Finite Group Databasebeta

← #17061821

Subgroups — #17061821

Order 4352.

1558 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762849C211yesmaximal, characteristic, complemented
2C2×#11762818C211yesmaximal, characteristic, complemented
3C2×#11762818C211yesmaximal, characteristic, complemented
4C2×#11762773C211yesmaximal, characteristic
5#14531167C211yesmaximal, characteristic
6#14529984C214yesmaximal, complemented
7#14529977C214yesmaximal, complemented
8#14528503C211yesmaximal, characteristic, complemented
9C22×C17×C22⋊Q8C211yesmaximal, characteristic
10C22×#63223C2211yescharacteristic
11C22×#63223C2211yescharacteristic
12C2×#10587612C2211yescharacteristic
13C2×#10587612C2211yescharacteristic
14#11762849—24no—
15C2×#10587478C2212yescomplemented
16C2×#10587479C2212yescomplemented
17#11762818—24no—
18#11762818—24no—
19C2×#10587506C2212yescomplemented
20C2×#10587506C2214yescomplemented
21C2×#10587506C2212yescomplemented
22#11762773—24no—
23C2×C22⋊#3622C2214yes—
24#11762672—21no—
25#11762669C2212yes—
26#11762669C2212yes—
27#11762665—21no—
28C4×C22⋊Dic17—22no—
29#11762571C2214yescomplemented
30C23×Q8×C17C2211yescharacteristic
31C24×C4×C17C2211yescharacteristic
32C2×C17×C22⋊Q8—24no—
33C2×C17×C22⋊Q8—24no—
34C2×C17×C22⋊Q8—22no—
35C2×C17×C22⋊Q8C2214yes—
36C22×C4⋊C4×C17—21no—
37C22×C4⋊C4×C17C2211yescharacteristic
38C22×C22⋊C4×C17—21no—
39C25×C17D411yescharacteristic
40C22×Q8×C17—22no—
41C22×Q8×C17—21no—
42C22×Q8×C17—22no—
43C22×Q8×C17—22no—
44C22×Q8×C17—22no—
45C22×Q8×C17—22no—
46C22×Q8×C17D414yes—
47C22×Q8×C17C2312yes—
48C23×C4×C17—21no—
49C23×C4×C17—21no—
50C23×C4×C17—22no—
51C23×C4×C17—22no—
52C23×C4×C17—22no—
53C23×C4×C17—22no—
54C23×C4×C17—22no—
55C23×C4×C17—22no—
56C23×C4×C17D411yescharacteristic
57C23×C4×C17D412yes—
58C23×C4×C17C2312yes—
59C23×C4×C17C2311yescharacteristic
60C17×C22⋊Q8—216no—
61C17×C22⋊Q8—28no—
62C17×C22⋊Q8—28no—
63C2×C4⋊C4×C17—28no—
64C2×C4⋊C4×C17—24no—
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17—22no—
67C2×C4⋊C4×C17D414yes—
68C2×C4⋊C4×C17C2312yes—
69C2×C22⋊C4×C17—28no—
70C2×C22⋊C4×C17—24no—
71C2×C22⋊Dic17—21no—
72C2×C22⋊Dic17—21no—
73C2×C22⋊Dic17—22no—
74C2×C22⋊Dic17—22no—
75#10587612—22no—
76#10587612—22no—
77#10587612—22no—
78#10587612—22no—
79C2×C4⋊Dic17C2312yes—
80C2×C4⋊Dic17C2311yescharacteristic
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4×Dic17—21no—
83C2×C4×Dic17—21no—
84C2×#63223—22no—
85C2×#63223—22no—
86C2×#63223—22no—
87C2×#63223—22no—
88C2×#63223—22no—
89C2×#63223D414yes—
90C22×#3622C2312yes—
91C22×#3622C2312yes—
92#10587506—28no—
93#10587506—24no—
94#10587506—24no—
95#10587502—22no—
96#10587502—22no—
97C22⋊#3622—28no—
98#10587479—24no—
99#10587478—24no—
100C24×C17—22no—
101C24×C17—24no—
102C24×C17—22no—
103C24×C17—22no—
104C24×C17—42no—
105C24×C17C2×D412yes—
106C24×C17C2×D411yescharacteristic
107C2×Q8×C17—24no—
108C2×Q8×C17—22no—
109C2×Q8×C17—28no—
110C2×Q8×C17—28no—
111C2×Q8×C17—22no—
112C2×Q8×C17—22no—
113C2×Q8×C17—22no—
114C2×Q8×C17—22no—
115C2×Q8×C17—44no—
116C2×Q8×C17—42no—
117C2×Q8×C17—42no—
118C2×Q8×C17—44no—
119C2×Q8×C17C2×D414yes—
120C22×C4×C17—22no—
121C22×C4×C17—24no—
122C22×C4×C17—24no—
123C22×C4×C17—22no—
124C22×C4×C17—22no—
125C22×C4×C17—22no—
126C22×C4×C17—22no—
127C22×C4×C17—24no—
128C22×C4×C17—24no—
129C22×C4×C17—24no—
130C22×C4×C17—22no—
131C22×C4×C17—22no—
132C22×C4×C17—22no—
133C22×C4×C17—22no—
134C22×C4×C17—22no—
135C22×C4×C17—22no—
136C22×C4×C17—44no—
137C22×C4×C17—42no—
138C22×C4×C17—42no—
139C22×C4×C17—44no—
140C22×C4×C17—44no—
141C22×C4×C17—44no—
142C22×C4×C17—44no—
143C22×C4×C17C2×D412yes—
144C22×C4×C17C2×D412yes—
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×D412yes—
147C22×C4×C17C2×D412yes—
148C22×C4×C17#4112yes—
149C22×C4×C17#4112yes—
150C22×C4×C17#4112yes—
151C22×C4×C17C2411yescharacteristic
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17—22no—
155C4⋊C4×C17—216no—
156C4⋊C4×C17—24no—
157C4⋊C4×C17—22no—
158C4⋊C4×C17C2×D414yes—
159C22⋊C4×C17—216no—
160C22⋊Dic17—44no—
161C22⋊Dic17—42no—
162C22⋊Dic17—42no—
163C22⋊Dic17—44no—
164C4⋊Dic17—21no—
165C4⋊Dic17—21no—
166C4⋊Dic17—22no—
167C4⋊Dic17—21no—
168C4⋊Dic17—21no—
169C4⋊Dic17—22no—
170C4×Dic17—42no—
171C4×Dic17—42no—
172#63223—28no—
173#63223—44no—
174C2×#3622—22no—
175C2×#3622—22no—
176C2×#3622—22no—
177C2×#3622—22no—
178C2×#3622—22no—
179C2×#3622C2×D414yes—
180#23481—171nomaximal, Sylow
181C23×C17—22no—
182C23×C17—22no—
183C23×C17—28no—
184C23×C17—22no—
185C23×C17—22no—
186C23×C17—22no—
187C23×C17—22no—
188C23×C17—22no—
189C23×C17—22no—
190C23×C17—48no—
191C23×C17—48no—
192C23×C17—44no—
193C23×C17—44no—
194C23×C17#12012yes—
195C23×C17#12112yes—
196C23×C17#12312yes—
197C23×C17#13714yes—
198C23×C17C22×D411yescharacteristic
199Q8×C17—24no—
200Q8×C17—24no—
201Q8×C17—42no—
202Q8×C17—42no—
203Q8×C17—48no—
204C2×C4×C17—24no—
205C2×C4×C17—24no—
206C2×C4×C17—22no—
207C2×C4×C17—22no—
208C2×C4×C17—22no—
209C2×C4×C17—48no—
210C2×C4×C17—42no—
211C2×C4×C17—42no—
212C2×C4×C17—44no—
213C2×C4×C17—44no—
214C2×C4×C17—48no—
215C2×C4×C17—42no—
216C2×C4×C17—42no—
217C2×C4×C17—42no—
218C2×C4×C17—42no—
219C2×C4×C17—48no—
220C2×C4×C17#12012yes—
221C2×C4×C17#12114yes—
222C2×C4×C17#12114yes—
223C2×C4×C17#12112yes—
224C2×C4×C17#12314yes—
225C2×C4×C17#12312yes—
226C2×C4×C17#12414yes—
227C2×C4×C17C22×D411yescharacteristic
228C2×C4×C17C22×D411yescharacteristic
229C2×C4×C17C22×D411yescharacteristic
230C2×C4×C17C2×#4111yescharacteristic
231C2×C4×C17C2×#4111yescharacteristic
232C2×C4×C17C2×#4111yescharacteristic
233C2×Dic17—41no—
234C2×Dic17—41no—
235C2×Dic17—42no—
236C2×Dic17—42no—
237C2×Dic17—42no—
238C2×Dic17—42no—
239C2×Dic17—42no—
240#3622—24no—
241#3622—42no—
242C22×C22⋊Q8D1711yescharacteristic
243C2×#428—171no—
244C2×#419—171no—
245C2×#419—171no—
246C2×#407—171no—
247#2280—171no—
248#2036—174no—
249#2035—174no—
250#1736—171no—
251C22×C17—24no—
252C22×C17—22no—
253C22×C17—22no—
254C22×C17—24no—
255C22×C17—24no—
256C22×C17—24no—
257C22×C17—24no—
258C22×C17—416no—
259C22×C17—42no—
260C22×C17—42no—
261C22×C17—42no—
262C22×C17—42no—
263C22×C17#39414yes—
264C22×C17C2×#12011yescharacteristic
265C22×C17C2×#12111yescharacteristic
266C22×C17C2×#12311yescharacteristic
267C22×C17C2×#13712yes—
268C22×C17#57512yes—
269C4×C17—42no—
270C4×C17—44no—
271C4×C17—44no—
272C4×C17#39414yes—
273C4×C17C2×#12111yescharacteristic
274C4×C17C2×#12111yescharacteristic
275C4×C17C2×#12311yescharacteristic
276C4×C17C2×#12411yescharacteristic
277Dic17—81no—
278Dic17—81no—
279Dic17—82no—
280C23×Q8C2×D1711yescharacteristic
281C24×C4C2×D1711yescharacteristic
282C22×M16—171no—
283C22×M16—171no—
284C2×C22⋊Q8—24no—
285C2×C22⋊Q8—22no—
286C2×C22⋊Q8—24no—
287C2×C22⋊Q8C2×D1714yes—
288C2×#117—171no—
289C2×#117—171no—
290C22×C4⋊C4—21no—
291C22×C4⋊C4C2×D1711yescharacteristic
292C22×C22⋊C4—21no—
293#428—344no—
294C2×#107—172no—
295C2×#106—172no—
296#419—344no—
297#419—344no—
298C2×#103—172no—
299C2×#103—174no—
300C2×#103—172no—
301#407—344no—
302C2×C22⋊C8—174no—
303#381—341no—
304#380—172no—
305#380—172no—
306#379—341no—
307C4×C22⋊C4—342no—
308#340—174no—
309C2×C17—24no—
310C2×C17—44no—
311C2×C17C2×#39411yescharacteristic
312C2×C17#300411yescharacteristic
313C2×C17#300511yescharacteristic
314C2×C17#305612yes—
315C2×C17#309512yes—
316C25C22⋊D1711yescharacteristic
317C22×Q8—21no—
318C22×Q8—22no—
319C22×Q8—22no—
320C22×Q8—22no—
321C22×Q8—22no—
322C22×Q8—22no—
323C22×Q8C22⋊D1714yes—
324C22×Q8C22×D1712yes—
325C23×C4—21no—
326C23×C4—21no—
327C23×C4—22no—
328C23×C4—22no—
329C23×C4—22no—
330C23×C4—22no—
331C23×C4—22no—
332C23×C4—22no—
333C23×C4C22⋊D1711yescharacteristic
334C23×C4C22⋊D1712yes—
335C23×C4C22×D1712yes—
336C23×C4C22×D1711yescharacteristic
337C2×M16—174no—
338C2×M16—342no—
339C2×M16—342no—
340C2×M16—342no—
341C2×M16—342no—
342C2×M16—342no—
343C22×C8—172no—
344C22×C8—172no—
345C22⋊Q8—216no—
346C22⋊Q8—28no—
347C22⋊Q8—28no—
348#117—342no—
349#117—342no—
350#117—342no—
351#117—342no—
352C2×C4⋊C4—28no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—172no—
357C2×C4⋊C4—171no—
358C2×C4⋊C4—171no—
359C2×C4⋊C4C22⋊D1714yes—
360C2×C4⋊C4C22×D1712yes—
361C2×C22⋊C4—28no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—341no—
364C2×C22⋊C4—341no—
365C2×C22⋊C4—342no—
366C2×C22⋊C4—342no—
367C2×C42—341no—
368C2×C42—341no—
369#107—344no—
370#106—344no—
371#103—348no—
372#103—344no—
373#103—344no—
374C22⋊C8—348no—
375#95—342no—
376#95—342no—
377C17#2348111yescharacteristic, complemented, Sylow
378C24—24no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—42no—
383C24C2×C22⋊D1712yes—
384C24C2×C22⋊D1711yescharacteristic
385C2×Q8—22no—
386C2×Q8—24no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—28no—
392C2×Q8—28no—
393C2×Q8—42no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—42no—
397C2×Q8C2×C22⋊D1714yes—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—44no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—341no—
422C22×C4—341no—
423C22×C4—342no—
424C22×C4D4×D1712yes—
425C22×C4#6325412yes—
426C22×C4#6325412yes—
427C22×C4#6325412yes—
428C22×C4C2×C22⋊D1712yes—
429C22×C4C2×C22⋊D1712yes—
430C22×C4C2×C22⋊D1712yes—
431C22×C4C2×C22⋊D1711yescharacteristic
432C22×C4C23×D1711yescharacteristic
433M16—348no—
434M16—684no—
435C2×C8—174no—
436C2×C8—342no—
437C2×C8—342no—
438C2×C8—342no—
439C2×C8—342no—
440C2×C8—342no—
441C4⋊C4—216no—
442C4⋊C4—22no—
443C4⋊C4—24no—
444C4⋊C4—342no—
445C4⋊C4—342no—
446C4⋊C4—341no—
447C4⋊C4—341no—
448C4⋊C4—341no—
449C4⋊C4—341no—
450C4⋊C4C2×C22⋊D1714yes—
451C22⋊C4—216no—
452C22⋊C4—684no—
453C22⋊C4—682no—
454C22⋊C4—684no—
455C22⋊C4—682no—
456C42—682no—
457C42—682no—
458C23—22no—
459C23—22no—
460C23—28no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—48no—
468C23—48no—
469C23—44no—
470C23—44no—
471C23#1058762112yes—
472C23#1058762512yes—
473C23#1058762814yes—
474C23#1058763912yes—
475C23C22×C22⋊D1711yescharacteristic
476Q8—24no—
477Q8—24no—
478Q8—48no—
479Q8—42no—
480Q8—42no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—48no—
487C2×C4—44no—
488C2×C4—48no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—48no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—681no—
498C2×C4—681no—
499C2×C4—682no—
500C2×C4—682no—
501C2×C4—682no—
502C2×C4—682no—
503C2×C4—682no—
504C2×C4#1058762112yes—
505C2×C4#1058762114yes—
506C2×C4#1058762214yes—
507C2×C4#1058762414yes—
508C2×C4#1058762512yes—
509C2×C4#1058762514yes—
510C2×C4#1058763912yes—
511C2×C4C2×D4×D1711yescharacteristic
512C2×C4C2×#6325411yescharacteristic
513C2×C4C2×#6325411yescharacteristic
514C2×C4C2×#6325411yescharacteristic
515C2×C4C22×C22⋊D1711yescharacteristic
516C2×C4C22×C22⋊D1711yescharacteristic
517C8—344no—
518C8—682no—
519C22—22no—
520C22—22no—
521C22—24no—
522C22—24no—
523C22—24no—
524C22—24no—
525C22—24no—
526C22—416no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22#1176272314yes—
532C22C2×#1058763911yescharacteristic
533C22C2×#1058762511yescharacteristic
534C22C2×#1058762111yescharacteristic
535C22C2×#1058762812yes—
536C22#1176360712yes—
537C4—44no—
538C4—42no—
539C4—44no—
540C4—1361no—
541C4—1361no—
542C4—1362no—
543C4#1176272314yes—
544C4C2×#1058762411yescharacteristic
545C4C2×#1058762511yescharacteristic
546C4C2×#1058762111yescharacteristic
547C4C2×#1058762211yescharacteristic
548C2—24no—
549C2—44no—
550C2C2×#1176272311yescharacteristic
551C2#1453801811yescharacteristic
552C2#1453802611yescharacteristic
553C2#1453841012yes—
554C2#1453872812yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC22×C17×C22⋊Q8C2
socleC23×C17C22×D4
prime radicalC25×C17D4
hypercenterC22×C22⋊Q8D17
hypocenterC17#23481
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#11762849C2semidirect
2. C2×#11762818C2semidirect
3. C2×#11762818C2semidirect
6. #14529984C2semidirect
7. #14529977C2semidirect
8. #14528503C2semidirect
15. C2×#10587478C22semidirect
16. C2×#10587479C22semidirect
19. C2×#10587506C22semidirect
20. C2×#10587506C22semidirect
21. C2×#10587506C22semidirect
29. #11762571C22semidirect
377. C17#23481semidirect

Maximal subgroups

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

subgroupindex
C2×#117628492
C2×#117628182
C2×#117627732
#145311672
#145299842
#145299772
#145285032
C22×C17×C22⋊Q82
#2348117