Finite Group Databasebeta

← #33950325

Subgroups — #33950325

Order 18688.

2604 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22624702C211yesmaximal, characteristic
2#22624644C211yesmaximal, characteristic, complemented
3#22624643C211yesmaximal, characteristic, complemented
4#22618844C216yesmaximal, complemented
5#22618837C216yesmaximal, complemented
6#22618830C216yesmaximal, complemented
7C2×#17988706C216yesmaximal, complemented
8#22618540C211yesmaximal, characteristic, complemented
9#22618479C211yesmaximal, characteristic, complemented
10#22618359C211yesmaximal, characteristic, complemented
11C73×#2901C211yesmaximal, characteristic, complemented
12#17989423C2211yescharacteristic
13#17989402C414yescomplemented
14#17989398C414yescomplemented
15#17989395C414yescomplemented
16#17989390C414yescomplemented
17#17989385C2216yes—
18C2×#14556611C2216yes—
19#17989354—24no—
20#17989354C2211yescharacteristic
21#17989354C2216yescomplemented
22#17989353C2216yescomplemented
23#17989352—24no—
24C2×#14556599C2216yescomplemented
25C2×#14556598—24no—
26C2×#14556598C2211yescharacteristic
27C2×#14556598C2211yescharacteristic
28C2×#14556598C2216yescomplemented
29#131×D73—24no—
30#17988706C2214yescomplemented
31#17988706C22112yescomplemented
32#17988706C22112yescomplemented
33#17988706C22112yescomplemented
34#17988701C22124yescomplemented
35#17988635C2216yescomplemented
36C2×#14556526C2216yescomplemented
37#17988605C2216yescomplemented
38C2×#14556519C2216yescomplemented
39#17988543C2213yescomplemented
40#17988543C2213yescomplemented
41C4×#11924012C2213yescomplemented
42C2×#14556502C2213yescomplemented
43#585×C73C2211yescharacteristic
44#568×C73C2211yescharacteristic
45C4×#41×C73C2211yescharacteristic
46#421×C73C2216yescomplemented
47C2×#104×C73C2216yescomplemented
48#14556639—21no—
49#14556638C2×C414yes—
50#14556637C2×C414yes—
51#14556636—23no—
52C2×#11924028C2316yes—
53#41×D73—24no—
54#14556622C2×C4112yescomplemented
55#14556621C2×C4112yescomplemented
56#14556617C2×C4112yescomplemented
57#14556616C2×C4112yescomplemented
58#14556611C2×C4112yescomplemented
59#14556611C2×C4112yescomplemented
60#14556611C23112yes—
61#14556611C2314yes—
62C2×#11924018C2×C416yescomplemented
63C2×#11924016C2×C416yescomplemented
64C2×#11924015C2×C416yescomplemented
65C2×#11924013C2×C416yescomplemented
66#14556599—26no—
67#14556599—26no—
68#14556599C23112yes—
69#14556599C2314yes—
70#14556598—212no—
71#14556598—26no—
72#14556598—26no—
73#14556598—24no—
74#14556598C2×C414yes—
75#14556598C2×C414yes—
76#14556598D414yes—
77#14556598D414yes—
78#14556598C23112yes—
79#14556598C2314yes—
80#14556597—212no—
81C2×#11924012—212no—
82C2×#11924012—26no—
83C2×#11924012C2316yes—
84C2×#11924012C2313yes—
85C2×#11924012C2313yes—
86C2×#11924011—26no—
87M16×D73—212no—
88C22×#10596831—26no—
89C22×#10596831C2313yes—
90C22×#10596831C2313yes—
91C2×C8×D73—26no—
92#14556526C2314yes—
93#14556526C23112yes—
94#14556519C23112yes—
95#14556519C2314yes—
96#14556502C2311yescharacteristic
97#14556502C2313yes—
98#14556502C2313yes—
99#14556502C2313yes—
100C4×#10596831C2316yes—
1012−1+4×C73—21no—
1022−1+4×C73C2×C414yes—
103H(5,𝔽2)×C73—23no—
104H(5,𝔽2)×C73C2×C414yes—
105C2×#41×C73—24no—
106C2×#41×C73C2316yes—
107C2×#41×C73C2311yescharacteristic
108#131×C73—24no—
109C2×M16×C73C2316yes—
110C4×Q8×C73—21no—
111C4×D4×C73—23no—
112#117×C73C2313yes—
113C2×C42×C73C2313yes—
114#104×C73C2314yes—
115#104×C73C23112yes—
116#11924029—24no—
117#11924028—26no—
118#11924028—26no—
119#11924028C22×C4112yes—
120#11924028C2414yes—
121#11924027—212no—
122C2×Dic146C22×C416yes—
123Q8×D73—24no—
124C2×D292C22×C416yes—
125C2×C22⋊D73—26no—
126D4×D73—212no—
127C2×C4×D73—26no—
128#11924018C22×C414yes—
129#11924018C22×C4112yes—
130#11924016C22×C414yes—
131#11924016C22×C4112yes—
132#11924015C22×C414yes—
133#11924015C22×C4112yes—
134#11924013C22×C414yes—
135#11924013C22×C4112yes—
136#11924012—26no—
137#11924012—24no—
138#11924012—212no—
139#11924012C22×C4112yes—
140#11924012C2×D4112yes—
141#11924012C2414yes—
142#11924011—212no—
143#11924011—24no—
144C2×#10596831—26no—
145C2×#10596831—24no—
146C2×#10596831—212no—
147C2×#10596831C22×C416yes—
148C2×#10596831C22×C416yes—
149C2×#10596831C2×D416yes—
150C2×#10596831C2×D416yes—
151C2×#10596831C2413yes—
152C2×#10596831C2411yescharacteristic
153C2×#10596831C2413yes—
154C2×#10596831C2411yescharacteristic
155C8×D73—24no—
156C8×D73—212no—
157#41×C73—24no—
158#41×C73—212no—
159#41×C73—26no—
160#41×C73—26no—
161#41×C73C22×C4112yes—
162#41×C73C22×C414yes—
163#41×C73C2×D414yes—
164#41×C73C2414yes—
165C2×Q8×C73—24no—
166C2×Q8×C73C22×C416yes—
167C2×Q8×C73#4111yescharacteristic
168C2×D4×C73—26no—
169C2×D4×C73—212no—
170C2×D4×C73C22×C416yes—
171C2×D4×C73#4113yes—
172C22×C4×C73—26no—
173C22×C4×C73—23no—
174C22×C4×C73C2413yes—
175M16×C73—26no—
176M16×C73C2414yes—
177C2×C8×C73—26no—
178C4⋊C4×C73—23no—
179C22⋊C4×C73—23no—
180C42×C73—23no—
181C42×C73C2411yescharacteristic
182C42×C73C2413yes—
183C23×C73—26no—
184C23×C73C2×#4113yes—
185C22×D73—26no—
186Q8×C73—26no—
187Q8×C73C4×D414yes—
188Q8×C73C23×C414yes—
189D4×C73—212no—
190D4×C73—24no—
191D4×C73—26no—
192D4×C73C4×D4112yes—
193D4×C73C23×C414yes—
194C2×C4×C73—26no—
195C2×C4×C73—212no—
196C2×C4×C73—24no—
197C2×C4×C73—21no—
198C2×C4×C73—23no—
199C2×C4×C73—83no—
200C2×C4×C73C23×C416yes—
201C2×C4×C73C22×D416yes—
202C2×C4×C73C2×#4113yes—
203C2×C4×C73C2511yescharacteristic
204C22⋊D73—212no—
205C22⋊D73—24no—
206C2×Dic73—26no—
207D292—26no—
208D292C23×C414yes—
209C4×D73—212no—
210C4×D73—24no—
211Dic146—26no—
212Dic146C23×C414yes—
213C8×C73—24no—
214#10596831—24no—
215#10596831C23×C411yescharacteristic
216#10596831C23×C411yescharacteristic
217#10596831C23×C416yes—
218#10596831C22×D411yescharacteristic
219#10596831C22×D416yes—
220#10596831C22×D411yescharacteristic
221C22×C73—24no—
222C22×C73—43no—
223C22×C73—86no—
224C22×C73C2×C4×D416yes—
225C22×C73C22×#4111yescharacteristic
226C2×D73—24no—
227C2×D73—86no—
228C4×C73—24no—
229C4×C73—43no—
230C4×C73—41no—
231C4×C73C2×C4×D416yes—
232C4×C73C24×C411yescharacteristic
233C4×C73C23×D411yescharacteristic
234Dic73—24no—
235#60184—731nomaximal, Sylow
236C2×C73—21no—
237C2×C73—26no—
238C2×C73—44no—
239C2×C73C22×C4×D411yescharacteristic
240D73—44no—
241#3584—731no—
242#3573—731no—
243#3573—731no—
244#2957—736no—
245#2956—736no—
246#2955—736no—
247C2×#444—736no—
248#2916—731no—
249#2901—731no—
250#2901D7311yescharacteristic, complemented
251#2875—731no—
252C73#6018411yescharacteristic, complemented, Sylow
253#585—731no—
254#585C2×D7311yescharacteristic
255#578—734no—
256#577—734no—
257#577—734no—
258#576—734no—
259#575—736no—
260C2×#135—736no—
261#568—731no—
262#568—736no—
263#568—736no—
264#568—1464no—
265#568—1464no—
266#568C2×D7311yescharacteristic
267C2×#131—731no—
268C2×#131—731no—
269C2×#131—736no—
270C2×#131—736no—
271C2×#131—1464no—
272C2×#131—1464no—
273C4×#41C2×D7311yescharacteristic
274#444—734no—
275#444—7312no—
276#444—7312no—
277#444—7312no—
278#443—7324no—
279#430—736no—
280C2×#108—736no—
281#421—736no—
282#421C2×D7316yescomplemented
283C2×#104—736no—
284C2×#104C2×D7316yescomplemented
285#405—733no—
286#405—733no—
287C4×M16—733no—
288C2×#97—733no—
2892−1+4—21no—
2902−1+4—734no—
2912−1+4—1461no—
2922−1+4C4×D7314yes—
293H(5,𝔽2)—23no—
294H(5,𝔽2)—734no—
295H(5,𝔽2)—1463no—
296H(5,𝔽2)C4×D7314yes—
297C2×#41—24no—
298C2×#41—736no—
299C2×#41—1464no—
300C2×#41C22×D7316yes—
301C2×#41C22×D7311yescharacteristic
302#137—7312no—
303#137—7312no—
304AGL(1,ℤ/8)—7312no—
305AGL(1,ℤ/8)—7312no—
306#135—7312no—
307#135—7312no—
308#135—734no—
309#135—7312no—
310C2×Q16—736no—
311C2×SD16—736no—
312C2×SD16—736no—
313C2×D8—736no—
314#131—24no—
315#131—734no—
316#131—734no—
317#131—734no—
318#131—734no—
319#131—734no—
320#131—7312no—
321#131—734no—
322#131—7312no—
323#131—14612no—
324#131—1466no—
325#131—1466no—
326#131—1466no—
327#131—1466no—
328#131—14612no—
329#131—1464no—
330C2×M16—733no—
331C2×M16—733no—
332C2×M16—736no—
333C2×M16—1466no—
334C2×M16—1466no—
335C2×M16—14612no—
336C2×M16—14612no—
337C2×M16C22×D7316yes—
338C22×C8—733no—
339C22×C8—733no—
340C22×C8—1466no—
341C22×C8—1466no—
342C4×Q8—21no—
343C4×D4—23no—
344#117C22×D7313yes—
345C2×C42C22×D7313yes—
346#108—734no—
347#108—7312no—
348#104—734no—
349#104—7312no—
350#104C22×D7314yes—
351#104C22×D73112yes—
352#97—731no—
353#97—733no—
354#97—733no—
355#97—733no—
356C4×C8—736no—
357#41—24no—
358#41—212no—
359#41—26no—
360#41—26no—
361#41—7312no—
362#41—734no—
363#41—1464no—
364#41—14612no—
365#41—1466no—
366#41—1466no—
367#41C2×C4×D73112yes—
368#41C2×C4×D7314yes—
369#41D4×D7314yes—
370#41C23×D7314yes—
371C2×Q8—24no—
372C2×Q8—736no—
373C2×Q8—1464no—
374C2×Q8C2×C4×D7316yes—
375C2×Q8#1192402711yescharacteristic
376C2×D4—26no—
377C2×D4—212no—
378C2×D4—736no—
379C2×D4—1466no—
380C2×D4—14612no—
381C2×D4C2×C4×D7316yes—
382C2×D4#1192402713yes—
383C22×C4—26no—
384C22×C4—23no—
385C22×C4—1466no—
386C22×C4C23×D7313yes—
387Q16—734no—
388Q16—7312no—
389SD16—734no—
390SD16—7312no—
391SD16—734no—
392SD16—7312no—
393D8—734no—
394D8—7312no—
395M16—26no—
396M16—7312no—
397M16—7312no—
398M16—734no—
399M16—1464no—
400M16—14612no—
401M16—1464no—
402M16—14612no—
403M16—1466no—
404M16C23×D7314yes—
405C2×C8—26no—
406C2×C8—736no—
407C2×C8—736no—
408C2×C8—736no—
409C2×C8—736no—
410C2×C8—733no—
411C2×C8—733no—
412C2×C8—731no—
413C2×C8—731no—
414C2×C8—1464no—
415C2×C8—14612no—
416C2×C8—1464no—
417C2×C8—14612no—
418C2×C8—1466no—
419C4⋊C4—23no—
420C22⋊C4—23no—
421C42—23no—
422C42C23×D7313yes—
423C42C23×D7311yescharacteristic
424C23—26no—
425C23—1466no—
426C23C2×#1192402713yes—
427Q8—26no—
428Q8—734no—
429Q8—1466no—
430Q8#1455655114yes—
431Q8C22×C4×D7314yes—
432D4—24no—
433D4—212no—
434D4—26no—
435D4—734no—
436D4—1464no—
437D4—14612no—
438D4—1466no—
439D4#14556551112yes—
440D4C22×C4×D7314yes—
441C2×C4—26no—
442C2×C4—24no—
443C2×C4—212no—
444C2×C4—21no—
445C2×C4—23no—
446C2×C4—83no—
447C2×C4—1466no—
448C2×C4—1464no—
449C2×C4—14612no—
450C2×C4C22×C4×D7316yes—
451C2×C4C2×D4×D7316yes—
452C2×C4C2×#1192402713yes—
453C2×C4C24×D7311yescharacteristic
454C8—24no—
455C8—731no—
456C8—731no—
457C8—736no—
458C8—731no—
459C8—731no—
460C8—736no—
461C8—1464no—
462C22—24no—
463C22—43no—
464C22—86no—
465C22—5846no—
466C22—1464no—
467C22C2×#1455655116yes—
468C22C22×#1192402711yescharacteristic
469C4—24no—
470C4—41no—
471C4—43no—
472C4—1464no—
473C4C2×#1455655116yes—
474C4C23×C4×D7311yescharacteristic
475C4C22×D4×D7311yescharacteristic
476C2—21no—
477C2—26no—
478C2—44no—
479C2—2924no—
480C2C22×#1455655111yescharacteristic
subgroupG/H
centerC4C22×D4×D73
commutatorC4×C73C24×C4
FrattiniC2×C4C24×D73
FittingC73×#2901C2
socleC2×C73C22×C4×D4
prime radical#22624702C2
cosocleC2×C4×C73C25
hypercenter#2901D73
hypocenterC73#60184
LCSG ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C2×#41 ◁ #2901

Complements

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

NH ≅ G/Ntype
2. #22624644C2semidirect
3. #22624643C2semidirect
4. #22618844C2semidirect
5. #22618837C2semidirect
6. #22618830C2semidirect
7. C2×#17988706C2semidirect
8. #22618540C2semidirect
9. #22618479C2semidirect
10. #22618359C2semidirect
11. C73×#2901C2semidirect
13. #17989402C4semidirect
14. #17989398C4semidirect
15. #17989395C4semidirect
16. #17989390C4semidirect
21. #17989354C22semidirect
22. #17989353C22semidirect
24. C2×#14556599C22semidirect
28. C2×#14556598C22semidirect
30. #17988706C22semidirect
31. #17988706C22semidirect
32. #17988706C22semidirect
33. #17988706C22semidirect
34. #17988701C22semidirect
35. #17988635C22semidirect
36. C2×#14556526C22semidirect
37. #17988605C22semidirect
38. C2×#14556519C22semidirect
39. #17988543C22semidirect
40. #17988543C22semidirect
41. C4×#11924012C22semidirect
42. C2×#14556502C22semidirect
46. #421×C73C22semidirect
47. C2×#104×C73C22semidirect
54. #14556622C2×C4semidirect
55. #14556621C2×C4semidirect
56. #14556617C2×C4semidirect
57. #14556616C2×C4semidirect
58. #14556611C2×C4semidirect
59. #14556611C2×C4semidirect
62. C2×#11924018C2×C4semidirect
63. C2×#11924016C2×C4semidirect
64. C2×#11924015C2×C4semidirect
65. C2×#11924013C2×C4semidirect
250. #2901D73semidirect
252. C73#60184semidirect
282. #421C2×D73semidirect
284. C2×#104C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226247022
#226246442
#226246432
#226188442
#226188372
#226188302
C2×#179887062
#226185402
#226184792
#226183592
C73×#29012
#6018473