Finite Group Databasebeta

← #17034221

Subgroups — #17034221

Order 4352.

1426 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×#117×D17C211yesmaximal, characteristic
2C2×#11762864C211yesmaximal, characteristic, complemented
3C2×#11762860C211yesmaximal, characteristic, complemented
4#14528851C212yesmaximal, complemented
5#14528767C212yesmaximal, complemented
6#14528766C212yesmaximal, complemented
7#14528693C212yesmaximal
8#14528553C211yesmaximal, characteristic, complemented
9#14528551C211yesmaximal, characteristic, complemented
10#14528550C211yesmaximal, characteristic, complemented
11C17×#1750C211yesmaximal, 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#11762864—24no—
29#11762860—24no—
30C2×#10587485C2212yescomplemented
31C2×#10587485C2212yescomplemented
32C2×C22⋊#3622C2212yes—
33C2×#10587491C2212yes—
34C2×#10587491C2212yes—
35C2×#10587491C2212yes—
36C2×#10587491C2212yes—
37C2×#10587474C2211yescharacteristic, complemented
38C2×C4×#3622C2211yescharacteristic, complemented
39#11762564C2212yescomplemented
40#11762563C2212yescomplemented
41#11762563C2212yescomplemented
42C2×C17×#117C2211yescharacteristic
43C2×C17×C22⋊C8C2212yes—
44C2×C17×#97C2211yescharacteristic, complemented
45C2×C4×C8×C17C2211yescharacteristic, complemented
46C17×#336C2212yescomplemented
47C24×D17C2×C411yescharacteristic
48C23×C4×C17C2311yescharacteristic
49C23×Dic17C2×C411yescharacteristic
50C22×C4×D17—24no—
51C22×C4×D17—22no—
52C22×C4×D17—22no—
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×D17C2311yescharacteristic
61C22×C4×D17C2311yescharacteristic
62C22×C8×C17C2312yes—
63C22×C8×C17C2312yes—
64C17×#117—22no—
65C17×#117—22no—
66C2×C4⋊C4×C17C2×C411yescharacteristic
67C2×C4⋊C4×C17C2×C411yescharacteristic
68C2×C22⋊C4×C17C2×C411yescharacteristic
69C2×C22⋊C4×C17C2×C411yescharacteristic
70C2×C42×C17C2311yescharacteristic
71C2×C42×C17C2311yescharacteristic
72C2×C22⋊Dic17C2×C411yescharacteristic
73C2×C22⋊Dic17C2×C411yescharacteristic
74C2×#63227C2×C411yescharacteristic
75C2×#63227C2×C411yescharacteristic
76C2×#63227C2×C411yescharacteristic
77C2×#63227C2×C411yescharacteristic
78#10587612—22no—
79#10587612—22no—
80C2×C4⋊Dic17C2×C411yescharacteristic
81C2×C4⋊Dic17C2×C411yescharacteristic
82C2×#63225C2×C411yescharacteristic
83C2×#63225C2×C411yescharacteristic
84C2×#63225C2×C411yescharacteristic
85C2×#63225C2×C411yescharacteristic
86C2×C4×Dic17C2×C411yescharacteristic
87C2×C4×Dic17C2×C411yescharacteristic
88C2×C4×Dic17C2×C411yescharacteristic
89C2×C4×Dic17C2×C411yescharacteristic
90C2×C4×Dic17C2311yescharacteristic
91C2×C4×Dic17C2311yescharacteristic
92C22×#3622C2312yes—
93C22×#3622C2312yes—
94#10587578—24no—
95#10587578—24no—
96C4⋊C4×D17—24no—
97C4⋊C4×D17—24no—
98C22⋊C4×D17—24no—
99C22⋊C4×D17—24no—
100#10587563—24no—
101#10587563—24no—
102#10587558—22no—
103#10587558—22no—
104#10587558—22no—
105#10587558—22no—
106C42×D17—22no—
107C42×D17—22no—
108C42×D17D414yes—
109C42×D17D414yes—
110C17×C22⋊C8—24no—
111C17×#97—22no—
112C4×C8×C17—22no—
113C22⋊#3622—24no—
114#10587491—24no—
115#10587491—24no—
116#10587491—24no—
117#10587491—24no—
118#10587485—24no—
119#10587485—24no—
120#10587474—22no—
121C4×#3622—22no—
122C24×C17C22×C411yescharacteristic
123C23×D17—22no—
124C23×D17—22no—
125C23×D17—22no—
126C23×D17—24no—
127C23×D17—42no—
128C23×D17C22×C411yescharacteristic
129C23×D17C22×C411yescharacteristic
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17—22no—
133C22×C4×C17C22×C411yescharacteristic
134C22×C4×C17C22×C411yescharacteristic
135C22×C4×C17C22×C411yescharacteristic
136C22×C4×C17C22×C411yescharacteristic
137C22×C4×C17C22×C411yescharacteristic
138C22×C4×C17#4114yes—
139C22×C4×C17C2411yescharacteristic
140C22×Dic17—22no—
141C22×Dic17—22no—
142C22×Dic17—22no—
143C22×Dic17C22×C411yescharacteristic
144C22×Dic17C22×C411yescharacteristic
145C22×Dic17C22×C411yescharacteristic
146C22×Dic17C22×C411yescharacteristic
147C22×Dic17C22×C411yescharacteristic
148C22×Dic17C22×C411yescharacteristic
149C2×C4×D17—216no—
150C2×C4×D17—216no—
151C2×C4×D17—22no—
152C2×C4×D17—22no—
153C2×C4×D17—24no—
154C2×C4×D17—22no—
155C2×C4×D17—22no—
156C2×C4×D17—21no—
157C2×C4×D17—21no—
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—24no—
166C2×C4×D17—42no—
167C2×C4×D17—42no—
168C2×C4×D17C22⋊C414yes—
169C2×C4×D17C22⋊C414yes—
170C2×C4×D17C2×D412yes—
171C2×C4×D17C2×D412yes—
172C2×C4×D17#4114yes—
173C2×C8×C17—22no—
174C2×C8×C17—22no—
175C2×C8×C17—22no—
176C2×C8×C17—22no—
177C2×C8×C17—22no—
178C2×C8×C17—22no—
179C4⋊C4×C17—22no—
180C4⋊C4×C17—22no—
181C22⋊C4×C17—22no—
182C22⋊C4×C17—22no—
183C42×C17—21no—
184C42×C17—21no—
185C42×C17C2×D412yes—
186C42×C17C2×D412yes—
187C22⋊Dic17—22no—
188C22⋊Dic17—22no—
189#63227—22no—
190#63227—22no—
191#63227—22no—
192#63227—22no—
193C4⋊Dic17—22no—
194C4⋊Dic17—22no—
195#63225—22no—
196#63225—22no—
197#63225—22no—
198#63225—22no—
199C4×Dic17—21no—
200C4×Dic17—21no—
201C4×Dic17—22no—
202C4×Dic17—22no—
203C4×Dic17C22⋊C414yes—
204C4×Dic17C22⋊C414yes—
205C4×Dic17C2×D412yes—
206C4×Dic17C2×D412yes—
207C2×#3622—22no—
208C2×#3622—22no—
209C2×#3622—22no—
210C2×#3622—22no—
211C2×#3622—22no—
212C2×#3622—22no—
213#20717—171nomaximal, Sylow
214C23×C17—21no—
215C23×C17—21no—
216C23×C17—22no—
217C23×C17—22no—
218C23×C17#11712yes—
219C23×C17C23×C411yescharacteristic
220C22×D17—21no—
221C22×D17—21no—
222C22×D17—22no—
223C22×D17—28no—
224C22×D17—21no—
225C22×D17—21no—
226C22×D17—22no—
227C22×D17—22no—
228C22×D17—48no—
229C22×D17—48no—
230C22×D17—44no—
231C22×D17—44no—
232C22×D17C2×C22⋊C412yes—
233C22×D17#11712yes—
234C22×D17#13114yes—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—22no—
238C2×C4×C17—22no—
239C2×C4×C17—24no—
240C2×C4×C17—24no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—22no—
244C2×C4×C17—22no—
245C2×C4×C17C2×C22⋊C412yes—
246C2×C4×C17C2×C22⋊C412yes—
247C2×C4×C17#11712yes—
248C2×C4×C17C22×D411yescharacteristic
249C2×C4×C17C22×D411yescharacteristic
250C2×C4×C17C2×#4112yes—
251C2×C4×C17C2×#4112yes—
252C2×Dic17—28no—
253C2×Dic17—21no—
254C2×Dic17—21no—
255C2×Dic17—21no—
256C2×Dic17—21no—
257C2×Dic17—22no—
258C2×Dic17—22no—
259C2×Dic17—21no—
260C2×Dic17—21no—
261C2×Dic17—22no—
262C2×Dic17—22no—
263C2×Dic17C2×C22⋊C412yes—
264C2×Dic17C2×C22⋊C412yes—
265C2×Dic17C2×C22⋊C412yes—
266C2×Dic17#11712yes—
267C4×D17—28no—
268C4×D17—28no—
269C4×D17—28no—
270C4×D17—24no—
271C4×D17—24no—
272C4×D17—24no—
273C4×D17—24no—
274C4×D17—44no—
275C4×D17—44no—
276C4×D17—44no—
277C8×C17—42no—
278C8×C17—42no—
279#3622—42no—
280#3622—42no—
281C22×#117—171no—
282C2×#432—171no—
283C2×#431—171no—
284#1817—172no—
285#1798—172no—
286#1798—172no—
287#1779—172no—
288#1750—171no—
289#1750—171no—
290#1750—171no—
291#1750D1711yescharacteristic, complemented
292C22×C17—24no—
293C22×C17—22no—
294C22×C17—41no—
295C22×C17—41no—
296C22×C17—42no—
297C22×C17C22×C22⋊C411yescharacteristic
298C22×C17C2×#11711yescharacteristic
299C22×C17C2×#11711yescharacteristic
300C22×C17C2×#13112yes—
301C22×C17#56812yes—
302C2×D17—22no—
303C2×D17—24no—
304C2×D17—24no—
305C2×D17—22no—
306C2×D17—24no—
307C2×D17—22no—
308C2×D17—416no—
309C2×D17—41no—
310C2×D17—41no—
311C2×D17—42no—
312C2×D17#40814yes—
313C4×C17—22no—
314C4×C17—22no—
315C4×C17—22no—
316C4×C17—41no—
317C4×C17—41no—
318C4×C17#52512yes—
319C4×C17#52512yes—
320Dic17—22no—
321Dic17—22no—
322Dic17—22no—
323Dic17—22no—
324Dic17—41no—
325Dic17—41no—
326C24×C4—171no—
327C2×#117—171no—
328C2×#117—171no—
329C2×#117—171no—
330C2×#117—171no—
331C2×#117—171no—
332C2×#117—171no—
333C2×#117—171no—
334C2×#117—344no—
335C2×#117—344no—
336C2×#117C2×D1711yescharacteristic
337C22×C4⋊C4—171no—
338C22×C4⋊C4—171no—
339C22×C22⋊C4—171no—
340C22×C22⋊C4—171no—
341C22×C42—171no—
342C22×C42—171no—
343#432—344no—
344#431—344no—
345C2×C4⋊C8—172no—
346C2×C4⋊C8—172no—
347C2×C22⋊C8—172no—
348C2×C22⋊C8—172no—
349C2×C22⋊C8—172no—
350C2×C22⋊C8—172no—
351C2×C22⋊C8—172no—
352C2×C22⋊C8C2×D1712yes—
353C2×#97—171no—
354C2×#97C2×D1711yescharacteristic, complemented
355C2×C4×C8—171no—
356C2×C4×C8C2×D1711yescharacteristic, complemented
357#336—172no—
358#336—172no—
359#336—172no—
360#336C2×D1712yescomplemented
361C2×C17—42no—
362C2×C17#228011yescharacteristic
363C2×C17C2×#40811yescharacteristic
364C2×C17#287911yescharacteristic
365C2×C17#292412yes—
366C2×C17#292412yes—
367D17—24no—
368D17—42no—
369C25—171no—
370C23×C4—171no—
371C23×C4—171no—
372C23×C4—171no—
373C23×C4—171no—
374C23×C4—171no—
375C23×C4—171no—
376C23×C4—171no—
377C23×C4—171no—
378C23×C4—171no—
379C23×C4—342no—
380C23×C4—342no—
381C23×C4—344no—
382C23×C4—344no—
383C23×C4C22×D1711yescharacteristic
384C22×C8—172no—
385C22×C8—172no—
386C22×C8C22×D1712yes—
387C22×C8C22×D1712yes—
388#117—22no—
389#117—22no—
390#117—342no—
391#117—342no—
392#117—344no—
393#117—344no—
394#117—344no—
395#117—344no—
396#117—342no—
397#117—342no—
398#117—342no—
399#117—342no—
400C2×C4⋊C4—171no—
401C2×C4⋊C4—171no—
402C2×C4⋊C4—171no—
403C2×C4⋊C4—171no—
404C2×C4⋊C4—171no—
405C2×C4⋊C4—171no—
406C2×C4⋊C4—344no—
407C2×C4⋊C4—344no—
408C2×C4⋊C4C4×D1711yescharacteristic
409C2×C4⋊C4C4×D1711yescharacteristic
410C2×C22⋊C4—171no—
411C2×C22⋊C4—171no—
412C2×C22⋊C4—171no—
413C2×C22⋊C4—171no—
414C2×C22⋊C4—171no—
415C2×C22⋊C4—171no—
416C2×C22⋊C4—344no—
417C2×C22⋊C4—344no—
418C2×C22⋊C4C4×D1711yescharacteristic
419C2×C22⋊C4C4×D1711yescharacteristic
420C2×C42—171no—
421C2×C42—171no—
422C2×C42—171no—
423C2×C42—174no—
424C2×C42—174no—
425C2×C42—171no—
426C2×C42—171no—
427C2×C42—171no—
428C2×C42—342no—
429C2×C42—342no—
430C2×C42C22×D1711yescharacteristic
431C2×C42C22×D1711yescharacteristic
432C4⋊C8—344no—
433C4⋊C8—344no—
434C22⋊C8—24no—
435C22⋊C8—344no—
436C22⋊C8—344no—
437C22⋊C8—344no—
438C22⋊C8—344no—
439C22⋊C8—344no—
440#97—22no—
441#97—342no—
442C4×C8—22no—
443C4×C8—342no—
444C17#2071711yescharacteristic, complemented, Sylow
445C24—171no—
446C24—171no—
447C24—342no—
448C24—342no—
449C24—344no—
450C24—342no—
451C24—682no—
452C24C2×C4×D1711yescharacteristic
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—171no—
457C22×C4—171no—
458C22×C4—171no—
459C22×C4—171no—
460C22×C4—172no—
461C22×C4—172no—
462C22×C4—174no—
463C22×C4—174no—
464C22×C4—174no—
465C22×C4—171no—
466C22×C4—171no—
467C22×C4—342no—
468C22×C4—342no—
469C22×C4—342no—
470C22×C4—344no—
471C22×C4—342no—
472C22×C4—3416no—
473C22×C4—3416no—
474C22×C4—342no—
475C22×C4—342no—
476C22×C4—344no—
477C22×C4—342no—
478C22×C4—342no—
479C22×C4—342no—
480C22×C4—342no—
481C22×C4—342no—
482C22×C4—341no—
483C22×C4—341no—
484C22×C4—342no—
485C22×C4—342no—
486C22×C4—342no—
487C22×C4—682no—
488C22×C4—682no—
489C22×C4C2×C4×D1711yescharacteristic
490C22×C4C2×C4×D1711yescharacteristic
491C22×C4C2×C4×D1711yescharacteristic
492C22×C4C2×C4×D1711yescharacteristic
493C22×C4C2×C4×D1711yescharacteristic
494C22×C4#6325214yes—
495C22×C4C23×D1711yescharacteristic
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—342no—
503C2×C8—342no—
504C2×C8—342no—
505C2×C8—342no—
506C2×C8—342no—
507C2×C8—342no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—342no—
511C4⋊C4—342no—
512C4⋊C4—342no—
513C4⋊C4—342no—
514C4⋊C4—342no—
515C4⋊C4—342no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—342no—
519C22⋊C4—342no—
520C22⋊C4—342no—
521C22⋊C4—342no—
522C22⋊C4—342no—
523C22⋊C4—342no—
524C42—21no—
525C42—21no—
526C42—172no—
527C42—172no—
528C42—174no—
529C42—174no—
530C42—342no—
531C42—342no—
532C42—341no—
533C42—341no—
534C42D4×D1712yes—
535C42D4×D1712yes—
536C23—21no—
537C23—21no—
538C23—22no—
539C23—22no—
540C23—172no—
541C23—174no—
542C23—172no—
543C23—348no—
544C23—341no—
545C23—341no—
546C23—342no—
547C23—341no—
548C23—341no—
549C23—342no—
550C23—342no—
551C23—688no—
552C23—684no—
553C23—688no—
554C23—684no—
555C23#1058755812yes—
556C23C22×C4×D1711yescharacteristic
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—172no—
568C2×C4—172no—
569C2×C4—172no—
570C2×C4—172no—
571C2×C4—348no—
572C2×C4—348no—
573C2×C4—348no—
574C2×C4—348no—
575C2×C4—341no—
576C2×C4—341no—
577C2×C4—341no—
578C2×C4—341no—
579C2×C4—342no—
580C2×C4—342no—
581C2×C4—344no—
582C2×C4—344no—
583C2×C4—344no—
584C2×C4—344no—
585C2×C4—341no—
586C2×C4—341no—
587C2×C4—342no—
588C2×C4—342no—
589C2×C4—684no—
590C2×C4—684no—
591C2×C4—684no—
592C2×C4#1058755812yes—
593C2×C4C22⋊C4×D1712yes—
594C2×C4C22⋊C4×D1712yes—
595C2×C4C2×#6325212yes—
596C2×C4C2×D4×D1711yescharacteristic
597C2×C4C2×D4×D1711yescharacteristic
598C2×C4#41×D1712yes—
599C8—42no—
600C8—42no—
601C8—682no—
602C8—682no—
603C22—22no—
604C22—24no—
605C22—41no—
606C22—41no—
607C22—42no—
608C22—174no—
609C22—342no—
610C22—342no—
611C22—344no—
612C22—344no—
613C22—342no—
614C22—344no—
615C22—6816no—
616C22—681no—
617C22—681no—
618C22—682no—
619C22C2×C22⋊C4×D1711yescharacteristic
620C22#117×D1711yescharacteristic
621C22C2×#1058755811yescharacteristic
622C22#131×D1712yes—
623C22#1176357412yes—
624C4—22no—
625C4—22no—
626C4—22no—
627C4—41no—
628C4—41no—
629C4—342no—
630C4—342no—
631C4—342no—
632C4—342no—
633C4—681no—
634C4—681no—
635C4#1176331812yes—
636C4#1176331812yes—
637C2—42no—
638C2—344no—
639C2—682no—
640C2#1453116511yescharacteristic
641C2D17×#40811yescharacteristic
642C2#1453722611yescharacteristic
643C2#1453743912yes—
644C2#1453744012yes—
subgroupG/H
center, FrattiniC22×C4C23×D17
commutator, socleC23×C17C23×C4
FittingC17×#1750C2
prime radicalC24×D17C2×C4
cosocleC22×C4×C17C24
hypercenter#1750D17
hypocenterC17#20717
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ #1750

Complements

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

NH ≅ G/Ntype
2. C2×#11762864C2semidirect
3. C2×#11762860C2semidirect
4. #14528851C2semidirect
5. #14528767C2semidirect
6. #14528766C2semidirect
8. #14528553C2semidirect
9. #14528551C2semidirect
10. #14528550C2semidirect
11. C17×#1750C2semidirect
30. C2×#10587485C22semidirect
31. C2×#10587485C22semidirect
37. C2×#10587474C22semidirect
38. C2×C4×#3622C22semidirect
39. #11762564C22semidirect
40. #11762563C22semidirect
41. #11762563C22semidirect
44. C2×C17×#97C22semidirect
45. C2×C4×C8×C17C22semidirect
46. C17×#336C22semidirect
291. #1750D17semidirect
354. C2×#97C2×D17semidirect
356. C2×C4×C8C2×D17semidirect
360. #336C2×D17semidirect
444. 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×#117628642
C2×#117628602
#145288512
#145287672
#145287662
#145286932
#145285532
#145285512
#145285502
C17×#17502
#2071717