Finite Group Databasebeta

← #17194792

Subgroups — #17194792

Order 4352.

2422 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587630C211yesmaximal, characteristic
2#14531872C214yesmaximal, complemented
3#14531841C212yesmaximal, complemented
4#14531838C212yesmaximal, complemented
5#14531662C212yesmaximal
6#14531656C212yesmaximal
7Q8×C22⋊Dic17C211yesmaximal, characteristic
8C22⋊C4×Dic34C211yesmaximal, characteristic
9#14531616C211yesmaximal, characteristic
10#14531527C214yesmaximal, complemented
11#14531401C212yesmaximal
12#14531353C212yesmaximal, complemented
13C4×#10587630C212yesmaximal, complemented
14#14531186C211yesmaximal, characteristic
15#14531139C212yesmaximal
16C4×#10587563C211yesmaximal, characteristic
17Q8×C22⋊C4×C17C211yesmaximal, characteristic
18C23×Dic34C2211yescharacteristic
19C2×#10587630C4116yescomplemented
20C2×#10587630C2211yescharacteristic
21C2×#10587630C2211yescharacteristic
22C2×#10587630C2211yescharacteristic
23C2×#10587630C2212yes—
24C2×#10587630C2212yes—
25C2×#10587630C2211yescharacteristic
26C2×Q8×Dic17C2212yes—
27C2×C4×Dic34C2212yes—
28C2×#10587573C2212yes—
29C2×#10587563C2211yescharacteristic
30C2×#10587563C2212yes—
31C2×#10587563C2211yescharacteristic
32C22×#63225C2212yes—
33C22×#63225C2212yes—
34C22×C4×Dic17C2211yescharacteristic
35#11762716C2212yes—
36#11762716C2214yes—
37#11762715C2212yes—
38#11762714C2212yes—
39#11762713C2214yes—
40#11762713C2214yes—
41#11762706C2214yes—
42#11762706C2214yes—
43#11762706C2214yes—
44#11762704C2214yes—
45#11762704C2214yes—
46#11762704C2214yes—
47#11762696C2214yes—
48#11762693C2214yes—
49#11762691C2214yes—
50#11762690C2214yes—
51#11762678C2212yes—
52#11762673C2212yes—
53#11762669C2211yescharacteristic
54#11762669C2212yes—
55#11762667C2212yes—
56#11762667C2212yes—
57#11762667C2212yes—
58C4⋊C4×Dic17C2212yes—
59C4×#63225C2212yes—
60C4×#63225C2214yes—
61C4×#63225C2214yes—
62C4×#63225C2214yes—
63#11762660C2214yes—
64#11762660C2214yes—
65#11762660C2214yes—
66#11762660C2212yes—
67#11762660C2212yes—
68C22⋊C4×Dic17C2212yes—
69C22⋊C4×Dic17C2212yes—
70C22⋊C4×Dic17C2211yescharacteristic
71C22⋊C4×Dic17C2211yescharacteristic
72C4×C22⋊Dic17C2211yescharacteristic
73C4×C22⋊Dic17C2212yes—
74#11762654C2212yes—
75C42×Dic17C2212yes—
76C23×Q8×C17C2211yescharacteristic
77C2×C4×Q8×C17C2212yes—
78C17×#391C2212yes—
79C17×#391C2214yes—
80C17×#380C2211yescharacteristic
81C17×#380C2212yes—
82C4×C22⋊C4×C17C2211yescharacteristic
83C4×C22⋊C4×C17C2212yes—
84C22×Q8×C17—44no—
85C22×Q8×C17C2×C418yes—
86C22×Q8×C17C2311yescharacteristic
87C22×Q8×C17C2311yescharacteristic
88C22×Q8×C17C2312yes—
89C23×C4×C17C2311yescharacteristic
90C23×C4×C17C2312yes—
91C23×Dic17C2311yescharacteristic
92C23×Dic17C2312yes—
93C23×Dic17C2311yescharacteristic
94C22×Dic34—44no—
95C22×Dic34C2×C418yes—
96C22×Dic34C2311yescharacteristic
97C22×Dic34C2311yescharacteristic
98C22×Dic34C2312yes—
99C4×Q8×C17—44no—
100C2×C4⋊C4×C17C2312yes—
101C2×C4⋊C4×C17C2314yes—
102C2×C22⋊C4×C17C2311yescharacteristic
103C2×C22⋊C4×C17C2312yes—
104C2×C22⋊C4×C17C2311yescharacteristic
105C2×C42×C17C2312yes—
106C2×C42×C17C2314yes—
107C2×C22⋊Dic17C2311yescharacteristic
108C2×C22⋊Dic17C2312yes—
109C2×C22⋊Dic17C2311yescharacteristic
110Q8×Dic17—44no—
111#10587630C2×C418yes—
112#10587630C2×C418yes—
113#10587630C2×C418yes—
114#10587630C2×C418yes—
115#10587630C2×C4116yes—
116#10587630C2×C4116yes—
117C2×C4⋊Dic17C2312yes—
118C2×C4⋊Dic17C2314yes—
119C2×#63225C2×C4116yes—
120C2×#63225C2×C4116yes—
121C2×#63225C2312yes—
122C2×#63225C2314yes—
123C2×#63225C2314yes—
124C2×#63225C2314yes—
125C2×#63225C2314yes—
126C2×#63225C2312yes—
127C2×#63225C2312yes—
128C2×#63225C2312yes—
129C2×#63225C2312yes—
130C2×#63225C2312yes—
131C2×#63225C2312yes—
132C2×#63225C2312yes—
133C2×#63225C2312yes—
134C2×#63225C2312yes—
135C2×C4×Dic17C2×C418yes—
136C2×C4×Dic17C2312yes—
137C2×C4×Dic17C2312yes—
138C2×C4×Dic17C2314yes—
139C2×C4×Dic17C2312yes—
140C2×C4×Dic17C2312yes—
141C2×C4×Dic17C2314yes—
142C2×C4×Dic17C2314yes—
143C2×C4×Dic17C2314yes—
144C2×C4×Dic17C2311yescharacteristic
145C2×C4×Dic17C2311yescharacteristic
146C2×C4×Dic17C2311yescharacteristic
147C2×C4×Dic17C2311yescharacteristic
148C2×C4×Dic17C2312yes—
149#10587573—44no—
150#10587563—24no—
151#10587563—28no—
152#10587563Q818yescomplemented
153C4×Dic34—44no—
154C17×#95C2314yes—
155C17×#95C2314yes—
156C17×#95C2312yes—
157C17×#95C2312yes—
158#10587502C2312yes—
159#10587502C2314yes—
160#10587502C2314yes—
161#10587502C2314yes—
162#10587502C2314yes—
163#10587502C2312yes—
164#10587502C2314yes—
165#10587502C2314yes—
166#10587502C2314yes—
167#10587502C2312yes—
168#10587502C2312yes—
169C24×C17C2411yescharacteristic
170C2×Q8×C17—24no—
171C2×Q8×C17—24no—
172C2×Q8×C17—44no—
173C2×Q8×C17—416no—
174C2×Q8×C17C22×C414yes—
175C2×Q8×C17C22×C414yes—
176C2×Q8×C17C22×C418yes—
177C22×C4×C17—24no—
178C22×C4×C17—44no—
179C22×C4×C17C22×C414yes—
180C22×C4×C17C22×C418yes—
181C22×C4×C17C2412yes—
182C22×C4×C17C2414yes—
183C22×C4×C17C2411yescharacteristic
184C22×C4×C17C2412yes—
185C22×C4×C17C2412yes—
186C22×C4×C17C2412yes—
187C22×C4×C17C2411yescharacteristic
188C22×Dic17—24no—
189C22×Dic17—44no—
190C22×Dic17C22×C414yes—
191C22×Dic17C22×C418yes—
192C22×Dic17C22×C414yes—
193C22×Dic17C2×Q814yes—
194C22×Dic17#4114yes—
195C22×Dic17C2412yes—
196C22×Dic17C2414yes—
197C22×Dic17C2412yes—
198C22×Dic17C2412yes—
199C22×Dic17C2411yescharacteristic
200C22×Dic17C2412yes—
201C22×Dic17C2411yescharacteristic
202C22×Dic17C2411yescharacteristic
203C22×Dic17C2411yescharacteristic
204C2×Dic34—24no—
205C2×Dic34—24no—
206C2×Dic34—416no—
207C2×Dic34—44no—
208C2×Dic34C22×C414yes—
209C2×Dic34C22×C414yes—
210C2×Dic34C22×C418yes—
211C4⋊C4×C17—24no—
212C4⋊C4×C17—44no—
213C22⋊C4×C17—22no—
214C22⋊C4×C17—24no—
215C22⋊C4×C17C2×Q814yes—
216C42×C17—24no—
217C42×C17—44no—
218C22⋊Dic17—22no—
219C22⋊Dic17—24no—
220C22⋊Dic17C2×Q814yes—
221C4⋊Dic17—24no—
222C4⋊Dic17—44no—
223#63225—24no—
224#63225—28no—
225#63225—28no—
226#63225—44no—
227#63225—44no—
228#63225C22×C418yes—
229#63225C22×C418yes—
230#63225C22×C418yes—
231#63225C22×C418yes—
232#63225C22×C418yes—
233#63225C22×C418yes—
234#63225C22×C418yes—
235#63225C22×C418yes—
236#63225C2×Q818yescomplemented
237C4×Dic17—24no—
238C4×Dic17—28no—
239C4×Dic17—24no—
240C4×Dic17—24no—
241C4×Dic17—24no—
242C4×Dic17—44no—
243C4×Dic17—44no—
244C4×Dic17—44no—
245C4×Dic17C22×C414yes—
246C4×Dic17C22×C414yes—
247C4×Dic17C22×C414yes—
248C4×Dic17C22×C414yes—
249C4×Dic17C2×Q818yescomplemented
250#34216—171nomaximal, Sylow
251C23×C17—22no—
252C23×C17—22no—
253C23×C17C23×C412yes—
254C23×C17C22×Q812yes—
255C23×C17C2×#4112yes—
256C23×C17C2511yescharacteristic
257Q8×C17—28no—
258Q8×C17—28no—
259Q8×C17—48no—
260C2×C4×C17—22no—
261C2×C4×C17—24no—
262C2×C4×C17—22no—
263C2×C4×C17—24no—
264C2×C4×C17—22no—
265C2×C4×C17—28no—
266C2×C4×C17—22no—
267C2×C4×C17—24no—
268C2×C4×C17—22no—
269C2×C4×C17—24no—
270C2×C4×C17—22no—
271C2×C4×C17—24no—
272C2×C4×C17—48no—
273C2×C4×C17C23×C412yes—
274C2×C4×C17C23×C414yes—
275C2×C4×C17C23×C414yes—
276C2×C4×C17C23×C412yes—
277C2×C4×C17C22×Q814yes—
278C2×C4×C17H(5,𝔽2)14yes—
279C2×Dic17—24no—
280C2×Dic17—24no—
281C2×Dic17—22no—
282C2×Dic17—22no—
283C2×Dic17—22no—
284C2×Dic17—28no—
285C2×Dic17—22no—
286C2×Dic17—24no—
287C2×Dic17—22no—
288C2×Dic17—24no—
289C2×Dic17—22no—
290C2×Dic17—24no—
291C2×Dic17—48no—
292C2×Dic17C4×Q8116yescomplemented
293C2×Dic17C23×C414yes—
294C2×Dic17C23×C414yes—
295C2×Dic17C23×C412yes—
296C2×Dic17C23×C412yes—
297C2×Dic17C23×C412yes—
298C2×Dic17C23×C412yes—
299C2×Dic17C22×Q814yes—
300C2×Dic17C22×Q812yes—
301C2×Dic17C22×Q812yes—
302C2×Dic17C2×#4112yes—
303C2×Dic17C2×#4112yes—
304C2×Dic172−1+414yes—
305Dic34—28no—
306Dic34—28no—
307Dic34—48no—
308C22×C4⋊Q8—171no—
309#2369—174no—
310#2365—172no—
311#2365—172no—
312#2345—172no—
313#2345—172no—
314Q8×C22⋊C4—171no—
315Q8×C22⋊C4—171no—
316Q8×C22⋊C4—171no—
317Q8×C22⋊C4D1711yescharacteristic
318#2331—174no—
319#2316—172no—
320C4×C4⋊Q8—172no—
321C4×C4⋊Q8—172no—
322#2285—171no—
323#2274—172no—
324C4×#117—171no—
325C22×C17—22no—
326C22×C17—22no—
327C22×C17—24no—
328C22×C17—24no—
329C22×C17C2×C4×Q814yes—
330C22×C17C24×C411yescharacteristic
331C22×C17C23×Q811yescharacteristic
332C22×C17C22×#4111yescharacteristic
333C22×C17C2×H(5,𝔽2)12yes—
334C22×C17C2×2−1+412yes—
335C4×C17—22no—
336C4×C17—24no—
337C4×C17—24no—
338C4×C17—24no—
339C4×C17—42no—
340C4×C17—44no—
341C4×C17#51814yes—
342Dic17—24no—
343Dic17—24no—
344Dic17—22no—
345Dic17—24no—
346Dic17—42no—
347Dic17—44no—
348Dic17C2×C4×Q814yes—
349Dic17C2×C4×Q814yes—
350Dic17#51914yes—
351C23×Q8—171no—
352C23×Q8C2×D1711yescharacteristic
353C2×C4⋊Q8—1716no—
354C2×C4⋊Q8—171no—
355C2×C4⋊Q8—171no—
356C2×C4⋊Q8—172no—
357C2×C4⋊Q8—171no—
358C2×C4⋊Q8—171no—
359C2×C4⋊Q8—172no—
360C2×C4×Q8—172no—
361C2×C4×Q8—172no—
362C2×C4×Q8—172no—
363C2×C4×Q8C2×D1712yes—
364C2×#117—171no—
365C2×#117—171no—
366C2×#117—172no—
367C22×C4⋊C4—172no—
368C22×C4⋊C4—172no—
369C22×C42—171no—
370#391—172no—
371#391—174no—
372#391—172no—
373#391—172no—
374#391—174no—
375#391—174no—
376#391C2×D1712yes—
377#391C2×D1714yes—
378#389—174no—
379#389—174no—
380#389—174no—
381#389—174no—
382#389—174no—
383#389—174no—
384#387—174no—
385#387—174no—
386#387—174no—
387#387—174no—
388#384—172no—
389#382—172no—
390#380—172no—
391#380—171no—
392#380—172no—
393#380—172no—
394#380—172no—
395#380C2×D1712yes—
396#380C2×D1711yescharacteristic
397C4×C4⋊C4—172no—
398C4×C4⋊C4—174no—
399C4×C4⋊C4—174no—
400C4×C4⋊C4—172no—
401C4×C4⋊C4—174no—
402C4×C4⋊C4—174no—
403C4×C4⋊C4—174no—
404C4×C4⋊C4—174no—
405C4×C4⋊C4—172no—
406C4×C4⋊C4—172no—
407C4×C22⋊C4—171no—
408C4×C22⋊C4—172no—
409C4×C22⋊C4—172no—
410C4×C22⋊C4—172no—
411C4×C22⋊C4—171no—
412C4×C22⋊C4—171no—
413C4×C22⋊C4C2×D1711yescharacteristic
414C4×C22⋊C4C2×D1712yes—
415#376—172no—
416C43—172no—
417C2×C17—24no—
418C2×C17C22×C4×Q811yescharacteristic
419C2×C17C2×#51811yescharacteristic
420C2×C17C2×#51911yescharacteristic
421C2×C17#350312yes—
422C2×C17#351112yes—
423C22×Q8—44no—
424C22×Q8—178no—
425C22×Q8—171no—
426C22×Q8—171no—
427C22×Q8—172no—
428C22×Q8—684no—
429C22×Q8C4×D1718yes—
430C22×Q8C22×D1711yescharacteristic
431C22×Q8C22×D1711yescharacteristic
432C22×Q8C22×D1712yes—
433C23×C4—171no—
434C23×C4—171no—
435C23×C4—172no—
436C23×C4C22×D1711yescharacteristic
437C23×C4C22×D1712yes—
438C4⋊Q8—178no—
439C4⋊Q8—178no—
440C4⋊Q8—178no—
441C4⋊Q8—178no—
442C4⋊Q8—1716no—
443C4⋊Q8—1716no—
444C4×Q8—44no—
445C4×Q8—684no—
446C4×Q8—684no—
447C4×Q8—684no—
448#117—178no—
449#117—344no—
450#117—348no—
451C2×C4⋊C4—1716no—
452C2×C4⋊C4—1716no—
453C2×C4⋊C4—172no—
454C2×C4⋊C4—172no—
455C2×C4⋊C4—172no—
456C2×C4⋊C4—172no—
457C2×C4⋊C4—172no—
458C2×C4⋊C4—172no—
459C2×C4⋊C4—172no—
460C2×C4⋊C4—172no—
461C2×C4⋊C4—172no—
462C2×C4⋊C4—174no—
463C2×C4⋊C4—172no—
464C2×C4⋊C4—172no—
465C2×C4⋊C4—174no—
466C2×C4⋊C4—174no—
467C2×C4⋊C4—174no—
468C2×C4⋊C4—174no—
469C2×C4⋊C4C22×D1712yes—
470C2×C4⋊C4C22×D1714yes—
471C2×C22⋊C4—171no—
472C2×C22⋊C4—171no—
473C2×C22⋊C4—172no—
474C2×C22⋊C4C22×D1711yescharacteristic
475C2×C22⋊C4C22×D1712yes—
476C2×C22⋊C4C22×D1711yescharacteristic
477C2×C42—178no—
478C2×C42—172no—
479C2×C42—171no—
480C2×C42—171no—
481C2×C42—171no—
482C2×C42—171no—
483C2×C42—172no—
484C2×C42—174no—
485C2×C42—172no—
486C2×C42—172no—
487C2×C42—172no—
488C2×C42—174no—
489C2×C42—174no—
490C2×C42—174no—
491C2×C42C22×D1712yes—
492C2×C42C22×D1714yes—
493#95—172no—
494#95—172no—
495#95—172no—
496#95—174no—
497#95—174no—
498#95—174no—
499#95—174no—
500#95—172no—
501#95—174no—
502#95—174no—
503#95—174no—
504#95C22×D1712yes—
505#95C22×D1714yes—
506#95C22×D1712yes—
507#95C22×D1714yes—
508C17#3421611yescharacteristic, complemented, Sylow
509C24C23×D1711yescharacteristic
510C2×Q8—24no—
511C2×Q8—24no—
512C2×Q8—44no—
513C2×Q8—416no—
514C2×Q8—174no—
515C2×Q8—174no—
516C2×Q8—178no—
517C2×Q8—344no—
518C2×Q8—344no—
519C2×Q8—684no—
520C2×Q8—6816no—
521C2×Q8C2×C4×D1714yes—
522C2×Q8C2×C4×D1714yes—
523C2×Q8C2×C4×D1718yes—
524C22×C4—24no—
525C22×C4—44no—
526C22×C4—174no—
527C22×C4—174no—
528C22×C4—174no—
529C22×C4—171no—
530C22×C4—171no—
531C22×C4—171no—
532C22×C4—174no—
533C22×C4—171no—
534C22×C4—172no—
535C22×C4—172no—
536C22×C4—172no—
537C22×C4—178no—
538C22×C4—172no—
539C22×C4—174no—
540C22×C4—344no—
541C22×C4—684no—
542C22×C4C2×C4×D1714yes—
543C22×C4C2×C4×D1718yes—
544C22×C4C23×D1712yes—
545C22×C4C23×D1714yes—
546C22×C4C23×D1711yescharacteristic
547C22×C4C23×D1712yes—
548C22×C4C23×D1712yes—
549C22×C4C23×D1712yes—
550C22×C4C23×D1711yescharacteristic
551C4⋊C4—24no—
552C4⋊C4—44no—
553C4⋊C4—178no—
554C4⋊C4—178no—
555C4⋊C4—178no—
556C4⋊C4—178no—
557C4⋊C4—178no—
558C4⋊C4—178no—
559C4⋊C4—178no—
560C4⋊C4—178no—
561C4⋊C4—178no—
562C4⋊C4—344no—
563C4⋊C4—348no—
564C4⋊C4—348no—
565C4⋊C4—344no—
566C4⋊C4—684no—
567C4⋊C4—684no—
568C4⋊C4—684no—
569C22⋊C4—22no—
570C22⋊C4—24no—
571C22⋊C4—174no—
572C22⋊C4—342no—
573C22⋊C4—344no—
574C22⋊C4Q8×D1714yes—
575C42—24no—
576C42—44no—
577C42—174no—
578C42—174no—
579C42—174no—
580C42—174no—
581C42—178no—
582C42—348no—
583C42—344no—
584C42—344no—
585C42—344no—
586C42—344no—
587C42—684no—
588C42—684no—
589C42—684no—
590C23—22no—
591C23—22no—
592C23C22×C4×D1712yes—
593C23C2×Q8×D1712yes—
594C23#41×D1712yes—
595C23C24×D1711yescharacteristic
596Q8—28no—
597Q8—28no—
598Q8—48no—
599Q8—348no—
600Q8—348no—
601Q8—688no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—28no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—48no—
615C2×C4—1716no—
616C2×C4—172no—
617C2×C4—172no—
618C2×C4—172no—
619C2×C4—172no—
620C2×C4—172no—
621C2×C4—172no—
622C2×C4—172no—
623C2×C4—174no—
624C2×C4—172no—
625C2×C4—174no—
626C2×C4—174no—
627C2×C4—174no—
628C2×C4—348no—
629C2×C4—342no—
630C2×C4—344no—
631C2×C4—344no—
632C2×C4—342no—
633C2×C4—342no—
634C2×C4—342no—
635C2×C4—344no—
636C2×C4—342no—
637C2×C4—344no—
638C2×C4—342no—
639C2×C4—344no—
640C2×C4—688no—
641C2×C4C22×C4×D1712yes—
642C2×C4C22×C4×D1714yes—
643C2×C4C22×C4×D1714yes—
644C2×C4C22×C4×D1712yes—
645C2×C4#1058769114yes—
646C2×C4C2×Q8×D1714yes—
647C22—24no—
648C22—24no—
649C22—22no—
650C22—22no—
651C22C4×Q8×D1714yes—
652C22C23×C4×D1711yescharacteristic
653C22C22×Q8×D1711yescharacteristic
654C22C2×#41×D1711yescharacteristic
655C22C2×#1058769112yes—
656C222−1+4×D1712yes—
657C4—24no—
658C4—22no—
659C4—24no—
660C4—24no—
661C4—42no—
662C4—44no—
663C4—174no—
664C4—174no—
665C4—174no—
666C4—344no—
667C4—344no—
668C4—344no—
669C4—342no—
670C4—682no—
671C4—684no—
672C4#1176328214yes—
673C2—24no—
674C2C2×C4×Q8×D1711yescharacteristic
675C2C2×#1176328211yescharacteristic
676C2D17×#51911yescharacteristic
677C2#1454259512yes—
678C2#1454269012yes—
subgroupG/H
center, FrattiniC23C24×D17
commutatorC22×C17C24×C4
FittingQ8×C22⋊C4×C17C2
socle, cosocleC23×C17C25
prime radicalC24×C17C24
hypercenterQ8×C22⋊C4D17
hypocenterC17#34216
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ Q8×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #14531872C2semidirect
3. #14531841C2semidirect
4. #14531838C2semidirect
10. #14531527C2semidirect
12. #14531353C2semidirect
13. C4×#10587630C2semidirect
19. C2×#10587630C4semidirect
152. #10587563Q8semidirect
236. #63225C2×Q8semidirect
249. C4×Dic17C2×Q8semidirect
292. C2×Dic17C4×Q8semidirect
508. C17#34216semidirect

Maximal subgroups

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

subgroupindex
C22×#105876302
#145318722
#145318412
#145318382
#145316622
#145316562
Q8×C22⋊Dic172
C22⋊C4×Dic342
#145316162
#145315272
#145314012
#145313532
C4×#105876302
#145311862
#145311392
C4×#105875632
Q8×C22⋊C4×C172
#3421617