Finite Group Databasebeta

← #17004429

Subgroups — #17004429

Order 4352.

598 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14538855C211yesmaximal, characteristic
2#14537878C211yesmaximal, characteristic, complemented
3#14527655C211yesmaximal, characteristic, complemented
4#14527641C211yesmaximal, characteristic, complemented
5#14527634C211yesmaximal, characteristic, complemented
6#14527620C211yesmaximal, characteristic, complemented
7#14527601C211yesmaximal, characteristic, complemented
8#14527596C211yesmaximal, characteristic, complemented
9C8×#63231C211yesmaximal, characteristic, complemented
10C8×#63229C211yesmaximal, characteristic, complemented
11#14527567C211yesmaximal, characteristic, complemented
12#14527218C211yesmaximal, characteristic
13#14527186C211yesmaximal, characteristic, complemented
14#14527084C211yesmaximal, characteristic, complemented
15C17×#2988C211yesmaximal, characteristic
16#11763366C2211yescharacteristic
17#11763150C411yescharacteristic
18#11763150C411yescharacteristic
19#11763094C411yescharacteristic
20#11763089C411yescharacteristic
21#11763019C411yescharacteristic
22#11763014C411yescharacteristic
23#11762971C411yescharacteristic
24#11762971C411yescharacteristic
25#11762939C2211yescharacteristic
26#11762939C2211yescharacteristic
27#11762938—21no—
28#11762938—21no—
29C8×Dic34C2211yescharacteristic
30C8×Dic34C2211yescharacteristic
31C4×#63231C2211yescharacteristic
32C4×#63231C2211yescharacteristic
33C4×#63229C2211yescharacteristic
34C4×#63229C2211yescharacteristic
35#11762867—21no—
36#11762867—21no—
37#11762866C2211yescharacteristic
38#11762862C2211yescharacteristic
39#11762862C2211yescharacteristic
40#11762561C2211yescharacteristic
41#11762561C2211yescharacteristic
42#11762559C2211yescharacteristic
43#11762559C2211yescharacteristic
44Q8⋊#3622C2211yescharacteristic
45Q8⋊#3622C2211yescharacteristic
46#11762539C2211yescharacteristic
47#11762539C2211yescharacteristic
48#11762539C2211yescharacteristic
49#11762539C2211yescharacteristic
50D4⋊#3622C2211yescharacteristic
51D4⋊#3622C2211yescharacteristic
52#11762531C2211yescharacteristic
53#11762531C2211yescharacteristic
54#11762530C2211yescharacteristic
55C8×#3622C2211yescharacteristic
56C4×#41×C17C2211yescharacteristic
57C8×Q8×C17C2211yescharacteristic
58C8×Q8×C17C2211yescharacteristic
59C8×D4×C17—21no—
60C8×D4×C17—21no—
61C8×D4×C17C2211yescharacteristic
62C8×D4×C17C2211yescharacteristic
63C17×#431—21no—
64C17×#431C2211yescharacteristic
65C2×C17×C4⋊C8—21no—
66C2×C17×C4⋊C8C2211yescharacteristic
67C2×#41×C17C2×C411yescharacteristic
68C22×C8×C17—21no—
69C22×C8×C17—21no—
70C22×C8×C17—21no—
71C4×Q8×C17C2×C411yescharacteristic
72C4×Q8×C17C2311yescharacteristic
73C4×D4×C17—21no—
74C4×D4×C17—21no—
75C4×D4×C17C2×C411yescharacteristic
76C4×D4×C17C2311yescharacteristic
77C17×#117—21no—
78C17×#117C2×C411yescharacteristic
79C2×C42×C17—21no—
80C2×C42×C17C2311yescharacteristic
81C2×#63231C2×C411yescharacteristic
82C2×#63231C2×C411yescharacteristic
83C2×#63229C2×C411yescharacteristic
84C2×#63229C2×C411yescharacteristic
85#10587612—21no—
86#10587610—21no—
87#10587610C2×C411yescharacteristic
88#10587610C2×C411yescharacteristic
89#10587556—21no—
90#10587555C2×C411yescharacteristic
91#10587555C2×C411yescharacteristic
92C4×Dic34—21no—
93C4×Dic34C2311yescharacteristic
94C4×Dic34C2311yescharacteristic
95C17×C4⋊C8—21no—
96C17×C4⋊C8—21no—
97C17×C4⋊C8—21no—
98C17×C4⋊C8D411yescharacteristic
99C17×C4⋊C8D411yescharacteristic
100C17×C4⋊C8C2311yescharacteristic
101C17×C4⋊C8C2311yescharacteristic
102C17×C22⋊C8—21no—
103C17×C22⋊C8—21no—
104C17×C22⋊C8D411yescharacteristic
105C17×C22⋊C8D411yescharacteristic
106C4×C8×C17—21no—
107C4×C8×C17—21no—
108C4×C8×C17C2311yescharacteristic
109C4×C8×C17C2311yescharacteristic
110#10587506C2×C411yescharacteristic
111#10587506C2×C411yescharacteristic
112#10587503C2×C411yescharacteristic
113#10587503C2×C411yescharacteristic
114C22⋊#3622—21no—
115C22⋊#3622—21no—
116C22⋊#3622C2×C411yescharacteristic
117C22⋊#3622C2×C411yescharacteristic
118#10587486—21no—
119#10587486—21no—
120#10587485—21no—
121#10587485—21no—
122#10587485—21no—
123#10587485—21no—
124C8×Dic17—21no—
125C8×Dic17—21no—
126#10587481C2×C411yescharacteristic
127#10587481C2×C411yescharacteristic
128#10587481C2×C411yescharacteristic
129#10587481C2×C411yescharacteristic
130#10587479C2×C411yescharacteristic
131#10587479C2×C411yescharacteristic
132#10587478C2×C411yescharacteristic
133#10587478C2×C411yescharacteristic
134C4⋊#3622—21no—
135C4⋊#3622C2×C411yescharacteristic
136C4⋊#3622C2×C411yescharacteristic
137C4⋊#3622C2311yescharacteristic
138C4⋊#3622C2311yescharacteristic
139#10587474—21no—
140C4×#3622C2311yescharacteristic
141C4×#3622C2311yescharacteristic
142C4×#3622C2311yescharacteristic
143C4×#3622C2311yescharacteristic
144#41×C17—22no—
145#41×C17—22no—
146C2×Q8×C17C22×C411yescharacteristic
147C2×D4×C17—21no—
148C2×D4×C17C22×C411yescharacteristic
149C22×C4×C17—21no—
150C22×C4×C17—21no—
151C22×C4×C17—21no—
152C22×C4×C17C22×C411yescharacteristic
153C22×C4×C17C2×D411yescharacteristic
154C22×C4×C17#4111yescharacteristic
155C2×Dic34—21no—
156C2×Dic34C22×C411yescharacteristic
157C2×Dic34C22×C411yescharacteristic
158C2×C8×C17—21no—
159C2×C8×C17—21no—
160C2×C8×C17—81no—
161C2×C8×C17—81no—
162C2×C8×C17—81no—
163C2×C8×C17C2×D411yescharacteristic
164C2×C8×C17C2×D411yescharacteristic
165C2×C8×C17C2×D411yescharacteristic
166C2×C8×C17C2×D411yescharacteristic
167C4⋊C4×C17—21no—
168C4⋊C4×C17—21no—
169C4⋊C4×C17C22×C411yescharacteristic
170C4⋊C4×C17C22×C411yescharacteristic
171C22⋊C4×C17—21no—
172C22⋊C4×C17—21no—
173C22⋊C4×C17—21no—
174C42×C17—21no—
175C42×C17—21no—
176C42×C17—21no—
177C42×C17C22×C411yescharacteristic
178C42×C17C2×D411yescharacteristic
179C42×C17#4111yescharacteristic
180C42×C17C2411yescharacteristic
181C22⋊Dic17—21no—
182C22⋊Dic17—21no—
183#63231—22no—
184#63231—22no—
185#63229—22no—
186#63229—22no—
187C4⋊Dic17—21no—
188C4⋊Dic17—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17C22×C411yescharacteristic
191C4⋊Dic17C22×C411yescharacteristic
192#63225—21no—
193#63225—21no—
194#63225—21no—
195#63225—21no—
196C4×Dic17—21no—
197C4×Dic17—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622—21no—
202C2×#3622C22×C411yescharacteristic
203C2×#3622C22×C411yescharacteristic
204C2×#3622C22×C411yescharacteristic
205C2×#3622C22×C411yescharacteristic
206#17921—171nomaximal, Sylow
207C23×C17—21no—
208C23×C17C4×D411yescharacteristic
209Q8×C17—21no—
210Q8×C17#13112yes—
211D4×C17—24no—
212D4×C17—21no—
213D4×C17#13112yes—
214C2×C4×C17—24no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17—81no—
226C2×C4×C17C4×D411yescharacteristic
227C2×C4×C17C4×D411yescharacteristic
228C2×C4×C17C4×D411yescharacteristic
229C2×C4×C17C23×C411yescharacteristic
230C2×C4×C17C22×D411yescharacteristic
231C2×C4×C17C2×#4111yescharacteristic
232C2×Dic17—21no—
233C2×Dic17—21no—
234C2×Dic17—21no—
235C2×Dic17—21no—
236Dic34—21no—
237Dic34—21no—
238Dic34—21no—
239Dic34—21no—
240Dic34—81no—
241C8×C17—21no—
242C8×C17—21no—
243C8×C17—21no—
244C8×C17—21no—
245C8×C17—41no—
246C8×C17—41no—
247#3622—21no—
248#3622—21no—
249#3622—21no—
250#3622—21no—
251#3622—41no—
252#3622—41no—
253#3622—41no—
254#3622—41no—
255#3107—171no—
256#2988D1711yescharacteristic
257#2985—171no—
258#1587—171no—
259#1585—171no—
260#1584—171no—
261#1582—171no—
262#1579—171no—
263SD16⋊C8—171no—
264C8×Q16—171no—
265C8×SD16—171no—
266#1573—171no—
267#1488—171no—
268#1481—171no—
269#1454—171no—
270C22×C17—21no—
271C22×C17—22no—
272C22×C17—21no—
273C22×C17—21no—
274C22×C17—41no—
275C22×C17—41no—
276C22×C17C2×C4×D411yescharacteristic
277C4×C17—21no—
278C4×C17—22no—
279C4×C17—21no—
280C4×C17—21no—
281C4×C17—21no—
282C4×C17—41no—
283C4×C17—41no—
284C4×C17C2×#13111yescharacteristic
285C4×C17C2×#13111yescharacteristic
286C4×C17#57711yescharacteristic
287C4×C17#57811yescharacteristic
288Dic17—41no—
289Dic17—41no—
290Dic17—41no—
291Dic17—41no—
292#533—171no—
293C4×#41C2×D1711yescharacteristic
294#485—171no—
295#485—171no—
296#475—171no—
297#474—171no—
298#462—171no—
299#461—171no—
300#452—171no—
301#452—171no—
302#446—171no—
303#446—171no—
304#446—341no—
305#446—341no—
306C8×Q8—171no—
307C8×Q8—171no—
308C8×Q8C2×D1711yescharacteristic
309C8×Q8C2×D1711yescharacteristic
310C4×Q16—171no—
311C4×Q16—171no—
312C4×SD16—171no—
313C4×SD16—171no—
314C8×D4—21no—
315C8×D4—21no—
316C8×D4C2×D1711yescharacteristic
317C8×D4C2×D1711yescharacteristic
318#433—341no—
319#433—341no—
320#432—171no—
321#431—21no—
322#431—171no—
323#431—171no—
324#431C2×D1711yescharacteristic
325C2×C4⋊C8—21no—
326C2×C4⋊C8C2×D1711yescharacteristic
327#335—171no—
328#335—171no—
329#334—171no—
330#334—171no—
331Q8⋊C8—171no—
332Q8⋊C8—171no—
333Q8⋊C8—171no—
334Q8⋊C8—171no—
335Q8⋊C8—171no—
336Q8⋊C8—171no—
337D4⋊C8—171no—
338D4⋊C8—171no—
339#322—171no—
340#322—171no—
341#322—171no—
342C82—171no—
343C2×C17—41no—
344C2×C17—42no—
345C2×C17#292911yescharacteristic
346C2×C17#294711yescharacteristic
347C2×C17#295711yescharacteristic
348C2×#41C4×D1711yescharacteristic
349C2×Q16—171no—
350C2×Q16—171no—
351C2×SD16—171no—
352C2×SD16—171no—
353C22×C8—21no—
354C22×C8—21no—
355C22×C8—21no—
356C4⋊Q8—171no—
357C4⋊Q8—171no—
358#125—341no—
359C22⋊Q8—171no—
360C22⋊Q8—171no—
361C22⋊Q8—341no—
362C4×Q8—171no—
363C4×Q8—171no—
364C4×Q8—341no—
365C4×Q8C4×D1711yescharacteristic
366C4×Q8C22×D1711yescharacteristic
367C4×D4—21no—
368C4×D4—21no—
369C4×D4C4×D1711yescharacteristic
370C4×D4C22×D1711yescharacteristic
371#117—21no—
372#117—341no—
373#117C4×D1711yescharacteristic
374C2×C42—21no—
375C2×C42C22×D1711yescharacteristic
376#107—171no—
377#107—171no—
378#106—171no—
379#106—171no—
380C4⋊C8—21no—
381C4⋊C8—21no—
382C4⋊C8—21no—
383C4⋊C8—171no—
384C4⋊C8—171no—
385C4⋊C8—171no—
386C4⋊C8—171no—
387C4⋊C8—341no—
388C4⋊C8—341no—
389C4⋊C8—341no—
390C4⋊C8—341no—
391C4⋊C8—341no—
392C4⋊C8C22⋊D1711yescharacteristic
393C4⋊C8C22⋊D1711yescharacteristic
394C4⋊C8C22×D1711yescharacteristic
395C4⋊C8C22×D1711yescharacteristic
396#103—171no—
397#103—171no—
398#103—171no—
399#103—171no—
400#103—171no—
401#103—171no—
402#102—171no—
403#102—171no—
404C22⋊C8—21no—
405C22⋊C8—21no—
406C22⋊C8—171no—
407C22⋊C8—171no—
408C22⋊C8—341no—
409C22⋊C8—341no—
410C22⋊C8C22⋊D1711yescharacteristic
411C22⋊C8C22⋊D1711yescharacteristic
412#97—341no—
413#97—341no—
414#97—341no—
415C4×C8—21no—
416C4×C8—21no—
417C4×C8—171no—
418C4×C8—171no—
419C4×C8—171no—
420C4×C8—171no—
421C4×C8—341no—
422C4×C8—341no—
423C4×C8C22×D1711yescharacteristic
424C4×C8C22×D1711yescharacteristic
425C17#1792111yescharacteristic, complemented, Sylow
426#41—22no—
427#41—22no—
428C2×Q8—171no—
429C2×Q8—171no—
430C2×Q8—341no—
431C2×Q8C2×C4×D1711yescharacteristic
432C2×D4—21no—
433C2×D4C2×C4×D1711yescharacteristic
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4C2×C4×D1711yescharacteristic
438C22×C4#6325211yescharacteristic
439C22×C4C2×C22⋊D1711yescharacteristic
440Q16—342no—
441Q16—342no—
442SD16—342no—
443SD16—342no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—81no—
447C2×C8—81no—
448C2×C8—81no—
449C2×C8—171no—
450C2×C8—171no—
451C2×C8—171no—
452C2×C8—171no—
453C2×C8—341no—
454C2×C8—341no—
455C2×C8—341no—
456C2×C8—341no—
457C2×C8C2×C22⋊D1711yescharacteristic
458C2×C8C2×C22⋊D1711yescharacteristic
459C2×C8C2×C22⋊D1711yescharacteristic
460C2×C8C2×C22⋊D1711yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—171no—
464C4⋊C4—171no—
465C4⋊C4—341no—
466C4⋊C4—341no—
467C4⋊C4—341no—
468C4⋊C4—341no—
469C4⋊C4—341no—
470C4⋊C4—341no—
471C4⋊C4—341no—
472C4⋊C4C2×C4×D1711yescharacteristic
473C4⋊C4C2×C4×D1711yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—341no—
478C22⋊C4—341no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—341no—
483C42—341no—
484C42C2×C4×D1711yescharacteristic
485C42#6325211yescharacteristic
486C42C2×C22⋊D1711yescharacteristic
487C42C23×D1711yescharacteristic
488C23—21no—
489C23C4×C22⋊D1711yescharacteristic
490Q8—21no—
491Q8—341no—
492Q8—341no—
493Q8—341no—
494Q8—341no—
495Q8—1361no—
496Q8#1058758712yes—
497D4—24no—
498D4—21no—
499D4#1058758712yes—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—81no—
512C2×C4—341no—
513C2×C4—341no—
514C2×C4—341no—
515C2×C4—341no—
516C2×C4C4×C22⋊D1711yescharacteristic
517C2×C4C4×C22⋊D1711yescharacteristic
518C2×C4C4×C22⋊D1711yescharacteristic
519C2×C4C22×C4×D1711yescharacteristic
520C2×C4C2×#6325211yescharacteristic
521C2×C4C22×C22⋊D1711yescharacteristic
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—41no—
527C8—41no—
528C8—341no—
529C8—341no—
530C8—341no—
531C8—341no—
532C8—681no—
533C8—681no—
534C8—681no—
535C8—681no—
536C22—21no—
537C22—22no—
538C22—21no—
539C22—21no—
540C22—41no—
541C22—41no—
542C22C2×C4×C22⋊D1711yescharacteristic
543C4—21no—
544C4—22no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—41no—
549C4—41no—
550C4—681no—
551C4—681no—
552C4—681no—
553C4—681no—
554C4C2×#1058758711yescharacteristic
555C4C2×#1058758711yescharacteristic
556C4#1176362011yescharacteristic
557C4#1176362411yescharacteristic
558C2—41no—
559C2—42no—
560C2#1453747811yescharacteristic
561C2#1453762311yescharacteristic
562C2#1453769911yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC42C23×D17
FittingC17×#2988C2
socleC22×C17C2×C4×D4
prime radicalC2×#41×C17C2×C4
cosocleC42×C17C24
hypercenter#2988D17
hypocenterC17#17921
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2988

Complements

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

NH ≅ G/Ntype
2. #14537878C2semidirect
3. #14527655C2semidirect
4. #14527641C2semidirect
5. #14527634C2semidirect
6. #14527620C2semidirect
7. #14527601C2semidirect
8. #14527596C2semidirect
9. C8×#63231C2semidirect
10. C8×#63229C2semidirect
11. #14527567C2semidirect
13. #14527186C2semidirect
14. #14527084C2semidirect
425. C17#17921semidirect

Maximal subgroups

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

subgroupindex
#145388552
#145378782
#145276552
#145276412
#145276342
#145276202
#145276012
#145275962
C8×#632312
C8×#632292
#145275672
#145272182
#145271862
#145270842
C17×#29882
#1792117