Finite Group Databasebeta

← #16951854

Subgroups — #16951854

Order 4352.

910 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587505C211yesmaximal, characteristic
2C2×#11762604C212yesmaximal, complemented
3#14528699C211yesmaximal, characteristic
4#14526909C212yesmaximal, complemented
5C17×#2435C211yesmaximal, characteristic
6C22×C22⋊Dic17—21no—
7C2×#10587505—24no—
8C2×#10587505—21no—
9C2×#10587505—21no—
10C2×#10587505C2212yes—
11C2×#10587505C2212yes—
12#11762665—21no—
13#11762604—24no—
14C23×D4×C17C2211yescharacteristic
15C2×C17×#124C411yescharacteristic, complemented
16C2×C17×#124C2211yescharacteristic
17C2×C17×C22⋊Q8C411yescharacteristic, complemented
18C17×#394—21no—
19C17×#394—21no—
20C17×#390C411yescharacteristic, complemented
21C17×#390C2211yescharacteristic
22C17×#379—21no—
23C17×#379—21no—
24C4×C22⋊C4×C17C411yescharacteristic, complemented
25C25×C17D411yescharacteristic
26C25×C17D411yescharacteristic
27C22×Q8×C17C2×C411yescharacteristic, complemented
28C22×D4×C17—22no—
29C22×D4×C17—22no—
30C22×D4×C17—41no—
31C22×D4×C17—41no—
32C22×D4×C17—42no—
33C22×D4×C17C2×C411yescharacteristic
34C22×D4×C17D411yescharacteristic
35C22×D4×C17D411yescharacteristic
36C22×D4×C17C2311yescharacteristic
37C23×C4×C17C2×C411yescharacteristic
38C23×Dic17—41no—
39C17×#124—22no—
40C17×#124—22no—
41C17×#124—42no—
42C17×C22⋊Q8—41no—
43C17×C22⋊Q8—41no—
44C2×C4⋊C4×C17—21no—
45C2×C4⋊C4×C17C2×C411yescharacteristic, complemented
46C2×C22⋊C4×C17—21no—
47C2×C22⋊C4×C17—21no—
48C2×C22⋊C4×C17—21no—
49C2×C22⋊C4×C17—21no—
50C2×C22⋊C4×C17—21no—
51C2×C22⋊C4×C17—21no—
52C2×C22⋊C4×C17—21no—
53C2×C22⋊C4×C17—21no—
54C2×C22⋊C4×C17—21no—
55C2×C22⋊C4×C17—21no—
56C2×C42×C17C2×C411yescharacteristic, complemented
57C2×C42×C17C2×C411yescharacteristic, complemented
58C2×C22⋊Dic17—24no—
59C2×C22⋊Dic17—22no—
60C2×C22⋊Dic17—22no—
61C2×C22⋊Dic17—41no—
62C2×C22⋊Dic17—42no—
63C2×C22⋊Dic17—41no—
64C2×C22⋊Dic17—41no—
65C17×#95—21no—
66#10587505—24no—
67#10587505—22no—
68#10587505—22no—
69#10587505—22no—
70#10587505—42no—
71#10587505D414yes—
72C24×C17—21no—
73C24×C17—22no—
74C24×C17—21no—
75C24×C17—21no—
76C24×C17—21no—
77C24×C17—21no—
78C24×C17—21no—
79C24×C17—22no—
80C24×C17—41no—
81C24×C17—42no—
82C24×C17—42no—
83C24×C17—81no—
84C24×C17—81no—
85C24×C17C22⋊C411yescharacteristic
86C24×C17C22⋊C411yescharacteristic
87C24×C17C22⋊C411yescharacteristic
88C24×C17C2×D411yescharacteristic
89C24×C17C2×D411yescharacteristic
90C2×Q8×C17—22no—
91C2×Q8×C17—41no—
92C2×Q8×C17—41no—
93C2×D4×C17—22no—
94C2×D4×C17—22no—
95C2×D4×C17—21no—
96C2×D4×C17—22no—
97C2×D4×C17—21no—
98C2×D4×C17—22no—
99C2×D4×C17—41no—
100C2×D4×C17—41no—
101C2×D4×C17—41no—
102C2×D4×C17—41no—
103C2×D4×C17—41no—
104C2×D4×C17—41no—
105C2×D4×C17—44no—
106C2×D4×C17—44no—
107C2×D4×C17—44no—
108C2×D4×C17—44no—
109C2×D4×C17C2×D412yes—
110C2×D4×C17#4112yes—
111C22×C4×C17—21no—
112C22×C4×C17—21no—
113C22×C4×C17—22no—
114C22×C4×C17—21no—
115C22×C4×C17—21no—
116C22×C4×C17—21no—
117C22×C4×C17—21no—
118C22×C4×C17—41no—
119C22×C4×C17C22⋊C411yescharacteristic
120C22×C4×C17C22×C411yescharacteristic
121C22×Dic17—44no—
122C22×Dic17—42no—
123C22×Dic17—42no—
124C22×Dic17—82no—
125C22×Dic17—82no—
126C4⋊C4×C17—41no—
127C4⋊C4×C17—42no—
128C22⋊C4×C17—42no—
129C22⋊C4×C17—42no—
130C22⋊C4×C17—41no—
131C22⋊C4×C17—41no—
132C22⋊C4×C17—42no—
133C22⋊C4×C17—41no—
134C22⋊C4×C17—42no—
135C22⋊C4×C17—41no—
136C22⋊C4×C17—81no—
137C22⋊C4×C17—81no—
138C22⋊C4×C17—81no—
139C22⋊C4×C17—81no—
140C42×C17—22no—
141C42×C17—41no—
142C22⋊Dic17—24no—
143C22⋊Dic17—44no—
144C22⋊Dic17—42no—
145C22⋊Dic17—42no—
146C22⋊Dic17—42no—
147C22⋊Dic17—82no—
148C22⋊Dic17—82no—
149#13090—171nomaximal, Sylow
150C23×C17—22no—
151C23×C17—22no—
152C23×C17—22no—
153C23×C17—22no—
154C23×C17—22no—
155C23×C17—21no—
156C23×C17—21no—
157C23×C17—21no—
158C23×C17—21no—
159C23×C17—21no—
160C23×C17—21no—
161C23×C17—21no—
162C23×C17—21no—
163C23×C17—21no—
164C23×C17—21no—
165C23×C17—21no—
166C23×C17—41no—
167C23×C17—42no—
168C23×C17—42no—
169C23×C17—42no—
170C23×C17—41no—
171C23×C17—41no—
172C23×C17—41no—
173C23×C17—41no—
174C23×C17—41no—
175C23×C17—41no—
176C23×C17—41no—
177C23×C17—41no—
178C23×C17—42no—
179C23×C17—42no—
180C23×C17—42no—
181C23×C17—81no—
182C23×C17—81no—
183C23×C17—81no—
184C23×C17—81no—
185C23×C17—84no—
186C23×C17—82no—
187C23×C17—82no—
188C23×C17—84no—
189C23×C17—84no—
190C23×C17#9911yescharacteristic
191C23×C17#9911yescharacteristic
192C23×C17#9912yes—
193C23×C17C2×C22⋊C411yescharacteristic
194C23×C17#12011yescharacteristic
195C23×C17#12112yes—
196C23×C17#12311yescharacteristic
197Q8×C17—82no—
198D4×C17—42no—
199D4×C17—42no—
200D4×C17—44no—
201D4×C17—44no—
202D4×C17—42no—
203D4×C17—42no—
204C2×C4×C17—21no—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—21no—
208C2×C4×C17—21no—
209C2×C4×C17—42no—
210C2×C4×C17—42no—
211C2×C4×C17—41no—
212C2×C4×C17—41no—
213C2×C4×C17—41no—
214C2×C4×C17—41no—
215C2×C4×C17—41no—
216C2×C4×C17—41no—
217C2×C4×C17—41no—
218C2×C4×C17—41no—
219C2×C4×C17—41no—
220C2×C4×C17—41no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17C4×D412yes—
224C2×Dic17—44no—
225C2×Dic17—88no—
226C2×Dic17—82no—
227C2×Dic17—82no—
228C2×Dic17—82no—
229C2×Dic17—82no—
230C2×Dic17—82no—
231C22×#99—171no—
232#2435D1711yescharacteristic
233C2×#354—172no—
234#1781—171no—
235#1393—172no—
236C22×C17—21no—
237C22×C17—21no—
238C22×C17—21no—
239C22×C17—21no—
240C22×C17—21no—
241C22×C17—21no—
242C22×C17—21no—
243C22×C17—22no—
244C22×C17—21no—
245C22×C17—22no—
246C22×C17—21no—
247C22×C17—42no—
248C22×C17—42no—
249C22×C17—42no—
250C22×C17—41no—
251C22×C17—42no—
252C22×C17—41no—
253C22×C17—44no—
254C22×C17—44no—
255C22×C17—42no—
256C22×C17—41no—
257C22×C17—41no—
258C22×C17—41no—
259C22×C17—41no—
260C22×C17—41no—
261C22×C17—41no—
262C22×C17—41no—
263C22×C17—41no—
264C22×C17—41no—
265C22×C17—81no—
266C22×C17—81no—
267C22×C17—81no—
268C22×C17—84no—
269C22×C17—81no—
270C22×C17—82no—
271C22×C17—82no—
272C22×C17—84no—
273C22×C17#35412yescomplemented
274C22×C17#38611yescharacteristic
275C22×C17C2×#9911yescharacteristic
276C22×C17C2×#9911yescharacteristic
277C4×C17—22no—
278C4×C17—41no—
279C4×C17—81no—
280C4×C17—81no—
281C4×C17—81no—
282C4×C17—81no—
283Dic17—84no—
284Dic17—162no—
285C23×D4C2×D1711yescharacteristic
286C2×#124Dic1711yescharacteristic, complemented
287C2×#124C2×D1711yescharacteristic
288C2×C22⋊Q8Dic1711yescharacteristic, complemented
289C22×C22⋊C4—341no—
290C2×#99—172no—
291C2×#99—172no—
292C2×#99—344no—
293C2×#99—341no—
294C2×#99—341no—
295#394—21no—
296#394—21no—
297#390Dic1711yescharacteristic, complemented
298#390C2×D1711yescharacteristic
299#379—21no—
300#379—21no—
301#379—341no—
302C4×C22⋊C4Dic1711yescharacteristic, complemented
303#354—344no—
304C2×C17—21no—
305C2×C17—21no—
306C2×C17—22no—
307C2×C17—41no—
308C2×C17—42no—
309C2×C17—41no—
310C2×C17—42no—
311C2×C17—41no—
312C2×C17—81no—
313C2×C17#189811yescharacteristic
314C2×C17C2×#35411yescharacteristic
315C2×C17#212711yescharacteristic
316C25C22⋊D1711yescharacteristic
317C25C22⋊D1711yescharacteristic
318C22×Q8C2×Dic1711yescharacteristic, complemented
319C22×D4—22no—
320C22×D4—22no—
321C22×D4—41no—
322C22×D4—41no—
323C22×D4—42no—
324C22×D4C2×Dic1711yescharacteristic
325C22×D4C22⋊D1711yescharacteristic
326C22×D4C22⋊D1711yescharacteristic
327C22×D4C22×D1711yescharacteristic
328C23×C4—681no—
329C23×C4C2×Dic1711yescharacteristic
330#124—22no—
331#124—22no—
332#124—42no—
333C22⋊Q8—41no—
334C22⋊Q8—41no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4C2×Dic1711yescharacteristic, complemented
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—344no—
348C2×C22⋊C4—342no—
349C2×C22⋊C4—342no—
350C2×C22⋊C4—681no—
351C2×C22⋊C4—682no—
352C2×C22⋊C4—681no—
353C2×C22⋊C4—681no—
354C2×C42C2×Dic1711yescharacteristic, complemented
355C2×C42C2×Dic1711yescharacteristic, complemented
356#99—174no—
357#99—344no—
358#99—342no—
359#99—342no—
360#99—342no—
361#99—682no—
362#95—21no—
363C17#1309011yescharacteristic, complemented, Sylow
364C24—21no—
365C24—22no—
366C24—21no—
367C24—21no—
368C24—21no—
369C24—21no—
370C24—21no—
371C24—22no—
372C24—41no—
373C24—42no—
374C24—42no—
375C24—81no—
376C24—81no—
377C24C22⋊Dic1711yescharacteristic
378C24C22⋊Dic1711yescharacteristic
379C24C22⋊Dic1711yescharacteristic
380C24C2×C22⋊D1711yescharacteristic
381C24C2×C22⋊D1711yescharacteristic
382C2×Q8—22no—
383C2×Q8—41no—
384C2×Q8—41no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4D4×D1712yes—
402C2×D4#6325412yes—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—41no—
411C22×C4—684no—
412C22×C4—682no—
413C22×C4—682no—
414C22×C4—1362no—
415C22×C4—1362no—
416C22×C4C22⋊Dic1711yescharacteristic
417C22×C4C22×Dic1711yescharacteristic
418C4⋊C4—41no—
419C4⋊C4—42no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—42no—
425C22⋊C4—41no—
426C22⋊C4—41no—
427C22⋊C4—42no—
428C22⋊C4—81no—
429C22⋊C4—81no—
430C22⋊C4—81no—
431C22⋊C4—81no—
432C22⋊C4—344no—
433C22⋊C4—684no—
434C22⋊C4—682no—
435C22⋊C4—682no—
436C22⋊C4—682no—
437C22⋊C4—1362no—
438C22⋊C4—1362no—
439C42—22no—
440C42—41no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—22no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—41no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—81no—
473C23—81no—
474C23—81no—
475C23—81no—
476C23—84no—
477C23—82no—
478C23—82no—
479C23—84no—
480C23—84no—
481C23#1058750511yescharacteristic
482C23#1058750511yescharacteristic
483C23#1058750512yes—
484C23#1058762111yescharacteristic
485C23#1058762311yescharacteristic
486C23#1058762512yes—
487C23C2×C22⋊Dic1711yescharacteristic
488Q8—82no—
489D4—42no—
490D4—42no—
491D4—44no—
492D4—44no—
493D4—42no—
494D4—42no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—684no—
515C2×C4—1362no—
516C2×C4—1362no—
517C2×C4—1362no—
518C2×C4—1368no—
519C2×C4—1362no—
520C2×C4—1362no—
521C2×C4D4×Dic1712yes—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—22no—
531C22—22no—
532C22—21no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—41no—
537C22—41no—
538C22—42no—
539C22—44no—
540C22—44no—
541C22—42no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—84no—
555C22—81no—
556C22—82no—
557C22—82no—
558C22—84no—
559C22#1176260412yescomplemented
560C22#1176268911yescharacteristic
561C22C2×#1058750511yescharacteristic
562C22C2×#1058750511yescharacteristic
563C4—22no—
564C4—41no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4—2722no—
570C4—1364no—
571C2—21no—
572C2—21no—
573C2—22no—
574C2—42no—
575C2—41no—
576C2—41no—
577C2—42no—
578C2—41no—
579C2—81no—
580C2#1452923811yescharacteristic
581C2C2×#1176260411yescharacteristic
582C2#1453045311yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC22×D4C22×D17
FittingC17×#2435C2
socleC22×C17C2×#99
prime radicalC23×D4×C17C22
cosocleC22×D4×C17C23
hypercenter#2435D17
hypocenterC17#13090
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #2435

Complements

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

NH ≅ G/Ntype
2. C2×#11762604C2semidirect
4. #14526909C2semidirect
15. C2×C17×#124C4semidirect
17. C2×C17×C22⋊Q8C4semidirect
20. C17×#390C4semidirect
24. C4×C22⋊C4×C17C4semidirect
27. C22×Q8×C17C2×C4semidirect
45. C2×C4⋊C4×C17C2×C4semidirect
56. C2×C42×C17C2×C4semidirect
57. C2×C42×C17C2×C4semidirect
273. C22×C17#354semidirect
286. C2×#124Dic17semidirect
288. C2×C22⋊Q8Dic17semidirect
297. #390Dic17semidirect
302. C4×C22⋊C4Dic17semidirect
318. C22×Q8C2×Dic17semidirect
336. C2×C4⋊C4C2×Dic17semidirect
354. C2×C42C2×Dic17semidirect
355. C2×C42C2×Dic17semidirect
363. C17#13090semidirect
559. C22#11762604semidirect

Maximal subgroups

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

subgroupindex
C22×#105875052
C2×#117626042
#145286992
#145269092
C17×#24352
#1309017