Finite Group Databasebeta

← #17187690

Subgroups — #17187690

Order 4352.

682 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD32×D17C211yesmaximal, characteristic
2#14537647C211yesmaximal, characteristic, complemented
3#14537631C211yesmaximal, characteristic
4#14530650C211yesmaximal, characteristic, complemented
5#14530649C211yesmaximal, characteristic, complemented
6#14530647C211yesmaximal, characteristic, complemented
7#14530639C211yesmaximal, characteristic, complemented
8C4×#10587499C211yesmaximal, characteristic, complemented
9C4×#10587498C211yesmaximal, characteristic, complemented
10#14530633C211yesmaximal, characteristic, complemented
11#14530565C211yesmaximal, characteristic, complemented
12#14530502C211yesmaximal, characteristic, complemented
13#14530498C211yesmaximal, characteristic
14#14530382C211yesmaximal, characteristic, complemented
15C17×#2175C211yesmaximal, characteristic, complemented
16C2×Q16×D17C2211yescharacteristic
17C2×D8×D17C2211yescharacteristic
18#11763285—21no—
19#11763281—21no—
20C2×#10587471C2211yescharacteristic
21C2×#10587498C2211yescharacteristic
22C2×#10587499C2211yescharacteristic
23SD32×D17C418yescomplemented
24C2×#10587469C2211yescharacteristic
25C2×C16×D17C2211yescharacteristic
26#11762919—21no—
27#11762918—21no—
28#11762914—21no—
29#11762913—21no—
30C4×Dic68C2211yescharacteristic, complemented
31Q16×Dic17C2211yescharacteristic, complemented
32#11762903C2211yescharacteristic, complemented
33C4×D136C2211yescharacteristic
34D8×Dic17C2211yescharacteristic
35#11762891C2211yescharacteristic
36#11762846C2211yescharacteristic
37#11762846C2211yescharacteristic
38#11762816—21no—
39#11762812—21no—
40#11762763C2211yescharacteristic
41#11762636C2211yescharacteristic, complemented
42#11762636C2211yescharacteristic, complemented
43#11762635C2211yescharacteristic, complemented
44#11762614C2211yescharacteristic, complemented
45#11762613C2211yescharacteristic, complemented
46#11762612C2211yescharacteristic, complemented
47#11762611C2211yescharacteristic
48#11762610C2211yescharacteristic
49#11762609C2211yescharacteristic
50#11762584C2211yescharacteristic, complemented
51C4×#63214C2211yescharacteristic, complemented
52C16×Dic17C2211yescharacteristic, complemented
53C2×C17×SD32C2211yescharacteristic
54C4×Q16×C17C2211yescharacteristic, complemented
55C4×D8×C17C2211yescharacteristic
56C17×#366C2211yescharacteristic, complemented
57C17×#358C2211yescharacteristic, complemented
58C17×#357C2211yescharacteristic
59C17×#346C2211yescharacteristic, complemented
60C2×Q8×D17—21no—
61C2×D4×D17—21no—
62C2×Q16×C17C2311yescharacteristic
63C2×D8×C17C2311yescharacteristic
64C4×Q8×C17—21no—
65C4×D4×C17—21no—
66Q8×Dic17—21no—
67C2×#63231—21no—
68D4×Dic17—21no—
69C2×#63228—21no—
70C4×C22⋊D17—41no—
71C2×#63227—41no—
72Q16×D17—22no—
73Q16×D17C2×C414yescomplemented
74D8×D17—22no—
75D8×D17C2×C414yes—
76C2×Dic68C2311yescharacteristic
77C2×D136C2311yescharacteristic
78C2×#63218D411yescharacteristic
79C2×C8×D17C2311yescharacteristic
80#10587579—21no—
81#10587578—21no—
82#10587578—21no—
83#10587578—41no—
84#10587573—21no—
85#10587573—41no—
86#10587567—41no—
87C22⋊C4×D17—41no—
88C4×D68—21no—
89#10587558—41no—
90#10587558D411yescharacteristic
91C4×Dic34—21no—
92C4×Dic34—41no—
93C17×SD32C2×C414yescomplemented
94C2×C16×C17C2311yescharacteristic
95C2×C16×C17C2311yescharacteristic
96C17×#107C2311yescharacteristic
97C17×#107C2311yescharacteristic
98C17×#103—21no—
99C17×#102—21no—
100C4×C8×C17C2311yescharacteristic
101#10587506—21no—
102#10587503—21no—
103#10587499C2×C414yescomplemented
104#10587498C2×C414yescomplemented
105#10587492—21no—
106#10587489C2311yescharacteristic
107#10587489C2311yescharacteristic
108#10587487—21no—
109#10587486D411yescharacteristic
110C8×Dic17C2311yescharacteristic
111C2×#63214C2311yescharacteristic
112C2×#63214C2311yescharacteristic
113#10587481—21no—
114#10587480—21no—
115#10587479—21no—
116#10587479—21no—
117#10587474D411yescharacteristic
118#10587471C2×C414yescomplemented
119#10587469—22no—
120C16×D17C2×C414yescomplemented
121C23×D17—41no—
122C2×Q8×C17—21no—
123C2×D4×C17—21no—
124C22×C4×C17—41no—
125C2×C22⋊D17—41no—
126C22×Dic17—41no—
127Q8×D17—24no—
128Q8×D17—42no—
129D4×D17—24no—
130D4×D17—42no—
131C2×D68—21no—
132C2×C4×D17—41no—
133C2×C4×D17—41no—
134C2×C4×D17—41no—
135C2×C4×D17C2×D411yescharacteristic
136C2×Dic34—21no—
137C2×Dic34—41no—
138Q16×C17—21no—
139Q16×C17C22×C412yes—
140D8×C17—21no—
141D8×C17C22×C412yes—
142C2×C8×C17C2×D411yescharacteristic
143C2×C8×C17C2411yescharacteristic
144C4⋊C4×C17—21no—
145C4⋊C4×C17—21no—
146C4⋊C4×C17—41no—
147C22⋊C4×C17—41no—
148C42×C17—41no—
149C42×C17C2×D411yescharacteristic
150C22⋊Dic17—41no—
151#63231—24no—
152#63228—24no—
153#63227—41no—
154#63227—41no—
155#63227—41no—
156#63227—41no—
157#63227D811yescharacteristic
158#63227D811yescharacteristic
159C4⋊Dic17—21no—
160C4⋊Dic17—21no—
161C4⋊Dic17—41no—
162#63225—41no—
163#63225—41no—
164#63225D811yescharacteristic
165#63225D811yescharacteristic
166C4×Dic17—41no—
167C4×Dic17—41no—
168C4×Dic17—41no—
169C4×Dic17—41no—
170C4×Dic17—41no—
171C4×Dic17C2×D411yescharacteristic
172C2×#3622C2×D411yescharacteristic
173C2×#3622C2×D411yescharacteristic
174Dic68—21no—
175Dic68C22×C412yes—
176D136—21no—
177D136C22×C412yes—
178#63218—22no—
179C8×D17C22×C412yes—
180C8×D17#4112yes—
181C16×C17—21no—
182C16×C17C22×C412yes—
183#63214—21no—
184#63214C22×C412yes—
185#33457—171nomaximal, Sylow
186C23×C17—41no—
187C22×D17—41no—
188C22×D17—44no—
189C22×D17—161no—
190C22×D17—161no—
191C22×D17C2×D811yescharacteristic
192Q8×C17—22no—
193Q8×C17—41no—
194D4×C17—22no—
195D4×C17—41no—
196C2×C4×C17—41no—
197C2×C4×C17—41no—
198C2×C4×C17—41no—
199C2×C4×C17—161no—
200C2×C4×C17C2×D811yescharacteristic
201C2×C4×C17C2×D811yescharacteristic
202C2×C4×C17C22×D411yescharacteristic
203C22⋊D17—44no—
204C2×Dic17—41no—
205C2×Dic17—41no—
206C2×Dic17—41no—
207C2×Dic17—161no—
208C2×Dic17C2×D811yescharacteristic
209C2×Dic17C2×D811yescharacteristic
210C2×Dic17C2×D811yescharacteristic
211D68—22no—
212D68—41no—
213C4×D17—44no—
214C4×D17—161no—
215C4×D17—161no—
216C4×D17C4×D412yes—
217C4×D17#13512yes—
218Dic34—22no—
219Dic34—41no—
220Dic34—44no—
221C8×C17—21no—
222C8×C17C23×C411yescharacteristic
223C8×C17C2×#4111yescharacteristic
224#3622—21no—
225#3622C4×D412yes—
226C22×SD32—171no—
227#2950—171no—
228#2948—171no—
229#2175—171no—
230#2175—171no—
231#2175—171no—
232#2175D1711yescharacteristic, complemented
233C4×SD32—171no—
234C4×SD32—171no—
235C4×SD32—171no—
236C4×SD32—171no—
237#2158—171no—
238#2141—171no—
239#2140—171no—
240#2108—171no—
241C22×C17—42no—
242C22×C17—81no—
243C22×C17—81no—
244C22×C17C22×D811yescharacteristic
245C2×D17—21no—
246C2×D17—21no—
247C2×D17—42no—
248C2×D17—81no—
249C2×D17—81no—
250C2×D17—164no—
251C2×D17C4×D812yes—
252C4×C17—21no—
253C4×C17—21no—
254C4×C17—42no—
255C4×C17—81no—
256C4×C17—81no—
257C4×C17C2×C4×D411yescharacteristic
258C4×C17C2×#13511yescharacteristic
259Dic17—21no—
260Dic17—21no—
261Dic17—42no—
262Dic17—81no—
263Dic17—81no—
264Dic17C4×D812yes—
265C22×Q16—171no—
266C22×D8—171no—
267#519—341no—
268#518—341no—
269C2×SD32—171no—
270C2×SD32—171no—
271C2×SD32—178no—
272C2×SD32—171no—
273C2×SD32C2×D1711yescharacteristic
274C2×M32—171no—
275C22×C16—171no—
276#442—341no—
277#442—341no—
278#441—341no—
279#441—341no—
280C4×Q16—171no—
281C4×Q16—171no—
282C4×Q16—171no—
283C4×Q16C2×D1711yescharacteristic, complemented
284C4×D8—171no—
285C4×D8—171no—
286C4×D8—171no—
287C4×D8C2×D1711yescharacteristic
288#427—171no—
289#427—171no—
290#419—341no—
291#418—341no—
292#405—171no—
293#367—171no—
294#367—171no—
295#366—171no—
296#366C2×D1711yescharacteristic, complemented
297#358—171no—
298#358—171no—
299#358—171no—
300#358C2×D1711yescharacteristic, complemented
301#357—171no—
302#357—171no—
303#357—171no—
304#357C2×D1711yescharacteristic
305#346—171no—
306#346C2×D1711yescharacteristic, complemented
307C4×C16—171no—
308C4×C16—171no—
309C2×C17—82no—
310C2×C17C2×C4×D811yescharacteristic
311C2×C17#341611yescharacteristic
312C2×C17#341611yescharacteristic
313D17—22no—
314D17—82no—
315C22×Q8—341no—
316C22×D4—341no—
317C2×Q16—174no—
318C2×Q16—171no—
319C2×Q16—342no—
320C2×Q16—341no—
321C2×Q16C22×D1711yescharacteristic
322C2×D8—174no—
323C2×D8—171no—
324C2×D8—342no—
325C2×D8—341no—
326C2×D8C22×D1711yescharacteristic
327C2×M16—171no—
328C22×C8—171no—
329C4×Q8—21no—
330C4×Q8—341no—
331C4×Q8—341no—
332C4×Q8—341no—
333C4×Q8—681no—
334C4×Q8—681no—
335C4×D4—21no—
336C4×D4—341no—
337C4×D4—341no—
338C4×D4—341no—
339C4×D4—681no—
340C4×D4—681no—
341#117—171no—
342#117—341no—
343#117—341no—
344#117—681no—
345#117—681no—
346C2×C22⋊C4—681no—
347C2×C22⋊C4—681no—
348SD32—174no—
349SD32—174no—
350SD32—174no—
351SD32C4×D1714yescomplemented
352M32—342no—
353C2×C16—171no—
354C2×C16—174no—
355C2×C16—171no—
356C2×C16C22×D1711yescharacteristic
357C2×C16C22×D1711yescharacteristic
358#107—171no—
359#107—171no—
360#107C22×D1711yescharacteristic
361#107C22×D1711yescharacteristic
362#106—341no—
363#106—341no—
364#103—21no—
365#103—341no—
366#103—341no—
367#103—341no—
368#102—21no—
369#102—341no—
370#102—341no—
371#102—341no—
372#97—171no—
373#97—171no—
374C4×C8—171no—
375C4×C8C22×D1711yescharacteristic
376C17#3345711yescharacteristic, complemented, Sylow
377C24—681no—
378C2×Q8—21no—
379C2×Q8—344no—
380C2×Q8—341no—
381C2×Q8—681no—
382C2×Q8—682no—
383C2×D4—21no—
384C2×D4—344no—
385C2×D4—341no—
386C2×D4—682no—
387C2×D4—681no—
388C22×C4—41no—
389C22×C4—171no—
390C22×C4—681no—
391C22×C4—681no—
392C22×C4—681no—
393C22×C4—681no—
394Q16—21no—
395Q16—172no—
396Q16—344no—
397Q16—341no—
398Q16C2×C4×D1712yes—
399D8—21no—
400D8—172no—
401D8—344no—
402D8—341no—
403D8C2×C4×D1712yes—
404M16—342no—
405C2×C8—171no—
406C2×C8—172no—
407C2×C8—172no—
408C2×C8—171no—
409C2×C8D4×D1711yescharacteristic
410C2×C8C23×D1711yescharacteristic
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—41no—
414C4⋊C4—171no—
415C4⋊C4—171no—
416C4⋊C4—341no—
417C4⋊C4—341no—
418C4⋊C4—681no—
419C4⋊C4—681no—
420C4⋊C4—681no—
421C22⋊C4—41no—
422C22⋊C4—171no—
423C22⋊C4—171no—
424C22⋊C4—681no—
425C22⋊C4—681no—
426C22⋊C4—681no—
427C22⋊C4—681no—
428C22⋊C4—681no—
429C42—41no—
430C42—171no—
431C42—681no—
432C42—681no—
433C42—681no—
434C42—681no—
435C42—681no—
436C42D4×D1711yescharacteristic
437C16—21no—
438C16—172no—
439C16—341no—
440C16C2×C4×D1712yes—
441C23—41no—
442C23—2721no—
443C23—2721no—
444C23—171no—
445C23—684no—
446C23—681no—
447Q8—22no—
448Q8—41no—
449Q8—342no—
450Q8—684no—
451Q8—681no—
452D4—22no—
453D4—41no—
454D4—342no—
455D4—684no—
456D4—681no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—161no—
461C2×C4—2721no—
462C2×C4—2721no—
463C2×C4—2721no—
464C2×C4—171no—
465C2×C4—172no—
466C2×C4—172no—
467C2×C4—171no—
468C2×C4—171no—
469C2×C4—684no—
470C2×C4—681no—
471C2×C4—681no—
472C2×C4—681no—
473C2×C4D8×D1711yescharacteristic
474C2×C4D8×D1711yescharacteristic
475C2×C4C2×D4×D1711yescharacteristic
476C8—21no—
477C8—172no—
478C8—341no—
479C8C22×C4×D1711yescharacteristic
480C8#41×D1711yescharacteristic
481C22—42no—
482C22—81no—
483C22—81no—
484C22—2724no—
485C22—172no—
486C22—341no—
487C22—341no—
488C22—682no—
489C22—1361no—
490C22—1361no—
491C22C2×D8×D1711yescharacteristic
492C4—21no—
493C4—21no—
494C4—42no—
495C4—81no—
496C4—81no—
497C4—172no—
498C4—341no—
499C4—341no—
500C4—682no—
501C4—1361no—
502C4—1361no—
503C4C4×D4×D1711yescharacteristic
504C4#135×D1711yescharacteristic
505C2—82no—
506C2—342no—
507C2—1362no—
508C2C4×D8×D1711yescharacteristic
509C2#1454192911yescharacteristic
510C2#1454193011yescharacteristic
subgroupG/H
centerC22C2×D8×D17
commutatorC8×C17C23×C4
FrattiniC2×C8C23×D17
FittingC17×#2175C2
socleC22×C17C22×D8
prime radicalC2×D8×D17C22
cosocleC2×C8×C17C24
hypercenter#2175D17
hypocenterC17#33457
LCSG ▷ C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2175

Complements

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

NH ≅ G/Ntype
2. #14537647C2semidirect
4. #14530650C2semidirect
5. #14530649C2semidirect
6. #14530647C2semidirect
7. #14530639C2semidirect
8. C4×#10587499C2semidirect
9. C4×#10587498C2semidirect
10. #14530633C2semidirect
11. #14530565C2semidirect
12. #14530502C2semidirect
14. #14530382C2semidirect
15. C17×#2175C2semidirect
23. SD32×D17C4semidirect
30. C4×Dic68C22semidirect
31. Q16×Dic17C22semidirect
32. #11762903C22semidirect
41. #11762636C22semidirect
42. #11762636C22semidirect
43. #11762635C22semidirect
44. #11762614C22semidirect
45. #11762613C22semidirect
46. #11762612C22semidirect
50. #11762584C22semidirect
51. C4×#63214C22semidirect
52. C16×Dic17C22semidirect
54. C4×Q16×C17C22semidirect
56. C17×#366C22semidirect
57. C17×#358C22semidirect
59. C17×#346C22semidirect
73. Q16×D17C2×C4semidirect
93. C17×SD32C2×C4semidirect
103. #10587499C2×C4semidirect
104. #10587498C2×C4semidirect
118. #10587471C2×C4semidirect
120. C16×D17C2×C4semidirect
232. #2175D17semidirect
283. C4×Q16C2×D17semidirect
296. #366C2×D17semidirect
300. #358C2×D17semidirect
306. #346C2×D17semidirect
351. SD32C4×D17semidirect
376. C17#33457semidirect

Maximal subgroups

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

subgroupindex
C2×SD32×D172
#145376472
#145376312
#145306502
#145306492
#145306472
#145306392
C4×#105874992
C4×#105874982
#145306332
#145305652
#145305022
#145304982
#145303822
C17×#21752
#3345717