Finite Group Databasebeta

← #17051554

Subgroups — #17051554

Order 4352.

1010 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762808C211yesmaximal, characteristic
2#14531615C211yesmaximal, characteristic
3#14529473C211yesmaximal, characteristic, complemented
4#14529459C211yesmaximal, characteristic, complemented
5#14529425C211yesmaximal, characteristic, complemented
6#14529418C211yesmaximal, characteristic, complemented
7#14529044C211yesmaximal, characteristic, complemented
8#14529030C211yesmaximal, characteristic, complemented
9#14528870C211yesmaximal, characteristic, complemented
10#14528815C211yesmaximal, characteristic, complemented
11#14528722C211yesmaximal, characteristic
12#14528612C211yesmaximal, characteristic, complemented
13C4×#10587480C211yesmaximal, characteristic, complemented
14#14528568C211yesmaximal, characteristic, complemented
15C17×#2345C211yesmaximal, characteristic, complemented
16C22×#63252C2211yescharacteristic
17C22×#63223C2211yescharacteristic
18C2×C4×D68C2211yescharacteristic
19C2×C4×D68C2211yescharacteristic
20C2×#10587478C2211yescharacteristic
21C2×#10587479C2211yescharacteristic
22#11762808C418yescomplemented
23C2×#10587481C2211yescharacteristic
24C2×#10587481C2211yescharacteristic
25C2×#10587481C2211yescharacteristic
26C2×#10587481C2211yescharacteristic
27C2×#10587480C2211yescharacteristic
28C2×#10587480C2211yescharacteristic
29C2×#10587480C2211yescharacteristic
30C2×#10587480C2211yescharacteristic
31C2×C22⋊#3622C2211yescharacteristic
32C2×#10587474C2211yescharacteristic
33C2×C4×#3622C2211yescharacteristic
34#11762715C2211yescharacteristic
35#11762715C2211yescharacteristic
36#11762685—21no—
37#11762685—21no—
38#11762671—21no—
39#11762669C2211yescharacteristic
40#95×D17—21no—
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762564C2211yescharacteristic
48#11762564C2211yescharacteristic
49C22×C4⋊C4×C17C2211yescharacteristic
50C17×#389C2211yescharacteristic
51C17×#389C2211yescharacteristic
52C17×#387—21no—
53C17×#387—21no—
54C17×#380—21no—
55C17×#380C2211yescharacteristic
56C4×C4⋊C4×C17C2211yescharacteristic
57C4×C4⋊C4×C17C2211yescharacteristic
58C4×C22⋊C4×C17—21no—
59C4×C22⋊C4×C17C2211yescharacteristic
60C23×C4×C17—21no—
61C23×C4×C17C2311yescharacteristic
62C22×C22⋊D17—21no—
63C2×#63252—22no—
64C2×#63252—22no—
65C2×#63252—22no—
66C2×#63252—41no—
67C2×#63252—41no—
68C2×#63252C2×C414yes—
69C22×D68C2311yescharacteristic
70C22×C4×D17—21no—
71C22×C4×D17—21no—
72C22×C4×D17—21no—
73C22×Dic34C2311yescharacteristic
74C2×C4⋊C4×C17—21no—
75C2×C4⋊C4×C17—22no—
76C2×C4⋊C4×C17—21no—
77C2×C4⋊C4×C17—21no—
78C2×C4⋊C4×C17C2×C414yescomplemented
79C2×C4⋊C4×C17C2311yescharacteristic
80C2×C4⋊C4×C17C2311yescharacteristic
81C2×C4⋊C4×C17C2311yescharacteristic
82C2×C4⋊C4×C17C2311yescharacteristic
83C2×C22⋊C4×C17—21no—
84C2×C22⋊C4×C17D411yescharacteristic
85C2×C22⋊C4×C17D411yescharacteristic
86C2×C42×C17—21no—
87C2×C42×C17—21no—
88C2×C42×C17C2311yescharacteristic
89C2×C42×C17C2311yescharacteristic
90C2×C22⋊Dic17—21no—
91C2×#63227—21no—
92C2×#63227—21no—
93C2×C4⋊Dic17C2311yescharacteristic
94C2×C4⋊Dic17C2311yescharacteristic
95C2×#63223—41no—
96C2×#63223—41no—
97C2×#63223C2×C414yescomplemented
98C22×#3622C2311yescharacteristic
99C22×#3622C2311yescharacteristic
100C22×#3622C2311yescharacteristic
101C22×#3622C2311yescharacteristic
102C4×D68—22no—
103C4×D68—22no—
104C4×D68—41no—
105C4×D68—41no—
106C17×#95—21no—
107C17×#95—21no—
108C17×#95—21no—
109C17×#95—21no—
110C17×#95—21no—
111C17×#95D411yescharacteristic
112C17×#95D411yescharacteristic
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118C22⋊#3622—22no—
119#10587481—41no—
120#10587481—41no—
121#10587481C2×C414yescomplemented
122#10587481C2×C414yescomplemented
123#10587480—22no—
124#10587480—22no—
125#10587480C2×C414yes—
126#10587480C2×C414yes—
127#10587479—41no—
128#10587478—41no—
129#10587474—41no—
130C4×#3622—41no—
131C24×C17C2×D411yescharacteristic
132C23×D17—21no—
133C22×C4×C17—21no—
134C22×C4×C17—21no—
135C22×C4×C17—21no—
136C22×C4×C17—21no—
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17—24no—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17—81no—
143C22×C4×C17—81no—
144C22×C4×C17C22×C412yes—
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17#4112yes—
151C22×C4×C17C2411yescharacteristic
152C2×C22⋊D17—24no—
153C2×C22⋊D17—42no—
154C2×C22⋊D17—42no—
155C22×Dic17—21no—
156C22×Dic17—21no—
157C22×Dic17—21no—
158#63252—28no—
159#63252—44no—
160#63252—44no—
161#63252—44no—
162C2×D68—21no—
163C2×D68—21no—
164C2×D68—21no—
165C2×D68C22×C412yes—
166C2×D68#4112yes—
167C2×D68#4112yes—
168C2×C4×D17—24no—
169C2×C4×D17—42no—
170C2×C4×D17—42no—
171C2×C4×D17—42no—
172C2×C4×D17—42no—
173C2×C4×D17—42no—
174C2×C4×D17—42no—
175C2×C4×D17—81no—
176C2×C4×D17—81no—
177C2×Dic34—21no—
178C2×Dic34—21no—
179C2×Dic34—21no—
180C2×Dic34—21no—
181C2×Dic34—21no—
182C2×Dic34C22×C412yes—
183C4⋊C4×C17—24no—
184C4⋊C4×C17—21no—
185C4⋊C4×C17—21no—
186C4⋊C4×C17—21no—
187C4⋊C4×C17—21no—
188C4⋊C4×C17—21no—
189C4⋊C4×C17—21no—
190C4⋊C4×C17—81no—
191C4⋊C4×C17—81no—
192C4⋊C4×C17C22×C412yes—
193C4⋊C4×C17C22×C412yes—
194C22⋊C4×C17—22no—
195C22⋊C4×C17—41no—
196C22⋊C4×C17—42no—
197C42×C17—21no—
198C42×C17—21no—
199C42×C17—21no—
200C42×C17—21no—
201C42×C17—81no—
202C42×C17—81no—
203C22⋊Dic17—81no—
204#63227—42no—
205#63227—42no—
206C4⋊Dic17—21no—
207C4⋊Dic17—21no—
208C4⋊Dic17—21no—
209C4⋊Dic17—21no—
210#63223—44no—
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#22578—171nomaximal, Sylow
224C23×C17—21no—
225C23×C17—21no—
226C23×C17—21no—
227C23×C17—21no—
228C23×C17—21no—
229C23×C17—21no—
230C23×C17C4×D412yes—
231C23×C17C22×D411yescharacteristic
232C22×D17—22no—
233C22×D17—22no—
234C22×D17—22no—
235C22×D17—41no—
236C22×D17—41no—
237C22×D17—41no—
238C22×D17—41no—
239C2×C4×C17—21no—
240C2×C4×C17—21no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—22no—
244C2×C4×C17—22no—
245C2×C4×C17—22no—
246C2×C4×C17—22no—
247C2×C4×C17—22no—
248C2×C4×C17—22no—
249C2×C4×C17—21no—
250C2×C4×C17—21no—
251C2×C4×C17—21no—
252C2×C4×C17—21no—
253C2×C4×C17—21no—
254C2×C4×C17—21no—
255C2×C4×C17—21no—
256C2×C4×C17—21no—
257C2×C4×C17—21no—
258C2×C4×C17—41no—
259C2×C4×C17—41no—
260C2×C4×C17—41no—
261C2×C4×C17—41no—
262C2×C4×C17—41no—
263C2×C4×C17—41no—
264C2×C4×C17—41no—
265C2×C4×C17—41no—
266C2×C4×C17—41no—
267C2×C4×C17—41no—
268C2×C4×C17—84no—
269C2×C4×C17C4×D412yes—
270C2×C4×C17#13712yes—
271C2×C4×C17C23×C411yescharacteristic
272C2×C4×C17C2×#4111yescharacteristic
273C2×C4×C17C2×#4111yescharacteristic
274C2×C4×C17C2×#4111yescharacteristic
275C2×C4×C17H(5,𝔽2)11yescharacteristic
276C2×C4×C17H(5,𝔽2)11yescharacteristic
277C22⋊D17—48no—
278C2×Dic17—22no—
279C2×Dic17—41no—
280C2×Dic17—41no—
281C2×Dic17—41no—
282C2×Dic17—41no—
283C2×Dic17—41no—
284C2×Dic17—41no—
285C2×Dic17—41no—
286C2×Dic17—41no—
287D68—22no—
288D68—22no—
289D68—22no—
290D68#11714yes—
291C4×D17—48no—
292C4×D17—84no—
293C4×D17—84no—
294Dic34—22no—
295Dic34—22no—
296Dic34—41no—
297Dic34—41no—
298#3622—22no—
299#3622—22no—
300#3622—41no—
301#3622—41no—
302C2×#417—171no—
303#2345D1711yescharacteristic, complemented
304#2339—171no—
305#1935—171no—
306#1933—171no—
307#1928—171no—
308#1927—171no—
309#1864—171no—
310#1862—171no—
311#1823—171no—
312#1809—171no—
313#1788—171no—
314#1762—171no—
315C4×#102—171no—
316#1753—171no—
317C22×C17—22no—
318C22×C17—22no—
319C22×C17—22no—
320C22×C17—41no—
321C22×C17—41no—
322C22×C17—41no—
323C22×C17—41no—
324C22×C17C2×C4×D411yescharacteristic
325C22×C17#54611yescharacteristic
326C22×C17#54611yescharacteristic
327C22×C17C2×#13711yescharacteristic
328C22×C17#57511yescharacteristic
329C22×C17#57611yescharacteristic
330C22×C17#57711yescharacteristic
331C2×D17—24no—
332C2×D17—42no—
333C2×D17—42no—
334C2×D17—42no—
335C2×D17—42no—
336C2×D17—42no—
337C2×D17—42no—
338C4×C17—22no—
339C4×C17—22no—
340C4×C17—41no—
341C4×C17—42no—
342C4×C17—42no—
343C4×C17—41no—
344C4×C17—41no—
345C4×C17—81no—
346C4×C17—81no—
347C4×C17C2×#11711yescharacteristic
348C4×C17#51811yescharacteristic
349C4×C17#55111yescharacteristic
350C4×C17#55111yescharacteristic
351Dic17—42no—
352Dic17—81no—
353Dic17—81no—
354C22×#41—171no—
355C22×M16—171no—
356C2×C4×D4—171no—
357C2×C4×D4—171no—
358C22×C4⋊C4C2×D1711yescharacteristic
359C2×#107—171no—
360C2×#106—171no—
361#417—178no—
362C2×#103—171no—
363C2×#103—171no—
364C2×#103—171no—
365C2×#103—171no—
366C2×#102—171no—
367C2×#102—171no—
368C2×#102—171no—
369C2×#102—171no—
370C2×C22⋊C8—171no—
371C2×#97—171no—
372C2×C4×C8—171no—
373#391—171no—
374#391—171no—
375#389C2×D1711yescharacteristic
376#389C2×D1711yescharacteristic
377#387—21no—
378#387—21no—
379#386—341no—
380#386—341no—
381#381—341no—
382#380—21no—
383#380—171no—
384#380C2×D1711yescharacteristic
385C4×C4⋊C4C2×D1711yescharacteristic
386C4×C4⋊C4C2×D1711yescharacteristic
387C4×C22⋊C4—21no—
388C4×C22⋊C4C2×D1711yescharacteristic
389C2×#95—341no—
390#340—171no—
391#340—171no—
392#340—171no—
393#340—171no—
394#340—171no—
395#340—171no—
396#336—171no—
397#336—171no—
398C2×C17—42no—
399C2×C17#234211yescharacteristic
400C2×C17C4×#13711yescharacteristic
401C2×C17#294611yescharacteristic
402C2×C17#330711yescharacteristic
403C2×C17#331411yescharacteristic
404C2×C17#332211yescharacteristic
405C2×C17#333111yescharacteristic
406D17—44no—
407C2×#41—174no—
408C2×#41—342no—
409C2×#41—342no—
410C2×#41—342no—
411C2×#41—681no—
412C2×#41—681no—
413C22×Q8—171no—
414C22×D4—171no—
415C22×D4—341no—
416C23×C4—21no—
417C23×C4—341no—
418C23×C4—341no—
419C23×C4—341no—
420C23×C4C22×D1711yescharacteristic
421C2×M16—174no—
422C2×M16—681no—
423C2×M16—681no—
424C22×C8—171no—
425C22×C8—171no—
426C22×C8—171no—
427C22×C8—171no—
428C4×D4—342no—
429C4×D4—342no—
430C4×D4—681no—
431C4×D4—681no—
432C2×C4⋊C4—22no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—171no—
437C2×C4⋊C4—171no—
438C2×C4⋊C4C4×D1714yescomplemented
439C2×C4⋊C4C22×D1711yescharacteristic
440C2×C4⋊C4C22×D1711yescharacteristic
441C2×C4⋊C4C22×D1711yescharacteristic
442C2×C4⋊C4C22×D1711yescharacteristic
443C2×C22⋊C4—21no—
444C2×C22⋊C4—341no—
445C2×C22⋊C4—341no—
446C2×C22⋊C4—341no—
447C2×C22⋊C4C22⋊D1711yescharacteristic
448C2×C22⋊C4C22⋊D1711yescharacteristic
449C2×C42—21no—
450C2×C42—21no—
451C2×C42C22×D1711yescharacteristic
452C2×C42C22×D1711yescharacteristic
453#107—681no—
454#106—681no—
455#103—174no—
456#103—174no—
457#103—681no—
458#103—681no—
459#102—174no—
460#102—174no—
461#102—342no—
462#102—342no—
463C22⋊C8—342no—
464#97—681no—
465C4×C8—681no—
466#95—21no—
467#95—21no—
468#95—21no—
469#95—21no—
470#95—21no—
471#95—341no—
472#95—341no—
473#95—341no—
474#95—341no—
475#95—341no—
476#95C22⋊D1711yescharacteristic
477#95C22⋊D1711yescharacteristic
478C17#2257811yescharacteristic, complemented, Sylow
479C24—341no—
480C24C2×C22⋊D1711yescharacteristic
481#41—348no—
482#41—684no—
483#41—684no—
484#41—684no—
485C2×Q8—172no—
486C2×Q8—341no—
487C2×Q8—341no—
488C2×Q8—341no—
489C2×Q8—341no—
490C2×Q8—341no—
491C2×D4—172no—
492C2×D4—172no—
493C2×D4—172no—
494C2×D4—344no—
495C2×D4—341no—
496C2×D4—341no—
497C2×D4—341no—
498C2×D4—682no—
499C2×D4—682no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—24no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—81no—
510C22×C4—81no—
511C22×C4—341no—
512C22×C4—344no—
513C22×C4—341no—
514C22×C4—341no—
515C22×C4—682no—
516C22×C4—682no—
517C22×C4—682no—
518C22×C4—682no—
519C22×C4—682no—
520C22×C4—682no—
521C22×C4—1361no—
522C22×C4—1361no—
523C22×C4C2×C4×D1712yes—
524C22×C4#6325212yes—
525C22×C4C2×C22⋊D1711yescharacteristic
526C22×C4C2×C22⋊D1711yescharacteristic
527C22×C4C2×C22⋊D1711yescharacteristic
528C22×C4C2×C22⋊D1711yescharacteristic
529C22×C4C2×C22⋊D1711yescharacteristic
530C22×C4C23×D1711yescharacteristic
531M16—684no—
532C2×C8—172no—
533C2×C8—172no—
534C2×C8—341no—
535C2×C8—341no—
536C2×C8—341no—
537C2×C8—341no—
538C2×C8—341no—
539C2×C8—341no—
540C2×C8—341no—
541C2×C8—341no—
542C2×C8—341no—
543C2×C8—341no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—24no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—81no—
552C4⋊C4—81no—
553C4⋊C4—341no—
554C4⋊C4—341no—
555C4⋊C4—341no—
556C4⋊C4—341no—
557C4⋊C4C2×C4×D1712yes—
558C4⋊C4C2×C4×D1712yes—
559C22⋊C4—22no—
560C22⋊C4—41no—
561C22⋊C4—42no—
562C22⋊C4—682no—
563C22⋊C4—682no—
564C22⋊C4—1361no—
565C42—21no—
566C42—21no—
567C42—21no—
568C42—21no—
569C42—81no—
570C42—81no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—342no—
578C23—342no—
579C23—342no—
580C23—681no—
581C23—681no—
582C23—681no—
583C23—681no—
584C23C4×C22⋊D1712yes—
585C23C22×C22⋊D1711yescharacteristic
586Q8—342no—
587Q8—342no—
588Q8—681no—
589Q8—681no—
590D4—174no—
591D4—342no—
592D4—342no—
593D4—342no—
594D4—688no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—84no—
625C2×C4—342no—
626C2×C4—688no—
627C2×C4—681no—
628C2×C4—681no—
629C2×C4—681no—
630C2×C4—681no—
631C2×C4—681no—
632C2×C4—681no—
633C2×C4—681no—
634C2×C4—681no—
635C2×C4—1364no—
636C2×C4—1364no—
637C2×C4C4×C22⋊D1712yes—
638C2×C4#1058762812yes—
639C2×C4C22×C4×D1711yescharacteristic
640C2×C4C2×#6325211yescharacteristic
641C2×C4#41×D1711yescharacteristic
642C2×C4#41×D1711yescharacteristic
643C2×C4#1058769611yescharacteristic
644C2×C4#1058769611yescharacteristic
645C8—342no—
646C8—342no—
647C8—681no—
648C8—681no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—41no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—344no—
657C22—682no—
658C22—682no—
659C22—682no—
660C22—682no—
661C22—682no—
662C22—682no—
663C22C2×C4×C22⋊D1711yescharacteristic
664C22#1176344711yescharacteristic
665C22#1176344711yescharacteristic
666C22C2×#1058762811yescharacteristic
667C22#1176360711yescharacteristic
668C22#1176361211yescharacteristic
669C22#1176362011yescharacteristic
670C4—22no—
671C4—22no—
672C4—41no—
673C4—42no—
674C4—42no—
675C4—41no—
676C4—41no—
677C4—81no—
678C4—81no—
679C4—682no—
680C4—1361no—
681C4—1361no—
682C4#117×D1711yescharacteristic
683C4#1176328311yescharacteristic
684C4#1176348311yescharacteristic
685C4#1176348311yescharacteristic
686C2—42no—
687C2—684no—
688C2#1453163411yescharacteristic
689C2C4×#1058762811yescharacteristic
690C2#1453761611yescharacteristic
691C2#1454075411yescharacteristic
692C2#1454085911yescharacteristic
693C2#1454097911yescharacteristic
694C2#1454109411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2345C2
socleC23×C17C22×D4
prime radicalC22×#63252C22
cosocleC22×C4×C17C24
hypercenter#2345D17
hypocenterC17#22578
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2345

Complements

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

NH ≅ G/Ntype
3. #14529473C2semidirect
4. #14529459C2semidirect
5. #14529425C2semidirect
6. #14529418C2semidirect
7. #14529044C2semidirect
8. #14529030C2semidirect
9. #14528870C2semidirect
10. #14528815C2semidirect
12. #14528612C2semidirect
13. C4×#10587480C2semidirect
14. #14528568C2semidirect
15. C17×#2345C2semidirect
22. #11762808C4semidirect
78. C2×C4⋊C4×C17C2×C4semidirect
97. C2×#63223C2×C4semidirect
121. #10587481C2×C4semidirect
122. #10587481C2×C4semidirect
303. #2345D17semidirect
438. C2×C4⋊C4C4×D17semidirect
478. C17#22578semidirect

Maximal subgroups

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

subgroupindex
C2×#117628082
#145316152
#145294732
#145294592
#145294252
#145294182
#145290442
#145290302
#145288702
#145288152
#145287222
#145286122
C4×#105874802
#145285682
C17×#23452
#2257817