Finite Group Databasebeta

← #2230×D73 #33240710

Subgroups — #2230×D73 #33240710

Order 18688.

558 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#498×D73C211yesmaximal, characteristic, complemented
2#443×D73C211yesmaximal, characteristic
3#22612103C211yesmaximal, characteristic, complemented
4#22612102C211yesmaximal, characteristic, complemented
5#22612101C212yesmaximal, complemented
6#22612100C212yesmaximal, complemented
7#22612099C211yesmaximal, characteristic, complemented
8#365×D73C212yesmaximal, complemented
9#364×D73C211yesmaximal, characteristic, complemented
10#360×D73C212yesmaximal
11C73×#2230C211yesmaximal, characteristic, complemented
12#135×D73C2211yescharacteristic
13#131×D73—21no—
14C4⋊Q8×D73—21no—
15M32×D73C2212yes—
16#17988985C2211yescharacteristic, complemented
17#17988984—21no—
18#17988984C2212yescomplemented
19#17988983—21no—
20#17988705C2211yescharacteristic
21#17988704C2211yescharacteristic
22#17988703C2211yescharacteristic
23#17988702—21no—
24#17988701—21no—
25#108×D73C2211yescharacteristic
26#106×D73—21no—
27#106×D73C2212yes—
28#104×D73—21no—
29C4×C8×D73C2211yescharacteristic
30#17988411C2212yescomplemented
31#17988410C2212yescomplemented
32#17988409C2212yescomplemented
33#17988408C2211yescharacteristic, complemented
34#17988407C2212yescomplemented
35#17988398C2212yes—
36#17988397C2212yes—
37#17988396C2212yes—
38#498×C73C2211yescharacteristic, complemented
39#443×C73C2211yescharacteristic
40#365×C73C2212yescomplemented
41#364×C73C2211yescharacteristic, complemented
42#360×C73C2212yes—
43#41×D73—21no—
44C2×Q8×D73—41no—
45#14556614Q811yescharacteristic, complemented
46#14556613C2311yescharacteristic
47#14556612—21no—
48#14556611—21no—
49#14556610Q811yescharacteristic, complemented
50Q16×D73Q811yescharacteristic, complemented
51SD16×D73—21no—
52D8×D73Q811yescharacteristic, complemented
53#14556599—21no—
54#14556599—41no—
55#14556598—21no—
56#14556597—21no—
57#14556597—41no—
58M16×D73—21no—
59M16×D73—41no—
60C2×C8×D73—41no—
61C2×C8×D73D411yescharacteristic
62C2×C8×D73C2311yescharacteristic
63#14556591—21no—
64#14556590—22no—
65#14556590—41no—
66#14556589—41no—
67C4⋊C4×D73—22no—
68C4⋊C4×D73—41no—
69C42×D73D411yescharacteristic
70#14556531C2312yes—
71#14556530—22no—
72C16×D73—22no—
73#14556527C2311yescharacteristic
74#14556526—21no—
75#14556523—21no—
76#14556523C2312yes—
77#14556522—22no—
78#14556522—22no—
79#14556519—21no—
80#14556518—21no—
81#14556517—21no—
82C4×#10596831D411yescharacteristic
83C8×Dic73D411yescharacteristic
84C8×Dic73C2311yescharacteristic
85#135×C73C2311yescharacteristic
86#131×C73—21no—
87C4⋊Q8×C73—21no—
88M32×C73C2312yes—
89#108×C73C2311yescharacteristic
90#106×C73—21no—
91#106×C73C2312yes—
92#104×C73—21no—
93C4×C8×C73C2311yescharacteristic
94#11924029—21no—
95#11924028—21no—
96#11924028—41no—
97#11924027—21no—
98#11924027—41no—
99C2×Dic146—42no—
100C2×Dic146—41no—
101Q8×D73—21no—
102Q8×D73—84no—
103D4×D73—21no—
104D4×D73—41no—
105C2×C4×D73—21no—
106C2×C4×D73—41no—
107C2×C4×D73—42no—
108C2×C4×D73C2×D411yescharacteristic
109Dic292C2×Q811yescharacteristic, complemented
110#11924018—21no—
111#11924017—21no—
112#11924016—21no—
113#11924015—21no—
114D584C2×Q811yescharacteristic, complemented
115#11924013—21no—
116#11924012—21no—
117#11924012—41no—
118#11924011—41no—
119#11924011—41no—
120#11924011—41no—
121C2×#10596831—41no—
122C2×#10596831C2×D411yescharacteristic
123C2×#10596831C2×D411yescharacteristic
124C8×D73—41no—
125C8×D73—41no—
126C8×D73—41no—
127C8×D73SD1611yescharacteristic
128C8×D73SD1611yescharacteristic
129C8×D73SD1611yescharacteristic
130C8×D73SD1611yescharacteristic
131C8×D73C2×Q811yescharacteristic
132C8×D73C2×Q811yescharacteristic
133C8×D73#4111yescharacteristic
134C8×D73#4111yescharacteristic
135C4⋊Dic73—22no—
136C4⋊Dic73—41no—
137#11924007—42no—
138#11924007—42no—
139C4×Dic73—21no—
140C4×Dic73C2×D411yescharacteristic
141#11924003—22no—
142#41×C73—21no—
143C2×Q8×C73—41no—
144Q16×C73C2×Q811yescharacteristic, complemented
145SD16×C73—21no—
146D8×C73C2×Q811yescharacteristic, complemented
147M16×C73—21no—
148M16×C73—41no—
149C2×C8×C73—41no—
150C2×C8×C73C2×D411yescharacteristic
151C2×C8×C73C2411yescharacteristic
152C4⋊C4×C73—22no—
153C4⋊C4×C73—41no—
154C42×C73C2×D411yescharacteristic
155C16×C73—22no—
156C22×D73—41no—
157C22×D73AGL(1,ℤ/8)11yescharacteristic
158Q8×C73—21no—
159Q8×C73—82no—
160D4×C73—21no—
161D4×C73—41no—
162C2×C4×C73—21no—
163C2×C4×C73—41no—
164C2×C4×C73—42no—
165C2×C4×C73C22×D411yescharacteristic
166C22⋊D73—41no—
167C22⋊D73—41no—
168C22⋊D73—41no—
169C2×Dic73—21no—
170C2×Dic73—41no—
171C2×Dic73—42no—
172C2×Dic73AGL(1,ℤ/8)11yescharacteristic
173D292—21no—
174D292—41no—
175C4×D73—24no—
176C4×D73—41no—
177C4×D73—41no—
178C4×D73—41no—
179C4×D73—84no—
180C4×D73C22⋊Q811yescharacteristic
181C4×D73C22⋊Q811yescharacteristic
182C4×D73C2×SD1611yescharacteristic
183C4×D73C2×SD1611yescharacteristic
184Dic146—21no—
185Dic146—41no—
186Dic146—84no—
187Dic146—82no—
188C8×C73—41no—
189C8×C73C2×SD1611yescharacteristic
190C8×C73C2×SD1611yescharacteristic
191C8×C73C22×Q811yescharacteristic
192C8×C73C2×#4111yescharacteristic
193#10596831—41no—
194#10596831C22⋊Q811yescharacteristic
195#10596831C22⋊Q811yescharacteristic
196#10596831C2×SD1611yescharacteristic
197#10596831C2×SD1611yescharacteristic
198C22×C73—41no—
199C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
200C2×D73—22no—
201C2×D73—41no—
202C2×D73—82no—
203C2×D73#47611yescharacteristic
204C2×D73#47611yescharacteristic
205C4×C73—22no—
206C4×C73—41no—
207C4×C73—82no—
208C4×C73C2×C22⋊Q811yescharacteristic
209C4×C73C22×SD1611yescharacteristic
210Dic73—22no—
211Dic73—41no—
212Dic73—82no—
213Dic73#47611yescharacteristic
214Dic73#47611yescharacteristic
215C2×#2230—731nomaximal, Sylow
216C2×C73—21no—
217C2×C73—81no—
218C2×C73C2×#47611yescharacteristic
219D73—21no—
220D73—81no—
221D73#223012yesdirect product factor
222C2×#498—731no—
223C2×#443—731no—
224#2230—731no—
225#2230—731no—
226#2230—732no—
227#2230—731no—
228#2230—732no—
229#2230D7311yescharacteristic, direct product factor
230C2×#365—732no—
231C2×#364—731no—
232C2×#360—732no—
233C73C2×#223011yescharacteristic, complemented, Sylow
234C2×#135—731no—
235C2×#131—1461no—
236C2×C4⋊Q8—1461no—
237C2×M32—732no—
238#498—732no—
239#498—731no—
240#498—1461no—
241#498—1461no—
242#498C2×D7311yescharacteristic, complemented
243#443—731no—
244#443—731no—
245#443—731no—
246#443—1461no—
247#443—1461no—
248#443C2×D7311yescharacteristic
249C2×#108—731no—
250C2×#106—732no—
251C2×#106—1461no—
252C2×#104—1461no—
253C2×C4×C8—731no—
254#365—732no—
255#365—732no—
256#365—732no—
257#365C2×D7312yescomplemented
258#364—732no—
259#364—731no—
260#364C2×D7311yescharacteristic, complemented
261#360—732no—
262#360—732no—
263#360—732no—
264#360C2×D7312yes—
265C2×#41—1461no—
266C22×Q8—2921no—
267#135—731no—
268#135—731no—
269#135—731no—
270#135—1461no—
271#135—1461no—
272#135C22×D7311yescharacteristic
273C2×Q16—731no—
274C2×SD16—1461no—
275C2×D8—731no—
276#131—21no—
277#131—2921no—
278#131—2921no—
279#131—1461no—
280#131—1461no—
281#131—1461no—
282C2×M16—2921no—
283C2×M16—1461no—
284C22×C8—2921no—
285C22×C8—731no—
286C22×C8—731no—
287C4⋊Q8—21no—
288C4⋊Q8—2921no—
289C4⋊Q8—2921no—
290C4⋊Q8—1462no—
291C4⋊Q8—1461no—
292C2×C4⋊C4—2921no—
293C2×C4⋊C4—1462no—
294C2×C42—731no—
295M32—732no—
296M32—1462no—
297M32C22×D7312yes—
298C2×C16—1462no—
299#108—731no—
300#108—1461no—
301#108C22×D7311yescharacteristic
302#106—21no—
303#106—732no—
304#106—1462no—
305#106—1462no—
306#106—1461no—
307#106C22×D7312yes—
308#104—21no—
309#104—1461no—
310#104—1461no—
311#104—1461no—
312C4×C8—731no—
313C4×C8—731no—
314C4×C8—731no—
315C4×C8C22×D7311yescharacteristic
316#41—21no—
317#41—2921no—
318#41—2921no—
319#41—1461no—
320#41—1461no—
321#41—1461no—
322C2×Q8—41no—
323C2×Q8—2921no—
324C2×Q8—2922no—
325C2×Q8—5844no—
326C2×Q8—1461no—
327C2×D4—2921no—
328C2×D4—1461no—
329C22×C4—2921no—
330C22×C4—2922no—
331C22×C4—731no—
332C22×C4—1461no—
333Q16—731no—
334Q16—1461no—
335Q16Q8×D7311yescharacteristic, complemented
336SD16—21no—
337SD16—1461no—
338SD16—1461no—
339SD16—1461no—
340D8—731no—
341D8—1461no—
342D8Q8×D7311yescharacteristic, complemented
343M16—21no—
344M16—41no—
345M16—2921no—
346M16—2921no—
347M16—2921no—
348M16—2921no—
349M16—1461no—
350C2×C8—41no—
351C2×C8—2921no—
352C2×C8—2921no—
353C2×C8—2921no—
354C2×C8—2921no—
355C2×C8—731no—
356C2×C8—731no—
357C2×C8—731no—
358C2×C8—731no—
359C2×C8—731no—
360C2×C8—731no—
361C2×C8—731no—
362C2×C8—731no—
363C2×C8—731no—
364C2×C8—731no—
365C2×C8D4×D7311yescharacteristic
366C2×C8C23×D7311yescharacteristic
367C4⋊C4—22no—
368C4⋊C4—41no—
369C4⋊C4—2921no—
370C4⋊C4—2922no—
371C4⋊C4—2922no—
372C4⋊C4—1462no—
373C42—731no—
374C42—1461no—
375C42D4×D7311yescharacteristic
376C16—22no—
377C16—1462no—
378C23—2921no—
379C23—731no—
380Q8—21no—
381Q8—82no—
382Q8—2921no—
383Q8—5842no—
384Q8—5844no—
385Q8—1461no—
386D4—21no—
387D4—41no—
388D4—2921no—
389D4—2921no—
390D4—2921no—
391D4—2921no—
392D4—1461no—
393C2×C4—21no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—2921no—
397C2×C4—2921no—
398C2×C4—2921no—
399C2×C4—2921no—
400C2×C4—2922no—
401C2×C4—5844no—
402C2×C4—731no—
403C2×C4—731no—
404C2×C4—731no—
405C2×C4—731no—
406C2×C4—731no—
407C2×C4—1464no—
408C2×C4—1461no—
409C2×C4C2×D4×D7311yescharacteristic
410C8—41no—
411C8—2921no—
412C8—731no—
413C8—731no—
414C8—731no—
415C8—731no—
416C8SD16×D7311yescharacteristic
417C8SD16×D7311yescharacteristic
418C8C2×Q8×D7311yescharacteristic
419C8#41×D7311yescharacteristic
420C22—41no—
421C22—2921no—
422C22—5842no—
423C22—731no—
424C22—731no—
425C22—1462no—
426C22AGL(1,ℤ/8)×D7311yescharacteristic
427C4—22no—
428C4—41no—
429C4—82no—
430C4—2921no—
431C4—5842no—
432C4—731no—
433C4—731no—
434C4—1462no—
435C4C22⋊Q8×D7311yescharacteristic
436C4C2×SD16×D7311yescharacteristic
437C2—21no—
438C2—81no—
439C2—5841no—
440C2—732no—
441C2—1461no—
442C2#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
hypocenterC73C2×#2230
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. #498×D73C2semidirect
3. #22612103C2semidirect
4. #22612102C2semidirect
5. #22612101C2semidirect
6. #22612100C2semidirect
7. #22612099C2semidirect
8. #365×D73C2semidirect
9. #364×D73C2semidirect
11. C73×#2230C2semidirect
16. #17988985C22semidirect
18. #17988984C22semidirect
30. #17988411C22semidirect
31. #17988410C22semidirect
32. #17988409C22semidirect
33. #17988408C22semidirect
34. #17988407C22semidirect
38. #498×C73C22semidirect
40. #365×C73C22semidirect
41. #364×C73C22semidirect
45. #14556614Q8semidirect
49. #14556610Q8semidirect
50. Q16×D73Q8semidirect
52. D8×D73Q8semidirect
109. Dic292C2×Q8semidirect
114. D584C2×Q8semidirect
144. Q16×C73C2×Q8semidirect
146. D8×C73C2×Q8semidirect
221. D73#2230direct (N × H)
229. #2230D73direct (N × H)
233. C73C2×#2230semidirect
242. #498C2×D73semidirect
257. #365C2×D73semidirect
260. #364C2×D73semidirect
335. Q16Q8×D73semidirect
342. D8Q8×D73semidirect

Maximal subgroups

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

subgroupindex
#498×D732
#443×D732
#226121032
#226121022
#226121012
#226121002
#226120992
#365×D732
#364×D732
#360×D732
C73×#22302
C2×#223073