Finite Group Databasebeta

← #17037722

Subgroups — #17037722

Order 4352.

982 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762850C211yesmaximal, characteristic
2C2×#11762848C211yesmaximal, characteristic
3C2×#11762766C211yesmaximal, characteristic
4#14531406C211yesmaximal, characteristic
5#14531399C211yesmaximal, characteristic
6#14529527C211yesmaximal, characteristic, complemented
7#14529526C211yesmaximal, characteristic, complemented
8#14529521C211yesmaximal, characteristic, complemented
9#14529518C211yesmaximal, characteristic, complemented
10#14529514C211yesmaximal, characteristic, complemented
11#14529510C211yesmaximal, characteristic, complemented
12#14529509C211yesmaximal, characteristic, complemented
13#14528499C211yesmaximal, characteristic
14D17×#340C211yesmaximal, characteristic
15C17×#1943C211yesmaximal, characteristic, complemented
16C22×#63218C2211yescharacteristic
17C22×C8×D17C2211yescharacteristic
18C2×#10587578C2211yescharacteristic
19C2×#10587578C2211yescharacteristic
20C2×#10587558C2211yescharacteristic
21C2×C4⋊C4×D17C2211yescharacteristic
22C2×C4⋊C4×D17C2211yescharacteristic
23#11762850C418yescomplemented
24#11762848—24no—
25C2×#10587489C2211yescharacteristic
26C2×#10587478C2211yescharacteristic
27C2×#10587478C2211yescharacteristic
28C2×#10587488C2211yescharacteristic
29C2×#10587479C2211yescharacteristic
30C2×#10587479C2211yescharacteristic
31#11762766—24no—
32C2×#10587486C2211yescharacteristic
33C2×C4×#3622C2211yescharacteristic
34C2×C8×Dic17C2211yescharacteristic
35#11762708C2211yescharacteristic
36#11762708C2211yescharacteristic
37#11762707C2211yescharacteristic
38#11762707C2211yescharacteristic
39#11762706—21no—
40#11762705C2211yescharacteristic
41#11762705C2211yescharacteristic
42#11762704—21no—
43#11762683—21no—
44#11762680—21no—
45#11762678—21no—
46#11762673—21no—
47#11762668—21no—
48#11762666—21no—
49#11762572C2211yescharacteristic
50#11762572C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762571C2211yescharacteristic
53#11762571C2211yescharacteristic
54#11762571C2211yescharacteristic
55C2×C17×#107C2211yescharacteristic
56C2×C17×#106C2211yescharacteristic
57C2×C17×#97C2211yescharacteristic
58C17×#389C2211yescharacteristic
59C17×#389C2211yescharacteristic
60C17×#340C2211yescharacteristic
61C17×#340C2211yescharacteristic
62C22×C4×D17—21no—
63C22×C4×D17—21no—
64C22×C4×D17C2311yescharacteristic
65C22×C8×C17C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C2×C4⋊C4×C17—21no—
68C2×C4⋊C4×C17—21no—
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C42×C17C2311yescharacteristic
74C2×#63227—21no—
75C2×#63227—21no—
76C2×#63227D411yescharacteristic
77C2×#63227D411yescharacteristic
78C2×C4⋊Dic17—21no—
79C2×C4⋊Dic17—21no—
80C2×C4⋊Dic17C2311yescharacteristic
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×C4⋊Dic17C2311yescharacteristic
84C2×#63225—21no—
85C2×#63225—21no—
86C2×#63225—21no—
87C2×#63225—21no—
88C2×#63225—21no—
89C2×#63225—21no—
90C2×#63225D411yescharacteristic
91C2×#63225D411yescharacteristic
92C2×C4×Dic17—21no—
93C2×C4×Dic17—21no—
94C2×C4×Dic17C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96C22×#3622C2311yescharacteristic
97C2×#63218C2×C414yes—
98C2×#63218D414yes—
99C2×#63218Q814yes—
100C2×C8×D17—22no—
101C2×C8×D17—22no—
102C2×C8×D17Q814yes—
103#10587578—22no—
104#10587578—22no—
105#10587578—22no—
106#10587578C2×C414yes—
107C4⋊C4×D17—22no—
108C4⋊C4×D17—22no—
109C4⋊C4×D17—22no—
110C4⋊C4×D17C2×C414yes—
111#10587558—22no—
112#10587558—22no—
113C17×#107C2×C414yescomplemented
114C17×#106—22no—
115C17×#97—22no—
116C17×#95—21no—
117C17×#95—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587489—22no—
125#10587488C2×C414yescomplemented
126#10587486—22no—
127C8×Dic17—22no—
128#10587479—22no—
129#10587479C2×C414yescomplemented
130#10587478—22no—
131#10587478C2×C414yescomplemented
132C4×#3622—22no—
133C23×D17C2×D411yescharacteristic
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—21no—
137C22×C4×C17—21no—
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2411yescharacteristic
141C22×Dic17—21no—
142C22×Dic17—21no—
143C22×Dic17—21no—
144C22×Dic17—21no—
145C22×Dic17C2×D411yescharacteristic
146C22×Dic17C2×D411yescharacteristic
147C22×Dic17C2×D411yescharacteristic
148C2×C4×D17—24no—
149C2×C4×D17—42no—
150C2×C4×D17—81no—
151C2×C4×D17—81no—
152C2×C4×D17—81no—
153C2×C4×D17—81no—
154C2×C4×D17C22×C412yes—
155C2×C4×D17C2×D412yes—
156C2×C4×D17C2×Q812yes—
157C2×C4×D17C2×Q812yes—
158C2×C4×D17#4112yes—
159C2×C4×D17#4112yes—
160C2×C8×C17—21no—
161C2×C8×C17—21no—
162C2×C8×C17C22×C412yes—
163C2×C8×C17C2×D412yes—
164C2×C8×C17C2×Q812yes—
165C2×C8×C17C2×Q812yes—
166C4⋊C4×C17—21no—
167C4⋊C4×C17—21no—
168C4⋊C4×C17—21no—
169C4⋊C4×C17—21no—
170C4⋊C4×C17—21no—
171C4⋊C4×C17—21no—
172C4⋊C4×C17—81no—
173C4⋊C4×C17—81no—
174C4⋊C4×C17C22×C412yes—
175C4⋊C4×C17C22×C412yes—
176C42×C17—21no—
177C42×C17—21no—
178#63227—24no—
179#63227—22no—
180#63227—22no—
181#63227—42no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17—21no—
185C4⋊Dic17—21no—
186C4⋊Dic17—21no—
187C4⋊Dic17—21no—
188C4⋊Dic17—81no—
189C4⋊Dic17—81no—
190C4⋊Dic17C22×C412yes—
191C4⋊Dic17C22×C412yes—
192#63225—24no—
193#63225—22no—
194#63225—22no—
195#63225—42no—
196#63225—81no—
197#63225—81no—
198#63225—81no—
199#63225—81no—
200C4×Dic17—21no—
201C4×Dic17—21no—
202C4×Dic17—24no—
203C4×Dic17—42no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622C22×C412yes—
207C2×#3622C2×D412yes—
208C2×#3622C2×Q812yes—
209C2×#3622C2×Q812yes—
210#63218C4⋊C4116yescomplemented
211C8×D17—28no—
212#21160—171nomaximal, Sylow
213C23×C17C22×D411yescharacteristic
214C22×D17—21no—
215C22×D17—21no—
216C22×D17—21no—
217C22×D17—21no—
218C22×D17C4×D412yes—
219C22×D17#12112yes—
220C22×D17C4⋊Q812yes—
221C2×C4×C17—22no—
222C2×C4×C17—22no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—41no—
230C2×C4×C17—41no—
231C2×C4×C17—41no—
232C2×C4×C17—41no—
233C2×C4×C17—41no—
234C2×C4×C17—41no—
235C2×C4×C17—41no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17—41no—
239C2×C4×C17C23×C411yescharacteristic
240C2×C4×C17C22×D411yescharacteristic
241C2×C4×C17C22×Q811yescharacteristic
242C2×C4×C17C22×Q811yescharacteristic
243C2×C4×C17C2×#4111yescharacteristic
244C2×C4×C17C2×#4111yescharacteristic
245C2×Dic17—22no—
246C2×Dic17—22no—
247C2×Dic17—21no—
248C2×Dic17—21no—
249C2×Dic17—21no—
250C2×Dic17—21no—
251C2×Dic17—21no—
252C2×Dic17—21no—
253C2×Dic17—41no—
254C2×Dic17—41no—
255C2×Dic17—41no—
256C2×Dic17—41no—
257C2×Dic17—41no—
258C2×Dic17—41no—
259C2×Dic17—41no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263C2×Dic17C4×D412yes—
264C2×Dic17#12112yes—
265C2×Dic17C4⋊Q812yes—
266C4×D17—84no—
267C4×D17—84no—
268C4×D17C2×C4⋊C414yes—
269C4×D17C4×Q814yes—
270C4×D17C22⋊Q814yes—
271C4×D17#12514yes—
272C8×C17—22no—
273C8×C17C2×C4⋊C414yes—
274#3622—22no—
275#3622C2×C4⋊C414yes—
276C2×#428—171no—
277C2×#428—171no—
278C2×#405—171no—
279#2317—171no—
280#2316—171no—
281#1944—171no—
282#1944—171no—
283#1943—171no—
284#1943D1711yescharacteristic, complemented
285#1942—171no—
286#1942—171no—
287#1941—171no—
288#1941—171no—
289#1735—171no—
290C2×#340—171no—
291C22×C17C2×C4×D411yescharacteristic
292C22×C17C2×#12111yescharacteristic
293C22×C17C2×C4⋊Q811yescharacteristic
294C22×C17#57611yescharacteristic
295C22×C17#57711yescharacteristic
296C22×C17#57711yescharacteristic
297C22×C17#57811yescharacteristic
298C2×D17—22no—
299C2×D17—22no—
300C2×D17—22no—
301C2×D17—22no—
302C2×D17—22no—
303C2×D17—22no—
304C2×D17#38914yes—
305C4×C17—42no—
306C4×C17—42no—
307C4×C17—41no—
308C4×C17—41no—
309C4×C17—81no—
310C4×C17—81no—
311C4×C17C22×C4⋊C411yescharacteristic
312C4×C17C2×C4×Q811yescharacteristic
313C4×C17C2×C22⋊Q811yescharacteristic
314C4×C17C2×#12511yescharacteristic
315Dic17—42no—
316Dic17—42no—
317Dic17—41no—
318Dic17—41no—
319Dic17—81no—
320Dic17—81no—
321Dic17#38914yes—
322C22×M16—171no—
323C23×C8—171no—
324C2×#117—171no—
325C2×#117—171no—
326C2×#117—171no—
327C22×C4⋊C4—171no—
328C22×C4⋊C4—171no—
329#428—178no—
330#428—344no—
331C2×#107—171no—
332C2×#107—171no—
333C2×#107—171no—
334C2×#107C2×D1711yescharacteristic
335C2×#106—171no—
336C2×#106—171no—
337C2×#106—171no—
338C2×#106C2×D1711yescharacteristic
339#405—344no—
340C2×#97—171no—
341C2×#97C2×D1711yescharacteristic
342C2×C4×C8—171no—
343C2×C4×C8—171no—
344#389—171no—
345#389—171no—
346#389—171no—
347#389—171no—
348#389—171no—
349#389—171no—
350#389—341no—
351#389—341no—
352#389C2×D1711yescharacteristic
353#389C2×D1711yescharacteristic
354#385—341no—
355#385—341no—
356#384—341no—
357#382—341no—
358#380—341no—
359#380—341no—
360#340—171no—
361#340—171no—
362#340—171no—
363#340—171no—
364#340—171no—
365#340—171no—
366#340C2×D1711yescharacteristic
367#340C2×D1711yescharacteristic
368C2×C17C2×#38911yescharacteristic
369C2×C17#294611yescharacteristic
370C2×C17#294711yescharacteristic
371C2×C17#306211yescharacteristic
372C2×C17#306311yescharacteristic
373C2×C17#316311yescharacteristic
374C2×C17#316411yescharacteristic
375D17—24no—
376C23×C4—171no—
377C23×C4—341no—
378C23×C4—341no—
379C2×M16—174no—
380C2×M16—174no—
381C2×M16—174no—
382C22×C8—171no—
383C22×C8—171no—
384C22×C8—174no—
385C22×C8—342no—
386C22×C8—342no—
387C22×C8C22×D1711yescharacteristic
388C22×C8C22×D1711yescharacteristic
389#117—174no—
390#117—342no—
391#117—342no—
392#117—342no—
393#117—342no—
394#117—342no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—171no—
398C2×C4⋊C4—171no—
399C2×C4⋊C4—174no—
400C2×C4⋊C4—171no—
401C2×C4⋊C4—171no—
402C2×C4⋊C4—171no—
403C2×C4⋊C4—171no—
404C2×C4⋊C4—342no—
405C2×C4⋊C4—342no—
406C2×C4⋊C4—342no—
407C2×C4⋊C4—341no—
408C2×C4⋊C4—341no—
409C2×C4⋊C4—341no—
410C2×C4⋊C4—341no—
411C2×C4⋊C4—341no—
412C2×C4⋊C4—341no—
413C2×C4⋊C4—341no—
414C2×C4⋊C4—341no—
415C2×C4⋊C4C22×D1711yescharacteristic
416C2×C4⋊C4C22×D1711yescharacteristic
417C2×C4⋊C4C22×D1711yescharacteristic
418C2×C4⋊C4C22×D1711yescharacteristic
419C2×C22⋊C4—171no—
420C2×C22⋊C4—171no—
421C2×C22⋊C4—341no—
422C2×C22⋊C4—341no—
423C2×C42—171no—
424C2×C42—341no—
425C2×C42—341no—
426C2×C42C22×D1711yescharacteristic
427#107—174no—
428#107—342no—
429#107—342no—
430#107C4×D1714yescomplemented
431#106—22no—
432#106—174no—
433#106—174no—
434#106—342no—
435#97—22no—
436#97—342no—
437C4×C8—342no—
438C4×C8—342no—
439#95—21no—
440#95—21no—
441#95—341no—
442#95—341no—
443#95—341no—
444#95—341no—
445#95—341no—
446#95—341no—
447C17#2116011yescharacteristic, complemented, Sylow
448C24—171no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—171no—
454C22×C4—172no—
455C22×C4—172no—
456C22×C4—172no—
457C22×C4—172no—
458C22×C4—172no—
459C22×C4—172no—
460C22×C4—171no—
461C22×C4—171no—
462C22×C4—344no—
463C22×C4—341no—
464C22×C4—341no—
465C22×C4—341no—
466C22×C4—341no—
467C22×C4—682no—
468C22×C4—1361no—
469C22×C4—1361no—
470C22×C4—1361no—
471C22×C4—1361no—
472C22×C4D4×D1711yescharacteristic
473C22×C4D4×D1711yescharacteristic
474C22×C4C23×D1711yescharacteristic
475M16—1716no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—172no—
479C2×C8—172no—
480C2×C8—172no—
481C2×C8—172no—
482C2×C8—348no—
483C2×C8—341no—
484C2×C8—341no—
485C2×C8C2×C4×D1712yes—
486C2×C8D4×D1712yes—
487C2×C8Q8×D1712yes—
488C2×C8Q8×D1712yes—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—81no—
496C4⋊C4—81no—
497C4⋊C4—172no—
498C4⋊C4—172no—
499C4⋊C4—344no—
500C4⋊C4—342no—
501C4⋊C4—342no—
502C4⋊C4—341no—
503C4⋊C4—341no—
504C4⋊C4—341no—
505C4⋊C4—341no—
506C4⋊C4—341no—
507C4⋊C4—341no—
508C4⋊C4—682no—
509C4⋊C4—1361no—
510C4⋊C4—1361no—
511C4⋊C4—1361no—
512C4⋊C4—1361no—
513C4⋊C4—1361no—
514C4⋊C4—1361no—
515C4⋊C4C2×C4×D1712yes—
516C4⋊C4C2×C4×D1712yes—
517C22⋊C4—342no—
518C22⋊C4—342no—
519C22⋊C4—344no—
520C22⋊C4—682no—
521C42—21no—
522C42—21no—
523C42—344no—
524C42—341no—
525C42—341no—
526C42—682no—
527C23—172no—
528C23—172no—
529C23—172no—
530C23—341no—
531C23—341no—
532C23—341no—
533C23—341no—
534C23C2×D4×D1711yescharacteristic
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—172no—
554C2×C4—172no—
555C2×C4—172no—
556C2×C4—174no—
557C2×C4—174no—
558C2×C4—174no—
559C2×C4—174no—
560C2×C4—342no—
561C2×C4—342no—
562C2×C4—341no—
563C2×C4—341no—
564C2×C4—341no—
565C2×C4—341no—
566C2×C4—341no—
567C2×C4—341no—
568C2×C4—681no—
569C2×C4—681no—
570C2×C4—681no—
571C2×C4—681no—
572C2×C4—681no—
573C2×C4—681no—
574C2×C4—681no—
575C2×C4—681no—
576C2×C4—681no—
577C2×C4—681no—
578C2×C4—1364no—
579C2×C4—1364no—
580C2×C4C22×C4×D1711yescharacteristic
581C2×C4C2×D4×D1711yescharacteristic
582C2×C4C2×Q8×D1711yescharacteristic
583C2×C4C2×Q8×D1711yescharacteristic
584C2×C4#41×D1711yescharacteristic
585C2×C4#41×D1711yescharacteristic
586C8—22no—
587C8—174no—
588C8—342no—
589C8C4⋊C4×D1714yes—
590C22—174no—
591C22—342no—
592C22—342no—
593C22—342no—
594C22—342no—
595C22—342no—
596C22—342no—
597C22C4×D4×D1711yescharacteristic
598C22#121×D1711yescharacteristic
599C22C4⋊Q8×D1711yescharacteristic
600C22#1176361311yescharacteristic
601C22#1176361811yescharacteristic
602C22#1176361911yescharacteristic
603C22#1176362511yescharacteristic
604C4—42no—
605C4—42no—
606C4—41no—
607C4—41no—
608C4—81no—
609C4—81no—
610C4—174no—
611C4—682no—
612C4—682no—
613C4—681no—
614C4—681no—
615C4—1361no—
616C4—1361no—
617C4C2×C4⋊C4×D1711yescharacteristic
618C4C4×Q8×D1711yescharacteristic
619C4C22⋊Q8×D1711yescharacteristic
620C4#125×D1711yescharacteristic
621C2—344no—
622C2D17×#38911yescharacteristic
623C2#1453761311yescharacteristic
624C2#1453762611yescharacteristic
625C2#1453845911yescharacteristic
626C2#1453847011yescharacteristic
627C2#1453931111yescharacteristic
628C2#1453931911yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1943C2
socleC23×C17C22×D4
prime radicalC23×D17C2×D4
cosocleC22×C4×C17C24
hypercenter#1943D17
hypocenterC17#21160
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1943

Complements

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

NH ≅ G/Ntype
6. #14529527C2semidirect
7. #14529526C2semidirect
8. #14529521C2semidirect
9. #14529518C2semidirect
10. #14529514C2semidirect
11. #14529510C2semidirect
12. #14529509C2semidirect
15. C17×#1943C2semidirect
23. #11762850C4semidirect
113. C17×#107C2×C4semidirect
125. #10587488C2×C4semidirect
129. #10587479C2×C4semidirect
131. #10587478C2×C4semidirect
210. #63218C4⋊C4semidirect
284. #1943D17semidirect
430. #107C4×D17semidirect
447. C17#21160semidirect

Maximal subgroups

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

subgroupindex
C2×#117628502
C2×#117628482
C2×#117627662
#145314062
#145313992
#145295272
#145295262
#145295212
#145295182
#145295142
#145295102
#145295092
#145284992
D17×#3402
C17×#19432
#2116017