Finite Group Databasebeta

← #16956212

Subgroups — #16956212

Order 4352.

2438 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587563C211yesmaximal, characteristic
2C23×#63225C211yesmaximal, characteristic
3C2×#11762672C211yesmaximal, characteristic
4C2×#11762670C211yesmaximal, characteristic
5#14527035C214yesmaximal, complemented
6#14527015C214yesmaximal, complemented
7#14527006C211yesmaximal, characteristic
8#14527006C211yesmaximal, characteristic
9C2×C17×#381C211yesmaximal, characteristic
10C24×Dic17C2211yescharacteristic
11C24×Dic17C2211yescharacteristic
12C2×#10587563—24no—
13C2×#10587563—24no—
14C2×#10587563—24no—
15C22×#63225—24no—
16C22×#63225—24no—
17C22×#63225C418yescomplemented
18C22×#63225C2212yes—
19C22×#63225C2212yes—
20C22×#63225C2212yes—
21C22×C22⋊Dic17C2211yescharacteristic
22C22×C22⋊Dic17C2211yescharacteristic
23C22×C4×Dic17C2212yes—
24#11762672—24no—
25#11762670—24no—
26C2×#10587502C2214yes—
27C2×#10587502C2214yes—
28C2×#10587502C2212yes—
29C2×#10587502C2214yes—
30C2×#10587502C2212yes—
31C24×C4×C17C2211yescharacteristic
32C22×C22⋊C4×C17C2211yescharacteristic
33C22×C22⋊C4×C17C2211yescharacteristic
34C17×#381—24no—
35C2×C17×#95C2214yes—
36C25×C17C2311yescharacteristic
37C23×C4×C17—22no—
38C23×C4×C17—22no—
39C23×C4×C17—22no—
40C23×C4×C17—22no—
41C23×C4×C17—22no—
42C23×C4×C17—22no—
43C23×C4×C17C2×C414yes—
44C23×C4×C17C2312yes—
45C23×C4×C17C2312yes—
46C23×C4×C17C2312yes—
47C23×Dic17—22no—
48C23×Dic17—22no—
49C23×Dic17—22no—
50C23×Dic17—22no—
51C23×Dic17—22no—
52C23×Dic17—22no—
53C23×Dic17—28no—
54C23×Dic17C2×C414yes—
55C23×Dic17C2×C414yes—
56C23×Dic17D414yes—
57C23×Dic17D414yes—
58C23×Dic17C2312yes—
59C23×Dic17C2312yes—
60C23×Dic17C2311yescharacteristic
61C23×Dic17C2311yescharacteristic
62C23×Dic17C2312yes—
63C2×C22⋊C4×C17—22no—
64C2×C22⋊C4×C17—22no—
65C2×C22⋊C4×C17—22no—
66C2×C22⋊C4×C17—22no—
67C2×C22⋊C4×C17—22no—
68C2×C22⋊C4×C17D414yes—
69C2×C22⋊Dic17—22no—
70C2×C22⋊Dic17—22no—
71C2×C22⋊Dic17—22no—
72C2×C22⋊Dic17—22no—
73C2×C22⋊Dic17—22no—
74C2×C22⋊Dic17D414yes—
75C2×#63225—24no—
76C2×#63225—232no—
77C2×#63225—24no—
78C2×#63225—24no—
79C2×#63225—24no—
80C2×#63225—24no—
81C2×#63225—24no—
82C2×#63225—24no—
83C2×#63225C2×C418yes—
84C2×#63225C2×C418yes—
85C2×C4×Dic17—24no—
86C2×C4×Dic17—24no—
87C2×C4×Dic17—24no—
88#10587563—232no—
89C17×#95—28no—
90#10587502—24no—
91#10587502—24no—
92#10587502—28no—
93#10587502—28no—
94#10587502—28no—
95C24×C17—24no—
96C24×C17—24no—
97C24×C17C22×C412yes—
98C24×C17C2×D412yes—
99C24×C17C2×D412yes—
100C24×C17C2×D412yes—
101C24×C17C2×D412yes—
102C24×C17#4112yes—
103C24×C17#4112yes—
104C24×C17C2411yescharacteristic
105C22×C4×C17—28no—
106C22×C4×C17—28no—
107C22×C4×C17—28no—
108C22×C4×C17—22no—
109C22×C4×C17—22no—
110C22×C4×C17—22no—
111C22×C4×C17—22no—
112C22×C4×C17—22no—
113C22×C4×C17—22no—
114C22×C4×C17—22no—
115C22×C4×C17—22no—
116C22×C4×C17—24no—
117C22×C4×C17—24no—
118C22×C4×C17—22no—
119C22×C4×C17—22no—
120C22×C4×C17—24no—
121C22×C4×C17—22no—
122C22×C4×C17—24no—
123C22×C4×C17—44no—
124C22×C4×C17—44no—
125C22×C4×C17—44no—
126C22×C4×C17—44no—
127C22×C4×C17C22×C414yes—
128C22×C4×C17C2×D414yes—
129C22×Dic17—216no—
130C22×Dic17—216no—
131C22×Dic17—28no—
132C22×Dic17—28no—
133C22×Dic17—28no—
134C22×Dic17—28no—
135C22×Dic17—28no—
136C22×Dic17—28no—
137C22×Dic17—28no—
138C22×Dic17—21no—
139C22×Dic17—21no—
140C22×Dic17—22no—
141C22×Dic17—22no—
142C22×Dic17—21no—
143C22×Dic17—21no—
144C22×Dic17—21no—
145C22×Dic17—21no—
146C22×Dic17—21no—
147C22×Dic17—21no—
148C22×Dic17—24no—
149C22×Dic17—216no—
150C22×Dic17—22no—
151C22×Dic17—22no—
152C22×Dic17—24no—
153C22×Dic17—24no—
154C22×Dic17—22no—
155C22×Dic17—22no—
156C22×Dic17—24no—
157C22×Dic17—22no—
158C22×Dic17—24no—
159C22×Dic17C22⋊C4116yes—
160C22×Dic17C22×C412yes—
161C22×Dic17C22×C412yes—
162C22×Dic17C22×C414yes—
163C22×Dic17C2×D414yes—
164C22×Dic17C2×D412yes—
165C22×Dic17C2×D412yes—
166C22×Dic17C2×D412yes—
167C22×Dic17C2×D412yes—
168C22×Dic17#4118yes—
169C22⋊C4×C17—28no—
170C22⋊C4×C17—44no—
171C22⋊Dic17—28no—
172C22⋊Dic17—44no—
173#63225—216no—
174#63225—216no—
175#63225—216no—
176C4×Dic17—216no—
177#13870—171nomaximal, Sylow
178C23×C17—24no—
179C23×C17—24no—
180C23×C17—24no—
181C23×C17—24no—
182C23×C17—24no—
183C23×C17—24no—
184C23×C17—28no—
185C23×C17—28no—
186C23×C17—28no—
187C23×C17—28no—
188C23×C17C2×C22⋊C414yes—
189C23×C17C4×D414yes—
190C23×C17C4×D414yes—
191C23×C17#12014yes—
192C23×C17#12114yes—
193C23×C17#12114yes—
194C23×C17#12314yes—
195C23×C17C23×C411yescharacteristic
196C23×C17C22×D411yescharacteristic
197C23×C17C22×D411yescharacteristic
198C23×C17C22×D411yescharacteristic
199C23×C17C22×D411yescharacteristic
200C23×C17C2×#4111yescharacteristic
201C23×C17C2×#4111yescharacteristic
202C23×C17C2×#4114yes—
203C23×C172−1+414yes—
204C2×C4×C17—24no—
205C2×C4×C17—24no—
206C2×C4×C17—24no—
207C2×C4×C17—24no—
208C2×C4×C17—28no—
209C2×C4×C17—42no—
210C2×C4×C17—42no—
211C2×C4×C17—42no—
212C2×C4×C17—42no—
213C2×C4×C17—42no—
214C2×C4×C17—42no—
215C2×C4×C17—44no—
216C2×C4×C17—416no—
217C2×C4×C17—42no—
218C2×C4×C17—42no—
219C2×C4×C17—42no—
220C2×C4×C17—44no—
221C2×C4×C17—44no—
222C2×Dic17—232no—
223C2×Dic17—24no—
224C2×Dic17—24no—
225C2×Dic17—22no—
226C2×Dic17—22no—
227C2×Dic17—22no—
228C2×Dic17—22no—
229C2×Dic17—22no—
230C2×Dic17—22no—
231C2×Dic17—22no—
232C2×Dic17—22no—
233C2×Dic17—22no—
234C2×Dic17—22no—
235C2×Dic17—22no—
236C2×Dic17—22no—
237C2×Dic17—28no—
238C2×Dic17—28no—
239C2×Dic17—232no—
240C2×Dic17—24no—
241C2×Dic17—28no—
242C2×Dic17—42no—
243C2×Dic17—42no—
244C2×Dic17—44no—
245C2×Dic17—42no—
246C2×Dic17—42no—
247C2×Dic17—42no—
248C2×Dic17—44no—
249C2×Dic17—44no—
250C2×Dic17C2×C22⋊C414yes—
251C2×Dic17C2×C22⋊C414yes—
252C2×Dic17#11718yescomplemented
253C22×#117—171no—
254C23×C4⋊C4—171no—
255C2×#381—171no—
256C2×#381—171no—
257C2×#381D1711yescharacteristic
258#1441—174no—
259#1436—174no—
260C22⋊#95—171no—
261C22⋊#95—171no—
262C22×C17—24no—
263C22×C17—28no—
264C22×C17—28no—
265C22×C17—28no—
266C22×C17—28no—
267C22×C17—28no—
268C22×C17—28no—
269C22×C17—24no—
270C22×C17#38618yes—
271C22×C17C22×C22⋊C411yescharacteristic
272C22×C17C2×#11712yes—
273C22×C17C2×C4×D411yescharacteristic
274C22×C17C2×C4×D411yescharacteristic
275C22×C17#51912yes—
276C22×C17C2×#12011yescharacteristic
277C22×C17C2×#12111yescharacteristic
278C22×C17C2×#12111yescharacteristic
279C22×C17C2×#12311yescharacteristic
280C22×C17#52512yes—
281C22×C17#52512yes—
282C22×C17#53612yes—
283C22×C17#53612yes—
284C22×C17#54714yes—
285C22×C17#54714yes—
286C22×C17#55214yes—
287C22×C17#55214yes—
288C4×C17—44no—
289C4×C17—44no—
290C4×C17—82no—
291Dic17—28no—
292Dic17—24no—
293Dic17—24no—
294Dic17—44no—
295Dic17—82no—
296C24×C4—171no—
297C24×C4—171no—
298C24×C4C2×D1711yescharacteristic
299C2×#117—344no—
300C2×#117—344no—
301C2×#117—344no—
302C22×C4⋊C4—178no—
303C22×C4⋊C4—172no—
304C22×C4⋊C4—172no—
305C22×C4⋊C4—172no—
306C22×C4⋊C4—344no—
307C22×C4⋊C4—344no—
308C22×C22⋊C4—171no—
309C22×C22⋊C4—171no—
310C22×C22⋊C4C2×D1711yescharacteristic
311C22×C22⋊C4C2×D1711yescharacteristic
312C22×C42—172no—
313#381—24no—
314#381—344no—
315#381—344no—
316C2×#95—174no—
317C2×#95—174no—
318C2×#95—172no—
319C2×#95—174no—
320C2×#95—172no—
321C2×#95C2×D1714yes—
322C2×C17—28no—
323C2×C17#228011yescharacteristic
324C2×C17C2×#38611yescharacteristic
325C2×C17#231011yescharacteristic
326C2×C17#234412yes—
327C2×C17#234412yes—
328C2×C17#240612yes—
329C2×C17#241312yes—
330C2×C17#241312yes—
331C2×C17#248412yes—
332C25C22×D1711yescharacteristic
333C23×C4—22no—
334C23×C4—22no—
335C23×C4—22no—
336C23×C4—22no—
337C23×C4—22no—
338C23×C4—22no—
339C23×C4—174no—
340C23×C4—174no—
341C23×C4—174no—
342C23×C4—171no—
343C23×C4—172no—
344C23×C4—171no—
345C23×C4—174no—
346C23×C4—172no—
347C23×C4—172no—
348C23×C4—348no—
349C23×C4—342no—
350C23×C4—342no—
351C23×C4—342no—
352C23×C4—342no—
353C23×C4—342no—
354C23×C4—342no—
355C23×C4C4×D1714yes—
356C23×C4C22×D1712yes—
357C23×C4C22×D1712yes—
358C23×C4C22×D1712yes—
359#117—3432no—
360C2×C4⋊C4—178no—
361C2×C4⋊C4—178no—
362C2×C4⋊C4—344no—
363C2×C4⋊C4—344no—
364C2×C4⋊C4—344no—
365C2×C4⋊C4—344no—
366C2×C4⋊C4—3432no—
367C2×C4⋊C4—344no—
368C2×C4⋊C4—344no—
369C2×C4⋊C4—344no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—174no—
376C2×C22⋊C4—342no—
377C2×C22⋊C4—342no—
378C2×C22⋊C4—342no—
379C2×C22⋊C4—342no—
380C2×C22⋊C4—342no—
381C2×C22⋊C4C22⋊D1714yes—
382C2×C42—344no—
383C2×C42—344no—
384C2×C42—344no—
385#95—28no—
386#95—348no—
387#95—348no—
388#95—348no—
389#95—344no—
390#95—344no—
391C17#1387011yescharacteristic, complemented, Sylow
392C24—24no—
393C24—24no—
394C24C2×C4×D1712yes—
395C24#6325212yes—
396C24D4×D1712yes—
397C24D4×D1712yes—
398C24D4×D1712yes—
399C24#6325412yes—
400C24C2×C22⋊D1712yes—
401C24C23×D1711yescharacteristic
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—172no—
425C22×C4—172no—
426C22×C4—174no—
427C22×C4—172no—
428C22×C4—172no—
429C22×C4—172no—
430C22×C4—172no—
431C22×C4—178no—
432C22×C4—1716no—
433C22×C4—174no—
434C22×C4—3416no—
435C22×C4—3416no—
436C22×C4—3416no—
437C22×C4—341no—
438C22×C4—341no—
439C22×C4—341no—
440C22×C4—342no—
441C22×C4—342no—
442C22×C4—341no—
443C22×C4—341no—
444C22×C4—341no—
445C22×C4—344no—
446C22×C4—341no—
447C22×C4—341no—
448C22×C4—348no—
449C22×C4—348no—
450C22×C4—348no—
451C22×C4—348no—
452C22×C4—348no—
453C22×C4—348no—
454C22×C4—348no—
455C22×C4—342no—
456C22×C4—344no—
457C22×C4—344no—
458C22×C4—342no—
459C22×C4—342no—
460C22×C4—342no—
461C22×C4—342no—
462C22×C4—344no—
463C22×C4—344no—
464C22×C4C2×C4×D1714yes—
465C22×C4C2×C22⋊D1714yes—
466C4⋊C4—3416no—
467C4⋊C4—3416no—
468C4⋊C4—3416no—
469C22⋊C4—28no—
470C22⋊C4—44no—
471C22⋊C4—348no—
472C22⋊C4—684no—
473C42—3416no—
474C23—28no—
475C23—28no—
476C23—28no—
477C23—28no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23C22⋊C4×D1714yes—
485C23#1058756714yes—
486C23#1058756914yes—
487C23#1058757014yes—
488C23C4×C22⋊D1714yes—
489C23#1058762314yes—
490C23#1058762414yes—
491C23C22×C4×D1711yescharacteristic
492C23C2×#6325211yescharacteristic
493C23C2×D4×D1711yescharacteristic
494C23C2×D4×D1711yescharacteristic
495C23C2×D4×D1711yescharacteristic
496C23C2×#6325411yescharacteristic
497C23#41×D1714yes—
498C23#1058769714yes—
499C23C22×C22⋊D1711yescharacteristic
500C2×C4—28no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—416no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—174no—
519C2×C4—174no—
520C2×C4—178no—
521C2×C4—3432no—
522C2×C4—344no—
523C2×C4—344no—
524C2×C4—342no—
525C2×C4—342no—
526C2×C4—342no—
527C2×C4—342no—
528C2×C4—342no—
529C2×C4—342no—
530C2×C4—342no—
531C2×C4—342no—
532C2×C4—342no—
533C2×C4—342no—
534C2×C4—342no—
535C2×C4—342no—
536C2×C4—348no—
537C2×C4—348no—
538C2×C4—3432no—
539C2×C4—348no—
540C2×C4—344no—
541C2×C4—684no—
542C2×C4—682no—
543C2×C4—682no—
544C2×C4—684no—
545C2×C4—684no—
546C2×C4—682no—
547C2×C4—682no—
548C2×C4—682no—
549C22—28no—
550C22—24no—
551C22—28no—
552C22—28no—
553C22—28no—
554C22—24no—
555C22—28no—
556C22—28no—
557C22#1176268518yes—
558C22C2×C22⋊C4×D1711yescharacteristic
559C22#117×D1712yes—
560C22C2×#1058756711yescharacteristic
561C22C2×C4×C22⋊D1711yescharacteristic
562C22#1176328612yes—
563C22C2×#1058762311yescharacteristic
564C22C2×#1058762411yescharacteristic
565C22C2×#1058757011yescharacteristic
566C22C2×#1058756911yescharacteristic
567C22#1176331712yes—
568C22#1176331712yes—
569C22#1176338112yes—
570C22#1176338112yes—
571C22#1176345714yes—
572C22#1176345814yes—
573C22#1176349214yes—
574C22#1176349614yes—
575C4—44no—
576C4—44no—
577C4—82no—
578C4—348no—
579C4—344no—
580C4—344no—
581C4—684no—
582C4—1362no—
583C2—28no—
584C2#1453116411yescharacteristic
585C2C2×#1176268511yescharacteristic
586C2#1453136911yescharacteristic
587C2#1453164812yes—
588C2#1453165412yes—
589C2#1453211612yes—
590C2#1453218712yes—
591C2#1453219312yes—
592C2#1453306812yes—
subgroupG/H
center, FrattiniC24C23×D17
commutatorC23×C17C23×C4
FittingC2×C17×#381C2
socle, cosocleC24×C17C24
prime radicalC25×C17C23
hypercenterC2×#381D17
hypocenterC17#13870
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C24 ◁ C2×#381

Complements

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

NH ≅ G/Ntype
5. #14527035C2semidirect
6. #14527015C2semidirect
17. C22×#63225C4semidirect
252. C2×Dic17#117semidirect
391. C17#13870semidirect

Maximal subgroups

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

subgroupindex
C22×#105875632
C23×#632252
C2×#117626722
C2×#117626702
#145270352
#145270152
#145270062
C2×C17×#3812
#1387017