Finite Group Databasebeta

← #17036134

Subgroups — #17036134

Order 4352.

1718 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763380C211yesmaximal, characteristic
2C2×#11763286C211yesmaximal, characteristic, complemented
3C2×#11762783C211yesmaximal, characteristic
4#14531368C211yesmaximal, characteristic, complemented
5#14529239C218yesmaximal, complemented
6#14528516C211yesmaximal, characteristic, complemented
7#14528516C211yesmaximal, characteristic, complemented
8C2×C17×#410C211yesmaximal, characteristic
9C23×Dic34C2211yescharacteristic
10#11763380—22no—
11#11763380—22no—
12#11763380—22no—
13#11763380—22no—
14C2×#10587630C412yescomplemented
15C2×#10587622C412yescomplemented
16C2×#10587621C2214yes—
17C2×#10587610C414yescomplemented
18#11763286—24no—
19#11763286—24no—
20C2×C4×Dic34—22no—
21C2×#10587573—22no—
22C2×#10587612C2211yescharacteristic
23C2×#10587612C2211yescharacteristic
24C2×#10587612C2211yescharacteristic
25C2×#10587563C2214yes—
26#11762783—24no—
27C2×#10587505C2214yes—
28#11762715—22no—
29#11762714—22no—
30#11762670C2214yes—
31#11762575C2214yes—
32#11762575C2214yes—
33C22×#41×C17C2211yescharacteristic
34C2×C17×#117C2211yescharacteristic
35C2×C17×#117C2211yescharacteristic
36C17×#410—24no—
37C2×C17×#99C2214yes—
38C2×#41×C17—22no—
39C2×#41×C17—22no—
40C2×#41×C17—21no—
41C2×#41×C17—21no—
42C2×#41×C17—21no—
43C2×#41×C17—21no—
44C2×#41×C17—22no—
45C2×#41×C17—22no—
46C22×Q8×C17C2×C411yescharacteristic
47C22×D4×C17C2×C411yescharacteristic
48C22×D4×C17C2312yes—
49C23×C4×C17C2×C412yes—
50C23×C4×C17C2311yescharacteristic
51C23×Dic17C2312yes—
52C22×Dic34—22no—
53C22×Dic34—22no—
54C22×Dic34—22no—
55C22×Dic34—22no—
56C22×Dic34C2×C412yes—
57C22×Dic34C2×C412yes—
58C22×Dic34D412yes—
59C22×Dic34D412yes—
60C22×Dic34D412yes—
61C22×Dic34D412yes—
62C17×#117—22no—
63C17×#117—22no—
64C17×#117—22no—
65C17×#117D414yes—
66C2×C4⋊C4×C17—21no—
67C2×C4⋊C4×C17—21no—
68C2×C22⋊C4×C17C2312yes—
69C2×C22⋊C4×C17C2312yes—
70C2×C42×C17—21no—
71C2×C42×C17—21no—
72C2×C22⋊Dic17C2×C418yes—
73C2×C22⋊Dic17C2312yes—
74C2×C22⋊Dic17C2312yes—
75C2×C22⋊Dic17C2312yes—
76#10587630—24no—
77#10587630—24no—
78#10587622—24no—
79#10587622—24no—
80#10587621—28no—
81#10587621—28no—
82#10587612—22no—
83#10587612—21no—
84#10587612—21no—
85#10587612—21no—
86#10587612—21no—
87#10587612—22no—
88#10587612—22no—
89#10587612D414yes—
90C2×C4⋊Dic17—21no—
91C2×C4⋊Dic17—21no—
92C2×C4⋊Dic17C2×C412yes—
93#10587610—28no—
94#10587610—28no—
95C2×#63225—24no—
96C2×#63225C2×C418yes—
97C2×C4×Dic17—21no—
98C2×C4×Dic17—21no—
99C2×C4×Dic17—24no—
100C2×C4×Dic17C2×C412yes—
101#10587573—216no—
102#10587563—28no—
103#10587563—28no—
104C4×Dic34—216no—
105C17×#99—28no—
106#10587505—28no—
107#10587502—28no—
108C24×C17C22×C412yes—
109C24×C17C2411yescharacteristic
110#41×C17—42no—
111#41×C17—42no—
112#41×C17—42no—
113#41×C17—42no—
114#41×C17—42no—
115#41×C17—42no—
116#41×C17—42no—
117#41×C17—42no—
118C2×Q8×C17—21no—
119C2×Q8×C17—21no—
120C2×Q8×C17—22no—
121C2×Q8×C17—22no—
122C2×D4×C17—22no—
123C2×D4×C17—22no—
124C2×D4×C17—22no—
125C2×D4×C17—22no—
126C2×D4×C17—22no—
127C2×D4×C17—22no—
128C2×D4×C17—21no—
129C2×D4×C17—21no—
130C2×D4×C17—22no—
131C2×D4×C17—22no—
132C22×C4×C17—22no—
133C22×C4×C17—22no—
134C22×C4×C17—22no—
135C22×C4×C17—24no—
136C22×C4×C17—24no—
137C22×C4×C17—22no—
138C22×C4×C17C22×C412yes—
139C22×C4×C17C22×C411yescharacteristic
140C22×C4×C17C22×C411yescharacteristic
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2×D411yescharacteristic
144C22×C4×C17C2×D411yescharacteristic
145C22×C4×C17C2×D412yes—
146C22×C4×C17C2×D412yes—
147C22×C4×C17#4112yes—
148C22×C4×C17#4112yes—
149C22×Dic17—22no—
150C22×Dic17—22no—
151C22×Dic17—22no—
152C22×Dic17—22no—
153C22×Dic17—22no—
154C22×Dic17—22no—
155C22×Dic17C22×C414yes—
156C22×Dic17C22×C414yes—
157C22×Dic17C2×D414yes—
158C22×Dic17C2×D414yes—
159C2×Dic34—24no—
160C2×Dic34—24no—
161C2×Dic34—24no—
162C2×Dic34—24no—
163C2×Dic34—24no—
164C2×Dic34—24no—
165C2×Dic34—24no—
166C2×Dic34—24no—
167C2×Dic34—24no—
168C2×Dic34—24no—
169C2×Dic34—24no—
170C2×Dic34—24no—
171C2×Dic34C22⋊C414yes—
172C2×Dic34C22⋊C414yes—
173C2×Dic34C22⋊C414yes—
174C2×Dic34C22⋊C414yes—
175C4⋊C4×C17—24no—
176C4⋊C4×C17—42no—
177C22⋊C4×C17—22no—
178C22⋊C4×C17—22no—
179C22⋊C4×C17—22no—
180C22⋊C4×C17C2×D414yes—
181C42×C17—24no—
182C42×C17—42no—
183C22⋊Dic17—22no—
184C22⋊Dic17—22no—
185C22⋊Dic17—22no—
186C22⋊Dic17—216no—
187C22⋊Dic17—22no—
188C22⋊Dic17—22no—
189C22⋊Dic17C2×D414yes—
190C4⋊Dic17—22no—
191C4⋊Dic17—22no—
192C4⋊Dic17—24no—
193C4⋊Dic17—42no—
194#63225—216no—
195#63225—216no—
196C4×Dic17—216no—
197C4×Dic17—22no—
198C4×Dic17—22no—
199C4×Dic17—24no—
200C4×Dic17—42no—
201#20986—171nomaximal, Sylow
202C23×C17—22no—
203C23×C17—22no—
204C23×C17—22no—
205C23×C17—42no—
206C23×C17—44no—
207C23×C17C23×C411yescharacteristic
208C23×C17C22×D411yescharacteristic
209C23×C17C22×D411yescharacteristic
210C23×C17C22×D411yescharacteristic
211C23×C17C22×D411yescharacteristic
212C23×C17C2×#4111yescharacteristic
213C23×C17C2×#4111yescharacteristic
214Q8×C17—42no—
215Q8×C17—42no—
216D4×C17—42no—
217D4×C17—42no—
218D4×C17—42no—
219D4×C17—42no—
220D4×C17—42no—
221D4×C17—42no—
222C2×C4×C17—22no—
223C2×C4×C17—22no—
224C2×C4×C17—22no—
225C2×C4×C17—24no—
226C2×C4×C17—44no—
227C2×C4×C17—44no—
228C2×C4×C17—44no—
229C2×C4×C17—44no—
230C2×C4×C17—42no—
231C2×C4×C17C2×C22⋊C411yescharacteristic
232C2×C4×C17C2×C22⋊C411yescharacteristic
233C2×C4×C17C2×C22⋊C411yescharacteristic
234C2×C4×C17C2×C22⋊C411yescharacteristic
235C2×C4×C17C4×D412yes—
236C2×C4×C17C4×D412yes—
237C2×C4×C17C4×D412yes—
238C2×C4×C17C4×D412yes—
239C2×C4×C17#12012yes—
240C2×C4×C17#12012yes—
241C2×C4×C17#12112yes—
242C2×C4×C17#12112yes—
243C2×C4×C17#12112yes—
244C2×C4×C17#12112yes—
245C2×C4×C17#12312yes—
246C2×C4×C17#12312yes—
247C2×Dic17—24no—
248C2×Dic17—24no—
249C2×Dic17—24no—
250C2×Dic17—24no—
251C2×Dic17—24no—
252C2×Dic17—24no—
253C2×Dic17—24no—
254C2×Dic17—24no—
255C2×Dic17—44no—
256C2×Dic17—44no—
257C2×Dic17—44no—
258C2×Dic17—42no—
259C2×Dic17C2×C22⋊C418yes—
260Dic34—28no—
261Dic34—28no—
262Dic34—28no—
263Dic34—28no—
264C2×#536—171no—
265C2×#519—171no—
266C2×#410—171no—
267C2×#410D1711yescharacteristic
268#2310—171no—
269#1899—178no—
270#1740—171no—
271#1740—171no—
272C22×C17—22no—
273C22×C17—22no—
274C22×C17—22no—
275C22×C17—41no—
276C22×C17—41no—
277C22×C17—41no—
278C22×C17—41no—
279C22×C17—44no—
280C22×C17—42no—
281C22×C17—82no—
282C22×C17—82no—
283C22×C17C22×C22⋊C411yescharacteristic
284C22×C17C2×C4×D411yescharacteristic
285C22×C17C2×C4×D411yescharacteristic
286C22×C17C2×#12011yescharacteristic
287C22×C17C2×#12111yescharacteristic
288C22×C17C2×#12111yescharacteristic
289C22×C17C2×#12311yescharacteristic
290C4×C17—44no—
291C4×C17—42no—
292C4×C17—82no—
293C4×C17#38612yes—
294C4×C17#38612yes—
295C4×C17#38612yes—
296C4×C17#38612yes—
297Dic17—28no—
298Dic17—44no—
299Dic17—44no—
300Dic17—82no—
301C22×#41C2×D1711yescharacteristic
302C23×Q8—171no—
303#536—342no—
304#536—342no—
305#536—342no—
306#536—342no—
307C2×C4⋊Q8—172no—
308C2×#124—172no—
309C2×#123—174no—
310C2×C22⋊Q8—174no—
311#519—344no—
312#519—344no—
313C2×C4×Q8—342no—
314C2×C4×Q8—342no—
315C2×#117—171no—
316C2×#117—171no—
317C2×#117—171no—
318C2×#117—174no—
319C2×#117C2×D1711yescharacteristic
320C2×#117C2×D1711yescharacteristic
321#410—24no—
322#410—344no—
323C2×#99—174no—
324C2×#99C2×D1714yes—
325#391—342no—
326#391—342no—
327#381—174no—
328#342—174no—
329#342—174no—
330C2×C17—21no—
331C2×C17—21no—
332C2×C17—21no—
333C2×C17—21no—
334C2×C17—21no—
335C2×C17—21no—
336C2×C17—82no—
337C2×C17C2×#38611yescharacteristic
338C2×C17#288412yes—
339C2×#41—22no—
340C2×#41—21no—
341C2×#41—21no—
342C2×#41—21no—
343C2×#41—21no—
344C2×#41—22no—
345C2×#41—22no—
346C2×#41—22no—
347C22×Q8—172no—
348C22×Q8—172no—
349C22×Q8—172no—
350C22×Q8—172no—
351C22×Q8—172no—
352C22×Q8—172no—
353C22×Q8—342no—
354C22×Q8—342no—
355C22×Q8—342no—
356C22×Q8—342no—
357C22×Q8C4×D1711yescharacteristic
358C22×D4C4×D1711yescharacteristic
359C22×D4C22×D1712yes—
360C23×C4—172no—
361C23×C4C4×D1712yes—
362C23×C4C22×D1711yescharacteristic
363C4⋊Q8—344no—
364C4⋊Q8—344no—
365#124—344no—
366#124—344no—
367#123—348no—
368#123—348no—
369C22⋊Q8—348no—
370C22⋊Q8—348no—
371C4×Q8—3416no—
372C4×Q8—3416no—
373#117—22no—
374#117—22no—
375#117—22no—
376#117—174no—
377#117—348no—
378#117—341no—
379#117—341no—
380#117—341no—
381#117—341no—
382#117—342no—
383#117—348no—
384#117—342no—
385#117—342no—
386#117C22⋊D1714yes—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—172no—
390C2×C4⋊C4—178no—
391C2×C4⋊C4—344no—
392C2×C4⋊C4—341no—
393C2×C4⋊C4—341no—
394C2×C22⋊C4—172no—
395C2×C22⋊C4—178no—
396C2×C22⋊C4—172no—
397C2×C22⋊C4—172no—
398C2×C22⋊C4C22×D1712yes—
399C2×C22⋊C4C22×D1712yes—
400C2×C42—21no—
401C2×C42—21no—
402C2×C42—172no—
403C2×C42—344no—
404C2×C42—341no—
405C2×C42—341no—
406#99—28no—
407#99—348no—
408#95—348no—
409C17#2098611yescharacteristic, complemented, Sylow
410C24C2×C4×D1712yes—
411C24C23×D1711yescharacteristic
412#41—42no—
413#41—42no—
414#41—42no—
415#41—42no—
416#41—42no—
417#41—42no—
418#41—42no—
419#41—42no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—174no—
425C2×Q8—174no—
426C2×Q8—174no—
427C2×Q8—174no—
428C2×Q8—344no—
429C2×Q8—344no—
430C2×Q8—344no—
431C2×Q8—344no—
432C2×Q8—344no—
433C2×Q8—344no—
434C2×Q8—344no—
435C2×Q8—344no—
436C2×Q8—344no—
437C2×Q8—344no—
438C2×Q8—344no—
439C2×Q8—344no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—22no—
449C2×D4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—174no—
457C22×C4—174no—
458C22×C4—174no—
459C22×C4—174no—
460C22×C4—342no—
461C22×C4—342no—
462C22×C4—342no—
463C22×C4—342no—
464C22×C4—342no—
465C22×C4—342no—
466C22×C4C2×C4×D1712yes—
467C22×C4C2×C4×D1711yescharacteristic
468C22×C4C2×C4×D1711yescharacteristic
469C22×C4#6325212yes—
470C22×C4D4×D1711yescharacteristic
471C22×C4D4×D1711yescharacteristic
472C22×C4D4×D1711yescharacteristic
473C22×C4D4×D1711yescharacteristic
474C22×C4D4×D1712yes—
475C22×C4#6325412yes—
476C22×C4C2×C22⋊D1712yes—
477C4⋊C4—24no—
478C4⋊C4—42no—
479C4⋊C4—3416no—
480C4⋊C4—3416no—
481C4⋊C4—342no—
482C4⋊C4—342no—
483C4⋊C4—344no—
484C4⋊C4—682no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—174no—
489C22⋊C4—3416no—
490C22⋊C4—342no—
491C22⋊C4—342no—
492C22⋊C4—342no—
493C22⋊C4—342no—
494C22⋊C4—342no—
495C22⋊C4C2×C22⋊D1714yes—
496C42—24no—
497C42—42no—
498C42—3416no—
499C42—342no—
500C42—342no—
501C42—344no—
502C42—682no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—42no—
507C23—44no—
508C23C22×C4×D1711yescharacteristic
509C23C2×#6325211yescharacteristic
510C23C2×D4×D1711yescharacteristic
511C23C2×D4×D1711yescharacteristic
512C23C2×D4×D1711yescharacteristic
513C23C2×#6325411yescharacteristic
514C23C22×C22⋊D1711yescharacteristic
515Q8—42no—
516Q8—42no—
517Q8—348no—
518Q8—348no—
519Q8—348no—
520Q8—348no—
521D4—42no—
522D4—42no—
523D4—42no—
524D4—42no—
525D4—42no—
526D4—42no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—178no—
537C2×C4—344no—
538C2×C4—344no—
539C2×C4—344no—
540C2×C4—344no—
541C2×C4—344no—
542C2×C4—344no—
543C2×C4—344no—
544C2×C4—344no—
545C2×C4—684no—
546C2×C4—684no—
547C2×C4—684no—
548C2×C4—682no—
549C2×C4C22⋊C4×D1711yescharacteristic
550C2×C4C22⋊C4×D1711yescharacteristic
551C2×C4C22⋊C4×D1711yescharacteristic
552C2×C4C22⋊C4×D1711yescharacteristic
553C2×C4#1058756712yes—
554C2×C4#1058756712yes—
555C2×C4#1058756912yes—
556C2×C4#1058756912yes—
557C2×C4#1058757012yes—
558C2×C4#1058757012yes—
559C2×C4C4×C22⋊D1712yes—
560C2×C4C4×C22⋊D1712yes—
561C2×C4#1058762312yes—
562C2×C4#1058762312yes—
563C2×C4#1058762412yes—
564C2×C4#1058762412yes—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—44no—
573C22—42no—
574C22—82no—
575C22—82no—
576C22C2×C22⋊C4×D1711yescharacteristic
577C22C2×#1058756711yescharacteristic
578C22C2×C4×C22⋊D1711yescharacteristic
579C22C2×#1058762311yescharacteristic
580C22C2×#1058762411yescharacteristic
581C22C2×#1058757011yescharacteristic
582C22C2×#1058756911yescharacteristic
583C4—44no—
584C4—42no—
585C4—82no—
586C4—348no—
587C4—684no—
588C4—684no—
589C4—1362no—
590C4#1176268512yes—
591C4#1176268512yes—
592C4#1176268512yes—
593C4#1176268512yes—
594C2—21no—
595C2—21no—
596C2—21no—
597C2—21no—
598C2—21no—
599C2—21no—
600C2—82no—
601C2C2×#1176268511yescharacteristic
602C2#1453724612yes—
subgroupG/H
centerC22C2×C22⋊C4×D17
commutatorC23×C17C23×C4
FrattiniC24C23×D17
FittingC2×C17×#410C2
socleC22×C17C22×C22⋊C4
prime radicalC22×#41×C17C22
cosocleC24×C17C24
hypercenterC2×#410D17
hypocenterC17#20986
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
2. C2×#11763286C2semidirect
4. #14531368C2semidirect
5. #14529239C2semidirect
6. #14528516C2semidirect
7. #14528516C2semidirect
14. C2×#10587630C4semidirect
15. C2×#10587622C4semidirect
17. C2×#10587610C4semidirect
409. C17#20986semidirect

Maximal subgroups

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

subgroupindex
C2×#117633802
C2×#117632862
C2×#117627832
#145313682
#145292392
#145285162
C2×C17×#4102
#2098617