Finite Group Databasebeta

← #33950327

Subgroups — #33950327

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#22624642C211yesmaximal, characteristic, complemented
3#22618846C216yesmaximal, complemented
4#22618838C216yesmaximal, complemented
5#22618832C216yesmaximal, complemented
6C2×#17988707C216yesmaximal, complemented
7#22618540C211yesmaximal, characteristic, complemented
8#22618478C211yesmaximal, characteristic, complemented
9#22618476C211yesmaximal, characteristic, complemented
10#22618357C211yesmaximal, characteristic, complemented
11C73×#3573C211yesmaximal, characteristic, complemented
12#17989423C2211yescharacteristic
13#17989402C414yescomplemented
14#17989398C414yescomplemented
15#17989395C414yescomplemented
16#17989390C414yescomplemented
17#17989385C2216yes—
18C2×#14556611C2216yes—
19#17989354C2211yescharacteristic
20#17989353C2216yescomplemented
21#17989352—24no—
22C2×#14556598C2211yescharacteristic
23C2×#14556597C2216yescomplemented
24#131×D73—24no—
25#41×Dic73C2211yescharacteristic
26#17988707C2214yescomplemented
27#17988707C22112yescomplemented
28#17988707C22112yescomplemented
29#17988707C22112yescomplemented
30#17988702C22124yescomplemented
31#17988635C2216yescomplemented
32C2×#14556526C2216yescomplemented
33#17988604C2216yescomplemented
34#17988603C2216yescomplemented
35C2×#14556518C2216yescomplemented
36C2×#14556517C2216yescomplemented
37#17988545C2213yescomplemented
38#17988542C2213yescomplemented
39M16×Dic73C2213yescomplemented
40C2×#14556501C2213yescomplemented
41#585×C73C2211yescharacteristic
42#568×C73—24no—
43#568×C73C2211yescharacteristic
44#568×C73C2216yescomplemented
45C2×#131×C73—24no—
46C2×#131×C73C2211yescharacteristic
47C2×#131×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—212no—
67#14556598—24no—
68#14556598C2×C414yes—
69#14556598C2×C414yes—
70#14556597—26no—
71#14556597—26no—
72#14556597C23112yes—
73#14556597C2314yes—
74C2×#11924012C2316yes—
75C2×#11924012C2313yes—
76C2×#11924011—26no—
77M16×D73—212no—
78C22×#10596831C2313yes—
79C2×C8×D73—26no—
80Q8×Dic73—21no—
81D4×Dic73—23no—
82#14556550C2313yes—
83C2×C4×Dic73C2313yes—
84#14556526C2314yes—
85#14556526C23112yes—
86#14556518C23112yes—
87#14556518C2314yes—
88#14556517C23112yes—
89#14556517C2314yes—
90#14556501C2311yescharacteristic
91#14556501C2313yes—
92#14556501C2313yes—
93#14556501C2313yes—
94C8×Dic73C2316yes—
952−1+4×C73—21no—
962−1+4×C73C2×C414yes—
97H(5,𝔽2)×C73—23no—
98H(5,𝔽2)×C73C2×C414yes—
99C2×#41×C73—24no—
100C2×#41×C73C2316yes—
101C2×#41×C73C2311yescharacteristic
102#131×C73—24no—
103#131×C73—212no—
104#131×C73—26no—
105#131×C73—26no—
106#131×C73D414yes—
107#131×C73D414yes—
108#131×C73C23112yes—
109#131×C73C2314yes—
110C2×M16×C73—212no—
111C2×M16×C73—26no—
112C2×M16×C73C2316yes—
113C2×M16×C73C2313yes—
114C22×C8×C73—26no—
115C22×C8×C73C2313yes—
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—
127C22×Dic73—23no—
128C2×C4×D73—26no—
129#11924018C22×C414yes—
130#11924018C22×C4112yes—
131#11924016C22×C414yes—
132#11924016C22×C4112yes—
133#11924015C22×C414yes—
134#11924015C22×C4112yes—
135#11924013C22×C414yes—
136#11924013C22×C4112yes—
137#11924012—26no—
138#11924012C22×C4112yes—
139#11924012C2414yes—
140#11924011—212no—
141#11924011—24no—
142C2×#10596831—26no—
143C2×#10596831C22×C416yes—
144C2×#10596831C22×C416yes—
145C2×#10596831C2413yes—
146C2×#10596831C2411yescharacteristic
147C8×D73—24no—
148C8×D73—212no—
149C4⋊Dic73—23no—
150C22⋊Dic73—23no—
151C4×Dic73—23no—
152C4×Dic73C2413yes—
153C4×Dic73C2411yescharacteristic
154#41×C73—24no—
155#41×C73—212no—
156#41×C73—26no—
157#41×C73—26no—
158#41×C73C22×C4112yes—
159#41×C73C22×C414yes—
160#41×C73C2×D414yes—
161#41×C73C2414yes—
162C2×Q8×C73—24no—
163C2×Q8×C73C22×C416yes—
164C2×Q8×C73#4111yescharacteristic
165C2×D4×C73—26no—
166C2×D4×C73—212no—
167C2×D4×C73C22×C416yes—
168C2×D4×C73#4113yes—
169C22×C4×C73—26no—
170C22×C4×C73C2413yes—
171M16×C73—26no—
172M16×C73—212no—
173M16×C73—24no—
174M16×C73C2×D4112yes—
175M16×C73C2414yes—
176C2×C8×C73—26no—
177C2×C8×C73—24no—
178C2×C8×C73—212no—
179C2×C8×C73C2×D416yes—
180C2×C8×C73C2×D416yes—
181C2×C8×C73C2411yescharacteristic
182C2×C8×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×C73C23×C416yes—
198C2×C4×C73C22×D416yes—
199C2×C4×C73C2×#4113yes—
200C2×C4×C73C2511yescharacteristic
201C22⋊D73—212no—
202C22⋊D73—24no—
203C2×Dic73—26no—
204C2×Dic73—21no—
205C2×Dic73—23no—
206C2×Dic73—83no—
207D292—26no—
208D292C23×C414yes—
209C4×D73—212no—
210C4×D73—24no—
211Dic146—26no—
212Dic146C23×C414yes—
213C8×C73—24no—
214C8×C73C22×D411yescharacteristic
215C8×C73C22×D416yes—
216C8×C73C22×D411yescharacteristic
217#10596831—24no—
218#10596831C23×C411yescharacteristic
219#10596831C23×C411yescharacteristic
220#10596831C23×C416yes—
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×C73C2×C4×D416yes—
230C4×C73C24×C411yescharacteristic
231C4×C73C23×D411yescharacteristic
232Dic73—24no—
233Dic73—43no—
234Dic73—41no—
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#3573D7311yescharacteristic, complemented
244#2957—736no—
245#2956—736no—
246#2955—736no—
247C2×#444—736no—
248#2916—731no—
249#2901—731no—
250#2901—731no—
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—24no—
262#568—731no—
263#568—736no—
264#568—1464no—
265#568C2×D7311yescharacteristic
266#568C2×D7316yescomplemented
267C2×#131—24no—
268C2×#131—731no—
269C2×#131—736no—
270C2×#131—1464no—
271C2×#131C2×D7311yescharacteristic
272C2×#131C2×D7316yescomplemented
273C4×#41—731no—
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#421—736no—
283C2×#104—736no—
284C2×#104—736no—
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—212no—
316#131—26no—
317#131—26no—
318#131—734no—
319#131—734no—
320#131—734no—
321#131—7312no—
322#131—1464no—
323#131—14612no—
324#131—1466no—
325#131—1466no—
326#131C22⋊D7314yes—
327#131C22⋊D7314yes—
328#131C22×D7314yes—
329#131C22×D73112yes—
330C2×M16—212no—
331C2×M16—26no—
332C2×M16—733no—
333C2×M16—736no—
334C2×M16—14612no—
335C2×M16—1466no—
336C2×M16C22×D7316yes—
337C2×M16C22×D7313yes—
338C22×C8—26no—
339C22×C8—733no—
340C22×C8—1466no—
341C22×C8C22×D7313yes—
342C4×Q8—1461no—
343C4×D4—1463no—
344#117—733no—
345C2×C42—733no—
346#108—734no—
347#108—7312no—
348#104—734no—
349#104—7312no—
350#104—734no—
351#104—7312no—
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#41C2×C22⋊D7314yes—
370#41C23×D7314yes—
371C2×Q8—24no—
372C2×Q8—736no—
373C2×Q8—1464no—
374C2×Q8C2×C4×D7316yes—
375C2×Q8#1192402811yescharacteristic
376C2×D4—26no—
377C2×D4—212no—
378C2×D4—736no—
379C2×D4—1466no—
380C2×D4—14612no—
381C2×D4C2×C4×D7316yes—
382C2×D4#1192402813yes—
383C22×C4—26no—
384C22×C4—1463no—
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—212no—
397M16—24no—
398M16—7312no—
399M16—734no—
400M16—1466no—
401M16—1464no—
402M16—14612no—
403M16C2×C22⋊D73112yes—
404M16C23×D7314yes—
405C2×C8—26no—
406C2×C8—24no—
407C2×C8—212no—
408C2×C8—736no—
409C2×C8—736no—
410C2×C8—733no—
411C2×C8—731no—
412C2×C8—1466no—
413C2×C8—1464no—
414C2×C8—14612no—
415C2×C8C2×C22⋊D7316yes—
416C2×C8C2×C22⋊D7316yes—
417C2×C8C23×D7313yes—
418C2×C8C23×D7311yescharacteristic
419C4⋊C4—1463no—
420C22⋊C4—1463no—
421C42—733no—
422C42—731no—
423C42—1463no—
424C23—26no—
425C23—1466no—
426C23C2×#1192402813yes—
427Q8—26no—
428Q8—734no—
429Q8—1466no—
430Q8C4×C22⋊D7314yes—
431Q8C22×C4×D7314yes—
432D4—24no—
433D4—212no—
434D4—26no—
435D4—734no—
436D4—1464no—
437D4—14612no—
438D4—1466no—
439D4C4×C22⋊D73112yes—
440D4C22×C4×D7314yes—
441C2×C4—26no—
442C2×C4—24no—
443C2×C4—212no—
444C2×C4—5843no—
445C2×C4—1461no—
446C2×C4—1463no—
447C2×C4—1466no—
448C2×C4—1464no—
449C2×C4—14612no—
450C2×C4C22×C4×D7316yes—
451C2×C4C22×C22⋊D7316yes—
452C2×C4C2×#1192402813yes—
453C2×C4C24×D7311yescharacteristic
454C8—24no—
455C8—731no—
456C8—731no—
457C8—736no—
458C8—1464no—
459C8C22×C22⋊D7311yescharacteristic
460C8C22×C22⋊D7316yes—
461C8C22×C22⋊D7311yescharacteristic
462C22—24no—
463C22—43no—
464C22—86no—
465C22—5846no—
466C22—1464no—
467C22C2×C4×C22⋊D7316yes—
468C22C22×#1192402811yescharacteristic
469C4—24no—
470C4—2921no—
471C4—2923no—
472C4—1464no—
473C4C2×C4×C22⋊D7316yes—
474C4C23×C4×D7311yescharacteristic
475C4C23×C22⋊D7311yescharacteristic
476C2—21no—
477C2—26no—
478C2—44no—
479C2—2924no—
480C2C22×C4×C22⋊D7311yescharacteristic
subgroupG/H
centerC4C23×C22⋊D73
commutatorC4×C73C24×C4
FrattiniC2×C4C24×D73
FittingC73×#3573C2
socleC2×C73C22×C4×D4
prime radical#22624702C2
cosocleC2×C4×C73C25
hypercenter#3573D73
hypocenterC73#60184
LCSG ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C2×#131 ◁ #3573

Complements

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

NH ≅ G/Ntype
2. #22624642C2semidirect
3. #22618846C2semidirect
4. #22618838C2semidirect
5. #22618832C2semidirect
6. C2×#17988707C2semidirect
7. #22618540C2semidirect
8. #22618478C2semidirect
9. #22618476C2semidirect
10. #22618357C2semidirect
11. C73×#3573C2semidirect
13. #17989402C4semidirect
14. #17989398C4semidirect
15. #17989395C4semidirect
16. #17989390C4semidirect
20. #17989353C22semidirect
23. C2×#14556597C22semidirect
26. #17988707C22semidirect
27. #17988707C22semidirect
28. #17988707C22semidirect
29. #17988707C22semidirect
30. #17988702C22semidirect
31. #17988635C22semidirect
32. C2×#14556526C22semidirect
33. #17988604C22semidirect
34. #17988603C22semidirect
35. C2×#14556518C22semidirect
36. C2×#14556517C22semidirect
37. #17988545C22semidirect
38. #17988542C22semidirect
39. M16×Dic73C22semidirect
40. C2×#14556501C22semidirect
44. #568×C73C22semidirect
47. C2×#131×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
243. #3573D73semidirect
252. C73#60184semidirect
266. #568C2×D73semidirect
272. C2×#131C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226247022
#226246422
#226188462
#226188382
#226188322
C2×#179887072
#226185402
#226184782
#226184762
#226183572
C73×#35732
#6018473