Finite Group Databasebeta

← #17033807

Subgroups — #17033807

Order 4352.

1450 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D17C211yesmaximal, characteristic
2C2×#11762831C211yesmaximal, characteristic
3C2×#11762816C212yesmaximal
4#14531189C211yesmaximal, characteristic
5#14528798C212yesmaximal, complemented
6#14528797C212yesmaximal, complemented
7#14528790C211yesmaximal, characteristic, complemented
8#14528789C211yesmaximal, characteristic, complemented
9#14528786C211yesmaximal, characteristic, complemented
10#14528502C212yesmaximal
11C17×#1803C211yesmaximal, characteristic, complemented
12C22×Q8×D17C2211yescharacteristic
13C22×#63218C2212yes—
14C2×Q8×Dic17C2211yescharacteristic
15C2×C4×Dic34—21no—
16C2×C4×Dic34C2211yescharacteristic
17C2×#10587573—21no—
18C2×#10587573C2211yescharacteristic
19C4×Q8×D17—24no—
20C4×Q8×D17C418yescomplemented
21C2×#10587578C2212yes—
22C2×C4⋊C4×D17—21no—
23C2×C4⋊C4×D17C2211yescharacteristic
24C2×C42×D17—21no—
25C2×C42×D17C2211yescharacteristic
26#11762831—24no—
27C2×C4⋊#3622C2211yescharacteristic
28C2×#10587485C2212yes—
29#11762816—28no—
30C2×#10587487C2212yes—
31C2×#10587506C2212yes—
32C2×#10587481C2212yes—
33#11762679C2211yescharacteristic
34#11762674C2212yes—
35#11762666—21no—
36#11762666—21no—
37C4⋊C4×Dic17—21no—
38C4⋊C4×Dic17—21no—
39#11762572C2212yes—
40#11762571C2212yes—
41#11762571C2212yes—
42C2×C4×Q8×C17C2211yescharacteristic
43C2×C17×C4⋊C8C2211yescharacteristic
44C2×C17×#103C2212yes—
45C17×#384C2211yescharacteristic
46C17×#340C2212yes—
47C22×Q8×C17C2311yescharacteristic
48C2×Q8×D17—22no—
49C2×Q8×D17—22no—
50C2×Q8×D17—22no—
51C2×Q8×D17C2×C414yes—
52C2×Q8×D17D414yes—
53C2×Q8×D17Q814yes—
54C22×C4×D17—21no—
55C22×C4×D17—21no—
56C22×C4×D17D411yescharacteristic
57C22×C4×D17D411yescharacteristic
58C22×C4×D17C2311yescharacteristic
59C22×Dic34—21no—
60C22×Dic34C2311yescharacteristic
61C22×C8×C17C2312yes—
62C4×Q8×C17—22no—
63C4×Q8×C17C2×C414yes—
64C2×C4⋊C4×C17—21no—
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17—21no—
67C2×C4⋊C4×C17C2312yes—
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C42×C17—21no—
70C2×C42×C17C2311yescharacteristic
71Q8×Dic17—22no—
72Q8×Dic17C2×C414yes—
73C2×#63227—22no—
74C2×C4⋊Dic17—21no—
75C2×C4⋊Dic17—21no—
76C2×C4⋊Dic17—21no—
77C2×C4⋊Dic17C2312yes—
78C2×C4⋊Dic17C2311yescharacteristic
79C2×#63225—21no—
80C2×#63225—21no—
81C2×#63225—21no—
82C2×C4×Dic17—21no—
83C2×C4×Dic17—21no—
84C2×C4×Dic17—21no—
85C2×C4×Dic17—22no—
86C2×C4×Dic17D411yescharacteristic
87C2×C4×Dic17D411yescharacteristic
88C2×C4×Dic17C2311yescharacteristic
89C22×#3622C2312yes—
90C2×#63218—24no—
91C2×#63218—24no—
92C2×#63218—24no—
93#10587578—24no—
94#10587578—24no—
95C4⋊C4×D17—28no—
96C4⋊C4×D17—22no—
97C4⋊C4×D17C2×C414yes—
98#10587573—28no—
99#10587573—22no—
100#10587573C2×C414yes—
101C42×D17—28no—
102C42×D17C2×C414yes—
103C42×D17D414yes—
104C4×Dic34—28no—
105C4×Dic34—22no—
106C4×Dic34C2×C414yes—
107C17×C4⋊C8—22no—
108C17×#103—24no—
109#10587506—24no—
110#10587502—22no—
111#10587502—22no—
112#10587487—24no—
113#10587485—24no—
114#10587481—24no—
115C4⋊#3622—22no—
116C23×D17C2×D411yescharacteristic
117C2×Q8×C17—21no—
118C2×Q8×C17—21no—
119C2×Q8×C17—21no—
120C2×Q8×C17C22×C412yes—
121C2×Q8×C17C2×D412yes—
122C2×Q8×C17C2×Q812yes—
123C22×C4×C17—21no—
124C22×C4×C17—21no—
125C22×C4×C17—22no—
126C22×C4×C17C2×D411yescharacteristic
127C22×C4×C17C2×D411yescharacteristic
128C22×C4×C17C2411yescharacteristic
129C22×Dic17—21no—
130C22×Dic17—21no—
131C22×Dic17—22no—
132C22×Dic17C2×D411yescharacteristic
133C22×Dic17C2×D411yescharacteristic
134C22×Dic17C2×D411yescharacteristic
135Q8×D17—28no—
136Q8×D17—28no—
137Q8×D17—28no—
138Q8×D17C4⋊C4116yescomplemented
139C2×C4×D17—24no—
140C2×C4×D17—24no—
141C2×C4×D17—24no—
142C2×C4×D17—22no—
143C2×C4×D17—22no—
144C2×C4×D17—24no—
145C2×C4×D17—41no—
146C2×C4×D17—41no—
147C2×C4×D17—42no—
148C2×C4×D17—42no—
149C2×C4×D17C22⋊C414yes—
150C2×C4×D17C22⋊C414yes—
151C2×C4×D17C22×C412yes—
152C2×C4×D17C2×D412yes—
153C2×C4×D17C2×D412yes—
154C2×C4×D17C2×Q812yes—
155C2×C4×D17#4112yes—
156C2×C4×D17#4112yes—
157C2×Dic34—21no—
158C2×Dic34—21no—
159C2×Dic34—21no—
160C2×Dic34—24no—
161C2×Dic34—24no—
162C2×Dic34—24no—
163C2×Dic34C22×C412yes—
164C2×Dic34C2×D412yes—
165C2×Dic34C2×Q812yes—
166C2×C8×C17—22no—
167C2×C8×C17—22no—
168C2×C8×C17—22no—
169C4⋊C4×C17—24no—
170C4⋊C4×C17—21no—
171C4⋊C4×C17—22no—
172C4⋊C4×C17—22no—
173C4⋊C4×C17—42no—
174C4⋊C4×C17—42no—
175C4⋊C4×C17C22×C412yes—
176C42×C17—24no—
177C42×C17C22×C412yes—
178C42×C17C2×D412yes—
179#63227—44no—
180C4⋊Dic17—21no—
181C4⋊Dic17—24no—
182C4⋊Dic17—22no—
183C4⋊Dic17—22no—
184C4⋊Dic17—42no—
185C4⋊Dic17—42no—
186C4⋊Dic17C22×C412yes—
187#63225—24no—
188#63225—24no—
189#63225—24no—
190C4×Dic17—24no—
191C4×Dic17—24no—
192C4×Dic17—24no—
193C4×Dic17—44no—
194C4×Dic17C22⋊C414yes—
195C4×Dic17C22⋊C414yes—
196C4×Dic17C22×C412yes—
197C4×Dic17C2×D412yes—
198C2×#3622—22no—
199C2×#3622—22no—
200C2×#3622—22no—
201#63218—48no—
202#20647—171nomaximal, Sylow
203C23×C17C22×D411yescharacteristic
204C22×D17—21no—
205C22×D17—21no—
206C22×D17—21no—
207C22×D17—21no—
208C22×D17C2×C22⋊C412yes—
209C22×D17#12112yes—
210C22×D17C22⋊Q812yes—
211Q8×C17—22no—
212Q8×C17—22no—
213Q8×C17—22no—
214Q8×C17C2×C4⋊C414yes—
215C2×C4×C17—22no—
216C2×C4×C17—22no—
217C2×C4×C17—22no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—22no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17—42no—
224C2×C4×C17—42no—
225C2×C4×C17—42no—
226C2×C4×C17C2×C22⋊C412yes—
227C2×C4×C17C2×C22⋊C412yes—
228C2×C4×C17#13712yes—
229C2×C4×C17#13712yes—
230C2×C4×C17C23×C411yescharacteristic
231C2×C4×C17C22×D411yescharacteristic
232C2×C4×C17C22×D411yescharacteristic
233C2×C4×C17C22×Q811yescharacteristic
234C2×C4×C17C2×#4111yescharacteristic
235C2×C4×C17C2×#4111yescharacteristic
236C2×Dic17—22no—
237C2×Dic17—22no—
238C2×Dic17—22no—
239C2×Dic17—21no—
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—21no—
243C2×Dic17—22no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—42no—
247C2×Dic17—42no—
248C2×Dic17—42no—
249C2×Dic17C2×C22⋊C412yes—
250C2×Dic17C2×C22⋊C412yes—
251C2×Dic17C2×C22⋊C412yes—
252C2×Dic17#12112yes—
253C2×Dic17C22⋊Q812yes—
254C4×D17—216no—
255C4×D17—28no—
256C4×D17—44no—
257C4×D17—48no—
258C4×D17C2×C4⋊C414yes—
259C4×D17#11714yes—
260C4×D17#12114yes—
261C4×D17C22⋊Q814yes—
262Dic34—22no—
263Dic34—22no—
264Dic34—22no—
265Dic34—216no—
266Dic34C2×C4⋊C414yes—
267C8×C17—42no—
268#3622—42no—
269C22×C4×Q8—171no—
270C2×#423—171no—
271C2×#419—172no—
272#2285—171no—
273#1805—172no—
274#1805—172no—
275#1803—171no—
276#1803—171no—
277#1803—171no—
278#1803D1711yescharacteristic, complemented
279#1736—172no—
280C22×C17C22×C22⋊C411yescharacteristic
281C22×C17C2×#12111yescharacteristic
282C22×C17C2×C22⋊Q811yescharacteristic
283C22×C17C2×#13711yescharacteristic
284C22×C17C2×#13711yescharacteristic
285C22×C17#57511yescharacteristic
286C22×C17#57511yescharacteristic
287C2×D17—22no—
288C2×D17—22no—
289C2×D17—22no—
290C2×D17—22no—
291C2×D17—22no—
292C2×D17—22no—
293C2×D17#38014yes—
294C4×C17—22no—
295C4×C17—24no—
296C4×C17—42no—
297C4×C17—82no—
298C4×C17#41914yescomplemented
299C4×C17C22×C4⋊C411yescharacteristic
300C4×C17C2×#11711yescharacteristic
301C4×C17C2×#12111yescharacteristic
302C4×C17C2×C22⋊Q811yescharacteristic
303Dic17—22no—
304Dic17—22no—
305Dic17—24no—
306Dic17—42no—
307Dic17—82no—
308Dic17#38014yes—
309C23×Q8—171no—
310C22×M16—172no—
311C2×C4×Q8—171no—
312C2×C4×Q8—178no—
313C2×C4×Q8—171no—
314C2×C4×Q8—171no—
315C2×C4×Q8—341no—
316C2×C4×Q8—341no—
317C2×C4×Q8—344no—
318C2×C4×Q8C2×D1711yescharacteristic
319C2×#117—172no—
320C22×C4⋊C4—171no—
321C22×C4⋊C4—341no—
322C22×C42—171no—
323C22×C42—341no—
324#423—344no—
325C2×C4⋊C8—172no—
326C2×C4⋊C8—171no—
327C2×C4⋊C8C2×D1711yescharacteristic
328#419—348no—
329C2×#103—172no—
330C2×#103—172no—
331C2×#103—172no—
332C2×#103C2×D1712yes—
333#384—171no—
334#384C2×D1711yescharacteristic
335#382—172no—
336#380—341no—
337#380—341no—
338C4×C4⋊C4—341no—
339C4×C4⋊C4—341no—
340#340—172no—
341#340—172no—
342#340—172no—
343#340C2×D1712yes—
344C2×C17C2×#38011yescharacteristic
345C2×C17C2×#41911yescharacteristic
346C2×C17#289611yescharacteristic
347C2×C17#303711yescharacteristic
348C2×C17#303711yescharacteristic
349C2×C17#307811yescharacteristic
350C2×C17#307811yescharacteristic
351D17—24no—
352C22×Q8—174no—
353C22×Q8—174no—
354C22×Q8—174no—
355C22×Q8—171no—
356C22×Q8—341no—
357C22×Q8—342no—
358C22×Q8—342no—
359C22×Q8—342no—
360C22×Q8C22×D1711yescharacteristic
361C23×C4—171no—
362C23×C4—171no—
363C23×C4—171no—
364C23×C4—341no—
365C23×C4—341no—
366C2×M16—344no—
367C2×M16—344no—
368C2×M16—344no—
369C22×C8—172no—
370C22×C8C22×D1712yes—
371C4×Q8—22no—
372C4×Q8—174no—
373C4×Q8—174no—
374C4×Q8—174no—
375C4×Q8—348no—
376C4×Q8—342no—
377C4×Q8—348no—
378C4×Q8—342no—
379C4×Q8—342no—
380C4×Q8C4×D1714yes—
381#117—344no—
382#117—344no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—171no—
387C2×C4⋊C4—174no—
388C2×C4⋊C4—172no—
389C2×C4⋊C4—342no—
390C2×C4⋊C4—348no—
391C2×C4⋊C4—341no—
392C2×C4⋊C4—341no—
393C2×C4⋊C4—341no—
394C2×C4⋊C4—341no—
395C2×C4⋊C4—341no—
396C2×C4⋊C4—341no—
397C2×C4⋊C4C22×D1712yes—
398C2×C4⋊C4C22×D1711yescharacteristic
399C2×C22⋊C4—342no—
400C2×C42—21no—
401C2×C42—171no—
402C2×C42—171no—
403C2×C42—171no—
404C2×C42—174no—
405C2×C42—174no—
406C2×C42—348no—
407C2×C42—341no—
408C2×C42—341no—
409C2×C42—341no—
410C2×C42—342no—
411C2×C42C22×D1711yescharacteristic
412C4⋊C8—22no—
413C4⋊C8—344no—
414C4⋊C8—342no—
415#103—24no—
416#103—344no—
417#103—344no—
418#103—344no—
419#95—342no—
420#95—342no—
421C17#2064711yescharacteristic, complemented, Sylow
422C24—171no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—1716no—
427C2×Q8—172no—
428C2×Q8—172no—
429C2×Q8—172no—
430C2×Q8—344no—
431C2×Q8—344no—
432C2×Q8—344no—
433C2×Q8—348no—
434C2×Q8—348no—
435C2×Q8—348no—
436C2×Q8—341no—
437C2×Q8—341no—
438C2×Q8—341no—
439C2×Q8C2×C4×D1712yes—
440C2×Q8D4×D1712yes—
441C2×Q8Q8×D1712yes—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—171no—
446C22×C4—172no—
447C22×C4—172no—
448C22×C4—172no—
449C22×C4—172no—
450C22×C4—172no—
451C22×C4—172no—
452C22×C4—171no—
453C22×C4—171no—
454C22×C4—174no—
455C22×C4—174no—
456C22×C4—344no—
457C22×C4—344no—
458C22×C4—344no—
459C22×C4—344no—
460C22×C4—341no—
461C22×C4—342no—
462C22×C4—342no—
463C22×C4—341no—
464C22×C4—342no—
465C22×C4—681no—
466C22×C4—681no—
467C22×C4—682no—
468C22×C4—682no—
469C22×C4D4×D1711yescharacteristic
470C22×C4D4×D1711yescharacteristic
471C22×C4C23×D1711yescharacteristic
472M16—688no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—342no—
477C2×C8—342no—
478C2×C8—342no—
479C4⋊C4—21no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—172no—
486C4⋊C4—344no—
487C4⋊C4—344no—
488C4⋊C4—344no—
489C4⋊C4—341no—
490C4⋊C4—344no—
491C4⋊C4—342no—
492C4⋊C4—342no—
493C4⋊C4—682no—
494C4⋊C4—682no—
495C4⋊C4C2×C4×D1712yes—
496C22⋊C4—684no—
497C42—24no—
498C42—172no—
499C42—172no—
500C42—174no—
501C42—174no—
502C42—344no—
503C42—344no—
504C42—344no—
505C42—684no—
506C42C2×C4×D1712yes—
507C42D4×D1712yes—
508C23—172no—
509C23—172no—
510C23—172no—
511C23—341no—
512C23—341no—
513C23—341no—
514C23—341no—
515C23C2×D4×D1711yescharacteristic
516Q8—22no—
517Q8—22no—
518Q8—22no—
519Q8—174no—
520Q8—3416no—
521Q8—342no—
522Q8—342no—
523Q8—342no—
524Q8C4⋊C4×D1714yes—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—172no—
537C2×C4—172no—
538C2×C4—172no—
539C2×C4—174no—
540C2×C4—174no—
541C2×C4—174no—
542C2×C4—174no—
543C2×C4—172no—
544C2×C4—172no—
545C2×C4—3416no—
546C2×C4—342no—
547C2×C4—342no—
548C2×C4—342no—
549C2×C4—341no—
550C2×C4—341no—
551C2×C4—341no—
552C2×C4—341no—
553C2×C4—348no—
554C2×C4—342no—
555C2×C4—684no—
556C2×C4—688no—
557C2×C4—681no—
558C2×C4—681no—
559C2×C4—682no—
560C2×C4—682no—
561C2×C4—682no—
562C2×C4C22⋊C4×D1712yes—
563C2×C4C22⋊C4×D1712yes—
564C2×C4#1058760112yes—
565C2×C4#1058760412yes—
566C2×C4C22×C4×D1711yescharacteristic
567C2×C4C2×D4×D1711yescharacteristic
568C2×C4C2×D4×D1711yescharacteristic
569C2×C4C2×Q8×D1711yescharacteristic
570C2×C4#41×D1711yescharacteristic
571C2×C4#41×D1711yescharacteristic
572C8—42no—
573C8—682no—
574C22—174no—
575C22—342no—
576C22—342no—
577C22—342no—
578C22—342no—
579C22—342no—
580C22—342no—
581C22C2×C22⋊C4×D1711yescharacteristic
582C22#121×D1711yescharacteristic
583C22C22⋊Q8×D1711yescharacteristic
584C22C2×#1058760111yescharacteristic
585C22C2×#1058760411yescharacteristic
586C22#1176360511yescharacteristic
587C22#1176360811yescharacteristic
588C4—24no—
589C4—22no—
590C4—42no—
591C4—82no—
592C4—174no—
593C4—342no—
594C4—344no—
595C4—342no—
596C4—682no—
597C4—1362no—
598C4#1176281614yescomplemented
599C4C2×C4⋊C4×D1711yescharacteristic
600C4#117×D1711yescharacteristic
601C4#121×D1711yescharacteristic
602C4C22⋊Q8×D1711yescharacteristic
603C2—344no—
604C2D17×#38011yescharacteristic
605C2C2×#1176281611yescharacteristic
606C2#1453730011yescharacteristic
607C2#1453825111yescharacteristic
608C2#1453825711yescharacteristic
609C2#1453858411yescharacteristic
610C2#1453859011yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1803C2
socleC23×C17C22×D4
prime radicalC23×D17C2×D4
cosocleC22×C4×C17C24
hypercenter#1803D17
hypocenterC17#20647
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
5. #14528798C2semidirect
6. #14528797C2semidirect
7. #14528790C2semidirect
8. #14528789C2semidirect
9. #14528786C2semidirect
11. C17×#1803C2semidirect
20. C4×Q8×D17C4semidirect
138. Q8×D17C4⋊C4semidirect
278. #1803D17semidirect
298. C4×C17#419semidirect
421. C17#20647semidirect
598. C4#11762816semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D172
C2×#117628312
C2×#117628162
#145311892
#145287982
#145287972
#145287902
#145287892
#145287862
#145285022
C17×#18032
#2064717