Finite Group Databasebeta

← #17097256

Subgroups — #17097256

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×#11763113C211yesmaximal, characteristic, complemented
2C2×#11762849C211yesmaximal, characteristic
3#14531809C211yesmaximal, characteristic
4#14530335C211yesmaximal, characteristic, complemented
5#14530280C211yesmaximal, characteristic, complemented
6#14530176C211yesmaximal, characteristic, complemented
7#14530119C211yesmaximal, characteristic, complemented
8#14529550C211yesmaximal, characteristic, complemented
9#14529359C211yesmaximal, characteristic, complemented
10#14529215C211yesmaximal, characteristic
11#14529091C211yesmaximal, characteristic
12#14529051C211yesmaximal, characteristic, complemented
13#14528981C211yesmaximal, characteristic
14#14528969C211yesmaximal, characteristic
15C17×#2442C211yesmaximal, characteristic
16C22×#63223C2211yescharacteristic
17C2×#10587610C2211yescharacteristic
18C2×C4×Dic34C2211yescharacteristic
19C2×#10587612C2211yescharacteristic
20#11763113—24no—
21#11762849—24no—
22C2×#10587478C2211yescharacteristic
23C2×#10587478C2211yescharacteristic
24C2×#10587478C2211yescharacteristic
25C2×#10587479C2211yescharacteristic
26C2×#10587479C2211yescharacteristic
27C2×#10587479C2211yescharacteristic
28C2×C4⋊#3622C2211yescharacteristic
29C2×C4⋊#3622C2211yescharacteristic
30C2×#10587481C2211yescharacteristic
31C2×#10587481C2211yescharacteristic
32C2×#10587481C2211yescharacteristic
33C2×#10587481C2211yescharacteristic
34C2×C22⋊#3622C2211yescharacteristic
35#11762715C2211yescharacteristic
36#11762707C2211yescharacteristic
37#11762703—21no—
38#11762697—21no—
39#11762691—21no—
40C4×C4⋊Dic17C2211yescharacteristic
41#11762660—21no—
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762564C2211yescharacteristic
49#11762564C2211yescharacteristic
50C2×C17×#125C2211yescharacteristic
51C22×C4⋊C4×C17C2211yescharacteristic
52C17×#400—21no—
53C17×#398—21no—
54C17×#389C2211yescharacteristic
55C17×#389C2211yescharacteristic
56C17×#385—21no—
57C17×#382C2211yescharacteristic
58C17×#380—21no—
59C17×#380C2211yescharacteristic
60C4×C22⋊C4×C17C2211yescharacteristic
61C23×C4×C17—21no—
62C23×C4×C17C2311yescharacteristic
63C22×Dic34C2311yescharacteristic
64C17×#125—22no—
65C17×#125—41no—
66C2×C4⋊C4×C17—22no—
67C2×C4⋊C4×C17—21no—
68C2×C4⋊C4×C17—21no—
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17—41no—
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C22⋊C4×C17—21no—
78C2×C22⋊C4×C17D411yescharacteristic
79C2×C22⋊C4×C17D411yescharacteristic
80C2×C42×C17C2311yescharacteristic
81C2×C42×C17C2311yescharacteristic
82C2×C22⋊Dic17—21no—
83C2×C22⋊Dic17—21no—
84#10587612—22no—
85#10587612—41no—
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89#10587610—22no—
90#10587610Q814yes—
91C2×#63225—21no—
92C2×#63225—21no—
93C2×#63225—21no—
94C2×C4×Dic17—21no—
95C2×C4×Dic17—21no—
96C2×C4×Dic17—21no—
97C2×#63223—22no—
98C2×#63223—22no—
99C2×#63223—22no—
100C22×#3622C2311yescharacteristic
101C22×#3622C2311yescharacteristic
102C22×#3622C2311yescharacteristic
103C22×#3622C2311yescharacteristic
104C4×Dic34—22no—
105C4×Dic34—41no—
106C17×#95—21no—
107C17×#95—21no—
108C17×#95—21no—
109C17×#95—21no—
110C17×#95—21no—
111C17×#95D411yescharacteristic
112C17×#95D411yescharacteristic
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117C22⋊#3622—22no—
118#10587481—22no—
119#10587481—22no—
120#10587481—22no—
121#10587481—41no—
122#10587479—22no—
123#10587479—41no—
124#10587479D414yes—
125#10587478—22no—
126#10587478—22no—
127#10587478—22no—
128C4⋊#3622—22no—
129C4⋊#3622—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×D411yescharacteristic
144C22×C4×C17C2×D411yescharacteristic
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×Q812yes—
148C22×C4×C17#4112yes—
149C22×C4×C17C2411yescharacteristic
150C22×Dic17—21no—
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—21no—
154C2×Dic34—21no—
155C2×Dic34—21no—
156C2×Dic34—21no—
157C2×Dic34—21no—
158C2×Dic34—21no—
159C2×Dic34C2×Q812yes—
160C4⋊C4×C17—21no—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—21no—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—42no—
168C4⋊C4×C17—42no—
169C4⋊C4×C17—42no—
170C4⋊C4×C17—81no—
171C4⋊C4×C17—81no—
172C4⋊C4×C17C2×D412yes—
173C22⋊C4×C17—41no—
174C22⋊C4×C17—41no—
175C22⋊C4×C17—42no—
176C42×C17—21no—
177C42×C17—21no—
178C42×C17—21no—
179C42×C17—21no—
180C22⋊Dic17—24no—
181C22⋊Dic17—42no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17—21no—
185C4⋊Dic17—21no—
186C4⋊Dic17C2×D412yes—
187C4⋊Dic17C2×Q812yes—
188#63225—24no—
189#63225—42no—
190#63225—42no—
191C4×Dic17—42no—
192C4×Dic17—42no—
193C4×Dic17—42no—
194#63223—28no—
195C2×#3622—21no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622—21no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622C2×D412yes—
206C2×#3622#4112yes—
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—21no—
223C2×C4×C17—21no—
224C2×C4×C17—22no—
225C2×C4×C17—21no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—41no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17—41no—
239C2×C4×C17—41no—
240C2×C4×C17—41no—
241C2×C4×C17—41no—
242C2×C4×C17—41no—
243C2×C4×C17—41no—
244C2×C4×C17—41no—
245C2×C4×C17—41no—
246C2×C4×C17—84no—
247C2×C4×C17C22⋊Q812yes—
248C2×C4×C17C22×D411yescharacteristic
249C2×C4×C17C22×Q811yescharacteristic
250C2×C4×C17C2×#4111yescharacteristic
251C2×C4×C17C2×#4111yescharacteristic
252C2×C4×C17H(5,𝔽2)11yescharacteristic
253C2×C4×C172−1+411yescharacteristic
254C2×Dic17—22no—
255C2×Dic17—22no—
256C2×Dic17—22no—
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—
270#3622—22no—
271#3622—22no—
272#3622—41no—
273#3622—41no—
274C2×#479—171no—
275C2×#428—171no—
276#2442D1711yescharacteristic
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—41no—
305C4×C17—41no—
306C4×C17—41no—
307C4×C17—41no—
308C4×C17—42no—
309C4×C17—81no—
310C4×C17—81no—
311C4×C17—81no—
312C4×C17#52511yescharacteristic
313C4×C17#53911yescharacteristic
314C4×C17D4×Q811yescharacteristic
315C4×C17#55411yescharacteristic
316Dic17—42no—
317Dic17—42no—
318Dic17—42no—
319Dic17—81no—
320C22×M16—171no—
321C2×#125C2×D1711yescharacteristic
322C2×C22⋊Q8—171no—
323C2×C4×Q8—171no—
324C2×#117—171no—
325C22×C4⋊C4C2×D1711yescharacteristic
326#479—344no—
327#428—344no—
328C2×#107—171no—
329C2×#107—171no—
330C2×#107—171no—
331C2×#106—171no—
332C2×#106—171no—
333C2×#106—171no—
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—21no—
342#398—21no—
343#391—171no—
344#389—171no—
345#389C2×D1711yescharacteristic
346#389C2×D1711yescharacteristic
347#388—341no—
348#388—341no—
349#387—341no—
350#385—21no—
351#382C2×D1711yescharacteristic
352#380—21no—
353#380C2×D1711yescharacteristic
354C4×C4⋊C4—171no—
355C4×C4⋊C4—341no—
356C4×C22⋊C4C2×D1711yescharacteristic
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—342no—
377C2×M16—342no—
378C2×M16—342no—
379C22×C8—171no—
380C22×C8—171no—
381C22×C8—171no—
382C22×C8—171no—
383#125—22no—
384#125—41no—
385C22⋊Q8—174no—
386C22⋊Q8—342no—
387C4×Q8—342no—
388C4×Q8—681no—
389#117—342no—
390#117—681no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—41no—
398C2×C4⋊C4—171no—
399C2×C4⋊C4—171no—
400C2×C4⋊C4—171no—
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—341no—
410C2×C22⋊C4—341no—
411C2×C22⋊C4C22⋊D1711yescharacteristic
412C2×C22⋊C4C22⋊D1711yescharacteristic
413C2×C42—341no—
414C2×C42—341no—
415C2×C42—341no—
416C2×C42C22×D1711yescharacteristic
417C2×C42C22×D1711yescharacteristic
418#107—342no—
419#107—342no—
420#107—342no—
421#106—174no—
422#106—342no—
423#106—681no—
424C4⋊C8—342no—
425C4⋊C8—342no—
426#103—342no—
427#103—342no—
428#103—342no—
429#103—681no—
430C22⋊C8—342no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—341no—
437#95—341no—
438#95—341no—
439#95—341no—
440#95C22⋊D1711yescharacteristic
441#95C22⋊D1711yescharacteristic
442C17#2636711yescharacteristic, complemented, Sylow
443C24C2×C22⋊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—341no—
462C22×C4—341no—
463C22×C4—341no—
464C22×C4—341no—
465C22×C4Q8×D1712yes—
466C22×C4#6325612yes—
467C22×C4C2×C22⋊D1711yescharacteristic
468C22×C4C2×C22⋊D1711yescharacteristic
469C22×C4C2×C22⋊D1711yescharacteristic
470C22×C4C2×C22⋊D1711yescharacteristic
471C22×C4C2×C22⋊D1711yescharacteristic
472C22×C4C23×D1711yescharacteristic
473M16—348no—
474C2×C8—172no—
475C2×C8—172no—
476C2×C8—341no—
477C2×C8—341no—
478C2×C8—341no—
479C2×C8—341no—
480C2×C8—341no—
481C2×C8—341no—
482C2×C8—341no—
483C2×C8—341no—
484C2×C8—341no—
485C2×C8—341no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—81no—
497C4⋊C4—81no—
498C4⋊C4—172no—
499C4⋊C4—172no—
500C4⋊C4—341no—
501C4⋊C4—341no—
502C4⋊C4—344no—
503C4⋊C4—341no—
504C4⋊C4—341no—
505C4⋊C4—682no—
506C4⋊C4—682no—
507C4⋊C4D4×D1712yes—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—42no—
511C22⋊C4—344no—
512C22⋊C4—682no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—682no—
518C42—682no—
519C42—682no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23#1058763212yes—
527C23C22×C22⋊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—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—84no—
563C2×C4—342no—
564C2×C4—342no—
565C2×C4—342no—
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#1058763212yes—
576C2×C4C2×D4×D1711yescharacteristic
577C2×C4#1058769111yescharacteristic
578C2×C4C2×Q8×D1711yescharacteristic
579C2×C4C2×#6325611yescharacteristic
580C2×C4#41×D1711yescharacteristic
581C2×C4#1058769711yescharacteristic
582C8—342no—
583C8—342no—
584C8—681no—
585C8—681no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22C2×#1058763211yescharacteristic
594C22D4×C22⋊D1711yescharacteristic
595C22#1176345811yescharacteristic
596C22#1176360711yescharacteristic
597C22#1176360711yescharacteristic
598C22#1176362011yescharacteristic
599C22#1176362411yescharacteristic
600C4—22no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—42no—
606C4—81no—
607C4—81no—
608C4—81no—
609C4—682no—
610C4—682no—
611C4—682no—
612C4—1361no—
613C4#1176332211yescharacteristic
614C4#1176340811yescharacteristic
615C4#1176347211yescharacteristic
616C4#1176351311yescharacteristic
617C2—42no—
618C2#1453258911yescharacteristic
619C2#1453860211yescharacteristic
620C2#1453862011yescharacteristic
621C2#1454034011yescharacteristic
622C2#1454041111yescharacteristic
623C2#1454131411yescharacteristic
624C2#1454133811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2442C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#2442D17
hypocenterC17#26367
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2442

Complements

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

NH ≅ G/Ntype
1. C2×#11763113C2semidirect
4. #14530335C2semidirect
5. #14530280C2semidirect
6. #14530176C2semidirect
7. #14530119C2semidirect
8. #14529550C2semidirect
9. #14529359C2semidirect
12. #14529051C2semidirect
442. C17#26367semidirect

Maximal subgroups

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

subgroupindex
C2×#117631132
C2×#117628492
#145318092
#145303352
#145302802
#145301762
#145301192
#145295502
#145293592
#145292152
#145290912
#145290512
#145289812
#145289692
C17×#24422
#2636717