Finite Group Databasebeta

← #16933454

Subgroups — #16933454

Order 4352.

742 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#14537083C211yesmaximal, characteristic
2#14528762C211yesmaximal, characteristic, complemented
3C2×#11762547C211yesmaximal, characteristic, complemented
4C2×#11762545C211yesmaximal, characteristic, complemented
5C2×#11762543C211yesmaximal, characteristic
6#95×Dic17C211yesmaximal, characteristic, complemented
7C2×C17×#327C211yesmaximal, characteristic
8C2×#10587622C411yescharacteristic, complemented
9C2×#10587610C2211yescharacteristic
10C22×C4×Dic17—21no—
11C22×C4×Dic17C2211yescharacteristic
12C2×#10587485—21no—
13C2×C22⋊#3622C2211yescharacteristic
14#11762714C411yescharacteristic, complemented
15#11762712C411yescharacteristic, complemented
16#11762703—21no—
17#11762703—21no—
18#11762689—21no—
19#11762689—21no—
20#11762674C411yescharacteristic, complemented
21C2×#10587502—21no—
22C2×#10587502C2211yescharacteristic
23#11762563—21no—
24#11762547—22no—
25#11762545—22no—
26#11762543—22no—
27C2×C17×#121C2211yescharacteristic
28C2×C17×C22⋊C8C2211yescharacteristic
29C2×C17×#95C2211yescharacteristic
30C17×#327—22no—
31C22×D4×C17—21no—
32C22×D4×C17C2×C411yescharacteristic
33C23×C4×C17—21no—
34C23×C4×C17C2311yescharacteristic
35C23×Dic17—21no—
36C23×Dic17D411yescharacteristic
37C23×Dic17D411yescharacteristic
38C22×Dic34C2×C411yescharacteristic
39C22×C8×C17—21no—
40C17×#121—21no—
41C17×#121—21no—
42C17×#121—21no—
43C17×#121—21no—
44C2×C4⋊C4×C17C2×C411yescharacteristic
45C2×C22⋊C4×C17—21no—
46C2×C22⋊Dic17—21no—
47C2×C22⋊Dic17—21no—
48C2×C22⋊Dic17—21no—
49C2×C22⋊Dic17—21no—
50C2×C22⋊Dic17—21no—
51#10587622—42no—
52C2×C4⋊Dic17C2×C411yescharacteristic
53#10587610—21no—
54#10587610—21no—
55#10587610—21no—
56#10587610—21no—
57C2×#63225—21no—
58C2×C4×Dic17—24no—
59C2×C4×Dic17—42no—
60C2×C4×Dic17—42no—
61C2×C4×Dic17—42no—
62C2×C4×Dic17C2×C411yescharacteristic
63C2×C4×Dic17C2×C411yescharacteristic
64C2×C4×Dic17D411yescharacteristic
65C2×C4×Dic17D411yescharacteristic
66C22×#3622—21no—
67C17×C22⋊C8—21no—
68C17×C22⋊C8—21no—
69C17×#95—21no—
70C17×#95D412yes—
71#10587502—24no—
72#10587502—21no—
73#10587502—21no—
74#10587502—21no—
75#10587502—21no—
76#10587502—21no—
77#10587502D412yes—
78C22⋊#3622—21no—
79C22⋊#3622—21no—
80#10587485—42no—
81C24×C17—21no—
82C24×C17C2×D411yescharacteristic
83C2×D4×C17—21no—
84C2×D4×C17—21no—
85C2×D4×C17—41no—
86C2×D4×C17—41no—
87C2×D4×C17—41no—
88C2×D4×C17—41no—
89C2×D4×C17—41no—
90C2×D4×C17—41no—
91C2×D4×C17—41no—
92C2×D4×C17—41no—
93C22×C4×C17—22no—
94C22×C4×C17—22no—
95C22×C4×C17—22no—
96C22×C4×C17—21no—
97C22×C4×C17—41no—
98C22×C4×C17—41no—
99C22×C4×C17—41no—
100C22×C4×C17—41no—
101C22×C4×C17—41no—
102C22×C4×C17—41no—
103C22×C4×C17C22×C411yescharacteristic
104C22×C4×C17C2×D411yescharacteristic
105C22×C4×C17C2×D411yescharacteristic
106C22×C4×C17C2×D411yescharacteristic
107C22×C4×C17#4111yescharacteristic
108C22×C4×C17#4111yescharacteristic
109C22×Dic17—21no—
110C22×Dic17—21no—
111C22×Dic17—22no—
112C22×Dic17—22no—
113C22×Dic17—22no—
114C22×Dic17—22no—
115C22×Dic17—22no—
116C22×Dic17—21no—
117C22×Dic17—21no—
118C22×Dic17—22no—
119C22×Dic17—21no—
120C22×Dic17—21no—
121C22×Dic17—41no—
122C22×Dic17—41no—
123C22×Dic17—41no—
124C22×Dic17—41no—
125C22×Dic17—41no—
126C22×Dic17C22⋊C411yescharacteristic
127C22×Dic17C22⋊C411yescharacteristic
128C22×Dic17C22⋊C411yescharacteristic
129C22×Dic17C22⋊C411yescharacteristic
130C2×Dic34—21no—
131C2×Dic34—21no—
132C2×Dic34—41no—
133C2×Dic34—41no—
134C2×C8×C17—41no—
135C2×C8×C17—41no—
136C2×C8×C17—41no—
137C4⋊C4×C17—21no—
138C4⋊C4×C17—21no—
139C22⋊C4×C17—41no—
140C22⋊C4×C17—41no—
141C22⋊Dic17—42no—
142C22⋊Dic17—41no—
143C22⋊Dic17—41no—
144C22⋊Dic17—42no—
145C22⋊Dic17—42no—
146C22⋊Dic17—42no—
147C4⋊Dic17—21no—
148C4⋊Dic17—21no—
149#63225—41no—
150#63225—41no—
151C4×Dic17—22no—
152C4×Dic17—22no—
153C4×Dic17—41no—
154C4×Dic17—41no—
155C4×Dic17—48no—
156C2×#3622—41no—
157C2×#3622—41no—
158C2×#3622—41no—
159#9713—171nomaximal, Sylow
160C23×C17—21no—
161C23×C17—21no—
162C23×C17—21no—
163C23×C17—21no—
164C23×C17—41no—
165C23×C17—41no—
166C23×C17—41no—
167C23×C17—41no—
168C23×C17—41no—
169C23×C17—41no—
170C23×C17—41no—
171C23×C17C2×C22⋊C411yescharacteristic
172C23×C17#12111yescharacteristic
173C23×C17#12111yescharacteristic
174C23×C17#13511yescharacteristic
175C23×C17#13511yescharacteristic
176C23×C17AGL(1,ℤ/8)11yescharacteristic
177C23×C17#13711yescharacteristic
178D4×C17—81no—
179D4×C17—81no—
180D4×C17—81no—
181D4×C17—81no—
182D4×C17—81no—
183D4×C17—81no—
184C2×C4×C17—21no—
185C2×C4×C17—21no—
186C2×C4×C17—21no—
187C2×C4×C17—21no—
188C2×C4×C17—24no—
189C2×C4×C17—41no—
190C2×C4×C17—41no—
191C2×C4×C17—41no—
192C2×C4×C17—41no—
193C2×C4×C17—42no—
194C2×C4×C17—42no—
195C2×C4×C17—42no—
196C2×C4×C17—42no—
197C2×C4×C17—42no—
198C2×C4×C17—42no—
199C2×C4×C17—41no—
200C2×C4×C17—41no—
201C2×C4×C17—41no—
202C2×C4×C17C4×D411yescharacteristic
203C2×C4×C17C4×D411yescharacteristic
204C2×C4×C17#12411yescharacteristic
205C2×C4×C17#12711yescharacteristic
206C2×Dic17—24no—
207C2×Dic17—24no—
208C2×Dic17—21no—
209C2×Dic17—21no—
210C2×Dic17—21no—
211C2×Dic17—21no—
212C2×Dic17—21no—
213C2×Dic17—21no—
214C2×Dic17—24no—
215C2×Dic17—21no—
216C2×Dic17—21no—
217C2×Dic17—21no—
218C2×Dic17—42no—
219C2×Dic17—42no—
220C2×Dic17—42no—
221C2×Dic17—42no—
222C2×Dic17—42no—
223C2×Dic17—42no—
224C2×Dic17—42no—
225C2×Dic17—42no—
226C2×Dic17—41no—
227C2×Dic17—41no—
228C2×Dic17—41no—
229C2×Dic17—41no—
230C2×Dic17—41no—
231C2×Dic17—41no—
232C2×Dic17#9912yes—
233C2×Dic17#10412yes—
234C2×Dic17#10412yes—
235Dic34—81no—
236Dic34—81no—
237C8×C17—81no—
238#3622—81no—
239#2850—171no—
240#1797—171no—
241C2×#328—171no—
242C2×#328—171no—
243C2×#327—171no—
244C2×#327D1711yescharacteristic
245C4×#95—171no—
246C22×C17—21no—
247C22×C17—21no—
248C22×C17—21no—
249C22×C17—21no—
250C22×C17—21no—
251C22×C17—21no—
252C22×C17—21no—
253C22×C17—21no—
254C22×C17—21no—
255C22×C17—21no—
256C22×C17—21no—
257C22×C17—21no—
258C22×C17—81no—
259C22×C17—81no—
260C22×C17—81no—
261C22×C17—81no—
262C22×C17—81no—
263C22×C17—81no—
264C22×C17—81no—
265C22×C17#39011yescharacteristic
266C22×C17C2×#9911yescharacteristic
267C22×C17#41011yescharacteristic
268C22×C17#41611yescharacteristic
269C22×C17#41711yescharacteristic
270C22×C17C2×#10411yescharacteristic
271C22×C17C2×#10411yescharacteristic
272C22×C17#46311yescharacteristic
273C22×C17#46311yescharacteristic
274C22×C17#46411yescharacteristic
275C22×C17#46411yescharacteristic
276C4×C17—21no—
277C4×C17—21no—
278C4×C17—21no—
279C4×C17—21no—
280C4×C17—44no—
281C4×C17—81no—
282Dic17—22no—
283Dic17—22no—
284Dic17—22no—
285Dic17—44no—
286Dic17—41no—
287Dic17—81no—
288Dic17—81no—
289C2×#124—171no—
290C2×C22⋊Q8—171no—
291C2×#121C2×D1711yescharacteristic
292C22×C42—171no—
293C22×C42—341no—
294C2×C4⋊C8—341no—
295C2×C22⋊C8—171no—
296C2×C22⋊C8C2×D1711yescharacteristic
297#391—171no—
298#390—171no—
299#388—341no—
300#388—341no—
301#386—341no—
302#386—341no—
303#382—171no—
304C2×#95—171no—
305C2×#95—341no—
306C2×#95C2×D1711yescharacteristic
307#336—341no—
308#328—342no—
309#328—342no—
310#327—22no—
311#327—342no—
312C2×C17—21no—
313C2×C17—21no—
314C2×C17—21no—
315C2×C17—21no—
316C2×C17—161no—
317C2×C17#150011yescharacteristic
318C2×C17#150111yescharacteristic
319C2×C17#150411yescharacteristic
320C2×C17#150911yescharacteristic
321C2×C17#196211yescharacteristic
322C2×C17#196411yescharacteristic
323C2×C17#198311yescharacteristic
324C22×Q8—171no—
325C22×D4—21no—
326C22×D4C4×D1711yescharacteristic
327C23×C4—21no—
328C23×C4—171no—
329C23×C4—171no—
330C23×C4—341no—
331C23×C4C22×D1711yescharacteristic
332C22×C8—21no—
333C22×C8—341no—
334#124—682no—
335C22⋊Q8—341no—
336C22⋊Q8—341no—
337C22⋊Q8—341no—
338C22⋊Q8—341no—
339#121—21no—
340#121—21no—
341#121—21no—
342#121—21no—
343C2×C4⋊C4—171no—
344C2×C4⋊C4—341no—
345C2×C4⋊C4C4×D1711yescharacteristic
346C2×C22⋊C4—21no—
347C2×C22⋊C4—341no—
348C2×C22⋊C4—341no—
349C2×C22⋊C4—341no—
350C2×C22⋊C4—341no—
351C2×C22⋊C4—341no—
352C2×C42—171no—
353C2×C42—171no—
354C2×C42—171no—
355C2×C42—171no—
356C2×C42—344no—
357C2×C42—682no—
358C2×C42—682no—
359C2×C42—682no—
360C4⋊C8—682no—
361C22⋊C8—21no—
362C22⋊C8—21no—
363C22⋊C8—341no—
364C22⋊C8—341no—
365#95—21no—
366#95—172no—
367#95—341no—
368#95—341no—
369#95—341no—
370#95—341no—
371#95—344no—
372#95—341no—
373#95C22⋊D1712yes—
374C17#971311yescharacteristic, complemented, Sylow
375C24—21no—
376C24D4×D1711yescharacteristic
377C2×Q8—341no—
378C2×Q8—341no—
379C2×Q8—681no—
380C2×Q8—681no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—41no—
384C2×D4—41no—
385C2×D4—41no—
386C2×D4—41no—
387C2×D4—41no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—41no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—171no—
402C22×C4—171no—
403C22×C4—171no—
404C22×C4—171no—
405C22×C4—342no—
406C22×C4—342no—
407C22×C4—342no—
408C22×C4—341no—
409C22×C4—341no—
410C22×C4—342no—
411C22×C4—342no—
412C22×C4—342no—
413C22×C4—341no—
414C22×C4—341no—
415C22×C4—341no—
416C22×C4—341no—
417C22×C4—681no—
418C22×C4—681no—
419C22×C4—681no—
420C22×C4—681no—
421C22×C4—681no—
422C22×C4C2×C4×D1711yescharacteristic
423C22×C4#6325211yescharacteristic
424C22×C4D4×D1711yescharacteristic
425C22×C4D4×D1711yescharacteristic
426C22×C4#6325411yescharacteristic
427C22×C4C2×C22⋊D1711yescharacteristic
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—681no—
432C2×C8—681no—
433C2×C8—681no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—341no—
437C4⋊C4—341no—
438C4⋊C4—681no—
439C4⋊C4—681no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—682no—
443C22⋊C4—682no—
444C22⋊C4—682no—
445C22⋊C4—682no—
446C22⋊C4—681no—
447C22⋊C4—681no—
448C42—342no—
449C42—342no—
450C42—681no—
451C42—681no—
452C42—688no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23C22⋊C4×D1711yescharacteristic
465C23#1058757011yescharacteristic
466C23#1058759711yescharacteristic
467C23#1058759811yescharacteristic
468C23#1058760111yescharacteristic
469C23#1058760211yescharacteristic
470C23#1058762511yescharacteristic
471Q8—1361no—
472Q8—1361no—
473D4—81no—
474D4—81no—
475D4—81no—
476D4—81no—
477D4—81no—
478D4—81no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—24no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—172no—
498C2×C4—172no—
499C2×C4—172no—
500C2×C4—344no—
501C2×C4—344no—
502C2×C4—341no—
503C2×C4—341no—
504C2×C4—341no—
505C2×C4—341no—
506C2×C4—341no—
507C2×C4—341no—
508C2×C4—344no—
509C2×C4—341no—
510C2×C4—341no—
511C2×C4—341no—
512C2×C4—682no—
513C2×C4—682no—
514C2×C4—682no—
515C2×C4—682no—
516C2×C4—682no—
517C2×C4—682no—
518C2×C4—682no—
519C2×C4—682no—
520C2×C4—681no—
521C2×C4—681no—
522C2×C4—681no—
523C2×C4—681no—
524C2×C4—681no—
525C2×C4—681no—
526C2×C4#1058756711yescharacteristic
527C2×C4#1058757111yescharacteristic
528C2×C4C4×C22⋊D1711yescharacteristic
529C2×C4#1058762611yescharacteristic
530C8—81no—
531C8—1361no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22—81no—
551C22#1176270911yescharacteristic
552C22#99×D1711yescharacteristic
553C22#1176278111yescharacteristic
554C22#1176280211yescharacteristic
555C22#1176280711yescharacteristic
556C22#104×D1711yescharacteristic
557C22#104×D1711yescharacteristic
558C22#1176302211yescharacteristic
559C22#1176302311yescharacteristic
560C22#1176302911yescharacteristic
561C22#1176303011yescharacteristic
562C4—21no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—44no—
567C4—81no—
568C4—342no—
569C4—342no—
570C4—342no—
571C4—684no—
572C4—681no—
573C4—1361no—
574C4—1361no—
575C2—21no—
576C2—21no—
577C2—21no—
578C2—21no—
579C2—161no—
580C2#1452726711yescharacteristic
581C2#1452727211yescharacteristic
582C2#1452728511yescharacteristic
583C2#1452730611yescharacteristic
584C2#1452961011yescharacteristic
585C2#1452961811yescharacteristic
586C2#1452971511yescharacteristic
subgroupG/H
centerC23C22⋊C4×D17
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC2×C17×#327C2
socleC23×C17C2×C22⋊C4
prime radicalC2×C17×#121C22
cosocleC23×C4×C17C23
hypercenterC2×#327D17
hypocenterC17#9713
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
2. #14528762C2semidirect
3. C2×#11762547C2semidirect
4. C2×#11762545C2semidirect
6. #95×Dic17C2semidirect
8. C2×#10587622C4semidirect
14. #11762714C4semidirect
15. #11762712C4semidirect
20. #11762674C4semidirect
374. C17#9713semidirect

Maximal subgroups

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

subgroupindex
#145370832
#145287622
C2×#117625472
C2×#117625452
C2×#117625432
#95×Dic172
C2×C17×#3272
#971317