Finite Group Databasebeta

← #17963121

Subgroups — #17963121

Order 4352.

2438 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587677C211yesmaximal, characteristic
2C2×#11763752C211yesmaximal, characteristic
3C2×#11763752C212yesmaximal
4#14543969C214yesmaximal, complemented
5#14543963C214yesmaximal, complemented
6#14543958C212yesmaximal
7C22×C4⋊C4×D17C211yesmaximal, characteristic
8C2×#10587677—24no—
9C2×#10587677—28no—
10C22×C4⋊#595C2212yes—
11C22×C22⋊#595C2212yes—
12C22×C22⋊#595C2212yes—
13C22×C4×#595C2212yes—
14#11763752—24no—
15#11763752—28no—
16C2×(C2×#595)⋊C4C2212yes—
17C2×(C2×#595)⋊C4C2218yes—
18C2×(C2×#595)⋊C4C2212yes—
19C2×(C2×#595)⋊C4C2218yes—
20C23×C4×D17C2211yescharacteristic
21C23×C4×D17C2211yescharacteristic
22C23×C4×D17C2211yescharacteristic
23C22×C4⋊Dic17C411yescharacteristic, complemented
24C22×#63225C411yescharacteristic, complemented
25C22×#63225C411yescharacteristic, complemented
26C2×C4⋊C4×D17—24no—
27C2×C4⋊C4×D17—24no—
28C2×C4⋊C4×D17C412yescomplemented
29C2×C4⋊C4×D17C412yescomplemented
30C2×C4⋊C4×D17C2212yes—
31C2×C4⋊C4×D17C2212yes—
32C22×C4⋊C4×C17C411yescharacteristic, complemented
33C24×D17C2311yescharacteristic
34C23×#595C2314yes—
35C23×#595C2314yes—
36C23×C4×C17C2×C411yescharacteristic
37C23×C4×C17C2×C411yescharacteristic
38C23×C4×C17C2×C411yescharacteristic
39C23×Dic17C2×C411yescharacteristic
40C23×Dic17C2×C411yescharacteristic
41C23×Dic17C2×C411yescharacteristic
42C22×C4×D17—22no—
43C22×C4×D17—22no—
44C22×C4×D17—24no—
45C22×C4×D17—24no—
46C22×C4×D17—28no—
47C22×C4×D17—24no—
48C22×C4×D17—22no—
49C22×C4×D17—22no—
50C22×C4×D17—24no—
51C22×C4×D17C2×C412yes—
52C22×C4×D17C2×C412yes—
53C22×C4×D17C2×C411yescharacteristic
54C22×C4×D17C2×C411yescharacteristic
55C22×C4×D17D414yes—
56C22×C4×D17D414yes—
57C22×C4×D17C2312yes—
58C22×C4×D17C2311yescharacteristic
59C22×C4×D17C2311yescharacteristic
60C22×C4×D17C2312yes—
61C2×C22⋊#595—24no—
62C2×C22⋊#595—24no—
63C2×C22⋊#595—24no—
64C2×C22⋊#595—24no—
65C2×C22⋊#595—24no—
66C2×C22⋊#595D418yes—
67#10587677—232no—
68C2×C4⋊#595—28no—
69C2×C4⋊#595—24no—
70C2×C4×#595—28no—
71C2×C4×#595—24no—
72C2×C4⋊C4×C17—24no—
73C2×C4⋊C4×C17C2×C412yes—
74C2×C4⋊C4×C17C2×C412yes—
75C2×C4⋊Dic17—24no—
76C2×C4⋊Dic17C2×C412yes—
77C2×C4⋊Dic17C2×C412yes—
78C2×#63225—24no—
79C2×#63225—24no—
80C2×#63225C2×C412yes—
81C2×#63225C2×C412yes—
82C2×#63225C2×C412yes—
83C2×#63225C2×C412yes—
84C4⋊C4×D17—28no—
85C4⋊C4×D17—28no—
86C4⋊C4×D17—24no—
87C4⋊C4×D17—24no—
88C4⋊C4×D17—24no—
89C4⋊C4×D17—24no—
90(C2×#595)⋊C4—216no—
91(C2×#595)⋊C4—24no—
92(C2×#595)⋊C4—24no—
93(C2×#595)⋊C4—216no—
94C24×C17C22×C411yescharacteristic
95C23×D17—24no—
96C23×D17—24no—
97C23×D17C22×C411yescharacteristic
98C23×D17C2×D412yes—
99C23×D17C2×D412yes—
100C23×D17C2×D414yes—
101C23×D17#4114yes—
102C23×D17C2411yescharacteristic
103C22×#595—24no—
104C22×#595—24no—
105C22×#595—28no—
106C22×#595—28no—
107C22×#595—24no—
108C22×#595—24no—
109C22×#595—24no—
110C22×#595—28no—
111C22×#595—28no—
112C22×#595C2×D418yes—
113C22×C4×C17—22no—
114C22×C4×C17—24no—
115C22×C4×C17—24no—
116C22×C4×C17—22no—
117C22×C4×C17—24no—
118C22×C4×C17C22⋊C414yes—
119C22×C4×C17C22×C412yes—
120C22×C4×C17C22×C411yescharacteristic
121C22×C4×C17C22×C411yescharacteristic
122C22×C4×C17C22×C412yes—
123C22×Dic17—22no—
124C22×Dic17—24no—
125C22×Dic17—24no—
126C22×Dic17—22no—
127C22×Dic17—24no—
128C22×Dic17C22⋊C414yes—
129C22×Dic17C22×C412yes—
130C22×Dic17C22×C411yescharacteristic
131C22×Dic17C22×C411yescharacteristic
132C22×Dic17C22×C412yes—
133C2×C4×D17—216no—
134C2×C4×D17—216no—
135C2×C4×D17—24no—
136C2×C4×D17—24no—
137C2×C4×D17—24no—
138C2×C4×D17—28no—
139C2×C4×D17—216no—
140C2×C4×D17—28no—
141C2×C4×D17—22no—
142C2×C4×D17—22no—
143C2×C4×D17—22no—
144C2×C4×D17—22no—
145C2×C4×D17—22no—
146C2×C4×D17—22no—
147C2×C4×D17—24no—
148C2×C4×D17—24no—
149C2×C4×D17—24no—
150C2×C4×D17—24no—
151C2×C4×D17—28no—
152C2×C4×D17—24no—
153C2×C4×D17—22no—
154C2×C4×D17—22no—
155C2×C4×D17—24no—
156C2×C4×D17—44no—
157C2×C4×D17—48no—
158C2×C4×D17—44no—
159C2×C4×D17C22⋊C414yes—
160C2×C4×D17C22⋊C414yes—
161C2×C4×D17C2×D412yes—
162C2×C4×D17C2×D412yes—
163C2×C4×D17C2×D412yes—
164C2×C4×D17C2×D412yes—
165C2×C4×D17#4118yes—
166C22⋊#595—216no—
167C22⋊#595—48no—
168C4⋊#595—216no—
169C4×#595—216no—
170C4⋊C4×C17—24no—
171C4⋊C4×C17—24no—
172C4⋊Dic17—24no—
173C4⋊Dic17—24no—
174#63225—24no—
175#63225—24no—
176#63225—24no—
177#63225—24no—
178#13870—171nomaximal, Sylow
179C23×C17—22no—
180C23×C17—22no—
181C23×C17C2×C22⋊C412yes—
182C23×C17C4×D414yes—
183C23×C17C23×C411yescharacteristic
184C22×D17—22no—
185C22×D17—22no—
186C22×D17—24no—
187C22×D17—24no—
188C22×D17—24no—
189C22×D17—24no—
190C22×D17—216no—
191C22×D17—216no—
192C22×D17C2×C22⋊C412yes—
193C22×D17C4×D414yes—
194C22×D17#12014yes—
195C22×D17#12118yes—
196C22×D17#12314yes—
197C22×D17C22×D412yes—
198C22×D17C22×D411yescharacteristic
199C22×D17C22×D411yescharacteristic
200C22×D17C2×#4112yes—
201C22×D17C2×#4114yes—
202C22×D172−1+414yes—
203C2×#595—28no—
204C2×#595—216no—
205C2×#595—44no—
206C2×#595—44no—
207C2×#595—48no—
208C2×#595—44no—
209C2×#595—44no—
210C2×#595—44no—
211C2×#595—48no—
212C2×#595—48no—
213C2×C4×C17—28no—
214C2×C4×C17—22no—
215C2×C4×C17—22no—
216C2×C4×C17—22no—
217C2×C4×C17—22no—
218C2×C4×C17—28no—
219C2×C4×C17—22no—
220C2×C4×C17—24no—
221C2×C4×C17—44no—
222C2×C4×C17C2×C22⋊C412yes—
223C2×C4×C17C2×C22⋊C412yes—
224C2×C4×C17#11714yescomplemented
225C2×Dic17—28no—
226C2×Dic17—22no—
227C2×Dic17—22no—
228C2×Dic17—22no—
229C2×Dic17—22no—
230C2×Dic17—28no—
231C2×Dic17—22no—
232C2×Dic17—24no—
233C2×Dic17—44no—
234C2×Dic17C2×C22⋊C412yes—
235C2×Dic17C2×C22⋊C412yes—
236C2×Dic17#11714yescomplemented
237C4×D17—216no—
238C4×D17—22no—
239C4×D17—22no—
240C4×D17—24no—
241C4×D17—24no—
242C4×D17—22no—
243C4×D17—22no—
244C4×D17—28no—
245C4×D17—28no—
246C4×D17—28no—
247C4×D17—28no—
248C4×D17—42no—
249C4×D17—44no—
250C4×D17—42no—
251C4×D17—48no—
252C22×#117—171no—
253C23×C4⋊C4—171no—
254C2×#381—172no—
255C2×#381—171no—
256#1441—174no—
257#1436—174no—
258C22⋊#95—172no—
259C22×C17—22no—
260C22×C17—28no—
261C22×C17—22no—
262C22×C17#38614yes—
263C22×C17C22×C22⋊C411yescharacteristic
264C22×C17C2×#11712yes—
265C22×C17C2×C4×D412yes—
266C22×C17#51912yes—
267C2×D17—22no—
268C2×D17—28no—
269C2×D17—22no—
270C2×D17—28no—
271C2×D17—216no—
272C2×D17—28no—
273C2×D17#38614yes—
274C2×D17C2×#12011yescharacteristic
275C2×D17C2×#12112yes—
276C2×D17C2×#12311yescharacteristic
277C2×D17#52512yes—
278C2×D17#52512yes—
279C2×D17#53612yes—
280C2×D17#53612yes—
281C2×D17#54718yes—
282C2×D17#55218yes—
283#595—48no—
284#595—84no—
285C4×C17—24no—
286C4×C17—22no—
287C4×C17—22no—
288C4×C17—42no—
289Dic17—24no—
290Dic17—22no—
291Dic17—22no—
292Dic17—42no—
293C24×C4—171no—
294C24×C4—171no—
295C24×C4—171no—
296C2×#117—348no—
297C2×#117—344no—
298C22×C4⋊C4—171no—
299C22×C4⋊C4—171no—
300C22×C4⋊C4—171no—
301C22×C4⋊C4—172no—
302C22×C4⋊C4—172no—
303C22×C4⋊C4—172no—
304C22×C4⋊C4—172no—
305C22×C4⋊C4—172no—
306C22×C4⋊C4—344no—
307C22×C4⋊C4—344no—
308C22×C4⋊C4#59511yescharacteristic, complemented
309C22×C22⋊C4—172no—
310C22×C22⋊C4—172no—
311C22×C42—172no—
312#381—348no—
313#381—344no—
314C2×#95—178no—
315C2×#95—172no—
316C2×#95—178no—
317C2×#95—172no—
318C2×C17—24no—
319C2×C17#228011yescharacteristic
320C2×C17C2×#38611yescharacteristic
321C2×C17#231011yescharacteristic
322C2×C17#234414yes—
323D17—24no—
324D17#240612yes—
325D17#241314yes—
326D17#248412yes—
327C25—171no—
328C23×C4—171no—
329C23×C4—174no—
330C23×C4—174no—
331C23×C4—171no—
332C23×C4—171no—
333C23×C4—171no—
334C23×C4—172no—
335C23×C4—171no—
336C23×C4—174no—
337C23×C4—171no—
338C23×C4—171no—
339C23×C4—172no—
340C23×C4—172no—
341C23×C4—172no—
342C23×C4—174no—
343C23×C4—348no—
344C23×C4—342no—
345C23×C4—342no—
346C23×C4—344no—
347C23×C4—344no—
348C23×C4—342no—
349C23×C4—342no—
350C23×C4—344no—
351C23×C4—344no—
352C23×C4C2×#59511yescharacteristic
353C23×C4C2×#59511yescharacteristic
354C23×C4C2×#59511yescharacteristic
355#117—3432no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—172no—
358C2×C4⋊C4—172no—
359C2×C4⋊C4—172no—
360C2×C4⋊C4—172no—
361C2×C4⋊C4—172no—
362C2×C4⋊C4—172no—
363C2×C4⋊C4—348no—
364C2×C4⋊C4—348no—
365C2×C4⋊C4—344no—
366C2×C4⋊C4—344no—
367C2×C4⋊C4—344no—
368C2×C4⋊C4—344no—
369C2×C4⋊C4—344no—
370C2×C4⋊C4—344no—
371C2×C4⋊C4—344no—
372C2×C4⋊C4—348no—
373C2×C4⋊C4—344no—
374C2×C4⋊C4C2×#59512yes—
375C2×C4⋊C4C2×#59512yes—
376C2×C22⋊C4—178no—
377C2×C22⋊C4—344no—
378C2×C22⋊C4—344no—
379C2×C22⋊C4—344no—
380C2×C22⋊C4—344no—
381C2×C22⋊C4—344no—
382C2×C42—344no—
383C2×C42—348no—
384#95—344no—
385#95—344no—
386#95—3416no—
387#95—3416no—
388C17#1387011yescharacteristic, complemented, Sylow
389C24—171no—
390C24—174no—
391C24—174no—
392C24—172no—
393C24—172no—
394C24—171no—
395C24—344no—
396C24—344no—
397C24C22×#59511yescharacteristic
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—174no—
404C22×C4—171no—
405C22×C4—171no—
406C22×C4—172no—
407C22×C4—178no—
408C22×C4—174no—
409C22×C4—174no—
410C22×C4—172no—
411C22×C4—172no—
412C22×C4—172no—
413C22×C4—172no—
414C22×C4—178no—
415C22×C4—172no—
416C22×C4—344no—
417C22×C4—342no—
418C22×C4—342no—
419C22×C4—342no—
420C22×C4—344no—
421C22×C4—342no—
422C22×C4—348no—
423C22×C4—3416no—
424C22×C4—348no—
425C22×C4—344no—
426C22×C4—342no—
427C22×C4—342no—
428C22×C4—344no—
429C22×C4—344no—
430C22×C4—3416no—
431C22×C4—3416no—
432C22×C4—344no—
433C22×C4—342no—
434C22×C4—342no—
435C22×C4—344no—
436C22×C4—344no—
437C22×C4—348no—
438C22×C4—344no—
439C22×C4—344no—
440C22×C4—342no—
441C22×C4—344no—
442C22×C4—348no—
443C22×C4—344no—
444C22×C4—344no—
445C22×C4—348no—
446C22×C4—344no—
447C22×C4—342no—
448C22×C4—344no—
449C22×C4—344no—
450C22×C4—348no—
451C22×C4—344no—
452C22×C4—348no—
453C22×C4—684no—
454C22×C4—688no—
455C22×C4—684no—
456C22×C4C22⋊#59514yes—
457C22×C4C22×#59512yes—
458C22×C4C22×#59512yes—
459C22×C4C22×#59511yescharacteristic
460C22×C4C22×#59511yescharacteristic
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—344no—
464C4⋊C4—344no—
465C4⋊C4—344no—
466C4⋊C4—344no—
467C4⋊C4—3416no—
468C4⋊C4—344no—
469C4⋊C4—344no—
470C22⋊C4—3416no—
471C22⋊C4—688no—
472C42—3416no—
473C23—22no—
474C23—22no—
475C23—171no—
476C23—171no—
477C23—174no—
478C23—172no—
479C23—172no—
480C23—174no—
481C23—174no—
482C23—178no—
483C23—174no—
484C23—172no—
485C23—174no—
486C23—344no—
487C23—344no—
488C23—344no—
489C23—344no—
490C23—3416no—
491C23—3416no—
492C23—342no—
493C23—342no—
494C23D4×#59514yes—
495C23C2×C22⋊#59512yes—
496C23C23×#59511yescharacteristic
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—44no—
506C2×C4—172no—
507C2×C4—172no—
508C2×C4—174no—
509C2×C4—342no—
510C2×C4—342no—
511C2×C4—348no—
512C2×C4—348no—
513C2×C4—344no—
514C2×C4—344no—
515C2×C4—342no—
516C2×C4—342no—
517C2×C4—348no—
518C2×C4—348no—
519C2×C4—3416no—
520C2×C4—348no—
521C2×C4—348no—
522C2×C4—342no—
523C2×C4—342no—
524C2×C4—342no—
525C2×C4—342no—
526C2×C4—348no—
527C2×C4—3416no—
528C2×C4—342no—
529C2×C4—344no—
530C2×C4—682no—
531C2×C4—684no—
532C2×C4—682no—
533C2×C4—688no—
534C2×C4—684no—
535C2×C4—684no—
536C2×C4—688no—
537C2×C4—684no—
538C2×C4—684no—
539C2×C4—684no—
540C2×C4—684no—
541C2×C4—688no—
542C2×C4—688no—
543C2×C4#1058767714yescomplemented
544C2×C4C2×C22⋊#59512yes—
545C2×C4C2×C22⋊#59512yes—
546C22—22no—
547C22—22no—
548C22—28no—
549C22—172no—
550C22—172no—
551C22—171no—
552C22—172no—
553C22—171no—
554C22—172no—
555C22—172no—
556C22—178no—
557C22—178no—
558C22—174no—
559C22—348no—
560C22—3416no—
561C22—348no—
562C22—342no—
563C22—342no—
564C22—348no—
565C22#1176375714yes—
566C22C22×C22⋊#59511yescharacteristic
567C22C2×#1058767712yes—
568C22C2×D4×#59512yes—
569C22#1176387712yes—
570C4—24no—
571C4—22no—
572C4—22no—
573C4—42no—
574C4—344no—
575C4—342no—
576C4—342no—
577C4—688no—
578C4—682no—
579C4—1364no—
580C2—24no—
581C2—172no—
582C2—174no—
583C2—172no—
584C2—344no—
585C2#1454522811yescharacteristic
586C2C2×#1176375711yescharacteristic
587C2#1454525811yescharacteristic
588C2#1454529214yes—
subgroupG/H
center, FrattiniC23C23×#595
commutator, socleC23×C17C23×C4
FittingC22×C4⋊C4×C17C4
prime radicalC24×D17C23
cosocleC23×D17C24
hypercenterC22×C4⋊C4#595
hypocenterC17#13870
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #14543969C2semidirect
5. #14543963C2semidirect
23. C22×C4⋊Dic17C4semidirect
24. C22×#63225C4semidirect
25. C22×#63225C4semidirect
28. C2×C4⋊C4×D17C4semidirect
29. C2×C4⋊C4×D17C4semidirect
32. C22×C4⋊C4×C17C4semidirect
224. C2×C4×C17#117semidirect
236. C2×Dic17#117semidirect
308. C22×C4⋊C4#595semidirect
388. C17#13870semidirect
543. C2×C4#10587677semidirect

Maximal subgroups

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

subgroupindex
C22×#105876772
C2×#117637522
#145439692
#145439632
#145439582
C22×C4⋊C4×D172
#1387017