Finite Group Databasebeta

← #17042005

Subgroups — #17042005

Order 4352.

936 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#125×Dic17C211yesmaximal, characteristic, complemented
2C4×D4×Dic17C211yesmaximal, characteristic
3#14529742C211yesmaximal, characteristic, complemented
4#14529554C211yesmaximal, characteristic, complemented
5#14529371C211yesmaximal, characteristic, complemented
6#14529357C211yesmaximal, characteristic, complemented
7#14529047C211yesmaximal, characteristic
8#14529033C211yesmaximal, characteristic
9#14528794C211yesmaximal, characteristic
10#14528684C211yesmaximal, characteristic, complemented
11#14528683C211yesmaximal, characteristic, complemented
12#14528639C211yesmaximal, characteristic, complemented
13#14528611C211yesmaximal, characteristic
14#14528611C211yesmaximal, characteristic
15C2×C17×#478C211yesmaximal, characteristic
16C2×D4×Dic17C2211yescharacteristic
17C2×D4×Dic17C2211yescharacteristic
18C22×C4×Dic17—21no—
19C2×#10587489C2211yescharacteristic
20C2×#10587488C2211yescharacteristic
21C2×C4⋊#3622C2211yescharacteristic
22C2×#10587503C2211yescharacteristic
23C2×#10587503C2211yescharacteristic
24C2×#10587486C2211yescharacteristic
25C2×#10587486C2211yescharacteristic
26#11762708C2211yescharacteristic
27#11762708C2211yescharacteristic
28#11762695—21no—
29#11762695—21no—
30#11762675C2211yescharacteristic
31C4⋊C4×Dic17—21no—
32C4⋊C4×Dic17—21no—
33C4⋊C4×Dic17C2211yescharacteristic
34C4⋊C4×Dic17C2211yescharacteristic
35C4⋊C4×Dic17C2211yescharacteristic
36C4⋊C4×Dic17C2211yescharacteristic
37C4×C4⋊Dic17C2211yescharacteristic
38C22⋊C4×Dic17—21no—
39C22⋊C4×Dic17—21no—
40C4×C22⋊Dic17—21no—
41C42×Dic17C2211yescharacteristic
42#11762572C2211yescharacteristic
43#11762572C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C2×C17×#125C2211yescharacteristic
53C2×C4×D4×C17C2211yescharacteristic
54C17×#478C418yescomplemented
55C2×C17×#107C2211yescharacteristic
56C2×C17×#106C2211yescharacteristic
57C2×C17×C4⋊C8C2211yescharacteristic
58C2×C17×#102C2211yescharacteristic
59C2×C17×#102C2211yescharacteristic
60C22×D4×C17C2311yescharacteristic
61C23×C4×C17—21no—
62C23×Dic17—21no—
63C23×Dic17—21no—
64C22×C8×C17C2311yescharacteristic
65C22×C8×C17C2311yescharacteristic
66C17×#125—22no—
67C17×#125C2×C414yescomplemented
68C4×D4×C17—22no—
69C4×D4×C17C2×C414yes—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C22⋊C4×C17—21no—
76C2×C42×C17C2311yescharacteristic
77C2×C22⋊Dic17—21no—
78C2×C22⋊Dic17—21no—
79D4×Dic17—22no—
80D4×Dic17—22no—
81D4×Dic17—22no—
82D4×Dic17Q814yes—
83C2×C4⋊Dic17—21no—
84C2×C4⋊Dic17—21no—
85C2×C4⋊Dic17C2311yescharacteristic
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4×Dic17—21no—
90C2×C4×Dic17—21no—
91C2×C4×Dic17—21no—
92C2×C4×Dic17—21no—
93C2×C4×Dic17—21no—
94C2×C4×Dic17—44no—
95C2×C4×Dic17D411yescharacteristic
96C2×C4×Dic17D411yescharacteristic
97C2×C4×Dic17D411yescharacteristic
98C2×C4×Dic17D411yescharacteristic
99C2×C4×Dic17C2311yescharacteristic
100C2×C4×Dic17C2311yescharacteristic
101C22×#3622C2311yescharacteristic
102C22×#3622C2311yescharacteristic
103C17×#107C2×C414yescomplemented
104C17×#106C2×C414yescomplemented
105C17×C4⋊C8C2×C414yescomplemented
106C17×#102C2×C414yes—
107C17×#102C2×C414yes—
108#10587503—22no—
109#10587503—22no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587489—22no—
120#10587488—22no—
121#10587486—22no—
122#10587486—22no—
123C4⋊#3622—22no—
124C24×C17—21no—
125C2×D4×C17—21no—
126C2×D4×C17—21no—
127C2×D4×C17—21no—
128C2×D4×C17C22×C412yes—
129C2×D4×C17C2×Q812yes—
130C2×D4×C17#4112yes—
131C22×C4×C17—21no—
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—24no—
135C22×C4×C17—42no—
136C22×C4×C17—42no—
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2411yescharacteristic
140C22×Dic17—24no—
141C22×Dic17—21no—
142C22×Dic17—21no—
143C22×Dic17—24no—
144C22×Dic17—21no—
145C22×Dic17—21no—
146C22×Dic17—42no—
147C22×Dic17—42no—
148C22×Dic17—42no—
149C22×Dic17—42no—
150C22×Dic17C2×D411yescharacteristic
151C22×Dic17C2×D411yescharacteristic
152C22×Dic17C2×D411yescharacteristic
153C22×Dic17C2×D411yescharacteristic
154C2×C8×C17—21no—
155C2×C8×C17—21no—
156C2×C8×C17—21no—
157C2×C8×C17—21no—
158C2×C8×C17C22×C412yes—
159C2×C8×C17C22×C412yes—
160C4⋊C4×C17—24no—
161C4⋊C4×C17—24no—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17C22×C412yes—
166C4⋊C4×C17C22×C412yes—
167C4⋊C4×C17C22×C412yes—
168C22⋊C4×C17—24no—
169C42×C17C22×C412yes—
170C42×C17#4112yes—
171C22⋊Dic17—24no—
172C22⋊Dic17—24no—
173C4⋊Dic17—21no—
174C4⋊Dic17—21no—
175C4⋊Dic17—21no—
176C4⋊Dic17—21no—
177C4⋊Dic17—21no—
178C4⋊Dic17—21no—
179C4⋊Dic17—42no—
180C4⋊Dic17—42no—
181C4⋊Dic17C2×Q812yes—
182C4⋊Dic17#4112yes—
183C4×Dic17—21no—
184C4×Dic17—21no—
185C4×Dic17—22no—
186C4×Dic17—41no—
187C4×Dic17—41no—
188C4×Dic17—41no—
189C4×Dic17—42no—
190C4×Dic17—42no—
191C4×Dic17—42no—
192C4×Dic17—42no—
193C4×Dic17—42no—
194C4×Dic17C2×Q812yes—
195C4×Dic17#4112yes—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622—21no—
202#21617—171nomaximal, Sylow
203C23×C17—22no—
204C23×C17—22no—
205C23×C17—22no—
206C23×C17—41no—
207C23×C17—41no—
208C23×C17—41no—
209C23×C17—41no—
210C23×C17C22×D411yescharacteristic
211D4×C17—22no—
212D4×C17—22no—
213D4×C17—22no—
214D4×C17C4×Q814yes—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—22no—
220C2×C4×C17—22no—
221C2×C4×C17—22no—
222C2×C4×C17—48no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17C4×D412yes—
230C2×C4×C17C4×D412yes—
231C2×C4×C17C23×C411yescharacteristic
232C2×C4×C17C22×Q811yescharacteristic
233C2×C4×C17C2×#4111yescharacteristic
234C2×C4×C17C2×#4111yescharacteristic
235C2×C4×C17C2×#4111yescharacteristic
236C2×C4×C17C2×#4111yescharacteristic
237C2×Dic17—21no—
238C2×Dic17—21no—
239C2×Dic17—21no—
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—21no—
243C2×Dic17—22no—
244C2×Dic17—21no—
245C2×Dic17—22no—
246C2×Dic17—21no—
247C2×Dic17—21no—
248C2×Dic17—48no—
249C2×Dic17—48no—
250C2×Dic17—41no—
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×Dic17C22⋊Q812yes—
261C2×Dic17C22⋊Q812yes—
262C2×Dic17AGL(1,ℤ/8)12yes—
263C8×C17—22no—
264C8×C17—22no—
265#3622—41no—
266#3622—41no—
267C2×#478D1711yescharacteristic
268C4×#125—171no—
269C42×D4—171no—
270#1989—171no—
271#1949—171no—
272#1920—171no—
273#1918—171no—
274#1864—171no—
275#1862—171no—
276#1804—171no—
277#1776—171no—
278#1776—171no—
279#1767—171no—
280#1762—171no—
281#1762—171no—
282C22×C17—24no—
283C22×C17—42no—
284C22×C17—42no—
285C22×C17—42no—
286C22×C17—42no—
287C22×C17—42no—
288C22×C17—42no—
289C22×C17C2×C4×D411yescharacteristic
290C22×C17C2×C22⋊Q811yescharacteristic
291C22×C17#52511yescharacteristic
292C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C17#57511yescharacteristic
294C22×C17#57511yescharacteristic
295C22×C17#57511yescharacteristic
296C4×C17—22no—
297C4×C17—22no—
298C4×C17—42no—
299C4×C17—42no—
300C4×C17—42no—
301C4×C17C2×C4×Q811yescharacteristic
302C4×C17C4×#4111yescharacteristic
303C4×C17#52911yescharacteristic
304C4×C17#53311yescharacteristic
305Dic17—22no—
306Dic17—22no—
307Dic17—22no—
308Dic17—42no—
309Dic17—41no—
310Dic17—42no—
311Dic17—81no—
312Dic17—81no—
313C2×#125C2×D1711yescharacteristic
314C2×C4×D4—171no—
315C2×C4×D4—171no—
316C2×C4×D4C2×D1711yescharacteristic
317C22×C42—341no—
318#478Dic1718yescomplemented
319C2×#107—171no—
320C2×#107C2×D1711yescharacteristic
321C2×#106—171no—
322C2×#106C2×D1711yescharacteristic
323C2×C4⋊C8—171no—
324C2×C4⋊C8C2×D1711yescharacteristic
325C2×#102—171no—
326C2×#102—171no—
327C2×#102C2×D1711yescharacteristic
328C2×#102C2×D1711yescharacteristic
329C2×#97—171no—
330C2×#97—171no—
331#389—171no—
332#389—171no—
333#387—341no—
334#387—341no—
335#382—171no—
336C4×C4⋊C4—171no—
337C4×C4⋊C4—171no—
338C4×C4⋊C4—171no—
339C4×C4⋊C4—171no—
340C4×C4⋊C4—171no—
341C4×C4⋊C4—341no—
342C4×C4⋊C4—341no—
343C4×C22⋊C4—341no—
344C4×C22⋊C4—341no—
345C4×C22⋊C4—341no—
346C43—171no—
347#340—171no—
348#340—171no—
349#340—171no—
350#340—171no—
351#340—171no—
352#340—171no—
353#340—171no—
354#340—171no—
355#336—171no—
356#336—171no—
357C2×C17—44no—
358C2×C17C4×C22⋊Q811yescharacteristic
359C2×C17#294311yescharacteristic
360C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
361C2×C17#307711yescharacteristic
362C2×C17#308011yescharacteristic
363C2×C17#311111yescharacteristic
364C2×C17#311411yescharacteristic
365C22×D4C22×D1711yescharacteristic
366C23×C4—21no—
367C23×C4—341no—
368C23×C4—341no—
369C22×C8—171no—
370C22×C8—171no—
371C22×C8C22×D1711yescharacteristic
372C22×C8C22×D1711yescharacteristic
373#125—22no—
374#125C2×Dic1714yescomplemented
375C4×D4—22no—
376C4×D4—174no—
377C4×D4—342no—
378C4×D4—342no—
379C4×D4—342no—
380C4×D4C2×Dic1714yes—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—171no—
384C2×C4⋊C4—171no—
385C2×C4⋊C4—171no—
386C2×C4⋊C4—171no—
387C2×C4⋊C4—341no—
388C2×C4⋊C4—341no—
389C2×C4⋊C4C22×D1711yescharacteristic
390C2×C4⋊C4C22×D1711yescharacteristic
391C2×C4⋊C4C22×D1711yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—341no—
394C2×C22⋊C4—341no—
395C2×C42—171no—
396C2×C42—171no—
397C2×C42—171no—
398C2×C42—171no—
399C2×C42—171no—
400C2×C42—171no—
401C2×C42—341no—
402C2×C42—341no—
403C2×C42—341no—
404C2×C42—341no—
405C2×C42—341no—
406C2×C42—684no—
407C2×C42C22×D1711yescharacteristic
408#107—342no—
409#107C2×Dic1714yescomplemented
410#106—342no—
411#106C2×Dic1714yescomplemented
412C4⋊C8—342no—
413C4⋊C8C2×Dic1714yescomplemented
414#102—342no—
415#102—342no—
416#102C2×Dic1714yes—
417#102C2×Dic1714yes—
418#97—342no—
419#97—342no—
420#95—341no—
421#95—341no—
422#95—341no—
423#95—341no—
424#95—341no—
425#95—341no—
426#95—341no—
427#95—341no—
428#95—341no—
429C17#2161711yescharacteristic, complemented, Sylow
430C24—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4Q8×D1712yes—
435C2×D4#6325612yes—
436C2×D4C22×Dic1712yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—171no—
444C22×C4—171no—
445C22×C4—171no—
446C22×C4—171no—
447C22×C4—344no—
448C22×C4—344no—
449C22×C4—341no—
450C22×C4—341no—
451C22×C4—341no—
452C22×C4—341no—
453C22×C4—682no—
454C22×C4—682no—
455C22×C4—682no—
456C22×C4—682no—
457C22×C4D4×D1711yescharacteristic
458C22×C4D4×D1711yescharacteristic
459C22×C4C23×D1711yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—341no—
465C2×C8—341no—
466C2×C8—341no—
467C2×C8—341no—
468C2×C8—341no—
469C2×C8—341no—
470C2×C8C22×Dic1712yes—
471C2×C8C22×Dic1712yes—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—172no—
478C4⋊C4—172no—
479C4⋊C4—341no—
480C4⋊C4—341no—
481C4⋊C4—341no—
482C4⋊C4—341no—
483C4⋊C4—341no—
484C4⋊C4—341no—
485C4⋊C4—682no—
486C4⋊C4—682no—
487C4⋊C4C22×Dic1712yes—
488C4⋊C4C22×Dic1712yes—
489C4⋊C4C22×Dic1712yes—
490C22⋊C4—24no—
491C22⋊C4—344no—
492C22⋊C4—344no—
493C42—172no—
494C42—172no—
495C42—342no—
496C42—341no—
497C42—341no—
498C42—681no—
499C42—681no—
500C42—681no—
501C42—682no—
502C42—682no—
503C42—682no—
504C42—682no—
505C42—682no—
506C42#6325412yes—
507C42C22×Dic1712yes—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23C2×D4×D1711yescharacteristic
516D4—22no—
517D4—22no—
518D4—22no—
519D4Q8×Dic1714yes—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—48no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—172no—
535C2×C4—172no—
536C2×C4—172no—
537C2×C4—341no—
538C2×C4—341no—
539C2×C4—341no—
540C2×C4—341no—
541C2×C4—341no—
542C2×C4—341no—
543C2×C4—342no—
544C2×C4—341no—
545C2×C4—342no—
546C2×C4—341no—
547C2×C4—341no—
548C2×C4—688no—
549C2×C4—688no—
550C2×C4—681no—
551C2×C4—681no—
552C2×C4—681no—
553C2×C4—681no—
554C2×C4—681no—
555C2×C4—681no—
556C2×C4—681no—
557C2×C4—681no—
558C2×C4—681no—
559C2×C4—681no—
560C2×C4D4×Dic1712yes—
561C2×C4D4×Dic1712yes—
562C2×C4C2×#6325411yescharacteristic
563C2×C4C2×Q8×D1711yescharacteristic
564C2×C4C2×#6325611yescharacteristic
565C2×C4#41×D1711yescharacteristic
566C2×C4#41×D1711yescharacteristic
567C2×C4C23×Dic1711yescharacteristic
568C8—22no—
569C8—22no—
570C8—681no—
571C8—681no—
572C22—24no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22C2×D4×Dic1711yescharacteristic
580C22C22⋊Q8×D1711yescharacteristic
581C22#1176332211yescharacteristic
582C22AGL(1,ℤ/8)×D1711yescharacteristic
583C22#1176360511yescharacteristic
584C22#1176360611yescharacteristic
585C22#1176360611yescharacteristic
586C4—22no—
587C4—22no—
588C4—42no—
589C4—42no—
590C4—42no—
591C4—342no—
592C4—342no—
593C4—342no—
594C4—682no—
595C4—681no—
596C4—682no—
597C4—1361no—
598C4—1361no—
599C4C2×Q8×Dic1711yescharacteristic
600C4#41×Dic1711yescharacteristic
601C4#1176334511yescharacteristic
602C4#1176336111yescharacteristic
603C2—44no—
604C2C22⋊Q8×Dic1711yescharacteristic
605C2#1453758511yescharacteristic
606C2AGL(1,ℤ/8)×Dic1711yescharacteristic
607C2#1453857911yescharacteristic
608C2#1453860111yescharacteristic
609C2#1453889811yescharacteristic
610C2#1453892711yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C17×#478C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenterC2×#478D17
hypocenterC17#21617
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
1. #125×Dic17C2semidirect
3. #14529742C2semidirect
4. #14529554C2semidirect
5. #14529371C2semidirect
6. #14529357C2semidirect
10. #14528684C2semidirect
11. #14528683C2semidirect
12. #14528639C2semidirect
54. C17×#478C4semidirect
67. C17×#125C2×C4semidirect
103. C17×#107C2×C4semidirect
104. C17×#106C2×C4semidirect
105. C17×C4⋊C8C2×C4semidirect
318. #478Dic17semidirect
374. #125C2×Dic17semidirect
409. #107C2×Dic17semidirect
411. #106C2×Dic17semidirect
413. C4⋊C8C2×Dic17semidirect
429. C17#21617semidirect

Maximal subgroups

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

subgroupindex
#125×Dic172
C4×D4×Dic172
#145297422
#145295542
#145293712
#145293572
#145290472
#145290332
#145287942
#145286842
#145286832
#145286392
#145286112
C2×C17×#4782
#2161717