Finite Group Databasebeta

← #17002135

Subgroups — #17002135

Order 4352.

1002 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14537813C211yesmaximal, characteristic, complemented
2#14531757C211yesmaximal, characteristic, complemented
3#14527340C212yesmaximal, complemented
4#14527322C212yesmaximal, complemented
5#14527317C211yesmaximal, characteristic, complemented
6#14527313C211yesmaximal, characteristic, complemented
7#14527309C212yesmaximal, complemented
8#14527304C211yesmaximal, characteristic, complemented
9#14527300C211yesmaximal, characteristic, complemented
10#14527275C212yesmaximal, complemented
11C17×#3505C211yesmaximal, characteristic
12#11762881—22no—
13#11762881—22no—
14#11762866C2212yescomplemented
15#11762773C2211yescharacteristic
16#11762773C2211yescharacteristic
17C2×C22⋊#3622C2212yescomplemented
18#11762712—21no—
19#11762702—21no—
20#11762702—21no—
21#11762675C2211yescharacteristic, complemented
22#11762672C2211yescharacteristic
23#11762672C2211yescharacteristic, complemented
24C4⋊C4×Dic17—21no—
25C4×C4⋊Dic17C2211yescharacteristic, complemented
26C4×C22⋊Dic17C2211yescharacteristic
27C4×C22⋊Dic17C2211yescharacteristic, complemented
28#11762546C2212yescomplemented
29#11762546C2212yescomplemented
30#11762546C2212yescomplemented
31#11762546C2212yescomplemented
32#11762543C2212yes—
33#11762543C2212yes—
34#11762543C2212yes—
35#11762543C2212yes—
36C17×#553—22no—
37C17×#553C412yescomplemented
38C17×#552C412yescomplemented
39C17×#551—22no—
40C17×#548C412yescomplemented
41C17×#546C412yescomplemented
42C17×#539—21no—
43C17×#538—21no—
44C17×#534—21no—
45C17×#529C2212yes—
46C17×#525C2212yes—
47C2×C17×C22⋊Q8C2211yescharacteristic
48C2×C17×#121C2211yescharacteristic
49C17×#518—21no—
50C2×C4×D4×C17C2211yescharacteristic
51C2×#41×C17C2×C412yes—
52C22×Q8×C17C2×C411yescharacteristic
53C22×D4×C17—21no—
54C22×D4×C17C2×C411yescharacteristic
55C22×D4×C17D411yescharacteristic
56C23×C4×C17C2311yescharacteristic
57C23×C4×C17C2311yescharacteristic
58C17×#126—22no—
59C17×#126—22no—
60C17×#126—22no—
61C17×#125—22no—
62C17×#125C2×C412yes—
63C17×#124—22no—
64C17×#124C2×C412yes—
65C17×#123—22no—
66C17×#123—22no—
67C17×#123—22no—
68C17×#123—22no—
69C17×#123—22no—
70C17×C22⋊Q8—21no—
71C17×C22⋊Q8—21no—
72C17×C22⋊Q8—22no—
73C17×C22⋊Q8C2×C412yes—
74C17×C22⋊Q8C2×C412yes—
75C17×C22⋊Q8C2312yes—
76C17×#121—22no—
77C17×#121—22no—
78C17×#121—21no—
79C17×#121—21no—
80C17×#121—22no—
81C17×#121C2×C412yes—
82C17×#121C2×C412yes—
83C17×#121C2312yes—
84C17×#120—22no—
85C17×#120—22no—
86C4×Q8×C17C2×C412yes—
87C4×D4×C17—22no—
88C4×D4×C17—22no—
89C4×D4×C17—22no—
90C4×D4×C17—22no—
91C4×D4×C17—22no—
92C4×D4×C17C2×C412yes—
93C4×D4×C17C2×C412yes—
94C4×D4×C17C2×C412yes—
95C4×D4×C17D412yes—
96C4×D4×C17D412yes—
97C17×#117—22no—
98C17×#117—22no—
99C17×#117C2×C412yes—
100C2×C4⋊C4×C17—21no—
101C2×C4⋊C4×C17C2×C411yescharacteristic
102C2×C4⋊C4×C17C2×C411yescharacteristic
103C2×C4⋊C4×C17D411yescharacteristic
104C2×C22⋊C4×C17—21no—
105C2×C22⋊C4×C17—21no—
106C2×C22⋊C4×C17—22no—
107C2×C22⋊C4×C17D411yescharacteristic
108C2×C22⋊C4×C17D411yescharacteristic
109C2×C42×C17C2311yescharacteristic
110C2×C22⋊Dic17—21no—
111C2×C22⋊Dic17—21no—
112C2×C22⋊Dic17—21no—
113C2×C22⋊Dic17—21no—
114C2×C4⋊Dic17—21no—
115C2×C4⋊Dic17—21no—
116C2×C4×Dic17—21no—
117C2×C4×Dic17—21no—
118C2×C4×Dic17—21no—
119C2×#63223—21no—
120C2×#63223—21no—
121C22×#3622—22no—
122#10587502—21no—
123#10587502—21no—
124#10587502—21no—
125#10587502C2311yescharacteristic
126#10587502C2311yescharacteristic
127#10587502C2311yescharacteristic
128#10587502C2311yescharacteristic
129C22⋊#3622—22no—
130C22⋊#3622—21no—
131C22⋊#3622—21no—
132C22⋊#3622C2312yes—
133C22⋊#3622C2312yes—
134C4⋊#3622—22no—
135#10587474—22no—
136C24×C17—21no—
137C24×C17C2×D411yescharacteristic
138C24×C17C2×D411yescharacteristic
139#41×C17—82no—
140C2×Q8×C17—21no—
141C2×Q8×C17—81no—
142C2×Q8×C17C22×C412yes—
143C2×D4×C17—22no—
144C2×D4×C17—22no—
145C2×D4×C17—22no—
146C2×D4×C17—22no—
147C2×D4×C17—21no—
148C2×D4×C17—41no—
149C2×D4×C17—41no—
150C2×D4×C17—42no—
151C2×D4×C17—42no—
152C2×D4×C17—81no—
153C2×D4×C17C22⋊C412yes—
154C2×D4×C17C22⋊C412yes—
155C2×D4×C17C22×C412yes—
156C22×C4×C17—21no—
157C22×C4×C17—21no—
158C22×C4×C17—21no—
159C22×C4×C17—22no—
160C22×C4×C17—22no—
161C22×C4×C17—81no—
162C22×C4×C17—81no—
163C22×C4×C17C22×C411yescharacteristic
164C22×C4×C17C22×C412yes—
165C22×C4×C17C22×C411yescharacteristic
166C22×C4×C17C2×D411yescharacteristic
167C22×C4×C17C2×D411yescharacteristic
168C22×C4×C17C2×D411yescharacteristic
169C22×C4×C17C2×D411yescharacteristic
170C22×C4×C17C2×D411yescharacteristic
171C22×C4×C17C2×D412yes—
172C22×C4×C17C2×D411yescharacteristic
173C22×C4×C17C2411yescharacteristic
174C22×Dic17—21no—
175C22×Dic17—21no—
176C22×Dic17—21no—
177C22×Dic17—21no—
178C4⋊C4×C17—21no—
179C4⋊C4×C17—22no—
180C4⋊C4×C17—22no—
181C4⋊C4×C17—22no—
182C4⋊C4×C17—22no—
183C4⋊C4×C17—22no—
184C4⋊C4×C17—21no—
185C4⋊C4×C17—22no—
186C4⋊C4×C17C22⋊C412yes—
187C4⋊C4×C17C22⋊C412yes—
188C4⋊C4×C17C22×C412yes—
189C4⋊C4×C17C22×C412yes—
190C22⋊C4×C17—22no—
191C22⋊C4×C17—22no—
192C22⋊C4×C17—22no—
193C22⋊C4×C17—22no—
194C22⋊C4×C17—22no—
195C22⋊C4×C17—22no—
196C22⋊C4×C17—22no—
197C22⋊C4×C17—22no—
198C22⋊C4×C17—22no—
199C22⋊C4×C17—22no—
200C22⋊C4×C17C22⋊C412yes—
201C22⋊C4×C17C22⋊C412yes—
202C22⋊C4×C17C22⋊C412yes—
203C22⋊C4×C17C22⋊C412yes—
204C42×C17—22no—
205C42×C17—22no—
206C42×C17C22×C412yes—
207C42×C17C2×D412yes—
208C22⋊Dic17—42no—
209C22⋊Dic17—42no—
210C22⋊Dic17—81no—
211C22⋊Dic17—81no—
212C4⋊Dic17—42no—
213C4⋊Dic17—42no—
214C4×Dic17—42no—
215C4×Dic17—81no—
216C4×Dic17—81no—
217#63223—81no—
218#63223—81no—
219C2×#3622—22no—
220C2×#3622—22no—
221C2×#3622—82no—
222#17625—171nomaximal, Sylow
223C23×C17—21no—
224C23×C17—21no—
225C23×C17—21no—
226C23×C17—21no—
227C23×C17—22no—
228C23×C17—41no—
229C23×C17—41no—
230C23×C17—41no—
231C23×C17—41no—
232C23×C17—42no—
233C23×C17—81no—
234C23×C17—81no—
235C23×C17C2×C22⋊C411yescharacteristic
236C23×C17C2×C22⋊C412yes—
237C23×C17C2×C22⋊C411yescharacteristic
238C23×C17C22×D411yescharacteristic
239Q8×C17—42no—
240Q8×C17—81no—
241D4×C17—42no—
242D4×C17—42no—
243D4×C17—42no—
244D4×C17—44no—
245D4×C17—42no—
246D4×C17—42no—
247D4×C17—81no—
248D4×C17—82no—
249D4×C17—82no—
250C2×C4×C17—21no—
251C2×C4×C17—21no—
252C2×C4×C17—22no—
253C2×C4×C17—22no—
254C2×C4×C17—22no—
255C2×C4×C17—41no—
256C2×C4×C17—41no—
257C2×C4×C17—41no—
258C2×C4×C17—42no—
259C2×C4×C17—42no—
260C2×C4×C17—41no—
261C2×C4×C17—41no—
262C2×C4×C17—41no—
263C2×C4×C17—41no—
264C2×C4×C17—41no—
265C2×C4×C17—42no—
266C2×C4×C17—82no—
267C2×C4×C17—82no—
268C2×C4×C17—81no—
269C2×C4×C17—81no—
270C2×C4×C17C2×C22⋊C412yes—
271C2×C4×C17C2×C22⋊C412yes—
272C2×C4×C17C2×C22⋊C412yes—
273C2×C4×C17C2×C22⋊C411yescharacteristic
274C2×C4×C17C2×C22⋊C411yescharacteristic
275C2×C4×C17C23×C411yescharacteristic
276C2×C4×C17C22×D411yescharacteristic
277C2×Dic17—22no—
278C2×Dic17—41no—
279C2×Dic17—41no—
280C2×Dic17—41no—
281C2×Dic17—41no—
282C2×Dic17—41no—
283C2×Dic17—41no—
284C2×Dic17—41no—
285C2×Dic17—41no—
286C2×Dic17—41no—
287C2×Dic17—41no—
288C2×Dic17—41no—
289#3622—42no—
290#3622—42no—
291#3505D1711yescharacteristic
292#2973—171no—
293#2357—171no—
294#1516—172no—
295#1512—172no—
296#1511—171no—
297#1510—171no—
298#1509—172no—
299#1508—171no—
300#1507—171no—
301#1501—172no—
302C22×C17—21no—
303C22×C17—21no—
304C22×C17—21no—
305C22×C17—22no—
306C22×C17—21no—
307C22×C17—21no—
308C22×C17—21no—
309C22×C17—22no—
310C22×C17—21no—
311C22×C17—22no—
312C22×C17—21no—
313C22×C17—42no—
314C22×C17—42no—
315C22×C17—42no—
316C22×C17—81no—
317C22×C17—81no—
318C22×C17—81no—
319C22×C17—81no—
320C22×C17—162no—
321C22×C17C22×C22⋊C411yescharacteristic
322C22×C17#57511yescharacteristic
323C22×C17#57511yescharacteristic
324C4×C17—21no—
325C4×C17—21no—
326C4×C17—22no—
327C4×C17—21no—
328C4×C17—21no—
329C4×C17—22no—
330C4×C17—22no—
331C4×C17—42no—
332C4×C17—42no—
333C4×C17—42no—
334Dic17—42no—
335Dic17—81no—
336Dic17—81no—
337Dic17—81no—
338#553—22no—
339#553Dic1712yescomplemented
340#552Dic1712yescomplemented
341#551—22no—
342#548Dic1712yescomplemented
343#546Dic1712yescomplemented
344#539—21no—
345#538—21no—
346#534—21no—
347#529C2×D1712yes—
348#525C2×D1712yes—
349C2×C22⋊Q8C2×D1711yescharacteristic
350C2×#121C2×D1711yescharacteristic
351#518—21no—
352C2×C4×D4C2×D1711yescharacteristic
353#435—342no—
354#435—342no—
355#432—172no—
356#407—171no—
357#407—171no—
358C2×C22⋊C8—172no—
359#390—341no—
360#388—341no—
361#388—341no—
362#382—171no—
363#381—171no—
364#381—171no—
365C4×C4⋊C4—171no—
366C4×C4⋊C4—341no—
367C4×C22⋊C4—171no—
368C4×C22⋊C4—171no—
369#328—172no—
370#328—172no—
371#328—172no—
372#328—172no—
373#327—172no—
374#327—172no—
375#327—172no—
376#327—172no—
377C2×C17—21no—
378C2×C17—21no—
379C2×C17—41no—
380C2×C17—42no—
381C2×C17—41no—
382C2×C17—82no—
383C2×C17#288311yescharacteristic
384C2×C17#289611yescharacteristic
385C2×C17#290111yescharacteristic
386C2×#41C2×Dic1712yes—
387C22×Q8C2×Dic1711yescharacteristic
388C22×D4—21no—
389C22×D4C2×Dic1711yescharacteristic
390C22×D4C22⋊D1711yescharacteristic
391C23×C4C22×D1711yescharacteristic
392C23×C4C22×D1711yescharacteristic
393C2×M16—341no—
394C2×M16—341no—
395C22×C8—342no—
396#126—22no—
397#126—22no—
398#126—22no—
399#125—22no—
400#125C2×Dic1712yes—
401#124—22no—
402#124C2×Dic1712yes—
403#123—22no—
404#123—22no—
405#123—22no—
406#123—22no—
407#123—22no—
408C22⋊Q8—21no—
409C22⋊Q8—21no—
410C22⋊Q8—22no—
411C22⋊Q8C2×Dic1712yes—
412C22⋊Q8C2×Dic1712yes—
413C22⋊Q8C22×D1712yes—
414#121—22no—
415#121—22no—
416#121—21no—
417#121—21no—
418#121—22no—
419#121C2×Dic1712yes—
420#121C2×Dic1712yes—
421#121C22×D1712yes—
422#120—22no—
423#120—22no—
424C4×Q8C2×Dic1712yes—
425C4×D4—22no—
426C4×D4—22no—
427C4×D4—22no—
428C4×D4—22no—
429C4×D4—22no—
430C4×D4C2×Dic1712yes—
431C4×D4C2×Dic1712yes—
432C4×D4C2×Dic1712yes—
433C4×D4C22⋊D1712yes—
434C4×D4C22⋊D1712yes—
435#117—22no—
436#117—22no—
437#117C2×Dic1712yes—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—341no—
440C2×C4⋊C4—341no—
441C2×C4⋊C4C2×Dic1711yescharacteristic
442C2×C4⋊C4C2×Dic1711yescharacteristic
443C2×C4⋊C4C22⋊D1711yescharacteristic
444C2×C22⋊C4—21no—
445C2×C22⋊C4—21no—
446C2×C22⋊C4—22no—
447C2×C22⋊C4—341no—
448C2×C22⋊C4—341no—
449C2×C22⋊C4—341no—
450C2×C22⋊C4—341no—
451C2×C22⋊C4C22⋊D1711yescharacteristic
452C2×C22⋊C4C22⋊D1711yescharacteristic
453C2×C42—341no—
454C2×C42—341no—
455C2×C42—341no—
456C2×C42C22×D1711yescharacteristic
457C4⋊C8—342no—
458C22⋊C8—172no—
459C22⋊C8—172no—
460C22⋊C8—342no—
461C22⋊C8—341no—
462C22⋊C8—341no—
463#97—342no—
464#95—171no—
465#95—171no—
466#95—171no—
467#95—171no—
468#95—341no—
469#95—341no—
470#95—341no—
471C17#1762511yescharacteristic, complemented, Sylow
472C24—21no—
473C24C2×C22⋊D1711yescharacteristic
474C24C2×C22⋊D1711yescharacteristic
475#41—82no—
476C2×Q8—21no—
477C2×Q8—81no—
478C2×Q8C22×Dic1712yes—
479C2×D4—22no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—21no—
484C2×D4—41no—
485C2×D4—41no—
486C2×D4—42no—
487C2×D4—42no—
488C2×D4—81no—
489C2×D4C22⋊Dic1712yes—
490C2×D4C22⋊Dic1712yes—
491C2×D4C22×Dic1712yes—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—81no—
498C22×C4—81no—
499C22×C4—341no—
500C22×C4—341no—
501C22×C4—341no—
502C22×C4—341no—
503C22×C4C22×Dic1711yescharacteristic
504C22×C4C22×Dic1711yescharacteristic
505C22×C4C22×Dic1712yes—
506C22×C4C2×C22⋊D1711yescharacteristic
507C22×C4C2×C22⋊D1711yescharacteristic
508C22×C4C2×C22⋊D1711yescharacteristic
509C22×C4C2×C22⋊D1712yes—
510C22×C4C2×C22⋊D1711yescharacteristic
511C22×C4C2×C22⋊D1711yescharacteristic
512C22×C4C2×C22⋊D1711yescharacteristic
513C22×C4C23×D1711yescharacteristic
514M16—1361no—
515M16—1361no—
516C2×C8—342no—
517C2×C8—342no—
518C2×C8—1362no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—682no—
528C4⋊C4—682no—
529C4⋊C4C22⋊Dic1712yes—
530C4⋊C4C22⋊Dic1712yes—
531C4⋊C4C22×Dic1712yes—
532C4⋊C4C22×Dic1712yes—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—682no—
544C22⋊C4—682no—
545C22⋊C4—1361no—
546C22⋊C4—1361no—
547C22⋊C4C22⋊Dic1712yes—
548C22⋊C4C22⋊Dic1712yes—
549C22⋊C4C22⋊Dic1712yes—
550C22⋊C4C22⋊Dic1712yes—
551C42—22no—
552C42—22no—
553C42—682no—
554C42—1361no—
555C42—1361no—
556C42C22×Dic1712yes—
557C42C2×C22⋊D1712yes—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—22no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—42no—
568C23—81no—
569C23—81no—
570C23C2×C22⋊Dic1711yescharacteristic
571C23C2×C22⋊Dic1712yes—
572C23C2×C22⋊Dic1711yescharacteristic
573C23C22×C22⋊D1711yescharacteristic
574Q8—42no—
575Q8—81no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—44no—
580D4—42no—
581D4—42no—
582D4—81no—
583D4—82no—
584D4—82no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—342no—
606C2×C4—681no—
607C2×C4—681no—
608C2×C4—681no—
609C2×C4—681no—
610C2×C4—681no—
611C2×C4—681no—
612C2×C4—681no—
613C2×C4—681no—
614C2×C4—681no—
615C2×C4—681no—
616C2×C4—681no—
617C2×C4C2×C22⋊Dic1712yes—
618C2×C4C2×C22⋊Dic1712yes—
619C2×C4C2×C22⋊Dic1712yes—
620C2×C4C2×C22⋊Dic1711yescharacteristic
621C2×C4C2×C22⋊Dic1711yescharacteristic
622C2×C4C23×Dic1711yescharacteristic
623C2×C4C22×C22⋊D1711yescharacteristic
624C8—682no—
625C8—682no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—22no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—22no—
634C22—21no—
635C22—22no—
636C22—21no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—162no—
645C22C22×C22⋊Dic1711yescharacteristic
646C22#1176360711yescharacteristic
647C22#1176360711yescharacteristic
648C4—21no—
649C4—21no—
650C4—22no—
651C4—21no—
652C4—21no—
653C4—22no—
654C4—22no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—682no—
659C4—1361no—
660C4—1361no—
661C4—1361no—
662C2—21no—
663C2—21no—
664C2—41no—
665C2—41no—
666C2—42no—
667C2—82no—
668C2#1453724511yescharacteristic
669C2#1453730311yescharacteristic
670C2#1453733111yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
Fitting, prime radicalC17×#3505C2
socleC22×C17C22×C22⋊C4
cosocleC22×C4×C17C24
hypercenter#3505D17
hypocenterC17#17625
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #3505

Complements

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

NH ≅ G/Ntype
1. #14537813C2semidirect
2. #14531757C2semidirect
3. #14527340C2semidirect
4. #14527322C2semidirect
5. #14527317C2semidirect
6. #14527313C2semidirect
7. #14527309C2semidirect
8. #14527304C2semidirect
9. #14527300C2semidirect
10. #14527275C2semidirect
14. #11762866C22semidirect
17. C2×C22⋊#3622C22semidirect
21. #11762675C22semidirect
23. #11762672C22semidirect
25. C4×C4⋊Dic17C22semidirect
27. C4×C22⋊Dic17C22semidirect
28. #11762546C22semidirect
29. #11762546C22semidirect
30. #11762546C22semidirect
31. #11762546C22semidirect
37. C17×#553C4semidirect
38. C17×#552C4semidirect
40. C17×#548C4semidirect
41. C17×#546C4semidirect
339. #553Dic17semidirect
340. #552Dic17semidirect
342. #548Dic17semidirect
343. #546Dic17semidirect
471. C17#17625semidirect

Maximal subgroups

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

subgroupindex
#145378132
#145317572
#145273402
#145273222
#145273172
#145273132
#145273092
#145273042
#145273002
#145272752
C17×#35052
#1762517