Finite Group Databasebeta

← #17093367

Subgroups — #17093367

Order 4352.

678 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14534018C211yesmaximal, characteristic, complemented
2#14530312C211yesmaximal, characteristic
3#14530305C211yesmaximal, characteristic
4#14530158C211yesmaximal, characteristic
5#14530158C211yesmaximal, characteristic
6#14529552C211yesmaximal, characteristic, complemented
7#14529550C211yesmaximal, characteristic, complemented
8#14529547C211yesmaximal, characteristic, complemented
9#14529503C211yesmaximal, characteristic, complemented
10#14529376C211yesmaximal, characteristic, complemented
11#14529355C211yesmaximal, characteristic, complemented
12#14528946C211yesmaximal, characteristic, complemented
13#14528794C211yesmaximal, characteristic
14C4×#10587503C211yesmaximal, characteristic
15C17×#2357C211yesmaximal, characteristic
16C2×#10587478C2211yescharacteristic
17C2×#10587478C2211yescharacteristic
18C2×#10587478C2211yescharacteristic
19C2×#10587479C2211yescharacteristic
20C2×#10587479C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×C4⋊#3622C2211yescharacteristic
23C2×C4⋊#3622C2211yescharacteristic
24C2×C4⋊#3622C2211yescharacteristic
25C2×#10587503C2211yescharacteristic
26C2×#10587503C2211yescharacteristic
27C2×#10587503C2211yescharacteristic
28C2×#10587503C2211yescharacteristic
29C2×C4×#3622C2211yescharacteristic
30#11762748—21no—
31#11762742—21no—
32#11762707C2211yescharacteristic
33#11762707C2211yescharacteristic
34#11762696—21no—
35#11762692—21no—
36#11762679C2211yescharacteristic
37#11762675C2211yescharacteristic
38#11762675C2211yescharacteristic
39C4×C4⋊Dic17C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48C2×C4×D4×C17C2211yescharacteristic
49C17×#390C2211yescharacteristic
50C17×#390C2211yescharacteristic
51C17×#388—21no—
52C17×#388—21no—
53C17×#382C2211yescharacteristic
54C17×#381—21no—
55C4×C4⋊C4×C17C2211yescharacteristic
56C4×C4⋊C4×C17C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
58C4×C22⋊C4×C17—21no—
59C22×D4×C17C2311yescharacteristic
60C23×C4×C17—21no—
61C4×D4×C17—22no—
62C4×D4×C17Q814yes—
63C2×C4⋊C4×C17D411yescharacteristic
64C2×C4⋊C4×C17D411yescharacteristic
65C2×C4⋊C4×C17C2311yescharacteristic
66C2×C4⋊C4×C17C2311yescharacteristic
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C22⋊C4×C17—21no—
69C2×C22⋊C4×C17—21no—
70C2×C22⋊C4×C17—21no—
71C2×C22⋊C4×C17—21no—
72C2×C22⋊C4×C17—21no—
73C2×C42×C17—21no—
74C2×C42×C17—21no—
75C2×C42×C17C2311yescharacteristic
76C2×C42×C17C2311yescharacteristic
77C2×C42×C17C2311yescharacteristic
78C2×C4⋊Dic17—21no—
79C2×C4⋊Dic17—21no—
80C2×C4⋊Dic17C2311yescharacteristic
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×#63225—21no—
85C2×#63225—21no—
86C2×C4×Dic17—21no—
87C22×#3622C2311yescharacteristic
88C22×#3622C2311yescharacteristic
89C22×#3622C2311yescharacteristic
90C22×#3622C2311yescharacteristic
91C17×#95—21no—
92C17×#95—21no—
93C17×#95—21no—
94C17×#95—21no—
95C17×#95D411yescharacteristic
96C17×#95D411yescharacteristic
97#10587503—22no—
98#10587503—22no—
99#10587503—41no—
100#10587503—41no—
101#10587502—21no—
102#10587502—21no—
103#10587502—21no—
104#10587502—21no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587479—41no—
109#10587479—41no—
110#10587479—41no—
111#10587478—22no—
112#10587478—22no—
113#10587478—41no—
114C4⋊#3622—22no—
115C4⋊#3622—22no—
116C4⋊#3622—22no—
117C4×#3622—22no—
118C24×C17—21no—
119C2×D4×C17—21no—
120C2×D4×C17—21no—
121C2×D4×C17—21no—
122C2×D4×C17—21no—
123C2×D4×C17—21no—
124C2×D4×C17C2×Q812yes—
125C22×C4×C17—21no—
126C22×C4×C17—24no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—81no—
130C22×C4×C17—81no—
131C22×C4×C17C2×D411yescharacteristic
132C22×C4×C17C2×D411yescharacteristic
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17C2×D411yescharacteristic
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2411yescharacteristic
138C22×Dic17—21no—
139C22×Dic17—21no—
140C22×Dic17—21no—
141C22×Dic17—21no—
142C4⋊C4×C17—22no—
143C4⋊C4×C17—21no—
144C4⋊C4×C17—21no—
145C4⋊C4×C17—21no—
146C4⋊C4×C17—21no—
147C4⋊C4×C17—21no—
148C4⋊C4×C17—41no—
149C4⋊C4×C17C2×Q812yes—
150C22⋊C4×C17—24no—
151C22⋊C4×C17—42no—
152C22⋊C4×C17—42no—
153C22⋊C4×C17—81no—
154C22⋊C4×C17—81no—
155C42×C17—21no—
156C42×C17—21no—
157C42×C17—21no—
158C42×C17—21no—
159C42×C17—81no—
160C42×C17—81no—
161C42×C17C2×Q812yes—
162C42×C17#4112yes—
163C4⋊Dic17—21no—
164C4⋊Dic17—21no—
165C4⋊Dic17—21no—
166C4⋊Dic17—21no—
167C4⋊Dic17—21no—
168C4⋊Dic17—21no—
169C4⋊Dic17—21no—
170C4⋊Dic17—21no—
171C4⋊Dic17—42no—
172C4⋊Dic17—81no—
173#63225—42no—
174#63225—81no—
175C4×Dic17—81no—
176C2×#3622—21no—
177C2×#3622—21no—
178C2×#3622—21no—
179C2×#3622—21no—
180C2×#3622—21no—
181C2×#3622—21no—
182C2×#3622—21no—
183C2×#3622—21no—
184C2×#3622—21no—
185C2×#3622—21no—
186C2×#3622#4112yes—
187C2×#3622#4112yes—
188#26094—171nomaximal, Sylow
189C23×C17—22no—
190C23×C17—41no—
191C23×C17—41no—
192C23×C17—41no—
193C23×C17—41no—
194C23×C17—41no—
195C23×C17—41no—
196C23×C17C22×D411yescharacteristic
197D4×C17—22no—
198D4×C17—22no—
199D4×C17—41no—
200D4×C17—41no—
201C2×C4×C17—21no—
202C2×C4×C17—21no—
203C2×C4×C17—21no—
204C2×C4×C17—21no—
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—22no—
208C2×C4×C17—21no—
209C2×C4×C17—21no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—21no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—41no—
218C2×C4×C17—41no—
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×C17C22⋊Q812yes—
227C2×C4×C17C22⋊Q812yes—
228C2×C4×C17#13512yes—
229C2×C4×C17C22×Q811yescharacteristic
230C2×C4×C17C2×#4111yescharacteristic
231C2×C4×C17C2×#4111yescharacteristic
232C2×C4×C17C2×#4111yescharacteristic
233C2×C4×C17H(5,𝔽2)11yescharacteristic
234C2×C4×C172−1+411yescharacteristic
235C2×Dic17—41no—
236C2×Dic17—41no—
237C2×Dic17—41no—
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—
247#3622—22no—
248#3622—22no—
249#3622—41no—
250#3622—41no—
251#2570—171no—
252#2357D1711yescharacteristic
253#2093—171no—
254#2092—171no—
255#2065—171no—
256#2065—171no—
257#1949—171no—
258#1949—171no—
259#1948—171no—
260#1939—171no—
261#1921—171no—
262#1918—171no—
263#1846—171no—
264#1804—171no—
265C4×#102—171no—
266C22×C17—42no—
267C22×C17—42no—
268C22×C17—42no—
269C22×C17—81no—
270C22×C17—81no—
271C22×C17—81no—
272C22×C17—81no—
273C22×C17C2×C22⋊Q811yescharacteristic
274C22×C17#54711yescharacteristic
275C22×C17#55011yescharacteristic
276C22×C17C2×#13511yescharacteristic
277C22×C17#57511yescharacteristic
278C22×C17#57611yescharacteristic
279C22×C17#57711yescharacteristic
280C4×C17—22no—
281C4×C17—22no—
282C4×C17—22no—
283C4×C17—42no—
284C4×C17—41no—
285C4×C17—41no—
286C4×C17—41no—
287C4×C17—81no—
288C4×C17—81no—
289C4×C17#53311yescharacteristic
290C4×C17#54411yescharacteristic
291C4×C17#55211yescharacteristic
292C4×C17#55311yescharacteristic
293Dic17—81no—
294Dic17—81no—
295Dic17—81no—
296Dic17—81no—
297C2×C4×D4C2×D1711yescharacteristic
298C2×#107—171no—
299C2×#107—171no—
300C2×#107—171no—
301C2×#106—171no—
302C2×#106—171no—
303C2×#106—171no—
304C2×C4⋊C8—171no—
305C2×C4⋊C8—171no—
306C2×C4⋊C8—171no—
307C2×#102—171no—
308C2×#102—171no—
309C2×#102—171no—
310C2×#102—171no—
311C2×C4×C8—171no—
312#400—341no—
313#398—341no—
314#390C2×D1711yescharacteristic
315#390C2×D1711yescharacteristic
316#389—171no—
317#389—171no—
318#388—21no—
319#388—21no—
320#387—341no—
321#387—341no—
322#384—171no—
323#382—171no—
324#382—171no—
325#382C2×D1711yescharacteristic
326#381—21no—
327C4×C4⋊C4—171no—
328C4×C4⋊C4C2×D1711yescharacteristic
329C4×C4⋊C4C2×D1711yescharacteristic
330C4×C4⋊C4C2×D1711yescharacteristic
331C4×C22⋊C4—21no—
332#340—171no—
333#340—171no—
334#340—171no—
335#340—171no—
336#340—171no—
337#340—171no—
338#336—171no—
339#336—171no—
340C2×C17—82no—
341C2×C17#244111yescharacteristic
342C2×C17#307611yescharacteristic
343C2×C17#308111yescharacteristic
344C2×C17#333711yescharacteristic
345C2×C17#335511yescharacteristic
346C2×C17#339611yescharacteristic
347C2×C17#340111yescharacteristic
348C22×D4C22×D1711yescharacteristic
349C23×C4—21no—
350C22×C8—171no—
351C22×C8—171no—
352C22×C8—171no—
353C22×C8—171no—
354C4×D4—22no—
355C4×D4Dic3414yes—
356C2×C4⋊C4—171no—
357C2×C4⋊C4—171no—
358C2×C4⋊C4—171no—
359C2×C4⋊C4—171no—
360C2×C4⋊C4—341no—
361C2×C4⋊C4—341no—
362C2×C4⋊C4—341no—
363C2×C4⋊C4—341no—
364C2×C4⋊C4C22⋊D1711yescharacteristic
365C2×C4⋊C4C22⋊D1711yescharacteristic
366C2×C4⋊C4C22×D1711yescharacteristic
367C2×C4⋊C4C22×D1711yescharacteristic
368C2×C4⋊C4C22×D1711yescharacteristic
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C42—21no—
375C2×C42—21no—
376C2×C42—341no—
377C2×C42C22×D1711yescharacteristic
378C2×C42C22×D1711yescharacteristic
379C2×C42C22×D1711yescharacteristic
380#107—342no—
381#107—342no—
382#107—681no—
383#106—681no—
384#106—681no—
385#106—681no—
386C4⋊C8—342no—
387C4⋊C8—342no—
388C4⋊C8—342no—
389#102—342no—
390#102—342no—
391#102—681no—
392#102—681no—
393C4×C8—342no—
394#95—21no—
395#95—21no—
396#95—21no—
397#95—21no—
398#95—341no—
399#95—341no—
400#95—341no—
401#95—341no—
402#95—341no—
403#95—341no—
404#95—341no—
405#95C22⋊D1711yescharacteristic
406#95C22⋊D1711yescharacteristic
407C17#2609411yescharacteristic, complemented, Sylow
408C24—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4C2×Dic3412yes—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—341no—
422C22×C4—341no—
423C22×C4—341no—
424C22×C4—341no—
425C22×C4C2×C22⋊D1711yescharacteristic
426C22×C4C2×C22⋊D1711yescharacteristic
427C22×C4C2×C22⋊D1711yescharacteristic
428C22×C4C2×C22⋊D1711yescharacteristic
429C22×C4C2×C22⋊D1711yescharacteristic
430C22×C4C2×C22⋊D1711yescharacteristic
431C22×C4C23×D1711yescharacteristic
432C2×C8—172no—
433C2×C8—172no—
434C2×C8—341no—
435C2×C8—341no—
436C2×C8—341no—
437C2×C8—341no—
438C2×C8—341no—
439C2×C8—341no—
440C2×C8—341no—
441C2×C8—341no—
442C2×C8—341no—
443C2×C8—341no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—41no—
451C4⋊C4—341no—
452C4⋊C4—341no—
453C4⋊C4—341no—
454C4⋊C4—341no—
455C4⋊C4—341no—
456C4⋊C4—341no—
457C4⋊C4—341no—
458C4⋊C4—341no—
459C4⋊C4—682no—
460C4⋊C4—682no—
461C4⋊C4—1361no—
462C4⋊C4—1361no—
463C4⋊C4C2×Dic3412yes—
464C22⋊C4—24no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—81no—
468C22⋊C4—81no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—81no—
474C42—81no—
475C42—1361no—
476C42C2×Dic3412yes—
477C42#6325212yes—
478C23—22no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23C22×C22⋊D1711yescharacteristic
486D4—22no—
487D4—22no—
488D4—41no—
489D4—41no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—84no—
515C2×C4—681no—
516C2×C4—681no—
517C2×C4—681no—
518C2×C4—681no—
519C2×C4—681no—
520C2×C4—681no—
521C2×C4—681no—
522C2×C4—681no—
523C2×C4—681no—
524C2×C4—681no—
525C2×C4—681no—
526C2×C4—681no—
527C2×C4#1058761012yes—
528C2×C4#1058761012yes—
529C2×C4#1058763612yes—
530C2×C4C22×Dic3411yescharacteristic
531C2×C4C2×#6325211yescharacteristic
532C2×C4#41×D1711yescharacteristic
533C2×C4#41×D1711yescharacteristic
534C2×C4#1058769611yescharacteristic
535C2×C4#1058769711yescharacteristic
536C8—342no—
537C8—342no—
538C8—681no—
539C8—681no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22C2×#1058761011yescharacteristic
548C22#1176345811yescharacteristic
549C22#1176347811yescharacteristic
550C22C2×#1058763611yescharacteristic
551C22#1176360711yescharacteristic
552C22#1176361211yescharacteristic
553C22#1176362011yescharacteristic
554C4—22no—
555C4—22no—
556C4—22no—
557C4—41no—
558C4—41no—
559C4—41no—
560C4—42no—
561C4—81no—
562C4—81no—
563C4—1361no—
564C4—1361no—
565C4—1361no—
566C4—1361no—
567C4#1176336211yescharacteristic
568C4#1176343711yescharacteristic
569C4#1176349811yescharacteristic
570C4#1176350311yescharacteristic
571C2—82no—
572C2#1453249111yescharacteristic
573C2#1453857211yescharacteristic
574C2#1453861511yescharacteristic
575C2#1454115711yescharacteristic
576C2#1454138411yescharacteristic
577C2#1454179211yescharacteristic
578C2#1454184411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2357C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
hypercenter#2357D17
hypocenterC17#26094
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2357

Complements

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

NH ≅ G/Ntype
1. #14534018C2semidirect
6. #14529552C2semidirect
7. #14529550C2semidirect
8. #14529547C2semidirect
9. #14529503C2semidirect
10. #14529376C2semidirect
11. #14529355C2semidirect
12. #14528946C2semidirect
407. C17#26094semidirect

Maximal subgroups

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

subgroupindex
#145340182
#145303122
#145303052
#145301582
#145295522
#145295502
#145295472
#145295032
#145293762
#145293552
#145289462
#145287942
C4×#105875032
C17×#23572
#2609417