Finite Group Databasebeta

← #17037442

Subgroups — #17037442

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×#11762910C211yesmaximal, characteristic
2#14531421C211yesmaximal, characteristic
3#14531149C211yesmaximal, characteristic, complemented
4#14529408C211yesmaximal, characteristic
5#14529407C211yesmaximal, characteristic
6#14529375C211yesmaximal, characteristic, complemented
7#14529356C211yesmaximal, characteristic, complemented
8#14529141C211yesmaximal, characteristic
9#14529050C211yesmaximal, characteristic, complemented
10#14529038C211yesmaximal, characteristic, complemented
11#14528680C211yesmaximal, characteristic, complemented
12#103×Dic17C211yesmaximal, characteristic, complemented
13#102×Dic17C211yesmaximal, characteristic
14#14528534C211yesmaximal, characteristic, complemented
15C17×#1880C211yesmaximal, characteristic
16C2×Q8×Dic17C2211yescharacteristic
17C2×Q8×Dic17C2211yescharacteristic
18C2×D4×Dic17C2211yescharacteristic
19#11762910C418yescomplemented
20C2×#10587489C2211yescharacteristic
21C2×#10587478C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×#10587506C2211yescharacteristic
24C2×#10587506C2211yescharacteristic
25C2×#10587503C2211yescharacteristic
26C2×#10587503C2211yescharacteristic
27C2×#10587486C2211yescharacteristic
28C2×C8×Dic17C2211yescharacteristic
29#11762712C2211yescharacteristic
30#11762707C2211yescharacteristic
31#11762705C2211yescharacteristic
32#11762703—21no—
33#11762697—21no—
34#11762670—21no—
35#11762667—21no—
36C4⋊C4×Dic17—21no—
37C4⋊C4×Dic17—21no—
38C4⋊C4×Dic17C2211yescharacteristic
39C4⋊C4×Dic17C2211yescharacteristic
40C4⋊C4×Dic17C2211yescharacteristic
41C4×C4⋊Dic17—21no—
42C4×C4⋊Dic17C2211yescharacteristic
43#11762655—21no—
44#11762655C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762564C2211yescharacteristic
51#11762563C2211yescharacteristic
52#11762563C2211yescharacteristic
53C22×SD16×C17C2211yescharacteristic
54C2×C4×Q8×C17C2211yescharacteristic
55C2×C17×#103C2211yescharacteristic
56C2×C17×#102C2211yescharacteristic
57C17×#390C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#336C2211yescharacteristic
60C22×Q8×C17C2311yescharacteristic
61C22×D4×C17C2311yescharacteristic
62C23×Dic17—21no—
63C2×SD16×C17—22no—
64C2×SD16×C17—22no—
65C2×SD16×C17C2×C414yes—
66C22×C8×C17C2311yescharacteristic
67C22×C8×C17C2311yescharacteristic
68C4×Q8×C17—22no—
69C4×Q8×C17—22no—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C22⋊C4×C17—21no—
74C2×C22⋊C4×C17—21no—
75C2×C42×C17—21no—
76C2×C42×C17C2311yescharacteristic
77C2×C22⋊Dic17—21no—
78C2×C22⋊Dic17—21no—
79C2×C22⋊Dic17—21no—
80Q8×Dic17—22no—
81Q8×Dic17—22no—
82Q8×Dic17—22no—
83Q8×Dic17C2×C414yes—
84D4×Dic17—22no—
85D4×Dic17C2×C414yes—
86C2×C4⋊Dic17—21no—
87C2×C4⋊Dic17—21no—
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×#63225—21no—
93C2×#63225—21no—
94C2×C4×Dic17—21no—
95C2×C4×Dic17—21no—
96C2×C4×Dic17—21no—
97C2×C4×Dic17—21no—
98C2×C4×Dic17—21no—
99C2×C4×Dic17—21no—
100C2×C4×Dic17C2311yescharacteristic
101C2×C4×Dic17C2311yescharacteristic
102C22×#3622C2311yescharacteristic
103C22×#3622C2311yescharacteristic
104C17×#103—22no—
105C17×#102—22no—
106#10587506—22no—
107#10587506C2×C414yes—
108#10587503—22no—
109#10587503C2×C414yes—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502D411yescharacteristic
117#10587502D411yescharacteristic
118#10587502D411yescharacteristic
119#10587502D411yescharacteristic
120#10587489C2×C414yes—
121#10587486C2×C414yes—
122C8×Dic17—22no—
123#10587479—22no—
124#10587478—22no—
125C24×C17—21no—
126C2×Q8×C17—21no—
127C2×Q8×C17—21no—
128C2×Q8×C17—21no—
129C2×Q8×C17C22×C412yes—
130C2×Q8×C17#4112yes—
131C2×Q8×C17#4112yes—
132C2×D4×C17—21no—
133C2×D4×C17—21no—
134C2×D4×C17—21no—
135C2×D4×C17—21no—
136C2×D4×C17—21no—
137C2×D4×C17C22×C412yes—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2411yescharacteristic
144C22×Dic17—21no—
145C22×Dic17—21no—
146C22×Dic17—21no—
147C22×Dic17—24no—
148C22×Dic17—21no—
149C22×Dic17—81no—
150C22×Dic17—81no—
151C22×Dic17C2×D411yescharacteristic
152C22×Dic17C2×D411yescharacteristic
153C22×Dic17C2×D411yescharacteristic
154C22×Dic17C2×D411yescharacteristic
155SD16×C17—28no—
156C2×C8×C17—21no—
157C2×C8×C17—21no—
158C2×C8×C17—21no—
159C2×C8×C17—21no—
160C2×C8×C17—21no—
161C2×C8×C17C22×C412yes—
162C4⋊C4×C17—24no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17#4112yes—
167C22⋊C4×C17—81no—
168C22⋊C4×C17—81no—
169C42×C17—24no—
170C42×C17—21no—
171C42×C17—21no—
172C22⋊Dic17—24no—
173C22⋊Dic17—81no—
174C22⋊Dic17—81no—
175C4⋊Dic17—24no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—24no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17C22×C412yes—
183C4⋊Dic17C22×C412yes—
184C4⋊Dic17#4112yes—
185#63225—81no—
186#63225—81no—
187C4×Dic17—24no—
188C4×Dic17—21no—
189C4×Dic17—21no—
190C4×Dic17—24no—
191C4×Dic17—81no—
192C4×Dic17—81no—
193C4×Dic17—81no—
194C4×Dic17—81no—
195C4×Dic17C22×C412yes—
196C4×Dic17#4112yes—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622—21no—
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
212Q8×C17—22no—
213Q8×C17—22no—
214Q8×C17—22no—
215Q8×C17#11714yes—
216D4×C17—22no—
217D4×C17—22no—
218D4×C17—41no—
219D4×C17—41no—
220C2×C4×C17—22no—
221C2×C4×C17—22no—
222C2×C4×C17—22no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—22no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—41no—
231C2×C4×C17—41no—
232C2×C4×C17—41no—
233C2×C4×C17—41no—
234C2×C4×C17—41no—
235C2×C4×C17C23×C411yescharacteristic
236C2×C4×C17C2×#4111yescharacteristic
237C2×C4×C17C2×#4111yescharacteristic
238C2×C4×C17C2×#4111yescharacteristic
239C2×C4×C17C2×#4111yescharacteristic
240C2×C4×C17C2×#4111yescharacteristic
241C2×Dic17—21no—
242C2×Dic17—21no—
243C2×Dic17—21no—
244C2×Dic17—21no—
245C2×Dic17—22no—
246C2×Dic17—22no—
247C2×Dic17—22no—
248C2×Dic17—21no—
249C2×Dic17—21no—
250C2×Dic17—21no—
251C2×Dic17—21no—
252C2×Dic17—21no—
253C2×Dic17—21no—
254C2×Dic17—41no—
255C2×Dic17—41no—
256C2×Dic17—41no—
257C2×Dic17—41no—
258C2×Dic17—41no—
259C2×Dic17—41no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263C2×Dic17—84no—
264C2×Dic17C4×D412yes—
265C2×Dic17C4×D412yes—
266C8×C17—22no—
267C8×C17—41no—
268#3622—22no—
269#3622—41no—
270C2×#440—171no—
271#2319—171no—
272#2276—171no—
273#1926—171no—
274#1925—171no—
275#1921—171no—
276#1918—171no—
277#1880—171no—
278#1880D1711yescharacteristic
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—24no—
300C4×C17—41no—
301C4×C17—42no—
302C4×C17—41no—
303C4×C17—81no—
304C4×C17C2×#11711yescharacteristic
305C4×C17C4×#4111yescharacteristic
306C4×C17#52911yescharacteristic
307C4×C17#52911yescharacteristic
308Dic17—22no—
309Dic17—22no—
310Dic17—41no—
311Dic17—42no—
312Dic17—41no—
313Dic17—42no—
314Dic17—42no—
315Dic17—81no—
316C22×SD16C2×D1711yescharacteristic
317C2×C4×Q8—171no—
318C2×C4×Q8—171no—
319C2×C4×Q8C2×D1711yescharacteristic
320C2×C4×D4—171no—
321#440—178no—
322C2×#107—171no—
323C2×#107—171no—
324C2×#106—171no—
325C2×#103—171no—
326C2×#103—171no—
327C2×#103C2×D1711yescharacteristic
328C2×#102—171no—
329C2×#102—171no—
330C2×#102C2×D1711yescharacteristic
331C2×#97—171no—
332C2×C4×C8—171no—
333#390—171no—
334#390C2×D1711yescharacteristic
335#389—171no—
336#389—171no—
337#388—341no—
338#388—341no—
339#381—341no—
340#380—341no—
341C4×C4⋊C4—171no—
342C4×C4⋊C4—171no—
343C4×C4⋊C4—171no—
344C4×C4⋊C4—171no—
345C4×C4⋊C4—341no—
346C4×C4⋊C4—341no—
347C4×C4⋊C4—341no—
348#376—171no—
349#376—341no—
350#340—171no—
351#340—171no—
352#340—171no—
353#340—171no—
354#340—171no—
355#340C2×D1711yescharacteristic
356#336—171no—
357#336—171no—
358#336—171no—
359#336C2×D1711yescharacteristic
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×Q8C22×D1711yescharacteristic
369C22×D4C22×D1711yescharacteristic
370C23×C4—341no—
371C2×SD16—22no—
372C2×SD16—22no—
373C2×SD16C4×D1714yes—
374C22×C8—171no—
375C22×C8—171no—
376C22×C8C22×D1711yescharacteristic
377C22×C8C22×D1711yescharacteristic
378C4×Q8—22no—
379C4×Q8—22no—
380C4×Q8—174no—
381C4×Q8—342no—
382C4×Q8—342no—
383C4×Q8—342no—
384C4×D4—174no—
385C4×D4—342no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—171no—
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⋊C4—341no—
395C2×C4⋊C4C22×D1711yescharacteristic
396C2×C4⋊C4C22×D1711yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—341no—
400C2×C22⋊C4—341no—
401C2×C22⋊C4—341no—
402C2×C42—21no—
403C2×C42—171no—
404C2×C42—171no—
405C2×C42—341no—
406C2×C42—341no—
407C2×C42—341no—
408C2×C42—341no—
409C2×C42—341no—
410C2×C42—341no—
411C2×C42C22×D1711yescharacteristic
412#107—174no—
413#107—342no—
414#106—342no—
415#103—22no—
416#103—174no—
417#103—342no—
418#102—22no—
419#102—174no—
420#102—342no—
421#97—174no—
422C4×C8—342no—
423#95—171no—
424#95—171no—
425#95—171no—
426#95—171no—
427#95—341no—
428#95—341no—
429#95—341no—
430#95—341no—
431#95—341no—
432#95—341no—
433C17#2112811yescharacteristic, complemented, Sylow
434C24—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8C2×C4×D1712yes—
439C2×Q8#6325412yes—
440C2×Q8#6325412yes—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4C2×C4×D1712yes—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—171no—
451C22×C4—171no—
452C22×C4—171no—
453C22×C4—171no—
454C22×C4—344no—
455C22×C4—341no—
456C22×C4—341no—
457C22×C4—341no—
458C22×C4—341no—
459C22×C4—1361no—
460C22×C4—1361no—
461C22×C4D4×D1711yescharacteristic
462C22×C4D4×D1711yescharacteristic
463C22×C4C23×D1711yescharacteristic
464SD16—28no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—172no—
471C2×C8—341no—
472C2×C8—341no—
473C2×C8—341no—
474C2×C8—341no—
475C2×C8—341no—
476C2×C8C2×C4×D1712yes—
477C4⋊C4—24no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—172no—
482C4⋊C4—172no—
483C4⋊C4—172no—
484C4⋊C4—344no—
485C4⋊C4—344no—
486C4⋊C4—341no—
487C4⋊C4—341no—
488C4⋊C4—341no—
489C4⋊C4—341no—
490C4⋊C4—341no—
491C4⋊C4—1361no—
492C4⋊C4—1361no—
493C4⋊C4#6325412yes—
494C22⋊C4—81no—
495C22⋊C4—81no—
496C22⋊C4—344no—
497C22⋊C4—1361no—
498C22⋊C4—1361no—
499C42—24no—
500C42—21no—
501C42—21no—
502C42—172no—
503C42—172no—
504C42—344no—
505C42—344no—
506C42—341no—
507C42—341no—
508C42—1361no—
509C42—1361no—
510C42—1361no—
511C42—1361no—
512C23—22no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23C2×D4×D1711yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—22no—
523Q8#1058756314yes—
524D4—22no—
525D4—22no—
526D4—41no—
527D4—41no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—172no—
544C2×C4—172no—
545C2×C4—341no—
546C2×C4—341no—
547C2×C4—341no—
548C2×C4—341no—
549C2×C4—342no—
550C2×C4—342no—
551C2×C4—341no—
552C2×C4—342no—
553C2×C4—341no—
554C2×C4—341no—
555C2×C4—341no—
556C2×C4—341no—
557C2×C4—341no—
558C2×C4—681no—
559C2×C4—681no—
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×C4C22×C4×D1711yescharacteristic
569C2×C4C2×#6325411yescharacteristic
570C2×C4C2×#6325411yescharacteristic
571C2×C4C2×#6325411yescharacteristic
572C2×C4#41×D1711yescharacteristic
573C2×C4#41×D1711yescharacteristic
574C8—22no—
575C8—41no—
576C8—342no—
577C8—681no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22C4×D4×D1711yescharacteristic
586C22#1176331511yescharacteristic
587C22#1176331911yescharacteristic
588C22#1176361111yescharacteristic
589C22#1176361511yescharacteristic
590C22#1176361911yescharacteristic
591C22#1176362511yescharacteristic
592C4—24no—
593C4—41no—
594C4—42no—
595C4—41no—
596C4—81no—
597C4—342no—
598C4—342no—
599C4—681no—
600C4—682no—
601C4—681no—
602C4—682no—
603C4—682no—
604C4—1361no—
605C4C2×#1058756311yescharacteristic
606C4C4×#6325411yescharacteristic
607C4#1176333611yescharacteristic
608C4#1176334011yescharacteristic
609C2—82no—
610C2#1453122411yescharacteristic
611C2#1453761111yescharacteristic
612C2#1453762411yescharacteristic
613C2#1453897611yescharacteristic
614C2#1453900611yescharacteristic
615C2#1453904511yescharacteristic
616C2#1453907311yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1880C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenter#1880D17
hypocenterC17#21128
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1880

Complements

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

NH ≅ G/Ntype
3. #14531149C2semidirect
6. #14529375C2semidirect
7. #14529356C2semidirect
9. #14529050C2semidirect
10. #14529038C2semidirect
11. #14528680C2semidirect
12. #103×Dic17C2semidirect
14. #14528534C2semidirect
19. #11762910C4semidirect
433. C17#21128semidirect

Maximal subgroups

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

subgroupindex
C2×#117629102
#145314212
#145311492
#145294082
#145294072
#145293752
#145293562
#145291412
#145290502
#145290382
#145286802
#103×Dic172
#102×Dic172
#145285342
C17×#18802
#2112817