Finite Group Databasebeta

← #17045093

Subgroups — #17045093

Order 4352.

774 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531221C211yesmaximal, characteristic
2#14529395C211yesmaximal, characteristic, complemented
3#14529388C211yesmaximal, characteristic, complemented
4#14529346C211yesmaximal, characteristic, complemented
5#14529339C211yesmaximal, characteristic, complemented
6#14529081C211yesmaximal, characteristic
7#14529076C211yesmaximal, characteristic
8#14528967C211yesmaximal, characteristic, complemented
9#14528962C211yesmaximal, characteristic, complemented
10#14528928C211yesmaximal, characteristic, complemented
11#14528902C211yesmaximal, characteristic, complemented
12#14528851C211yesmaximal, characteristic, complemented
13#14528763C211yesmaximal, characteristic, complemented
14#14528550C211yesmaximal, characteristic, complemented
15C17×#1729C211yesmaximal, characteristic, complemented
16C2×C4×C22⋊D17C2211yescharacteristic
17C2×C42×D17C2211yescharacteristic
18C2×#10587485C2211yescharacteristic
19C2×#10587485C2211yescharacteristic
20C2×#10587485C2211yescharacteristic
21C2×#10587485C2211yescharacteristic
22C2×C22⋊#3622C2211yescharacteristic
23C2×C22⋊#3622C2211yescharacteristic
24C2×#10587491C2211yescharacteristic
25C2×#10587491C2211yescharacteristic
26C2×#10587491C2211yescharacteristic
27C2×#10587491C2211yescharacteristic
28C2×#10587474C2211yescharacteristic
29C2×C4×#3622C2211yescharacteristic
30#11762697C411yescharacteristic
31#11762697C411yescharacteristic
32#11762690C411yescharacteristic
33#11762690C411yescharacteristic
34#11762685C411yescharacteristic
35#11762685C411yescharacteristic
36#11762680C411yescharacteristic
37#11762680C411yescharacteristic
38C4×#63225C2211yescharacteristic
39C4×#63227C2211yescharacteristic
40C4×C22⋊Dic17C2211yescharacteristic
41#11762655C2211yescharacteristic
42#11762564C2211yescharacteristic
43#11762564C2211yescharacteristic
44#11762563C2211yescharacteristic
45#11762563C2211yescharacteristic
46#11762563C2211yescharacteristic
47#11762563C2211yescharacteristic
48#11762563C2211yescharacteristic
49#11762563C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C22×M16×C17C2211yescharacteristic
53C23×C8×C17C2211yescharacteristic
54C2×C17×#117C2211yescharacteristic
55C17×#336C2211yescharacteristic
56C17×#336C2211yescharacteristic
57C17×#336C2211yescharacteristic
58C17×#336C2211yescharacteristic
59C23×C4×C17C2311yescharacteristic
60C22×C22⋊D17C2×C411yescharacteristic
61C22×C4×D17C2×C411yescharacteristic
62C22×C4×D17C2×C411yescharacteristic
63C22×C4×D17C2311yescharacteristic
64C2×M16×C17—22no—
65C2×M16×C17—41no—
66C2×M16×C17—41no—
67C22×C8×C17—22no—
68C22×C8×C17—41no—
69C22×C8×C17—41no—
70C22×C8×C17C2311yescharacteristic
71C22×C8×C17C2311yescharacteristic
72C22×C8×C17C2311yescharacteristic
73C22×C8×C17C2311yescharacteristic
74C17×#117—41no—
75C17×#117—41no—
76C2×C4⋊C4×C17C2×C411yescharacteristic
77C2×C4⋊C4×C17C2×C411yescharacteristic
78C2×C22⋊C4×C17C2×C411yescharacteristic
79C2×C22⋊C4×C17C2×C411yescharacteristic
80C2×C42×C17C2311yescharacteristic
81C2×C42×C17C2311yescharacteristic
82C2×C22⋊Dic17C2×C411yescharacteristic
83C2×C22⋊Dic17C2×C411yescharacteristic
84C2×C22⋊Dic17C2×C411yescharacteristic
85C4×C22⋊D17—22no—
86C4×C22⋊D17—22no—
87C2×#63227C2×C411yescharacteristic
88C2×#63227C2×C411yescharacteristic
89C2×#63227C2×C411yescharacteristic
90C2×#63225C2×C411yescharacteristic
91C2×#63225C2×C411yescharacteristic
92C2×#63225C2×C411yescharacteristic
93C2×C4×Dic17C2×C411yescharacteristic
94C2×C4×Dic17C2×C411yescharacteristic
95C2×C4×Dic17C2311yescharacteristic
96C2×C4×Dic17C2311yescharacteristic
97C2×C4×Dic17C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99C22×#3622C2311yescharacteristic
100C22×#3622C2311yescharacteristic
101C22×#3622C2311yescharacteristic
102C42×D17—41no—
103C42×D17—41no—
104#10587502C2×C411yescharacteristic
105#10587502C2×C411yescharacteristic
106#10587502C2×C411yescharacteristic
107#10587502C2×C411yescharacteristic
108#10587502C2×C411yescharacteristic
109#10587502C2×C411yescharacteristic
110#10587502C2×C411yescharacteristic
111#10587502C2×C411yescharacteristic
112#10587502C2×C411yescharacteristic
113#10587502C2×C411yescharacteristic
114C22⋊#3622—22no—
115C22⋊#3622—22no—
116#10587491—41no—
117#10587491—41no—
118#10587491—41no—
119#10587491—41no—
120#10587485—41no—
121#10587485—41no—
122#10587485—41no—
123#10587485—41no—
124#10587474—41no—
125C4×#3622—41no—
126C24×C17C22×C411yescharacteristic
127C23×D17C22×C411yescharacteristic
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17C22×C411yescharacteristic
133C22×C4×C17C22×C411yescharacteristic
134C22×C4×C17C22×C411yescharacteristic
135C22×C4×C17C22×C411yescharacteristic
136C22×C4×C17C22×C411yescharacteristic
137C22×C4×C17#4112yes—
138C22×C4×C17#4112yes—
139C22×C4×C17C2411yescharacteristic
140C2×C22⋊D17—22no—
141C2×C22⋊D17—22no—
142C2×C22⋊D17—41no—
143C22×Dic17C22×C411yescharacteristic
144C22×Dic17C22×C411yescharacteristic
145C22×Dic17C22×C411yescharacteristic
146C22×Dic17C22×C411yescharacteristic
147C22×Dic17C22×C411yescharacteristic
148C22×Dic17C22×C411yescharacteristic
149C22×Dic17C22×C411yescharacteristic
150C2×C4×D17—22no—
151C2×C4×D17—21no—
152C2×C4×D17—21no—
153C2×C4×D17—21no—
154C2×C4×D17—21no—
155C2×C4×D17—21no—
156C2×C4×D17—21no—
157C2×C4×D17—22no—
158C2×C4×D17—41no—
159C2×C4×D17—41no—
160C2×C4×D17—41no—
161C2×C4×D17—41no—
162M16×C17—44no—
163C2×C8×C17—21no—
164C2×C8×C17—21no—
165C2×C8×C17—21no—
166C2×C8×C17—21no—
167C2×C8×C17—21no—
168C2×C8×C17—21no—
169C2×C8×C17—21no—
170C2×C8×C17—21no—
171C2×C8×C17—21no—
172C2×C8×C17—21no—
173C2×C8×C17—21no—
174C2×C8×C17—21no—
175C2×C8×C17—44no—
176C4⋊C4×C17—41no—
177C4⋊C4×C17—41no—
178C22⋊C4×C17—41no—
179C22⋊C4×C17—41no—
180C42×C17—21no—
181C42×C17—21no—
182C42×C17—21no—
183C42×C17—21no—
184C22⋊Dic17—22no—
185C22⋊Dic17—22no—
186C22⋊Dic17—41no—
187#63227—22no—
188#63227—22no—
189#63227—41no—
190#63225—22no—
191#63225—22no—
192#63225—41no—
193C4×Dic17—21no—
194C4×Dic17—21no—
195C4×Dic17—21no—
196C4×Dic17—21no—
197C4×Dic17—21no—
198C4×Dic17—21no—
199C4×Dic17—41no—
200C4×Dic17—41no—
201C2×#3622—21no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622—21no—
207C2×#3622—21no—
208C2×#3622—21no—
209C2×#3622—21no—
210C2×#3622—21no—
211C2×#3622—21no—
212C2×#3622—21no—
213#22006—171nomaximal, Sylow
214C23×C17—21no—
215C23×C17—21no—
216C23×C17—21no—
217C23×C17—21no—
218C23×C17—21no—
219C23×C17—21no—
220C23×C17#11712yes—
221C23×C17C23×C411yescharacteristic
222C22×D17—21no—
223C22×D17—21no—
224C22×D17—21no—
225C22×D17—21no—
226C22×D17—21no—
227C22×D17—21no—
228C22×D17#13112yes—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—22no—
232C2×C4×C17—22no—
233C2×C4×C17—22no—
234C2×C4×C17—22no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—21no—
244C2×C4×C17—21no—
245C2×C4×C17—21no—
246C2×C4×C17—21no—
247C2×C4×C17#11712yes—
248C2×C4×C17C2×#4111yescharacteristic
249C2×C4×C17C2×#4111yescharacteristic
250C2×C4×C17H(5,𝔽2)11yescharacteristic
251C2×C4×C17H(5,𝔽2)11yescharacteristic
252C2×C4×C17H(5,𝔽2)11yescharacteristic
253C2×C4×C172−1+411yescharacteristic
254C22⋊D17—44no—
255C2×Dic17—21no—
256C2×Dic17—21no—
257C2×Dic17—21no—
258C2×Dic17—21no—
259C2×Dic17—21no—
260C2×Dic17—21no—
261C2×Dic17—21no—
262C2×Dic17—21no—
263C2×Dic17—21no—
264C2×Dic17—21no—
265C2×Dic17—21no—
266C2×Dic17—21no—
267C2×Dic17—21no—
268C2×Dic17—21no—
269C2×Dic17—21no—
270C2×Dic17—21no—
271C2×Dic17—21no—
272C2×Dic17—21no—
273C2×Dic17—21no—
274C2×Dic17—21no—
275C2×Dic17#13112yes—
276C4×D17—44no—
277C4×D17—44no—
278C4×D17—41no—
279C4×D17—41no—
280C4×D17—41no—
281C4×D17—41no—
282C8×C17—41no—
283C8×C17—41no—
284C8×C17—41no—
285C8×C17—41no—
286#3622—41no—
287#3622—41no—
288#3622—41no—
289#3622—41no—
290#2290—171no—
291#1924—171no—
292#1923—171no—
293#1917—171no—
294#1916—171no—
295#1870—171no—
296#1869—171no—
297#1851—171no—
298#1850—171no—
299#1840—171no—
300#1832—171no—
301#1817—171no—
302#1797—171no—
303#1750—171no—
304#1729D1711yescharacteristic, complemented
305C22×C17—22no—
306C22×C17—22no—
307C22×C17—22no—
308C22×C17—41no—
309C22×C17—41no—
310C22×C17—41no—
311C22×C17—41no—
312C22×C17C2×#11711yescharacteristic
313C22×C17#51811yescharacteristic
314C22×C17#52011yescharacteristic
315C22×C17C2×#13111yescharacteristic
316C22×C17C2×#13111yescharacteristic
317C22×C17#56811yescharacteristic
318C22×C17#56811yescharacteristic
319C2×D17—22no—
320C2×D17—22no—
321C2×D17—22no—
322C2×D17—41no—
323C2×D17—41no—
324C2×D17—41no—
325C2×D17—41no—
326C4×C17—22no—
327C4×C17—41no—
328C4×C17—41no—
329C4×C17—41no—
330C4×C17—41no—
331C4×C17#53811yescharacteristic
332C4×C17#53811yescharacteristic
333C4×C17#53911yescharacteristic
334C4×C17#53911yescharacteristic
335Dic17—22no—
336Dic17—41no—
337Dic17—41no—
338Dic17—41no—
339Dic17—41no—
340Dic17—41no—
341Dic17—41no—
342C22×M16C2×D1711yescharacteristic
343C23×C8C2×D1711yescharacteristic
344C2×C4×D4—171no—
345C2×#117C2×D1711yescharacteristic
346C22×C42—171no—
347C2×C4⋊C8—171no—
348C2×C4⋊C8—171no—
349C2×C4⋊C8—171no—
350C2×C4⋊C8—171no—
351C2×C22⋊C8—171no—
352C2×C22⋊C8—171no—
353C2×C22⋊C8—171no—
354C2×C22⋊C8—171no—
355C2×C22⋊C8—171no—
356C2×C22⋊C8—171no—
357C2×#97—171no—
358C2×C4×C8—171no—
359#388—171no—
360#388—171no—
361#387—171no—
362#387—171no—
363#386—171no—
364#386—171no—
365#385—171no—
366#385—171no—
367C4×C4⋊C4—171no—
368C4×C22⋊C4—171no—
369C4×C22⋊C4—171no—
370#376—171no—
371#336—171no—
372#336—171no—
373#336—171no—
374#336—171no—
375#336—171no—
376#336—171no—
377#336—171no—
378#336—171no—
379#336—171no—
380#336—171no—
381#336C2×D1711yescharacteristic
382#336C2×D1711yescharacteristic
383#336C2×D1711yescharacteristic
384#336C2×D1711yescharacteristic
385C2×C17—42no—
386C2×C17#231811yescharacteristic
387C2×C17#292411yescharacteristic
388C2×C17#292511yescharacteristic
389C2×C17#297711yescharacteristic
390C2×C17#297811yescharacteristic
391C2×C17#299211yescharacteristic
392C2×C17#299311yescharacteristic
393D17—42no—
394C22×D4—171no—
395C23×C4—171no—
396C23×C4—171no—
397C23×C4—171no—
398C23×C4C22×D1711yescharacteristic
399C2×M16—22no—
400C2×M16—41no—
401C2×M16—41no—
402C22×C8—22no—
403C22×C8—41no—
404C22×C8—41no—
405C22×C8—171no—
406C22×C8—171no—
407C22×C8—171no—
408C22×C8—171no—
409C22×C8C22×D1711yescharacteristic
410C22×C8C22×D1711yescharacteristic
411C22×C8C22×D1711yescharacteristic
412C22×C8C22×D1711yescharacteristic
413C4×D4—342no—
414C4×D4—342no—
415#117—41no—
416#117—41no—
417C2×C4⋊C4—171no—
418C2×C4⋊C4—171no—
419C2×C4⋊C4—171no—
420C2×C4⋊C4C4×D1711yescharacteristic
421C2×C4⋊C4C4×D1711yescharacteristic
422C2×C22⋊C4—171no—
423C2×C22⋊C4—171no—
424C2×C22⋊C4—171no—
425C2×C22⋊C4—171no—
426C2×C22⋊C4—171no—
427C2×C22⋊C4—171no—
428C2×C22⋊C4C4×D1711yescharacteristic
429C2×C22⋊C4C4×D1711yescharacteristic
430C2×C42—171no—
431C2×C42—171no—
432C2×C42—171no—
433C2×C42—171no—
434C2×C42—171no—
435C2×C42—681no—
436C2×C42—681no—
437C2×C42C22×D1711yescharacteristic
438C2×C42C22×D1711yescharacteristic
439C4⋊C8—681no—
440C4⋊C8—681no—
441C4⋊C8—681no—
442C4⋊C8—681no—
443C22⋊C8—342no—
444C22⋊C8—342no—
445C22⋊C8—681no—
446C22⋊C8—681no—
447C22⋊C8—681no—
448C22⋊C8—681no—
449#97—681no—
450C4×C8—681no—
451#95—171no—
452#95—171no—
453#95—171no—
454#95—171no—
455#95—171no—
456#95—171no—
457#95—171no—
458#95—171no—
459#95—171no—
460#95—171no—
461C17#2200611yescharacteristic, complemented, Sylow
462C24—171no—
463C24C2×C4×D1711yescharacteristic
464C2×D4—342no—
465C2×D4—342no—
466C2×D4—681no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—171no—
472C22×C4—171no—
473C22×C4—171no—
474C22×C4—171no—
475C22×C4—171no—
476C22×C4—171no—
477C22×C4—171no—
478C22×C4—342no—
479C22×C4—342no—
480C22×C4—341no—
481C22×C4—341no—
482C22×C4—341no—
483C22×C4—341no—
484C22×C4—341no—
485C22×C4—341no—
486C22×C4—681no—
487C22×C4—681no—
488C22×C4—681no—
489C22×C4—681no—
490C22×C4C2×C4×D1711yescharacteristic
491C22×C4C2×C4×D1711yescharacteristic
492C22×C4C2×C4×D1711yescharacteristic
493C22×C4C2×C4×D1711yescharacteristic
494C22×C4C2×C4×D1711yescharacteristic
495C22×C4#6325212yes—
496C22×C4#6325212yes—
497C22×C4C23×D1711yescharacteristic
498M16—44no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—44no—
512C2×C8—341no—
513C2×C8—341no—
514C2×C8—341no—
515C2×C8—341no—
516C2×C8—341no—
517C2×C8—341no—
518C2×C8—341no—
519C2×C8—341no—
520C2×C8—341no—
521C2×C8—341no—
522C2×C8—341no—
523C2×C8—341no—
524C4⋊C4—41no—
525C4⋊C4—41no—
526C4⋊C4—342no—
527C4⋊C4—342no—
528C4⋊C4—681no—
529C22⋊C4—41no—
530C22⋊C4—41no—
531C22⋊C4—342no—
532C22⋊C4—342no—
533C22⋊C4—342no—
534C22⋊C4—342no—
535C22⋊C4—681no—
536C22⋊C4—681no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—341no—
542C42—341no—
543C42—341no—
544C42—341no—
545C42—341no—
546C42—341no—
547C42—681no—
548C42—681no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—172no—
556C23—341no—
557C23—341no—
558C23—341no—
559C23—341no—
560C23—341no—
561C23—341no—
562C23#1058755812yes—
563C23C22×C4×D1711yescharacteristic
564D4—684no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—172no—
584C2×C4—341no—
585C2×C4—341no—
586C2×C4—341no—
587C2×C4—341no—
588C2×C4—341no—
589C2×C4—341no—
590C2×C4—341no—
591C2×C4—341no—
592C2×C4—341no—
593C2×C4—341no—
594C2×C4—341no—
595C2×C4—341no—
596C2×C4—341no—
597C2×C4—341no—
598C2×C4—341no—
599C2×C4—341no—
600C2×C4—341no—
601C2×C4—341no—
602C2×C4—341no—
603C2×C4—341no—
604C2×C4—684no—
605C2×C4—684no—
606C2×C4—681no—
607C2×C4—681no—
608C2×C4—681no—
609C2×C4—681no—
610C2×C4#1058755812yes—
611C2×C4C2×#6325211yescharacteristic
612C2×C4C2×#6325211yescharacteristic
613C2×C4#1058769111yescharacteristic
614C2×C4#1058769111yescharacteristic
615C2×C4#1058769111yescharacteristic
616C2×C4#1058769411yescharacteristic
617C8—41no—
618C8—41no—
619C8—41no—
620C8—41no—
621C8—681no—
622C8—681no—
623C8—681no—
624C8—681no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—342no—
633C22—342no—
634C22—342no—
635C22—681no—
636C22—681no—
637C22—681no—
638C22—681no—
639C22C2×#1058755811yescharacteristic
640C22#1176328211yescharacteristic
641C22#1176329011yescharacteristic
642C22#131×D1711yescharacteristic
643C22#131×D1711yescharacteristic
644C22#1176357411yescharacteristic
645C22#1176357411yescharacteristic
646C4—22no—
647C4—41no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—342no—
652C4—681no—
653C4—681no—
654C4—681no—
655C4—681no—
656C4—681no—
657C4—681no—
658C4#1176339711yescharacteristic
659C4#1176339711yescharacteristic
660C4#1176340611yescharacteristic
661C4#1176340611yescharacteristic
662C2—42no—
663C2—682no—
664C2#1453141811yescharacteristic
665C2#1453743911yescharacteristic
666C2#1453744811yescharacteristic
667C2#1453783611yescharacteristic
668C2#1453784211yescharacteristic
669C2#1453792111yescharacteristic
670C2#1453793111yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D17
commutator, socleC23×C17C23×C4
FittingC17×#1729C2
prime radicalC22×C22⋊D17C2×C4
cosocleC22×C4×C17C24
hypercenter#1729D17
hypocenterC17#22006
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ #1729

Complements

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

NH ≅ G/Ntype
2. #14529395C2semidirect
3. #14529388C2semidirect
4. #14529346C2semidirect
5. #14529339C2semidirect
8. #14528967C2semidirect
9. #14528962C2semidirect
10. #14528928C2semidirect
11. #14528902C2semidirect
12. #14528851C2semidirect
13. #14528763C2semidirect
14. #14528550C2semidirect
15. C17×#1729C2semidirect
304. #1729D17semidirect
461. C17#22006semidirect

Maximal subgroups

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

subgroupindex
#145312212
#145293952
#145293882
#145293462
#145293392
#145290812
#145290762
#145289672
#145289622
#145289282
#145289022
#145288512
#145287632
#145285502
C17×#17292
#2200617