Finite Group Databasebeta

← #17194496

Subgroups — #17194496

Order 4352.

2470 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×Dic17C211yesmaximal, characteristic
2#14531740C212yesmaximal, complemented
3#14531711C214yesmaximal, complemented
4D4×#63225C212yesmaximal, complemented
5#14531671C211yesmaximal, characteristic, complemented
6#14531667C211yesmaximal, characteristic, complemented
7#14531646C212yesmaximal, complemented
8#14531640C212yesmaximal, complemented
9#14531614C211yesmaximal, characteristic
10#14531429C212yesmaximal, complemented
11#14531410C214yesmaximal, complemented
12#14531186C211yesmaximal, characteristic, complemented
13#14531178C211yesmaximal, characteristic, complemented
14#14531175C212yesmaximal, complemented
15C2×#11762673C212yesmaximal, complemented
16#14531145C212yesmaximal, complemented
17C17×#2339C211yesmaximal, characteristic
18#41×Dic17C4116yescomplemented
19C2×Q8×Dic17C2211yescharacteristic
20C2×Q8×Dic17C2211yescharacteristic
21C2×D4×Dic17C2212yes—
22C2×D4×Dic17C2211yescharacteristic
23C2×D4×Dic17C2211yescharacteristic
24C2×D4×Dic17C2212yes—
25C2×D4×Dic17C2212yes—
26C2×#10587612C2212yes—
27C2×#10587612C2211yescharacteristic
28C2×#10587563C2212yes—
29C22×#63225C2214yes—
30C22×C4×Dic17C2212yes—
31C22×C4×Dic17C2211yescharacteristic
32#11762716C2212yes—
33#11762707C2212yes—
34#11762705C2212yes—
35#11762703C2214yes—
36#11762697C2214yes—
37#11762694C2214yes—
38#11762694C2214yes—
39#11762692C2214yes—
40#11762692C2214yes—
41#11762689C2214yes—
42#11762682C2214yes—
43#11762681C2214yes—
44#11762678C2212yes—
45#11762676C2211yescharacteristic
46#11762676C2211yescharacteristic
47#11762676C2214yes—
48#11762673C2212yes—
49#11762673C2212yes—
50#11762673C2214yes—
51#11762673C2211yescharacteristic
52#11762673C2212yes—
53#11762673C2214yes—
54#11762673C2211yescharacteristic
55#11762672C2212yes—
56#11762670C2212yes—
57#11762669C2211yescharacteristic
58#11762667C2212yes—
59#11762667C2214yes—
60#11762667C2212yes—
61C4⋊C4×Dic17C2212yes—
62C4⋊C4×Dic17C2212yes—
63C4×#63225C2212yes—
64C4×C4⋊Dic17C2212yes—
65#11762660C2212yes—
66#11762660C2214yes—
67C22⋊C4×Dic17C2214yes—
68C22⋊C4×Dic17C2214yes—
69C4×C22⋊Dic17C2211yescharacteristic
70C4×C22⋊Dic17C2214yes—
71#11762655C2212yes—
72#11762654C2212yes—
73#11762654C2212yes—
74C2×#10587502C2212yes—
75C2×#10587502C2214yes—
76C22×#41×C17C2211yescharacteristic
77C2×C4×D4×C17C2212yes—
78C17×#391C2212yes—
79C17×#386C2214yes—
80C17×#381C2212yes—
81C17×#380C2211yescharacteristic
82C4×C22⋊C4×C17C2211yescharacteristic
83C2×C17×#95C2212yes—
84C2×#41×C17—28no—
85C2×#41×C17C2×C418yes—
86C22×Q8×C17C2311yescharacteristic
87C22×D4×C17C2312yes—
88C22×D4×C17C2311yescharacteristic
89C23×C4×C17C2312yes—
90C23×C4×C17C2311yescharacteristic
91C23×C4×C17C2314yes—
92C23×Dic17C2312yes—
93C23×Dic17C2311yescharacteristic
94C23×Dic17C2314yes—
95C23×Dic17C2312yes—
96C23×Dic17C2311yescharacteristic
97C4×D4×C17—28no—
98C2×C4⋊C4×C17C2312yes—
99C2×C22⋊C4×C17C2314yes—
100C2×C22⋊C4×C17C2312yes—
101C2×C22⋊C4×C17C2312yes—
102C2×C42×C17C2312yes—
103C2×C22⋊Dic17C2314yes—
104C2×C22⋊Dic17C2312yes—
105C2×C22⋊Dic17C2312yes—
106C2×C22⋊Dic17C2312yes—
107C2×C22⋊Dic17C2312yes—
108C2×C22⋊Dic17C2312yes—
109Q8×Dic17C2×C418yescomplemented
110Q8×Dic17C2×C418yescomplemented
111D4×Dic17—28no—
112D4×Dic17C2×C418yes—
113D4×Dic17C2×C418yes—
114D4×Dic17C2×C4116yescomplemented
115D4×Dic17C2×C4116yescomplemented
116#10587612C2×C4116yescomplemented
117#10587612C2×C418yescomplemented
118C2×C4⋊Dic17C2312yes—
119C2×C4⋊Dic17C2312yes—
120C2×C4⋊Dic17C2312yes—
121C2×C4⋊Dic17C2311yescharacteristic
122C2×C4⋊Dic17C2311yescharacteristic
123C2×#63225—48no—
124C2×#63225C2312yes—
125C2×#63225C2314yes—
126C2×#63225C2312yes—
127C2×#63225C2314yes—
128C2×#63225C2314yes—
129C2×#63225C2314yes—
130C2×C4×Dic17C2×C4116yescomplemented
131C2×C4×Dic17C2×C418yescomplemented
132C2×C4×Dic17C2312yes—
133C2×C4×Dic17C2314yes—
134C2×C4×Dic17C2314yes—
135C2×C4×Dic17C2314yes—
136C2×C4×Dic17C2311yescharacteristic
137C2×C4×Dic17C2312yes—
138C2×C4×Dic17C2311yescharacteristic
139C2×C4×Dic17C2311yescharacteristic
140C2×C4×Dic17C2312yes—
141C2×C4×Dic17C2311yescharacteristic
142C2×C4×Dic17C2312yes—
143#10587563—44no—
144C17×#95C2314yes—
145C17×#95C2314yes—
146C17×#95C2311yescharacteristic
147C17×#95C2312yes—
148C17×#95C2311yescharacteristic
149#10587502C2314yes—
150#10587502C2314yes—
151#10587502C2312yes—
152#10587502C2312yes—
153#10587502C2314yes—
154#10587502C2314yes—
155#10587502C2312yes—
156#10587502C2312yes—
157#10587502C2314yes—
158#10587502C2314yes—
159#10587502C2312yes—
160#10587502C2311yescharacteristic
161#10587502C2312yes—
162#10587502C2314yes—
163#10587502C2311yescharacteristic
164#10587502C2312yes—
165C24×C17C2411yescharacteristic
166C24×C17C2412yes—
167#41×C17—232no—
168C2×Q8×C17—24no—
169C2×Q8×C17C22×C414yes—
170C2×D4×C17—28no—
171C2×D4×C17C22×C418yes—
172C2×D4×C17C22×C414yes—
173C2×D4×C17#4118yes—
174C22×C4×C17—28no—
175C22×C4×C17—24no—
176C22×C4×C17—28no—
177C22×C4×C17—44no—
178C22×C4×C17—44no—
179C22×C4×C17C22×C418yes—
180C22×C4×C17C22×C414yes—
181C22×C4×C17C2412yes—
182C22×C4×C17C2414yes—
183C22×C4×C17C2412yes—
184C22×C4×C17C2411yescharacteristic
185C22×C4×C17C2412yes—
186C22×C4×C17C2411yescharacteristic
187C22×Dic17—22no—
188C22×Dic17—24no—
189C22×Dic17—24no—
190C22×Dic17—22no—
191C22×Dic17—28no—
192C22×Dic17—44no—
193C22×Dic17—42no—
194C22×Dic17—44no—
195C22×Dic17—44no—
196C22×Dic17C22×C418yes—
197C22×Dic17C22×C414yes—
198C22×Dic17C22×C418yes—
199C22×Dic17C22×C414yes—
200C22×Dic17C2412yes—
201C22×Dic17C2414yes—
202C22×Dic17C2412yes—
203C22×Dic17C2411yescharacteristic
204C22×Dic17C2411yescharacteristic
205C22×Dic17C2412yes—
206C22×Dic17C2412yes—
207C22×Dic17C2411yescharacteristic
208C22×Dic17C2411yescharacteristic
209C4⋊C4×C17—24no—
210C22⋊C4×C17—28no—
211C22⋊C4×C17—42no—
212C22⋊C4×C17—42no—
213C42×C17—24no—
214C22⋊Dic17—28no—
215C22⋊Dic17—42no—
216C22⋊Dic17—42no—
217C22⋊Dic17C22×C418yes—
218C22⋊Dic17C22×C418yes—
219C22⋊Dic17C22×C418yes—
220C4⋊Dic17—24no—
221C4⋊Dic17C22×C414yes—
222C4⋊Dic17C22×C414yes—
223C4⋊Dic17C22×C418yes—
224C4⋊Dic17C22×C418yes—
225#63225—28no—
226#63225—24no—
227#63225—24no—
228#63225—44no—
229#63225—44no—
230#63225—44no—
231C4×Dic17—24no—
232C4×Dic17—44no—
233C4×Dic17—44no—
234C4×Dic17—44no—
235C4×Dic17C22×C414yes—
236C4×Dic17C22×C418yes—
237C4×Dic17C22×C414yes—
238C4×Dic17C22×C414yes—
239C4×Dic17C22×C414yes—
240C4×Dic17C22×C418yes—
241C4×Dic17C22×C418yes—
242#34191—171nomaximal, Sylow
243C23×C17—22no—
244C23×C17—22no—
245C23×C17—24no—
246C23×C17—22no—
247C23×C17—21no—
248C23×C17—21no—
249C23×C17—22no—
250C23×C17C23×C412yes—
251C23×C17C23×C414yes—
252C23×C17C2×#4118yes—
253C23×C17C2511yescharacteristic
254Q8×C17—28no—
255D4×C17—216no—
256D4×C17#117116yescomplemented
257C2×C4×C17—216no—
258C2×C4×C17—22no—
259C2×C4×C17—24no—
260C2×C4×C17—28no—
261C2×C4×C17—22no—
262C2×C4×C17—22no—
263C2×C4×C17—24no—
264C2×C4×C17—24no—
265C2×C4×C17—22no—
266C2×C4×C17—22no—
267C2×C4×C17—22no—
268C2×C4×C17—22no—
269C2×C4×C17—24no—
270C2×C4×C17—88no—
271C2×C4×C17C23×C412yes—
272C2×C4×C17C23×C414yes—
273C2×C4×C17C23×C412yes—
274C2×C4×C17C2×#4114yes—
275C2×Dic17—21no—
276C2×Dic17—22no—
277C2×Dic17—21no—
278C2×Dic17—22no—
279C2×Dic17—22no—
280C2×Dic17—24no—
281C2×Dic17—22no—
282C2×Dic17—216no—
283C2×Dic17—28no—
284C2×Dic17—22no—
285C2×Dic17—24no—
286C2×Dic17—22no—
287C2×Dic17—24no—
288C2×Dic17—22no—
289C2×Dic17—22no—
290C2×Dic17—22no—
291C2×Dic17—22no—
292C2×Dic17—24no—
293C2×Dic17—48no—
294C2×Dic17—44no—
295C2×Dic17—88no—
296C2×Dic17C23×C412yes—
297C2×Dic17C23×C414yes—
298C2×Dic17C23×C412yes—
299C2×Dic17C23×C414yes—
300C2×Dic17C23×C412yes—
301C2×Dic17C23×C412yes—
302C2×Dic17H(5,𝔽2)12yes—
303C2×Dic172−1+412yes—
304C2×C4×#41—171no—
305#2355—172no—
306#2352—174no—
307D4×C4⋊C4—172no—
308#2346—171no—
309#2346—171no—
310#2343—172no—
311#2343—172no—
312#2339—171no—
313#2339D1711yescharacteristic
314#2320—172no—
315#2318—174no—
316#2285—171no—
317#2283—172no—
318#2283—171no—
319C2×#382—172no—
320#2275—172no—
321C22×C17—24no—
322C22×C17—24no—
323C22×C17—28no—
324C22×C17—28no—
325C22×C17—22no—
326C22×C17—22no—
327C22×C17—22no—
328C22×C17—42no—
329C22×C17—42no—
330C22×C17C2×#11718yes—
331C22×C17C24×C411yescharacteristic
332C22×C17C22×#4112yes—
333C22×C17C2×H(5,𝔽2)11yescharacteristic
334C22×C17C2×2−1+411yescharacteristic
335C22×C17#58512yes—
336C4×C17—22no—
337C4×C17—24no—
338C4×C17—44no—
339C4×C17—42no—
340C4×C17—42no—
341C4×C17C2×#11714yes—
342Dic17—22no—
343Dic17—24no—
344Dic17—22no—
345Dic17—24no—
346Dic17—22no—
347Dic17—44no—
348Dic17—42no—
349Dic17—42no—
350Dic17#52014yes—
351C22×#41C2×D1711yescharacteristic
352C4×#41—1716no—
353C2×C4×Q8—171no—
354C2×C4×Q8—171no—
355C2×C4×D4—172no—
356C2×C4×D4—171no—
357C2×C4×D4—172no—
358C2×C4×D4—171no—
359C2×C4×D4—172no—
360C2×C4×D4C2×D1712yes—
361C2×#117—172no—
362C2×#117—171no—
363C2×#117—172no—
364C22×C4⋊C4—174no—
365C22×C42—172no—
366C22×C42—171no—
367#391—172no—
368#391C2×D1712yes—
369#389—172no—
370#389—172no—
371#388—174no—
372#388—174no—
373#387—174no—
374#387—174no—
375#387—174no—
376#387—174no—
377#386—174no—
378#386C2×D1714yes—
379#385—174no—
380#385—174no—
381#384—172no—
382#383—174no—
383#383—171no—
384#383—171no—
385#382—174no—
386#382—172no—
387#382—172no—
388#382—174no—
389#382—171no—
390#382—172no—
391#382—171no—
392#381—172no—
393#381—172no—
394#381C2×D1712yes—
395#380—171no—
396#380—172no—
397#380—174no—
398#380—172no—
399#380C2×D1711yescharacteristic
400C4×C4⋊C4—172no—
401C4×C4⋊C4—172no—
402C4×C4⋊C4—172no—
403C4×C4⋊C4—172no—
404C4×C4⋊C4—172no—
405C4×C4⋊C4—174no—
406C4×C22⋊C4—171no—
407C4×C22⋊C4—174no—
408C4×C22⋊C4—174no—
409C4×C22⋊C4—174no—
410C4×C22⋊C4C2×D1711yescharacteristic
411#376—172no—
412#376—172no—
413#376—172no—
414C2×#95—174no—
415C2×#95—172no—
416C2×#95C2×D1712yes—
417C2×C17—28no—
418C2×C17—42no—
419C2×C17C22×#11711yescharacteristic
420C2×C17C2×#52011yescharacteristic
421C2×C17#342811yescharacteristic
422C2×C17#351412yes—
423C2×C17#351512yes—
424C2×#41—28no—
425C2×#41C4×D1718yes—
426C22×Q8C22×D1711yescharacteristic
427C22×D4C22×D1712yes—
428C22×D4C22×D1711yescharacteristic
429C23×C4—172no—
430C23×C4—171no—
431C23×C4—172no—
432C23×C4—171no—
433C23×C4—174no—
434C23×C4C22×D1712yes—
435C23×C4C22×D1711yescharacteristic
436C23×C4C22×D1714yes—
437C4×Q8—178no—
438C4×Q8—178no—
439C4×D4—28no—
440C4×D4—178no—
441C4×D4—178no—
442C4×D4—1716no—
443C4×D4—1716no—
444C4×D4—348no—
445#117—1716no—
446#117—178no—
447#117—684no—
448C2×C4⋊C4—171no—
449C2×C4⋊C4—172no—
450C2×C4⋊C4—172no—
451C2×C4⋊C4—171no—
452C2×C4⋊C4—172no—
453C2×C4⋊C4—174no—
454C2×C4⋊C4—172no—
455C2×C4⋊C4—172no—
456C2×C4⋊C4—174no—
457C2×C4⋊C4—174no—
458C2×C4⋊C4—174no—
459C2×C4⋊C4—688no—
460C2×C4⋊C4C22×D1712yes—
461C2×C22⋊C4—172no—
462C2×C22⋊C4—172no—
463C2×C22⋊C4—172no—
464C2×C22⋊C4—174no—
465C2×C22⋊C4—172no—
466C2×C22⋊C4—172no—
467C2×C22⋊C4C22×D1714yes—
468C2×C22⋊C4C22×D1712yes—
469C2×C22⋊C4C22×D1712yes—
470C2×C42—1716no—
471C2×C42—178no—
472C2×C42—172no—
473C2×C42—171no—
474C2×C42—171no—
475C2×C42—172no—
476C2×C42—171no—
477C2×C42—172no—
478C2×C42—171no—
479C2×C42—172no—
480C2×C42—174no—
481C2×C42—174no—
482C2×C42—174no—
483C2×C42C22×D1712yes—
484#95—172no—
485#95—171no—
486#95—172no—
487#95—174no—
488#95—174no—
489#95—174no—
490#95—174no—
491#95—171no—
492#95—172no—
493#95—172no—
494#95—172no—
495#95—174no—
496#95—174no—
497#95—172no—
498#95—172no—
499#95—174no—
500#95C22×D1711yescharacteristic
501#95C22×D1712yes—
502#95C22×D1714yes—
503#95C22×D1711yescharacteristic
504#95C22×D1714yes—
505C17#3419111yescharacteristic, complemented, Sylow
506C24C23×D1711yescharacteristic
507C24C23×D1712yes—
508#41—232no—
509C2×Q8—24no—
510C2×Q8C2×C4×D1714yes—
511C2×D4—28no—
512C2×D4C2×C4×D1718yes—
513C2×D4C2×C4×D1714yes—
514C2×D4#6325418yes—
515C22×C4—28no—
516C22×C4—24no—
517C22×C4—28no—
518C22×C4—44no—
519C22×C4—44no—
520C22×C4—171no—
521C22×C4—174no—
522C22×C4—178no—
523C22×C4—171no—
524C22×C4—171no—
525C22×C4—172no—
526C22×C4—172no—
527C22×C4—171no—
528C22×C4—174no—
529C22×C4—178no—
530C22×C4—172no—
531C22×C4—172no—
532C22×C4—174no—
533C22×C4—344no—
534C22×C4—342no—
535C22×C4—344no—
536C22×C4—342no—
537C22×C4—348no—
538C22×C4—684no—
539C22×C4—682no—
540C22×C4—684no—
541C22×C4—684no—
542C22×C4C2×C4×D1718yes—
543C22×C4C2×C4×D1714yes—
544C22×C4C23×D1712yes—
545C22×C4C23×D1712yes—
546C22×C4C23×D1714yes—
547C22×C4C23×D1711yescharacteristic
548C22×C4C23×D1712yes—
549C22×C4C23×D1711yescharacteristic
550C4⋊C4—24no—
551C4⋊C4—178no—
552C4⋊C4—178no—
553C4⋊C4—174no—
554C4⋊C4—174no—
555C4⋊C4—344no—
556C4⋊C4—348no—
557C4⋊C4—344no—
558C4⋊C4—344no—
559C4⋊C4—684no—
560C4⋊C4—684no—
561C4⋊C4—684no—
562C22⋊C4—28no—
563C22⋊C4—42no—
564C22⋊C4—42no—
565C22⋊C4—178no—
566C22⋊C4—178no—
567C22⋊C4—178no—
568C22⋊C4—348no—
569C22⋊C4—682no—
570C22⋊C4—682no—
571C42—24no—
572C42—174no—
573C42—178no—
574C42—174no—
575C42—178no—
576C42—174no—
577C42—178no—
578C42—174no—
579C42—344no—
580C42—684no—
581C42—684no—
582C42—684no—
583C23—22no—
584C23—22no—
585C23—24no—
586C23—22no—
587C23—21no—
588C23—21no—
589C23—22no—
590C23C22×C4×D1712yes—
591C23C22×C4×D1714yes—
592C23C2×#6325418yes—
593C23C24×D1711yescharacteristic
594Q8—28no—
595D4—216no—
596D4#10587563116yescomplemented
597C2×C4—216no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—88no—
611C2×C4—172no—
612C2×C4—172no—
613C2×C4—172no—
614C2×C4—172no—
615C2×C4—172no—
616C2×C4—174no—
617C2×C4—174no—
618C2×C4—172no—
619C2×C4—3416no—
620C2×C4—348no—
621C2×C4—341no—
622C2×C4—342no—
623C2×C4—341no—
624C2×C4—342no—
625C2×C4—344no—
626C2×C4—342no—
627C2×C4—342no—
628C2×C4—342no—
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—688no—
638C2×C4—684no—
639C2×C4—1368no—
640C2×C4C22×C4×D1712yes—
641C2×C4C22×C4×D1714yes—
642C2×C4C22×C4×D1712yes—
643C2×C4C2×#6325414yes—
644C22—28no—
645C22—24no—
646C22—28no—
647C22—24no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—42no—
652C22—42no—
653C22C2×#1058756318yes—
654C22C23×C4×D1711yescharacteristic
655C22C22×#6325412yes—
656C22H(5,𝔽2)×D1711yescharacteristic
657C222−1+4×D1711yescharacteristic
658C22#1176364512yes—
659C4—22no—
660C4—24no—
661C4—42no—
662C4—44no—
663C4—42no—
664C4—174no—
665C4—342no—
666C4—344no—
667C4—342no—
668C4—344no—
669C4—342no—
670C4—682no—
671C4—684no—
672C4—682no—
673C4C2×#1058756314yes—
674C2—28no—
675C2—42no—
676C2C22×#1058756311yescharacteristic
677C2D17×#52011yescharacteristic
678C2#1454198311yescharacteristic
679C2#1454273212yes—
680C2#1454274912yes—
subgroupG/H
center, FrattiniC23C24×D17
commutatorC22×C17C24×C4
FittingC17×#2339C2
socle, cosocleC23×C17C25
prime radicalC22×#41×C17C22
hypercenter#2339D17
hypocenterC17#34191
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ #2339

Complements

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

NH ≅ G/Ntype
2. #14531740C2semidirect
3. #14531711C2semidirect
4. D4×#63225C2semidirect
5. #14531671C2semidirect
6. #14531667C2semidirect
7. #14531646C2semidirect
8. #14531640C2semidirect
10. #14531429C2semidirect
11. #14531410C2semidirect
12. #14531186C2semidirect
13. #14531178C2semidirect
14. #14531175C2semidirect
15. C2×#11762673C2semidirect
16. #14531145C2semidirect
18. #41×Dic17C4semidirect
109. Q8×Dic17C2×C4semidirect
110. Q8×Dic17C2×C4semidirect
114. D4×Dic17C2×C4semidirect
115. D4×Dic17C2×C4semidirect
116. #10587612C2×C4semidirect
117. #10587612C2×C4semidirect
130. C2×C4×Dic17C2×C4semidirect
131. C2×C4×Dic17C2×C4semidirect
256. D4×C17#117semidirect
505. C17#34191semidirect
596. D4#10587563semidirect

Maximal subgroups

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

subgroupindex
C2×#41×Dic172
#145317402
#145317112
D4×#632252
#145316712
#145316672
#145316462
#145316402
#145316142
#145314292
#145314102
#145311862
#145311782
#145311752
C2×#117626732
#145311452
C17×#23392
#3419117