Finite Group Databasebeta

← #17056120

Subgroups — #17056120

Order 4352.

926 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763088C211yesmaximal, characteristic
2#14531837C211yesmaximal, characteristic, complemented
3#14531685C211yesmaximal, characteristic
4#14529743C211yesmaximal, characteristic, complemented
5#14529539C211yesmaximal, characteristic, complemented
6#14529516C211yesmaximal, characteristic, complemented
7#14529357C211yesmaximal, characteristic, complemented
8#14529352C211yesmaximal, characteristic, complemented
9#14529030C211yesmaximal, characteristic
10#14529028C211yesmaximal, characteristic
11C8×#63225C211yesmaximal, characteristic, complemented
12C4×#10587492C211yesmaximal, characteristic
13#14528598C211yesmaximal, characteristic
14#14528598C211yesmaximal, characteristic
15C17×#1942C211yesmaximal, characteristic, complemented
16C2×#10587574C2211yescharacteristic
17C2×#10587573—21no—
18C2×C4×D68C2211yescharacteristic
19C2×#10587579C2211yescharacteristic
20C2×#10587579C2211yescharacteristic
21C2×C4⋊C4×D17—21no—
22#11763088C418yescomplemented
23C2×#10587478C2211yescharacteristic
24C2×#10587478C2211yescharacteristic
25C2×#10587485C2211yescharacteristic
26C2×#10587492C2211yescharacteristic
27C2×#10587492C2211yescharacteristic
28C2×#10587480C2211yescharacteristic
29C2×#10587480C2211yescharacteristic
30C2×C8×Dic17C2211yescharacteristic
31C2×C8×Dic17C2211yescharacteristic
32#11762713—21no—
33#11762708C2211yescharacteristic
34#11762705C2211yescharacteristic
35#11762691—21no—
36#11762690—21no—
37#11762683—21no—
38#11762680—21no—
39#11762678C2211yescharacteristic
40#11762668—21no—
41C4⋊C4×Dic17C2211yescharacteristic
42C4⋊C4×Dic17C2211yescharacteristic
43C4×#63225C2211yescharacteristic
44C4×C4⋊Dic17C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762564C2211yescharacteristic
52#11762563C2211yescharacteristic
53C2×C17×#107C2211yescharacteristic
54C2×C17×#107C2211yescharacteristic
55C2×C4×C8×C17C2211yescharacteristic
56C17×#389C2211yescharacteristic
57C17×#389C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#340C2211yescharacteristic
60C22×D68C2311yescharacteristic
61C22×C4×D17—21no—
62C22×C4×D17—21no—
63C22×C4×D17—21no—
64C22×Dic34—21no—
65C22×C8×C17C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C2×C4⋊C4×C17—21no—
68C2×C4⋊C4×C17—21no—
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C42×C17C2311yescharacteristic
74C2×#63227—21no—
75C2×#63227—21no—
76C2×#63227—21no—
77C2×C4⋊Dic17—21no—
78C2×C4⋊Dic17C2311yescharacteristic
79C2×C4⋊Dic17C2311yescharacteristic
80C2×C4⋊Dic17C2311yescharacteristic
81C2×#63225—21no—
82C2×#63225—21no—
83C2×#63225—21no—
84C2×#63225—21no—
85C2×#63225D411yescharacteristic
86C2×#63225D411yescharacteristic
87C2×C4×Dic17—21no—
88C2×C4×Dic17—21no—
89C2×C4×Dic17—21no—
90C2×C4×Dic17C2311yescharacteristic
91C2×C4×Dic17C2311yescharacteristic
92C22×#3622C2311yescharacteristic
93C22×#3622C2311yescharacteristic
94#10587579—22no—
95#10587579—22no—
96#10587579—41no—
97#10587579C2×C414yes—
98C4⋊C4×D17—44no—
99#10587574—22no—
100#10587574C2×C414yescomplemented
101#10587573—44no—
102C4×D68—22no—
103C4×D68—41no—
104C17×#107—41no—
105C17×#107C2×C414yescomplemented
106C4×C8×C17—41no—
107C17×#95—21no—
108C17×#95—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502D411yescharacteristic
115#10587502D411yescharacteristic
116#10587492C2×C414yes—
117#10587492Q814yescomplemented
118#10587485C2×C414yescomplemented
119C8×Dic17—22no—
120C8×Dic17—41no—
121#10587480—22no—
122#10587480C2×C414yes—
123#10587478—41no—
124#10587478C2×C414yescomplemented
125C23×D17—21no—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17C2×D411yescharacteristic
131C22×C4×C17C2×D411yescharacteristic
132C22×C4×C17C2411yescharacteristic
133C22×Dic17—21no—
134C22×Dic17—21no—
135C22×Dic17—21no—
136C22×Dic17C2×D411yescharacteristic
137C22×Dic17C2×D411yescharacteristic
138C22×Dic17C2×D411yescharacteristic
139C22×Dic17C2×D411yescharacteristic
140C2×D68—21no—
141C2×D68—21no—
142C2×D68—21no—
143C2×D68C22×C412yes—
144C2×D68C2×Q812yes—
145C2×D68#4112yes—
146C2×C4×D17—24no—
147C2×C4×D17—42no—
148C2×C4×D17—42no—
149C2×C4×D17—42no—
150C2×C4×D17—42no—
151C2×C4×D17—42no—
152C2×C4×D17—42no—
153C2×C4×D17—81no—
154C2×C4×D17—81no—
155C2×Dic34—24no—
156C2×Dic34—42no—
157C2×Dic34—42no—
158C2×C8×C17—21no—
159C2×C8×C17—21no—
160C2×C8×C17—21no—
161C2×C8×C17—21no—
162C2×C8×C17C22×C412yes—
163C2×C8×C17C2×Q812yes—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—21no—
168C4⋊C4×C17—21no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17—42no—
171C4⋊C4×C17—42no—
172C4⋊C4×C17C22×C412yes—
173C4⋊C4×C17C22×C412yes—
174C42×C17—21no—
175C42×C17—21no—
176#63227—24no—
177#63227—42no—
178#63227—42no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—42no—
184C4⋊Dic17C22×C412yes—
185C4⋊Dic17C2×Q812yes—
186#63225—24no—
187#63225—22no—
188#63225—42no—
189#63225—42no—
190#63225—81no—
191#63225D814yes—
192C4×Dic17—21no—
193C4×Dic17—21no—
194C4×Dic17—42no—
195C4×Dic17—42no—
196C4×Dic17—42no—
197C4×Dic17C22×C412yes—
198C4×Dic17#4112yes—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622—21no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622C22×C412yes—
205#22962—171nomaximal, Sylow
206C23×C17C22×D411yescharacteristic
207C22×D17—22no—
208C22×D17—22no—
209C22×D17—22no—
210C22×D17—41no—
211C22×D17—41no—
212C22×D17—41no—
213C22×D17—41no—
214C2×C4×C17—21no—
215C2×C4×C17—21no—
216C2×C4×C17—22no—
217C2×C4×C17—22no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—41no—
230C2×C4×C17—41no—
231C2×C4×C17C2×D812yes—
232C2×C4×C17C23×C411yescharacteristic
233C2×C4×C17C22×Q811yescharacteristic
234C2×C4×C17C2×#4111yescharacteristic
235C2×C4×C17C2×#4111yescharacteristic
236C2×C4×C17H(5,𝔽2)11yescharacteristic
237C2×C4×C172−1+411yescharacteristic
238C2×Dic17—21no—
239C2×Dic17—21no—
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—22no—
243C2×Dic17—21no—
244C2×Dic17—22no—
245C2×Dic17—21no—
246C2×Dic17—21no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17C4×D412yes—
255C2×Dic17C4×D412yes—
256C2×Dic17C2×D812yes—
257C2×Dic17C2×D812yes—
258C2×Dic17#13512yes—
259D68—22no—
260D68—22no—
261D68—22no—
262D68C4×Q814yescomplemented
263C4×D17—48no—
264C4×D17—84no—
265C4×D17—84no—
266Dic34—48no—
267C8×C17—22no—
268C8×C17—22no—
269#3622—22no—
270#3622—41no—
271C2×#474—171no—
272#2365—171no—
273D4×C4⋊C4—171no—
274#1990—171no—
275#1947—171no—
276#1942—171no—
277#1942D1711yescharacteristic, complemented
278#1918—171no—
279#1918—171no—
280#1862—171no—
281#1862—171no—
282C8×C4⋊C4—171no—
283C4×#102—171no—
284C4×#102—171no—
285C4×#102—171no—
286C22×C17C2×C4×D411yescharacteristic
287C22×C17#54611yescharacteristic
288C22×C17D4×Q811yescharacteristic
289C22×C17C22×D811yescharacteristic
290C22×C17C2×#13511yescharacteristic
291C22×C17#57711yescharacteristic
292C22×C17#57811yescharacteristic
293C2×D17—24no—
294C2×D17—42no—
295C2×D17—42no—
296C2×D17—42no—
297C2×D17—42no—
298C2×D17—42no—
299C2×D17—42no—
300C4×C17—22no—
301C4×C17—42no—
302C4×C17—42no—
303C4×C17—41no—
304C4×C17—81no—
305C4×C17—81no—
306C4×C17C2×C4×Q811yescharacteristic
307C4×C17#52011yescharacteristic
308C4×C17#55211yescharacteristic
309C4×C17#55411yescharacteristic
310Dic17—22no—
311Dic17—22no—
312Dic17—42no—
313Dic17—42no—
314Dic17—41no—
315Dic17—81no—
316Dic17C4×D814yes—
317C2×C4⋊Q8—171no—
318C2×C4×Q8—341no—
319C2×C4×D4—171no—
320C2×C4×D4—171no—
321C2×C4×D4—171no—
322C22×C4⋊C4—341no—
323#474—178no—
324C2×#107—171no—
325C2×#107—171no—
326C2×#107C2×D1711yescharacteristic
327C2×#107C2×D1711yescharacteristic
328C2×C4⋊C8—171no—
329C2×#102—171no—
330C2×#102—171no—
331C2×#102—171no—
332C2×#102—171no—
333C2×C4×C8—171no—
334C2×C4×C8—171no—
335C2×C4×C8C2×D1711yescharacteristic
336#391—341no—
337#389—171no—
338#389—171no—
339#389C2×D1711yescharacteristic
340#389C2×D1711yescharacteristic
341#387—341no—
342#387—341no—
343#385—341no—
344#385—341no—
345#384—171no—
346#380—341no—
347C4×C4⋊C4—171no—
348C4×C4⋊C4—171no—
349C4×C4⋊C4—171no—
350C4×C4⋊C4—171no—
351#340—171no—
352#340—171no—
353#340—171no—
354#340—171no—
355#340—171no—
356#340—171no—
357#340C2×D1711yescharacteristic
358#340C2×D1711yescharacteristic
359#336—171no—
360#336—171no—
361C2×C17#236011yescharacteristic
362C2×C17C2×C4×D811yescharacteristic
363C2×C17#294711yescharacteristic
364C2×C17#329411yescharacteristic
365C2×C17#330211yescharacteristic
366C2×C17Q8×D811yescharacteristic
367C2×C17#338111yescharacteristic
368D17—44no—
369C22×Q8—341no—
370C22×D4—171no—
371C23×C4—341no—
372C23×C4—341no—
373C23×C4—341no—
374C22×C8—171no—
375C22×C8—171no—
376C22×C8C22×D1711yescharacteristic
377C22×C8C22×D1711yescharacteristic
378C4⋊Q8—174no—
379C4⋊Q8—342no—
380C4×Q8—684no—
381C4×D4—174no—
382C4×D4—342no—
383C4×D4—342no—
384C4×D4—342no—
385C4×D4—681no—
386C4×D4—681no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—171no—
390C2×C4⋊C4—171no—
391C2×C4⋊C4—171no—
392C2×C4⋊C4—171no—
393C2×C4⋊C4—171no—
394C2×C4⋊C4—341no—
395C2×C4⋊C4—341no—
396C2×C4⋊C4—341no—
397C2×C4⋊C4—341no—
398C2×C4⋊C4—341no—
399C2×C4⋊C4—684no—
400C2×C4⋊C4C22×D1711yescharacteristic
401C2×C4⋊C4C22×D1711yescharacteristic
402C2×C4⋊C4C22×D1711yescharacteristic
403C2×C4⋊C4C22×D1711yescharacteristic
404C2×C22⋊C4—341no—
405C2×C22⋊C4—341no—
406C2×C22⋊C4—341no—
407C2×C42—171no—
408C2×C42—171no—
409C2×C42—341no—
410C2×C42—341no—
411C2×C42—341no—
412C2×C42C22×D1711yescharacteristic
413#107—41no—
414#107—174no—
415#107—681no—
416#107C4×D1714yescomplemented
417C4⋊C8—174no—
418#102—174no—
419#102—174no—
420#102—174no—
421#102—342no—
422C4×C8—41no—
423C4×C8—342no—
424C4×C8—681no—
425#95—21no—
426#95—21no—
427#95—171no—
428#95—171no—
429#95—341no—
430#95—341no—
431#95—341no—
432#95—341no—
433#95—341no—
434C17#2296211yescharacteristic, complemented, Sylow
435C24—341no—
436C2×Q8—344no—
437C2×Q8—682no—
438C2×Q8—682no—
439C2×D4—172no—
440C2×D4—172no—
441C2×D4—172no—
442C2×D4—341no—
443C2×D4—341no—
444C2×D4—341no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—171no—
450C22×C4—171no—
451C22×C4—171no—
452C22×C4—171no—
453C22×C4—344no—
454C22×C4—341no—
455C22×C4—341no—
456C22×C4—341no—
457C22×C4—682no—
458C22×C4—682no—
459C22×C4—682no—
460C22×C4—682no—
461C22×C4—682no—
462C22×C4—682no—
463C22×C4—1361no—
464C22×C4—1361no—
465C22×C4D4×D1711yescharacteristic
466C22×C4D4×D1711yescharacteristic
467C22×C4C23×D1711yescharacteristic
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—172no—
473C2×C8—341no—
474C2×C8—341no—
475C2×C8—341no—
476C2×C8—341no—
477C2×C8—341no—
478C2×C8C2×C4×D1712yes—
479C2×C8Q8×D1712yes—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—174no—
489C4⋊C4—172no—
490C4⋊C4—172no—
491C4⋊C4—342no—
492C4⋊C4—344no—
493C4⋊C4—341no—
494C4⋊C4—341no—
495C4⋊C4—341no—
496C4⋊C4—341no—
497C4⋊C4—682no—
498C4⋊C4—682no—
499C4⋊C4—682no—
500C4⋊C4—1361no—
501C4⋊C4C2×C4×D1712yes—
502C4⋊C4C2×C4×D1712yes—
503C22⋊C4—344no—
504C22⋊C4—682no—
505C22⋊C4—682no—
506C42—21no—
507C42—21no—
508C42—172no—
509C42—172no—
510C42—341no—
511C42—341no—
512C42—682no—
513C42—682no—
514C42—682no—
515C23—342no—
516C23—342no—
517C23—342no—
518C23—681no—
519C23—681no—
520C23—681no—
521C23—681no—
522C23C2×D4×D1711yescharacteristic
523Q8—688no—
524D4—174no—
525D4—342no—
526D4—342no—
527D4—342no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—172no—
546C2×C4—172no—
547C2×C4—172no—
548C2×C4—172no—
549C2×C4—172no—
550C2×C4—341no—
551C2×C4—341no—
552C2×C4—341no—
553C2×C4—341no—
554C2×C4—342no—
555C2×C4—342no—
556C2×C4—341no—
557C2×C4—341no—
558C2×C4—341no—
559C2×C4—688no—
560C2×C4—681no—
561C2×C4—681no—
562C2×C4—681no—
563C2×C4—681no—
564C2×C4—681no—
565C2×C4—681no—
566C2×C4—681no—
567C2×C4—1364no—
568C2×C4—1364no—
569C2×C4D8×D1712yes—
570C2×C4C22×C4×D1711yescharacteristic
571C2×C4#1058769111yescharacteristic
572C2×C4C2×Q8×D1711yescharacteristic
573C2×C4#1058769411yescharacteristic
574C2×C4#41×D1711yescharacteristic
575C2×C4#41×D1711yescharacteristic
576C8—22no—
577C8—22no—
578C8—342no—
579C8—681no—
580C22—344no—
581C22—682no—
582C22—682no—
583C22—682no—
584C22—682no—
585C22—682no—
586C22—682no—
587C22C4×D4×D1711yescharacteristic
588C22#1176344811yescharacteristic
589C22#1176347411yescharacteristic
590C22C2×D8×D1711yescharacteristic
591C22#135×D1711yescharacteristic
592C22#1176361611yescharacteristic
593C22#1176362211yescharacteristic
594C4—22no—
595C4—42no—
596C4—42no—
597C4—41no—
598C4—81no—
599C4—81no—
600C4—174no—
601C4—342no—
602C4—342no—
603C4—682no—
604C4—682no—
605C4—681no—
606C4—1361no—
607C4C4×Q8×D1711yescharacteristic
608C4#1176329011yescharacteristic
609C4#1176349111yescharacteristic
610C4#1176351311yescharacteristic
611C2—684no—
612C2#1453178611yescharacteristic
613C2C4×D8×D1711yescharacteristic
614C2#1453762211yescharacteristic
615C2#1454059211yescharacteristic
616C2#1454069611yescharacteristic
617C2#1454163611yescharacteristic
618C2#1454165611yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1942C2
socleC23×C17C22×D4
prime radicalC22×D68C23
cosocleC22×C4×C17C24
hypercenter#1942D17
hypocenterC17#22962
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1942

Complements

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

NH ≅ G/Ntype
2. #14531837C2semidirect
4. #14529743C2semidirect
5. #14529539C2semidirect
6. #14529516C2semidirect
7. #14529357C2semidirect
8. #14529352C2semidirect
11. C8×#63225C2semidirect
15. C17×#1942C2semidirect
22. #11763088C4semidirect
100. #10587574C2×C4semidirect
105. C17×#107C2×C4semidirect
117. #10587492Q8semidirect
118. #10587485C2×C4semidirect
124. #10587478C2×C4semidirect
262. D68C4×Q8semidirect
277. #1942D17semidirect
416. #107C4×D17semidirect
434. C17#22962semidirect

Maximal subgroups

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

subgroupindex
C2×#117630882
#145318372
#145316852
#145297432
#145295392
#145295162
#145293572
#145293522
#145290302
#145290282
C8×#632252
C4×#105874922
#145285982
C17×#19422
#2296217