Finite Group Databasebeta

← #17088738

Subgroups — #17088738

Order 4352.

1096 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763219C211yesmaximal, characteristic, complemented
2C2×#11763155C211yesmaximal, characteristic, complemented
3#14534605C211yesmaximal, characteristic, complemented
4#14529970C212yesmaximal, complemented
5#14529963C212yesmaximal, complemented
6#14529750C211yesmaximal, characteristic, complemented
7#14529609C211yesmaximal, characteristic, complemented
8#14529045C212yesmaximal, complemented
9#14529031C212yesmaximal, complemented
10#14528550C211yesmaximal, characteristic, complemented
11C2×C17×#540C211yesmaximal, characteristic
12C2×#10587555C2211yescharacteristic
13#11763219—24no—
14#11763155—24no—
15C2×#10587478C2212yes—
16C2×#10587479C2212yes—
17C2×#10587503C2212yes—
18C2×#10587503C2212yes—
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×#540—24no—
35C17×#540—24no—
36C2×C17×#127C2211yescharacteristic
37C2×C17×#125C2211yescharacteristic
38C2×C17×#123—21no—
39C2×C17×#123—21no—
40C2×C17×#121—21no—
41C2×C17×#121—21no—
42C2×C17×#121C2211yescharacteristic
43C2×C17×#121C2211yescharacteristic
44C2×C4×D4×C17C2211yescharacteristic
45C2×C4×D4×C17C2211yescharacteristic
46C2×C17×#117C2211yescharacteristic
47C22×D4×C17—21no—
48C22×D4×C17—21no—
49C22×D4×C17—21no—
50C22×D4×C17—21no—
51C22×D4×C17C2311yescharacteristic
52C22×D4×C17C2311yescharacteristic
53C23×C4×C17—21no—
54C23×C4×C17—21no—
55C23×C4×C17C2311yescharacteristic
56C22×Dic34—21no—
57C17×#127—22no—
58C17×#127D414yes—
59C17×#125—22no—
60C17×#125D414yes—
61C17×#123—28no—
62C17×#123—28no—
63C17×#121—28no—
64C17×#121—28no—
65C17×#121—22no—
66C17×#121—22no—
67C17×#121—22no—
68C17×#121—22no—
69C4×D4×C17—22no—
70C4×D4×C17—22no—
71C4×D4×C17—22no—
72C4×D4×C17—22no—
73C17×#117—22no—
74C17×#117—22no—
75C2×C4⋊C4×C17—21no—
76C2×C4⋊C4×C17—21no—
77C2×C4⋊C4×C17D411yescharacteristic
78C2×C4⋊C4×C17D411yescharacteristic
79C2×C4⋊C4×C17C2311yescharacteristic
80C2×C4⋊C4×C17C2311yescharacteristic
81C2×C22⋊C4×C17—21no—
82C2×C22⋊C4×C17—21no—
83C2×C22⋊C4×C17—21no—
84C2×C22⋊C4×C17—21no—
85C2×C22⋊C4×C17D411yescharacteristic
86C2×C22⋊C4×C17D411yescharacteristic
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#10587503—24no—
99#10587503—24no—
100#10587503—24no—
101#10587503—24no—
102#10587502—22no—
103#10587502—21no—
104#10587502—22no—
105#10587502—21no—
106C22⋊#3622—24no—
107#10587479—24no—
108#10587478—24no—
109#10587474—22no—
110C4×#3622—22no—
111C24×C17—21no—
112C24×C17—21no—
113C24×C17C2×D411yescharacteristic
114C2×D4×C17—24no—
115C2×D4×C17—24no—
116C2×D4×C17—24no—
117C2×D4×C17—24no—
118C2×D4×C17—21no—
119C2×D4×C17—21no—
120C2×D4×C17—22no—
121C2×D4×C17—22no—
122C2×D4×C17—22no—
123C2×D4×C17—22no—
124C2×D4×C17—82no—
125C2×D4×C17—82no—
126C2×D4×C17—82no—
127C2×D4×C17—82no—
128C2×D4×C17C2×D412yes—
129C2×D4×C17C2×D412yes—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17—24no—
133C22×C4×C17—24no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—22no—
137C22×C4×C17—22no—
138C22×C4×C17—82no—
139C22×C4×C17—82no—
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17C2×D411yescharacteristic
145C22×C4×C17C2411yescharacteristic
146C22×Dic17—22no—
147C22×Dic17—22no—
148C2×Dic34—42no—
149C2×Dic34—82no—
150C4⋊C4×C17—22no—
151C4⋊C4×C17—22no—
152C4⋊C4×C17—24no—
153C4⋊C4×C17—24no—
154C4⋊C4×C17—21no—
155C4⋊C4×C17—21no—
156C4⋊C4×C17C2×D412yes—
157C4⋊C4×C17C2×D412yes—
158C22⋊C4×C17—24no—
159C22⋊C4×C17—24no—
160C22⋊C4×C17—24no—
161C22⋊C4×C17—24no—
162C22⋊C4×C17—22no—
163C22⋊C4×C17—22no—
164C42×C17—21no—
165C42×C17—21no—
166C42×C17C2×D412yes—
167C42×C17C2×D412yes—
168C22⋊Dic17—82no—
169C4⋊Dic17—22no—
170C4⋊Dic17—22no—
171C4⋊Dic17—22no—
172C4⋊Dic17—22no—
173C4⋊Dic17—42no—
174#63225—82no—
175C2×#3622—22no—
176C2×#3622—22no—
177C2×#3622—22no—
178C2×#3622—22no—
179C2×#3622—22no—
180C2×#3622#4114yes—
181#25685—171nomaximal, Sylow
182C23×C17—21no—
183C23×C17—22no—
184C23×C17—22no—
185C23×C17—21no—
186C23×C17—21no—
187C23×C17—22no—
188C23×C17—22no—
189C23×C17—41no—
190C23×C17—41no—
191C23×C17—41no—
192C23×C17—41no—
193C23×C17—42no—
194C23×C17—42no—
195C23×C17—42no—
196C23×C17—42no—
197C23×C17C22×D411yescharacteristic
198D4×C17—24no—
199D4×C17—24no—
200D4×C17—42no—
201D4×C17—42no—
202D4×C17—84no—
203D4×C17—84no—
204D4×C17—84no—
205D4×C17—84no—
206C2×C4×C17—22no—
207C2×C4×C17—22no—
208C2×C4×C17—21no—
209C2×C4×C17—21no—
210C2×C4×C17—21no—
211C2×C4×C17—22no—
212C2×C4×C17—22no—
213C2×C4×C17—21no—
214C2×C4×C17—21no—
215C2×C4×C17—22no—
216C2×C4×C17—22no—
217C2×C4×C17—42no—
218C2×C4×C17—42no—
219C2×C4×C17—42no—
220C2×C4×C17—42no—
221C2×C4×C17—84no—
222C2×C4×C17—84no—
223C2×C4×C17#12012yes—
224C2×C4×C17#12012yes—
225C2×C4×C17C22×D411yescharacteristic
226C2×C4×C17C22×D411yescharacteristic
227C2×C4×C17C2×#4112yes—
228C2×C4×C172−1+412yes—
229C2×Dic17—42no—
230C2×Dic17—42no—
231C2×Dic17—42no—
232C2×Dic17—42no—
233C2×Dic17—42no—
234C2×Dic17—42no—
235Dic34—84no—
236#3622—24no—
237#3622—42no—
238C2×#540D1711yescharacteristic
239C2×#501—171no—
240C2×#486—171no—
241#2628—171no—
242#2034—172no—
243#2033—172no—
244#1991—171no—
245#1961—171no—
246#1864—172no—
247#1862—172no—
248#1750—171no—
249C22×C17—42no—
250C22×C17—41no—
251C22×C17—42no—
252C22×C17—42no—
253C22×C17—42no—
254C22×C17—42no—
255C22×C17—41no—
256C22×C17—42no—
257C22×C17—42no—
258C22×C17—41no—
259C22×C17—42no—
260C22×C17—82no—
261C22×C17—82no—
262C22×C17—82no—
263C22×C17—82no—
264C22×C17C2×#12011yescharacteristic
265C22×C17#53211yescharacteristic
266C22×C17#53611yescharacteristic
267C22×C17#57612yes—
268C22×C17#57712yes—
269C4×C17—22no—
270C4×C17—22no—
271C4×C17—41no—
272C4×C17—41no—
273C4×C17—42no—
274C4×C17—41no—
275C4×C17—42no—
276C4×C17#54712yes—
277C4×C17#54812yes—
278Dic17—82no—
279Dic17—82no—
280#540—24no—
281#540—24no—
282C2×C4⋊Q8—171no—
283C2×#127C2×D1711yescharacteristic
284C2×#125C2×D1711yescharacteristic
285C2×#123—21no—
286C2×#123—21no—
287C2×#121—21no—
288C2×#121—21no—
289C2×#121C2×D1711yescharacteristic
290C2×#121C2×D1711yescharacteristic
291C2×C4×D4C2×D1711yescharacteristic
292C2×C4×D4C2×D1711yescharacteristic
293C2×#117C2×D1711yescharacteristic
294#501—344no—
295#486—344no—
296C2×#107—172no—
297C2×#106—172no—
298C2×#102—172no—
299C2×#102—172no—
300C2×#102—172no—
301C2×#102—172no—
302C2×C22⋊C8—172no—
303C2×#97—171no—
304C2×C4×C8—171no—
305#400—341no—
306#399—341no—
307#395—341no—
308#393—341no—
309#389—172no—
310#382—171no—
311#380—172no—
312#340—172no—
313#340—172no—
314#336—172no—
315C2×C17—81no—
316C2×C17—82no—
317C2×C17—82no—
318C2×C17#243211yescharacteristic
319C2×C17#300811yescharacteristic
320C2×C17#300911yescharacteristic
321C2×C17#317912yes—
322C2×C17#324112yes—
323C22×Q8—341no—
324C22×D4—21no—
325C22×D4—21no—
326C22×D4—21no—
327C22×D4—21no—
328C22×D4C22×D1711yescharacteristic
329C22×D4C22×D1711yescharacteristic
330C23×C4—21no—
331C23×C4—21no—
332C23×C4C22×D1711yescharacteristic
333C22×C8—172no—
334C22×C8—172no—
335C4⋊Q8—342no—
336C4⋊Q8—342no—
337#127—22no—
338#127C22⋊D1714yes—
339#125—22no—
340#125C22⋊D1714yes—
341#123—28no—
342#123—28no—
343#121—28no—
344#121—28no—
345#121—22no—
346#121—22no—
347#121—22no—
348#121—22no—
349C4×D4—22no—
350C4×D4—22no—
351C4×D4—22no—
352C4×D4—22no—
353#117—22no—
354#117—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—172no—
358C2×C4⋊C4—172no—
359C2×C4⋊C4—341no—
360C2×C4⋊C4—342no—
361C2×C4⋊C4C22⋊D1711yescharacteristic
362C2×C4⋊C4C22⋊D1711yescharacteristic
363C2×C4⋊C4C22×D1711yescharacteristic
364C2×C4⋊C4C22×D1711yescharacteristic
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—342no—
370C2×C22⋊C4C22⋊D1711yescharacteristic
371C2×C22⋊C4C22⋊D1711yescharacteristic
372C2×C42C22×D1711yescharacteristic
373C2×C42C22×D1711yescharacteristic
374#107—344no—
375#106—344no—
376#102—344no—
377#102—344no—
378#102—344no—
379#102—344no—
380C22⋊C8—344no—
381#97—342no—
382C4×C8—342no—
383#95—342no—
384#95—341no—
385#95—341no—
386#95—342no—
387C17#2568511yescharacteristic, complemented, Sylow
388C24—21no—
389C24—21no—
390C24C2×C22⋊D1711yescharacteristic
391C2×Q8—682no—
392C2×Q8—1362no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—82no—
404C2×D4—82no—
405C2×D4—82no—
406C2×D4—82no—
407C2×D4C2×C22⋊D1712yes—
408C2×D4C2×C22⋊D1712yes—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—82no—
418C22×C4—82no—
419C22×C4—342no—
420C22×C4—342no—
421C22×C4C2×C22⋊D1711yescharacteristic
422C22×C4C2×C22⋊D1711yescharacteristic
423C22×C4C2×C22⋊D1711yescharacteristic
424C22×C4C2×C22⋊D1711yescharacteristic
425C22×C4C2×C22⋊D1711yescharacteristic
426C22×C4C23×D1711yescharacteristic
427C2×C8—174no—
428C2×C8—342no—
429C2×C8—342no—
430C2×C8—342no—
431C2×C8—342no—
432C2×C8—342no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—342no—
440C4⋊C4—342no—
441C4⋊C4—342no—
442C4⋊C4—342no—
443C4⋊C4—682no—
444C4⋊C4—1362no—
445C4⋊C4C2×C22⋊D1712yes—
446C4⋊C4C2×C22⋊D1712yes—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—1362no—
454C42—21no—
455C42—21no—
456C42C2×C22⋊D1712yes—
457C42C2×C22⋊D1712yes—
458C23—21no—
459C23—22no—
460C23—22no—
461C23—21no—
462C23—22no—
463C23—22no—
464C23—21no—
465C23—41no—
466C23—41no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—41no—
472C23—41no—
473C23C22×C22⋊D1711yescharacteristic
474Q8—1364no—
475D4—24no—
476D4—24no—
477D4—42no—
478D4—42no—
479D4—84no—
480D4—84no—
481D4—84no—
482D4—84no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—84no—
499C2×C4—84no—
500C2×C4—682no—
501C2×C4—682no—
502C2×C4—682no—
503C2×C4—682no—
504C2×C4—682no—
505C2×C4—682no—
506C2×C4#1058763912yes—
507C2×C4#1058763912yes—
508C2×C4#41×D1712yes—
509C2×C4#1058769712yes—
510C2×C4C22×C22⋊D1711yescharacteristic
511C2×C4C22×C22⋊D1711yescharacteristic
512C8—344no—
513C8—682no—
514C22—41no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—41no—
519C22—42no—
520C22—42no—
521C22—41no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22C2×#1058763911yescharacteristic
530C22#1176335511yescharacteristic
531C22#1176338011yescharacteristic
532C22#1176361212yes—
533C22#1176362012yes—
534C4—22no—
535C4—22no—
536C4—41no—
537C4—41no—
538C4—42no—
539C4—41no—
540C4—42no—
541C4—1362no—
542C4—1362no—
543C4#1176345812yes—
544C4#1176346612yes—
545C2—81no—
546C2—82no—
547C2—82no—
548C2#1453240811yescharacteristic
549C2#1453805211yescharacteristic
550C2#1453806111yescharacteristic
551C2#1453946912yes—
552C2#1454003112yes—
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
Fitting, prime radicalC2×C17×#540C2
socleC23×C17C22×D4
hypercenterC2×#540D17
hypocenterC17#25685
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#540

Complements

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

NH ≅ G/Ntype
1. C2×#11763219C2semidirect
2. C2×#11763155C2semidirect
3. #14534605C2semidirect
4. #14529970C2semidirect
5. #14529963C2semidirect
6. #14529750C2semidirect
7. #14529609C2semidirect
8. #14529045C2semidirect
9. #14529031C2semidirect
10. #14528550C2semidirect
387. C17#25685semidirect

Maximal subgroups

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

subgroupindex
C2×#117632192
C2×#117631552
#145346052
#145299702
#145299632
#145297502
#145296092
#145290452
#145290312
#145285502
C2×C17×#5402
#2568517