Finite Group Databasebeta

← #34030297

Subgroups — #34030297

Order 18688.

652 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D73⋊#444C211yesmaximal, characteristic, complemented
2#22626685C211yesmaximal, characteristic, complemented
3#22626343C212yesmaximal, complemented
4#22626341C212yesmaximal, complemented
5#22626339C212yesmaximal, complemented
6#22626337C212yesmaximal, complemented
7#22626278C211yesmaximal, characteristic, complemented
8#22626258C211yesmaximal, characteristic, complemented
9#22626251C211yesmaximal, characteristic, complemented
10C2×#17989478C211yesmaximal, characteristic, complemented
11#22623064C211yesmaximal, characteristic
12D73⋊#131—21no—
13#41×#64408—21no—
14C2×#14556666C2211yescharacteristic
15#17989634C2212yes—
16#17989633C2212yes—
17#17989632—22no—
18#17989628C2212yescomplemented
19#17989627C2212yescomplemented
20#17989626—22no—
21D73⋊#108C2211yescharacteristic
22#17989597C2211yescharacteristic
23D73⋊#104—21no—
24#17989577—21no—
25C2×#14556645C2211yescharacteristic
26#17989512C2212yescomplemented
27#17989511C2212yescomplemented
28#17989506C2212yescomplemented
29#17989505C2212yescomplemented
30#17989504C2212yes—
31#17989503C2212yes—
32#17989478C2212yescomplemented
33#17989478C2212yescomplemented
34#135×D73C2211yescharacteristic
35#135×D73C2211yescharacteristic
36#17989029C411yescharacteristic, complemented
37#17989029C2211yescharacteristic
38#17989026C411yescharacteristic, complemented
39#17989026C2211yescharacteristic
40#17989024C411yescharacteristic, complemented
41#17989024C2211yescharacteristic
42#17989021C411yescharacteristic, complemented
43#17989021C2211yescharacteristic
44#17989019C411yescharacteristic, complemented
45#17989017C411yescharacteristic, complemented
46#17989017C411yescharacteristic, complemented
47C2×#14556530C2211yescharacteristic
48#508×C73C411yescharacteristic, complemented
49#14556679—22no—
50#14556678—22no—
51#14556678—42no—
52D73⋊M16—21no—
53D73⋊M16—41no—
54C2×#11924036—41no—
55C2×#11924036D411yescharacteristic
56Q8×#64408—22no—
57D4×#64408—22no—
58D4×#64408—42no—
59#14556670—21no—
60#14556670—41no—
61C2×C4×#64408—41no—
62C2×C4×#64408D411yescharacteristic
63#14556666—21no—
64#14556666C2312yes—
65C2×#11924031—21no—
66#14556662—21no—
67#14556661C2312yes—
68#14556660—21no—
69#14556659C2312yes—
70#14556656—22no—
71#14556655—22no—
72#14556654—22no—
73#14556653—22no—
74#14556645D412yes—
75#14556645C2312yes—
76#41×D73—21no—
77#41×D73—21no—
78#14556614C2×C411yescharacteristic
79#14556614C2311yescharacteristic
80#14556613C2×C411yescharacteristic
81#14556613C2×C411yescharacteristic, complemented
82#14556612—21no—
83#14556612—21no—
84#14556611—21no—
85#14556611—21no—
86#14556610C2×C411yescharacteristic
87#14556610C2311yescharacteristic
88Q16×D73C2×C411yescharacteristic, complemented
89Q16×D73C2311yescharacteristic
90SD16×D73—21no—
91SD16×D73—21no—
92D8×D73C2×C411yescharacteristic, complemented
93D8×D73C2311yescharacteristic
94C2×C8×D73C2311yescharacteristic
95Dic584C2×C411yescharacteristic, complemented
96#14556537C2×C411yescharacteristic, complemented
97#14556537C2×C411yescharacteristic, complemented
98#14556536C2×C411yescharacteristic, complemented
99#14556536C2×C411yescharacteristic, complemented
100#14556535C2×C411yescharacteristic, complemented
101#14556535C2×C411yescharacteristic, complemented
102#14556534C2×C411yescharacteristic, complemented
103#14556534C2×C411yescharacteristic, complemented
104D1168C2×C411yescharacteristic, complemented
105#14556532C2×C411yescharacteristic, complemented
106#14556532C2×C411yescharacteristic, complemented
107#14556530C2×C411yescharacteristic, complemented
108#14556530C2×C411yescharacteristic, complemented
109#14556530C2311yescharacteristic
110#14556530C2311yescharacteristic
111C2×#11924003C2×C411yescharacteristic, complemented
112#135×C73C2×C411yescharacteristic
113#135×C73C2×C411yescharacteristic, complemented
114Q32×C73C2×C411yescharacteristic, complemented
115SD32×C73C2×C411yescharacteristic, complemented
116SD32×C73C2×C411yescharacteristic, complemented
117D16×C73C2×C411yescharacteristic, complemented
118C2×C16×C73C2×C411yescharacteristic, complemented
119C22×#64408—21no—
120C22×#64408—42no—
121#11924038—22no—
122#11924038—42no—
123#11924037—42no—
124#11924037—41no—
125#11924036—21no—
126#11924036—42no—
127#11924036—41no—
128#11924036C2×D412yes—
129C2×#10596833—21no—
130C2×#10596833—42no—
131C4⋊#64408—22no—
132C4⋊#64408—42no—
133C22⋊#64408—42no—
134C22⋊#64408—41no—
135C4×#64408—21no—
136C4×#64408—42no—
137C4×#64408—41no—
138C4×#64408C2×D412yes—
139#11924031—22no—
140#11924029—21no—
141#11924029—21no—
142#11924028—21no—
143#11924028—21no—
144#11924028—41no—
145#11924028—41no—
146#11924027—21no—
147#11924027—21no—
148#11924027—41no—
149#11924027—41no—
150Q8×D73—21no—
151Q8×D73—21no—
152D4×D73—21no—
153D4×D73—21no—
154D4×D73—41no—
155D4×D73—41no—
156C2×C4×D73—41no—
157C2×C4×D73—41no—
158C2×C4×D73C2×D411yescharacteristic
159Dic292C22×C411yescharacteristic
160Dic292C22×C411yescharacteristic
161#11924018—21no—
162#11924018—21no—
163#11924017—21no—
164#11924017—21no—
165#11924016—21no—
166#11924016—21no—
167#11924015—21no—
168#11924015—21no—
169D584C22×C411yescharacteristic
170D584C22×C411yescharacteristic
171#11924013—21no—
172#11924013—21no—
173C2×#10596831#4111yescharacteristic
174C8×D73C22×C411yescharacteristic
175C8×D73C2×D411yescharacteristic
176C8×D73#4111yescharacteristic
177C8×D73C2411yescharacteristic
178#11924003C22×C411yescharacteristic
179#11924003C22×C411yescharacteristic
180#41×C73—21no—
181#41×C73—21no—
182Q16×C73C22×C411yescharacteristic
183Q16×C73C22×C411yescharacteristic
184SD16×C73—21no—
185SD16×C73—21no—
186D8×C73C22×C411yescharacteristic
187D8×C73C22×C411yescharacteristic
188C2×C8×C73C22×C411yescharacteristic
189C16×C73C22×C411yescharacteristic
190C16×C73C22×C411yescharacteristic
191C22×D73—41no—
192C22×D73—41no—
193C22×D73AGL(1,ℤ/8)11yescharacteristic
194C2×#64408—22no—
195C2×#64408—42no—
196C2×#64408—81no—
197C2×#64408—84no—
198Q8×C73—21no—
199Q8×C73—21no—
200D4×C73—21no—
201D4×C73—21no—
202D4×C73—41no—
203D4×C73—41no—
204C2×C4×C73—41no—
205C2×C4×C73—41no—
206C2×C4×C73C4×D411yescharacteristic
207C22⋊D73—41no—
208C22⋊D73—41no—
209C22⋊D73—41no—
210C22⋊D73—41no—
211C22⋊D73—41no—
212C22⋊D73—41no—
213C2×Dic73—41no—
214C2×Dic73—41no—
215C2×Dic73AGL(1,ℤ/8)11yescharacteristic
216D292—21no—
217D292—21no—
218D292—41no—
219D292—41no—
220C4×D73—41no—
221C4×D73—41no—
222C4×D73—41no—
223C4×D73—41no—
224C4×D73—41no—
225C4×D73—41no—
226C4×D73C4×D411yescharacteristic
227C4×D73AGL(1,ℤ/8)11yescharacteristic
228C4×D73AGL(1,ℤ/8)11yescharacteristic
229C4×D73C22×D411yescharacteristic
230Dic146—21no—
231Dic146—21no—
232Dic146—41no—
233Dic146—41no—
234#10596833—22no—
235#10596833—42no—
236C8×C73C4×D411yescharacteristic
237C8×C73C23×C411yescharacteristic
238#10596831C4×D411yescharacteristic
239#10596831C2×#4111yescharacteristic
240C22×C73—41no—
241C22×C73—41no—
242C22×C73#44211yescharacteristic
243C2×D73—41no—
244C2×D73—41no—
245C2×D73—41no—
246C2×D73—82no—
247C2×D73—82no—
248C2×D73#44211yescharacteristic
249C2×D73C2×AGL(1,ℤ/8)11yescharacteristic
250#64408—42no—
251#64408—82no—
252C4×C73—41no—
253C4×C73—41no—
254C4×C73#44211yescharacteristic
255C4×C73C2×C4×D411yescharacteristic
256Dic73—41no—
257Dic73—41no—
258Dic73#44211yescharacteristic
259Dic73C2×AGL(1,ℤ/8)11yescharacteristic
260#33468—731nomaximal, Sylow
261C2×C73—21no—
262C2×C73—81no—
263C2×C73—81no—
264C2×C73C2×#44211yescharacteristic
265D73—21no—
266D73—21no—
267D73—81no—
268D73—81no—
269#3414—731no—
270C2×#444—731no—
271#2943—731no—
272AGL(1,ℤ/16)—732no—
273AGL(1,ℤ/16)—732no—
274#2180—732no—
275#2180—732no—
276#2155—731no—
277#2148—731no—
278#2145—731no—
279C2×#347—731no—
280C73#3346811yescharacteristic, complemented, Sylow
281C2×#135—731no—
282C2×#135—731no—
283C2×#131—1461no—
284C4×#41—1461no—
285#510—731no—
286#510—731no—
287#510—731no—
288#510—731no—
289#509—731no—
290#509—731no—
291#509—731no—
292#509—731no—
293#508—731no—
294#508—731no—
295#508—731no—
296#508#6440811yescharacteristic, complemented
297C2×M32—731no—
298C2×M32—731no—
299#444—732no—
300#444—732no—
301#444—1462no—
302#442—732no—
303#441—732no—
304#440—1462no—
305C2×#108—731no—
306#427—731no—
307C2×#104—1461no—
308#416—1461no—
309C2×#97—731no—
310#365—732no—
311#365—732no—
312#362—732no—
313#361—732no—
314#360—732no—
315#360—732no—
316#347—732no—
317#347—732no—
318C2×#41—1461no—
319C2×#41—1461no—
320#135—731no—
321#135—731no—
322#135—731no—
323#135—731no—
324#135—731no—
325#135—731no—
326#135—1461no—
327#135—1461no—
328#135—1461no—
329#135—1461no—
330#135C2×#6440811yescharacteristic
331#135C2×#6440811yescharacteristic, complemented
332C2×Q16—731no—
333C2×Q16—731no—
334C2×SD16—1461no—
335C2×SD16—1461no—
336C2×D8—731no—
337C2×D8—731no—
338#131—2922no—
339#131—1462no—
340#131—1462no—
341C2×M16—2921no—
342C2×M16—1461no—
343C22×C8—2921no—
344C22×C8—731no—
345C22×C8—731no—
346C4×Q8—1462no—
347C4×D4—2922no—
348C4×D4—1462no—
349#117—2921no—
350#117—1461no—
351C2×C42—2921no—
352C2×C42—731no—
353Q32—731no—
354Q32—731no—
355Q32—731no—
356Q32C2×#6440811yescharacteristic, complemented
357SD32—731no—
358SD32—731no—
359SD32—731no—
360SD32—731no—
361SD32—731no—
362SD32—731no—
363SD32C2×#6440811yescharacteristic, complemented
364SD32C2×#6440811yescharacteristic, complemented
365D16—731no—
366D16—731no—
367D16—731no—
368D16C2×#6440811yescharacteristic, complemented
369M32—731no—
370M32—731no—
371M32—731no—
372M32—731no—
373M32—732no—
374M32—1461no—
375C2×C16—731no—
376C2×C16—1461no—
377C2×C16C2×#6440811yescharacteristic, complemented
378#108—732no—
379#108—1461no—
380#107—1461no—
381#106—732no—
382#104—1462no—
383#104—1462no—
384#103—1462no—
385#102—1462no—
386#97—732no—
387#97—732no—
388#41—21no—
389#41—21no—
390#41—2921no—
391#41—2921no—
392#41—2921no—
393#41—2921no—
394#41—1461no—
395#41—1461no—
396#41—1461no—
397#41—1461no—
398#41—1461no—
399#41—1461no—
400C2×Q8—1461no—
401C2×Q8—1461no—
402C2×D4—2921no—
403C2×D4—2921no—
404C2×D4—1461no—
405C2×D4—1461no—
406C22×C4—2921no—
407C22×C4—2921no—
408C22×C4—2922no—
409C22×C4—731no—
410C22×C4—1461no—
411Q16—731no—
412Q16—731no—
413Q16—1461no—
414Q16—1461no—
415Q16C22×#6440811yescharacteristic
416Q16C22×#6440811yescharacteristic
417SD16—21no—
418SD16—21no—
419SD16—1461no—
420SD16—1461no—
421SD16—1461no—
422SD16—1461no—
423SD16—1461no—
424SD16—1461no—
425D8—731no—
426D8—731no—
427D8—1461no—
428D8—1461no—
429D8C22×#6440811yescharacteristic
430D8C22×#6440811yescharacteristic
431M16—2921no—
432M16—2922no—
433M16—2922no—
434M16—1462no—
435C2×C8—2921no—
436C2×C8—2922no—
437C2×C8—2922no—
438C2×C8—731no—
439C2×C8—731no—
440C2×C8—731no—
441C2×C8—731no—
442C2×C8—731no—
443C2×C8—732no—
444C2×C8—1461no—
445C2×C8—1461no—
446C2×C8C22×#6440811yescharacteristic
447C4⋊C4—2922no—
448C4⋊C4—1462no—
449C22⋊C4—2921no—
450C22⋊C4—2922no—
451C42—2921no—
452C42—2922no—
453C42—732no—
454C42—1461no—
455C16—731no—
456C16—731no—
457C16—1462no—
458C16C22×#6440811yescharacteristic
459C16C22×#6440811yescharacteristic
460C23—2921no—
461C23—2921no—
462C23—731no—
463Q8—21no—
464Q8—21no—
465Q8—2921no—
466Q8—2921no—
467Q8—1461no—
468Q8—1461no—
469D4—21no—
470D4—21no—
471D4—41no—
472D4—41no—
473D4—2921no—
474D4—2921no—
475D4—2921no—
476D4—2921no—
477D4—2921no—
478D4—2921no—
479D4—2921no—
480D4—2921no—
481D4—1461no—
482D4—1461no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—2921no—
486C2×C4—2921no—
487C2×C4—2921no—
488C2×C4—2921no—
489C2×C4—2921no—
490C2×C4—2921no—
491C2×C4—2921no—
492C2×C4—2921no—
493C2×C4—2922no—
494C2×C4—5841no—
495C2×C4—5844no—
496C2×C4—731no—
497C2×C4—731no—
498C2×C4—731no—
499C2×C4—731no—
500C2×C4—731no—
501C2×C4—1462no—
502C2×C4D4×#6440811yescharacteristic
503C8—2922no—
504C8—731no—
505C8—731no—
506C8—1462no—
507C8D4×#6440811yescharacteristic
508C8C23×#6440811yescharacteristic
509C22—41no—
510C22—41no—
511C22—2921no—
512C22—2921no—
513C22—2921no—
514C22—5842no—
515C22—5842no—
516C22—731no—
517C22—731no—
518C22#1798962811yescharacteristic
519C4—41no—
520C4—41no—
521C4—2922no—
522C4—2921no—
523C4—2921no—
524C4—5842no—
525C4—731no—
526C4—731no—
527C4#1798962811yescharacteristic
528C4C2×D4×#6440811yescharacteristic
529C2—21no—
530C2—81no—
531C2—81no—
532C2—5841no—
533C2—5841no—
534C2—1461no—
535C2—1461no—
536C2C2×#1798962811yescharacteristic
subgroupG/H
centerC4#17989628
commutatorC8×C73C23×C4
FrattiniC8C23×#64408
Fitting#508×C73C4
socleC2×C73C2×#442
prime radical#22623064C2
cosocleC8×D73C24
hypercenter#508#64408
hypocenterC73#33468
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. D73⋊#444C2semidirect
2. #22626685C2semidirect
3. #22626343C2semidirect
4. #22626341C2semidirect
5. #22626339C2semidirect
6. #22626337C2semidirect
7. #22626278C2semidirect
8. #22626258C2semidirect
9. #22626251C2semidirect
10. C2×#17989478C2semidirect
18. #17989628C22semidirect
19. #17989627C22semidirect
26. #17989512C22semidirect
27. #17989511C22semidirect
28. #17989506C22semidirect
29. #17989505C22semidirect
32. #17989478C22semidirect
33. #17989478C22semidirect
36. #17989029C4semidirect
38. #17989026C4semidirect
40. #17989024C4semidirect
42. #17989021C4semidirect
44. #17989019C4semidirect
45. #17989017C4semidirect
46. #17989017C4semidirect
48. #508×C73C4semidirect
81. #14556613C2×C4semidirect
88. Q16×D73C2×C4semidirect
92. D8×D73C2×C4semidirect
95. Dic584C2×C4semidirect
96. #14556537C2×C4semidirect
97. #14556537C2×C4semidirect
98. #14556536C2×C4semidirect
99. #14556536C2×C4semidirect
100. #14556535C2×C4semidirect
101. #14556535C2×C4semidirect
102. #14556534C2×C4semidirect
103. #14556534C2×C4semidirect
104. D1168C2×C4semidirect
105. #14556532C2×C4semidirect
106. #14556532C2×C4semidirect
107. #14556530C2×C4semidirect
108. #14556530C2×C4semidirect
111. C2×#11924003C2×C4semidirect
113. #135×C73C2×C4semidirect
114. Q32×C73C2×C4semidirect
115. SD32×C73C2×C4semidirect
116. SD32×C73C2×C4semidirect
117. D16×C73C2×C4semidirect
118. C2×C16×C73C2×C4semidirect
280. C73#33468semidirect
296. #508#64408semidirect
331. #135C2×#64408semidirect
356. Q32C2×#64408semidirect
363. SD32C2×#64408semidirect
364. SD32C2×#64408semidirect
368. D16C2×#64408semidirect
377. C2×C16C2×#64408semidirect

Maximal subgroups

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

subgroupindex
D73⋊#4442
#226266852
#226263432
#226263412
#226263392
#226263372
#226262782
#226262582
#226262512
C2×#179894782
#226230642
#3346873