Finite Group Databasebeta

← #17088690

Subgroups — #17088690

Order 4352.

1032 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763212C211yesmaximal, characteristic, complemented
2C2×#11763168C211yesmaximal, characteristic, complemented
3#14534605C211yesmaximal, characteristic, complemented
4#14529984C212yesmaximal, complemented
5#14529963C212yesmaximal
6#14529758C211yesmaximal, characteristic, complemented
7#14529596C211yesmaximal, characteristic, complemented
8#14529038C212yesmaximal, complemented
9#14529031C212yesmaximal, complemented
10#14528550C211yesmaximal, characteristic, complemented
11C2×C17×#542C211yesmaximal, characteristic
12C2×#10587555C2211yescharacteristic
13#11763212—24no—
14#11763168—24no—
15C2×#10587478C2212yes—
16C2×#10587478C2212yes—
17C2×#10587506C2212yes—
18C2×#10587506C2212yes—
19C2×#10587503C2212yes—
20C2×#10587503C2212yes—
21C2×C22⋊#3622C2212yes—
22C2×#10587474C2211yescharacteristic
23C2×C4×#3622C2211yescharacteristic
24#11762748—21no—
25#11762745—21no—
26#11762729—21no—
27#11762721—21no—
28#11762707C2212yes—
29#11762675C2211yescharacteristic
30#11762669C2212yes—
31#11762571C2212yes—
32#11762571C2212yes—
33#11762564C2212yes—
34C17×#542—24no—
35C17×#542—24no—
36C2×C17×C4⋊Q8C2211yescharacteristic
37C2×C17×#126—21no—
38C2×C17×#124—21no—
39C2×C17×#124C2211yescharacteristic
40C2×C17×#123—21no—
41C2×C17×C22⋊Q8—21no—
42C2×C17×C22⋊Q8C2211yescharacteristic
43C2×C17×#121C2211yescharacteristic
44C2×C4×Q8×C17C2211yescharacteristic
45C2×C4×D4×C17C2211yescharacteristic
46C2×C17×#117C2211yescharacteristic
47C22×Q8×C17—21no—
48C22×Q8×C17C2311yescharacteristic
49C22×D4×C17—21no—
50C22×D4×C17C2311yescharacteristic
51C23×C4×C17—21no—
52C23×C4×C17C2311yescharacteristic
53C22×Dic34—21no—
54C17×C4⋊Q8—22no—
55C17×C4⋊Q8D414yes—
56C17×#126—28no—
57C17×#124—28no—
58C17×#124—22no—
59C17×#124D414yes—
60C17×#123—28no—
61C17×C22⋊Q8—28no—
62C17×C22⋊Q8—22no—
63C17×C22⋊Q8—22no—
64C17×#121—22no—
65C17×#121—22no—
66C4×Q8×C17—22no—
67C4×Q8×C17—22no—
68C4×D4×C17—22no—
69C4×D4×C17—22no—
70C17×#117—22no—
71C17×#117—22no—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17—21no—
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17D411yescharacteristic
76C2×C4⋊C4×C17D411yescharacteristic
77C2×C4⋊C4×C17C2311yescharacteristic
78C2×C4⋊C4×C17C2311yescharacteristic
79C2×C22⋊C4×C17—21no—
80C2×C22⋊C4×C17—21no—
81C2×C22⋊C4×C17—21no—
82C2×C22⋊C4×C17—21no—
83C2×C22⋊C4×C17—21no—
84C2×C22⋊C4×C17D411yescharacteristic
85C2×C22⋊C4×C17D411yescharacteristic
86C2×C42×C17—21no—
87C2×C42×C17C2311yescharacteristic
88C2×C42×C17C2311yescharacteristic
89C2×C22⋊Dic17—22no—
90C2×C4⋊Dic17—21no—
91C2×C4⋊Dic17C2312yes—
92C2×C4⋊Dic17C2312yes—
93C2×#63225—22no—
94C22×#3622C2312yes—
95C22×#3622C2312yes—
96#10587555—22no—
97#10587555—22no—
98#10587506—24no—
99#10587506—24no—
100#10587503—24no—
101#10587503—24no—
102#10587502—22no—
103#10587502—21no—
104#10587502—22no—
105#10587502—21no—
106C22⋊#3622—24no—
107#10587478—24no—
108#10587478—24no—
109#10587474—22no—
110C4×#3622—22no—
111C24×C17—21no—
112C24×C17C2×D411yescharacteristic
113C2×Q8×C17—24no—
114C2×Q8×C17—21no—
115C2×Q8×C17—22no—
116C2×Q8×C17—22no—
117C2×Q8×C17—82no—
118C2×Q8×C17C2×D412yes—
119C2×D4×C17—24no—
120C2×D4×C17—21no—
121C2×D4×C17—22no—
122C2×D4×C17—22no—
123C2×D4×C17—82no—
124C2×D4×C17C2×D412yes—
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17—24no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—22no—
132C22×C4×C17—22no—
133C22×C4×C17—82no—
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17C2×D411yescharacteristic
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2411yescharacteristic
140C22×Dic17—22no—
141C22×Dic17—22no—
142C2×Dic34—42no—
143C2×Dic34—82no—
144C4⋊C4×C17—24no—
145C4⋊C4×C17—24no—
146C4⋊C4×C17—22no—
147C4⋊C4×C17—24no—
148C4⋊C4×C17—22no—
149C4⋊C4×C17—21no—
150C4⋊C4×C17—21no—
151C4⋊C4×C17C2×D412yes—
152C4⋊C4×C17C2×D412yes—
153C22⋊C4×C17—24no—
154C22⋊C4×C17—24no—
155C22⋊C4×C17—24no—
156C22⋊C4×C17—24no—
157C22⋊C4×C17—22no—
158C22⋊C4×C17—24no—
159C22⋊C4×C17—22no—
160C42×C17—24no—
161C42×C17—21no—
162C42×C17—21no—
163C42×C17C2×D412yes—
164C42×C17C2×D412yes—
165C22⋊Dic17—82no—
166C4⋊Dic17—22no—
167C4⋊Dic17—22no—
168C4⋊Dic17—22no—
169C4⋊Dic17—22no—
170C4⋊Dic17—42no—
171#63225—82no—
172C2×#3622—22no—
173C2×#3622—22no—
174C2×#3622—22no—
175C2×#3622—22no—
176C2×#3622—22no—
177C2×#3622#4114yes—
178#25681—171nomaximal, Sylow
179C23×C17—21no—
180C23×C17—22no—
181C23×C17—21no—
182C23×C17—21no—
183C23×C17—22no—
184C23×C17—22no—
185C23×C17—41no—
186C23×C17—41no—
187C23×C17—42no—
188C23×C17—42no—
189C23×C17C22×D411yescharacteristic
190Q8×C17—24no—
191Q8×C17—42no—
192Q8×C17—84no—
193D4×C17—24no—
194D4×C17—42no—
195D4×C17—84no—
196C2×C4×C17—22no—
197C2×C4×C17—22no—
198C2×C4×C17—21no—
199C2×C4×C17—22no—
200C2×C4×C17—21no—
201C2×C4×C17—22no—
202C2×C4×C17—21no—
203C2×C4×C17—22no—
204C2×C4×C17—21no—
205C2×C4×C17—21no—
206C2×C4×C17—22no—
207C2×C4×C17—22no—
208C2×C4×C17—41no—
209C2×C4×C17—41no—
210C2×C4×C17—42no—
211C2×C4×C17—42no—
212C2×C4×C17—42no—
213C2×C4×C17—42no—
214C2×C4×C17—84no—
215C2×C4×C17#12012yes—
216C2×C4×C17#12012yes—
217C2×C4×C17C22×D411yescharacteristic
218C2×C4×C17C22×D411yescharacteristic
219C2×C4×C17C2×#4112yes—
220C2×C4×C172−1+412yes—
221C2×Dic17—42no—
222C2×Dic17—42no—
223C2×Dic17—42no—
224C2×Dic17—42no—
225C2×Dic17—42no—
226C2×Dic17—42no—
227Dic34—84no—
228#3622—24no—
229#3622—42no—
230C2×#542D1711yescharacteristic
231C2×#500—171no—
232C2×#489—171no—
233#2628—171no—
234#2036—172no—
235#2033—172no—
236#1993—171no—
237#1958—171no—
238#1863—172no—
239#1862—172no—
240#1750—171no—
241C22×C17—42no—
242C22×C17—41no—
243C22×C17—42no—
244C22×C17—42no—
245C22×C17—42no—
246C22×C17—41no—
247C22×C17—41no—
248C22×C17—42no—
249C22×C17—82no—
250C22×C17—82no—
251C22×C17C2×#12011yescharacteristic
252C22×C17#53211yescharacteristic
253C22×C17#53611yescharacteristic
254C22×C17#57612yes—
255C22×C17#57812yes—
256C4×C17—22no—
257C4×C17—22no—
258C4×C17—41no—
259C4×C17—42no—
260C4×C17—41no—
261C4×C17—42no—
262C4×C17—42no—
263C4×C17—41no—
264C4×C17#54712yes—
265C4×C17#54812yes—
266Dic17—82no—
267Dic17—82no—
268#542—24no—
269#542—24no—
270C2×C4⋊Q8—171no—
271C2×C4⋊Q8C2×D1711yescharacteristic
272C2×#126—21no—
273C2×#124—21no—
274C2×#124C2×D1711yescharacteristic
275C2×#123—21no—
276C2×C22⋊Q8—21no—
277C2×C22⋊Q8C2×D1711yescharacteristic
278C2×#121C2×D1711yescharacteristic
279C2×C4×Q8C2×D1711yescharacteristic
280C2×C4×D4C2×D1711yescharacteristic
281C2×#117C2×D1711yescharacteristic
282#500—344no—
283#489—344no—
284C2×#107—172no—
285C2×#107—172no—
286C2×#103—172no—
287C2×#103—172no—
288C2×#102—172no—
289C2×#102—172no—
290C2×C22⋊C8—172no—
291C2×#97—171no—
292C2×C4×C8—171no—
293#400—341no—
294#399—341no—
295#395—341no—
296#393—341no—
297#389—172no—
298#382—171no—
299#380—172no—
300#340—172no—
301#340—172no—
302#336—172no—
303C2×C17—81no—
304C2×C17—82no—
305C2×C17#243211yescharacteristic
306C2×C17#300811yescharacteristic
307C2×C17#301111yescharacteristic
308C2×C17#317512yes—
309C2×C17#324312yes—
310C22×Q8—21no—
311C22×Q8—341no—
312C22×Q8C22×D1711yescharacteristic
313C22×D4—21no—
314C22×D4C22×D1711yescharacteristic
315C23×C4—21no—
316C23×C4C22×D1711yescharacteristic
317C22×C8—172no—
318C22×C8—172no—
319C4⋊Q8—22no—
320C4⋊Q8—342no—
321C4⋊Q8—342no—
322C4⋊Q8C22⋊D1714yes—
323#126—28no—
324#124—28no—
325#124—22no—
326#124C22⋊D1714yes—
327#123—28no—
328C22⋊Q8—28no—
329C22⋊Q8—22no—
330C22⋊Q8—22no—
331#121—22no—
332#121—22no—
333C4×Q8—22no—
334C4×Q8—22no—
335C4×D4—22no—
336C4×D4—22no—
337#117—22no—
338#117—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—172no—
343C2×C4⋊C4—172no—
344C2×C4⋊C4—341no—
345C2×C4⋊C4—342no—
346C2×C4⋊C4C22⋊D1711yescharacteristic
347C2×C4⋊C4C22⋊D1711yescharacteristic
348C2×C4⋊C4C22×D1711yescharacteristic
349C2×C4⋊C4C22×D1711yescharacteristic
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—342no—
356C2×C22⋊C4C22⋊D1711yescharacteristic
357C2×C22⋊C4C22⋊D1711yescharacteristic
358C2×C42—21no—
359C2×C42C22×D1711yescharacteristic
360C2×C42C22×D1711yescharacteristic
361#107—344no—
362#107—344no—
363#103—344no—
364#103—344no—
365#102—344no—
366#102—344no—
367C22⋊C8—344no—
368#97—342no—
369C4×C8—342no—
370#95—342no—
371#95—341no—
372#95—341no—
373#95—342no—
374C17#2568111yescharacteristic, complemented, Sylow
375C24—21no—
376C24C2×C22⋊D1711yescharacteristic
377C2×Q8—21no—
378C2×Q8—24no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—82no—
382C2×Q8—682no—
383C2×Q8—1362no—
384C2×Q8C2×C22⋊D1712yes—
385C2×D4—24no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—21no—
389C2×D4—82no—
390C2×D4C2×C22⋊D1712yes—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—82no—
400C22×C4—342no—
401C22×C4—342no—
402C22×C4C2×C22⋊D1711yescharacteristic
403C22×C4C2×C22⋊D1711yescharacteristic
404C22×C4C2×C22⋊D1711yescharacteristic
405C22×C4C2×C22⋊D1711yescharacteristic
406C22×C4C2×C22⋊D1711yescharacteristic
407C22×C4C23×D1711yescharacteristic
408C2×C8—174no—
409C2×C8—342no—
410C2×C8—342no—
411C2×C8—342no—
412C2×C8—342no—
413C2×C8—342no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—342no—
422C4⋊C4—342no—
423C4⋊C4—342no—
424C4⋊C4—342no—
425C4⋊C4—682no—
426C4⋊C4—1362no—
427C4⋊C4C2×C22⋊D1712yes—
428C4⋊C4C2×C22⋊D1712yes—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—24no—
433C22⋊C4—24no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—1362no—
437C42—24no—
438C42—21no—
439C42—21no—
440C42C2×C22⋊D1712yes—
441C42C2×C22⋊D1712yes—
442C23—21no—
443C23—22no—
444C23—21no—
445C23—22no—
446C23—22no—
447C23—21no—
448C23—41no—
449C23—42no—
450C23—42no—
451C23—41no—
452C23C22×C22⋊D1711yescharacteristic
453Q8—24no—
454Q8—42no—
455Q8—84no—
456Q8—1364no—
457D4—24no—
458D4—42no—
459D4—84no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—84no—
479C2×C4—682no—
480C2×C4—682no—
481C2×C4—682no—
482C2×C4—682no—
483C2×C4—682no—
484C2×C4—682no—
485C2×C4#1058763912yes—
486C2×C4#1058763912yes—
487C2×C4#41×D1712yes—
488C2×C4#1058769712yes—
489C2×C4C22×C22⋊D1711yescharacteristic
490C2×C4C22×C22⋊D1711yescharacteristic
491C8—344no—
492C8—682no—
493C22—42no—
494C22—41no—
495C22—42no—
496C22—42no—
497C22—41no—
498C22—41no—
499C22—42no—
500C22—42no—
501C22—82no—
502C22—82no—
503C22C2×#1058763911yescharacteristic
504C22#1176335511yescharacteristic
505C22#1176338011yescharacteristic
506C22#1176361212yes—
507C22#1176362412yes—
508C4—22no—
509C4—22no—
510C4—41no—
511C4—42no—
512C4—41no—
513C4—42no—
514C4—42no—
515C4—41no—
516C4—1362no—
517C4—1362no—
518C4#1176345812yes—
519C4#1176346612yes—
520C2—81no—
521C2—82no—
522C2#1453240811yescharacteristic
523C2#1453805211yescharacteristic
524C2#1453807911yescharacteristic
525C2#1453941712yes—
526C2#1454006112yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#542C2
socleC23×C17C22×D4
prime radicalC2×C17×#121C22
hypercenterC2×#542D17
hypocenterC17#25681
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#542

Complements

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

NH ≅ G/Ntype
1. C2×#11763212C2semidirect
2. C2×#11763168C2semidirect
3. #14534605C2semidirect
4. #14529984C2semidirect
6. #14529758C2semidirect
7. #14529596C2semidirect
8. #14529038C2semidirect
9. #14529031C2semidirect
10. #14528550C2semidirect
374. C17#25681semidirect

Maximal subgroups

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

subgroupindex
C2×#117632122
C2×#117631682
#145346052
#145299842
#145299632
#145297582
#145295962
#145290382
#145290312
#145285502
C2×C17×#5422
#2568117