Finite Group Databasebeta

← #17898665

Subgroups — #17898665

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#14543497C211yesmaximal, characteristic, complemented
3#14543496C211yesmaximal, characteristic, complemented
4#14537697C216yesmaximal, complemented
5#14537690C216yesmaximal, complemented
6#14537683C216yesmaximal, complemented
7C2×#11762928C216yesmaximal, complemented
8#14537393C211yesmaximal, characteristic, complemented
9#14537332C211yesmaximal, characteristic, complemented
10#14537212C211yesmaximal, characteristic, complemented
11C17×#2901C211yesmaximal, characteristic, complemented
12#11763645C2211yescharacteristic
13#11763624C414yescomplemented
14#11763620C414yescomplemented
15#11763617C414yescomplemented
16#11763612C414yescomplemented
17#11763607C2216yes—
18C2×#10587636C2216yes—
19#11763576—24no—
20#11763576C2211yescharacteristic
21#11763576C2216yescomplemented
22#11763575C2216yescomplemented
23#11763574—24no—
24C2×#10587593C2216yescomplemented
25C2×#10587634—24no—
26C2×#10587634C2211yescharacteristic
27C2×#10587634C2211yescharacteristic
28C2×#10587634C2216yescomplemented
29#131×D17—24no—
30#11762928C2214yescomplemented
31#11762928C22112yescomplemented
32#11762928C22112yescomplemented
33#11762928C22112yescomplemented
34#11762923C22124yescomplemented
35#11762857C2216yescomplemented
36C2×#10587493C2216yescomplemented
37#11762827C2216yescomplemented
38C2×#10587476C2216yescomplemented
39#11762765C2213yescomplemented
40#11762765C2213yescomplemented
41C4×#63223C2213yescomplemented
42C2×#10587474C2213yescomplemented
43C17×#585C2211yescharacteristic
44C17×#568C2211yescharacteristic
45C4×#41×C17C2211yescharacteristic
46C17×#421C2216yescomplemented
47C2×C17×#104C2216yescomplemented
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—
62C2×M16×C17C2316yes—
63C4×Q8×C17—21no—
64C4×D4×C17—23no—
65C17×#117C2313yes—
66C2×C42×C17C2313yes—
67#10587637C2×C4112yescomplemented
68#10587636C2×C4112yescomplemented
69#10587636C2×C4112yescomplemented
70#10587636C2314yes—
71#10587636C23112yes—
72#10587635C2×C4112yescomplemented
73#10587634—212no—
74#10587634—26no—
75#10587634—26no—
76#10587634—24no—
77#10587634C2×C414yes—
78#10587634C2×C414yes—
79#10587634D414yes—
80#10587634D414yes—
81#10587634C2314yes—
82#10587634C23112yes—
83C2×#63231C2×C416yescomplemented
84#10587628C2×C4112yescomplemented
85C2×#63230C2×C416yescomplemented
86C2×#63229C2×C416yescomplemented
87#10587618C2×C4112yescomplemented
88C2×#63228C2×C416yescomplemented
89C2×#63223—212no—
90C2×#63223—26no—
91C2×#63223C2316yes—
92C2×#63223C2313yes—
93C2×#63223C2313yes—
94C22×#3622—26no—
95C22×#3622C2313yes—
96C22×#3622C2313yes—
97#10587593—26no—
98#10587593—26no—
99#10587593C2314yes—
100#10587593C23112yes—
101M16×D17—212no—
102#10587587—212no—
103C2×#63218—26no—
104C2×C8×D17—26no—
105C17×#104C2314yes—
106C17×#104C23112yes—
107#10587493C2314yes—
108#10587493C23112yes—
109#10587476C2314yes—
110#10587476C23112yes—
111#10587474C2311yescharacteristic
112#10587474C2313yes—
113#10587474C2313yes—
114#10587474C2313yes—
115C4×#3622C2316yes—
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×C17—23no—
133C22×C4×C17C2413yes—
134C2×C22⋊D17—26no—
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×C17C2414yes—
148C2×C8×C17—26no—
149C4⋊C4×C17—23no—
150C22⋊C4×C17—23no—
151C42×C17—23no—
152C42×C17C2413yes—
153C42×C17C2411yescharacteristic
154#63231C22×C4112yes—
155#63231C22×C414yes—
156#63230C22×C414yes—
157#63230C22×C4112yes—
158#63229C22×C4112yes—
159#63229C22×C414yes—
160#63228C22×C414yes—
161#63228C22×C4112yes—
162#63223—26no—
163#63223—24no—
164#63223—212no—
165#63223C22×C4112yes—
166#63223C2×D4112yes—
167#63223C2414yes—
168C2×#3622—26no—
169C2×#3622—24no—
170C2×#3622—212no—
171C2×#3622C22×C416yes—
172C2×#3622C22×C416yes—
173C2×#3622C2×D416yes—
174C2×#3622C2×D416yes—
175C2×#3622C2413yes—
176C2×#3622C2411yescharacteristic
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×C17—21no—
199C2×C4×C17—23no—
200C2×C4×C17—83no—
201C2×C4×C17C23×C416yes—
202C2×C4×C17C22×D416yes—
203C2×C4×C17C2×#4113yes—
204C2×C4×C17C2511yescharacteristic
205C22⋊D17—212no—
206C22⋊D17—24no—
207C2×Dic17—26no—
208D68—26no—
209D68C23×C414yes—
210C4×D17—212no—
211C4×D17—24no—
212Dic34—26no—
213Dic34C23×C414yes—
214C8×C17—24no—
215#3622—24no—
216#3622C23×C411yescharacteristic
217#3622C23×C411yescharacteristic
218#3622C23×C416yes—
219#3622C22×D411yescharacteristic
220#3622C22×D416yes—
221#3622C22×D411yescharacteristic
222#3584—171no—
223#3573—171no—
224#3573—171no—
225#2957—176no—
226#2956—176no—
227#2955—176no—
228C2×#444—176no—
229#2916—171no—
230#2901—171no—
231#2901D1711yescharacteristic, complemented
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×C17—43no—
242C4×C17—41no—
243C4×C17C2×C4×D416yes—
244C4×C17C24×C411yescharacteristic
245C4×C17C23×D411yescharacteristic
246Dic17—24no—
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—171no—
256#568—176no—
257#568—176no—
258#568—344no—
259#568—344no—
260#568C2×D1711yescharacteristic
261C2×#131—171no—
262C2×#131—171no—
263C2×#131—176no—
264C2×#131—176no—
265C2×#131—344no—
266C2×#131—344no—
267C4×#41C2×D1711yescharacteristic
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#421C2×D1716yescomplemented
277C2×#104—176no—
278C2×#104C2×D1716yescomplemented
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—174no—
315#131—174no—
316#131—174no—
317#131—174no—
318#131—174no—
319#131—1712no—
320#131—174no—
321#131—1712no—
322#131—3412no—
323#131—346no—
324#131—346no—
325#131—346no—
326#131—346no—
327#131—3412no—
328#131—344no—
329C2×M16—173no—
330C2×M16—173no—
331C2×M16—176no—
332C2×M16—346no—
333C2×M16—346no—
334C2×M16—3412no—
335C2×M16—3412no—
336C2×M16C22×D1716yes—
337C22×C8—173no—
338C22×C8—173no—
339C22×C8—346no—
340C22×C8—346no—
341C4×Q8—21no—
342C4×D4—23no—
343#117C22×D1713yes—
344C2×C42C22×D1713yes—
345#108—174no—
346#108—1712no—
347#104—174no—
348#104—1712no—
349#104C22×D1714yes—
350#104C22×D17112yes—
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#41D4×D1714yes—
370#41C23×D1714yes—
371C2×Q8—24no—
372C2×Q8—176no—
373C2×Q8—344no—
374C2×Q8C2×C4×D1716yes—
375C2×Q8#6325411yescharacteristic
376C2×D4—26no—
377C2×D4—212no—
378C2×D4—176no—
379C2×D4—346no—
380C2×D4—3412no—
381C2×D4C2×C4×D1716yes—
382C2×D4#6325413yes—
383C22×C4—26no—
384C22×C4—23no—
385C22×C4—346no—
386C22×C4C23×D1713yes—
387Q16—174no—
388Q16—1712no—
389SD16—174no—
390SD16—1712no—
391SD16—174no—
392SD16—1712no—
393D8—174no—
394D8—1712no—
395M16—26no—
396M16—1712no—
397M16—1712no—
398M16—174no—
399M16—344no—
400M16—3412no—
401M16—344no—
402M16—3412no—
403M16—346no—
404M16C23×D1714yes—
405C2×C8—26no—
406C2×C8—176no—
407C2×C8—176no—
408C2×C8—176no—
409C2×C8—176no—
410C2×C8—173no—
411C2×C8—173no—
412C2×C8—171no—
413C2×C8—171no—
414C2×C8—344no—
415C2×C8—3412no—
416C2×C8—344no—
417C2×C8—3412no—
418C2×C8—346no—
419C4⋊C4—23no—
420C22⋊C4—23no—
421C42—23no—
422C42C23×D1713yes—
423C42C23×D1711yescharacteristic
424C23—26no—
425C23—346no—
426C23C2×#6325413yes—
427Q8—26no—
428Q8—174no—
429Q8—346no—
430Q8#1058756714yes—
431Q8C22×C4×D1714yes—
432D4—212no—
433D4—24no—
434D4—26no—
435D4—174no—
436D4—3412no—
437D4—344no—
438D4—346no—
439D4#10587567112yes—
440D4C22×C4×D1714yes—
441C2×C4—26no—
442C2×C4—212no—
443C2×C4—24no—
444C2×C4—21no—
445C2×C4—23no—
446C2×C4—83no—
447C2×C4—346no—
448C2×C4—3412no—
449C2×C4—344no—
450C2×C4C22×C4×D1716yes—
451C2×C4C2×D4×D1716yes—
452C2×C4C2×#6325413yes—
453C2×C4C24×D1711yescharacteristic
454C8—24no—
455C8—171no—
456C8—171no—
457C8—176no—
458C8—171no—
459C8—171no—
460C8—176no—
461C8—344no—
462C22—24no—
463C22—43no—
464C22—86no—
465C22—344no—
466C22—1366no—
467C22C2×#1058756716yes—
468C22C22×#6325411yescharacteristic
469C4—24no—
470C4—41no—
471C4—43no—
472C4—344no—
473C4C2×#1058756716yes—
474C4C23×C4×D1711yescharacteristic
475C4C22×D4×D1711yescharacteristic
476C2—21no—
477C2—26no—
478C2—44no—
479C2—684no—
480C2C22×#1058756711yescharacteristic
subgroupG/H
centerC4C22×D4×D17
commutatorC4×C17C24×C4
FrattiniC2×C4C24×D17
FittingC17×#2901C2
socleC2×C17C22×C4×D4
prime radical#14543555C2
cosocleC2×C4×C17C25
hypercenter#2901D17
hypocenterC17#60184
LCSG ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C4 ◁ C2×#41 ◁ #2901

Complements

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

NH ≅ G/Ntype
2. #14543497C2semidirect
3. #14543496C2semidirect
4. #14537697C2semidirect
5. #14537690C2semidirect
6. #14537683C2semidirect
7. C2×#11762928C2semidirect
8. #14537393C2semidirect
9. #14537332C2semidirect
10. #14537212C2semidirect
11. C17×#2901C2semidirect
13. #11763624C4semidirect
14. #11763620C4semidirect
15. #11763617C4semidirect
16. #11763612C4semidirect
21. #11763576C22semidirect
22. #11763575C22semidirect
24. C2×#10587593C22semidirect
28. C2×#10587634C22semidirect
30. #11762928C22semidirect
31. #11762928C22semidirect
32. #11762928C22semidirect
33. #11762928C22semidirect
34. #11762923C22semidirect
35. #11762857C22semidirect
36. C2×#10587493C22semidirect
37. #11762827C22semidirect
38. C2×#10587476C22semidirect
39. #11762765C22semidirect
40. #11762765C22semidirect
41. C4×#63223C22semidirect
42. C2×#10587474C22semidirect
46. C17×#421C22semidirect
47. C2×C17×#104C22semidirect
67. #10587637C2×C4semidirect
68. #10587636C2×C4semidirect
69. #10587636C2×C4semidirect
72. #10587635C2×C4semidirect
83. C2×#63231C2×C4semidirect
84. #10587628C2×C4semidirect
85. C2×#63230C2×C4semidirect
86. C2×#63229C2×C4semidirect
87. #10587618C2×C4semidirect
88. C2×#63228C2×C4semidirect
231. #2901D17semidirect
276. #421C2×D17semidirect
278. C2×#104C2×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
#145434972
#145434962
#145376972
#145376902
#145376832
C2×#117629282
#145373932
#145373322
#145372122
C17×#29012
#6018417