Finite Group Databasebeta

← #17070399

Subgroups — #17070399

Order 4352.

1150 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763096C211yesmaximal, characteristic, complemented
2C2×#11763086C211yesmaximal, characteristic, complemented
3C4×#10587616C211yesmaximal, characteristic
4#14530119C212yesmaximal, complemented
5#14529927C212yesmaximal, complemented
6#14529737C211yesmaximal, characteristic, complemented
7#14529733C211yesmaximal, characteristic, complemented
8#14529106C212yesmaximal
9#14528983C212yesmaximal, complemented
10#14528764C211yesmaximal, characteristic, complemented
11C2×C17×#533C211yesmaximal, characteristic, complemented
12C2×#10587616C2211yescharacteristic
13C2×C4×D68C2211yescharacteristic
14C2×C4×C22⋊D17—21no—
15#11763096—24no—
16#11763086—24no—
17C2×#10587478C2212yes—
18C2×#10587479C2212yes—
19C2×C4⋊#3622C2211yescharacteristic
20C2×C4⋊#3622C2211yescharacteristic
21C2×#10587480C2212yes—
22C2×#10587480C2212yes—
23C2×C22⋊#3622C2212yes—
24#11762711C2211yescharacteristic
25#11762707C2211yescharacteristic
26#11762700—21no—
27#11762688—21no—
28#11762669C2211yescharacteristic
29C4×C4⋊Dic17C2211yescharacteristic
30C4×#63227—21no—
31#11762571C2212yes—
32#11762571C2212yes—
33#11762571C2212yes—
34#11762571C2212yes—
35#11762564C2212yes—
36C17×#533—24no—
37C17×#533—24no—
38C2×C17×C4⋊Q8C2211yescharacteristic
39C2×C17×C4⋊Q8C2211yescharacteristic
40C2×C17×#125C2212yes—
41C2×C17×C22⋊Q8—21no—
42C2×C17×C22⋊Q8—21no—
43C2×C4×Q8×C17—21no—
44C2×C4×Q8×C17—21no—
45C2×C17×#117C2212yes—
46C22×C42×C17C2211yescharacteristic
47C22×Q8×C17—21no—
48C22×Q8×C17—21no—
49C23×C4×C17D411yescharacteristic
50C23×C4×C17D411yescharacteristic
51C23×C4×C17C2311yescharacteristic
52C22×C22⋊D17—21no—
53C22×D68C2311yescharacteristic
54C22×C4×D17—21no—
55C17×C4⋊Q8—22no—
56C17×C4⋊Q8—22no—
57C17×C4⋊Q8D414yescomplemented
58C17×C4⋊Q8D414yescomplemented
59C17×#125—24no—
60C17×#125—24no—
61C17×C22⋊Q8—28no—
62C17×C22⋊Q8—28no—
63C4×Q8×C17—28no—
64C4×Q8×C17—28no—
65C17×#117—24no—
66C17×#117—24no—
67C2×C4⋊C4×C17—22no—
68C2×C4⋊C4×C17—22no—
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17C2312yes—
72C2×C4⋊C4×C17C2312yes—
73C2×C22⋊C4×C17—22no—
74C2×C42×C17—22no—
75C2×C42×C17—22no—
76C2×C42×C17—22no—
77C2×C42×C17D411yescharacteristic
78C2×C42×C17D411yescharacteristic
79C2×C42×C17C2311yescharacteristic
80C2×C42×C17C2311yescharacteristic
81C2×C22⋊Dic17—21no—
82#10587616—22no—
83#10587616Q814yescomplemented
84C4×C22⋊D17—44no—
85C2×#63227—21no—
86C2×#63227—21no—
87C2×#63227—21no—
88C2×#63227—21no—
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×#63225—21no—
93C2×C4×Dic17—21no—
94C22×#3622C2312yes—
95C22×#3622C2312yes—
96C4×D68—22no—
97C4×D68—22no—
98#10587502—21no—
99#10587502—21no—
100#10587502—21no—
101C22⋊#3622—24no—
102#10587480—24no—
103#10587480—24no—
104#10587479—24no—
105#10587478—24no—
106C4⋊#3622—22no—
107C4⋊#3622—22no—
108C24×C17C2×D411yescharacteristic
109C23×D17—21no—
110C2×Q8×C17—24no—
111C2×Q8×C17—24no—
112C2×Q8×C17—42no—
113C2×Q8×C17—42no—
114C2×Q8×C17—42no—
115C2×Q8×C17—42no—
116C22×C4×C17—21no—
117C22×C4×C17—21no—
118C22×C4×C17—21no—
119C22×C4×C17—21no—
120C22×C4×C17—22no—
121C22×C4×C17—22no—
122C22×C4×C17—22no—
123C22×C4×C17—22no—
124C22×C4×C17—22no—
125C22×C4×C17—22no—
126C22×C4×C17—41no—
127C22×C4×C17—41no—
128C22×C4×C17C2×D411yescharacteristic
129C22×C4×C17C2×D411yescharacteristic
130C22×C4×C17C2×D411yescharacteristic
131C22×C4×C17C2×D411yescharacteristic
132C22×C4×C17C2×D411yescharacteristic
133C22×C4×C17C2×Q812yes—
134C22×C4×C17#4112yes—
135C22×C4×C17C2411yescharacteristic
136C2×C22⋊D17—24no—
137C2×C22⋊D17—42no—
138C2×C22⋊D17—42no—
139C22×Dic17—21no—
140C22×Dic17—21no—
141C22×Dic17—21no—
142C2×D68—21no—
143C2×D68—21no—
144C2×D68—21no—
145C2×D68—21no—
146C2×D68—21no—
147C2×D68C2×Q812yes—
148C2×C4×D17—42no—
149C2×C4×D17—42no—
150C2×C4×D17—42no—
151C4⋊C4×C17—28no—
152C4⋊C4×C17—28no—
153C4⋊C4×C17—24no—
154C4⋊C4×C17—24no—
155C4⋊C4×C17—22no—
156C4⋊C4×C17—22no—
157C4⋊C4×C17C2×D414yes—
158C4⋊C4×C17C2×D414yes—
159C22⋊C4×C17—28no—
160C42×C17—28no—
161C42×C17—22no—
162C42×C17—22no—
163C42×C17—21no—
164C42×C17—21no—
165C42×C17C2×D412yes—
166C42×C17C2×D412yes—
167C22⋊Dic17—42no—
168#63227—24no—
169#63227—42no—
170#63227—42no—
171#63227—42no—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17—21no—
175C4⋊Dic17—21no—
176C4⋊Dic17—21no—
177C4⋊Dic17C2×Q812yes—
178#63225—42no—
179C4×Dic17—42no—
180C2×#3622—22no—
181C2×#3622—22no—
182C2×#3622—22no—
183C2×#3622—22no—
184C2×#3622—22no—
185C2×#3622—22no—
186#24300—171nomaximal, Sylow
187C23×C17—21no—
188C23×C17—21no—
189C23×C17—21no—
190C23×C17—21no—
191C23×C17—21no—
192C23×C17—21no—
193C23×C17C22⋊Q812yes—
194C23×C17C22×D411yescharacteristic
195C22×D17—22no—
196C22×D17—41no—
197C22×D17—41no—
198C22×D17—41no—
199C22×D17—41no—
200C22×D17—41no—
201C22×D17—41no—
202Q8×C17—48no—
203Q8×C17—48no—
204C2×C4×C17—21no—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—21no—
208C2×C4×C17—22no—
209C2×C4×C17—22no—
210C2×C4×C17—22no—
211C2×C4×C17—22no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—24no—
215C2×C4×C17—24no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—44no—
221C2×C4×C17—42no—
222C2×C4×C17—42no—
223C2×C4×C17—42no—
224C2×C4×C17—42no—
225C2×C4×C17—44no—
226C2×C4×C17#12012yes—
227C2×C4×C17#12012yes—
228C2×C4×C17C22⋊Q812yes—
229C2×C4×C17#13512yes—
230C2×C4×C17AGL(1,ℤ/8)12yes—
231C2×C4×C17C22×D411yescharacteristic
232C2×C4×C17C22×D411yescharacteristic
233C2×C4×C17C22×Q811yescharacteristic
234C2×C4×C17C2×#4111yescharacteristic
235C2×C4×C172−1+411yescharacteristic
236C2×C4×C172−1+411yescharacteristic
237C22⋊D17—48no—
238C2×Dic17—22no—
239C2×Dic17—41no—
240C2×Dic17—41no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247D68—22no—
248D68—22no—
249D68—41no—
250D68—41no—
251C4×D17—84no—
252#3622—42no—
253#3622—42no—
254C2×#533D1711yescharacteristic, complemented
255C2×#476—171no—
256C2×#474—171no—
257C4×#121—171no—
258#2059—172no—
259#2025—172no—
260#1988—171no—
261#1987—171no—
262#1875—172no—
263#1854—172no—
264#1797—171no—
265C22×C17—22no—
266C22×C17—22no—
267C22×C17—22no—
268C22×C17—41no—
269C22×C17—41no—
270C22×C17—41no—
271C22×C17—41no—
272C22×C17C2×#12011yescharacteristic
273C22×C17C2×C22⋊Q811yescharacteristic
274C22×C17#53611yescharacteristic
275C22×C17C2×#13511yescharacteristic
276C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C17#57511yescharacteristic
278C22×C17#57511yescharacteristic
279C2×D17—42no—
280C2×D17—42no—
281C2×D17—42no—
282C2×D17—81no—
283C2×D17—81no—
284C2×D17—81no—
285C2×D17—81no—
286C4×C17—22no—
287C4×C17—22no—
288C4×C17—41no—
289C4×C17—44no—
290C4×C17—41no—
291C4×C17—44no—
292C4×C17#44914yes—
293C4×C17#54811yescharacteristic
294C4×C17#54811yescharacteristic
295C4×C17D4×Q811yescharacteristic
296C4×C17D4×Q811yescharacteristic
297Dic17—42no—
298Dic17—81no—
299Dic17—81no—
300#533—24no—
301#533—24no—
302C2×C4⋊Q8C2×D1711yescharacteristic
303C2×C4⋊Q8C2×D1711yescharacteristic
304C2×#125C2×D1712yes—
305C2×C22⋊Q8—21no—
306C2×C22⋊Q8—21no—
307C2×#121—171no—
308C2×C4×Q8—21no—
309C2×C4×Q8—21no—
310C2×C4×D4—171no—
311C2×C4×D4—341no—
312C2×#117C2×D1712yes—
313C22×C42C2×D1711yescharacteristic
314#476—344no—
315#474—344no—
316C2×#107—172no—
317C2×#106—172no—
318C2×C4⋊C8—171no—
319C2×C4⋊C8—171no—
320C2×#102—172no—
321C2×#102—172no—
322C2×C22⋊C8—172no—
323#390—171no—
324#389—171no—
325#388—341no—
326#386—341no—
327#380—171no—
328C4×C4⋊C4—171no—
329C4×C22⋊C4—341no—
330#340—172no—
331#340—172no—
332#340—172no—
333#340—172no—
334#336—172no—
335C2×C17—42no—
336C2×C17#240911yescharacteristic
337C2×C17C2×#44911yescharacteristic
338C2×C17#300511yescharacteristic
339C2×C17#307611yescharacteristic
340C2×C17#307711yescharacteristic
341C2×C17#323311yescharacteristic
342C2×C17#323511yescharacteristic
343D17—82no—
344C22×Q8—21no—
345C22×Q8—21no—
346C22×D4—171no—
347C22×D4—341no—
348C23×C4—341no—
349C23×C4C22⋊D1711yescharacteristic
350C23×C4C22⋊D1711yescharacteristic
351C23×C4C22×D1711yescharacteristic
352C22×C8—172no—
353C22×C8—172no—
354C4⋊Q8—22no—
355C4⋊Q8—22no—
356C4⋊Q8C22⋊D1714yescomplemented
357C4⋊Q8C22⋊D1714yescomplemented
358#125—24no—
359#125—24no—
360C22⋊Q8—28no—
361C22⋊Q8—28no—
362#121—174no—
363#121—342no—
364C4×Q8—28no—
365C4×Q8—28no—
366C4×D4—342no—
367C4×D4—342no—
368C4×D4—684no—
369#117—24no—
370#117—24no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—171no—
376C2×C4⋊C4—171no—
377C2×C4⋊C4—171no—
378C2×C4⋊C4—341no—
379C2×C4⋊C4C22×D1712yes—
380C2×C4⋊C4C22×D1712yes—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—341no—
383C2×C22⋊C4—341no—
384C2×C22⋊C4—341no—
385C2×C22⋊C4—341no—
386C2×C22⋊C4—341no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—341no—
391C2×C42C22⋊D1711yescharacteristic
392C2×C42C22⋊D1711yescharacteristic
393C2×C42C22×D1711yescharacteristic
394C2×C42C22×D1711yescharacteristic
395#107—344no—
396#106—344no—
397C4⋊C8—342no—
398C4⋊C8—342no—
399#102—344no—
400#102—344no—
401C22⋊C8—344no—
402#95—341no—
403#95—341no—
404#95—341no—
405C17#2430011yescharacteristic, complemented, Sylow
406C24—341no—
407C24C2×C22⋊D1711yescharacteristic
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×D4—172no—
415C2×D4—344no—
416C2×D4—341no—
417C2×D4—341no—
418C2×D4—341no—
419C2×D4—341no—
420C2×D4—341no—
421C2×D4—682no—
422C2×D4—682no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—341no—
436C22×C4—341no—
437C22×C4—341no—
438C22×C4—682no—
439C22×C4—682no—
440C22×C4—682no—
441C22×C4Q8×D1712yes—
442C22×C4#6325612yes—
443C22×C4C2×C22⋊D1711yescharacteristic
444C22×C4C2×C22⋊D1711yescharacteristic
445C22×C4C2×C22⋊D1711yescharacteristic
446C22×C4C2×C22⋊D1711yescharacteristic
447C22×C4C2×C22⋊D1711yescharacteristic
448C22×C4C23×D1711yescharacteristic
449C2×C8—342no—
450C2×C8—342no—
451C2×C8—342no—
452C2×C8—342no—
453C2×C8—342no—
454C2×C8—342no—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—172no—
462C4⋊C4—341no—
463C4⋊C4—341no—
464C4⋊C4—341no—
465C4⋊C4—341no—
466C4⋊C4—341no—
467C4⋊C4—682no—
468C4⋊C4C2×C22⋊D1714yes—
469C4⋊C4C2×C22⋊D1714yes—
470C22⋊C4—28no—
471C22⋊C4—344no—
472C22⋊C4—682no—
473C22⋊C4—682no—
474C22⋊C4—682no—
475C22⋊C4—682no—
476C42—28no—
477C42—21no—
478C42—21no—
479C42—22no—
480C42—22no—
481C42—682no—
482C42C2×C22⋊D1712yes—
483C42C2×C22⋊D1712yes—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—342no—
491C23—681no—
492C23—681no—
493C23—681no—
494C23—681no—
495C23—681no—
496C23—681no—
497C23#1058763212yes—
498C23C22×C22⋊D1711yescharacteristic
499Q8—48no—
500Q8—48no—
501D4—342no—
502D4—342no—
503D4—688no—
504D4—681no—
505D4—681no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—342no—
529C2×C4—681no—
530C2×C4—681no—
531C2×C4—681no—
532C2×C4—681no—
533C2×C4—681no—
534C2×C4—681no—
535C2×C4—681no—
536C2×C4—681no—
537C2×C4—1364no—
538C2×C4#1058763212yes—
539C2×C4#1058763512yes—
540C2×C4#1058763612yes—
541C2×C4#1058763912yes—
542C2×C4#1058763912yes—
543C2×C4C2×Q8×D1711yescharacteristic
544C2×C4C2×#6325611yescharacteristic
545C2×C4#1058769411yescharacteristic
546C2×C4#1058769411yescharacteristic
547C2×C4C22×C22⋊D1711yescharacteristic
548C2×C4C22×C22⋊D1711yescharacteristic
549C8—682no—
550C8—682no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—682no—
559C22—682no—
560C22—682no—
561C22—1361no—
562C22—1361no—
563C22—1361no—
564C22—1361no—
565C22C2×#1058763911yescharacteristic
566C22C2×#1058763211yescharacteristic
567C22#1176338311yescharacteristic
568C22C2×#1058763611yescharacteristic
569C22C2×#1058763511yescharacteristic
570C22#1176360711yescharacteristic
571C22#1176360711yescharacteristic
572C4—22no—
573C4—22no—
574C4—41no—
575C4—44no—
576C4—41no—
577C4—44no—
578C4—682no—
579C4—1361no—
580C4—1361no—
581C4#1176295414yes—
582C4#1176347011yescharacteristic
583C4#1176347011yescharacteristic
584C4Q8×C22⋊D1711yescharacteristic
585C4Q8×C22⋊D1711yescharacteristic
586C2—42no—
587C2—1362no—
588C2#1453214311yescharacteristic
589C2C2×#1176295411yescharacteristic
590C2#1453802611yescharacteristic
591C2#1453856611yescharacteristic
592C2#1453857711yescharacteristic
593C2#1453996311yescharacteristic
594C2#1453998211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#533C2
socleC23×C17C22×D4
prime radicalC2×#10587616C22
hypercenterC2×#533D17
hypocenterC17#24300
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
1. C2×#11763096C2semidirect
2. C2×#11763086C2semidirect
4. #14530119C2semidirect
5. #14529927C2semidirect
6. #14529737C2semidirect
7. #14529733C2semidirect
9. #14528983C2semidirect
10. #14528764C2semidirect
11. C2×C17×#533C2semidirect
57. C17×C4⋊Q8D4semidirect
58. C17×C4⋊Q8D4semidirect
83. #10587616Q8semidirect
254. C2×#533D17semidirect
356. C4⋊Q8C22⋊D17semidirect
357. C4⋊Q8C22⋊D17semidirect
405. C17#24300semidirect

Maximal subgroups

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

subgroupindex
C2×#117630962
C2×#117630862
C4×#105876162
#145301192
#145299272
#145297372
#145297332
#145291062
#145289832
#145287642
C2×C17×#5332
#2430017