Finite Group Databasebeta

← #17051910

Subgroups — #17051910

Order 4352.

780 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762867C211yesmaximal, characteristic
2C2×#11762863C211yesmaximal, characteristic
3#14531268C211yesmaximal, characteristic
4#14529505C211yesmaximal, characteristic, complemented
5#14529500C211yesmaximal, characteristic, complemented
6#14529347C211yesmaximal, characteristic, complemented
7#14529347C211yesmaximal, characteristic, complemented
8#14529347C211yesmaximal, characteristic, complemented
9#14529340C211yesmaximal, characteristic, complemented
10#14528853C211yesmaximal, characteristic
11#14528850C211yesmaximal, characteristic
12#14528661C211yesmaximal, characteristic, complemented
13#14528654C211yesmaximal, characteristic, complemented
14#14528550C211yesmaximal, characteristic
15C17×#1798C211yesmaximal, characteristic
16C2×#10587612C2211yescharacteristic
17C22×C4×Dic17C2211yescharacteristic
18#11762867—24no—
19#11762863—24no—
20C2×#10587485C2211yescharacteristic
21C2×#10587485C2211yescharacteristic
22C2×#10587485C2211yescharacteristic
23C2×#10587485C2211yescharacteristic
24C2×C22⋊#3622C2211yescharacteristic
25C2×C22⋊#3622C2211yescharacteristic
26C2×#10587474C2211yescharacteristic
27C2×#10587486C2211yescharacteristic
28C2×#10587486C2211yescharacteristic
29C2×#10587486C2211yescharacteristic
30C2×C4×#3622C2211yescharacteristic
31C2×C8×Dic17C2211yescharacteristic
32#11762708C411yescharacteristic
33#11762708C411yescharacteristic
34#11762708C411yescharacteristic
35#11762708C411yescharacteristic
36#11762684C411yescharacteristic
37#11762684C411yescharacteristic
38#11762684C411yescharacteristic
39#11762684C411yescharacteristic
40C4×C4⋊Dic17C2211yescharacteristic
41C4×C4⋊Dic17C2211yescharacteristic
42C4×C4⋊Dic17C2211yescharacteristic
43C4×C4⋊Dic17C2211yescharacteristic
44#11762564C2211yescharacteristic
45#11762564C2211yescharacteristic
46#11762563C2211yescharacteristic
47#11762563C2211yescharacteristic
48#11762563C2211yescharacteristic
49#11762563C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52#11762563C2211yescharacteristic
53#11762563C2211yescharacteristic
54C2×C17×#117C2211yescharacteristic
55C2×C17×C4⋊C8C2211yescharacteristic
56C2×C17×C4⋊C8C2211yescharacteristic
57C2×C17×C22⋊C8C2211yescharacteristic
58C2×C17×C22⋊C8C2211yescharacteristic
59C17×#336C2211yescharacteristic
60C17×#336C2211yescharacteristic
61C23×C4×C17C2311yescharacteristic
62C23×Dic17C2×C411yescharacteristic
63C23×Dic17C2×C411yescharacteristic
64C22×C8×C17C2311yescharacteristic
65C22×C8×C17C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C22×C8×C17C2311yescharacteristic
68C17×#117—22no—
69C17×#117—22no—
70C2×C4⋊C4×C17C2×C411yescharacteristic
71C2×C4⋊C4×C17C2×C411yescharacteristic
72C2×C22⋊C4×C17C2×C411yescharacteristic
73C2×C22⋊C4×C17C2×C411yescharacteristic
74C2×C42×C17C2311yescharacteristic
75C2×C42×C17C2311yescharacteristic
76C2×C22⋊Dic17C2×C411yescharacteristic
77C2×C22⋊Dic17C2×C411yescharacteristic
78#10587612—22no—
79#10587612—22no—
80C2×C4⋊Dic17C2×C411yescharacteristic
81C2×C4⋊Dic17C2×C411yescharacteristic
82C2×C4⋊Dic17C2×C411yescharacteristic
83C2×C4⋊Dic17C2×C411yescharacteristic
84C2×C4⋊Dic17C2×C411yescharacteristic
85C2×C4⋊Dic17C2×C411yescharacteristic
86C2×C4⋊Dic17C2×C411yescharacteristic
87C2×C4⋊Dic17C2×C411yescharacteristic
88C2×C4⋊Dic17C2×C411yescharacteristic
89C2×C4⋊Dic17C2×C411yescharacteristic
90C2×C4×Dic17—22no—
91C2×C4×Dic17—22no—
92C2×C4×Dic17C2×C411yescharacteristic
93C2×C4×Dic17C2×C411yescharacteristic
94C2×C4×Dic17C2311yescharacteristic
95C2×C4×Dic17C2311yescharacteristic
96C2×C4×Dic17C2311yescharacteristic
97C2×C4×Dic17C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99C22×#3622C2311yescharacteristic
100C22×#3622C2311yescharacteristic
101C22×#3622C2311yescharacteristic
102C17×C4⋊C8—22no—
103C17×C4⋊C8—22no—
104C17×C22⋊C8—22no—
105C17×C22⋊C8—22no—
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⋊#3622D414yes—
116#10587486—22no—
117#10587486—22no—
118#10587486—22no—
119#10587485—22no—
120#10587485—22no—
121#10587485—22no—
122#10587485—22no—
123C8×Dic17—22no—
124#10587474—22no—
125C4×#3622—22no—
126C24×C17C22×C411yescharacteristic
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17C22×C411yescharacteristic
132C22×C4×C17C22×C411yescharacteristic
133C22×C4×C17C22×C411yescharacteristic
134C22×C4×C17C22×C411yescharacteristic
135C22×C4×C17C22×C411yescharacteristic
136C22×C4×C17C2×D412yes—
137C22×C4×C17#4112yes—
138C22×C4×C17C2411yescharacteristic
139C22×Dic17—22no—
140C22×Dic17—22no—
141C22×Dic17—22no—
142C22×Dic17—41no—
143C22×Dic17—41no—
144C22×Dic17—41no—
145C22×Dic17C22×C411yescharacteristic
146C22×Dic17C22×C411yescharacteristic
147C22×Dic17C22×C411yescharacteristic
148C22×Dic17C22×C411yescharacteristic
149C22×Dic17C22×C411yescharacteristic
150C22×Dic17C22×C411yescharacteristic
151C22×Dic17C22×C411yescharacteristic
152C22×Dic17C22×C411yescharacteristic
153C2×C8×C17—21no—
154C2×C8×C17—21no—
155C2×C8×C17—21no—
156C2×C8×C17—21no—
157C2×C8×C17—21no—
158C2×C8×C17—21no—
159C2×C8×C17—21no—
160C2×C8×C17—21no—
161C2×C8×C17—21no—
162C2×C8×C17—21no—
163C2×C8×C17—21no—
164C2×C8×C17—21no—
165C4⋊C4×C17—22no—
166C4⋊C4×C17—22no—
167C22⋊C4×C17—22no—
168C22⋊C4×C17—22no—
169C42×C17—21no—
170C42×C17—21no—
171C42×C17#4112yes—
172C42×C17#4112yes—
173C22⋊Dic17—22no—
174C22⋊Dic17—22no—
175C4⋊Dic17—22no—
176C4⋊Dic17—22no—
177C4⋊Dic17—22no—
178C4⋊Dic17—22no—
179C4⋊Dic17—41no—
180C4⋊Dic17—41no—
181C4⋊Dic17—41no—
182C4⋊Dic17—41no—
183C4⋊Dic17—41no—
184C4⋊Dic17—41no—
185C4×Dic17—22no—
186C4×Dic17—22no—
187C4×Dic17—21no—
188C4×Dic17—21no—
189C4×Dic17—21no—
190C4×Dic17—21no—
191C4×Dic17—21no—
192C4×Dic17—21no—
193C4×Dic17—21no—
194C4×Dic17—21no—
195C2×#3622—21no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622—21no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622C2×D412yes—
206C2×#3622C2×D412yes—
207#22605—171nomaximal, Sylow
208C23×C17—21no—
209C23×C17—21no—
210C23×C17—21no—
211C23×C17—21no—
212C23×C17—21no—
213C23×C17—21no—
214C23×C17C4×D412yes—
215C23×C17C23×C411yescharacteristic
216C2×C4×C17—21no—
217C2×C4×C17—21no—
218C2×C4×C17—21no—
219C2×C4×C17—21no—
220C2×C4×C17—21no—
221C2×C4×C17—21no—
222C2×C4×C17—22no—
223C2×C4×C17—22no—
224C2×C4×C17—22no—
225C2×C4×C17—22no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17#11712yes—
233C2×C4×C17#11712yes—
234C2×C4×C17C4×D412yes—
235C2×C4×C17C22×D411yescharacteristic
236C2×C4×C17C2×#4111yescharacteristic
237C2×C4×C17C2×#4111yescharacteristic
238C2×C4×C17C2×#4111yescharacteristic
239C2×C4×C172−1+411yescharacteristic
240C2×C4×C172−1+411yescharacteristic
241C2×Dic17—21no—
242C2×Dic17—21no—
243C2×Dic17—21no—
244C2×Dic17—21no—
245C2×Dic17—21no—
246C2×Dic17—21no—
247C2×Dic17—21no—
248C2×Dic17—21no—
249C2×Dic17—21no—
250C2×Dic17—21no—
251C2×Dic17—21no—
252C2×Dic17—21no—
253C2×Dic17—21no—
254C2×Dic17—21no—
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—44no—
265C2×Dic17—44no—
266C2×Dic17#13112yes—
267C8×C17—41no—
268C8×C17—41no—
269C8×C17—41no—
270C8×C17—41no—
271#3622—22no—
272#3622—22no—
273#3622—41no—
274#3622—41no—
275C2×#433—171no—
276C2×#432—171no—
277#2296—171no—
278#1940—171no—
279#1939—171no—
280#1917—171no—
281#1917—171no—
282#1917—171no—
283#1916—171no—
284#1818—171no—
285#1817—171no—
286#1798D1711yescharacteristic
287#1771—171no—
288#1770—171no—
289#1750—171no—
290C22×C17—22no—
291C22×C17—22no—
292C22×C17—22no—
293C22×C17—41no—
294C22×C17—41no—
295C22×C17—41no—
296C22×C17—41no—
297C22×C17C2×#11711yescharacteristic
298C22×C17C2×C4×D411yescharacteristic
299C22×C17#51911yescharacteristic
300C22×C17C2×#13111yescharacteristic
301C22×C17#56811yescharacteristic
302C22×C17#56811yescharacteristic
303C22×C17#56811yescharacteristic
304C4×C17—22no—
305C4×C17—22no—
306C4×C17—22no—
307C4×C17—41no—
308C4×C17—41no—
309C4×C17#54711yescharacteristic
310C4×C17#54711yescharacteristic
311C4×C17#55211yescharacteristic
312C4×C17#55211yescharacteristic
313Dic17—22no—
314Dic17—41no—
315Dic17—41no—
316Dic17—41no—
317Dic17—41no—
318Dic17—41no—
319Dic17—41no—
320Dic17—41no—
321C2×#117—171no—
322C2×#117C2×D1711yescharacteristic
323C22×C42—171no—
324#433—344no—
325#432—344no—
326C2×C4⋊C8—171no—
327C2×C4⋊C8—171no—
328C2×C4⋊C8—171no—
329C2×C4⋊C8—171no—
330C2×C4⋊C8C2×D1711yescharacteristic
331C2×C4⋊C8C2×D1711yescharacteristic
332C2×C22⋊C8—171no—
333C2×C22⋊C8—171no—
334C2×C22⋊C8C2×D1711yescharacteristic
335C2×C22⋊C8C2×D1711yescharacteristic
336C2×#97—171no—
337C2×#97—171no—
338C2×#97—171no—
339C2×#97—171no—
340C2×C4×C8—171no—
341C2×C4×C8—171no—
342#389—171no—
343#389—171no—
344#389—171no—
345#389—171no—
346#385—171no—
347#385—171no—
348#385—171no—
349#385—171no—
350C4×C4⋊C4—171no—
351C4×C4⋊C4—171no—
352C4×C4⋊C4—171no—
353C4×C4⋊C4—171no—
354#336—171no—
355#336—171no—
356#336—171no—
357#336—171no—
358#336—171no—
359#336—171no—
360#336—171no—
361#336—171no—
362#336—171no—
363#336—171no—
364#336C2×D1711yescharacteristic
365#336C2×D1711yescharacteristic
366C2×C17—42no—
367C2×C17#234411yescharacteristic
368C2×C17#292311yescharacteristic
369C2×C17#292511yescharacteristic
370C2×C17#293011yescharacteristic
371C2×C17#293311yescharacteristic
372C2×C17#298511yescharacteristic
373C2×C17#298711yescharacteristic
374C23×C4—171no—
375C23×C4—171no—
376C23×C4C22×D1711yescharacteristic
377C22×C8—171no—
378C22×C8—171no—
379C22×C8—171no—
380C22×C8—171no—
381C22×C8C22×D1711yescharacteristic
382C22×C8C22×D1711yescharacteristic
383C22×C8C22×D1711yescharacteristic
384C22×C8C22×D1711yescharacteristic
385#117—22no—
386#117—22no—
387#117—342no—
388#117—342no—
389C2×C4⋊C4—171no—
390C2×C4⋊C4—171no—
391C2×C4⋊C4—171no—
392C2×C4⋊C4—171no—
393C2×C4⋊C4—171no—
394C2×C4⋊C4—171no—
395C2×C4⋊C4—171no—
396C2×C4⋊C4—171no—
397C2×C4⋊C4—171no—
398C2×C4⋊C4—171no—
399C2×C4⋊C4C4×D1711yescharacteristic
400C2×C4⋊C4C4×D1711yescharacteristic
401C2×C22⋊C4—171no—
402C2×C22⋊C4—171no—
403C2×C22⋊C4C4×D1711yescharacteristic
404C2×C22⋊C4C4×D1711yescharacteristic
405C2×C42—171no—
406C2×C42—171no—
407C2×C42—171no—
408C2×C42—171no—
409C2×C42—171no—
410C2×C42—171no—
411C2×C42—342no—
412C2×C42—342no—
413C2×C42C22×D1711yescharacteristic
414C2×C42C22×D1711yescharacteristic
415C4⋊C8—22no—
416C4⋊C8—22no—
417C4⋊C8—342no—
418C4⋊C8—342no—
419C4⋊C8—342no—
420C4⋊C8—342no—
421C22⋊C8—22no—
422C22⋊C8—22no—
423C22⋊C8—174no—
424C22⋊C8—342no—
425#97—342no—
426#97—342no—
427#97—342no—
428#97—342no—
429C4×C8—342no—
430C4×C8—342no—
431#95—171no—
432#95—171no—
433#95—171no—
434#95—171no—
435#95—171no—
436#95—171no—
437#95—171no—
438#95—171no—
439C17#2260511yescharacteristic, complemented, Sylow
440C24C2×C4×D1711yescharacteristic
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—171no—
446C22×C4—171no—
447C22×C4—171no—
448C22×C4—171no—
449C22×C4—171no—
450C22×C4—171no—
451C22×C4—171no—
452C22×C4—171no—
453C22×C4—342no—
454C22×C4—342no—
455C22×C4—342no—
456C22×C4—681no—
457C22×C4—681no—
458C22×C4—681no—
459C22×C4C2×C4×D1711yescharacteristic
460C22×C4C2×C4×D1711yescharacteristic
461C22×C4C2×C4×D1711yescharacteristic
462C22×C4C2×C4×D1711yescharacteristic
463C22×C4C2×C4×D1711yescharacteristic
464C22×C4D4×D1712yes—
465C22×C4#6325612yes—
466C22×C4C23×D1711yescharacteristic
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—172no—
480C2×C8—172no—
481C2×C8—341no—
482C2×C8—341no—
483C2×C8—341no—
484C2×C8—341no—
485C2×C8—341no—
486C2×C8—341no—
487C2×C8—341no—
488C2×C8—341no—
489C2×C8—341no—
490C2×C8—341no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—342no—
494C4⋊C4—342no—
495C4⋊C4—342no—
496C4⋊C4—342no—
497C4⋊C4—681no—
498C4⋊C4—681no—
499C4⋊C4—681no—
500C4⋊C4—681no—
501C4⋊C4—681no—
502C4⋊C4—681no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—342no—
506C22⋊C4—342no—
507C42—21no—
508C42—21no—
509C42—341no—
510C42—341no—
511C42—341no—
512C42—341no—
513C42—341no—
514C42—341no—
515C42—341no—
516C42—341no—
517C42—342no—
518C42—342no—
519C42#6325412yes—
520C42#6325412yes—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23#1058757912yes—
528C23C22×C4×D1711yescharacteristic
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—172no—
546C2×C4—341no—
547C2×C4—341no—
548C2×C4—341no—
549C2×C4—341no—
550C2×C4—341no—
551C2×C4—341no—
552C2×C4—341no—
553C2×C4—341no—
554C2×C4—341no—
555C2×C4—341no—
556C2×C4—341no—
557C2×C4—341no—
558C2×C4—341no—
559C2×C4—341no—
560C2×C4—341no—
561C2×C4—341no—
562C2×C4—341no—
563C2×C4—341no—
564C2×C4—341no—
565C2×C4—341no—
566C2×C4—341no—
567C2×C4—341no—
568C2×C4—341no—
569C2×C4—684no—
570C2×C4—684no—
571C2×C4#1058756312yes—
572C2×C4#1058756312yes—
573C2×C4#1058757912yes—
574C2×C4C2×D4×D1711yescharacteristic
575C2×C4C2×#6325411yescharacteristic
576C2×C4C2×#6325411yescharacteristic
577C2×C4C2×#6325611yescharacteristic
578C2×C4#1058769711yescharacteristic
579C2×C4#1058769711yescharacteristic
580C8—41no—
581C8—41no—
582C8—41no—
583C8—41no—
584C8—342no—
585C8—342no—
586C8—681no—
587C8—681no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22C2×#1058756311yescharacteristic
596C22C2×#1058757911yescharacteristic
597C22#1176328611yescharacteristic
598C22#131×D1711yescharacteristic
599C22#1176357411yescharacteristic
600C22#1176357411yescharacteristic
601C22#1176357411yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—41no—
606C4—41no—
607C4—342no—
608C4—681no—
609C4—681no—
610C4—681no—
611C4—681no—
612C4—681no—
613C4—681no—
614C4—681no—
615C4#1176345911yescharacteristic
616C4#1176345911yescharacteristic
617C4#1176350111yescharacteristic
618C4#1176350111yescharacteristic
619C2—42no—
620C2#1453165511yescharacteristic
621C2#1453743111yescharacteristic
622C2#1453744611yescharacteristic
623C2#1453748611yescharacteristic
624C2#1453750811yescharacteristic
625C2#1453787611yescharacteristic
626C2#1453788811yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D17
commutator, socleC23×C17C23×C4
FittingC17×#1798C2
prime radicalC24×C17C22×C4
cosocleC22×C4×C17C24
hypercenter#1798D17
hypocenterC17#22605
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ #1798

Complements

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

NH ≅ G/Ntype
4. #14529505C2semidirect
5. #14529500C2semidirect
6. #14529347C2semidirect
7. #14529347C2semidirect
8. #14529347C2semidirect
9. #14529340C2semidirect
12. #14528661C2semidirect
13. #14528654C2semidirect
439. C17#22605semidirect

Maximal subgroups

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

subgroupindex
C2×#117628672
C2×#117628632
#145312682
#145295052
#145295002
#145293472
#145293402
#145288532
#145288502
#145286612
#145286542
#145285502
C17×#17982
#2260517