Finite Group Databasebeta

← #17039173

Subgroups — #17039173

Order 4352.

870 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762859C211yesmaximal, characteristic
2#14531179C211yesmaximal, characteristic
3#117×Dic17C211yesmaximal, characteristic
4#14529555C211yesmaximal, characteristic, complemented
5#14529553C211yesmaximal, characteristic, complemented
6#14529551C211yesmaximal, characteristic, complemented
7#14529549C211yesmaximal, characteristic, complemented
8#14528985C211yesmaximal, characteristic
9#14528983C211yesmaximal, characteristic
10#14528981C211yesmaximal, characteristic
11#107×Dic17C211yesmaximal, characteristic, complemented
12#14528676C211yesmaximal, characteristic, complemented
13#106×Dic17C211yesmaximal, characteristic, complemented
14#14528672C211yesmaximal, characteristic, complemented
15C17×#1854C211yesmaximal, characteristic
16C2×#10587612C2211yescharacteristic
17C22×C4×Dic17C2211yescharacteristic
18#11762859C418yescomplemented
19C2×#10587489C2211yescharacteristic
20C2×#10587478C2211yescharacteristic
21C2×#10587478C2211yescharacteristic
22C2×#10587488C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×#10587479C2211yescharacteristic
25C2×#10587485C2211yescharacteristic
26C2×#10587485C2211yescharacteristic
27C2×C22⋊#3622C2211yescharacteristic
28C2×C8×Dic17C2211yescharacteristic
29C2×C8×Dic17C2211yescharacteristic
30#11762676—21no—
31#11762674C2211yescharacteristic
32#11762674C2211yescharacteristic
33#11762670—21no—
34#11762669C2211yescharacteristic
35C4⋊C4×Dic17C2211yescharacteristic
36C4⋊C4×Dic17C2211yescharacteristic
37C4⋊C4×Dic17C2211yescharacteristic
38C4⋊C4×Dic17C2211yescharacteristic
39C4⋊C4×Dic17C2211yescharacteristic
40C4⋊C4×Dic17C2211yescharacteristic
41C22⋊C4×Dic17—21no—
42C22⋊C4×Dic17—21no—
43C22⋊C4×Dic17—21no—
44#11762655—21no—
45#11762654—21no—
46C42×Dic17—21no—
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762571C2211yescharacteristic
53C2×C17×#117C2211yescharacteristic
54C2×C17×#107C2211yescharacteristic
55C2×C17×#106C2211yescharacteristic
56C2×C17×C22⋊C8C2211yescharacteristic
57C17×#380C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#340C2211yescharacteristic
60C23×C4×C17C2311yescharacteristic
61C23×Dic17D411yescharacteristic
62C23×Dic17D411yescharacteristic
63C22×C8×C17C2311yescharacteristic
64C22×C8×C17C2311yescharacteristic
65C17×#117—22no—
66C17×#117—22no—
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C22⋊C4×C17—21no—
72C2×C22⋊C4×C17—21no—
73C2×C42×C17—21no—
74C2×C22⋊Dic17—21no—
75C2×C22⋊Dic17—21no—
76#10587612—22no—
77#10587612—22no—
78C2×C4⋊Dic17C2311yescharacteristic
79C2×C4⋊Dic17C2311yescharacteristic
80C2×C4⋊Dic17C2311yescharacteristic
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4×Dic17—21no—
83C2×C4×Dic17—21no—
84C2×C4×Dic17—21no—
85C2×C4×Dic17—21no—
86C2×C4×Dic17—21no—
87C2×C4×Dic17—21no—
88C2×C4×Dic17—21no—
89C2×C4×Dic17C2×C414yes—
90C2×C4×Dic17D411yescharacteristic
91C2×C4×Dic17D411yescharacteristic
92C2×C4×Dic17Q814yes—
93C2×C4×Dic17C2311yescharacteristic
94C2×C4×Dic17C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96C22×#3622C2311yescharacteristic
97C17×#107—22no—
98C17×#106—22no—
99C17×C22⋊C8C2×C414yes—
100C17×#95—21no—
101#10587502—21no—
102#10587502—21no—
103#10587502—21no—
104#10587502—21no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112C22⋊#3622C2×C414yes—
113#10587489—22no—
114#10587488—22no—
115#10587485C2×C414yes—
116#10587485C2×C414yes—
117C8×Dic17C2×C414yes—
118C8×Dic17C2×C414yes—
119#10587479—22no—
120#10587479—22no—
121#10587478—22no—
122#10587478—22no—
123C24×C17C2×D411yescharacteristic
124C22×C4×C17—21no—
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17C22×C412yes—
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17C2×Q812yes—
135C22×C4×C17C2411yescharacteristic
136C22×Dic17—22no—
137C22×Dic17—22no—
138C22×Dic17—21no—
139C22×Dic17—21no—
140C22×Dic17—21no—
141C22×Dic17—21no—
142C22×Dic17—41no—
143C22×Dic17—41no—
144C22×Dic17C4⋊C414yes—
145C22×Dic17C4⋊C414yes—
146C22×Dic17C2×D411yescharacteristic
147C22×Dic17C2×D411yescharacteristic
148C22×Dic17C2×D411yescharacteristic
149C22×Dic17C2×D411yescharacteristic
150C2×C8×C17—21no—
151C2×C8×C17—21no—
152C2×C8×C17—21no—
153C2×C8×C17—21no—
154C2×C8×C17C22×C412yes—
155C2×C8×C17C22×C412yes—
156C4⋊C4×C17—21no—
157C4⋊C4×C17—21no—
158C4⋊C4×C17—21no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—21no—
161C4⋊C4×C17—21no—
162C4⋊C4×C17#4112yes—
163C4⋊C4×C17#4112yes—
164C22⋊C4×C17—24no—
165C22⋊C4×C17—42no—
166C42×C17—24no—
167C22⋊Dic17—24no—
168C22⋊Dic17—42no—
169C4⋊Dic17—21no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17—21no—
175C4⋊Dic17#4112yes—
176C4⋊Dic17#4112yes—
177C4×Dic17—24no—
178C4×Dic17—42no—
179C4×Dic17—42no—
180C4×Dic17—42no—
181C4×Dic17—42no—
182C4×Dic17—42no—
183C4×Dic17—42no—
184C4×Dic17C4⋊C414yes—
185C4×Dic17C4⋊C414yes—
186C4×Dic17C22×C412yes—
187C4×Dic17C22×C412yes—
188C4×Dic17C2×Q812yes—
189C4×Dic17C2×Q812yes—
190C2×#3622—21no—
191C2×#3622—21no—
192C2×#3622—21no—
193C2×#3622—21no—
194C2×#3622C22×C412yes—
195C2×#3622C22×C412yes—
196#21360—171nomaximal, Sylow
197C23×C17—21no—
198C23×C17—21no—
199C23×C17—21no—
200C23×C17—21no—
201C23×C17—21no—
202C23×C17—21no—
203C23×C17C2×C4⋊C412yes—
204C23×C17C22×D411yescharacteristic
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—22no—
208C2×C4×C17—22no—
209C2×C4×C17—22no—
210C2×C4×C17—22no—
211C2×C4×C17—22no—
212C2×C4×C17—22no—
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×C17C2×C4⋊C412yes—
224C2×C4×C17C23×C411yescharacteristic
225C2×C4×C17C22×Q811yescharacteristic
226C2×C4×C17C2×#4111yescharacteristic
227C2×C4×C17C2×#4111yescharacteristic
228C2×C4×C17C2×#4111yescharacteristic
229C2×C4×C17C2×#4111yescharacteristic
230C2×Dic17—21no—
231C2×Dic17—21no—
232C2×Dic17—21no—
233C2×Dic17—21no—
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—28no—
237C2×Dic17—22no—
238C2×Dic17—22no—
239C2×Dic17—44no—
240C2×Dic17—41no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17C2×C4⋊C412yes—
251C2×Dic17C2×C4⋊C412yes—
252C2×Dic17C2×C4⋊C412yes—
253C2×Dic17C2×C4⋊C412yes—
254C2×Dic17#13512yes—
255C2×Dic17#13512yes—
256C8×C17—22no—
257C8×C17—22no—
258#3622—22no—
259#3622—22no—
260C2×#431—171no—
261#2283—171no—
262C4×#117—171no—
263#1949—171no—
264#1949—171no—
265#1949—171no—
266#1949—171no—
267#1854—171no—
268#1854—171no—
269#1854—171no—
270#1854D1711yescharacteristic
271C4×#107—171no—
272C4×#107—171no—
273C4×#106—171no—
274C4×#106—171no—
275C22×C17—22no—
276C22×C17—22no—
277C22×C17—22no—
278C22×C17—41no—
279C22×C17—41no—
280C22×C17—41no—
281C22×C17—41no—
282C22×C17C22×C4⋊C411yescharacteristic
283C22×C17#52511yescharacteristic
284C22×C17#52511yescharacteristic
285C22×C17C2×#13511yescharacteristic
286C22×C17C2×#13511yescharacteristic
287C22×C17#57511yescharacteristic
288C22×C17#57511yescharacteristic
289C4×C17—22no—
290C4×C17—42no—
291C4×C17—42no—
292C4×C17—81no—
293C4×C17—81no—
294C4×C17C4×#4111yescharacteristic
295C4×C17C4×#4111yescharacteristic
296C4×C17#53311yescharacteristic
297C4×C17#53311yescharacteristic
298Dic17—22no—
299Dic17—22no—
300Dic17—22no—
301Dic17—42no—
302Dic17—42no—
303Dic17—81no—
304Dic17—81no—
305Dic17#42714yes—
306C2×#117—171no—
307C2×#117C2×D1711yescharacteristic
308C22×C42—171no—
309#431—178no—
310C2×#107—171no—
311C2×#107—171no—
312C2×#107—171no—
313C2×#107C2×D1711yescharacteristic
314C2×#106—171no—
315C2×#106—171no—
316C2×#106—171no—
317C2×#106C2×D1711yescharacteristic
318C2×C4⋊C8—171no—
319C2×C4⋊C8—171no—
320C2×C22⋊C8—171no—
321C2×C22⋊C8C2×D1711yescharacteristic
322C2×C4×C8—171no—
323C2×C4×C8—171no—
324#383—341no—
325#382—171no—
326#382—171no—
327#381—341no—
328#380—171no—
329#380C2×D1711yescharacteristic
330C4×C4⋊C4—171no—
331C4×C4⋊C4—171no—
332C4×C4⋊C4—171no—
333C4×C4⋊C4—171no—
334C4×C4⋊C4—171no—
335C4×C4⋊C4—171no—
336C4×C22⋊C4—341no—
337C4×C22⋊C4—341no—
338C4×C22⋊C4—341no—
339#376—341no—
340#376—341no—
341C43—341no—
342#340—171no—
343#340—171no—
344#340—171no—
345#340—171no—
346#340—171no—
347#340—171no—
348#340C2×D1711yescharacteristic
349#340C2×D1711yescharacteristic
350C2×C17—42no—
351C2×C17#229611yescharacteristic
352C2×C17C2×#42711yescharacteristic
353C2×C17#291111yescharacteristic
354C2×C17#310211yescharacteristic
355C2×C17#310211yescharacteristic
356C2×C17#310211yescharacteristic
357C2×C17#310211yescharacteristic
358C23×C4—171no—
359C23×C4—171no—
360C23×C4C22×D1711yescharacteristic
361C22×C8—171no—
362C22×C8—171no—
363C22×C8C22×D1711yescharacteristic
364C22×C8C22×D1711yescharacteristic
365#117—22no—
366#117—22no—
367#117—342no—
368#117—342no—
369C2×C4⋊C4—171no—
370C2×C4⋊C4—171no—
371C2×C4⋊C4—171no—
372C2×C4⋊C4—171no—
373C2×C4⋊C4C22×D1711yescharacteristic
374C2×C4⋊C4C22×D1711yescharacteristic
375C2×C4⋊C4C22×D1711yescharacteristic
376C2×C4⋊C4C22×D1711yescharacteristic
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—341no—
380C2×C22⋊C4—341no—
381C2×C42—21no—
382C2×C42—174no—
383C2×C42—171no—
384C2×C42—171no—
385C2×C42—174no—
386C2×C42—171no—
387C2×C42—171no—
388C2×C42—341no—
389C2×C42—341no—
390C2×C42—341no—
391C2×C42—341no—
392C2×C42—341no—
393C2×C42—341no—
394C2×C42—341no—
395#107—22no—
396#107—342no—
397#107—342no—
398#107—342no—
399#106—22no—
400#106—342no—
401#106—342no—
402#106—342no—
403C4⋊C8—174no—
404C4⋊C8—174no—
405C22⋊C8—174no—
406C22⋊C8C4×D1714yes—
407C4×C8—174no—
408C4×C8—174no—
409#95—21no—
410#95—341no—
411#95—341no—
412#95—341no—
413#95—341no—
414#95—341no—
415#95—341no—
416#95—341no—
417#95—341no—
418#95—341no—
419#95—341no—
420#95—341no—
421C17#2136011yescharacteristic, complemented, Sylow
422C24D4×D1711yescharacteristic
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—171no—
432C22×C4—171no—
433C22×C4—171no—
434C22×C4—171no—
435C22×C4—174no—
436C22×C4—174no—
437C22×C4—342no—
438C22×C4—342no—
439C22×C4—341no—
440C22×C4—341no—
441C22×C4—341no—
442C22×C4—341no—
443C22×C4—681no—
444C22×C4—681no—
445C22×C4C2×C4×D1712yes—
446C22×C4D4×D1711yescharacteristic
447C22×C4Q8×D1712yes—
448C22×C4C23×D1711yescharacteristic
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—172no—
454C2×C8—172no—
455C2×C8—341no—
456C2×C8—341no—
457C2×C8—341no—
458C2×C8—341no—
459C2×C8C2×C4×D1712yes—
460C2×C8C2×C4×D1712yes—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—172no—
468C4⋊C4—172no—
469C4⋊C4—341no—
470C4⋊C4—341no—
471C4⋊C4—341no—
472C4⋊C4—341no—
473C4⋊C4—341no—
474C4⋊C4—341no—
475C4⋊C4#6325412yes—
476C4⋊C4#6325412yes—
477C22⋊C4—24no—
478C22⋊C4—42no—
479C22⋊C4—344no—
480C22⋊C4—682no—
481C42—24no—
482C42—172no—
483C42—172no—
484C42—172no—
485C42—172no—
486C42—174no—
487C42—174no—
488C42—344no—
489C42—682no—
490C42—682no—
491C42—682no—
492C42—682no—
493C42—682no—
494C42—682no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23C4⋊C4×D1712yes—
502C23C2×D4×D1711yescharacteristic
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—172no—
522C2×C4—172no—
523C2×C4—172no—
524C2×C4—172no—
525C2×C4—172no—
526C2×C4—172no—
527C2×C4—348no—
528C2×C4—341no—
529C2×C4—341no—
530C2×C4—341no—
531C2×C4—341no—
532C2×C4—341no—
533C2×C4—341no—
534C2×C4—342no—
535C2×C4—342no—
536C2×C4—684no—
537C2×C4—681no—
538C2×C4—681no—
539C2×C4—681no—
540C2×C4—681no—
541C2×C4—681no—
542C2×C4—681no—
543C2×C4—681no—
544C2×C4—681no—
545C2×C4—681no—
546C2×C4—681no—
547C2×C4C4⋊C4×D1712yes—
548C2×C4C22×C4×D1711yescharacteristic
549C2×C4C2×#6325411yescharacteristic
550C2×C4C2×#6325411yescharacteristic
551C2×C4C2×Q8×D1711yescharacteristic
552C2×C4#41×D1711yescharacteristic
553C2×C4#41×D1711yescharacteristic
554C8—22no—
555C8—22no—
556C8—342no—
557C8—342no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22C2×C4⋊C4×D1711yescharacteristic
566C22#1176331511yescharacteristic
567C22#1176331711yescharacteristic
568C22#135×D1711yescharacteristic
569C22#135×D1711yescharacteristic
570C22#1176360511yescharacteristic
571C22#1176360811yescharacteristic
572C4—22no—
573C4—42no—
574C4—42no—
575C4—81no—
576C4—81no—
577C4—174no—
578C4—342no—
579C4—342no—
580C4—342no—
581C4—682no—
582C4—682no—
583C4—1361no—
584C4—1361no—
585C4C4×#6325411yescharacteristic
586C4C4×#6325411yescharacteristic
587C4#1176336111yescharacteristic
588C4#1176336111yescharacteristic
589C2—42no—
590C2#1453126311yescharacteristic
591C2D17×#42711yescharacteristic
592C2#1453737111yescharacteristic
593C2#1453879511yescharacteristic
594C2#1453879711yescharacteristic
595C2#1453880311yescharacteristic
596C2#1453880511yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1854C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
cosocleC22×C4×C17C24
hypercenter#1854D17
hypocenterC17#21360
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
4. #14529555C2semidirect
5. #14529553C2semidirect
6. #14529551C2semidirect
7. #14529549C2semidirect
11. #107×Dic17C2semidirect
12. #14528676C2semidirect
13. #106×Dic17C2semidirect
14. #14528672C2semidirect
18. #11762859C4semidirect
421. C17#21360semidirect

Maximal subgroups

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

subgroupindex
C2×#117628592
#145311792
#117×Dic172
#145295552
#145295532
#145295512
#145295492
#145289852
#145289832
#145289812
#107×Dic172
#145286762
#106×Dic172
#145286722
C17×#18542
#2136017