Finite Group Databasebeta

← #33259175

Subgroups — #33259175

Order 18688.

2042 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618241C211yesmaximal, characteristic, complemented
2#22618239C212yesmaximal, complemented
3#22615903C214yesmaximal, complemented
4#22615154C214yesmaximal, complemented
5#22615153C214yesmaximal, complemented
6#22613658C212yesmaximal, complemented
7#22613657C212yesmaximal, complemented
8#22613653C212yesmaximal, complemented
9#22613651C212yesmaximal, complemented
10#22612542C214yesmaximal
11#380×D73C212yesmaximal
12C2×C4×#11924005C211yesmaximal, characteristic
13C73×#2851C211yesmaximal, characteristic, complemented
14C23×C4×D73C2211yescharacteristic
15C2×#14556583C2211yescharacteristic, complemented
16C2×#14556582C2212yescomplemented
17C2×C4⋊C4×D73C2212yes—
18C2×C4⋊C4×D73C2212yes—
19C22×#11924005C2212yes—
20C2×C22⋊C4×D73C2214yes—
21C22×C4×Dic73C2211yescharacteristic
22#17988528C2214yescomplemented
23#17988528C2214yescomplemented
24#17988527C2214yescomplemented
25#17988525C2214yescomplemented
26#17988521C2214yescomplemented
27#17988520C2214yescomplemented
28#17988518C2214yescomplemented
29#17988517C2214yescomplemented
30#17988486C2212yescomplemented
31#17988485C2212yescomplemented
32#17988483C2212yescomplemented
33#17988474C2214yescomplemented
34#17988471C2214yescomplemented
35#17988470C2214yescomplemented
36#17988462C2214yes—
37#17988461C2214yes—
38#17988461C2214yes—
39#17988460C2214yes—
40#17988459C2214yes—
41#17988458C2214yes—
42#17988458C2214yes—
43#17988453C2211yescharacteristic, complemented
44#17988452C2212yescomplemented
45#17988447C2212yes—
46#17988446C2214yes—
47#17988446C2214yes—
48#17988446C2214yes—
49#17988445C2214yes—
50#17988444C2214yes—
51#17988444C2214yes—
52#17988444C2212yes—
53#17988444C2212yes—
54C4×#11924005C2212yes—
55C4×#11924005C2212yes—
56C4×#11924005C2212yes—
57C4×#11924005C2212yes—
58C2×#14556498C2212yes—
59#95×D73C2214yes—
60C2×#125×C73C2211yescharacteristic, complemented
61C22×C42×C73C2211yescharacteristic
62#389×C73C2212yescomplemented
63#387×C73C2214yescomplemented
64#385×C73C2214yes—
65#382×C73C2211yescharacteristic, complemented
66#380×C73C2212yes—
67C24×D73C2311yescharacteristic
68C23×Dic73C2312yes—
69C23×Dic73C2311yescharacteristic
70C22×C4×D73—42no—
71C22×C4×D73—42no—
72C22×C4×D73Q818yes—
73C22×C4×D73C2312yes—
74C22×C4×D73C2314yes—
75C22×C4×D73C2314yes—
76C22×C4×D73C2312yes—
77#14556583—42no—
78#14556582—44no—
79C2×C4⋊Dic73C2312yes—
80C2×C4⋊Dic73C2312yes—
81C2×C4⋊Dic73C2314yes—
82C2×C4⋊Dic73C2314yes—
83C2×#11924007C2314yes—
84C2×#11924007C2314yes—
85C2×#11924007C2314yes—
86C2×#11924007C2314yes—
87C2×#11924007C2314yes—
88C2×#11924007C2314yes—
89C4⋊C4×D73—44no—
90C4⋊C4×D73—44no—
91C2×C22⋊Dic73C2314yes—
92C2×#11924005—28no—
93C2×#11924005C2312yes—
94C2×#11924005C2312yes—
95C2×#11924005C2312yes—
96C2×#11924005C2312yes—
97C2×#11924005C2314yes—
98C2×#11924005C2314yes—
99C22⋊C4×D73—216no—
100C2×C4×Dic73—42no—
101C2×C4×Dic73C2312yes—
102C2×C4×Dic73C2311yescharacteristic
103C2×C4×Dic73C2311yescharacteristic
104#14556498C2314yes—
105#14556498C2314yes—
106#14556498C2314yes—
107#14556498C2312yes—
108#14556498C2312yes—
109#14556498C2312yes—
110#14556498C2312yes—
111#14556498C2314yes—
112#14556498C2314yes—
113#14556498C2314yes—
114#14556498C2314yes—
115#14556498C2314yes—
116#14556498C2314yes—
117C23×C4×C73C2311yescharacteristic
118C23×C4×C73C2312yes—
119#125×C73—42no—
120C2×C4⋊C4×C73C2312yes—
121C2×C4⋊C4×C73C2312yes—
122C2×C4⋊C4×C73C2314yes—
123C2×C4⋊C4×C73C2314yes—
124C2×C22⋊C4×C73C2314yes—
125C2×C42×C73—42no—
126C2×C42×C73C2311yescharacteristic
127C2×C42×C73C2311yescharacteristic
128C2×C42×C73C2312yes—
129#95×C73C2314yes—
130#95×C73C2314yes—
131#95×C73C2314yes—
132C23×D73—41no—
133C23×D73—41no—
134C23×D73—42no—
135C23×D73C2×Q814yes—
136C23×D73#4118yes—
137C23×D73C2412yes—
138C22×Dic73—24no—
139C22×Dic73—42no—
140C22×Dic73—41no—
141C22×Dic73—41no—
142C22×Dic73—42no—
143C22×Dic73C2×Q814yes—
144C22×Dic73C2412yes—
145C22×Dic73C2412yes—
146C22×Dic73C2414yes—
147C22×Dic73C2414yes—
148C22×Dic73C2412yes—
149C2×C4×D73—24no—
150C2×C4×D73—24no—
151C2×C4×D73—28no—
152C2×C4×D73—28no—
153C2×C4×D73—416no—
154C2×C4×D73—42no—
155C2×C4×D73—42no—
156C2×C4×D73—44no—
157C2×C4×D73—44no—
158C2×C4×D73—44no—
159C2×C4×D73—44no—
160C2×C4×D73C2×Q818yes—
161C2×C4×D73C2×Q818yes—
162C4⋊Dic73—24no—
163C4⋊Dic73—24no—
164C4⋊Dic73—44no—
165C4⋊Dic73—44no—
166#11924007—44no—
167#11924007—44no—
168#11924007—44no—
169#11924007—44no—
170#11924007—44no—
171#11924007—44no—
172C22⋊Dic73—28no—
173#11924005—28no—
174#11924005—28no—
175#11924005—24no—
176#11924005—24no—
177#11924005—24no—
178#11924005—24no—
179C4×Dic73—24no—
180C4×Dic73—41no—
181C4×Dic73—41no—
182C24×C73C2411yescharacteristic
183C22×C4×C73—22no—
184C22×C4×C73—22no—
185C22×C4×C73—24no—
186C22×C4×C73—42no—
187C22×C4×C73—42no—
188C22×C4×C73C2×Q814yes—
189C22×C4×C73C2414yes—
190C22×C4×C73C2412yes—
191C22×C4×C73C2412yes—
192C22×C4×C73C2414yes—
193C22×C4×C73C2411yescharacteristic
194C22×C4×C73C2411yescharacteristic
195C4⋊C4×C73—24no—
196C4⋊C4×C73—24no—
197C4⋊C4×C73—44no—
198C4⋊C4×C73—44no—
199C22⋊C4×C73—28no—
200C42×C73—22no—
201C42×C73—22no—
202C42×C73—42no—
203C23×C73—21no—
204C23×C73—21no—
205C23×C73—22no—
206C23×C73C22×Q812yes—
207C23×C73C2×#4114yes—
208C23×C73C2511yescharacteristic
209C22×D73—22no—
210C22×D73—22no—
211C22×D73—22no—
212C22×D73—22no—
213C22×D73—44no—
214C22×D73—44no—
215C22×D73—48no—
216C22×D73—48no—
217C22×D73#125116yescomplemented
218C22×D73C22×Q814yes—
219C22×D73C2×#4114yes—
220C22×D73C2×#4114yes—
221C2×C4×C73—21no—
222C2×C4×C73—21no—
223C2×C4×C73—28no—
224C2×C4×C73—24no—
225C2×C4×C73—24no—
226C2×C4×C73—22no—
227C2×C4×C73—22no—
228C2×C4×C73—22no—
229C2×C4×C73—22no—
230C2×C4×C73—24no—
231C2×C4×C73—24no—
232C2×C4×C73—22no—
233C2×C4×C73—24no—
234C2×C4×C73—24no—
235C2×C4×C73—22no—
236C2×C4×C73—48no—
237C2×C4×C73C22×Q812yes—
238C2×C4×C73C22×Q812yes—
239C2×C4×C73H(5,𝔽2)12yes—
240C2×C4×C73H(5,𝔽2)12yes—
241C2×Dic73—22no—
242C2×Dic73—22no—
243C2×Dic73—22no—
244C2×Dic73—22no—
245C2×Dic73—24no—
246C2×Dic73—24no—
247C2×Dic73—22no—
248C2×Dic73—22no—
249C2×Dic73—24no—
250C2×Dic73—24no—
251C2×Dic73—24no—
252C2×Dic73—22no—
253C2×Dic73—24no—
254C2×Dic73—22no—
255C2×Dic73—44no—
256C2×Dic73—48no—
257C2×Dic73C22×Q814yes—
258C4×D73—216no—
259C4×D73—48no—
260C4×D73—88no—
261C4×D73—88no—
262C22×C73—22no—
263C22×C73—22no—
264C22×C73—24no—
265C22×C73—24no—
266C22×C73C2×#12514yes—
267C22×C73C23×Q811yescharacteristic
268C22×C73C22×#4112yes—
269C22×C73C2×H(5,𝔽2)11yescharacteristic
270C22×C73C2×H(5,𝔽2)11yescharacteristic
271C22×C73#58512yes—
272C2×D73—24no—
273C2×D73—24no—
274C2×D73—24no—
275C2×D73—24no—
276C2×D73—24no—
277C2×D73—24no—
278C2×D73—416no—
279C2×D73C2×#12518yes—
280C4×C73—22no—
281C4×C73—44no—
282C4×C73—44no—
283C4×C73—42no—
284C4×C73—42no—
285C4×C73#54314yes—
286Dic73—24no—
287Dic73—44no—
288Dic73—44no—
289Dic73—42no—
290Dic73—42no—
291#35020—731nomaximal, Sylow
292C2×C73—24no—
293C2×C73C22×#12511yescharacteristic
294C2×C73C2×#54311yescharacteristic
295C2×C73#345811yescharacteristic
296C2×C73#350512yes—
297C2×C73#350512yes—
298D73—28no—
299#2851—731no—
300#2851—732no—
301#2851D7311yescharacteristic, complemented
302#2646—734no—
303#2569—734no—
304#2569—734no—
305#2442—732no—
306#2442—732no—
307#2442—732no—
308#2442—732no—
309#2315—734no—
310C2×#380—732no—
311C2×C4×C22⋊C4—731no—
312C73#3502011yescharacteristic, complemented, Sylow
313C24×C4—731no—
314C2×#125—732no—
315C2×#125—731no—
316C2×#125C2×D7311yescharacteristic, complemented
317C22×C4⋊C4—732no—
318C22×C4⋊C4—732no—
319C22×C22⋊C4—732no—
320C22×C22⋊C4—734no—
321C22×C42—731no—
322C22×C42C2×D7311yescharacteristic
323#400—734no—
324#400—734no—
325#400—734no—
326#400—734no—
327#398—734no—
328#398—734no—
329#398—734no—
330#398—734no—
331#389—732no—
332#389—732no—
333#389—732no—
334#389C2×D7312yescomplemented
335#387—734no—
336#387—734no—
337#387—734no—
338#387C2×D7314yescomplemented
339#385—734no—
340#385—734no—
341#385—734no—
342#385—734no—
343#385—734no—
344#385—734no—
345#385—734no—
346#385C2×D7314yes—
347#382—732no—
348#382—731no—
349#382C2×D7311yescharacteristic, complemented
350#380—732no—
351#380—734no—
352#380—734no—
353#380—732no—
354#380—732no—
355#380—734no—
356#380—734no—
357#380—734no—
358#380—734no—
359#380C2×D7312yes—
360C4×C22⋊C4—732no—
361C4×C22⋊C4—732no—
362C4×C22⋊C4—732no—
363C4×C22⋊C4—732no—
364C2×#95—732no—
365C2×#95—734no—
366C25—731no—
367C23×C4—2922no—
368C23×C4—2922no—
369C23×C4—731no—
370C23×C4—738no—
371C23×C4—732no—
372C23×C4—732no—
373C23×C4—734no—
374C23×C4—734no—
375C23×C4—732no—
376C23×C4C22×D7311yescharacteristic
377C23×C4C22×D7312yes—
378#125—42no—
379#125—2922no—
380#125—2924no—
381C2×C4⋊C4—2924no—
382C2×C4⋊C4—2924no—
383C2×C4⋊C4—734no—
384C2×C4⋊C4—734no—
385C2×C4⋊C4—734no—
386C2×C4⋊C4—734no—
387C2×C4⋊C4—734no—
388C2×C4⋊C4—734no—
389C2×C4⋊C4—734no—
390C2×C4⋊C4—734no—
391C2×C4⋊C4—732no—
392C2×C4⋊C4—732no—
393C2×C4⋊C4C22×D7312yes—
394C2×C4⋊C4C22×D7312yes—
395C2×C4⋊C4C22×D7314yes—
396C2×C4⋊C4C22×D7314yes—
397C2×C22⋊C4—734no—
398C2×C22⋊C4—732no—
399C2×C22⋊C4—732no—
400C2×C22⋊C4—732no—
401C2×C22⋊C4—732no—
402C2×C22⋊C4—734no—
403C2×C22⋊C4—734no—
404C2×C22⋊C4—14616no—
405C2×C22⋊C4—1468no—
406C2×C22⋊C4C22×D7314yes—
407C2×C42—42no—
408C2×C42—2922no—
409C2×C42—731no—
410C2×C42—731no—
411C2×C42—732no—
412C2×C42C22×D7311yescharacteristic
413C2×C42C22×D7311yescharacteristic
414C2×C42C22×D7312yes—
415#95—732no—
416#95—732no—
417#95—732no—
418#95—732no—
419#95—734no—
420#95—734no—
421#95—734no—
422#95—734no—
423#95—734no—
424#95—734no—
425#95—734no—
426#95—734no—
427#95—734no—
428#95C22×D7314yes—
429#95C22×D7314yes—
430#95C22×D7314yes—
431C24—2921no—
432C24—2921no—
433C24—2922no—
434C24—734no—
435C24—738no—
436C24—732no—
437C24C23×D7311yescharacteristic
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—2922no—
444C22×C4—2922no—
445C22×C4—29216no—
446C22×C4—2924no—
447C22×C4—2924no—
448C22×C4—2924no—
449C22×C4—2924no—
450C22×C4—2922no—
451C22×C4—2921no—
452C22×C4—2921no—
453C22×C4—2922no—
454C22×C4—734no—
455C22×C4—738no—
456C22×C4—732no—
457C22×C4—738no—
458C22×C4—732no—
459C22×C4—732no—
460C22×C4—734no—
461C22×C4—734no—
462C22×C4—1468no—
463C22×C4—1468no—
464C22×C4—1464no—
465C22×C4—1464no—
466C22×C4—1464no—
467C22×C4Q8×D7314yes—
468C22×C4C23×D7312yes—
469C22×C4C23×D7312yes—
470C22×C4C23×D7314yes—
471C22×C4C23×D7314yes—
472C22×C4C23×D7311yescharacteristic
473C22×C4C23×D7311yescharacteristic
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—2924no—
479C4⋊C4—2924no—
480C4⋊C4—2924no—
481C4⋊C4—2924no—
482C4⋊C4—2924no—
483C4⋊C4—2924no—
484C4⋊C4—2924no—
485C4⋊C4—2924no—
486C4⋊C4—1464no—
487C4⋊C4—1464no—
488C22⋊C4—28no—
489C22⋊C4—1468no—
490C22⋊C4—1468no—
491C22⋊C4—1464no—
492C22⋊C4—1464no—
493C22⋊C4—1464no—
494C22⋊C4—1464no—
495C22⋊C4—1468no—
496C42—22no—
497C42—22no—
498C42—42no—
499C42—2921no—
500C42—2921no—
501C42—1464no—
502C23—21no—
503C23—21no—
504C23—22no—
505C23—2924no—
506C23—2928no—
507C23—2928no—
508C23—2924no—
509C23—7316no—
510C23—734no—
511C23—734no—
512C23—734no—
513C23—1462no—
514C23—1462no—
515C23—1462no—
516C23—1462no—
517C23C2×Q8×D7312yes—
518C23#41×D7314yes—
519C23C24×D7311yescharacteristic
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—28no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—48no—
536C2×C4—2928no—
537C2×C4—2928no—
538C2×C4—2924no—
539C2×C4—5848no—
540C2×C4—5848no—
541C2×C4—734no—
542C2×C4—1464no—
543C2×C4—1462no—
544C2×C4—1462no—
545C2×C4—1464no—
546C2×C4—1464no—
547C2×C4—1462no—
548C2×C4—1462no—
549C2×C4—1464no—
550C2×C4—1464no—
551C2×C4—1462no—
552C2×C4—1462no—
553C2×C4—1464no—
554C2×C4—1462no—
555C2×C4—1462no—
556C2×C4—14616no—
557C2×C4C2×Q8×D7312yes—
558C2×C4C2×Q8×D7312yes—
559C2×C4#1455663612yes—
560C2×C4#1455663712yes—
561C22—24no—
562C22—24no—
563C22—22no—
564C22—22no—
565C22—29216no—
566C22—738no—
567C22—1464no—
568C22—1464no—
569C22—1464no—
570C22—1464no—
571C22—1464no—
572C22—1464no—
573C22#125×D7314yes—
574C22C22×Q8×D7311yescharacteristic
575C22C2×#41×D7312yes—
576C22C2×#1455663611yescharacteristic
577C22C2×#1455663711yescharacteristic
578C22#1798942312yes—
579C4—22no—
580C4—42no—
581C4—42no—
582C4—44no—
583C4—44no—
584C4—2924no—
585C4—2922no—
586C4—2922no—
587C4—2924no—
588C4—1464no—
589C4#1798921214yes—
590C2—24no—
591C2—1468no—
592C2C2×#125×D7311yescharacteristic
593C2C2×#1798921211yescharacteristic
594C2#2262333411yescharacteristic
595C2#2262375912yes—
596C2#2262376612yes—
subgroupG/H
center, FrattiniC23C24×D73
commutator, socle, cosocleC23×C73C25
FittingC73×#2851C2
prime radicalC24×D73C23
hypercenter#2851D73
hypocenterC73#35020
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ #2851

Complements

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

NH ≅ G/Ntype
1. #22618241C2semidirect
2. #22618239C2semidirect
3. #22615903C2semidirect
4. #22615154C2semidirect
5. #22615153C2semidirect
6. #22613658C2semidirect
7. #22613657C2semidirect
8. #22613653C2semidirect
9. #22613651C2semidirect
13. C73×#2851C2semidirect
15. C2×#14556583C22semidirect
16. C2×#14556582C22semidirect
22. #17988528C22semidirect
23. #17988528C22semidirect
24. #17988527C22semidirect
25. #17988525C22semidirect
26. #17988521C22semidirect
27. #17988520C22semidirect
28. #17988518C22semidirect
29. #17988517C22semidirect
30. #17988486C22semidirect
31. #17988485C22semidirect
32. #17988483C22semidirect
33. #17988474C22semidirect
34. #17988471C22semidirect
35. #17988470C22semidirect
43. #17988453C22semidirect
44. #17988452C22semidirect
60. C2×#125×C73C22semidirect
62. #389×C73C22semidirect
63. #387×C73C22semidirect
65. #382×C73C22semidirect
217. C22×D73#125semidirect
301. #2851D73semidirect
312. C73#35020semidirect
316. C2×#125C2×D73semidirect
334. #389C2×D73semidirect
338. #387C2×D73semidirect
349. #382C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226182412
#226182392
#226159032
#226151542
#226151532
#226136582
#226136572
#226136532
#226136512
#226125422
#380×D732
C2×C4×#119240052
C73×#28512
#3502073