Finite Group Databasebeta

← #17061939

Subgroups — #17061939

Order 4352.

1362 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14537373C211yesmaximal, characteristic, complemented
2C2×#11762826C211yesmaximal, characteristic
3#14537303C211yesmaximal, characteristic
4C22×#10587507C211yesmaximal, characteristic, complemented
5#14531370C211yesmaximal, characteristic, complemented
6#14529998C218yesmaximal, complemented
7#14528493C211yesmaximal, characteristic, complemented
8C2×C17×#536C211yesmaximal, characteristic
9C22×#63223C2211yescharacteristic
10C22×#63223C2211yescharacteristic
11C2×#10587612C2211yescharacteristic
12C2×#10587612C2211yescharacteristic
13#11762852C2214yescomplemented
14#11762849—22no—
15#11762845—22no—
16#11762826—24no—
17C2×#10587508C2214yes—
18#11762818—21no—
19#11762814—21no—
20#11762810C2214yes—
21#11762805—22no—
22C2×#10587507—24no—
23C2×#10587507C2214yescomplemented
24#11762716—22no—
25#11762716C2214yescomplemented
26#11762672—22no—
27#11762568C2214yescomplemented
28C22×#41×C17C2211yescharacteristic
29C23×Q8×C17C2211yescharacteristic
30C17×#536—22no—
31C17×#536—22no—
32C17×#536—22no—
33C17×#536C2214yes—
34C2×C17×C4⋊Q8—21no—
35C2×C17×#124—21no—
36C2×C17×#123—22no—
37C2×C17×C22⋊Q8—22no—
38C2×C17×#117C2211yescharacteristic
39C2×#41×C17—21no—
40C2×#41×C17—21no—
41C2×#41×C17—22no—
42C2×#41×C17—22no—
43C2×#41×C17—22no—
44C2×#41×C17—22no—
45C2×#41×C17D412yes—
46C2×#41×C17C2312yes—
47C22×Q8×C17—21no—
48C22×Q8×C17—21no—
49C22×Q8×C17—21no—
50C22×Q8×C17—21no—
51C22×Q8×C17—22no—
52C22×Q8×C17—22no—
53C22×Q8×C17—22no—
54C22×Q8×C17—22no—
55C22×Q8×C17D411yescharacteristic
56C22×Q8×C17D412yes—
57C22×Q8×C17C2312yes—
58C22×D4×C17—21no—
59C22×D4×C17D411yescharacteristic
60C23×C4×C17—21no—
61C23×C4×C17D412yes—
62C23×C4×C17C2311yescharacteristic
63C17×C4⋊Q8—24no—
64C17×C4⋊Q8—24no—
65C17×#124—24no—
66C17×#124—24no—
67C17×#123—28no—
68C17×#123—28no—
69C17×C22⋊Q8—28no—
70C17×C22⋊Q8—28no—
71C17×#117—21no—
72C17×#117—21no—
73C17×#117D412yes—
74C17×#117C2312yes—
75C2×C4⋊C4×C17—24no—
76C2×C4⋊C4×C17D412yes—
77C2×C22⋊C4×C17—24no—
78C2×C22⋊C4×C17—21no—
79C2×C42×C17—21no—
80C2×C22⋊Dic17—21no—
81C2×C22⋊Dic17—21no—
82#10587612—22no—
83#10587612—21no—
84#10587612—21no—
85#10587612—21no—
86#10587612—21no—
87#10587612—22no—
88C2×C4⋊Dic17—21no—
89C2×C4⋊Dic17C2312yes—
90C2×C4×Dic17—21no—
91C2×C4×Dic17C2312yes—
92C2×#63223—21no—
93C2×#63223—21no—
94C2×#63223—22no—
95C2×#63223—22no—
96C2×#63223—22no—
97C2×#63223D414yes—
98C2×#63223C2312yes—
99C2×#63223C2312yes—
100C22×#3622—21no—
101C22×#3622—21no—
102#10587508—28no—
103#10587507—28no—
104#10587506—24no—
105#10587503—24no—
106#10587502—28no—
107#10587479—24no—
108#10587478—24no—
109C24×C17—21no—
110C24×C17C2×D411yescharacteristic
111#41×C17—24no—
112#41×C17—24no—
113#41×C17—22no—
114#41×C17—22no—
115#41×C17—22no—
116#41×C17—22no—
117#41×C17—42no—
118#41×C17—42no—
119#41×C17—42no—
120#41×C17—42no—
121C2×Q8×C17—22no—
122C2×Q8×C17—22no—
123C2×Q8×C17—21no—
124C2×Q8×C17—24no—
125C2×Q8×C17—24no—
126C2×Q8×C17—22no—
127C2×Q8×C17—22no—
128C2×Q8×C17—22no—
129C2×Q8×C17—22no—
130C2×Q8×C17—22no—
131C2×Q8×C17—22no—
132C2×Q8×C17—22no—
133C2×Q8×C17—22no—
134C2×Q8×C17—42no—
135C2×Q8×C17—42no—
136C2×Q8×C17—42no—
137C2×Q8×C17—42no—
138C2×Q8×C17—42no—
139C2×Q8×C17—42no—
140C2×Q8×C17—42no—
141C2×Q8×C17—42no—
142C2×Q8×C17C2×D412yes—
143C2×D4×C17—22no—
144C2×D4×C17—22no—
145C2×D4×C17—21no—
146C2×D4×C17—22no—
147C2×D4×C17—22no—
148C2×D4×C17—42no—
149C2×D4×C17—42no—
150C2×D4×C17C2×D412yes—
151C22×C4×C17—22no—
152C22×C4×C17—21no—
153C22×C4×C17—22no—
154C22×C4×C17—22no—
155C22×C4×C17—24no—
156C22×C4×C17—24no—
157C22×C4×C17—22no—
158C22×C4×C17—22no—
159C22×C4×C17—42no—
160C22×C4×C17—42no—
161C22×C4×C17C2×D412yes—
162C22×C4×C17C2×D414yes—
163C22×C4×C17C2×D411yescharacteristic
164C22×C4×C17C2×D411yescharacteristic
165C22×C4×C17C2×D411yescharacteristic
166C22×C4×C17C2×D411yescharacteristic
167C22×C4×C17C2×D411yescharacteristic
168C22×C4×C17#4112yes—
169C22×C4×C17#4112yes—
170C22×C4×C17#4112yes—
171C22×C4×C17C2411yescharacteristic
172C22×Dic17—22no—
173C22×Dic17—22no—
174C4⋊C4×C17—216no—
175C4⋊C4×C17—22no—
176C4⋊C4×C17C2×D414yes—
177C22⋊C4×C17—216no—
178C22⋊C4×C17—22no—
179C22⋊C4×C17—22no—
180C42×C17—22no—
181C42×C17—22no—
182C22⋊Dic17—41no—
183C22⋊Dic17—41no—
184C22⋊Dic17—42no—
185C4⋊Dic17—22no—
186C4⋊Dic17—22no—
187C4⋊Dic17—42no—
188C4×Dic17—22no—
189C4×Dic17—22no—
190C4×Dic17—41no—
191C4×Dic17—41no—
192#63223—24no—
193#63223—22no—
194#63223—22no—
195#63223—22no—
196#63223—42no—
197#63223C2×D414yes—
198C2×#3622—24no—
199C2×#3622—22no—
200C2×#3622—22no—
201C2×#3622—42no—
202#23494—171nomaximal, Sylow
203C23×C17—22no—
204C23×C17—42no—
205C23×C17—44no—
206C23×C17—42no—
207C23×C17#12011yescharacteristic
208C23×C17#12011yescharacteristic
209C23×C17#12112yes—
210C23×C17#12312yes—
211C23×C17C22×D411yescharacteristic
212Q8×C17—22no—
213Q8×C17—22no—
214Q8×C17—42no—
215Q8×C17—42no—
216Q8×C17—42no—
217Q8×C17—42no—
218Q8×C17—44no—
219Q8×C17—44no—
220Q8×C17—42no—
221D4×C17—22no—
222D4×C17—22no—
223D4×C17—42no—
224D4×C17—42no—
225D4×C17—44no—
226D4×C17—42no—
227C2×C4×C17—24no—
228C2×C4×C17—24no—
229C2×C4×C17—22no—
230C2×C4×C17—24no—
231C2×C4×C17—24no—
232C2×C4×C17—44no—
233C2×C4×C17—44no—
234C2×C4×C17—44no—
235C2×C4×C17—42no—
236C2×C4×C17—42no—
237C2×C4×C17—42no—
238C2×C4×C17—44no—
239C2×C4×C17#12011yescharacteristic
240C2×C4×C17#12011yescharacteristic
241C2×C4×C17#12112yes—
242C2×C4×C17#12112yes—
243C2×C4×C17#12112yes—
244C2×C4×C17#12112yes—
245C2×C4×C17#12112yes—
246C2×C4×C17#12312yes—
247C2×C4×C17#12312yes—
248C2×C4×C17#12312yes—
249C2×C4×C17#12412yes—
250C2×C4×C17#12412yes—
251C2×C4×C17C22×D411yescharacteristic
252C2×C4×C17C22×D411yescharacteristic
253C2×C4×C17C22×D411yescharacteristic
254C2×C4×C17C2×#4111yescharacteristic
255C2×C4×C17C2×#4111yescharacteristic
256C2×C4×C17C2×#4111yescharacteristic
257C2×Dic17—44no—
258C2×Dic17—44no—
259C2×Dic17—42no—
260C2×Dic17—42no—
261#3622—24no—
262#3622—42no—
263C2×#536D1711yescharacteristic
264#2911—171no—
265C2×#421—171no—
266#2896—171no—
267C22×#101—171no—
268#2310—171no—
269#2038—178no—
270#1733—171no—
271C22×C17—22no—
272C22×C17—22no—
273C22×C17—22no—
274C22×C17—42no—
275C22×C17—42no—
276C22×C17—42no—
277C22×C17—44no—
278C22×C17—84no—
279C22×C17#39412yes—
280C22×C17#39412yes—
281C22×C17C2×#12011yescharacteristic
282C22×C17C2×#12111yescharacteristic
283C22×C17C2×#12311yescharacteristic
284C4×C17—44no—
285C4×C17—42no—
286C4×C17—44no—
287C4×C17#39412yes—
288C4×C17#39412yes—
289C4×C17C2×#12111yescharacteristic
290C4×C17C2×#12111yescharacteristic
291C4×C17C2×#12311yescharacteristic
292C4×C17C2×#12411yescharacteristic
293Dic17—82no—
294Dic17—82no—
295C22×#41C2×D1711yescharacteristic
296C23×Q8C2×D1711yescharacteristic
297C22×M16—171no—
298C22×M16—171no—
299#536—22no—
300#536—22no—
301#536—22no—
302#536C2×D1714yes—
303C2×C4⋊Q8—21no—
304C2×#124—21no—
305C2×#123—22no—
306C2×C22⋊Q8—22no—
307C2×#117—171no—
308C2×#117—171no—
309C2×#117C2×D1711yescharacteristic
310#428—174no—
311#428—342no—
312#427—342no—
313#421—344no—
314C2×#104—174no—
315#419—341no—
316#418—341no—
317#417—174no—
318#416—342no—
319C2×#101—174no—
320C2×#101—344no—
321#391—174no—
322#391—342no—
323#381—342no—
324#339—174no—
325C2×C17—21no—
326C2×C17—22no—
327C2×C17—21no—
328C2×C17—22no—
329C2×C17—82no—
330C2×C17C2×#39411yescharacteristic
331C2×C17#302012yes—
332C2×#41—21no—
333C2×#41—21no—
334C2×#41—22no—
335C2×#41—22no—
336C2×#41—22no—
337C2×#41—22no—
338C2×#41C22⋊D1712yes—
339C2×#41C22×D1712yes—
340C22×Q8—22no—
341C22×Q8—22no—
342C22×Q8—22no—
343C22×Q8—22no—
344C22×Q8—21no—
345C22×Q8—21no—
346C22×Q8—21no—
347C22×Q8—21no—
348C22×Q8C22⋊D1711yescharacteristic
349C22×Q8C22⋊D1712yes—
350C22×Q8C22×D1712yes—
351C22×D4—21no—
352C22×D4C22⋊D1711yescharacteristic
353C23×C4—21no—
354C23×C4C22⋊D1712yes—
355C23×C4C22×D1711yescharacteristic
356C2×M16—174no—
357C2×M16—172no—
358C2×M16—172no—
359C2×M16—342no—
360C2×M16—341no—
361C2×M16—341no—
362C2×M16—342no—
363C2×M16—342no—
364C22×C8—341no—
365C22×C8—341no—
366C4⋊Q8—24no—
367C4⋊Q8—24no—
368#124—24no—
369#124—24no—
370#123—28no—
371#123—28no—
372C22⋊Q8—28no—
373C22⋊Q8—28no—
374#117—21no—
375#117—21no—
376#117—342no—
377#117—342no—
378#117—341no—
379#117—341no—
380#117—341no—
381#117—341no—
382#117C22⋊D1712yes—
383#117C22×D1712yes—
384C2×C4⋊C4—24no—
385C2×C4⋊C4—172no—
386C2×C4⋊C4—341no—
387C2×C4⋊C4C22⋊D1712yes—
388C2×C22⋊C4—24no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—341no—
391C2×C22⋊C4—341no—
392C2×C42—21no—
393C2×C42—172no—
394C2×C42—341no—
395#107—344no—
396#106—344no—
397#104—348no—
398#103—344no—
399#102—344no—
400#101—348no—
401#95—348no—
402C17#2349411yescharacteristic, complemented, Sylow
403C24—21no—
404C24C2×C22⋊D1711yescharacteristic
405#41—22no—
406#41—22no—
407#41—22no—
408#41—22no—
409#41—24no—
410#41—24no—
411#41—42no—
412#41—42no—
413#41—42no—
414#41—42no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—24no—
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—21no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8C2×C22⋊D1712yes—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—21no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4C2×C22⋊D1712yes—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—342no—
456C22×C4—342no—
457C22×C4D4×D1712yes—
458C22×C4#6325412yes—
459C22×C4#6325412yes—
460C22×C4#6325412yes—
461C22×C4C2×C22⋊D1714yes—
462C22×C4C2×C22⋊D1711yescharacteristic
463C22×C4C2×C22⋊D1711yescharacteristic
464C22×C4C2×C22⋊D1711yescharacteristic
465C22×C4C2×C22⋊D1711yescharacteristic
466C22×C4C2×C22⋊D1711yescharacteristic
467C22×C4C23×D1711yescharacteristic
468M16—174no—
469M16—342no—
470M16—344no—
471M16—342no—
472M16—342no—
473M16—682no—
474C2×C8—342no—
475C2×C8—344no—
476C2×C8—342no—
477C2×C8—682no—
478C4⋊C4—216no—
479C4⋊C4—22no—
480C4⋊C4—342no—
481C4⋊C4—342no—
482C4⋊C4—682no—
483C4⋊C4C2×C22⋊D1714yes—
484C22⋊C4—216no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—682no—
488C22⋊C4—681no—
489C22⋊C4—681no—
490C42—22no—
491C42—22no—
492C42—342no—
493C42—342no—
494C42—681no—
495C42—681no—
496C23—22no—
497C23—42no—
498C23—44no—
499C23—42no—
500C23#1058762112yes—
501C23#1058762512yes—
502C23#1058763911yescharacteristic
503C23#1058763911yescharacteristic
504C23C22×C22⋊D1711yescharacteristic
505Q8—22no—
506Q8—22no—
507Q8—44no—
508Q8—44no—
509Q8—42no—
510Q8—42no—
511Q8—42no—
512Q8—42no—
513Q8—42no—
514D4—22no—
515D4—22no—
516D4—44no—
517D4—42no—
518D4—42no—
519D4—42no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—684no—
533C2×C4—684no—
534C2×C4—682no—
535C2×C4—682no—
536C2×C4#1058762112yes—
537C2×C4#1058762112yes—
538C2×C4#1058762112yes—
539C2×C4#1058762212yes—
540C2×C4#1058762212yes—
541C2×C4#1058762412yes—
542C2×C4#1058762412yes—
543C2×C4#1058762512yes—
544C2×C4#1058762512yes—
545C2×C4#1058762512yes—
546C2×C4#1058763911yescharacteristic
547C2×C4#1058763911yescharacteristic
548C2×C4C2×D4×D1711yescharacteristic
549C2×C4C2×#6325411yescharacteristic
550C2×C4C2×#6325411yescharacteristic
551C2×C4C2×#6325411yescharacteristic
552C2×C4C22×C22⋊D1711yescharacteristic
553C2×C4C22×C22⋊D1711yescharacteristic
554C8—344no—
555C8—682no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—44no—
563C22—84no—
564C22#1176272312yes—
565C22#1176272312yes—
566C22C2×#1058763911yescharacteristic
567C22C2×#1058762511yescharacteristic
568C22C2×#1058762111yescharacteristic
569C4—44no—
570C4—42no—
571C4—44no—
572C4—1362no—
573C4—1362no—
574C4#1176272312yes—
575C4#1176272312yes—
576C4C2×#1058762411yescharacteristic
577C4C2×#1058762511yescharacteristic
578C4C2×#1058762111yescharacteristic
579C4C2×#1058762211yescharacteristic
580C2—21no—
581C2—22no—
582C2—22no—
583C2—21no—
584C2—82no—
585C2C2×#1176272311yescharacteristic
586C2#1453814812yes—
subgroupG/H
centerC22C2×#10587639
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#536C2
socleC22×C17C2×#120
prime radicalC22×#41×C17C22
hypercenterC2×#536D17
hypocenterC17#23494
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
1. #14537373C2semidirect
4. C22×#10587507C2semidirect
5. #14531370C2semidirect
6. #14529998C2semidirect
7. #14528493C2semidirect
13. #11762852C22semidirect
23. C2×#10587507C22semidirect
25. #11762716C22semidirect
27. #11762568C22semidirect
402. C17#23494semidirect

Maximal subgroups

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

subgroupindex
#145373732
C2×#117628262
#145373032
C22×#105875072
#145313702
#145299982
#145284932
C2×C17×#5362
#2349417