Finite Group Databasebeta

← #17056173

Subgroups — #17056173

Order 4352.

846 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531884C211yesmaximal, characteristic, complemented
2#14531694C211yesmaximal, characteristic
3#14529750C211yesmaximal, characteristic, complemented
4#14529554C211yesmaximal, characteristic, complemented
5#14529518C211yesmaximal, characteristic, complemented
6#14529513C211yesmaximal, characteristic, complemented
7#14529378C211yesmaximal, characteristic, complemented
8#14529371C211yesmaximal, characteristic, complemented
9#14529047C211yesmaximal, characteristic
10#14529033C211yesmaximal, characteristic
11#14528649C211yesmaximal, characteristic, complemented
12#102×Dic17C211yesmaximal, characteristic
13#102×Dic17C211yesmaximal, characteristic
14C4×#10587503C211yesmaximal, characteristic
15C2×C17×#478C211yesmaximal, characteristic
16C2×D4×Dic17C2211yescharacteristic
17C2×D4×Dic17C2211yescharacteristic
18C2×#10587489C2211yescharacteristic
19C2×#10587488C2211yescharacteristic
20C2×#10587503C2211yescharacteristic
21C2×#10587503C2211yescharacteristic
22C2×C4×#3622C2211yescharacteristic
23C2×C8×Dic17C2211yescharacteristic
24C2×C8×Dic17C2211yescharacteristic
25#11762708—21no—
26#11762708C2211yescharacteristic
27#11762708C2211yescharacteristic
28#11762708C2211yescharacteristic
29#11762708C2211yescharacteristic
30#11762695—21no—
31#11762695—21no—
32#11762684—21no—
33#11762684—21no—
34#11762675C2211yescharacteristic
35#11762669—21no—
36C4⋊C4×Dic17—21no—
37C4⋊C4×Dic17C2211yescharacteristic
38C4⋊C4×Dic17C2211yescharacteristic
39C4×C4⋊Dic17C2211yescharacteristic
40C4×C4⋊Dic17C2211yescharacteristic
41C2×#10587502—21no—
42#11762572C2211yescharacteristic
43#11762572C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C2×C17×#125C2211yescharacteristic
53C2×C4×D4×C17C2211yescharacteristic
54C17×#478C418yescomplemented
55C2×C17×#107C2211yescharacteristic
56C2×C17×#106C2211yescharacteristic
57C2×C17×C4⋊C8C2211yescharacteristic
58C2×C17×#102C2211yescharacteristic
59C2×C17×#102C2211yescharacteristic
60C22×D4×C17C2311yescharacteristic
61C23×C4×C17—21no—
62C23×Dic17—21no—
63C23×Dic17—21no—
64C22×C8×C17C2311yescharacteristic
65C22×C8×C17C2311yescharacteristic
66C17×#125—22no—
67C17×#125C2×C414yescomplemented
68C4×D4×C17—22no—
69C4×D4×C17C2×C414yes—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C22⋊C4×C17—21no—
76C2×C42×C17C2311yescharacteristic
77C2×C22⋊Dic17—21no—
78C2×C22⋊Dic17—21no—
79D4×Dic17—22no—
80D4×Dic17—22no—
81D4×Dic17—41no—
82D4×Dic17—41no—
83C2×C4⋊Dic17—21no—
84C2×C4⋊Dic17—21no—
85C2×C4⋊Dic17—21no—
86C2×C4⋊Dic17—21no—
87C2×C4⋊Dic17D411yescharacteristic
88C2×C4⋊Dic17D411yescharacteristic
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×C4×Dic17—21no—
94C2×C4×Dic17—21no—
95C2×C4×Dic17—21no—
96C2×C4×Dic17C2311yescharacteristic
97C2×C4×Dic17C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99C22×#3622C2311yescharacteristic
100C17×#107C2×C414yescomplemented
101C17×#106C2×C414yescomplemented
102C17×C4⋊C8C2×C414yescomplemented
103C17×#102C2×C414yes—
104C17×#102C2×C414yes—
105#10587503—22no—
106#10587503Q814yes—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502D411yescharacteristic
117#10587502D411yescharacteristic
118#10587489—41no—
119#10587488—41no—
120C8×Dic17—41no—
121C8×Dic17—41no—
122C4×#3622—22no—
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—21no—
133C22×C4×C17—24no—
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—21no—
143C22×Dic17—42no—
144C22×Dic17—42no—
145C22×Dic17—42no—
146C22×Dic17—42no—
147C22×Dic17—81no—
148C22×Dic17—81no—
149C22×Dic17C2×D411yescharacteristic
150C22×Dic17C2×D411yescharacteristic
151C22×Dic17C2×D411yescharacteristic
152C22×Dic17C2×D411yescharacteristic
153C2×C8×C17—21no—
154C2×C8×C17—21no—
155C2×C8×C17—21no—
156C2×C8×C17—21no—
157C2×C8×C17C22×C412yes—
158C2×C8×C17C22×C412yes—
159C4⋊C4×C17—24no—
160C4⋊C4×C17—24no—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17C22×C412yes—
165C4⋊C4×C17C22×C412yes—
166C4⋊C4×C17C22×C412yes—
167C22⋊C4×C17—24no—
168C42×C17C22×C412yes—
169C42×C17#4112yes—
170C22⋊Dic17—42no—
171C22⋊Dic17—42no—
172C4⋊Dic17—21no—
173C4⋊Dic17—22no—
174C4⋊Dic17—22no—
175C4⋊Dic17—21no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—42no—
182C4⋊Dic17—81no—
183C4⋊Dic17—81no—
184C4⋊Dic17—81no—
185C4⋊Dic17C2×Q812yes—
186C4×Dic17—21no—
187C4×Dic17—21no—
188C4×Dic17—21no—
189C4×Dic17—21no—
190C4×Dic17—42no—
191C4×Dic17—81no—
192C4×Dic17—81no—
193C2×#3622—21no—
194C2×#3622—21no—
195C2×#3622—21no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622C2×Q812yes—
199#22966—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—21no—
213C2×C4×C17—21no—
214C2×C4×C17—21no—
215C2×C4×C17—21no—
216C2×C4×C17—22no—
217C2×C4×C17—22no—
218C2×C4×C17—22no—
219C2×C4×C17—41no—
220C2×C4×C17—41no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17—84no—
226C2×C4×C17C4×D412yes—
227C2×C4×C17C4×D412yes—
228C2×C4×C17C23×C411yescharacteristic
229C2×C4×C17C22×Q811yescharacteristic
230C2×C4×C17C2×#4111yescharacteristic
231C2×C4×C17C2×#4111yescharacteristic
232C2×C4×C17H(5,𝔽2)11yescharacteristic
233C2×C4×C172−1+411yescharacteristic
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—21no—
239C2×Dic17—21no—
240C2×Dic17—22no—
241C2×Dic17—21no—
242C2×Dic17—21no—
243C2×Dic17—21no—
244C2×Dic17—21no—
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—41no—
255C2×Dic17—41no—
256C2×Dic17—84no—
257C2×Dic17—84no—
258C2×Dic17#13512yes—
259C2×Dic17#13512yes—
260C8×C17—22no—
261C8×C17—22no—
262#3622—22no—
263#3622—41no—
264C2×#478D1711yescharacteristic
265#2370—171no—
266#2349—171no—
267#1991—171no—
268#1949—171no—
269#1942—171no—
270#1941—171no—
271#1921—171no—
272#1920—171no—
273#1864—171no—
274#1862—171no—
275C8×C4⋊C4—171no—
276C4×#102—171no—
277C4×#102—171no—
278C4×#102—171no—
279C22×C17—24no—
280C22×C17—42no—
281C22×C17—42no—
282C22×C17—42no—
283C22×C17—42no—
284C22×C17—42no—
285C22×C17—42no—
286C22×C17C2×C4×D411yescharacteristic
287C22×C17#54611yescharacteristic
288C22×C17D4×Q811yescharacteristic
289C22×C17C2×#13511yescharacteristic
290C22×C17C2×#13511yescharacteristic
291C22×C17#57611yescharacteristic
292C22×C17#57711yescharacteristic
293C4×C17—22no—
294C4×C17—22no—
295C4×C17—42no—
296C4×C17—42no—
297C4×C17—42no—
298C4×C17C2×C4×Q811yescharacteristic
299C4×C17#52011yescharacteristic
300C4×C17#55211yescharacteristic
301C4×C17#55411yescharacteristic
302Dic17—22no—
303Dic17—22no—
304Dic17—42no—
305Dic17—41no—
306Dic17—41no—
307Dic17—81no—
308Dic17—81no—
309Dic17—81no—
310C2×#125C2×D1711yescharacteristic
311C2×C4×D4—171no—
312C2×C4×D4—171no—
313C2×C4×D4C2×D1711yescharacteristic
314#478Dic1718yescomplemented
315C2×#107—171no—
316C2×#107C2×D1711yescharacteristic
317C2×#106—171no—
318C2×#106C2×D1711yescharacteristic
319C2×C4⋊C8C2×D1711yescharacteristic
320C2×#102—171no—
321C2×#102—171no—
322C2×#102C2×D1711yescharacteristic
323C2×#102C2×D1711yescharacteristic
324C2×C4×C8—171no—
325C2×C4×C8—171no—
326C2×C4×C8—171no—
327#389—171no—
328#389—171no—
329#389—171no—
330#389—171no—
331#389—341no—
332#387—341no—
333#387—341no—
334#385—341no—
335#385—341no—
336#382—171no—
337#380—341no—
338C4×C4⋊C4—171no—
339C4×C4⋊C4—171no—
340C4×C4⋊C4—171no—
341C4×C4⋊C4—171no—
342C4×C4⋊C4—341no—
343C2×#95—341no—
344#340—171no—
345#340—171no—
346#340—171no—
347#340—171no—
348#340—171no—
349#340—171no—
350#340—171no—
351#340—171no—
352#336—171no—
353#336—171no—
354C2×C17—44no—
355C2×C17#236011yescharacteristic
356C2×C17C4×#13511yescharacteristic
357C2×C17#294611yescharacteristic
358C2×C17#329611yescharacteristic
359C2×C17#329711yescharacteristic
360C2×C17#338011yescharacteristic
361C2×C17#338111yescharacteristic
362C22×D4C22×D1711yescharacteristic
363C23×C4—21no—
364C23×C4—341no—
365C23×C4—341no—
366C22×C8—171no—
367C22×C8—171no—
368C22×C8C22×D1711yescharacteristic
369C22×C8C22×D1711yescharacteristic
370#125—22no—
371#125C2×Dic1714yescomplemented
372C4×D4—22no—
373C4×D4—342no—
374C4×D4—342no—
375C4×D4—681no—
376C4×D4—681no—
377C4×D4C2×Dic1714yes—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—171no—
381C2×C4⋊C4—171no—
382C2×C4⋊C4—171no—
383C2×C4⋊C4—171no—
384C2×C4⋊C4—171no—
385C2×C4⋊C4—171no—
386C2×C4⋊C4—341no—
387C2×C4⋊C4—341no—
388C2×C4⋊C4—341no—
389C2×C4⋊C4—341no—
390C2×C4⋊C4C22×D1711yescharacteristic
391C2×C4⋊C4C22×D1711yescharacteristic
392C2×C4⋊C4C22×D1711yescharacteristic
393C2×C22⋊C4—21no—
394C2×C22⋊C4—341no—
395C2×C22⋊C4—341no—
396C2×C42—171no—
397C2×C42—171no—
398C2×C42—341no—
399C2×C42—341no—
400C2×C42—341no—
401C2×C42C22×D1711yescharacteristic
402#107—681no—
403#107C2×Dic1714yescomplemented
404#106—681no—
405#106C2×Dic1714yescomplemented
406C4⋊C8C2×Dic1714yescomplemented
407#102—174no—
408#102—342no—
409#102C2×Dic1714yes—
410#102C2×Dic1714yes—
411C4×C8—342no—
412C4×C8—681no—
413C4×C8—681no—
414#95—171no—
415#95—171no—
416#95—341no—
417#95—341no—
418#95—341no—
419#95—341no—
420#95—341no—
421#95—341no—
422#95—341no—
423#95—341no—
424#95—341no—
425C17#2296611yescharacteristic, complemented, Sylow
426C24—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4Q8×D1712yes—
431C2×D4#6325612yes—
432C2×D4C22×Dic1712yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—81no—
438C22×C4—81no—
439C22×C4—171no—
440C22×C4—171no—
441C22×C4—171no—
442C22×C4—171no—
443C22×C4—341no—
444C22×C4—341no—
445C22×C4—341no—
446C22×C4—341no—
447C22×C4—682no—
448C22×C4—682no—
449C22×C4—682no—
450C22×C4—682no—
451C22×C4—1361no—
452C22×C4—1361no—
453C22×C4D4×D1711yescharacteristic
454C22×C4D4×D1711yescharacteristic
455C22×C4C23×D1711yescharacteristic
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—172no—
461C2×C8—341no—
462C2×C8—341no—
463C2×C8—341no—
464C2×C8—341no—
465C2×C8—341no—
466C2×C8C22×Dic1712yes—
467C2×C8C22×Dic1712yes—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—172no—
474C4⋊C4—342no—
475C4⋊C4—342no—
476C4⋊C4—341no—
477C4⋊C4—341no—
478C4⋊C4—341no—
479C4⋊C4—341no—
480C4⋊C4—341no—
481C4⋊C4—341no—
482C4⋊C4—341no—
483C4⋊C4—682no—
484C4⋊C4—1361no—
485C4⋊C4—1361no—
486C4⋊C4—1361no—
487C4⋊C4C22×Dic1712yes—
488C4⋊C4C22×Dic1712yes—
489C4⋊C4C22×Dic1712yes—
490C22⋊C4—24no—
491C22⋊C4—682no—
492C22⋊C4—682no—
493C42—341no—
494C42—341no—
495C42—341no—
496C42—341no—
497C42—682no—
498C42—1361no—
499C42—1361no—
500C42#6325412yes—
501C42C22×Dic1712yes—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23C2×D4×D1711yescharacteristic
510D4—22no—
511D4—22no—
512D4—22no—
513D4Q8×Dic1714yes—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—84no—
528C2×C4—172no—
529C2×C4—172no—
530C2×C4—341no—
531C2×C4—341no—
532C2×C4—341no—
533C2×C4—341no—
534C2×C4—342no—
535C2×C4—341no—
536C2×C4—341no—
537C2×C4—341no—
538C2×C4—341no—
539C2×C4—341no—
540C2×C4—341no—
541C2×C4—681no—
542C2×C4—681no—
543C2×C4—681no—
544C2×C4—681no—
545C2×C4—681no—
546C2×C4—681no—
547C2×C4—681no—
548C2×C4—681no—
549C2×C4—681no—
550C2×C4—681no—
551C2×C4—681no—
552C2×C4—1364no—
553C2×C4—1364no—
554C2×C4D4×Dic1712yes—
555C2×C4D4×Dic1712yes—
556C2×C4C2×#6325411yescharacteristic
557C2×C4C2×Q8×D1711yescharacteristic
558C2×C4C2×#6325611yescharacteristic
559C2×C4#1058769611yescharacteristic
560C2×C4#1058769711yescharacteristic
561C2×C4C23×Dic1711yescharacteristic
562C8—22no—
563C8—22no—
564C8—342no—
565C8—681no—
566C22—24no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22C2×D4×Dic1711yescharacteristic
574C22#1176345011yescharacteristic
575C22#1176347211yescharacteristic
576C22#135×D1711yescharacteristic
577C22#135×D1711yescharacteristic
578C22#1176361311yescharacteristic
579C22#1176361911yescharacteristic
580C4—22no—
581C4—22no—
582C4—42no—
583C4—42no—
584C4—42no—
585C4—342no—
586C4—342no—
587C4—681no—
588C4—682no—
589C4—681no—
590C4—1361no—
591C4—1361no—
592C4—1361no—
593C4C2×Q8×Dic1711yescharacteristic
594C4#1176329211yescharacteristic
595C4#1176350111yescharacteristic
596C4#1176351711yescharacteristic
597C2—44no—
598C2#1453179311yescharacteristic
599C2#135×Dic1711yescharacteristic
600C2#1453761711yescharacteristic
601C2#1454061611yescharacteristic
602C2#1454063111yescharacteristic
603C2#1454164611yescharacteristic
604C2#1454165411yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C17×#478C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenterC2×#478D17
hypocenterC17#22966
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
1. #14531884C2semidirect
3. #14529750C2semidirect
4. #14529554C2semidirect
5. #14529518C2semidirect
6. #14529513C2semidirect
7. #14529378C2semidirect
8. #14529371C2semidirect
11. #14528649C2semidirect
54. C17×#478C4semidirect
67. C17×#125C2×C4semidirect
100. C17×#107C2×C4semidirect
101. C17×#106C2×C4semidirect
102. C17×C4⋊C8C2×C4semidirect
314. #478Dic17semidirect
371. #125C2×Dic17semidirect
403. #107C2×Dic17semidirect
405. #106C2×Dic17semidirect
406. C4⋊C8C2×Dic17semidirect
425. C17#22966semidirect

Maximal subgroups

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

subgroupindex
#145318842
#145316942
#145297502
#145295542
#145295182
#145295132
#145293782
#145293712
#145290472
#145290332
#145286492
#102×Dic172
C4×#105875032
C2×C17×#4782
#2296617