Finite Group Databasebeta

← #17091493

Subgroups — #17091493

Order 4352.

946 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763119C211yesmaximal, characteristic, complemented
2C22×#10587487C211yesmaximal, characteristic
3#14532509C211yesmaximal, characteristic
4#14531616C211yesmaximal, characteristic
5#14530333C211yesmaximal, characteristic, complemented
6#14530180C211yesmaximal, characteristic, complemented
7#14530160C211yesmaximal, characteristic, complemented
8#14529934C211yesmaximal, characteristic, complemented
9#14529049C211yesmaximal, characteristic, complemented
10#14529039C211yesmaximal, characteristic, complemented
11#14528971C211yesmaximal, characteristic
12#14528962C211yesmaximal, characteristic
13#14528798C211yesmaximal, characteristic, complemented
14#14528718C211yesmaximal, characteristic
15C17×#1807C211yesmaximal, characteristic
16C23×Dic34C2211yescharacteristic
17C2×#10587576C2211yescharacteristic
18C2×#10587573C2211yescharacteristic
19C2×#10587573C2211yescharacteristic
20C22×C4⋊Dic17C2211yescharacteristic
21#11763119—24no—
22C2×#10587478C2211yescharacteristic
23C2×#10587478C2211yescharacteristic
24C2×#10587485C2211yescharacteristic
25C2×#10587485C2211yescharacteristic
26C2×#10587487—24no—
27C2×#10587487C2211yescharacteristic
28C2×#10587487C2211yescharacteristic
29C2×#10587487C2211yescharacteristic
30C2×#10587487C2211yescharacteristic
31C2×#10587481C2211yescharacteristic
32C2×#10587481C2211yescharacteristic
33C2×#10587481C2211yescharacteristic
34C2×#10587481C2211yescharacteristic
35C2×C22⋊#3622C2211yescharacteristic
36#11762751—21no—
37#11762741—21no—
38#11762714C2211yescharacteristic
39#11762714C2211yescharacteristic
40#11762713—21no—
41#11762713—21no—
42#11762708C2211yescharacteristic
43#11762705C2211yescharacteristic
44#11762684—21no—
45#11762674C2211yescharacteristic
46#11762667—21no—
47#11762667—21no—
48C22⋊C4×Dic17—21no—
49C4×C22⋊Dic17C2211yescharacteristic
50#11762571C2211yescharacteristic
51#11762571C2211yescharacteristic
52#11762563C2211yescharacteristic
53#11762563C2211yescharacteristic
54C23×C8×C17C2211yescharacteristic
55C2×C17×#106C2211yescharacteristic
56C2×C17×#106C2211yescharacteristic
57C17×#380C2211yescharacteristic
58C17×#380C2211yescharacteristic
59C17×#340C2211yescharacteristic
60C17×#340C2211yescharacteristic
61C23×C4×C17C2311yescharacteristic
62C23×Dic17—21no—
63C23×Dic17—21no—
64C22×Dic34—22no—
65C22×Dic34—22no—
66C22×Dic34—21no—
67C22×Dic34—22no—
68C22×Dic34—41no—
69C22×Dic34—41no—
70C22×Dic34Q814yes—
71C22×Dic34C2311yescharacteristic
72C22×Dic34C2311yescharacteristic
73C22×C8×C17—22no—
74C22×C8×C17—22no—
75C22×C8×C17—41no—
76C22×C8×C17C2311yescharacteristic
77C22×C8×C17C2311yescharacteristic
78C2×C4⋊C4×C17C2311yescharacteristic
79C2×C4⋊C4×C17C2311yescharacteristic
80C2×C4⋊C4×C17C2311yescharacteristic
81C2×C4⋊C4×C17C2311yescharacteristic
82C2×C22⋊C4×C17—21no—
83C2×C22⋊C4×C17—21no—
84C2×C22⋊Dic17D411yescharacteristic
85C2×C22⋊Dic17D411yescharacteristic
86C2×C4⋊Dic17—21no—
87C2×C4⋊Dic17—21no—
88C2×C4⋊Dic17—21no—
89C2×C4⋊Dic17—22no—
90C2×C4⋊Dic17—41no—
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×#63225—21no—
94C2×#63225—21no—
95C2×#63225—21no—
96C2×#63225—21no—
97C2×C4×Dic17—21no—
98C2×C4×Dic17—21no—
99C2×C4×Dic17C2311yescharacteristic
100C2×C4×Dic17C2311yescharacteristic
101C22×#3622C2311yescharacteristic
102C22×#3622C2311yescharacteristic
103#10587576—22no—
104#10587576—41no—
105#10587573—22no—
106#10587573—22no—
107#10587573—41no—
108#10587573—41no—
109C17×#106—22no—
110C17×#106—41no—
111C17×#95—21no—
112C17×#95—21no—
113#10587502—21no—
114#10587502—21no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502D411yescharacteristic
122#10587502D411yescharacteristic
123C22⋊#3622—22no—
124#10587487—22no—
125#10587487—22no—
126#10587487—22no—
127#10587487D414yes—
128#10587485—22no—
129#10587485—41no—
130#10587481—22no—
131#10587481—22no—
132#10587481—41no—
133#10587481—41no—
134#10587478—22no—
135#10587478—41no—
136C24×C17C2×D411yescharacteristic
137C22×C4×C17—21no—
138C22×C4×C17—21no—
139C22×C4×C17—21no—
140C22×C4×C17—21no—
141C22×C4×C17—21no—
142C22×C4×C17—21no—
143C22×C4×C17—21no—
144C22×C4×C17—21no—
145C22×C4×C17C2×D411yescharacteristic
146C22×C4×C17C2×Q812yes—
147C22×C4×C17#4112yes—
148C22×C4×C17C2411yescharacteristic
149C22×Dic17—21no—
150C22×Dic17—24no—
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17—42no—
155C22×Dic17—42no—
156C22×Dic17—42no—
157C22×Dic17—42no—
158C22×Dic17—81no—
159C22×Dic17C2×D411yescharacteristic
160C22×Dic17C2×D411yescharacteristic
161C22×Dic17C2×D411yescharacteristic
162C22×Dic17C2×D411yescharacteristic
163C2×Dic34—21no—
164C2×Dic34—21no—
165C2×Dic34—21no—
166C2×Dic34—21no—
167C2×Dic34—21no—
168C2×Dic34—21no—
169C2×Dic34—21no—
170C2×Dic34—28no—
171C2×Dic34—24no—
172C2×Dic34—21no—
173C2×Dic34—44no—
174C2×Dic34—44no—
175C2×Dic34—44no—
176C2×Dic34—42no—
177C2×Dic34—42no—
178C2×Dic34C2×D412yes—
179C2×Dic34C2×Q812yes—
180C2×Dic34C2×Q812yes—
181C2×Dic34#4112yes—
182C2×C8×C17—21no—
183C2×C8×C17—21no—
184C2×C8×C17—21no—
185C2×C8×C17—21no—
186C2×C8×C17—21no—
187C2×C8×C17—44no—
188C2×C8×C17C2×D412yes—
189C4⋊C4×C17—21no—
190C4⋊C4×C17—21no—
191C4⋊C4×C17—21no—
192C4⋊C4×C17—21no—
193C4⋊C4×C17—21no—
194C4⋊C4×C17—21no—
195C4⋊C4×C17—21no—
196C4⋊C4×C17—21no—
197C22⋊C4×C17—42no—
198C22⋊C4×C17—42no—
199C22⋊Dic17—22no—
200C22⋊Dic17—22no—
201C4⋊Dic17—21no—
202C4⋊Dic17—21no—
203C4⋊Dic17—21no—
204C4⋊Dic17—42no—
205C4⋊Dic17—42no—
206C4⋊Dic17—81no—
207C4⋊Dic17C2×D412yes—
208#63225—42no—
209#63225—42no—
210#63225—42no—
211#63225—81no—
212C4×Dic17—21no—
213C4×Dic17—21no—
214C4×Dic17—21no—
215C4×Dic17—21no—
216C4×Dic17—42no—
217C4×Dic17—42no—
218C2×#3622—21no—
219C2×#3622—21no—
220C2×#3622—21no—
221C2×#3622—21no—
222C2×#3622—21no—
223C2×#3622—21no—
224#25957—171nomaximal, Sylow
225C23×C17—21no—
226C23×C17—21no—
227C23×C17—21no—
228C23×C17—21no—
229C23×C17C22⋊Q812yes—
230C23×C17#13512yes—
231C23×C17#13712yes—
232C23×C17C22×D411yescharacteristic
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—22no—
236C2×C4×C17—22no—
237C2×C4×C17—22no—
238C2×C4×C17—22no—
239C2×C4×C17—41no—
240C2×C4×C17—41no—
241C2×C4×C17—41no—
242C2×C4×C17—41no—
243C2×C4×C17—41no—
244C2×C4×C17—41no—
245C2×C4×C17—41no—
246C2×C4×C17—41no—
247C2×C4×C17—41no—
248C2×C4×C17—41no—
249C2×C4×C17—41no—
250C2×C4×C17—41no—
251C2×C4×C17C22⋊Q812yes—
252C2×C4×C17C22×D411yescharacteristic
253C2×C4×C17C22×Q811yescharacteristic
254C2×C4×C17C2×#4111yescharacteristic
255C2×C4×C17C2×#4111yescharacteristic
256C2×C4×C17H(5,𝔽2)11yescharacteristic
257C2×C4×C17H(5,𝔽2)11yescharacteristic
258C2×Dic17—22no—
259C2×Dic17—22no—
260C2×Dic17—22no—
261C2×Dic17—22no—
262C2×Dic17—22no—
263C2×Dic17—21no—
264C2×Dic17—21no—
265C2×Dic17—21no—
266C2×Dic17—21no—
267C2×Dic17—21no—
268C2×Dic17—21no—
269C2×Dic17—21no—
270C2×Dic17—21no—
271C2×Dic17—21no—
272C2×Dic17—21no—
273C2×Dic17—21no—
274C2×Dic17—21no—
275C2×Dic17—48no—
276C2×Dic17—41no—
277C2×Dic17—41no—
278C2×Dic17—41no—
279C2×Dic17—41no—
280C2×Dic17—41no—
281C2×Dic17—41no—
282C2×Dic17—41no—
283C2×Dic17—84no—
284Dic34—22no—
285Dic34—22no—
286Dic34—22no—
287Dic34—22no—
288Dic34—22no—
289Dic34—48no—
290Dic34—41no—
291Dic34—41no—
292Dic34C22⋊Q814yes—
293C8×C17—22no—
294C8×C17—41no—
295#3622—41no—
296#3622—41no—
297C2×#479—171no—
298C22×#103—171no—
299#2442—171no—
300Q8×C22⋊C4—171no—
301#2097—171no—
302#2068—171no—
303#2066—171no—
304#2026—171no—
305#1865—171no—
306#1863—171no—
307#1852—171no—
308#1850—171no—
309#1807D1711yescharacteristic
310#1805—171no—
311#1787—171no—
312C22×C17—22no—
313C22×C17—22no—
314C22×C17—22no—
315C22×C17—22no—
316C22×C17—22no—
317C22×C17—22no—
318C22×C17#47914yescomplemented
319C22×C17C2×C22⋊Q811yescharacteristic
320C22×C17D4211yescharacteristic
321C22×C17#54611yescharacteristic
322C22×C17C2×#13511yescharacteristic
323C22×C17C2×#13711yescharacteristic
324C22×C17#57711yescharacteristic
325C22×C17#57811yescharacteristic
326C4×C17—22no—
327C4×C17—81no—
328C4×C17—81no—
329C4×C17—81no—
330C4×C17—81no—
331C4×C17C2×C22⋊Q811yescharacteristic
332C4×C17#54311yescharacteristic
333C4×C17#54611yescharacteristic
334C4×C17#55111yescharacteristic
335Dic17—24no—
336Dic17—42no—
337Dic17—41no—
338Dic17—41no—
339Dic17—42no—
340Dic17—41no—
341Dic17—41no—
342Dic17—81no—
343C23×Q8—171no—
344C23×C8C2×D1711yescharacteristic
345C2×#125—171no—
346C2×C4×Q8—171no—
347C2×C4×Q8—171no—
348C22×C4⋊C4—171no—
349#479—344no—
350C2×#107—171no—
351C2×#107—171no—
352C2×#106C2×D1711yescharacteristic
353C2×#106C2×D1711yescharacteristic
354C2×C4⋊C8—171no—
355C2×C4⋊C8—171no—
356C2×#103—171no—
357C2×#103—171no—
358C2×#103—171no—
359C2×#103—171no—
360C2×#103—171no—
361C2×#103—171no—
362C2×#103—171no—
363C2×#103—171no—
364C2×#103—344no—
365C2×C22⋊C8—171no—
366#400—341no—
367#398—341no—
368#391—171no—
369#391—171no—
370#391—341no—
371#391—341no—
372#389—171no—
373#389—171no—
374#385—341no—
375#382—171no—
376#380—341no—
377#380—341no—
378#380C2×D1711yescharacteristic
379#380C2×D1711yescharacteristic
380C4×C22⋊C4—171no—
381C4×C22⋊C4—341no—
382#340—171no—
383#340—171no—
384#340C2×D1711yescharacteristic
385#340C2×D1711yescharacteristic
386#336—171no—
387#336—171no—
388C2×C17—24no—
389C2×C17#243711yescharacteristic
390C2×C17C2×#47911yescharacteristic
391C2×C17#308211yescharacteristic
392C2×C17#327011yescharacteristic
393C2×C17#328211yescharacteristic
394C2×C17#330011yescharacteristic
395C2×C17#332011yescharacteristic
396C22×Q8—174no—
397C22×Q8—171no—
398C22×Q8—171no—
399C22×Q8—342no—
400C22×Q8—342no—
401C22×Q8—342no—
402C22×Q8—341no—
403C22×Q8—681no—
404C22×Q8—681no—
405C23×C4—341no—
406C23×C4—341no—
407C23×C4C22×D1711yescharacteristic
408C22×C8—22no—
409C22×C8—22no—
410C22×C8—41no—
411C22×C8—171no—
412C22×C8—171no—
413C22×C8C22×D1711yescharacteristic
414C22×C8C22×D1711yescharacteristic
415#125—342no—
416#125—681no—
417C4×Q8—342no—
418C4×Q8—342no—
419C4×Q8—681no—
420C4×Q8—681no—
421C2×C4⋊C4—171no—
422C2×C4⋊C4—171no—
423C2×C4⋊C4—342no—
424C2×C4⋊C4—341no—
425C2×C4⋊C4—341no—
426C2×C4⋊C4—341no—
427C2×C4⋊C4—341no—
428C2×C4⋊C4—341no—
429C2×C4⋊C4—341no—
430C2×C4⋊C4—341no—
431C2×C4⋊C4—681no—
432C2×C4⋊C4C22×D1711yescharacteristic
433C2×C4⋊C4C22×D1711yescharacteristic
434C2×C4⋊C4C22×D1711yescharacteristic
435C2×C4⋊C4C22×D1711yescharacteristic
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—171no—
439C2×C22⋊C4—171no—
440C2×C42—171no—
441C2×C42—171no—
442C2×C42—341no—
443C2×C42—341no—
444#107—342no—
445#107—681no—
446#106—22no—
447#106—41no—
448C4⋊C8—342no—
449C4⋊C8—681no—
450#103—174no—
451#103—342no—
452#103—342no—
453#103—342no—
454#103—342no—
455#103—342no—
456#103—681no—
457#103—681no—
458C22⋊C8—342no—
459#95—21no—
460#95—21no—
461#95—171no—
462#95—171no—
463#95—341no—
464#95—341no—
465#95—341no—
466#95—341no—
467#95—341no—
468#95—341no—
469#95—341no—
470#95—341no—
471C17#2595711yescharacteristic, complemented, Sylow
472C24D4×D1711yescharacteristic
473C2×Q8—172no—
474C2×Q8—172no—
475C2×Q8—172no—
476C2×Q8—172no—
477C2×Q8—348no—
478C2×Q8—341no—
479C2×Q8—341no—
480C2×Q8—341no—
481C2×Q8—341no—
482C2×Q8—344no—
483C2×Q8—341no—
484C2×Q8—341no—
485C2×Q8—341no—
486C2×Q8—341no—
487C2×Q8—682no—
488C2×Q8—682no—
489C2×Q8—684no—
490C2×Q8—684no—
491C2×Q8—684no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—171no—
501C22×C4—171no—
502C22×C4—171no—
503C22×C4—171no—
504C22×C4—341no—
505C22×C4—341no—
506C22×C4—341no—
507C22×C4—344no—
508C22×C4—341no—
509C22×C4—682no—
510C22×C4—682no—
511C22×C4—682no—
512C22×C4—682no—
513C22×C4—1361no—
514C22×C4#6325212yes—
515C22×C4D4×D1711yescharacteristic
516C22×C4Q8×D1712yes—
517C22×C4C23×D1711yescharacteristic
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—44no—
524C2×C8—341no—
525C2×C8—341no—
526C2×C8—341no—
527C2×C8—341no—
528C2×C8—341no—
529C2×C8—341no—
530C2×C8D4×D1712yes—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—172no—
540C4⋊C4—341no—
541C4⋊C4—341no—
542C4⋊C4—341no—
543C4⋊C4—682no—
544C4⋊C4—682no—
545C4⋊C4—682no—
546C4⋊C4—682no—
547C4⋊C4—682no—
548C4⋊C4—1361no—
549C4⋊C4—1361no—
550C22⋊C4—42no—
551C22⋊C4—42no—
552C22⋊C4—342no—
553C22⋊C4—342no—
554C42—341no—
555C42—341no—
556C42—341no—
557C42—341no—
558C42—682no—
559C42—682no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23#1058758212yes—
565C23#1058760112yes—
566C23#1058760212yes—
567C23C2×D4×D1711yescharacteristic
568Q8—174no—
569Q8—342no—
570Q8—342no—
571Q8—342no—
572Q8—342no—
573Q8—342no—
574Q8—688no—
575Q8—681no—
576Q8—681no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—342no—
596C2×C4—342no—
597C2×C4—342no—
598C2×C4—342no—
599C2×C4—341no—
600C2×C4—341no—
601C2×C4—341no—
602C2×C4—341no—
603C2×C4—341no—
604C2×C4—341no—
605C2×C4—341no—
606C2×C4—341no—
607C2×C4—341no—
608C2×C4—341no—
609C2×C4—342no—
610C2×C4—341no—
611C2×C4—341no—
612C2×C4—688no—
613C2×C4—681no—
614C2×C4—681no—
615C2×C4—681no—
616C2×C4—681no—
617C2×C4—681no—
618C2×C4—681no—
619C2×C4—681no—
620C2×C4—1364no—
621C2×C4#1058758212yes—
622C2×C4C2×#6325211yescharacteristic
623C2×C4C2×D4×D1711yescharacteristic
624C2×C4#1058769111yescharacteristic
625C2×C4C2×Q8×D1711yescharacteristic
626C2×C4#41×D1711yescharacteristic
627C2×C4#1058769611yescharacteristic
628C8—22no—
629C8—41no—
630C8—681no—
631C8—681no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22#1176311914yescomplemented
639C22C2×#1058758211yescharacteristic
640C22#1176344311yescharacteristic
641C22#1176344811yescharacteristic
642C22C2×#1058760211yescharacteristic
643C22C2×#1058760111yescharacteristic
644C22#1176361611yescharacteristic
645C22#1176362311yescharacteristic
646C4—22no—
647C4—81no—
648C4—81no—
649C4—81no—
650C4—81no—
651C4—344no—
652C4—681no—
653C4—682no—
654C4—681no—
655C4—681no—
656C4—681no—
657C4—682no—
658C4—1361no—
659C4C22⋊Q8×D1711yescharacteristic
660C4#1176343411yescharacteristic
661C4#1176344911yescharacteristic
662C4#1176348311yescharacteristic
663C2—24no—
664C2#1453245211yescharacteristic
665C2C2×#1176311911yescharacteristic
666C2#1453862911yescharacteristic
667C2#1454035111yescharacteristic
668C2#1454048811yescharacteristic
669C2#1454067711yescharacteristic
670C2#1454096111yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#1807C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#1807D17
hypocenterC17#25957
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1807

Complements

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

NH ≅ G/Ntype
1. C2×#11763119C2semidirect
5. #14530333C2semidirect
6. #14530180C2semidirect
7. #14530160C2semidirect
8. #14529934C2semidirect
9. #14529049C2semidirect
10. #14529039C2semidirect
13. #14528798C2semidirect
318. C22×C17#479semidirect
471. C17#25957semidirect
638. C22#11763119semidirect

Maximal subgroups

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

subgroupindex
C2×#117631192
C22×#105874872
#145325092
#145316162
#145303332
#145301802
#145301602
#145299342
#145290492
#145290392
#145289712
#145289622
#145287982
#145287182
C17×#18072
#2595717