Finite Group Databasebeta

← #17040724

Subgroups — #17040724

Order 4352.

1008 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762910C211yesmaximal, characteristic
2C2×SD16×Dic17C211yesmaximal, characteristic
3#14531504C211yesmaximal, characteristic, complemented
4#14531145C211yesmaximal, characteristic
5#14529708C211yesmaximal, characteristic, complemented
6#14529554C211yesmaximal, characteristic, complemented
7#14529427C211yesmaximal, characteristic, complemented
8#14529420C211yesmaximal, characteristic, complemented
9#14529120C211yesmaximal, characteristic
10#14529120C211yesmaximal, characteristic
11#14528794C211yesmaximal, characteristic
12#14528643C211yesmaximal, characteristic, complemented
13#14528616C211yesmaximal, characteristic, complemented
14#103×Dic17C211yesmaximal, characteristic, complemented
15C2×C17×#461C211yesmaximal, characteristic
16C2×Q8×Dic17C2211yescharacteristic
17C2×Q8×Dic17C2211yescharacteristic
18C2×D4×Dic17C2211yescharacteristic
19C2×D4×Dic17C2211yescharacteristic
20#11762910—24no—
21SD16×Dic17—24no—
22C2×#10587489C2211yescharacteristic
23C2×#10587488C2211yescharacteristic
24C2×C4⋊#3622C2211yescharacteristic
25C2×#10587506C2211yescharacteristic
26C2×#10587506C2211yescharacteristic
27C2×#10587506C2211yescharacteristic
28C2×#10587506C2211yescharacteristic
29C2×#10587503C2211yescharacteristic
30C2×#10587503C2211yescharacteristic
31C2×#10587486C2211yescharacteristic
32C2×C8×Dic17C2211yescharacteristic
33#11762716C2211yescharacteristic
34#11762716C2211yescharacteristic
35#11762695—21no—
36#11762695—21no—
37#11762695—21no—
38#11762695—21no—
39#11762675C2211yescharacteristic
40C4⋊C4×Dic17C2211yescharacteristic
41C4⋊C4×Dic17C2211yescharacteristic
42C4×C4⋊Dic17C2211yescharacteristic
43C22⋊C4×Dic17—21no—
44C22⋊C4×Dic17—21no—
45C4×C22⋊Dic17—21no—
46#11762655C2211yescharacteristic
47C2×#10587502—21no—
48#11762572C2211yescharacteristic
49#11762572C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762563C2211yescharacteristic
53#11762563C2211yescharacteristic
54C22×SD16×C17C2211yescharacteristic
55C22×SD16×C17C2211yescharacteristic
56C2×C17×C4⋊Q8C2211yescharacteristic
57C2×C4×D4×C17C2211yescharacteristic
58C17×#461C418yescomplemented
59C2×C17×C4⋊C8C2211yescharacteristic
60C2×C17×#103C2211yescharacteristic
61C2×C17×#103C2211yescharacteristic
62C22×Q8×C17C2311yescharacteristic
63C22×Q8×C17C2311yescharacteristic
64C22×D4×C17C2311yescharacteristic
65C23×C4×C17—21no—
66C23×Dic17—21no—
67C23×Dic17—21no—
68C2×SD16×C17—22no—
69C2×SD16×C17—22no—
70C2×SD16×C17—22no—
71C2×SD16×C17—22no—
72C2×SD16×C17C2×C414yes—
73C2×SD16×C17C2×C414yes—
74C22×C8×C17C2311yescharacteristic
75C22×C8×C17C2311yescharacteristic
76C17×C4⋊Q8—22no—
77C17×C4⋊Q8C2×C414yescomplemented
78C4×D4×C17—22no—
79C4×D4×C17C2×C414yes—
80C2×C4⋊C4×C17—21no—
81C2×C4⋊C4×C17—21no—
82C2×C4⋊C4×C17C2311yescharacteristic
83C2×C22⋊C4×C17—21no—
84C2×C42×C17C2311yescharacteristic
85C2×C22⋊Dic17—21no—
86C2×C22⋊Dic17—21no—
87Q8×Dic17—22no—
88Q8×Dic17—22no—
89Q8×Dic17—22no—
90Q8×Dic17—22no—
91D4×Dic17—22no—
92D4×Dic17—22no—
93D4×Dic17—22no—
94D4×Dic17D414yes—
95C2×C4⋊Dic17—21no—
96C2×C4⋊Dic17—21no—
97C2×C4⋊Dic17C2311yescharacteristic
98C2×C4⋊Dic17C2311yescharacteristic
99C2×C4⋊Dic17C2311yescharacteristic
100C2×C4⋊Dic17C2311yescharacteristic
101C2×C4×Dic17—21no—
102C2×C4×Dic17—21no—
103C2×C4×Dic17—21no—
104C2×C4×Dic17—21no—
105C2×C4×Dic17—21no—
106C2×C4×Dic17C2311yescharacteristic
107C2×C4×Dic17C2311yescharacteristic
108C22×#3622C2311yescharacteristic
109C22×#3622C2311yescharacteristic
110C17×C4⋊C8C2×C414yescomplemented
111C17×#103C2×C414yescomplemented
112C17×#103C2×C414yescomplemented
113#10587506—22no—
114#10587506—22no—
115#10587506—22no—
116#10587506—22no—
117#10587503—22no—
118#10587503—22no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587502—21no—
125#10587502—21no—
126#10587502—21no—
127#10587502—21no—
128#10587502D411yescharacteristic
129#10587502D411yescharacteristic
130#10587502D411yescharacteristic
131#10587502D411yescharacteristic
132#10587489—22no—
133#10587488—22no—
134#10587486—22no—
135C8×Dic17—22no—
136C4⋊#3622—22no—
137C24×C17—21no—
138C2×Q8×C17—21no—
139C2×Q8×C17—21no—
140C2×Q8×C17—21no—
141C2×Q8×C17—21no—
142C2×Q8×C17—21no—
143C2×Q8×C17—21no—
144C2×Q8×C17—21no—
145C2×Q8×C17—21no—
146C2×Q8×C17—21no—
147C2×Q8×C17—21no—
148C2×Q8×C17C22×C412yes—
149C2×Q8×C17C22×C412yes—
150C2×D4×C17—21no—
151C2×D4×C17—21no—
152C2×D4×C17—21no—
153C2×D4×C17C22×C412yes—
154C2×D4×C17C2×D412yes—
155C2×D4×C17#4112yes—
156C22×C4×C17—21no—
157C22×C4×C17—21no—
158C22×C4×C17—21no—
159C22×C4×C17—24no—
160C22×C4×C17—81no—
161C22×C4×C17—81no—
162C22×C4×C17C2×D411yescharacteristic
163C22×C4×C17C2×D411yescharacteristic
164C22×C4×C17C2411yescharacteristic
165C22×Dic17—21no—
166C22×Dic17—21no—
167C22×Dic17—24no—
168C22×Dic17—24no—
169C22×Dic17—21no—
170C22×Dic17—21no—
171C22×Dic17—81no—
172C22×Dic17—81no—
173C22×Dic17—81no—
174C22×Dic17—81no—
175C22×Dic17C2×D411yescharacteristic
176C22×Dic17C2×D411yescharacteristic
177C22×Dic17C2×D411yescharacteristic
178C22×Dic17C2×D411yescharacteristic
179SD16×C17—28no—
180SD16×C17—28no—
181C2×C8×C17—21no—
182C2×C8×C17—21no—
183C2×C8×C17—21no—
184C2×C8×C17—21no—
185C2×C8×C17C22×C412yes—
186C2×C8×C17C22×C412yes—
187C4⋊C4×C17—24no—
188C4⋊C4×C17—24no—
189C4⋊C4×C17—21no—
190C4⋊C4×C17C22×C412yes—
191C22⋊C4×C17—24no—
192C42×C17C22×C412yes—
193C42×C17#4112yes—
194C22⋊Dic17—24no—
195C22⋊Dic17—24no—
196C4⋊Dic17—21no—
197C4⋊Dic17—21no—
198C4⋊Dic17—21no—
199C4⋊Dic17—21no—
200C4⋊Dic17—21no—
201C4⋊Dic17—21no—
202C4⋊Dic17—42no—
203C4⋊Dic17—42no—
204C4⋊Dic17C2×D412yes—
205C4⋊Dic17#4112yes—
206C4×Dic17—21no—
207C4×Dic17—21no—
208C4×Dic17—42no—
209C4×Dic17—42no—
210C4×Dic17—81no—
211C4×Dic17—81no—
212C4×Dic17—81no—
213C4×Dic17C2×D412yes—
214C4×Dic17#4112yes—
215C2×#3622—21no—
216C2×#3622—21no—
217C2×#3622—21no—
218C2×#3622—21no—
219C2×#3622—21no—
220C2×#3622—21no—
221#21514—171nomaximal, Sylow
222C23×C17—22no—
223C23×C17—22no—
224C23×C17—22no—
225C23×C17—41no—
226C23×C17—41no—
227C23×C17—41no—
228C23×C17—41no—
229C23×C17C22×D411yescharacteristic
230Q8×C17—22no—
231Q8×C17—22no—
232Q8×C17—22no—
233Q8×C17—22no—
234Q8×C17—41no—
235Q8×C17—41no—
236Q8×C17—41no—
237Q8×C17—41no—
238D4×C17—22no—
239D4×C17—22no—
240D4×C17—22no—
241D4×C17C4×D414yes—
242C2×C4×C17—21no—
243C2×C4×C17—21no—
244C2×C4×C17—21no—
245C2×C4×C17—21no—
246C2×C4×C17—22no—
247C2×C4×C17—22no—
248C2×C4×C17—22no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17—41no—
252C2×C4×C17—41no—
253C2×C4×C17—41no—
254C2×C4×C17—41no—
255C2×C4×C17—84no—
256C2×C4×C17C4×D412yes—
257C2×C4×C17C4×D412yes—
258C2×C4×C17C23×C411yescharacteristic
259C2×C4×C17C22×D411yescharacteristic
260C2×C4×C17C2×#4111yescharacteristic
261C2×C4×C17C2×#4111yescharacteristic
262C2×C4×C17C2×#4111yescharacteristic
263C2×C4×C17C2×#4111yescharacteristic
264C2×Dic17—21no—
265C2×Dic17—21no—
266C2×Dic17—21no—
267C2×Dic17—21no—
268C2×Dic17—22no—
269C2×Dic17—21no—
270C2×Dic17—22no—
271C2×Dic17—21no—
272C2×Dic17—21no—
273C2×Dic17—21no—
274C2×Dic17—21no—
275C2×Dic17—21no—
276C2×Dic17—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17—41no—
280C2×Dic17—41no—
281C2×Dic17—41no—
282C2×Dic17—41no—
283C2×Dic17—41no—
284C2×Dic17—41no—
285C2×Dic17—41no—
286C2×Dic17—84no—
287C2×Dic17—84no—
288C2×Dic17#12112yes—
289C2×Dic17#12112yes—
290C8×C17—22no—
291C8×C17—22no—
292#3622—41no—
293#3622—41no—
294C2×#461D1711yescharacteristic
295C2×#440—171no—
296C2×C4×SD16—171no—
297#2327—171no—
298#2275—171no—
299#1981—171no—
300#1949—171no—
301#1928—171no—
302#1927—171no—
303#1877—171no—
304#1877—171no—
305#1804—171no—
306#1768—171no—
307#1763—171no—
308C4×#103—171no—
309C22×C17—24no—
310C22×C17—42no—
311C22×C17—42no—
312C22×C17—42no—
313C22×C17—42no—
314C22×C17—42no—
315C22×C17—42no—
316C22×C17C2×C4×D411yescharacteristic
317C22×C17C2×#12111yescharacteristic
318C22×C17#52511yescharacteristic
319C22×C17#57711yescharacteristic
320C22×C17#57711yescharacteristic
321C22×C17#57711yescharacteristic
322C22×C17#57811yescharacteristic
323C4×C17—22no—
324C4×C17—22no—
325C4×C17—42no—
326C4×C17—42no—
327C4×C17—42no—
328C4×C17C2×C4×D411yescharacteristic
329C4×C17C4×#4111yescharacteristic
330C4×C17#52911yescharacteristic
331C4×C17#53211yescharacteristic
332Dic17—22no—
333Dic17—22no—
334Dic17—41no—
335Dic17—41no—
336Dic17—42no—
337Dic17—42no—
338Dic17—81no—
339Dic17—81no—
340C22×SD16C2×D1711yescharacteristic
341C22×SD16C2×D1711yescharacteristic
342C2×C4⋊Q8C2×D1711yescharacteristic
343C2×C4×Q8—171no—
344C2×C4×Q8—171no—
345C2×C4×D4—171no—
346C2×C4×D4—171no—
347C2×C4×D4C2×D1711yescharacteristic
348#461Dic1718yescomplemented
349#440—344no—
350C4×SD16—344no—
351C2×#107—171no—
352C2×#106—171no—
353C2×C4⋊C8—171no—
354C2×C4⋊C8C2×D1711yescharacteristic
355C2×#103—171no—
356C2×#103—171no—
357C2×#103—171no—
358C2×#103—171no—
359C2×#103C2×D1711yescharacteristic
360C2×#103C2×D1711yescharacteristic
361C2×#102—171no—
362C2×#102—171no—
363C2×#97—171no—
364C2×C4×C8—171no—
365#391—171no—
366#391—171no—
367#387—341no—
368#387—341no—
369#387—341no—
370#387—341no—
371#382—171no—
372C4×C4⋊C4—171no—
373C4×C4⋊C4—171no—
374C4×C4⋊C4—171no—
375C4×C22⋊C4—341no—
376C4×C22⋊C4—341no—
377C4×C22⋊C4—341no—
378#376—171no—
379C2×#95—341no—
380#340—171no—
381#340—171no—
382#340—171no—
383#340—171no—
384#336—171no—
385#336—171no—
386C2×C17—44no—
387C2×C17C4×#12111yescharacteristic
388C2×C17#294711yescharacteristic
389C2×C17#294911yescharacteristic
390C2×C17#304211yescharacteristic
391C2×C17#304611yescharacteristic
392C2×C17#312011yescharacteristic
393C2×C17#312311yescharacteristic
394C22×Q8C22×D1711yescharacteristic
395C22×Q8C22×D1711yescharacteristic
396C22×D4C22×D1711yescharacteristic
397C23×C4—21no—
398C23×C4—341no—
399C23×C4—341no—
400C2×SD16—22no—
401C2×SD16—22no—
402C2×SD16—22no—
403C2×SD16—22no—
404C2×SD16C2×Dic1714yes—
405C2×SD16C2×Dic1714yes—
406C22×C8—171no—
407C22×C8—171no—
408C22×C8C22×D1711yescharacteristic
409C22×C8C22×D1711yescharacteristic
410C4⋊Q8—22no—
411C4⋊Q8C2×Dic1714yescomplemented
412C4×Q8—342no—
413C4×Q8—342no—
414C4×Q8—342no—
415C4×Q8—342no—
416C4×D4—22no—
417C4×D4—174no—
418C4×D4—342no—
419C4×D4—342no—
420C4×D4—342no—
421C4×D4C2×Dic1714yes—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—171no—
425C2×C4⋊C4—171no—
426C2×C4⋊C4—171no—
427C2×C4⋊C4—171no—
428C2×C4⋊C4—341no—
429C2×C4⋊C4—341no—
430C2×C4⋊C4C22×D1711yescharacteristic
431C2×C22⋊C4—21no—
432C2×C22⋊C4—341no—
433C2×C22⋊C4—341no—
434C2×C42—171no—
435C2×C42—171no—
436C2×C42—341no—
437C2×C42—341no—
438C2×C42—341no—
439C2×C42—341no—
440C2×C42—341no—
441C2×C42C22×D1711yescharacteristic
442#107—342no—
443#106—342no—
444C4⋊C8—342no—
445C4⋊C8C2×Dic1714yescomplemented
446#103—342no—
447#103—342no—
448#103—342no—
449#103—342no—
450#103C2×Dic1714yescomplemented
451#103C2×Dic1714yescomplemented
452#102—342no—
453#102—342no—
454#97—342no—
455C4×C8—342no—
456#95—171no—
457#95—171no—
458#95—171no—
459#95—171no—
460#95—341no—
461#95—341no—
462#95—341no—
463#95—341no—
464#95—341no—
465#95—341no—
466#95—341no—
467#95—341no—
468#95—341no—
469C17#2151411yescharacteristic, complemented, Sylow
470C24—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8C22×Dic1712yes—
482C2×Q8C22×Dic1712yes—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4D4×D1712yes—
487C2×D4#6325412yes—
488C2×D4C22×Dic1712yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—24no—
493C22×C4—81no—
494C22×C4—81no—
495C22×C4—171no—
496C22×C4—171no—
497C22×C4—171no—
498C22×C4—171no—
499C22×C4—344no—
500C22×C4—341no—
501C22×C4—341no—
502C22×C4—344no—
503C22×C4—341no—
504C22×C4—341no—
505C22×C4—1361no—
506C22×C4—1361no—
507C22×C4—1361no—
508C22×C4—1361no—
509C22×C4D4×D1711yescharacteristic
510C22×C4D4×D1711yescharacteristic
511C22×C4C23×D1711yescharacteristic
512SD16—28no—
513SD16—28no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—341no—
519C2×C8—341no—
520C2×C8—341no—
521C2×C8—341no—
522C2×C8—341no—
523C2×C8—341no—
524C2×C8C22×Dic1712yes—
525C2×C8C22×Dic1712yes—
526C4⋊C4—21no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—172no—
530C4⋊C4—172no—
531C4⋊C4—341no—
532C4⋊C4—341no—
533C4⋊C4—341no—
534C4⋊C4—341no—
535C4⋊C4—341no—
536C4⋊C4—341no—
537C4⋊C4—682no—
538C4⋊C4—682no—
539C4⋊C4C22×Dic1712yes—
540C22⋊C4—24no—
541C22⋊C4—344no—
542C22⋊C4—344no—
543C42—172no—
544C42—172no—
545C42—341no—
546C42—341no—
547C42—682no—
548C42—682no—
549C42—1361no—
550C42—1361no—
551C42—1361no—
552C42#6325412yes—
553C42C22×Dic1712yes—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23C2×D4×D1711yescharacteristic
562Q8—22no—
563Q8—22no—
564Q8—22no—
565Q8—22no—
566Q8—41no—
567Q8—41no—
568Q8—41no—
569Q8—41no—
570D4—22no—
571D4—22no—
572D4—22no—
573D4D4×Dic1714yes—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—84no—
588C2×C4—172no—
589C2×C4—172no—
590C2×C4—341no—
591C2×C4—341no—
592C2×C4—341no—
593C2×C4—341no—
594C2×C4—342no—
595C2×C4—341no—
596C2×C4—341no—
597C2×C4—342no—
598C2×C4—341no—
599C2×C4—341no—
600C2×C4—341no—
601C2×C4—341no—
602C2×C4—681no—
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—1364no—
613C2×C4—1364no—
614C2×C4D4×Dic1712yes—
615C2×C4D4×Dic1712yes—
616C2×C4C2×D4×D1711yescharacteristic
617C2×C4C2×#6325411yescharacteristic
618C2×C4C2×#6325411yescharacteristic
619C2×C4#41×D1711yescharacteristic
620C2×C4#41×D1711yescharacteristic
621C2×C4C23×Dic1711yescharacteristic
622C8—22no—
623C8—22no—
624C8—681no—
625C8—681no—
626C22—24no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22C2×D4×Dic1711yescharacteristic
634C22#121×D1711yescharacteristic
635C22#1176331911yescharacteristic
636C22#1176361511yescharacteristic
637C22#1176361511yescharacteristic
638C22#1176361811yescharacteristic
639C22#1176362511yescharacteristic
640C4—22no—
641C4—22no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—342no—
646C4—342no—
647C4—681no—
648C4—682no—
649C4—682no—
650C4—681no—
651C4—1361no—
652C4—1361no—
653C4C2×D4×Dic1711yescharacteristic
654C4#41×Dic1711yescharacteristic
655C4#1176334011yescharacteristic
656C4#1176335411yescharacteristic
657C2—44no—
658C2#121×Dic1711yescharacteristic
659C2#1453762811yescharacteristic
660C2#1453764411yescharacteristic
661C2#1453829111yescharacteristic
662C2#1453832911yescharacteristic
663C2#1453899511yescharacteristic
664C2#1453903411yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C17×#461C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenterC2×#461D17
hypocenterC17#21514
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
3. #14531504C2semidirect
5. #14529708C2semidirect
6. #14529554C2semidirect
7. #14529427C2semidirect
8. #14529420C2semidirect
12. #14528643C2semidirect
13. #14528616C2semidirect
14. #103×Dic17C2semidirect
58. C17×#461C4semidirect
77. C17×C4⋊Q8C2×C4semidirect
110. C17×C4⋊C8C2×C4semidirect
111. C17×#103C2×C4semidirect
112. C17×#103C2×C4semidirect
348. #461Dic17semidirect
411. C4⋊Q8C2×Dic17semidirect
445. C4⋊C8C2×Dic17semidirect
450. #103C2×Dic17semidirect
451. #103C2×Dic17semidirect
469. C17#21514semidirect

Maximal subgroups

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

subgroupindex
C2×#117629102
C2×SD16×Dic172
#145315042
#145311452
#145297082
#145295542
#145294272
#145294202
#145291202
#145287942
#145286432
#145286162
#103×Dic172
C2×C17×#4612
#2151417