Finite Group Databasebeta

← #17037385

Subgroups — #17037385

Order 4352.

922 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762907C211yesmaximal, characteristic
2#14531248C211yesmaximal, characteristic
3#14529411C211yesmaximal, characteristic
4#14529404C211yesmaximal, characteristic
5#14529376C211yesmaximal, characteristic, complemented
6#14529355C211yesmaximal, characteristic, complemented
7#14529139C211yesmaximal, characteristic
8#14529139C211yesmaximal, characteristic
9#14529052C211yesmaximal, characteristic, complemented
10#14529038C211yesmaximal, characteristic, complemented
11#14528682C211yesmaximal, characteristic, complemented
12#14528614C211yesmaximal, characteristic, complemented
13#14528612C211yesmaximal, characteristic
14#14528533C211yesmaximal, characteristic, complemented
15C42×Q8×C17C211yesmaximal, characteristic, complemented
16C22×#63230C2211yescharacteristic
17C2×C4×D68C2211yescharacteristic
18C2×#10587558—21no—
19#11762907C418yescomplemented
20C2×#10587478C2211yescharacteristic
21C2×#10587478C2211yescharacteristic
22C2×#10587479C2211yescharacteristic
23C2×#10587506C2211yescharacteristic
24C2×#10587506C2211yescharacteristic
25C2×#10587506C2211yescharacteristic
26C2×#10587480C2211yescharacteristic
27C2×#10587480C2211yescharacteristic
28C2×#10587480C2211yescharacteristic
29C2×#10587474C2211yescharacteristic
30C2×#10587474C2211yescharacteristic
31#11762711C2211yescharacteristic
32#11762711C2211yescharacteristic
33#11762707C2211yescharacteristic
34#11762707C2211yescharacteristic
35#11762700—21no—
36#11762700—21no—
37C4×C4⋊Dic17C2211yescharacteristic
38C4×#63227—21no—
39#11762571C2211yescharacteristic
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762564C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48#11762564C2211yescharacteristic
49C2×C4×Q8×C17C2211yescharacteristic
50C2×C4×Q8×C17C2211yescharacteristic
51C2×C4×Q8×C17C2211yescharacteristic
52C4×C4⋊C4×C17—21no—
53C4×C4⋊C4×C17—21no—
54C4×C4⋊C4×C17—21no—
55C4×C4⋊C4×C17C2211yescharacteristic
56C4×C4⋊C4×C17C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
58C43×C17—21no—
59C43×C17C2211yescharacteristic
60C22×Q8×C17C2311yescharacteristic
61C22×D68C2311yescharacteristic
62C22×C4×D17—21no—
63C4×Q8×C17—22no—
64C4×Q8×C17—22no—
65C4×Q8×C17—22no—
66C4×Q8×C17—22no—
67C4×Q8×C17—22no—
68C4×Q8×C17C2×C414yes—
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17—21no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C42×C17—21no—
76C2×C42×C17—21no—
77C2×C42×C17—21no—
78C2×C42×C17—21no—
79C2×C42×C17—21no—
80C2×C42×C17—21no—
81C2×C42×C17D411yescharacteristic
82C2×C42×C17D411yescharacteristic
83C2×C42×C17D411yescharacteristic
84C2×C42×C17D411yescharacteristic
85C2×C42×C17C2311yescharacteristic
86C2×C42×C17C2311yescharacteristic
87C2×C42×C17C2311yescharacteristic
88C2×#63230—22no—
89C2×#63230—22no—
90C2×#63230C2×C414yes—
91C2×#63227—21no—
92C2×#63227—21no—
93C2×#63227—21no—
94C2×#63227—21no—
95C2×#63227—21no—
96C2×C4⋊Dic17C2311yescharacteristic
97C2×C4⋊Dic17C2311yescharacteristic
98C2×C4⋊Dic17C2311yescharacteristic
99C2×#63225—21no—
100C2×#63225—21no—
101C2×C4×Dic17—21no—
102C22×#3622C2311yescharacteristic
103C22×#3622C2311yescharacteristic
104C22×#3622C2311yescharacteristic
105C22×#3622C2311yescharacteristic
106C4×D68—22no—
107C4×D68C2×C414yes—
108#10587558—44no—
109C17×#95—21no—
110C17×#95—21no—
111C17×#95—21no—
112#10587506—22no—
113#10587506—22no—
114#10587506C2×C414yes—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587480—22no—
119#10587480—22no—
120#10587480C2×C414yes—
121#10587479—22no—
122#10587478—22no—
123#10587478C2×C414yes—
124#10587474—22no—
125#10587474C2×C414yes—
126C23×D17—21no—
127C2×Q8×C17—21no—
128C2×Q8×C17—21no—
129C2×Q8×C17—21no—
130C2×Q8×C17C22×C412yes—
131C2×Q8×C17#4112yes—
132C2×Q8×C17#4112yes—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—21no—
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2×D411yescharacteristic
141C22×C4×C17C2×D411yescharacteristic
142C22×C4×C17C2×D411yescharacteristic
143C22×C4×C17C2411yescharacteristic
144C22×Dic17—21no—
145C22×Dic17—21no—
146C22×Dic17—21no—
147C2×D68—21no—
148C2×D68—21no—
149C2×D68—21no—
150C2×D68—21no—
151C2×D68—21no—
152C2×D68C22×C412yes—
153C2×C4×D17—24no—
154C2×C4×D17—42no—
155C2×C4×D17—42no—
156C4⋊C4×C17—24no—
157C4⋊C4×C17—24no—
158C4⋊C4×C17—24no—
159C4⋊C4×C17—21no—
160C4⋊C4×C17—21no—
161C4⋊C4×C17—21no—
162C4⋊C4×C17C22×C412yes—
163C4⋊C4×C17#4112yes—
164C4⋊C4×C17#4112yes—
165C42×C17—24no—
166C42×C17—24no—
167C42×C17—24no—
168C42×C17—22no—
169C42×C17—22no—
170C42×C17—21no—
171C42×C17—21no—
172C42×C17—21no—
173C42×C17—21no—
174C42×C17—41no—
175C42×C17—41no—
176C42×C17—42no—
177C42×C17—42no—
178C42×C17—42no—
179C42×C17C22×C412yes—
180C42×C17#4112yes—
181#63230—28no—
182#63227—24no—
183#63227—42no—
184#63227—42no—
185#63227—42no—
186#63227—42no—
187C4⋊Dic17—21no—
188C4⋊Dic17—21no—
189C4⋊Dic17—21no—
190C4⋊Dic17—21no—
191C4⋊Dic17—21no—
192C4⋊Dic17C22×C412yes—
193#63225—42no—
194#63225—42no—
195C4×Dic17—42no—
196C2×#3622—21no—
197C2×#3622—21no—
198C2×#3622—21no—
199C2×#3622—21no—
200C2×#3622—21no—
201C2×#3622—21no—
202C2×#3622—21no—
203C2×#3622—21no—
204C2×#3622—21no—
205C2×#3622—21no—
206C2×#3622C22×C412yes—
207C2×#3622C22×C412yes—
208#21124—171nomaximal, Sylow
209C23×C17C22×D411yescharacteristic
210C22×D17—22no—
211C22×D17—41no—
212C22×D17—41no—
213C22×D17—41no—
214C22×D17—41no—
215C22×D17—41no—
216C22×D17—41no—
217Q8×C17—22no—
218Q8×C17—22no—
219Q8×C17—22no—
220Q8×C17#11714yescomplemented
221C2×C4×C17—22no—
222C2×C4×C17—21no—
223C2×C4×C17—21no—
224C2×C4×C17—22no—
225C2×C4×C17—22no—
226C2×C4×C17—21no—
227C2×C4×C17—21no—
228C2×C4×C17—21no—
229C2×C4×C17—21no—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—22no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
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—41no—
242C2×C4×C17—41no—
243C2×C4×C17—41no—
244C2×C4×C17—41no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17C4×D412yes—
248C2×C4×C17C4×D412yes—
249C2×C4×C17AGL(1,ℤ/8)12yes—
250C2×C4×C17#13712yes—
251C2×C4×C17C23×C411yescharacteristic
252C2×C4×C17C2×#4111yescharacteristic
253C2×C4×C17C2×#4111yescharacteristic
254C2×C4×C17C2×#4111yescharacteristic
255C2×C4×C17C2×#4111yescharacteristic
256C2×C4×C17C2×#4111yescharacteristic
257C2×Dic17—22no—
258C2×Dic17—41no—
259C2×Dic17—41no—
260C2×Dic17—41no—
261C2×Dic17—41no—
262C2×Dic17—41no—
263C2×Dic17—41no—
264C2×Dic17—41no—
265C2×Dic17—41no—
266D68—22no—
267D68—22no—
268D68—41no—
269D68—41no—
270C4×D17—48no—
271#3622—22no—
272#3622—22no—
273#3622—41no—
274#3622—41no—
275C2×#440—171no—
276#2293—171no—
277C42×Q8D1711yescharacteristic, complemented
278#1926—171no—
279#1925—171no—
280#1921—171no—
281#1918—171no—
282#1880—171no—
283#1880—171no—
284#1865—171no—
285#1863—171no—
286#1776—171no—
287#1763—171no—
288#1762—171no—
289#1745—171no—
290C22×C17C2×C4×D411yescharacteristic
291C22×C17#52511yescharacteristic
292C22×C17#52511yescharacteristic
293C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
294C22×C17C2×#13711yescharacteristic
295C22×C17#57511yescharacteristic
296C22×C17#57511yescharacteristic
297C2×D17—42no—
298C2×D17—42no—
299C2×D17—42no—
300C2×D17—81no—
301C2×D17—81no—
302C2×D17—81no—
303C2×D17—81no—
304C4×C17—22no—
305C4×C17—22no—
306C4×C17—22no—
307C4×C17—22no—
308C4×C17—24no—
309C4×C17—41no—
310C4×C17—42no—
311C4×C17—41no—
312C4×C17—42no—
313C4×C17—42no—
314C4×C17C2×#11711yescharacteristic
315C4×C17C4×#4111yescharacteristic
316C4×C17#52911yescharacteristic
317C4×C17#52911yescharacteristic
318Dic17—42no—
319Dic17—81no—
320Dic17—81no—
321C22×SD16—171no—
322C2×C4×Q8C2×D1711yescharacteristic
323C2×C4×Q8C2×D1711yescharacteristic
324C2×C4×Q8C2×D1711yescharacteristic
325C2×C4×D4—171no—
326C2×#117—341no—
327#440—178no—
328C2×#107—171no—
329C2×#107—171no—
330C2×#106—171no—
331C2×#103—171no—
332C2×#103—171no—
333C2×#103—171no—
334C2×#102—171no—
335C2×#102—171no—
336C2×#102—171no—
337C2×#97—171no—
338C2×#97—171no—
339#390—171no—
340#390—171no—
341#389—171no—
342#389—171no—
343#388—341no—
344#388—341no—
345C4×C4⋊C4—21no—
346C4×C4⋊C4—21no—
347C4×C4⋊C4—21no—
348C4×C4⋊C4—171no—
349C4×C4⋊C4C2×D1711yescharacteristic
350C4×C4⋊C4C2×D1711yescharacteristic
351C4×C4⋊C4C2×D1711yescharacteristic
352C4×C22⋊C4—341no—
353C43—21no—
354C43C2×D1711yescharacteristic
355#340—171no—
356#340—171no—
357#340—171no—
358#340—171no—
359#340—171no—
360#340—171no—
361#336—171no—
362#336—171no—
363#336—171no—
364#336—171no—
365C2×C17#229011yescharacteristic
366C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
367C2×C17C4×#13711yescharacteristic
368C2×C17#311111yescharacteristic
369C2×C17#311211yescharacteristic
370C2×C17#311511yescharacteristic
371C2×C17#311711yescharacteristic
372D17—82no—
373C22×Q8C22×D1711yescharacteristic
374C22×D4—171no—
375C23×C4—341no—
376C2×SD16—174no—
377C2×SD16—342no—
378C2×SD16—342no—
379C22×C8—171no—
380C22×C8—171no—
381C22×C8—171no—
382C22×C8—171no—
383C4×Q8—22no—
384C4×Q8—22no—
385C4×Q8—22no—
386C4×Q8—22no—
387C4×Q8—22no—
388C4×Q8C4×D1714yes—
389C4×D4—174no—
390C4×D4—342no—
391#117—684no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—171no—
396C2×C4⋊C4—171no—
397C2×C4⋊C4—171no—
398C2×C4⋊C4—341no—
399C2×C4⋊C4—341no—
400C2×C4⋊C4C22×D1711yescharacteristic
401C2×C4⋊C4C22×D1711yescharacteristic
402C2×C4⋊C4C22×D1711yescharacteristic
403C2×C22⋊C4—341no—
404C2×C22⋊C4—341no—
405C2×C22⋊C4—341no—
406C2×C22⋊C4—341no—
407C2×C22⋊C4—341no—
408C2×C42—21no—
409C2×C42—21no—
410C2×C42—21no—
411C2×C42—21no—
412C2×C42—21no—
413C2×C42—21no—
414C2×C42—341no—
415C2×C42C22⋊D1711yescharacteristic
416C2×C42C22⋊D1711yescharacteristic
417C2×C42C22⋊D1711yescharacteristic
418C2×C42C22⋊D1711yescharacteristic
419C2×C42C22×D1711yescharacteristic
420C2×C42C22×D1711yescharacteristic
421C2×C42C22×D1711yescharacteristic
422#107—174no—
423#107—342no—
424#106—342no—
425#103—174no—
426#103—342no—
427#103—342no—
428#102—174no—
429#102—342no—
430#102—342no—
431#97—174no—
432#97—342no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—341no—
437#95—341no—
438#95—341no—
439C17#2112411yescharacteristic, complemented, Sylow
440C24—341no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8C2×C4×D1712yes—
445C2×Q8#6325212yes—
446C2×Q8#6325212yes—
447C2×D4—172no—
448C2×D4—341no—
449C2×D4—341no—
450C2×D4—341no—
451C2×D4—341no—
452C2×D4—341no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—341no—
458C22×C4—344no—
459C22×C4—341no—
460C22×C4—341no—
461C22×C4—682no—
462C22×C4—682no—
463C22×C4C2×C22⋊D1711yescharacteristic
464C22×C4C2×C22⋊D1711yescharacteristic
465C22×C4C2×C22⋊D1711yescharacteristic
466C22×C4C2×C22⋊D1711yescharacteristic
467C22×C4C2×C22⋊D1711yescharacteristic
468C22×C4C2×C22⋊D1711yescharacteristic
469C22×C4C23×D1711yescharacteristic
470SD16—348no—
471C2×C8—172no—
472C2×C8—172no—
473C2×C8—341no—
474C2×C8—341no—
475C2×C8—341no—
476C2×C8—341no—
477C2×C8—341no—
478C2×C8—341no—
479C2×C8—341no—
480C2×C8—341no—
481C2×C8—341no—
482C2×C8—341no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—24no—
489C4⋊C4—172no—
490C4⋊C4—341no—
491C4⋊C4—341no—
492C4⋊C4—341no—
493C4⋊C4—341no—
494C4⋊C4—341no—
495C4⋊C4—682no—
496C4⋊C4—682no—
497C4⋊C4C2×C4×D1712yes—
498C4⋊C4#6325212yes—
499C4⋊C4#6325212yes—
500C22⋊C4—344no—
501C22⋊C4—682no—
502C22⋊C4—682no—
503C22⋊C4—682no—
504C22⋊C4—682no—
505C42—24no—
506C42—24no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—24no—
512C42—22no—
513C42—22no—
514C42—41no—
515C42—41no—
516C42—42no—
517C42—42no—
518C42—42no—
519C42—682no—
520C42C2×C4×D1712yes—
521C42#6325212yes—
522C23—342no—
523C23—681no—
524C23—681no—
525C23—681no—
526C23—681no—
527C23—681no—
528C23—681no—
529C23C22×C22⋊D1711yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—22no—
533Q8#1058755814yescomplemented
534D4—342no—
535D4—342no—
536D4—681no—
537D4—681no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—342no—
565C2×C4—688no—
566C2×C4—681no—
567C2×C4—681no—
568C2×C4—681no—
569C2×C4—681no—
570C2×C4—681no—
571C2×C4—681no—
572C2×C4—681no—
573C2×C4—681no—
574C2×C4C4×C22⋊D1712yes—
575C2×C4C4×C22⋊D1712yes—
576C2×C4#1058762812yes—
577C2×C4#1058763512yes—
578C2×C4C22×C4×D1711yescharacteristic
579C2×C4C2×#6325211yescharacteristic
580C2×C4C2×#6325211yescharacteristic
581C2×C4C2×#6325211yescharacteristic
582C2×C4C2×#6325211yescharacteristic
583C2×C4C2×#6325211yescharacteristic
584C8—342no—
585C8—342no—
586C8—681no—
587C8—681no—
588C22—682no—
589C22—682no—
590C22—682no—
591C22—1361no—
592C22—1361no—
593C22—1361no—
594C22—1361no—
595C22C2×C4×C22⋊D1711yescharacteristic
596C22#1176332011yescharacteristic
597C22#1176332011yescharacteristic
598C22C2×#1058763511yescharacteristic
599C22C2×#1058762811yescharacteristic
600C22#1176360711yescharacteristic
601C22#1176360711yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—24no—
606C4—22no—
607C4—42no—
608C4—41no—
609C4—42no—
610C4—41no—
611C4—42no—
612C4—682no—
613C4—1361no—
614C4—1361no—
615C4C2×#1058755811yescharacteristic
616C4C4×#6325211yescharacteristic
617C4#1176334311yescharacteristic
618C4#1176334311yescharacteristic
619C2—1362no—
620C2#1453122711yescharacteristic
621C2C4×#1058763511yescharacteristic
622C2C4×#1058762811yescharacteristic
623C2#1453889311yescharacteristic
624C2#1453890611yescharacteristic
625C2#1453893511yescharacteristic
626C2#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#21124
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
5. #14529376C2semidirect
6. #14529355C2semidirect
9. #14529052C2semidirect
10. #14529038C2semidirect
11. #14528682C2semidirect
12. #14528614C2semidirect
14. #14528533C2semidirect
15. C42×Q8×C17C2semidirect
19. #11762907C4semidirect
220. Q8×C17#117semidirect
277. C42×Q8D17semidirect
439. C17#21124semidirect
533. Q8#10587558semidirect

Maximal subgroups

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

subgroupindex
C2×#117629072
#145312482
#145294112
#145294042
#145293762
#145293552
#145291392
#145290522
#145290382
#145286822
#145286142
#145286122
#145285332
C42×Q8×C172
#2112417