Finite Group Databasebeta

← #17092313

Subgroups — #17092313

Order 4352.

732 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531651C211yesmaximal, characteristic
2#14530346C211yesmaximal, characteristic, complemented
3#14530285C211yesmaximal, characteristic, complemented
4#14530275C211yesmaximal, characteristic, complemented
5#14530119C211yesmaximal, characteristic, complemented
6#14529742C211yesmaximal, characteristic, complemented
7#14529545C211yesmaximal, characteristic, complemented
8#14529373C211yesmaximal, characteristic, complemented
9#14529352C211yesmaximal, characteristic, complemented
10#14528983C211yesmaximal, characteristic
11#14528969C211yesmaximal, characteristic
12#14528872C211yesmaximal, characteristic
13#14528815C211yesmaximal, characteristic
14#14528505C211yesmaximal, characteristic
15C17×#2441C211yesmaximal, characteristic
16C22×#63223C2211yescharacteristic
17C2×#10587563—21no—
18C22×C4⋊Dic17C2211yescharacteristic
19C2×#10587478C2211yescharacteristic
20C2×#10587478C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×C4⋊#3622C2211yescharacteristic
24C2×C4⋊#3622C2211yescharacteristic
25C2×C22⋊#3622C2211yescharacteristic
26#11762707C2211yescharacteristic
27#11762707C2211yescharacteristic
28#11762694—21no—
29#11762694—21no—
30#11762670—21no—
31#11762669C2211yescharacteristic
32C4×C4⋊Dic17C2211yescharacteristic
33C4×C4⋊Dic17C2211yescharacteristic
34#11762571C2211yescharacteristic
35#11762571C2211yescharacteristic
36#11762571C2211yescharacteristic
37#11762571C2211yescharacteristic
38#11762571C2211yescharacteristic
39#11762571C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48C2×C17×#125C2211yescharacteristic
49C2×C17×#117C2211yescharacteristic
50C17×#399—21no—
51C17×#396—21no—
52C17×#389—21no—
53C17×#389C2211yescharacteristic
54C17×#389C2211yescharacteristic
55C17×#387—21no—
56C17×#384C2211yescharacteristic
57C17×#380C2211yescharacteristic
58C4×C22⋊C4×C17C2211yescharacteristic
59C23×C4×C17C2311yescharacteristic
60C23×Dic17—21no—
61C17×#125—22no—
62C17×#125—41no—
63C17×#117—22no—
64C17×#117—41no—
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17—21no—
67C2×C4⋊C4×C17—21no—
68C2×C4⋊C4×C17—21no—
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C22⋊C4×C17—21no—
76C2×C22⋊C4×C17—21no—
77C2×C22⋊C4×C17D411yescharacteristic
78C2×C22⋊C4×C17D411yescharacteristic
79C2×C42×C17—21no—
80C2×C42×C17C2311yescharacteristic
81C2×C42×C17C2311yescharacteristic
82C2×C22⋊Dic17—21no—
83C2×C4⋊Dic17—21no—
84C2×C4⋊Dic17—22no—
85C2×C4⋊Dic17Q814yes—
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4⋊Dic17C2311yescharacteristic
90C2×#63225—21no—
91C2×#63225—21no—
92C2×C4×Dic17—21no—
93C2×C4×Dic17—21no—
94C2×#63223—22no—
95C2×#63223—22no—
96C2×#63223—41no—
97C22×#3622C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99C22×#3622C2311yescharacteristic
100C22×#3622C2311yescharacteristic
101#10587563—44no—
102C17×#95—21no—
103C17×#95—21no—
104C17×#95—21no—
105C17×#95D411yescharacteristic
106C17×#95D411yescharacteristic
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112C22⋊#3622—22no—
113#10587479—41no—
114#10587479—41no—
115#10587478—22no—
116#10587478—22no—
117C4⋊#3622—22no—
118C4⋊#3622—41no—
119C24×C17C2×D411yescharacteristic
120C22×C4×C17—21no—
121C22×C4×C17—21no—
122C22×C4×C17—21no—
123C22×C4×C17—21no—
124C22×C4×C17—21no—
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17C2×D411yescharacteristic
129C22×C4×C17C2×D411yescharacteristic
130C22×C4×C17C2×D411yescharacteristic
131C22×C4×C17C2×D411yescharacteristic
132C22×C4×C17C2×D411yescharacteristic
133C22×C4×C17C2×Q812yes—
134C22×C4×C17#4112yes—
135C22×C4×C17C2411yescharacteristic
136C22×Dic17—24no—
137C22×Dic17—21no—
138C22×Dic17—21no—
139C22×Dic17—21no—
140C22×Dic17—21no—
141C22×Dic17—42no—
142C22×Dic17—42no—
143C4⋊C4×C17—21no—
144C4⋊C4×C17—21no—
145C4⋊C4×C17—21no—
146C4⋊C4×C17—21no—
147C4⋊C4×C17—21no—
148C4⋊C4×C17—21no—
149C4⋊C4×C17—21no—
150C4⋊C4×C17—42no—
151C4⋊C4×C17—42no—
152C4⋊C4×C17—42no—
153C4⋊C4×C17—42no—
154C4⋊C4×C17—81no—
155C4⋊C4×C17—81no—
156C4⋊C4×C17#4112yes—
157C22⋊C4×C17—22no—
158C22⋊C4×C17—41no—
159C22⋊C4×C17—42no—
160C22⋊C4×C17—81no—
161C42×C17—21no—
162C42×C17—21no—
163C42×C17—21no—
164C42×C17—21no—
165C42×C17—42no—
166C22⋊Dic17—42no—
167C4⋊Dic17—24no—
168C4⋊Dic17—21no—
169C4⋊Dic17—21no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17C2×Q812yes—
175C4⋊Dic17C2×Q812yes—
176#63225—42no—
177#63225—42no—
178C4×Dic17—42no—
179C4×Dic17—42no—
180#63223—44no—
181C2×#3622—21no—
182C2×#3622—21no—
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#4112yes—
193#26014—171nomaximal, Sylow
194C23×C17—21no—
195C23×C17—21no—
196C23×C17—21no—
197C23×C17—21no—
198C23×C17—21no—
199C23×C17—21no—
200C23×C17C22⋊Q812yes—
201C23×C17C22×D411yescharacteristic
202C2×C4×C17—21no—
203C2×C4×C17—21no—
204C2×C4×C17—21no—
205C2×C4×C17—21no—
206C2×C4×C17—22no—
207C2×C4×C17—22no—
208C2×C4×C17—22no—
209C2×C4×C17—22no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—22no—
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—41no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
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×C17C22⋊Q812yes—
232C2×C4×C17AGL(1,ℤ/8)12yes—
233C2×C4×C17C22×Q811yescharacteristic
234C2×C4×C17C2×#4111yescharacteristic
235C2×C4×C17C2×#4111yescharacteristic
236C2×C4×C17C2×#4111yescharacteristic
237C2×C4×C17H(5,𝔽2)11yescharacteristic
238C2×C4×C172−1+411yescharacteristic
239C2×Dic17—22no—
240C2×Dic17—22no—
241C2×Dic17—48no—
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—
251C2×Dic17—41no—
252#3622—22no—
253#3622—41no—
254#3622—41no—
255#3622—41no—
256#2441D1711yescharacteristic
257#2344—171no—
258#2100—171no—
259#2088—171no—
260#2086—171no—
261#2059—171no—
262#1989—171no—
263#1948—171no—
264#1921—171no—
265#1918—171no—
266#1854—171no—
267#1851—171no—
268#1824—171no—
269#1809—171no—
270#1736—171no—
271C22×C17—22no—
272C22×C17—22no—
273C22×C17—22no—
274C22×C17—41no—
275C22×C17—41no—
276C22×C17—41no—
277C22×C17—41no—
278C22×C17C2×C22⋊Q811yescharacteristic
279C22×C17#54611yescharacteristic
280C22×C17#54711yescharacteristic
281C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
282C22×C17#57511yescharacteristic
283C22×C17#57711yescharacteristic
284C22×C17#57811yescharacteristic
285C4×C17—22no—
286C4×C17—22no—
287C4×C17—41no—
288C4×C17—42no—
289C4×C17—41no—
290C4×C17—41no—
291C4×C17—81no—
292C4×C17—81no—
293C4×C17—81no—
294C4×C17#53311yescharacteristic
295C4×C17#54411yescharacteristic
296C4×C17#55111yescharacteristic
297C4×C17#55211yescharacteristic
298Dic17—42no—
299Dic17—42no—
300Dic17—81no—
301Dic17—81no—
302C22×M16—171no—
303C2×#125C2×D1711yescharacteristic
304C2×#117—341no—
305C2×#117C2×D1711yescharacteristic
306C22×C4⋊C4—171no—
307C2×#107—171no—
308C2×#107—171no—
309C2×#106—171no—
310C2×#106—171no—
311C2×C4⋊C8—171no—
312C2×C4⋊C8—171no—
313C2×C22⋊C8—171no—
314#399—21no—
315#396—21no—
316#389—21no—
317#389—171no—
318#389—171no—
319#389C2×D1711yescharacteristic
320#389C2×D1711yescharacteristic
321#387—21no—
322#387—341no—
323#387—341no—
324#384C2×D1711yescharacteristic
325#381—341no—
326#380—171no—
327#380C2×D1711yescharacteristic
328C4×C4⋊C4—171no—
329C4×C4⋊C4—171no—
330C4×C22⋊C4C2×D1711yescharacteristic
331#340—171no—
332#340—171no—
333#340—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#340—171no—
343#336—171no—
344#336—171no—
345C2×C17—42no—
346C2×C17#243811yescharacteristic
347C2×C17#307711yescharacteristic
348C2×C17#308211yescharacteristic
349C2×C17#331711yescharacteristic
350C2×C17#333011yescharacteristic
351C2×C17#334511yescharacteristic
352C2×C17#335211yescharacteristic
353C23×C4—341no—
354C23×C4C22×D1711yescharacteristic
355C2×M16—342no—
356C2×M16—342no—
357C2×M16—681no—
358C22×C8—171no—
359C22×C8—171no—
360C22×C8—171no—
361C22×C8—171no—
362#125—22no—
363#125—41no—
364#117—22no—
365#117—41no—
366#117—684no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—174no—
374C2×C4⋊C4—171no—
375C2×C4⋊C4—171no—
376C2×C4⋊C4—171no—
377C2×C4⋊C4—171no—
378C2×C4⋊C4—342no—
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×C4⋊C4C22×D1711yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—341no—
389C2×C22⋊C4C22⋊D1711yescharacteristic
390C2×C22⋊C4C22⋊D1711yescharacteristic
391C2×C42—21no—
392C2×C42—341no—
393C2×C42—341no—
394C2×C42C22×D1711yescharacteristic
395C2×C42C22×D1711yescharacteristic
396#107—342no—
397#107—342no—
398#106—681no—
399#106—681no—
400C4⋊C8—342no—
401C4⋊C8—681no—
402C22⋊C8—342no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—341no—
407#95—341no—
408#95—341no—
409#95—341no—
410#95—341no—
411#95C22⋊D1711yescharacteristic
412#95C22⋊D1711yescharacteristic
413C17#2601411yescharacteristic, complemented, Sylow
414C24C2×C22⋊D1711yescharacteristic
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—341no—
424C22×C4—341no—
425C22×C4—344no—
426C22×C4—341no—
427C22×C4—341no—
428C22×C4—682no—
429C22×C4—682no—
430C22×C4Q8×D1712yes—
431C22×C4#6325612yes—
432C22×C4C2×C22⋊D1711yescharacteristic
433C22×C4C2×C22⋊D1711yescharacteristic
434C22×C4C2×C22⋊D1711yescharacteristic
435C22×C4C2×C22⋊D1711yescharacteristic
436C22×C4C2×C22⋊D1711yescharacteristic
437C22×C4C23×D1711yescharacteristic
438M16—684no—
439C2×C8—172no—
440C2×C8—341no—
441C2×C8—341no—
442C2×C8—341no—
443C2×C8—341no—
444C2×C8—341no—
445C2×C8—341no—
446C2×C8—341no—
447C2×C8—341no—
448C2×C8—341no—
449C2×C8—341no—
450C2×C8—341no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—81no—
463C4⋊C4—81no—
464C4⋊C4—172no—
465C4⋊C4—172no—
466C4⋊C4—341no—
467C4⋊C4—341no—
468C4⋊C4—344no—
469C4⋊C4—341no—
470C4⋊C4—341no—
471C4⋊C4—341no—
472C4⋊C4—341no—
473C4⋊C4—682no—
474C4⋊C4—682no—
475C4⋊C4#6325412yes—
476C22⋊C4—22no—
477C22⋊C4—41no—
478C22⋊C4—42no—
479C22⋊C4—81no—
480C22⋊C4—682no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—42no—
486C42—682no—
487C42—682no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23#1058763212yes—
495C23C22×C22⋊D1711yescharacteristic
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—342no—
526C2×C4—342no—
527C2×C4—688no—
528C2×C4—681no—
529C2×C4—681no—
530C2×C4—681no—
531C2×C4—681no—
532C2×C4—681no—
533C2×C4—681no—
534C2×C4—681no—
535C2×C4—681no—
536C2×C4—681no—
537C2×C4—681no—
538C2×C4#1058763212yes—
539C2×C4#1058763512yes—
540C2×C4C2×#6325411yescharacteristic
541C2×C4#1058769111yescharacteristic
542C2×C4C2×Q8×D1711yescharacteristic
543C2×C4C2×#6325611yescharacteristic
544C2×C4#41×D1711yescharacteristic
545C2×C4#1058769711yescharacteristic
546C8—342no—
547C8—681no—
548C8—681no—
549C8—681no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22C2×#1058763211yescharacteristic
558C22#1176345311yescharacteristic
559C22#1176345811yescharacteristic
560C22C2×#1058763511yescharacteristic
561C22#1176360711yescharacteristic
562C22#1176362011yescharacteristic
563C22#1176362411yescharacteristic
564C4—22no—
565C4—22no—
566C4—41no—
567C4—42no—
568C4—41no—
569C4—41no—
570C4—81no—
571C4—81no—
572C4—81no—
573C4—682no—
574C4—682no—
575C4—1361no—
576C4—1361no—
577C4#1176336111yescharacteristic
578C4#1176343811yescharacteristic
579C4#1176348611yescharacteristic
580C4#1176350111yescharacteristic
581C2—42no—
582C2#1453245611yescharacteristic
583C2#1453857711yescharacteristic
584C2#1453862011yescharacteristic
585C2#1454091011yescharacteristic
586C2#1454108911yescharacteristic
587C2#1454124811yescharacteristic
588C2#1454135311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2441C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#2441D17
hypocenterC17#26014
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2441

Complements

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

NH ≅ G/Ntype
2. #14530346C2semidirect
3. #14530285C2semidirect
4. #14530275C2semidirect
5. #14530119C2semidirect
6. #14529742C2semidirect
7. #14529545C2semidirect
8. #14529373C2semidirect
9. #14529352C2semidirect
413. C17#26014semidirect

Maximal subgroups

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

subgroupindex
#145316512
#145303462
#145302852
#145302752
#145301192
#145297422
#145295452
#145293732
#145293522
#145289832
#145289692
#145288722
#145288152
#145285052
C17×#24412
#2601417