Finite Group Databasebeta

← #17068094

Subgroups — #17068094

Order 4352.

1126 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763120C211yesmaximal, characteristic, complemented
2C2×#11762852C211yesmaximal, characteristic, complemented
3#14533101C211yesmaximal, characteristic, complemented
4#14530109C212yesmaximal, complemented
5#14529963C212yesmaximal, complemented
6#14529223C211yesmaximal, characteristic, complemented
7#14529082C211yesmaximal, characteristic, complemented
8#14529045C212yesmaximal, complemented
9#14528983C212yesmaximal, complemented
10#14528879C211yesmaximal, characteristic, complemented
11C2×C17×#555C211yesmaximal, characteristic
12C22×#63223C2211yescharacteristic
13C22×C4⋊Dic17C2211yescharacteristic
14#11763120—24no—
15#11762852—24no—
16C2×#10587478C2212yes—
17C2×#10587478C2212yes—
18C2×#10587479C2212yes—
19C2×#10587503C2212yes—
20C2×#10587503C2212yes—
21C2×C22⋊#3622C2211yescharacteristic
22C2×C22⋊#3622C2212yes—
23#11762748—21no—
24#11762745—21no—
25#11762707C2212yes—
26#11762689—21no—
27#11762681—21no—
28#11762669C2211yescharacteristic
29#11762669C2212yes—
30#11762571C2212yes—
31#11762571C2212yes—
32#11762571C2212yes—
33#11762564C2212yes—
34C17×#555—24no—
35C17×#555—24no—
36C2×C17×C4⋊Q8C2211yescharacteristic
37C2×C17×#124—21no—
38C2×C17×#124—21no—
39C2×C17×#121—21no—
40C2×C17×#121C2211yescharacteristic
41C2×C17×#121C2211yescharacteristic
42C2×C4×D4×C17C2211yescharacteristic
43C2×C4×D4×C17C2211yescharacteristic
44C2×C17×#117—21no—
45C2×C17×#117C2211yescharacteristic
46C2×C17×#117C2211yescharacteristic
47C22×Q8×C17—21no—
48C22×D4×C17—21no—
49C22×D4×C17—21no—
50C22×D4×C17D411yescharacteristic
51C22×D4×C17C2311yescharacteristic
52C23×C4×C17—21no—
53C23×C4×C17C2311yescharacteristic
54C23×C4×C17C2311yescharacteristic
55C23×Dic17—21no—
56C17×C4⋊Q8—22no—
57C17×C4⋊Q8—22no—
58C17×#124—28no—
59C17×#124—28no—
60C17×#121—28no—
61C17×#121—22no—
62C17×#121—22no—
63C17×#121—22no—
64C17×#121D414yes—
65C4×D4×C17—22no—
66C4×D4×C17—22no—
67C4×D4×C17—22no—
68C4×D4×C17—22no—
69C17×#117—28no—
70C17×#117—22no—
71C17×#117—22no—
72C17×#117—22no—
73C17×#117D414yes—
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17D411yescharacteristic
76C2×C4⋊C4×C17C2311yescharacteristic
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×C17—21no—
88C2×C42×C17C2311yescharacteristic
89C2×C22⋊Dic17—21no—
90C2×C22⋊Dic17—22no—
91C2×C4⋊Dic17—21no—
92C2×C4⋊Dic17—22no—
93C2×C4⋊Dic17—22no—
94C2×C4⋊Dic17C2312yes—
95C2×C4⋊Dic17C2312yes—
96C2×#63225—22no—
97C2×#63223—24no—
98C2×#63223—22no—
99C22×#3622C2312yes—
100C22×#3622C2312yes—
101#10587503—24no—
102#10587503—24no—
103#10587502—22no—
104#10587502—21no—
105#10587502—22no—
106C22⋊#3622—24no—
107C22⋊#3622—22no—
108#10587479—24no—
109#10587478—24no—
110#10587478—24no—
111C24×C17—21no—
112C24×C17C2×D411yescharacteristic
113C24×C17C2×D411yescharacteristic
114C2×Q8×C17—24no—
115C2×Q8×C17—82no—
116C2×D4×C17—22no—
117C2×D4×C17—24no—
118C2×D4×C17—24no—
119C2×D4×C17—22no—
120C2×D4×C17—21no—
121C2×D4×C17—22no—
122C2×D4×C17—22no—
123C2×D4×C17—41no—
124C2×D4×C17—82no—
125C2×D4×C17—82no—
126C2×D4×C17C2×D412yes—
127C22×C4×C17—22no—
128C22×C4×C17—22no—
129C22×C4×C17—24no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—22no—
136C22×C4×C17—22no—
137C22×C4×C17—42no—
138C22×C4×C17—42no—
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2×D412yes—
143C22×C4×C17C2×D412yes—
144C22×C4×C17C2×D411yescharacteristic
145C22×C4×C17C2411yescharacteristic
146C22×Dic17—22no—
147C22×Dic17—22no—
148C22×Dic17—42no—
149C22×Dic17—82no—
150C4⋊C4×C17—24no—
151C4⋊C4×C17—22no—
152C4⋊C4×C17—21no—
153C4⋊C4×C17—21no—
154C4⋊C4×C17—21no—
155C4⋊C4×C17C2×D412yes—
156C4⋊C4×C17C2×D412yes—
157C4⋊C4×C17C2×D412yes—
158C22⋊C4×C17—24no—
159C22⋊C4×C17—24no—
160C22⋊C4×C17—24no—
161C22⋊C4×C17—22no—
162C22⋊C4×C17—24no—
163C22⋊C4×C17—22no—
164C22⋊C4×C17—24no—
165C42×C17—24no—
166C42×C17—24no—
167C42×C17—21no—
168C42×C17—21no—
169C22⋊Dic17—42no—
170C22⋊Dic17—82no—
171C4⋊Dic17—22no—
172C4⋊Dic17—22no—
173C4⋊Dic17—22no—
174C4⋊Dic17—22no—
175C4⋊Dic17—42no—
176#63225—82no—
177#63223—28no—
178C2×#3622—22no—
179C2×#3622—22no—
180C2×#3622—22no—
181C2×#3622—22no—
182C2×#3622—22no—
183C2×#3622#4114yes—
184#24105—171nomaximal, Sylow
185C23×C17—22no—
186C23×C17—21no—
187C23×C17—21no—
188C23×C17—22no—
189C23×C17—21no—
190C23×C17—22no—
191C23×C17—22no—
192C23×C17—21no—
193C23×C17—21no—
194C23×C17—22no—
195C23×C17—42no—
196C23×C17—41no—
197C23×C17—41no—
198C23×C17—42no—
199C23×C17#12012yes—
200C23×C17C22×D411yescharacteristic
201Q8×C17—84no—
202D4×C17—24no—
203D4×C17—44no—
204D4×C17—41no—
205D4×C17—41no—
206D4×C17—84no—
207D4×C17—84no—
208C2×C4×C17—21no—
209C2×C4×C17—22no—
210C2×C4×C17—21no—
211C2×C4×C17—22no—
212C2×C4×C17—22no—
213C2×C4×C17—24no—
214C2×C4×C17—21no—
215C2×C4×C17—22no—
216C2×C4×C17—21no—
217C2×C4×C17—22no—
218C2×C4×C17—22no—
219C2×C4×C17—22no—
220C2×C4×C17—22no—
221C2×C4×C17—48no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17—42no—
225C2×C4×C17—42no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17—42no—
229C2×C4×C17#12012yes—
230C2×C4×C17#13712yes—
231C2×C4×C17C22×D411yescharacteristic
232C2×C4×C17C22×D411yescharacteristic
233C2×C4×C17C2×#4112yes—
234C2×C4×C172−1+412yes—
235C2×Dic17—42no—
236C2×Dic17—42no—
237C2×Dic17—42no—
238C2×Dic17—42no—
239C2×Dic17—42no—
240C2×Dic17—42no—
241C2×Dic17—84no—
242#3622—24no—
243#3622—42no—
244C2×#555D1711yescharacteristic
245C2×#480—171no—
246C2×#428—171no—
247#2487—171no—
248#2057—172no—
249#2033—172no—
250#1895—171no—
251#1870—171no—
252#1864—172no—
253#1854—172no—
254#1825—171no—
255C22×C17—22no—
256C22×C17—22no—
257C22×C17—22no—
258C22×C17—42no—
259C22×C17—42no—
260C22×C17—42no—
261C22×C17—42no—
262C22×C17—41no—
263C22×C17—42no—
264C22×C17—42no—
265C22×C17—41no—
266C22×C17—41no—
267C22×C17—42no—
268C22×C17—82no—
269C22×C17—82no—
270C22×C17C2×#12011yescharacteristic
271C22×C17#54712yes—
272C22×C17C2×#13711yescharacteristic
273C22×C17#57511yescharacteristic
274C22×C17#57612yes—
275C4×C17—22no—
276C4×C17—22no—
277C4×C17—41no—
278C4×C17—42no—
279C4×C17—41no—
280C4×C17—42no—
281C4×C17—42no—
282C4×C17#52511yescharacteristic
283C4×C17#53611yescharacteristic
284C4×C17#54712yes—
285Dic17—82no—
286Dic17—82no—
287C22×M16—171no—
288#555—24no—
289#555—24no—
290C2×C4⋊Q8C2×D1711yescharacteristic
291C2×#124—21no—
292C2×#124—21no—
293C2×#121—21no—
294C2×#121C2×D1711yescharacteristic
295C2×#121C2×D1711yescharacteristic
296C2×C4×D4C2×D1711yescharacteristic
297C2×C4×D4C2×D1711yescharacteristic
298C2×#117—21no—
299C2×#117C2×D1711yescharacteristic
300C2×#117C2×D1711yescharacteristic
301C22×C4⋊C4—171no—
302#480—344no—
303#428—344no—
304C2×#107—172no—
305C2×#107—172no—
306C2×#106—172no—
307C2×#102—172no—
308C2×#102—172no—
309C2×C22⋊C8—171no—
310C2×C22⋊C8—172no—
311#400—341no—
312#399—341no—
313#389—172no—
314#386—341no—
315#385—341no—
316#380—171no—
317#380—172no—
318#340—172no—
319#340—172no—
320#340—172no—
321#336—172no—
322C2×C17—42no—
323C2×C17—81no—
324C2×C17—82no—
325C2×C17#240611yescharacteristic
326C2×C17#300711yescharacteristic
327C2×C17#300811yescharacteristic
328C2×C17#334712yes—
329C2×C17#335712yes—
330C22×Q8—21no—
331C22×D4—21no—
332C22×D4—21no—
333C22×D4C22⋊D1711yescharacteristic
334C22×D4C22×D1711yescharacteristic
335C23×C4—21no—
336C23×C4—341no—
337C23×C4C22×D1711yescharacteristic
338C23×C4C22×D1711yescharacteristic
339C2×M16—344no—
340C2×M16—342no—
341C22×C8—172no—
342C22×C8—172no—
343C4⋊Q8—22no—
344C4⋊Q8—22no—
345#124—28no—
346#124—28no—
347#121—28no—
348#121—22no—
349#121—22no—
350#121—22no—
351#121C22⋊D1714yes—
352C4×D4—22no—
353C4×D4—22no—
354C4×D4—22no—
355C4×D4—22no—
356#117—28no—
357#117—22no—
358#117—22no—
359#117—22no—
360#117C22⋊D1714yes—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—172no—
363C2×C4⋊C4—172no—
364C2×C4⋊C4—342no—
365C2×C4⋊C4—342no—
366C2×C4⋊C4—341no—
367C2×C4⋊C4—342no—
368C2×C4⋊C4C22⋊D1711yescharacteristic
369C2×C4⋊C4C22×D1711yescharacteristic
370C2×C4⋊C4C22×D1711yescharacteristic
371C2×C4⋊C4C22×D1711yescharacteristic
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—341no—
378C2×C22⋊C4—342no—
379C2×C22⋊C4C22⋊D1711yescharacteristic
380C2×C22⋊C4C22⋊D1711yescharacteristic
381C2×C42—21no—
382C2×C42—21no—
383C2×C42C22×D1711yescharacteristic
384#107—344no—
385#107—344no—
386#106—344no—
387#102—344no—
388#102—344no—
389C22⋊C8—344no—
390C22⋊C8—342no—
391#95—342no—
392#95—341no—
393#95—342no—
394C17#2410511yescharacteristic, complemented, Sylow
395C24—21no—
396C24C2×C22⋊D1711yescharacteristic
397C24C2×C22⋊D1711yescharacteristic
398C2×Q8—24no—
399C2×Q8—82no—
400C2×D4—22no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—41no—
408C2×D4—82no—
409C2×D4—82no—
410C2×D4C2×C22⋊D1712yes—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—342no—
424C22×C4—342no—
425C22×C4—682no—
426C22×C4—1362no—
427C22×C4C2×C22⋊D1711yescharacteristic
428C22×C4C2×C22⋊D1711yescharacteristic
429C22×C4C2×C22⋊D1712yes—
430C22×C4C2×C22⋊D1712yes—
431C22×C4C2×C22⋊D1711yescharacteristic
432C22×C4C2×C22⋊D1711yescharacteristic
433C22×C4C23×D1711yescharacteristic
434M16—348no—
435C2×C8—174no—
436C2×C8—342no—
437C2×C8—342no—
438C2×C8—342no—
439C2×C8—342no—
440C2×C8—342no—
441C4⋊C4—24no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—342no—
447C4⋊C4—342no—
448C4⋊C4—342no—
449C4⋊C4—342no—
450C4⋊C4—682no—
451C4⋊C4—1362no—
452C4⋊C4C2×C22⋊D1712yes—
453C4⋊C4C2×C22⋊D1712yes—
454C4⋊C4C2×C22⋊D1712yes—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—22no—
460C22⋊C4—24no—
461C22⋊C4—22no—
462C22⋊C4—682no—
463C22⋊C4—1362no—
464C42—24no—
465C42—21no—
466C42—21no—
467C42—24no—
468C23—22no—
469C23—21no—
470C23—21no—
471C23—22no—
472C23—22no—
473C23—21no—
474C23—22no—
475C23—21no—
476C23—21no—
477C23—22no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—42no—
482C23#1058763912yes—
483C23C22×C22⋊D1711yescharacteristic
484Q8—84no—
485D4—24no—
486D4—44no—
487D4—41no—
488D4—41no—
489D4—84no—
490D4—84no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—48no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—682no—
513C2×C4—682no—
514C2×C4—682no—
515C2×C4—682no—
516C2×C4—682no—
517C2×C4—682no—
518C2×C4—1364no—
519C2×C4#1058763712yes—
520C2×C4#1058763912yes—
521C2×C4#41×D1712yes—
522C2×C4#1058769712yes—
523C2×C4C22×C22⋊D1711yescharacteristic
524C2×C4C22×C22⋊D1711yescharacteristic
525C8—344no—
526C8—682no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—42no—
531C22—42no—
532C22—41no—
533C22—42no—
534C22—42no—
535C22—41no—
536C22—41no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—82no—
541C22—82no—
542C22C2×#1058763911yescharacteristic
543C22#1176345812yes—
544C22C2×#1058763711yescharacteristic
545C22#1176360711yescharacteristic
546C22#1176361212yes—
547C4—22no—
548C4—22no—
549C4—41no—
550C4—42no—
551C4—41no—
552C4—42no—
553C4—42no—
554C4—1362no—
555C4—1362no—
556C4#1176331611yescharacteristic
557C4#1176338011yescharacteristic
558C4#1176345812yes—
559C2—42no—
560C2—81no—
561C2—82no—
562C2#1453211511yescharacteristic
563C2#1453804411yescharacteristic
564C2#1453805211yescharacteristic
565C2#1454128412yes—
566C2#1454140412yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
Fitting, prime radicalC2×C17×#555C2
socleC23×C17C22×D4
hypercenterC2×#555D17
hypocenterC17#24105
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. C2×#11763120C2semidirect
2. C2×#11762852C2semidirect
3. #14533101C2semidirect
4. #14530109C2semidirect
5. #14529963C2semidirect
6. #14529223C2semidirect
7. #14529082C2semidirect
8. #14529045C2semidirect
9. #14528983C2semidirect
10. #14528879C2semidirect
394. C17#24105semidirect

Maximal subgroups

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

subgroupindex
C2×#117631202
C2×#117628522
#145331012
#145301092
#145299632
#145292232
#145290822
#145290452
#145289832
#145288792
C2×C17×#5552
#2410517