Finite Group Databasebeta

← #17037439

Subgroups — #17037439

Order 4352.

862 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762909C211yesmaximal, characteristic
2#14531147C211yesmaximal, characteristic, complemented
3#14529408C211yesmaximal, characteristic
4#14529401C211yesmaximal, characteristic
5#14529373C211yesmaximal, characteristic, complemented
6#14529352C211yesmaximal, characteristic, complemented
7#14529142C211yesmaximal, characteristic
8#14529142C211yesmaximal, characteristic
9#14529051C211yesmaximal, characteristic, complemented
10#14529037C211yesmaximal, characteristic, complemented
11#14528682C211yesmaximal, characteristic, complemented
12C4×#10587481C211yesmaximal, characteristic, complemented
13C4×#10587503C211yesmaximal, characteristic
14#14528536C211yesmaximal, characteristic, complemented
15C17×#2319C211yesmaximal, characteristic
16C22×#63229C2211yescharacteristic
17C2×C4×Dic34C2211yescharacteristic
18C2×C4×Dic34C2211yescharacteristic
19C2×C4×Dic34C2211yescharacteristic
20#11762909C418yescomplemented
21C2×#10587478C2211yescharacteristic
22C2×#10587478C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×#10587481C2211yescharacteristic
25C2×#10587481C2211yescharacteristic
26C2×#10587481C2211yescharacteristic
27C2×#10587503C2211yescharacteristic
28C2×#10587503C2211yescharacteristic
29C2×#10587503C2211yescharacteristic
30C2×#10587474C2211yescharacteristic
31C2×C4×#3622C2211yescharacteristic
32C4×C4⋊Dic17C2211yescharacteristic
33C4×C4⋊Dic17C2211yescharacteristic
34C4×C4⋊Dic17C2211yescharacteristic
35#11762660—21no—
36#11762660—21no—
37#11762660—21no—
38#11762654—21no—
39#11762571C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762564C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48#11762564C2211yescharacteristic
49C2×C4×D4×C17C2211yescharacteristic
50C17×#390C2211yescharacteristic
51C17×#390C2211yescharacteristic
52C17×#389C2211yescharacteristic
53C17×#389C2211yescharacteristic
54C17×#388—21no—
55C17×#388—21no—
56C17×#381—21no—
57C17×#380—21no—
58C4×C4⋊C4×C17C2211yescharacteristic
59C17×#376C2211yescharacteristic
60C22×D4×C17C2311yescharacteristic
61C23×C4×C17—21no—
62C22×Dic34C2311yescharacteristic
63C4×D4×C17—22no—
64C4×D4×C17C2×C414yes—
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17—21no—
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C22⋊C4×C17—21no—
71C2×C22⋊C4×C17—21no—
72C2×C22⋊C4×C17—21no—
73C2×C22⋊C4×C17—21no—
74C2×C22⋊C4×C17—21no—
75C2×C42×C17—21no—
76C2×C42×C17C2311yescharacteristic
77C2×C42×C17C2311yescharacteristic
78C2×C42×C17C2311yescharacteristic
79C2×#63229—22no—
80C2×#63229—22no—
81C2×#63229C2×C414yes—
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×#63225—21no—
86C2×#63225—21no—
87C2×#63225—21no—
88C2×C4×Dic17—21no—
89C2×C4×Dic17—21no—
90C2×C4×Dic17—21no—
91C2×C4×Dic17—21no—
92C2×C4×Dic17—21no—
93C2×C4×Dic17—21no—
94C22×#3622C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96C22×#3622C2311yescharacteristic
97C22×#3622C2311yescharacteristic
98C4×Dic34—22no—
99C4×Dic34—22no—
100C4×Dic34—22no—
101C4×Dic34—22no—
102C4×Dic34—22no—
103C4×Dic34C2×C414yes—
104C17×#95—21no—
105C17×#95—21no—
106C17×#95—21no—
107C17×#95D411yescharacteristic
108C17×#95D411yescharacteristic
109C17×#95D411yescharacteristic
110C17×#95D411yescharacteristic
111#10587503—22no—
112#10587503—22no—
113#10587503C2×C414yes—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587481—22no—
118#10587481—22no—
119#10587481C2×C414yes—
120#10587479—22no—
121#10587478—22no—
122#10587478C2×C414yes—
123#10587474C2×C414yes—
124C4×#3622—22no—
125C24×C17—21no—
126C2×D4×C17—21no—
127C2×D4×C17—21no—
128C2×D4×C17—21no—
129C2×D4×C17—21no—
130C2×D4×C17—21no—
131C2×D4×C17C22×C412yes—
132C22×C4×C17—21no—
133C22×C4×C17—24no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—81no—
137C22×C4×C17—81no—
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17C2411yescharacteristic
145C22×Dic17—21no—
146C22×Dic17—21no—
147C22×Dic17—21no—
148C22×Dic17—21no—
149C2×Dic34—21no—
150C2×Dic34—21no—
151C2×Dic34—21no—
152C2×Dic34C22×C412yes—
153C2×Dic34#4112yes—
154C2×Dic34#4112yes—
155C4⋊C4×C17—21no—
156C4⋊C4×C17—21no—
157C4⋊C4×C17—21no—
158C4⋊C4×C17—21no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—81no—
161C4⋊C4×C17—81no—
162C4⋊C4×C17C22×C412yes—
163C22⋊C4×C17—24no—
164C22⋊C4×C17—81no—
165C22⋊C4×C17—81no—
166C22⋊C4×C17—81no—
167C22⋊C4×C17—81no—
168C42×C17—21no—
169C42×C17—21no—
170C42×C17—21no—
171C42×C17—21no—
172C42×C17—81no—
173C42×C17C22×C412yes—
174C42×C17#4112yes—
175#63229—28no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17C22×C412yes—
180C4⋊Dic17#4112yes—
181C4⋊Dic17#4112yes—
182#63225—24no—
183#63225—24no—
184#63225—24no—
185C4×Dic17—24no—
186C4×Dic17—24no—
187C4×Dic17—24no—
188C4×Dic17—81no—
189C4×Dic17—81no—
190C4×Dic17—81no—
191C2×#3622—21no—
192C2×#3622—21no—
193C2×#3622—21no—
194C2×#3622—21no—
195C2×#3622—21no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622C22×C412yes—
202C2×#3622C22×C412yes—
203#21128—171nomaximal, Sylow
204C23×C17—22no—
205C23×C17—41no—
206C23×C17—41no—
207C23×C17—41no—
208C23×C17—41no—
209C23×C17—41no—
210C23×C17—41no—
211C23×C17C22×D411yescharacteristic
212D4×C17—22no—
213D4×C17—22no—
214D4×C17—41no—
215D4×C17—41no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—22no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—41no—
234C2×C4×C17—41no—
235C2×C4×C17—41no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17—41no—
239C2×C4×C17—41no—
240C2×C4×C17—41no—
241C2×C4×C17—84no—
242C2×C4×C17C4×D412yes—
243C2×C4×C17C4×D412yes—
244C2×C4×C17C23×C411yescharacteristic
245C2×C4×C17C2×#4111yescharacteristic
246C2×C4×C17C2×#4111yescharacteristic
247C2×C4×C17C2×#4111yescharacteristic
248C2×C4×C17C2×#4111yescharacteristic
249C2×C4×C17C2×#4111yescharacteristic
250C2×Dic17—22no—
251C2×Dic17—22no—
252C2×Dic17—22no—
253C2×Dic17—22no—
254C2×Dic17—22no—
255C2×Dic17—22no—
256C2×Dic17—41no—
257C2×Dic17—41no—
258C2×Dic17—41no—
259C2×Dic17—41no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262Dic34—22no—
263Dic34—22no—
264Dic34—22no—
265Dic34#11714yes—
266#3622—22no—
267#3622—22no—
268#3622—41no—
269#3622—41no—
270C2×#440—171no—
271#2319D1711yescharacteristic
272#2276—171no—
273#1926—171no—
274#1925—171no—
275#1921—171no—
276#1918—171no—
277#1880—171no—
278#1880—171no—
279#1865—171no—
280#1863—171no—
281#1776—171no—
282C4×#103—171no—
283C4×#102—171no—
284#1746—171no—
285C22×C17—42no—
286C22×C17—42no—
287C22×C17—42no—
288C22×C17—81no—
289C22×C17—81no—
290C22×C17—81no—
291C22×C17—81no—
292C22×C17C2×C4×D411yescharacteristic
293C22×C17#52511yescharacteristic
294C22×C17#52511yescharacteristic
295C22×C17#57611yescharacteristic
296C22×C17#57711yescharacteristic
297C22×C17#57711yescharacteristic
298C22×C17#57811yescharacteristic
299C4×C17—22no—
300C4×C17—22no—
301C4×C17—41no—
302C4×C17—41no—
303C4×C17—41no—
304C4×C17—41no—
305C4×C17—42no—
306C4×C17—81no—
307C4×C17—81no—
308C4×C17C2×#11711yescharacteristic
309C4×C17C4×#4111yescharacteristic
310C4×C17#52911yescharacteristic
311C4×C17#52911yescharacteristic
312Dic17—24no—
313Dic17—42no—
314Dic17—42no—
315Dic17—42no—
316C22×SD16—171no—
317C2×C4×Q8—171no—
318C2×C4×Q8—171no—
319C2×C4×Q8—171no—
320C2×C4×D4C2×D1711yescharacteristic
321#440—178no—
322C2×#107—171no—
323C2×#107—171no—
324C2×#106—171no—
325C2×#103—171no—
326C2×#103—171no—
327C2×#103—171no—
328C2×#102—171no—
329C2×#102—171no—
330C2×#102—171no—
331C2×#97—171no—
332C2×C4×C8—171no—
333#390C2×D1711yescharacteristic
334#390C2×D1711yescharacteristic
335#389C2×D1711yescharacteristic
336#389C2×D1711yescharacteristic
337#388—21no—
338#388—21no—
339#381—21no—
340#380—21no—
341C4×C4⋊C4—171no—
342C4×C4⋊C4—171no—
343C4×C4⋊C4—171no—
344C4×C4⋊C4—341no—
345C4×C4⋊C4—341no—
346C4×C4⋊C4—341no—
347C4×C4⋊C4C2×D1711yescharacteristic
348#376—341no—
349#376C2×D1711yescharacteristic
350#340—171no—
351#340—171no—
352#340—171no—
353#340—171no—
354#340—171no—
355#340—171no—
356#336—171no—
357#336—171no—
358#336—171no—
359#336—171no—
360C2×C17—82no—
361C2×C17#229011yescharacteristic
362C2×C17#294611yescharacteristic
363C2×C17#294711yescharacteristic
364C2×C17#311911yescharacteristic
365C2×C17#312111yescharacteristic
366C2×C17#312411yescharacteristic
367C2×C17#312611yescharacteristic
368C22×Q8—171no—
369C22×D4C22×D1711yescharacteristic
370C23×C4—21no—
371C2×SD16—174no—
372C2×SD16—342no—
373C2×SD16—342no—
374C22×C8—171no—
375C22×C8—171no—
376C22×C8—171no—
377C22×C8—171no—
378C4×Q8—174no—
379C4×Q8—342no—
380C4×Q8—342no—
381C4×Q8—342no—
382C4×Q8—342no—
383C4×Q8—342no—
384C4×D4—22no—
385C4×D4C4×D1714yes—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—171no—
389C2×C4⋊C4—171no—
390C2×C4⋊C4—171no—
391C2×C4⋊C4—341no—
392C2×C4⋊C4—341no—
393C2×C4⋊C4—341no—
394C2×C4⋊C4C22×D1711yescharacteristic
395C2×C4⋊C4C22×D1711yescharacteristic
396C2×C4⋊C4C22×D1711yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C42—21no—
403C2×C42—341no—
404C2×C42—341no—
405C2×C42—341no—
406C2×C42—341no—
407C2×C42—341no—
408C2×C42—341no—
409C2×C42C22×D1711yescharacteristic
410C2×C42C22×D1711yescharacteristic
411C2×C42C22×D1711yescharacteristic
412#107—174no—
413#107—342no—
414#106—342no—
415#103—174no—
416#103—342no—
417#103—342no—
418#102—174no—
419#102—342no—
420#102—342no—
421#97—174no—
422C4×C8—342no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—341no—
427#95—341no—
428#95—341no—
429#95C22⋊D1711yescharacteristic
430#95C22⋊D1711yescharacteristic
431#95C22⋊D1711yescharacteristic
432#95C22⋊D1711yescharacteristic
433C17#2112811yescharacteristic, complemented, Sylow
434C24—21no—
435C2×Q8—172no—
436C2×Q8—172no—
437C2×Q8—172no—
438C2×Q8—341no—
439C2×Q8—341no—
440C2×Q8—341no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4C2×C4×D1712yes—
447C22×C4—21no—
448C22×C4—24no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—81no—
452C22×C4—81no—
453C22×C4—341no—
454C22×C4—341no—
455C22×C4—341no—
456C22×C4—341no—
457C22×C4C2×C22⋊D1711yescharacteristic
458C22×C4C2×C22⋊D1711yescharacteristic
459C22×C4C2×C22⋊D1711yescharacteristic
460C22×C4C2×C22⋊D1711yescharacteristic
461C22×C4C2×C22⋊D1711yescharacteristic
462C22×C4C2×C22⋊D1711yescharacteristic
463C22×C4C23×D1711yescharacteristic
464SD16—348no—
465C2×C8—172no—
466C2×C8—172no—
467C2×C8—341no—
468C2×C8—341no—
469C2×C8—341no—
470C2×C8—341no—
471C2×C8—341no—
472C2×C8—341no—
473C2×C8—341no—
474C2×C8—341no—
475C2×C8—341no—
476C2×C8—341no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—81no—
483C4⋊C4—81no—
484C4⋊C4—172no—
485C4⋊C4—172no—
486C4⋊C4—172no—
487C4⋊C4—344no—
488C4⋊C4—344no—
489C4⋊C4—341no—
490C4⋊C4—341no—
491C4⋊C4—341no—
492C4⋊C4—344no—
493C4⋊C4C2×C4×D1712yes—
494C22⋊C4—24no—
495C22⋊C4—81no—
496C22⋊C4—81no—
497C22⋊C4—81no—
498C22⋊C4—81no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—81no—
504C42—344no—
505C42—344no—
506C42—344no—
507C42—1361no—
508C42—1361no—
509C42—1361no—
510C42C2×C4×D1712yes—
511C42#6325212yes—
512C23—22no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23C22×C22⋊D1711yescharacteristic
520Q8—174no—
521Q8—342no—
522Q8—342no—
523Q8—342no—
524D4—22no—
525D4—22no—
526D4—41no—
527D4—41no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—84no—
554C2×C4—342no—
555C2×C4—342no—
556C2×C4—342no—
557C2×C4—342no—
558C2×C4—342no—
559C2×C4—342no—
560C2×C4—681no—
561C2×C4—681no—
562C2×C4—681no—
563C2×C4—681no—
564C2×C4—681no—
565C2×C4—681no—
566C2×C4C4×C22⋊D1712yes—
567C2×C4C4×C22⋊D1712yes—
568C2×C4C22×C4×D1711yescharacteristic
569C2×C4C2×#6325211yescharacteristic
570C2×C4#41×D1711yescharacteristic
571C2×C4#41×D1711yescharacteristic
572C2×C4#41×D1711yescharacteristic
573C2×C4#41×D1711yescharacteristic
574C8—342no—
575C8—342no—
576C8—681no—
577C8—681no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22C2×C4×C22⋊D1711yescharacteristic
586C22#1176331611yescharacteristic
587C22#1176331611yescharacteristic
588C22#1176361211yescharacteristic
589C22#1176362011yescharacteristic
590C22#1176362011yescharacteristic
591C22#1176362411yescharacteristic
592C4—22no—
593C4—22no—
594C4—41no—
595C4—41no—
596C4—42no—
597C4—41no—
598C4—41no—
599C4—81no—
600C4—81no—
601C4—344no—
602C4—682no—
603C4—682no—
604C4—682no—
605C4#117×D1711yescharacteristic
606C4#1176327711yescharacteristic
607C4#1176333711yescharacteristic
608C4#1176333711yescharacteristic
609C2—82no—
610C2#1453122111yescharacteristic
611C2#1453761611yescharacteristic
612C2#1453762311yescharacteristic
613C2#1453897711yescharacteristic
614C2#1453900711yescharacteristic
615C2#1453904011yescharacteristic
616C2#1453907011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2319C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenter#2319D17
hypocenterC17#21128
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ #376 ◁ #2319

Complements

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

NH ≅ G/Ntype
2. #14531147C2semidirect
5. #14529373C2semidirect
6. #14529352C2semidirect
9. #14529051C2semidirect
10. #14529037C2semidirect
11. #14528682C2semidirect
12. C4×#10587481C2semidirect
14. #14528536C2semidirect
20. #11762909C4semidirect
433. C17#21128semidirect

Maximal subgroups

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

subgroupindex
C2×#117629092
#145311472
#145294082
#145294012
#145293732
#145293522
#145291422
#145290512
#145290372
#145286822
C4×#105874812
C4×#105875032
#145285362
C17×#23192
#2112817