Finite Group Databasebeta

← #17898667

Subgroups — #17898667

Order 4352.

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#14543555C211yesmaximal, characteristic
2#14543495C211yesmaximal, characteristic, complemented
3#14537699C216yesmaximal, complemented
4#14537691C216yesmaximal, complemented
5#14537685C216yesmaximal, complemented
6C2×#11762929C216yesmaximal, complemented
7#14537393C211yesmaximal, characteristic, complemented
8#14537331C211yesmaximal, characteristic, complemented
9#14537329C211yesmaximal, characteristic, complemented
10#14537210C211yesmaximal, characteristic, complemented
11C17×#3573C211yesmaximal, characteristic, complemented
12#11763645C2211yescharacteristic
13#11763624C414yescomplemented
14#11763620C414yescomplemented
15#11763617C414yescomplemented
16#11763612C414yescomplemented
17#11763607C2216yes—
18C2×#10587636C2216yes—
19#11763576C2211yescharacteristic
20#11763575C2216yescomplemented
21#11763574—24no—
22C2×#10587634C2211yescharacteristic
23C2×#10587587C2216yescomplemented
24#131×D17—24no—
25#41×Dic17C2211yescharacteristic
26#11762929C2214yescomplemented
27#11762929C22112yescomplemented
28#11762929C22112yescomplemented
29#11762929C22112yescomplemented
30#11762924C22124yescomplemented
31#11762857C2216yescomplemented
32C2×#10587493C2216yescomplemented
33#11762826C2216yescomplemented
34#11762825C2216yescomplemented
35C2×#10587508C2216yescomplemented
36C2×#10587496C2216yescomplemented
37#11762767C2213yescomplemented
38#11762764C2213yescomplemented
39M16×Dic17C2213yescomplemented
40C2×#10587486C2213yescomplemented
41C17×#585C2211yescharacteristic
42C17×#568—24no—
43C17×#568C2211yescharacteristic
44C17×#568C2216yescomplemented
45C2×C17×#131—24no—
46C2×C17×#131C2211yescharacteristic
47C2×C17×#131C2216yescomplemented
48C17×2−1+4—21no—
49C17×2−1+4C2×C414yes—
50C17×H(5,𝔽2)—23no—
51C17×H(5,𝔽2)C2×C414yes—
52C2×#41×C17—24no—
53C2×#41×C17C2316yes—
54C2×#41×C17C2311yescharacteristic
55#10587697C2×C414yes—
56#10587696C2×C414yes—
57#41×D17—24no—
58#10587694—21no—
59#10587691—23no—
60C2×#63252C2316yes—
61C17×#131—24no—
62C17×#131—212no—
63C17×#131—26no—
64C17×#131—26no—
65C17×#131D414yes—
66C17×#131D414yes—
67C17×#131C2314yes—
68C17×#131C23112yes—
69C2×M16×C17—212no—
70C2×M16×C17—26no—
71C2×M16×C17C2316yes—
72C2×M16×C17C2313yes—
73C22×C8×C17—26no—
74C22×C8×C17C2313yes—
75#10587637C2×C4112yescomplemented
76#10587636C2×C4112yescomplemented
77#10587636C2×C4112yescomplemented
78#10587636C2314yes—
79#10587636C23112yes—
80#10587635C2×C4112yescomplemented
81#10587634—24no—
82#10587634C2×C414yes—
83#10587634C2×C414yes—
84Q8×Dic17—21no—
85C2×#63231C2×C416yescomplemented
86#10587628C2×C4112yescomplemented
87C2×#63230C2×C416yescomplemented
88D4×Dic17—23no—
89C2×#63229C2×C416yescomplemented
90#10587618C2×C4112yescomplemented
91C2×#63228C2×C416yescomplemented
92#10587612C2313yes—
93C2×C4×Dic17C2313yes—
94C2×#63223C2316yes—
95C2×#63223C2313yes—
96C22×#3622C2313yes—
97#10587593—212no—
98M16×D17—212no—
99#10587587—26no—
100#10587587—26no—
101#10587587C2314yes—
102#10587587C23112yes—
103C2×#63218—26no—
104C2×C8×D17—26no—
105#10587508C2314yes—
106#10587508C23112yes—
107#10587496C2314yes—
108#10587496C23112yes—
109#10587493C2314yes—
110#10587493C23112yes—
111#10587486C2311yescharacteristic
112#10587486C2313yes—
113#10587486C2313yes—
114#10587486C2313yes—
115C8×Dic17C2316yes—
116#41×C17—24no—
117#41×C17—212no—
118#41×C17—26no—
119#41×C17—26no—
120#41×C17C22×C4112yes—
121#41×C17C22×C414yes—
122#41×C17C2×D414yes—
123#41×C17C2414yes—
124C2×Q8×C17—24no—
125C2×Q8×C17C22×C416yes—
126C2×Q8×C17#4111yescharacteristic
127C2×D4×C17—26no—
128C2×D4×C17—212no—
129C2×D4×C17C22×C416yes—
130C2×D4×C17#4113yes—
131C22×C4×C17—26no—
132C22×C4×C17C2413yes—
133C2×C22⋊D17—26no—
134C22×Dic17—23no—
135#63256—24no—
136Q8×D17—24no—
137#63254—212no—
138D4×D17—212no—
139#63252—26no—
140#63252—26no—
141#63252C22×C4112yes—
142#63252C2414yes—
143C2×D68C22×C416yes—
144C2×C4×D17—26no—
145C2×Dic34C22×C416yes—
146M16×C17—26no—
147M16×C17—212no—
148M16×C17—24no—
149M16×C17C2×D4112yes—
150M16×C17C2414yes—
151C2×C8×C17—26no—
152C2×C8×C17—24no—
153C2×C8×C17—212no—
154C2×C8×C17C2×D416yes—
155C2×C8×C17C2×D416yes—
156C2×C8×C17C2411yescharacteristic
157C2×C8×C17C2413yes—
158C22⋊Dic17—23no—
159#63231C22×C4112yes—
160#63231C22×C414yes—
161#63230C22×C414yes—
162#63230C22×C4112yes—
163#63229C22×C4112yes—
164#63229C22×C414yes—
165#63228C22×C414yes—
166#63228C22×C4112yes—
167C4⋊Dic17—23no—
168C4×Dic17—23no—
169C4×Dic17C2413yes—
170C4×Dic17C2411yescharacteristic
171#63223—26no—
172#63223C22×C4112yes—
173#63223C2414yes—
174C2×#3622—26no—
175C2×#3622C22×C416yes—
176C2×#3622C22×C416yes—
177C2×#3622C2413yes—
178C2×#3622C2411yescharacteristic
179#63218—24no—
180#63218—212no—
181C8×D17—24no—
182C8×D17—212no—
183#60184—171nomaximal, Sylow
184C23×C17—26no—
185C23×C17C2×#4113yes—
186C22×D17—26no—
187Q8×C17—26no—
188Q8×C17C4×D414yes—
189Q8×C17C23×C414yes—
190D4×C17—212no—
191D4×C17—24no—
192D4×C17—26no—
193D4×C17C4×D4112yes—
194D4×C17C23×C414yes—
195C2×C4×C17—26no—
196C2×C4×C17—212no—
197C2×C4×C17—24no—
198C2×C4×C17C23×C416yes—
199C2×C4×C17C22×D416yes—
200C2×C4×C17C2×#4113yes—
201C2×C4×C17C2511yescharacteristic
202C22⋊D17—212no—
203C22⋊D17—24no—
204C2×Dic17—26no—
205C2×Dic17—23no—
206C2×Dic17—21no—
207C2×Dic17—83no—
208D68—26no—
209D68C23×C414yes—
210C4×D17—212no—
211C4×D17—24no—
212Dic34—26no—
213Dic34C23×C414yes—
214C8×C17—24no—
215C8×C17C22×D411yescharacteristic
216C8×C17C22×D416yes—
217C8×C17C22×D411yescharacteristic
218#3622—24no—
219#3622C23×C411yescharacteristic
220#3622C23×C411yescharacteristic
221#3622C23×C416yes—
222#3584—171no—
223#3573—171no—
224#3573D1711yescharacteristic, complemented
225#2957—176no—
226#2956—176no—
227#2955—176no—
228C2×#444—176no—
229#2916—171no—
230#2901—171no—
231#2901—171no—
232#2875—171no—
233C22×C17—24no—
234C22×C17—43no—
235C22×C17—86no—
236C22×C17C2×C4×D416yes—
237C22×C17C22×#4111yescharacteristic
238C2×D17—24no—
239C2×D17—86no—
240C4×C17—24no—
241C4×C17C2×C4×D416yes—
242C4×C17C24×C411yescharacteristic
243C4×C17C23×D411yescharacteristic
244Dic17—24no—
245Dic17—43no—
246Dic17—41no—
247#585—171no—
248#585C2×D1711yescharacteristic
249#578—174no—
250#577—174no—
251#577—174no—
252#576—174no—
253#575—176no—
254C2×#135—176no—
255#568—24no—
256#568—171no—
257#568—176no—
258#568—344no—
259#568C2×D1711yescharacteristic
260#568C2×D1716yescomplemented
261C2×#131—24no—
262C2×#131—171no—
263C2×#131—176no—
264C2×#131—344no—
265C2×#131C2×D1711yescharacteristic
266C2×#131C2×D1716yescomplemented
267C4×#41—171no—
268#444—174no—
269#444—1712no—
270#444—1712no—
271#444—1712no—
272#443—1724no—
273#430—176no—
274C2×#108—176no—
275#421—176no—
276#421—176no—
277C2×#104—176no—
278C2×#104—176no—
279#405—173no—
280#405—173no—
281C4×M16—173no—
282C2×#97—173no—
283C2×C17—21no—
284C2×C17—26no—
285C2×C17—44no—
286C2×C17C22×C4×D411yescharacteristic
287D17—44no—
2882−1+4—21no—
2892−1+4—174no—
2902−1+4—341no—
2912−1+4C4×D1714yes—
292H(5,𝔽2)—23no—
293H(5,𝔽2)—174no—
294H(5,𝔽2)—343no—
295H(5,𝔽2)C4×D1714yes—
296C2×#41—24no—
297C2×#41—176no—
298C2×#41—344no—
299C2×#41C22×D1716yes—
300C2×#41C22×D1711yescharacteristic
301#137—1712no—
302#137—1712no—
303AGL(1,ℤ/8)—1712no—
304AGL(1,ℤ/8)—1712no—
305#135—1712no—
306#135—1712no—
307#135—174no—
308#135—1712no—
309C2×Q16—176no—
310C2×SD16—176no—
311C2×SD16—176no—
312C2×D8—176no—
313#131—24no—
314#131—212no—
315#131—26no—
316#131—26no—
317#131—174no—
318#131—174no—
319#131—174no—
320#131—1712no—
321#131—344no—
322#131—3412no—
323#131—346no—
324#131—346no—
325#131C22⋊D1714yes—
326#131C22⋊D1714yes—
327#131C22×D1714yes—
328#131C22×D17112yes—
329C2×M16—212no—
330C2×M16—26no—
331C2×M16—173no—
332C2×M16—176no—
333C2×M16—3412no—
334C2×M16—346no—
335C2×M16C22×D1716yes—
336C2×M16C22×D1713yes—
337C22×C8—26no—
338C22×C8—173no—
339C22×C8—346no—
340C22×C8C22×D1713yes—
341C4×Q8—341no—
342C4×D4—343no—
343#117—173no—
344C2×C42—173no—
345#108—174no—
346#108—1712no—
347#104—174no—
348#104—1712no—
349#104—174no—
350#104—1712no—
351#97—171no—
352#97—173no—
353#97—173no—
354#97—173no—
355C4×C8—176no—
356C17#6018411yescharacteristic, complemented, Sylow
357#41—24no—
358#41—212no—
359#41—26no—
360#41—26no—
361#41—1712no—
362#41—174no—
363#41—344no—
364#41—3412no—
365#41—346no—
366#41—346no—
367#41C2×C4×D17112yes—
368#41C2×C4×D1714yes—
369#41C2×C22⋊D1714yes—
370#41C23×D1714yes—
371C2×Q8—24no—
372C2×Q8—176no—
373C2×Q8—344no—
374C2×Q8C2×C4×D1716yes—
375C2×Q8#6325211yescharacteristic
376C2×D4—26no—
377C2×D4—212no—
378C2×D4—176no—
379C2×D4—346no—
380C2×D4—3412no—
381C2×D4C2×C4×D1716yes—
382C2×D4#6325213yes—
383C22×C4—26no—
384C22×C4—346no—
385C22×C4—343no—
386C22×C4C23×D1713yes—
387Q16—174no—
388Q16—1712no—
389SD16—174no—
390SD16—1712no—
391SD16—174no—
392SD16—1712no—
393D8—174no—
394D8—1712no—
395M16—26no—
396M16—212no—
397M16—24no—
398M16—1712no—
399M16—174no—
400M16—346no—
401M16—344no—
402M16—3412no—
403M16C2×C22⋊D17112yes—
404M16C23×D1714yes—
405C2×C8—26no—
406C2×C8—24no—
407C2×C8—212no—
408C2×C8—176no—
409C2×C8—176no—
410C2×C8—173no—
411C2×C8—171no—
412C2×C8—346no—
413C2×C8—344no—
414C2×C8—3412no—
415C2×C8C2×C22⋊D1716yes—
416C2×C8C2×C22⋊D1716yes—
417C2×C8C23×D1713yes—
418C2×C8C23×D1711yescharacteristic
419C4⋊C4—343no—
420C22⋊C4—343no—
421C42—173no—
422C42—171no—
423C42—343no—
424C23—26no—
425C23—346no—
426C23C2×#6325213yes—
427Q8—26no—
428Q8—174no—
429Q8—346no—
430Q8C4×C22⋊D1714yes—
431Q8C22×C4×D1714yes—
432D4—212no—
433D4—24no—
434D4—26no—
435D4—174no—
436D4—3412no—
437D4—344no—
438D4—346no—
439D4C4×C22⋊D17112yes—
440D4C22×C4×D1714yes—
441C2×C4—26no—
442C2×C4—212no—
443C2×C4—24no—
444C2×C4—346no—
445C2×C4—3412no—
446C2×C4—344no—
447C2×C4—341no—
448C2×C4—343no—
449C2×C4—1363no—
450C2×C4C22×C4×D1716yes—
451C2×C4C2×#6325213yes—
452C2×C4C22×C22⋊D1716yes—
453C2×C4C24×D1711yescharacteristic
454C8—24no—
455C8—171no—
456C8—171no—
457C8—176no—
458C8—344no—
459C8C22×C22⋊D1711yescharacteristic
460C8C22×C22⋊D1716yes—
461C8C22×C22⋊D1711yescharacteristic
462C22—24no—
463C22—43no—
464C22—86no—
465C22—344no—
466C22—1366no—
467C22C2×C4×C22⋊D1716yes—
468C22C22×#6325211yescharacteristic
469C4—24no—
470C4—344no—
471C4—681no—
472C4—683no—
473C4C2×C4×C22⋊D1716yes—
474C4C23×C4×D1711yescharacteristic
475C4C23×C22⋊D1711yescharacteristic
476C2—21no—
477C2—26no—
478C2—44no—
479C2—684no—
480C2C22×C4×C22⋊D1711yescharacteristic
subgroupG/H
centerC4C23×C22⋊D17
commutatorC4×C17C24×C4
FrattiniC2×C4C24×D17
FittingC17×#3573C2
socleC2×C17C22×C4×D4
prime radical#14543555C2
cosocleC2×C4×C17C25
hypercenter#3573D17
hypocenterC17#60184
LCSG ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C4 ◁ C2×#131 ◁ #3573

Complements

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

NH ≅ G/Ntype
2. #14543495C2semidirect
3. #14537699C2semidirect
4. #14537691C2semidirect
5. #14537685C2semidirect
6. C2×#11762929C2semidirect
7. #14537393C2semidirect
8. #14537331C2semidirect
9. #14537329C2semidirect
10. #14537210C2semidirect
11. C17×#3573C2semidirect
13. #11763624C4semidirect
14. #11763620C4semidirect
15. #11763617C4semidirect
16. #11763612C4semidirect
20. #11763575C22semidirect
23. C2×#10587587C22semidirect
26. #11762929C22semidirect
27. #11762929C22semidirect
28. #11762929C22semidirect
29. #11762929C22semidirect
30. #11762924C22semidirect
31. #11762857C22semidirect
32. C2×#10587493C22semidirect
33. #11762826C22semidirect
34. #11762825C22semidirect
35. C2×#10587508C22semidirect
36. C2×#10587496C22semidirect
37. #11762767C22semidirect
38. #11762764C22semidirect
39. M16×Dic17C22semidirect
40. C2×#10587486C22semidirect
44. C17×#568C22semidirect
47. C2×C17×#131C22semidirect
75. #10587637C2×C4semidirect
76. #10587636C2×C4semidirect
77. #10587636C2×C4semidirect
80. #10587635C2×C4semidirect
85. C2×#63231C2×C4semidirect
86. #10587628C2×C4semidirect
87. C2×#63230C2×C4semidirect
89. C2×#63229C2×C4semidirect
90. #10587618C2×C4semidirect
91. C2×#63228C2×C4semidirect
224. #3573D17semidirect
260. #568C2×D17semidirect
266. C2×#131C2×D17semidirect
356. C17#60184semidirect

Maximal subgroups

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

subgroupindex
#145435552
#145434952
#145376992
#145376912
#145376852
C2×#117629292
#145373932
#145373312
#145373292
#145372102
C17×#35732
#6018417