Finite Group Databasebeta

← #17034207

Subgroups — #17034207

Order 4352.

1974 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C42×D17C211yesmaximal, characteristic
2C2×#11762865C212yesmaximal, complemented
3#14528851C212yesmaximal, complemented
4#14528764C211yesmaximal, characteristic, complemented
5#14528761C212yesmaximal, complemented
6#14528693C212yesmaximal
7#14528548C212yesmaximal, complemented
8#14528545C212yesmaximal, complemented
9C17×#1797C211yesmaximal, characteristic, complemented
10C23×C4×D17C411yescharacteristic
11C23×C4×D17C411yescharacteristic
12C23×C4×D17C2211yescharacteristic
13C22×C4×Dic17C411yescharacteristic
14C22×C4×Dic17C411yescharacteristic
15C22×C4×Dic17C2211yescharacteristic
16C2×C42×D17—28no—
17C2×C42×D17C411yescharacteristic
18C2×C42×D17C411yescharacteristic
19C2×C42×D17C411yescharacteristic
20C2×C42×D17C411yescharacteristic
21C2×C42×D17C2211yescharacteristic
22C2×C42×D17C2211yescharacteristic
23C2×C42×D17C2211yescharacteristic
24C2×C42×D17C2211yescharacteristic
25#11762865—28no—
26C2×C4⋊#3622C2212yescomplemented
27C2×C22⋊#3622C2212yes—
28C2×#10587491C2214yes—
29C2×#10587491C2214yes—
30C2×#10587486C2214yescomplemented
31#11762564C2212yescomplemented
32#11762563C2214yescomplemented
33C22×C42×C17C2211yescharacteristic
34C2×C17×C4⋊C8C2212yescomplemented
35C2×C17×C22⋊C8C2212yes—
36C17×#336C2212yescomplemented
37C24×D17C2×C411yescharacteristic
38C23×C4×C17C2×C411yescharacteristic
39C23×C4×C17C2×C411yescharacteristic
40C23×C4×C17C2311yescharacteristic
41C23×Dic17C2×C411yescharacteristic
42C23×Dic17C2×C411yescharacteristic
43C23×Dic17C2×C411yescharacteristic
44C22×C4×D17—24no—
45C22×C4×D17—24no—
46C22×C4×D17—22no—
47C22×C4×D17—22no—
48C22×C4×D17—24no—
49C22×C4×D17—24no—
50C22×C4×D17—24no—
51C22×C4×D17—24no—
52C22×C4×D17—24no—
53C22×C4×D17—24no—
54C22×C4×D17C2×C411yescharacteristic
55C22×C4×D17C2×C411yescharacteristic
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×D17C2×C411yescharacteristic
63C22×C4×D17C2×C411yescharacteristic
64C22×C4×D17C2311yescharacteristic
65C22×C4×D17C2311yescharacteristic
66C22×C8×C17C2314yes—
67C2×C42×C17—24no—
68C2×C42×C17C2×C411yescharacteristic
69C2×C42×C17C2×C411yescharacteristic
70C2×C42×C17C2311yescharacteristic
71C2×C42×C17C2311yescharacteristic
72C2×C4×Dic17—24no—
73C2×C4×Dic17—24no—
74C2×C4×Dic17—24no—
75C2×C4×Dic17C2×C411yescharacteristic
76C2×C4×Dic17C2×C411yescharacteristic
77C2×C4×Dic17C2×C411yescharacteristic
78C2×C4×Dic17C2×C411yescharacteristic
79C2×C4×Dic17C2×C411yescharacteristic
80C2×C4×Dic17C2×C411yescharacteristic
81C2×C4×Dic17C2×C411yescharacteristic
82C2×C4×Dic17C2×C411yescharacteristic
83C2×C4×Dic17C2×C411yescharacteristic
84C2×C4×Dic17C2×C411yescharacteristic
85C2×C4×Dic17C2311yescharacteristic
86C2×C4×Dic17C2311yescharacteristic
87C22×#3622C2314yes—
88C42×D17—24no—
89C42×D17—24no—
90C42×D17—24no—
91C42×D17—24no—
92C42×D17—24no—
93C42×D17—24no—
94C42×D17—24no—
95C42×D17D418yes—
96C17×C4⋊C8—24no—
97C17×C22⋊C8—24no—
98C22⋊#3622—24no—
99#10587491—28no—
100#10587491—28no—
101#10587486—28no—
102C4⋊#3622—24no—
103C24×C17C22×C411yescharacteristic
104C23×D17—22no—
105C23×D17—22no—
106C23×D17—22no—
107C23×D17—22no—
108C23×D17—22no—
109C23×D17—22no—
110C23×D17M1614yes—
111C23×D17C22×C411yescharacteristic
112C23×D17C22×C411yescharacteristic
113C22×C4×C17—22no—
114C22×C4×C17—22no—
115C22×C4×C17—22no—
116C22×C4×C17—22no—
117C22×C4×C17—22no—
118C22×C4×C17—22no—
119C22×C4×C17—22no—
120C22×C4×C17M1614yes—
121C22×C4×C17C22×C411yescharacteristic
122C22×C4×C17C22×C411yescharacteristic
123C22×C4×C17C22×C411yescharacteristic
124C22×C4×C17C22×C411yescharacteristic
125C22×C4×C17C22×C411yescharacteristic
126C22×C4×C17#4112yes—
127C22×C4×C17#4112yes—
128C22×C4×C17C2411yescharacteristic
129C22×Dic17—22no—
130C22×Dic17—22no—
131C22×Dic17—22no—
132C22×Dic17—22no—
133C22×Dic17—22no—
134C22×Dic17—22no—
135C22×Dic17—22no—
136C22×Dic17—22no—
137C22×Dic17—22no—
138C22×Dic17C22×C411yescharacteristic
139C22×Dic17C22×C411yescharacteristic
140C22×Dic17C22×C411yescharacteristic
141C22×Dic17C22×C411yescharacteristic
142C22×Dic17C22×C411yescharacteristic
143C22×Dic17C22×C411yescharacteristic
144C2×C4×D17—232no—
145C2×C4×D17—216no—
146C2×C4×D17—216no—
147C2×C4×D17—232no—
148C2×C4×D17—22no—
149C2×C4×D17—22no—
150C2×C4×D17—22no—
151C2×C4×D17—22no—
152C2×C4×D17—22no—
153C2×C4×D17—22no—
154C2×C4×D17—22no—
155C2×C4×D17—22no—
156C2×C4×D17—22no—
157C2×C4×D17—22no—
158C2×C4×D17—22no—
159C2×C4×D17—22no—
160C2×C4×D17—22no—
161C2×C4×D17—22no—
162C2×C4×D17—22no—
163C2×C4×D17—22no—
164C2×C4×D17—22no—
165C2×C4×D17—22no—
166C2×C4×D17—22no—
167C2×C4×D17—22no—
168C2×C4×D17—22no—
169C2×C4×D17—21no—
170C2×C4×D17—21no—
171C2×C4×D17—21no—
172C2×C4×D17—21no—
173C2×C4×D17—22no—
174C2×C4×D17—22no—
175C2×C4×D17—22no—
176C2×C4×D17—22no—
177C2×C4×D17—22no—
178C2×C4×D17—22no—
179C2×C4×D17—22no—
180C2×C4×D17C22⋊C414yes—
181C2×C4×D17C22⋊C414yes—
182C2×C4×D17M1614yes—
183C2×C4×D17M1614yes—
184C2×C4×D17C2×D414yes—
185C2×C4×D17#4114yes—
186C2×C8×C17—24no—
187C2×C8×C17—24no—
188C2×C8×C17—24no—
189C42×C17—22no—
190C42×C17—22no—
191C42×C17—22no—
192C42×C17C2×D414yes—
193C4×Dic17—22no—
194C4×Dic17—22no—
195C4×Dic17—22no—
196C4×Dic17—22no—
197C4×Dic17—22no—
198C4×Dic17—22no—
199C4×Dic17—22no—
200C4×Dic17—22no—
201C4×Dic17—22no—
202C4×Dic17C22⋊C414yes—
203C4×Dic17C22⋊C414yes—
204C4×Dic17C2×D414yes—
205C2×#3622—24no—
206C2×#3622—24no—
207C2×#3622—24no—
208#20715—171nomaximal, Sylow
209C23×C17—21no—
210C23×C17—21no—
211C23×C17—21no—
212C23×C17—21no—
213C23×C17#11712yes—
214C23×C17C2×M1612yes—
215C23×C17C2×M1612yes—
216C23×C17C23×C411yescharacteristic
217C22×D17—28no—
218C22×D17—28no—
219C22×D17—28no—
220C22×D17—28no—
221C22×D17—21no—
222C22×D17—21no—
223C22×D17—21no—
224C22×D17—21no—
225C22×D17—21no—
226C22×D17—21no—
227C22×D17—21no—
228C22×D17—21no—
229C22×D17—28no—
230C22×D17—28no—
231C22×D17—28no—
232C22×D17C2×C22⋊C412yes—
233C22×D17#11712yes—
234C22×D17C2×M1612yes—
235C22×D17C2×M1612yes—
236C22×D17C2×M1612yes—
237C22×D17C2×M1612yes—
238C2×C4×C17—28no—
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—24no—
248C2×C4×C17—24no—
249C2×C4×C17—28no—
250C2×C4×C17C2×C22⋊C412yes—
251C2×C4×C17C2×C22⋊C412yes—
252C2×C4×C17#11712yes—
253C2×C4×C17C2×M1612yes—
254C2×C4×C17C2×M1612yes—
255C2×C4×C17C2×M1612yes—
256C2×C4×C17C2×M1612yes—
257C2×C4×C17C22×D412yes—
258C2×C4×C17C2×#4111yescharacteristic
259C2×C4×C17C2×#4111yescharacteristic
260C2×C4×C17C2×#4112yes—
261C2×Dic17—28no—
262C2×Dic17—28no—
263C2×Dic17—28no—
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—21no—
276C2×Dic17C2×C22⋊C412yes—
277C2×Dic17C2×C22⋊C412yes—
278C2×Dic17C2×C22⋊C412yes—
279C2×Dic17#11712yes—
280C2×Dic17C2×M1612yes—
281C2×Dic17C2×M1612yes—
282C4×D17—28no—
283C4×D17—28no—
284C4×D17—28no—
285C4×D17—28no—
286C4×D17—28no—
287C4×D17—28no—
288C4×D17—28no—
289C4×D17—28no—
290C4×D17—24no—
291C4×D17—24no—
292C4×D17—24no—
293C4×D17—24no—
294C4×D17—28no—
295C4×D17—28no—
296C8×C17—44no—
297#3622—44no—
298C23×C42—171no—
299C2×#432—172no—
300#1817—172no—
301#1797—171no—
302#1797—172no—
303#1797D1711yescharacteristic, complemented
304#1779—172no—
305#1749—172no—
306#1749—172no—
307C22×C17—22no—
308C22×C17—22no—
309C22×C17—22no—
310C22×C17—22no—
311C22×C17—22no—
312C22×C17—22no—
313C22×C17#43214yescomplemented
314C22×C17C22×C22⋊C411yescharacteristic
315C22×C17C2×#11711yescharacteristic
316C22×C17C2×#11711yescharacteristic
317C22×C17C22×M1611yescharacteristic
318C22×C17C22×M1611yescharacteristic
319C22×C17C22×M1611yescharacteristic
320C22×C17C22×M1611yescharacteristic
321C2×D17—232no—
322C2×D17—22no—
323C2×D17—22no—
324C2×D17—22no—
325C2×D17—22no—
326C2×D17—22no—
327C2×D17—22no—
328C2×D17—22no—
329C2×D17—22no—
330C2×D17—22no—
331C2×D17—22no—
332C2×D17—22no—
333C2×D17—22no—
334C2×D17#40714yes—
335C2×D17#43214yescomplemented
336C4×C17—22no—
337C4×C17—22no—
338C4×C17—22no—
339C4×C17—22no—
340C4×C17#40714yes—
341C4×C17#52512yes—
342C4×C17#52512yes—
343Dic17—22no—
344Dic17—22no—
345Dic17—22no—
346Dic17—22no—
347Dic17—22no—
348Dic17—22no—
349C24×C4—171no—
350C24×C4—171no—
351C24×C4—171no—
352C22×C42—171no—
353C22×C42—171no—
354C22×C42—171no—
355C22×C42—171no—
356C22×C42—171no—
357C22×C42—171no—
358C22×C42—171no—
359C22×C42—171no—
360C22×C42—171no—
361C22×C42—171no—
362C22×C42—171no—
363C22×C42—348no—
364C22×C42C2×D1711yescharacteristic
365#432—348no—
366C2×C4⋊C8—172no—
367C2×C4⋊C8C2×D1712yescomplemented
368C2×C22⋊C8—172no—
369C2×C22⋊C8—174no—
370C2×C22⋊C8—174no—
371C2×C22⋊C8C2×D1712yes—
372C2×#97—174no—
373#336—174no—
374#336—172no—
375#336C2×D1712yescomplemented
376C2×C17—24no—
377C2×C17#228011yescharacteristic
378C2×C17C2×#40711yescharacteristic
379C2×C17C2×#40711yescharacteristic
380C2×C17C2×#43211yescharacteristic
381C2×C17C2×#43211yescharacteristic
382C2×C17#292011yescharacteristic
383C2×C17#292011yescharacteristic
384D17—24no—
385D17—24no—
386C25—171no—
387C23×C4—171no—
388C23×C4—171no—
389C23×C4—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—171no—
400C23×C4—171no—
401C23×C4—171no—
402C23×C4—344no—
403C23×C4—344no—
404C23×C4—344no—
405C23×C4—344no—
406C23×C4—344no—
407C23×C4—344no—
408C23×C4—344no—
409C23×C4—342no—
410C23×C4—342no—
411C23×C4—344no—
412C23×C4C4×D1711yescharacteristic
413C23×C4C4×D1711yescharacteristic
414C23×C4C22×D1711yescharacteristic
415C22×C8—174no—
416C22×C8C22×D1714yes—
417C2×C42—24no—
418C2×C42—171no—
419C2×C42—171no—
420C2×C42—171no—
421C2×C42—178no—
422C2×C42—171no—
423C2×C42—171no—
424C2×C42—171no—
425C2×C42—171no—
426C2×C42—171no—
427C2×C42—171no—
428C2×C42—171no—
429C2×C42—171no—
430C2×C42—171no—
431C2×C42—344no—
432C2×C42—344no—
433C2×C42—344no—
434C2×C42—344no—
435C2×C42—344no—
436C2×C42—344no—
437C2×C42—344no—
438C2×C42—344no—
439C2×C42—344no—
440C2×C42—344no—
441C2×C42C4×D1711yescharacteristic
442C2×C42C4×D1711yescharacteristic
443C2×C42C22×D1711yescharacteristic
444C2×C42C22×D1711yescharacteristic
445C4⋊C8—24no—
446C4⋊C8—344no—
447C22⋊C8—24no—
448C22⋊C8—348no—
449C22⋊C8—348no—
450C22⋊C8—344no—
451#97—348no—
452C17#2071511yescharacteristic, complemented, Sylow
453C24—171no—
454C24—171no—
455C24—174no—
456C24—342no—
457C24—342no—
458C24—342no—
459C24—342no—
460C24—342no—
461C24—342no—
462C24C2×C4×D1711yescharacteristic
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—171no—
471C22×C4—171no—
472C22×C4—171no—
473C22×C4—171no—
474C22×C4—171no—
475C22×C4—171no—
476C22×C4—174no—
477C22×C4—174no—
478C22×C4—174no—
479C22×C4—174no—
480C22×C4—174no—
481C22×C4—174no—
482C22×C4—342no—
483C22×C4—342no—
484C22×C4—342no—
485C22×C4—342no—
486C22×C4—342no—
487C22×C4—342no—
488C22×C4—342no—
489C22×C4—342no—
490C22×C4—342no—
491C22×C4—342no—
492C22×C4—342no—
493C22×C4—342no—
494C22×C4—342no—
495C22×C4—342no—
496C22×C4—342no—
497C22×C4—342no—
498C22×C4—3432no—
499C22×C4—3432no—
500C22×C4—3416no—
501C22×C4—3416no—
502C22×C4—342no—
503C22×C4—342no—
504C22×C4—342no—
505C22×C4—342no—
506C22×C4—342no—
507C22×C4—342no—
508C22×C4—342no—
509C22×C4—342no—
510C22×C4—342no—
511C22×C4—342no—
512C22×C4—341no—
513C22×C4—341no—
514C22×C4—341no—
515C22×C4—341no—
516C22×C4—342no—
517C22×C4—342no—
518C22×C4—342no—
519C22×C4—342no—
520C22×C4—342no—
521C22×C4—342no—
522C22×C4—342no—
523C22×C4—342no—
524C22×C4—342no—
525C22×C4—342no—
526C22×C4—342no—
527C22×C4#6321814yes—
528C22×C4C2×C4×D1711yescharacteristic
529C22×C4C2×C4×D1711yescharacteristic
530C22×C4C2×C4×D1711yescharacteristic
531C22×C4C2×C4×D1711yescharacteristic
532C22×C4C2×C4×D1711yescharacteristic
533C22×C4#6325412yes—
534C22×C4#6325612yes—
535C22×C4C23×D1711yescharacteristic
536C2×C8—24no—
537C2×C8—24no—
538C2×C8—24no—
539C2×C8—344no—
540C2×C8—344no—
541C2×C8—344no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—174no—
546C42—174no—
547C42—174no—
548C42—342no—
549C42—342no—
550C42—342no—
551C42—342no—
552C42—342no—
553C42—342no—
554C42—342no—
555C42—342no—
556C42—342no—
557C42D4×D1714yes—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—172no—
563C23—172no—
564C23—172no—
565C23—172no—
566C23—172no—
567C23—172no—
568C23—348no—
569C23—348no—
570C23—348no—
571C23—348no—
572C23—348no—
573C23—348no—
574C23—341no—
575C23—341no—
576C23—341no—
577C23—341no—
578C23—341no—
579C23—341no—
580C23—341no—
581C23—341no—
582C23—348no—
583C23#1058757812yes—
584C23C2×#6321812yes—
585C23M16×D1712yes—
586C23C22×C4×D1711yescharacteristic
587C2×C4—28no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—28no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—172no—
600C2×C4—172no—
601C2×C4—172no—
602C2×C4—172no—
603C2×C4—172no—
604C2×C4—172no—
605C2×C4—348no—
606C2×C4—348no—
607C2×C4—348no—
608C2×C4—348no—
609C2×C4—348no—
610C2×C4—348no—
611C2×C4—348no—
612C2×C4—348no—
613C2×C4—348no—
614C2×C4—348no—
615C2×C4—348no—
616C2×C4—341no—
617C2×C4—341no—
618C2×C4—341no—
619C2×C4—341no—
620C2×C4—341no—
621C2×C4—341no—
622C2×C4—341no—
623C2×C4—341no—
624C2×C4—341no—
625C2×C4—341no—
626C2×C4—341no—
627C2×C4—341no—
628C2×C4—348no—
629C2×C4—348no—
630C2×C4—344no—
631C2×C4—344no—
632C2×C4—344no—
633C2×C4—344no—
634C2×C4C22⋊C4×D1712yes—
635C2×C4C22⋊C4×D1712yes—
636C2×C4#1058757812yes—
637C2×C4C2×#6321812yes—
638C2×C4C2×#6321812yes—
639C2×C4M16×D1712yes—
640C2×C4M16×D1712yes—
641C2×C4C2×D4×D1712yes—
642C2×C4C2×#6325411yescharacteristic
643C2×C4C2×#6325611yescharacteristic
644C2×C4#41×D1712yes—
645C8—44no—
646C8—684no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—174no—
654C22—174no—
655C22—3432no—
656C22—342no—
657C22—342no—
658C22—342no—
659C22—342no—
660C22—342no—
661C22—342no—
662C22—342no—
663C22—342no—
664C22—342no—
665C22—342no—
666C22—342no—
667C22—342no—
668C22#1176286514yescomplemented
669C22C2×C22⋊C4×D1711yescharacteristic
670C22#117×D1711yescharacteristic
671C22C2×#1058757811yescharacteristic
672C22C2×M16×D1711yescharacteristic
673C22C2×M16×D1711yescharacteristic
674C22C2×M16×D1711yescharacteristic
675C22C22×#6321811yescharacteristic
676C4—22no—
677C4—22no—
678C4—22no—
679C4—22no—
680C4—342no—
681C4—342no—
682C4—342no—
683C4—342no—
684C4—342no—
685C4—342no—
686C4#1176277114yes—
687C4#1176331912yes—
688C4#1176332212yes—
689C2—24no—
690C2—344no—
691C2—344no—
692C2#1453116611yescharacteristic
693C2D17×#40711yescharacteristic
694C2C2×#1176277111yescharacteristic
695C2D17×#43211yescharacteristic
696C2C2×#1176286511yescharacteristic
697C2#1453741211yescharacteristic
698C2#1453741611yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D17
commutator, socleC23×C17C23×C4
FittingC17×#1797C2
prime radicalC24×D17C2×C4
cosocleC22×C4×C17C24
hypercenter#1797D17
hypocenterC17#20715
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ #1797

Complements

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

NH ≅ G/Ntype
2. C2×#11762865C2semidirect
3. #14528851C2semidirect
4. #14528764C2semidirect
5. #14528761C2semidirect
7. #14528548C2semidirect
8. #14528545C2semidirect
9. C17×#1797C2semidirect
26. C2×C4⋊#3622C22semidirect
30. C2×#10587486C22semidirect
31. #11762564C22semidirect
32. #11762563C22semidirect
34. C2×C17×C4⋊C8C22semidirect
36. C17×#336C22semidirect
303. #1797D17semidirect
313. C22×C17#432semidirect
335. C2×D17#432semidirect
367. C2×C4⋊C8C2×D17semidirect
375. #336C2×D17semidirect
452. C17#20715semidirect
668. C22#11762865semidirect

Maximal subgroups

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

subgroupindex
C22×C42×D172
C2×#117628652
#145288512
#145287642
#145287612
#145286932
#145285482
#145285452
C17×#17972
#2071517