Finite Group Databasebeta

← #33240729

Subgroups — #33240729

Order 18688.

514 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620456C211yesmaximal, characteristic, complemented
2#22618843C211yesmaximal, characteristic
3#22612108C211yesmaximal, characteristic, complemented
4#22612107C211yesmaximal, characteristic, complemented
5#22612106C211yesmaximal, characteristic, complemented
6#22612105C211yesmaximal, characteristic, complemented
7#22612101C211yesmaximal, characteristic, complemented
8#22612100C211yesmaximal, characteristic, complemented
9#22612099C211yesmaximal, characteristic, complemented
10#22611702C211yesmaximal, characteristic, complemented
11#365×D73C211yesmaximal, characteristic, complemented
12#22611695C211yesmaximal, characteristic, complemented
13#22611665C211yesmaximal, characteristic
14#360×D73C211yesmaximal, characteristic
15C73×#2230C211yesmaximal, characteristic, complemented
16#135×D73C2211yescharacteristic
17#17989352—21no—
18#17989214—21no—
19M32×D73C2211yescharacteristic
20M32×D73C2211yescharacteristic
21#17988994—21no—
22#17988993—21no—
23#17988989C2211yescharacteristic, complemented
24#17988987C2211yescharacteristic, complemented
25#17988984C2211yescharacteristic, complemented
26#17988707—21no—
27#17988706—21no—
28#17988705C2211yescharacteristic
29#17988704C2211yescharacteristic
30#17988703C2211yescharacteristic
31#108×D73C2211yescharacteristic
32#17988626—21no—
33#17988622C2211yescharacteristic
34#106×D73C2211yescharacteristic
35#17988602—21no—
36#17988541C2211yescharacteristic
37#17988411C2211yescharacteristic, complemented
38#17988411C2211yescharacteristic, complemented
39#17988410C2211yescharacteristic, complemented
40#17988410C2211yescharacteristic, complemented
41#17988409C2211yescharacteristic, complemented
42#17988409C2211yescharacteristic, complemented
43#17988408C2211yescharacteristic, complemented
44#17988407C2211yescharacteristic, complemented
45#17988407C2211yescharacteristic, complemented
46#17988398C2211yescharacteristic
47#17988398C2211yescharacteristic
48#17988397C2211yescharacteristic
49#17988397C2211yescharacteristic
50#17988396C2211yescharacteristic
51#17988396C2211yescharacteristic
52#498×C73C2211yescharacteristic, complemented
53#443×C73C2211yescharacteristic
54#365×C73C2211yescharacteristic, complemented
55#365×C73C2211yescharacteristic, complemented
56#364×C73C2211yescharacteristic, complemented
57#360×C73C2211yescharacteristic
58#360×C73C2211yescharacteristic
59#41×D73—21no—
60#14556614Q811yescharacteristic, complemented
61#14556613C2311yescharacteristic
62#14556612—21no—
63#14556611—21no—
64#14556610Q811yescharacteristic, complemented
65Q16×D73Q811yescharacteristic, complemented
66SD16×D73—21no—
67D8×D73Q811yescharacteristic, complemented
68#14556599—21no—
69#14556599—41no—
70#14556598—21no—
71#14556597—21no—
72#14556597—41no—
73C2×#11924011—41no—
74C2×#11924011D411yescharacteristic
75M16×D73—21no—
76M16×D73—41no—
77C2×C8×D73C2311yescharacteristic
78#14556590—21no—
79#14556589—41no—
80#14556583—21no—
81#14556582—21no—
82#14556581—41no—
83#14556569—41no—
84#14556567—41no—
85#14556549—21no—
86#14556548D411yescharacteristic
87C4⋊C4×D73—21no—
88#14556531C2311yescharacteristic
89#14556531C2311yescharacteristic
90#14556530—21no—
91#14556530—21no—
92C16×D73—21no—
93C16×D73—21no—
94#14556527C2311yescharacteristic
95#14556526—21no—
96#14556525—21no—
97#14556524—21no—
98#14556524—21no—
99#14556523C2311yescharacteristic
100#14556523C2311yescharacteristic
101#14556522—21no—
102#14556522—21no—
103#14556519—21no—
104#14556518—21no—
105#14556517—21no—
106#14556502D411yescharacteristic
107#14556501D411yescharacteristic
108C8×Dic73C2311yescharacteristic
109#135×C73C2311yescharacteristic
110#131×C73—21no—
111C4⋊Q8×C73—21no—
112M32×C73C2311yescharacteristic
113M32×C73C2311yescharacteristic
114#108×C73C2311yescharacteristic
115#106×C73—21no—
116#106×C73C2311yescharacteristic
117#106×C73C2311yescharacteristic
118#104×C73—21no—
119C4×C8×C73C2311yescharacteristic
120#11924029—21no—
121#11924028—21no—
122#11924028—41no—
123#11924027—21no—
124#11924027—41no—
125C2×Dic146—41no—
126Q8×D73—21no—
127D4×D73—21no—
128D4×D73—41no—
129C2×C4×D73—41no—
130C2×C4×D73—41no—
131C2×C4×D73C2×D411yescharacteristic
132Dic292C2×Q811yescharacteristic
133#11924018—21no—
134#11924017—21no—
135#11924016—21no—
136#11924015—21no—
137D584C2×Q811yescharacteristic
138#11924013—21no—
139#11924012—21no—
140#11924012—41no—
141#11924011—41no—
142#11924011—41no—
143#11924011—41no—
144#11924011SD1611yescharacteristic
145#11924011SD1611yescharacteristic
146#11924011SD1611yescharacteristic
147#11924011SD1611yescharacteristic
148C2×#10596831—41no—
149C2×#10596831C2×D411yescharacteristic
150C2×#10596831C2×D411yescharacteristic
151C8×D73—41no—
152C8×D73—41no—
153C8×D73—41no—
154C8×D73C2×Q811yescharacteristic
155C8×D73C2×Q811yescharacteristic
156C8×D73#4111yescharacteristic
157C8×D73#4111yescharacteristic
158C4⋊Dic73—21no—
159C4⋊Dic73—21no—
160C4⋊Dic73—41no—
161#11924007—21no—
162#11924007—41no—
163#11924007—41no—
164#11924007—41no—
165#11924007—41no—
166#11924007—41no—
167#11924005—21no—
168#11924005—41no—
169C4×Dic73—41no—
170C4×Dic73C2×D411yescharacteristic
171#11924003—21no—
172#11924003—21no—
173#41×C73—21no—
174C2×Q8×C73—41no—
175Q16×C73C2×Q811yescharacteristic
176SD16×C73—21no—
177D8×C73C2×Q811yescharacteristic
178M16×C73—21no—
179M16×C73—41no—
180C2×C8×C73—41no—
181C2×C8×C73C2×D411yescharacteristic
182C2×C8×C73C2411yescharacteristic
183C4⋊C4×C73—21no—
184C4⋊C4×C73—21no—
185C4⋊C4×C73—41no—
186C42×C73C2×D411yescharacteristic
187C16×C73—21no—
188C16×C73—21no—
189C22×D73—41no—
190C22×D73AGL(1,ℤ/8)11yescharacteristic
191Q8×C73—21no—
192Q8×C73—161no—
193D4×C73—21no—
194D4×C73—41no—
195C2×C4×C73—21no—
196C2×C4×C73—41no—
197C2×C4×C73—41no—
198C2×C4×C73—41no—
199C2×C4×C73C22×D411yescharacteristic
200C22⋊D73—41no—
201C22⋊D73—41no—
202C22⋊D73—41no—
203C2×Dic73—21no—
204C2×Dic73—41no—
205C2×Dic73—41no—
206C2×Dic73—41no—
207C2×Dic73AGL(1,ℤ/8)11yescharacteristic
208D292—21no—
209D292—41no—
210C4×D73—41no—
211C4×D73—41no—
212C4×D73—41no—
213C4×D73—161no—
214C4×D73C22⋊Q811yescharacteristic
215C4×D73C22⋊Q811yescharacteristic
216C4×D73C2×SD1611yescharacteristic
217C4×D73C2×SD1611yescharacteristic
218Dic146—21no—
219Dic146—41no—
220Dic146—161no—
221C8×C73—41no—
222C8×C73C2×SD1611yescharacteristic
223C8×C73C2×SD1611yescharacteristic
224C8×C73C22×Q811yescharacteristic
225C8×C73C2×#4111yescharacteristic
226#10596831—41no—
227#10596831C22⋊Q811yescharacteristic
228#10596831C22⋊Q811yescharacteristic
229#10596831C2×SD1611yescharacteristic
230#10596831C2×SD1611yescharacteristic
231C22×C73—41no—
232C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
233C2×D73—41no—
234C2×D73—41no—
235C2×D73—161no—
236C2×D73#47611yescharacteristic
237C2×D73#47611yescharacteristic
238C4×C73—41no—
239C4×C73—41no—
240C4×C73—81no—
241C4×C73—81no—
242C4×C73C2×C22⋊Q811yescharacteristic
243C4×C73C22×SD1611yescharacteristic
244Dic73—41no—
245Dic73—41no—
246Dic73—81no—
247Dic73—81no—
248Dic73#47611yescharacteristic
249Dic73#47611yescharacteristic
250#33638—731nomaximal, Sylow
251C2×C73—21no—
252C2×C73—81no—
253C2×C73C2×#47611yescharacteristic
254D73—21no—
255D73—21no—
256D73—81no—
257#3163—731no—
258#2957—731no—
259#2231—731no—
260#2231—731no—
261#2231—731no—
262#2231—731no—
263#2230—731no—
264#2230—731no—
265#2230—731no—
266#2230D7311yescharacteristic, complemented
267#2155—731no—
268C2×#365—731no—
269#2153—731no—
270#2145—731no—
271C2×#360—731no—
272C73#3363811yescharacteristic, complemented, Sylow
273C2×#135—731no—
274#568—1461no—
275#544—1461no—
276C2×M32—731no—
277C2×M32—731no—
278#501—1461no—
279#501—1461no—
280#499—731no—
281#499—731no—
282#498—731no—
283#498C2×D7311yescharacteristic, complemented
284#444—1461no—
285#444—1461no—
286#443—731no—
287#443—731no—
288#443—731no—
289#443C2×D7311yescharacteristic
290C2×#108—731no—
291#428—1461no—
292#427—731no—
293C2×#106—731no—
294#421—1461no—
295#405—731no—
296#365—731no—
297#365—731no—
298#365—731no—
299#365—731no—
300#365—731no—
301#365—731no—
302#365C2×D7311yescharacteristic, complemented
303#365C2×D7311yescharacteristic, complemented
304#364—731no—
305#364—731no—
306#364—731no—
307#364C2×D7311yescharacteristic, complemented
308#360—731no—
309#360—731no—
310#360—731no—
311#360—731no—
312#360—731no—
313#360—731no—
314#360C2×D7311yescharacteristic
315#360C2×D7311yescharacteristic
316C2×#41—1461no—
317#135—731no—
318#135—731no—
319#135—731no—
320#135—1461no—
321#135—1461no—
322#135C22×D7311yescharacteristic
323C2×Q16—731no—
324C2×SD16—1461no—
325C2×D8—731no—
326#131—21no—
327#131—2921no—
328#131—2921no—
329#131—1461no—
330#131—1461no—
331#131—1461no—
332C2×M16—2921no—
333C2×M16—2921no—
334C2×M16—731no—
335C2×M16—1461no—
336C22×C8—731no—
337C4⋊Q8—21no—
338C4⋊Q8—2921no—
339C4⋊Q8—1461no—
340#125—2921no—
341#125—1461no—
342#125—1461no—
343C22⋊Q8—2921no—
344C22⋊Q8—2921no—
345#117—731no—
346#117—1461no—
347C2×C4⋊C4—1461no—
348M32—731no—
349M32—731no—
350M32—1461no—
351M32—1461no—
352M32C22×D7311yescharacteristic
353M32C22×D7311yescharacteristic
354C2×C16—1461no—
355C2×C16—1461no—
356#108—731no—
357#108—1461no—
358#108C22×D7311yescharacteristic
359#107—1461no—
360#107—1461no—
361#107—1461no—
362#106—21no—
363#106—731no—
364#106—731no—
365#106—1461no—
366#106—1461no—
367#106C22×D7311yescharacteristic
368#106C22×D7311yescharacteristic
369#104—21no—
370#104—1461no—
371#104—1461no—
372#104—1461no—
373#97—731no—
374#97—731no—
375C4×C8—731no—
376C4×C8C22×D7311yescharacteristic
377#41—21no—
378#41—2921no—
379#41—2921no—
380#41—1461no—
381#41—1461no—
382#41—1461no—
383C2×Q8—41no—
384C2×Q8—2921no—
385C2×Q8—1461no—
386C2×D4—2921no—
387C2×D4—1461no—
388C22×C4—2921no—
389C22×C4—2921no—
390C22×C4—731no—
391Q16—731no—
392Q16—1461no—
393Q16Q8×D7311yescharacteristic
394SD16—21no—
395SD16—1461no—
396SD16—1461no—
397SD16—1461no—
398D8—731no—
399D8—1461no—
400D8Q8×D7311yescharacteristic
401M16—21no—
402M16—41no—
403M16—2921no—
404M16—2921no—
405M16—2921no—
406M16—2921no—
407M16—731no—
408M16—731no—
409M16—731no—
410M16—731no—
411M16—1461no—
412C2×C8—41no—
413C2×C8—2921no—
414C2×C8—2921no—
415C2×C8—2921no—
416C2×C8—2921no—
417C2×C8—731no—
418C2×C8—731no—
419C2×C8—731no—
420C2×C8—731no—
421C2×C8—731no—
422C2×C8—731no—
423C2×C8D4×D7311yescharacteristic
424C2×C8C23×D7311yescharacteristic
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—41no—
428C4⋊C4—2921no—
429C4⋊C4—2921no—
430C4⋊C4—2921no—
431C4⋊C4—2921no—
432C4⋊C4—2921no—
433C4⋊C4—2921no—
434C4⋊C4—1461no—
435C4⋊C4—1461no—
436C4⋊C4—1461no—
437C22⋊C4—2921no—
438C22⋊C4—1461no—
439C42—2921no—
440C42—731no—
441C42D4×D7311yescharacteristic
442C16—21no—
443C16—21no—
444C16—1461no—
445C16—1461no—
446C23—2921no—
447C23—731no—
448Q8—21no—
449Q8—161no—
450Q8—2921no—
451Q8—11681no—
452Q8—1461no—
453D4—21no—
454D4—41no—
455D4—2921no—
456D4—2921no—
457D4—2921no—
458D4—2921no—
459D4—1461no—
460C2×C4—21no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—2921no—
465C2×C4—2921no—
466C2×C4—2921no—
467C2×C4—2921no—
468C2×C4—2921no—
469C2×C4—2921no—
470C2×C4—731no—
471C2×C4—731no—
472C2×C4—731no—
473C2×C4—731no—
474C2×C4—731no—
475C2×C4—11681no—
476C2×C4—1461no—
477C2×C4C2×D4×D7311yescharacteristic
478C8—41no—
479C8—2921no—
480C8—731no—
481C8—731no—
482C8—731no—
483C8—731no—
484C8SD16×D7311yescharacteristic
485C8SD16×D7311yescharacteristic
486C8C2×Q8×D7311yescharacteristic
487C8#41×D7311yescharacteristic
488C22—41no—
489C22—2921no—
490C22—2921no—
491C22—731no—
492C22—731no—
493C22—11681no—
494C22AGL(1,ℤ/8)×D7311yescharacteristic
495C4—41no—
496C4—41no—
497C4—81no—
498C4—81no—
499C4—2921no—
500C4—2921no—
501C4—5841no—
502C4—5841no—
503C4—731no—
504C4—731no—
505C4C22⋊Q8×D7311yescharacteristic
506C4C2×SD16×D7311yescharacteristic
507C2—21no—
508C2—81no—
509C2—5841no—
510C2—1461no—
511C2—1461no—
512C2#476×D7311yescharacteristic
subgroupG/H
centerC2#476×D73
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2230C2
socleC2×C73C2×#476
prime radical#135×D73C22
hypercenter#2230D73
hypocenterC73#33638
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2230

Complements

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

NH ≅ G/Ntype
1. #22620456C2semidirect
3. #22612108C2semidirect
4. #22612107C2semidirect
5. #22612106C2semidirect
6. #22612105C2semidirect
7. #22612101C2semidirect
8. #22612100C2semidirect
9. #22612099C2semidirect
10. #22611702C2semidirect
11. #365×D73C2semidirect
12. #22611695C2semidirect
15. C73×#2230C2semidirect
23. #17988989C22semidirect
24. #17988987C22semidirect
25. #17988984C22semidirect
37. #17988411C22semidirect
38. #17988411C22semidirect
39. #17988410C22semidirect
40. #17988410C22semidirect
41. #17988409C22semidirect
42. #17988409C22semidirect
43. #17988408C22semidirect
44. #17988407C22semidirect
45. #17988407C22semidirect
52. #498×C73C22semidirect
54. #365×C73C22semidirect
55. #365×C73C22semidirect
56. #364×C73C22semidirect
60. #14556614Q8semidirect
64. #14556610Q8semidirect
65. Q16×D73Q8semidirect
67. D8×D73Q8semidirect
266. #2230D73semidirect
272. C73#33638semidirect
283. #498C2×D73semidirect
302. #365C2×D73semidirect
303. #365C2×D73semidirect
307. #364C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226204562
#226188432
#226121082
#226121072
#226121062
#226121052
#226121012
#226121002
#226120992
#226117022
#365×D732
#226116952
#226116652
#360×D732
C73×#22302
#3363873