Finite Group Databasebeta

← #33240469

Subgroups — #33240469

Order 18688.

686 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989026C211yesmaximal, characteristic, complemented
2C22×#14556530C211yesmaximal, characteristic, complemented
3#22619510C211yesmaximal, characteristic
4#467×D73C211yesmaximal, characteristic, complemented
5#22612043C211yesmaximal, characteristic, complemented
6#22612041C211yesmaximal, characteristic, complemented
7#22612040C211yesmaximal, characteristic, complemented
8#22612038C211yesmaximal, characteristic, complemented
9#22612025C211yesmaximal, characteristic, complemented
10#22612022C211yesmaximal, characteristic, complemented
11#22612020C211yesmaximal, characteristic, complemented
12#22611718C211yesmaximal, characteristic, complemented
13#22611649C211yesmaximal, characteristic, complemented
14#22611640C211yesmaximal, characteristic, complemented
15C73×#2217C211yesmaximal, characteristic, complemented
16C2×#14556610C2211yescharacteristic
17C2×Q16×D73C2211yescharacteristic
18C22×C8×D73C2211yescharacteristic
19#17989097—21no—
20C22⋊Q8×D73—21no—
21#17989026—24no—
22C2×Dic584C2211yescharacteristic, complemented
23C2×#14556537C2211yescharacteristic, complemented
24C2×#14556535C2211yescharacteristic, complemented
25C2×#14556530—24no—
26C2×#14556530C2211yescharacteristic
27C2×#14556530C2211yescharacteristic
28C2×#14556530C2211yescharacteristic
29C2×#14556530C2211yescharacteristic
30C22×#11924003C2211yescharacteristic
31#17988829C2211yescharacteristic
32#17988829C2211yescharacteristic
33#17988828C2211yescharacteristic
34#17988828C2211yescharacteristic
35#17988827—21no—
36#17988826—21no—
37#17988825C2211yescharacteristic
38#17988820C2211yescharacteristic
39#17988815—21no—
40#17988814—21no—
41#17988624C2211yescharacteristic
42#107×D73C2211yescharacteristic
43#17988580—21no—
44#103×D73—21no—
45#17988414C2211yescharacteristic, complemented
46#17988413C2211yescharacteristic, complemented
47#17988412C2211yescharacteristic, complemented
48#17988392C2211yescharacteristic, complemented
49#17988392C2211yescharacteristic, complemented
50#17988391C2211yescharacteristic, complemented
51#17988390C2211yescharacteristic, complemented
52#17988390C2211yescharacteristic, complemented
53#17988388C2211yescharacteristic, complemented
54C2×SD32×C73C2211yescharacteristic, complemented
55C22×C16×C73C2211yescharacteristic
56#467×C73C2211yescharacteristic
57#466×C73C2211yescharacteristic
58#367×C73C2211yescharacteristic, complemented
59#358×C73C2211yescharacteristic, complemented
60#357×C73C2211yescharacteristic, complemented
61C2×#11924027—21no—
62C2×Q8×D73—21no—
63C22×C4×D73D411yescharacteristic
64#14556610—22no—
65#14556610—22no—
66C2×Dic292C2311yescharacteristic
67C2×Dic292C2311yescharacteristic
68C2×#11924018—21no—
69Q16×D73—22no—
70Q16×D73—22no—
71C2×#11924016—21no—
72C22×#10596831D411yescharacteristic
73C2×C8×D73—22no—
74C2×C8×D73—22no—
75C2×C8×D73D411yescharacteristic
76C2×C8×D73D411yescharacteristic
77C2×C8×D73C2311yescharacteristic
78C2×C8×D73C2311yescharacteristic
79#14556573—41no—
80#14556571—21no—
81#14556571—21no—
82#14556570—21no—
83#14556570—21no—
84#14556569—41no—
85#14556568—41no—
86#14556567—21no—
87#14556562—21no—
88#14556559—41no—
89D4×Dic73—41no—
90#14556551—41no—
91#14556549—21no—
92C4⋊C4×D73—21no—
93C4⋊C4×D73—41no—
94C22⋊C4×D73—41no—
95Dic584—22no—
96#14556537—22no—
97#14556535—22no—
98#14556530—22no—
99#14556530—22no—
100#14556530—22no—
101#14556530D414yescomplemented
102C2×#11924003—22no—
103C2×#11924003C2311yescharacteristic
104C2×#11924003C2311yescharacteristic
105#14556525C2311yescharacteristic
106#14556525C2311yescharacteristic
107#14556524—21no—
108#14556522—21no—
109#14556516—21no—
110#14556516—21no—
111#14556515—21no—
112#14556514—21no—
113#14556514—21no—
114#14556512—21no—
115C2×Q16×C73C2311yescharacteristic
116C2×D8×C73C2311yescharacteristic
117C22×C8×C73C2311yescharacteristic
118C22⋊Q8×C73—21no—
119#121×C73—21no—
120SD32×C73—22no—
121C2×C16×C73—22no—
122C2×C16×C73C2311yescharacteristic
123C2×C16×C73C2311yescharacteristic
124#107×C73C2311yescharacteristic
125#107×C73C2311yescharacteristic
126#103×C73—21no—
127#102×C73—21no—
128C23×D73D811yescharacteristic
129#11924027—42no—
130#11924027—42no—
131C2×Dic146—21no—
132C2×Dic146—21no—
133C2×Dic146—41no—
134Q8×D73—42no—
135Q8×D73—42no—
136C2×C22⋊D73—41no—
137C2×C22⋊D73—41no—
138C22×Dic73—41no—
139C22×Dic73D811yescharacteristic
140C2×C4×D73—22no—
141C2×C4×D73—22no—
142C2×C4×D73—41no—
143C2×C4×D73—41no—
144C2×C4×D73D811yescharacteristic
145C2×C4×D73D811yescharacteristic
146C2×C4×D73C2×D411yescharacteristic
147C2×C4×D73C2×D411yescharacteristic
148Dic292—21no—
149Dic292—21no—
150Dic292—21no—
151Dic292—21no—
152#11924018—42no—
153#11924016—42no—
154C2×#10596831—22no—
155C2×#10596831C2×D411yescharacteristic
156C2×#10596831C2×D411yescharacteristic
157C8×D73—22no—
158C8×D73—21no—
159C8×D73—21no—
160C8×D73—22no—
161C8×D73C2×D412yes—
162C8×D73#4112yes—
163C4⋊Dic73—21no—
164C4⋊Dic73—21no—
165C4⋊Dic73—41no—
166C4⋊Dic73—41no—
167#11924007—41no—
168#11924007—41no—
169#11924007—41no—
170#11924007—41no—
171C22⋊Dic73—41no—
172C22⋊Dic73—41no—
173C22⋊Dic73—41no—
174#11924005—41no—
175#11924005—41no—
176#11924005—41no—
177#11924005—41no—
178C4×Dic73—41no—
179#11924003—21no—
180#11924003C2×D412yes—
181C2×Q8×C73—21no—
182C2×D4×C73—21no—
183C2×D4×C73—41no—
184C22×C4×C73C2×D411yescharacteristic
185Q16×C73—21no—
186Q16×C73—21no—
187D8×C73—21no—
188D8×C73—21no—
189C2×C8×C73—21no—
190C2×C8×C73—21no—
191C2×C8×C73C2×D411yescharacteristic
192C2×C8×C73C2411yescharacteristic
193C4⋊C4×C73—21no—
194C4⋊C4×C73—21no—
195C4⋊C4×C73—41no—
196C22⋊C4×C73—41no—
197C22⋊C4×C73—41no—
198C16×C73—21no—
199C16×C73C2×D412yes—
200C23×C73—41no—
201C23×C73C2×D811yescharacteristic
202C22×D73—22no—
203C22×D73—22no—
204C22×D73—41no—
205C22×D73C2×D811yescharacteristic
206C22×D73C2×D811yescharacteristic
207Q8×C73—41no—
208Q8×C73—41no—
209D4×C73—41no—
210D4×C73—41no—
211D4×C73—82no—
212C2×C4×C73—21no—
213C2×C4×C73—21no—
214C2×C4×C73—41no—
215C2×C4×C73—41no—
216C2×C4×C73—41no—
217C2×C4×C73C2×D811yescharacteristic
218C2×C4×C73C22×D411yescharacteristic
219C22⋊D73—82no—
220C22⋊D73—82no—
221C2×Dic73—22no—
222C2×Dic73—41no—
223C2×Dic73—41no—
224C2×Dic73—41no—
225C2×Dic73—41no—
226C2×Dic73—82no—
227C2×Dic73C2×D811yescharacteristic
228C2×Dic73C2×D811yescharacteristic
229C4×D73—22no—
230C4×D73—21no—
231C4×D73—21no—
232C4×D73—22no—
233C4×D73—82no—
234C4×D73—82no—
235C4×D73#12112yes—
236C4×D73#13512yes—
237Dic146—41no—
238Dic146—41no—
239Dic146—41no—
240Dic146—41no—
241Dic146—82no—
242C8×C73—21no—
243C8×C73C22×D411yescharacteristic
244C8×C73C2×#4111yescharacteristic
245#10596831—21no—
246#10596831#12112yes—
247C22×C73—21no—
248C22×C73—21no—
249C22×C73—81no—
250C22×C73—81no—
251C22×C73—81no—
252C22×C73#51012yescomplemented
253C22×C73C22×D811yescharacteristic
254C2×D73—21no—
255C2×D73—21no—
256C2×D73—21no—
257C2×D73—21no—
258C2×D73—21no—
259C2×D73—44no—
260C2×D73#46612yes—
261C4×C73—21no—
262C4×C73—81no—
263C4×C73—81no—
264C4×C73—81no—
265C4×C73C2×#12111yescharacteristic
266C4×C73C2×#13511yescharacteristic
267Dic73—21no—
268Dic73—81no—
269Dic73—81no—
270Dic73—81no—
271Dic73—81no—
272Dic73#46612yes—
273#33606—731nomaximal, Sylow
274C2×C73—22no—
275C2×C73—161no—
276C2×C73C2×#46611yescharacteristic
277C2×C73C2×#51011yescharacteristic
278C2×C73#341411yescharacteristic
279D73—22no—
280D73—41no—
281C2×#510—731no—
282C22×M32—731no—
283#3050—731no—
284C2×#467—731no—
285#2220—731no—
286#2220—731no—
287#2219—731no—
288#2219—731no—
289#2217—731no—
290#2217D7311yescharacteristic, complemented
291#2216—731no—
292#2216—731no—
293#2159—731no—
294#2141—731no—
295#2139—731no—
296C73#3360611yescharacteristic, complemented, Sylow
297C2×#135—731no—
298C22×Q16—731no—
299C23×C8—731no—
300#525—1461no—
301C2×C22⋊Q8—1461no—
302#510—1464no—
303C2×Q32—731no—
304C2×Q32—731no—
305C2×SD32—731no—
306C2×SD32C2×D7311yescharacteristic, complemented
307C2×M32—731no—
308C2×M32—731no—
309C2×M32—731no—
310C2×M32—731no—
311C2×M32—1464no—
312C22×C16—731no—
313C22×C16C2×D7311yescharacteristic
314#467—731no—
315#467—731no—
316#467—731no—
317#467—731no—
318#467—731no—
319#467—1461no—
320#467—1461no—
321#467C2×D7311yescharacteristic
322#466—731no—
323#466C2×D7311yescharacteristic
324#465—1461no—
325#465—1461no—
326#427—731no—
327C2×#107—731no—
328#416—1461no—
329C2×#103—1461no—
330#367—731no—
331#367C2×D7311yescharacteristic, complemented
332#366—731no—
333#366—731no—
334#358—731no—
335#358—731no—
336#358—731no—
337#358—731no—
338#358—731no—
339#358C2×D7311yescharacteristic, complemented
340#357—731no—
341#357C2×D7311yescharacteristic, complemented
342C2×#41—1461no—
343C22×Q8—1461no—
344C23×C4—731no—
345#135—1462no—
346#135—1462no—
347C2×Q16—731no—
348C2×Q16—731no—
349C2×Q16—1462no—
350C2×Q16—1462no—
351C2×Q16—1461no—
352C2×Q16C22×D7311yescharacteristic
353C2×SD16—1461no—
354C2×D8C22×D7311yescharacteristic
355C22×C8—731no—
356C22×C8—731no—
357C22×C8—731no—
358C22×C8—731no—
359C22×C8—731no—
360C22×C8—1462no—
361C22×C8—1462no—
362C22×C8C22×D7311yescharacteristic
363#123—2921no—
364C22⋊Q8—21no—
365C22⋊Q8—2921no—
366C22⋊Q8—2921no—
367C22⋊Q8—1461no—
368C22⋊Q8—1461no—
369C22⋊Q8—1461no—
370C22⋊Q8—1461no—
371C22⋊Q8—1461no—
372#121—21no—
373#121—1461no—
374#120—2921no—
375C4×D4—2921no—
376C4×D4—2921no—
377#117—1461no—
378C2×C4⋊C4—2921no—
379C2×C4⋊C4—1461no—
380C2×C22⋊C4—2921no—
381Q32—1462no—
382Q32—1462no—
383SD32—22no—
384SD32—1462no—
385M32—734no—
386M32—1462no—
387M32—1462no—
388M32—1462no—
389C2×C16—22no—
390C2×C16—731no—
391C2×C16—731no—
392C2×C16—1462no—
393C2×C16C22×D7311yescharacteristic
394C2×C16C22×D7311yescharacteristic
395#107—731no—
396#107—731no—
397#107—1461no—
398#107C22×D7311yescharacteristic
399#107C22×D7311yescharacteristic
400#106—1461no—
401#103—21no—
402#103—1461no—
403#103—1461no—
404#103—1461no—
405#103—1461no—
406#103—1461no—
407#102—21no—
408#102—1461no—
409C24—731no—
410#41—2922no—
411#41—2922no—
412C2×Q8—21no—
413C2×Q8—2921no—
414C2×Q8—2922no—
415C2×Q8—2922no—
416C2×Q8—1461no—
417C2×Q8—1461no—
418C2×D4—21no—
419C2×D4—41no—
420C2×D4—2921no—
421C2×D4—2921no—
422C22×C4—2921no—
423C22×C4—2921no—
424C22×C4—2921no—
425C22×C4—731no—
426C22×C4—731no—
427C22×C4—731no—
428C22×C4—731no—
429C22×C4—731no—
430C22×C4—1462no—
431C22×C4—1462no—
432C22×C4D4×D7311yescharacteristic
433Q16—21no—
434Q16—21no—
435Q16—2922no—
436Q16—1461no—
437Q16—1461no—
438Q16—1461no—
439Q16—1461no—
440SD16—2922no—
441D8—21no—
442D8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—731no—
446C2×C8—732no—
447C2×C8—732no—
448C2×C8—731no—
449C2×C8—1462no—
450C2×C8—1462no—
451C2×C8—1462no—
452C2×C8—1461no—
453C2×C8—1461no—
454C2×C8D4×D7311yescharacteristic
455C2×C8C23×D7311yescharacteristic
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—41no—
459C4⋊C4—2921no—
460C4⋊C4—2921no—
461C4⋊C4—2921no—
462C4⋊C4—2921no—
463C4⋊C4—2921no—
464C4⋊C4—2921no—
465C4⋊C4—1461no—
466C4⋊C4—1461no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—2921no—
470C22⋊C4—2921no—
471C22⋊C4—2921no—
472C22⋊C4—2921no—
473C22⋊C4—2921no—
474C22⋊C4—2921no—
475C22⋊C4—2921no—
476C42—2921no—
477C16—21no—
478C16—732no—
479C16—1461no—
480C16D4×D7312yes—
481C23—41no—
482C23—2921no—
483C23—731no—
484C23—731no—
485C23—1462no—
486C23—1462no—
487C23D8×D7311yescharacteristic
488Q8—41no—
489Q8—41no—
490Q8—2921no—
491Q8—2921no—
492Q8—2921no—
493Q8—2921no—
494Q8—5842no—
495D4—41no—
496D4—41no—
497D4—82no—
498D4—5842no—
499D4—5842no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—2921no—
506C2×C4—2921no—
507C2×C4—2921no—
508C2×C4—2921no—
509C2×C4—5842no—
510C2×C4—5842no—
511C2×C4—5842no—
512C2×C4—731no—
513C2×C4—732no—
514C2×C4—732no—
515C2×C4—731no—
516C2×C4—1462no—
517C2×C4—1462no—
518C2×C4—1462no—
519C2×C4—1461no—
520C2×C4—1461no—
521C2×C4D8×D7311yescharacteristic
522C2×C4C2×D4×D7311yescharacteristic
523C8—21no—
524C8—732no—
525C8—1461no—
526C8C2×D4×D7311yescharacteristic
527C8#41×D7311yescharacteristic
528C22—21no—
529C22—21no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22—2924no—
534C22—732no—
535C22—1461no—
536C22—1461no—
537C22—1461no—
538C22—1461no—
539C22—1461no—
540C22#1798902612yescomplemented
541C22C2×D8×D7311yescharacteristic
542C4—21no—
543C4—81no—
544C4—81no—
545C4—81no—
546C4—5841no—
547C4—5841no—
548C4—5841no—
549C4—5841no—
550C4—732no—
551C4—1461no—
552C4#121×D7311yescharacteristic
553C4#135×D7311yescharacteristic
554C2—22no—
555C2—161no—
556C2—2921no—
557C2—1462no—
558C2#466×D7311yescharacteristic
559C2C2×#1798902611yescharacteristic
560C2#2262306411yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2217C2
socleC22×C73C22×D8
prime radical#22619510C2
hypercenter#2217D73
hypocenterC73#33606
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2217

Complements

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

NH ≅ G/Ntype
1. C2×#17989026C2semidirect
2. C22×#14556530C2semidirect
4. #467×D73C2semidirect
5. #22612043C2semidirect
6. #22612041C2semidirect
7. #22612040C2semidirect
8. #22612038C2semidirect
9. #22612025C2semidirect
10. #22612022C2semidirect
11. #22612020C2semidirect
12. #22611718C2semidirect
13. #22611649C2semidirect
14. #22611640C2semidirect
15. C73×#2217C2semidirect
22. C2×Dic584C22semidirect
23. C2×#14556537C22semidirect
24. C2×#14556535C22semidirect
45. #17988414C22semidirect
46. #17988413C22semidirect
47. #17988412C22semidirect
48. #17988392C22semidirect
49. #17988392C22semidirect
50. #17988391C22semidirect
51. #17988390C22semidirect
52. #17988390C22semidirect
53. #17988388C22semidirect
54. C2×SD32×C73C22semidirect
58. #367×C73C22semidirect
59. #358×C73C22semidirect
60. #357×C73C22semidirect
101. #14556530D4semidirect
252. C22×C73#510semidirect
290. #2217D73semidirect
296. C73#33606semidirect
306. C2×SD32C2×D73semidirect
331. #367C2×D73semidirect
339. #358C2×D73semidirect
341. #357C2×D73semidirect
540. C22#17989026semidirect

Maximal subgroups

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

subgroupindex
C2×#179890262
C22×#145565302
#226195102
#467×D732
#226120432
#226120412
#226120402
#226120382
#226120252
#226120222
#226120202
#226117182
#226116492
#226116402
C73×#22172
#3360673