Finite Group Databasebeta

← #17954500

Subgroups — #17954500

Order 4352.

964 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊#3624C211yesmaximal, characteristic
2#14544606C211yesmaximal, characteristic
3C2×#11763654C212yesmaximal, complemented
4#14543622C212yesmaximal, complemented
5C2×#11762684C211yesmaximal, characteristic
6C23×#3624—21no—
7C2×C22⋊#3624—24no—
8C2×C22⋊#3624—21no—
9C2×C22⋊#3624—21no—
10C2×C22⋊#3624—21no—
11C2×C22⋊#3624C2212yes—
12C2×C22⋊#3624C2212yes—
13#11763654—24no—
14C24×Dic17C2211yescharacteristic
15C22×C4⋊Dic17C411yescharacteristic
16C22×C4⋊Dic17C2211yescharacteristic
17C22×C22⋊Dic17C411yescharacteristic
18#11762684—21no—
19#11762684—21no—
20#11762684—21no—
21#11762684—21no—
22C2×#10587502C411yescharacteristic
23C2×#10587502C2211yescharacteristic
24C22×C22⋊C4×C17C411yescharacteristic
25C25×C17C2×C411yescharacteristic
26C23×C4×C17C2×C411yescharacteristic
27C23×C4×C17C2×C411yescharacteristic
28C23×Dic17—22no—
29C23×Dic17—22no—
30C23×Dic17—42no—
31C23×Dic17—42no—
32C23×Dic17C2×C411yescharacteristic
33C23×Dic17C2×C411yescharacteristic
34C23×Dic17C2×C411yescharacteristic
35C23×Dic17D411yescharacteristic
36C23×Dic17D411yescharacteristic
37C23×Dic17D411yescharacteristic
38C23×Dic17D411yescharacteristic
39C23×Dic17C2311yescharacteristic
40C22×#3624—24no—
41C22×#3624—22no—
42C22×#3624—22no—
43C22×#3624—42no—
44C22×#3624—42no—
45C2×C22⋊C4×C17—21no—
46C2×C22⋊C4×C17—21no—
47C2×C22⋊C4×C17—22no—
48C2×C22⋊C4×C17C814yescomplemented
49C2×C22⋊Dic17—21no—
50C2×C22⋊Dic17—21no—
51C2×C22⋊Dic17—42no—
52C2×C4⋊Dic17—21no—
53C2×C4⋊Dic17—21no—
54C2×C4⋊Dic17—22no—
55C2×C4⋊Dic17—21no—
56C2×C4⋊Dic17—21no—
57C2×C4⋊Dic17—22no—
58C2×C4⋊Dic17—42no—
59C22⋊#3624—24no—
60C22⋊#3624—22no—
61C22⋊#3624—22no—
62C22⋊#3624—22no—
63C22⋊#3624—42no—
64C22⋊#3624—42no—
65C22⋊#3624D414yes—
66#10587502—21no—
67#10587502—21no—
68#10587502—21no—
69#10587502—21no—
70C24×C17—22no—
71C24×C17—22no—
72C24×C17—22no—
73C24×C17—22no—
74C24×C17—41no—
75C24×C17—41no—
76C24×C17C22⋊C411yescharacteristic
77C24×C17C22⋊C411yescharacteristic
78C24×C17C2×C812yes—
79C24×C17M1612yes—
80C24×C17C22×C411yescharacteristic
81C22×C4×C17—21no—
82C22×C4×C17—21no—
83C22×C4×C17—22no—
84C22×C4×C17—21no—
85C22×C4×C17—21no—
86C22×C4×C17—22no—
87C22×C4×C17—41no—
88C22×C4×C17—41no—
89C22×C4×C17C2×C812yescomplemented
90C22×C4×C17C2×C812yescomplemented
91C22×Dic17—22no—
92C22×Dic17—22no—
93C22×Dic17—21no—
94C22×Dic17—21no—
95C22×Dic17—22no—
96C22×Dic17—22no—
97C22×Dic17—22no—
98C22×Dic17—22no—
99C22×Dic17—21no—
100C22×Dic17—22no—
101C22×Dic17—22no—
102C22×Dic17—21no—
103C22×Dic17—21no—
104C22×Dic17—21no—
105C22×Dic17—22no—
106C22×Dic17—22no—
107C22×Dic17—41no—
108C22×Dic17—41no—
109C22×Dic17—41no—
110C22×Dic17—41no—
111C22×Dic17—41no—
112C22×Dic17—41no—
113C22×Dic17—41no—
114C22×Dic17—48no—
115C22×Dic17—48no—
116C22×Dic17—41no—
117C22×Dic17—41no—
118C22×Dic17—41no—
119C22×Dic17—41no—
120C22×Dic17C22⋊C411yescharacteristic
121C22×Dic17C22⋊C411yescharacteristic
122C22×Dic17C2×D411yescharacteristic
123C22×Dic17C2×D411yescharacteristic
124C22×Dic17C2×D412yes—
125C22×Dic17#4112yes—
126C2×#3624—24no—
127C2×#3624—48no—
128C2×#3624—42no—
129C2×#3624—42no—
130C2×#3624—42no—
131C2×#3624—42no—
132C2×#3624—42no—
133C22⋊C4×C17—24no—
134C22⋊C4×C17—24no—
135C22⋊Dic17—42no—
136C22⋊Dic17—42no—
137C4⋊Dic17—42no—
138C4⋊Dic17—42no—
139C4⋊Dic17—41no—
140C4⋊Dic17—42no—
141C4⋊Dic17—41no—
142#9081—171nomaximal, Sylow
143C23×C17—21no—
144C23×C17—21no—
145C23×C17—21no—
146C23×C17—21no—
147C23×C17—21no—
148C23×C17—21no—
149C23×C17—21no—
150C23×C17—21no—
151C23×C17—24no—
152C23×C17—24no—
153C23×C17—21no—
154C23×C17—21no—
155C23×C17—21no—
156C23×C17—21no—
157C23×C17—21no—
158C23×C17—21no—
159C23×C17—24no—
160C23×C17—24no—
161C23×C17—42no—
162C23×C17—42no—
163C23×C17—44no—
164C23×C17—44no—
165C23×C17—44no—
166C23×C17—44no—
167C23×C17C22⋊C812yes—
168C23×C17C22⋊C812yes—
169C23×C17#9912yes—
170C23×C17#10112yes—
171C23×C17C2×C22⋊C411yescharacteristic
172C23×C17C4×D412yes—
173C23×C17C22×C811yescharacteristic
174C23×C17C2×M1611yescharacteristic
175C23×C17#13112yes—
176C2×C4×C17—22no—
177C2×C4×C17—22no—
178C2×C4×C17—24no—
179C2×C4×C17—42no—
180C2×C4×C17—41no—
181C2×C4×C17—41no—
182C2×C4×C17—41no—
183C2×C4×C17—41no—
184C2×C4×C17—42no—
185C2×C4×C17—42no—
186C2×Dic17—24no—
187C2×Dic17—24no—
188C2×Dic17—21no—
189C2×Dic17—21no—
190C2×Dic17—21no—
191C2×Dic17—21no—
192C2×Dic17—21no—
193C2×Dic17—21no—
194C2×Dic17—21no—
195C2×Dic17—21no—
196C2×Dic17—21no—
197C2×Dic17—42no—
198C2×Dic17—42no—
199C2×Dic17—44no—
200C2×Dic17—44no—
201C2×Dic17—44no—
202C2×Dic17—42no—
203C2×Dic17—42no—
204C2×Dic17—41no—
205C2×Dic17—41no—
206C2×Dic17—41no—
207C2×Dic17—41no—
208C2×Dic17—41no—
209C2×Dic17—41no—
210C2×Dic17—82no—
211C2×Dic17—82no—
212C2×Dic17#9912yes—
213C2×Dic17#12011yescharacteristic
214C2×Dic17#12112yes—
215C2×Dic17#12311yescharacteristic
216#3624—44no—
217#3624—82no—
218C22×C22⋊C8—171no—
219C2×#385—171no—
220#1779—171no—
221C2×#324—172no—
222#1280—172no—
223C22×C17—22no—
224C22×C17—22no—
225C22×C17—22no—
226C22×C17—22no—
227C22×C17—22no—
228C22×C17—21no—
229C22×C17—21no—
230C22×C17—21no—
231C22×C17—21no—
232C22×C17—22no—
233C22×C17—22no—
234C22×C17—22no—
235C22×C17—22no—
236C22×C17—22no—
237C22×C17—22no—
238C22×C17—22no—
239C22×C17—22no—
240C22×C17—48no—
241C22×C17—48no—
242C22×C17—41no—
243C22×C17—41no—
244C22×C17—41no—
245C22×C17—41no—
246C22×C17#32414yescomplemented
247C22×C17#38611yescharacteristic
248C22×C17C2×C22⋊C811yescharacteristic
249C22×C17#40811yescharacteristic
250C22×C17C2×#9911yescharacteristic
251C22×C17C2×#9911yescharacteristic
252C22×C17C2×#10111yescharacteristic
253C22×C17#41311yescharacteristic
254C22×C17C8×D412yes—
255C22×C17#43512yes—
256C4×C17—42no—
257C4×C17—42no—
258Dic17—21no—
259Dic17—22no—
260Dic17—22no—
261Dic17—21no—
262Dic17—41no—
263Dic17—81no—
264Dic17—81no—
265C24×C4—171no—
266C23×C8—341no—
267C22×C4⋊C4—171no—
268C22×C4⋊C4—171no—
269C22×C22⋊C4—171no—
270C22×C22⋊C4#59511yescharacteristic
271C2×C22⋊C8—172no—
272C2×C22⋊C8—172no—
273C2×C22⋊C8—344no—
274C2×C22⋊C8—341no—
275C2×C22⋊C8—341no—
276C2×C22⋊C8—341no—
277#385—341no—
278#385—341no—
279#385—341no—
280#385—341no—
281C2×#95—171no—
282C2×#95—171no—
283#324—344no—
284C2×C17—24no—
285C2×C17—22no—
286C2×C17—22no—
287C2×C17—42no—
288C2×C17C2×#32411yescharacteristic
289C2×C17#145911yescharacteristic
290C2×C17#146811yescharacteristic
291C2×C17#147211yescharacteristic
292C2×C17#186911yescharacteristic
293C2×C17#189811yescharacteristic
294C2×C17#190111yescharacteristic
295C25C2×#59511yescharacteristic
296C23×C4—171no—
297C23×C4—171no—
298C23×C4—171no—
299C23×C4—171no—
300C23×C4—171no—
301C23×C4—171no—
302C23×C4—171no—
303C23×C4—171no—
304C23×C4—342no—
305C23×C4—342no—
306C23×C4—682no—
307C23×C4—682no—
308C23×C4C2×#59511yescharacteristic
309C23×C4C2×#59511yescharacteristic
310C22×C8—342no—
311C22×C8—344no—
312C22×C8—342no—
313C22×C8—682no—
314C22×C8—682no—
315C2×C4⋊C4—342no—
316C2×C4⋊C4—341no—
317C2×C4⋊C4—341no—
318C2×C4⋊C4—341no—
319C2×C4⋊C4—341no—
320C2×C4⋊C4—342no—
321C2×C4⋊C4—682no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—341no—
326C2×C22⋊C4—341no—
327C2×C22⋊C4—682no—
328C2×C22⋊C4#362414yescomplemented
329C22⋊C8—174no—
330C22⋊C8—342no—
331C22⋊C8—344no—
332C22⋊C8—342no—
333C22⋊C8—342no—
334C22⋊C8—682no—
335C22⋊C8—682no—
336#95—341no—
337#95—341no—
338#95—341no—
339#95—341no—
340C17#908111yescharacteristic, complemented, Sylow
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—41no—
346C24—41no—
347C24C2×#362412yes—
348C24#6324712yes—
349C24C22⋊#59511yescharacteristic
350C24C22⋊#59511yescharacteristic
351C24C22×#59511yescharacteristic
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—171no—
361C22×C4—171no—
362C22×C4—171no—
363C22×C4—172no—
364C22×C4—172no—
365C22×C4—171no—
366C22×C4—341no—
367C22×C4—342no—
368C22×C4—341no—
369C22×C4—341no—
370C22×C4—342no—
371C22×C4—342no—
372C22×C4—342no—
373C22×C4—342no—
374C22×C4—341no—
375C22×C4—342no—
376C22×C4—342no—
377C22×C4—342no—
378C22×C4—341no—
379C22×C4—341no—
380C22×C4—342no—
381C22×C4—342no—
382C22×C4—681no—
383C22×C4—681no—
384C22×C4—681no—
385C22×C4—681no—
386C22×C4—681no—
387C22×C4—681no—
388C22×C4—681no—
389C22×C4—688no—
390C22×C4—688no—
391C22×C4—681no—
392C22×C4—681no—
393C22×C4—681no—
394C22×C4—681no—
395C22×C4C2×#362412yescomplemented
396C22×C4C2×#362412yescomplemented
397C2×C8—344no—
398C2×C8—688no—
399C2×C8—682no—
400C2×C8—682no—
401C2×C8—682no—
402C2×C8—682no—
403C2×C8—682no—
404C4⋊C4—681no—
405C4⋊C4—682no—
406C4⋊C4—681no—
407C4⋊C4—682no—
408C4⋊C4—682no—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—682no—
412C22⋊C4—682no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—24no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—42no—
432C23—42no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23#1058753912yes—
438C23#1058754512yes—
439C23C22⋊#362412yes—
440C23C22⋊#362412yes—
441C23#1058767812yes—
442C23D4×#59512yes—
443C23C22×#362411yescharacteristic
444C23C2×#6324711yescharacteristic
445C23C2×C22⋊#59511yescharacteristic
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—171no—
457C2×C4—172no—
458C2×C4—171no—
459C2×C4—172no—
460C2×C4—344no—
461C2×C4—344no—
462C2×C4—341no—
463C2×C4—341no—
464C2×C4—341no—
465C2×C4—341no—
466C2×C4—341no—
467C2×C4—341no—
468C2×C4—341no—
469C2×C4—341no—
470C2×C4—341no—
471C2×C4—682no—
472C2×C4—682no—
473C2×C4—684no—
474C2×C4—684no—
475C2×C4—684no—
476C2×C4—682no—
477C2×C4—682no—
478C2×C4—681no—
479C2×C4—681no—
480C2×C4—681no—
481C2×C4—681no—
482C2×C4—681no—
483C2×C4—681no—
484C2×C4—1362no—
485C2×C4—1362no—
486C8—684no—
487C8—1362no—
488C22—22no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—48no—
506C22—48no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22#1176365414yescomplemented
512C22#1176375711yescharacteristic
513C22C2×C22⋊#362411yescharacteristic
514C22#1176378211yescharacteristic
515C22D17⋊#9911yescharacteristic
516C22C2×#1058753911yescharacteristic
517C22C2×#1058754511yescharacteristic
518C22#1176379611yescharacteristic
519C22D4×#362412yes—
520C22#1176384012yes—
521C4—42no—
522C4—42no—
523C4—341no—
524C4—342no—
525C4—341no—
526C4—342no—
527C4—681no—
528C4—1361no—
529C4—1361no—
530C2—22no—
531C2—22no—
532C2—24no—
533C2—42no—
534C2C2×#1176365411yescharacteristic
535C2#1454402011yescharacteristic
536C2#1454405111yescharacteristic
537C2#1454406311yescharacteristic
538C2#1454475711yescharacteristic
539C2#1454480611yescharacteristic
540C2#1454481111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC23×C17C22×C8
FrattiniC24C22×#595
FittingC22×C22⋊C4×C17C4
socleC23×C17C2×C22⋊C4
prime radicalC25×C17C2×C4
cosocleC23×Dic17C23
hypercenterC22×C22⋊C4#595
hypocenterC17#9081
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C25 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#11763654C2semidirect
4. #14543622C2semidirect
48. C2×C22⋊C4×C17C8semidirect
89. C22×C4×C17C2×C8semidirect
90. C22×C4×C17C2×C8semidirect
246. C22×C17#324semidirect
328. C2×C22⋊C4#3624semidirect
340. C17#9081semidirect
395. C22×C4C2×#3624semidirect
396. C22×C4C2×#3624semidirect
511. C22#11763654semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊#36242
#145446062
C2×#117636542
#145436222
C2×#117626842
#908117