Finite Group Databasebeta

← #17019725

Subgroups — #17019725

Order 4352.

704 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14539722C211yesmaximal, characteristic, complemented
2#14538264C211yesmaximal, characteristic
3#14537895C211yesmaximal, characteristic, complemented
4#14528202C212yesmaximal, complemented
5#14528195C212yesmaximal, complemented
6#14528134C212yesmaximal, complemented
7#14528127C212yesmaximal, complemented
8#14527554C211yesmaximal, characteristic, complemented
9#14527474C211yesmaximal, characteristic, complemented
10C2×#11762552C211yesmaximal, characteristic, complemented
11C17×#3076C211yesmaximal, characteristic, complemented
12C2×#10587635—21no—
13C2×#10587560C2211yescharacteristic
14#11763183C2212yes—
15#11763157C2212yes—
16#11763009—21no—
17#11763009—21no—
18#11763009—21no—
19#11763009C2212yes—
20#11763004—21no—
21#11763004C2212yes—
22#11762959—22no—
23Q8×#3622C2212yescomplemented
24D4×#3622—22no—
25D4×#3622C2212yescomplemented
26#11762862—21no—
27#11762862C2211yescharacteristic
28#11762833C2211yescharacteristic
29C2×C4⋊#3622—21no—
30C2×C4⋊#3622C2211yescharacteristic
31#11762813—21no—
32#11762559C2212yescomplemented
33#11762559C2212yescomplemented
34#11762558C2212yescomplemented
35#11762555C2212yescomplemented
36#11762552C2212yes—
37#11762552C2212yes—
38C17×#533C2211yescharacteristic
39C4×#41×C17C2211yescharacteristic
40C17×#479—21no—
41C17×#478—21no—
42C17×#477C2211yescharacteristic, complemented
43C17×#476C2211yescharacteristic, complemented
44C17×#475C2211yescharacteristic, complemented
45C17×#474C2211yescharacteristic, complemented
46C17×#427—21no—
47C2×C17×C4⋊C8C2211yescharacteristic
48C17×#416—21no—
49C2×#41×C17D411yescharacteristic
50C22×D68—21no—
51C22×D68—21no—
52C22×D68—21no—
53C22×C8×C17—21no—
54C17×C4⋊Q8C2311yescharacteristic
55C17×C4⋊Q8C2311yescharacteristic
56C17×#125—21no—
57C17×C22⋊Q8—21no—
58C17×C22⋊Q8—21no—
59C4×Q8×C17—21no—
60C4×Q8×C17—21no—
61C4×Q8×C17D411yescharacteristic
62C4×Q8×C17C2311yescharacteristic
63C4×D4×C17—21no—
64C4×D4×C17—21no—
65C4×D4×C17D411yescharacteristic
66C4×D4×C17C2311yescharacteristic
67C17×#117—21no—
68C17×#117—21no—
69C17×#117D411yescharacteristic
70C2×C42×C17—21no—
71C2×C42×C17C2311yescharacteristic
72#10587635—44no—
73C2×#63230—22no—
74C2×#63230—22no—
75C2×#63230—22no—
76C2×#63228—22no—
77C2×#63223—21no—
78C22×#3622—22no—
79C22×#3622—21no—
80#10587560—21no—
81#10587560—21no—
82#10587560—21no—
83#10587560C2312yes—
84C17×#107—21no—
85C17×#107—21no—
86C17×#106—21no—
87C17×#106—21no—
88C17×C4⋊C8—21no—
89C17×C4⋊C8C2311yescharacteristic
90C17×C4⋊C8C2311yescharacteristic
91C17×#103—21no—
92C17×#103—21no—
93C17×#102—21no—
94C17×#102—21no—
95C22⋊#3622—22no—
96C22⋊#3622D412yes—
97#10587480—22no—
98#10587480—22no—
99#10587480—22no—
100#10587480—22no—
101C4⋊#3622—22no—
102C4⋊#3622—21no—
103C4⋊#3622—21no—
104C4⋊#3622D411yescharacteristic
105C4⋊#3622D411yescharacteristic
106C4⋊#3622C2312yes—
107C4⋊#3622C2312yes—
108C4×#3622—22no—
109C4×#3622C2312yes—
110C23×D17—41no—
111#41×C17—22no—
112#41×C17SD1614yescomplemented
113C2×Q8×C17—21no—
114C2×Q8×C17—21no—
115C2×Q8×C17C2×D411yescharacteristic
116C2×D4×C17—21no—
117C2×D4×C17C2×D411yescharacteristic
118C22×C4×C17—21no—
119C22×C4×C17—21no—
120C22×C4×C17—21no—
121C22×C4×C17C2×D411yescharacteristic
122C22×C4×C17C2×D411yescharacteristic
123C22×C4×C17#4111yescharacteristic
124C2×D68—22no—
125C2×D68—22no—
126C2×D68—22no—
127C2×D68—42no—
128C2×D68—42no—
129C2×D68—41no—
130C2×D68—41no—
131C2×D68—41no—
132C2×D68—81no—
133C2×D68—81no—
134C2×D68—82no—
135C2×C8×C17—21no—
136C2×C8×C17—21no—
137C2×C8×C17—42no—
138C4⋊C4×C17—21no—
139C4⋊C4×C17—21no—
140C4⋊C4×C17—21no—
141C4⋊C4×C17—21no—
142C4⋊C4×C17—21no—
143C4⋊C4×C17—21no—
144C4⋊C4×C17—41no—
145C4⋊C4×C17—41no—
146C4⋊C4×C17C2×D411yescharacteristic
147C4⋊C4×C17C2×D411yescharacteristic
148C22⋊C4×C17—21no—
149C22⋊C4×C17—21no—
150C22⋊C4×C17—21no—
151C22⋊C4×C17—41no—
152C42×C17—21no—
153C42×C17—21no—
154C42×C17—21no—
155C42×C17—41no—
156C42×C17C2×D411yescharacteristic
157C42×C17C2×D411yescharacteristic
158C42×C17#4111yescharacteristic
159C42×C17C2411yescharacteristic
160#63230—44no—
161#63230—82no—
162#63230—82no—
163#63228—44no—
164#63223—42no—
165C2×#3622—22no—
166C2×#3622—22no—
167C2×#3622—82no—
168C2×#3622—81no—
169C2×#3622C2×D412yes—
170C2×#3622C2×D412yes—
171#19409—171nomaximal, Sylow
172C23×C17—21no—
173C23×C17#12111yescharacteristic
174C22×D17—42no—
175C22×D17—82no—
176C22×D17—82no—
177C22×D17—82no—
178Q8×C17—21no—
179Q8×C17—41no—
180Q8×C17—41no—
181Q8×C17—81no—
182Q8×C17C2×SD1612yes—
183D4×C17—24no—
184D4×C17—21no—
185D4×C17C2×SD1612yes—
186C2×C4×C17—21no—
187C2×C4×C17—21no—
188C2×C4×C17—21no—
189C2×C4×C17—24no—
190C2×C4×C17—21no—
191C2×C4×C17—21no—
192C2×C4×C17—41no—
193C2×C4×C17—41no—
194C2×C4×C17—41no—
195C2×C4×C17—41no—
196C2×C4×C17—41no—
197C2×C4×C17—81no—
198C2×C4×C17#12111yescharacteristic
199C2×C4×C17#12111yescharacteristic
200C2×C4×C17#12111yescharacteristic
201C2×C4×C17C2×SD1612yes—
202C2×C4×C17#13512yes—
203C2×C4×C17C22×D411yescharacteristic
204C2×C4×C17C22×D411yescharacteristic
205C2×C4×C17C2×#4111yescharacteristic
206D68—42no—
207D68—42no—
208D68—42no—
209D68—42no—
210D68—82no—
211D68—82no—
212D68—82no—
213D68—82no—
214C8×C17—41no—
215C8×C17—41no—
216#3622—22no—
217#3622—22no—
218#3622—42no—
219#3622—42no—
220#3208—171no—
221#3076D1711yescharacteristic, complemented
222#3038—171no—
223#2988—171no—
224#1678—172no—
225#1677—172no—
226#1666—172no—
227#1665—172no—
228#1570—171no—
229#1550—171no—
230C2×#331—171no—
231C22×C17—21no—
232C22×C17—21no—
233C22×C17—22no—
234C22×C17—41no—
235C22×C17—41no—
236C22×C17#46512yes—
237C22×C17C2×#12111yescharacteristic
238C2×D17—82no—
239C2×D17—82no—
240C2×D17—82no—
241C2×D17—164no—
242C4×C17—21no—
243C4×C17—22no—
244C4×C17—21no—
245C4×C17—21no—
246C4×C17—41no—
247C4×C17—41no—
248C4×C17—81no—
249C4×C17—81no—
250C4×C17#46512yes—
251C4×C17C22×SD1611yescharacteristic
252C4×C17C2×#13511yescharacteristic
253C4×C17#57611yescharacteristic
254C4×C17#57711yescharacteristic
255C2×AGL(1,ℤ/8)—341no—
256#533C2×D1711yescharacteristic
257C2×#127—171no—
258C4×#41C2×D1711yescharacteristic
259#492—172no—
260#486—172no—
261#479—21no—
262#478—21no—
263#477C2×D1711yescharacteristic, complemented
264#476C2×D1711yescharacteristic, complemented
265#475C2×D1711yescharacteristic, complemented
266#474C2×D1711yescharacteristic, complemented
267#460—172no—
268#460—341no—
269#460—341no—
270#460—341no—
271#459—172no—
272#459—341no—
273#450—342no—
274C8×Q8—172no—
275C8×D4—172no—
276C8×D4—342no—
277#431—171no—
278#431—341no—
279#427—21no—
280#423—171no—
281C2×C4⋊C8—171no—
282C2×C4⋊C8—341no—
283C2×C4⋊C8C2×D1711yescharacteristic
284#418—341no—
285#416—21no—
286#334—172no—
287#334—172no—
288#333—172no—
289#332—172no—
290#331—172no—
291#331—172no—
292C2×C17—22no—
293C2×C17—42no—
294C2×C17#304511yescharacteristic
295C2×C17C2×#46511yescharacteristic
296C2×C17#306811yescharacteristic
297D17—162no—
298C2×#41C22⋊D1711yescharacteristic
299C22×D4—341no—
300C22×D4—341no—
301C22×D4—341no—
302AGL(1,ℤ/8)—684no—
303C2×SD16—342no—
304C2×SD16—342no—
305C2×SD16—342no—
306C2×D8—342no—
307C2×M16—341no—
308C22×C8—21no—
309C22×C8—342no—
310C22×C8—341no—
311C4⋊Q8C22×D1711yescharacteristic
312C4⋊Q8C22×D1711yescharacteristic
313#127—172no—
314#127—341no—
315#127—341no—
316#127—341no—
317#125—21no—
318C22⋊Q8—21no—
319C22⋊Q8—21no—
320C4×Q8—21no—
321C4×Q8—21no—
322C4×Q8C22⋊D1711yescharacteristic
323C4×Q8C22×D1711yescharacteristic
324C4×D4—21no—
325C4×D4—21no—
326C4×D4C22⋊D1711yescharacteristic
327C4×D4C22×D1711yescharacteristic
328#117—21no—
329#117—21no—
330#117C22⋊D1711yescharacteristic
331C2×C42—21no—
332C2×C42C22×D1711yescharacteristic
333#107—21no—
334#107—21no—
335#106—21no—
336#106—21no—
337C4⋊C8—21no—
338C4⋊C8—171no—
339C4⋊C8—171no—
340C4⋊C8—172no—
341C4⋊C8—172no—
342C4⋊C8—342no—
343C4⋊C8—341no—
344C4⋊C8—341no—
345C4⋊C8C22×D1711yescharacteristic
346C4⋊C8C22×D1711yescharacteristic
347#103—21no—
348#103—21no—
349#102—21no—
350#102—21no—
351#102—342no—
352#102—342no—
353#102—342no—
354#102—342no—
355C22⋊C8—172no—
356C22⋊C8—342no—
357C4×C8—172no—
358C4×C8—342no—
359C17#1940911yescharacteristic, complemented, Sylow
360C24—681no—
361#41—22no—
362#41#6323014yescomplemented
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8C2×C22⋊D1711yescharacteristic
366C2×D4—21no—
367C2×D4—342no—
368C2×D4—342no—
369C2×D4—342no—
370C2×D4—681no—
371C2×D4—681no—
372C2×D4—681no—
373C2×D4—682no—
374C2×D4—682no—
375C2×D4—1361no—
376C2×D4—1361no—
377C2×D4—1362no—
378C2×D4C2×C22⋊D1711yescharacteristic
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4D4×D1711yescharacteristic
383C22×C4#6325411yescharacteristic
384C22×C4C2×C22⋊D1711yescharacteristic
385SD16—684no—
386SD16—1362no—
387SD16—1362no—
388D8—684no—
389M16—682no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—42no—
393C2×C8—172no—
394C2×C8—172no—
395C2×C8—342no—
396C2×C8—342no—
397C2×C8—1362no—
398C2×C8—1361no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4C2×C22⋊D1711yescharacteristic
408C4⋊C4C2×C22⋊D1711yescharacteristic
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—41no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—41no—
417C42D4×D1711yescharacteristic
418C42#6325411yescharacteristic
419C42C2×C22⋊D1711yescharacteristic
420C42C23×D1711yescharacteristic
421C23—21no—
422C23—682no—
423C23—1362no—
424C23—1362no—
425C23—1362no—
426C23#1058762511yescharacteristic
427Q8—21no—
428Q8—41no—
429Q8—41no—
430Q8—81no—
431Q8C2×#6323012yes—
432D4—24no—
433D4—21no—
434D4—682no—
435D4—682no—
436D4—682no—
437D4—682no—
438D4—1362no—
439D4—1362no—
440D4—1362no—
441D4—1362no—
442D4C2×#6323012yes—
443C2×C4—21no—
444C2×C4—24no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—81no—
455C2×C4#1058762511yescharacteristic
456C2×C4#1058762511yescharacteristic
457C2×C4#1058762511yescharacteristic
458C2×C4C2×#6323012yes—
459C2×C4#1058763612yes—
460C2×C4C2×D4×D1711yescharacteristic
461C2×C4C2×#6325411yescharacteristic
462C2×C4C22×C22⋊D1711yescharacteristic
463C8—41no—
464C8—41no—
465C8—342no—
466C8—342no—
467C8—682no—
468C8—682no—
469C22—21no—
470C22—21no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22—2724no—
475C22—1362no—
476C22—1362no—
477C22—1362no—
478C22#1176304012yes—
479C22C2×#1058762511yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—22no—
484C4—41no—
485C4—41no—
486C4—81no—
487C4—81no—
488C4#1176304012yes—
489C4C22×#6323011yescharacteristic
490C4C2×#1058763611yescharacteristic
491C4#1176361111yescharacteristic
492C4#1176361611yescharacteristic
493C2—22no—
494C2—42no—
495C2—2722no—
496C2#1453832111yescharacteristic
497C2C2×#1176304011yescharacteristic
498C2#1453851411yescharacteristic
subgroupG/H
centerC22C2×#10587625
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingC17×#3076C2
socleC22×C17C2×#121
prime radical#14538264C2
hypercenter#3076D17
hypocenterC17#19409
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3076

Complements

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

NH ≅ G/Ntype
1. #14539722C2semidirect
3. #14537895C2semidirect
4. #14528202C2semidirect
5. #14528195C2semidirect
6. #14528134C2semidirect
7. #14528127C2semidirect
8. #14527554C2semidirect
9. #14527474C2semidirect
10. C2×#11762552C2semidirect
11. C17×#3076C2semidirect
23. Q8×#3622C22semidirect
25. D4×#3622C22semidirect
32. #11762559C22semidirect
33. #11762559C22semidirect
34. #11762558C22semidirect
35. #11762555C22semidirect
42. C17×#477C22semidirect
43. C17×#476C22semidirect
44. C17×#475C22semidirect
45. C17×#474C22semidirect
112. #41×C17SD16semidirect
221. #3076D17semidirect
263. #477C2×D17semidirect
264. #476C2×D17semidirect
265. #475C2×D17semidirect
266. #474C2×D17semidirect
359. C17#19409semidirect
362. #41#63230semidirect

Maximal subgroups

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

subgroupindex
#145397222
#145382642
#145378952
#145282022
#145281952
#145281342
#145281272
#145275542
#145274742
C2×#117625522
C17×#30762
#1940917