Finite Group Databasebeta

← #17093427

Subgroups — #17093427

Order 4352.

708 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#14534018C211yesmaximal, characteristic, complemented
2#14530312C211yesmaximal, characteristic
3#14530305C211yesmaximal, characteristic
4#14530172C211yesmaximal, characteristic
5#14530158C211yesmaximal, characteristic
6#14529552C211yesmaximal, characteristic, complemented
7#14529542C211yesmaximal, characteristic, complemented
8#14529540C211yesmaximal, characteristic, complemented
9#14529508C211yesmaximal, characteristic, complemented
10#14529376C211yesmaximal, characteristic, complemented
11#14529369C211yesmaximal, characteristic, complemented
12#14528950C211yesmaximal, characteristic, complemented
13#14528794C211yesmaximal, characteristic
14#14528609C211yesmaximal, characteristic
15C17×#2354C211yesmaximal, characteristic
16C2×#10587478C2211yescharacteristic
17C2×#10587478C2211yescharacteristic
18C2×#10587478C2211yescharacteristic
19C2×#10587478C2211yescharacteristic
20C2×#10587479C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×C4⋊#3622C2211yescharacteristic
23C2×C4⋊#3622C2211yescharacteristic
24C2×C4⋊#3622C2211yescharacteristic
25C2×#10587503C2211yescharacteristic
26C2×#10587503C2211yescharacteristic
27C2×#10587503C2211yescharacteristic
28C2×#10587503C2211yescharacteristic
29C2×#10587474C2211yescharacteristic
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
49C2×C17×#117—21no—
50C17×#390C2211yescharacteristic
51C17×#390C2211yescharacteristic
52C17×#388—21no—
53C17×#388—21no—
54C17×#384C2211yescharacteristic
55C17×#380—21no—
56C4×C4⋊C4×C17C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
58C4×C4⋊C4×C17C2211yescharacteristic
59C22×D4×C17C2311yescharacteristic
60C23×C4×C17—21no—
61C4×D4×C17—22no—
62C4×D4×C17Q814yes—
63C17×#117—44no—
64C2×C4⋊C4×C17—21no—
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17D411yescharacteristic
67C2×C4⋊C4×C17D411yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C22⋊C4×C17—21no—
72C2×C22⋊C4×C17—21no—
73C2×C22⋊C4×C17—21no—
74C2×C22⋊C4×C17—21no—
75C2×C22⋊C4×C17—21no—
76C2×C42×C17—21no—
77C2×C42×C17—21no—
78C2×C42×C17C2311yescharacteristic
79C2×C42×C17C2311yescharacteristic
80C2×C42×C17C2311yescharacteristic
81C2×C4⋊Dic17—21no—
82C2×C4⋊Dic17—21no—
83C2×C4⋊Dic17C2311yescharacteristic
84C2×C4⋊Dic17C2311yescharacteristic
85C2×C4⋊Dic17C2311yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×#63225—21no—
88C2×#63225—21no—
89C2×C4×Dic17—21no—
90C22×#3622C2311yescharacteristic
91C22×#3622C2311yescharacteristic
92C22×#3622C2311yescharacteristic
93C22×#3622C2311yescharacteristic
94C17×#95—21no—
95C17×#95—21no—
96C17×#95D411yescharacteristic
97C17×#95D411yescharacteristic
98#10587503—22no—
99#10587503—22no—
100#10587503—41no—
101#10587503—41no—
102#10587502—21no—
103#10587502—21no—
104#10587502—21no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587479—22no—
110#10587479—41no—
111#10587478—22no—
112#10587478—41no—
113#10587478—41no—
114#10587478—41no—
115C4⋊#3622—22no—
116C4⋊#3622—22no—
117C4⋊#3622—22no—
118#10587474—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×C17C2×Q812yes—
126C22×C4×C17—21no—
127C22×C4×C17—24no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—42no—
131C22×C4×C17—42no—
132C22×C4×C17C2×D411yescharacteristic
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17C2×D411yescharacteristic
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2411yescharacteristic
139C22×Dic17—21no—
140C22×Dic17—21no—
141C22×Dic17—21no—
142C22×Dic17—21no—
143C4⋊C4×C17—22no—
144C4⋊C4×C17—21no—
145C4⋊C4×C17—21no—
146C4⋊C4×C17—21no—
147C4⋊C4×C17—21no—
148C4⋊C4×C17—21no—
149C4⋊C4×C17—41no—
150C4⋊C4×C17—42no—
151C4⋊C4×C17—81no—
152C4⋊C4×C17C2×Q812yes—
153C22⋊C4×C17—24no—
154C22⋊C4×C17—42no—
155C22⋊C4×C17—42no—
156C22⋊C4×C17—81no—
157C22⋊C4×C17—81no—
158C42×C17—21no—
159C42×C17—21no—
160C42×C17—21no—
161C42×C17—21no—
162C42×C17—42no—
163C42×C17—42no—
164C42×C17C2×Q812yes—
165C42×C17#4112yes—
166C4⋊Dic17—21no—
167C4⋊Dic17—21no—
168C4⋊Dic17—21no—
169C4⋊Dic17—21no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17—42no—
175C4⋊Dic17—81no—
176#63225—42no—
177#63225—81no—
178C4×Dic17—81no—
179C2×#3622—21no—
180C2×#3622—21no—
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#4112yes—
190C2×#3622#4112yes—
191#26098—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—21no—
209C2×C4×C17—21no—
210C2×C4×C17—22no—
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—48no—
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×C17C22⋊Q812yes—
230C2×C4×C17C22⋊Q812yes—
231C2×C4×C17AGL(1,ℤ/8)12yes—
232C2×C4×C17C22×Q811yescharacteristic
233C2×C4×C17C2×#4111yescharacteristic
234C2×C4×C17C2×#4111yescharacteristic
235C2×C4×C17C2×#4111yescharacteristic
236C2×C4×C17H(5,𝔽2)11yescharacteristic
237C2×C4×C172−1+411yescharacteristic
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—
250#3622—22no—
251#3622—22no—
252#3622—41no—
253#3622—41no—
254#2570—171no—
255#2354D1711yescharacteristic
256#2093—171no—
257#2092—171no—
258#2067—171no—
259#2065—171no—
260#1949—171no—
261#1947—171no—
262#1947—171no—
263#1940—171no—
264#1921—171no—
265#1920—171no—
266#1847—171no—
267#1804—171no—
268#1762—171no—
269C22×C17—42no—
270C22×C17—42no—
271C22×C17—42no—
272C22×C17—81no—
273C22×C17—81no—
274C22×C17—81no—
275C22×C17—81no—
276C22×C17C2×C22⋊Q811yescharacteristic
277C22×C17#54711yescharacteristic
278C22×C17#55011yescharacteristic
279C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C17#57511yescharacteristic
281C22×C17#57611yescharacteristic
282C22×C17#57811yescharacteristic
283C4×C17—22no—
284C4×C17—22no—
285C4×C17—22no—
286C4×C17—42no—
287C4×C17—41no—
288C4×C17—41no—
289C4×C17—41no—
290C4×C17—81no—
291C4×C17—81no—
292C4×C17#53311yescharacteristic
293C4×C17#54411yescharacteristic
294C4×C17#55211yescharacteristic
295C4×C17#55311yescharacteristic
296Dic17—81no—
297Dic17—81no—
298Dic17—81no—
299Dic17—81no—
300C2×C4×D4C2×D1711yescharacteristic
301C2×#117—21no—
302C2×#107—171no—
303C2×#107—171no—
304C2×#107—171no—
305C2×#107—171no—
306C2×#106—171no—
307C2×#106—171no—
308C2×C4⋊C8—171no—
309C2×C4⋊C8—171no—
310C2×C4⋊C8—171no—
311C2×#102—171no—
312C2×#102—171no—
313C2×#102—171no—
314C2×#102—171no—
315C2×#97—171no—
316#400—341no—
317#398—341no—
318#390C2×D1711yescharacteristic
319#390C2×D1711yescharacteristic
320#389—171no—
321#389—171no—
322#388—21no—
323#388—21no—
324#387—341no—
325#387—341no—
326#384—171no—
327#384C2×D1711yescharacteristic
328#382—171no—
329#382—171no—
330#380—21no—
331C4×C4⋊C4—171no—
332C4×C4⋊C4C2×D1711yescharacteristic
333C4×C4⋊C4C2×D1711yescharacteristic
334C4×C4⋊C4C2×D1711yescharacteristic
335#340—171no—
336#340—171no—
337#340—171no—
338#340—171no—
339#340—171no—
340#340—171no—
341#336—171no—
342#336—171no—
343C2×C17—82no—
344C2×C17#244111yescharacteristic
345C2×C17#307711yescharacteristic
346C2×C17#308311yescharacteristic
347C2×C17#334311yescharacteristic
348C2×C17#334711yescharacteristic
349C2×C17#339911yescharacteristic
350C2×C17#340211yescharacteristic
351C22×D4C22×D1711yescharacteristic
352C23×C4—21no—
353C22×C8—171no—
354C22×C8—171no—
355C22×C8—171no—
356C22×C8—171no—
357C4×D4—22no—
358C4×D4Dic3414yes—
359#117—44no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—171no—
363C2×C4⋊C4—171no—
364C2×C4⋊C4—171no—
365C2×C4⋊C4—171no—
366C2×C4⋊C4—341no—
367C2×C4⋊C4—341no—
368C2×C4⋊C4—341no—
369C2×C4⋊C4—341no—
370C2×C4⋊C4C22⋊D1711yescharacteristic
371C2×C4⋊C4C22⋊D1711yescharacteristic
372C2×C4⋊C4C22×D1711yescharacteristic
373C2×C4⋊C4C22×D1711yescharacteristic
374C2×C4⋊C4C22×D1711yescharacteristic
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42—341no—
383C2×C42C22×D1711yescharacteristic
384C2×C42C22×D1711yescharacteristic
385C2×C42C22×D1711yescharacteristic
386#107—342no—
387#107—681no—
388#107—681no—
389#107—681no—
390#106—342no—
391#106—681no—
392C4⋊C8—342no—
393C4⋊C8—342no—
394C4⋊C8—342no—
395#102—342no—
396#102—342no—
397#102—681no—
398#102—681no—
399#97—342no—
400#95—21no—
401#95—21no—
402#95—341no—
403#95—341no—
404#95—341no—
405#95—341no—
406#95—341no—
407#95—341no—
408#95—341no—
409#95C22⋊D1711yescharacteristic
410#95C22⋊D1711yescharacteristic
411C17#2609811yescharacteristic, complemented, Sylow
412C24—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4C2×Dic3412yes—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—341no—
426C22×C4—341no—
427C22×C4—341no—
428C22×C4—341no—
429C22×C4C2×C22⋊D1711yescharacteristic
430C22×C4C2×C22⋊D1711yescharacteristic
431C22×C4C2×C22⋊D1711yescharacteristic
432C22×C4C2×C22⋊D1711yescharacteristic
433C22×C4C2×C22⋊D1711yescharacteristic
434C22×C4C2×C22⋊D1711yescharacteristic
435C22×C4C23×D1711yescharacteristic
436C2×C8—172no—
437C2×C8—172no—
438C2×C8—341no—
439C2×C8—341no—
440C2×C8—341no—
441C2×C8—341no—
442C2×C8—341no—
443C2×C8—341no—
444C2×C8—341no—
445C2×C8—341no—
446C2×C8—341no—
447C2×C8—341no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—41no—
455C4⋊C4—42no—
456C4⋊C4—81no—
457C4⋊C4—341no—
458C4⋊C4—341no—
459C4⋊C4—341no—
460C4⋊C4—341no—
461C4⋊C4—341no—
462C4⋊C4—341no—
463C4⋊C4—341no—
464C4⋊C4—341no—
465C4⋊C4—682no—
466C4⋊C4—682no—
467C4⋊C4—1361no—
468C4⋊C4—1361no—
469C4⋊C4C2×Dic3412yes—
470C22⋊C4—24no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—81no—
474C22⋊C4—81no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—42no—
480C42—42no—
481C42—1361no—
482C42C2×Dic3412yes—
483C42#6325212yes—
484C23—22no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23C22×C22⋊D1711yescharacteristic
492D4—22no—
493D4—22no—
494D4—41no—
495D4—41no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—48no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—681no—
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#1058761012yes—
534C2×C4#1058761012yes—
535C2×C4#1058761812yes—
536C2×C4C22×Dic3411yescharacteristic
537C2×C4C2×#6325211yescharacteristic
538C2×C4#41×D1711yescharacteristic
539C2×C4#41×D1711yescharacteristic
540C2×C4#1058769611yescharacteristic
541C2×C4#1058769711yescharacteristic
542C8—342no—
543C8—342no—
544C8—681no—
545C8—681no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22C2×#1058761011yescharacteristic
554C22#1176345811yescharacteristic
555C22#1176347811yescharacteristic
556C22C2×#1058761811yescharacteristic
557C22#1176360711yescharacteristic
558C22#1176361211yescharacteristic
559C22#1176362411yescharacteristic
560C4—22no—
561C4—22no—
562C4—22no—
563C4—42no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—81no—
568C4—81no—
569C4—1361no—
570C4—1361no—
571C4—1361no—
572C4—1361no—
573C4#1176336211yescharacteristic
574C4#1176343711yescharacteristic
575C4#1176349811yescharacteristic
576C4#1176350311yescharacteristic
577C2—82no—
578C2#1453249111yescharacteristic
579C2#1453858111yescharacteristic
580C2#1453863511yescharacteristic
581C2#1454121811yescharacteristic
582C2#1454128411yescharacteristic
583C2#1454182611yescharacteristic
584C2#1454185911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2354C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
hypercenter#2354D17
hypocenterC17#26098
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2354

Complements

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

NH ≅ G/Ntype
1. #14534018C2semidirect
6. #14529552C2semidirect
7. #14529542C2semidirect
8. #14529540C2semidirect
9. #14529508C2semidirect
10. #14529376C2semidirect
11. #14529369C2semidirect
12. #14528950C2semidirect
411. C17#26098semidirect

Maximal subgroups

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

subgroupindex
#145340182
#145303122
#145303052
#145301722
#145301582
#145295522
#145295422
#145295402
#145295082
#145293762
#145293692
#145289502
#145287942
#145286092
C17×#23542
#2609817