Finite Group Databasebeta

← #17050952

Subgroups — #17050952

Order 4352.

1110 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763515C211yesmaximal, characteristic
2C2×#11762938C211yesmaximal, characteristic, complemented
3C4⋊C4×Dic34C211yesmaximal, characteristic, complemented
4#14529754C211yesmaximal, characteristic, complemented
5#14529751C211yesmaximal, characteristic, complemented
6#14529743C211yesmaximal, characteristic, complemented
7#14529738C211yesmaximal, characteristic, complemented
8#14529730C211yesmaximal, characteristic, complemented
9#14528785C211yesmaximal, characteristic
10#14528783C211yesmaximal, characteristic
11#14528783C211yesmaximal, characteristic
12#14528612C211yesmaximal, characteristic
13#14528608C211yesmaximal, characteristic
14#14528598C211yesmaximal, characteristic
15C17×#1989C211yesmaximal, characteristic, complemented
16#11763515—24no—
17#11763515C418yescomplemented
18C2×#10587574C2211yescharacteristic
19C2×#10587576C2211yescharacteristic
20C2×#10587583—21no—
21C2×#10587582—21no—
22C2×#10587582—21no—
23C2×C4×Dic34—21no—
24C2×C4×Dic34C2211yescharacteristic
25C2×C4×D68C2211yescharacteristic
26C2×#10587579C2211yescharacteristic
27C2×#10587579C2211yescharacteristic
28C2×C4⋊C4×D17—21no—
29#11762938—24no—
30C2×#10587485C2211yescharacteristic
31C2×#10587485C2211yescharacteristic
32C2×#10587492C2211yescharacteristic
33C2×#10587492C2211yescharacteristic
34C2×#10587480C2211yescharacteristic
35C2×#10587480C2211yescharacteristic
36C2×#10587474C2211yescharacteristic
37C2×#10587486C2211yescharacteristic
38C2×C8×Dic17C2211yescharacteristic
39#11762707—21no—
40#11762706—21no—
41#11762706—21no—
42#11762679C2211yescharacteristic
43#11762678C2211yescharacteristic
44#11762678C2211yescharacteristic
45C4⋊C4×Dic17C2211yescharacteristic
46C4⋊C4×Dic17C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762571C2211yescharacteristic
53#11762571C2211yescharacteristic
54#11762571C2211yescharacteristic
55C2×C17×#125C2211yescharacteristic
56C2×C17×C4⋊C8C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#340C2211yescharacteristic
60C17×#340C2211yescharacteristic
61C17×#340C2211yescharacteristic
62C22×D68C2311yescharacteristic
63C22×C4×D17—21no—
64C22×C4×D17—21no—
65C22×C4×D17—21no—
66C22×Dic34—21no—
67C22×Dic34D411yescharacteristic
68C22×C8×C17C2311yescharacteristic
69C22×C8×C17C2311yescharacteristic
70C17×#125—22no—
71C17×#125C2×C414yes—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17—21no—
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C4⋊C4×C17C2311yescharacteristic
78C2×C42×C17—21no—
79C2×C42×C17C2311yescharacteristic
80C2×#63227—21no—
81C2×#63227—21no—
82C2×#63227—21no—
83C2×C4⋊Dic17—21no—
84C2×C4⋊Dic17—21no—
85C2×C4⋊Dic17—21no—
86C2×C4⋊Dic17D411yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4⋊Dic17C2311yescharacteristic
90C2×#63225—21no—
91C2×#63225—21no—
92C2×#63225—21no—
93C2×#63225—21no—
94C2×#63225—21no—
95C2×#63225—21no—
96C2×#63225D411yescharacteristic
97C2×#63225D411yescharacteristic
98C2×C4×Dic17—21no—
99C2×C4×Dic17—21no—
100C2×C4×Dic17C2311yescharacteristic
101C2×C4×Dic17C2311yescharacteristic
102C22×#3622C2311yescharacteristic
103C22×#3622C2311yescharacteristic
104#10587583—28no—
105#10587582—28no—
106#10587582—28no—
107#10587579—22no—
108#10587579—22no—
109#10587579C2×C414yes—
110#10587579C2×C414yes—
111C4⋊C4×D17—28no—
112#10587576—22no—
113#10587576C2×C414yes—
114#10587574—22no—
115#10587574C2×C414yes—
116C4×D68—22no—
117C4×D68C2×C414yes—
118C4×Dic34—44no—
119C4×Dic34C2×C414yes—
120C4×Dic34D414yes—
121C17×C4⋊C8—22no—
122C17×#95—21no—
123#10587502—21no—
124#10587502—21no—
125#10587492—22no—
126#10587492—22no—
127#10587486—22no—
128#10587485—22no—
129#10587485—22no—
130C8×Dic17—22no—
131#10587480—22no—
132#10587480Q814yescomplemented
133#10587474—22no—
134C23×D17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—21no—
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17C2411yescharacteristic
142C22×Dic17—21no—
143C22×Dic17—21no—
144C22×Dic17—21no—
145C22×Dic17C2×D411yescharacteristic
146C22×Dic17C2×D411yescharacteristic
147C22×Dic17C2×D411yescharacteristic
148C22×Dic17C2×D411yescharacteristic
149C2×D68—21no—
150C2×D68—21no—
151C2×D68—21no—
152C2×D68C22×C412yes—
153C2×D68C2×Q812yes—
154C2×D68#4112yes—
155C2×C4×D17—24no—
156C2×C4×D17—24no—
157C2×C4×D17—24no—
158C2×C4×D17—42no—
159C2×C4×D17—42no—
160C2×C4×D17—81no—
161C2×C4×D17—81no—
162C2×C4×D17—81no—
163C2×C4×D17—81no—
164C2×Dic34—22no—
165C2×Dic34—22no—
166C2×Dic34—24no—
167C2×Dic34—81no—
168C2×Dic34—81no—
169C2×Dic34C22⋊C414yes—
170C2×C8×C17—21no—
171C2×C8×C17—21no—
172C2×C8×C17—21no—
173C2×C8×C17—21no—
174C2×C8×C17—21no—
175C2×C8×C17—21no—
176C4⋊C4×C17—24no—
177C4⋊C4×C17—24no—
178C4⋊C4×C17—21no—
179C4⋊C4×C17—21no—
180C4⋊C4×C17—21no—
181C4⋊C4×C17—21no—
182C4⋊C4×C17—21no—
183C4⋊C4×C17C22×C412yes—
184C4⋊C4×C17C22×C412yes—
185C4⋊C4×C17C2×Q812yes—
186C42×C17—42no—
187C42×C17C22×C412yes—
188C42×C17C2×D412yes—
189#63227—24no—
190#63227—24no—
191#63227—24no—
192C4⋊Dic17—21no—
193C4⋊Dic17—24no—
194C4⋊Dic17—21no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—42no—
199C4⋊Dic17—81no—
200C4⋊Dic17C22⋊C414yes—
201C4⋊Dic17C22×C412yes—
202#63225—24no—
203#63225—24no—
204#63225—42no—
205#63225—42no—
206#63225—81no—
207#63225—81no—
208#63225C22⋊C414yes—
209#63225C22⋊C414yes—
210C4×Dic17—42no—
211C4×Dic17—42no—
212C4×Dic17C22×C412yes—
213C4×Dic17C22×C412yes—
214C4×Dic17C2×D412yes—
215C4×Dic17C2×D412yes—
216C2×#3622—21no—
217C2×#3622—21no—
218C2×#3622—21no—
219C2×#3622—21no—
220C2×#3622—21no—
221C2×#3622C2×Q812yes—
222#22522—171nomaximal, Sylow
223C23×C17C22×D411yescharacteristic
224C22×D17—22no—
225C22×D17—22no—
226C22×D17—22no—
227C22×D17—41no—
228C22×D17—41no—
229C22×D17—41no—
230C22×D17—41no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—22no—
236C2×C4×C17—22no—
237C2×C4×C17—22no—
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—41no—
247C2×C4×C17C2×C22⋊C412yes—
248C2×C4×C17C2×C22⋊C412yes—
249C2×C4×C17C23×C411yescharacteristic
250C2×C4×C17C22×D411yescharacteristic
251C2×C4×C17C22×Q811yescharacteristic
252C2×C4×C17C2×#4111yescharacteristic
253C2×C4×C172−1+411yescharacteristic
254C2×C4×C172−1+411yescharacteristic
255C2×Dic17—21no—
256C2×Dic17—21no—
257C2×Dic17—21no—
258C2×Dic17—21no—
259C2×Dic17—21no—
260C2×Dic17—21no—
261C2×Dic17—22no—
262C2×Dic17—41no—
263C2×Dic17—41no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266C2×Dic17—41no—
267C2×Dic17—41no—
268C2×Dic17—41no—
269C2×Dic17—41no—
270C2×Dic17C2×C22⋊C412yes—
271C2×Dic17C2×C22⋊C412yes—
272C2×Dic17C2×C22⋊C412yes—
273C2×Dic17C2×C22⋊C412yes—
274C2×Dic17AGL(1,ℤ/8)12yes—
275C2×Dic17#13712yes—
276D68—22no—
277D68—22no—
278D68—22no—
279D68C4×Q814yescomplemented
280C4×D17—48no—
281C4×D17—84no—
282C4×D17—84no—
283Dic34—28no—
284Dic34—84no—
285C8×C17—41no—
286C8×C17—41no—
287#3622—22no—
288#3622—41no—
289C2×#554—171no—
290C2×#446—171no—
291Q8×C4⋊C4—171no—
292#1992—171no—
293#1992—171no—
294#1990—171no—
295#1989—171no—
296#1989D1711yescharacteristic, complemented
297#1987—171no—
298#1802—171no—
299#1802—171no—
300#1802—171no—
301#1762—171no—
302#1762—171no—
303C4×#102—171no—
304C22×C17C22×C22⋊C411yescharacteristic
305C22×C17#54811yescharacteristic
306C22×C17D4×Q811yescharacteristic
307C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
308C22×C17C2×#13711yescharacteristic
309C22×C17#57611yescharacteristic
310C22×C17#57711yescharacteristic
311C2×D17—24no—
312C2×D17—42no—
313C2×D17—42no—
314C2×D17—42no—
315C2×D17—42no—
316C2×D17—42no—
317C2×D17—42no—
318C4×C17—22no—
319C4×C17—22no—
320C4×C17—42no—
321C4×C17—42no—
322C4×C17—42no—
323C4×C17—81no—
324C4×C17C2×C4×Q811yescharacteristic
325C4×C17#51911yescharacteristic
326C4×C17#54811yescharacteristic
327C4×C17D4×Q811yescharacteristic
328Dic17—22no—
329Dic17—22no—
330Dic17—22no—
331Dic17—42no—
332Dic17—81no—
333Dic17—81no—
334Dic17#41714yes—
335#554—178no—
336#554—344no—
337C2×C4⋊Q8—171no—
338C2×#125—171no—
339C2×#125C2×D1711yescharacteristic
340C2×C22⋊Q8—341no—
341C2×C22⋊Q8—341no—
342C2×C22⋊Q8—341no—
343C2×C4×Q8—171no—
344C2×C4×Q8—341no—
345C2×C4×D4—171no—
346C2×C4×D4—171no—
347C2×C4×D4—171no—
348C22×C4⋊C4—341no—
349#446—344no—
350C2×C4⋊C8—171no—
351C2×C4⋊C8—171no—
352C2×C4⋊C8C2×D1711yescharacteristic
353C2×#102—171no—
354C2×#102—171no—
355C2×#102—171no—
356C2×#102—171no—
357C2×#97—171no—
358C2×#97—171no—
359C2×C4×C8—171no—
360#389—341no—
361#389—341no—
362#389—341no—
363#384—171no—
364#384—171no—
365#384—171no—
366C4×C4⋊C4—171no—
367C4×C4⋊C4—171no—
368C4×C4⋊C4C2×D1711yescharacteristic
369#340—171no—
370#340—171no—
371#340—171no—
372#340—171no—
373#340—171no—
374#340—171no—
375#340—171no—
376#340—171no—
377#340C2×D1711yescharacteristic
378#340C2×D1711yescharacteristic
379#340C2×D1711yescharacteristic
380#340C2×D1711yescharacteristic
381C2×C17Q8×C22⋊C411yescharacteristic
382C2×C17C2×#41711yescharacteristic
383C2×C17#289711yescharacteristic
384C2×C17#336611yescharacteristic
385C2×C17#337211yescharacteristic
386C2×C17#338411yescharacteristic
387C2×C17#338511yescharacteristic
388D17—44no—
389C22×Q8—171no—
390C22×Q8—341no—
391C22×D4—171no—
392C23×C4—341no—
393C23×C4—341no—
394C23×C4—341no—
395C22×C8—171no—
396C22×C8—171no—
397C22×C8C22×D1711yescharacteristic
398C22×C8C22×D1711yescharacteristic
399C4⋊Q8—174no—
400C4⋊Q8—342no—
401#125—22no—
402#125—174no—
403#125—342no—
404#125C4×D1714yes—
405C22⋊Q8—348no—
406C22⋊Q8—348no—
407C22⋊Q8—348no—
408C4×Q8—174no—
409C4×Q8—174no—
410C4×Q8—684no—
411C4×D4—174no—
412C4×D4—174no—
413C4×D4—174no—
414C4×D4—342no—
415C4×D4—342no—
416C4×D4—342no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—171no—
420C2×C4⋊C4—171no—
421C2×C4⋊C4—171no—
422C2×C4⋊C4—171no—
423C2×C4⋊C4—171no—
424C2×C4⋊C4—171no—
425C2×C4⋊C4—348no—
426C2×C4⋊C4—341no—
427C2×C4⋊C4—341no—
428C2×C4⋊C4—341no—
429C2×C4⋊C4—341no—
430C2×C4⋊C4—341no—
431C2×C4⋊C4—341no—
432C2×C4⋊C4—341no—
433C2×C4⋊C4—341no—
434C2×C4⋊C4—341no—
435C2×C4⋊C4C22×D1711yescharacteristic
436C2×C4⋊C4C22×D1711yescharacteristic
437C2×C4⋊C4C22×D1711yescharacteristic
438C2×C4⋊C4C22×D1711yescharacteristic
439C2×C22⋊C4—341no—
440C2×C22⋊C4—341no—
441C2×C22⋊C4—341no—
442C2×C42—21no—
443C2×C42—171no—
444C2×C42—171no—
445C2×C42—341no—
446C2×C42—341no—
447C2×C42C22×D1711yescharacteristic
448C4⋊C8—22no—
449C4⋊C8—342no—
450C4⋊C8—342no—
451#102—174no—
452#102—342no—
453#102—342no—
454#102—342no—
455#97—342no—
456#97—342no—
457C4×C8—342no—
458#95—21no—
459#95—341no—
460#95—341no—
461C17#2252211yescharacteristic, complemented, Sylow
462C24—341no—
463C2×Q8—174no—
464C2×Q8—342no—
465C2×Q8—342no—
466C2×Q8—344no—
467C2×Q8—1361no—
468C2×Q8—1361no—
469C2×D4—172no—
470C2×D4—172no—
471C2×D4—172no—
472C2×D4—341no—
473C2×D4—341no—
474C2×D4—341no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—171no—
480C22×C4—171no—
481C22×C4—171no—
482C22×C4—171no—
483C22×C4—344no—
484C22×C4—344no—
485C22×C4—341no—
486C22×C4—344no—
487C22×C4—341no—
488C22×C4—341no—
489C22×C4—682no—
490C22×C4—682no—
491C22×C4—1361no—
492C22×C4—1361no—
493C22×C4—1361no—
494C22×C4—1361no—
495C22×C4D4×D1711yescharacteristic
496C22×C4D4×D1711yescharacteristic
497C22×C4C23×D1711yescharacteristic
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—172no—
505C2×C8—341no—
506C2×C8—341no—
507C2×C8—341no—
508C2×C8—341no—
509C2×C8—341no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—174no—
518C4⋊C4—174no—
519C4⋊C4—174no—
520C4⋊C4—172no—
521C4⋊C4—344no—
522C4⋊C4—344no—
523C4⋊C4—344no—
524C4⋊C4—341no—
525C4⋊C4—341no—
526C4⋊C4—341no—
527C4⋊C4—341no—
528C4⋊C4—341no—
529C4⋊C4—682no—
530C4⋊C4—682no—
531C4⋊C4—682no—
532C4⋊C4—1361no—
533C4⋊C4—1361no—
534C4⋊C4—1361no—
535C4⋊C4C2×C4×D1712yes—
536C4⋊C4C2×C4×D1712yes—
537C4⋊C4Q8×D1712yes—
538C22⋊C4—344no—
539C22⋊C4—344no—
540C22⋊C4—344no—
541C42—42no—
542C42—172no—
543C42—172no—
544C42—172no—
545C42—172no—
546C42—682no—
547C42—682no—
548C42C2×C4×D1712yes—
549C42D4×D1712yes—
550C23—342no—
551C23—342no—
552C23—342no—
553C23—681no—
554C23—681no—
555C23—681no—
556C23—681no—
557C23C2×D4×D1711yescharacteristic
558Q8—348no—
559Q8—1364no—
560D4—174no—
561D4—342no—
562D4—342no—
563D4—342no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—172no—
581C2×C4—172no—
582C2×C4—172no—
583C2×C4—172no—
584C2×C4—172no—
585C2×C4—172no—
586C2×C4—341no—
587C2×C4—341no—
588C2×C4—341no—
589C2×C4—341no—
590C2×C4—341no—
591C2×C4—341no—
592C2×C4—342no—
593C2×C4—688no—
594C2×C4—681no—
595C2×C4—681no—
596C2×C4—681no—
597C2×C4—681no—
598C2×C4—681no—
599C2×C4—681no—
600C2×C4—681no—
601C2×C4—681no—
602C2×C4—1364no—
603C2×C4—1364no—
604C2×C4C22⋊C4×D1712yes—
605C2×C4C22⋊C4×D1712yes—
606C2×C4C22×C4×D1711yescharacteristic
607C2×C4C2×D4×D1711yescharacteristic
608C2×C4C2×Q8×D1711yescharacteristic
609C2×C4#1058769411yescharacteristic
610C2×C4#41×D1711yescharacteristic
611C2×C4#1058769711yescharacteristic
612C8—41no—
613C8—41no—
614C8—342no—
615C8—681no—
616C22—344no—
617C22—682no—
618C22—682no—
619C22—682no—
620C22—682no—
621C22—682no—
622C22—682no—
623C22C2×C22⋊C4×D1711yescharacteristic
624C22#1176346911yescharacteristic
625C22#1176347211yescharacteristic
626C22AGL(1,ℤ/8)×D1711yescharacteristic
627C22#137×D1711yescharacteristic
628C22#1176361111yescharacteristic
629C22#1176361611yescharacteristic
630C4—22no—
631C4—22no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—81no—
636C4—174no—
637C4—342no—
638C4—342no—
639C4—342no—
640C4—682no—
641C4—1361no—
642C4—1361no—
643C4C4×Q8×D1711yescharacteristic
644C4#1176328511yescharacteristic
645C4#1176346711yescharacteristic
646C4#1176347411yescharacteristic
647C2—684no—
648C2#1453161811yescharacteristic
649C2D17×#41711yescharacteristic
650C2#1453730611yescharacteristic
651C2#1454149411yescharacteristic
652C2#1454156811yescharacteristic
653C2#1454168611yescharacteristic
654C2#1454169711yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1989C2
socleC23×C17C22×D4
prime radicalC22×D68C23
cosocleC22×C4×C17C24
hypercenter#1989D17
hypocenterC17#22522
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1989

Complements

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

NH ≅ G/Ntype
2. C2×#11762938C2semidirect
3. C4⋊C4×Dic34C2semidirect
4. #14529754C2semidirect
5. #14529751C2semidirect
6. #14529743C2semidirect
7. #14529738C2semidirect
8. #14529730C2semidirect
15. C17×#1989C2semidirect
17. #11763515C4semidirect
132. #10587480Q8semidirect
279. D68C4×Q8semidirect
296. #1989D17semidirect
461. C17#22522semidirect

Maximal subgroups

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

subgroupindex
C2×#117635152
C2×#117629382
C4⋊C4×Dic342
#145297542
#145297512
#145297432
#145297382
#145297302
#145287852
#145287832
#145286122
#145286082
#145285982
C17×#19892
#2252217