Finite Group Databasebeta

← #17040581

Subgroups — #17040581

Order 4352.

1092 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763012C211yesmaximal, characteristic
2C2×#11762909C211yesmaximal, characteristic
3C4×#10587630C211yesmaximal, characteristic, complemented
4C4×D4×Dic17C211yesmaximal, characteristic
5#14529705C211yesmaximal, characteristic, complemented
6#14529539C211yesmaximal, characteristic, complemented
7#14529428C211yesmaximal, characteristic, complemented
8#14529424C211yesmaximal, characteristic, complemented
9#14529120C211yesmaximal, characteristic
10#14529116C211yesmaximal, characteristic
11#14528782C211yesmaximal, characteristic
12#14528638C211yesmaximal, characteristic, complemented
13#14528615C211yesmaximal, characteristic, complemented
14#14528614C211yesmaximal, characteristic, complemented
15C2×C17×#440C211yesmaximal, characteristic
16C22×#63229C2211yescharacteristic
17C2×#10587630C2211yescharacteristic
18C2×C4×Dic34C2211yescharacteristic
19C2×D4×Dic17C2211yescharacteristic
20C2×D4×Dic17C2211yescharacteristic
21C22×C4×Dic17—21no—
22#11763012C418yescomplemented
23#11762909—24no—
24C2×#10587478C2211yescharacteristic
25C2×#10587485C2211yescharacteristic
26C2×#10587485C2211yescharacteristic
27C2×#10587487C2211yescharacteristic
28C2×#10587487C2211yescharacteristic
29C2×#10587506C2211yescharacteristic
30C2×#10587506C2211yescharacteristic
31C2×#10587481C2211yescharacteristic
32C2×#10587503C2211yescharacteristic
33C2×#10587474C2211yescharacteristic
34#11762716C2211yescharacteristic
35#11762714C2211yescharacteristic
36#11762704—21no—
37#11762704—21no—
38#11762678C2211yescharacteristic
39C4⋊C4×Dic17C2211yescharacteristic
40C4⋊C4×Dic17C2211yescharacteristic
41C4×#63225—21no—
42C4×#63225—21no—
43C4×C4⋊Dic17C2211yescharacteristic
44C22⋊C4×Dic17—21no—
45C22⋊C4×Dic17—21no—
46C4×C22⋊Dic17—21no—
47C42×Dic17C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762572C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762563C2211yescharacteristic
53#11762563C2211yescharacteristic
54C22×SD16×C17C2211yescharacteristic
55C2×C4×Q8×C17C2211yescharacteristic
56C2×C4×D4×C17C2211yescharacteristic
57C17×#440—24no—
58C2×C17×#107C2211yescharacteristic
59C2×C17×#103C2211yescharacteristic
60C2×C17×#102C2211yescharacteristic
61C2×C17×#97C2211yescharacteristic
62C22×Q8×C17C2311yescharacteristic
63C22×D4×C17C2311yescharacteristic
64C23×C4×C17—21no—
65C23×Dic17—21no—
66C23×Dic17—21no—
67C22×Dic34C2311yescharacteristic
68C2×SD16×C17—22no—
69C2×SD16×C17—22no—
70C2×SD16×C17C2×C414yes—
71C22×C8×C17C2311yescharacteristic
72C22×C8×C17C2311yescharacteristic
73C4×Q8×C17—22no—
74C4×Q8×C17—22no—
75C4×D4×C17—22no—
76C4×D4×C17D414yes—
77C2×C4⋊C4×C17—21no—
78C2×C4⋊C4×C17C2311yescharacteristic
79C2×C4⋊C4×C17C2311yescharacteristic
80C2×C22⋊C4×C17—21no—
81C2×C42×C17—21no—
82C2×C42×C17C2311yescharacteristic
83C2×C22⋊Dic17—21no—
84C2×C22⋊Dic17—21no—
85#10587630—22no—
86#10587630C2×C414yescomplemented
87D4×Dic17—22no—
88D4×Dic17—22no—
89D4×Dic17—22no—
90D4×Dic17C2×C414yes—
91C2×#63229—22no—
92C2×#63229—22no—
93C2×#63229C2×C414yes—
94C2×C4⋊Dic17C2311yescharacteristic
95C2×C4⋊Dic17C2311yescharacteristic
96C2×C4⋊Dic17C2311yescharacteristic
97C2×#63225—21no—
98C2×#63225—21no—
99C2×#63225—21no—
100C2×#63225—21no—
101C2×#63225—21no—
102C2×C4×Dic17—21no—
103C2×C4×Dic17—21no—
104C2×C4×Dic17—21no—
105C2×C4×Dic17—21no—
106C2×C4×Dic17—44no—
107C2×C4×Dic17D411yescharacteristic
108C2×C4×Dic17D411yescharacteristic
109C2×C4×Dic17D411yescharacteristic
110C2×C4×Dic17D411yescharacteristic
111C2×C4×Dic17C2311yescharacteristic
112C2×C4×Dic17C2311yescharacteristic
113C22×#3622C2311yescharacteristic
114C22×#3622C2311yescharacteristic
115C4×Dic34—22no—
116C4×Dic34—22no—
117C17×#107—22no—
118C17×#103—22no—
119C17×#102—22no—
120C17×#97—22no—
121#10587506—22no—
122#10587506C2×C414yescomplemented
123#10587503—22no—
124#10587502—21no—
125#10587502—21no—
126#10587502—21no—
127#10587502—21no—
128#10587502—21no—
129#10587502—21no—
130#10587502—21no—
131#10587487—22no—
132#10587487C2×C414yescomplemented
133#10587485—22no—
134#10587485C2×C414yescomplemented
135#10587481—22no—
136#10587478—22no—
137#10587474—22no—
138C24×C17—21no—
139C2×Q8×C17—21no—
140C2×Q8×C17—21no—
141C2×Q8×C17—21no—
142C2×Q8×C17—21no—
143C2×Q8×C17—21no—
144C2×Q8×C17C22×C412yes—
145C2×D4×C17—21no—
146C2×D4×C17—21no—
147C2×D4×C17—21no—
148C2×D4×C17C22×C412yes—
149C2×D4×C17C2×D412yes—
150C2×D4×C17#4112yes—
151C22×C4×C17—21no—
152C22×C4×C17—21no—
153C22×C4×C17—24no—
154C22×C4×C17—21no—
155C22×C4×C17—42no—
156C22×C4×C17—42no—
157C22×C4×C17C2×D411yescharacteristic
158C22×C4×C17C2×D411yescharacteristic
159C22×C4×C17C2411yescharacteristic
160C22×Dic17—24no—
161C22×Dic17—21no—
162C22×Dic17—21no—
163C22×Dic17—21no—
164C22×Dic17—24no—
165C22×Dic17—21no—
166C22×Dic17—42no—
167C22×Dic17—42no—
168C22×Dic17—42no—
169C22×Dic17—42no—
170C22×Dic17C2×D411yescharacteristic
171C22×Dic17C2×D411yescharacteristic
172C22×Dic17C2×D411yescharacteristic
173C22×Dic17C2×D411yescharacteristic
174C2×Dic34—21no—
175C2×Dic34—21no—
176C2×Dic34—21no—
177C2×Dic34—21no—
178C2×Dic34—21no—
179C2×Dic34C22×C412yes—
180SD16×C17—28no—
181C2×C8×C17—21no—
182C2×C8×C17—21no—
183C2×C8×C17—21no—
184C2×C8×C17—21no—
185C2×C8×C17—21no—
186C2×C8×C17C22×C412yes—
187C4⋊C4×C17—21no—
188C4⋊C4×C17—21no—
189C4⋊C4×C17—21no—
190C4⋊C4×C17—42no—
191C4⋊C4×C17C2×D412yes—
192C22⋊C4×C17—24no—
193C42×C17—42no—
194C42×C17C2×D412yes—
195C42×C17#4112yes—
196C22⋊Dic17—24no—
197C22⋊Dic17—24no—
198#63229—28no—
199C4⋊Dic17—21no—
200C4⋊Dic17—21no—
201C4⋊Dic17—21no—
202C4⋊Dic17—21no—
203C4⋊Dic17C22×C412yes—
204C4⋊Dic17#4112yes—
205#63225—24no—
206#63225—24no—
207#63225—42no—
208#63225—42no—
209#63225—42no—
210C4×Dic17—21no—
211C4×Dic17—21no—
212C4×Dic17—22no—
213C4×Dic17—22no—
214C4×Dic17—22no—
215C4×Dic17—41no—
216C4×Dic17—42no—
217C4×Dic17—42no—
218C4×Dic17—42no—
219C4×Dic17—42no—
220C4×Dic17C22×C412yes—
221C4×Dic17#4112yes—
222C2×#3622—21no—
223C2×#3622—21no—
224C2×#3622—21no—
225C2×#3622—21no—
226C2×#3622—21no—
227C2×#3622C22×C412yes—
228#21504—171nomaximal, Sylow
229C23×C17—22no—
230C23×C17—22no—
231C23×C17—22no—
232C23×C17—41no—
233C23×C17—41no—
234C23×C17—41no—
235C23×C17—41no—
236C23×C17C22×D411yescharacteristic
237Q8×C17—22no—
238Q8×C17—22no—
239Q8×C17—41no—
240Q8×C17—41no—
241D4×C17—22no—
242D4×C17—22no—
243D4×C17—22no—
244D4×C17C4×D414yes—
245C2×C4×C17—21no—
246C2×C4×C17—21no—
247C2×C4×C17—21no—
248C2×C4×C17—21no—
249C2×C4×C17—22no—
250C2×C4×C17—22no—
251C2×C4×C17—48no—
252C2×C4×C17—41no—
253C2×C4×C17—41no—
254C2×C4×C17—41no—
255C2×C4×C17—41no—
256C2×C4×C17—41no—
257C2×C4×C17—41no—
258C2×C4×C17—41no—
259C2×C4×C17#12112yes—
260C2×C4×C17#12112yes—
261C2×C4×C17C23×C411yescharacteristic
262C2×C4×C17C22×D411yescharacteristic
263C2×C4×C17C2×#4111yescharacteristic
264C2×C4×C17C2×#4111yescharacteristic
265C2×C4×C17C2×#4111yescharacteristic
266C2×C4×C17C2×#4111yescharacteristic
267C2×Dic17—21no—
268C2×Dic17—21no—
269C2×Dic17—21no—
270C2×Dic17—21no—
271C2×Dic17—22no—
272C2×Dic17—22no—
273C2×Dic17—21no—
274C2×Dic17—22no—
275C2×Dic17—21no—
276C2×Dic17—21no—
277C2×Dic17—48no—
278C2×Dic17—41no—
279C2×Dic17—41no—
280C2×Dic17—41no—
281C2×Dic17—41no—
282C2×Dic17—41no—
283C2×Dic17—41no—
284C2×Dic17—48no—
285C2×Dic17—41no—
286C2×Dic17—41no—
287C2×Dic17—41no—
288C2×Dic17C4×D412yes—
289C2×Dic17C4×D412yes—
290C2×Dic17AGL(1,ℤ/8)12yes—
291C2×Dic17#13712yes—
292C2×Dic17#13712yes—
293Dic34—22no—
294Dic34—22no—
295Dic34—41no—
296Dic34—41no—
297C8×C17—22no—
298C8×C17—41no—
299#3622—22no—
300#3622—41no—
301C2×#461—171no—
302C2×#440—171no—
303C2×#440D1711yescharacteristic
304C4×C4⋊Q8—171no—
305C42×D4—171no—
306#1981—171no—
307#1947—171no—
308#1928—171no—
309#1928—171no—
310#1877—171no—
311#1877—171no—
312#1802—171no—
313#1767—171no—
314#1763—171no—
315#1763—171no—
316C22×C17—24no—
317C22×C17—42no—
318C22×C17—42no—
319C22×C17—42no—
320C22×C17—42no—
321C22×C17—42no—
322C22×C17—42no—
323C22×C17C2×C4×D411yescharacteristic
324C22×C17C2×#12111yescharacteristic
325C22×C17#52511yescharacteristic
326C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
327C22×C17C2×#13711yescharacteristic
328C22×C17C2×#13711yescharacteristic
329C22×C17#57511yescharacteristic
330C4×C17—22no—
331C4×C17—22no—
332C4×C17—42no—
333C4×C17—42no—
334C4×C17—81no—
335C4×C17C2×C4×D411yescharacteristic
336C4×C17C4×#4111yescharacteristic
337C4×C17#52911yescharacteristic
338C4×C17#53211yescharacteristic
339Dic17—22no—
340Dic17—22no—
341Dic17—42no—
342Dic17—41no—
343Dic17—42no—
344Dic17—42no—
345Dic17—81no—
346Dic17#44014yescomplemented
347C22×SD16—171no—
348C22×SD16C2×D1711yescharacteristic
349C2×C4⋊Q8—171no—
350C2×C4×Q8—171no—
351C2×C4×Q8C2×D1711yescharacteristic
352C2×C4×D4—171no—
353C2×C4×D4—171no—
354C2×C4×D4C2×D1711yescharacteristic
355C22×C42—341no—
356#461—178no—
357#440—24no—
358#440—344no—
359C2×#107—171no—
360C2×#107C2×D1711yescharacteristic
361C2×C4⋊C8—171no—
362C2×C4⋊C8—171no—
363C2×#103—171no—
364C2×#103—171no—
365C2×#103—171no—
366C2×#103—171no—
367C2×#103—171no—
368C2×#103C2×D1711yescharacteristic
369C2×#102—171no—
370C2×#102C2×D1711yescharacteristic
371C2×#97—171no—
372C2×#97C2×D1711yescharacteristic
373#391—171no—
374#391—171no—
375#389—341no—
376#389—341no—
377#384—171no—
378C4×C4⋊C4—171no—
379C4×C4⋊C4—171no—
380C4×C4⋊C4—171no—
381C4×C4⋊C4—341no—
382C4×C4⋊C4—341no—
383C4×C22⋊C4—341no—
384C4×C22⋊C4—341no—
385C4×C22⋊C4—341no—
386C43—171no—
387#340—171no—
388#340—171no—
389#340—171no—
390#340—171no—
391#336—171no—
392#336—171no—
393C2×C17—44no—
394C2×C17C4×#12111yescharacteristic
395C2×C17C2×#44011yescharacteristic
396C2×C17C4×#13711yescharacteristic
397C2×C17#303611yescharacteristic
398C2×C17#303911yescharacteristic
399C2×C17#311011yescharacteristic
400C2×C17#311611yescharacteristic
401C22×Q8—171no—
402C22×Q8C22×D1711yescharacteristic
403C22×D4C22×D1711yescharacteristic
404C23×C4—21no—
405C23×C4—341no—
406C23×C4—341no—
407C2×SD16—22no—
408C2×SD16—22no—
409C2×SD16—174no—
410C2×SD16—342no—
411C2×SD16—342no—
412C2×SD16C4×D1714yes—
413C22×C8—171no—
414C22×C8—171no—
415C22×C8C22×D1711yescharacteristic
416C22×C8C22×D1711yescharacteristic
417C4⋊Q8—174no—
418C4⋊Q8—342no—
419C4×Q8—22no—
420C4×Q8—22no—
421C4×Q8—342no—
422C4×Q8—342no—
423C4×D4—22no—
424C4×D4—174no—
425C4×D4—342no—
426C4×D4—342no—
427C4×D4—342no—
428C4×D4C22⋊D1714yes—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—171no—
431C2×C4⋊C4—171no—
432C2×C4⋊C4—171no—
433C2×C4⋊C4—341no—
434C2×C4⋊C4—341no—
435C2×C4⋊C4—341no—
436C2×C4⋊C4—341no—
437C2×C4⋊C4—341no—
438C2×C4⋊C4C22×D1711yescharacteristic
439C2×C4⋊C4C22×D1711yescharacteristic
440C2×C22⋊C4—21no—
441C2×C22⋊C4—341no—
442C2×C22⋊C4—341no—
443C2×C42—21no—
444C2×C42—171no—
445C2×C42—171no—
446C2×C42—171no—
447C2×C42—171no—
448C2×C42—171no—
449C2×C42—171no—
450C2×C42—341no—
451C2×C42—341no—
452C2×C42—341no—
453C2×C42—341no—
454C2×C42—684no—
455C2×C42C22×D1711yescharacteristic
456#107—22no—
457#107—342no—
458C4⋊C8—174no—
459C4⋊C8—342no—
460#103—22no—
461#103—174no—
462#103—174no—
463#103—342no—
464#103—342no—
465#103—342no—
466#102—22no—
467#102—342no—
468#97—22no—
469#97—342no—
470#95—341no—
471#95—341no—
472#95—341no—
473#95—341no—
474#95—341no—
475#95—341no—
476#95—341no—
477C17#2150411yescharacteristic, complemented, Sylow
478C24—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—172no—
485C2×Q8—341no—
486C2×Q8—341no—
487C2×Q8—341no—
488C2×Q8—341no—
489C2×Q8—341no—
490C2×Q8C2×C4×D1712yes—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4C2×C4×D1712yes—
495C2×D4#6325212yes—
496C2×D4C2×C22⋊D1712yes—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—24no—
500C22×C4—21no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—171no—
504C22×C4—171no—
505C22×C4—171no—
506C22×C4—171no—
507C22×C4—344no—
508C22×C4—344no—
509C22×C4—341no—
510C22×C4—341no—
511C22×C4—341no—
512C22×C4—341no—
513C22×C4—682no—
514C22×C4—682no—
515C22×C4—682no—
516C22×C4—682no—
517C22×C4D4×D1711yescharacteristic
518C22×C4D4×D1711yescharacteristic
519C22×C4C23×D1711yescharacteristic
520SD16—28no—
521SD16—348no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—172no—
528C2×C8—341no—
529C2×C8—341no—
530C2×C8—341no—
531C2×C8—341no—
532C2×C8—341no—
533C2×C8C2×C4×D1712yes—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—42no—
538C4⋊C4—172no—
539C4⋊C4—172no—
540C4⋊C4—344no—
541C4⋊C4—344no—
542C4⋊C4—341no—
543C4⋊C4—341no—
544C4⋊C4—341no—
545C4⋊C4—341no—
546C4⋊C4—682no—
547C4⋊C4—682no—
548C4⋊C4—682no—
549C4⋊C4C2×C22⋊D1712yes—
550C22⋊C4—24no—
551C22⋊C4—344no—
552C22⋊C4—344no—
553C42—42no—
554C42—172no—
555C42—172no—
556C42—342no—
557C42—342no—
558C42—342no—
559C42—341no—
560C42—341no—
561C42—681no—
562C42—682no—
563C42—682no—
564C42—682no—
565C42—682no—
566C42#6325412yes—
567C42C2×C22⋊D1712yes—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23C2×D4×D1711yescharacteristic
576Q8—22no—
577Q8—22no—
578Q8—41no—
579Q8—41no—
580Q8—342no—
581Q8—342no—
582Q8—681no—
583Q8—681no—
584D4—22no—
585D4—22no—
586D4—22no—
587D4C4×C22⋊D1714yes—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—48no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—172no—
603C2×C4—172no—
604C2×C4—172no—
605C2×C4—172no—
606C2×C4—172no—
607C2×C4—341no—
608C2×C4—341no—
609C2×C4—341no—
610C2×C4—341no—
611C2×C4—342no—
612C2×C4—342no—
613C2×C4—341no—
614C2×C4—342no—
615C2×C4—341no—
616C2×C4—341no—
617C2×C4—688no—
618C2×C4—688no—
619C2×C4—681no—
620C2×C4—681no—
621C2×C4—681no—
622C2×C4—681no—
623C2×C4—681no—
624C2×C4—681no—
625C2×C4—681no—
626C2×C4—681no—
627C2×C4—681no—
628C2×C4#1058762512yes—
629C2×C4#1058762512yes—
630C2×C4C22×C4×D1711yescharacteristic
631C2×C4C2×#6325211yescharacteristic
632C2×C4C2×#6325411yescharacteristic
633C2×C4#41×D1711yescharacteristic
634C2×C4#41×D1711yescharacteristic
635C2×C4C22×C22⋊D1711yescharacteristic
636C8—22no—
637C8—41no—
638C8—342no—
639C8—681no—
640C22—24no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22C4×D4×D1711yescharacteristic
648C22C2×#1058762511yescharacteristic
649C22#1176331811yescharacteristic
650C22AGL(1,ℤ/8)×D1711yescharacteristic
651C22#137×D1711yescharacteristic
652C22#137×D1711yescharacteristic
653C22#1176360511yescharacteristic
654C4—22no—
655C4—22no—
656C4—42no—
657C4—42no—
658C4—81no—
659C4—174no—
660C4—342no—
661C4—342no—
662C4—682no—
663C4—682no—
664C4—681no—
665C4—682no—
666C4—1361no—
667C4C2×C4×C22⋊D1711yescharacteristic
668C4C4×#6325411yescharacteristic
669C4#1176333911yescharacteristic
670C4#1176335511yescharacteristic
671C2—44no—
672C2C4×#1058762511yescharacteristic
673C2D17×#44011yescharacteristic
674C2#1453760611yescharacteristic
675C2#1453824511yescharacteristic
676C2#1453826911yescharacteristic
677C2#1453888111yescharacteristic
678C2#1453894211yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C17×#440C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenterC2×#440D17
hypocenterC17#21504
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
3. C4×#10587630C2semidirect
5. #14529705C2semidirect
6. #14529539C2semidirect
7. #14529428C2semidirect
8. #14529424C2semidirect
12. #14528638C2semidirect
13. #14528615C2semidirect
14. #14528614C2semidirect
22. #11763012C4semidirect
86. #10587630C2×C4semidirect
122. #10587506C2×C4semidirect
132. #10587487C2×C4semidirect
134. #10587485C2×C4semidirect
346. Dic17#440semidirect
477. C17#21504semidirect

Maximal subgroups

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

subgroupindex
C2×#117630122
C2×#117629092
C4×#105876302
C4×D4×Dic172
#145297052
#145295392
#145294282
#145294242
#145291202
#145291162
#145287822
#145286382
#145286152
#145286142
C2×C17×#4402
#2150417