Finite Group Databasebeta

← #17044248

Subgroups — #17044248

Order 4352.

1162 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763341C211yesmaximal, characteristic
2C2×#11762831C211yesmaximal, characteristic, complemented
3#14531189C211yesmaximal, characteristic, complemented
4#14529738C212yesmaximal, complemented
5#14529715C212yesmaximal, complemented
6#14529219C212yesmaximal
7#14529216C212yesmaximal, complemented
8#14528790C211yesmaximal, characteristic, complemented
9#14528786C211yesmaximal, characteristic, complemented
10#14528784C211yesmaximal, characteristic, complemented
11C17×#1802C211yesmaximal, characteristic, complemented
12C22×#63218C2212yes—
13#11763341—24no—
14#11763341C418yescomplemented
15C2×#10587565—21no—
16C2×#10587576C2211yescharacteristic
17C2×#10587622C2211yescharacteristic
18C2×#10587569—21no—
19C2×#10587583C2211yescharacteristic
20C2×#10587624C2211yescharacteristic
21C2×C4×Dic34C2211yescharacteristic
22C2×C4×C22⋊D17—21no—
23C2×#10587612—21no—
24C2×#10587578—21no—
25C2×C42×D17C2211yescharacteristic
26#11762831—24no—
27C2×C4⋊#3622C2211yescharacteristic
28C2×#10587485C2212yes—
29C2×#10587487C2212yes—
30C2×#10587481C2212yes—
31C2×#10587503C2212yes—
32#11762679C2211yescharacteristic
33#11762674—21no—
34#11762666—21no—
35#11762666C2212yes—
36C4⋊C4×Dic17—21no—
37C4⋊C4×Dic17C2212yes—
38#11762572C2212yes—
39#11762571C2212yes—
40#11762571C2212yes—
41C2×C4×D4×C17C2211yescharacteristic
42C2×C17×C4⋊C8C2211yescharacteristic
43C2×C17×#102C2212yes—
44C17×#384C2211yescharacteristic
45C17×#340C2212yes—
46C22×D4×C17C2311yescharacteristic
47C23×C4×C17—21no—
48C22×C22⋊D17—21no—
49C22×C4×D17D411yescharacteristic
50C22×C4×D17D411yescharacteristic
51C22×C4×D17C2311yescharacteristic
52C22×Dic34C2311yescharacteristic
53C22×C8×C17C2312yes—
54C4×D4×C17—22no—
55C4×D4×C17C2×C414yes—
56C2×C4⋊C4×C17—21no—
57C2×C4⋊C4×C17—21no—
58C2×C4⋊C4×C17C2312yes—
59C2×C4⋊C4×C17C2311yescharacteristic
60C2×C22⋊C4×C17—21no—
61C2×C42×C17C2311yescharacteristic
62C2×C22⋊Dic17—21no—
63C2×C22⋊Dic17—21no—
64C2×C22⋊Dic17—21no—
65#10587624—22no—
66#10587624C2×C414yes—
67#10587622—22no—
68#10587622C2×C414yes—
69C4×C22⋊D17—28no—
70C2×#63227—21no—
71C2×#63227—22no—
72#10587612—28no—
73C2×C4⋊Dic17—21no—
74C2×C4⋊Dic17—21no—
75C2×C4⋊Dic17—21no—
76C2×C4⋊Dic17—21no—
77C2×C4⋊Dic17C2312yes—
78C2×C4⋊Dic17C2311yescharacteristic
79C2×#63225—21no—
80C2×#63225—21no—
81C2×C4×Dic17—21no—
82C2×C4×Dic17—22no—
83C2×C4×Dic17D411yescharacteristic
84C2×C4×Dic17D411yescharacteristic
85C2×C4×Dic17C2311yescharacteristic
86C22×#3622C2312yes—
87C2×#63218—24no—
88C2×#63218—42no—
89C2×#63218—42no—
90#10587583—22no—
91#10587583C2×C414yes—
92#10587578—44no—
93#10587576—22no—
94#10587576C2×C414yes—
95#10587569—28no—
96#10587565—28no—
97C42×D17C2×C414yes—
98C42×D17D414yes—
99C4×Dic34—22no—
100C4×Dic34C2×C414yes—
101C17×C4⋊C8—22no—
102C17×#102—42no—
103#10587503—42no—
104#10587502—22no—
105#10587502—22no—
106#10587487—42no—
107#10587485—24no—
108#10587481—42no—
109C4⋊#3622—22no—
110C24×C17—21no—
111C23×D17C2×D411yescharacteristic
112C2×D4×C17—21no—
113C2×D4×C17—21no—
114C2×D4×C17—21no—
115C2×D4×C17—21no—
116C2×D4×C17—21no—
117C2×D4×C17C22×C412yes—
118C22×C4×C17—21no—
119C22×C4×C17—24no—
120C22×C4×C17—22no—
121C22×C4×C17—42no—
122C22×C4×C17—42no—
123C22×C4×C17C2×D411yescharacteristic
124C22×C4×C17C2×D411yescharacteristic
125C22×C4×C17C2411yescharacteristic
126C2×C22⋊D17—24no—
127C2×C22⋊D17—42no—
128C2×C22⋊D17—42no—
129C22×Dic17—21no—
130C22×Dic17—21no—
131C22×Dic17—22no—
132C22×Dic17C2×D411yescharacteristic
133C22×Dic17C2×D411yescharacteristic
134C22×Dic17C2×D411yescharacteristic
135C2×C4×D17—22no—
136C2×C4×D17—22no—
137C2×C4×D17—21no—
138C2×C4×D17—21no—
139C2×C4×D17—21no—
140C2×C4×D17—21no—
141C2×C4×D17—41no—
142C2×C4×D17—41no—
143C2×C4×D17C22⋊C414yes—
144C2×C4×D17C22⋊C414yes—
145C2×C4×D17C22×C412yes—
146C2×C4×D17C2×D412yes—
147C2×Dic34—21no—
148C2×Dic34—21no—
149C2×Dic34—21no—
150C2×Dic34—21no—
151C2×Dic34—21no—
152C2×Dic34C22×C412yes—
153C2×C8×C17—22no—
154C2×C8×C17—22no—
155C2×C8×C17—22no—
156C4⋊C4×C17—21no—
157C4⋊C4×C17—22no—
158C4⋊C4×C17—22no—
159C4⋊C4×C17—42no—
160C4⋊C4×C17—42no—
161C4⋊C4×C17C22×C412yes—
162C22⋊C4×C17—24no—
163C42×C17C22×C412yes—
164C42×C17C2×D412yes—
165C22⋊Dic17—24no—
166C22⋊Dic17—24no—
167C22⋊Dic17—24no—
168#63227—24no—
169#63227—44no—
170C4⋊Dic17—21no—
171C4⋊Dic17—24no—
172C4⋊Dic17—24no—
173C4⋊Dic17—22no—
174C4⋊Dic17—22no—
175C4⋊Dic17—42no—
176C4⋊Dic17—42no—
177C4⋊Dic17C22×C412yes—
178#63225—24no—
179#63225—24no—
180C4×Dic17—24no—
181C4×Dic17—44no—
182C4×Dic17C22⋊C414yes—
183C4×Dic17C22⋊C414yes—
184C4×Dic17C22×C412yes—
185C4×Dic17C2×D412yes—
186C2×#3622—22no—
187C2×#3622—22no—
188C2×#3622—22no—
189#63218—48no—
190#21890—171nomaximal, Sylow
191C23×C17—22no—
192C23×C17—41no—
193C23×C17—41no—
194C23×C17—41no—
195C23×C17—41no—
196C23×C17—41no—
197C23×C17—41no—
198C23×C17C22×D411yescharacteristic
199C22×D17—21no—
200C22×D17—21no—
201C22×D17—21no—
202C22×D17—21no—
203C22×D17—21no—
204C22×D17—21no—
205C22×D17C2×C22⋊C412yes—
206D4×C17—22no—
207D4×C17—22no—
208D4×C17—41no—
209D4×C17—41no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—22no—
213C2×C4×C17—41no—
214C2×C4×C17—41no—
215C2×C4×C17—48no—
216C2×C4×C17—42no—
217C2×C4×C17—42no—
218C2×C4×C17—42no—
219C2×C4×C17C2×C22⋊C412yes—
220C2×C4×C17C2×C22⋊C412yes—
221C2×C4×C17AGL(1,ℤ/8)12yes—
222C2×C4×C17#13712yes—
223C2×C4×C17C23×C411yescharacteristic
224C2×C4×C17C22×D411yescharacteristic
225C2×C4×C17H(5,𝔽2)11yescharacteristic
226C2×C4×C17H(5,𝔽2)11yescharacteristic
227C2×C4×C172−1+411yescharacteristic
228C2×C4×C172−1+411yescharacteristic
229C22⋊D17—48no—
230C2×Dic17—21no—
231C2×Dic17—21no—
232C2×Dic17—21no—
233C2×Dic17—21no—
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—22no—
237C2×Dic17—22no—
238C2×Dic17—41no—
239C2×Dic17—41no—
240C2×Dic17—41no—
241C2×Dic17—41no—
242C2×Dic17—42no—
243C2×Dic17—42no—
244C2×Dic17—42no—
245C2×Dic17C2×C22⋊C412yes—
246C2×Dic17C2×C22⋊C412yes—
247C2×Dic17C2×C22⋊C412yes—
248C4×D17—22no—
249C4×D17—22no—
250C4×D17—22no—
251C4×D17—22no—
252C4×D17—28no—
253C4×D17—44no—
254Dic34—22no—
255Dic34—22no—
256Dic34—41no—
257Dic34—41no—
258C8×C17—42no—
259#3622—42no—
260C2×#529—171no—
261C2×#423—171no—
262#2285—171no—
263#1989—172no—
264#1983—172no—
265#1895—172no—
266#1894—172no—
267#1803—171no—
268#1803—171no—
269#1802—171no—
270#1802D1711yescharacteristic, complemented
271C22×C17—42no—
272C22×C17—42no—
273C22×C17—42no—
274C22×C17—81no—
275C22×C17—81no—
276C22×C17—81no—
277C22×C17—81no—
278C22×C17C22×C22⋊C411yescharacteristic
279C22×C17#53711yescharacteristic
280C22×C17#53711yescharacteristic
281C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C17C2×#13711yescharacteristic
283C22×C17#57511yescharacteristic
284C22×C17#57511yescharacteristic
285C2×D17—22no—
286C2×D17—22no—
287C2×D17—22no—
288C2×D17—41no—
289C2×D17—41no—
290C2×D17—41no—
291C2×D17—41no—
292C4×C17—22no—
293C4×C17—42no—
294C4×C17—82no—
295C4×C17#41714yes—
296C4×C17#52011yescharacteristic
297C4×C17#52011yescharacteristic
298C4×C17#53511yescharacteristic
299C4×C17#53611yescharacteristic
300Dic17—22no—
301Dic17—22no—
302Dic17—22no—
303Dic17—42no—
304Dic17—42no—
305Dic17—82no—
306C22×M16—172no—
307#529—178no—
308#529—344no—
309C2×#126—341no—
310C2×#125—171no—
311C2×#124—171no—
312C2×#123—341no—
313C2×C22⋊Q8—171no—
314C2×#121—171no—
315C2×C4×Q8—171no—
316C2×C4×D4—341no—
317C2×C4×D4C2×D1711yescharacteristic
318C2×#117—341no—
319C2×#117—341no—
320C22×C42—171no—
321#423—344no—
322C2×C4⋊C8—172no—
323C2×C4⋊C8—171no—
324C2×C4⋊C8C2×D1711yescharacteristic
325C2×#103—172no—
326C2×#103—172no—
327C2×#102—172no—
328C2×#102C2×D1712yes—
329#384—171no—
330#384C2×D1711yescharacteristic
331#382—341no—
332#380—172no—
333#380—341no—
334C4×C4⋊C4—172no—
335C4×C4⋊C4—341no—
336#340—172no—
337#340—172no—
338#340—172no—
339#340C2×D1712yes—
340C2×C17—82no—
341C2×C17#231111yescharacteristic
342C2×C17C2×#41711yescharacteristic
343C2×C17#289611yescharacteristic
344C2×C17#325111yescharacteristic
345C2×C17#325111yescharacteristic
346C2×C17#325211yescharacteristic
347C2×C17#325211yescharacteristic
348D17—42no—
349C22×Q8—171no—
350C22×D4—341no—
351C22×D4C22×D1711yescharacteristic
352C23×C4—21no—
353C23×C4—171no—
354C23×C4—171no—
355C23×C4—171no—
356C2×M16—344no—
357C2×M16—682no—
358C2×M16—682no—
359C22×C8—172no—
360C22×C8C22×D1712yes—
361#126—348no—
362#125—174no—
363#125—342no—
364#124—174no—
365#124—342no—
366#123—348no—
367C22⋊Q8—174no—
368C22⋊Q8—342no—
369#121—174no—
370#121—342no—
371C4×Q8—174no—
372C4×Q8—342no—
373C4×D4—22no—
374C4×D4—348no—
375C4×D4C4×D1714yes—
376#117—348no—
377#117—684no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—171no—
381C2×C4⋊C4—172no—
382C2×C4⋊C4—341no—
383C2×C4⋊C4—341no—
384C2×C4⋊C4—341no—
385C2×C4⋊C4—341no—
386C2×C4⋊C4—341no—
387C2×C4⋊C4—341no—
388C2×C4⋊C4C22×D1712yes—
389C2×C4⋊C4C22×D1711yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—341no—
392C2×C22⋊C4—341no—
393C2×C22⋊C4—341no—
394C2×C22⋊C4—341no—
395C2×C22⋊C4—342no—
396C2×C42—171no—
397C2×C42—171no—
398C2×C42—171no—
399C2×C42—174no—
400C2×C42—174no—
401C2×C42—341no—
402C2×C42—342no—
403C2×C42C22×D1711yescharacteristic
404C4⋊C8—22no—
405C4⋊C8—344no—
406C4⋊C8—342no—
407#103—682no—
408#103—682no—
409#102—42no—
410#102—682no—
411#95—342no—
412#95—342no—
413C17#2189011yescharacteristic, complemented, Sylow
414C24—21no—
415C24—171no—
416C2×Q8—172no—
417C2×Q8—341no—
418C2×Q8—341no—
419C2×Q8—341no—
420C2×Q8—341no—
421C2×Q8—341no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—344no—
428C2×D4—682no—
429C2×D4—682no—
430C2×D4C2×C4×D1712yes—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—171no—
437C22×C4—171no—
438C22×C4—171no—
439C22×C4—172no—
440C22×C4—172no—
441C22×C4—174no—
442C22×C4—174no—
443C22×C4—341no—
444C22×C4—341no—
445C22×C4—341no—
446C22×C4—341no—
447C22×C4—342no—
448C22×C4—342no—
449C22×C4—341no—
450C22×C4—341no—
451C22×C4—342no—
452C22×C4—681no—
453C22×C4—681no—
454C22×C4D4×D1711yescharacteristic
455C22×C4D4×D1711yescharacteristic
456C22×C4C23×D1711yescharacteristic
457M16—688no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—342no—
462C2×C8—342no—
463C2×C8—342no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—172no—
470C4⋊C4—344no—
471C4⋊C4—344no—
472C4⋊C4—344no—
473C4⋊C4—341no—
474C4⋊C4—344no—
475C4⋊C4—342no—
476C4⋊C4—342no—
477C4⋊C4—682no—
478C4⋊C4—682no—
479C4⋊C4C2×C4×D1712yes—
480C22⋊C4—24no—
481C22⋊C4—344no—
482C22⋊C4—344no—
483C22⋊C4—344no—
484C22⋊C4—344no—
485C22⋊C4—684no—
486C42—172no—
487C42—172no—
488C42—174no—
489C42—174no—
490C42—344no—
491C42—684no—
492C42C2×C4×D1712yes—
493C42D4×D1712yes—
494C23—22no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—172no—
502C23—341no—
503C23—341no—
504C23—341no—
505C23—341no—
506C23—341no—
507C23—341no—
508C23C2×D4×D1711yescharacteristic
509Q8—342no—
510Q8—342no—
511Q8—681no—
512Q8—681no—
513D4—22no—
514D4—22no—
515D4—41no—
516D4—41no—
517D4—688no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—48no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—172no—
528C2×C4—172no—
529C2×C4—172no—
530C2×C4—341no—
531C2×C4—341no—
532C2×C4—341no—
533C2×C4—341no—
534C2×C4—341no—
535C2×C4—341no—
536C2×C4—342no—
537C2×C4—342no—
538C2×C4—348no—
539C2×C4—342no—
540C2×C4—342no—
541C2×C4—342no—
542C2×C4—342no—
543C2×C4—684no—
544C2×C4—681no—
545C2×C4—681no—
546C2×C4—681no—
547C2×C4—681no—
548C2×C4—682no—
549C2×C4—682no—
550C2×C4—682no—
551C2×C4C22⋊C4×D1712yes—
552C2×C4C22⋊C4×D1712yes—
553C2×C4#1058759712yes—
554C2×C4#1058760112yes—
555C2×C4C22×C4×D1711yescharacteristic
556C2×C4C2×D4×D1711yescharacteristic
557C2×C4#1058769111yescharacteristic
558C2×C4#1058769411yescharacteristic
559C2×C4#1058769611yescharacteristic
560C2×C4#1058769711yescharacteristic
561C8—42no—
562C8—682no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—342no—
571C22—342no—
572C22—342no—
573C22—681no—
574C22—681no—
575C22—681no—
576C22—681no—
577C22C2×C22⋊C4×D1711yescharacteristic
578C22#1176338811yescharacteristic
579C22#1176338911yescharacteristic
580C22C2×#1058759711yescharacteristic
581C22C2×#1058760111yescharacteristic
582C22#1176360511yescharacteristic
583C22#1176360811yescharacteristic
584C4—22no—
585C4—42no—
586C4—82no—
587C4—342no—
588C4—342no—
589C4—342no—
590C4—682no—
591C4—682no—
592C4—1362no—
593C4#1176280714yes—
594C4#1176328911yescharacteristic
595C4#1176329011yescharacteristic
596C4#1176337511yescharacteristic
597C4#1176338211yescharacteristic
598C2—82no—
599C2—682no—
600C2#1453137311yescharacteristic
601C2C2×#1176280711yescharacteristic
602C2#1453730011yescharacteristic
603C2#1454014211yescharacteristic
604C2#1454014811yescharacteristic
605C2#1454015311yescharacteristic
606C2#1454015911yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1802C2
socleC23×C17C22×D4
prime radicalC2×#10587624C22
cosocleC22×C4×C17C24
hypercenter#1802D17
hypocenterC17#21890
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. C2×#11762831C2semidirect
3. #14531189C2semidirect
4. #14529738C2semidirect
5. #14529715C2semidirect
7. #14529216C2semidirect
8. #14528790C2semidirect
9. #14528786C2semidirect
10. #14528784C2semidirect
11. C17×#1802C2semidirect
14. #11763341C4semidirect
270. #1802D17semidirect
413. C17#21890semidirect

Maximal subgroups

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

subgroupindex
C2×#117633412
C2×#117628312
#145311892
#145297382
#145297152
#145292192
#145292162
#145287902
#145287862
#145287842
C17×#18022
#2189017