Finite Group Databasebeta

← #17088482

Subgroups — #17088482

Order 4352.

1048 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763168C211yesmaximal, characteristic
2C2×#11763104C211yesmaximal, characteristic, complemented
3#14533769C211yesmaximal, characteristic, complemented
4#14530027C212yesmaximal, complemented
5#14530020C212yesmaximal
6#14529725C211yesmaximal, characteristic, complemented
7#14529717C211yesmaximal, characteristic
8#14529038C212yesmaximal, complemented
9#14529031C212yesmaximal
10#14528614C211yesmaximal, characteristic, complemented
11C2×C17×#556C211yesmaximal, characteristic
12C2×#10587555C2211yescharacteristic
13C2×#10587630—21no—
14#11763168—24no—
15#11763104—24no—
16C2×#10587478C2212yes—
17C2×C4⋊#3622C2211yescharacteristic
18C2×#10587506C2212yes—
19C2×#10587506C2212yes—
20C2×#10587506C2212yes—
21C2×#10587506C2212yes—
22C2×#10587503C2212yes—
23C2×#10587503C2212yes—
24C2×#10587474C2211yescharacteristic
25#11762739—21no—
26#11762739—21no—
27#11762714—21no—
28#11762707C2212yes—
29#11762707C2212yes—
30C4×C4⋊Dic17C2211yescharacteristic
31#11762571C2212yes—
32#11762564C2212yes—
33#11762564C2212yes—
34C17×#556—24no—
35C17×#556—24no—
36C2×C17×C4⋊Q8C2211yescharacteristic
37C2×C17×#126—21no—
38C2×C17×#126—21no—
39C2×C17×#124C2211yescharacteristic
40C2×C17×#124C2211yescharacteristic
41C2×C17×C22⋊Q8—21no—
42C2×C4×Q8×C17C2211yescharacteristic
43C2×C4×Q8×C17C2211yescharacteristic
44C2×C4×Q8×C17C2211yescharacteristic
45C2×C4×D4×C17C2211yescharacteristic
46C2×C17×#117—21no—
47C22×Q8×C17D411yescharacteristic
48C22×Q8×C17C2311yescharacteristic
49C22×Q8×C17C2311yescharacteristic
50C22×D4×C17C2311yescharacteristic
51C23×C4×C17—21no—
52C22×Dic34—21no—
53C17×C4⋊Q8—22no—
54C17×C4⋊Q8—22no—
55C17×#126—28no—
56C17×#126—28no—
57C17×#124—22no—
58C17×#124—22no—
59C17×#124—22no—
60C17×#124D414yes—
61C17×C22⋊Q8—28no—
62C4×Q8×C17—22no—
63C4×Q8×C17—22no—
64C4×Q8×C17—22no—
65C4×Q8×C17—22no—
66C4×Q8×C17—22no—
67C4×Q8×C17D414yes—
68C4×D4×C17—22no—
69C4×D4×C17—22no—
70C17×#117—28no—
71C2×C4⋊C4×C17—21no—
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×C17D411yescharacteristic
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×C42×C17—21no—
85C2×C42×C17—21no—
86C2×C42×C17C2311yescharacteristic
87C2×C42×C17C2311yescharacteristic
88C2×C42×C17C2311yescharacteristic
89#10587630—44no—
90C2×C4⋊Dic17—21no—
91C2×C4⋊Dic17C2312yes—
92C2×C4⋊Dic17C2312yes—
93C2×#63225—22no—
94C2×#63225—22no—
95C2×C4×Dic17—21no—
96C22×#3622C2312yes—
97C22×#3622C2312yes—
98#10587555—22no—
99#10587555—22no—
100#10587506—24no—
101#10587506—24no—
102#10587506—24no—
103#10587506—24no—
104#10587503—24no—
105#10587503—24no—
106#10587502—22no—
107#10587502—22no—
108#10587502—21no—
109#10587478—24no—
110C4⋊#3622—22no—
111#10587474—22no—
112C24×C17—21no—
113C2×Q8×C17—22no—
114C2×Q8×C17—22no—
115C2×Q8×C17—21no—
116C2×Q8×C17—21no—
117C2×Q8×C17—22no—
118C2×Q8×C17—22no—
119C2×Q8×C17—22no—
120C2×Q8×C17—22no—
121C2×Q8×C17—41no—
122C2×Q8×C17C2×D412yes—
123C2×Q8×C17C2×D412yes—
124C2×D4×C17—21no—
125C2×D4×C17—22no—
126C2×D4×C17—22no—
127C2×D4×C17C2×D412yes—
128C22×C4×C17—24no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17—42no—
133C22×C4×C17—42no—
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17C2×D411yescharacteristic
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2411yescharacteristic
141C22×Dic17—22no—
142C22×Dic17—22no—
143C2×Dic34—42no—
144C2×Dic34—42no—
145C2×Dic34—42no—
146C4⋊C4×C17—24no—
147C4⋊C4×C17—24no—
148C4⋊C4×C17—24no—
149C4⋊C4×C17—24no—
150C4⋊C4×C17—22no—
151C4⋊C4×C17—22no—
152C4⋊C4×C17—22no—
153C4⋊C4×C17—21no—
154C4⋊C4×C17C2×D412yes—
155C22⋊C4×C17—24no—
156C22⋊C4×C17—24no—
157C22⋊C4×C17—24no—
158C22⋊C4×C17—24no—
159C22⋊C4×C17—24no—
160C42×C17—24no—
161C42×C17—24no—
162C42×C17—21no—
163C42×C17—21no—
164C42×C17—21no—
165C42×C17—21no—
166C42×C17C2×D412yes—
167C42×C17C2×D412yes—
168C4⋊Dic17—22no—
169C4⋊Dic17—22no—
170C4⋊Dic17—22no—
171C4⋊Dic17—42no—
172C4⋊Dic17#4114yes—
173#63225—44no—
174#63225—44no—
175C4×Dic17—42no—
176C2×#3622—22no—
177C2×#3622—22no—
178C2×#3622—22no—
179C2×#3622—22no—
180C2×#3622—22no—
181C2×#3622—22no—
182#25661—171nomaximal, Sylow
183C23×C17—22no—
184C23×C17—42no—
185C23×C17—41no—
186C23×C17—41no—
187C23×C17—42no—
188C23×C17C22×D411yescharacteristic
189Q8×C17—22no—
190Q8×C17—22no—
191Q8×C17—24no—
192Q8×C17—44no—
193Q8×C17—41no—
194Q8×C17—41no—
195Q8×C17—42no—
196D4×C17—24no—
197D4×C17—41no—
198D4×C17—41no—
199C2×C4×C17—21no—
200C2×C4×C17—21no—
201C2×C4×C17—21no—
202C2×C4×C17—21no—
203C2×C4×C17—22no—
204C2×C4×C17—21no—
205C2×C4×C17—22no—
206C2×C4×C17—21no—
207C2×C4×C17—22no—
208C2×C4×C17—22no—
209C2×C4×C17—21no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—22no—
213C2×C4×C17—22no—
214C2×C4×C17—48no—
215C2×C4×C17—42no—
216C2×C4×C17—42no—
217C2×C4×C17—41no—
218C2×C4×C17—41no—
219C2×C4×C17#12012yes—
220C2×C4×C17#12012yes—
221C2×C4×C17AGL(1,ℤ/8)12yes—
222C2×C4×C17C22×D411yescharacteristic
223C2×C4×C17C22×D411yescharacteristic
224C2×C4×C17C2×#4112yes—
225C2×C4×C172−1+412yes—
226C2×Dic17—24no—
227C2×Dic17—42no—
228C2×Dic17—42no—
229C2×Dic17—42no—
230C2×Dic17—42no—
231C2×Dic17—42no—
232Dic34—84no—
233#3622—42no—
234#3622—42no—
235C2×#556D1711yescharacteristic
236C2×#489—171no—
237C2×#477—171no—
238#2546—171no—
239#2043—172no—
240#2042—172no—
241#1985—171no—
242#1983—171no—
243#1863—172no—
244#1862—172no—
245#1763—171no—
246C22×C17—42no—
247C22×C17—42no—
248C22×C17—42no—
249C22×C17—82no—
250C22×C17—82no—
251C22×C17C2×#12011yescharacteristic
252C22×C17#54812yes—
253C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
254C22×C17#57511yescharacteristic
255C22×C17#57812yes—
256C4×C17—22no—
257C4×C17—22no—
258C4×C17—22no—
259C4×C17—42no—
260C4×C17—42no—
261C4×C17—41no—
262C4×C17—42no—
263C4×C17—41no—
264C4×C17—41no—
265C4×C17#53211yescharacteristic
266C4×C17#53611yescharacteristic
267C4×C17#54712yes—
268Dic17—44no—
269Dic17—82no—
270#556—24no—
271#556—24no—
272C2×C4⋊Q8—171no—
273C2×C4⋊Q8—341no—
274C2×C4⋊Q8C2×D1711yescharacteristic
275C2×#126—21no—
276C2×#126—21no—
277C2×#124C2×D1711yescharacteristic
278C2×#124C2×D1711yescharacteristic
279C2×C22⋊Q8—21no—
280C2×C4×Q8C2×D1711yescharacteristic
281C2×C4×Q8C2×D1711yescharacteristic
282C2×C4×Q8C2×D1711yescharacteristic
283C2×C4×D4C2×D1711yescharacteristic
284C2×#117—21no—
285#489—344no—
286#477—344no—
287C2×#107—172no—
288C2×C4⋊C8—171no—
289C2×#103—172no—
290C2×#103—172no—
291C2×#103—172no—
292C2×#103—172no—
293C2×#102—172no—
294C2×#102—172no—
295C2×#97—171no—
296#398—341no—
297#398—341no—
298#391—341no—
299#389—172no—
300#389—172no—
301C4×C4⋊C4—171no—
302#340—172no—
303#336—172no—
304#336—172no—
305C2×C17—82no—
306C2×C17#243211yescharacteristic
307C2×C17#300611yescharacteristic
308C2×C17#301111yescharacteristic
309C2×C17#336712yes—
310C2×C17#337712yes—
311C22×Q8—341no—
312C22×Q8C22⋊D1711yescharacteristic
313C22×Q8C22×D1711yescharacteristic
314C22×Q8C22×D1711yescharacteristic
315C22×D4C22×D1711yescharacteristic
316C23×C4—21no—
317C22×C8—172no—
318C22×C8—172no—
319C4⋊Q8—22no—
320C4⋊Q8—22no—
321C4⋊Q8—342no—
322C4⋊Q8—342no—
323C4⋊Q8—684no—
324#126—28no—
325#126—28no—
326#124—22no—
327#124—22no—
328#124—22no—
329#124C22⋊D1714yes—
330C22⋊Q8—28no—
331C4×Q8—22no—
332C4×Q8—22no—
333C4×Q8—22no—
334C4×Q8—22no—
335C4×Q8—22no—
336C4×Q8C22⋊D1714yes—
337C4×D4—22no—
338C4×D4—22no—
339#117—28no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—172no—
345C2×C4⋊C4—172no—
346C2×C4⋊C4—341no—
347C2×C4⋊C4—342no—
348C2×C4⋊C4—342no—
349C2×C4⋊C4C22⋊D1711yescharacteristic
350C2×C4⋊C4C22⋊D1711yescharacteristic
351C2×C4⋊C4C22⋊D1711yescharacteristic
352C2×C4⋊C4C22×D1711yescharacteristic
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42—341no—
361C2×C42C22×D1711yescharacteristic
362C2×C42C22×D1711yescharacteristic
363C2×C42C22×D1711yescharacteristic
364#107—344no—
365C4⋊C8—342no—
366#103—344no—
367#103—344no—
368#103—344no—
369#103—344no—
370#102—344no—
371#102—344no—
372#97—342no—
373#95—341no—
374#95—342no—
375#95—342no—
376C17#2566111yescharacteristic, complemented, Sylow
377C24—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—41no—
387C2×Q8—682no—
388C2×Q8—682no—
389C2×Q8—682no—
390C2×Q8C2×C22⋊D1712yes—
391C2×Q8C2×C22⋊D1712yes—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—21no—
395C2×D4C2×C22⋊D1712yes—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—342no—
403C22×C4—342no—
404C22×C4C2×C22⋊D1711yescharacteristic
405C22×C4C2×C22⋊D1711yescharacteristic
406C22×C4C2×C22⋊D1711yescharacteristic
407C22×C4C2×C22⋊D1711yescharacteristic
408C22×C4C2×C22⋊D1711yescharacteristic
409C22×C4C2×C22⋊D1711yescharacteristic
410C22×C4C23×D1711yescharacteristic
411C2×C8—342no—
412C2×C8—342no—
413C2×C8—342no—
414C2×C8—342no—
415C2×C8—342no—
416C2×C8—342no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—21no—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—174no—
426C4⋊C4—342no—
427C4⋊C4—342no—
428C4⋊C4—342no—
429C4⋊C4—684no—
430C4⋊C4—684no—
431C4⋊C4—682no—
432C4⋊C4C2×C22⋊D1712yes—
433C22⋊C4—24no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C42—24no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—24no—
444C42—682no—
445C42C2×C22⋊D1712yes—
446C42C2×C22⋊D1712yes—
447C23—22no—
448C23—42no—
449C23—41no—
450C23—41no—
451C23—42no—
452C23C22×C22⋊D1711yescharacteristic
453Q8—24no—
454Q8—22no—
455Q8—22no—
456Q8—44no—
457Q8—42no—
458Q8—41no—
459Q8—41no—
460Q8—1364no—
461D4—24no—
462D4—41no—
463D4—41no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—48no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—344no—
485C2×C4—682no—
486C2×C4—682no—
487C2×C4—682no—
488C2×C4—682no—
489C2×C4—682no—
490C2×C4#1058763512yes—
491C2×C4#1058763912yes—
492C2×C4#1058763912yes—
493C2×C4#41×D1712yes—
494C2×C4#1058769712yes—
495C2×C4C22×C22⋊D1711yescharacteristic
496C2×C4C22×C22⋊D1711yescharacteristic
497C8—682no—
498C8—682no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—82no—
503C22—82no—
504C22C2×#1058763911yescharacteristic
505C22#1176346612yes—
506C22C2×#1058763511yescharacteristic
507C22#1176360711yescharacteristic
508C22#1176362412yes—
509C4—22no—
510C4—22no—
511C4—22no—
512C4—41no—
513C4—42no—
514C4—42no—
515C4—42no—
516C4—41no—
517C4—41no—
518C4—684no—
519C4—1362no—
520C4#1176335511yescharacteristic
521C4#1176338011yescharacteristic
522C4#1176345812yes—
523C2—82no—
524C2#1453240811yescharacteristic
525C2#1453803611yescharacteristic
526C2#1453807911yescharacteristic
527C2#1454149812yes—
528C2#1454162012yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#556C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
hypercenterC2×#556D17
hypocenterC17#25661
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
2. C2×#11763104C2semidirect
3. #14533769C2semidirect
4. #14530027C2semidirect
6. #14529725C2semidirect
8. #14529038C2semidirect
10. #14528614C2semidirect
376. C17#25661semidirect

Maximal subgroups

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

subgroupindex
C2×#117631682
C2×#117631042
#145337692
#145300272
#145300202
#145297252
#145297172
#145290382
#145290312
#145286142
C2×C17×#5562
#2566117