Finite Group Databasebeta

← #17042091

Subgroups — #17042091

Order 4352.

876 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763097C211yesmaximal, characteristic
2#14531531C211yesmaximal, characteristic, complemented
3#14531145C211yesmaximal, characteristic
4#14529734C211yesmaximal, characteristic, complemented
5#14529549C211yesmaximal, characteristic, complemented
6#14529366C211yesmaximal, characteristic, complemented
7#14529363C211yesmaximal, characteristic, complemented
8#14529045C211yesmaximal, characteristic
9#14528792C211yesmaximal, characteristic
10#14528683C211yesmaximal, characteristic, complemented
11#14528672C211yesmaximal, characteristic, complemented
12#14528640C211yesmaximal, characteristic, complemented
13#14528611C211yesmaximal, characteristic
14#102×Dic17C211yesmaximal, characteristic
15C17×#1862C211yesmaximal, characteristic
16C2×#10587574C2211yescharacteristic
17C2×D4×Dic17C2211yescharacteristic
18C2×D4×Dic17C2211yescharacteristic
19#11763097C418yescomplemented
20C2×#10587488C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×#10587485C2211yescharacteristic
24C2×#10587485C2211yescharacteristic
25C2×#10587503C2211yescharacteristic
26C2×#10587503C2211yescharacteristic
27C2×#10587486C2211yescharacteristic
28C2×C8×Dic17C2211yescharacteristic
29#11762714—21no—
30#11762713—21no—
31#11762707C2211yescharacteristic
32#11762691—21no—
33#11762690—21no—
34#11762674C2211yescharacteristic
35C4⋊C4×Dic17C2211yescharacteristic
36C4⋊C4×Dic17C2211yescharacteristic
37C4⋊C4×Dic17C2211yescharacteristic
38C4⋊C4×Dic17C2211yescharacteristic
39C4×C4⋊Dic17C2211yescharacteristic
40C22⋊C4×Dic17—21no—
41C22⋊C4×Dic17—21no—
42C4×C22⋊Dic17—21no—
43#11762655C2211yescharacteristic
44C2×#10587502—21no—
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762564C2211yescharacteristic
53C2×C4×D4×C17C2211yescharacteristic
54C2×C17×#107C2211yescharacteristic
55C2×C17×#102C2211yescharacteristic
56C2×C17×#102C2211yescharacteristic
57C17×#389C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#336C2211yescharacteristic
60C22×D4×C17C2311yescharacteristic
61C23×C4×C17—21no—
62C23×Dic17—21no—
63C23×Dic17—21no—
64C22×Dic34—21no—
65C22×C8×C17C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C4×D4×C17—22no—
68C4×D4×C17—22no—
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C22⋊C4×C17—21no—
74C2×C42×C17C2311yescharacteristic
75C2×C22⋊Dic17—21no—
76C2×C22⋊Dic17—21no—
77D4×Dic17—22no—
78D4×Dic17—22no—
79D4×Dic17C2×C414yes—
80D4×Dic17Q814yescomplemented
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×#63225—21no—
86C2×#63225—21no—
87C2×C4×Dic17—21no—
88C2×C4×Dic17—21no—
89C2×C4×Dic17—21no—
90C2×C4×Dic17—21no—
91C2×C4×Dic17—21no—
92C2×C4×Dic17C2311yescharacteristic
93C2×C4×Dic17C2311yescharacteristic
94C22×#3622C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96#10587574—22no—
97#10587574C2×C414yescomplemented
98C17×#107—22no—
99C17×#102—22no—
100C17×#102C2×C414yes—
101C17×#95—21no—
102#10587503—22no—
103#10587503C2×C414yes—
104#10587502—21no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502D411yescharacteristic
113#10587502D411yescharacteristic
114#10587502D411yescharacteristic
115#10587502D411yescharacteristic
116#10587488C2×C414yescomplemented
117#10587486—22no—
118#10587485—22no—
119#10587485C2×C414yescomplemented
120C8×Dic17—22no—
121#10587479—22no—
122#10587479C2×C414yescomplemented
123C24×C17—21no—
124C2×D4×C17—21no—
125C2×D4×C17—21no—
126C2×D4×C17—21no—
127C2×D4×C17C22×C412yes—
128C2×D4×C17C2×Q812yes—
129C2×D4×C17#4112yes—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17—24no—
133C22×C4×C17—21no—
134C22×C4×C17—81no—
135C22×C4×C17—81no—
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2411yescharacteristic
139C22×Dic17—21no—
140C22×Dic17—21no—
141C22×Dic17—21no—
142C22×Dic17—24no—
143C22×Dic17—24no—
144C22×Dic17—21no—
145C22×Dic17—81no—
146C22×Dic17—81no—
147C22×Dic17—81no—
148C22×Dic17—81no—
149C22×Dic17C2×D411yescharacteristic
150C22×Dic17C2×D411yescharacteristic
151C22×Dic17C2×D411yescharacteristic
152C22×Dic17C2×D411yescharacteristic
153C2×Dic34—24no—
154C2×Dic34—81no—
155C2×Dic34—81no—
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—21no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—81no—
167C4⋊C4×C17C22×C412yes—
168C4⋊C4×C17#4112yes—
169C22⋊C4×C17—24no—
170C42×C17—21no—
171C42×C17—21no—
172C22⋊Dic17—24no—
173C22⋊Dic17—24no—
174C4⋊Dic17—21no—
175C4⋊Dic17—21no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17C22×C412yes—
180C4⋊Dic17C22×C412yes—
181C4⋊Dic17C2×Q812yes—
182#63225—24no—
183#63225—81no—
184C4×Dic17—81no—
185C4×Dic17—81no—
186C4×Dic17—81no—
187C4×Dic17—81no—
188C4×Dic17—81no—
189C4×Dic17C22×C412yes—
190C4×Dic17C2×Q812yes—
191C4×Dic17#4112yes—
192C4×Dic17#4112yes—
193C2×#3622—21no—
194C2×#3622—21no—
195C2×#3622—21no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622C22×C412yes—
199#21623—171nomaximal, Sylow
200C23×C17—22no—
201C23×C17—22no—
202C23×C17—22no—
203C23×C17—41no—
204C23×C17—41no—
205C23×C17—41no—
206C23×C17—41no—
207C23×C17C22×D411yescharacteristic
208D4×C17—22no—
209D4×C17—22no—
210D4×C17—22no—
211D4×C17C4×Q814yes—
212C2×C4×C17—22no—
213C2×C4×C17—21no—
214C2×C4×C17—21no—
215C2×C4×C17—22no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—41no—
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—84no—
228C2×C4×C17C23×C411yescharacteristic
229C2×C4×C17C22×Q811yescharacteristic
230C2×C4×C17C2×#4111yescharacteristic
231C2×C4×C17C2×#4111yescharacteristic
232C2×C4×C17C2×#4111yescharacteristic
233C2×C4×C17C2×#4111yescharacteristic
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—21no—
239C2×Dic17—21no—
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—22no—
243C2×Dic17—22no—
244C2×Dic17—22no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17—84no—
255C2×Dic17—84no—
256C2×Dic17C4×D412yes—
257C2×Dic17C4×D412yes—
258C2×Dic17C22⋊Q812yes—
259C2×Dic17C22⋊Q812yes—
260Dic34—84no—
261C8×C17—22no—
262C8×C17—41no—
263#3622—22no—
264#3622—41no—
265C2×#476—171no—
266#2331—171no—
267#2275—171no—
268#1988—171no—
269#1949—171no—
270#1920—171no—
271#1919—171no—
272#1864—171no—
273#1862D1711yescharacteristic
274#1804—171no—
275#1776—171no—
276C4×#106—171no—
277#1768—171no—
278#1762—171no—
279C4×#102—171no—
280C22×C17—24no—
281C22×C17—42no—
282C22×C17—42no—
283C22×C17—42no—
284C22×C17—42no—
285C22×C17—42no—
286C22×C17—42no—
287C22×C17C2×C4×D411yescharacteristic
288C22×C17C2×C22⋊Q811yescharacteristic
289C22×C17#52511yescharacteristic
290C22×C17#57611yescharacteristic
291C22×C17#57711yescharacteristic
292C22×C17#57711yescharacteristic
293C22×C17#57711yescharacteristic
294C4×C17—41no—
295C4×C17—42no—
296C4×C17—41no—
297C4×C17—42no—
298C4×C17—81no—
299C4×C17C2×C4×Q811yescharacteristic
300C4×C17C4×#4111yescharacteristic
301C4×C17#52911yescharacteristic
302C4×C17#53311yescharacteristic
303Dic17—22no—
304Dic17—22no—
305Dic17—22no—
306Dic17—22no—
307Dic17—42no—
308Dic17—42no—
309Dic17—42no—
310Dic17—81no—
311C2×C4⋊Q8—171no—
312C2×C4×D4—171no—
313C2×C4×D4—171no—
314C2×C4×D4C2×D1711yescharacteristic
315#476—178no—
316C2×#107C2×D1711yescharacteristic
317C2×#106—171no—
318C2×#106—171no—
319C2×#106—171no—
320C2×C4⋊C8—171no—
321C2×C4⋊C8—171no—
322C2×#102—171no—
323C2×#102—171no—
324C2×#102C2×D1711yescharacteristic
325C2×#102C2×D1711yescharacteristic
326C2×#97—171no—
327C2×C4×C8—171no—
328#391—341no—
329#391—341no—
330#389—171no—
331#389C2×D1711yescharacteristic
332#387—341no—
333#387—341no—
334#382—171no—
335C4×C4⋊C4—171no—
336C4×C4⋊C4—171no—
337C4×C4⋊C4—171no—
338C4×C4⋊C4—171no—
339C4×C4⋊C4—171no—
340C4×C22⋊C4—341no—
341C4×C22⋊C4—341no—
342C4×C22⋊C4—341no—
343#376—171no—
344C2×#95—341no—
345#340—171no—
346#340—171no—
347#340—171no—
348#340—171no—
349#340—171no—
350#340—171no—
351#340—171no—
352#340C2×D1711yescharacteristic
353#336—171no—
354#336C2×D1711yescharacteristic
355C2×C17—44no—
356C2×C17C4×C22⋊Q811yescharacteristic
357C2×C17#294611yescharacteristic
358C2×C17#294911yescharacteristic
359C2×C17#308111yescharacteristic
360C2×C17#308411yescharacteristic
361C2×C17#311811yescharacteristic
362C2×C17#312511yescharacteristic
363C22×Q8—341no—
364C22×D4C22×D1711yescharacteristic
365C23×C4—21no—
366C23×C4—341no—
367C23×C4—341no—
368C22×C8—171no—
369C22×C8—171no—
370C22×C8C22×D1711yescharacteristic
371C22×C8C22×D1711yescharacteristic
372C4⋊Q8—174no—
373C4⋊Q8—342no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4—174no—
377C4×D4—174no—
378C4×D4—342no—
379C4×D4—342no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—171no—
382C2×C4⋊C4—171no—
383C2×C4⋊C4—171no—
384C2×C4⋊C4—171no—
385C2×C4⋊C4—341no—
386C2×C4⋊C4—341no—
387C2×C4⋊C4C22×D1711yescharacteristic
388C2×C4⋊C4C22×D1711yescharacteristic
389C2×C4⋊C4C22×D1711yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—341no—
392C2×C22⋊C4—341no—
393C2×C42—171no—
394C2×C42—171no—
395C2×C42—341no—
396C2×C42—341no—
397C2×C42—341no—
398C2×C42—341no—
399C2×C42—341no—
400C2×C42C22×D1711yescharacteristic
401#107—22no—
402#106—174no—
403#106—174no—
404#106—342no—
405C4⋊C8—174no—
406C4⋊C8—342no—
407#102—22no—
408#102—174no—
409#102—342no—
410#102C4×D1714yes—
411#97—342no—
412C4×C8—342no—
413#95—21no—
414#95—171no—
415#95—171no—
416#95—171no—
417#95—171no—
418#95—341no—
419#95—341no—
420#95—341no—
421#95—341no—
422#95—341no—
423#95—341no—
424#95—341no—
425#95—341no—
426C17#2162311yescharacteristic, complemented, Sylow
427C24—21no—
428C2×Q8—344no—
429C2×Q8—1361no—
430C2×Q8—1361no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4C2×C4×D1712yes—
435C2×D4#6325412yes—
436C2×D4Q8×D1712yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—171no—
444C22×C4—171no—
445C22×C4—171no—
446C22×C4—171no—
447C22×C4—344no—
448C22×C4—344no—
449C22×C4—341no—
450C22×C4—341no—
451C22×C4—341no—
452C22×C4—341no—
453C22×C4—1361no—
454C22×C4—1361no—
455C22×C4—1361no—
456C22×C4—1361no—
457C22×C4D4×D1711yescharacteristic
458C22×C4D4×D1711yescharacteristic
459C22×C4C23×D1711yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—172no—
466C2×C8—341no—
467C2×C8—341no—
468C2×C8—341no—
469C2×C8—341no—
470C2×C8—341no—
471C2×C8C2×C4×D1712yes—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—81no—
477C4⋊C4—172no—
478C4⋊C4—172no—
479C4⋊C4—172no—
480C4⋊C4—344no—
481C4⋊C4—341no—
482C4⋊C4—341no—
483C4⋊C4—341no—
484C4⋊C4—341no—
485C4⋊C4—341no—
486C4⋊C4—1361no—
487C4⋊C4C2×C4×D1712yes—
488C4⋊C4#6325412yes—
489C22⋊C4—24no—
490C22⋊C4—344no—
491C22⋊C4—344no—
492C42—21no—
493C42—21no—
494C42—172no—
495C42—172no—
496C42—172no—
497C42—172no—
498C42—1361no—
499C42—1361no—
500C42—1361no—
501C42—1361no—
502C42—1361no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23C2×D4×D1711yescharacteristic
511Q8—1364no—
512D4—22no—
513D4—22no—
514D4—22no—
515D4#1058757314yes—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—84no—
532C2×C4—172no—
533C2×C4—172no—
534C2×C4—172no—
535C2×C4—172no—
536C2×C4—341no—
537C2×C4—341no—
538C2×C4—341no—
539C2×C4—341no—
540C2×C4—341no—
541C2×C4—341no—
542C2×C4—341no—
543C2×C4—341no—
544C2×C4—342no—
545C2×C4—342no—
546C2×C4—342no—
547C2×C4—681no—
548C2×C4—681no—
549C2×C4—681no—
550C2×C4—681no—
551C2×C4—681no—
552C2×C4—681no—
553C2×C4—681no—
554C2×C4—681no—
555C2×C4—681no—
556C2×C4—1364no—
557C2×C4—1364no—
558C2×C4C22×C4×D1711yescharacteristic
559C2×C4C2×#6325411yescharacteristic
560C2×C4C2×#6325411yescharacteristic
561C2×C4C2×Q8×D1711yescharacteristic
562C2×C4#41×D1711yescharacteristic
563C2×C4#41×D1711yescharacteristic
564C8—22no—
565C8—41no—
566C8—342no—
567C8—681no—
568C22—24no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22C4×D4×D1711yescharacteristic
576C22C22⋊Q8×D1711yescharacteristic
577C22#1176331511yescharacteristic
578C22#1176361011yescharacteristic
579C22#1176361611yescharacteristic
580C22#1176361811yescharacteristic
581C22#1176361911yescharacteristic
582C4—41no—
583C4—42no—
584C4—41no—
585C4—42no—
586C4—81no—
587C4—342no—
588C4—342no—
589C4—342no—
590C4—342no—
591C4—682no—
592C4—682no—
593C4—682no—
594C4—1361no—
595C4C2×#1058757311yescharacteristic
596C4C4×#6325411yescharacteristic
597C4#1176333611yescharacteristic
598C4#1176336111yescharacteristic
599C2—44no—
600C2#1453131611yescharacteristic
601C2#1453760911yescharacteristic
602C2#1453764011yescharacteristic
603C2#1453860811yescharacteristic
604C2#1453864711yescharacteristic
605C2#1453896111yescharacteristic
606C2#1453905411yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1862C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenter#1862D17
hypocenterC17#21623
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1862

Complements

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

NH ≅ G/Ntype
2. #14531531C2semidirect
4. #14529734C2semidirect
5. #14529549C2semidirect
6. #14529366C2semidirect
7. #14529363C2semidirect
10. #14528683C2semidirect
11. #14528672C2semidirect
12. #14528640C2semidirect
19. #11763097C4semidirect
80. D4×Dic17Q8semidirect
97. #10587574C2×C4semidirect
116. #10587488C2×C4semidirect
119. #10587485C2×C4semidirect
122. #10587479C2×C4semidirect
426. C17#21623semidirect

Maximal subgroups

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

subgroupindex
C2×#117630972
#145315312
#145311452
#145297342
#145295492
#145293662
#145293632
#145290452
#145287922
#145286832
#145286722
#145286402
#145286112
#102×Dic172
C17×#18622
#2162317