Finite Group Databasebeta

← #16929909

Subgroups — #16929909

Order 4352.

1030 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14537066C211yesmaximal, characteristic
2C2×#11762544C212yesmaximal, complemented
3#14527007C211yesmaximal, characteristic, complemented
4#14526664C212yesmaximal, complemented
5C2×C17×#407C211yesmaximal, characteristic, complemented
6C2×#10587616C411yescharacteristic, complemented
7C2×#10587616C411yescharacteristic, complemented
8C2×#10587616C2211yescharacteristic
9C2×#10587639—21no—
10C22×C22⋊Dic17—21no—
11#11762688—21no—
12#11762688—21no—
13#11762688—21no—
14#11762669C411yescharacteristic, complemented
15#11762669C411yescharacteristic, complemented
16#11762669C2211yescharacteristic
17C2×#10587502—21no—
18C2×#10587502C2212yes—
19#11762544—24no—
20C24×C4×C17C2211yescharacteristic
21C22×M16×C17—21no—
22C17×#407—24no—
23C2×C17×C22⋊C8—21no—
24C2×C17×C22⋊C8C2212yes—
25C25×C17D411yescharacteristic
26C23×C4×C17—22no—
27C23×C4×C17—22no—
28C23×C4×C17—42no—
29C23×C4×C17—42no—
30C23×C4×C17C2×C411yescharacteristic
31C23×C4×C17C2×C411yescharacteristic
32C23×C4×C17D411yescharacteristic
33C23×C4×C17D411yescharacteristic
34C23×C4×C17D411yescharacteristic
35C23×C4×C17C2311yescharacteristic
36C22×C22⋊D17—21no—
37C22×C22⋊D17—21no—
38C22×C22⋊D17—21no—
39C23×Dic17—22no—
40C22×D68C2×C411yescharacteristic
41C2×M16×C17—24no—
42C2×M16×C17—42no—
43C2×M16×C17—42no—
44C22×C8×C17—22no—
45#10587639—44no—
46C2×C22⋊Dic17—24no—
47C2×C22⋊Dic17—21no—
48C2×C22⋊Dic17—21no—
49C2×C22⋊Dic17—21no—
50C2×C22⋊Dic17—42no—
51C2×C22⋊Dic17—42no—
52#10587616—22no—
53#10587616—22no—
54#10587616—42no—
55#10587616—42no—
56C2×#63227—21no—
57C2×#63227—21no—
58C2×#63227—21no—
59C2×C4⋊Dic17C2×C411yescharacteristic
60C2×C4⋊Dic17C2×C411yescharacteristic
61C2×C4⋊Dic17C2×C411yescharacteristic
62C17×C22⋊C8—24no—
63C17×C22⋊C8—22no—
64C17×C22⋊C8D414yescomplemented
65#10587502—21no—
66#10587502—21no—
67#10587502—21no—
68#10587502—22no—
69#10587502—22no—
70#10587502—42no—
71C24×C17—22no—
72C24×C17—22no—
73C24×C17—21no—
74C24×C17—21no—
75C24×C17—22no—
76C24×C17—22no—
77C24×C17—41no—
78C24×C17—41no—
79C24×C17C22⋊C411yescharacteristic
80C24×C17C22⋊C411yescharacteristic
81C24×C17C2×D411yescharacteristic
82C23×D17—21no—
83C22×C4×C17—22no—
84C22×C4×C17—22no—
85C22×C4×C17—21no—
86C22×C4×C17—21no—
87C22×C4×C17—22no—
88C22×C4×C17—22no—
89C22×C4×C17—22no—
90C22×C4×C17—22no—
91C22×C4×C17—22no—
92C22×C4×C17—22no—
93C22×C4×C17—21no—
94C22×C4×C17—21no—
95C22×C4×C17—41no—
96C22×C4×C17—41no—
97C22×C4×C17—41no—
98C22×C4×C17—41no—
99C22×C4×C17—41no—
100C22×C4×C17—41no—
101C22×C4×C17—41no—
102C22×C4×C17—48no—
103C22×C4×C17—48no—
104C22×C4×C17C22⋊C411yescharacteristic
105C22×C4×C17C22⋊C411yescharacteristic
106C22×C4×C17C22×C411yescharacteristic
107C22×C4×C17C2×D411yescharacteristic
108C22×C4×C17C2×D412yes—
109C22×C4×C17#4112yes—
110C2×C22⋊D17—42no—
111C2×C22⋊D17—42no—
112C2×C22⋊D17—42no—
113C2×C22⋊D17—42no—
114C2×C22⋊D17—42no—
115C2×C22⋊D17—42no—
116C2×C22⋊D17—42no—
117C2×C22⋊D17—41no—
118C2×C22⋊D17—42no—
119C2×C22⋊D17—41no—
120C22×Dic17—24no—
121C22×Dic17—21no—
122C22×Dic17—21no—
123C22×Dic17—21no—
124C22×Dic17—42no—
125C22×Dic17—42no—
126C22×Dic17—42no—
127C22×Dic17—42no—
128C22×Dic17—42no—
129C22×Dic17—42no—
130C2×D68—22no—
131C2×D68—41no—
132C2×D68—41no—
133M16×C17—48no—
134C2×C8×C17—24no—
135C2×C8×C17—42no—
136C2×C8×C17—42no—
137C22⋊Dic17—48no—
138C22⋊Dic17—42no—
139C22⋊Dic17—42no—
140C22⋊Dic17—42no—
141#63227—42no—
142#63227—42no—
143#63227—41no—
144#63227—41no—
145C4⋊Dic17—22no—
146C4⋊Dic17—41no—
147C4⋊Dic17—41no—
148#9117—171nomaximal, Sylow
149C23×C17—21no—
150C23×C17—21no—
151C23×C17—21no—
152C23×C17—21no—
153C23×C17—24no—
154C23×C17—24no—
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—21no—
167C23×C17—44no—
168C23×C17—44no—
169C23×C17—44no—
170C23×C17—44no—
171C23×C17—42no—
172C23×C17—42no—
173C23×C17—42no—
174C23×C17—42no—
175C23×C17#9912yes—
176C23×C17#10412yes—
177C23×C17C2×C22⋊C411yescharacteristic
178C23×C17#12112yes—
179C23×C17AGL(1,ℤ/8)11yescharacteristic
180C23×C17AGL(1,ℤ/8)12yes—
181C23×C17#13711yescharacteristic
182C22×D17—41no—
183C22×D17—41no—
184C22×D17—41no—
185C22×D17—41no—
186C22×D17—41no—
187C22×D17—41no—
188C22×D17—41no—
189C2×C4×C17—24no—
190C2×C4×C17—24no—
191C2×C4×C17—21no—
192C2×C4×C17—21no—
193C2×C4×C17—21no—
194C2×C4×C17—21no—
195C2×C4×C17—21no—
196C2×C4×C17—21no—
197C2×C4×C17—21no—
198C2×C4×C17—21no—
199C2×C4×C17—21no—
200C2×C4×C17—42no—
201C2×C4×C17—42no—
202C2×C4×C17—44no—
203C2×C4×C17—44no—
204C2×C4×C17—44no—
205C2×C4×C17—42no—
206C2×C4×C17—42no—
207C2×C4×C17#9912yes—
208C2×C4×C17C4×D412yes—
209C2×C4×C17#12011yescharacteristic
210C2×C4×C17#12311yescharacteristic
211C22⋊D17—84no—
212C22⋊D17—84no—
213C22⋊D17—82no—
214C22⋊D17—82no—
215C2×Dic17—44no—
216C2×Dic17—44no—
217C2×Dic17—44no—
218C2×Dic17—41no—
219C2×Dic17—41no—
220C2×Dic17—41no—
221C2×Dic17—41no—
222C2×Dic17—41no—
223C2×Dic17—41no—
224C2×Dic17—41no—
225C2×Dic17—41no—
226C2×Dic17—41no—
227C2×Dic17—82no—
228C2×Dic17—82no—
229C2×Dic17—82no—
230C2×Dic17—82no—
231D68—82no—
232C8×C17—44no—
233C2×#407D1711yescharacteristic, complemented
234#2847—171no—
235C2×#327—172no—
236C22⋊#95—171no—
237#1294—172no—
238C22×C17—22no—
239C22×C17—22no—
240C22×C17—21no—
241C22×C17—21no—
242C22×C17—22no—
243C22×C17—22no—
244C22×C17—21no—
245C22×C17—21no—
246C22×C17—22no—
247C22×C17—22no—
248C22×C17—22no—
249C22×C17—22no—
250C22×C17—22no—
251C22×C17—22no—
252C22×C17—48no—
253C22×C17—48no—
254C22×C17—41no—
255C22×C17—41no—
256C22×C17—41no—
257C22×C17—41no—
258C22×C17—41no—
259C22×C17—41no—
260C22×C17—41no—
261C22×C17—41no—
262C22×C17#38611yescharacteristic
263C22×C17C2×#9911yescharacteristic
264C22×C17C2×#9911yescharacteristic
265C22×C17#41711yescharacteristic
266C22×C17#41811yescharacteristic
267C22×C17C2×#10411yescharacteristic
268C22×C17#42111yescharacteristic
269C22×C17#46812yes—
270C22×C17#46912yes—
271C2×D17—81no—
272C2×D17—81no—
273C2×D17—81no—
274C2×D17—81no—
275C2×D17—81no—
276C2×D17—81no—
277C2×D17—81no—
278C4×C17—21no—
279C4×C17—22no—
280C4×C17—22no—
281C4×C17—21no—
282C4×C17—41no—
283Dic17—84no—
284Dic17—81no—
285Dic17—81no—
286Dic17—81no—
287C24×C4C2×D1711yescharacteristic
288C22×M16—21no—
289C2×#121—171no—
290C2×#121—171no—
291C2×#121—171no—
292C2×#120—341no—
293C22×C22⋊C4—341no—
294#407—24no—
295C2×C22⋊C8—21no—
296C2×C22⋊C8C2×D1712yes—
297#386—341no—
298#386—341no—
299#386—341no—
300#380—171no—
301#380—171no—
302#380—171no—
303C2×#95—172no—
304C2×#95—341no—
305#327—344no—
306C2×C17—22no—
307C2×C17—22no—
308C2×C17—42no—
309C2×C17—42no—
310C2×C17#150411yescharacteristic
311C2×C17#150711yescharacteristic
312C2×C17#151011yescharacteristic
313C2×C17#151311yescharacteristic
314C2×C17#187111yescharacteristic
315C2×C17#189511yescharacteristic
316C2×C17#189811yescharacteristic
317D17—161no—
318C25D6811yescharacteristic
319C22×D4—171no—
320C22×D4—341no—
321C22×D4—341no—
322C22×D4—341no—
323C23×C4—22no—
324C23×C4—22no—
325C23×C4—42no—
326C23×C4—42no—
327C23×C4—342no—
328C23×C4C4×D1711yescharacteristic
329C23×C4C4×D1711yescharacteristic
330C23×C4D6811yescharacteristic
331C23×C4C22⋊D1711yescharacteristic
332C23×C4C22⋊D1711yescharacteristic
333C23×C4C22×D1711yescharacteristic
334C2×M16—24no—
335C2×M16—42no—
336C2×M16—42no—
337C22×C8—22no—
338#121—342no—
339#121—342no—
340#121—682no—
341#121—682no—
342#120—684no—
343C2×C4⋊C4—171no—
344C2×C4⋊C4—171no—
345C2×C4⋊C4—171no—
346C2×C22⋊C4—344no—
347C2×C22⋊C4—341no—
348C2×C22⋊C4—341no—
349C2×C22⋊C4—341no—
350C2×C22⋊C4—341no—
351C2×C22⋊C4—341no—
352C2×C22⋊C4—341no—
353C2×C22⋊C4—682no—
354C2×C22⋊C4—682no—
355C22⋊C8—24no—
356C22⋊C8—22no—
357C22⋊C8C22⋊D1714yescomplemented
358#95—341no—
359#95—341no—
360#95—341no—
361#95—342no—
362#95—342no—
363#95—682no—
364C17#911711yescharacteristic, complemented, Sylow
365C24—22no—
366C24—22no—
367C24—21no—
368C24—21no—
369C24—22no—
370C24—22no—
371C24—41no—
372C24—41no—
373C24—341no—
374C24#6322711yescharacteristic
375C24#6322711yescharacteristic
376C24C2×D6811yescharacteristic
377C2×D4—342no—
378C2×D4—681no—
379C2×D4—681no—
380C2×D4—682no—
381C2×D4—682no—
382C2×D4—682no—
383C2×D4—682no—
384C2×D4—682no—
385C2×D4—682no—
386C2×D4—682no—
387C2×D4—681no—
388C2×D4—681no—
389C2×D4—682no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—344no—
412C22×C4—341no—
413C22×C4—341no—
414C22×C4—341no—
415C22×C4—682no—
416C22×C4—682no—
417C22×C4—682no—
418C22×C4—682no—
419C22×C4—682no—
420C22×C4—682no—
421C22×C4#6322711yescharacteristic
422C22×C4#6322711yescharacteristic
423C22×C4C2×C4×D1711yescharacteristic
424C22×C4#6325212yes—
425C22×C4C2×C22⋊D1711yescharacteristic
426C22×C4C2×C22⋊D1712yes—
427M16—48no—
428C2×C8—24no—
429C2×C8—42no—
430C2×C8—42no—
431C4⋊C4—342no—
432C4⋊C4—681no—
433C4⋊C4—681no—
434C22⋊C4—682no—
435C22⋊C4—688no—
436C22⋊C4—682no—
437C22⋊C4—682no—
438C22⋊C4—682no—
439C22⋊C4—682no—
440C22⋊C4—681no—
441C22⋊C4—681no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—24no—
447C23—24no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—681no—
469C23—681no—
470C23—681no—
471C23—681no—
472C23—681no—
473C23—681no—
474C23—681no—
475C23#1058747712yes—
476C23#1058749612yes—
477C23#1058759412yes—
478C23#1058759411yescharacteristic
479C23#1058759511yescharacteristic
480C23C2×#6322711yescharacteristic
481C23#1058761612yes—
482D4—1362no—
483D4—1364no—
484D4—1364no—
485D4—1362no—
486D4—1362no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—684no—
506C2×C4—684no—
507C2×C4—684no—
508C2×C4—681no—
509C2×C4—681no—
510C2×C4—681no—
511C2×C4—681no—
512C2×C4—681no—
513C2×C4—681no—
514C2×C4—681no—
515C2×C4—681no—
516C2×C4—681no—
517C2×C4—1362no—
518C2×C4—1362no—
519C2×C4—1362no—
520C2×C4—1362no—
521C2×C4#1058747712yes—
522C2×C4C4×C22⋊D1712yes—
523C2×C4#1058761511yescharacteristic
524C2×C4#1058763911yescharacteristic
525C8—44no—
526C22—21no—
527C22—21no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—21no—
532C22—21no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—48no—
541C22—48no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—1361no—
551C22—1361no—
552C22—1361no—
553C22—1361no—
554C22—1361no—
555C22—1361no—
556C22—1361no—
557C22#1176268811yescharacteristic
558C22C2×#1058747711yescharacteristic
559C22C2×#1058747711yescharacteristic
560C22#1176281111yescharacteristic
561C22#1176281511yescharacteristic
562C22C2×#1058749611yescharacteristic
563C22#1176282511yescharacteristic
564C22#1176305512yes—
565C22#1176306212yes—
566C4—21no—
567C4—22no—
568C4—21no—
569C4—22no—
570C4—41no—
571C4—1364no—
572C4—1361no—
573C4—1361no—
574C4—1361no—
575C2—22no—
576C2—22no—
577C2—42no—
578C2—42no—
579C2—2721no—
580C2#1452728911yescharacteristic
581C2#1452730111yescharacteristic
582C2#1452731411yescharacteristic
583C2#1452732811yescharacteristic
584C2#1452908611yescharacteristic
585C2#1452922211yescharacteristic
586C2#1452923711yescharacteristic
subgroupG/H
centerC23C2×#63227
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC2×C17×#407C2
socleC23×C17C2×C22⋊C4
prime radical#14537066C2
cosocleC23×C4×C17C23
hypercenterC2×#407D17
hypocenterC17#9117
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#407

Complements

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

NH ≅ G/Ntype
2. C2×#11762544C2semidirect
3. #14527007C2semidirect
4. #14526664C2semidirect
5. C2×C17×#407C2semidirect
6. C2×#10587616C4semidirect
7. C2×#10587616C4semidirect
14. #11762669C4semidirect
15. #11762669C4semidirect
64. C17×C22⋊C8D4semidirect
233. C2×#407D17semidirect
357. C22⋊C8C22⋊D17semidirect
364. C17#9117semidirect

Maximal subgroups

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

subgroupindex
#145370662
C2×#117625442
#145270072
#145266642
C2×C17×#4072
#911717