Finite Group Databasebeta

← #17041998

Subgroups — #17041998

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×#11763106C211yesmaximal, characteristic
2C42×D68C211yesmaximal, characteristic
3#14529742C211yesmaximal, characteristic, complemented
4#14529550C211yesmaximal, characteristic, complemented
5#14529366C211yesmaximal, characteristic, complemented
6#14529352C211yesmaximal, characteristic, complemented
7#14529044C211yesmaximal, characteristic
8#14529030C211yesmaximal, characteristic
9#14528791C211yesmaximal, characteristic
10#14528682C211yesmaximal, characteristic, complemented
11#14528682C211yesmaximal, characteristic, complemented
12#14528637C211yesmaximal, characteristic, complemented
13#14528612C211yesmaximal, characteristic
14#14528612C211yesmaximal, characteristic
15C4×C17×#125C211yesmaximal, characteristic, complemented
16C2×C4×D68C2211yescharacteristic
17C2×C4×D68C2211yescharacteristic
18C2×C4×D68C2211yescharacteristic
19C2×C42×D17—21no—
20#11763106C418yescomplemented
21C2×#10587478C2211yescharacteristic
22C2×#10587478C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×#10587479C2211yescharacteristic
25C2×C4⋊#3622C2211yescharacteristic
26C2×C4⋊#3622C2211yescharacteristic
27C2×#10587480C2211yescharacteristic
28C2×#10587480C2211yescharacteristic
29C2×#10587480C2211yescharacteristic
30C2×#10587480C2211yescharacteristic
31C2×#10587474C2211yescharacteristic
32C2×#10587474C2211yescharacteristic
33C4×C4⋊Dic17C2211yescharacteristic
34C4×C4⋊Dic17C2211yescharacteristic
35C4×C4⋊Dic17C2211yescharacteristic
36C4×#63227—21no—
37C4×#63227—21no—
38C4×#63227—21no—
39#11762571C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762564C2211yescharacteristic
48#11762564C2211yescharacteristic
49C2×C17×#125C2211yescharacteristic
50C17×#389C2211yescharacteristic
51C17×#389C2211yescharacteristic
52C17×#387—21no—
53C17×#387—21no—
54C17×#382C2211yescharacteristic
55C4×C4⋊C4×C17—21no—
56C4×C4⋊C4×C17—21no—
57C4×C4⋊C4×C17C2211yescharacteristic
58C4×C4⋊C4×C17C2211yescharacteristic
59C43×C17C2211yescharacteristic
60C22×D68C2311yescharacteristic
61C22×C4×D17—21no—
62C22×C4×D17—21no—
63C22×C4×D17—21no—
64C17×#125—22no—
65C17×#125C2×C414yescomplemented
66C2×C4⋊C4×C17—21no—
67C2×C4⋊C4×C17—21no—
68C2×C4⋊C4×C17—21no—
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C42×C17—21no—
75C2×C42×C17—21no—
76C2×C42×C17D411yescharacteristic
77C2×C42×C17D411yescharacteristic
78C2×C42×C17D411yescharacteristic
79C2×C42×C17D411yescharacteristic
80C2×C42×C17C2311yescharacteristic
81C2×C42×C17C2311yescharacteristic
82C2×C42×C17C2311yescharacteristic
83C2×#63227—21no—
84C2×#63227—21no—
85C2×#63227—21no—
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4×Dic17—21no—
90C2×C4×Dic17—21no—
91C2×C4×Dic17—21no—
92C22×#3622C2311yescharacteristic
93C22×#3622C2311yescharacteristic
94C22×#3622C2311yescharacteristic
95C22×#3622C2311yescharacteristic
96C4×D68—22no—
97C4×D68—22no—
98C4×D68—22no—
99C4×D68—22no—
100C4×D68C2×C414yes—
101C4×D68Q814yes—
102C42×D17—44no—
103C17×#95—21no—
104C17×#95—21no—
105C17×#95—21no—
106C17×#95—21no—
107C17×#95—21no—
108C17×#95—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587480—22no—
113#10587480—22no—
114#10587480C2×C414yes—
115#10587480C2×C414yes—
116#10587479—22no—
117#10587479C2×C414yescomplemented
118#10587478—22no—
119#10587478C2×C414yescomplemented
120C4⋊#3622—22no—
121C4⋊#3622C2×C414yescomplemented
122#10587474—22no—
123#10587474—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×C17C2×D411yescharacteristic
132C22×C4×C17C2×D411yescharacteristic
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17C2411yescharacteristic
136C22×Dic17—21no—
137C22×Dic17—21no—
138C22×Dic17—21no—
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—
154C4⋊C4×C17—24no—
155C4⋊C4×C17—24no—
156C4⋊C4×C17—21no—
157C4⋊C4×C17—21no—
158C4⋊C4×C17—21no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—21no—
161C4⋊C4×C17—21no—
162C4⋊C4×C17—42no—
163C4⋊C4×C17—42no—
164C4⋊C4×C17C22×C412yes—
165C4⋊C4×C17C22×C412yes—
166C42×C17—22no—
167C42×C17—21no—
168C42×C17—21no—
169C42×C17—41no—
170C42×C17—41no—
171C42×C17—41no—
172C42×C17—42no—
173C42×C17—42no—
174C42×C17C22×C412yes—
175C42×C17C2×Q812yes—
176C42×C17#4112yes—
177C42×C17#4112yes—
178#63227—24no—
179#63227—24no—
180#63227—24no—
181C4⋊Dic17—21no—
182C4⋊Dic17—21no—
183C4⋊Dic17—21no—
184C4⋊Dic17C22×C412yes—
185C4⋊Dic17C2×Q812yes—
186C4⋊Dic17#4112yes—
187C4×Dic17—42no—
188C4×Dic17—42no—
189C4×Dic17—42no—
190C2×#3622—21no—
191C2×#3622—21no—
192C2×#3622—21no—
193C2×#3622—21no—
194C2×#3622—21no—
195C2×#3622—21no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622C22×C412yes—
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—21no—
214C2×C4×C17—21no—
215C2×C4×C17—21no—
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—22no—
222C2×C4×C17—22no—
223C2×C4×C17—21no—
224C2×C4×C17—21no—
225C2×C4×C17—21no—
226C2×C4×C17—41no—
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×C17C4×D412yes—
237C2×C4×C17C4×D412yes—
238C2×C4×C17C22⋊Q812yes—
239C2×C4×C17C22⋊Q812yes—
240C2×C4×C17AGL(1,ℤ/8)12yes—
241C2×C4×C17C23×C411yescharacteristic
242C2×C4×C17C22×Q811yescharacteristic
243C2×C4×C17C2×#4111yescharacteristic
244C2×C4×C17C2×#4111yescharacteristic
245C2×C4×C17C2×#4111yescharacteristic
246C2×C4×C17C2×#4111yescharacteristic
247C2×Dic17—22no—
248C2×Dic17—22no—
249C2×Dic17—22no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17—41no—
255C2×Dic17—41no—
256D68—22no—
257D68—22no—
258D68—22no—
259D68C4×Q814yes—
260C4×D17—48no—
261C4×D17—48no—
262C4×D17—48no—
263#3622—22no—
264#3622—22no—
265#3622—41no—
266#3622—41no—
267C2×#478—171no—
268C4×#125D1711yescharacteristic, complemented
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×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—22no—
298C4×C17—22no—
299C4×C17—22no—
300C4×C17—22no—
301C4×C17—41no—
302C4×C17—42no—
303C4×C17—42no—
304C4×C17—81no—
305C4×C17—81no—
306C4×C17C2×C4×Q811yescharacteristic
307C4×C17C4×#4111yescharacteristic
308C4×C17#52911yescharacteristic
309C4×C17#53311yescharacteristic
310Dic17—42no—
311Dic17—42no—
312Dic17—42no—
313C2×#125C2×D1711yescharacteristic
314C2×C4×D4—171no—
315C2×C4×D4—171no—
316C2×C4×D4—171no—
317C22×C42—341no—
318#478—178no—
319C2×#107—171no—
320C2×#107—171no—
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#389C2×D1711yescharacteristic
332#389C2×D1711yescharacteristic
333#387—21no—
334#387—21no—
335#382C2×D1711yescharacteristic
336C4×C4⋊C4—21no—
337C4×C4⋊C4—21no—
338C4×C4⋊C4—171no—
339C4×C4⋊C4—171no—
340C4×C4⋊C4—171no—
341C4×C4⋊C4C2×D1711yescharacteristic
342C4×C4⋊C4C2×D1711yescharacteristic
343C4×C22⋊C4—341no—
344C4×C22⋊C4—341no—
345C4×C22⋊C4—341no—
346C43C2×D1711yescharacteristic
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×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×C8—171no—
372C22×C8—171no—
373#125—22no—
374#125C4×D1714yescomplemented
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—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—171no—
386C2×C4⋊C4—171no—
387C2×C4⋊C4—171no—
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—21no—
397C2×C42—341no—
398C2×C42—341no—
399C2×C42—341no—
400C2×C42—684no—
401C2×C42C22⋊D1711yescharacteristic
402C2×C42C22⋊D1711yescharacteristic
403C2×C42C22⋊D1711yescharacteristic
404C2×C42C22⋊D1711yescharacteristic
405C2×C42C22×D1711yescharacteristic
406C2×C42C22×D1711yescharacteristic
407C2×C42C22×D1711yescharacteristic
408#107—174no—
409#107—342no—
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—21no—
423#95—21no—
424#95—21no—
425#95—21no—
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—344no—
442C22×C4—341no—
443C22×C4—341no—
444C22×C4—341no—
445C22×C4—344no—
446C22×C4—344no—
447C22×C4—682no—
448C22×C4—682no—
449C22×C4—682no—
450C22×C4—682no—
451C22×C4—682no—
452C22×C4—682no—
453C22×C4C2×C22⋊D1711yescharacteristic
454C22×C4C2×C22⋊D1711yescharacteristic
455C22×C4C2×C22⋊D1711yescharacteristic
456C22×C4C2×C22⋊D1711yescharacteristic
457C22×C4C2×C22⋊D1711yescharacteristic
458C22×C4C2×C22⋊D1711yescharacteristic
459C22×C4C23×D1711yescharacteristic
460C2×C8—172no—
461C2×C8—172no—
462C2×C8—341no—
463C2×C8—341no—
464C2×C8—341no—
465C2×C8—341no—
466C2×C8—341no—
467C2×C8—341no—
468C2×C8—341no—
469C2×C8—341no—
470C2×C8—341no—
471C2×C8—341no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—172no—
483C4⋊C4—172no—
484C4⋊C4—172no—
485C4⋊C4—341no—
486C4⋊C4—341no—
487C4⋊C4—341no—
488C4⋊C4C2×C4×D1712yes—
489C4⋊C4C2×C4×D1712yes—
490C22⋊C4—344no—
491C22⋊C4—344no—
492C22⋊C4—344no—
493C42—21no—
494C42—21no—
495C42—22no—
496C42—41no—
497C42—41no—
498C42—41no—
499C42—42no—
500C42—42no—
501C42—682no—
502C42—682no—
503C42—682no—
504C42C2×C4×D1712yes—
505C42#6325212yes—
506C42Q8×D1712yes—
507C42#6325612yes—
508C23—342no—
509C23—342no—
510C23—342no—
511C23—681no—
512C23—681no—
513C23—681no—
514C23—681no—
515C23C22×C22⋊D1711yescharacteristic
516D4—174no—
517D4—342no—
518D4—342no—
519D4—342no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—342no—
546C2×C4—342no—
547C2×C4—342no—
548C2×C4—688no—
549C2×C4—688no—
550C2×C4—688no—
551C2×C4—681no—
552C2×C4—681no—
553C2×C4—681no—
554C2×C4—681no—
555C2×C4—681no—
556C2×C4—681no—
557C2×C4C4×C22⋊D1712yes—
558C2×C4C4×C22⋊D1712yes—
559C2×C4#1058763212yes—
560C2×C4#1058763212yes—
561C2×C4#1058763512yes—
562C2×C4C22×C4×D1711yescharacteristic
563C2×C4C2×#6325211yescharacteristic
564C2×C4C2×Q8×D1711yescharacteristic
565C2×C4C2×#6325611yescharacteristic
566C2×C4#41×D1711yescharacteristic
567C2×C4#41×D1711yescharacteristic
568C8—342no—
569C8—342no—
570C8—681no—
571C8—681no—
572C22—344no—
573C22—682no—
574C22—682no—
575C22—682no—
576C22—682no—
577C22—682no—
578C22—682no—
579C22C2×C4×C22⋊D1711yescharacteristic
580C22C2×#1058763211yescharacteristic
581C22#1176331611yescharacteristic
582C22C2×#1058763511yescharacteristic
583C22#1176360711yescharacteristic
584C22#1176360711yescharacteristic
585C22#1176360711yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—22no—
590C4—22no—
591C4—42no—
592C4—42no—
593C4—41no—
594C4—81no—
595C4—81no—
596C4—682no—
597C4—682no—
598C4—682no—
599C4C4×Q8×D1711yescharacteristic
600C4C4×#6325611yescharacteristic
601C4#1176334611yescharacteristic
602C4#1176336311yescharacteristic
603C2—684no—
604C2C4×#1058763211yescharacteristic
605C2#1453758211yescharacteristic
606C2C4×#1058763511yescharacteristic
607C2#1453857711yescharacteristic
608C2#1453860211yescharacteristic
609C2#1453888711yescharacteristic
610C2#1453891811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC4×C17×#125C2
socleC23×C17C22×D4
prime radicalC22×D68C23
cosocleC22×C4×C17C24
hypercenterC4×#125D17
hypocenterC17#21617
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C43 ◁ C4×#125

Complements

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

NH ≅ G/Ntype
3. #14529742C2semidirect
4. #14529550C2semidirect
5. #14529366C2semidirect
6. #14529352C2semidirect
10. #14528682C2semidirect
11. #14528682C2semidirect
12. #14528637C2semidirect
15. C4×C17×#125C2semidirect
20. #11763106C4semidirect
65. C17×#125C2×C4semidirect
117. #10587479C2×C4semidirect
119. #10587478C2×C4semidirect
121. C4⋊#3622C2×C4semidirect
268. C4×#125D17semidirect
374. #125C4×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×#117631062
C42×D682
#145297422
#145295502
#145293662
#145293522
#145290442
#145290302
#145287912
#145286822
#145286372
#145286122
C4×C17×#1252
#2161717