Finite Group Databasebeta

← #17070925

Subgroups — #17070925

Order 4352.

1064 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763106C211yesmaximal, characteristic
2C2×#11762907C211yesmaximal, characteristic
3#14533702C211yesmaximal, characteristic
4#14530119C211yesmaximal, characteristic, complemented
5#14530109C211yesmaximal, characteristic, complemented
6#14529927C211yesmaximal, characteristic
7#14529917C211yesmaximal, characteristic
8#14529754C211yesmaximal, characteristic, complemented
9#14529725C211yesmaximal, characteristic, complemented
10#14529139C211yesmaximal, characteristic
11#14529139C211yesmaximal, characteristic
12#14529052C211yesmaximal, characteristic, complemented
13#14529038C211yesmaximal, characteristic, complemented
14#14528789C211yesmaximal, characteristic, complemented
15C2×C17×#557C211yesmaximal, characteristic, complemented
16C22×#63230C2211yescharacteristic
17C2×#10587632—21no—
18C2×C4×D68C2211yescharacteristic
19#11763106—24no—
20#11762907—24no—
21C2×#10587478C2211yescharacteristic
22C2×#10587478C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×C4⋊#3622C2211yescharacteristic
25C2×#10587506C2211yescharacteristic
26C2×#10587506C2211yescharacteristic
27C2×#10587506C2211yescharacteristic
28C2×#10587480C2211yescharacteristic
29C2×#10587480C2211yescharacteristic
30C2×#10587480C2211yescharacteristic
31C2×#10587474C2211yescharacteristic
32#11762731—21no—
33#11762731—21no—
34#11762711C2211yescharacteristic
35#11762711C2211yescharacteristic
36#11762707C2211yescharacteristic
37#11762707C2211yescharacteristic
38#11762679C2211yescharacteristic
39#11762666—21no—
40#11762571C2211yescharacteristic
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762564C2211yescharacteristic
47#11762564C2211yescharacteristic
48#11762564C2211yescharacteristic
49#11762564C2211yescharacteristic
50C17×#557—24no—
51C17×#557—24no—
52C2×C17×C4⋊Q8—21no—
53C2×C17×C4⋊Q8C2211yescharacteristic
54C2×C17×#125—21no—
55C2×C17×#125—21no—
56C2×C17×#125C2211yescharacteristic
57C2×C17×#125C2211yescharacteristic
58C2×C4×Q8×C17—21no—
59C2×C4×Q8×C17C2211yescharacteristic
60C2×C4×Q8×C17C2211yescharacteristic
61C2×C4×Q8×C17C2211yescharacteristic
62C2×C4×Q8×C17C2211yescharacteristic
63C22×Q8×C17—21no—
64C22×Q8×C17D411yescharacteristic
65C22×Q8×C17C2311yescharacteristic
66C22×D68C2311yescharacteristic
67C22×C4×D17—21no—
68C17×C4⋊Q8—28no—
69C17×C4⋊Q8—22no—
70C17×C4⋊Q8—22no—
71C17×#125—28no—
72C17×#125—28no—
73C17×#125—22no—
74C17×#125—22no—
75C17×#125—22no—
76C17×#125D414yes—
77C4×Q8×C17—28no—
78C4×Q8×C17—22no—
79C4×Q8×C17—22no—
80C4×Q8×C17—22no—
81C4×Q8×C17—22no—
82C4×Q8×C17—22no—
83C4×Q8×C17—22no—
84C4×Q8×C17—22no—
85C4×Q8×C17D414yes—
86C2×C4⋊C4×C17—21no—
87C2×C4⋊C4×C17—21no—
88C2×C4⋊C4×C17—21no—
89C2×C4⋊C4×C17—21no—
90C2×C4⋊C4×C17—21no—
91C2×C4⋊C4×C17—21no—
92C2×C4⋊C4×C17—21no—
93C2×C4⋊C4×C17—21no—
94C2×C4⋊C4×C17D411yescharacteristic
95C2×C4⋊C4×C17D411yescharacteristic
96C2×C4⋊C4×C17D411yescharacteristic
97C2×C4⋊C4×C17C2311yescharacteristic
98C2×C4⋊C4×C17C2311yescharacteristic
99C2×C4⋊C4×C17C2311yescharacteristic
100C2×C42×C17—21no—
101C2×C42×C17—21no—
102C2×C42×C17—21no—
103C2×C42×C17C2311yescharacteristic
104C2×C42×C17C2311yescharacteristic
105C2×C42×C17C2311yescharacteristic
106#10587632—44no—
107C2×#63230—22no—
108C2×#63230—22no—
109C2×#63230Q814yescomplemented
110C2×#63227—21no—
111C2×#63227—21no—
112C2×#63227—21no—
113C2×#63227—21no—
114C2×#63227—21no—
115C2×C4⋊Dic17—21no—
116C2×C4⋊Dic17—21no—
117C2×C4⋊Dic17C2311yescharacteristic
118C2×C4⋊Dic17C2311yescharacteristic
119C2×C4⋊Dic17C2311yescharacteristic
120C2×#63225—21no—
121C2×#63225—21no—
122C22×#3622C2311yescharacteristic
123C22×#3622C2311yescharacteristic
124C22×#3622C2311yescharacteristic
125C22×#3622C2311yescharacteristic
126C4×D68—22no—
127C4×D68—22no—
128#10587506—22no—
129#10587506—22no—
130#10587506—22no—
131#10587502—21no—
132#10587502—21no—
133#10587480—22no—
134#10587480—22no—
135#10587480—22no—
136#10587479—22no—
137#10587478—22no—
138#10587478—22no—
139C4⋊#3622—22no—
140#10587474—22no—
141C23×D17—21no—
142C2×Q8×C17—24no—
143C2×Q8×C17—22no—
144C2×Q8×C17—22no—
145C2×Q8×C17—21no—
146C2×Q8×C17—21no—
147C2×Q8×C17—21no—
148C2×Q8×C17—41no—
149C2×Q8×C17—42no—
150C2×Q8×C17—42no—
151C2×Q8×C17C2×D412yes—
152C2×Q8×C17C2×Q812yes—
153C2×Q8×C17#4112yes—
154C22×C4×C17—21no—
155C22×C4×C17—21no—
156C22×C4×C17—21no—
157C22×C4×C17—21no—
158C22×C4×C17C2×D411yescharacteristic
159C22×C4×C17C2×D411yescharacteristic
160C22×C4×C17C2×D411yescharacteristic
161C22×C4×C17C2×D411yescharacteristic
162C22×C4×C17C2×D411yescharacteristic
163C22×C4×C17C2×D411yescharacteristic
164C22×C4×C17C2411yescharacteristic
165C22×Dic17—21no—
166C22×Dic17—21no—
167C22×Dic17—21no—
168C2×D68—21no—
169C2×D68—21no—
170C2×D68—21no—
171C2×D68—21no—
172C2×D68—21no—
173C2×D68C2×Q812yes—
174C2×C4×D17—42no—
175C2×C4×D17—42no—
176C2×C4×D17—42no—
177C4⋊C4×C17—24no—
178C4⋊C4×C17—24no—
179C4⋊C4×C17—24no—
180C4⋊C4×C17—24no—
181C4⋊C4×C17—24no—
182C4⋊C4×C17—22no—
183C4⋊C4×C17—22no—
184C4⋊C4×C17—24no—
185C4⋊C4×C17—24no—
186C4⋊C4×C17—24no—
187C4⋊C4×C17—22no—
188C4⋊C4×C17—21no—
189C4⋊C4×C17—21no—
190C4⋊C4×C17—21no—
191C4⋊C4×C17C2×D412yes—
192C4⋊C4×C17C2×D412yes—
193C4⋊C4×C17C2×D412yes—
194C42×C17—24no—
195C42×C17—24no—
196C42×C17—24no—
197C42×C17—21no—
198C42×C17—21no—
199C42×C17—21no—
200C42×C17—21no—
201C42×C17C2×D412yes—
202C42×C17C2×D412yes—
203#63230—28no—
204#63227—42no—
205#63227—42no—
206#63227—42no—
207#63227—81no—
208#63227—81no—
209C4⋊Dic17—21no—
210C4⋊Dic17—21no—
211C4⋊Dic17—21no—
212C4⋊Dic17—21no—
213C4⋊Dic17—21no—
214C4⋊Dic17—21no—
215C4⋊Dic17—42no—
216C4⋊Dic17—81no—
217#63225—42no—
218#63225—42no—
219C2×#3622—21no—
220C2×#3622—21no—
221C2×#3622—21no—
222C2×#3622—21no—
223C2×#3622—21no—
224C2×#3622—21no—
225C2×#3622—21no—
226C2×#3622—21no—
227C2×#3622—21no—
228C2×#3622—21no—
229C2×#3622—21no—
230C2×#3622C2×Q812yes—
231#24346—171nomaximal, Sylow
232C23×C17C22×D411yescharacteristic
233C22×D17—22no—
234C22×D17—41no—
235C22×D17—41no—
236C22×D17—41no—
237C22×D17—41no—
238C22×D17—41no—
239C22×D17—41no—
240Q8×C17—22no—
241Q8×C17—22no—
242Q8×C17—22no—
243Q8×C17—44no—
244Q8×C17—48no—
245Q8×C17C22⋊Q814yescomplemented
246C2×C4×C17—22no—
247C2×C4×C17—21no—
248C2×C4×C17—22no—
249C2×C4×C17—22no—
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—22no—
258C2×C4×C17—21no—
259C2×C4×C17—22no—
260C2×C4×C17—22no—
261C2×C4×C17—21no—
262C2×C4×C17—21no—
263C2×C4×C17—21no—
264C2×C4×C17—21no—
265C2×C4×C17—21no—
266C2×C4×C17—21no—
267C2×C4×C17—41no—
268C2×C4×C17—41no—
269C2×C4×C17—41no—
270C2×C4×C17—41no—
271C2×C4×C17—41no—
272C2×C4×C17—41no—
273C2×C4×C17#12012yes—
274C2×C4×C17#12012yes—
275C2×C4×C17AGL(1,ℤ/8)12yes—
276C2×C4×C17C22×D411yescharacteristic
277C2×C4×C17C22×D411yescharacteristic
278C2×C4×C17C22×Q811yescharacteristic
279C2×C4×C17C2×#4111yescharacteristic
280C2×C4×C172−1+411yescharacteristic
281C2×C4×C172−1+411yescharacteristic
282C2×Dic17—41no—
283C2×Dic17—41no—
284C2×Dic17—41no—
285C2×Dic17—41no—
286C2×Dic17—41no—
287C2×Dic17—41no—
288C2×Dic17—41no—
289C2×Dic17—41no—
290C2×Dic17—41no—
291D68—22no—
292D68—22no—
293D68—41no—
294D68—41no—
295C4×D17—84no—
296#3622—22no—
297#3622—41no—
298#3622—41no—
299#3622—41no—
300C2×#557D1711yescharacteristic, complemented
301C2×#478—171no—
302C2×#440—171no—
303#2538—171no—
304#2059—171no—
305#2057—171no—
306#2025—171no—
307#2023—171no—
308#1992—171no—
309#1985—171no—
310#1880—171no—
311#1880—171no—
312#1865—171no—
313#1863—171no—
314#1803—171no—
315C22×C17C2×#12011yescharacteristic
316C22×C17#54811yescharacteristic
317C22×C17D4×Q811yescharacteristic
318C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
319C22×C17#57511yescharacteristic
320C22×C17#57711yescharacteristic
321C22×C17#57811yescharacteristic
322C2×D17—42no—
323C2×D17—42no—
324C2×D17—42no—
325C2×D17—81no—
326C2×D17—81no—
327C2×D17—81no—
328C2×D17—81no—
329C4×C17—24no—
330C4×C17—22no—
331C4×C17—22no—
332C4×C17—22no—
333C4×C17—41no—
334C4×C17—42no—
335C4×C17—42no—
336C4×C17—41no—
337C4×C17—41no—
338C4×C17—42no—
339C4×C17C2×C22⋊Q811yescharacteristic
340C4×C17#53611yescharacteristic
341C4×C17#54811yescharacteristic
342C4×C17D4×Q811yescharacteristic
343Dic17—81no—
344Dic17—81no—
345Dic17—81no—
346C22×SD16—171no—
347#557—24no—
348#557—24no—
349C2×C4⋊Q8—21no—
350C2×C4⋊Q8C2×D1711yescharacteristic
351C2×#125—21no—
352C2×#125—21no—
353C2×#125C2×D1711yescharacteristic
354C2×#125C2×D1711yescharacteristic
355C2×C22⋊Q8—341no—
356C2×C4×Q8—21no—
357C2×C4×Q8C2×D1711yescharacteristic
358C2×C4×Q8C2×D1711yescharacteristic
359C2×C4×Q8C2×D1711yescharacteristic
360C2×C4×Q8C2×D1711yescharacteristic
361C2×C4×D4—171no—
362#478—344no—
363#440—344no—
364C2×#107—171no—
365C2×#107—171no—
366C2×#106—171no—
367C2×C4⋊C8—171no—
368C2×#103—171no—
369C2×#103—171no—
370C2×#103—171no—
371C2×#102—171no—
372C2×#102—171no—
373C2×#102—171no—
374C2×#97—171no—
375#396—341no—
376#396—341no—
377#390—171no—
378#390—171no—
379#389—171no—
380#389—171no—
381#384—171no—
382#380—341no—
383#340—171no—
384#340—171no—
385#340—171no—
386#340—171no—
387#340—171no—
388#340—171no—
389#336—171no—
390#336—171no—
391#336—171no—
392#336—171no—
393C2×C17#240911yescharacteristic
394C2×C17#300611yescharacteristic
395C2×C17#301311yescharacteristic
396C2×C17#336811yescharacteristic
397C2×C17#337711yescharacteristic
398C2×C17#338611yescharacteristic
399C2×C17#338811yescharacteristic
400D17—82no—
401C22×Q8—21no—
402C22×Q8C22⋊D1711yescharacteristic
403C22×Q8C22×D1711yescharacteristic
404C22×D4—171no—
405C23×C4—341no—
406C2×SD16—174no—
407C2×SD16—342no—
408C2×SD16—342no—
409C22×C8—171no—
410C22×C8—171no—
411C22×C8—171no—
412C22×C8—171no—
413C4⋊Q8—28no—
414C4⋊Q8—22no—
415C4⋊Q8—22no—
416#125—28no—
417#125—28no—
418#125—22no—
419#125—22no—
420#125—22no—
421#125C22⋊D1714yes—
422C22⋊Q8—684no—
423C4×Q8—28no—
424C4×Q8—22no—
425C4×Q8—22no—
426C4×Q8—22no—
427C4×Q8—22no—
428C4×Q8—22no—
429C4×Q8—22no—
430C4×Q8—22no—
431C4×Q8C22⋊D1714yes—
432C4×D4—342no—
433C4×D4—342no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—21no—
440C2×C4⋊C4—21no—
441C2×C4⋊C4—21no—
442C2×C4⋊C4—171no—
443C2×C4⋊C4—171no—
444C2×C4⋊C4—171no—
445C2×C4⋊C4—341no—
446C2×C4⋊C4—341no—
447C2×C4⋊C4—341no—
448C2×C4⋊C4—341no—
449C2×C4⋊C4C22⋊D1711yescharacteristic
450C2×C4⋊C4C22⋊D1711yescharacteristic
451C2×C4⋊C4C22⋊D1711yescharacteristic
452C2×C4⋊C4C22×D1711yescharacteristic
453C2×C4⋊C4C22×D1711yescharacteristic
454C2×C4⋊C4C22×D1711yescharacteristic
455C2×C22⋊C4—341no—
456C2×C22⋊C4—341no—
457C2×C22⋊C4—341no—
458C2×C22⋊C4—341no—
459C2×C22⋊C4—341no—
460C2×C42—21no—
461C2×C42—21no—
462C2×C42—21no—
463C2×C42C22×D1711yescharacteristic
464C2×C42C22×D1711yescharacteristic
465C2×C42C22×D1711yescharacteristic
466#107—342no—
467#107—342no—
468#106—342no—
469C4⋊C8—342no—
470#103—342no—
471#103—342no—
472#103—342no—
473#102—342no—
474#102—342no—
475#102—342no—
476#97—342no—
477#95—341no—
478#95—341no—
479C17#2434611yescharacteristic, complemented, Sylow
480C24—341no—
481C2×Q8—24no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—22no—
486C2×Q8—22no—
487C2×Q8—41no—
488C2×Q8—42no—
489C2×Q8—42no—
490C2×Q8Q8×D1712yes—
491C2×Q8#6325612yes—
492C2×Q8C2×C22⋊D1712yes—
493C2×D4—172no—
494C2×D4—341no—
495C2×D4—341no—
496C2×D4—341no—
497C2×D4—341no—
498C2×D4—341no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—341no—
504C22×C4—341no—
505C22×C4—341no—
506C22×C4—682no—
507C22×C4—682no—
508C22×C4—682no—
509C22×C4C2×C22⋊D1711yescharacteristic
510C22×C4C2×C22⋊D1711yescharacteristic
511C22×C4C2×C22⋊D1711yescharacteristic
512C22×C4C2×C22⋊D1711yescharacteristic
513C22×C4C2×C22⋊D1711yescharacteristic
514C22×C4C2×C22⋊D1711yescharacteristic
515C22×C4C23×D1711yescharacteristic
516SD16—348no—
517C2×C8—172no—
518C2×C8—341no—
519C2×C8—341no—
520C2×C8—341no—
521C2×C8—341no—
522C2×C8—341no—
523C2×C8—341no—
524C2×C8—341no—
525C2×C8—341no—
526C2×C8—341no—
527C2×C8—341no—
528C2×C8—341no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—24no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—24no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—341no—
544C4⋊C4—341no—
545C4⋊C4—341no—
546C4⋊C4—341no—
547C4⋊C4—341no—
548C4⋊C4—341no—
549C4⋊C4—682no—
550C4⋊C4—682no—
551C4⋊C4—682no—
552C4⋊C4—1361no—
553C4⋊C4C2×C22⋊D1712yes—
554C4⋊C4C2×C22⋊D1712yes—
555C4⋊C4C2×C22⋊D1712yes—
556C22⋊C4—682no—
557C22⋊C4—682no—
558C22⋊C4—682no—
559C22⋊C4—1361no—
560C22⋊C4—1361no—
561C42—24no—
562C42—24no—
563C42—21no—
564C42—21no—
565C42—21no—
566C42—21no—
567C42—24no—
568C42C2×C22⋊D1712yes—
569C42C2×C22⋊D1712yes—
570C23—342no—
571C23—681no—
572C23—681no—
573C23—681no—
574C23—681no—
575C23—681no—
576C23—681no—
577C23C22×C22⋊D1711yescharacteristic
578Q8—22no—
579Q8—22no—
580Q8—22no—
581Q8—44no—
582Q8—48no—
583Q8#1058763214yescomplemented
584D4—342no—
585D4—342no—
586D4—681no—
587D4—681no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—681no—
616C2×C4—681no—
617C2×C4—681no—
618C2×C4—681no—
619C2×C4—681no—
620C2×C4—681no—
621C2×C4—681no—
622C2×C4—681no—
623C2×C4—681no—
624C2×C4—1364no—
625C2×C4#1058763512yes—
626C2×C4#1058763912yes—
627C2×C4#1058763912yes—
628C2×C4C2×Q8×D1711yescharacteristic
629C2×C4C2×#6325611yescharacteristic
630C2×C4#1058769411yescharacteristic
631C2×C4#1058769411yescharacteristic
632C2×C4C22×C22⋊D1711yescharacteristic
633C2×C4C22×C22⋊D1711yescharacteristic
634C8—342no—
635C8—681no—
636C8—681no—
637C8—681no—
638C22—682no—
639C22—682no—
640C22—682no—
641C22—1361no—
642C22—1361no—
643C22—1361no—
644C22—1361no—
645C22C2×#1058763911yescharacteristic
646C22#1176347011yescharacteristic
647C22Q8×C22⋊D1711yescharacteristic
648C22C2×#1058763511yescharacteristic
649C22#1176360711yescharacteristic
650C22#1176361711yescharacteristic
651C22#1176362411yescharacteristic
652C4—24no—
653C4—22no—
654C4—22no—
655C4—22no—
656C4—41no—
657C4—42no—
658C4—42no—
659C4—41no—
660C4—41no—
661C4—42no—
662C4—1361no—
663C4—1361no—
664C4—1361no—
665C4C2×#1058763211yescharacteristic
666C4#1176338311yescharacteristic
667C4#1176347011yescharacteristic
668C4Q8×C22⋊D1711yescharacteristic
669C2—1362no—
670C2#1453214311yescharacteristic
671C2#1453803611yescharacteristic
672C2#1453809911yescharacteristic
673C2#1454151311yescharacteristic
674C2#1454162611yescharacteristic
675C2#1454170111yescharacteristic
676C2#1454172311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC2×C17×#557C2
socleC23×C17C22×D4
prime radicalC22×D68C23
hypercenterC2×#557D17
hypocenterC17#24346
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
4. #14530119C2semidirect
5. #14530109C2semidirect
8. #14529754C2semidirect
9. #14529725C2semidirect
12. #14529052C2semidirect
13. #14529038C2semidirect
14. #14528789C2semidirect
15. C2×C17×#557C2semidirect
109. C2×#63230Q8semidirect
245. Q8×C17C22⋊Q8semidirect
300. C2×#557D17semidirect
479. C17#24346semidirect
583. Q8#10587632semidirect

Maximal subgroups

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

subgroupindex
C2×#117631062
C2×#117629072
#145337022
#145301192
#145301092
#145299272
#145299172
#145297542
#145297252
#145291392
#145290522
#145290382
#145287892
C2×C17×#5572
#2434617