Finite Group Databasebeta

← #17137442

Subgroups — #17137442

Order 4352.

724 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14537151C211yesmaximal, characteristic
2#14531261C211yesmaximal, characteristic, complemented
3#14530282C211yesmaximal, characteristic, complemented
4#14530178C211yesmaximal, characteristic, complemented
5#14530171C211yesmaximal, characteristic
6#14530102C211yesmaximal, characteristic
7#14529638C211yesmaximal, characteristic
8#14529517C211yesmaximal, characteristic, complemented
9#14529467C211yesmaximal, characteristic
10#14529460C211yesmaximal, characteristic, complemented
11#14529405C211yesmaximal, characteristic
12#14529370C211yesmaximal, characteristic, complemented
13#14529138C211yesmaximal, characteristic
14#14529041C211yesmaximal, characteristic, complemented
15C4×C17×C4⋊C8C211yesmaximal, characteristic, complemented
16C22×#63219C2211yescharacteristic
17C2×#10587562—21no—
18C2×C4×Dic34C2211yescharacteristic
19C2×#10587489C2211yescharacteristic
20C2×#10587489C2211yescharacteristic
21C2×#10587488C2211yescharacteristic
22C2×#10587487C2211yescharacteristic
23C2×#10587487C2211yescharacteristic
24C2×#10587487C2211yescharacteristic
25C2×#10587492C2211yescharacteristic
26C2×#10587492C2211yescharacteristic
27C2×#10587492C2211yescharacteristic
28#11762715C2211yescharacteristic
29#11762715C2211yescharacteristic
30#11762711C2211yescharacteristic
31#11762711C2211yescharacteristic
32#11762711C2211yescharacteristic
33#11762707C2211yescharacteristic
34#11762707C2211yescharacteristic
35#11762707C2211yescharacteristic
36#11762707C2211yescharacteristic
37#11762707C2211yescharacteristic
38#11762700—21no—
39#11762700—21no—
40#11762700—21no—
41#11762693—21no—
42#11762693—21no—
43C4×#63225—21no—
44C4×C4⋊Dic17C2211yescharacteristic
45#11762655—21no—
46#11762572C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762572C2211yescharacteristic
50#11762572C2211yescharacteristic
51#11762572C2211yescharacteristic
52C2×C17×C4⋊C8C2211yescharacteristic
53C2×C17×C4⋊C8C2211yescharacteristic
54C2×C4×C8×C17C2211yescharacteristic
55C2×C4×C8×C17C2211yescharacteristic
56C43×C17C2211yescharacteristic
57C17×#336C2211yescharacteristic
58C17×#336C2211yescharacteristic
59C22×D68C2311yescharacteristic
60C22×Dic34C2311yescharacteristic
61C22×C8×C17C2311yescharacteristic
62C22×C8×C17C2311yescharacteristic
63C22×C8×C17C2311yescharacteristic
64C22×C8×C17C2311yescharacteristic
65C2×C42×C17D411yescharacteristic
66C2×C42×C17D411yescharacteristic
67C2×C42×C17D411yescharacteristic
68C2×C42×C17D411yescharacteristic
69C2×C42×C17C2311yescharacteristic
70C2×C42×C17C2311yescharacteristic
71C2×C42×C17C2311yescharacteristic
72C2×#63227—21no—
73C2×#63227—21no—
74C2×#63227—21no—
75C2×#63227—21no—
76C2×#63227—21no—
77C2×#63227—21no—
78C2×C4⋊Dic17C2311yescharacteristic
79C2×C4⋊Dic17C2311yescharacteristic
80C2×C4⋊Dic17C2311yescharacteristic
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×#63225—21no—
85C2×#63225—21no—
86C2×#63225—21no—
87C2×#63225—21no—
88C2×#63225—21no—
89C2×#63225—21no—
90C2×C4×Dic17—21no—
91C2×C4×Dic17—21no—
92C2×#63219—22no—
93C2×#63219—22no—
94C2×#63219—41no—
95#10587562—44no—
96C4×Dic34—22no—
97C4×Dic34—22no—
98C17×C4⋊C8—22no—
99C17×C4⋊C8—22no—
100C4×C8×C17—41no—
101C4×C8×C17—41no—
102#10587502—21no—
103#10587502—21no—
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#10587492—22no—
113#10587492—22no—
114#10587492—41no—
115#10587489—22no—
116#10587489—41no—
117#10587488—22no—
118#10587487—22no—
119#10587487—22no—
120#10587487—41no—
121C23×D17—21no—
122C22×C4×C17C2×D411yescharacteristic
123C22×C4×C17C2×D411yescharacteristic
124C22×C4×C17C2×D411yescharacteristic
125C22×C4×C17C2×D411yescharacteristic
126C22×C4×C17C2×D411yescharacteristic
127C22×C4×C17C2×D411yescharacteristic
128C22×C4×C17C2411yescharacteristic
129C22×Dic17—21no—
130C22×Dic17—21no—
131C22×Dic17—21no—
132C22×Dic17—21no—
133C22×Dic17—21no—
134C22×Dic17—21no—
135C22×Dic17—21no—
136C2×D68—21no—
137C2×D68—21no—
138C2×D68—21no—
139C2×D68—21no—
140C2×D68—21no—
141C2×D68—21no—
142C2×Dic34—21no—
143C2×Dic34—21no—
144C2×Dic34—21no—
145C2×Dic34—21no—
146C2×Dic34—21no—
147C2×Dic34#4112yes—
148C2×C8×C17—21no—
149C2×C8×C17—21no—
150C2×C8×C17—21no—
151C2×C8×C17—21no—
152C2×C8×C17—21no—
153C2×C8×C17—21no—
154C2×C8×C17—21no—
155C2×C8×C17—21no—
156C2×C8×C17—21no—
157C2×C8×C17—21no—
158C2×C8×C17#4112yes—
159C2×C8×C17#4112yes—
160C42×C17—22no—
161C42×C17—22no—
162C42×C17—22no—
163C42×C17—22no—
164C42×C17—21no—
165C42×C17—21no—
166C42×C17—21no—
167C42×C17—21no—
168C42×C17—21no—
169C42×C17—21no—
170#63227—42no—
171#63227—42no—
172#63227—42no—
173#63227—42no—
174#63227—42no—
175#63227—42no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17—21no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17—21no—
185C4⋊Dic17—21no—
186C4⋊Dic17—21no—
187C4⋊Dic17#4112yes—
188#63225—24no—
189#63225—42no—
190#63225—42no—
191#63225—42no—
192#63225—42no—
193#63225—42no—
194C4×Dic17—24no—
195C4×Dic17—42no—
196#63219—44no—
197#29435—171nomaximal, Sylow
198C23×C17C22×D411yescharacteristic
199C22×D17—41no—
200C22×D17—41no—
201C22×D17—41no—
202C22×D17—41no—
203C22×D17—41no—
204C22×D17—41no—
205C22×D17—41no—
206C2×C4×C17—21no—
207C2×C4×C17—21no—
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—21no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17#13512yes—
221C2×C4×C17#13512yes—
222C2×C4×C17AGL(1,ℤ/8)12yes—
223C2×C4×C17#13712yes—
224C2×C4×C17C2×#4111yescharacteristic
225C2×C4×C17C2×#4111yescharacteristic
226C2×C4×C17C2×#4111yescharacteristic
227C2×C4×C17C2×#4111yescharacteristic
228C2×C4×C17H(5,𝔽2)11yescharacteristic
229C2×C4×C172−1+411yescharacteristic
230C2×Dic17—22no—
231C2×Dic17—22no—
232C2×Dic17—41no—
233C2×Dic17—41no—
234C2×Dic17—41no—
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—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251D68—41no—
252D68—41no—
253D68—41no—
254D68—41no—
255Dic34—22no—
256Dic34—22no—
257Dic34—41no—
258Dic34—41no—
259C8×C17—22no—
260C8×C17—22no—
261C8×C17—41no—
262C8×C17—41no—
263#2860—171no—
264#2295—171no—
265#2088—171no—
266#2068—171no—
267#2067—171no—
268#2055—171no—
269#1968—171no—
270#1942—171no—
271#1934—171no—
272#1933—171no—
273#1925—171no—
274#1920—171no—
275#1880—171no—
276#1863—171no—
277C4×C4⋊C8D1711yescharacteristic, complemented
278C22×C17#52511yescharacteristic
279C22×C17#53211yescharacteristic
280C22×C17#53711yescharacteristic
281C22×C17C2×#13511yescharacteristic
282C22×C17C2×#13511yescharacteristic
283C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C17C2×#13711yescharacteristic
285C2×D17—81no—
286C2×D17—81no—
287C2×D17—81no—
288C2×D17—81no—
289C2×D17—81no—
290C2×D17—81no—
291C2×D17—81no—
292C4×C17—22no—
293C4×C17—22no—
294C4×C17—22no—
295C4×C17—22no—
296C4×C17—41no—
297C4×C17—41no—
298C4×C17#55111yescharacteristic
299C4×C17#55211yescharacteristic
300C4×C17#55311yescharacteristic
301C4×C17#55611yescharacteristic
302Dic17—42no—
303Dic17—42no—
304Dic17—81no—
305Dic17—81no—
306Dic17—81no—
307Dic17—81no—
308Dic17—81no—
309C22×SD16—171no—
310C2×#126—341no—
311C2×C4×Q8—171no—
312C2×#107—171no—
313C2×#107—171no—
314C2×#106—171no—
315C2×C4⋊C8C2×D1711yescharacteristic
316C2×C4⋊C8C2×D1711yescharacteristic
317C2×#103—171no—
318C2×#103—171no—
319C2×#103—171no—
320C2×#102—171no—
321C2×#102—171no—
322C2×#102—171no—
323C2×C4×C8C2×D1711yescharacteristic
324C2×C4×C8C2×D1711yescharacteristic
325#391—171no—
326#391—171no—
327#390—171no—
328#390—171no—
329#390—171no—
330#389—171no—
331#389—171no—
332#389—171no—
333#389—171no—
334#389—171no—
335#388—341no—
336#388—341no—
337#388—341no—
338#387—341no—
339#387—341no—
340C4×C4⋊C4—171no—
341C4×C4⋊C4—341no—
342#376—341no—
343C43C2×D1711yescharacteristic
344#340—171no—
345#340—171no—
346#340—171no—
347#340—171no—
348#340—171no—
349#340—171no—
350#336C2×D1711yescharacteristic
351#336C2×D1711yescharacteristic
352C2×C17#260111yescharacteristic
353C2×C17#310611yescharacteristic
354C2×C17#310811yescharacteristic
355C2×C17#316811yescharacteristic
356C2×C17#318211yescharacteristic
357C2×C17#324711yescharacteristic
358C2×C17#324811yescharacteristic
359D17—161no—
360C22×Q8—171no—
361C22×D4—171no—
362C2×SD16—342no—
363C2×SD16—342no—
364C2×SD16—681no—
365C22×C8C22×D1711yescharacteristic
366C22×C8C22×D1711yescharacteristic
367C22×C8C22×D1711yescharacteristic
368C22×C8C22×D1711yescharacteristic
369#126—684no—
370C4×Q8—342no—
371C4×Q8—342no—
372C2×C4⋊C4—171no—
373C2×C4⋊C4—171no—
374C2×C4⋊C4—171no—
375C2×C4⋊C4—171no—
376C2×C4⋊C4—171no—
377C2×C4⋊C4—171no—
378C2×C4⋊C4—341no—
379C2×C4⋊C4—341no—
380C2×C4⋊C4—341no—
381C2×C4⋊C4—341no—
382C2×C4⋊C4—341no—
383C2×C4⋊C4—341no—
384C2×C22⋊C4—341no—
385C2×C22⋊C4—341no—
386C2×C22⋊C4—341no—
387C2×C22⋊C4—341no—
388C2×C22⋊C4—341no—
389C2×C22⋊C4—341no—
390C2×C42—341no—
391C2×C42—341no—
392C2×C42D6811yescharacteristic
393C2×C42D6811yescharacteristic
394C2×C42D6811yescharacteristic
395C2×C42D6811yescharacteristic
396C2×C42C22×D1711yescharacteristic
397C2×C42C22×D1711yescharacteristic
398C2×C42C22×D1711yescharacteristic
399#107—342no—
400#107—681no—
401#106—342no—
402C4⋊C8—22no—
403C4⋊C8—22no—
404#103—342no—
405#103—342no—
406#103—681no—
407#102—342no—
408#102—342no—
409#102—681no—
410C4×C8—41no—
411C4×C8—41no—
412#95—341no—
413#95—341no—
414#95—341no—
415#95—341no—
416#95—341no—
417#95—341no—
418#95—341no—
419#95—341no—
420#95—341no—
421#95—341no—
422C17#2943511yescharacteristic, complemented, Sylow
423C24—341no—
424C2×Q8—172no—
425C2×Q8—341no—
426C2×Q8—341no—
427C2×Q8—341no—
428C2×Q8—341no—
429C2×Q8—341no—
430C2×D4—341no—
431C2×D4—341no—
432C2×D4—341no—
433C2×D4—341no—
434C2×D4—341no—
435C2×D4—341no—
436C22×C4—341no—
437C22×C4—341no—
438C22×C4—341no—
439C22×C4—341no—
440C22×C4—341no—
441C22×C4—341no—
442C22×C4—341no—
443C22×C4C2×D6811yescharacteristic
444C22×C4C2×D6811yescharacteristic
445C22×C4C2×D6811yescharacteristic
446C22×C4C2×D6811yescharacteristic
447C22×C4C2×D6811yescharacteristic
448C22×C4C2×D6811yescharacteristic
449C22×C4C23×D1711yescharacteristic
450SD16—684no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8#6325212yes—
462C2×C8#6325212yes—
463C4⋊C4—172no—
464C4⋊C4—344no—
465C4⋊C4—341no—
466C4⋊C4—341no—
467C4⋊C4—341no—
468C4⋊C4—341no—
469C4⋊C4—341no—
470C4⋊C4—341no—
471C4⋊C4—341no—
472C4⋊C4—341no—
473C4⋊C4—341no—
474C4⋊C4—341no—
475C4⋊C4—341no—
476C4⋊C4—682no—
477C4⋊C4—682no—
478C4⋊C4—682no—
479C4⋊C4—682no—
480C4⋊C4—682no—
481C22⋊C4—682no—
482C22⋊C4—682no—
483C22⋊C4—682no—
484C22⋊C4—682no—
485C22⋊C4—682no—
486C22⋊C4—682no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—344no—
498C42—682no—
499C23—681no—
500C23—681no—
501C23—681no—
502C23—681no—
503C23—681no—
504C23—681no—
505C23—681no—
506C23C22×D6811yescharacteristic
507Q8—342no—
508Q8—342no—
509Q8—681no—
510Q8—681no—
511D4—681no—
512D4—681no—
513D4—681no—
514D4—681no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—342no—
530C2×C4—342no—
531C2×C4—681no—
532C2×C4—681no—
533C2×C4—681no—
534C2×C4—681no—
535C2×C4—681no—
536C2×C4—681no—
537C2×C4—681no—
538C2×C4—681no—
539C2×C4—681no—
540C2×C4—681no—
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#1058759012yes—
551C2×C4#1058759012yes—
552C2×C4#1058759412yes—
553C2×C4#1058759512yes—
554C2×C4C2×#6325211yescharacteristic
555C2×C4C2×#6325211yescharacteristic
556C2×C4#1058769111yescharacteristic
557C2×C4#1058769411yescharacteristic
558C2×C4#41×D1711yescharacteristic
559C2×C4#41×D1711yescharacteristic
560C8—22no—
561C8—22no—
562C8—41no—
563C8—41no—
564C22—1361no—
565C22—1361no—
566C22—1361no—
567C22—1361no—
568C22—1361no—
569C22—1361no—
570C22—1361no—
571C22#1176332111yescharacteristic
572C22#1176335911yescharacteristic
573C22#1176339011yescharacteristic
574C22C2×#1058759011yescharacteristic
575C22C2×#1058759011yescharacteristic
576C22C2×#1058759411yescharacteristic
577C22C2×#1058759511yescharacteristic
578C4—22no—
579C4—22no—
580C4—22no—
581C4—22no—
582C4—41no—
583C4—41no—
584C4—682no—
585C4—682no—
586C4—1361no—
587C4—1361no—
588C4—1361no—
589C4—1361no—
590C4—1361no—
591C4#1176348811yescharacteristic
592C4#1176350011yescharacteristic
593C4#1176350811yescharacteristic
594C4#1176353111yescharacteristic
595C2—2721no—
596C2#1453437511yescharacteristic
597C2#1453884511yescharacteristic
598C2#1453886611yescharacteristic
599C2#1453935411yescharacteristic
600C2#1453950211yescharacteristic
601C2#1454011111yescharacteristic
602C2#1454012211yescharacteristic
subgroupG/H
centerC23C22×D68
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC4×C17×C4⋊C8C2
socleC23×C17C22×D4
prime radicalC22×D68C23
hypercenterC4×C4⋊C8D17
hypocenterC17#29435
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C43 ◁ C4×C4⋊C8

Complements

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

NH ≅ G/Ntype
2. #14531261C2semidirect
3. #14530282C2semidirect
4. #14530178C2semidirect
8. #14529517C2semidirect
10. #14529460C2semidirect
12. #14529370C2semidirect
14. #14529041C2semidirect
15. C4×C17×C4⋊C8C2semidirect
277. C4×C4⋊C8D17semidirect
422. C17#29435semidirect

Maximal subgroups

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

subgroupindex
#145371512
#145312612
#145302822
#145301782
#145301712
#145301022
#145296382
#145295172
#145294672
#145294602
#145294052
#145293702
#145291382
#145290412
C4×C17×C4⋊C82
#2943517