Finite Group Databasebeta

← #17046575

Subgroups — #17046575

Order 4352.

732 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531526C211yesmaximal, characteristic, complemented
2#14531427C211yesmaximal, characteristic
3#14529742C211yesmaximal, characteristic, complemented
4#14529543C211yesmaximal, characteristic, complemented
5#14529528C211yesmaximal, characteristic, complemented
6#14529518C211yesmaximal, characteristic, complemented
7#14529483C211yesmaximal, characteristic
8#14529469C211yesmaximal, characteristic
9#14529406C211yesmaximal, characteristic
10#14529406C211yesmaximal, characteristic
11#14529378C211yesmaximal, characteristic, complemented
12#14529364C211yesmaximal, characteristic, complemented
13#14528784C211yesmaximal, characteristic
14#14528643C211yesmaximal, characteristic, complemented
15C2×C17×#478C211yesmaximal, characteristic
16C2×#10587489C2211yescharacteristic
17C2×#10587489C2211yescharacteristic
18C2×C4⋊#3622C2211yescharacteristic
19C2×#10587503C2211yescharacteristic
20C2×#10587503C2211yescharacteristic
21C2×#10587486C2211yescharacteristic
22C2×C8×Dic17C2211yescharacteristic
23#11762712C2211yescharacteristic
24#11762712C2211yescharacteristic
25#11762708—21no—
26#11762708—21no—
27#11762708C2211yescharacteristic
28#11762708C2211yescharacteristic
29#11762708C2211yescharacteristic
30#11762708C2211yescharacteristic
31#11762702—21no—
32#11762702—21no—
33#11762695—21no—
34#11762695—21no—
35#11762679C2211yescharacteristic
36#11762672—21no—
37#11762669—21no—
38C4⋊C4×Dic17C2211yescharacteristic
39C4⋊C4×Dic17C2211yescharacteristic
40C4×C4⋊Dic17C2211yescharacteristic
41#11762655C2211yescharacteristic
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—
62C22×C8×C17C2311yescharacteristic
63C22×C8×C17C2311yescharacteristic
64C17×#125—22no—
65C17×#125C2×C414yes—
66C4×D4×C17—22no—
67C4×D4×C17C2×C414yes—
68C2×C4⋊C4×C17—21no—
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—
77C2×C22⋊Dic17—21no—
78C2×C22⋊Dic17—21no—
79C2×C4⋊Dic17—21no—
80C2×C4⋊Dic17—21no—
81C2×C4⋊Dic17—21no—
82C2×C4⋊Dic17—21no—
83C2×C4⋊Dic17—21no—
84C2×C4⋊Dic17—21no—
85C2×C4⋊Dic17C2311yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4×Dic17—21no—
90C2×C4×Dic17—21no—
91C2×C4×Dic17—21no—
92C2×C4×Dic17C2311yescharacteristic
93C2×C4×Dic17C2311yescharacteristic
94C22×#3622C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96C17×#107C2×C414yes—
97C17×#106C2×C414yes—
98C17×C4⋊C8C2×C414yes—
99C17×#102C2×C414yes—
100C17×#102C2×C414yes—
101#10587503—41no—
102#10587503—41no—
103#10587502—21no—
104#10587502—21no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502D411yescharacteristic
111#10587502D411yescharacteristic
112#10587502D411yescharacteristic
113#10587502D411yescharacteristic
114#10587489—41no—
115#10587489—41no—
116#10587486—41no—
117C8×Dic17—41no—
118C4⋊#3622—22no—
119C24×C17—21no—
120C2×D4×C17—21no—
121C2×D4×C17—21no—
122C2×D4×C17—21no—
123C2×D4×C17—21no—
124C2×D4×C17—21no—
125C2×D4×C17C22×C412yes—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—24no—
130C22×C4×C17—81no—
131C22×C4×C17—81no—
132C22×C4×C17C2×D411yescharacteristic
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17C2411yescharacteristic
135C22×Dic17—21no—
136C22×Dic17—21no—
137C22×Dic17—21no—
138C22×Dic17—21no—
139C22×Dic17C2×D411yescharacteristic
140C22×Dic17C2×D411yescharacteristic
141C22×Dic17C2×D411yescharacteristic
142C22×Dic17C2×D411yescharacteristic
143C2×C8×C17—21no—
144C2×C8×C17—21no—
145C2×C8×C17—21no—
146C2×C8×C17—21no—
147C2×C8×C17C22×C412yes—
148C2×C8×C17C22×C412yes—
149C4⋊C4×C17—24no—
150C4⋊C4×C17—24no—
151C4⋊C4×C17—21no—
152C4⋊C4×C17—21no—
153C4⋊C4×C17—21no—
154C4⋊C4×C17C22×C412yes—
155C4⋊C4×C17C22×C412yes—
156C4⋊C4×C17C22×C412yes—
157C22⋊C4×C17—24no—
158C42×C17C22×C412yes—
159C42×C17#4112yes—
160C22⋊Dic17—81no—
161C22⋊Dic17—81no—
162C22⋊Dic17—81no—
163C22⋊Dic17—81no—
164C4⋊Dic17—21no—
165C4⋊Dic17—21no—
166C4⋊Dic17—21no—
167C4⋊Dic17—21no—
168C4⋊Dic17—21no—
169C4⋊Dic17—21no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—81no—
173C4⋊Dic17—81no—
174C4⋊Dic17—81no—
175C4⋊Dic17—81no—
176C4⋊Dic17—81no—
177C4⋊Dic17—81no—
178C4×Dic17—21no—
179C4×Dic17—21no—
180C4×Dic17—21no—
181C4×Dic17—21no—
182C4×Dic17—81no—
183C4×Dic17—81no—
184C4×Dic17—81no—
185C2×#3622—21no—
186C2×#3622—21no—
187C2×#3622—21no—
188C2×#3622—21no—
189C2×#3622—21no—
190C2×#3622—21no—
191#22126—171nomaximal, Sylow
192C23×C17—22no—
193C23×C17—41no—
194C23×C17—41no—
195C23×C17—41no—
196C23×C17—41no—
197C23×C17—41no—
198C23×C17—41no—
199C23×C17C22×D411yescharacteristic
200D4×C17—22no—
201D4×C17—22no—
202D4×C17—41no—
203D4×C17—41no—
204C2×C4×C17—21no—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—21no—
208C2×C4×C17—22no—
209C2×C4×C17—22no—
210C2×C4×C17—22no—
211C2×C4×C17—41no—
212C2×C4×C17—41no—
213C2×C4×C17—41no—
214C2×C4×C17—41no—
215C2×C4×C17—41no—
216C2×C4×C17—41no—
217C2×C4×C17—84no—
218C2×C4×C17C4×D412yes—
219C2×C4×C17C4×D412yes—
220C2×C4×C17C23×C411yescharacteristic
221C2×C4×C17C2×#4111yescharacteristic
222C2×C4×C17H(5,𝔽2)11yescharacteristic
223C2×C4×C172−1+411yescharacteristic
224C2×C4×C172−1+411yescharacteristic
225C2×C4×C172−1+411yescharacteristic
226C2×Dic17—21no—
227C2×Dic17—21no—
228C2×Dic17—21no—
229C2×Dic17—21no—
230C2×Dic17—21no—
231C2×Dic17—21no—
232C2×Dic17—21no—
233C2×Dic17—21no—
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—41no—
239C2×Dic17—41no—
240C2×Dic17—41no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C8×C17—22no—
251C8×C17—22no—
252#3622—41no—
253#3622—41no—
254C2×#478D1711yescharacteristic
255#2330—171no—
256#2319—171no—
257#1989—171no—
258#1947—171no—
259#1944—171no—
260#1942—171no—
261#1936—171no—
262#1934—171no—
263#1925—171no—
264#1925—171no—
265#1921—171no—
266#1919—171no—
267#1802—171no—
268#1768—171no—
269C22×C17—42no—
270C22×C17—42no—
271C22×C17—42no—
272C22×C17—81no—
273C22×C17—81no—
274C22×C17—81no—
275C22×C17—81no—
276C22×C17C2×C4×D411yescharacteristic
277C22×C17#53611yescharacteristic
278C22×C17#53711yescharacteristic
279C22×C17#57611yescharacteristic
280C22×C17#57611yescharacteristic
281C22×C17#57711yescharacteristic
282C22×C17#57811yescharacteristic
283C4×C17—22no—
284C4×C17—22no—
285C4×C17—42no—
286C4×C17—42no—
287C4×C17—42no—
288C4×C17#51911yescharacteristic
289C4×C17#52011yescharacteristic
290C4×C17#54111yescharacteristic
291C4×C17#54211yescharacteristic
292Dic17—41no—
293Dic17—41no—
294Dic17—41no—
295Dic17—41no—
296Dic17—81no—
297Dic17—81no—
298Dic17—81no—
299Dic17—81no—
300C2×#125C2×D1711yescharacteristic
301C2×C4×D4C2×D1711yescharacteristic
302#478Dic1718yescomplemented
303C2×#107—171no—
304C2×#107—171no—
305C2×#107C2×D1711yescharacteristic
306C2×#106C2×D1711yescharacteristic
307C2×C4⋊C8—171no—
308C2×C4⋊C8C2×D1711yescharacteristic
309C2×#102—171no—
310C2×#102—171no—
311C2×#102C2×D1711yescharacteristic
312C2×#102C2×D1711yescharacteristic
313C2×#97—171no—
314C2×C4×C8—171no—
315#390—171no—
316#390—171no—
317#389—171no—
318#389—171no—
319#389—171no—
320#389—171no—
321#389—341no—
322#389—341no—
323#388—341no—
324#388—341no—
325#387—341no—
326#387—341no—
327#384—171no—
328#381—341no—
329#380—341no—
330C4×C4⋊C4—171no—
331C4×C4⋊C4—171no—
332C4×C4⋊C4—171no—
333#376—171no—
334#340—171no—
335#340—171no—
336#340—171no—
337#340—171no—
338#340—171no—
339#340—171no—
340#340—171no—
341#340—171no—
342#336—171no—
343#336—171no—
344C2×C17—82no—
345C2×C17#232311yescharacteristic
346C2×C17#294611yescharacteristic
347C2×C17#295111yescharacteristic
348C2×C17#324111yescharacteristic
349C2×C17#324311yescharacteristic
350C2×C17#325811yescharacteristic
351C2×C17#325911yescharacteristic
352C22×D4C22×D1711yescharacteristic
353C23×C4—21no—
354C22×C8—171no—
355C22×C8—171no—
356C22×C8C22×D1711yescharacteristic
357C22×C8C22×D1711yescharacteristic
358#125—22no—
359#125C2×Dic1714yes—
360C4×D4—22no—
361C4×D4C2×Dic1714yes—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—171no—
365C2×C4⋊C4—171no—
366C2×C4⋊C4—171no—
367C2×C4⋊C4—171no—
368C2×C4⋊C4—341no—
369C2×C4⋊C4—341no—
370C2×C4⋊C4—341no—
371C2×C4⋊C4—341no—
372C2×C4⋊C4—341no—
373C2×C4⋊C4—341no—
374C2×C4⋊C4C22×D1711yescharacteristic
375C2×C4⋊C4C22×D1711yescharacteristic
376C2×C4⋊C4C22×D1711yescharacteristic
377C2×C22⋊C4—21no—
378C2×C22⋊C4—341no—
379C2×C22⋊C4—341no—
380C2×C22⋊C4—341no—
381C2×C22⋊C4—341no—
382C2×C42—171no—
383C2×C42—171no—
384C2×C42—341no—
385C2×C42—341no—
386C2×C42—341no—
387C2×C42C22×D1711yescharacteristic
388#107—681no—
389#107—681no—
390#107C2×Dic1714yes—
391#106C2×Dic1714yes—
392C4⋊C8—342no—
393C4⋊C8C2×Dic1714yes—
394#102—681no—
395#102—681no—
396#102C2×Dic1714yes—
397#102C2×Dic1714yes—
398#97—681no—
399C4×C8—681no—
400#95—171no—
401#95—171no—
402#95—171no—
403#95—171no—
404#95—341no—
405#95—341no—
406#95—341no—
407#95—341no—
408#95—341no—
409#95—341no—
410#95—341no—
411C17#2212611yescharacteristic, complemented, Sylow
412C24—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4C22×Dic1712yes—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—81no—
424C22×C4—81no—
425C22×C4—171no—
426C22×C4—171no—
427C22×C4—171no—
428C22×C4—171no—
429C22×C4—341no—
430C22×C4—341no—
431C22×C4—341no—
432C22×C4—341no—
433C22×C4D4×D1711yescharacteristic
434C22×C4D4×D1711yescharacteristic
435C22×C4C23×D1711yescharacteristic
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—341no—
441C2×C8—341no—
442C2×C8—341no—
443C2×C8—341no—
444C2×C8—341no—
445C2×C8—341no—
446C2×C8C22×Dic1712yes—
447C2×C8C22×Dic1712yes—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—341no—
454C4⋊C4—341no—
455C4⋊C4—341no—
456C4⋊C4—341no—
457C4⋊C4—341no—
458C4⋊C4—341no—
459C4⋊C4—341no—
460C4⋊C4—341no—
461C4⋊C4—1361no—
462C4⋊C4—1361no—
463C4⋊C4—1361no—
464C4⋊C4—1361no—
465C4⋊C4—1361no—
466C4⋊C4—1361no—
467C4⋊C4C22×Dic1712yes—
468C4⋊C4C22×Dic1712yes—
469C4⋊C4C22×Dic1712yes—
470C22⋊C4—24no—
471C22⋊C4—1361no—
472C22⋊C4—1361no—
473C22⋊C4—1361no—
474C22⋊C4—1361no—
475C42—341no—
476C42—341no—
477C42—341no—
478C42—341no—
479C42—1361no—
480C42—1361no—
481C42—1361no—
482C42#6325412yes—
483C42C22×Dic1712yes—
484C23—22no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23C2×D4×D1711yescharacteristic
492D4—22no—
493D4—22no—
494D4—41no—
495D4—41no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—84no—
510C2×C4—341no—
511C2×C4—341no—
512C2×C4—341no—
513C2×C4—341no—
514C2×C4—341no—
515C2×C4—341no—
516C2×C4—341no—
517C2×C4—341no—
518C2×C4—341no—
519C2×C4—341no—
520C2×C4—341no—
521C2×C4—341no—
522C2×C4—681no—
523C2×C4—681no—
524C2×C4—681no—
525C2×C4—681no—
526C2×C4—681no—
527C2×C4—681no—
528C2×C4—681no—
529C2×C4—681no—
530C2×C4—681no—
531C2×C4—681no—
532C2×C4—681no—
533C2×C4—681no—
534C2×C4D4×Dic1712yes—
535C2×C4D4×Dic1712yes—
536C2×C4C2×#6325411yescharacteristic
537C2×C4#1058769411yescharacteristic
538C2×C4#1058769411yescharacteristic
539C2×C4#1058769611yescharacteristic
540C2×C4#1058769711yescharacteristic
541C2×C4C23×Dic1711yescharacteristic
542C8—22no—
543C8—22no—
544C8—681no—
545C8—681no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22C2×D4×Dic1711yescharacteristic
554C22#1176338211yescharacteristic
555C22#1176338911yescharacteristic
556C22#1176361111yescharacteristic
557C22#1176361311yescharacteristic
558C22#1176361911yescharacteristic
559C22#1176362311yescharacteristic
560C4—22no—
561C4—22no—
562C4—42no—
563C4—42no—
564C4—42no—
565C4—681no—
566C4—681no—
567C4—681no—
568C4—681no—
569C4—1361no—
570C4—1361no—
571C4—1361no—
572C4—1361no—
573C4#1176328711yescharacteristic
574C4#1176329211yescharacteristic
575C4#1176341911yescharacteristic
576C4#1176343211yescharacteristic
577C2—82no—
578C2#1453146811yescharacteristic
579C2#1453761711yescharacteristic
580C2#1453766011yescharacteristic
581C2#1454003411yescharacteristic
582C2#1454006611yescharacteristic
583C2#1454021411yescharacteristic
584C2#1454022211yescharacteristic
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#22126
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. #14531526C2semidirect
3. #14529742C2semidirect
4. #14529543C2semidirect
5. #14529528C2semidirect
6. #14529518C2semidirect
11. #14529378C2semidirect
12. #14529364C2semidirect
14. #14528643C2semidirect
54. C17×#478C4semidirect
302. #478Dic17semidirect
411. C17#22126semidirect

Maximal subgroups

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

subgroupindex
#145315262
#145314272
#145297422
#145295432
#145295282
#145295182
#145294832
#145294692
#145294062
#145293782
#145293642
#145287842
#145286432
C2×C17×#4782
#2212617