Finite Group Databasebeta

← #17050994

Subgroups — #17050994

Order 4352.

1066 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763520C211yesmaximal, characteristic
2C2×#11762937C211yesmaximal, characteristic, complemented
3#14531736C211yesmaximal, characteristic, complemented
4#14529751C211yesmaximal, characteristic, complemented
5#14529747C211yesmaximal, characteristic, complemented
6#14529740C211yesmaximal, characteristic, complemented
7#14529738C211yesmaximal, characteristic, complemented
8#14529737C211yesmaximal, characteristic, complemented
9#14528793C211yesmaximal, characteristic
10#14528791C211yesmaximal, characteristic
11#14528783C211yesmaximal, characteristic
12#14528610C211yesmaximal, characteristic
13#14528608C211yesmaximal, characteristic
14#14528598C211yesmaximal, characteristic
15C17×#1993C211yesmaximal, characteristic, complemented
16#11763520—24no—
17#11763520C418yescomplemented
18C2×#10587576C2211yescharacteristic
19C2×#10587576C2211yescharacteristic
20C2×#10587583—21no—
21C2×#10587632—21no—
22C2×#10587632—21no—
23C2×C4×Dic34C2211yescharacteristic
24C2×C4×D68C2211yescharacteristic
25C2×#10587579C2211yescharacteristic
26C2×#10587579C2211yescharacteristic
27C2×C4⋊C4×D17—21no—
28#11762937—24no—
29C2×C4⋊#3622C2211yescharacteristic
30C2×#10587485C2211yescharacteristic
31C2×#10587485C2211yescharacteristic
32C2×#10587492C2211yescharacteristic
33C2×#10587492C2211yescharacteristic
34C2×#10587480C2211yescharacteristic
35C2×#10587480C2211yescharacteristic
36C2×#10587486C2211yescharacteristic
37C2×C8×Dic17C2211yescharacteristic
38#11762713—21no—
39#11762694—21no—
40#11762691—21no—
41#11762678C2211yescharacteristic
42#11762674C2211yescharacteristic
43C4⋊C4×Dic17C2211yescharacteristic
44C4⋊C4×Dic17C2211yescharacteristic
45C4×C4⋊Dic17C2211yescharacteristic
46#11762660—21no—
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762571C2211yescharacteristic
53#11762571C2211yescharacteristic
54#11762571C2211yescharacteristic
55C2×C17×C4⋊Q8C2211yescharacteristic
56C2×C17×#97C2211yescharacteristic
57C17×#382C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#340C2211yescharacteristic
60C17×#340C2211yescharacteristic
61C17×#340C2211yescharacteristic
62C22×Q8×C17—21no—
63C22×D68C2311yescharacteristic
64C22×C4×D17—21no—
65C22×C4×D17—21no—
66C22×C4×D17—21no—
67C22×Dic34D411yescharacteristic
68C22×C8×C17C2311yescharacteristic
69C22×C8×C17C2311yescharacteristic
70C17×C4⋊Q8—22no—
71C17×C4⋊Q8C2×C414yes—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C42×C17C2311yescharacteristic
78#10587632—28no—
79#10587632—28no—
80C2×#63227—21no—
81C2×#63227—21no—
82C2×#63227—21no—
83C2×C4⋊Dic17—21no—
84C2×C4⋊Dic17—21no—
85C2×C4⋊Dic17D411yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×#63225—21no—
90C2×#63225—21no—
91C2×#63225—21no—
92C2×#63225—21no—
93C2×#63225D411yescharacteristic
94C2×#63225D411yescharacteristic
95C2×C4×Dic17—21no—
96C2×C4×Dic17—21no—
97C2×C4×Dic17—21no—
98C2×C4×Dic17C2311yescharacteristic
99C2×C4×Dic17C2311yescharacteristic
100C22×#3622C2311yescharacteristic
101C22×#3622C2311yescharacteristic
102#10587583—28no—
103#10587579—22no—
104#10587579—22no—
105#10587579C2×C414yes—
106#10587579C2×C414yes—
107C4⋊C4×D17—28no—
108#10587576—22no—
109#10587576—22no—
110#10587576C2×C414yes—
111#10587576C2×C414yes—
112C4×D68—22no—
113C4×D68C2×C414yes—
114C4×Dic34C2×C414yes—
115C4×Dic34D414yes—
116C17×#97—22no—
117C17×#95—21no—
118C17×#95—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587492—22no—
125#10587492Q814yescomplemented
126#10587486—22no—
127#10587485—22no—
128#10587485—22no—
129C8×Dic17—22no—
130#10587480—22no—
131#10587480—22no—
132C4⋊#3622—22no—
133C23×D17—21no—
134C2×Q8×C17—24no—
135C2×Q8×C17—81no—
136C2×Q8×C17—81no—
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2411yescharacteristic
144C22×Dic17—21no—
145C22×Dic17—21no—
146C22×Dic17—21no—
147C22×Dic17C2×D411yescharacteristic
148C22×Dic17C2×D411yescharacteristic
149C22×Dic17C2×D411yescharacteristic
150C22×Dic17C2×D411yescharacteristic
151C2×D68—21no—
152C2×D68—21no—
153C2×D68—21no—
154C2×D68C22×C412yes—
155C2×D68C2×Q812yes—
156C2×D68#4112yes—
157C2×C4×D17—24no—
158C2×C4×D17—24no—
159C2×C4×D17—24no—
160C2×C4×D17—42no—
161C2×C4×D17—42no—
162C2×C4×D17—81no—
163C2×C4×D17—81no—
164C2×C4×D17—81no—
165C2×C4×D17—81no—
166C2×Dic34—41no—
167C2×Dic34—41no—
168C2×Dic34C22⋊C414yes—
169C2×C8×C17—21no—
170C2×C8×C17—21no—
171C2×C8×C17—21no—
172C2×C8×C17—21no—
173C2×C8×C17—21no—
174C2×C8×C17C2×Q812yes—
175C4⋊C4×C17—24no—
176C4⋊C4×C17—21no—
177C4⋊C4×C17—21no—
178C4⋊C4×C17—21no—
179C4⋊C4×C17—21no—
180C4⋊C4×C17—21no—
181C4⋊C4×C17—21no—
182C4⋊C4×C17C22×C412yes—
183C4⋊C4×C17C22×C412yes—
184C42×C17C22×C412yes—
185C42×C17C2×D412yes—
186#63227—24no—
187#63227—24no—
188#63227—24no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17—24no—
192C4⋊Dic17—24no—
193C4⋊Dic17—21no—
194C4⋊Dic17—21no—
195C4⋊Dic17C22⋊C414yes—
196C4⋊Dic17C22×C412yes—
197C4⋊Dic17C2×Q812yes—
198#63225—24no—
199#63225—24no—
200#63225—42no—
201#63225—81no—
202#63225C22⋊C414yes—
203#63225C22⋊C414yes—
204C4×Dic17—42no—
205C4×Dic17—81no—
206C4×Dic17—81no—
207C4×Dic17C22×C412yes—
208C4×Dic17C22×C412yes—
209C4×Dic17C2×D412yes—
210C4×Dic17C2×D412yes—
211C2×#3622—21no—
212C2×#3622—21no—
213C2×#3622—21no—
214C2×#3622—21no—
215C2×#3622—21no—
216C2×#3622—21no—
217#22526—171nomaximal, Sylow
218C23×C17C22×D411yescharacteristic
219C22×D17—22no—
220C22×D17—22no—
221C22×D17—22no—
222C22×D17—41no—
223C22×D17—41no—
224C22×D17—41no—
225C22×D17—41no—
226Q8×C17—84no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—22no—
232C2×C4×C17—22no—
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×C17—41no—
240C2×C4×C17—41no—
241C2×C4×C17—41no—
242C2×C4×C17—41no—
243C2×C4×C17C2×C22⋊C412yes—
244C2×C4×C17C2×C22⋊C412yes—
245C2×C4×C17C23×C411yescharacteristic
246C2×C4×C17C22×D411yescharacteristic
247C2×C4×C17C22×Q811yescharacteristic
248C2×C4×C17C2×#4111yescharacteristic
249C2×C4×C172−1+411yescharacteristic
250C2×C4×C172−1+411yescharacteristic
251C2×Dic17—21no—
252C2×Dic17—21no—
253C2×Dic17—21no—
254C2×Dic17—21no—
255C2×Dic17—21no—
256C2×Dic17—21no—
257C2×Dic17—21no—
258C2×Dic17—21no—
259C2×Dic17—22no—
260C2×Dic17—22no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263C2×Dic17—41no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266C2×Dic17—41no—
267C2×Dic17—41no—
268C2×Dic17C2×C22⋊C412yes—
269C2×Dic17C2×C22⋊C412yes—
270C2×Dic17C2×C22⋊C412yes—
271C2×Dic17C2×C22⋊C412yes—
272D68—22no—
273D68—22no—
274D68—22no—
275D68C4×Q814yes—
276C4×D17—48no—
277C4×D17—84no—
278C4×D17—84no—
279Dic34—44no—
280C8×C17—22no—
281C8×C17—41no—
282#3622—41no—
283#3622—41no—
284C2×#554—171no—
285C2×#446—171no—
286#2355—171no—
287#1993D1711yescharacteristic, complemented
288#1992—171no—
289#1991—171no—
290#1989—171no—
291#1989—171no—
292#1988—171no—
293#1804—171no—
294#1804—171no—
295#1802—171no—
296#1762—171no—
297#1762—171no—
298C4×#102—171no—
299C22×C17C22×C22⋊C411yescharacteristic
300C22×C17#54811yescharacteristic
301C22×C17D4×Q811yescharacteristic
302C22×C17#57511yescharacteristic
303C22×C17#57511yescharacteristic
304C22×C17#57611yescharacteristic
305C22×C17#57711yescharacteristic
306C2×D17—24no—
307C2×D17—42no—
308C2×D17—42no—
309C2×D17—42no—
310C2×D17—42no—
311C2×D17—42no—
312C2×D17—42no—
313C4×C17—22no—
314C4×C17—22no—
315C4×C17—42no—
316C4×C17—42no—
317C4×C17—81no—
318C4×C17—81no—
319C4×C17C2×C4×Q811yescharacteristic
320C4×C17#51911yescharacteristic
321C4×C17#54811yescharacteristic
322C4×C17D4×Q811yescharacteristic
323Dic17—22no—
324Dic17—22no—
325Dic17—22no—
326Dic17—22no—
327Dic17—42no—
328Dic17—42no—
329Dic17—81no—
330#554—178no—
331#554—344no—
332C2×C4⋊Q8C2×D1711yescharacteristic
333C2×#125—171no—
334C2×#125—171no—
335C2×C22⋊Q8—341no—
336C2×C22⋊Q8—341no—
337C2×C22⋊Q8—341no—
338C2×C4×Q8—171no—
339C2×C4×D4—171no—
340C2×C4×D4—171no—
341C2×C4×D4—171no—
342C22×C4⋊C4—341no—
343#446—344no—
344C2×C4⋊C8—171no—
345C2×C4⋊C8—171no—
346C2×C4⋊C8—171no—
347C2×#102—171no—
348C2×#102—171no—
349C2×#102—171no—
350C2×#102—171no—
351C2×#97—171no—
352C2×#97C2×D1711yescharacteristic
353C2×C4×C8—171no—
354#391—341no—
355#387—341no—
356#387—341no—
357#384—171no—
358#382—171no—
359#382C2×D1711yescharacteristic
360C4×C4⋊C4—171no—
361C4×C4⋊C4—171no—
362C4×C4⋊C4—171no—
363C4×C4⋊C4—341no—
364#340—171no—
365#340—171no—
366#340—171no—
367#340—171no—
368#340—171no—
369#340—171no—
370#340—171no—
371#340—171no—
372#340C2×D1711yescharacteristic
373#340C2×D1711yescharacteristic
374#340C2×D1711yescharacteristic
375#340C2×D1711yescharacteristic
376C2×C17Q8×C22⋊C411yescharacteristic
377C2×C17#289611yescharacteristic
378C2×C17#289711yescharacteristic
379C2×C17#337011yescharacteristic
380C2×C17#337611yescharacteristic
381C2×C17#338811yescharacteristic
382C2×C17#338911yescharacteristic
383D17—44no—
384C22×Q8—21no—
385C22×Q8—171no—
386C22×D4—171no—
387C23×C4—341no—
388C23×C4—341no—
389C23×C4—341no—
390C22×C8—171no—
391C22×C8—171no—
392C22×C8C22×D1711yescharacteristic
393C22×C8C22×D1711yescharacteristic
394C4⋊Q8—22no—
395C4⋊Q8C4×D1714yes—
396#125—174no—
397#125—174no—
398#125—342no—
399#125—342no—
400C22⋊Q8—348no—
401C22⋊Q8—348no—
402C22⋊Q8—348no—
403C4×Q8—174no—
404C4×Q8—174no—
405C4×D4—174no—
406C4×D4—174no—
407C4×D4—174no—
408C4×D4—342no—
409C4×D4—342no—
410C4×D4—342no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—171no—
413C2×C4⋊C4—171no—
414C2×C4⋊C4—171no—
415C2×C4⋊C4—171no—
416C2×C4⋊C4—171no—
417C2×C4⋊C4—171no—
418C2×C4⋊C4—348no—
419C2×C4⋊C4—341no—
420C2×C4⋊C4—341no—
421C2×C4⋊C4—341no—
422C2×C4⋊C4—341no—
423C2×C4⋊C4—341no—
424C2×C4⋊C4—341no—
425C2×C4⋊C4C22×D1711yescharacteristic
426C2×C4⋊C4C22×D1711yescharacteristic
427C2×C4⋊C4C22×D1711yescharacteristic
428C2×C4⋊C4C22×D1711yescharacteristic
429C2×C22⋊C4—341no—
430C2×C22⋊C4—341no—
431C2×C22⋊C4—341no—
432C2×C42—171no—
433C2×C42—171no—
434C2×C42—341no—
435C2×C42—341no—
436C2×C42—341no—
437C2×C42C22×D1711yescharacteristic
438C4⋊C8—342no—
439C4⋊C8—342no—
440C4⋊C8—342no—
441#102—174no—
442#102—342no—
443#102—342no—
444#102—342no—
445#97—22no—
446#97—342no—
447C4×C8—342no—
448#95—21no—
449#95—21no—
450#95—341no—
451#95—341no—
452#95—341no—
453#95—341no—
454#95—341no—
455C17#2252611yescharacteristic, complemented, Sylow
456C24—341no—
457C2×Q8—24no—
458C2×Q8—81no—
459C2×Q8—81no—
460C2×Q8—174no—
461C2×Q8—681no—
462C2×Q8—681no—
463C2×D4—172no—
464C2×D4—172no—
465C2×D4—172no—
466C2×D4—341no—
467C2×D4—341no—
468C2×D4—341no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—171no—
474C22×C4—171no—
475C22×C4—171no—
476C22×C4—171no—
477C22×C4—344no—
478C22×C4—344no—
479C22×C4—341no—
480C22×C4—341no—
481C22×C4—344no—
482C22×C4—341no—
483C22×C4—682no—
484C22×C4—682no—
485C22×C4—1361no—
486C22×C4—1361no—
487C22×C4—1361no—
488C22×C4—1361no—
489C22×C4D4×D1711yescharacteristic
490C22×C4D4×D1711yescharacteristic
491C22×C4C23×D1711yescharacteristic
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—341no—
498C2×C8—341no—
499C2×C8—341no—
500C2×C8—341no—
501C2×C8—341no—
502C2×C8—341no—
503C2×C8Q8×D1712yes—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—24no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—174no—
512C4⋊C4—174no—
513C4⋊C4—174no—
514C4⋊C4—172no—
515C4⋊C4—172no—
516C4⋊C4—344no—
517C4⋊C4—344no—
518C4⋊C4—344no—
519C4⋊C4—344no—
520C4⋊C4—341no—
521C4⋊C4—341no—
522C4⋊C4—341no—
523C4⋊C4—341no—
524C4⋊C4—682no—
525C4⋊C4—1361no—
526C4⋊C4C2×C4×D1712yes—
527C4⋊C4C2×C4×D1712yes—
528C22⋊C4—344no—
529C22⋊C4—344no—
530C22⋊C4—344no—
531C42—172no—
532C42—172no—
533C42—172no—
534C42—172no—
535C42—682no—
536C42—1361no—
537C42—1361no—
538C42C2×C4×D1712yes—
539C42D4×D1712yes—
540C23—342no—
541C23—342no—
542C23—342no—
543C23—681no—
544C23—681no—
545C23—681no—
546C23—681no—
547C23C2×D4×D1711yescharacteristic
548Q8—84no—
549Q8—684no—
550D4—174no—
551D4—342no—
552D4—342no—
553D4—342no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—172no—
571C2×C4—172no—
572C2×C4—172no—
573C2×C4—172no—
574C2×C4—341no—
575C2×C4—341no—
576C2×C4—341no—
577C2×C4—341no—
578C2×C4—341no—
579C2×C4—341no—
580C2×C4—341no—
581C2×C4—341no—
582C2×C4—342no—
583C2×C4—342no—
584C2×C4—688no—
585C2×C4—681no—
586C2×C4—681no—
587C2×C4—681no—
588C2×C4—681no—
589C2×C4—681no—
590C2×C4—681no—
591C2×C4—681no—
592C2×C4—1364no—
593C2×C4—1364no—
594C2×C4C22⋊C4×D1712yes—
595C2×C4C22⋊C4×D1712yes—
596C2×C4C22×C4×D1711yescharacteristic
597C2×C4C2×D4×D1711yescharacteristic
598C2×C4C2×Q8×D1711yescharacteristic
599C2×C4#1058769411yescharacteristic
600C2×C4#41×D1711yescharacteristic
601C2×C4#1058769711yescharacteristic
602C8—22no—
603C8—41no—
604C8—681no—
605C8—681no—
606C22—344no—
607C22—682no—
608C22—682no—
609C22—682no—
610C22—682no—
611C22—682no—
612C22—682no—
613C22C2×C22⋊C4×D1711yescharacteristic
614C22#1176346711yescharacteristic
615C22#1176347411yescharacteristic
616C22#1176360811yescharacteristic
617C22#1176360811yescharacteristic
618C22#1176361311yescharacteristic
619C22#1176361911yescharacteristic
620C4—22no—
621C4—22no—
622C4—42no—
623C4—42no—
624C4—81no—
625C4—81no—
626C4—342no—
627C4—342no—
628C4—342no—
629C4—342no—
630C4—682no—
631C4—682no—
632C4—1361no—
633C4C4×Q8×D1711yescharacteristic
634C4#1176328511yescharacteristic
635C4#1176346911yescharacteristic
636C4#1176347211yescharacteristic
637C2—684no—
638C2#1453161811yescharacteristic
639C2#1453730211yescharacteristic
640C2#1453730911yescharacteristic
641C2#1454153611yescharacteristic
642C2#1454161011yescharacteristic
643C2#1454172811yescharacteristic
644C2#1454173911yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1993C2
socleC23×C17C22×D4
prime radicalC22×D68C23
cosocleC22×C4×C17C24
hypercenter#1993D17
hypocenterC17#22526
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
2. C2×#11762937C2semidirect
3. #14531736C2semidirect
4. #14529751C2semidirect
5. #14529747C2semidirect
6. #14529740C2semidirect
7. #14529738C2semidirect
8. #14529737C2semidirect
15. C17×#1993C2semidirect
17. #11763520C4semidirect
125. #10587492Q8semidirect
287. #1993D17semidirect
455. C17#22526semidirect

Maximal subgroups

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

subgroupindex
C2×#117635202
C2×#117629372
#145317362
#145297512
#145297472
#145297402
#145297382
#145297372
#145287932
#145287912
#145287832
#145286102
#145286082
#145285982
C17×#19932
#2252617