Finite Group Databasebeta

← #17097254

Subgroups — #17097254

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×#11763116C211yesmaximal, characteristic, complemented
2C2×#11762852C211yesmaximal, characteristic
3#14532508C211yesmaximal, characteristic
4#14530332C211yesmaximal, characteristic, complemented
5#14530280C211yesmaximal, characteristic, complemented
6#14530179C211yesmaximal, characteristic, complemented
7#14530122C211yesmaximal, characteristic, complemented
8#14529552C211yesmaximal, characteristic, complemented
9#14529362C211yesmaximal, characteristic, complemented
10#14529218C211yesmaximal, characteristic
11#14529092C211yesmaximal, characteristic
12#14529052C211yesmaximal, characteristic, complemented
13#14528983C211yesmaximal, characteristic
14#14528969C211yesmaximal, characteristic
15C17×#2362C211yesmaximal, characteristic
16C22×#63223C2211yescharacteristic
17C2×#10587556C2211yescharacteristic
18C22×C4⋊Dic17C2211yescharacteristic
19#11763116—24no—
20#11762852—24no—
21C2×#10587478C2211yescharacteristic
22C2×#10587478C2211yescharacteristic
23C2×#10587478C2211yescharacteristic
24C2×#10587479C2211yescharacteristic
25C2×#10587479C2211yescharacteristic
26C2×#10587479C2211yescharacteristic
27C2×C4⋊#3622C2211yescharacteristic
28C2×C4⋊#3622C2211yescharacteristic
29C2×#10587506C2211yescharacteristic
30C2×#10587506C2211yescharacteristic
31C2×#10587506C2211yescharacteristic
32C2×#10587506C2211yescharacteristic
33C2×C22⋊#3622C2211yescharacteristic
34#11762751—21no—
35#11762743—21no—
36#11762707C2211yescharacteristic
37#11762707C2211yescharacteristic
38#11762681—21no—
39#11762675C2211yescharacteristic
40#11762669C2211yescharacteristic
41#11762667—21no—
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762564C2211yescharacteristic
49#11762564C2211yescharacteristic
50C2×C17×C22⋊Q8C2211yescharacteristic
51C2×C4×Q8×C17C2211yescharacteristic
52C2×C17×#117C2211yescharacteristic
53C17×#391C2211yescharacteristic
54C17×#389C2211yescharacteristic
55C17×#388—21no—
56C17×#388—21no—
57C17×#387—21no—
58C4×C4⋊C4×C17—21no—
59C4×C4⋊C4×C17C2211yescharacteristic
60C4×C22⋊C4×C17C2211yescharacteristic
61C22×Q8×C17C2311yescharacteristic
62C23×C4×C17C2311yescharacteristic
63C23×Dic17—21no—
64C17×C22⋊Q8—22no—
65C17×C22⋊Q8Q814yes—
66C4×Q8×C17—22no—
67C4×Q8×C17—41no—
68C17×#117—22no—
69C17×#117—41no—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C22⋊C4×C17—21no—
77C2×C22⋊C4×C17—21no—
78C2×C22⋊C4×C17D411yescharacteristic
79C2×C22⋊C4×C17D411yescharacteristic
80C2×C42×C17—21no—
81C2×C42×C17—21no—
82C2×C42×C17—21no—
83C2×C42×C17C2311yescharacteristic
84C2×C42×C17C2311yescharacteristic
85C2×C22⋊Dic17—21no—
86C2×C4⋊Dic17—21no—
87C2×C4⋊Dic17—22no—
88C2×C4⋊Dic17—41no—
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×#63225—21no—
94C2×#63225—21no—
95C2×#63225—21no—
96C2×#63225—21no—
97C2×#63223—22no—
98C2×#63223—22no—
99C2×#63223—22no—
100C22×#3622C2311yescharacteristic
101C22×#3622C2311yescharacteristic
102C22×#3622C2311yescharacteristic
103C22×#3622C2311yescharacteristic
104#10587556—22no—
105#10587556—41no—
106C17×#95—21no—
107C17×#95—21no—
108C17×#95—21no—
109C17×#95—21no—
110C17×#95D411yescharacteristic
111C17×#95D411yescharacteristic
112#10587506—22no—
113#10587506—22no—
114#10587506—22no—
115#10587506—41no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121C22⋊#3622—22no—
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
131C2×Q8×C17—21no—
132C2×Q8×C17—21no—
133C2×Q8×C17—21no—
134C2×Q8×C17—21no—
135C2×Q8×C17—21no—
136C2×Q8×C17C2×Q812yes—
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17—21no—
143C22×C4×C17—21no—
144C22×C4×C17—21no—
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2×Q812yes—
151C22×C4×C17#4112yes—
152C22×C4×C17C2411yescharacteristic
153C22×Dic17—21no—
154C22×Dic17—21no—
155C22×Dic17—21no—
156C22×Dic17—21no—
157C22×Dic17—42no—
158C22×Dic17—42no—
159C22×Dic17—81no—
160C4⋊C4×C17—24no—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—42no—
166C4⋊C4×C17—42no—
167C4⋊C4×C17C2×D412yes—
168C4⋊C4×C17C2×Q812yes—
169C22⋊C4×C17—24no—
170C22⋊C4×C17—41no—
171C22⋊C4×C17—41no—
172C22⋊C4×C17—42no—
173C42×C17—21no—
174C42×C17—21no—
175C42×C17—21no—
176C42×C17—21no—
177C42×C17—42no—
178C42×C17—42no—
179C42×C17—42no—
180C22⋊Dic17—42no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17—21no—
185C4⋊Dic17—21no—
186C4⋊Dic17—21no—
187C4⋊Dic17—21no—
188C4⋊Dic17—42no—
189C4⋊Dic17C2×D412yes—
190#63225—42no—
191#63225—42no—
192#63225—81no—
193#63225—81no—
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
216Q8×C17—22no—
217Q8×C17—22no—
218Q8×C17—41no—
219Q8×C17—41no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—22no—
223C2×C4×C17—22no—
224C2×C4×C17—22no—
225C2×C4×C17—22no—
226C2×C4×C17—22no—
227C2×C4×C17—22no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—22no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—41no—
242C2×C4×C17—41no—
243C2×C4×C17—41no—
244C2×C4×C17—41no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—41no—
248C2×C4×C17—41no—
249C2×C4×C17—41no—
250C2×C4×C17C22⋊Q812yes—
251C2×C4×C17C22×D411yescharacteristic
252C2×C4×C17C22×Q811yescharacteristic
253C2×C4×C17C2×#4111yescharacteristic
254C2×C4×C17C2×#4111yescharacteristic
255C2×C4×C17H(5,𝔽2)11yescharacteristic
256C2×C4×C172−1+411yescharacteristic
257C2×Dic17—22no—
258C2×Dic17—41no—
259C2×Dic17—41no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263C2×Dic17—41no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266C2×Dic17—41no—
267C2×Dic17—41no—
268C2×Dic17—41no—
269C2×Dic17—84no—
270#3622—22no—
271#3622—22no—
272#3622—41no—
273#3622—41no—
274C2×#479—171no—
275C2×#428—171no—
276#2442—171no—
277#2362D1711yescharacteristic
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—41no—
306C4×C17—41no—
307C4×C17—42no—
308C4×C17—41no—
309C4×C17—42no—
310C4×C17—41no—
311C4×C17—81no—
312C4×C17#52511yescharacteristic
313C4×C17#53911yescharacteristic
314C4×C17D4×Q811yescharacteristic
315C4×C17#55411yescharacteristic
316Dic17—42no—
317Dic17—81no—
318Dic17—81no—
319Dic17—81no—
320C22×M16—171no—
321C2×#125—171no—
322C2×C22⋊Q8C2×D1711yescharacteristic
323C2×C4×Q8C2×D1711yescharacteristic
324C2×#117C2×D1711yescharacteristic
325C22×C4⋊C4—171no—
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—341no—
342#398—341no—
343#391C2×D1711yescharacteristic
344#389—171no—
345#389—171no—
346#389C2×D1711yescharacteristic
347#388—21no—
348#388—21no—
349#387—21no—
350#385—341no—
351#382—171no—
352#380—171no—
353#380—341no—
354C4×C4⋊C4—21no—
355C4×C4⋊C4C2×D1711yescharacteristic
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×Q8C22×D1711yescharacteristic
374C23×C4—341no—
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—342no—
384#125—681no—
385C22⋊Q8—22no—
386C22⋊Q8Dic3414yes—
387C4×Q8—22no—
388C4×Q8—41no—
389#117—22no—
390#117—41no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—171no—
395C2×C4⋊C4—171no—
396C2×C4⋊C4—171no—
397C2×C4⋊C4—171no—
398C2×C4⋊C4—342no—
399C2×C4⋊C4—341no—
400C2×C4⋊C4—341no—
401C2×C4⋊C4—341no—
402C2×C4⋊C4—341no—
403C2×C4⋊C4—341no—
404C2×C4⋊C4—681no—
405C2×C4⋊C4C22×D1711yescharacteristic
406C2×C4⋊C4C22×D1711yescharacteristic
407C2×C4⋊C4C22×D1711yescharacteristic
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—341no—
411C2×C22⋊C4C22⋊D1711yescharacteristic
412C2×C22⋊C4C22⋊D1711yescharacteristic
413C2×C42—21no—
414C2×C42—21no—
415C2×C42—21no—
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—341no—
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—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8C2×Dic3412yes—
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—341no—
459C22×C4—341no—
460C22×C4—341no—
461C22×C4—341no—
462C22×C4—682no—
463C22×C4—682no—
464C22×C4—1361no—
465C22×C4C2×Dic3412yes—
466C22×C4#6325212yes—
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—24no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—172no—
494C4⋊C4—341no—
495C4⋊C4—341no—
496C4⋊C4—341no—
497C4⋊C4—341no—
498C4⋊C4—341no—
499C4⋊C4—341no—
500C4⋊C4—341no—
501C4⋊C4—682no—
502C4⋊C4—682no—
503C4⋊C4—682no—
504C4⋊C4—1361no—
505C4⋊C4—1361no—
506C4⋊C4C2×Dic3412yes—
507C4⋊C4D4×D1712yes—
508C22⋊C4—24no—
509C22⋊C4—41no—
510C22⋊C4—41no—
511C22⋊C4—42no—
512C22⋊C4—682no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—42no—
518C42—42no—
519C42—42no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23#1058761012yes—
527C23C22×C22⋊D1711yescharacteristic
528Q8—22no—
529Q8—22no—
530Q8—41no—
531Q8—41no—
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—22no—
541C2×C4—22no—
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—21no—
551C2×C4—21no—
552C2×C4—22no—
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—342no—
563C2×C4—681no—
564C2×C4—681no—
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—1364no—
575C2×C4#1058761012yes—
576C2×C4C22×Dic3411yescharacteristic
577C2×C4C2×#6325211yescharacteristic
578C2×C4C2×D4×D1711yescharacteristic
579C2×C4#1058769111yescharacteristic
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×#1058761011yescharacteristic
594C22D4×C22⋊D1711yescharacteristic
595C22#1176345811yescharacteristic
596C22#1176360711yescharacteristic
597C22#1176360711yescharacteristic
598C22#1176361711yescharacteristic
599C22#1176362411yescharacteristic
600C4—22no—
601C4—41no—
602C4—41no—
603C4—42no—
604C4—42no—
605C4—41no—
606C4—42no—
607C4—41no—
608C4—81no—
609C4—682no—
610C4—1361no—
611C4—1361no—
612C4—1361no—
613C4#1176331811yescharacteristic
614C4#1176340111yescharacteristic
615C4D4×Dic3411yescharacteristic
616C4#1176351811yescharacteristic
617C2—42no—
618C2#1453258711yescharacteristic
619C2#1453860611yescharacteristic
620C2#1453862611yescharacteristic
621C2#1454033511yescharacteristic
622C2#1454041611yescharacteristic
623C2#1454130811yescharacteristic
624C2#1454134411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#2362C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#2362D17
hypocenterC17#26367
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
1. C2×#11763116C2semidirect
4. #14530332C2semidirect
5. #14530280C2semidirect
6. #14530179C2semidirect
7. #14530122C2semidirect
8. #14529552C2semidirect
9. #14529362C2semidirect
12. #14529052C2semidirect
442. C17#26367semidirect

Maximal subgroups

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

subgroupindex
C2×#117631162
C2×#117628522
#145325082
#145303322
#145302802
#145301792
#145301222
#145295522
#145293622
#145292182
#145290922
#145290522
#145289832
#145289692
C17×#23622
#2636717