Finite Group Databasebeta

← #33253148

Subgroups — #33253148

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#22615961C214yesmaximal, complemented
2#22615916C214yesmaximal, complemented
3#22615906C211yesmaximal, characteristic, complemented
4#22615892C212yesmaximal, complemented
5#22615762C211yesmaximal, characteristic, complemented
6#22615723C212yesmaximal, complemented
7#22615682C211yesmaximal, characteristic
8#22613187C214yesmaximal, complemented
9#22613174C214yesmaximal, complemented
10#127×Dic73C211yesmaximal, characteristic, complemented
11#125×Dic73C211yesmaximal, characteristic, complemented
12#22612447C212yesmaximal, complemented
13#22612393C212yesmaximal, complemented
14#22612270C211yesmaximal, characteristic, complemented
15C73×#2641C211yesmaximal, characteristic, complemented
16C2×#14556562C2212yes—
17C2×D4×Dic73C2212yes—
18C2×D4×Dic73C2214yes—
19C2×#14556551C2214yes—
20C2×#14556548C2211yescharacteristic
21C2×#14556548C2211yescharacteristic
22C2×#14556547C2214yes—
23C2×C42×D73C2211yescharacteristic
24#17988529C2214yes—
25#17988523C2214yes—
26#17988521C2214yes—
27#17988520C2214yes—
28#17988515C2214yes—
29#17988515C2214yes—
30#17988515C2214yes—
31#17988512C2214yes—
32#17988510C2214yes—
33#17988509C2214yes—
34#17988503C2214yes—
35#17988502C2214yes—
36#17988501C2214yes—
37#17988495C2214yes—
38#17988490C2212yes—
39#17988490C2214yes—
40#17988490C2212yes—
41#17988486C2212yes—
42#17988485C2212yes—
43#17988485C2212yes—
44#17988483C2212yes—
45#17988481C2214yes—
46#17988481C2214yes—
47#17988475C2214yes—
48#17988473C2214yes—
49#17988463C2214yes—
50#17988457C2211yescharacteristic
51#17988453C2211yescharacteristic
52#17988444C2212yes—
53C4⋊C4×Dic73C2212yes—
54C4⋊C4×Dic73C2214yes—
55#17988438C2212yes—
56#17988438C2212yes—
57C22⋊C4×Dic73C2214yes—
58C22⋊C4×Dic73C2214yes—
59#17988434C2212yes—
60#17988434C2212yes—
61#17988433C2211yescharacteristic
62C42×Dic73C2211yescharacteristic
63C42×Dic73C2211yescharacteristic
64C2×#127×C73C2211yescharacteristic
65C2×#125×C73C2211yescharacteristic
66#394×C73C2214yes—
67#394×C73C2214yes—
68#390×C73C2212yes—
69#390×C73C2212yes—
70#376×C73C2211yescharacteristic
71C22×C22⋊D73C2314yes—
72C23×Dic73C2314yes—
73C22×C4×D73C2311yescharacteristic
74C22×C4×D73C2311yescharacteristic
75C22×C4×D73C2311yescharacteristic
76#14556562—28no—
77D4×Dic73—28no—
78D4×Dic73—48no—
79#14556551—48no—
80#14556548—24no—
81#14556548—24no—
82#14556547—48no—
83C2×C4⋊Dic73C2312yes—
84C2×C4⋊Dic73C2312yes—
85C2×C4⋊Dic73C2314yes—
86C2×C4⋊Dic73C2314yes—
87C2×#11924007C2312yes—
88C2×#11924007C2312yes—
89C2×#11924007C2314yes—
90C2×#11924007C2314yes—
91C2×#11924007C2314yes—
92C2×C22⋊Dic73C2314yes—
93C2×C22⋊Dic73C2314yes—
94C2×C22⋊Dic73C2314yes—
95C2×C22⋊Dic73C2314yes—
96C2×C22⋊Dic73C2314yes—
97C2×C22⋊Dic73C2314yes—
98C2×#11924005C2312yes—
99C2×#11924005C2312yes—
100C2×#11924005C2314yes—
101C2×C4×Dic73C2311yescharacteristic
102C2×C4×Dic73C2311yescharacteristic
103C2×C4×Dic73C2312yes—
104C2×C4×Dic73C2312yes—
105C2×C4×Dic73C2311yescharacteristic
106C2×C4×Dic73C2314yes—
107C2×C4×Dic73C2311yescharacteristic
108C2×C4×Dic73C2312yes—
109C2×C4×Dic73C2312yes—
110C2×C4×Dic73C2311yescharacteristic
111C42×D73—24no—
112#14556498C2312yes—
113#14556498C2312yes—
114#14556498C2314yes—
115#14556498C2314yes—
116#14556498C2314yes—
117#14556498C2314yes—
118#14556498C2314yes—
119#14556498C2314yes—
120#14556498C2314yes—
121C22×D4×C73C2312yes—
122C22×D4×C73C2314yes—
123#127×C73—24no—
124#125×C73—24no—
125C2×C4⋊C4×C73C2312yes—
126C2×C4⋊C4×C73C2314yes—
127C2×C22⋊C4×C73C2314yes—
128C2×C22⋊C4×C73C2314yes—
129C2×C22⋊C4×C73C2314yes—
130C2×C22⋊C4×C73C2314yes—
131C2×C42×C73C2311yescharacteristic
132C2×C42×C73C2311yescharacteristic
133C2×C42×C73C2311yescharacteristic
134#95×C73C2312yes—
135#95×C73C2312yes—
136C23×D73C2411yescharacteristic
137C2×C22⋊D73—28no—
138C2×C22⋊D73—44no—
139C2×C22⋊D73—44no—
140C22×Dic73—28no—
141C22×Dic73—28no—
142C22×Dic73—28no—
143C22×Dic73C2414yes—
144C22×Dic73C2412yes—
145C22×Dic73C2412yes—
146C22×Dic73C2414yes—
147C22×Dic73C2411yescharacteristic
148C22×Dic73C2411yescharacteristic
149C22×Dic73C2411yescharacteristic
150C2×C4×D73—22no—
151C2×C4×D73—22no—
152C2×C4×D73—22no—
153C2×C4×D73—22no—
154C2×C4×D73—24no—
155C2×C4×D73—42no—
156C2×C4×D73#4114yes—
157C4⋊Dic73—24no—
158C4⋊Dic73—24no—
159C4⋊Dic73—44no—
160C4⋊Dic73—44no—
161#11924007—24no—
162#11924007—24no—
163#11924007—44no—
164#11924007—44no—
165#11924007—44no—
166C22⋊Dic73—28no—
167C22⋊Dic73—28no—
168C22⋊Dic73—44no—
169C22⋊Dic73—44no—
170C22⋊Dic73—44no—
171C22⋊Dic73—44no—
172#11924005—24no—
173#11924005—24no—
174#11924005—44no—
175C4×Dic73—22no—
176C4×Dic73—22no—
177C4×Dic73—22no—
178C4×Dic73—22no—
179C4×Dic73—24no—
180C4×Dic73—28no—
181C4×Dic73—24no—
182C4×Dic73—42no—
183C4×Dic73—42no—
184C4×Dic73#4114yes—
185C24×C73C2414yes—
186C2×D4×C73—28no—
187C2×D4×C73—24no—
188C2×D4×C73—24no—
189C2×D4×C73—24no—
190C2×D4×C73—44no—
191C2×D4×C73—44no—
192C22×C4×C73C2411yescharacteristic
193C22×C4×C73C2412yes—
194C22×C4×C73C2411yescharacteristic
195C22×C4×C73C2414yes—
196C22×C4×C73C2412yes—
197C22×C4×C73C2411yescharacteristic
198C4⋊C4×C73—28no—
199C4⋊C4×C73—24no—
200C22⋊C4×C73—44no—
201C22⋊C4×C73—44no—
202C22⋊C4×C73—44no—
203C22⋊C4×C73—44no—
204C42×C73—22no—
205C42×C73—22no—
206C42×C73#4114yes—
207C23×C73—24no—
208C23×C73—24no—
209C23×C73—24no—
210C23×C73—24no—
211C23×C73—24no—
212C23×C73—24no—
213C23×C73—24no—
214C23×C73C2511yescharacteristic
215C22×D73—21no—
216C22×D73—21no—
217C22×D73—22no—
218C22×D73—22no—
219C22×D73C2×#4112yes—
220D4×C73—416no—
221D4×C73—48no—
222C2×C4×C73—22no—
223C2×C4×C73—22no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
226C2×C4×C73—22no—
227C2×C4×C73—24no—
228C2×C4×C73—24no—
229C2×C4×C73—22no—
230C2×C4×C73—22no—
231C2×C4×C73—22no—
232C2×C4×C73—24no—
233C2×C4×C73—22no—
234C2×C4×C73C2×#4112yes—
235C2×C4×C73C2×#4112yes—
236C2×C4×C73C2×#4114yes—
237C2×C4×C73C2×#4112yes—
238C22⋊D73—416no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—22no—
242C2×Dic73—22no—
243C2×Dic73—22no—
244C2×Dic73—24no—
245C2×Dic73—24no—
246C2×Dic73—22no—
247C2×Dic73—22no—
248C2×Dic73—24no—
249C2×Dic73—24no—
250C2×Dic73—24no—
251C2×Dic73—22no—
252C2×Dic73—22no—
253C2×Dic73—24no—
254C2×Dic73—416no—
255C2×Dic73C2×#4112yes—
256C2×Dic73C2×#4112yes—
257C2×Dic73C2×#4112yes—
258C2×Dic73H(5,𝔽2)14yes—
259C4×D73—28no—
260C4×D73—44no—
261C4×D73—44no—
262C22×C73—44no—
263C22×C73—44no—
264C22×C73—44no—
265C22×C73—44no—
266C22×C73—44no—
267C22×C73—44no—
268C22×C73—44no—
269C22×C73C22×#4111yescharacteristic
270C22×C73C22×#4111yescharacteristic
271C22×C73C22×#4111yescharacteristic
272C22×C73C2×H(5,𝔽2)12yes—
273C22×C73#58512yes—
274C2×D73—22no—
275C2×D73—24no—
276C2×D73—41no—
277C2×D73—41no—
278C2×D73—42no—
279C4×C73—24no—
280C4×C73—41no—
281C4×C73—41no—
282C4×C73—44no—
283C4×C73—42no—
284C4×C73#52914yes—
285Dic73—22no—
286Dic73—22no—
287Dic73—44no—
288Dic73—42no—
289Dic73—42no—
290Dic73—44no—
291Dic73#54014yes—
292#34621—731nomaximal, Sylow
293C2×C73—84no—
294C2×C73C2×#52911yescharacteristic
295C2×C73C2×#54011yescharacteristic
296C2×C73#345511yescharacteristic
297C2×C73#346012yes—
298C2×C73#346012yes—
299D73—42no—
300#2651—734no—
301#2648—734no—
302#2647—731no—
303#2644—732no—
304#2641D7311yescharacteristic, complemented
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×#127C2×D7311yescharacteristic
317C2×#125C2×D7311yescharacteristic
318C2×#121—732no—
319C2×C4×D4—734no—
320C2×C4×D4—732no—
321C2×C4×D4—734no—
322C2×#117—734no—
323C2×#117—731no—
324C2×#117—731no—
325C22×C42—731no—
326#400—734no—
327#399—734no—
328#398—734no—
329#398—734no—
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#394C2×D7314yes—
340#394C2×D7314yes—
341#392—734no—
342#390—732no—
343#390—734no—
344#390—732no—
345#390C2×D7312yes—
346#390C2×D7312yes—
347#389—732no—
348#389—732no—
349#389—732no—
350#389—732no—
351#388—734no—
352#388—734no—
353#388—734no—
354#387—734no—
355#386—734no—
356#384—731no—
357#382—731no—
358#380—732no—
359C4×C4⋊C4—734no—
360C4×C4⋊C4—732no—
361C4×C4⋊C4—732no—
362C4×C4⋊C4—732no—
363C4×C22⋊C4—734no—
364C4×C22⋊C4—734no—
365C4×C22⋊C4—732no—
366C4×C22⋊C4—732no—
367#376—731no—
368#376C2×D7311yescharacteristic
369C43—731no—
370C43—731no—
371C22×D4—734no—
372C22×D4C22×D7312yes—
373C22×D4C22×D7314yes—
374C23×C4—731no—
375C23×C4—731no—
376C23×C4—734no—
377C23×C4—731no—
378#127—24no—
379#125—24no—
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—734no—
390C2×C4⋊C4—734no—
391C2×C4⋊C4—734no—
392C2×C4⋊C4—732no—
393C2×C4⋊C4—732no—
394C2×C4⋊C4—732no—
395C2×C4⋊C4—732no—
396C2×C4⋊C4C22×D7312yes—
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—732no—
406C2×C22⋊C4—732no—
407C2×C22⋊C4C22×D7314yes—
408C2×C22⋊C4C22×D7314yes—
409C2×C22⋊C4C22×D7314yes—
410C2×C22⋊C4C22×D7314yes—
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#95—734no—
432#95—734no—
433#95—734no—
434#95C22×D7312yes—
435#95C22×D7312yes—
436C24—731no—
437C24C23×D7314yes—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—28no—
441C2×D4—24no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4—2924no—
445C2×D4—2924no—
446C2×D4—1468no—
447C22×C4—2922no—
448C22×C4—731no—
449C22×C4—731no—
450C22×C4—731no—
451C22×C4—734no—
452C22×C4—734no—
453C22×C4—732no—
454C22×C4—732no—
455C22×C4—734no—
456C22×C4—1462no—
457C22×C4—1462no—
458C22×C4—1464no—
459C22×C4—1468no—
460C22×C4—1468no—
461C22×C4—1468no—
462C22×C4—1462no—
463C22×C4—1462no—
464C22×C4C23×D7311yescharacteristic
465C22×C4C23×D7314yes—
466C22×C4C23×D7312yes—
467C22×C4C23×D7311yescharacteristic
468C22×C4C23×D7312yes—
469C22×C4C23×D7311yescharacteristic
470C4⋊C4—28no—
471C4⋊C4—24no—
472C4⋊C4—2924no—
473C4⋊C4—2924no—
474C4⋊C4—2924no—
475C4⋊C4—2924no—
476C4⋊C4—2924no—
477C4⋊C4—1464no—
478C4⋊C4—1464no—
479C4⋊C4—1464no—
480C4⋊C4—1464no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—2924no—
486C22⋊C4—2924no—
487C22⋊C4—2924no—
488C22⋊C4—2924no—
489C22⋊C4—2924no—
490C22⋊C4—1464no—
491C22⋊C4—1468no—
492C22⋊C4—1468no—
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#1192402714yes—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—732no—
515C23—1461no—
516C23—1461no—
517C23—1462no—
518C23—1462no—
519C23C24×D7311yescharacteristic
520D4—416no—
521D4—48no—
522D4—29216no—
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—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—2924no—
536C2×C4—2924no—
537C2×C4—29216no—
538C2×C4—732no—
539C2×C4—734no—
540C2×C4—732no—
541C2×C4—732no—
542C2×C4—1462no—
543C2×C4—1464no—
544C2×C4—1464no—
545C2×C4—1462no—
546C2×C4—1462no—
547C2×C4—1462no—
548C2×C4—1464no—
549C2×C4—1464no—
550C2×C4—1464no—
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×#1192402714yes—
561C2×C4C2×#1192402712yes—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—44no—
569C22—2921no—
570C22—2921no—
571C22—2922no—
572C22—1464no—
573C22—1462no—
574C22C2×#41×D7311yescharacteristic
575C22C2×#41×D7311yescharacteristic
576C22C22×#1192402711yescharacteristic
577C22H(5,𝔽2)×D7312yes—
578C22#1798942212yes—
579C4—24no—
580C4—41no—
581C4—41no—
582C4—44no—
583C4—42no—
584C4—2924no—
585C4—2922no—
586C4—2924no—
587C4—2922no—
588C4—734no—
589C4—1462no—
590C4—1462no—
591C4#1798911614yes—
592C2—84no—
593C2—2922no—
594C2C2×#1798911611yescharacteristic
595C2#540×D7311yescharacteristic
596C2#2262331511yescharacteristic
597C2#2262334412yes—
598C2#2262334612yes—
subgroupG/H
center, FrattiniC23C24×D73
commutator, socle, cosocleC23×C73C25
FittingC73×#2641C2
prime radical#22615682C2
hypercenter#2641D73
hypocenterC73#34621
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ #2641

Complements

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

NH ≅ G/Ntype
1. #22615961C2semidirect
2. #22615916C2semidirect
3. #22615906C2semidirect
4. #22615892C2semidirect
5. #22615762C2semidirect
6. #22615723C2semidirect
8. #22613187C2semidirect
9. #22613174C2semidirect
10. #127×Dic73C2semidirect
11. #125×Dic73C2semidirect
12. #22612447C2semidirect
13. #22612393C2semidirect
14. #22612270C2semidirect
15. C73×#2641C2semidirect
304. #2641D73semidirect
315. C73#34621semidirect

Maximal subgroups

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

subgroupindex
#226159612
#226159162
#226159062
#226158922
#226157622
#226157232
#226156822
#226131872
#226131742
#127×Dic732
#125×Dic732
#226124472
#226123932
#226122702
C73×#26412
#3462173