Finite Group Databasebeta

← #17043768

Subgroups — #17043768

Order 4352.

1474 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587564C211yesmaximal, characteristic
2#14531162C212yesmaximal
3#14529767C218yesmaximal, complemented
4#14528858C211yesmaximal, characteristic
5#14528856C212yesmaximal
6C17×#1819C211yesmaximal, characteristic
7C23×Dic34C411yescharacteristic, complemented
8C24×Dic17C2211yescharacteristic
9C2×#10587564—22no—
10C2×#10587564—22no—
11C2×#10587564—22no—
12C2×#10587564—22no—
13C2×#10587564C414yescomplemented
14C2×#10587564C2214yes—
15C2×#10587563—22no—
16C22×C4⋊Dic17C411yescharacteristic, complemented
17C22×#63225C411yescharacteristic, complemented
18C22×#63225C411yescharacteristic, complemented
19C22×C22⋊Dic17C2211yescharacteristic
20#11762670—22no—
21#11762667—22no—
22#11762667C2214yes—
23#11762665C2212yes—
24C22⋊C4×Dic17—22no—
25C22⋊C4×Dic17C2214yes—
26#11762576C2214yes—
27#11762575C2218yes—
28C22×C22⋊C4×C17C2211yescharacteristic
29C17×#379C2212yes—
30C17×#342C2214yes—
31C25×C17C2311yescharacteristic
32C23×C4×C17C2×C411yescharacteristic
33C23×C4×C17C2×C411yescharacteristic
34C23×Dic17—44no—
35C23×Dic17C2×C411yescharacteristic
36C23×Dic17C2×C411yescharacteristic
37C23×Dic17C2×C411yescharacteristic
38C23×Dic17C2×C411yescharacteristic
39C23×Dic17C2×C412yes—
40C23×Dic17D412yes—
41C23×Dic17D412yes—
42C23×Dic17D412yes—
43C23×Dic17D412yes—
44C23×Dic17C2312yes—
45C22×Dic34—21no—
46C22×Dic34—22no—
47C22×Dic34—21no—
48C22×Dic34—44no—
49C22×Dic34C2×C414yes—
50C2×C22⋊C4×C17—22no—
51C2×C22⋊C4×C17—22no—
52C2×C22⋊C4×C17—22no—
53C2×C22⋊C4×C17—22no—
54C2×C22⋊C4×C17C2×C412yes—
55C2×C22⋊C4×C17C2314yes—
56C2×C22⋊C4×C17C2312yes—
57C2×C22⋊Dic17—22no—
58C2×C22⋊Dic17—22no—
59C2×C22⋊Dic17—22no—
60C2×C22⋊Dic17—22no—
61C2×C22⋊Dic17C2×C412yes—
62C2×C22⋊Dic17C2314yes—
63C2×C22⋊Dic17C2312yes—
64C2×C4⋊Dic17—22no—
65C2×C4⋊Dic17—21no—
66C2×C4⋊Dic17—21no—
67C2×C4⋊Dic17C2×C414yes—
68C2×#63225—21no—
69C2×#63225—22no—
70C2×#63225—22no—
71C2×#63225—21no—
72C2×#63225—21no—
73C2×#63225—21no—
74C2×#63225—24no—
75C2×#63225C2×C414yes—
76C2×#63225C2×C414yes—
77C2×C4×Dic17—24no—
78#10587564—24no—
79#10587564—24no—
80#10587564—24no—
81#10587564—24no—
82#10587564—24no—
83#10587564—24no—
84#10587564—24no—
85#10587564—24no—
86#10587563—48no—
87#10587502—24no—
88#10587502—24no—
89C24×C17—24no—
90C24×C17—42no—
91C24×C17—42no—
92C24×C17C22×C411yescharacteristic
93C24×C17C22×C411yescharacteristic
94C24×C17C2×D412yes—
95C24×C17C2×D412yes—
96C24×C17C2411yescharacteristic
97C22×C4×C17—21no—
98C22×C4×C17—21no—
99C22×C4×C17—21no—
100C22×C4×C17—21no—
101C22×C4×C17—24no—
102C22×C4×C17—42no—
103C22×C4×C17—42no—
104C22×C4×C17C22×C412yes—
105C22×C4×C17C22×C412yes—
106C22×Dic17—24no—
107C22×Dic17—28no—
108C22×Dic17—21no—
109C22×Dic17—21no—
110C22×Dic17—21no—
111C22×Dic17—21no—
112C22×Dic17—24no—
113C22×Dic17—44no—
114C22×Dic17—44no—
115C22×Dic17—44no—
116C22×Dic17—44no—
117C22×Dic17—42no—
118C22×Dic17—44no—
119C22×Dic17—42no—
120C22×Dic17—42no—
121C22×Dic17C22⋊C411yescharacteristic
122C22×Dic17C22⋊C412yes—
123C22×Dic17C22⋊C412yes—
124C22×Dic17C22⋊C412yes—
125C22×Dic17C22⋊C412yes—
126C22×Dic17C22⋊C412yes—
127C22×Dic17C22⋊C412yes—
128C22×Dic17C22⋊C411yescharacteristic
129C22×Dic17C22⋊C411yescharacteristic
130C22×Dic17C22⋊C411yescharacteristic
131C22×Dic17C22×C412yes—
132C22×Dic17C22×C412yes—
133C22×Dic17C22×C412yes—
134C22×Dic17C22×C412yes—
135C22×Dic17C2×D412yes—
136C22×Dic17C2×D412yes—
137C22×Dic17C2×D412yes—
138C22×Dic17C2×D412yes—
139C2×Dic34—24no—
140C2×Dic34—24no—
141C2×Dic34—44no—
142C2×Dic34—48no—
143C2×Dic34—44no—
144C2×Dic34—48no—
145C22⋊C4×C17—24no—
146C22⋊C4×C17—24no—
147C22⋊C4×C17—28no—
148C22⋊C4×C17—44no—
149C22⋊C4×C17—44no—
150C22⋊Dic17—24no—
151C22⋊Dic17—24no—
152C22⋊Dic17—28no—
153C22⋊Dic17—44no—
154C22⋊Dic17—44no—
155C4⋊Dic17—24no—
156C4⋊Dic17—24no—
157#63225—24no—
158#63225—24no—
159#63225—24no—
160#63225—24no—
161#63225—48no—
162C4×Dic17—48no—
163#21825—171nomaximal, Sylow
164C23×C17—22no—
165C23×C17—22no—
166C23×C17—22no—
167C23×C17—22no—
168C23×C17—24no—
169C23×C17—24no—
170C23×C17—24no—
171C23×C17—24no—
172C23×C17—41no—
173C23×C17—42no—
174C23×C17—41no—
175C23×C17—41no—
176C23×C17—42no—
177C23×C17—41no—
178C23×C17—416no—
179C23×C17C2×C22⋊C412yes—
180C23×C17C2×C22⋊C411yescharacteristic
181C23×C17C2×C22⋊C411yescharacteristic
182C23×C17C23×C411yescharacteristic
183C23×C17C22×D412yes—
184C23×C172−1+414yes—
185C2×C4×C17—22no—
186C2×C4×C17—22no—
187C2×C4×C17—42no—
188C2×C4×C17—42no—
189C2×C4×C17—42no—
190C2×C4×C17—42no—
191C2×C4×C17—42no—
192C2×C4×C17—42no—
193C2×C4×C17—44no—
194C2×C4×C17—44no—
195C2×C4×C17—44no—
196C2×Dic17—22no—
197C2×Dic17—24no—
198C2×Dic17—24no—
199C2×Dic17—24no—
200C2×Dic17—24no—
201C2×Dic17—24no—
202C2×Dic17—24no—
203C2×Dic17—24no—
204C2×Dic17—22no—
205C2×Dic17—22no—
206C2×Dic17—42no—
207C2×Dic17—42no—
208C2×Dic17—41no—
209C2×Dic17—41no—
210C2×Dic17—42no—
211C2×Dic17—42no—
212C2×Dic17—42no—
213C2×Dic17—42no—
214C2×Dic17—42no—
215C2×Dic17—42no—
216C2×Dic17—42no—
217C2×Dic17—42no—
218C2×Dic17—42no—
219C2×Dic17—44no—
220C2×Dic17—44no—
221C2×Dic17—44no—
222C2×Dic17#9914yes—
223C2×Dic17C2×C22⋊C412yes—
224C2×Dic17C2×C22⋊C412yes—
225C2×Dic17C2×C22⋊C412yes—
226C2×Dic17C2×C22⋊C412yes—
227Dic34—48no—
228Dic34—48no—
229C22×C22⋊Q8—171no—
230#2280—172no—
231#1996—178no—
232#1819—172no—
233#1819—171no—
234#1819D1711yescharacteristic
235C22×C17—28no—
236C22×C17—21no—
237C22×C17—21no—
238C22×C17—21no—
239C22×C17—21no—
240C22×C17—22no—
241C22×C17—21no—
242C22×C17—21no—
243C22×C17—22no—
244C22×C17—22no—
245C22×C17—22no—
246C22×C17—22no—
247C22×C17—44no—
248C22×C17—48no—
249C22×C17—48no—
250C22×C17—44no—
251C22×C17C2×#9912yes—
252C22×C17#41012yes—
253C22×C17C22×C22⋊C411yescharacteristic
254C22×C17#51912yes—
255C22×C17#53614yes—
256C4×C17—42no—
257C4×C17—42no—
258C4×C17—84no—
259Dic17—22no—
260Dic17—22no—
261Dic17—22no—
262Dic17—42no—
263Dic17—42no—
264Dic17—84no—
265Dic17C2×#9914yes—
266C23×Q8—171no—
267C24×C4—171no—
268C2×C22⋊Q8—174no—
269C2×C22⋊Q8—174no—
270C2×C22⋊Q8—342no—
271C2×C22⋊Q8—342no—
272C2×C22⋊Q8—342no—
273C2×C22⋊Q8—342no—
274C2×#117—342no—
275C22×C4⋊C4—171no—
276C22×C4⋊C4—171no—
277C22×C4⋊C4—171no—
278C22×C22⋊C4—171no—
279C22×C22⋊C4C2×D1711yescharacteristic
280#381—342no—
281#380—174no—
282#380—342no—
283#379—172no—
284#379C2×D1712yes—
285C4×C22⋊C4—174no—
286C4×C22⋊C4—342no—
287#342—178no—
288#342—174no—
289#342C2×D1714yes—
290C2×C17—24no—
291C2×C17—22no—
292C2×C17—22no—
293C2×C17—22no—
294C2×C17—42no—
295C2×C17#231011yescharacteristic
296C2×C17C22×#9911yescharacteristic
297C2×C17C2×#41011yescharacteristic
298C25C22×D1711yescharacteristic
299C22×Q8—174no—
300C22×Q8—341no—
301C22×Q8—342no—
302C22×Q8—341no—
303C22×Q8—684no—
304C23×C4—171no—
305C23×C4—172no—
306C23×C4—171no—
307C23×C4—172no—
308C23×C4—172no—
309C23×C4—172no—
310C23×C4—172no—
311C23×C4—172no—
312C23×C4—171no—
313C23×C4—171no—
314C23×C4—684no—
315C23×C4C4×D1711yescharacteristic
316C23×C4C4×D1711yescharacteristic
317C22⋊Q8—344no—
318C22⋊Q8—344no—
319C22⋊Q8—344no—
320C22⋊Q8—344no—
321C22⋊Q8—344no—
322C22⋊Q8—344no—
323C22⋊Q8—344no—
324C22⋊Q8—344no—
325#117—688no—
326C2×C4⋊C4—174no—
327C2×C4⋊C4—174no—
328C2×C4⋊C4—174no—
329C2×C4⋊C4—341no—
330C2×C4⋊C4—342no—
331C2×C4⋊C4—341no—
332C2×C4⋊C4—342no—
333C2×C4⋊C4—341no—
334C2×C4⋊C4—341no—
335C2×C4⋊C4—342no—
336C2×C4⋊C4—341no—
337C2×C4⋊C4—341no—
338C2×C4⋊C4—344no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—172no—
344C2×C22⋊C4—172no—
345C2×C22⋊C4—174no—
346C2×C22⋊C4—342no—
347C2×C22⋊C4—342no—
348C2×C22⋊C4—342no—
349C2×C22⋊C4—342no—
350C2×C22⋊C4C4×D1712yes—
351C2×C22⋊C4C22×D1714yes—
352C2×C22⋊C4C22×D1712yes—
353C2×C42—344no—
354#95—344no—
355#95—344no—
356C17#2182511yescharacteristic, complemented, Sylow
357C24—24no—
358C24—42no—
359C24—42no—
360C24C2×C4×D1711yescharacteristic
361C24C2×C4×D1711yescharacteristic
362C24D4×D1712yes—
363C24D4×D1712yes—
364C24C23×D1711yescharacteristic
365C2×Q8—344no—
366C2×Q8—344no—
367C2×Q8—684no—
368C2×Q8—688no—
369C2×Q8—688no—
370C2×Q8—684no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—24no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—171no—
379C22×C4—172no—
380C22×C4—172no—
381C22×C4—172no—
382C22×C4—172no—
383C22×C4—172no—
384C22×C4—171no—
385C22×C4—172no—
386C22×C4—172no—
387C22×C4—172no—
388C22×C4—172no—
389C22×C4—172no—
390C22×C4—171no—
391C22×C4—171no—
392C22×C4—172no—
393C22×C4—172no—
394C22×C4—172no—
395C22×C4—172no—
396C22×C4—348no—
397C22×C4—344no—
398C22×C4—341no—
399C22×C4—341no—
400C22×C4—341no—
401C22×C4—341no—
402C22×C4—344no—
403C22×C4—684no—
404C22×C4—684no—
405C22×C4—684no—
406C22×C4—684no—
407C22×C4—684no—
408C22×C4—682no—
409C22×C4—682no—
410C22×C4—682no—
411C22×C4C2×C4×D1712yes—
412C22×C4C2×C4×D1712yes—
413C4⋊C4—344no—
414C4⋊C4—344no—
415C4⋊C4—344no—
416C4⋊C4—344no—
417C4⋊C4—344no—
418C4⋊C4—344no—
419C4⋊C4—688no—
420C22⋊C4—24no—
421C22⋊C4—24no—
422C22⋊C4—28no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—344no—
426C22⋊C4—344no—
427C22⋊C4—348no—
428C22⋊C4—684no—
429C22⋊C4—684no—
430C42—688no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—24no—
439C23—41no—
440C23—42no—
441C23—41no—
442C23—41no—
443C23—42no—
444C23—41no—
445C23—416no—
446C23C22⋊C4×D1712yes—
447C23C22⋊C4×D1711yescharacteristic
448C23C22⋊C4×D1711yescharacteristic
449C23C22×C4×D1711yescharacteristic
450C23C2×D4×D1712yes—
451C23#1058769714yes—
452Q8—688no—
453Q8—688no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—172no—
466C2×C4—174no—
467C2×C4—172no—
468C2×C4—172no—
469C2×C4—172no—
470C2×C4—344no—
471C2×C4—344no—
472C2×C4—344no—
473C2×C4—344no—
474C2×C4—344no—
475C2×C4—344no—
476C2×C4—342no—
477C2×C4—344no—
478C2×C4—342no—
479C2×C4—342no—
480C2×C4—682no—
481C2×C4—682no—
482C2×C4—682no—
483C2×C4—682no—
484C2×C4—682no—
485C2×C4—681no—
486C2×C4—681no—
487C2×C4—682no—
488C2×C4—682no—
489C2×C4—682no—
490C2×C4—682no—
491C2×C4—682no—
492C2×C4—682no—
493C2×C4—684no—
494C2×C4—684no—
495C2×C4—684no—
496C22—28no—
497C22—21no—
498C22—22no—
499C22—21no—
500C22—21no—
501C22—22no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—44no—
509C22—48no—
510C22—48no—
511C22—44no—
512C22#99×D1712yes—
513C22#1176278112yes—
514C22C2×C22⋊C4×D1711yescharacteristic
515C22#1176328612yes—
516C22#1176338114yes—
517C4—42no—
518C4—42no—
519C4—84no—
520C4—174no—
521C4—342no—
522C4—342no—
523C4—342no—
524C4—682no—
525C4—682no—
526C4—1364no—
527C2—24no—
528C2—22no—
529C2—22no—
530C2—22no—
531C2—42no—
532C2#1453136911yescharacteristic
533C2C2×#99×D1711yescharacteristic
534C2C2×#1176278111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D17
commutatorC23×C17C23×C4
FrattiniC24C23×D17
FittingC17×#1819C2
socleC22×C17C22×C22⋊C4
prime radicalC25×C17C23
cosocleC24×C17C24
hypercenter#1819D17
hypocenterC17#21825
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C25 ◁ #1819

Complements

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

NH ≅ G/Ntype
3. #14529767C2semidirect
7. C23×Dic34C4semidirect
13. C2×#10587564C4semidirect
16. C22×C4⋊Dic17C4semidirect
17. C22×#63225C4semidirect
18. C22×#63225C4semidirect
356. C17#21825semidirect

Maximal subgroups

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

subgroupindex
C22×#105875642
#145311622
#145297672
#145288582
#145288562
C17×#18192
#2182517