Finite Group Databasebeta

← #17107395

Subgroups — #17107395

Order 4352.

700 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14533206C211yesmaximal, characteristic
2#14530192C211yesmaximal, characteristic, complemented
3#14530165C211yesmaximal, characteristic, complemented
4#14529984C211yesmaximal, characteristic
5#14529984C211yesmaximal, characteristic
6#14529542C211yesmaximal, characteristic, complemented
7#14529519C211yesmaximal, characteristic, complemented
8#14529399C211yesmaximal, characteristic
9#14529373C211yesmaximal, characteristic, complemented
10#14529355C211yesmaximal, characteristic, complemented
11#14529038C211yesmaximal, characteristic, complemented
12#14528983C211yesmaximal, characteristic
13#14528879C211yesmaximal, characteristic
14#14528614C211yesmaximal, characteristic, complemented
15C17×#2362C211yesmaximal, characteristic
16C2×#10587565—21no—
17C2×#10587478C2211yescharacteristic
18C2×#10587478C2211yescharacteristic
19C2×#10587478C2211yescharacteristic
20C2×#10587479C2211yescharacteristic
21C2×C4⋊#3622C2211yescharacteristic
22C2×#10587506C2211yescharacteristic
23C2×#10587506C2211yescharacteristic
24C2×#10587506C2211yescharacteristic
25C2×#10587506C2211yescharacteristic
26C2×C22⋊#3622C2211yescharacteristic
27C2×C22⋊#3622C2211yescharacteristic
28C2×#10587474C2211yescharacteristic
29#11762749—21no—
30#11762734—21no—
31#11762707C2211yescharacteristic
32#11762707C2211yescharacteristic
33#11762694—21no—
34#11762679C2211yescharacteristic
35#11762669C2211yescharacteristic
36#11762669C2211yescharacteristic
37C4×C4⋊Dic17C2211yescharacteristic
38#11762571C2211yescharacteristic
39#11762571C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48C2×C17×C22⋊Q8C2211yescharacteristic
49C2×C4×Q8×C17C2211yescharacteristic
50C2×C17×#117C2211yescharacteristic
51C17×#391C2211yescharacteristic
52C17×#389C2211yescharacteristic
53C17×#388—21no—
54C17×#388—21no—
55C17×#387—21no—
56C4×C4⋊C4×C17—21no—
57C4×C4⋊C4×C17C2211yescharacteristic
58C4×C22⋊C4×C17C2211yescharacteristic
59C22×Q8×C17C2311yescharacteristic
60C23×C4×C17C2311yescharacteristic
61C17×C22⋊Q8—22no—
62C17×C22⋊Q8—22no—
63C4×Q8×C17—22no—
64C4×Q8×C17—41no—
65C17×#117—22no—
66C17×#117—41no—
67C2×C4⋊C4×C17—21no—
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×C22⋊C4×C17—21no—
75C2×C22⋊C4×C17D411yescharacteristic
76C2×C22⋊C4×C17D411yescharacteristic
77C2×C42×C17—21no—
78C2×C42×C17—21no—
79C2×C42×C17—21no—
80C2×C42×C17C2311yescharacteristic
81C2×C42×C17C2311yescharacteristic
82C2×C22⋊Dic17—21no—
83C2×C22⋊Dic17—21no—
84C2×C4⋊Dic17—21no—
85C2×C4⋊Dic17—21no—
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4⋊Dic17C2311yescharacteristic
90C2×#63225—21no—
91C2×#63225—21no—
92C2×C4×Dic17—21no—
93C22×#3622C2311yescharacteristic
94C22×#3622C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96C22×#3622C2311yescharacteristic
97#10587565—44no—
98C17×#95—21no—
99C17×#95—21no—
100C17×#95—21no—
101C17×#95—21no—
102C17×#95D411yescharacteristic
103C17×#95D411yescharacteristic
104#10587506—22no—
105#10587506—22no—
106#10587506—22no—
107#10587506—41no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113C22⋊#3622—22no—
114C22⋊#3622—41no—
115#10587479—41no—
116#10587478—22no—
117#10587478—22no—
118#10587478—22no—
119C4⋊#3622—22no—
120#10587474—41no—
121C24×C17C2×D411yescharacteristic
122C2×Q8×C17—21no—
123C2×Q8×C17—21no—
124C2×Q8×C17—21no—
125C2×Q8×C17—21no—
126C2×Q8×C17—21no—
127C2×Q8×C17#4112yes—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—21no—
137C22×C4×C17—21no—
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2411yescharacteristic
144C22×Dic17—21no—
145C22×Dic17—21no—
146C22×Dic17—21no—
147C22×Dic17—21no—
148C4⋊C4×C17—24no—
149C4⋊C4×C17—21no—
150C4⋊C4×C17—21no—
151C4⋊C4×C17—21no—
152C4⋊C4×C17—21no—
153C4⋊C4×C17—42no—
154C4⋊C4×C17—42no—
155C4⋊C4×C17#4112yes—
156C4⋊C4×C17#4112yes—
157C22⋊C4×C17—24no—
158C22⋊C4×C17—22no—
159C22⋊C4×C17—41no—
160C22⋊C4×C17—42no—
161C42×C17—21no—
162C42×C17—21no—
163C42×C17—21no—
164C42×C17—21no—
165C42×C17—42no—
166C42×C17—42no—
167C42×C17—42no—
168C22⋊Dic17—42no—
169C22⋊Dic17—42no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17—21no—
175C4⋊Dic17—21no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—42no—
179C4⋊Dic17—42no—
180#63225—42no—
181#63225—42no—
182C4×Dic17—42no—
183C2×#3622—21no—
184C2×#3622—21no—
185C2×#3622—21no—
186C2×#3622—21no—
187C2×#3622—21no—
188C2×#3622—21no—
189C2×#3622—21no—
190C2×#3622—21no—
191C2×#3622—21no—
192C2×#3622—21no—
193C2×#3622—21no—
194C2×#3622#4112yes—
195#27058—171nomaximal, Sylow
196C23×C17—21no—
197C23×C17—21no—
198C23×C17—21no—
199C23×C17—21no—
200C23×C17—21no—
201C23×C17—21no—
202C23×C17—21no—
203C23×C17C22×D411yescharacteristic
204Q8×C17—22no—
205Q8×C17—22no—
206Q8×C17—41no—
207Q8×C17—41no—
208C2×C4×C17—21no—
209C2×C4×C17—21no—
210C2×C4×C17—22no—
211C2×C4×C17—22no—
212C2×C4×C17—21no—
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—21no—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—41no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—41no—
230C2×C4×C17—41no—
231C2×C4×C17—41no—
232C2×C4×C17—41no—
233C2×C4×C17—41no—
234C2×C4×C17—41no—
235C2×C4×C17—41no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17AGL(1,ℤ/8)12yes—
239C2×C4×C17C2×#4111yescharacteristic
240C2×C4×C17C2×#4111yescharacteristic
241C2×C4×C17C2×#4111yescharacteristic
242C2×C4×C17C2×#4111yescharacteristic
243C2×C4×C17H(5,𝔽2)11yescharacteristic
244C2×C4×C172−1+411yescharacteristic
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—41no—
257#3622—22no—
258#3622—41no—
259#3622—41no—
260#3622—41no—
261#2496—171no—
262#2362D1711yescharacteristic
263#2072—171no—
264#2066—171no—
265#2036—171no—
266#2036—171no—
267#1947—171no—
268#1943—171no—
269#1924—171no—
270#1921—171no—
271#1918—171no—
272#1863—171no—
273#1854—171no—
274#1825—171no—
275#1763—171no—
276C22×C17—41no—
277C22×C17—41no—
278C22×C17—41no—
279C22×C17—41no—
280C22×C17—41no—
281C22×C17—41no—
282C22×C17—41no—
283C22×C17#52511yescharacteristic
284C22×C17#54611yescharacteristic
285C22×C17#54711yescharacteristic
286C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
287C22×C17#57511yescharacteristic
288C22×C17#57811yescharacteristic
289C22×C17#57811yescharacteristic
290C4×C17—22no—
291C4×C17—41no—
292C4×C17—42no—
293C4×C17—41no—
294C4×C17—42no—
295C4×C17—41no—
296C4×C17—42no—
297C4×C17—41no—
298C4×C17—81no—
299C4×C17#52911yescharacteristic
300C4×C17#53911yescharacteristic
301C4×C17#55311yescharacteristic
302C4×C17#55611yescharacteristic
303Dic17—81no—
304Dic17—81no—
305Dic17—81no—
306Dic17—81no—
307C2×#126—341no—
308C2×C22⋊Q8C2×D1711yescharacteristic
309C2×C4×Q8C2×D1711yescharacteristic
310C2×#117C2×D1711yescharacteristic
311C2×#107—171no—
312C2×#107—171no—
313C2×#107—171no—
314C2×#106—171no—
315C2×C4⋊C8—171no—
316C2×#103—171no—
317C2×#103—171no—
318C2×#103—171no—
319C2×#103—171no—
320C2×C22⋊C8—171no—
321C2×C22⋊C8—171no—
322C2×#97—171no—
323#400—341no—
324#397—341no—
325#391C2×D1711yescharacteristic
326#389—171no—
327#389—171no—
328#389C2×D1711yescharacteristic
329#388—21no—
330#388—21no—
331#387—21no—
332#387—341no—
333#384—171no—
334#380—171no—
335#380—171no—
336C4×C4⋊C4—21no—
337C4×C4⋊C4—171no—
338C4×C4⋊C4C2×D1711yescharacteristic
339C4×C22⋊C4C2×D1711yescharacteristic
340#340—171no—
341#340—171no—
342#340—171no—
343#340—171no—
344#340—171no—
345#340—171no—
346#340—171no—
347#340—171no—
348#336—171no—
349#336—171no—
350C2×C17—81no—
351C2×C17#246611yescharacteristic
352C2×C17#311511yescharacteristic
353C2×C17#312711yescharacteristic
354C2×C17#331311yescharacteristic
355C2×C17#333011yescharacteristic
356C2×C17#334311yescharacteristic
357C2×C17#334911yescharacteristic
358C22×Q8C22×D1711yescharacteristic
359C23×C4C22×D1711yescharacteristic
360C22×C8—171no—
361C22×C8—171no—
362C22×C8—171no—
363C22×C8—171no—
364#126—684no—
365C22⋊Q8—22no—
366C22⋊Q8—22no—
367C4×Q8—22no—
368C4×Q8—41no—
369#117—22no—
370#117—41no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
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⋊C4C22×D1711yescharacteristic
383C2×C4⋊C4C22×D1711yescharacteristic
384C2×C4⋊C4C22×D1711yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—341no—
388C2×C22⋊C4—341no—
389C2×C22⋊C4C22⋊D1711yescharacteristic
390C2×C22⋊C4C22⋊D1711yescharacteristic
391C2×C42—21no—
392C2×C42—21no—
393C2×C42—21no—
394C2×C42—341no—
395C2×C42C22×D1711yescharacteristic
396C2×C42C22×D1711yescharacteristic
397#107—342no—
398#107—342no—
399#107—342no—
400#106—681no—
401C4⋊C8—342no—
402#103—342no—
403#103—342no—
404#103—342no—
405#103—681no—
406C22⋊C8—342no—
407C22⋊C8—681no—
408#97—681no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—341no—
414#95—341no—
415#95—341no—
416#95—341no—
417#95—341no—
418#95C22⋊D1711yescharacteristic
419#95C22⋊D1711yescharacteristic
420C17#2705811yescharacteristic, complemented, Sylow
421C24C2×C22⋊D1711yescharacteristic
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8#6325212yes—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—341no—
439C22×C4—341no—
440C22×C4—341no—
441C22×C4—341no—
442C22×C4C2×C22⋊D1711yescharacteristic
443C22×C4C2×C22⋊D1711yescharacteristic
444C22×C4C2×C22⋊D1711yescharacteristic
445C22×C4C2×C22⋊D1711yescharacteristic
446C22×C4C2×C22⋊D1711yescharacteristic
447C22×C4C23×D1711yescharacteristic
448C2×C8—172no—
449C2×C8—341no—
450C2×C8—341no—
451C2×C8—341no—
452C2×C8—341no—
453C2×C8—341no—
454C2×C8—341no—
455C2×C8—341no—
456C2×C8—341no—
457C2×C8—341no—
458C2×C8—341no—
459C2×C8—341no—
460C4⋊C4—24no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—42no—
466C4⋊C4—42no—
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—682no—
476C4⋊C4—682no—
477C4⋊C4—682no—
478C4⋊C4—682no—
479C4⋊C4#6325212yes—
480C4⋊C4#6325412yes—
481C22⋊C4—24no—
482C22⋊C4—22no—
483C22⋊C4—41no—
484C22⋊C4—42no—
485C22⋊C4—682no—
486C22⋊C4—682no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—42no—
492C42—42no—
493C42—42no—
494C42—682no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23C22×C22⋊D1711yescharacteristic
503Q8—22no—
504Q8—22no—
505Q8—41no—
506Q8—41no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
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—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
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#1058763512yes—
550C2×C4C2×#6325211yescharacteristic
551C2×C4C2×#6325211yescharacteristic
552C2×C4C2×#6325411yescharacteristic
553C2×C4#1058769111yescharacteristic
554C2×C4#41×D1711yescharacteristic
555C2×C4#1058769711yescharacteristic
556C8—342no—
557C8—681no—
558C8—681no—
559C8—681no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22#1176332011yescharacteristic
568C22#1176345311yescharacteristic
569C22#1176345811yescharacteristic
570C22C2×#1058763511yescharacteristic
571C22#1176360711yescharacteristic
572C22#1176362411yescharacteristic
573C22#1176362411yescharacteristic
574C4—22no—
575C4—41no—
576C4—42no—
577C4—41no—
578C4—42no—
579C4—41no—
580C4—42no—
581C4—41no—
582C4—81no—
583C4—1361no—
584C4—1361no—
585C4—1361no—
586C4—1361no—
587C4#1176333911yescharacteristic
588C4#1176340111yescharacteristic
589C4#1176350911yescharacteristic
590C4#1176352911yescharacteristic
591C2—81no—
592C2#1453281411yescharacteristic
593C2#1453893511yescharacteristic
594C2#1453908411yescharacteristic
595C2#1454085011yescharacteristic
596C2#1454108911yescharacteristic
597C2#1454122411yescharacteristic
598C2#1454130811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2362C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#2362D17
hypocenterC17#27058
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
2. #14530192C2semidirect
3. #14530165C2semidirect
6. #14529542C2semidirect
7. #14529519C2semidirect
9. #14529373C2semidirect
10. #14529355C2semidirect
11. #14529038C2semidirect
14. #14528614C2semidirect
420. C17#27058semidirect

Maximal subgroups

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

subgroupindex
#145332062
#145301922
#145301652
#145299842
#145295422
#145295192
#145293992
#145293732
#145293552
#145290382
#145289832
#145288792
#145286142
C17×#23622
#2705817