Finite Group Databasebeta

← #17037832

Subgroups — #17037832

Order 4352.

990 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×#11762848C211yesmaximal, characteristic
2C2×#11762844C211yesmaximal, characteristic
3C2×#11762834C211yesmaximal, characteristic
4#14531405C211yesmaximal, characteristic
5#14531138C211yesmaximal, characteristic
6#14529554C211yesmaximal, characteristic, complemented
7#14529553C211yesmaximal, characteristic, complemented
8#14529552C211yesmaximal, characteristic, complemented
9#14529551C211yesmaximal, characteristic, complemented
10#14529544C211yesmaximal, characteristic, complemented
11#14529541C211yesmaximal, characteristic, complemented
12#14529540C211yesmaximal, characteristic, complemented
13#14528502C211yesmaximal, characteristic
14D17×#340C211yesmaximal, characteristic
15C17×#1948C211yesmaximal, characteristic, complemented
16C22×#63218C2211yescharacteristic
17C22×C8×D17C2211yescharacteristic
18C2×#10587578C2211yescharacteristic
19C2×#10587578C2211yescharacteristic
20C2×#10587558C2211yescharacteristic
21C2×C4⋊C4×D17C2211yescharacteristic
22C2×C4⋊C4×D17C2211yescharacteristic
23#11762848—24no—
24#11762844—24no—
25C2×#10587489C2211yescharacteristic
26C2×#10587478C2211yescharacteristic
27C2×#10587478C2211yescharacteristic
28C2×#10587478C2211yescharacteristic
29C2×#10587488C2211yescharacteristic
30C2×#10587479C2211yescharacteristic
31#11762834C418yescomplemented
32C2×C4⋊#3622C2211yescharacteristic
33C2×#10587485C2211yescharacteristic
34C2×#10587485C2211yescharacteristic
35#11762704—21no—
36#11762704—21no—
37#11762680—21no—
38#11762680—21no—
39#11762678C2211yescharacteristic
40#11762675C2211yescharacteristic
41#11762674C2211yescharacteristic
42C4⋊C4×Dic17C2211yescharacteristic
43C4⋊C4×Dic17C2211yescharacteristic
44C4×#63225—21no—
45C4×C4⋊Dic17C2211yescharacteristic
46#11762660—21no—
47C4×#63227—21no—
48#11762656—21no—
49#11762572C2211yescharacteristic
50#11762572C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762571C2211yescharacteristic
53#11762571C2211yescharacteristic
54#11762571C2211yescharacteristic
55C2×C17×#106C2211yescharacteristic
56C2×C17×#106C2211yescharacteristic
57C2×C17×C4⋊C8C2211yescharacteristic
58C17×#384C2211yescharacteristic
59C4×C4⋊C4×C17C2211yescharacteristic
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×C17—21no—
74C2×C42×C17C2311yescharacteristic
75C2×#63227—21no—
76C2×#63227—21no—
77C2×#63227D411yescharacteristic
78C2×#63227D411yescharacteristic
79C2×C4⋊Dic17C2311yescharacteristic
80C2×C4⋊Dic17C2311yescharacteristic
81C2×C4⋊Dic17C2311yescharacteristic
82C2×C4⋊Dic17C2311yescharacteristic
83C2×#63225—21no—
84C2×#63225—21no—
85C2×#63225—21no—
86C2×#63225—21no—
87C2×#63225D411yescharacteristic
88C2×#63225D411yescharacteristic
89C2×C4×Dic17—21no—
90C2×C4×Dic17—21no—
91C2×C4×Dic17—21no—
92C2×C4×Dic17—21no—
93C2×C4×Dic17—21no—
94C2×C4×Dic17C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96C22×#3622C2311yescharacteristic
97C2×#63218—22no—
98C2×#63218—22no—
99C2×#63218C2×C414yes—
100C2×C8×D17—22no—
101C2×C8×D17—22no—
102C2×C8×D17C2×C414yes—
103#10587578—22no—
104#10587578—22no—
105#10587578—22no—
106#10587578D414yes—
107C4⋊C4×D17—22no—
108C4⋊C4×D17—22no—
109C4⋊C4×D17—22no—
110C4⋊C4×D17Q814yes—
111#10587558C2×C414yes—
112#10587558Q814yes—
113C17×#106—22no—
114C17×#106—22no—
115C17×C4⋊C8C2×C414yescomplemented
116C17×#95—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587489—22no—
125#10587488—22no—
126#10587485C2×C414yescomplemented
127#10587485C2×C414yescomplemented
128#10587479—22no—
129#10587478—22no—
130#10587478—22no—
131#10587478—22no—
132C4⋊#3622C2×C414yescomplemented
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×C17—21no—
163C2×C8×C17—21no—
164C2×C8×C17C22×C412yes—
165C2×C8×C17C22×C412yes—
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×C17C2×D412yes—
175C4⋊C4×C17C2×Q812yes—
176C42×C17—81no—
177C42×C17C22×C412yes—
178C42×C17C2×Q812yes—
179#63227—24no—
180#63227—42no—
181#63227C4⋊C414yes—
182#63227C4⋊C414yes—
183C4⋊Dic17—21no—
184C4⋊Dic17—21no—
185C4⋊Dic17—21no—
186C4⋊Dic17—21no—
187C4⋊Dic17—21no—
188C4⋊Dic17—21no—
189C4⋊Dic17C2×D412yes—
190C4⋊Dic17C2×Q812yes—
191#63225—24no—
192#63225—42no—
193#63225—81no—
194#63225—81no—
195#63225C4⋊C414yes—
196#63225C4⋊C414yes—
197C4×Dic17—24no—
198C4×Dic17—42no—
199C4×Dic17—81no—
200C4×Dic17—81no—
201C4×Dic17—81no—
202C4×Dic17C22×C412yes—
203C4×Dic17C2×Q812yes—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622—21no—
207C2×#3622—21no—
208C2×#3622C22×C412yes—
209C2×#3622C22×C412yes—
210#63218—28no—
211C8×D17—28no—
212#21175—171nomaximal, Sylow
213C23×C17C22×D411yescharacteristic
214C22×D17—21no—
215C22×D17—21no—
216C22×D17—21no—
217C22×D17—21no—
218C22×D17C2×C4⋊C412yes—
219C22×D17#12112yes—
220C22×D17C22⋊Q812yes—
221C2×C4×C17—21no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—22no—
226C2×C4×C17—22no—
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—41no—
235C2×C4×C17—41no—
236C2×C4×C17—41no—
237C2×C4×C17C2×C4⋊C412yes—
238C2×C4×C17C2×C4⋊C412yes—
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—21no—
246C2×Dic17—21no—
247C2×Dic17—21no—
248C2×Dic17—21no—
249C2×Dic17—22no—
250C2×Dic17—22no—
251C2×Dic17—41no—
252C2×Dic17—41no—
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×Dic17C2×C4⋊C412yes—
262C2×Dic17C2×C4⋊C412yes—
263C2×Dic17C2×C4⋊C412yes—
264C2×Dic17#12112yes—
265C2×Dic17C22⋊Q812yes—
266C4×D17—84no—
267C4×D17—84no—
268C4×D17C4×D414yes—
269C4×D17C4×Q814yes—
270C4×D17#12514yes—
271C4×D17C4⋊Q814yes—
272C8×C17—22no—
273C8×C17—22no—
274#3622—22no—
275#3622—22no—
276C2×#428—171no—
277C2×#427—171no—
278C2×#424—171no—
279#2317—171no—
280#2274—171no—
281#1949—171no—
282#1949—171no—
283#1949—171no—
284#1949—171no—
285#1948—171no—
286#1948D1711yescharacteristic, complemented
287#1947—171no—
288#1947—171no—
289#1736—171no—
290C2×#340—171no—
291C22×C17C22×C4⋊C411yescharacteristic
292C22×C17C2×#12111yescharacteristic
293C22×C17C2×C22⋊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—22no—
306C4×C17—22no—
307C4×C17—42no—
308C4×C17—42no—
309C4×C17—81no—
310C4×C17—81no—
311C4×C17C2×C4×D411yescharacteristic
312C4×C17C2×C4×Q811yescharacteristic
313C4×C17C2×#12511yescharacteristic
314C4×C17C2×C4⋊Q811yescharacteristic
315Dic17—22no—
316Dic17—22no—
317Dic17—42no—
318Dic17—42no—
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—344no—
330#427—344no—
331C2×#107—171no—
332C2×#107—171no—
333C2×#107—171no—
334C2×#107—171no—
335C2×#106—171no—
336C2×#106—171no—
337C2×#106C2×D1711yescharacteristic
338C2×#106C2×D1711yescharacteristic
339#424—178no—
340C2×C4⋊C8—171no—
341C2×C4⋊C8—171no—
342C2×C4⋊C8—171no—
343C2×C4⋊C8C2×D1711yescharacteristic
344#389—341no—
345#389—341no—
346#385—341no—
347#385—341no—
348#384—171no—
349#384C2×D1711yescharacteristic
350#382—171no—
351#382—171no—
352C4×C4⋊C4—171no—
353C4×C4⋊C4—171no—
354C4×C4⋊C4—171no—
355C4×C4⋊C4—341no—
356C4×C4⋊C4—341no—
357C4×C4⋊C4C2×D1711yescharacteristic
358C4×C22⋊C4—341no—
359C4×C22⋊C4—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#291211yescharacteristic
370C2×C17#291311yescharacteristic
371C2×C17#304511yescharacteristic
372C2×C17#304611yescharacteristic
373C2×C17#308111yescharacteristic
374C2×C17#308211yescharacteristic
375D17—24no—
376C23×C4—171no—
377C23×C4—341no—
378C23×C4—341no—
379C2×M16—174no—
380C2×M16—342no—
381C2×M16—342no—
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—174no—
391#117—174no—
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—171no—
400C2×C4⋊C4—171no—
401C2×C4⋊C4—174no—
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⋊C4C22×D1711yescharacteristic
412C2×C4⋊C4C22×D1711yescharacteristic
413C2×C4⋊C4C22×D1711yescharacteristic
414C2×C4⋊C4C22×D1711yescharacteristic
415C2×C22⋊C4—171no—
416C2×C22⋊C4—171no—
417C2×C22⋊C4—341no—
418C2×C22⋊C4—341no—
419C2×C42—21no—
420C2×C42—171no—
421C2×C42—341no—
422C2×C42—341no—
423C2×C42—341no—
424C2×C42—341no—
425C2×C42—341no—
426C2×C42C22×D1711yescharacteristic
427#107—342no—
428#107—342no—
429#107—342no—
430#107—342no—
431#106—22no—
432#106—22no—
433#106—342no—
434#106—342no—
435C4⋊C8—174no—
436C4⋊C8—174no—
437C4⋊C8—174no—
438C4⋊C8C4×D1714yescomplemented
439#95—21no—
440#95—341no—
441#95—341no—
442#95—341no—
443#95—341no—
444#95—341no—
445#95—341no—
446#95—341no—
447C17#2117511yescharacteristic, 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—348no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—172no—
481C2×C8—172no—
482C2×C8—348no—
483C2×C8—341no—
484C2×C8—341no—
485C2×C8—341no—
486C2×C8—341no—
487C2×C8C2×C4×D1712yes—
488C2×C8C2×C4×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—174no—
498C4⋊C4—174no—
499C4⋊C4—172no—
500C4⋊C4—172no—
501C4⋊C4—344no—
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⋊C4D4×D1712yes—
512C4⋊C4Q8×D1712yes—
513C22⋊C4—174no—
514C22⋊C4—174no—
515C22⋊C4—344no—
516C22⋊C4—682no—
517C42—81no—
518C42—172no—
519C42—172no—
520C42—344no—
521C42—682no—
522C42—1361no—
523C42—1361no—
524C42—1361no—
525C42C2×C4×D1712yes—
526C42Q8×D1712yes—
527C23—172no—
528C23—172no—
529C23—172no—
530C23—341no—
531C23—341no—
532C23—341no—
533C23—341no—
534C23C2×D4×D1711yescharacteristic
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
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—41no—
551C2×C4—172no—
552C2×C4—172no—
553C2×C4—172no—
554C2×C4—174no—
555C2×C4—174no—
556C2×C4—174no—
557C2×C4—174no—
558C2×C4—172no—
559C2×C4—172no—
560C2×C4—341no—
561C2×C4—341no—
562C2×C4—341no—
563C2×C4—341no—
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—681no—
576C2×C4—1364no—
577C2×C4—1364no—
578C2×C4C4⋊C4×D1712yes—
579C2×C4C4⋊C4×D1712yes—
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—22no—
588C8—342no—
589C8—342no—
590C22—174no—
591C22—342no—
592C22—342no—
593C22—342no—
594C22—342no—
595C22—342no—
596C22—342no—
597C22C2×C4⋊C4×D1711yescharacteristic
598C22#121×D1711yescharacteristic
599C22C22⋊Q8×D1711yescharacteristic
600C22#1176361311yescharacteristic
601C22#1176361511yescharacteristic
602C22#1176361511yescharacteristic
603C22#1176362511yescharacteristic
604C4—22no—
605C4—22no—
606C4—42no—
607C4—42no—
608C4—81no—
609C4—81no—
610C4—174no—
611C4—342no—
612C4—342no—
613C4—682no—
614C4—682no—
615C4—1361no—
616C4—1361no—
617C4C4×D4×D1711yescharacteristic
618C4C4×Q8×D1711yescharacteristic
619C4#125×D1711yescharacteristic
620C4C4⋊Q8×D1711yescharacteristic
621C2—344no—
622C2D17×#38911yescharacteristic
623C2#1453737511yescharacteristic
624C2#1453738311yescharacteristic
625C2#1453832011yescharacteristic
626C2#1453832911yescharacteristic
627C2#1453861011yescharacteristic
628C2#1453861911yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1948C2
socleC23×C17C22×D4
prime radicalC23×D17C2×D4
cosocleC22×C4×C17C24
hypercenter#1948D17
hypocenterC17#21175
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
6. #14529554C2semidirect
7. #14529553C2semidirect
8. #14529552C2semidirect
9. #14529551C2semidirect
10. #14529544C2semidirect
11. #14529541C2semidirect
12. #14529540C2semidirect
15. C17×#1948C2semidirect
31. #11762834C4semidirect
115. C17×C4⋊C8C2×C4semidirect
126. #10587485C2×C4semidirect
127. #10587485C2×C4semidirect
132. C4⋊#3622C2×C4semidirect
286. #1948D17semidirect
438. C4⋊C8C4×D17semidirect
447. C17#21175semidirect

Maximal subgroups

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

subgroupindex
C2×#117628482
C2×#117628442
C2×#117628342
#145314052
#145311382
#145295542
#145295532
#145295522
#145295512
#145295442
#145295412
#145295402
#145285022
D17×#3402
C17×#19482
#2117517