Finite Group Databasebeta

← #33253140

Subgroups — #33253140

Order 18688.

1918 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22615963C214yesmaximal, complemented
2#22615918C214yesmaximal, complemented
3#22615894C212yesmaximal, complemented
4#22615874C211yesmaximal, characteristic, complemented
5#22615764C211yesmaximal, characteristic, complemented
6#22615725C212yesmaximal, complemented
7#22615684C211yesmaximal, characteristic
8#22613197C214yesmaximal, complemented
9#22613172C214yesmaximal, complemented
10#22612493C211yesmaximal, characteristic, complemented
11#22612486C211yesmaximal, characteristic, complemented
12#22612439C212yesmaximal, complemented
13#22612389C212yesmaximal, complemented
14#22612270C211yesmaximal, characteristic, complemented
15C73×#2647C211yesmaximal, characteristic, complemented
16C2×#14556588C2211yescharacteristic
17C2×#14556583C2211yescharacteristic
18C2×#14556566C2212yes—
19C2×#14556553C2212yes—
20C2×#14556553C2214yes—
21C2×#14556553C2214yes—
22C2×#14556549C2214yes—
23C2×#14556548C2211yescharacteristic
24C2×#14556548C2211yescharacteristic
25C2×C42×D73C2211yescharacteristic
26#17988522C2214yes—
27#17988517C2214yes—
28#17988516C2214yes—
29#17988516C2214yes—
30#17988514C2214yes—
31#17988513C2214yes—
32#17988509C2214yes—
33#17988509C2214yes—
34#17988505C2214yes—
35#17988505C2214yes—
36#17988505C2214yes—
37#17988505C2214yes—
38#17988504C2214yes—
39#17988497C2214yes—
40#17988489C2212yes—
41#17988489C2212yes—
42#17988488C2212yes—
43#17988488C2214yes—
44#17988488C2212yes—
45#17988486C2212yes—
46#17988483C2212yes—
47#17988479C2214yes—
48#17988477C2214yes—
49#17988477C2214yes—
50#17988469C2214yes—
51#17988465C2214yes—
52#17988444C2212yes—
53C4⋊C4×Dic73C2212yes—
54#17988438C2214yes—
55#17988438C2212yes—
56#17988438C2212yes—
57#17988434C2212yes—
58#17988434C2212yes—
59#17988434C2214yes—
60#17988434C2214yes—
61#17988433C2211yescharacteristic
62C42×Dic73C2211yescharacteristic
63C42×Dic73C2211yescharacteristic
64#400×C73C2214yes—
65#398×C73C2214yes—
66#389×C73C2212yes—
67#389×C73C2212yes—
68#384×C73C2211yescharacteristic
69#382×C73C2211yescharacteristic
70#376×C73C2211yescharacteristic
71C22×D292C2312yes—
72C22×D292C2314yes—
73C22×D292C2314yes—
74C22×C4×D73C2311yescharacteristic
75C22×C4×D73C2314yes—
76C22×C4×D73C2311yescharacteristic
77C22×C4×D73C2311yescharacteristic
78#14556588—24no—
79#14556583—24no—
80#14556566—28no—
81#14556553—28no—
82#14556553—48no—
83#14556553—48no—
84#14556549—48no—
85#14556548—24no—
86#14556548—24no—
87C2×C4⋊Dic73C2312yes—
88C2×C4⋊Dic73C2314yes—
89C2×#11924007C2312yes—
90C2×#11924007C2312yes—
91C2×#11924007C2314yes—
92C2×#11924007C2314yes—
93C2×#11924005C2312yes—
94C2×#11924005C2312yes—
95C2×#11924005C2314yes—
96C2×#11924005C2314yes—
97C2×#11924005C2314yes—
98C2×#11924005C2314yes—
99C2×#11924005C2314yes—
100C2×#11924005C2314yes—
101C2×#11924005C2314yes—
102C2×#11924005C2314yes—
103C2×#11924005C2314yes—
104C2×#11924005C2314yes—
105C2×#11924005C2314yes—
106C2×C4×Dic73C2311yescharacteristic
107C2×C4×Dic73C2311yescharacteristic
108C2×C4×Dic73C2312yes—
109C2×C4×Dic73C2312yes—
110C2×C4×Dic73C2311yescharacteristic
111C2×C4×Dic73C2314yes—
112C2×C4×Dic73C2311yescharacteristic
113C2×C4×Dic73C2312yes—
114C2×C4×Dic73C2312yes—
115C2×C4×Dic73C2311yescharacteristic
116C42×D73—24no—
117#14556498C2312yes—
118#14556498C2312yes—
119#14556498C2314yes—
120#14556498C2314yes—
121#14556498C2314yes—
122#14556498C2314yes—
123C2×C4⋊C4×C73C2312yes—
124C2×C4⋊C4×C73C2312yes—
125C2×C4⋊C4×C73C2314yes—
126C2×C4⋊C4×C73C2314yes—
127C2×C4⋊C4×C73C2314yes—
128C2×C42×C73C2311yescharacteristic
129C2×C42×C73C2311yescharacteristic
130C2×C42×C73C2311yescharacteristic
131#95×C73C2314yes—
132#95×C73C2314yes—
133#95×C73C2314yes—
134#95×C73C2312yes—
135#95×C73C2312yes—
136C23×D73C2414yes—
137C23×D73C2411yescharacteristic
138C2×D292—28no—
139C2×D292—24no—
140C2×D292—24no—
141C2×D292—28no—
142C2×D292—24no—
143C2×D292—44no—
144C2×D292—44no—
145C2×D292—44no—
146C2×D292—44no—
147C22×Dic73C2412yes—
148C22×Dic73C2412yes—
149C22×Dic73C2414yes—
150C22×Dic73C2411yescharacteristic
151C22×Dic73C2411yescharacteristic
152C22×Dic73C2411yescharacteristic
153C2×C4×D73—22no—
154C2×C4×D73—22no—
155C2×C4×D73—22no—
156C2×C4×D73—22no—
157C2×C4×D73—28no—
158C2×C4×D73—28no—
159C2×C4×D73—24no—
160C2×C4×D73—28no—
161C2×C4×D73—42no—
162C2×C4×D73#4114yes—
163C4⋊Dic73—24no—
164C4⋊Dic73—44no—
165#11924007—24no—
166#11924007—28no—
167#11924007—24no—
168#11924007—44no—
169#11924005—24no—
170#11924005—24no—
171#11924005—28no—
172#11924005—28no—
173#11924005—44no—
174#11924005—44no—
175#11924005—44no—
176#11924005—44no—
177#11924005—44no—
178#11924005—44no—
179#11924005—44no—
180#11924005—44no—
181#11924005—44no—
182C4×Dic73—22no—
183C4×Dic73—22no—
184C4×Dic73—22no—
185C4×Dic73—22no—
186C4×Dic73—24no—
187C4×Dic73—28no—
188C4×Dic73—24no—
189C4×Dic73—42no—
190C4×Dic73—42no—
191C4×Dic73#4114yes—
192C22×C4×C73C2411yescharacteristic
193C22×C4×C73C2412yes—
194C22×C4×C73C2414yes—
195C22×C4×C73C2411yescharacteristic
196C22×C4×C73C2414yes—
197C22×C4×C73C2412yes—
198C22×C4×C73C2411yescharacteristic
199C4⋊C4×C73—24no—
200C4⋊C4×C73—24no—
201C4⋊C4×C73—44no—
202C4⋊C4×C73—44no—
203C4⋊C4×C73—44no—
204C42×C73—22no—
205C42×C73—22no—
206C42×C73#4114yes—
207C23×C73C2511yescharacteristic
208C22×D73—21no—
209C22×D73—21no—
210C22×D73—22no—
211C22×D73—22no—
212C22×D73—24no—
213C22×D73—24no—
214C22×D73—24no—
215C22×D73—24no—
216C22×D73—24no—
217C22×D73—24no—
218C22×D73—24no—
219C22×D73C2×#4112yes—
220C2×C4×C73—22no—
221C2×C4×C73—22no—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—22no—
225C2×C4×C73—24no—
226C2×C4×C73—24no—
227C2×C4×C73—22no—
228C2×C4×C73—24no—
229C2×C4×C73—22no—
230C2×C4×C73—24no—
231C2×C4×C73—24no—
232C2×C4×C73—22no—
233C2×C4×C73—22no—
234C2×C4×C73—24no—
235C2×C4×C73C2×#4112yes—
236C2×C4×C73C2×#4112yes—
237C2×C4×C73C2×#4114yes—
238C2×C4×C73C2×#4112yes—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—22no—
242C2×Dic73—22no—
243C2×Dic73—22no—
244C2×Dic73—24no—
245C2×Dic73—22no—
246C2×Dic73—24no—
247C2×Dic73—22no—
248C2×Dic73—22no—
249C2×Dic73—22no—
250C2×Dic73—24no—
251C2×Dic73C2×#4112yes—
252C2×Dic73C2×#4112yes—
253C2×Dic73C2×#4112yes—
254C2×Dic73H(5,𝔽2)14yes—
255D292—416no—
256D292—416no—
257D292—48no—
258C4×D73—28no—
259C4×D73—44no—
260C4×D73—44no—
261C4×D73—416no—
262C22×C73C22×#4111yescharacteristic
263C22×C73C22×#4111yescharacteristic
264C22×C73C22×#4111yescharacteristic
265C22×C73C2×H(5,𝔽2)12yes—
266C22×C73#58512yes—
267C2×D73—22no—
268C2×D73—24no—
269C2×D73—41no—
270C2×D73—41no—
271C2×D73—42no—
272C2×D73—44no—
273C2×D73—44no—
274C2×D73—44no—
275C2×D73—44no—
276C2×D73—44no—
277C2×D73—44no—
278C2×D73—44no—
279C4×C73—24no—
280C4×C73—41no—
281C4×C73—44no—
282C4×C73—41no—
283C4×C73—44no—
284C4×C73—42no—
285C4×C73#52914yes—
286Dic73—22no—
287Dic73—22no—
288Dic73—42no—
289Dic73—42no—
290Dic73—44no—
291Dic73#54014yes—
292#34621—731nomaximal, Sylow
293C2×C73C2×#52911yescharacteristic
294C2×C73C2×#54011yescharacteristic
295C2×C73#345511yescharacteristic
296C2×C73#346012yes—
297C2×C73#346012yes—
298D73—42no—
299D73—84no—
300#2651—734no—
301#2648—734no—
302#2647D7311yescharacteristic, complemented
303#2644—732no—
304#2641—731no—
305#2629—731no—
306#2624—732no—
307#2619—731no—
308#2395—734no—
309#2392—734no—
310C4×#127—731no—
311C4×#125—731no—
312C4×#121—732no—
313#2293—732no—
314C4×#117—731no—
315C73#3462111yescharacteristic, complemented, Sylow
316C2×#127—731no—
317C2×#125—731no—
318C2×#121—732no—
319C2×C4×D4—734no—
320C2×C4×D4—734no—
321C2×C4×D4—732no—
322C2×#117—734no—
323C2×#117—731no—
324C2×#117—731no—
325C22×C42—731no—
326#400C2×D7314yes—
327#399—734no—
328#398—734no—
329#398C2×D7314yes—
330#397—734no—
331#397—734no—
332#397—734no—
333#397—734no—
334#396—734no—
335#396—734no—
336#394—734no—
337#394—734no—
338#394—734no—
339#394—734no—
340#394—734no—
341#392—734no—
342#390—732no—
343#390—734no—
344#390—732no—
345#390—732no—
346#390—732no—
347#389—732no—
348#389—732no—
349#389C2×D7312yes—
350#389C2×D7312yes—
351#388—734no—
352#388—734no—
353#388—734no—
354#387—734no—
355#386—734no—
356#384C2×D7311yescharacteristic
357#382C2×D7311yescharacteristic
358#380—732no—
359C4×C4⋊C4—732no—
360C4×C4⋊C4—734no—
361C4×C4⋊C4—732no—
362C4×C4⋊C4—732no—
363C4×C22⋊C4—732no—
364C4×C22⋊C4—734no—
365C4×C22⋊C4—732no—
366C4×C22⋊C4—734no—
367#376—731no—
368#376C2×D7311yescharacteristic
369C43—731no—
370C43—731no—
371C22×D4—734no—
372C22×D4—734no—
373C22×D4—732no—
374C23×C4—731no—
375C23×C4—731no—
376C23×C4—734no—
377C23×C4—731no—
378#127—1464no—
379#125—1464no—
380#121—1468no—
381C4×D4—2928no—
382C4×D4—2928no—
383C4×D4—1468no—
384#117—2928no—
385#117—1464no—
386#117—1464no—
387C2×C4⋊C4—734no—
388C2×C4⋊C4—734no—
389C2×C4⋊C4—732no—
390C2×C4⋊C4—732no—
391C2×C4⋊C4—732no—
392C2×C4⋊C4—734no—
393C2×C4⋊C4C22×D7312yes—
394C2×C4⋊C4C22×D7312yes—
395C2×C4⋊C4C22×D7314yes—
396C2×C4⋊C4C22×D7314yes—
397C2×C4⋊C4C22×D7314yes—
398C2×C22⋊C4—734no—
399C2×C22⋊C4—734no—
400C2×C22⋊C4—734no—
401C2×C22⋊C4—734no—
402C2×C22⋊C4—734no—
403C2×C22⋊C4—734no—
404C2×C22⋊C4—734no—
405C2×C22⋊C4—734no—
406C2×C22⋊C4—734no—
407C2×C22⋊C4—732no—
408C2×C22⋊C4—732no—
409C2×C22⋊C4—734no—
410C2×C22⋊C4—734no—
411C2×C42—732no—
412C2×C42—732no—
413C2×C42—731no—
414C2×C42—732no—
415C2×C42—734no—
416C2×C42—731no—
417C2×C42—731no—
418C2×C42—732no—
419C2×C42—731no—
420C2×C42—731no—
421C2×C42—1464no—
422C2×C42C22×D7311yescharacteristic
423C2×C42C22×D7311yescharacteristic
424C2×C42C22×D7311yescharacteristic
425#95—732no—
426#95—732no—
427#95—734no—
428#95—734no—
429#95—734no—
430#95—734no—
431#95C22×D7312yes—
432#95C22×D7312yes—
433#95C22×D7314yes—
434#95C22×D7314yes—
435#95C22×D7314yes—
436C24—731no—
437C24—734no—
438C2×D4—2924no—
439C2×D4—2924no—
440C2×D4—2924no—
441C2×D4—2924no—
442C2×D4—1468no—
443C2×D4—1464no—
444C2×D4—1464no—
445C2×D4—1468no—
446C2×D4—1464no—
447C22×C4—2922no—
448C22×C4—731no—
449C22×C4—731no—
450C22×C4—731no—
451C22×C4—734no—
452C22×C4—732no—
453C22×C4—732no—
454C22×C4—734no—
455C22×C4—1462no—
456C22×C4—1462no—
457C22×C4—1468no—
458C22×C4—1468no—
459C22×C4—1464no—
460C22×C4—1468no—
461C22×C4—1462no—
462C22×C4—1462no—
463C22×C4C23×D7311yescharacteristic
464C22×C4C23×D7314yes—
465C22×C4C23×D7311yescharacteristic
466C22×C4C23×D7314yes—
467C22×C4C23×D7312yes—
468C22×C4C23×D7312yes—
469C22×C4C23×D7311yescharacteristic
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—2924no—
476C4⋊C4—2924no—
477C4⋊C4—1464no—
478C4⋊C4—1464no—
479C4⋊C4—1468no—
480C4⋊C4—1464no—
481C22⋊C4—2924no—
482C22⋊C4—2924no—
483C22⋊C4—2924no—
484C22⋊C4—2924no—
485C22⋊C4—2924no—
486C22⋊C4—2924no—
487C22⋊C4—2924no—
488C22⋊C4—2924no—
489C22⋊C4—2924no—
490C22⋊C4—1468no—
491C22⋊C4—1468no—
492C22⋊C4—1464no—
493C22⋊C4—1464no—
494C42—22no—
495C42—22no—
496C42—2922no—
497C42—2922no—
498C42—734no—
499C42—1462no—
500C42—1462no—
501C42—1464no—
502C42—1468no—
503C42—1464no—
504C42—1462no—
505C42—1462no—
506C42#1192402914yes—
507C23—732no—
508C23—1464no—
509C23—1464no—
510C23—1464no—
511C23—1464no—
512C23—1464no—
513C23—1464no—
514C23—1464no—
515C23—1461no—
516C23—1461no—
517C23—1462no—
518C23—1462no—
519C23C24×D7311yescharacteristic
520D4—29216no—
521D4—29216no—
522D4—2928no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—2924no—
539C2×C4—29216no—
540C2×C4—2924no—
541C2×C4—732no—
542C2×C4—734no—
543C2×C4—732no—
544C2×C4—732no—
545C2×C4—1462no—
546C2×C4—1462no—
547C2×C4—1464no—
548C2×C4—1462no—
549C2×C4—1464no—
550C2×C4—1462no—
551C2×C4—1462no—
552C2×C4—1462no—
553C2×C4—1464no—
554C2×C4—1461no—
555C2×C4—1461no—
556C2×C4—1462no—
557C2×C4—1468no—
558C2×C4#41×D7312yes—
559C2×C4#41×D7312yes—
560C2×C4C2×#1192402914yes—
561C2×C4C2×#1192402912yes—
562C22—2921no—
563C22—2921no—
564C22—2922no—
565C22—2924no—
566C22—2924no—
567C22—2924no—
568C22—2924no—
569C22—2924no—
570C22—2924no—
571C22—2924no—
572C22—1464no—
573C22—1462no—
574C22C2×#41×D7311yescharacteristic
575C22C2×#41×D7311yescharacteristic
576C22C22×#1192402911yescharacteristic
577C22H(5,𝔽2)×D7312yes—
578C22#1798942412yes—
579C4—24no—
580C4—41no—
581C4—44no—
582C4—41no—
583C4—44no—
584C4—42no—
585C4—2922no—
586C4—2922no—
587C4—2924no—
588C4—734no—
589C4—1462no—
590C4—1462no—
591C4#1798912214yes—
592C2—2922no—
593C2—5844no—
594C2C2×#1798912211yescharacteristic
595C2#540×D7311yescharacteristic
596C2#2262332211yescharacteristic
597C2#2262335112yes—
598C2#2262335412yes—
subgroupG/H
center, FrattiniC23C24×D73
commutator, socle, cosocleC23×C73C25
FittingC73×#2647C2
prime radical#22615684C2
hypercenter#2647D73
hypocenterC73#34621
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ #2647

Complements

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

NH ≅ G/Ntype
1. #22615963C2semidirect
2. #22615918C2semidirect
3. #22615894C2semidirect
4. #22615874C2semidirect
5. #22615764C2semidirect
6. #22615725C2semidirect
8. #22613197C2semidirect
9. #22613172C2semidirect
10. #22612493C2semidirect
11. #22612486C2semidirect
12. #22612439C2semidirect
13. #22612389C2semidirect
14. #22612270C2semidirect
15. C73×#2647C2semidirect
302. #2647D73semidirect
315. C73#34621semidirect

Maximal subgroups

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

subgroupindex
#226159632
#226159182
#226158942
#226158742
#226157642
#226157252
#226156842
#226131972
#226131722
#226124932
#226124862
#226124392
#226123892
#226122702
C73×#26472
#3462173