yes—
156C2×C4×D73#4114yes—
157C4⋊Dic73—24no—
158C4⋊Dic73—44no—
159C4⋊Dic73C22×C418yes—
160#11924007—28no—
161#11924007—24no—
162#11924007—44no—
163#11924007—44no—
164#11924007C22×C4116yes—
165#11924007C22×C4116yes—
166#11924007C2×Q814yes—
167#11924007C2×Q814yes—
168#11924005—24no—
169#11924005C22×C4116yes—
170#11924005C2×Q814yes—
171#11924005C2×Q814yes—
172C4×Dic73—22no—
173C4×Dic73—44no—
174C4×Dic73—42no—
175C4×Dic73—44no—
176C4×Dic73—42no—
177C4×Dic73—44no—
178C4×Dic73C22×C418yes—
179C4×Dic73C22×C414yes—
180C4×Dic73C22×C4116yes—
181C4×Dic73C2×Q814yes—
182C22×C4×C73C2414yes—
183C22×C4×C73C2412yes—
184C22×C4×C73C2412yes—
185C22×C4×C73C2414yes—
186C22×C4×C73C2411yescharacteristic
187C22×C4×C73C2411yescharacteristic
188C22×C4×C73C2411yescharacteristic
189C4⋊C4×C73—24no—
190C4⋊C4×C73—44no—
191C4⋊C4×C73C22×C418yes—
192C4⋊C4×C73C22×C4116yes—
193C42×C73—22no—
194C42×C73—42no—
195C42×C73—44no—
196C42×C73—42no—
197C42×C73C22×C414yes—
198C42×C73C2×Q814yes—
199C23×C73C2511yescharacteristic
200C22×D73—22no—
201C22×D73—22no—
202C22×D73C23×C412yes—
203C22×D73C22×Q812yes—
204C22×D73C2×#4112yes—
205C2×C4×C73—22no—
206C2×C4×C73—22no—
207C2×C4×C73—22no—
208C2×C4×C73—22no—
209C2×C4×C73—24no—
210C2×C4×C73—24no—
211C2×C4×C73—22no—
212C2×C4×C73—24no—
213C2×C4×C73—22no—
214C2×C4×C73—22no—
215C2×C4×C73—24no—
216C2×C4×C73—24no—
217C2×C4×C73C23×C414yes—
218C2×C4×C73C23×C418yes—
219C2×C4×C73C23×C412yes—
220C2×C4×C73C22×Q812yes—
221C2×C4×C73C22×Q812yes—
222C2×C4×C73C22×Q812yes—
223C2×C4×C73C2×#4112yes—
224C2×Dic73—22no—
225C2×Dic73—22no—
226C2×Dic73—24no—
227C2×Dic73—24no—
228C2×Dic73—22no—
229C2×Dic73—22no—
230C2×Dic73—22no—
231C2×Dic73—24no—
232C2×Dic73—22no—
233C2×Dic73—24no—
234C2×Dic73—22no—
235C2×Dic73—24no—
236C2×Dic73C23×C414yes—
237C2×Dic73C23×C418yes—
238C2×Dic73C23×C412yes—
239C2×Dic73C22×Q812yes—
240C2×Dic73C22×Q812yes—
241C2×Dic73C22×Q812yes—
242C2×Dic73C2×#4112yes—
243C4×D73—48no—
244C4×D73—88no—
245C4×D73C4×Q8116yes—
246Dic146—416no—
247C22×C73C24×C411yescharacteristic
248C22×C73C23×Q811yescharacteristic
249C22×C73C22×#4111yescharacteristic
250C22×C73#58512yes—
251C22×C73#58512yes—
252C2×D73—24no—
253C2×D73—22no—
254C2×D73—22no—
255C2×D73—24no—
256C2×D73C2×C4×Q814yes—
257C4×C73—28no—
258C4×C73—24no—
259C4×C73—22no—
260C4×C73—22no—
261C4×C73—44no—
262C4×C73—42no—
263C4×C73C2×C4×Q814yes—
264Dic73—24no—
265Dic73—28no—
266Dic73—22no—
267Dic73—22no—
268Dic73—44no—
269Dic73—42no—
270Dic73C2×C4×Q814yes—
271#34215—731nomaximal, Sylow
272C2×C73C22×C4×Q811yescharacteristic
273C2×C73#342812yes—
274C2×C73#350411yescharacteristic
275C2×C73#350411yescharacteristic
276C2×C73#351012yes—
277D73—24no—
278C2×#533—731no—
279#2368—734no—
280#2368—734no—
281#2355—734no—
282#2355—734no—
283#2331—732no—
284#2331—732no—
285#2317—732no—
286C4×#125—731no—
287C4×#125—731no—
288C4×#125—731no—
289C4×#125D7311yescharacteristic, complemented
290C2×#382—731no—
291#2274—732no—
292C4×#117—731no—
293C73#3421511yescharacteristic, complemented, Sylow
294#533—7316no—
295C2×C4⋊Q8—732no—
296C2×#125—731no—
297C2×#125C2×D7311yescharacteristic
298C2×C22⋊Q8—734no—
299C2×C4×Q8—734no—
300C2×#117—732no—
301C2×#117—731no—
302C2×#117—731no—
303C22×C4⋊C4—732no—
304C22×C42—731no—
305#391—734no—
306#391—734no—
307#391—734no—
308#391—734no—
309#389—732no—
310#389—732no—
311#389—732no—
312#389C2×D7312yes—
313#387—734no—
314#387—734no—
315#387—734no—
316#387—734no—
317#387—734no—
318#387—734no—
319#387—734no—
320#387C2×D7314yes—
321#385—734no—
322#385—734no—
323#384—734no—
324#382—732no—
325#382—731no—
326#382—734no—
327#382—734no—
328#382C2×D7311yescharacteristic
329C4×C4⋊C4—732no—
330C4×C4⋊C4—732no—
331C4×C4⋊C4—734no—
332C4×C4⋊C4—732no—
333C4×C4⋊C4—734no—
334C4×C4⋊C4—732no—
335C4×C4⋊C4—732no—
336C4×C4⋊C4—734no—
337C4×C4⋊C4—734no—
338C4×C4⋊C4—734no—
339C4×C4⋊C4C2×D7312yes—
340C4×C4⋊C4C2×D7314yes—
341C4×C22⋊C4—732no—
342C4×C22⋊C4—732no—
343C4×C22⋊C4—734no—
344C4×C22⋊C4—734no—
345#376—731no—
346#376—731no—
347C2×#95—734no—
348C43—731no—
349C43C2×D7311yescharacteristic
350C22×Q8—734no—
351C23×C4—732no—
352C23×C4—734no—
353C23×C4—731no—
354C4⋊Q8—7316no—
355#125—738no—
356#125C4×D7318yescomplemented
357C22⋊Q8—7332no—
358C4×Q8—7332no—
359#117—7316no—
360#117—738no—
361#117—1464no—
362C2×C4⋊C4—734no—
363C2×C4⋊C4—734no—
364C2×C4⋊C4—734no—
365C2×C4⋊C4—732no—
366C2×C4⋊C4—731no—
367C2×C4⋊C4—731no—
368C2×C4⋊C4—734no—
369C2×C4⋊C4—734no—
370C2×C4⋊C4—734no—
371C2×C4⋊C4—732no—
372C2×C4⋊C4—732no—
373C2×C4⋊C4—1468no—
374C2×C4⋊C4C22×D7312yes—
375C2×C4⋊C4C22×D7314yes—
376C2×C4⋊C4C22×D7312yes—
377C2×C4⋊C4C22×D7314yes—
378C2×C22⋊C4—734no—
379C2×C22⋊C4—731no—
380C2×C22⋊C4—731no—
381C2×C22⋊C4—732no—
382C2×C42—732no—
383C2×C42—732no—
384C2×C42—734no—
385C2×C42—734no—
386C2×C42—734no—
387C2×C42—734no—
388C2×C42—731no—
389C2×C42—732no—
390C2×C42—731no—
391C2×C42—738no—
392C2×C42—731no—
393C2×C42C22×D7311yescharacteristic
394C2×C42C22×D7312yes—
395C2×C42C22×D7312yes—
396C2×C42C22×D7314yes—
397C2×C42C22×D7311yescharacteristic
398C2×C42C22×D7311yescharacteristic
399#95—734no—
400#95—734no—
401#95—734no—
402#95—732no—
403#95—732no—
404#95—732no—
405#95—732no—
406#95—734no—
407#95—734no—
408#95—734no—
409#95—734no—
410#95—734no—
411#95—734no—
412#95C22×D7314yes—
413#95C22×D7314yes—
414#95C22×D7314yes—
415C24—731no—
416C2×Q8—2924no—
417C2×Q8—2924no—
418C2×Q8—7316no—
419C22×C4—2922no—
420C22×C4—2922no—
421C22×C4—2924no—
422C22×C4—2924no—
423C22×C4—738no—
424C22×C4—731no—
425C22×C4—734no—
426C22×C4—734no—
427C22×C4—734no—
428C22×C4—732no—
429C22×C4—731no—
430C22×C4—731no—
431C22×C4—734no—
432C22×C4—734no—
433C22×C4—732no—
434C22×C4—1468no—
435C22×C4C23×D7314yes—
436C22×C4C23×D7312yes—
437C22×C4C23×D7312yes—
438C22×C4C23×D7314yes—
439C22×C4C23×D7311yescharacteristic
440C22×C4C23×D7311yescharacteristic
441C22×C4C23×D7311yescharacteristic
442C4⋊C4—24no—
443C4⋊C4—44no—
444C4⋊C4—2924no—
445C4⋊C4—2924no—
446C4⋊C4—2924no—
447C4⋊C4—7316no—
448C4⋊C4—7316no—
449C4⋊C4—738no—
450C4⋊C4—734no—
451C4⋊C4—734no—
452C4⋊C4—1468no—
453C4⋊C4—1464no—
454C4⋊C4—1464no—
455C4⋊C4C2×C4×D7318yes—
456C4⋊C4C2×C4×D73116yes—
457C22⋊C4—7316no—
458C22⋊C4—734no—
459C22⋊C4—734no—
460C22⋊C4—1464no—
461C42—22no—
462C42—42no—
463C42—44no—
464C42—42no—
465C42—2924no—
466C42—2922no—
467C42—2922no—
468C42—2924no—
469C42—2924no—
470C42—738no—
471C42—734no—
472C42—7316no—
473C42—734no—
474C42—1462no—
475C42C2×C4×D7314yes—
476C42Q8×D7314yes—
477C23—732no—
478C23—732no—
479C23—732no—
480C23—1462no—
481C23—1462no—
482C23C24×D7311yescharacteristic
483Q8—29216no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—2928no—
497C2×C4—5848no—
498C2×C4—732no—
499C2×C4—732no—
500C2×C4—732no—
501C2×C4—7316no—
502C2×C4—734no—
503C2×C4—738no—
504C2×C4—732no—
505C2×C4—732no—
506C2×C4—1462no—
507C2×C4—1464no—
508C2×C4—1462no—
509C2×C4—1462no—
510C2×C4—1464no—
511C2×C4—1464no—
512C2×C4—1462no—
513C2×C4—1462no—
514C2×C4—1464no—
515C2×C4—1464no—
516C2×C4—1462no—
517C2×C4—1462no—
518C2×C4C22×C4×D7314yes—
519C2×C4C22×C4×D7318yes—
520C2×C4C22×C4×D7312yes—
521C2×C4C2×Q8×D7312yes—
522C2×C4C2×Q8×D7312yes—
523C2×C4C2×Q8×D7312yes—
524C2×C4#41×D7312yes—
525C22—734no—
526C22—1462no—
527C22—1462no—
528C22—1464no—
529C22—1464no—
530C22C23×C4×D7311yescharacteristic
531C22C22×Q8×D7311yescharacteristic
532C22C2×#41×D7311yescharacteristic
533C22#1798942312yes—
534C22#1798942312yes—
535C4—22no—
536C4—22no—
537C4—24no—
538C4—28no—
539C4—44no—
540C4—42no—
541C4—2924no—
542C4—2922no—
543C4—734no—
544C4—1462no—
545C4—1462no—
546C4—1464no—
547C4—1468no—
548C4C4×Q8×D7314yes—
549C2—1464no—
550C2C2×C4×Q8×D7311yescharacteristic
551C2#2262313012yes—
552C2#2262375211yescharacteristic
553C2#2262375511yescharacteristic
554C2#2262382012yes—
subgroupG/H
center, FrattiniC23C24×D73
commutatorC22×C73C24×C4
FittingC4×#125×C73C2
socle, cosocleC23×C73C25
prime radicalC23×D73C24
hypercenterC4×#125D73
hypocenterC73#34215
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×#125

Complements

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

NH ≅ G/Ntype
4. #22612885C2semidirect
5. #22612884C2semidirect
6. #22612678C2semidirect
7. #22612675C2semidirect
9. #125×Dic73C2semidirect
10. C4×#14556583C2semidirect
11. #22612486C2semidirect
15. C4×#125×C73C2semidirect
16. #17989143C4semidirect
76. #14556590C2×C4semidirect
77. #14556583C2×C4semidirect
79. #14556556C2×C4semidirect
82. #14556548Q8semidirect
123. #125×C73C2×C4semidirect
289. C4×#125D73semidirect
293. C73#34215semidirect
356. #125C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179891432
#226130152
#226130082
#226128852
#226128842
#226126782
#226126752
#226125532
#125×Dic732
C4×#145565832
#226124862
#382×D732
#226122852
C4×#145565482
C4×#125×C732
#3421573