Finite Group Databasebeta

← #17052682

Subgroups — #17052682

Order 4352.

982 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762912C211yesmaximal, characteristic
2C2×#11762907C211yesmaximal, characteristic
3C2×#11762894C211yesmaximal, characteristic
4Q8×C4⋊Dic17C211yesmaximal, characteristic, complemented
5#14531692C211yesmaximal, characteristic
6#14529521C211yesmaximal, characteristic, complemented
7#14529516C211yesmaximal, characteristic, complemented
8#14529050C211yesmaximal, characteristic, complemented
9#14529046C211yesmaximal, characteristic
10#14529036C211yesmaximal, characteristic, complemented
11#14529032C211yesmaximal, characteristic
12#14528953C211yesmaximal, characteristic, complemented
13#14528785C211yesmaximal, characteristic
14#14528661C211yesmaximal, characteristic, complemented
15C17×#1805C211yesmaximal, characteristic, complemented
16C22×#63230C2211yescharacteristic
17C2×Q8×Dic17C2211yescharacteristic
18C2×Q8×Dic17C2211yescharacteristic
19C2×C4×D68C2211yescharacteristic
20C2×#10587579C2211yescharacteristic
21C2×#10587579C2211yescharacteristic
22#11762912—24no—
23#11762907C418yescomplemented
24#11762894—24no—
25C2×#10587478C2211yescharacteristic
26C2×#10587478C2211yescharacteristic
27C2×#10587478C2211yescharacteristic
28C2×#10587479C2211yescharacteristic
29C2×#10587479C2211yescharacteristic
30C2×#10587506C2211yescharacteristic
31C2×#10587492C2211yescharacteristic
32C2×#10587492C2211yescharacteristic
33C2×#10587480C2211yescharacteristic
34C2×#10587474C2211yescharacteristic
35C2×#10587486C2211yescharacteristic
36C2×C8×Dic17C2211yescharacteristic
37#11762708—21no—
38#11762708—21no—
39#11762708C2211yescharacteristic
40#11762708C2211yescharacteristic
41#11762705C2211yescharacteristic
42#11762705C2211yescharacteristic
43#11762683—21no—
44#11762683—21no—
45#11762679—21no—
46#11762679C2211yescharacteristic
47#11762668—21no—
48C4×C4⋊Dic17—21no—
49C4×C4⋊Dic17C2211yescharacteristic
50#95×D17—21no—
51#11762572C2211yescharacteristic
52#11762572C2211yescharacteristic
53#11762563C2211yescharacteristic
54#11762563C2211yescharacteristic
55C2×C4×Q8×C17C2211yescharacteristic
56C2×C17×C4⋊C8C2211yescharacteristic
57C2×C17×#103C2211yescharacteristic
58C2×C17×#103C2211yescharacteristic
59C17×#382C2211yescharacteristic
60C17×#340C2211yescharacteristic
61C17×#340C2211yescharacteristic
62C22×Q8×C17C2311yescharacteristic
63C22×D68C2311yescharacteristic
64C22×C4×D17—21no—
65C22×C4×D17—21no—
66C22×C4×D17—21no—
67C22×C8×C17C2311yescharacteristic
68C22×C8×C17C2311yescharacteristic
69C4×Q8×C17—22no—
70C4×Q8×C17C2×C414yescomplemented
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C42×C17—21no—
76C2×C42×C17C2311yescharacteristic
77Q8×Dic17—22no—
78Q8×Dic17—22no—
79Q8×Dic17—41no—
80Q8×Dic17—41no—
81C2×#63230C2×C414yes—
82C2×#63230D414yes—
83C2×#63230Q814yes—
84C2×#63227—21no—
85C2×#63227—21no—
86C2×#63227—21no—
87C2×C4⋊Dic17—21no—
88C2×C4⋊Dic17—21no—
89C2×C4⋊Dic17—21no—
90C2×C4⋊Dic17—21no—
91C2×C4⋊Dic17—21no—
92C2×C4⋊Dic17—21no—
93C2×C4⋊Dic17D411yescharacteristic
94C2×C4⋊Dic17D411yescharacteristic
95C2×C4⋊Dic17C2311yescharacteristic
96C2×C4⋊Dic17C2311yescharacteristic
97C2×C4⋊Dic17C2311yescharacteristic
98C2×#63225—21no—
99C2×#63225—21no—
100C2×C4×Dic17—21no—
101C2×C4×Dic17—21no—
102C2×C4×Dic17C2311yescharacteristic
103C2×C4×Dic17C2311yescharacteristic
104C22×#3622C2311yescharacteristic
105C22×#3622C2311yescharacteristic
106#10587579—22no—
107#10587579—22no—
108#10587579—41no—
109#10587579—41no—
110C4×D68—22no—
111C4×D68C2×C414yes—
112C17×C4⋊C8—22no—
113C17×#103—22no—
114C17×#103—22no—
115C17×#95—21no—
116C17×#95—21no—
117#10587506C2×C414yescomplemented
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502D411yescharacteristic
123#10587502D411yescharacteristic
124#10587492—22no—
125#10587492—22no—
126#10587486—22no—
127C8×Dic17—22no—
128#10587480C2×C414yes—
129#10587479—22no—
130#10587479—41no—
131#10587478—22no—
132#10587478—41no—
133#10587478C2×C414yescomplemented
134#10587474C2×C414yescomplemented
135C23×D17—21no—
136C2×Q8×C17—21no—
137C2×Q8×C17—21no—
138C2×Q8×C17—21no—
139C2×Q8×C17C22×C412yes—
140C2×Q8×C17C2×D412yes—
141C2×Q8×C17C2×Q812yes—
142C22×C4×C17—21no—
143C22×C4×C17—21no—
144C22×C4×C17—21no—
145C22×C4×C17—21no—
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2411yescharacteristic
149C22×Dic17—21no—
150C22×Dic17—21no—
151C22×Dic17—21no—
152C22×Dic17C2×D411yescharacteristic
153C22×Dic17C2×D411yescharacteristic
154C22×Dic17C2×D411yescharacteristic
155C22×Dic17C2×D411yescharacteristic
156C2×D68—21no—
157C2×D68—21no—
158C2×D68—21no—
159C2×D68C22×C412yes—
160C2×D68C2×D412yes—
161C2×D68C2×Q812yes—
162C2×C4×D17—24no—
163C2×C4×D17—42no—
164C2×C4×D17—42no—
165C2×C4×D17—42no—
166C2×C4×D17—42no—
167C2×C4×D17—81no—
168C2×C4×D17—81no—
169C2×C4×D17—81no—
170C2×C4×D17—81no—
171C2×C8×C17—21no—
172C2×C8×C17—21no—
173C2×C8×C17—21no—
174C2×C8×C17—21no—
175C2×C8×C17—21no—
176C2×C8×C17—21no—
177C4⋊C4×C17—24no—
178C4⋊C4×C17—21no—
179C4⋊C4×C17—21no—
180C4⋊C4×C17—21no—
181C4⋊C4×C17—21no—
182C4⋊C4×C17—21no—
183C4⋊C4×C17C22×C412yes—
184C42×C17—24no—
185C42×C17C22×C412yes—
186C42×C17#4112yes—
187#63230C4⋊C4116yescomplemented
188#63227—24no—
189#63227—42no—
190#63227—42no—
191C4⋊Dic17—21no—
192C4⋊Dic17—22no—
193C4⋊Dic17—21no—
194C4⋊Dic17—21no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—41no—
198C4⋊Dic17—42no—
199C4⋊Dic17—42no—
200C4⋊Dic17—42no—
201C4⋊Dic17—81no—
202C4⋊Dic17—81no—
203C4⋊Dic17—81no—
204C4⋊Dic17C22×C412yes—
205#63225—81no—
206#63225—81no—
207C4×Dic17—21no—
208C4×Dic17—21no—
209C4×Dic17—21no—
210C4×Dic17—21no—
211C4×Dic17—42no—
212C4×Dic17—42no—
213C2×#3622—21no—
214C2×#3622—21no—
215C2×#3622C22×C412yes—
216C2×#3622C22×C412yes—
217C2×#3622C2×D412yes—
218C2×#3622C2×Q812yes—
219#22678—171nomaximal, Sylow
220C23×C17C22×D411yescharacteristic
221C22×D17—22no—
222C22×D17—22no—
223C22×D17—22no—
224C22×D17—41no—
225C22×D17—41no—
226C22×D17—41no—
227C22×D17—41no—
228Q8×C17—22no—
229Q8×C17—22no—
230Q8×C17—22no—
231Q8×C17C2×C4⋊C414yes—
232C2×C4×C17—22no—
233C2×C4×C17—22no—
234C2×C4×C17—22no—
235C2×C4×C17—21no—
236C2×C4×C17—21no—
237C2×C4×C17—21no—
238C2×C4×C17—21no—
239C2×C4×C17—22no—
240C2×C4×C17—41no—
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×C17—41no—
248C2×C4×C17C4×D412yes—
249C2×C4×C17C4×D412yes—
250C2×C4×C17C23×C411yescharacteristic
251C2×C4×C17C22×D411yescharacteristic
252C2×C4×C17C22×Q811yescharacteristic
253C2×C4×C17C2×#4111yescharacteristic
254C2×C4×C17H(5,𝔽2)11yescharacteristic
255C2×C4×C172−1+411yescharacteristic
256C2×Dic17—21no—
257C2×Dic17—21no—
258C2×Dic17—22no—
259C2×Dic17—21no—
260C2×Dic17—21no—
261C2×Dic17—21no—
262C2×Dic17—21no—
263C2×Dic17—21no—
264C2×Dic17—21no—
265C2×Dic17—21no—
266C2×Dic17—21no—
267C2×Dic17—21no—
268C2×Dic17—41no—
269C2×Dic17—41no—
270C2×Dic17—41no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273C2×Dic17—41no—
274C2×Dic17—41no—
275C2×Dic17—41no—
276C2×Dic17#13712yes—
277D68—22no—
278D68—22no—
279D68—22no—
280D68C2×C4⋊C414yes—
281C4×D17—84no—
282C4×D17—84no—
283C4×D17—84no—
284C8×C17—41no—
285C8×C17—41no—
286#3622—22no—
287#3622C2×C4⋊C414yes—
288C2×#440—171no—
289C2×#440—171no—
290C2×C4×SD16—171no—
291Q8×C4⋊C4—171no—
292#2349—171no—
293#1943—171no—
294#1942—171no—
295#1865—171no—
296#1864—171no—
297#1863—171no—
298#1862—171no—
299#1847—171no—
300#1805D1711yescharacteristic, complemented
301#1802—171no—
302#1771—171no—
303C22×C17C2×C4×D411yescharacteristic
304C22×C17D4211yescharacteristic
305C22×C17D4×Q811yescharacteristic
306C22×C17C2×#13711yescharacteristic
307C22×C17#57511yescharacteristic
308C22×C17#57611yescharacteristic
309C22×C17#57711yescharacteristic
310C2×D17—24no—
311C2×D17—42no—
312C2×D17—42no—
313C2×D17—42no—
314C2×D17—42no—
315C2×D17—42no—
316C2×D17—42no—
317C4×C17—24no—
318C4×C17—22no—
319C4×C17—22no—
320C4×C17—42no—
321C4×C17—81no—
322C4×C17—81no—
323C4×C17C22×C4⋊C411yescharacteristic
324C4×C17#52011yescharacteristic
325C4×C17#54711yescharacteristic
326C4×C17#55411yescharacteristic
327Dic17—22no—
328Dic17—42no—
329Dic17—41no—
330Dic17—41no—
331Dic17—41no—
332Dic17—81no—
333Dic17—81no—
334C22×SD16—171no—
335C2×C4×Q8—171no—
336C2×C4×Q8—171no—
337C2×C4×Q8C2×D1711yescharacteristic
338C2×C4×D4—171no—
339C2×C4×D4—171no—
340C2×C4×D4—171no—
341#440—178no—
342#440—344no—
343C4×SD16—344no—
344C2×#107—171no—
345C2×#107—171no—
346C2×#107—171no—
347C2×#106—171no—
348C2×#106—171no—
349C2×C4⋊C8C2×D1711yescharacteristic
350C2×#103—171no—
351C2×#103C2×D1711yescharacteristic
352C2×#103C2×D1711yescharacteristic
353C2×#102—171no—
354C2×#102—171no—
355C2×#102—171no—
356C2×#97—171no—
357C2×#97—171no—
358C2×C4×C8—171no—
359#389—171no—
360#389—171no—
361#389—171no—
362#389—171no—
363#389—341no—
364#389—341no—
365#385—341no—
366#385—341no—
367#384—171no—
368#384—341no—
369#382C2×D1711yescharacteristic
370#380—341no—
371C4×C4⋊C4—171no—
372C4×C4⋊C4—341no—
373C2×#95—341no—
374#340—171no—
375#340—171no—
376#340C2×D1711yescharacteristic
377#340C2×D1711yescharacteristic
378#336—171no—
379#336—171no—
380C2×C17D4×C4⋊C411yescharacteristic
381C2×C17C4×#13711yescharacteristic
382C2×C17#294611yescharacteristic
383C2×C17#326611yescharacteristic
384C2×C17#326811yescharacteristic
385C2×C17#338511yescharacteristic
386C2×C17#338811yescharacteristic
387D17—44no—
388C22×Q8C22×D1711yescharacteristic
389C22×D4—171no—
390C23×C4—341no—
391C23×C4—341no—
392C23×C4—341no—
393C2×SD16—174no—
394C2×SD16—174no—
395C2×SD16—174no—
396C22×C8—171no—
397C22×C8—171no—
398C22×C8C22×D1711yescharacteristic
399C22×C8C22×D1711yescharacteristic
400C4×Q8—22no—
401C4×Q8—342no—
402C4×Q8—342no—
403C4×Q8—681no—
404C4×Q8—681no—
405C4×Q8C4×D1714yescomplemented
406C4×D4—174no—
407C4×D4—342no—
408C4×D4—342no—
409C4×D4—342no—
410C4×D4—681no—
411C4×D4—681no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—171no—
414C2×C4⋊C4—171no—
415C2×C4⋊C4—171no—
416C2×C4⋊C4—171no—
417C2×C4⋊C4—171no—
418C2×C4⋊C4—341no—
419C2×C4⋊C4—341no—
420C2×C4⋊C4—341no—
421C2×C4⋊C4—341no—
422C2×C4⋊C4—341no—
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×C42—21no—
433C2×C42—171no—
434C2×C42—171no—
435C2×C42—341no—
436C2×C42—341no—
437C2×C42C22×D1711yescharacteristic
438#107—174no—
439#107—342no—
440#107—681no—
441#106—342no—
442#106—681no—
443C4⋊C8—22no—
444#103—22no—
445#103—22no—
446#103—174no—
447#102—174no—
448#102—342no—
449#102—342no—
450#97—174no—
451#97—342no—
452C4×C8—342no—
453#95—21no—
454#95—21no—
455#95—171no—
456#95—171no—
457#95—341no—
458#95—341no—
459#95—341no—
460#95—341no—
461C17#2267811yescharacteristic, complemented, Sylow
462C24—341no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8C2×C4×D1712yes—
467C2×Q8D4×D1712yes—
468C2×Q8Q8×D1712yes—
469C2×D4—172no—
470C2×D4—172no—
471C2×D4—172no—
472C2×D4—341no—
473C2×D4—341no—
474C2×D4—341no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—171no—
480C22×C4—171no—
481C22×C4—171no—
482C22×C4—171no—
483C22×C4—341no—
484C22×C4—344no—
485C22×C4—341no—
486C22×C4—341no—
487C22×C4—682no—
488C22×C4—682no—
489C22×C4—682no—
490C22×C4—682no—
491C22×C4—1361no—
492C22×C4—1361no—
493C22×C4—1361no—
494C22×C4—1361no—
495C22×C4D4×D1711yescharacteristic
496C22×C4D4×D1711yescharacteristic
497C22×C4C23×D1711yescharacteristic
498SD16—1716no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—172no—
506C2×C8—172no—
507C2×C8—172no—
508C2×C8—172no—
509C2×C8—341no—
510C2×C8—341no—
511C4⋊C4—21no—
512C4⋊C4—24no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—172no—
518C4⋊C4—342no—
519C4⋊C4—341no—
520C4⋊C4—341no—
521C4⋊C4—341no—
522C4⋊C4—341no—
523C4⋊C4—341no—
524C4⋊C4—681no—
525C4⋊C4—682no—
526C4⋊C4—682no—
527C4⋊C4—682no—
528C4⋊C4—1361no—
529C4⋊C4—1361no—
530C4⋊C4—1361no—
531C4⋊C4—1361no—
532C4⋊C4—1361no—
533C4⋊C4C2×C4×D1712yes—
534C22⋊C4—344no—
535C22⋊C4—682no—
536C22⋊C4—682no—
537C42—24no—
538C42—341no—
539C42—341no—
540C42—341no—
541C42—341no—
542C42—682no—
543C42—682no—
544C42C2×C4×D1712yes—
545C42#6325412yes—
546C23—342no—
547C23—342no—
548C23—342no—
549C23—681no—
550C23—681no—
551C23—681no—
552C23—681no—
553C23C2×D4×D1711yescharacteristic
554Q8—22no—
555Q8—22no—
556Q8—22no—
557Q8C4⋊C4×D1714yes—
558D4—174no—
559D4—342no—
560D4—342no—
561D4—342no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—172no—
579C2×C4—341no—
580C2×C4—341no—
581C2×C4—342no—
582C2×C4—341no—
583C2×C4—341no—
584C2×C4—341no—
585C2×C4—341no—
586C2×C4—341no—
587C2×C4—341no—
588C2×C4—341no—
589C2×C4—341no—
590C2×C4—341no—
591C2×C4—681no—
592C2×C4—681no—
593C2×C4—681no—
594C2×C4—681no—
595C2×C4—681no—
596C2×C4—681no—
597C2×C4—681no—
598C2×C4—681no—
599C2×C4—1364no—
600C2×C4—1364no—
601C2×C4—1364no—
602C2×C4#1058756712yes—
603C2×C4#1058756712yes—
604C2×C4C22×C4×D1711yescharacteristic
605C2×C4C2×D4×D1711yescharacteristic
606C2×C4C2×#6325411yescharacteristic
607C2×C4C2×Q8×D1711yescharacteristic
608C2×C4#1058769611yescharacteristic
609C2×C4#1058769711yescharacteristic
610C8—41no—
611C8—41no—
612C8—174no—
613C8—342no—
614C22—344no—
615C22—682no—
616C22—682no—
617C22—682no—
618C22—682no—
619C22—682no—
620C22—682no—
621C22C2×#1058756711yescharacteristic
622C22#1176344311yescharacteristic
623C22#1176347211yescharacteristic
624C22#137×D1711yescharacteristic
625C22#1176360811yescharacteristic
626C22#1176361111yescharacteristic
627C22#1176361611yescharacteristic
628C4—24no—
629C4—22no—
630C4—22no—
631C4—42no—
632C4—81no—
633C4—81no—
634C4—342no—
635C4—681no—
636C4—681no—
637C4—681no—
638C4—682no—
639C4—1361no—
640C4—1361no—
641C4C2×C4⋊C4×D1711yescharacteristic
642C4#1176328911yescharacteristic
643C4#1176345911yescharacteristic
644C4#1176351711yescharacteristic
645C2—684no—
646C2#1453168011yescharacteristic
647C2#1453759911yescharacteristic
648C2#1453761911yescharacteristic
649C2#1454028911yescharacteristic
650C2#1454031911yescharacteristic
651C2#1454169711yescharacteristic
652C2#1454173211yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1805C2
socleC23×C17C22×D4
prime radicalC22×D68C23
cosocleC22×C4×C17C24
hypercenter#1805D17
hypocenterC17#22678
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
4. Q8×C4⋊Dic17C2semidirect
6. #14529521C2semidirect
7. #14529516C2semidirect
8. #14529050C2semidirect
10. #14529036C2semidirect
12. #14528953C2semidirect
14. #14528661C2semidirect
15. C17×#1805C2semidirect
23. #11762907C4semidirect
70. C4×Q8×C17C2×C4semidirect
117. #10587506C2×C4semidirect
133. #10587478C2×C4semidirect
134. #10587474C2×C4semidirect
187. #63230C4⋊C4semidirect
300. #1805D17semidirect
405. C4×Q8C4×D17semidirect
461. C17#22678semidirect

Maximal subgroups

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

subgroupindex
C2×#117629122
C2×#117629072
C2×#117628942
Q8×C4⋊Dic172
#145316922
#145295212
#145295162
#145290502
#145290462
#145290362
#145290322
#145289532
#145287852
#145286612
C17×#18052
#2267817