Finite Group Databasebeta

← #17043911

Subgroups — #17043911

Order 4352.

1170 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763337C211yesmaximal, characteristic
2C2×#11762861C211yesmaximal, characteristic, complemented
3C2×C4⋊C8×D17C211yesmaximal, characteristic, complemented
4#14529718C212yesmaximal, complemented
5#14529576C212yesmaximal, complemented
6#14529075C212yesmaximal, complemented
7#14529073C212yesmaximal, complemented
8#14528768C211yesmaximal, characteristic, complemented
9#14528765C211yesmaximal, characteristic, complemented
10#14528765C211yesmaximal, characteristic, complemented
11C17×#1798C211yesmaximal, characteristic, complemented
12C22×C8×D17C2212yes—
13#11763337—24no—
14#11763337—24no—
15C2×#10587584C411yescharacteristic
16C2×#10587565C411yescharacteristic
17C2×#10587575C411yescharacteristic
18C2×#10587571C411yescharacteristic
19C2×#10587580C411yescharacteristic
20C2×#10587569C411yescharacteristic
21C2×#10587582C411yescharacteristic
22C2×#10587570C411yescharacteristic
23C2×C4×Dic34C2211yescharacteristic
24C2×C4×D68C2211yescharacteristic
25C2×C4×C22⋊D17C2211yescharacteristic
26C2×C4×C22⋊D17C2211yescharacteristic
27C2×#10587558C2211yescharacteristic
28C2×C42×D17C2211yescharacteristic
29#11762861—24no—
30C2×C4⋊#3622C2211yescharacteristic
31C2×C4⋊#3622C2211yescharacteristic
32C2×#10587485C2212yes—
33C4⋊C8×D17—24no—
34C2×C22⋊#3622C2212yes—
35C2×#10587491C2212yes—
36C2×#10587491C2212yes—
37C2×#10587491C2212yes—
38C2×C8×Dic17C2212yes—
39#11762564C2212yes—
40#11762563C2212yes—
41#11762563C2212yes—
42C2×C17×#117C2211yescharacteristic
43C2×C17×C4⋊C8C2211yescharacteristic
44C2×C17×C4⋊C8C2211yescharacteristic
45C2×C17×C22⋊C8C2212yes—
46C17×#336C2212yes—
47C23×C4×C17C2311yescharacteristic
48C22×C22⋊D17C2×C411yescharacteristic
49C22×C22⋊D17C2×C411yescharacteristic
50C22×D68C2×C411yescharacteristic
51C22×C4×D17C2×C411yescharacteristic
52C22×C4×D17C2×C411yescharacteristic
53C22×C4×D17C2311yescharacteristic
54C22×C4×D17C2311yescharacteristic
55C22×Dic34C2×C411yescharacteristic
56C22×C8×C17C2312yes—
57C22×C8×C17C2312yes—
58C17×#117—22no—
59C17×#117—22no—
60C2×C4⋊C4×C17C2×C411yescharacteristic
61C2×C4⋊C4×C17C2×C411yescharacteristic
62C2×C22⋊C4×C17C2×C411yescharacteristic
63C2×C22⋊C4×C17C2×C411yescharacteristic
64C2×C42×C17C2311yescharacteristic
65C2×C42×C17C2311yescharacteristic
66C2×C22⋊Dic17C2×C411yescharacteristic
67C2×C22⋊Dic17C2×C411yescharacteristic
68C4×C22⋊D17—22no—
69C4×C22⋊D17—22no—
70C4×C22⋊D17—22no—
71C4×C22⋊D17—22no—
72C2×#63227C2×C411yescharacteristic
73C2×#63227C2×C411yescharacteristic
74C2×#63227C2×C411yescharacteristic
75C2×#63227C2×C411yescharacteristic
76C2×#63227C2×C411yescharacteristic
77C2×#63227C2×C411yescharacteristic
78C2×C4⋊Dic17C2×C411yescharacteristic
79C2×C4⋊Dic17C2×C411yescharacteristic
80C2×#63225C2×C411yescharacteristic
81C2×#63225C2×C411yescharacteristic
82C2×#63225C2×C411yescharacteristic
83C2×#63225C2×C411yescharacteristic
84C2×#63225C2×C411yescharacteristic
85C2×#63225C2×C411yescharacteristic
86C2×C4×Dic17C2×C411yescharacteristic
87C2×C4×Dic17C2×C411yescharacteristic
88C2×C4×Dic17C2311yescharacteristic
89C2×C4×Dic17C2311yescharacteristic
90C22×#3622C2312yes—
91C22×#3622C2312yes—
92C2×C8×D17—24no—
93C2×C8×D17—42no—
94C2×C8×D17—42no—
95#10587584—24no—
96#10587582—24no—
97#10587580—24no—
98#10587575—24no—
99#10587571—24no—
100#10587570—24no—
101#10587569—24no—
102#10587565—24no—
103C4×D68—22no—
104C4×D68—22no—
105#10587558—22no—
106#10587558—22no—
107C42×D17D414yes—
108C42×D17D414yes—
109C4×Dic34—22no—
110C4×Dic34—22no—
111C17×C4⋊C8—22no—
112C17×C4⋊C8—22no—
113C17×C22⋊C8—42no—
114C22⋊#3622—42no—
115#10587491—24no—
116#10587491—42no—
117#10587491—42no—
118#10587485—24no—
119C8×Dic17—24no—
120C4⋊#3622—22no—
121C4⋊#3622—22no—
122C24×C17C22×C411yescharacteristic
123C23×D17C22×C411yescharacteristic
124C23×D17C22×C411yescharacteristic
125C22×C4×C17—21no—
126C22×C4×C17—21no—
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×C17C2411yescharacteristic
137C2×C22⋊D17—22no—
138C2×C22⋊D17—24no—
139C2×C22⋊D17—22no—
140C2×C22⋊D17—24no—
141C22×Dic17C22×C411yescharacteristic
142C22×Dic17C22×C411yescharacteristic
143C22×Dic17C22×C411yescharacteristic
144C22×Dic17C22×C411yescharacteristic
145C22×Dic17C22×C411yescharacteristic
146C22×Dic17C22×C411yescharacteristic
147C2×D68—22no—
148C2×D68—24no—
149C2×C4×D17—24no—
150C2×C4×D17—21no—
151C2×C4×D17—21no—
152C2×C4×D17—21no—
153C2×C4×D17—21no—
154C2×C4×D17—24no—
155C2×C4×D17—21no—
156C2×C4×D17—21no—
157C2×C4×D17—21no—
158C2×C4×D17—21no—
159C2×C4×D17—21no—
160C2×C4×D17—21no—
161C2×C4×D17C22⋊C414yes—
162C2×C4×D17C22⋊C414yes—
163C2×C4×D17C2×D412yes—
164C2×C4×D17C2×D412yes—
165C2×Dic34—22no—
166C2×Dic34—24no—
167C2×C8×C17—22no—
168C2×C8×C17—22no—
169C2×C8×C17—22no—
170C2×C8×C17—22no—
171C2×C8×C17—22no—
172C2×C8×C17—22no—
173C4⋊C4×C17—22no—
174C4⋊C4×C17—22no—
175C22⋊C4×C17—22no—
176C22⋊C4×C17—22no—
177C42×C17—21no—
178C42×C17—21no—
179C42×C17C2×D412yes—
180C42×C17C2×D412yes—
181C22⋊Dic17—22no—
182C22⋊Dic17—22no—
183#63227—22no—
184#63227—22no—
185#63227—22no—
186#63227—22no—
187#63227—22no—
188#63227—22no—
189C4⋊Dic17—22no—
190C4⋊Dic17—22no—
191#63225—22no—
192#63225—22no—
193#63225—22no—
194#63225—22no—
195#63225—22no—
196#63225—22no—
197C4×Dic17—21no—
198C4×Dic17—21no—
199C4×Dic17C22⋊C414yes—
200C4×Dic17C22⋊C414yes—
201C4×Dic17C2×D412yes—
202C4×Dic17C2×D412yes—
203C2×#3622—22no—
204C2×#3622—22no—
205C2×#3622—22no—
206C2×#3622—22no—
207C2×#3622—22no—
208C2×#3622—22no—
209C8×D17—48no—
210#21848—171nomaximal, Sylow
211C23×C17—21no—
212C23×C17—21no—
213C23×C17—21no—
214C23×C17—22no—
215C23×C17—22no—
216C23×C17C23×C411yescharacteristic
217C22×D17—21no—
218C22×D17—21no—
219C22×D17—22no—
220C22×D17—21no—
221C22×D17—21no—
222C22×D17—21no—
223C22×D17—22no—
224C22×D17—22no—
225C22×D17C2×C22⋊C412yes—
226C22×D17#13114yes—
227C2×C4×C17—22no—
228C2×C4×C17—22no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—22no—
235C2×C4×C17—22no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17—41no—
239C2×C4×C17—41no—
240C2×C4×C17C2×C22⋊C412yes—
241C2×C4×C17C2×C22⋊C412yes—
242C2×C4×C17C22×D411yescharacteristic
243C2×C4×C17C22×D411yescharacteristic
244C2×C4×C17H(5,𝔽2)11yescharacteristic
245C2×C4×C17H(5,𝔽2)11yescharacteristic
246C2×C4×C172−1+411yescharacteristic
247C2×C4×C172−1+411yescharacteristic
248C22⋊D17—44no—
249C22⋊D17—44no—
250C2×Dic17—21no—
251C2×Dic17—21no—
252C2×Dic17—22no—
253C2×Dic17—21no—
254C2×Dic17—21no—
255C2×Dic17—21no—
256C2×Dic17—21no—
257C2×Dic17—21no—
258C2×Dic17—22no—
259C2×Dic17—22no—
260C2×Dic17C2×C22⋊C412yes—
261C2×Dic17C2×C22⋊C412yes—
262C2×Dic17C2×C22⋊C412yes—
263C2×Dic17#13114yes—
264D68—44no—
265C4×D17—28no—
266C4×D17—22no—
267C4×D17—22no—
268C4×D17—22no—
269C4×D17—22no—
270C4×D17—28no—
271C4×D17—41no—
272C4×D17—41no—
273C4×D17—41no—
274C4×D17—41no—
275Dic34—44no—
276C8×C17—42no—
277C8×C17—42no—
278#3622—42no—
279#3622—42no—
280C2×#529—171no—
281C2×#431—171no—
282C22×C4⋊C8—171no—
283#1984—172no—
284#1954—172no—
285#1869—172no—
286#1869—172no—
287#1798—171no—
288#1798—171no—
289#1798—171no—
290#1798D1711yescharacteristic, complemented
291C22×C17—41no—
292C22×C17—41no—
293C22×C17—41no—
294C22×C17—42no—
295C22×C17—42no—
296C22×C17C22×C22⋊C411yescharacteristic
297C22×C17#52011yescharacteristic
298C22×C17#52011yescharacteristic
299C22×C17C2×#13112yes—
300C22×C17C2×#13112yes—
301C2×D17—22no—
302C2×D17—22no—
303C2×D17—24no—
304C2×D17—24no—
305C2×D17—41no—
306C2×D17—41no—
307C2×D17—41no—
308C2×D17—42no—
309C2×D17—42no—
310C2×D17#40814yes—
311C4×C17—22no—
312C4×C17—22no—
313C4×C17—41no—
314C4×C17—41no—
315C4×C17—41no—
316C4×C17#53511yescharacteristic
317C4×C17#53611yescharacteristic
318C4×C17#53711yescharacteristic
319C4×C17#53711yescharacteristic
320Dic17—22no—
321Dic17—22no—
322Dic17—41no—
323Dic17—41no—
324Dic17—41no—
325Dic17#40814yes—
326C23×C8—172no—
327#529—344no—
328#529—344no—
329C2×#126—171no—
330C2×#126—171no—
331C2×#125—171no—
332C2×#124—171no—
333C2×#123—171no—
334C2×#123—171no—
335C2×C22⋊Q8—171no—
336C2×#121—171no—
337C2×C4×Q8—171no—
338C2×C4×D4—171no—
339C2×C4×D4—171no—
340C2×C4×D4—171no—
341C2×#117—171no—
342C2×#117C2×D1711yescharacteristic
343C22×C42—171no—
344#431—344no—
345C2×C4⋊C8—172no—
346C2×C4⋊C8—171no—
347C2×C4⋊C8—171no—
348C2×C4⋊C8—344no—
349C2×C4⋊C8C2×D1711yescharacteristic
350C2×C4⋊C8C2×D1711yescharacteristic
351C2×C22⋊C8—172no—
352C2×C22⋊C8—172no—
353C2×C22⋊C8—172no—
354C2×C22⋊C8—172no—
355C2×C22⋊C8C2×D1712yes—
356C2×C4×C8—172no—
357#336—172no—
358#336—172no—
359#336—172no—
360#336C2×D1712yes—
361C2×C17—81no—
362C2×C17#231111yescharacteristic
363C2×C17C2×#40811yescharacteristic
364C2×C17C2×#40811yescharacteristic
365C2×C17#299412yes—
366C2×C17#299412yes—
367D17—24no—
368D17—81no—
369C22×Q8—171no—
370C22×D4—171no—
371C22×D4—171no—
372C22×D4—171no—
373C23×C4—171no—
374C23×C4—171no—
375C23×C4—171no—
376C23×C4—171no—
377C23×C4C22×D1711yescharacteristic
378C22×C8—172no—
379C22×C8—172no—
380C22×C8—344no—
381C22×C8—682no—
382C22×C8—682no—
383C22×C8C22×D1712yes—
384C22×C8C22×D1712yes—
385#126—344no—
386#126—344no—
387#125—344no—
388#124—344no—
389#123—344no—
390#123—344no—
391C22⋊Q8—344no—
392#121—344no—
393C4×Q8—342no—
394C4×Q8—342no—
395C4×D4—342no—
396C4×D4—342no—
397C4×D4—342no—
398C4×D4—342no—
399C4×D4—342no—
400C4×D4—342no—
401#117—22no—
402#117—22no—
403#117—342no—
404#117—342no—
405C2×C4⋊C4—171no—
406C2×C4⋊C4—171no—
407C2×C4⋊C4—171no—
408C2×C4⋊C4—171no—
409C2×C4⋊C4—171no—
410C2×C4⋊C4—171no—
411C2×C4⋊C4—171no—
412C2×C4⋊C4—171no—
413C2×C4⋊C4C4×D1711yescharacteristic
414C2×C4⋊C4C4×D1711yescharacteristic
415C2×C22⋊C4—171no—
416C2×C22⋊C4—171no—
417C2×C22⋊C4—171no—
418C2×C22⋊C4—171no—
419C2×C22⋊C4—171no—
420C2×C22⋊C4—171no—
421C2×C22⋊C4—171no—
422C2×C22⋊C4—171no—
423C2×C22⋊C4C4×D1711yescharacteristic
424C2×C22⋊C4C4×D1711yescharacteristic
425C2×C42—171no—
426C2×C42—171no—
427C2×C42—171no—
428C2×C42—174no—
429C2×C42—174no—
430C2×C42—171no—
431C2×C42C22×D1711yescharacteristic
432C2×C42C22×D1711yescharacteristic
433C4⋊C8—22no—
434C4⋊C8—22no—
435C4⋊C8—344no—
436C4⋊C8—342no—
437C4⋊C8—342no—
438C22⋊C8—42no—
439C22⋊C8—344no—
440C22⋊C8—682no—
441C22⋊C8—682no—
442C22⋊C8—682no—
443C4×C8—344no—
444C17#2184811yescharacteristic, complemented, Sylow
445C24—171no—
446C24—171no—
447C24C2×C4×D1711yescharacteristic
448C2×Q8—342no—
449C2×Q8—344no—
450C2×D4—342no—
451C2×D4—344no—
452C2×D4—342no—
453C2×D4—344no—
454C2×D4—342no—
455C2×D4—344no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—171no—
463C22×C4—171no—
464C22×C4—171no—
465C22×C4—172no—
466C22×C4—172no—
467C22×C4—174no—
468C22×C4—174no—
469C22×C4—171no—
470C22×C4—171no—
471C22×C4—171no—
472C22×C4—344no—
473C22×C4—344no—
474C22×C4—341no—
475C22×C4—341no—
476C22×C4—341no—
477C22×C4—341no—
478C22×C4—341no—
479C22×C4—341no—
480C22×C4—341no—
481C22×C4—341no—
482C22×C4—341no—
483C22×C4—341no—
484C22×C4C2×C4×D1711yescharacteristic
485C22×C4C2×C4×D1711yescharacteristic
486C22×C4C2×C4×D1711yescharacteristic
487C22×C4C2×C4×D1711yescharacteristic
488C22×C4C2×C4×D1711yescharacteristic
489C22×C4C23×D1711yescharacteristic
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—342no—
497C2×C8—342no—
498C2×C8—342no—
499C2×C8—342no—
500C2×C8—342no—
501C2×C8—342no—
502C2×C8—688no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—342no—
506C4⋊C4—342no—
507C4⋊C4—342no—
508C4⋊C4—342no—
509C4⋊C4—342no—
510C4⋊C4—342no—
511C4⋊C4—342no—
512C4⋊C4—342no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—342no—
516C22⋊C4—342no—
517C22⋊C4—342no—
518C22⋊C4—342no—
519C22⋊C4—342no—
520C22⋊C4—342no—
521C22⋊C4—342no—
522C22⋊C4—342no—
523C42—21no—
524C42—21no—
525C42—172no—
526C42—172no—
527C42—174no—
528C42—174no—
529C42—341no—
530C42—341no—
531C42D4×D1712yes—
532C42D4×D1712yes—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—22no—
537C23—22no—
538C23—172no—
539C23—174no—
540C23—341no—
541C23—341no—
542C23—342no—
543C23—341no—
544C23—341no—
545C23—341no—
546C23—342no—
547C23—342no—
548C23C22×C4×D1711yescharacteristic
549Q8—684no—
550D4—684no—
551D4—684no—
552D4—684no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—172no—
567C2×C4—174no—
568C2×C4—172no—
569C2×C4—172no—
570C2×C4—348no—
571C2×C4—348no—
572C2×C4—341no—
573C2×C4—341no—
574C2×C4—342no—
575C2×C4—342no—
576C2×C4—342no—
577C2×C4—342no—
578C2×C4—342no—
579C2×C4—341no—
580C2×C4—341no—
581C2×C4—341no—
582C2×C4—341no—
583C2×C4—341no—
584C2×C4—342no—
585C2×C4—342no—
586C2×C4—681no—
587C2×C4—681no—
588C2×C4—681no—
589C2×C4—681no—
590C2×C4C22⋊C4×D1712yes—
591C2×C4C22⋊C4×D1712yes—
592C2×C4C2×D4×D1711yescharacteristic
593C2×C4C2×D4×D1711yescharacteristic
594C2×C4#1058769111yescharacteristic
595C2×C4#1058769411yescharacteristic
596C2×C4#1058769611yescharacteristic
597C2×C4#1058769711yescharacteristic
598C8—42no—
599C8—42no—
600C8—682no—
601C8—682no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—42no—
606C22—42no—
607C22—174no—
608C22—342no—
609C22—342no—
610C22—344no—
611C22—344no—
612C22—681no—
613C22—681no—
614C22—681no—
615C22—682no—
616C22—682no—
617C22C2×C22⋊C4×D1711yescharacteristic
618C22#1176328911yescharacteristic
619C22#1176329011yescharacteristic
620C22#131×D1712yes—
621C22#131×D1712yes—
622C4—22no—
623C4—22no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—174no—
628C4—342no—
629C4—342no—
630C4—681no—
631C4—681no—
632C4—681no—
633C4#1176337511yescharacteristic
634C4#1176338211yescharacteristic
635C4#1176338811yescharacteristic
636C4#1176338911yescharacteristic
637C2—81no—
638C2—344no—
639C2—1361no—
640C2#1453137311yescharacteristic
641C2D17×#40811yescharacteristic
642C2D17×#40811yescharacteristic
643C2#1453793812yes—
644C2#1453794212yes—
subgroupG/H
center, FrattiniC22×C4C23×D17
commutator, socleC23×C17C23×C4
FittingC17×#1798C2
prime radicalC2×#10587570C4
cosocleC22×C4×C17C24
hypercenter#1798D17
hypocenterC17#21848
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ #1798

Complements

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

NH ≅ G/Ntype
2. C2×#11762861C2semidirect
3. C2×C4⋊C8×D17C2semidirect
4. #14529718C2semidirect
5. #14529576C2semidirect
6. #14529075C2semidirect
7. #14529073C2semidirect
8. #14528768C2semidirect
9. #14528765C2semidirect
10. #14528765C2semidirect
11. C17×#1798C2semidirect
290. #1798D17semidirect
444. C17#21848semidirect

Maximal subgroups

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

subgroupindex
C2×#117633372
C2×#117628612
C2×C4⋊C8×D172
#145297182
#145295762
#145290752
#145290732
#145287682
#145287652
C17×#17982
#2184817