Finite Group Databasebeta

← #17189149

Subgroups — #17189149

Order 4352.

632 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D17×#466C211yesmaximal, characteristic
2D17×#427C211yesmaximal, characteristic, complemented
3#14530976C211yesmaximal, characteristic, complemented
4#14530975C211yesmaximal, characteristic, complemented
5#14530972C211yesmaximal, characteristic, complemented
6#14530971C211yesmaximal, characteristic, complemented
7#14530968C211yesmaximal, characteristic, complemented
8#14530966C211yesmaximal, characteristic, complemented
9#14530963C211yesmaximal, characteristic, complemented
10#14530571C211yesmaximal, characteristic, complemented
11#14530569C211yesmaximal, characteristic, complemented
12#14530498C211yesmaximal, characteristic, complemented
13#14530498C211yesmaximal, characteristic, complemented
14#14530393C211yesmaximal, characteristic, complemented
15C17×#2234C211yesmaximal, characteristic, complemented
16C2×D8×D17C2211yescharacteristic
17C22×C8×D17C2211yescharacteristic
18#121×D17—21no—
19#117×D17—21no—
20C2×#10587469C2211yescharacteristic
21C2×#10587469C2211yescharacteristic
22#11763046C2211yescharacteristic
23#11763045C2211yescharacteristic
24#11763044—21no—
25#11763043—21no—
26#11763042C2211yescharacteristic
27#11762847C2211yescharacteristic, complemented
28#11762846C2211yescharacteristic
29#11762846C2211yescharacteristic
30#11762845—21no—
31#11762844—21no—
32#107×D17C2211yescharacteristic
33#107×D17C2211yescharacteristic, complemented
34#107×D17C2211yescharacteristic, complemented
35#106×D17—21no—
36#102×D17—21no—
37#11762637C2211yescharacteristic, complemented
38#11762636C2211yescharacteristic, complemented
39#11762636C2211yescharacteristic, complemented
40#11762635C2211yescharacteristic, complemented
41#11762634C2211yescharacteristic, complemented
42#11762634C2211yescharacteristic, complemented
43#11762611C2211yescharacteristic, complemented
44#11762611C2211yescharacteristic, complemented
45#11762610C2211yescharacteristic, complemented
46#11762610C2211yescharacteristic, complemented
47#11762609C2211yescharacteristic, complemented
48#11762609C2211yescharacteristic, complemented
49C22⋊#63214C2211yescharacteristic, complemented
50#11762588C2211yescharacteristic, complemented
51#11762588C2211yescharacteristic, complemented
52C17×#466C2211yescharacteristic
53C17×#427C2211yescharacteristic, complemented
54C17×#367C2211yescharacteristic, complemented
55C17×#366C2211yescharacteristic, complemented
56C17×#357C2211yescharacteristic, complemented
57C17×#357C2211yescharacteristic, complemented
58C17×C22⋊C16C2211yescharacteristic, complemented
59C2×D4×D17—21no—
60C2×D4×D17—41no—
61C22×C4×D17D411yescharacteristic
62C2×D8×C17C2311yescharacteristic
63C22×C8×C17C2311yescharacteristic
64C17×#121—21no—
65C17×#117—21no—
66#10587625—41no—
67#10587624—21no—
68C2×#63228—21no—
69#10587616—21no—
70#10587612—21no—
71C22×#3622D411yescharacteristic
72D8×D17—22no—
73D8×D17—22no—
74C2×D136C2311yescharacteristic
75C2×C8×D17—22no—
76C2×C8×D17—22no—
77C2×C8×D17D411yescharacteristic
78C2×C8×D17D411yescharacteristic
79C2×C8×D17C2311yescharacteristic
80C2×C8×D17C2311yescharacteristic
81#10587581—21no—
82#10587578—21no—
83#10587578—21no—
84C4⋊C4×D17—21no—
85C4⋊C4×D17—21no—
86C4⋊C4×D17—21no—
87#10587570—41no—
88C22⋊C4×D17—41no—
89C22⋊C4×D17—41no—
90#10587563—41no—
91#10587558—41no—
92C42×D17—41no—
93C2×C16×C17C2311yescharacteristic
94C2×C16×C17C2311yescharacteristic
95C17×#107C2311yescharacteristic
96C17×#107C2311yescharacteristic
97C17×#107C2311yescharacteristic
98C17×#106—21no—
99C17×#102—21no—
100#10587503—21no—
101#10587492—21no—
102#10587489C2311yescharacteristic
103#10587489C2311yescharacteristic
104#10587489C2311yescharacteristic
105#10587488—21no—
106C2×#63214C2311yescharacteristic
107C2×#63214C2311yescharacteristic
108#10587480—21no—
109#10587479—21no—
110#10587479—21no—
111#10587478—21no—
112#10587478—21no—
113#10587478—21no—
114#10587469—22no—
115#10587469—22no—
116C23×D17—41no—
117C23×D17D811yescharacteristic
118C2×D4×C17—21no—
119C2×D4×C17—41no—
120C22×C4×C17C2×D411yescharacteristic
121C2×C22⋊D17—41no—
122C2×C22⋊D17—41no—
123C2×C22⋊D17—41no—
124C22×Dic17D811yescharacteristic
125D4×D17—42no—
126D4×D17—42no—
127D4×D17—84no—
128C2×D68—21no—
129C2×D68—41no—
130C2×C4×D17—22no—
131C2×C4×D17—22no—
132C2×C4×D17—41no—
133C2×C4×D17—41no—
134C2×C4×D17—41no—
135C2×C4×D17D811yescharacteristic
136C2×C4×D17D811yescharacteristic
137C2×C4×D17C2×D411yescharacteristic
138C2×C4×D17C2×D411yescharacteristic
139D8×C17—21no—
140D8×C17—21no—
141C2×C8×C17—21no—
142C2×C8×C17—21no—
143C2×C8×C17C2×D411yescharacteristic
144C2×C8×C17C2411yescharacteristic
145C4⋊C4×C17—21no—
146C4⋊C4×C17—21no—
147C4⋊C4×C17—21no—
148C22⋊C4×C17—41no—
149C22⋊C4×C17—41no—
150C42×C17—41no—
151C22⋊Dic17—41no—
152C22⋊Dic17—41no—
153#63228—42no—
154#63227—41no—
155#63227—41no—
156#63227—41no—
157#63227—41no—
158C4⋊Dic17—21no—
159C4⋊Dic17—21no—
160C4⋊Dic17—21no—
161#63225—41no—
162#63225—41no—
163#63225—41no—
164C4×Dic17—41no—
165C4×Dic17—41no—
166C4×Dic17—41no—
167C2×#3622—22no—
168C2×#3622C2×D411yescharacteristic
169C2×#3622C2×D411yescharacteristic
170D136—21no—
171D136—21no—
172C8×D17—22no—
173C8×D17—21no—
174C8×D17—21no—
175C8×D17—22no—
176C8×D17#4112yes—
177C8×D17#4112yes—
178C16×C17—21no—
179C16×C17—21no—
180#63214—21no—
181#63214—21no—
182#33648—171nomaximal, Sylow
183C23×C17—41no—
184C23×C17C2×D811yescharacteristic
185C22×D17—22no—
186C22×D17—22no—
187C22×D17—41no—
188C22×D17—41no—
189C22×D17—82no—
190C22×D17—82no—
191C22×D17—82no—
192C22×D17C2×D811yescharacteristic
193C22×D17C2×D811yescharacteristic
194D4×C17—41no—
195D4×C17—41no—
196D4×C17—82no—
197C2×C4×C17—21no—
198C2×C4×C17—21no—
199C2×C4×C17—41no—
200C2×C4×C17—41no—
201C2×C4×C17—41no—
202C2×C4×C17C2×D811yescharacteristic
203C2×C4×C17C22×D411yescharacteristic
204C22⋊D17—82no—
205C22⋊D17—82no—
206C22⋊D17—82no—
207C2×Dic17—22no—
208C2×Dic17—41no—
209C2×Dic17—41no—
210C2×Dic17—41no—
211C2×Dic17C2×D811yescharacteristic
212C2×Dic17C2×D811yescharacteristic
213D68—41no—
214D68—41no—
215D68—82no—
216C4×D17—22no—
217C4×D17—21no—
218C4×D17—21no—
219C4×D17—22no—
220C4×D17—82no—
221C4×D17—82no—
222C4×D17—82no—
223C4×D17#12312yes—
224C4×D17#13712yes—
225C8×C17—21no—
226C8×C17C2×#4111yescharacteristic
227C8×C17C2×#4111yescharacteristic
228#3622—21no—
229#3622#12312yes—
230C2×#466—171no—
231C2×#427—171no—
232#2234—171no—
233#2234—171no—
234#2234—171no—
235#2234D1711yescharacteristic, complemented
236#2233—171no—
237#2233—171no—
238#2232—171no—
239#2232—171no—
240#2159—171no—
241#2159—171no—
242#2140—171no—
243#2140—171no—
244#2112—171no—
245C22×C17—21no—
246C22×C17—21no—
247C22×C17—21no—
248C22×C17—81no—
249C22×C17—81no—
250C22×C17—81no—
251C22×C17C22×D811yescharacteristic
252C2×D17—21no—
253C2×D17—21no—
254C2×D17—21no—
255C2×D17—21no—
256C2×D17—44no—
257C2×D17—81no—
258C2×D17—81no—
259C2×D17—81no—
260C2×D17—164no—
261C2×D17#48012yes—
262C2×D17#50912yes—
263C4×C17—21no—
264C4×C17—81no—
265C4×C17—81no—
266C4×C17—81no—
267C4×C17C2×#12311yescharacteristic
268C4×C17C2×#13711yescharacteristic
269Dic17—21no—
270Dic17—81no—
271Dic17—81no—
272Dic17—81no—
273Dic17#48012yes—
274C22×D8—171no—
275C23×C8—171no—
276C2×#121—341no—
277C2×#117—341no—
278C2×M32—171no—
279C2×M32—171no—
280#466—171no—
281#466—171no—
282#466—171no—
283#466—341no—
284#466—341no—
285#466C2×D1711yescharacteristic
286#427—171no—
287#427—171no—
288#427—171no—
289#427—341no—
290#427—341no—
291#427C2×D1711yescharacteristic, complemented
292C2×#107—171no—
293C2×#107—171no—
294C2×#107—171no—
295C2×#106—341no—
296C2×#102—341no—
297#367—171no—
298#367—171no—
299#367—171no—
300#367C2×D1711yescharacteristic, complemented
301#366—171no—
302#366—171no—
303#366—171no—
304#366C2×D1711yescharacteristic, complemented
305#357—171no—
306#357—171no—
307#357—171no—
308#357—171no—
309#357—171no—
310#357—171no—
311#357C2×D1711yescharacteristic, complemented
312#357C2×D1711yescharacteristic, complemented
313C22⋊C16—171no—
314C22⋊C16—171no—
315C22⋊C16—171no—
316C22⋊C16C2×D1711yescharacteristic, complemented
317C2×C17—41no—
318C2×C17—161no—
319C2×C17C2×#48011yescharacteristic
320C2×C17C2×#50911yescharacteristic
321C2×C17#341411yescharacteristic
322D17—22no—
323D17—22no—
324D17—161no—
325C22×D4—341no—
326C22×D4—681no—
327C23×C4—171no—
328C2×D8—171no—
329C2×D8—342no—
330C2×D8—342no—
331C2×D8—341no—
332C2×D8C22×D1711yescharacteristic
333C22×C8—171no—
334C22×C8—171no—
335C22×C8—171no—
336C22×C8—171no—
337C22×C8—171no—
338C22×C8—342no—
339C22×C8—342no—
340C22×C8C22×D1711yescharacteristic
341#121—21no—
342#121—341no—
343#121—341no—
344#121—341no—
345#121—681no—
346#121—681no—
347#117—21no—
348#117—341no—
349#117—341no—
350#117—341no—
351#117—681no—
352#117—681no—
353C2×C4⋊C4—341no—
354C2×C4⋊C4—341no—
355C2×C4⋊C4—341no—
356C2×C22⋊C4—681no—
357C2×C22⋊C4—681no—
358C2×C42—681no—
359M32—342no—
360M32—342no—
361C2×C16—171no—
362C2×C16—171no—
363C2×C16C22×D1711yescharacteristic
364C2×C16C22×D1711yescharacteristic
365#107—171no—
366#107—171no—
367#107—171no—
368#107—341no—
369#107—341no—
370#107—341no—
371#107C22×D1711yescharacteristic
372#107C22×D1711yescharacteristic
373#107C22×D1711yescharacteristic
374#106—21no—
375#106—341no—
376#106—341no—
377#106—341no—
378#102—21no—
379#102—341no—
380#102—341no—
381#102—341no—
382C17#3364811yescharacteristic, complemented, Sylow
383C24—171no—
384C24—681no—
385C2×D4—21no—
386C2×D4—41no—
387C2×D4—341no—
388C2×D4—681no—
389C2×D4—682no—
390C2×D4—682no—
391C2×D4—681no—
392C2×D4—681no—
393C2×D4—681no—
394C2×D4—1364no—
395C22×C4—171no—
396C22×C4—171no—
397C22×C4—171no—
398C22×C4—171no—
399C22×C4—171no—
400C22×C4—342no—
401C22×C4—342no—
402C22×C4—681no—
403C22×C4—681no—
404C22×C4—681no—
405C22×C4D4×D1711yescharacteristic
406D8—21no—
407D8—21no—
408D8—341no—
409D8—341no—
410D8—682no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—171no—
414C2×C8—172no—
415C2×C8—172no—
416C2×C8—171no—
417C2×C8—342no—
418C2×C8—342no—
419C2×C8—342no—
420C2×C8—341no—
421C2×C8—341no—
422C2×C8D4×D1711yescharacteristic
423C2×C8C23×D1711yescharacteristic
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—341no—
428C4⋊C4—341no—
429C4⋊C4—341no—
430C4⋊C4—681no—
431C4⋊C4—681no—
432C4⋊C4—681no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—681no—
436C22⋊C4—681no—
437C22⋊C4—681no—
438C22⋊C4—681no—
439C22⋊C4—681no—
440C22⋊C4—681no—
441C42—41no—
442C42—681no—
443C42—681no—
444C42—681no—
445C16—21no—
446C16—21no—
447C16—341no—
448C16—341no—
449C23—41no—
450C23—171no—
451C23—171no—
452C23—342no—
453C23—342no—
454C23—681no—
455C23—681no—
456C23—1362no—
457C23—1362no—
458C23—1362no—
459C23D8×D1711yescharacteristic
460D4—41no—
461D4—41no—
462D4—82no—
463D4—681no—
464D4—681no—
465D4—1362no—
466D4—1362no—
467D4—1362no—
468D4—1362no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—171no—
475C2×C4—172no—
476C2×C4—172no—
477C2×C4—171no—
478C2×C4—342no—
479C2×C4—342no—
480C2×C4—342no—
481C2×C4—341no—
482C2×C4—341no—
483C2×C4—681no—
484C2×C4—681no—
485C2×C4—681no—
486C2×C4—1362no—
487C2×C4—1362no—
488C2×C4—1362no—
489C2×C4D8×D1711yescharacteristic
490C2×C4C2×D4×D1711yescharacteristic
491C8—21no—
492C8—172no—
493C8—341no—
494C8#41×D1711yescharacteristic
495C8#41×D1711yescharacteristic
496C22—21no—
497C22—21no—
498C22—21no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—2724no—
503C22—172no—
504C22—172no—
505C22—341no—
506C22—341no—
507C22—341no—
508C22—341no—
509C22—684no—
510C22—1361no—
511C22—1361no—
512C22—1361no—
513C22C2×D8×D1711yescharacteristic
514C4—21no—
515C4—81no—
516C4—81no—
517C4—81no—
518C4—172no—
519C4—341no—
520C4—1361no—
521C4—1361no—
522C4—1361no—
523C4#123×D1711yescharacteristic
524C4#137×D1711yescharacteristic
525C2—41no—
526C2—161no—
527C2—2721no—
528C2—342no—
529C2—342no—
530C2D17×#48011yescharacteristic
531C2D17×#50911yescharacteristic
532C2#1454191711yescharacteristic
subgroupG/H
centerC22C2×D8×D17
commutator, cosocleC2×C8×C17C24
FrattiniC2×C8C23×D17
FittingC17×#2234C2
socleC22×C17C22×D8
prime radicalD17×#466C2
hypercenter#2234D17
hypocenterC17#33648
LCSG ▷ C2×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2234

Complements

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

NH ≅ G/Ntype
2. D17×#427C2semidirect
3. #14530976C2semidirect
4. #14530975C2semidirect
5. #14530972C2semidirect
6. #14530971C2semidirect
7. #14530968C2semidirect
8. #14530966C2semidirect
9. #14530963C2semidirect
10. #14530571C2semidirect
11. #14530569C2semidirect
12. #14530498C2semidirect
13. #14530498C2semidirect
14. #14530393C2semidirect
15. C17×#2234C2semidirect
27. #11762847C22semidirect
33. #107×D17C22semidirect
34. #107×D17C22semidirect
37. #11762637C22semidirect
38. #11762636C22semidirect
39. #11762636C22semidirect
40. #11762635C22semidirect
41. #11762634C22semidirect
42. #11762634C22semidirect
43. #11762611C22semidirect
44. #11762611C22semidirect
45. #11762610C22semidirect
46. #11762610C22semidirect
47. #11762609C22semidirect
48. #11762609C22semidirect
49. C22⋊#63214C22semidirect
50. #11762588C22semidirect
51. #11762588C22semidirect
53. C17×#427C22semidirect
54. C17×#367C22semidirect
55. C17×#366C22semidirect
56. C17×#357C22semidirect
57. C17×#357C22semidirect
58. C17×C22⋊C16C22semidirect
235. #2234D17semidirect
291. #427C2×D17semidirect
300. #367C2×D17semidirect
304. #366C2×D17semidirect
311. #357C2×D17semidirect
312. #357C2×D17semidirect
316. C22⋊C16C2×D17semidirect
382. C17#33648semidirect

Maximal subgroups

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

subgroupindex
D17×#4662
D17×#4272
#145309762
#145309752
#145309722
#145309712
#145309682
#145309662
#145309632
#145305712
#145305692
#145304982
#145303932
C17×#22342
#3364817