Finite Group Databasebeta

← #17051197

Subgroups — #17051197

Order 4352.

1054 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763532C211yesmaximal, characteristic
2#14531741C211yesmaximal, characteristic, complemented
3#14529714C211yesmaximal, characteristic
4#14529714C211yesmaximal, characteristic
5#14529706C211yesmaximal, characteristic, complemented
6#14529604C211yesmaximal, characteristic, complemented
7#14529600C211yesmaximal, characteristic
8#14529595C211yesmaximal, characteristic
9#14528800C211yesmaximal, characteristic, complemented
10#14528795C211yesmaximal, characteristic
11#14528790C211yesmaximal, characteristic, complemented
12#14528615C211yesmaximal, characteristic, complemented
13#14528610C211yesmaximal, characteristic
14C4×#10587487C211yesmaximal, characteristic, complemented
15C2×C17×#446C211yesmaximal, characteristic, complemented
16#11763532—24no—
17#11763532C418yescomplemented
18C2×#10587555C2211yescharacteristic
19C2×#10587584—21no—
20C2×#10587584—21no—
21C2×#10587561C2211yescharacteristic
22C2×#10587561C2211yescharacteristic
23C2×#10587632—21no—
24C2×C4×Dic34C2211yescharacteristic
25C2×C4×Dic34C2211yescharacteristic
26C2×C4×D68C2211yescharacteristic
27C2×#10587578—21no—
28C2×#10587487C2211yescharacteristic
29C2×#10587487C2211yescharacteristic
30C2×#10587487C2211yescharacteristic
31C2×#10587487C2211yescharacteristic
32C2×#10587487C2211yescharacteristic
33C2×#10587487C2211yescharacteristic
34C2×#10587487C2211yescharacteristic
35C2×#10587487C2211yescharacteristic
36C2×#10587492C2211yescharacteristic
37C2×#10587492C2211yescharacteristic
38C2×#10587492C2211yescharacteristic
39C2×#10587492C2211yescharacteristic
40#11762716—21no—
41#11762695—21no—
42#11762695—21no—
43#11762679C2211yescharacteristic
44#11762675C2211yescharacteristic
45#11762675C2211yescharacteristic
46C4⋊C4×Dic17—21no—
47C4×C4⋊Dic17C2211yescharacteristic
48C4×C4⋊Dic17C2211yescharacteristic
49C4×C4⋊Dic17C2211yescharacteristic
50#11762572C2211yescharacteristic
51#11762572C2211yescharacteristic
52#11762572C2211yescharacteristic
53#11762572C2211yescharacteristic
54C2×C4×Q8×C17C2211yescharacteristic
55C17×#446—24no—
56C2×C17×C4⋊C8C2211yescharacteristic
57C2×C17×C4⋊C8C2211yescharacteristic
58C2×C17×C4⋊C8C2211yescharacteristic
59C2×C17×#97C2211yescharacteristic
60C2×C17×#97C2211yescharacteristic
61C2×C4×C8×C17C2211yescharacteristic
62C22×Q8×C17D411yescharacteristic
63C22×D68C2311yescharacteristic
64C22×C4×D17—21no—
65C22×Dic34C2311yescharacteristic
66C22×Dic34C2311yescharacteristic
67C22×C8×C17C2311yescharacteristic
68C22×C8×C17C2311yescharacteristic
69C22×C8×C17C2311yescharacteristic
70C22×C8×C17C2311yescharacteristic
71C4×Q8×C17C2×C414yes—
72C4×Q8×C17D414yes—
73C2×C4⋊C4×C17D411yescharacteristic
74C2×C4⋊C4×C17D411yescharacteristic
75C2×C4⋊C4×C17D411yescharacteristic
76C2×C42×C17C2311yescharacteristic
77C2×C42×C17C2311yescharacteristic
78C2×C42×C17C2311yescharacteristic
79#10587632—28no—
80C2×#63227—21no—
81C2×#63227—21no—
82C2×#63227—21no—
83C2×#63227—21no—
84C2×#63227—21no—
85C2×C4⋊Dic17—21no—
86C2×C4⋊Dic17—21no—
87C2×C4⋊Dic17—21no—
88C2×C4⋊Dic17—21no—
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×C4⋊Dic17C2311yescharacteristic
94C2×#63225—21no—
95C2×#63225—21no—
96C2×C4×Dic17—21no—
97C2×C4×Dic17—21no—
98C2×C4×Dic17—21no—
99C2×C4×Dic17—21no—
100C2×C4×Dic17—21no—
101#10587584—28no—
102#10587584—28no—
103#10587578—28no—
104#10587561—22no—
105#10587561—22no—
106#10587561C2×C414yes—
107#10587561C2×C414yes—
108C4×D68—22no—
109C4×D68C2×C414yes—
110#10587555—22no—
111#10587555C2×C414yes—
112C4×Dic34—22no—
113C4×Dic34—22no—
114C4×Dic34C2×C414yes—
115C4×Dic34C2×C414yes—
116C17×C4⋊C8—22no—
117C17×C4⋊C8—22no—
118C17×C4⋊C8—22no—
119C17×#97—22no—
120C17×#97—22no—
121C4×C8×C17—22no—
122#10587502—21no—
123#10587502—21no—
124#10587502—21no—
125#10587502—21no—
126#10587502—21no—
127#10587502—21no—
128#10587502—21no—
129#10587492—22no—
130#10587492—22no—
131#10587492—41no—
132#10587492—41no—
133#10587487—22no—
134#10587487—22no—
135#10587487—22no—
136#10587487—22no—
137#10587487—41no—
138#10587487—41no—
139#10587487—41no—
140#10587487—41no—
141C23×D17—21no—
142C2×Q8×C17—41no—
143C2×Q8×C17—41no—
144C2×Q8×C17C22⋊C414yes—
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×D411yescharacteristic
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2×D411yescharacteristic
151C22×C4×C17C2411yescharacteristic
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17—21no—
155C22×Dic17—21no—
156C22×Dic17—21no—
157C22×Dic17—21no—
158C22×Dic17—21no—
159C2×D68—21no—
160C2×D68—21no—
161C2×D68—21no—
162C2×D68—21no—
163C2×D68—21no—
164C2×D68C22×C412yes—
165C2×C4×D17—24no—
166C2×C4×D17—42no—
167C2×C4×D17—42no—
168C2×Dic34—21no—
169C2×Dic34—21no—
170C2×Dic34—21no—
171C2×Dic34—21no—
172C2×Dic34—21no—
173C2×Dic34—21no—
174C2×Dic34—21no—
175C2×Dic34—21no—
176C2×Dic34—21no—
177C2×Dic34—21no—
178C2×Dic34C22×C412yes—
179C2×Dic34C22×C412yes—
180C2×C8×C17—21no—
181C2×C8×C17—21no—
182C2×C8×C17—21no—
183C2×C8×C17—21no—
184C2×C8×C17—21no—
185C2×C8×C17—21no—
186C2×C8×C17—21no—
187C2×C8×C17—21no—
188C2×C8×C17—21no—
189C2×C8×C17—21no—
190C2×C8×C17—21no—
191C2×C8×C17#4112yes—
192C4⋊C4×C17C22⋊C414yes—
193C4⋊C4×C17C22⋊C414yes—
194C4⋊C4×C17C22⋊C414yes—
195C42×C17C22×C412yes—
196C42×C17C22×C412yes—
197C42×C17C22×C412yes—
198C42×C17C2×D412yes—
199C42×C17C2×D412yes—
200C42×C17C2×D412yes—
201#63227—24no—
202#63227—24no—
203#63227—24no—
204#63227—24no—
205#63227—24no—
206C4⋊Dic17—24no—
207C4⋊Dic17—24no—
208C4⋊Dic17—21no—
209C4⋊Dic17—21no—
210C4⋊Dic17—21no—
211C4⋊Dic17—21no—
212C4⋊Dic17—21no—
213C4⋊Dic17—21no—
214C4⋊Dic17—21no—
215C4⋊Dic17—21no—
216C4⋊Dic17—42no—
217C4⋊Dic17—81no—
218C4⋊Dic17C22×C412yes—
219C4⋊Dic17#4112yes—
220#63225—24no—
221#63225—24no—
222C4×Dic17—24no—
223C4×Dic17—24no—
224C4×Dic17—42no—
225C4×Dic17—81no—
226C4×Dic17—81no—
227#22548—171nomaximal, Sylow
228C23×C17C22×D411yescharacteristic
229C22×D17—22no—
230C22×D17—41no—
231C22×D17—41no—
232C22×D17—41no—
233C22×D17—41no—
234C22×D17—41no—
235C22×D17—41no—
236Q8×C17—44no—
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—21no—
246C2×C4×C17—21no—
247C2×C4×C17—21no—
248C2×C4×C17—21no—
249C2×C4×C17C2×C22⋊C412yes—
250C2×C4×C17C2×C22⋊C412yes—
251C2×C4×C17C2×C22⋊C412yes—
252C2×C4×C17C2×C22⋊C412yes—
253C2×C4×C17C2×C22⋊C412yes—
254C2×C4×C17C2×C22⋊C412yes—
255C2×C4×C17C23×C411yescharacteristic
256C2×C4×C17C22×D411yescharacteristic
257C2×C4×C17C2×#4111yescharacteristic
258C2×C4×C17C2×#4111yescharacteristic
259C2×C4×C17H(5,𝔽2)11yescharacteristic
260C2×C4×C172−1+411yescharacteristic
261C2×Dic17—22no—
262C2×Dic17—22no—
263C2×Dic17—22no—
264C2×Dic17—22no—
265C2×Dic17—41no—
266C2×Dic17—41no—
267C2×Dic17—41no—
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—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17—41no—
280C2×Dic17—41no—
281C2×Dic17—41no—
282D68—22no—
283D68—22no—
284D68—41no—
285D68—41no—
286C4×D17—48no—
287Dic34—22no—
288Dic34—22no—
289Dic34—22no—
290Dic34—22no—
291Dic34—41no—
292Dic34—41no—
293Dic34—41no—
294Dic34—41no—
295C8×C17—22no—
296C8×C17—41no—
297C8×C17—41no—
298C8×C17—41no—
299C2×#556—171no—
300C2×#446D1711yescharacteristic, complemented
301#2355—171no—
302#1983—171no—
303#1983—171no—
304#1981—171no—
305#1960—171no—
306#1959—171no—
307#1958—171no—
308#1805—171no—
309#1804—171no—
310#1803—171no—
311#1763—171no—
312#1762—171no—
313C4×#103—171no—
314C22×C17C22×C22⋊C411yescharacteristic
315C22×C17#54811yescharacteristic
316C22×C17#55011yescharacteristic
317C22×C17#57511yescharacteristic
318C22×C17#57511yescharacteristic
319C22×C17#57711yescharacteristic
320C22×C17#57811yescharacteristic
321C2×D17—42no—
322C2×D17—42no—
323C2×D17—42no—
324C2×D17—81no—
325C2×D17—81no—
326C2×D17—81no—
327C2×D17—81no—
328C4×C17—22no—
329C4×C17—22no—
330C4×C17—22no—
331C4×C17—22no—
332C4×C17—22no—
333C4×C17—22no—
334C4×C17C4×#4111yescharacteristic
335C4×C17#52011yescharacteristic
336C4×C17#54811yescharacteristic
337C4×C17#55011yescharacteristic
338Dic17—42no—
339Dic17—42no—
340Dic17—42no—
341Dic17—42no—
342Dic17—81no—
343Dic17—81no—
344Dic17—81no—
345#556—178no—
346#556—344no—
347C2×C4⋊Q8—171no—
348C2×#126—341no—
349C2×#126—341no—
350C2×#124—171no—
351C2×#124—171no—
352C2×C22⋊Q8—341no—
353C2×C4×Q8—171no—
354C2×C4×Q8—171no—
355C2×C4×Q8C2×D1711yescharacteristic
356C2×C4×D4—171no—
357C2×#117—341no—
358#446—24no—
359C2×C4⋊C8C2×D1711yescharacteristic
360C2×C4⋊C8C2×D1711yescharacteristic
361C2×C4⋊C8C2×D1711yescharacteristic
362C2×#103—171no—
363C2×#103—171no—
364C2×#103—171no—
365C2×#103—171no—
366C2×#103—171no—
367C2×#103—171no—
368C2×#103—171no—
369C2×#103—171no—
370C2×#102—171no—
371C2×#102—171no—
372C2×#102—171no—
373C2×#102—171no—
374C2×#97C2×D1711yescharacteristic
375C2×#97C2×D1711yescharacteristic
376C2×C4×C8C2×D1711yescharacteristic
377#391—341no—
378#387—341no—
379#387—341no—
380#384—171no—
381#382—171no—
382#382—171no—
383C4×C4⋊C4—171no—
384C4×C4⋊C4—171no—
385C4×C4⋊C4—171no—
386C4×C4⋊C4—341no—
387#340—171no—
388#340—171no—
389#340—171no—
390#340—171no—
391C2×C17#234111yescharacteristic
392C2×C17#289611yescharacteristic
393C2×C17#289811yescharacteristic
394C2×C17#337611yescharacteristic
395C2×C17#337711yescharacteristic
396C2×C17#340111yescharacteristic
397C2×C17#340211yescharacteristic
398D17—82no—
399C22×Q8—171no—
400C22×Q8—171no—
401C22×Q8D6811yescharacteristic
402C22×D4—171no—
403C23×C4—341no—
404C22×C8C22×D1711yescharacteristic
405C22×C8C22×D1711yescharacteristic
406C22×C8C22×D1711yescharacteristic
407C22×C8C22×D1711yescharacteristic
408C4⋊Q8—174no—
409C4⋊Q8—342no—
410#126—348no—
411#126—348no—
412#124—174no—
413#124—174no—
414#124—342no—
415#124—342no—
416C22⋊Q8—348no—
417C4×Q8—174no—
418C4×Q8—174no—
419C4×Q8—342no—
420C4×Q8—342no—
421C4×Q8C4×D1714yes—
422C4×Q8C22⋊D1714yes—
423C4×D4—174no—
424C4×D4—342no—
425#117—348no—
426C2×C4⋊C4—171no—
427C2×C4⋊C4—171no—
428C2×C4⋊C4—171no—
429C2×C4⋊C4—171no—
430C2×C4⋊C4—171no—
431C2×C4⋊C4—341no—
432C2×C4⋊C4—341no—
433C2×C4⋊C4—341no—
434C2×C4⋊C4—341no—
435C2×C4⋊C4—341no—
436C2×C4⋊C4—341no—
437C2×C4⋊C4D6811yescharacteristic
438C2×C4⋊C4D6811yescharacteristic
439C2×C4⋊C4D6811yescharacteristic
440C2×C22⋊C4—341no—
441C2×C22⋊C4—341no—
442C2×C22⋊C4—341no—
443C2×C22⋊C4—341no—
444C2×C22⋊C4—341no—
445C2×C42—341no—
446C2×C42—341no—
447C2×C42—341no—
448C2×C42—341no—
449C2×C42—341no—
450C2×C42C22×D1711yescharacteristic
451C2×C42C22×D1711yescharacteristic
452C2×C42C22×D1711yescharacteristic
453C4⋊C8—22no—
454C4⋊C8—22no—
455C4⋊C8—22no—
456#103—342no—
457#103—342no—
458#103—342no—
459#103—342no—
460#103—681no—
461#103—681no—
462#103—681no—
463#103—681no—
464#102—342no—
465#102—342no—
466#102—681no—
467#102—681no—
468#97—22no—
469#97—22no—
470C4×C8—22no—
471#95—341no—
472#95—341no—
473#95—341no—
474#95—341no—
475#95—341no—
476#95—341no—
477#95—341no—
478C17#2254811yescharacteristic, complemented, Sylow
479C24—341no—
480C2×Q8—41no—
481C2×Q8—41no—
482C2×Q8—172no—
483C2×Q8—172no—
484C2×Q8—341no—
485C2×Q8—341no—
486C2×Q8—341no—
487C2×Q8—341no—
488C2×Q8—341no—
489C2×Q8—341no—
490C2×Q8—341no—
491C2×Q8—341no—
492C2×Q8—341no—
493C2×Q8—341no—
494C2×Q8#6322714yes—
495C2×D4—172no—
496C2×D4—341no—
497C2×D4—341no—
498C2×D4—341no—
499C2×D4—341no—
500C2×D4—341no—
501C22×C4—341no—
502C22×C4—341no—
503C22×C4—341no—
504C22×C4—341no—
505C22×C4—344no—
506C22×C4—341no—
507C22×C4—341no—
508C22×C4—341no—
509C22×C4—682no—
510C22×C4—682no—
511C22×C4C2×D6811yescharacteristic
512C22×C4C2×D6811yescharacteristic
513C22×C4C2×D6811yescharacteristic
514C22×C4C2×D6811yescharacteristic
515C22×C4C2×D6811yescharacteristic
516C22×C4C2×D6811yescharacteristic
517C22×C4C23×D1711yescharacteristic
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8#6325612yes—
530C4⋊C4—172no—
531C4⋊C4—172no—
532C4⋊C4—341no—
533C4⋊C4—341no—
534C4⋊C4—344no—
535C4⋊C4—344no—
536C4⋊C4—344no—
537C4⋊C4—344no—
538C4⋊C4—341no—
539C4⋊C4—341no—
540C4⋊C4—341no—
541C4⋊C4—341no—
542C4⋊C4—341no—
543C4⋊C4—341no—
544C4⋊C4—682no—
545C4⋊C4—1361no—
546C4⋊C4#6322714yes—
547C4⋊C4#6322714yes—
548C4⋊C4#6322714yes—
549C22⋊C4—344no—
550C22⋊C4—344no—
551C22⋊C4—344no—
552C22⋊C4—344no—
553C22⋊C4—344no—
554C42—344no—
555C42—344no—
556C42—682no—
557C42—1361no—
558C42—1361no—
559C42C2×C4×D1712yes—
560C42C2×C4×D1712yes—
561C42C2×C4×D1712yes—
562C42C2×C22⋊D1712yes—
563C42C2×C22⋊D1712yes—
564C42C2×C22⋊D1712yes—
565C23—342no—
566C23—681no—
567C23—681no—
568C23—681no—
569C23—681no—
570C23—681no—
571C23—681no—
572C23C22×D6811yescharacteristic
573Q8—44no—
574Q8—342no—
575Q8—342no—
576Q8—342no—
577Q8—342no—
578Q8—681no—
579Q8—681no—
580Q8—681no—
581Q8—681no—
582D4—342no—
583D4—342no—
584D4—681no—
585D4—681no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—342no—
599C2×C4—342no—
600C2×C4—342no—
601C2×C4—342no—
602C2×C4—688no—
603C2×C4—681no—
604C2×C4—681no—
605C2×C4—681no—
606C2×C4—681no—
607C2×C4—681no—
608C2×C4—681no—
609C2×C4—681no—
610C2×C4—681no—
611C2×C4—681no—
612C2×C4—681no—
613C2×C4—681no—
614C2×C4—681no—
615C2×C4—681no—
616C2×C4—681no—
617C2×C4—681no—
618C2×C4—681no—
619C2×C4—681no—
620C2×C4C2×#6322712yes—
621C2×C4C2×#6322712yes—
622C2×C4C2×#6322712yes—
623C2×C4C2×#6322712yes—
624C2×C4C2×#6322712yes—
625C2×C4C2×#6322712yes—
626C2×C4C22×C4×D1711yescharacteristic
627C2×C4C2×#6325611yescharacteristic
628C2×C4#1058769411yescharacteristic
629C2×C4#41×D1711yescharacteristic
630C2×C4#1058769611yescharacteristic
631C2×C4C22×C22⋊D1711yescharacteristic
632C8—22no—
633C8—41no—
634C8—41no—
635C8—41no—
636C22—682no—
637C22—682no—
638C22—682no—
639C22—1361no—
640C22—1361no—
641C22—1361no—
642C22—1361no—
643C22C22×#6322711yescharacteristic
644C22#1176347111yescharacteristic
645C22#1176348111yescharacteristic
646C22#1176360911yescharacteristic
647C22#1176360911yescharacteristic
648C22#1176362111yescharacteristic
649C22#1176362611yescharacteristic
650C4—22no—
651C4—22no—
652C4—22no—
653C4—22no—
654C4—22no—
655C4—22no—
656C4—682no—
657C4—682no—
658C4—682no—
659C4—682no—
660C4—1361no—
661C4—1361no—
662C4—1361no—
663C4C4×#6325611yescharacteristic
664C4#1176329111yescharacteristic
665C4#1176347011yescharacteristic
666C4#1176347811yescharacteristic
667C2—1362no—
668C2#1453162211yescharacteristic
669C2#1453730411yescharacteristic
670C2#1453731811yescharacteristic
671C2#1454161311yescharacteristic
672C2#1454162811yescharacteristic
673C2#1454185211yescharacteristic
674C2#1454186511yescharacteristic
subgroupG/H
centerC23C22×D68
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C17×#446C2
socleC23×C17C22×D4
prime radicalC22×D68C23
cosocleC22×C4×C17C24
hypercenterC2×#446D17
hypocenterC17#22548
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#446

Complements

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

NH ≅ G/Ntype
2. #14531741C2semidirect
5. #14529706C2semidirect
6. #14529604C2semidirect
9. #14528800C2semidirect
11. #14528790C2semidirect
12. #14528615C2semidirect
14. C4×#10587487C2semidirect
15. C2×C17×#446C2semidirect
17. #11763532C4semidirect
300. C2×#446D17semidirect
478. C17#22548semidirect

Maximal subgroups

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

subgroupindex
C2×#117635322
#145317412
#145297142
#145297062
#145296042
#145296002
#145295952
#145288002
#145287952
#145287902
#145286152
#145286102
C4×#105874872
C2×C17×#4462
#2254817