Finite Group Databasebeta

← #17040633

Subgroups — #17040633

Order 4352.

1004 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763031C211yesmaximal, characteristic
2C2×#11762914C211yesmaximal, characteristic, complemented
3C2×#11762907C211yesmaximal, characteristic
4C4×#10587561C211yesmaximal, characteristic
5#14529716C211yesmaximal, characteristic
6#14529552C211yesmaximal, characteristic, complemented
7#14529425C211yesmaximal, characteristic, complemented
8#14529411C211yesmaximal, characteristic
9#14529139C211yesmaximal, characteristic
10#14529125C211yesmaximal, characteristic, complemented
11#14528799C211yesmaximal, characteristic, complemented
12#14528637C211yesmaximal, characteristic, complemented
13#14528617C211yesmaximal, characteristic, complemented
14#14528612C211yesmaximal, characteristic
15C42×Q8×C17C211yesmaximal, characteristic, complemented
16C22×#63231C2211yescharacteristic
17C22×#63230C2211yescharacteristic
18C2×#10587561C2211yescharacteristic
19C2×C4×Dic34C2211yescharacteristic
20C2×C4×D68C2211yescharacteristic
21#11763031C418yescomplemented
22#11762914—24no—
23#11762907—24no—
24C2×#10587478C2211yescharacteristic
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×#10587474C2211yescharacteristic
38#11762715C2211yescharacteristic
39#11762711C2211yescharacteristic
40#11762700—21no—
41#11762700—21no—
42#11762675C2211yescharacteristic
43C4×#63227—21no—
44C4×#63227—21no—
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
60C43×C17—21no—
61C43×C17C2211yescharacteristic
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×C17D411yescharacteristic
85C2×C42×C17D411yescharacteristic
86C2×C42×C17D411yescharacteristic
87C2×C42×C17D411yescharacteristic
88C2×C42×C17C2311yescharacteristic
89C2×C42×C17C2311yescharacteristic
90C2×C42×C17C2311yescharacteristic
91C2×#63231—22no—
92C2×#63231—22no—
93C2×#63231C2×C414yescomplemented
94C2×#63230—22no—
95C2×#63230—22no—
96C2×#63230C2×C414yes—
97C2×#63227—21no—
98C2×#63227—21no—
99C2×#63227—21no—
100C2×#63227—21no—
101C2×#63227—21no—
102C2×C4⋊Dic17C2311yescharacteristic
103C2×C4⋊Dic17C2311yescharacteristic
104C2×#63225—21no—
105C2×C4×Dic17—21no—
106C22×#3622C2311yescharacteristic
107C22×#3622C2311yescharacteristic
108C22×#3622C2311yescharacteristic
109C22×#3622C2311yescharacteristic
110#10587561—22no—
111#10587561C2×C414yes—
112C4×D68—22no—
113C4×D68—22no—
114C4×Dic34—22no—
115C4×Dic34—22no—
116C17×#95—21no—
117C17×#95—21no—
118C17×#95—21no—
119#10587506—22no—
120#10587506—22no—
121#10587502—21no—
122#10587502—21no—
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#10587478—22no—
133C4⋊#3622—22no—
134C4⋊#3622C2×C414yescomplemented
135#10587474—22no—
136#10587474—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—42no—
166C2×C4×D17—42no—
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—22no—
186C42×C17—21no—
187C42×C17—21no—
188C42×C17—41no—
189C42×C17—41no—
190C42×C17—41no—
191C42×C17—42no—
192C42×C17—42no—
193C42×C17—42no—
194C42×C17C22×C412yes—
195C42×C17C2×D412yes—
196C42×C17#4112yes—
197C42×C17#4112yes—
198#63231—28no—
199#63230—28no—
200#63227—24no—
201#63227—24no—
202#63227—42no—
203#63227—42no—
204#63227—42no—
205C4⋊Dic17—21no—
206C4⋊Dic17—21no—
207C4⋊Dic17—21no—
208C4⋊Dic17—21no—
209#63225—42no—
210C4×Dic17—42no—
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×#3622—21no—
220C2×#3622—21no—
221C2×#3622C22×C412yes—
222C2×#3622C22×C412yes—
223#21508—171nomaximal, Sylow
224C23×C17C22×D411yescharacteristic
225C22×D17—22no—
226C22×D17—41no—
227C22×D17—41no—
228C22×D17—41no—
229C22×D17—41no—
230C22×D17—41no—
231C22×D17—41no—
232Q8×C17—22no—
233Q8×C17—22no—
234Q8×C17—22no—
235Q8×C17C4×D414yes—
236C2×C4×C17—22no—
237C2×C4×C17—22no—
238C2×C4×C17—21no—
239C2×C4×C17—22no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—21no—
244C2×C4×C17—21no—
245C2×C4×C17—21no—
246C2×C4×C17—21no—
247C2×C4×C17—21no—
248C2×C4×C17—21no—
249C2×C4×C17—22no—
250C2×C4×C17—22no—
251C2×C4×C17—21no—
252C2×C4×C17—22no—
253C2×C4×C17—21no—
254C2×C4×C17—21no—
255C2×C4×C17—41no—
256C2×C4×C17—41no—
257C2×C4×C17—41no—
258C2×C4×C17—41no—
259C2×C4×C17—41no—
260C2×C4×C17—41no—
261C2×C4×C17C4×D412yes—
262C2×C4×C17C4×D412yes—
263C2×C4×C17#12112yes—
264C2×C4×C17#12112yes—
265C2×C4×C17#13712yes—
266C2×C4×C17C23×C411yescharacteristic
267C2×C4×C17C22×D411yescharacteristic
268C2×C4×C17C2×#4111yescharacteristic
269C2×C4×C17C2×#4111yescharacteristic
270C2×C4×C17C2×#4111yescharacteristic
271C2×C4×C17C2×#4111yescharacteristic
272C2×Dic17—22no—
273C2×Dic17—41no—
274C2×Dic17—41no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17—41no—
280C2×Dic17—41no—
281D68—22no—
282D68—22no—
283D68—41no—
284D68—41no—
285C4×D17—84no—
286Dic34—22no—
287Dic34—22no—
288Dic34—41no—
289Dic34—41no—
290#3622—22no—
291#3622—22no—
292#3622—41no—
293#3622—41no—
294C2×#464—171no—
295C2×#441—171no—
296C2×#440—171no—
297C4×#124—171no—
298C42×Q8D1711yescharacteristic, complemented
299#1983—171no—
300#1949—171no—
301#1928—171no—
302#1926—171no—
303#1880—171no—
304#1878—171no—
305#1805—171no—
306#1767—171no—
307#1763—171no—
308#1762—171no—
309C22×C17C2×C4×D411yescharacteristic
310C22×C17C2×#12111yescharacteristic
311C22×C17#52511yescharacteristic
312C22×C17C2×#13711yescharacteristic
313C22×C17#57511yescharacteristic
314C22×C17#57511yescharacteristic
315C22×C17#57511yescharacteristic
316C2×D17—42no—
317C2×D17—42no—
318C2×D17—42no—
319C2×D17—81no—
320C2×D17—81no—
321C2×D17—81no—
322C2×D17—81no—
323C4×C17—22no—
324C4×C17—22no—
325C4×C17—22no—
326C4×C17—22no—
327C4×C17—22no—
328C4×C17—24no—
329C4×C17—41no—
330C4×C17—42no—
331C4×C17—42no—
332C4×C17—42no—
333C4×C17C2×C4×D411yescharacteristic
334C4×C17C4×#4111yescharacteristic
335C4×C17#52911yescharacteristic
336C4×C17#53211yescharacteristic
337Dic17—42no—
338Dic17—81no—
339Dic17—81no—
340C22×Q16—171no—
341C22×SD16—171no—
342C2×#124—171no—
343C2×C4×Q8—171no—
344C2×C4×Q8C2×D1711yescharacteristic
345C2×C4×Q8C2×D1711yescharacteristic
346C2×C4×Q8C2×D1711yescharacteristic
347C2×C4×D4—171no—
348#464—178no—
349#441—344no—
350#440—344no—
351C2×#107—171no—
352C2×#106—171no—
353C2×C4⋊C8—171no—
354C2×C4⋊C8—171no—
355C2×#103—171no—
356C2×#103—171no—
357C2×#103—171no—
358C2×#103—171no—
359C2×#103—171no—
360C2×#102—171no—
361C2×#102—171no—
362C2×#102—171no—
363C2×#97—171no—
364C2×#97—171no—
365#391—171no—
366#390—171no—
367#388—341no—
368#388—341no—
369#382—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
376C4×C22⋊C4—341no—
377C4×C22⋊C4—341no—
378C43—21no—
379C43C2×D1711yescharacteristic
380#340—171no—
381#340—171no—
382#340—171no—
383#340—171no—
384#336—171no—
385#336—171no—
386C2×C17C4×#12111yescharacteristic
387C2×C17#294311yescharacteristic
388C2×C17C4×#13711yescharacteristic
389C2×C17#303711yescharacteristic
390C2×C17#304011yescharacteristic
391C2×C17#311111yescharacteristic
392C2×C17#311711yescharacteristic
393D17—82no—
394C22×Q8—171no—
395C22×Q8C22×D1711yescharacteristic
396C22×D4—171no—
397C23×C4—341no—
398C2×Q16—174no—
399C2×Q16—342no—
400C2×Q16—342no—
401C2×SD16—174no—
402C2×SD16—342no—
403C2×SD16—342no—
404C22×C8—171no—
405C22×C8—171no—
406C22×C8—171no—
407C22×C8—171no—
408#124—174no—
409#124—342no—
410C4×Q8—22no—
411C4×Q8—22no—
412C4×Q8—22no—
413C4×Q8—22no—
414C4×Q8—342no—
415C4×Q8—342no—
416C4×Q8C4×D1714yescomplemented
417C4×Q8D6814yes—
418C4×D4—342no—
419C4×D4—342no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—171no—
424C2×C4⋊C4—171no—
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
444C2×C42C22⋊D1711yescharacteristic
445C2×C42C22×D1711yescharacteristic
446C2×C42C22×D1711yescharacteristic
447C2×C42C22×D1711yescharacteristic
448#107—342no—
449#106—342no—
450C4⋊C8—174no—
451C4⋊C8—342no—
452#103—174no—
453#103—342no—
454#103—342no—
455#103—342no—
456#103—342no—
457#102—174no—
458#102—342no—
459#102—342no—
460#97—342no—
461#97—342no—
462#95—21no—
463#95—21no—
464#95—21no—
465#95—341no—
466#95—341no—
467#95—341no—
468#95—341no—
469C17#2150811yescharacteristic, complemented, Sylow
470C24—341no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—172no—
475C2×Q8—341no—
476C2×Q8—341no—
477C2×Q8—341no—
478C2×Q8—341no—
479C2×Q8—341no—
480C2×Q8C2×C4×D1712yes—
481C2×Q8C2×D6812yes—
482C2×Q8#6325212yes—
483C2×D4—172no—
484C2×D4—341no—
485C2×D4—341no—
486C2×D4—341no—
487C2×D4—341no—
488C2×D4—341no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—341no—
494C22×C4—341no—
495C22×C4—341no—
496C22×C4—682no—
497C22×C4—682no—
498C22×C4—682no—
499C22×C4C2×C22⋊D1711yescharacteristic
500C22×C4C2×C22⋊D1711yescharacteristic
501C22×C4C2×C22⋊D1711yescharacteristic
502C22×C4C2×C22⋊D1711yescharacteristic
503C22×C4C2×C22⋊D1711yescharacteristic
504C22×C4C2×C22⋊D1711yescharacteristic
505C22×C4C23×D1711yescharacteristic
506Q16—348no—
507SD16—348no—
508C2×C8—172no—
509C2×C8—172no—
510C2×C8—341no—
511C2×C8—341no—
512C2×C8—341no—
513C2×C8—341no—
514C2×C8—341no—
515C2×C8—341no—
516C2×C8—341no—
517C2×C8—341no—
518C2×C8—341no—
519C2×C8—341no—
520C4⋊C4—24no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—341no—
527C4⋊C4—341no—
528C4⋊C4—341no—
529C4⋊C4—341no—
530C4⋊C4—682no—
531C4⋊C4C2×C4×D1712yes—
532C4⋊C4C2×D6812yes—
533C4⋊C4#6325212yes—
534C22⋊C4—344no—
535C22⋊C4—344no—
536C22⋊C4—682no—
537C22⋊C4—682no—
538C22⋊C4—682no—
539C42—24no—
540C42—21no—
541C42—21no—
542C42—24no—
543C42—24no—
544C42—22no—
545C42—41no—
546C42—41no—
547C42—41no—
548C42—42no—
549C42—42no—
550C42—42no—
551C42—682no—
552C42C2×C4×D1712yes—
553C42C2×D6812yes—
554C42#6325212yes—
555C42#6325212yes—
556C23—342no—
557C23—681no—
558C23—681no—
559C23—681no—
560C23—681no—
561C23—681no—
562C23—681no—
563C23C22×C22⋊D1711yescharacteristic
564Q8—22no—
565Q8—22no—
566Q8—22no—
567Q8—342no—
568Q8—342no—
569Q8—681no—
570Q8—681no—
571Q8C4×D6814yes—
572D4—342no—
573D4—342no—
574D4—681no—
575D4—681no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—342no—
602C2×C4—681no—
603C2×C4—681no—
604C2×C4—681no—
605C2×C4—681no—
606C2×C4—681no—
607C2×C4—681no—
608C2×C4—681no—
609C2×C4—681no—
610C2×C4—1364no—
611C2×C4C4×C22⋊D1712yes—
612C2×C4C4×C22⋊D1712yes—
613C2×C4#1058761612yes—
614C2×C4#1058761612yes—
615C2×C4#1058762812yes—
616C2×C4C22×C4×D1711yescharacteristic
617C2×C4C22×D6811yescharacteristic
618C2×C4C2×#6325211yescharacteristic
619C2×C4C2×#6325211yescharacteristic
620C2×C4C2×#6325211yescharacteristic
621C2×C4C2×#6325211yescharacteristic
622C8—342no—
623C8—342no—
624C8—681no—
625C8—681no—
626C22—682no—
627C22—682no—
628C22—682no—
629C22—1361no—
630C22—1361no—
631C22—1361no—
632C22—1361no—
633C22C2×C4×C22⋊D1711yescharacteristic
634C22C2×#1058761611yescharacteristic
635C22#1176332011yescharacteristic
636C22C2×#1058762811yescharacteristic
637C22#1176360711yescharacteristic
638C22#1176360711yescharacteristic
639C22#1176360711yescharacteristic
640C4—22no—
641C4—22no—
642C4—22no—
643C4—22no—
644C4—22no—
645C4—24no—
646C4—42no—
647C4—42no—
648C4—41no—
649C4—42no—
650C4—682no—
651C4—1361no—
652C4—1361no—
653C4C2×C4×D6811yescharacteristic
654C4C4×#6325211yescharacteristic
655C4#1176334311yescharacteristic
656C4#1176335911yescharacteristic
657C2—1362no—
658C2C4×#1058761611yescharacteristic
659C2#1453758211yescharacteristic
660C2C4×#1058762811yescharacteristic
661C2#1453825611yescharacteristic
662C2#1453827811yescharacteristic
663C2#1453889311yescharacteristic
664C2#1453895511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC42×Q8×C17C2
socleC23×C17C22×D4
prime radicalC22×D68C23
cosocleC22×C4×C17C24
hypercenterC42×Q8D17
hypocenterC17#21508
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C43 ◁ C42×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#11762914C2semidirect
6. #14529552C2semidirect
7. #14529425C2semidirect
10. #14529125C2semidirect
11. #14528799C2semidirect
12. #14528637C2semidirect
13. #14528617C2semidirect
15. C42×Q8×C17C2semidirect
21. #11763031C4semidirect
70. C4×Q8×C17C2×C4semidirect
93. C2×#63231C2×C4semidirect
127. #10587481C2×C4semidirect
134. C4⋊#3622C2×C4semidirect
298. C42×Q8D17semidirect
416. C4×Q8C4×D17semidirect
469. C17#21508semidirect

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×#117629072
C4×#105875612
#145297162
#145295522
#145294252
#145294112
#145291392
#145291252
#145287992
#145286372
#145286172
#145286122
C42×Q8×C172
#2150817