Finite Group Databasebeta

← #17040674

Subgroups — #17040674

Order 4352.

980 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763029C211yesmaximal, characteristic
2C2×#11762914C211yesmaximal, characteristic, complemented
3#14531496C211yesmaximal, characteristic
4#14531149C211yesmaximal, characteristic, complemented
5#14529715C211yesmaximal, characteristic
6#14529544C211yesmaximal, characteristic, complemented
7#14529417C211yesmaximal, characteristic, complemented
8#14529414C211yesmaximal, characteristic
9#14529141C211yesmaximal, characteristic
10#14529123C211yesmaximal, characteristic, complemented
11#14528787C211yesmaximal, characteristic, complemented
12#14528642C211yesmaximal, characteristic, complemented
13#14528609C211yesmaximal, characteristic
14C4×#10587487C211yesmaximal, characteristic, complemented
15C2×C4×SD16×C17C211yesmaximal, characteristic
16C22×#63231C2211yescharacteristic
17C2×#10587622C2211yescharacteristic
18C2×Q8×Dic17C2211yescharacteristic
19C2×Q8×Dic17C2211yescharacteristic
20C2×C4×Dic34C2211yescharacteristic
21#11763029C418yescomplemented
22#11762914—24no—
23C2×#10587479C2211yescharacteristic
24C2×#10587485C2211yescharacteristic
25C2×#10587485C2211yescharacteristic
26C2×#10587487C2211yescharacteristic
27C2×#10587487C2211yescharacteristic
28C2×#10587506C2211yescharacteristic
29C2×#10587481C2211yescharacteristic
30C2×#10587503C2211yescharacteristic
31C2×#10587503C2211yescharacteristic
32C2×#10587474C2211yescharacteristic
33#11762714C2211yescharacteristic
34#11762712C2211yescharacteristic
35#11762697—21no—
36#11762697—21no—
37#11762682—21no—
38#11762682—21no—
39#11762678C2211yescharacteristic
40C4⋊C4×Dic17—21no—
41C4⋊C4×Dic17—21no—
42C4⋊C4×Dic17C2211yescharacteristic
43C4⋊C4×Dic17C2211yescharacteristic
44C4×C4⋊Dic17—21no—
45C4×C4⋊Dic17C2211yescharacteristic
46#11762655—21no—
47#11762655C2211yescharacteristic
48#11762572C2211yescharacteristic
49#11762572C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762563C2211yescharacteristic
53#11762563C2211yescharacteristic
54C22×SD16×C17C2211yescharacteristic
55C2×C4×Q8×C17C2211yescharacteristic
56C2×C4×D4×C17C2211yescharacteristic
57C4×SD16×C17—24no—
58C2×C17×#106C2211yescharacteristic
59C2×C17×#103C2211yescharacteristic
60C2×C17×#102C2211yescharacteristic
61C2×C4×C8×C17C2211yescharacteristic
62C22×Q8×C17C2311yescharacteristic
63C22×D4×C17C2311yescharacteristic
64C23×C4×C17—21no—
65C22×Dic34C2311yescharacteristic
66C2×SD16×C17—22no—
67C2×SD16×C17—22no—
68C2×SD16×C17C2×C414yes—
69C22×C8×C17C2311yescharacteristic
70C22×C8×C17C2311yescharacteristic
71C4×Q8×C17—22no—
72C4×Q8×C17D414yes—
73C4×D4×C17—22no—
74C4×D4×C17—22no—
75C2×C4⋊C4×C17—21no—
76C2×C4⋊C4×C17C2311yescharacteristic
77C2×C4⋊C4×C17C2311yescharacteristic
78C2×C22⋊C4×C17—21no—
79C2×C42×C17—21no—
80C2×C42×C17C2311yescharacteristic
81C2×C22⋊Dic17—21no—
82C2×C22⋊Dic17—21no—
83C2×C22⋊Dic17—21no—
84C2×C22⋊Dic17—21no—
85Q8×Dic17—22no—
86Q8×Dic17—22no—
87Q8×Dic17—22no—
88Q8×Dic17C2×C414yescomplemented
89C2×#63231—22no—
90C2×#63231—22no—
91C2×#63231C2×C414yescomplemented
92#10587622—22no—
93#10587622C2×C414yes—
94C2×C4⋊Dic17—21no—
95C2×C4⋊Dic17—21no—
96C2×C4⋊Dic17C2311yescharacteristic
97C2×C4⋊Dic17C2311yescharacteristic
98C2×C4⋊Dic17C2311yescharacteristic
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—21no—
107C2×C4×Dic17—21no—
108C2×C4×Dic17C2311yescharacteristic
109C2×C4×Dic17C2311yescharacteristic
110C22×#3622C2311yescharacteristic
111C22×#3622C2311yescharacteristic
112C4×Dic34—22no—
113C4×Dic34—22no—
114C17×#106—22no—
115C17×#103—22no—
116C17×#102—22no—
117C4×C8×C17—22no—
118#10587506—22no—
119#10587503—22no—
120#10587503C2×C414yes—
121#10587502—21no—
122#10587502—21no—
123#10587502—21no—
124#10587502—21no—
125#10587502—21no—
126#10587502D411yescharacteristic
127#10587502D411yescharacteristic
128#10587502D411yescharacteristic
129#10587502D411yescharacteristic
130#10587487—22no—
131#10587487C2×C414yescomplemented
132#10587485—22no—
133#10587485C2×C414yescomplemented
134#10587481—22no—
135#10587479—22no—
136#10587474—22no—
137C24×C17—21no—
138C2×Q8×C17—21no—
139C2×Q8×C17—21no—
140C2×Q8×C17—21no—
141C2×Q8×C17C22×C412yes—
142C2×Q8×C17C2×D412yes—
143C2×Q8×C17#4112yes—
144C2×D4×C17—21no—
145C2×D4×C17—21no—
146C2×D4×C17—21no—
147C2×D4×C17—21no—
148C2×D4×C17—21no—
149C2×D4×C17C22×C412yes—
150C22×C4×C17—21no—
151C22×C4×C17—21no—
152C22×C4×C17—21no—
153C22×C4×C17—42no—
154C22×C4×C17—81no—
155C22×C4×C17—81no—
156C22×C4×C17C2×D411yescharacteristic
157C22×C4×C17C2×D411yescharacteristic
158C22×C4×C17C2411yescharacteristic
159C22×Dic17—21no—
160C22×Dic17—21no—
161C22×Dic17—21no—
162C22×Dic17—21no—
163C22×Dic17C2×D411yescharacteristic
164C22×Dic17C2×D411yescharacteristic
165C22×Dic17C2×D411yescharacteristic
166C22×Dic17C2×D411yescharacteristic
167C2×Dic34—21no—
168C2×Dic34—21no—
169C2×Dic34—21no—
170C2×Dic34—21no—
171C2×Dic34—21no—
172C2×Dic34C22×C412yes—
173SD16×C17—28no—
174C2×C8×C17—21no—
175C2×C8×C17—21no—
176C2×C8×C17—21no—
177C2×C8×C17—21no—
178C2×C8×C17—21no—
179C2×C8×C17C22×C412yes—
180C4⋊C4×C17—24no—
181C4⋊C4×C17—21no—
182C4⋊C4×C17—21no—
183C4⋊C4×C17—21no—
184C4⋊C4×C17C2×D412yes—
185C22⋊C4×C17—42no—
186C42×C17—24no—
187C42×C17C2×D412yes—
188C42×C17#4112yes—
189C22⋊Dic17—24no—
190C22⋊Dic17—24no—
191C22⋊Dic17—81no—
192C22⋊Dic17—81no—
193#63231—28no—
194C4⋊Dic17—24no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—24no—
198C4⋊Dic17—21no—
199C4⋊Dic17—21no—
200C4⋊Dic17C22×C412yes—
201C4⋊Dic17#4112yes—
202#63225—42no—
203#63225—81no—
204#63225—81no—
205C4×Dic17—24no—
206C4×Dic17—21no—
207C4×Dic17—21no—
208C4×Dic17—24no—
209C4×Dic17—42no—
210C4×Dic17—81no—
211C4×Dic17—81no—
212C4×Dic17—81no—
213C4×Dic17C22×C412yes—
214C4×Dic17#4112yes—
215C2×#3622—21no—
216C2×#3622—21no—
217C2×#3622—21no—
218C2×#3622—21no—
219C2×#3622—21no—
220C2×#3622C22×C412yes—
221#21511—171nomaximal, Sylow
222C23×C17—22no—
223C23×C17—41no—
224C23×C17—41no—
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×C17C4×D414yes—
234D4×C17—22no—
235D4×C17—22no—
236D4×C17—41no—
237D4×C17—41no—
238C2×C4×C17—22no—
239C2×C4×C17—22no—
240C2×C4×C17—22no—
241C2×C4×C17—21no—
242C2×C4×C17—21no—
243C2×C4×C17—21no—
244C2×C4×C17—21no—
245C2×C4×C17—22no—
246C2×C4×C17—41no—
247C2×C4×C17—41no—
248C2×C4×C17—41no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17—84no—
252C2×C4×C17#12112yes—
253C2×C4×C17#12112yes—
254C2×C4×C17C23×C411yescharacteristic
255C2×C4×C17C22×D411yescharacteristic
256C2×C4×C17C2×#4111yescharacteristic
257C2×C4×C17C2×#4111yescharacteristic
258C2×C4×C17C2×#4111yescharacteristic
259C2×C4×C17C2×#4111yescharacteristic
260C2×Dic17—21no—
261C2×Dic17—21no—
262C2×Dic17—21no—
263C2×Dic17—21no—
264C2×Dic17—22no—
265C2×Dic17—22no—
266C2×Dic17—22no—
267C2×Dic17—21no—
268C2×Dic17—21no—
269C2×Dic17—21no—
270C2×Dic17—21no—
271C2×Dic17—21no—
272C2×Dic17—21no—
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—
282C2×Dic17C4×D412yes—
283C2×Dic17C4×D412yes—
284Dic34—22no—
285Dic34—22no—
286Dic34—41no—
287Dic34—41no—
288C8×C17—22no—
289C8×C17—41no—
290#3622—22no—
291#3622—41no—
292C2×#464—171no—
293C2×#441—171no—
294C2×C4×SD16D1711yescharacteristic
295#2326—171no—
296#2276—171no—
297#1983—171no—
298#1948—171no—
299#1927—171no—
300#1926—171no—
301#1880—171no—
302#1878—171no—
303#1803—171no—
304#1768—171no—
305#1762—171no—
306C4×#103—171no—
307C22×C17—42no—
308C22×C17—42no—
309C22×C17—42no—
310C22×C17—81no—
311C22×C17—81no—
312C22×C17—81no—
313C22×C17—81no—
314C22×C17C2×C4×D411yescharacteristic
315C22×C17C2×#12111yescharacteristic
316C22×C17#52511yescharacteristic
317C22×C17#57711yescharacteristic
318C22×C17#57711yescharacteristic
319C22×C17#57711yescharacteristic
320C22×C17#57811yescharacteristic
321C4×C17—22no—
322C4×C17—22no—
323C4×C17—24no—
324C4×C17—42no—
325C4×C17—81no—
326C4×C17C2×C4×D411yescharacteristic
327C4×C17C4×#4111yescharacteristic
328C4×C17#52911yescharacteristic
329C4×C17#53211yescharacteristic
330Dic17—22no—
331Dic17—22no—
332Dic17—41no—
333Dic17—41no—
334Dic17—42no—
335Dic17—42no—
336Dic17—42no—
337Dic17—81no—
338C22×Q16—171no—
339C22×SD16C2×D1711yescharacteristic
340C2×#124—171no—
341C2×C4×Q8—171no—
342C2×C4×Q8—171no—
343C2×C4×Q8—171no—
344C2×C4×Q8C2×D1711yescharacteristic
345C2×C4×D4C2×D1711yescharacteristic
346#464—178no—
347#441—344no—
348C4×SD16—24no—
349C2×#106—171no—
350C2×#106C2×D1711yescharacteristic
351C2×C4⋊C8—171no—
352C2×C4⋊C8—171no—
353C2×#103—171no—
354C2×#103—171no—
355C2×#103—171no—
356C2×#103—171no—
357C2×#103C2×D1711yescharacteristic
358C2×#102—171no—
359C2×#102—171no—
360C2×#102C2×D1711yescharacteristic
361C2×#97—171no—
362C2×C4×C8C2×D1711yescharacteristic
363#391—171no—
364#390—171no—
365#388—341no—
366#388—341no—
367#385—341no—
368#385—341no—
369#384—171no—
370C4×C4⋊C4—171no—
371C4×C4⋊C4—171no—
372C4×C4⋊C4—171no—
373C4×C4⋊C4—341no—
374C4×C4⋊C4—341no—
375C4×C4⋊C4—341no—
376#376—171no—
377#376—341no—
378#340—171no—
379#340—171no—
380#340—171no—
381#340—171no—
382#336—171no—
383#336—171no—
384C2×C17—82no—
385C2×C17C4×#12111yescharacteristic
386C2×C17#294711yescharacteristic
387C2×C17#295211yescharacteristic
388C2×C17#304311yescharacteristic
389C2×C17#304611yescharacteristic
390C2×C17#312111yescharacteristic
391C2×C17#312511yescharacteristic
392C22×Q8—171no—
393C22×Q8C22×D1711yescharacteristic
394C22×D4C22×D1711yescharacteristic
395C23×C4—21no—
396C2×Q16—174no—
397C2×Q16—342no—
398C2×Q16—342no—
399C2×SD16—22no—
400C2×SD16—22no—
401C2×SD16C4×D1714yes—
402C22×C8—171no—
403C22×C8—171no—
404C22×C8C22×D1711yescharacteristic
405C22×C8C22×D1711yescharacteristic
406#124—174no—
407#124—342no—
408C4×Q8—22no—
409C4×Q8—174no—
410C4×Q8—342no—
411C4×Q8—342no—
412C4×Q8—342no—
413C4×Q8—342no—
414C4×Q8—342no—
415C4×Q8C22⋊D1714yes—
416C4×D4—22no—
417C4×D4—22no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—171no—
420C2×C4⋊C4—171no—
421C2×C4⋊C4—171no—
422C2×C4⋊C4—341no—
423C2×C4⋊C4—341no—
424C2×C4⋊C4—341no—
425C2×C4⋊C4—341no—
426C2×C4⋊C4—341no—
427C2×C4⋊C4C22×D1711yescharacteristic
428C2×C4⋊C4C22×D1711yescharacteristic
429C2×C22⋊C4—21no—
430C2×C22⋊C4—341no—
431C2×C22⋊C4—341no—
432C2×C22⋊C4—341no—
433C2×C22⋊C4—341no—
434C2×C42—21no—
435C2×C42—171no—
436C2×C42—171no—
437C2×C42—341no—
438C2×C42—341no—
439C2×C42—341no—
440C2×C42—341no—
441C2×C42—341no—
442C2×C42—341no—
443C2×C42C22×D1711yescharacteristic
444#106—22no—
445#106—342no—
446C4⋊C8—174no—
447C4⋊C8—342no—
448#103—22no—
449#103—174no—
450#103—342no—
451#103—342no—
452#103—342no—
453#102—22no—
454#102—174no—
455#102—342no—
456#97—342no—
457C4×C8—22no—
458#95—171no—
459#95—171no—
460#95—171no—
461#95—171no—
462#95—341no—
463#95—341no—
464#95—341no—
465#95—341no—
466#95—341no—
467C17#2151111yescharacteristic, complemented, Sylow
468C24—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—172no—
473C2×Q8—341no—
474C2×Q8—341no—
475C2×Q8—341no—
476C2×Q8—341no—
477C2×Q8—341no—
478C2×Q8C2×C4×D1712yes—
479C2×Q8#6325212yes—
480C2×Q8C2×C22⋊D1712yes—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4C2×C4×D1712yes—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—42no—
491C22×C4—81no—
492C22×C4—81no—
493C22×C4—171no—
494C22×C4—171no—
495C22×C4—171no—
496C22×C4—171no—
497C22×C4—341no—
498C22×C4—341no—
499C22×C4—341no—
500C22×C4—341no—
501C22×C4D4×D1711yescharacteristic
502C22×C4D4×D1711yescharacteristic
503C22×C4C23×D1711yescharacteristic
504Q16—348no—
505SD16—28no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—172no—
512C2×C8—341no—
513C2×C8—341no—
514C2×C8—341no—
515C2×C8—341no—
516C2×C8—341no—
517C2×C8C2×C4×D1712yes—
518C4⋊C4—21no—
519C4⋊C4—24no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—172no—
523C4⋊C4—172no—
524C4⋊C4—344no—
525C4⋊C4—344no—
526C4⋊C4—341no—
527C4⋊C4—341no—
528C4⋊C4—341no—
529C4⋊C4—341no—
530C4⋊C4—682no—
531C4⋊C4—1361no—
532C4⋊C4—1361no—
533C4⋊C4C2×C22⋊D1712yes—
534C22⋊C4—42no—
535C22⋊C4—344no—
536C22⋊C4—344no—
537C22⋊C4—1361no—
538C22⋊C4—1361no—
539C42—24no—
540C42—172no—
541C42—172no—
542C42—344no—
543C42—344no—
544C42—341no—
545C42—341no—
546C42—682no—
547C42—1361no—
548C42—1361no—
549C42—1361no—
550C42#6325412yes—
551C42C2×C22⋊D1712yes—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23C2×D4×D1711yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—22no—
563Q8—342no—
564Q8—342no—
565Q8—681no—
566Q8—681no—
567Q8C4×C22⋊D1714yes—
568D4—22no—
569D4—22no—
570D4—41no—
571D4—41no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—84no—
586C2×C4—172no—
587C2×C4—172no—
588C2×C4—341no—
589C2×C4—341no—
590C2×C4—341no—
591C2×C4—341no—
592C2×C4—342no—
593C2×C4—342no—
594C2×C4—341no—
595C2×C4—342no—
596C2×C4—341no—
597C2×C4—341no—
598C2×C4—341no—
599C2×C4—341no—
600C2×C4—341no—
601C2×C4—681no—
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#1058762512yes—
611C2×C4#1058762512yes—
612C2×C4C22×C4×D1711yescharacteristic
613C2×C4C2×#6325211yescharacteristic
614C2×C4C2×#6325411yescharacteristic
615C2×C4#41×D1711yescharacteristic
616C2×C4#41×D1711yescharacteristic
617C2×C4C22×C22⋊D1711yescharacteristic
618C8—22no—
619C8—41no—
620C8—342no—
621C8—681no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22C4×D4×D1711yescharacteristic
630C22C2×#1058762511yescharacteristic
631C22#1176331811yescharacteristic
632C22#1176361511yescharacteristic
633C22#1176361511yescharacteristic
634C22#1176361611yescharacteristic
635C22#1176362311yescharacteristic
636C4—22no—
637C4—22no—
638C4—24no—
639C4—42no—
640C4—81no—
641C4—342no—
642C4—342no—
643C4—681no—
644C4—681no—
645C4—682no—
646C4—682no—
647C4—682no—
648C4—1361no—
649C4C2×C4×C22⋊D1711yescharacteristic
650C4C4×#6325411yescharacteristic
651C4#1176333911yescharacteristic
652C4#1176335511yescharacteristic
653C2—82no—
654C2C4×#1058762511yescharacteristic
655C2#1453762011yescharacteristic
656C2#1453766211yescharacteristic
657C2#1453830211yescharacteristic
658C2#1453833211yescharacteristic
659C2#1453900911yescharacteristic
660C2#1453906711yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C4×SD16×C17C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenterC2×C4×SD16D17
hypocenterC17#21511
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
2. C2×#11762914C2semidirect
4. #14531149C2semidirect
6. #14529544C2semidirect
7. #14529417C2semidirect
10. #14529123C2semidirect
11. #14528787C2semidirect
12. #14528642C2semidirect
14. C4×#10587487C2semidirect
21. #11763029C4semidirect
88. Q8×Dic17C2×C4semidirect
91. C2×#63231C2×C4semidirect
131. #10587487C2×C4semidirect
133. #10587485C2×C4semidirect
467. C17#21511semidirect

Maximal subgroups

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

subgroupindex
C2×#117630292
C2×#117629142
#145314962
#145311492
#145297152
#145295442
#145294172
#145294142
#145291412
#145291232
#145287872
#145286422
#145286092
C4×#105874872
C2×C4×SD16×C172
#2151117