Finite Group Databasebeta

← #17040678

Subgroups — #17040678

Order 4352.

980 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763031C211yesmaximal, characteristic
2C2×#11762914C211yesmaximal, characteristic, complemented
3C2×C4×#63230C211yesmaximal, characteristic
4#14531495C211yesmaximal, characteristic
5#14529716C211yesmaximal, characteristic
6#14529547C211yesmaximal, characteristic, complemented
7#14529418C211yesmaximal, characteristic, complemented
8#14529411C211yesmaximal, characteristic
9#14529139C211yesmaximal, characteristic
10#14529125C211yesmaximal, characteristic, complemented
11#14528789C211yesmaximal, characteristic, complemented
12#14528641C211yesmaximal, characteristic, complemented
13#14528612C211yesmaximal, characteristic
14C4×#10587481C211yesmaximal, characteristic, complemented
15C17×#2276C211yesmaximal, characteristic, complemented
16C22×#63231C2211yescharacteristic
17C22×#63230C2211yescharacteristic
18C2×#10587561C2211yescharacteristic
19C2×C4×Dic34C2211yescharacteristic
20C2×C4×D68C2211yescharacteristic
21#11763031C418yescomplemented
22#11762914—24no—
23C4×#63230—24no—
24C2×#10587479C2211yescharacteristic
25C2×#10587479C2211yescharacteristic
26C2×C4⋊#3622C2211yescharacteristic
27C2×C4⋊#3622C2211yescharacteristic
28C2×#10587506C2211yescharacteristic
29C2×#10587506C2211yescharacteristic
30C2×#10587481C2211yescharacteristic
31C2×#10587481C2211yescharacteristic
32C2×#10587481C2211yescharacteristic
33C2×#10587480C2211yescharacteristic
34C2×#10587480C2211yescharacteristic
35C2×#10587480C2211yescharacteristic
36C2×#10587474C2211yescharacteristic
37C2×C4×#3622C2211yescharacteristic
38#11762715C2211yescharacteristic
39#11762711C2211yescharacteristic
40#11762701—21no—
41#11762701—21no—
42#11762683—21no—
43#11762683—21no—
44#11762679C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762564C2211yescharacteristic
50#11762564C2211yescharacteristic
51C2×C4×Q8×C17C2211yescharacteristic
52C2×C4×Q8×C17C2211yescharacteristic
53C2×C4×Q8×C17C2211yescharacteristic
54C4×C4⋊C4×C17—21no—
55C4×C4⋊C4×C17—21no—
56C4×C4⋊C4×C17—21no—
57C4×C4⋊C4×C17C2211yescharacteristic
58C4×C4⋊C4×C17C2211yescharacteristic
59C4×C4⋊C4×C17C2211yescharacteristic
60C17×#376—21no—
61C17×#376C2211yescharacteristic
62C22×Q8×C17C2311yescharacteristic
63C22×D68C2311yescharacteristic
64C22×C4×D17—21no—
65C22×Dic34C2311yescharacteristic
66C4×Q8×C17—22no—
67C4×Q8×C17—22no—
68C4×Q8×C17—22no—
69C4×Q8×C17—22no—
70C4×Q8×C17C2×C414yescomplemented
71C4×Q8×C17D414yes—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17—21no—
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C4⋊C4×C17C2311yescharacteristic
78C2×C42×C17—21no—
79C2×C42×C17—21no—
80C2×C42×C17—21no—
81C2×C42×C17—21no—
82C2×C42×C17—21no—
83C2×C42×C17—21no—
84C2×C42×C17C2311yescharacteristic
85C2×C42×C17C2311yescharacteristic
86C2×C42×C17C2311yescharacteristic
87C2×#63231—22no—
88C2×#63231—22no—
89C2×#63231C2×C414yescomplemented
90C2×#63230—22no—
91C2×#63230—22no—
92C2×#63230C2×C414yes—
93C2×#63227—21no—
94C2×#63227—21no—
95C2×#63227—21no—
96C2×#63227—21no—
97C2×#63227—21no—
98C2×C4⋊Dic17—21no—
99C2×C4⋊Dic17—21no—
100C2×C4⋊Dic17C2311yescharacteristic
101C2×C4⋊Dic17C2311yescharacteristic
102C2×#63225—21no—
103C2×C4×Dic17—21no—
104C22×#3622C2311yescharacteristic
105C22×#3622C2311yescharacteristic
106C22×#3622C2311yescharacteristic
107C22×#3622C2311yescharacteristic
108#10587561—22no—
109#10587561C2×C414yes—
110C4×D68—22no—
111C4×D68—22no—
112C4×Dic34—22no—
113C4×Dic34—22no—
114C17×#95—21no—
115C17×#95—21no—
116C17×#95—21no—
117C17×#95D411yescharacteristic
118C17×#95D411yescharacteristic
119C17×#95D411yescharacteristic
120C17×#95D411yescharacteristic
121#10587506—22no—
122#10587506—22no—
123#10587502—21no—
124#10587502—21no—
125#10587481—22no—
126#10587481—22no—
127#10587481C2×C414yescomplemented
128#10587480—22no—
129#10587480—22no—
130#10587480C2×C414yes—
131#10587479—22no—
132#10587479—22no—
133C4⋊#3622—22no—
134C4⋊#3622C2×C414yescomplemented
135#10587474—22no—
136C4×#3622—22no—
137C23×D17—21no—
138C2×Q8×C17—21no—
139C2×Q8×C17—21no—
140C2×Q8×C17—21no—
141C2×Q8×C17C22×C412yes—
142C2×Q8×C17C2×D412yes—
143C2×Q8×C17#4112yes—
144C22×C4×C17—21no—
145C22×C4×C17—21no—
146C22×C4×C17—21no—
147C22×C4×C17—21no—
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2×D411yescharacteristic
151C22×C4×C17C2×D411yescharacteristic
152C22×C4×C17C2×D411yescharacteristic
153C22×C4×C17C2×D411yescharacteristic
154C22×C4×C17C2411yescharacteristic
155C22×Dic17—21no—
156C22×Dic17—21no—
157C22×Dic17—21no—
158C2×D68—21no—
159C2×D68—21no—
160C2×D68—21no—
161C2×D68—21no—
162C2×D68—21no—
163C2×D68C22×C412yes—
164C2×C4×D17—42no—
165C2×C4×D17—81no—
166C2×C4×D17—81no—
167C2×Dic34—21no—
168C2×Dic34—21no—
169C2×Dic34—21no—
170C2×Dic34—21no—
171C2×Dic34—21no—
172C2×Dic34C22×C412yes—
173C4⋊C4×C17—24no—
174C4⋊C4×C17—24no—
175C4⋊C4×C17—24no—
176C4⋊C4×C17—21no—
177C4⋊C4×C17—21no—
178C4⋊C4×C17—21no—
179C4⋊C4×C17C22×C412yes—
180C4⋊C4×C17C2×D412yes—
181C4⋊C4×C17#4112yes—
182C42×C17—24no—
183C42×C17—24no—
184C42×C17—24no—
185C42×C17—21no—
186C42×C17—21no—
187C42×C17—81no—
188C42×C17—81no—
189C42×C17—81no—
190C42×C17C22×C412yes—
191C42×C17C2×D412yes—
192C42×C17#4112yes—
193C42×C17#4112yes—
194#63231—28no—
195#63230—28no—
196#63227—24no—
197#63227—24no—
198#63227—42no—
199#63227—81no—
200#63227—81no—
201C4⋊Dic17—21no—
202C4⋊Dic17—21no—
203C4⋊Dic17—21no—
204C4⋊Dic17—21no—
205C4⋊Dic17—81no—
206C4⋊Dic17—81no—
207#63225—42no—
208C4×Dic17—42no—
209C2×#3622—21no—
210C2×#3622—21no—
211C2×#3622—21no—
212C2×#3622—21no—
213C2×#3622—21no—
214C2×#3622—21no—
215C2×#3622—21no—
216C2×#3622—21no—
217C2×#3622—21no—
218C2×#3622—21no—
219C2×#3622C22×C412yes—
220C2×#3622C22×C412yes—
221#21511—171nomaximal, Sylow
222C23×C17C22×D411yescharacteristic
223C22×D17—22no—
224C22×D17—41no—
225C22×D17—41no—
226C22×D17—41no—
227C22×D17—41no—
228C22×D17—41no—
229C22×D17—41no—
230Q8×C17—22no—
231Q8×C17—22no—
232Q8×C17—22no—
233Q8×C17C4×D414yes—
234C2×C4×C17—22no—
235C2×C4×C17—22no—
236C2×C4×C17—22no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—21no—
244C2×C4×C17—21no—
245C2×C4×C17—22no—
246C2×C4×C17—22no—
247C2×C4×C17—21no—
248C2×C4×C17—21no—
249C2×C4×C17—22no—
250C2×C4×C17—21no—
251C2×C4×C17—21no—
252C2×C4×C17—21no—
253C2×C4×C17—21no—
254C2×C4×C17—41no—
255C2×C4×C17—41no—
256C2×C4×C17—41no—
257C2×C4×C17—41no—
258C2×C4×C17—41no—
259C2×C4×C17—41no—
260C2×C4×C17C4×D412yes—
261C2×C4×C17C4×D412yes—
262C2×C4×C17#12112yes—
263C2×C4×C17#12112yes—
264C2×C4×C17C23×C411yescharacteristic
265C2×C4×C17C22×D411yescharacteristic
266C2×C4×C17C2×#4111yescharacteristic
267C2×C4×C17C2×#4111yescharacteristic
268C2×C4×C17C2×#4111yescharacteristic
269C2×C4×C17C2×#4111yescharacteristic
270C2×Dic17—22no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273C2×Dic17—41no—
274C2×Dic17—41no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279D68—22no—
280D68—22no—
281D68—41no—
282D68—41no—
283C4×D17—84no—
284Dic34—22no—
285Dic34—22no—
286Dic34—41no—
287Dic34—41no—
288#3622—22no—
289#3622—22no—
290#3622—41no—
291#3622—41no—
292C2×#464—171no—
293C2×#441—171no—
294C2×C4×SD16—171no—
295#2326—171no—
296#2276D1711yescharacteristic, complemented
297#1983—171no—
298#1948—171no—
299#1927—171no—
300#1926—171no—
301#1880—171no—
302#1878—171no—
303#1803—171no—
304#1768—171no—
305#1762—171no—
306C4×#103—171no—
307C22×C17C2×C4×D411yescharacteristic
308C22×C17C2×#12111yescharacteristic
309C22×C17#52511yescharacteristic
310C22×C17#57711yescharacteristic
311C22×C17#57711yescharacteristic
312C22×C17#57711yescharacteristic
313C22×C17#57811yescharacteristic
314C2×D17—42no—
315C2×D17—42no—
316C2×D17—42no—
317C2×D17—81no—
318C2×D17—81no—
319C2×D17—81no—
320C2×D17—81no—
321C4×C17—22no—
322C4×C17—22no—
323C4×C17—22no—
324C4×C17—22no—
325C4×C17—24no—
326C4×C17—41no—
327C4×C17—41no—
328C4×C17—42no—
329C4×C17—42no—
330C4×C17—42no—
331C4×C17C2×C4×D411yescharacteristic
332C4×C17C4×#4111yescharacteristic
333C4×C17#52911yescharacteristic
334C4×C17#53211yescharacteristic
335Dic17—42no—
336Dic17—81no—
337Dic17—81no—
338C22×Q16—171no—
339C22×SD16—171no—
340C2×#124—171no—
341C2×C4×Q8—171no—
342C2×C4×Q8C2×D1711yescharacteristic
343C2×C4×Q8C2×D1711yescharacteristic
344C2×C4×Q8C2×D1711yescharacteristic
345C2×C4×D4—171no—
346#464—178no—
347#441—344no—
348C4×SD16—344no—
349C2×#106—171no—
350C2×#106—171no—
351C2×C4⋊C8—171no—
352C2×C4⋊C8—171no—
353C2×#103—171no—
354C2×#103—171no—
355C2×#103—171no—
356C2×#103—171no—
357C2×#103—171no—
358C2×#102—171no—
359C2×#102—171no—
360C2×#102—171no—
361C2×#97—171no—
362C2×C4×C8—171no—
363#391—171no—
364#390—171no—
365#388—341no—
366#388—341no—
367#385—341no—
368#385—341no—
369#384—171no—
370C4×C4⋊C4—21no—
371C4×C4⋊C4—21no—
372C4×C4⋊C4—21no—
373C4×C4⋊C4C2×D1711yescharacteristic
374C4×C4⋊C4C2×D1711yescharacteristic
375C4×C4⋊C4C2×D1711yescharacteristic
376#376—21no—
377#376C2×D1711yescharacteristic
378#340—171no—
379#340—171no—
380#340—171no—
381#340—171no—
382#336—171no—
383#336—171no—
384C2×C17C4×#12111yescharacteristic
385C2×C17#294711yescharacteristic
386C2×C17#295211yescharacteristic
387C2×C17#304311yescharacteristic
388C2×C17#304611yescharacteristic
389C2×C17#312111yescharacteristic
390C2×C17#312511yescharacteristic
391D17—82no—
392C22×Q8—171no—
393C22×Q8C22×D1711yescharacteristic
394C22×D4—171no—
395C23×C4—341no—
396C2×Q16—174no—
397C2×Q16—342no—
398C2×Q16—342no—
399C2×SD16—174no—
400C2×SD16—342no—
401C2×SD16—342no—
402C22×C8—171no—
403C22×C8—171no—
404C22×C8—171no—
405C22×C8—171no—
406#124—174no—
407#124—342no—
408C4×Q8—22no—
409C4×Q8—22no—
410C4×Q8—22no—
411C4×Q8—22no—
412C4×Q8—342no—
413C4×Q8—342no—
414C4×Q8C4×D1714yescomplemented
415C4×Q8D6814yes—
416C4×D4—342no—
417C4×D4—342no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—171no—
422C2×C4⋊C4—171no—
423C2×C4⋊C4—341no—
424C2×C4⋊C4—341no—
425C2×C4⋊C4—341no—
426C2×C4⋊C4C22×D1711yescharacteristic
427C2×C4⋊C4C22×D1711yescharacteristic
428C2×C4⋊C4C22×D1711yescharacteristic
429C2×C22⋊C4—341no—
430C2×C22⋊C4—341no—
431C2×C22⋊C4—341no—
432C2×C22⋊C4—341no—
433C2×C22⋊C4—341no—
434C2×C42—21no—
435C2×C42—21no—
436C2×C42—21no—
437C2×C42—21no—
438C2×C42—21no—
439C2×C42—21no—
440C2×C42—341no—
441C2×C42C22×D1711yescharacteristic
442C2×C42C22×D1711yescharacteristic
443C2×C42C22×D1711yescharacteristic
444#106—342no—
445#106—342no—
446C4⋊C8—174no—
447C4⋊C8—342no—
448#103—174no—
449#103—342no—
450#103—342no—
451#103—342no—
452#103—342no—
453#102—174no—
454#102—342no—
455#102—342no—
456#97—342no—
457C4×C8—342no—
458#95—21no—
459#95—21no—
460#95—21no—
461#95—341no—
462#95—341no—
463#95C22⋊D1711yescharacteristic
464#95C22⋊D1711yescharacteristic
465#95C22⋊D1711yescharacteristic
466#95C22⋊D1711yescharacteristic
467C17#2151111yescharacteristic, complemented, Sylow
468C24—341no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—172no—
473C2×Q8—341no—
474C2×Q8—341no—
475C2×Q8—341no—
476C2×Q8—341no—
477C2×Q8—341no—
478C2×Q8C2×C4×D1712yes—
479C2×Q8C2×D6812yes—
480C2×Q8#6325212yes—
481C2×D4—172no—
482C2×D4—341no—
483C2×D4—341no—
484C2×D4—341no—
485C2×D4—341no—
486C2×D4—341no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—341no—
492C22×C4—341no—
493C22×C4—341no—
494C22×C4—682no—
495C22×C4—1361no—
496C22×C4—1361no—
497C22×C4C2×C22⋊D1711yescharacteristic
498C22×C4C2×C22⋊D1711yescharacteristic
499C22×C4C2×C22⋊D1711yescharacteristic
500C22×C4C2×C22⋊D1711yescharacteristic
501C22×C4C2×C22⋊D1711yescharacteristic
502C22×C4C2×C22⋊D1711yescharacteristic
503C22×C4C23×D1711yescharacteristic
504Q16—348no—
505SD16—348no—
506C2×C8—172no—
507C2×C8—172no—
508C2×C8—341no—
509C2×C8—341no—
510C2×C8—341no—
511C2×C8—341no—
512C2×C8—341no—
513C2×C8—341no—
514C2×C8—341no—
515C2×C8—341no—
516C2×C8—341no—
517C2×C8—341no—
518C4⋊C4—24no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—341no—
525C4⋊C4—341no—
526C4⋊C4—341no—
527C4⋊C4—341no—
528C4⋊C4—682no—
529C4⋊C4—1361no—
530C4⋊C4—1361no—
531C4⋊C4C2×C4×D1712yes—
532C4⋊C4C2×D6812yes—
533C4⋊C4#6325212yes—
534C22⋊C4—344no—
535C22⋊C4—344no—
536C22⋊C4—682no—
537C22⋊C4—1361no—
538C22⋊C4—1361no—
539C42—24no—
540C42—21no—
541C42—21no—
542C42—24no—
543C42—24no—
544C42—81no—
545C42—81no—
546C42—81no—
547C42—682no—
548C42C2×C4×D1712yes—
549C42C2×D6812yes—
550C42#6325212yes—
551C42#6325212yes—
552C23—342no—
553C23—681no—
554C23—681no—
555C23—681no—
556C23—681no—
557C23—681no—
558C23—681no—
559C23C22×C22⋊D1711yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—22no—
563Q8—342no—
564Q8—342no—
565Q8—681no—
566Q8—681no—
567Q8C4×D6814yes—
568D4—342no—
569D4—342no—
570D4—681no—
571D4—681no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—342no—
599C2×C4—681no—
600C2×C4—681no—
601C2×C4—681no—
602C2×C4—681no—
603C2×C4—681no—
604C2×C4—681no—
605C2×C4—681no—
606C2×C4—681no—
607C2×C4—1364no—
608C2×C4C4×C22⋊D1712yes—
609C2×C4C4×C22⋊D1712yes—
610C2×C4#1058761612yes—
611C2×C4#1058761612yes—
612C2×C4C22×C4×D1711yescharacteristic
613C2×C4C22×D6811yescharacteristic
614C2×C4C2×#6325211yescharacteristic
615C2×C4C2×#6325211yescharacteristic
616C2×C4C2×#6325211yescharacteristic
617C2×C4C2×#6325211yescharacteristic
618C8—342no—
619C8—342no—
620C8—681no—
621C8—681no—
622C22—682no—
623C22—682no—
624C22—682no—
625C22—1361no—
626C22—1361no—
627C22—1361no—
628C22—1361no—
629C22C2×C4×C22⋊D1711yescharacteristic
630C22C2×#1058761611yescharacteristic
631C22#1176332011yescharacteristic
632C22#1176361711yescharacteristic
633C22#1176361711yescharacteristic
634C22#1176362011yescharacteristic
635C22#1176362411yescharacteristic
636C4—22no—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—24no—
641C4—41no—
642C4—41no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—682no—
647C4—1361no—
648C4—1361no—
649C4C2×C4×D6811yescharacteristic
650C4C4×#6325211yescharacteristic
651C4#1176334311yescharacteristic
652C4#1176335911yescharacteristic
653C2—1362no—
654C2C4×#1058761611yescharacteristic
655C2#1453762711yescharacteristic
656C2#1453766511yescharacteristic
657C2#1453830511yescharacteristic
658C2#1453833411yescharacteristic
659C2#1453901311yescharacteristic
660C2#1453906111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2276C2
socleC23×C17C22×D4
prime radicalC22×D68C23
cosocleC22×C4×C17C24
hypercenter#2276D17
hypocenterC17#21511
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ #376 ◁ #2276

Complements

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

NH ≅ G/Ntype
2. C2×#11762914C2semidirect
6. #14529547C2semidirect
7. #14529418C2semidirect
10. #14529125C2semidirect
11. #14528789C2semidirect
12. #14528641C2semidirect
14. C4×#10587481C2semidirect
15. C17×#2276C2semidirect
21. #11763031C4semidirect
70. C4×Q8×C17C2×C4semidirect
89. C2×#63231C2×C4semidirect
127. #10587481C2×C4semidirect
134. C4⋊#3622C2×C4semidirect
296. #2276D17semidirect
414. C4×Q8C4×D17semidirect
467. C17#21511semidirect

Maximal subgroups

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

subgroupindex
C2×#117630312
C2×#117629142
C2×C4×#632302
#145314952
#145297162
#145295472
#145294182
#145294112
#145291392
#145291252
#145287892
#145286412
#145286122
C4×#105874812
C17×#22762
#2151117