Finite Group Databasebeta

← #17034222

Subgroups — #17034222

Order 4352.

1426 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#117×D17C211yesmaximal, characteristic
2C2×#11762865C211yesmaximal, characteristic, complemented
3C2×#11762861C211yesmaximal, characteristic, complemented
4#14528851C212yesmaximal, complemented
5#14528768C211yesmaximal, characteristic, complemented
6#14528765C211yesmaximal, characteristic, complemented
7#14528765C211yesmaximal, characteristic, complemented
8#14528693C212yesmaximal
9#14528552C212yesmaximal, complemented
10#14528549C212yesmaximal, complemented
11C17×#1798C211yesmaximal, characteristic, complemented
12C23×C4×D17C2211yescharacteristic
13C2×#10587612C2211yescharacteristic
14C2×#10587578C411yescharacteristic
15C2×#10587578C411yescharacteristic
16C2×#10587558C2211yescharacteristic
17C2×#10587558C2211yescharacteristic
18C2×#10587563C411yescharacteristic
19C2×#10587563C411yescharacteristic
20#117×D17—24no—
21#117×D17—24no—
22C2×C4⋊C4×D17C411yescharacteristic
23C2×C4⋊C4×D17C411yescharacteristic
24C2×C22⋊C4×D17C411yescharacteristic
25C2×C22⋊C4×D17C411yescharacteristic
26C2×C42×D17C2211yescharacteristic
27C2×C42×D17C2211yescharacteristic
28#11762865—24no—
29#11762861—24no—
30C2×C4⋊#3622C2211yescharacteristic, complemented
31C2×C4⋊#3622C2211yescharacteristic, complemented
32C2×C22⋊#3622C2212yes—
33C2×#10587491C2212yes—
34C2×#10587491C2212yes—
35C2×#10587491C2212yes—
36C2×#10587491C2212yes—
37C2×#10587486C2212yescomplemented
38C2×C8×Dic17C2212yescomplemented
39#11762564C2212yescomplemented
40#11762563C2212yescomplemented
41#11762563C2212yescomplemented
42C2×C17×#117C2211yescharacteristic
43C2×C17×C4⋊C8C2211yescharacteristic, complemented
44C2×C17×C4⋊C8C2211yescharacteristic, complemented
45C2×C17×C22⋊C8C2212yes—
46C17×#336C2212yescomplemented
47C24×D17C2×C411yescharacteristic
48C23×C4×C17C2311yescharacteristic
49C23×Dic17C2×C411yescharacteristic
50C22×C4×D17—22no—
51C22×C4×D17—22no—
52C22×C4×D17—22no—
53C22×C4×D17—22no—
54C22×C4×D17—22no—
55C22×C4×D17—22no—
56C22×C4×D17C2×C411yescharacteristic
57C22×C4×D17C2×C411yescharacteristic
58C22×C4×D17C2×C411yescharacteristic
59C22×C4×D17C2×C411yescharacteristic
60C22×C4×D17C2×C411yescharacteristic
61C22×C4×D17C2×C411yescharacteristic
62C22×C4×D17C2311yescharacteristic
63C22×C4×D17C2311yescharacteristic
64C22×C8×C17C2312yes—
65C22×C8×C17C2312yes—
66C17×#117—22no—
67C17×#117—22no—
68C2×C4⋊C4×C17C2×C411yescharacteristic
69C2×C4⋊C4×C17C2×C411yescharacteristic
70C2×C22⋊C4×C17C2×C411yescharacteristic
71C2×C22⋊C4×C17C2×C411yescharacteristic
72C2×C42×C17C2311yescharacteristic
73C2×C42×C17C2311yescharacteristic
74C2×C22⋊Dic17C2×C411yescharacteristic
75C2×C22⋊Dic17C2×C411yescharacteristic
76C2×#63227C2×C411yescharacteristic
77C2×#63227C2×C411yescharacteristic
78C2×#63227C2×C411yescharacteristic
79C2×#63227C2×C411yescharacteristic
80#10587612—22no—
81#10587612—22no—
82C2×C4⋊Dic17C2×C411yescharacteristic
83C2×C4⋊Dic17C2×C411yescharacteristic
84C2×#63225C2×C411yescharacteristic
85C2×#63225C2×C411yescharacteristic
86C2×#63225C2×C411yescharacteristic
87C2×#63225C2×C411yescharacteristic
88C2×C4×Dic17C2×C411yescharacteristic
89C2×C4×Dic17C2×C411yescharacteristic
90C2×C4×Dic17C2×C411yescharacteristic
91C2×C4×Dic17C2×C411yescharacteristic
92C2×C4×Dic17C2311yescharacteristic
93C2×C4×Dic17C2311yescharacteristic
94C22×#3622C2312yes—
95C22×#3622C2312yes—
96#10587578—24no—
97#10587578—24no—
98C4⋊C4×D17—24no—
99C4⋊C4×D17—24no—
100C22⋊C4×D17—24no—
101C22⋊C4×D17—24no—
102#10587563—24no—
103#10587563—24no—
104#10587558—22no—
105#10587558—22no—
106#10587558—22no—
107#10587558—22no—
108C42×D17—22no—
109C42×D17—22no—
110C42×D17D414yes—
111C42×D17D414yes—
112C17×C4⋊C8—22no—
113C17×C4⋊C8—22no—
114C17×C22⋊C8—24no—
115C22⋊#3622—24no—
116#10587491—24no—
117#10587491—24no—
118#10587491—24no—
119#10587491—24no—
120#10587486—24no—
121C8×Dic17—24no—
122C4⋊#3622—22no—
123C4⋊#3622—22no—
124C24×C17C22×C411yescharacteristic
125C23×D17—22no—
126C23×D17—22no—
127C23×D17—22no—
128C23×D17—24no—
129C23×D17—42no—
130C23×D17C22×C411yescharacteristic
131C23×D17C22×C411yescharacteristic
132C22×C4×C17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17C22×C411yescharacteristic
137C22×C4×C17C22×C411yescharacteristic
138C22×C4×C17C22×C411yescharacteristic
139C22×C4×C17C22×C411yescharacteristic
140C22×C4×C17C22×C411yescharacteristic
141C22×C4×C17#4112yes—
142C22×C4×C17#4112yes—
143C22×C4×C17C2411yescharacteristic
144C22×Dic17—22no—
145C22×Dic17—22no—
146C22×Dic17—22no—
147C22×Dic17C22×C411yescharacteristic
148C22×Dic17C22×C411yescharacteristic
149C22×Dic17C22×C411yescharacteristic
150C22×Dic17C22×C411yescharacteristic
151C22×Dic17C22×C411yescharacteristic
152C22×Dic17C22×C411yescharacteristic
153C2×C4×D17—28no—
154C2×C4×D17—28no—
155C2×C4×D17—28no—
156C2×C4×D17—28no—
157C2×C4×D17—22no—
158C2×C4×D17—22no—
159C2×C4×D17—24no—
160C2×C4×D17—22no—
161C2×C4×D17—22no—
162C2×C4×D17—21no—
163C2×C4×D17—21no—
164C2×C4×D17—21no—
165C2×C4×D17—21no—
166C2×C4×D17—21no—
167C2×C4×D17—21no—
168C2×C4×D17—21no—
169C2×C4×D17—21no—
170C2×C4×D17—22no—
171C2×C4×D17—22no—
172C2×C4×D17—22no—
173C2×C4×D17—22no—
174C2×C4×D17—24no—
175C2×C4×D17—42no—
176C2×C4×D17—42no—
177C2×C4×D17C22⋊C414yes—
178C2×C4×D17C22⋊C414yes—
179C2×C4×D17C2×D412yes—
180C2×C4×D17C2×D412yes—
181C2×C4×D17#4112yes—
182C2×C4×D17#4112yes—
183C2×C8×C17—22no—
184C2×C8×C17—22no—
185C2×C8×C17—22no—
186C2×C8×C17—22no—
187C2×C8×C17—22no—
188C2×C8×C17—22no—
189C4⋊C4×C17—22no—
190C4⋊C4×C17—22no—
191C22⋊C4×C17—22no—
192C22⋊C4×C17—22no—
193C42×C17—21no—
194C42×C17—21no—
195C42×C17C2×D412yes—
196C42×C17C2×D412yes—
197C22⋊Dic17—22no—
198C22⋊Dic17—22no—
199#63227—22no—
200#63227—22no—
201#63227—22no—
202#63227—22no—
203C4⋊Dic17—22no—
204C4⋊Dic17—22no—
205#63225—22no—
206#63225—22no—
207#63225—22no—
208#63225—22no—
209C4×Dic17—21no—
210C4×Dic17—21no—
211C4×Dic17—22no—
212C4×Dic17—22no—
213C4×Dic17C22⋊C414yes—
214C4×Dic17C22⋊C414yes—
215C4×Dic17C2×D412yes—
216C4×Dic17C2×D412yes—
217C2×#3622—22no—
218C2×#3622—22no—
219C2×#3622—22no—
220C2×#3622—22no—
221C2×#3622—22no—
222C2×#3622—22no—
223#20717—171nomaximal, Sylow
224C23×C17—21no—
225C23×C17—21no—
226C23×C17—22no—
227C23×C17—22no—
228C23×C17#11712yes—
229C23×C17C23×C411yescharacteristic
230C22×D17—21no—
231C22×D17—21no—
232C22×D17—22no—
233C22×D17—28no—
234C22×D17—21no—
235C22×D17—21no—
236C22×D17—22no—
237C22×D17—22no—
238C22×D17—48no—
239C22×D17—48no—
240C22×D17—44no—
241C22×D17—44no—
242C22×D17C2×C22⋊C412yes—
243C22×D17#11712yes—
244C22×D17#13114yes—
245C2×C4×C17—21no—
246C2×C4×C17—21no—
247C2×C4×C17—22no—
248C2×C4×C17—22no—
249C2×C4×C17—22no—
250C2×C4×C17—22no—
251C2×C4×C17—22no—
252C2×C4×C17—22no—
253C2×C4×C17—21no—
254C2×C4×C17—21no—
255C2×C4×C17—22no—
256C2×C4×C17—22no—
257C2×C4×C17C2×C22⋊C412yes—
258C2×C4×C17C2×C22⋊C412yes—
259C2×C4×C17#11712yes—
260C2×C4×C17C22×D411yescharacteristic
261C2×C4×C17C22×D411yescharacteristic
262C2×C4×C17C2×#4111yescharacteristic
263C2×C4×C17C2×#4111yescharacteristic
264C2×C4×C17C2×#4111yescharacteristic
265C2×C4×C17C2×#4111yescharacteristic
266C2×Dic17—28no—
267C2×Dic17—21no—
268C2×Dic17—21no—
269C2×Dic17—21no—
270C2×Dic17—21no—
271C2×Dic17—22no—
272C2×Dic17—22no—
273C2×Dic17—21no—
274C2×Dic17—21no—
275C2×Dic17—22no—
276C2×Dic17—22no—
277C2×Dic17C2×C22⋊C412yes—
278C2×Dic17C2×C22⋊C412yes—
279C2×Dic17C2×C22⋊C412yes—
280C2×Dic17#11712yes—
281C4×D17—28no—
282C4×D17—28no—
283C4×D17—28no—
284C4×D17—22no—
285C4×D17—22no—
286C4×D17—22no—
287C4×D17—22no—
288C4×D17—22no—
289C4×D17—22no—
290C4×D17—22no—
291C4×D17—22no—
292C4×D17—44no—
293C4×D17—44no—
294C4×D17—44no—
295C8×C17—42no—
296C8×C17—42no—
297#3622—42no—
298#3622—42no—
299C22×#117—171no—
300C2×#432—171no—
301C2×#431—171no—
302#1817—172no—
303#1798—171no—
304#1798—171no—
305#1798—171no—
306#1798D1711yescharacteristic, complemented
307#1779—172no—
308#1750—172no—
309#1750—172no—
310C22×C17—24no—
311C22×C17—22no—
312C22×C17—41no—
313C22×C17—41no—
314C22×C17—42no—
315C22×C17C22×C22⋊C411yescharacteristic
316C22×C17C2×#11711yescharacteristic
317C22×C17C2×#11711yescharacteristic
318C22×C17C2×#13112yes—
319C22×C17#56812yes—
320C2×D17—22no—
321C2×D17—24no—
322C2×D17—24no—
323C2×D17—22no—
324C2×D17—24no—
325C2×D17—22no—
326C2×D17—416no—
327C2×D17—41no—
328C2×D17—41no—
329C2×D17—42no—
330C2×D17#40814yes—
331C4×C17—22no—
332C4×C17—22no—
333C4×C17—22no—
334C4×C17—41no—
335C4×C17—41no—
336C4×C17#52511yescharacteristic
337C4×C17#52511yescharacteristic
338C4×C17#52511yescharacteristic
339C4×C17#52511yescharacteristic
340Dic17—22no—
341Dic17—22no—
342Dic17—22no—
343Dic17—22no—
344Dic17—41no—
345Dic17—41no—
346C24×C4—171no—
347C2×#117—171no—
348C2×#117—171no—
349C2×#117—171no—
350C2×#117—171no—
351C2×#117—171no—
352C2×#117—171no—
353C2×#117—171no—
354C2×#117—344no—
355C2×#117—344no—
356C2×#117C2×D1711yescharacteristic
357C22×C4⋊C4—171no—
358C22×C4⋊C4—171no—
359C22×C22⋊C4—171no—
360C22×C22⋊C4—171no—
361C22×C42—171no—
362C22×C42—171no—
363#432—344no—
364#431—344no—
365C2×C4⋊C8—171no—
366C2×C4⋊C8—171no—
367C2×C4⋊C8C2×D1711yescharacteristic, complemented
368C2×C4⋊C8C2×D1711yescharacteristic, complemented
369C2×C22⋊C8—172no—
370C2×C22⋊C8—172no—
371C2×C22⋊C8—172no—
372C2×C22⋊C8—172no—
373C2×C22⋊C8—172no—
374C2×C22⋊C8C2×D1712yes—
375C2×#97—172no—
376C2×C4×C8—172no—
377#336—172no—
378#336—172no—
379#336—172no—
380#336C2×D1712yescomplemented
381C2×C17—42no—
382C2×C17#228011yescharacteristic
383C2×C17C2×#40811yescharacteristic
384C2×C17#287911yescharacteristic
385C2×C17#292412yes—
386C2×C17#292412yes—
387D17—24no—
388D17—42no—
389C25—171no—
390C23×C4—171no—
391C23×C4—171no—
392C23×C4—171no—
393C23×C4—171no—
394C23×C4—171no—
395C23×C4—171no—
396C23×C4—171no—
397C23×C4—171no—
398C23×C4—171no—
399C23×C4—342no—
400C23×C4—342no—
401C23×C4—342no—
402C23×C4—342no—
403C23×C4—342no—
404C23×C4—342no—
405C23×C4C22×D1711yescharacteristic
406C22×C8—172no—
407C22×C8—172no—
408C22×C8C22×D1712yes—
409C22×C8C22×D1712yes—
410#117—22no—
411#117—22no—
412#117—342no—
413#117—342no—
414#117—344no—
415#117—344no—
416#117—344no—
417#117—344no—
418#117—342no—
419#117—342no—
420#117—342no—
421#117—342no—
422C2×C4⋊C4—171no—
423C2×C4⋊C4—171no—
424C2×C4⋊C4—171no—
425C2×C4⋊C4—171no—
426C2×C4⋊C4—171no—
427C2×C4⋊C4—171no—
428C2×C4⋊C4—344no—
429C2×C4⋊C4—344no—
430C2×C4⋊C4C4×D1711yescharacteristic
431C2×C4⋊C4C4×D1711yescharacteristic
432C2×C22⋊C4—171no—
433C2×C22⋊C4—171no—
434C2×C22⋊C4—171no—
435C2×C22⋊C4—171no—
436C2×C22⋊C4—171no—
437C2×C22⋊C4—171no—
438C2×C22⋊C4—344no—
439C2×C22⋊C4—344no—
440C2×C22⋊C4C4×D1711yescharacteristic
441C2×C22⋊C4C4×D1711yescharacteristic
442C2×C42—171no—
443C2×C42—171no—
444C2×C42—171no—
445C2×C42—174no—
446C2×C42—174no—
447C2×C42—171no—
448C2×C42—171no—
449C2×C42—171no—
450C2×C42—342no—
451C2×C42—342no—
452C2×C42C22×D1711yescharacteristic
453C2×C42C22×D1711yescharacteristic
454C4⋊C8—22no—
455C4⋊C8—22no—
456C4⋊C8—342no—
457C4⋊C8—342no—
458C22⋊C8—24no—
459C22⋊C8—344no—
460C22⋊C8—344no—
461C22⋊C8—344no—
462C22⋊C8—344no—
463C22⋊C8—344no—
464#97—344no—
465C4×C8—344no—
466C17#2071711yescharacteristic, complemented, Sylow
467C24—171no—
468C24—171no—
469C24—342no—
470C24—342no—
471C24—344no—
472C24—342no—
473C24—682no—
474C24C2×C4×D1711yescharacteristic
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—171no—
480C22×C4—171no—
481C22×C4—171no—
482C22×C4—171no—
483C22×C4—172no—
484C22×C4—172no—
485C22×C4—172no—
486C22×C4—172no—
487C22×C4—174no—
488C22×C4—174no—
489C22×C4—171no—
490C22×C4—171no—
491C22×C4—342no—
492C22×C4—342no—
493C22×C4—342no—
494C22×C4—344no—
495C22×C4—342no—
496C22×C4—348no—
497C22×C4—348no—
498C22×C4—348no—
499C22×C4—348no—
500C22×C4—342no—
501C22×C4—342no—
502C22×C4—344no—
503C22×C4—342no—
504C22×C4—342no—
505C22×C4—342no—
506C22×C4—341no—
507C22×C4—341no—
508C22×C4—341no—
509C22×C4—341no—
510C22×C4—341no—
511C22×C4—341no—
512C22×C4—341no—
513C22×C4—341no—
514C22×C4—342no—
515C22×C4—342no—
516C22×C4—682no—
517C22×C4—682no—
518C22×C4C2×C4×D1711yescharacteristic
519C22×C4C2×C4×D1711yescharacteristic
520C22×C4C2×C4×D1711yescharacteristic
521C22×C4C2×C4×D1711yescharacteristic
522C22×C4C2×C4×D1711yescharacteristic
523C22×C4#6325412yes—
524C22×C4#6325612yes—
525C22×C4C23×D1711yescharacteristic
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—342no—
533C2×C8—342no—
534C2×C8—342no—
535C2×C8—342no—
536C2×C8—342no—
537C2×C8—342no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—342no—
541C4⋊C4—342no—
542C4⋊C4—342no—
543C4⋊C4—342no—
544C4⋊C4—342no—
545C4⋊C4—342no—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—342no—
549C22⋊C4—342no—
550C22⋊C4—342no—
551C22⋊C4—342no—
552C22⋊C4—342no—
553C22⋊C4—342no—
554C42—21no—
555C42—21no—
556C42—172no—
557C42—172no—
558C42—174no—
559C42—174no—
560C42—342no—
561C42—342no—
562C42—341no—
563C42—341no—
564C42D4×D1712yes—
565C42D4×D1712yes—
566C23—21no—
567C23—21no—
568C23—22no—
569C23—22no—
570C23—172no—
571C23—174no—
572C23—172no—
573C23—348no—
574C23—341no—
575C23—341no—
576C23—342no—
577C23—341no—
578C23—341no—
579C23—342no—
580C23—342no—
581C23—688no—
582C23—684no—
583C23—688no—
584C23—684no—
585C23#1058757812yes—
586C23C22×C4×D1711yescharacteristic
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—172no—
600C2×C4—172no—
601C2×C4—172no—
602C2×C4—172no—
603C2×C4—348no—
604C2×C4—348no—
605C2×C4—348no—
606C2×C4—348no—
607C2×C4—341no—
608C2×C4—341no—
609C2×C4—341no—
610C2×C4—341no—
611C2×C4—342no—
612C2×C4—342no—
613C2×C4—342no—
614C2×C4—342no—
615C2×C4—342no—
616C2×C4—342no—
617C2×C4—342no—
618C2×C4—342no—
619C2×C4—342no—
620C2×C4—342no—
621C2×C4—341no—
622C2×C4—341no—
623C2×C4—342no—
624C2×C4—342no—
625C2×C4—684no—
626C2×C4—684no—
627C2×C4—684no—
628C2×C4C22⋊C4×D1712yes—
629C2×C4C22⋊C4×D1712yes—
630C2×C4#1058757812yes—
631C2×C4C2×D4×D1711yescharacteristic
632C2×C4C2×D4×D1711yescharacteristic
633C2×C4C2×#6325411yescharacteristic
634C2×C4C2×#6325611yescharacteristic
635C2×C4#41×D1711yescharacteristic
636C2×C4#41×D1711yescharacteristic
637C8—42no—
638C8—42no—
639C8—682no—
640C8—682no—
641C22—22no—
642C22—24no—
643C22—41no—
644C22—41no—
645C22—42no—
646C22—174no—
647C22—342no—
648C22—342no—
649C22—344no—
650C22—344no—
651C22—342no—
652C22—344no—
653C22—6816no—
654C22—681no—
655C22—681no—
656C22—682no—
657C22C2×C22⋊C4×D1711yescharacteristic
658C22#117×D1711yescharacteristic
659C22C2×#1058757811yescharacteristic
660C22#131×D1712yes—
661C22#1176357412yes—
662C4—22no—
663C4—22no—
664C4—22no—
665C4—41no—
666C4—41no—
667C4—342no—
668C4—342no—
669C4—342no—
670C4—342no—
671C4—681no—
672C4—681no—
673C4#1176331911yescharacteristic
674C4#1176331911yescharacteristic
675C4#1176332211yescharacteristic
676C4#1176332211yescharacteristic
677C2—42no—
678C2—344no—
679C2—682no—
680C2#1453116611yescharacteristic
681C2D17×#40811yescharacteristic
682C2#1453722611yescharacteristic
683C2#1453744012yes—
684C2#1453744112yes—
subgroupG/H
center, FrattiniC22×C4C23×D17
commutator, socleC23×C17C23×C4
FittingC17×#1798C2
prime radicalC24×D17C2×C4
cosocleC22×C4×C17C24
hypercenter#1798D17
hypocenterC17#20717
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ #1798

Complements

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

NH ≅ G/Ntype
2. C2×#11762865C2semidirect
3. C2×#11762861C2semidirect
4. #14528851C2semidirect
5. #14528768C2semidirect
6. #14528765C2semidirect
7. #14528765C2semidirect
9. #14528552C2semidirect
10. #14528549C2semidirect
11. C17×#1798C2semidirect
30. C2×C4⋊#3622C22semidirect
31. C2×C4⋊#3622C22semidirect
37. C2×#10587486C22semidirect
38. C2×C8×Dic17C22semidirect
39. #11762564C22semidirect
40. #11762563C22semidirect
41. #11762563C22semidirect
43. C2×C17×C4⋊C8C22semidirect
44. C2×C17×C4⋊C8C22semidirect
46. C17×#336C22semidirect
306. #1798D17semidirect
367. C2×C4⋊C8C2×D17semidirect
368. C2×C4⋊C8C2×D17semidirect
380. #336C2×D17semidirect
466. C17#20717semidirect

Maximal subgroups

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

subgroupindex
C2×#117×D172
C2×#117628652
C2×#117628612
#145288512
#145287682
#145287652
#145286932
#145285522
#145285492
C17×#17982
#2071717