Finite Group Databasebeta

← #16964420

Subgroups — #16964420

Order 4352.

1706 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762716C212yesmaximal, complemented
2C2×#11762684C212yesmaximal
3#14527049C212yesmaximal, complemented
4#14527041C212yesmaximal, complemented
5#14527029C211yesmaximal, characteristic
6#14527021C212yesmaximal
7#14527009C212yesmaximal, complemented
8#14527006C211yesmaximal, characteristic
9C2×C17×#393C211yesmaximal, characteristic
10C24×Dic17C2211yescharacteristic
11C22×C4⋊Dic17C2212yes—
12C22×C4⋊Dic17C2212yes—
13C22×C4⋊Dic17C2211yescharacteristic
14C22×C22⋊Dic17C2212yes—
15C22×C22⋊Dic17C2211yescharacteristic
16C22×C4×Dic17C2212yes—
17#11762716—28no—
18#11762684—28no—
19C2×#10587502C2212yes—
20C2×#10587502C2212yes—
21C2×#10587502C2212yes—
22C2×#10587502C2212yes—
23C2×#10587502C2212yes—
24C2×#10587502C2211yescharacteristic
25C2×#10587502C2212yes—
26C2×#10587502C2212yes—
27C2×#10587502C2212yes—
28C23×Q8×C17C2211yescharacteristic
29C22×C22⋊C4×C17C2212yes—
30C22×C22⋊C4×C17C2211yescharacteristic
31C17×#393C418yescomplemented
32C2×C17×#95C2212yes—
33C2×C17×#95C2211yescharacteristic
34C25×C17C2311yescharacteristic
35C22×Q8×C17—24no—
36C22×Q8×C17—24no—
37C22×Q8×C17—41no—
38C22×Q8×C17—41no—
39C22×Q8×C17C2×C414yescomplemented
40C23×C4×C17C2312yes—
41C23×C4×C17C2311yescharacteristic
42C23×C4×C17C2311yescharacteristic
43C23×C4×C17C2312yes—
44C23×Dic17—22no—
45C23×Dic17—24no—
46C23×Dic17—42no—
47C23×Dic17—41no—
48C23×Dic17—41no—
49C23×Dic17C2312yes—
50C23×Dic17C2312yes—
51C23×Dic17C2312yes—
52C23×Dic17C2312yes—
53C2×C22⋊C4×C17—24no—
54C2×C22⋊C4×C17—24no—
55C2×C22⋊C4×C17—41no—
56C2×C22⋊C4×C17—41no—
57C2×C22⋊C4×C17C2×C418yes—
58C2×C22⋊C4×C17C2×C414yes—
59C2×C22⋊Dic17—24no—
60C2×C22⋊Dic17—24no—
61C2×C22⋊Dic17—41no—
62C2×C22⋊Dic17—41no—
63C2×C22⋊Dic17—42no—
64C2×C22⋊Dic17Q814yescomplemented
65C2×C4⋊Dic17—24no—
66C2×C4⋊Dic17—24no—
67C2×C4⋊Dic17—24no—
68C2×C4⋊Dic17—42no—
69C2×C4⋊Dic17—42no—
70C2×C4⋊Dic17—42no—
71C2×C4⋊Dic17—41no—
72C2×C4⋊Dic17—42no—
73C2×C4×Dic17—24no—
74C2×C4×Dic17—24no—
75C2×C4×Dic17—42no—
76C17×#95C2×C414yescomplemented
77C17×#95C2×C418yescomplemented
78#10587502—24no—
79#10587502—24no—
80#10587502—24no—
81#10587502—24no—
82#10587502—42no—
83#10587502—42no—
84#10587502—41no—
85#10587502—42no—
86#10587502D418yes—
87C24×C17—22no—
88C24×C17—22no—
89C24×C17—21no—
90C24×C17—22no—
91C24×C17—21no—
92C24×C17—21no—
93C24×C17—21no—
94C24×C17—22no—
95C24×C17C22×C412yes—
96C24×C17C2×Q812yes—
97C24×C17#4112yes—
98C24×C17C2411yescharacteristic
99C2×Q8×C17—216no—
100C2×Q8×C17—44no—
101C2×Q8×C17—44no—
102C2×Q8×C17—48no—
103C2×Q8×C17—44no—
104C22×C4×C17—22no—
105C22×C4×C17—22no—
106C22×C4×C17—22no—
107C22×C4×C17—22no—
108C22×C4×C17—22no—
109C22×C4×C17—22no—
110C22×C4×C17—22no—
111C22×C4×C17—22no—
112C22×C4×C17—21no—
113C22×C4×C17—21no—
114C22×C4×C17—22no—
115C22×C4×C17—22no—
116C22×C4×C17—22no—
117C22×C4×C17—21no—
118C22×C4×C17—21no—
119C22×C4×C17—22no—
120C22×C4×C17—22no—
121C22×C4×C17—22no—
122C22×C4×C17C22×C412yes—
123C22×C4×C17C22×C414yes—
124C22×C4×C17C22×C412yes—
125C22×C4×C17C22×C414yes—
126C22×C4×C17C2×D414yes—
127C22×C4×C17#4114yes—
128C22×Dic17—22no—
129C22×Dic17—22no—
130C22×Dic17—22no—
131C22×Dic17—22no—
132C22×Dic17—22no—
133C22×Dic17—22no—
134C22×Dic17—22no—
135C22×Dic17—22no—
136C22×Dic17—22no—
137C22×Dic17—22no—
138C22×Dic17—22no—
139C22×Dic17—22no—
140C22×Dic17—22no—
141C22×Dic17—22no—
142C22×Dic17—22no—
143C22×Dic17—22no—
144C22×Dic17—22no—
145C22×Dic17—22no—
146C22×Dic17—22no—
147C22×Dic17—22no—
148C22×Dic17—22no—
149C22×Dic17—22no—
150C22×Dic17—22no—
151C22×Dic17—22no—
152C22×Dic17—22no—
153C22×Dic17—44no—
154C22×Dic17—48no—
155C22×Dic17—48no—
156C22×Dic17—44no—
157C22×Dic17—44no—
158C22×Dic17C2×D414yes—
159C22×Dic17C2×D414yes—
160C22×Dic17C2×Q814yes—
161C22⋊C4×C17—216no—
162C22⋊C4×C17—44no—
163C22⋊Dic17—44no—
164C22⋊Dic17—48no—
165C4⋊Dic17—48no—
166C4⋊Dic17—48no—
167C4⋊Dic17—44no—
168C4×Dic17—48no—
169#14575—171nomaximal, Sylow
170C23×C17—24no—
171C23×C17—22no—
172C23×C17—22no—
173C23×C17—24no—
174C23×C17—22no—
175C23×C17—22no—
176C23×C17—22no—
177C23×C17—22no—
178C23×C17—24no—
179C23×C17—24no—
180C23×C17—24no—
181C23×C17—24no—
182C23×C17—42no—
183C23×C17—42no—
184C23×C17—42no—
185C23×C17—42no—
186C23×C17—41no—
187C23×C17—42no—
188C23×C17—41no—
189C23×C17—41no—
190C23×C17—42no—
191C23×C17—41no—
192C23×C17C4×Q814yes—
193C23×C17C23×C411yescharacteristic
194C23×C17C22×D412yes—
195C23×C17C22×Q811yescharacteristic
196C23×C17C2×#4112yes—
197C23×C17C2×#4111yescharacteristic
198C23×C17H(5,𝔽2)11yescharacteristic
199C23×C17H(5,𝔽2)11yescharacteristic
200C23×C17H(5,𝔽2)11yescharacteristic
201C23×C17H(5,𝔽2)12yes—
202C23×C172−1+411yescharacteristic
203C23×C172−1+412yes—
204Q8×C17—416no—
205C2×C4×C17—24no—
206C2×C4×C17—24no—
207C2×C4×C17—24no—
208C2×C4×C17—24no—
209C2×C4×C17—24no—
210C2×C4×C17—24no—
211C2×C4×C17—24no—
212C2×C4×C17—24no—
213C2×C4×C17—22no—
214C2×C4×C17—22no—
215C2×C4×C17—24no—
216C2×C4×C17—24no—
217C2×C4×C17—24no—
218C2×C4×C17—42no—
219C2×C4×C17—42no—
220C2×C4×C17—42no—
221C2×C4×C17—42no—
222C2×C4×C17—41no—
223C2×C4×C17—42no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
226C2×C4×C17—42no—
227C2×C4×C17—41no—
228C2×C4×C17C4×D418yes—
229C2×Dic17—24no—
230C2×Dic17—24no—
231C2×Dic17—24no—
232C2×Dic17—24no—
233C2×Dic17—24no—
234C2×Dic17—24no—
235C2×Dic17—24no—
236C2×Dic17—24no—
237C2×Dic17—24no—
238C2×Dic17—416no—
239C2×Dic17—42no—
240C2×Dic17—42no—
241C2×Dic17—42no—
242C2×Dic17—42no—
243C2×Dic17—42no—
244C2×Dic17—42no—
245C2×Dic17—42no—
246C2×Dic17—42no—
247C2×Dic17—42no—
248C2×Dic17—42no—
249C2×Dic17—42no—
250C2×Dic17—42no—
251C2×Dic17—42no—
252C2×Dic17—42no—
253C2×Dic17—42no—
254C2×Dic17—42no—
255C2×Dic17—42no—
256C2×Dic17—42no—
257C2×Dic17—42no—
258C2×#393D1711yescharacteristic
259C2×#391—172no—
260C2×#385—172no—
261#1444—172no—
262#1442—172no—
263#1439—171no—
264#1437—172no—
265#1434—172no—
266C22⋊#95—171no—
267C22×C17—24no—
268C22×C17—28no—
269C22×C17—44no—
270C22×C17—42no—
271C22×C17—42no—
272C22×C17—42no—
273C22×C17—42no—
274C22×C17—44no—
275C22×C17—44no—
276C22×C17—42no—
277C22×C17—42no—
278C22×C17C2×C4×D412yes—
279C22×C17C2×C4×Q811yescharacteristic
280C22×C17#51811yescharacteristic
281C22×C17#52011yescharacteristic
282C22×C17#52012yes—
283C22×C17#53512yes—
284C22×C17#53612yes—
285C22×C17#53811yescharacteristic
286C22×C17#53911yescharacteristic
287C22×C17#54212yes—
288C22×C17#54212yes—
289C22×C17#54311yescharacteristic
290C22×C17#54411yescharacteristic
291C22×C17D4211yescharacteristic
292C22×C17#54612yes—
293C22×C17#54612yes—
294C22×C17D4×Q812yes—
295C22×C17#55111yescharacteristic
296C22×C17#55212yes—
297C22×C17#55412yes—
298C22×C17H(3,𝔽4)11yescharacteristic
299C22×C17#56212yes—
300C22×C17#56311yescharacteristic
301C4×C17—28no—
302C4×C17—44no—
303C4×C17—42no—
304C4×C17—42no—
305Dic17—44no—
306Dic17—44no—
307Dic17—44no—
308Dic17—82no—
309C23×Q8C2×D1711yescharacteristic
310C24×C4—171no—
311C22×C4⋊C4—172no—
312C22×C4⋊C4—172no—
313C22×C4⋊C4—171no—
314C22×C22⋊C4—171no—
315C22×C22⋊C4—172no—
316C22×C22⋊C4C2×D1712yes—
317C22×C22⋊C4C2×D1711yescharacteristic
318C22×C42—172no—
319#393Dic1718yescomplemented
320#391—348no—
321#385—348no—
322C2×#95—172no—
323C2×#95—172no—
324C2×#95—172no—
325C2×#95—172no—
326C2×#95—172no—
327C2×#95—172no—
328C2×#95—171no—
329C2×#95—172no—
330C2×#95—172no—
331C2×#95C2×D1712yes—
332C2×#95C2×D1711yescharacteristic
333C2×C17—44no—
334C2×C17#232612yes—
335C2×C17#232911yescharacteristic
336C2×C17#235811yescharacteristic
337C2×C17#236012yes—
338C2×C17#237911yescharacteristic
339C2×C17#267011yescharacteristic
340C2×C17#267711yescharacteristic
341C2×C17#268712yes—
342C2×C17#269212yes—
343C2×C17#276311yescharacteristic
344C2×C17#276411yescharacteristic
345C25C22×D1711yescharacteristic
346C22×Q8—24no—
347C22×Q8—24no—
348C22×Q8—41no—
349C22×Q8—41no—
350C22×Q8C2×Dic1714yescomplemented
351C23×C4—172no—
352C23×C4—172no—
353C23×C4—172no—
354C23×C4—172no—
355C23×C4—344no—
356C23×C4—342no—
357C23×C4—681no—
358C23×C4—682no—
359C23×C4—681no—
360C23×C4C22×D1712yes—
361C23×C4C22×D1711yescharacteristic
362C23×C4C22×D1711yescharacteristic
363C23×C4C22×D1712yes—
364C2×C4⋊C4—344no—
365C2×C4⋊C4—344no—
366C2×C4⋊C4—344no—
367C2×C4⋊C4—682no—
368C2×C4⋊C4—682no—
369C2×C4⋊C4—682no—
370C2×C4⋊C4—681no—
371C2×C4⋊C4—682no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—41no—
375C2×C22⋊C4—41no—
376C2×C22⋊C4—174no—
377C2×C22⋊C4—344no—
378C2×C22⋊C4—344no—
379C2×C22⋊C4—681no—
380C2×C22⋊C4—681no—
381C2×C22⋊C4—682no—
382C2×C22⋊C4C2×Dic1718yes—
383C2×C22⋊C4C2×Dic1714yes—
384C2×C42—344no—
385C2×C42—344no—
386C2×C42—682no—
387#95—178no—
388#95—344no—
389#95—344no—
390#95—344no—
391#95—344no—
392#95—681no—
393#95—682no—
394#95—682no—
395#95—682no—
396#95C2×Dic1714yescomplemented
397#95C2×Dic1718yescomplemented
398C17#1457511yescharacteristic, complemented, Sylow
399C24—22no—
400C24—22no—
401C24—21no—
402C24—21no—
403C24—22no—
404C24—21no—
405C24—22no—
406C24—21no—
407C24Q8×D1712yes—
408C24#6325612yes—
409C24C22×Dic1712yes—
410C24C23×D1711yescharacteristic
411C2×Q8—216no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—48no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—174no—
435C22×C4—174no—
436C22×C4—174no—
437C22×C4—342no—
438C22×C4—342no—
439C22×C4—342no—
440C22×C4—342no—
441C22×C4—342no—
442C22×C4—342no—
443C22×C4—342no—
444C22×C4—342no—
445C22×C4—342no—
446C22×C4—342no—
447C22×C4—342no—
448C22×C4—342no—
449C22×C4—342no—
450C22×C4—342no—
451C22×C4—342no—
452C22×C4—342no—
453C22×C4—342no—
454C22×C4—342no—
455C22×C4—342no—
456C22×C4—342no—
457C22×C4—342no—
458C22×C4—342no—
459C22×C4—342no—
460C22×C4—342no—
461C22×C4—342no—
462C22×C4—684no—
463C22×C4—688no—
464C22×C4—688no—
465C22×C4—684no—
466C22×C4—684no—
467C22×C4D4×D1714yes—
468C22×C4#6325414yes—
469C22×C4C22×Dic1714yes—
470C22×C4C22×Dic1712yes—
471C22×C4C22×Dic1712yes—
472C22×C4C22×Dic1714yes—
473C4⋊C4—688no—
474C4⋊C4—688no—
475C4⋊C4—684no—
476C22⋊C4—216no—
477C22⋊C4—44no—
478C22⋊C4—684no—
479C22⋊C4—688no—
480C42—688no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—22no—
486C23—22no—
487C23—24no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—41no—
494C23—41no—
495C23—42no—
496C23—41no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—41no—
502C23—42no—
503C23Q8×Dic1714yes—
504C23C2×D4×D1712yes—
505C23C2×#6325412yes—
506C23#1058769111yescharacteristic
507C23#1058769111yescharacteristic
508C23C2×Q8×D1711yescharacteristic
509C23C2×#6325611yescharacteristic
510C23#1058769612yes—
511C23#1058769611yescharacteristic
512C23#1058769712yes—
513C23#1058769711yescharacteristic
514C23C23×Dic1711yescharacteristic
515Q8—416no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—344no—
540C2×C4—344no—
541C2×C4—344no—
542C2×C4—344no—
543C2×C4—344no—
544C2×C4—344no—
545C2×C4—344no—
546C2×C4—344no—
547C2×C4—344no—
548C2×C4—6816no—
549C2×C4—682no—
550C2×C4—682no—
551C2×C4—682no—
552C2×C4—682no—
553C2×C4—682no—
554C2×C4—682no—
555C2×C4—682no—
556C2×C4—682no—
557C2×C4—682no—
558C2×C4—682no—
559C2×C4—682no—
560C2×C4—682no—
561C2×C4—682no—
562C2×C4—682no—
563C2×C4—682no—
564C2×C4—682no—
565C2×C4—682no—
566C2×C4—682no—
567C2×C4—682no—
568C2×C4D4×Dic1718yes—
569C22—24no—
570C22—28no—
571C22—42no—
572C22—42no—
573C22—44no—
574C22—42no—
575C22—44no—
576C22—42no—
577C22—44no—
578C22—42no—
579C22—42no—
580C22C2×D4×Dic1712yes—
581C22C2×Q8×Dic1711yescharacteristic
582C22#1176328411yescharacteristic
583C22#1176329212yes—
584C22#1176329211yescharacteristic
585C22#1176337712yes—
586C22#1176337912yes—
587C22#1176339911yescharacteristic
588C22#1176340811yescharacteristic
589C22#1176342812yes—
590C22#1176342812yes—
591C22#1176343411yescharacteristic
592C22#1176343811yescharacteristic
593C22#1176344111yescharacteristic
594C22#1176344612yes—
595C22#1176345012yes—
596C22#1176347212yes—
597C22#1176349011yescharacteristic
598C22#1176350112yes—
599C22#1176351712yes—
600C22#1176355111yescharacteristic
601C22#1176355312yes—
602C22#1176355911yescharacteristic
603C4—28no—
604C4—42no—
605C4—44no—
606C4—42no—
607C4—684no—
608C4—684no—
609C4—684no—
610C4—1362no—
611C2—44no—
612C2#1453149712yes—
613C2#1453151911yescharacteristic
614C2#1453176811yescharacteristic
615C2#1453179312yes—
616C2#1453195811yescharacteristic
617C2#1453500411yescharacteristic
618C2#1453509311yescharacteristic
619C2#1453520412yes—
620C2#1453527912yes—
621C2#1453620111yescharacteristic
622C2#1453621011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC2×C17×#393C2
socle, cosocleC24×C17C24
prime radicalC25×C17C23
hypercenterC2×#393D17
hypocenterC17#14575
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
1. C2×#11762716C2semidirect
3. #14527049C2semidirect
4. #14527041C2semidirect
7. #14527009C2semidirect
31. C17×#393C4semidirect
39. C22×Q8×C17C2×C4semidirect
64. C2×C22⋊Dic17Q8semidirect
76. C17×#95C2×C4semidirect
77. C17×#95C2×C4semidirect
319. #393Dic17semidirect
350. C22×Q8C2×Dic17semidirect
396. #95C2×Dic17semidirect
397. #95C2×Dic17semidirect
398. C17#14575semidirect

Maximal subgroups

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

subgroupindex
C2×#117627162
C2×#117626842
#145270492
#145270412
#145270292
#145270212
#145270092
#145270062
C2×C17×#3932
#1457517