Finite Group Databasebeta

← #17097260

Subgroups — #17097260

Order 4352.

804 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763119C211yesmaximal, characteristic, complemented
2#14532502C211yesmaximal, characteristic
3#14531811C211yesmaximal, characteristic
4#14530329C211yesmaximal, characteristic, complemented
5#14530278C211yesmaximal, characteristic, complemented
6#14530174C211yesmaximal, characteristic, complemented
7#14530120C211yesmaximal, characteristic, complemented
8#14529553C211yesmaximal, characteristic, complemented
9#14529364C211yesmaximal, characteristic, complemented
10#14529217C211yesmaximal, characteristic
11#14529091C211yesmaximal, characteristic
12#14529049C211yesmaximal, characteristic, complemented
13#14528983C211yesmaximal, characteristic
14#14528967C211yesmaximal, characteristic
15C2×C17×#428C211yesmaximal, characteristic
16C2×#10587576C2211yescharacteristic
17C2×#10587610C2211yescharacteristic
18C2×#10587573C2211yescharacteristic
19#11763119—24no—
20C2×#10587478C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22C2×#10587485C2211yescharacteristic
23C2×#10587485C2211yescharacteristic
24C2×#10587487C2211yescharacteristic
25C2×#10587487C2211yescharacteristic
26C2×#10587481C2211yescharacteristic
27C2×#10587481C2211yescharacteristic
28C2×C22⋊#3622C2211yescharacteristic
29#11762748—21no—
30#11762742—21no—
31#11762714C2211yescharacteristic
32#11762708C2211yescharacteristic
33#11762705C2211yescharacteristic
34#11762705C2211yescharacteristic
35#11762703—21no—
36#11762697—21no—
37#11762693—21no—
38#11762682—21no—
39#11762674C2211yescharacteristic
40#11762669—21no—
41#11762669C2211yescharacteristic
42C4⋊C4×Dic17C2211yescharacteristic
43C4×#63225—21no—
44C4×C22⋊Dic17C2211yescharacteristic
45#11762572C2211yescharacteristic
46#11762572C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762563C2211yescharacteristic
52#11762563C2211yescharacteristic
53C22×M16×C17C2211yescharacteristic
54C2×C17×#117C2211yescharacteristic
55C22×C4⋊C4×C17C2211yescharacteristic
56C17×#428—24no—
57C2×C17×#107C2211yescharacteristic
58C2×C17×#107C2211yescharacteristic
59C2×C17×#106C2211yescharacteristic
60C2×C17×#106C2211yescharacteristic
61C23×C4×C17—21no—
62C23×C4×C17C2311yescharacteristic
63C22×Dic34C2311yescharacteristic
64C2×M16×C17—22no—
65C2×M16×C17—22no—
66C2×M16×C17—22no—
67C22×C8×C17C2311yescharacteristic
68C22×C8×C17C2311yescharacteristic
69C17×#117—22no—
70C17×#117—41no—
71C2×C4⋊C4×C17—22no—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17—41no—
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C4⋊C4×C17C2311yescharacteristic
78C2×C22⋊C4×C17—21no—
79C2×C42×C17—21no—
80C2×C22⋊Dic17—21no—
81C2×C22⋊Dic17—21no—
82C2×C22⋊Dic17D411yescharacteristic
83C2×C22⋊Dic17D411yescharacteristic
84C2×C4⋊Dic17—21no—
85C2×C4⋊Dic17—21no—
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89#10587610—22no—
90#10587610Q814yes—
91C2×#63225—21no—
92C2×#63225—21no—
93C2×#63225—21no—
94C2×#63225—21no—
95C2×#63225—21no—
96C2×C4×Dic17—21no—
97C2×C4×Dic17—21no—
98C2×C4×Dic17C2311yescharacteristic
99C2×C4×Dic17C2311yescharacteristic
100C22×#3622C2311yescharacteristic
101C22×#3622C2311yescharacteristic
102#10587576—22no—
103#10587576—41no—
104#10587573—22no—
105#10587573—41no—
106C17×#107—22no—
107C17×#107—22no—
108C17×#106—22no—
109C17×#106D414yes—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502D411yescharacteristic
120#10587502D411yescharacteristic
121C22⋊#3622—22no—
122#10587487—22no—
123#10587487—22no—
124#10587485—22no—
125#10587485—22no—
126#10587481—22no—
127#10587481—41no—
128#10587479—41no—
129#10587478—22no—
130C24×C17C2×D411yescharacteristic
131C22×C4×C17—21no—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—21no—
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17—42no—
140C22×C4×C17—42no—
141C22×C4×C17—81no—
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2×Q812yes—
144C22×C4×C17#4112yes—
145C22×C4×C17C2411yescharacteristic
146C22×Dic17—21no—
147C22×Dic17—21no—
148C22×Dic17—21no—
149C22×Dic17—21no—
150C22×Dic17C2×D411yescharacteristic
151C22×Dic17C2×D411yescharacteristic
152C22×Dic17C2×D411yescharacteristic
153C22×Dic17C2×D411yescharacteristic
154C2×Dic34—21no—
155C2×Dic34—21no—
156C2×Dic34—21no—
157C2×Dic34—21no—
158C2×Dic34—21no—
159C2×Dic34C2×Q812yes—
160M16×C17—28no—
161C2×C8×C17—21no—
162C2×C8×C17—21no—
163C2×C8×C17—21no—
164C2×C8×C17—21no—
165C2×C8×C17C2×D412yes—
166C2×C8×C17#4112yes—
167C4⋊C4×C17—21no—
168C4⋊C4×C17—21no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17—21no—
171C4⋊C4×C17—21no—
172C4⋊C4×C17—21no—
173C4⋊C4×C17—42no—
174C4⋊C4×C17C2×D412yes—
175C4⋊C4×C17C2×D412yes—
176C22⋊C4×C17—42no—
177C42×C17—42no—
178C22⋊Dic17—24no—
179C22⋊Dic17—41no—
180C22⋊Dic17—41no—
181C22⋊Dic17—42no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17—21no—
185C4⋊Dic17—21no—
186C4⋊Dic17—21no—
187C4⋊Dic17—42no—
188C4⋊Dic17—81no—
189C4⋊Dic17C2×Q812yes—
190#63225—24no—
191#63225—42no—
192#63225—42no—
193#63225—42no—
194#63225—81no—
195C4×Dic17—21no—
196C4×Dic17—21no—
197C4×Dic17—21no—
198C4×Dic17—21no—
199C4×Dic17—42no—
200C4×Dic17—42no—
201C2×#3622—21no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622—21no—
207#26367—171nomaximal, Sylow
208C23×C17—21no—
209C23×C17—21no—
210C23×C17—21no—
211C23×C17—21no—
212C23×C17—21no—
213C23×C17—21no—
214C23×C17C22⋊Q812yes—
215C23×C17C22×D411yescharacteristic
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—22no—
219C2×C4×C17—22no—
220C2×C4×C17—22no—
221C2×C4×C17—22no—
222C2×C4×C17—22no—
223C2×C4×C17—22no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—41no—
230C2×C4×C17—41no—
231C2×C4×C17—41no—
232C2×C4×C17—41no—
233C2×C4×C17—41no—
234C2×C4×C17—84no—
235C2×C4×C17C22⋊Q812yes—
236C2×C4×C17C22×D411yescharacteristic
237C2×C4×C17C22×Q811yescharacteristic
238C2×C4×C17C2×#4111yescharacteristic
239C2×C4×C17C2×#4111yescharacteristic
240C2×C4×C17H(5,𝔽2)11yescharacteristic
241C2×C4×C172−1+411yescharacteristic
242C2×Dic17—22no—
243C2×Dic17—22no—
244C2×Dic17—21no—
245C2×Dic17—21no—
246C2×Dic17—21no—
247C2×Dic17—21no—
248C2×Dic17—21no—
249C2×Dic17—21no—
250C2×Dic17—21no—
251C2×Dic17—21no—
252C2×Dic17—21no—
253C2×Dic17—21no—
254C2×Dic17—21no—
255C2×Dic17—21no—
256C2×Dic17—41no—
257C2×Dic17—41no—
258C2×Dic17—41no—
259C2×Dic17—41no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263C2×Dic17—41no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266Dic34—22no—
267Dic34—22no—
268Dic34—41no—
269Dic34—41no—
270C8×C17—22no—
271C8×C17—22no—
272#3622—41no—
273#3622—41no—
274C2×#479—171no—
275C2×#428D1711yescharacteristic
276#2442—171no—
277#2362—171no—
278#2097—171no—
279#2087—171no—
280#2068—171no—
281#2059—171no—
282#1949—171no—
283#1919—171no—
284#1894—171no—
285#1872—171no—
286#1865—171no—
287#1854—171no—
288#1851—171no—
289C22×C17—22no—
290C22×C17—22no—
291C22×C17—22no—
292C22×C17—41no—
293C22×C17—41no—
294C22×C17—41no—
295C22×C17—41no—
296C22×C17C2×C22⋊Q811yescharacteristic
297C22×C17D4211yescharacteristic
298C22×C17#54711yescharacteristic
299C22×C17#57511yescharacteristic
300C22×C17#57511yescharacteristic
301C22×C17#57711yescharacteristic
302C22×C17#57811yescharacteristic
303C4×C17—22no—
304C4×C17—42no—
305C4×C17—42no—
306C4×C17—81no—
307C4×C17—81no—
308C4×C17#52511yescharacteristic
309C4×C17#53911yescharacteristic
310C4×C17D4×Q811yescharacteristic
311C4×C17#55411yescharacteristic
312Dic17—42no—
313Dic17—41no—
314Dic17—41no—
315Dic17—42no—
316Dic17—41no—
317Dic17—41no—
318Dic17—81no—
319Dic17—81no—
320C22×M16C2×D1711yescharacteristic
321C2×#125—171no—
322C2×C22⋊Q8—171no—
323C2×C4×Q8—171no—
324C2×#117C2×D1711yescharacteristic
325C22×C4⋊C4C2×D1711yescharacteristic
326#479—344no—
327#428—24no—
328C2×#107—171no—
329C2×#107C2×D1711yescharacteristic
330C2×#107C2×D1711yescharacteristic
331C2×#106—171no—
332C2×#106C2×D1711yescharacteristic
333C2×#106C2×D1711yescharacteristic
334C2×C4⋊C8—171no—
335C2×C4⋊C8—171no—
336C2×#103—171no—
337C2×#103—171no—
338C2×#103—171no—
339C2×#103—171no—
340C2×C22⋊C8—171no—
341#400—341no—
342#398—341no—
343#391—171no—
344#389—171no—
345#389—171no—
346#389—171no—
347#388—341no—
348#388—341no—
349#387—341no—
350#385—341no—
351#382—171no—
352#380—171no—
353#380—341no—
354C4×C4⋊C4—171no—
355C4×C4⋊C4—341no—
356C4×C22⋊C4—171no—
357#340—171no—
358#340—171no—
359#340—171no—
360#340—171no—
361#340—171no—
362#340—171no—
363#336—171no—
364#336—171no—
365C2×C17—42no—
366C2×C17#244911yescharacteristic
367C2×C17#308011yescharacteristic
368C2×C17#308211yescharacteristic
369C2×C17#326911yescharacteristic
370C2×C17#327511yescharacteristic
371C2×C17#334911yescharacteristic
372C2×C17#335111yescharacteristic
373C22×Q8—171no—
374C23×C4—21no—
375C23×C4C22×D1711yescharacteristic
376C2×M16—22no—
377C2×M16—22no—
378C2×M16—22no—
379C22×C8—171no—
380C22×C8—171no—
381C22×C8C22×D1711yescharacteristic
382C22×C8C22×D1711yescharacteristic
383#125—342no—
384#125—681no—
385C22⋊Q8—174no—
386C22⋊Q8—342no—
387C4×Q8—342no—
388C4×Q8—681no—
389#117—22no—
390#117—41no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—41no—
394C2×C4⋊C4—171no—
395C2×C4⋊C4—171no—
396C2×C4⋊C4—171no—
397C2×C4⋊C4—341no—
398C2×C4⋊C4—341no—
399C2×C4⋊C4—341no—
400C2×C4⋊C4—341no—
401C2×C4⋊C4—341no—
402C2×C4⋊C4—341no—
403C2×C4⋊C4—341no—
404C2×C4⋊C4C22×D1711yescharacteristic
405C2×C4⋊C4C22×D1711yescharacteristic
406C2×C4⋊C4C22×D1711yescharacteristic
407C2×C4⋊C4C22×D1711yescharacteristic
408C2×C22⋊C4—21no—
409C2×C22⋊C4—171no—
410C2×C22⋊C4—171no—
411C2×C22⋊C4—341no—
412C2×C22⋊C4—341no—
413C2×C42—21no—
414C2×C42—171no—
415C2×C42—171no—
416C2×C42—341no—
417C2×C42—341no—
418#107—22no—
419#107—22no—
420#107—342no—
421#106—22no—
422#106—681no—
423#106C22⋊D1714yes—
424C4⋊C8—342no—
425C4⋊C8—342no—
426#103—342no—
427#103—342no—
428#103—342no—
429#103—681no—
430C22⋊C8—342no—
431#95—171no—
432#95—171no—
433#95—341no—
434#95—341no—
435#95—341no—
436#95—341no—
437#95—341no—
438#95—341no—
439#95—341no—
440#95—341no—
441#95—341no—
442C17#2636711yescharacteristic, complemented, Sylow
443C24D4×D1711yescharacteristic
444C2×Q8—172no—
445C2×Q8—341no—
446C2×Q8—341no—
447C2×Q8—341no—
448C2×Q8—341no—
449C2×Q8—341no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—81no—
461C22×C4—171no—
462C22×C4—171no—
463C22×C4—171no—
464C22×C4—171no—
465C22×C4—341no—
466C22×C4—341no—
467C22×C4—341no—
468C22×C4—341no—
469C22×C4#6325212yes—
470C22×C4D4×D1711yescharacteristic
471C22×C4Q8×D1712yes—
472C22×C4C23×D1711yescharacteristic
473M16—28no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—341no—
479C2×C8—341no—
480C2×C8—341no—
481C2×C8—341no—
482C2×C8—341no—
483C2×C8—341no—
484C2×C8#6325212yes—
485C2×C8C2×C22⋊D1712yes—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—42no—
493C4⋊C4—172no—
494C4⋊C4—341no—
495C4⋊C4—344no—
496C4⋊C4—341no—
497C4⋊C4—341no—
498C4⋊C4—341no—
499C4⋊C4—341no—
500C4⋊C4—682no—
501C4⋊C4—682no—
502C4⋊C4—682no—
503C4⋊C4—682no—
504C4⋊C4—1361no—
505C4⋊C4—1361no—
506C4⋊C4C2×C22⋊D1712yes—
507C4⋊C4C2×C22⋊D1712yes—
508C22⋊C4—42no—
509C22⋊C4—344no—
510C22⋊C4—681no—
511C22⋊C4—681no—
512C22⋊C4—682no—
513C42—42no—
514C42—341no—
515C42—341no—
516C42—341no—
517C42—341no—
518C42—682no—
519C42—682no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23#1058758212yes—
527C23C2×D4×D1711yescharacteristic
528Q8—342no—
529Q8—342no—
530Q8—681no—
531Q8—681no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—84no—
551C2×C4—341no—
552C2×C4—341no—
553C2×C4—342no—
554C2×C4—342no—
555C2×C4—341no—
556C2×C4—341no—
557C2×C4—341no—
558C2×C4—341no—
559C2×C4—341no—
560C2×C4—341no—
561C2×C4—341no—
562C2×C4—341no—
563C2×C4—341no—
564C2×C4—341no—
565C2×C4—681no—
566C2×C4—681no—
567C2×C4—681no—
568C2×C4—681no—
569C2×C4—681no—
570C2×C4—681no—
571C2×C4—681no—
572C2×C4—681no—
573C2×C4—681no—
574C2×C4—681no—
575C2×C4#1058758212yes—
576C2×C4C2×#6325211yescharacteristic
577C2×C4C2×#6325211yescharacteristic
578C2×C4#1058769111yescharacteristic
579C2×C4C2×Q8×D1711yescharacteristic
580C2×C4#1058769411yescharacteristic
581C2×C4C22×C22⋊D1711yescharacteristic
582C8—22no—
583C8—22no—
584C8—681no—
585C8—681no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22C2×#1058758211yescharacteristic
594C22D4×C22⋊D1711yescharacteristic
595C22#1176346011yescharacteristic
596C22#1176360511yescharacteristic
597C22#1176360811yescharacteristic
598C22#1176361911yescharacteristic
599C22#1176362511yescharacteristic
600C4—22no—
601C4—42no—
602C4—42no—
603C4—81no—
604C4—81no—
605C4—681no—
606C4—681no—
607C4—681no—
608C4—682no—
609C4—681no—
610C4—682no—
611C4—1361no—
612C4—1361no—
613C4#1176332011yescharacteristic
614C4#1176340611yescharacteristic
615C4Q8×C22⋊D1711yescharacteristic
616C4#1176351411yescharacteristic
617C2—42no—
618C2#1453259311yescharacteristic
619C2#1453860511yescharacteristic
620C2#1453862411yescharacteristic
621C2#1454033311yescharacteristic
622C2#1454041511yescharacteristic
623C2#1454131511yescharacteristic
624C2#1454134511yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#428C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenterC2×#428D17
hypocenterC17#26367
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
1. C2×#11763119C2semidirect
4. #14530329C2semidirect
5. #14530278C2semidirect
6. #14530174C2semidirect
7. #14530120C2semidirect
8. #14529553C2semidirect
9. #14529364C2semidirect
12. #14529049C2semidirect
442. C17#26367semidirect

Maximal subgroups

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

subgroupindex
C2×#117631192
#145325022
#145318112
#145303292
#145302782
#145301742
#145301202
#145295532
#145293642
#145292172
#145290912
#145290492
#145289832
#145289672
C2×C17×#4282
#2636717