Finite Group Databasebeta

← #17012931

Subgroups — #17012931

Order 4352.

932 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14539905C211yesmaximal, characteristic
2#14538815C211yesmaximal, characteristic
3#14538256C211yesmaximal, characteristic
4#14528026C214yesmaximal, complemented
5#14527880C214yesmaximal, complemented
6#14527456C211yesmaximal, characteristic, complemented
7#14527213C211yesmaximal, characteristic, complemented
8#14527179C211yesmaximal, characteristic, complemented
9C2×C17×#557C211yesmaximal, characteristic, complemented
10C2×#10587628—21no—
11#11763359C2211yescharacteristic
12#11763212C2212yes—
13#11763157C2212yes—
14#11763127—21no—
15#11763122—22no—
16#11763094C2212yes—
17#11763086—22no—
18#11763031—22no—
19#11763019C2212yes—
20#11763009C2212yes—
21#11763009C2212yes—
22#11762948—21no—
23C4×#63230—22no—
24#11762862C2211yescharacteristic
25#11762862C2211yescharacteristic
26#11762833C2211yescharacteristic
27#11762808—21no—
28#11762555C2214yes—
29Q8⋊#3622C2214yes—
30Q8⋊#3622C2214yes—
31C17×#557—22no—
32C17×#557—22no—
33C17×#557—22no—
34C17×#557C2214yes—
35C2×C17×C4⋊Q8—21no—
36C2×C17×C4⋊Q8C2211yescharacteristic
37C2×C17×#125—22no—
38C2×C17×#125—21no—
39C2×C4×Q8×C17—21no—
40C2×C4×Q8×C17—21no—
41C2×C4×Q8×C17C2211yescharacteristic
42C2×C4×Q8×C17C2211yescharacteristic
43C22×Q8×C17—21no—
44C22×Q8×C17D411yescharacteristic
45C22×Q8×C17D411yescharacteristic
46C2×#63252—21no—
47C17×C4⋊Q8—24no—
48C17×C4⋊Q8—24no—
49C17×C4⋊Q8—21no—
50C17×C4⋊Q8—21no—
51C17×C4⋊Q8D412yes—
52C17×C4⋊Q8C2312yes—
53C17×#125—28no—
54C17×#125—24no—
55C17×#125—28no—
56C17×#125—24no—
57C4×Q8×C17—24no—
58C4×Q8×C17—24no—
59C4×Q8×C17—24no—
60C4×Q8×C17—24no—
61C4×Q8×C17—21no—
62C4×Q8×C17—21no—
63C4×Q8×C17—21no—
64C4×Q8×C17—21no—
65C4×Q8×C17D412yes—
66C4×Q8×C17D412yes—
67C4×Q8×C17C2312yes—
68C4×Q8×C17C2312yes—
69C2×C4⋊C4×C17—22no—
70C2×C4⋊C4×C17—22no—
71C2×C4⋊C4×C17—22no—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17—21no—
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17D411yescharacteristic
76C2×C4⋊C4×C17D411yescharacteristic
77C2×C4⋊C4×C17D412yes—
78C2×C42×C17—22no—
79C2×C42×C17—21no—
80C2×C42×C17—21no—
81C2×C42×C17C2311yescharacteristic
82C2×#63231—22no—
83#10587628—44no—
84C2×#63230—22no—
85C2×#63230—22no—
86#10587616—21no—
87#10587616—21no—
88C2×#63223—21no—
89#10587561—21no—
90#10587560C2311yescharacteristic
91C4×D68—21no—
92C4×D68—21no—
93#10587555C2311yescharacteristic
94#10587506—22no—
95C22⋊#3622—21no—
96C22⋊#3622—21no—
97#10587481—22no—
98#10587480—22no—
99#10587480—22no—
100#10587480—24no—
101#10587479—22no—
102#10587478—24no—
103C4⋊#3622—21no—
104C4⋊#3622—21no—
105C4⋊#3622C2312yes—
106C4⋊#3622C2312yes—
107C4×#3622—21no—
108C4×#3622C2312yes—
109C2×Q8×C17—24no—
110C2×Q8×C17—21no—
111C2×Q8×C17—21no—
112C2×Q8×C17—22no—
113C2×Q8×C17—22no—
114C2×Q8×C17—22no—
115C2×Q8×C17—22no—
116C2×Q8×C17—42no—
117C2×Q8×C17—42no—
118C2×Q8×C17C2×D412yes—
119C2×Q8×C17C2×D412yes—
120C22×C4×C17—21no—
121C22×C4×C17—22no—
122C22×C4×C17—21no—
123C22×C4×C17—21no—
124C22×C4×C17—21no—
125C22×C4×C17C2×D411yescharacteristic
126C22×C4×C17C2×D411yescharacteristic
127C22×C4×C17C2×D411yescharacteristic
128C2×C22⋊D17—41no—
129#63252—42no—
130#63252—42no—
131C2×D68—21no—
132C2×D68—21no—
133C2×D68—21no—
134C2×C4×D17—41no—
135C2×Dic34—21no—
136C4⋊C4×C17—28no—
137C4⋊C4×C17—28no—
138C4⋊C4×C17—28no—
139C4⋊C4×C17—22no—
140C4⋊C4×C17—22no—
141C4⋊C4×C17—22no—
142C4⋊C4×C17—21no—
143C4⋊C4×C17—22no—
144C4⋊C4×C17—21no—
145C4⋊C4×C17—22no—
146C4⋊C4×C17—22no—
147C4⋊C4×C17—22no—
148C4⋊C4×C17C2×D414yes—
149C4⋊C4×C17C2×D412yes—
150C4⋊C4×C17C2×D412yes—
151C42×C17—28no—
152C42×C17—22no—
153C42×C17—22no—
154C42×C17—22no—
155C42×C17—22no—
156C42×C17C2×D411yescharacteristic
157C42×C17C2×D411yescharacteristic
158C42×C17C2×D411yescharacteristic
159C42×C17C2411yescharacteristic
160#63231—44no—
161#63230—44no—
162#63230—44no—
163#63227—41no—
164#63227—41no—
165C4⋊Dic17—21no—
166C4⋊Dic17—21no—
167#63223—42no—
168C2×#3622—22no—
169C2×#3622—22no—
170C2×#3622—22no—
171#18760—171nomaximal, Sylow
172C23×C17#12011yescharacteristic
173C22×D17—41no—
174Q8×C17—24no—
175Q8×C17—22no—
176Q8×C17—22no—
177Q8×C17—22no—
178Q8×C17—42no—
179Q8×C17—42no—
180Q8×C17—42no—
181Q8×C17—44no—
182Q8×C17#13714yescomplemented
183C2×C4×C17—24no—
184C2×C4×C17—22no—
185C2×C4×C17—22no—
186C2×C4×C17—22no—
187C2×C4×C17—22no—
188C2×C4×C17—22no—
189C2×C4×C17—24no—
190C2×C4×C17—41no—
191C2×C4×C17—42no—
192C2×C4×C17—42no—
193C2×C4×C17—42no—
194C2×C4×C17—44no—
195C2×C4×C17#12011yescharacteristic
196C2×C4×C17#12011yescharacteristic
197C2×C4×C17#12011yescharacteristic
198C2×C4×C17#13512yes—
199C2×C4×C17AGL(1,ℤ/8)12yes—
200C2×C4×C17C22×D411yescharacteristic
201C2×C4×C17C22×D411yescharacteristic
202C2×C4×C17C22×D411yescharacteristic
203C22⋊D17—82no—
204C2×Dic17—41no—
205D68—42no—
206D68—41no—
207D68—41no—
208D68—42no—
209C4×D17—82no—
210Dic34—41no—
211Dic34—41no—
212#3622—42no—
213#3622—42no—
214#3622—42no—
215C2×#557D1711yescharacteristic, complemented
216#3226—171no—
217#3103—171no—
218#3037—171no—
219#1648—174no—
220#1626—174no—
221#1546—171no—
222#1487—171no—
223#1479—171no—
224C22×C17—21no—
225C22×C17—21no—
226C22×C17#44912yes—
227C22×C17C2×#12011yescharacteristic
228C2×D17—81no—
229C2×D17—81no—
230C2×D17—81no—
231C4×C17—24no—
232C4×C17—21no—
233C4×C17—21no—
234C4×C17—21no—
235C4×C17—42no—
236C4×C17—44no—
237C4×C17—42no—
238C4×C17—42no—
239C4×C17#44912yes—
240C4×C17C2×#13511yescharacteristic
241C4×C17C2×AGL(1,ℤ/8)11yescharacteristic
242C4×C17C2×#13712yes—
243Dic17—81no—
244C2×#137—341no—
245#557—22no—
246#557—22no—
247#557—22no—
248#557C2×D1714yes—
249#532—171no—
250C2×C4⋊Q8—21no—
251C2×C4⋊Q8C2×D1711yescharacteristic
252C2×#125—22no—
253C2×#125—21no—
254C2×C4×Q8—21no—
255C2×C4×Q8—21no—
256C2×C4×Q8C2×D1711yescharacteristic
257C2×C4×Q8C2×D1711yescharacteristic
258#500—172no—
259#486—172no—
260#481—341no—
261#480—342no—
262#475—172no—
263#474—342no—
264#464—342no—
265#462—172no—
266#460—172no—
267#460—172no—
268#448—341no—
269C4×SD16—342no—
270#431—171no—
271#431—171no—
272#423—171no—
273#417—341no—
274#332—174no—
275Q8⋊C8—174no—
276Q8⋊C8—174no—
277C2×C17—22no—
278C2×C17C2×#44911yescharacteristic
279C2×C17#300411yescharacteristic
280C2×C17#302011yescharacteristic
281D17—161no—
282C2×#41—341no—
283C22×Q8—21no—
284C22×Q8C22⋊D1711yescharacteristic
285C22×Q8C22⋊D1711yescharacteristic
286#137—684no—
287C2×Q16—342no—
288C2×SD16—342no—
289C2×SD16—342no—
290C2×M16—341no—
291C4⋊Q8—24no—
292C4⋊Q8—24no—
293C4⋊Q8—21no—
294C4⋊Q8—21no—
295C4⋊Q8—171no—
296C4⋊Q8C22⋊D1712yes—
297C4⋊Q8C22×D1712yes—
298#127—171no—
299#125—28no—
300#125—24no—
301#125—28no—
302#125—24no—
303#124—341no—
304#121—341no—
305#121—341no—
306C4×Q8—24no—
307C4×Q8—24no—
308C4×Q8—24no—
309C4×Q8—24no—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×Q8—21no—
313C4×Q8—21no—
314C4×Q8C22⋊D1712yes—
315C4×Q8C22⋊D1712yes—
316C4×Q8C22×D1712yes—
317C4×Q8C22×D1712yes—
318C4×D4—341no—
319C4×D4—341no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4C22⋊D1711yescharacteristic
327C2×C4⋊C4C22⋊D1711yescharacteristic
328C2×C4⋊C4C22⋊D1712yes—
329C2×C42—22no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42C22×D1711yescharacteristic
333#107—344no—
334#106—342no—
335C4⋊C8—172no—
336C4⋊C8—172no—
337C4⋊C8—341no—
338C4⋊C8—341no—
339#103—342no—
340#103—342no—
341#102—342no—
342#102—342no—
343#102—344no—
344C22⋊C8—341no—
345C22⋊C8—341no—
346C4×C8—172no—
347C4×C8—341no—
348C17#1876011yescharacteristic, complemented, Sylow
349#41—682no—
350#41—682no—
351C2×Q8—24no—
352C2×Q8—22no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—341no—
361C2×Q8C2×C22⋊D1712yes—
362C2×Q8C2×C22⋊D1712yes—
363C2×D4—341no—
364C2×D4—341no—
365C2×D4—341no—
366C2×D4—681no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—681no—
373C22×C4C2×C22⋊D1711yescharacteristic
374C22×C4C2×C22⋊D1711yescharacteristic
375C22×C4C2×C22⋊D1711yescharacteristic
376Q16—684no—
377SD16—684no—
378SD16—684no—
379M16—682no—
380C2×C8—342no—
381C2×C8—342no—
382C2×C8—342no—
383C4⋊C4—28no—
384C4⋊C4—28no—
385C4⋊C4—28no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—22no—
395C4⋊C4—341no—
396C4⋊C4—341no—
397C4⋊C4C2×C22⋊D1712yes—
398C4⋊C4C2×C22⋊D1712yes—
399C4⋊C4C2×C22⋊D1714yes—
400C22⋊C4—681no—
401C22⋊C4—681no—
402C42—28no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42C2×C22⋊D1711yescharacteristic
408C42C2×C22⋊D1711yescharacteristic
409C42C2×C22⋊D1711yescharacteristic
410C42C23×D1711yescharacteristic
411C23—681no—
412C23#1058763911yescharacteristic
413Q8—24no—
414Q8—22no—
415Q8—22no—
416Q8—22no—
417Q8—42no—
418Q8—44no—
419Q8—42no—
420Q8—42no—
421Q8—681no—
422Q8—681no—
423Q8#1058762814yescomplemented
424D4—682no—
425D4—681no—
426D4—681no—
427D4—682no—
428D4—1362no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—681no—
442C2×C4—1362no—
443C2×C4#1058763512yes—
444C2×C4#1058763612yes—
445C2×C4#1058763911yescharacteristic
446C2×C4#1058763911yescharacteristic
447C2×C4#1058763911yescharacteristic
448C2×C4C22×C22⋊D1711yescharacteristic
449C2×C4C22×C22⋊D1711yescharacteristic
450C2×C4C22×C22⋊D1711yescharacteristic
451C8—682no—
452C8—682no—
453C8—682no—
454C22—21no—
455C22—21no—
456C22—1361no—
457C22—1361no—
458C22—1361no—
459C22#1176295412yes—
460C22C2×#1058763911yescharacteristic
461C4—24no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—42no—
466C4—44no—
467C4—42no—
468C4—42no—
469C4—1361no—
470C4#1176295412yes—
471C4C2×#1058763611yescharacteristic
472C4C2×#1058763511yescharacteristic
473C4C2×#1058762812yes—
474C2—22no—
475C2—2721no—
476C2C2×#1176295411yescharacteristic
477C2#1453801811yescharacteristic
478C2#1453814811yescharacteristic
subgroupG/H
centerC22C2×#10587639
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
FittingC2×C17×#557C2
socleC22×C17C2×#120
prime radical#11763359C22
hypercenterC2×#557D17
hypocenterC17#18760
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
4. #14528026C2semidirect
5. #14527880C2semidirect
6. #14527456C2semidirect
7. #14527213C2semidirect
8. #14527179C2semidirect
9. C2×C17×#557C2semidirect
182. Q8×C17#137semidirect
215. C2×#557D17semidirect
348. C17#18760semidirect
423. Q8#10587628semidirect

Maximal subgroups

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

subgroupindex
#145399052
#145388152
#145382562
#145280262
#145278802
#145274562
#145272132
#145271792
C2×C17×#5572
#1876017