Finite Group Databasebeta

← #17194649

Subgroups — #17194649

Order 4352.

3038 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587583C211yesmaximal, characteristic
2C2×#11763286C211yesmaximal, characteristic, complemented
3C2×#117×D17C211yesmaximal, characteristic
4#14531810C214yesmaximal, complemented
5#14531788C214yesmaximal, complemented
6#14531655C214yesmaximal, complemented
7#14531630C214yesmaximal
8#14531464C214yesmaximal, complemented
9#14531368C211yesmaximal, characteristic, complemented
10#14531314C214yesmaximal, complemented
11#14531163C211yesmaximal, characteristic
12#14531141C211yesmaximal, characteristic, complemented
13C17×#2316C211yesmaximal, characteristic, complemented
14C23×Dic34C2211yescharacteristic
15C23×C4×D17C2211yescharacteristic
16C2×#10587583C4116yescomplemented
17C2×#10587583C2212yes—
18C2×#10587583C2212yes—
19C2×#10587583C2212yes—
20C2×#10587583C2212yes—
21#11763286—28no—
22C2×C4×Dic34C2214yescomplemented
23C2×#10587573C2214yescomplemented
24C2×#10587612C2211yescharacteristic
25C2×#10587612C2211yescharacteristic
26C2×#10587578C2212yes—
27C2×#10587558C2212yes—
28C2×#10587563C2214yes—
29C2×#10587563C2212yes—
30#117×D17—28no—
31C22×C4⋊Dic17C2212yes—
32C2×C4⋊C4×D17C2212yes—
33C22×#63227C2212yes—
34C2×C22⋊C4×D17C2212yes—
35C2×C42×D17C2212yes—
36#11762715C2214yescomplemented
37#11762714C2214yescomplemented
38#11762705C2218yescomplemented
39#11762699C2218yes—
40#11762698C2218yes—
41#11762692C2218yescomplemented
42#11762692C2218yescomplemented
43#11762680C2218yes—
44#11762670C2214yes—
45#11762670C2212yes—
46#11762666C2212yes—
47#11762666C2212yes—
48#11762664C2212yes—
49C4⋊C4×Dic17C2214yescomplemented
50C4⋊C4×Dic17C2214yescomplemented
51C4⋊C4×Dic17C2218yescomplemented
52C4×#63227C2212yes—
53C4×#63227C2212yes—
54C4×C22⋊Dic17C2214yes—
55#11762656C2218yes—
56C2×C17×#117C2211yescharacteristic
57C2×C17×#117C2211yescharacteristic
58C22×C4⋊C4×C17C2211yescharacteristic
59C17×#389C2214yescomplemented
60C17×#389C2214yescomplemented
61C17×#380C2214yes—
62C24×D17C2311yescharacteristic
63C23×C4×C17C2312yes—
64C23×C4×C17C2311yescharacteristic
65C23×Dic17C2312yes—
66C23×Dic17C2311yescharacteristic
67C22×C4×D17—24no—
68C22×C4×D17—24no—
69C22×C4×D17C2×C418yes—
70C22×C4×D17C2314yes—
71C22×C4×D17C2311yescharacteristic
72C22×C4×D17C2311yescharacteristic
73C22×C4×D17C2311yescharacteristic
74C22×C4×D17C2311yescharacteristic
75C22×Dic34—24no—
76C22×Dic34—24no—
77C22×Dic34C2×C418yescomplemented
78C22×Dic34C2312yes—
79C22×Dic34C2312yes—
80C17×#117—24no—
81C17×#117—24no—
82C2×C4⋊C4×C17C2×C418yescomplemented
83C2×C4⋊C4×C17C2312yes—
84C2×C4⋊C4×C17C2312yes—
85C2×C4⋊C4×C17C2318yes—
86C2×C4⋊C4×C17C2312yes—
87C2×C4⋊C4×C17C2312yes—
88C2×C22⋊C4×C17C2312yes—
89C2×C22⋊C4×C17C2312yes—
90C2×C42×C17C2312yes—
91C2×C42×C17C2312yes—
92C2×C22⋊Dic17C2312yes—
93C2×C22⋊Dic17C2312yes—
94C2×#63227C2×C4116yes—
95C2×#63227C2314yes—
96C2×#63227C2314yes—
97C2×#63227C2314yes—
98C2×#63227C2314yes—
99C2×#63227C2314yes—
100#10587612—24no—
101#10587612D418yes—
102C2×C4⋊Dic17C2×C4116yescomplemented
103C2×C4⋊Dic17C2312yes—
104C2×C4⋊Dic17C2314yes—
105C2×C4⋊Dic17C2314yes—
106C2×C4⋊Dic17C2312yes—
107C2×#63225C2314yes—
108C2×#63225C2318yes—
109C2×C4×Dic17C2314yes—
110C2×C4×Dic17C2318yes—
111C2×C4×Dic17C2312yes—
112C2×C4×Dic17C2318yes—
113C2×C4×Dic17C2312yes—
114#10587583C2×C4116yes—
115#10587583C2×C4116yes—
116#10587583C2×C4116yescomplemented
117#10587583C2×C4116yescomplemented
118#10587578—28no—
119C4⋊C4×D17—28no—
120#10587573—216no—
121C22⋊C4×D17—28no—
122#10587563—28no—
123#10587563—216no—
124#10587558—28no—
125C42×D17—28no—
126C4×Dic34—216no—
127C17×#95C2318yes—
128#10587502C2318yes—
129#10587502C2314yes—
130#10587502C2314yes—
131#10587502C2318yes—
132#10587502C2318yes—
133C24×C17C2411yescharacteristic
134C23×D17—22no—
135C23×D17—22no—
136C23×D17—24no—
137C23×D17—42no—
138C23×D17C22×C414yes—
139C23×D17C2411yescharacteristic
140C23×D17C2411yescharacteristic
141C22×C4×C17—44no—
142C22×C4×C17C22×C418yes—
143C22×C4×C17C22×C414yes—
144C22×C4×C17C2×D414yes—
145C22×C4×C17#4114yes—
146C22×C4×C17C2414yes—
147C22×C4×C17C2414yes—
148C22×C4×C17C2414yes—
149C22×C4×C17C2411yescharacteristic
150C22×C4×C17C2411yescharacteristic
151C22×Dic17—24no—
152C22×Dic17—22no—
153C22×Dic17—24no—
154C22×Dic17—22no—
155C22×Dic17C22×C418yes—
156C22×Dic17C22×C414yes—
157C22×Dic17C2414yes—
158C22×Dic17C2414yes—
159C22×Dic17C2411yescharacteristic
160C22×Dic17C2411yescharacteristic
161C22×Dic17C2414yes—
162C2×C4×D17—232no—
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—22no—
170C2×C4×D17—22no—
171C2×C4×D17—28no—
172C2×C4×D17—28no—
173C2×C4×D17—44no—
174C2×C4×D17C22×C414yes—
175C2×C4×D17C22×C414yes—
176C2×C4×D17C22×C414yes—
177C2×C4×D17C22×C414yes—
178C2×Dic34—232no—
179C2×Dic34—24no—
180C2×Dic34—24no—
181C2×Dic34—24no—
182C2×Dic34—24no—
183C2×Dic34C22×C418yes—
184C2×Dic34C22×C418yes—
185C4⋊C4×C17—24no—
186C4⋊C4×C17—24no—
187C4⋊C4×C17—48no—
188C4⋊C4×C17C22×C418yes—
189C4⋊C4×C17C22×C418yes—
190C22⋊C4×C17—24no—
191C22⋊C4×C17—24no—
192C42×C17—24no—
193C42×C17—24no—
194C22⋊Dic17—24no—
195C22⋊Dic17C2×D418yes—
196#63227—28no—
197#63227—28no—
198#63227—44no—
199#63227C22×C4116yes—
200#63227C22×C4116yes—
201C4⋊Dic17—24no—
202C4⋊Dic17C22×C4116yes—
203C4⋊Dic17C22×C4116yes—
204C4⋊Dic17C2×D418yes—
205#63225—216no—
206#63225—28no—
207C4×Dic17—24no—
208C4×Dic17—216no—
209C4×Dic17—28no—
210C4×Dic17—48no—
211C4×Dic17C2×D418yes—
212#34203—171nomaximal, Sylow
213C23×C17—22no—
214C23×C17—22no—
215C23×C17C23×C412yes—
216C23×C17C22×D412yes—
217C23×C17C2×#4112yes—
218C23×C17C2511yescharacteristic
219C22×D17—21no—
220C22×D17—21no—
221C22×D17—22no—
222C22×D17—28no—
223C22×D17—21no—
224C22×D17—21no—
225C22×D17—22no—
226C22×D17—22no—
227C22×D17—28no—
228C22×D17—44no—
229C22×D17—48no—
230C22×D17—48no—
231C22×D17C23×C412yes—
232C22×D17C23×C412yes—
233C22×D172−1+414yes—
234C2×C4×C17—28no—
235C2×C4×C17—24no—
236C2×C4×C17—24no—
237C2×C4×C17—24no—
238C2×C4×C17—24no—
239C2×C4×C17—24no—
240C2×C4×C17—24no—
241C2×C4×C17—48no—
242C2×C4×C17C4×D4116yes—
243C2×C4×C17C23×C418yes—
244C2×C4×C17C23×C412yes—
245C2×C4×C17C23×C412yes—
246C2×C4×C17C22×D412yes—
247C2×C4×C17C22×D412yes—
248C2×C4×C17C2×#4112yes—
249C2×C4×C17C2×#4112yes—
250C2×Dic17—216no—
251C2×Dic17—28no—
252C2×Dic17—24no—
253C2×Dic17—21no—
254C2×Dic17—21no—
255C2×Dic17—24no—
256C2×Dic17—21no—
257C2×Dic17—21no—
258C2×Dic17—24no—
259C2×Dic17—24no—
260C2×Dic17—24no—
261C2×Dic17—24no—
262C2×Dic17—24no—
263C2×Dic17C23×C418yes—
264C2×Dic17C23×C412yes—
265C2×Dic17C23×C412yes—
266C2×Dic17C22×D418yes—
267C4×D17—28no—
268C4×D17—28no—
269C4×D17—28no—
270C4×D17—28no—
271C4×D17—416no—
272Dic34—216no—
273Dic34—216no—
274C22×C22⋊Q8—171no—
275C2×#519—171no—
276C22×#117—171no—
277#2362—174no—
278#2360—174no—
279#2344—174no—
280#2342—174no—
281#2323—174no—
282#2316D1711yescharacteristic, complemented
283#2310—171no—
284C4×C22⋊Q8—174no—
285#2280—171no—
286#2274—171no—
287C22×C17—22no—
288C22×C17—22no—
289C22×C17—24no—
290C22×C17—24no—
291C22×C17C2×C4×D414yes—
292C22×C17C24×C411yescharacteristic
293C22×C17C23×D411yescharacteristic
294C22×C17C22×#4111yescharacteristic
295C22×C17C2×2−1+412yes—
296C22×C17#58512yes—
297C2×D17—22no—
298C2×D17—22no—
299C2×D17—24no—
300C2×D17—24no—
301C2×D17—22no—
302C2×D17—22no—
303C2×D17—22no—
304C2×D17—416no—
305C2×D17—42no—
306C2×D17—42no—
307C2×D17#51914yescomplemented
308C4×C17—28no—
309C4×C17—44no—
310C4×C17—44no—
311C4×C17C2×C4×D414yes—
312C4×C17C2×C4×D414yes—
313Dic17—28no—
314Dic17—22no—
315Dic17—22no—
316Dic17—28no—
317Dic17—44no—
318C23×Q8—171no—
319C24×C4—171no—
320C2×C22⋊Q8—1716no—
321C2×C22⋊Q8—172no—
322C2×C22⋊Q8—172no—
323C2×C22⋊Q8—172no—
324C2×C22⋊Q8—172no—
325#519—348no—
326C2×C4×Q8—174no—
327C2×C4×Q8—174no—
328C2×#117—171no—
329C2×#117—171no—
330C2×#117—172no—
331C2×#117—174no—
332C2×#117—172no—
333C2×#117—172no—
334C2×#117—348no—
335C2×#117C2×D1711yescharacteristic
336C2×#117C2×D1711yescharacteristic
337C22×C4⋊C4—172no—
338C22×C4⋊C4—172no—
339C22×C4⋊C4C2×D1711yescharacteristic
340C22×C22⋊C4—172no—
341C22×C22⋊C4—172no—
342C22×C42—172no—
343#391—174no—
344#391—174no—
345#389—178no—
346#389C2×D1714yescomplemented
347#389C2×D1714yescomplemented
348#388—178no—
349#388—178no—
350#387—178no—
351#387—178no—
352#385—178no—
353#381—174no—
354#381—172no—
355#380—172no—
356#380—172no—
357#380C2×D1714yes—
358#379—172no—
359C4×C4⋊C4—174no—
360C4×C4⋊C4—178no—
361C4×C4⋊C4—174no—
362C4×C22⋊C4—174no—
363C4×C22⋊C4—172no—
364C4×C22⋊C4—172no—
365C4×C22⋊C4—178no—
366C2×C17—24no—
367C2×C17C22×C4×D411yescharacteristic
368C2×C17C2×#51911yescharacteristic
369C2×C17#342811yescharacteristic
370C2×C17#348912yes—
371C2×C17#349512yes—
372D17—24no—
373D17—42no—
374C25—171no—
375C22×Q8—178no—
376C22×Q8—172no—
377C22×Q8—172no—
378C22×Q8—344no—
379C22×Q8—344no—
380C23×C4—172no—
381C23×C4—171no—
382C23×C4—178no—
383C23×C4—171no—
384C23×C4—171no—
385C23×C4—171no—
386C23×C4—171no—
387C23×C4—174no—
388C23×C4—344no—
389C23×C4—344no—
390C23×C4C22×D1712yes—
391C23×C4C22×D1711yescharacteristic
392C22⋊Q8—1716no—
393C22⋊Q8—1716no—
394C22⋊Q8—1716no—
395C22⋊Q8—1716no—
396C4×Q8—3416no—
397C4×Q8—3416no—
398#117—24no—
399#117—24no—
400#117—178no—
401#117—348no—
402#117—348no—
403#117—3416no—
404#117—348no—
405#117—344no—
406C2×C4⋊C4—1716no—
407C2×C4⋊C4—172no—
408C2×C4⋊C4—174no—
409C2×C4⋊C4—174no—
410C2×C4⋊C4—174no—
411C2×C4⋊C4—178no—
412C2×C4⋊C4—172no—
413C2×C4⋊C4—348no—
414C2×C4⋊C4C4×D1718yescomplemented
415C2×C4⋊C4C22×D1712yes—
416C2×C4⋊C4C22×D1712yes—
417C2×C4⋊C4C22×D1718yes—
418C2×C4⋊C4C22×D1712yes—
419C2×C4⋊C4C22×D1712yes—
420C2×C22⋊C4—172no—
421C2×C22⋊C4—1716no—
422C2×C22⋊C4—174no—
423C2×C22⋊C4—174no—
424C2×C22⋊C4—172no—
425C2×C22⋊C4—174no—
426C2×C22⋊C4—174no—
427C2×C22⋊C4—174no—
428C2×C22⋊C4—348no—
429C2×C22⋊C4C22×D1712yes—
430C2×C22⋊C4C22×D1712yes—
431C2×C42—172no—
432C2×C42—174no—
433C2×C42—178no—
434C2×C42—172no—
435C2×C42—178no—
436C2×C42—348no—
437C2×C42C22×D1712yes—
438C2×C42C22×D1712yes—
439#95—174no—
440#95—178no—
441#95—174no—
442#95—178no—
443#95—178no—
444#95C22×D1718yes—
445C17#3420311yescharacteristic, complemented, Sylow
446C24—171no—
447C24—171no—
448C24—174no—
449C24—344no—
450C24—342no—
451C24—342no—
452C24—682no—
453C24C23×D1711yescharacteristic
454C2×Q8—178no—
455C2×Q8—178no—
456C2×Q8—344no—
457C2×Q8—344no—
458C2×Q8—344no—
459C2×Q8—344no—
460C2×Q8—3432no—
461C22×C4—44no—
462C22×C4—178no—
463C22×C4—174no—
464C22×C4—171no—
465C22×C4—174no—
466C22×C4—171no—
467C22×C4—174no—
468C22×C4—174no—
469C22×C4—174no—
470C22×C4—174no—
471C22×C4—174no—
472C22×C4—174no—
473C22×C4—348no—
474C22×C4—348no—
475C22×C4—342no—
476C22×C4—344no—
477C22×C4—342no—
478C22×C4—344no—
479C22×C4—342no—
480C22×C4—342no—
481C22×C4—342no—
482C22×C4—342no—
483C22×C4—342no—
484C22×C4—342no—
485C22×C4—342no—
486C22×C4—342no—
487C22×C4—3432no—
488C22×C4—684no—
489C22×C4C2×C4×D1718yes—
490C22×C4C2×C4×D1714yes—
491C22×C4D4×D1714yes—
492C22×C4#6325614yes—
493C22×C4C23×D1714yes—
494C22×C4C23×D1714yes—
495C22×C4C23×D1714yes—
496C22×C4C23×D1711yescharacteristic
497C22×C4C23×D1711yescharacteristic
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—48no—
501C4⋊C4—1716no—
502C4⋊C4—1716no—
503C4⋊C4—178no—
504C4⋊C4—348no—
505C4⋊C4—3416no—
506C4⋊C4—344no—
507C4⋊C4C2×C4×D1718yes—
508C4⋊C4C2×C4×D1718yes—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—178no—
512C22⋊C4—1716no—
513C22⋊C4—1716no—
514C22⋊C4—348no—
515C22⋊C4—348no—
516C22⋊C4—344no—
517C22⋊C4—684no—
518C42—24no—
519C42—24no—
520C42—178no—
521C42—348no—
522C42—3416no—
523C42—344no—
524C42—688no—
525C23—22no—
526C23—22no—
527C23—172no—
528C23—174no—
529C23—172no—
530C23—348no—
531C23—341no—
532C23—341no—
533C23—342no—
534C23—348no—
535C23—341no—
536C23—341no—
537C23—342no—
538C23—342no—
539C23—684no—
540C23—688no—
541C23—688no—
542C23C22×C4×D1712yes—
543C23C2×D4×D1712yes—
544C23C2×#6325612yes—
545C23C24×D1711yescharacteristic
546Q8—3416no—
547Q8—3416no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—48no—
556C2×C4—178no—
557C2×C4—172no—
558C2×C4—172no—
559C2×C4—178no—
560C2×C4—348no—
561C2×C4—3416no—
562C2×C4—341no—
563C2×C4—344no—
564C2×C4—341no—
565C2×C4—341no—
566C2×C4—341no—
567C2×C4—344no—
568C2×C4—344no—
569C2×C4—344no—
570C2×C4—348no—
571C2×C4—348no—
572C2×C4—348no—
573C2×C4—348no—
574C2×C4—344no—
575C2×C4—344no—
576C2×C4—344no—
577C2×C4—6816no—
578C2×C4#10587579116yes—
579C2×C4C22×C4×D1718yes—
580C2×C4C22×C4×D1712yes—
581C2×C4C22×C4×D1712yes—
582C2×C4C2×D4×D1712yes—
583C2×C4C2×D4×D1712yes—
584C2×C4C2×#6325612yes—
585C2×C4C2×#6325612yes—
586C22—24no—
587C22—22no—
588C22—22no—
589C22—24no—
590C22—174no—
591C22—344no—
592C22—342no—
593C22—342no—
594C22—344no—
595C22—342no—
596C22—342no—
597C22—342no—
598C22—6816no—
599C22—682no—
600C22—682no—
601C22C2×#1058757914yes—
602C22C23×C4×D1711yescharacteristic
603C22C22×D4×D1711yescharacteristic
604C22C22×#6325611yescharacteristic
605C222−1+4×D1712yes—
606C22#1176364412yes—
607C4—28no—
608C4—44no—
609C4—44no—
610C4—348no—
611C4—342no—
612C4—342no—
613C4—348no—
614C4—684no—
615C4C2×#1058757914yes—
616C4C2×#1058757914yes—
617C2—24no—
618C2—344no—
619C2—682no—
620C2C22×#1058757911yescharacteristic
621C2D17×#51911yescharacteristic
622C2#1454198211yescharacteristic
623C2#1454245912yes—
624C2#1454253912yes—
subgroupG/H
center, FrattiniC23C24×D17
commutatorC22×C17C24×C4
FittingC17×#2316C2
socle, cosocleC23×C17C25
prime radicalC24×D17C23
hypercenter#2316D17
hypocenterC17#34203
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ #2316

Complements

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

NH ≅ G/Ntype
2. C2×#11763286C2semidirect
4. #14531810C2semidirect
5. #14531788C2semidirect
6. #14531655C2semidirect
8. #14531464C2semidirect
9. #14531368C2semidirect
10. #14531314C2semidirect
12. #14531141C2semidirect
13. C17×#2316C2semidirect
16. C2×#10587583C4semidirect
22. C2×C4×Dic34C22semidirect
23. C2×#10587573C22semidirect
36. #11762715C22semidirect
37. #11762714C22semidirect
38. #11762705C22semidirect
41. #11762692C22semidirect
42. #11762692C22semidirect
49. C4⋊C4×Dic17C22semidirect
50. C4⋊C4×Dic17C22semidirect
51. C4⋊C4×Dic17C22semidirect
59. C17×#389C22semidirect
60. C17×#389C22semidirect
77. C22×Dic34C2×C4semidirect
82. C2×C4⋊C4×C17C2×C4semidirect
102. C2×C4⋊Dic17C2×C4semidirect
116. #10587583C2×C4semidirect
117. #10587583C2×C4semidirect
282. #2316D17semidirect
307. C2×D17#519semidirect
346. #389C2×D17semidirect
347. #389C2×D17semidirect
414. C2×C4⋊C4C4×D17semidirect
445. C17#34203semidirect

Maximal subgroups

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

subgroupindex
C22×#105875832
C2×#117632862
C2×#117×D172
#145318102
#145317882
#145316552
#145316302
#145314642
#145313682
#145313142
#145311632
#145311412
C17×#23162
#3420317