Finite Group Databasebeta

← #17068266

Subgroups — #17068266

Order 4352.

1154 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763101C211yesmaximal, characteristic, complemented
2C2×C4×#63229C211yesmaximal, characteristic, complemented
3#14533706C211yesmaximal, characteristic, complemented
4#14529963C212yesmaximal, complemented
5#14529787C212yesmaximal, complemented
6#14529746C211yesmaximal, characteristic, complemented
7#14529579C211yesmaximal, characteristic, complemented
8#14529096C212yesmaximal, complemented
9#14528983C212yesmaximal, complemented
10#14528784C211yesmaximal, characteristic, complemented
11C2×C17×#555C211yesmaximal, characteristic
12C22×#63229C2211yescharacteristic
13C2×#10587610C2212yes—
14C2×C4×Dic34C2211yescharacteristic
15#11763101—24no—
16C4×#63229—24no—
17C2×#10587478C2212yes—
18C2×#10587479C2211yescharacteristic
19C2×C4⋊#3622C2211yescharacteristic
20C2×#10587481C2212yes—
21C2×#10587481C2211yescharacteristic
22C2×#10587503C2211yescharacteristic
23C2×#10587503C2212yes—
24C2×C22⋊#3622C2212yes—
25C2×C22⋊#3622C2212yes—
26C2×C4×#3622C2211yescharacteristic
27#11762732—22no—
28#11762712—21no—
29#11762705—21no—
30#11762679C2211yescharacteristic
31#11762669C2212yes—
32#11762571C2212yes—
33#11762571C2212yes—
34#11762571C2212yes—
35C17×#555—24no—
36C17×#555—24no—
37C2×C17×C4⋊Q8C2211yescharacteristic
38C2×C17×#124—22no—
39C2×C17×#121—21no—
40C2×C17×#121C2212yes—
41C2×C4×D4×C17C2211yescharacteristic
42C2×C4×D4×C17C2211yescharacteristic
43C2×C17×#117—21no—
44C2×C17×#117C2212yes—
45C22×Q8×C17—21no—
46C22×D4×C17—22no—
47C22×D4×C17D411yescharacteristic
48C22×D4×C17C2311yescharacteristic
49C23×C4×C17—21no—
50C23×C4×C17C2312yes—
51C22×Dic34C2311yescharacteristic
52C17×C4⋊Q8—22no—
53C17×C4⋊Q8D414yes—
54C17×#124—216no—
55C17×#121—28no—
56C17×#121—24no—
57C17×#121—24no—
58C4×D4×C17—22no—
59C4×D4×C17—22no—
60C4×D4×C17—22no—
61C4×D4×C17D414yes—
62C17×#117—28no—
63C17×#117—24no—
64C17×#117—24no—
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17D411yescharacteristic
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2312yes—
69C2×C22⋊C4×C17—22no—
70C2×C22⋊C4×C17—22no—
71C2×C22⋊C4×C17—21no—
72C2×C22⋊C4×C17D412yes—
73C2×C42×C17—22no—
74C2×C42×C17C2311yescharacteristic
75C2×C22⋊Dic17—22no—
76C2×C22⋊Dic17—22no—
77C2×#63229—22no—
78C2×#63229—22no—
79C2×#63229—22no—
80C2×C4⋊Dic17—21no—
81C2×C4⋊Dic17—21no—
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2312yes—
84#10587610—24no—
85#10587610—42no—
86C2×#63225—22no—
87C2×#63225—21no—
88C2×C4×Dic17—21no—
89C22×#3622C2311yescharacteristic
90C22×#3622C2312yes—
91C22×#3622C2311yescharacteristic
92C4×Dic34—22no—
93C4×Dic34—22no—
94#10587503—22no—
95#10587503—42no—
96#10587502—22no—
97C22⋊#3622—24no—
98C22⋊#3622—42no—
99#10587481—24no—
100#10587481—22no—
101#10587479—22no—
102#10587478—24no—
103C4⋊#3622—22no—
104C4×#3622—22no—
105C24×C17—21no—
106C24×C17C2×D412yes—
107C2×Q8×C17—24no—
108C2×Q8×C17—81no—
109C2×Q8×C17—81no—
110C2×D4×C17—28no—
111C2×D4×C17—22no—
112C2×D4×C17—22no—
113C2×D4×C17—22no—
114C2×D4×C17—21no—
115C2×D4×C17—21no—
116C2×D4×C17—21no—
117C2×D4×C17—21no—
118C2×D4×C17—21no—
119C2×D4×C17—82no—
120C2×D4×C17—82no—
121C2×D4×C17C2×D412yes—
122C22×C4×C17—21no—
123C22×C4×C17—22no—
124C22×C4×C17—24no—
125C22×C4×C17—22no—
126C22×C4×C17—22no—
127C22×C4×C17—22no—
128C22×C4×C17—22no—
129C22×C4×C17—22no—
130C22×C4×C17—22no—
131C22×C4×C17—42no—
132C22×C4×C17—42no—
133C22×C4×C17C2×D412yes—
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17C2×D411yescharacteristic
136C22×C4×C17C2411yescharacteristic
137C22×Dic17—21no—
138C22×Dic17—21no—
139C22×Dic17—22no—
140C2×Dic34—21no—
141C2×Dic34—21no—
142C2×Dic34—21no—
143C2×Dic34—21no—
144C2×Dic34—21no—
145C2×Dic34#4112yes—
146C4⋊C4×C17—24no—
147C4⋊C4×C17—22no—
148C4⋊C4×C17—22no—
149C4⋊C4×C17—21no—
150C4⋊C4×C17C2×D414yes—
151C4⋊C4×C17C2×D412yes—
152C22⋊C4×C17—28no—
153C22⋊C4×C17—28no—
154C22⋊C4×C17—24no—
155C22⋊C4×C17—24no—
156C42×C17—28no—
157C42×C17C2×D412yes—
158C42×C17C2×D412yes—
159C22⋊Dic17—44no—
160C22⋊Dic17—82no—
161#63229—28no—
162C4⋊Dic17—21no—
163C4⋊Dic17—21no—
164C4⋊Dic17—22no—
165C4⋊Dic17—22no—
166C4⋊Dic17—42no—
167C4⋊Dic17—81no—
168#63225—44no—
169#63225—42no—
170C4×Dic17—42no—
171C2×#3622—21no—
172C2×#3622—21no—
173C2×#3622—21no—
174C2×#3622—22no—
175C2×#3622—22no—
176C2×#3622—21no—
177C2×#3622—22no—
178C2×#3622—21no—
179C2×#3622#4112yes—
180#24121—171nomaximal, Sylow
181C23×C17—22no—
182C23×C17—22no—
183C23×C17—22no—
184C23×C17—22no—
185C23×C17—22no—
186C23×C17—22no—
187C23×C17—22no—
188C23×C17—22no—
189C23×C17—41no—
190C23×C17—41no—
191C23×C17—41no—
192C23×C17—41no—
193C23×C17—41no—
194C23×C17—41no—
195C23×C17C22×D411yescharacteristic
196Q8×C17—84no—
197D4×C17—22no—
198D4×C17—22no—
199D4×C17—41no—
200D4×C17—41no—
201D4×C17—44no—
202D4×C17—88no—
203C2×C4×C17—21no—
204C2×C4×C17—21no—
205C2×C4×C17—22no—
206C2×C4×C17—22no—
207C2×C4×C17—24no—
208C2×C4×C17—22no—
209C2×C4×C17—22no—
210C2×C4×C17—41no—
211C2×C4×C17—41no—
212C2×C4×C17—42no—
213C2×C4×C17—42no—
214C2×C4×C17—42no—
215C2×C4×C17—42no—
216C2×C4×C17—42no—
217C2×C4×C17—42no—
218C2×C4×C17—48no—
219C2×C4×C17#12012yes—
220C2×C4×C17#12012yes—
221C2×C4×C17#13512yes—
222C2×C4×C17#13712yes—
223C2×C4×C17C22×D411yescharacteristic
224C2×C4×C17C22×D411yescharacteristic
225C2×C4×C17C2×#4111yescharacteristic
226C2×C4×C17C2×#4111yescharacteristic
227C2×C4×C17H(5,𝔽2)11yescharacteristic
228C2×C4×C172−1+411yescharacteristic
229C2×Dic17—22no—
230C2×Dic17—41no—
231C2×Dic17—41no—
232C2×Dic17—41no—
233C2×Dic17—42no—
234C2×Dic17—42no—
235C2×Dic17—41no—
236C2×Dic17—42no—
237C2×Dic17—41no—
238Dic34—22no—
239Dic34—22no—
240Dic34—41no—
241Dic34—41no—
242#3622—22no—
243#3622—41no—
244#3622—42no—
245C2×#555D1711yescharacteristic
246C2×#477—171no—
247C2×C4×SD16—171no—
248#2538—171no—
249#2033—172no—
250#2000—172no—
251#1990—171no—
252#1954—171no—
253#1873—172no—
254#1854—172no—
255#1802—171no—
256C22×C17—42no—
257C22×C17—42no—
258C22×C17—42no—
259C22×C17—42no—
260C22×C17—42no—
261C22×C17—42no—
262C22×C17—42no—
263C22×C17—42no—
264C22×C17—42no—
265C22×C17—42no—
266C22×C17—81no—
267C22×C17—81no—
268C22×C17—81no—
269C22×C17—81no—
270C22×C17C2×#12011yescharacteristic
271C22×C17#54611yescharacteristic
272C22×C17#54711yescharacteristic
273C22×C17C2×#13511yescharacteristic
274C22×C17C2×#13711yescharacteristic
275C22×C17#57611yescharacteristic
276C22×C17#57711yescharacteristic
277C4×C17—22no—
278C4×C17—42no—
279C4×C17—42no—
280C4×C17—44no—
281C4×C17#45214yes—
282C4×C17#52511yescharacteristic
283C4×C17#53711yescharacteristic
284C4×C17#54811yescharacteristic
285C4×C17#55011yescharacteristic
286Dic17—42no—
287Dic17—81no—
288Dic17—82no—
289C22×SD16—171no—
290#555—24no—
291#555—24no—
292C2×C4⋊Q8C2×D1711yescharacteristic
293C2×#124—22no—
294C2×C22⋊Q8—172no—
295C2×#121—21no—
296C2×#121C2×D1712yes—
297C2×C4×Q8—171no—
298C2×C4×D4C2×D1711yescharacteristic
299C2×C4×D4C2×D1711yescharacteristic
300C2×#117—21no—
301C2×#117C2×D1712yes—
302#477—344no—
303C4×SD16—344no—
304C2×#107—172no—
305C2×#106—171no—
306C2×C4⋊C8—171no—
307C2×#103—172no—
308C2×#103—171no—
309C2×#102—171no—
310C2×#102—172no—
311C2×C22⋊C8—172no—
312C2×C22⋊C8—172no—
313C2×C4×C8—171no—
314#396—342no—
315#390—341no—
316#389—341no—
317#384—171no—
318#380—172no—
319#340—172no—
320#340—172no—
321#340—172no—
322C2×C17—82no—
323C2×C17—82no—
324C2×C17#240711yescharacteristic
325C2×C17C2×#45211yescharacteristic
326C2×C17#301211yescharacteristic
327C2×C17#330511yescharacteristic
328C2×C17#332911yescharacteristic
329C2×C17#334011yescharacteristic
330C2×C17#335711yescharacteristic
331C22×Q8—21no—
332C22×Q8—171no—
333C22×D4—22no—
334C22×D4C22⋊D1711yescharacteristic
335C22×D4C22×D1711yescharacteristic
336C23×C4—21no—
337C23×C4C22×D1712yes—
338C2×SD16—342no—
339C2×SD16—342no—
340C2×SD16—342no—
341C22×C8—171no—
342C22×C8—171no—
343C22×C8—172no—
344C4⋊Q8—22no—
345C4⋊Q8C22⋊D1714yes—
346#124—216no—
347C22⋊Q8—344no—
348C22⋊Q8—682no—
349#121—28no—
350#121—24no—
351#121—24no—
352C4×Q8—342no—
353C4×Q8—342no—
354C4×D4—22no—
355C4×D4—22no—
356C4×D4—22no—
357C4×D4C22⋊D1714yes—
358#117—28no—
359#117—24no—
360#117—24no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—171no—
363C2×C4⋊C4—172no—
364C2×C4⋊C4—341no—
365C2×C4⋊C4—342no—
366C2×C4⋊C4—341no—
367C2×C4⋊C4—341no—
368C2×C4⋊C4C22⋊D1711yescharacteristic
369C2×C4⋊C4C22×D1711yescharacteristic
370C2×C4⋊C4C22×D1712yes—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—342no—
375C2×C22⋊C4—342no—
376C2×C22⋊C4C22⋊D1712yes—
377C2×C42—22no—
378C2×C42—341no—
379C2×C42C22×D1711yescharacteristic
380#107—344no—
381#106—342no—
382C4⋊C8—342no—
383#103—342no—
384#103—344no—
385#102—342no—
386#102—682no—
387C22⋊C8—344no—
388C22⋊C8—682no—
389C4×C8—342no—
390#95—342no—
391C17#2412111yescharacteristic, complemented, Sylow
392C24—21no—
393C24C2×C22⋊D1712yes—
394C2×Q8—24no—
395C2×Q8—81no—
396C2×Q8—81no—
397C2×Q8—172no—
398C2×Q8—341no—
399C2×Q8—341no—
400C2×Q8—341no—
401C2×Q8—341no—
402C2×Q8—341no—
403C2×D4—28no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—82no—
413C2×D4—82no—
414C2×D4C2×C22⋊D1712yes—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—341no—
427C22×C4—341no—
428C22×C4—342no—
429C22×C4C2×C22⋊D1712yes—
430C22×C4C2×C22⋊D1711yescharacteristic
431C22×C4C2×C22⋊D1711yescharacteristic
432C22×C4C23×D1711yescharacteristic
433SD16—348no—
434C2×C8—172no—
435C2×C8—341no—
436C2×C8—341no—
437C2×C8—342no—
438C2×C8—341no—
439C2×C8—342no—
440C2×C8—341no—
441C2×C8—341no—
442C2×C8—342no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—342no—
448C4⋊C4—342no—
449C4⋊C4—341no—
450C4⋊C4—341no—
451C4⋊C4—684no—
452C4⋊C4—682no—
453C4⋊C4—682no—
454C4⋊C4—1361no—
455C4⋊C4C2×C22⋊D1712yes—
456C4⋊C4C2×C22⋊D1714yes—
457C22⋊C4—28no—
458C22⋊C4—28no—
459C22⋊C4—24no—
460C22⋊C4—24no—
461C22⋊C4—684no—
462C22⋊C4—1362no—
463C42—28no—
464C42—682no—
465C42C2×C22⋊D1712yes—
466C42C2×C22⋊D1712yes—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23C22×C22⋊D1711yescharacteristic
482Q8—84no—
483Q8—342no—
484Q8—342no—
485Q8—681no—
486Q8—681no—
487D4—22no—
488D4—22no—
489D4—41no—
490D4—41no—
491D4—44no—
492D4—88no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—48no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—342no—
510C2×C4—681no—
511C2×C4—681no—
512C2×C4—681no—
513C2×C4—682no—
514C2×C4—681no—
515C2×C4—682no—
516C2×C4—681no—
517C2×C4—682no—
518C2×C4#1058763612yes—
519C2×C4#1058763712yes—
520C2×C4#1058763912yes—
521C2×C4#1058763912yes—
522C2×C4#41×D1711yescharacteristic
523C2×C4#41×D1711yescharacteristic
524C2×C4#1058769611yescharacteristic
525C2×C4#1058769711yescharacteristic
526C2×C4C22×C22⋊D1711yescharacteristic
527C2×C4C22×C22⋊D1711yescharacteristic
528C8—342no—
529C8—682no—
530C8—681no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—81no—
545C22C2×#1058763911yescharacteristic
546C22#1176344711yescharacteristic
547C22#1176345811yescharacteristic
548C22C2×#1058763611yescharacteristic
549C22C2×#1058763711yescharacteristic
550C22#1176361211yescharacteristic
551C22#1176362011yescharacteristic
552C4—22no—
553C4—42no—
554C4—42no—
555C4—44no—
556C4—682no—
557C4—1361no—
558C4—1362no—
559C4#1176297114yes—
560C4#1176331611yescharacteristic
561C4#1176338711yescharacteristic
562C4#1176346611yescharacteristic
563C4#1176347811yescharacteristic
564C2—82no—
565C2—82no—
566C2#1453212311yescharacteristic
567C2C2×#1176297111yescharacteristic
568C2#1453808811yescharacteristic
569C2#1454072811yescharacteristic
570C2#1454107211yescharacteristic
571C2#1454119711yescharacteristic
572C2#1454140411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
Fitting, prime radicalC2×C17×#555C2
socleC23×C17C22×D4
hypercenterC2×#555D17
hypocenterC17#24121
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×#11763101C2semidirect
2. C2×C4×#63229C2semidirect
3. #14533706C2semidirect
4. #14529963C2semidirect
5. #14529787C2semidirect
6. #14529746C2semidirect
7. #14529579C2semidirect
8. #14529096C2semidirect
9. #14528983C2semidirect
10. #14528784C2semidirect
391. C17#24121semidirect

Maximal subgroups

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

subgroupindex
C2×#117631012
C2×C4×#632292
#145337062
#145299632
#145297872
#145297462
#145295792
#145290962
#145289832
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C2×C17×#5552
#2412117