Finite Group Databasebeta

← #17042003

Subgroups — #17042003

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
1C2×#11763112C211yesmaximal, characteristic
2#14531343C211yesmaximal, characteristic, complemented
3C4×#10587579C211yesmaximal, characteristic
4#14529738C211yesmaximal, characteristic, complemented
5#14529551C211yesmaximal, characteristic, complemented
6#14529369C211yesmaximal, characteristic, complemented
7#14529048C211yesmaximal, characteristic
8#14529030C211yesmaximal, characteristic
9#14528793C211yesmaximal, characteristic
10#14528683C211yesmaximal, characteristic, complemented
11#14528680C211yesmaximal, characteristic, complemented
12#14528636C211yesmaximal, characteristic, complemented
13#14528608C211yesmaximal, characteristic
14#14528608C211yesmaximal, characteristic
15C17×#1918C211yesmaximal, characteristic, complemented
16C2×#10587576C2211yescharacteristic
17C2×C4×D68C2211yescharacteristic
18C2×#10587579C2211yescharacteristic
19C2×#10587579C2211yescharacteristic
20C2×C42×D17—21no—
21#11763112C418yescomplemented
22C2×#10587478C2211yescharacteristic
23C2×#10587488C2211yescharacteristic
24C2×#10587479C2211yescharacteristic
25C2×#10587485C2211yescharacteristic
26C2×#10587485C2211yescharacteristic
27C2×#10587492C2211yescharacteristic
28C2×#10587492C2211yescharacteristic
29C2×#10587480C2211yescharacteristic
30C2×#10587480C2211yescharacteristic
31C2×#10587486C2211yescharacteristic
32C2×#10587486C2211yescharacteristic
33#11762707C2211yescharacteristic
34#11762694—21no—
35#11762690—21no—
36#11762674C2211yescharacteristic
37C4⋊C4×Dic17—21no—
38C4⋊C4×Dic17C2211yescharacteristic
39C4⋊C4×Dic17C2211yescharacteristic
40C4⋊C4×Dic17C2211yescharacteristic
41C4⋊C4×Dic17C2211yescharacteristic
42#11762660—21no—
43C4×#63227—21no—
44#11762656—21no—
45#11762656—21no—
46C42×Dic17C2211yescharacteristic
47#11762572C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762564C2211yescharacteristic
53C2×C17×#107C2211yescharacteristic
54C17×#389C2211yescharacteristic
55C4×C4⋊C4×C17C2211yescharacteristic
56C17×#340C2211yescharacteristic
57C17×#340C2211yescharacteristic
58C17×#340C2211yescharacteristic
59C17×#336C2211yescharacteristic
60C22×D68C2311yescharacteristic
61C22×C4×D17—21no—
62C22×C4×D17—21no—
63C22×C4×D17—21no—
64C22×C8×C17C2311yescharacteristic
65C22×C8×C17C2311yescharacteristic
66C2×C4⋊C4×C17—21no—
67C2×C4⋊C4×C17C2311yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C42×C17—21no—
72C2×C42×C17C2311yescharacteristic
73C2×#63227—21no—
74C2×#63227—21no—
75C2×#63227—21no—
76C2×C4⋊Dic17—21no—
77C2×C4⋊Dic17C2311yescharacteristic
78C2×C4⋊Dic17C2311yescharacteristic
79C2×C4⋊Dic17C2311yescharacteristic
80C2×#63225—21no—
81C2×#63225—21no—
82C2×C4×Dic17—21no—
83C2×C4×Dic17—21no—
84C2×C4×Dic17—21no—
85C2×C4×Dic17—21no—
86C2×C4×Dic17D411yescharacteristic
87C2×C4×Dic17D411yescharacteristic
88C2×C4×Dic17D411yescharacteristic
89C2×C4×Dic17D411yescharacteristic
90C2×C4×Dic17C2311yescharacteristic
91C2×C4×Dic17C2311yescharacteristic
92C22×#3622C2311yescharacteristic
93C22×#3622C2311yescharacteristic
94#10587579—22no—
95#10587579—22no—
96#10587579C2×C414yes—
97#10587579Q814yes—
98#10587576—22no—
99#10587576C2×C414yescomplemented
100C4×D68—22no—
101C4×D68—22no—
102C42×D17—44no—
103C17×#107C2×C414yescomplemented
104C17×#95—21no—
105C17×#95—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587492—22no—
114#10587492C2×C414yes—
115#10587488—22no—
116#10587486—22no—
117#10587486—22no—
118#10587485—22no—
119#10587485C2×C414yescomplemented
120#10587480—22no—
121#10587480C2×C414yes—
122#10587479C2×C414yescomplemented
123#10587478—22no—
124C23×D17—21no—
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17C2×D411yescharacteristic
130C22×C4×C17C2×D411yescharacteristic
131C22×C4×C17C2411yescharacteristic
132C22×Dic17—21no—
133C22×Dic17—21no—
134C22×Dic17—21no—
135C22×Dic17C2×D411yescharacteristic
136C22×Dic17C2×D411yescharacteristic
137C22×Dic17C2×D411yescharacteristic
138C22×Dic17C2×D411yescharacteristic
139C2×D68—21no—
140C2×D68—21no—
141C2×D68—21no—
142C2×D68C22×C412yes—
143C2×D68C2×Q812yes—
144C2×D68#4112yes—
145C2×C4×D17—24no—
146C2×C4×D17—24no—
147C2×C4×D17—24no—
148C2×C4×D17—42no—
149C2×C4×D17—42no—
150C2×C4×D17—42no—
151C2×C4×D17—42no—
152C2×C4×D17—42no—
153C2×C4×D17—42no—
154C2×C8×C17—21no—
155C2×C8×C17—21no—
156C2×C8×C17—21no—
157C2×C8×C17—21no—
158C2×C8×C17—21no—
159C2×C8×C17C22×C412yes—
160C4⋊C4×C17—21no—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—21no—
163C4⋊C4×C17—21no—
164C4⋊C4×C17—21no—
165C4⋊C4×C17—42no—
166C4⋊C4×C17C22×C412yes—
167C4⋊C4×C17C22×C412yes—
168C4⋊C4×C17C2×Q812yes—
169C42×C17—21no—
170C42×C17—21no—
171C42×C17—42no—
172#63227—24no—
173#63227—24no—
174#63227—24no—
175C4⋊Dic17—21no—
176C4⋊Dic17—21no—
177C4⋊Dic17—24no—
178C4⋊Dic17—21no—
179C4⋊Dic17—21no—
180C4⋊Dic17C22×C412yes—
181C4⋊Dic17#4112yes—
182#63225—24no—
183#63225—42no—
184C4×Dic17—22no—
185C4×Dic17—41no—
186C4×Dic17—41no—
187C4×Dic17—41no—
188C4×Dic17—42no—
189C4×Dic17—42no—
190C4×Dic17—42no—
191C4×Dic17—42no—
192C4×Dic17C22×C412yes—
193C4×Dic17C2×Q812yes—
194C4×Dic17#4112yes—
195C4×Dic17#4112yes—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622C22×C412yes—
202#21617—171nomaximal, Sylow
203C23×C17C22×D411yescharacteristic
204C22×D17—22no—
205C22×D17—22no—
206C22×D17—22no—
207C22×D17—41no—
208C22×D17—41no—
209C22×D17—41no—
210C22×D17—41no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—22no—
214C2×C4×C17—22no—
215C2×C4×C17—22no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—41no—
220C2×C4×C17—41no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17AGL(1,ℤ/8)12yes—
229C2×C4×C17C23×C411yescharacteristic
230C2×C4×C17C22×Q811yescharacteristic
231C2×C4×C17C2×#4111yescharacteristic
232C2×C4×C17C2×#4111yescharacteristic
233C2×C4×C17C2×#4111yescharacteristic
234C2×C4×C17C2×#4111yescharacteristic
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—21no—
239C2×Dic17—21no—
240C2×Dic17—21no—
241C2×Dic17—21no—
242C2×Dic17—21no—
243C2×Dic17—22no—
244C2×Dic17—22no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17C4×D412yes—
253C2×Dic17C4×D412yes—
254C2×Dic17C22⋊Q812yes—
255C2×Dic17C22⋊Q812yes—
256D68—22no—
257D68—22no—
258D68—22no—
259D68C4×Q814yes—
260C4×D17—48no—
261C4×D17—48no—
262C4×D17—48no—
263C8×C17—22no—
264C8×C17—41no—
265#3622—22no—
266#3622—41no—
267C2×#478—171no—
268C4×#125—171no—
269C42×D4—171no—
270#1989—171no—
271#1949—171no—
272#1920—171no—
273#1918D1711yescharacteristic, complemented
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×C17C2×C4×D411yescharacteristic
283C22×C17C2×C22⋊Q811yescharacteristic
284C22×C17#52511yescharacteristic
285C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C17#57511yescharacteristic
287C22×C17#57511yescharacteristic
288C22×C17#57511yescharacteristic
289C2×D17—24no—
290C2×D17—42no—
291C2×D17—42no—
292C2×D17—42no—
293C2×D17—42no—
294C2×D17—42no—
295C2×D17—42no—
296C4×C17—22no—
297C4×C17—42no—
298C4×C17—41no—
299C4×C17—42no—
300C4×C17—42no—
301C4×C17—81no—
302C4×C17C2×C4×Q811yescharacteristic
303C4×C17C4×#4111yescharacteristic
304C4×C17#52911yescharacteristic
305C4×C17#53311yescharacteristic
306Dic17—22no—
307Dic17—22no—
308Dic17—22no—
309Dic17—22no—
310Dic17—42no—
311Dic17—42no—
312Dic17—81no—
313C2×#125—171no—
314C2×C4×D4—171no—
315C2×C4×D4—171no—
316C2×C4×D4—171no—
317C22×C42—341no—
318#478—178no—
319C2×#107—171no—
320C2×#107C2×D1711yescharacteristic
321C2×#106—171no—
322C2×#106—171no—
323C2×C4⋊C8—171no—
324C2×C4⋊C8—171no—
325C2×#102—171no—
326C2×#102—171no—
327C2×#102—171no—
328C2×#102—171no—
329C2×#97—171no—
330C2×#97—171no—
331#389—171no—
332#389C2×D1711yescharacteristic
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—341no—
341C4×C4⋊C4—341no—
342C4×C4⋊C4C2×D1711yescharacteristic
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#340C2×D1711yescharacteristic
353#340C2×D1711yescharacteristic
354#340C2×D1711yescharacteristic
355#336—171no—
356#336C2×D1711yescharacteristic
357C2×C17C4×C22⋊Q811yescharacteristic
358C2×C17#294311yescharacteristic
359C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
360C2×C17#307711yescharacteristic
361C2×C17#308011yescharacteristic
362C2×C17#311111yescharacteristic
363C2×C17#311411yescharacteristic
364D17—44no—
365C22×D4—171no—
366C23×C4—341no—
367C23×C4—341no—
368C23×C4—341no—
369C22×C8—171no—
370C22×C8—171no—
371C22×C8C22×D1711yescharacteristic
372C22×C8C22×D1711yescharacteristic
373#125—174no—
374#125—342no—
375C4×D4—174no—
376C4×D4—174no—
377C4×D4—342no—
378C4×D4—342no—
379C4×D4—342no—
380C4×D4—342no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—171no—
383C2×C4⋊C4—171no—
384C2×C4⋊C4—171no—
385C2×C4⋊C4—341no—
386C2×C4⋊C4—341no—
387C2×C4⋊C4—341no—
388C2×C4⋊C4C22×D1711yescharacteristic
389C2×C4⋊C4C22×D1711yescharacteristic
390C2×C4⋊C4C22×D1711yescharacteristic
391C2×C4⋊C4C22×D1711yescharacteristic
392C2×C22⋊C4—341no—
393C2×C22⋊C4—341no—
394C2×C22⋊C4—341no—
395C2×C42—21no—
396C2×C42—171no—
397C2×C42—171no—
398C2×C42—171no—
399C2×C42—171no—
400C2×C42—171no—
401C2×C42—171no—
402C2×C42—341no—
403C2×C42—341no—
404C2×C42—341no—
405C2×C42—341no—
406C2×C42—684no—
407C2×C42C22×D1711yescharacteristic
408#107—342no—
409#107C4×D1714yescomplemented
410#106—174no—
411#106—342no—
412C4⋊C8—174no—
413C4⋊C8—342no—
414#102—174no—
415#102—174no—
416#102—342no—
417#102—342no—
418#97—342no—
419#97—342no—
420#95—21no—
421#95—21no—
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—341no—
431C2×D4—172no—
432C2×D4—172no—
433C2×D4—172no—
434C2×D4—341no—
435C2×D4—341no—
436C2×D4—341no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—171no—
442C22×C4—171no—
443C22×C4—171no—
444C22×C4—171no—
445C22×C4—344no—
446C22×C4—344no—
447C22×C4—344no—
448C22×C4—341no—
449C22×C4—341no—
450C22×C4—341no—
451C22×C4—682no—
452C22×C4—682no—
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—21no—
465C2×C8—172no—
466C2×C8—341no—
467C2×C8—341no—
468C2×C8—341no—
469C2×C8—341no—
470C2×C8—341no—
471C2×C8C2×C4×D1712yes—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—42no—
478C4⋊C4—172no—
479C4⋊C4—172no—
480C4⋊C4—344no—
481C4⋊C4—344no—
482C4⋊C4—341no—
483C4⋊C4—341no—
484C4⋊C4—341no—
485C4⋊C4—341no—
486C4⋊C4—682no—
487C4⋊C4C2×C4×D1712yes—
488C4⋊C4C2×C4×D1712yes—
489C4⋊C4Q8×D1712yes—
490C22⋊C4—344no—
491C22⋊C4—344no—
492C22⋊C4—344no—
493C42—21no—
494C42—21no—
495C42—42no—
496C42—172no—
497C42—172no—
498C42—172no—
499C42—172no—
500C42—342no—
501C42—681no—
502C42—681no—
503C42—681no—
504C42—682no—
505C42—682no—
506C42—682no—
507C42—682no—
508C23—342no—
509C23—342no—
510C23—342no—
511C23—681no—
512C23—681no—
513C23—681no—
514C23—681no—
515C23C2×D4×D1711yescharacteristic
516D4—174no—
517D4—342no—
518D4—342no—
519D4—342no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—172no—
538C2×C4—172no—
539C2×C4—172no—
540C2×C4—172no—
541C2×C4—341no—
542C2×C4—341no—
543C2×C4—341no—
544C2×C4—341no—
545C2×C4—341no—
546C2×C4—341no—
547C2×C4—341no—
548C2×C4—341no—
549C2×C4—342no—
550C2×C4—342no—
551C2×C4—688no—
552C2×C4—688no—
553C2×C4—688no—
554C2×C4—681no—
555C2×C4—681no—
556C2×C4—681no—
557C2×C4—681no—
558C2×C4—681no—
559C2×C4—681no—
560C2×C4—681no—
561C2×C4#1058759712yes—
562C2×C4C22×C4×D1711yescharacteristic
563C2×C4C2×Q8×D1711yescharacteristic
564C2×C4#41×D1711yescharacteristic
565C2×C4#41×D1711yescharacteristic
566C2×C4#41×D1711yescharacteristic
567C2×C4#41×D1711yescharacteristic
568C8—22no—
569C8—41no—
570C8—342no—
571C8—681no—
572C22—344no—
573C22—682no—
574C22—682no—
575C22—682no—
576C22—682no—
577C22—682no—
578C22—682no—
579C22C4×D4×D1711yescharacteristic
580C22C22⋊Q8×D1711yescharacteristic
581C22#1176331711yescharacteristic
582C22C2×#1058759711yescharacteristic
583C22#1176360611yescharacteristic
584C22#1176360811yescharacteristic
585C22#1176360811yescharacteristic
586C4—22no—
587C4—42no—
588C4—42no—
589C4—42no—
590C4—41no—
591C4—81no—
592C4—342no—
593C4—342no—
594C4—342no—
595C4—342no—
596C4—682no—
597C4—682no—
598C4—1361no—
599C4C4×Q8×D1711yescharacteristic
600C4#1176327711yescharacteristic
601C4#1176334211yescharacteristic
602C4#1176336511yescharacteristic
603C2—684no—
604C2#1453131811yescharacteristic
605C2#1453758311yescharacteristic
606C2C4×#1058759711yescharacteristic
607C2#1453857611yescharacteristic
608C2#1453860311yescharacteristic
609C2#1453889611yescharacteristic
610C2#1453891911yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1918C2
socleC23×C17C22×D4
prime radicalC22×D68C23
cosocleC22×C4×C17C24
hypercenter#1918D17
hypocenterC17#21617
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1918

Complements

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

NH ≅ G/Ntype
2. #14531343C2semidirect
4. #14529738C2semidirect
5. #14529551C2semidirect
6. #14529369C2semidirect
10. #14528683C2semidirect
11. #14528680C2semidirect
12. #14528636C2semidirect
15. C17×#1918C2semidirect
21. #11763112C4semidirect
99. #10587576C2×C4semidirect
103. C17×#107C2×C4semidirect
119. #10587485C2×C4semidirect
122. #10587479C2×C4semidirect
273. #1918D17semidirect
409. #107C4×D17semidirect
429. C17#21617semidirect

Maximal subgroups

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

subgroupindex
C2×#117631122
#145313432
C4×#105875792
#145297382
#145295512
#145293692
#145290482
#145290302
#145287932
#145286832
#145286802
#145286362
#145286082
C17×#19182
#2161717