Finite Group Databasebeta

← #17106941

Subgroups — #17106941

Order 4352.

690 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14532658C211yesmaximal, characteristic
2#14531882C211yesmaximal, characteristic, complemented
3#14530172C211yesmaximal, characteristic
4#14530124C211yesmaximal, characteristic, complemented
5#14530123C211yesmaximal, characteristic, complemented
6#14529928C211yesmaximal, characteristic
7#14529741C211yesmaximal, characteristic, complemented
8#14529528C211yesmaximal, characteristic, complemented
9#14529406C211yesmaximal, characteristic
10#14529361C211yesmaximal, characteristic, complemented
11#14529358C211yesmaximal, characteristic, complemented
12#14529351C211yesmaximal, characteristic, complemented
13#14529047C211yesmaximal, characteristic
14#14528611C211yesmaximal, characteristic
15C17×#1926C211yesmaximal, characteristic
16C2×#10587556C2211yescharacteristic
17C2×#10587576C2211yescharacteristic
18C2×D4×Dic17C2211yescharacteristic
19C2×#10587489C2211yescharacteristic
20C2×#10587478C2211yescharacteristic
21C2×#10587488C2211yescharacteristic
22C2×C4⋊#3622C2211yescharacteristic
23C2×#10587503C2211yescharacteristic
24C2×#10587503C2211yescharacteristic
25C2×#10587486C2211yescharacteristic
26#11762745—21no—
27#11762732—21no—
28#11762712C2211yescharacteristic
29#11762708C2211yescharacteristic
30#11762708C2211yescharacteristic
31#11762707C2211yescharacteristic
32#11762706—21no—
33#11762705C2211yescharacteristic
34#11762693—21no—
35#11762692—21no—
36#11762681—21no—
37#11762667—21no—
38C4⋊C4×Dic17C2211yescharacteristic
39C4⋊C4×Dic17C2211yescharacteristic
40C4×#63225C2211yescharacteristic
41C4×C4⋊Dic17—21no—
42#11762572C2211yescharacteristic
43#11762572C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762563C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C2×C17×#106C2211yescharacteristic
53C2×C17×#102C2211yescharacteristic
54C2×C17×#102C2211yescharacteristic
55C17×#390C2211yescharacteristic
56C4×C4⋊C4×C17C2211yescharacteristic
57C17×#340C2211yescharacteristic
58C17×#336C2211yescharacteristic
59C22×D4×C17C2311yescharacteristic
60C23×Dic17—21no—
61C22×C8×C17C2311yescharacteristic
62C22×C8×C17C2311yescharacteristic
63C2×C4⋊C4×C17C2311yescharacteristic
64C2×C4⋊C4×C17C2311yescharacteristic
65C2×C4⋊C4×C17C2311yescharacteristic
66C2×C22⋊C4×C17—21no—
67C2×C22⋊C4×C17—21no—
68C2×C42×C17—21no—
69C2×C42×C17C2311yescharacteristic
70C2×C22⋊Dic17—21no—
71C2×C22⋊Dic17—21no—
72C2×C22⋊Dic17—21no—
73D4×Dic17—22no—
74D4×Dic17—41no—
75C2×C4⋊Dic17—21no—
76C2×C4⋊Dic17—21no—
77C2×C4⋊Dic17—21no—
78C2×C4⋊Dic17C2311yescharacteristic
79C2×C4⋊Dic17C2311yescharacteristic
80C2×C4⋊Dic17C2311yescharacteristic
81C2×C4⋊Dic17C2311yescharacteristic
82C2×#63225—21no—
83C2×#63225—21no—
84C2×#63225—21no—
85C2×#63225—21no—
86C2×#63225—21no—
87C2×#63225D411yescharacteristic
88C2×#63225D411yescharacteristic
89C2×C4×Dic17—21no—
90C2×C4×Dic17—21no—
91C2×C4×Dic17C2311yescharacteristic
92C2×C4×Dic17C2311yescharacteristic
93C22×#3622C2311yescharacteristic
94C22×#3622C2311yescharacteristic
95#10587576—22no—
96#10587576—22no—
97#10587556—22no—
98#10587556—41no—
99C17×#106—22no—
100C17×#102—22no—
101C17×#102Q814yes—
102C17×#95—21no—
103#10587503—22no—
104#10587503—41no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502D411yescharacteristic
112#10587502D411yescharacteristic
113#10587489—41no—
114#10587488—22no—
115#10587486—41no—
116#10587478—22no—
117C4⋊#3622—41no—
118C24×C17—21no—
119C2×D4×C17—21no—
120C2×D4×C17—21no—
121C2×D4×C17—21no—
122C2×D4×C17—21no—
123C2×D4×C17—21no—
124C2×D4×C17C2×Q812yes—
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17C2×D411yescharacteristic
129C22×C4×C17C2×D411yescharacteristic
130C22×C4×C17C2411yescharacteristic
131C22×Dic17—21no—
132C22×Dic17—21no—
133C22×Dic17—21no—
134C22×Dic17—21no—
135C22×Dic17—42no—
136C22×Dic17—42no—
137C22×Dic17—81no—
138C22×Dic17C2×D411yescharacteristic
139C22×Dic17C2×D411yescharacteristic
140C22×Dic17C2×D411yescharacteristic
141C22×Dic17C2×D411yescharacteristic
142C2×C8×C17—21no—
143C2×C8×C17—21no—
144C2×C8×C17—21no—
145C2×C8×C17—21no—
146C2×C8×C17—21no—
147C2×C8×C17C2×Q812yes—
148C4⋊C4×C17—21no—
149C4⋊C4×C17—21no—
150C4⋊C4×C17—21no—
151C4⋊C4×C17—21no—
152C4⋊C4×C17C2×Q812yes—
153C4⋊C4×C17#4112yes—
154C22⋊C4×C17—81no—
155C22⋊C4×C17—81no—
156C42×C17—21no—
157C42×C17—21no—
158C42×C17—42no—
159C22⋊Dic17—42no—
160C22⋊Dic17—42no—
161C22⋊Dic17—81no—
162C4⋊Dic17—24no—
163C4⋊Dic17—21no—
164C4⋊Dic17—21no—
165C4⋊Dic17—21no—
166C4⋊Dic17—21no—
167C4⋊Dic17—21no—
168C4⋊Dic17—21no—
169C4⋊Dic17—42no—
170C4⋊Dic17—81no—
171C4⋊Dic17#4112yes—
172C4⋊Dic17#4112yes—
173#63225—24no—
174#63225—41no—
175#63225—41no—
176#63225—42no—
177#63225—42no—
178#63225—81no—
179#63225—81no—
180C4×Dic17—21no—
181C4×Dic17—21no—
182C4×Dic17—21no—
183C4×Dic17—21no—
184C4×Dic17—81no—
185C4×Dic17—81no—
186C2×#3622—21no—
187C2×#3622—21no—
188C2×#3622—21no—
189C2×#3622—21no—
190C2×#3622—21no—
191C2×#3622—21no—
192#27027—171nomaximal, Sylow
193C23×C17—22no—
194C23×C17—41no—
195C23×C17—41no—
196C23×C17—41no—
197C23×C17—41no—
198C23×C17—41no—
199C23×C17—41no—
200C23×C17C22×D411yescharacteristic
201D4×C17—22no—
202D4×C17—22no—
203D4×C17—41no—
204D4×C17—41no—
205C2×C4×C17—22no—
206C2×C4×C17—21no—
207C2×C4×C17—21no—
208C2×C4×C17—22no—
209C2×C4×C17—21no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—41no—
214C2×C4×C17—41no—
215C2×C4×C17—41no—
216C2×C4×C17—41no—
217C2×C4×C17—41no—
218C2×C4×C17—41no—
219C2×C4×C17—41no—
220C2×C4×C17C22×Q811yescharacteristic
221C2×C4×C17C2×#4111yescharacteristic
222C2×C4×C17C2×#4111yescharacteristic
223C2×C4×C17C2×#4111yescharacteristic
224C2×C4×C17H(5,𝔽2)11yescharacteristic
225C2×C4×C172−1+411yescharacteristic
226C2×Dic17—22no—
227C2×Dic17—21no—
228C2×Dic17—21no—
229C2×Dic17—21no—
230C2×Dic17—22no—
231C2×Dic17—21no—
232C2×Dic17—21no—
233C2×Dic17—21no—
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—21no—
239C2×Dic17—21no—
240C2×Dic17—41no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—84no—
251C8×C17—22no—
252C8×C17—41no—
253#3622—41no—
254#3622—41no—
255#2454—171no—
256#2370—171no—
257#2067—171no—
258#2059—171no—
259#2059—171no—
260#2025—171no—
261#1989—171no—
262#1944—171no—
263#1926D1711yescharacteristic
264#1925—171no—
265#1919—171no—
266#1918—171no—
267#1917—171no—
268#1864—171no—
269#1762—171no—
270C22×C17—42no—
271C22×C17—42no—
272C22×C17—42no—
273C22×C17—81no—
274C22×C17—81no—
275C22×C17—81no—
276C22×C17—81no—
277C22×C17#52511yescharacteristic
278C22×C17#54611yescharacteristic
279C22×C17D4×Q811yescharacteristic
280C22×C17#57511yescharacteristic
281C22×C17#57511yescharacteristic
282C22×C17#57611yescharacteristic
283C22×C17#57711yescharacteristic
284C4×C17—42no—
285C4×C17—42no—
286C4×C17—41no—
287C4×C17—41no—
288C4×C17—81no—
289C4×C17#53311yescharacteristic
290C4×C17#54211yescharacteristic
291C4×C17#55411yescharacteristic
292C4×C17#55611yescharacteristic
293Dic17—41no—
294Dic17—41no—
295Dic17—42no—
296Dic17—41no—
297Dic17—41no—
298Dic17—42no—
299Dic17—81no—
300Dic17—81no—
301C2×#125—171no—
302C2×#125—171no—
303C2×C4×D4—171no—
304C2×#107—171no—
305C2×#107—171no—
306C2×#106—171no—
307C2×#106C2×D1711yescharacteristic
308C2×C4⋊C8—171no—
309C2×#102—171no—
310C2×#102—171no—
311C2×#102C2×D1711yescharacteristic
312C2×#102C2×D1711yescharacteristic
313C2×#97—171no—
314#399—341no—
315#396—341no—
316#390—171no—
317#390C2×D1711yescharacteristic
318#389—171no—
319#389—171no—
320#389—171no—
321#389—171no—
322#389—341no—
323#387—341no—
324#387—341no—
325#385—341no—
326#380—341no—
327C4×C4⋊C4—171no—
328C4×C4⋊C4—171no—
329C4×C4⋊C4—171no—
330C4×C4⋊C4—341no—
331C4×C4⋊C4C2×D1711yescharacteristic
332#340—171no—
333#340—171no—
334#340—171no—
335#340—171no—
336#340—171no—
337#340—171no—
338#340—171no—
339#340C2×D1711yescharacteristic
340#336—171no—
341#336—171no—
342#336—171no—
343#336C2×D1711yescharacteristic
344C2×C17—82no—
345C2×C17#246511yescharacteristic
346C2×C17#311111yescharacteristic
347C2×C17#311911yescharacteristic
348C2×C17#330911yescharacteristic
349C2×C17#332311yescharacteristic
350C2×C17#338811yescharacteristic
351C2×C17#338911yescharacteristic
352C22×D4C22×D1711yescharacteristic
353C23×C4—341no—
354C22×C8—171no—
355C22×C8—171no—
356C22×C8C22×D1711yescharacteristic
357C22×C8C22×D1711yescharacteristic
358#125—342no—
359#125—342no—
360#125—342no—
361#125—681no—
362C4×D4—342no—
363C4×D4—681no—
364C2×C4⋊C4—171no—
365C2×C4⋊C4—171no—
366C2×C4⋊C4—171no—
367C2×C4⋊C4—171no—
368C2×C4⋊C4—171no—
369C2×C4⋊C4—171no—
370C2×C4⋊C4—341no—
371C2×C4⋊C4—341no—
372C2×C4⋊C4—341no—
373C2×C4⋊C4—341no—
374C2×C4⋊C4—341no—
375C2×C4⋊C4—341no—
376C2×C4⋊C4—341no—
377C2×C4⋊C4—341no—
378C2×C4⋊C4C22×D1711yescharacteristic
379C2×C4⋊C4C22×D1711yescharacteristic
380C2×C4⋊C4C22×D1711yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—341no—
384C2×C22⋊C4—341no—
385C2×C22⋊C4—341no—
386C2×C42—21no—
387C2×C42—171no—
388C2×C42—171no—
389C2×C42—341no—
390C2×C42—341no—
391C2×C42C22×D1711yescharacteristic
392#107—342no—
393#107—681no—
394#106—22no—
395#106—342no—
396C4⋊C8—681no—
397#102—22no—
398#102—342no—
399#102—681no—
400#102Dic3414yes—
401#97—681no—
402#95—21no—
403#95—171no—
404#95—171no—
405#95—341no—
406#95—341no—
407#95—341no—
408#95—341no—
409#95—341no—
410#95—341no—
411C17#2702711yescharacteristic, complemented, Sylow
412C24—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4C2×Dic3412yes—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—171no—
423C22×C4—171no—
424C22×C4—171no—
425C22×C4—171no—
426C22×C4—341no—
427C22×C4—341no—
428C22×C4—341no—
429C22×C4—341no—
430C22×C4—682no—
431C22×C4—682no—
432C22×C4—1361no—
433C22×C4D4×D1711yescharacteristic
434C22×C4D4×D1711yescharacteristic
435C22×C4C23×D1711yescharacteristic
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—341no—
442C2×C8—341no—
443C2×C8—341no—
444C2×C8—341no—
445C2×C8—341no—
446C2×C8—341no—
447C2×C8C2×Dic3412yes—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—172no—
453C4⋊C4—172no—
454C4⋊C4—344no—
455C4⋊C4—344no—
456C4⋊C4—341no—
457C4⋊C4—341no—
458C4⋊C4—341no—
459C4⋊C4—341no—
460C4⋊C4—341no—
461C4⋊C4—341no—
462C4⋊C4—681no—
463C4⋊C4—681no—
464C4⋊C4—682no—
465C4⋊C4—682no—
466C4⋊C4—682no—
467C4⋊C4—1361no—
468C4⋊C4—1361no—
469C4⋊C4—1361no—
470C4⋊C4C2×Dic3412yes—
471C4⋊C4#6325412yes—
472C22⋊C4—81no—
473C22⋊C4—81no—
474C22⋊C4—682no—
475C22⋊C4—682no—
476C22⋊C4—1361no—
477C42—21no—
478C42—21no—
479C42—42no—
480C42—341no—
481C42—341no—
482C42—341no—
483C42—341no—
484C42—1361no—
485C42—1361no—
486C23—22no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23C2×D4×D1711yescharacteristic
494D4—22no—
495D4—22no—
496D4—41no—
497D4—41no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—341no—
514C2×C4—341no—
515C2×C4—342no—
516C2×C4—341no—
517C2×C4—341no—
518C2×C4—341no—
519C2×C4—341no—
520C2×C4—341no—
521C2×C4—341no—
522C2×C4—341no—
523C2×C4—341no—
524C2×C4—341no—
525C2×C4—341no—
526C2×C4—342no—
527C2×C4—681no—
528C2×C4—681no—
529C2×C4—681no—
530C2×C4—681no—
531C2×C4—681no—
532C2×C4—681no—
533C2×C4—681no—
534C2×C4—681no—
535C2×C4—681no—
536C2×C4—681no—
537C2×C4—1364no—
538C2×C4C22×Dic3411yescharacteristic
539C2×C4C2×#6325411yescharacteristic
540C2×C4#41×D1711yescharacteristic
541C2×C4#41×D1711yescharacteristic
542C2×C4#1058769611yescharacteristic
543C2×C4#1058769711yescharacteristic
544C8—22no—
545C8—41no—
546C8—681no—
547C8—681no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22#1176331911yescharacteristic
556C22#1176345111yescharacteristic
557C22D4×Dic3411yescharacteristic
558C22#1176360511yescharacteristic
559C22#1176360611yescharacteristic
560C22#1176361311yescharacteristic
561C22#1176361511yescharacteristic
562C4—42no—
563C4—42no—
564C4—41no—
565C4—41no—
566C4—81no—
567C4—681no—
568C4—681no—
569C4—682no—
570C4—681no—
571C4—682no—
572C4—681no—
573C4—1361no—
574C4—1361no—
575C4#1176336211yescharacteristic
576C4#1176342511yescharacteristic
577C4#1176351611yescharacteristic
578C4#1176352811yescharacteristic
579C2—82no—
580C2#1453281011yescharacteristic
581C2#1453889011yescharacteristic
582C2#1453898211yescharacteristic
583C2#1454078811yescharacteristic
584C2#1454100311yescharacteristic
585C2#1454172511yescharacteristic
586C2#1454174311yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#1926C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
hypercenter#1926D17
hypocenterC17#27027
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1926

Complements

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

NH ≅ G/Ntype
2. #14531882C2semidirect
4. #14530124C2semidirect
5. #14530123C2semidirect
7. #14529741C2semidirect
8. #14529528C2semidirect
10. #14529361C2semidirect
11. #14529358C2semidirect
12. #14529351C2semidirect
411. C17#27027semidirect

Maximal subgroups

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

subgroupindex
#145326582
#145318822
#145301722
#145301242
#145301232
#145299282
#145297412
#145295282
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