Finite Group Databasebeta

← #17091463

Subgroups — #17091463

Order 4352.

990 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763093C211yesmaximal, characteristic, complemented
2C22×#10587487C211yesmaximal, characteristic
3#14532487C211yesmaximal, characteristic
4#14531616C211yesmaximal, characteristic
5#14530324C211yesmaximal, characteristic, complemented
6#14530166C211yesmaximal, characteristic, complemented
7#14530160C211yesmaximal, characteristic, complemented
8#14529924C211yesmaximal, characteristic, complemented
9#14529053C211yesmaximal, characteristic, complemented
10#14529049C211yesmaximal, characteristic, complemented
11#14528976C211yesmaximal, characteristic
12#14528962C211yesmaximal, characteristic
13#14528788C211yesmaximal, characteristic, complemented
14#14528718C211yesmaximal, characteristic
15C17×#1807C211yesmaximal, characteristic
16C23×Dic34C2211yescharacteristic
17C2×#10587574C2211yescharacteristic
18C2×#10587564—21no—
19C2×#10587573C2211yescharacteristic
20C2×#10587573C2211yescharacteristic
21C22×C4⋊Dic17C2211yescharacteristic
22#11763093—24no—
23C2×#10587479C2211yescharacteristic
24C2×#10587479C2211yescharacteristic
25C2×#10587485C2211yescharacteristic
26C2×#10587485C2211yescharacteristic
27C2×#10587487—24no—
28C2×#10587487C2211yescharacteristic
29C2×#10587487C2211yescharacteristic
30C2×#10587487C2211yescharacteristic
31C2×#10587487C2211yescharacteristic
32C2×#10587481C2211yescharacteristic
33C2×#10587481C2211yescharacteristic
34C2×#10587481C2211yescharacteristic
35C2×#10587481C2211yescharacteristic
36C2×C22⋊#3622C2211yescharacteristic
37#11762743—21no—
38#11762730—21no—
39#11762714C2211yescharacteristic
40#11762714C2211yescharacteristic
41#11762713—21no—
42#11762713—21no—
43#11762708C2211yescharacteristic
44#11762705C2211yescharacteristic
45#11762681—21no—
46#11762678C2211yescharacteristic
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—21no—
69C22×Dic34—41no—
70C22×Dic34—41no—
71C22×Dic34Q814yescomplemented
72C22×Dic34C2311yescharacteristic
73C22×Dic34C2311yescharacteristic
74C22×C8×C17—22no—
75C22×C8×C17—22no—
76C22×C8×C17—41no—
77C22×C8×C17C2311yescharacteristic
78C22×C8×C17C2311yescharacteristic
79C2×C4⋊C4×C17C2311yescharacteristic
80C2×C4⋊C4×C17C2311yescharacteristic
81C2×C4⋊C4×C17C2311yescharacteristic
82C2×C4⋊C4×C17C2311yescharacteristic
83C2×C22⋊C4×C17—21no—
84C2×C22⋊C4×C17—21no—
85C2×C22⋊Dic17D411yescharacteristic
86C2×C22⋊Dic17D411yescharacteristic
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×#63225—21no—
98C2×#63225—21no—
99C2×C4×Dic17—21no—
100C2×C4×Dic17—21no—
101C2×C4×Dic17C2311yescharacteristic
102C2×C4×Dic17C2311yescharacteristic
103C22×#3622C2311yescharacteristic
104C22×#3622C2311yescharacteristic
105#10587574—22no—
106#10587574—41no—
107#10587573—22no—
108#10587573—22no—
109#10587573—41no—
110#10587573—41no—
111#10587564—44no—
112C17×#106—22no—
113C17×#106—41no—
114C17×#95—21no—
115C17×#95—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502D411yescharacteristic
123#10587502D411yescharacteristic
124C22⋊#3622—22no—
125#10587487—22no—
126#10587487—22no—
127#10587487—22no—
128#10587487D414yes—
129#10587485—22no—
130#10587485—41no—
131#10587481—22no—
132#10587481—22no—
133#10587481—41no—
134#10587481—41no—
135#10587479—22no—
136#10587479—41no—
137C24×C17C2×D411yescharacteristic
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×C17—21no—
146C22×C4×C17C2×D411yescharacteristic
147C22×C4×C17C2×Q812yes—
148C22×C4×C17#4112yes—
149C22×C4×C17C2411yescharacteristic
150C22×Dic17—21no—
151C22×Dic17—24no—
152C22×Dic17—21no—
153C22×Dic17—21no—
154C22×Dic17—21no—
155C22×Dic17—42no—
156C22×Dic17—42no—
157C22×Dic17—42no—
158C22×Dic17—42no—
159C22×Dic17—81no—
160C22×Dic17C2×D411yescharacteristic
161C22×Dic17C2×D411yescharacteristic
162C22×Dic17C2×D411yescharacteristic
163C22×Dic17C2×D411yescharacteristic
164C2×Dic34—21no—
165C2×Dic34—21no—
166C2×Dic34—21no—
167C2×Dic34—21no—
168C2×Dic34—21no—
169C2×Dic34—21no—
170C2×Dic34—21no—
171C2×Dic34—28no—
172C2×Dic34—24no—
173C2×Dic34—21no—
174C2×Dic34—44no—
175C2×Dic34—44no—
176C2×Dic34—44no—
177C2×Dic34—42no—
178C2×Dic34—42no—
179C2×Dic34—42no—
180C2×Dic34—42no—
181C2×Dic34—81no—
182C2×Dic34C2×D412yes—
183C2×Dic34C2×Q812yes—
184C2×Dic34C2×Q812yes—
185C2×Dic34#4112yes—
186C2×C8×C17—21no—
187C2×C8×C17—21no—
188C2×C8×C17—21no—
189C2×C8×C17—21no—
190C2×C8×C17—21no—
191C2×C8×C17—44no—
192C2×C8×C17C2×D412yes—
193C4⋊C4×C17—21no—
194C4⋊C4×C17—21no—
195C4⋊C4×C17—21no—
196C4⋊C4×C17—21no—
197C4⋊C4×C17—21no—
198C4⋊C4×C17—21no—
199C4⋊C4×C17—21no—
200C4⋊C4×C17—21no—
201C22⋊C4×C17—42no—
202C22⋊C4×C17—42no—
203C22⋊Dic17—22no—
204C22⋊Dic17SD1614yes—
205C4⋊Dic17—21no—
206C4⋊Dic17—21no—
207C4⋊Dic17—21no—
208C4⋊Dic17—42no—
209C4⋊Dic17—42no—
210C4⋊Dic17C2×D412yes—
211#63225—42no—
212#63225—42no—
213#63225—42no—
214#63225—42no—
215#63225—42no—
216#63225—81no—
217C4×Dic17—21no—
218C4×Dic17—21no—
219C4×Dic17—21no—
220C4×Dic17—21no—
221C4×Dic17—42no—
222C4×Dic17—42no—
223C2×#3622—21no—
224C2×#3622—21no—
225C2×#3622—21no—
226C2×#3622—21no—
227C2×#3622—21no—
228C2×#3622—21no—
229#25955—171nomaximal, Sylow
230C23×C17—21no—
231C23×C17—21no—
232C23×C17—21no—
233C23×C17—21no—
234C23×C17C22⋊Q812yes—
235C23×C17C2×SD1612yes—
236C23×C17#13712yes—
237C23×C17C22×D411yescharacteristic
238C2×C4×C17—21no—
239C2×C4×C17—21no—
240C2×C4×C17—22no—
241C2×C4×C17—22no—
242C2×C4×C17—22no—
243C2×C4×C17—22no—
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×C17—41no—
252C2×C4×C17—41no—
253C2×C4×C17—41no—
254C2×C4×C17—41no—
255C2×C4×C17—41no—
256C2×C4×C17C22⋊Q812yes—
257C2×C4×C17C22×D411yescharacteristic
258C2×C4×C17C22×Q811yescharacteristic
259C2×C4×C17C2×#4111yescharacteristic
260C2×C4×C17C2×#4111yescharacteristic
261C2×C4×C17H(5,𝔽2)11yescharacteristic
262C2×C4×C17H(5,𝔽2)11yescharacteristic
263C2×Dic17—22no—
264C2×Dic17—22no—
265C2×Dic17—21no—
266C2×Dic17—22no—
267C2×Dic17—21no—
268C2×Dic17—21no—
269C2×Dic17—21no—
270C2×Dic17—22no—
271C2×Dic17—22no—
272C2×Dic17—21no—
273C2×Dic17—21no—
274C2×Dic17—21no—
275C2×Dic17—21no—
276C2×Dic17—21no—
277C2×Dic17—21no—
278C2×Dic17—48no—
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×Dic17C2×SD1612yes—
288C2×Dic17C2×SD1612yes—
289Dic34—22no—
290Dic34—22no—
291Dic34—22no—
292Dic34—22no—
293Dic34—22no—
294Dic34—48no—
295Dic34—41no—
296Dic34—41no—
297Dic34—84no—
298Dic34C22⋊Q814yescomplemented
299C8×C17—22no—
300C8×C17—41no—
301#3622—41no—
302#3622—41no—
303C2×#475—171no—
304C22×#103—171no—
305#2440—171no—
306Q8×C22⋊C4—171no—
307#2096—171no—
308#2066—171no—
309#2066—171no—
310#2024—171no—
311#1865—171no—
312#1865—171no—
313#1853—171no—
314#1850—171no—
315#1807D1711yescharacteristic
316#1803—171no—
317#1787—171no—
318C22×C17—22no—
319C22×C17—22no—
320C22×C17—22no—
321C22×C17—22no—
322C22×C17—22no—
323C22×C17—22no—
324C22×C17#47514yescomplemented
325C22×C17C2×C22⋊Q811yescharacteristic
326C22×C17D4211yescharacteristic
327C22×C17#54611yescharacteristic
328C22×C17C22×SD1611yescharacteristic
329C22×C17C2×#13711yescharacteristic
330C22×C17#57711yescharacteristic
331C22×C17#57811yescharacteristic
332C4×C17—22no—
333C4×C17—81no—
334C4×C17—81no—
335C4×C17—81no—
336C4×C17—81no—
337C4×C17C2×C22⋊Q811yescharacteristic
338C4×C17#54311yescharacteristic
339C4×C17#54611yescharacteristic
340C4×C17#55111yescharacteristic
341Dic17—24no—
342Dic17—22no—
343Dic17—22no—
344Dic17—42no—
345Dic17—41no—
346Dic17—42no—
347Dic17—41no—
348Dic17—81no—
349C23×Q8—171no—
350C23×C8C2×D1711yescharacteristic
351C2×C4⋊Q8—171no—
352C2×C22⋊Q8—341no—
353C2×C4×Q8—171no—
354C2×C4×Q8—171no—
355C22×C4⋊C4—171no—
356#475—344no—
357C2×#106—171no—
358C2×#106—171no—
359C2×#106C2×D1711yescharacteristic
360C2×#106C2×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×#103—171no—
369C2×#103—171no—
370C2×#103—171no—
371C2×#103—344no—
372C2×C22⋊C8—171no—
373#398—341no—
374#395—341no—
375#391—171no—
376#391—171no—
377#391—341no—
378#391—341no—
379#389—171no—
380#389—171no—
381#385—341no—
382#384—171no—
383#380—341no—
384#380C2×D1711yescharacteristic
385#380C2×D1711yescharacteristic
386C4×C22⋊C4—171no—
387C4×C22⋊C4—341no—
388#340—171no—
389#340—171no—
390#340C2×D1711yescharacteristic
391#340C2×D1711yescharacteristic
392#336—171no—
393#336—171no—
394C2×C17—24no—
395C2×C17#243711yescharacteristic
396C2×C17C2×#47511yescharacteristic
397C2×C17#308211yescharacteristic
398C2×C17#327011yescharacteristic
399C2×C17D4×SD1611yescharacteristic
400C2×C17#329811yescharacteristic
401C2×C17#331611yescharacteristic
402C22×Q8—174no—
403C22×Q8—171no—
404C22×Q8—171no—
405C22×Q8—341no—
406C22×Q8—342no—
407C22×Q8—342no—
408C22×Q8—342no—
409C22×Q8—341no—
410C22×Q8—681no—
411C22×Q8—681no—
412C23×C4—341no—
413C23×C4—341no—
414C23×C4C22×D1711yescharacteristic
415C22×C8—22no—
416C22×C8—22no—
417C22×C8—41no—
418C22×C8—171no—
419C22×C8—171no—
420C22×C8C22×D1711yescharacteristic
421C22×C8C22×D1711yescharacteristic
422C4⋊Q8—342no—
423C4⋊Q8—681no—
424C22⋊Q8—684no—
425C4×Q8—342no—
426C4×Q8—342no—
427C4×Q8—681no—
428C4×Q8—681no—
429C2×C4⋊C4—171no—
430C2×C4⋊C4—171no—
431C2×C4⋊C4—342no—
432C2×C4⋊C4—341no—
433C2×C4⋊C4—341no—
434C2×C4⋊C4—341no—
435C2×C4⋊C4—341no—
436C2×C4⋊C4—341no—
437C2×C4⋊C4—341no—
438C2×C4⋊C4—341no—
439C2×C4⋊C4—341no—
440C2×C4⋊C4—681no—
441C2×C4⋊C4C22×D1711yescharacteristic
442C2×C4⋊C4C22×D1711yescharacteristic
443C2×C4⋊C4C22×D1711yescharacteristic
444C2×C4⋊C4C22×D1711yescharacteristic
445C2×C22⋊C4—21no—
446C2×C22⋊C4—21no—
447C2×C22⋊C4—171no—
448C2×C22⋊C4—171no—
449C2×C42—171no—
450C2×C42—171no—
451C2×C42—341no—
452C2×C42—341no—
453#106—22no—
454#106—41no—
455#106—342no—
456#106—681no—
457C4⋊C8—342no—
458C4⋊C8—681no—
459#103—174no—
460#103—342no—
461#103—342no—
462#103—342no—
463#103—342no—
464#103—342no—
465#103—681no—
466#103—681no—
467C22⋊C8—342no—
468#95—21no—
469#95—21no—
470#95—171no—
471#95—171no—
472#95—341no—
473#95—341no—
474#95—341no—
475#95—341no—
476#95—341no—
477#95—341no—
478C17#2595511yescharacteristic, complemented, Sylow
479C24D4×D1711yescharacteristic
480C2×Q8—172no—
481C2×Q8—172no—
482C2×Q8—172no—
483C2×Q8—172no—
484C2×Q8—348no—
485C2×Q8—341no—
486C2×Q8—341no—
487C2×Q8—341no—
488C2×Q8—341no—
489C2×Q8—344no—
490C2×Q8—341no—
491C2×Q8—341no—
492C2×Q8—341no—
493C2×Q8—341no—
494C2×Q8—682no—
495C2×Q8—682no—
496C2×Q8—684no—
497C2×Q8—684no—
498C2×Q8—684no—
499C2×Q8—682no—
500C2×Q8—682no—
501C2×Q8—1361no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—171no—
511C22×C4—171no—
512C22×C4—171no—
513C22×C4—171no—
514C22×C4—341no—
515C22×C4—341no—
516C22×C4—341no—
517C22×C4—344no—
518C22×C4—341no—
519C22×C4—682no—
520C22×C4—682no—
521C22×C4—682no—
522C22×C4—682no—
523C22×C4—1361no—
524C22×C4#6325212yes—
525C22×C4D4×D1711yescharacteristic
526C22×C4Q8×D1712yes—
527C22×C4C23×D1711yescharacteristic
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—44no—
534C2×C8—341no—
535C2×C8—341no—
536C2×C8—341no—
537C2×C8—341no—
538C2×C8—341no—
539C2×C8—341no—
540C2×C8D4×D1712yes—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—172no—
550C4⋊C4—341no—
551C4⋊C4—341no—
552C4⋊C4—341no—
553C4⋊C4—682no—
554C4⋊C4—682no—
555C4⋊C4—682no—
556C4⋊C4—682no—
557C4⋊C4—682no—
558C4⋊C4—682no—
559C4⋊C4—682no—
560C4⋊C4—1361no—
561C22⋊C4—42no—
562C22⋊C4—42no—
563C22⋊C4—174no—
564C22⋊C4—342no—
565C42—341no—
566C42—341no—
567C42—341no—
568C42—341no—
569C42—682no—
570C42—682no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23#1058758212yes—
576C23SD16×D1712yes—
577C23#1058760112yes—
578C23C2×D4×D1711yescharacteristic
579Q8—174no—
580Q8—342no—
581Q8—342no—
582Q8—342no—
583Q8—342no—
584Q8—342no—
585Q8—688no—
586Q8—681no—
587Q8—681no—
588Q8—1364no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—172no—
608C2×C4—172no—
609C2×C4—342no—
610C2×C4—341no—
611C2×C4—341no—
612C2×C4—341no—
613C2×C4—342no—
614C2×C4—341no—
615C2×C4—342no—
616C2×C4—342no—
617C2×C4—341no—
618C2×C4—341no—
619C2×C4—341no—
620C2×C4—341no—
621C2×C4—341no—
622C2×C4—342no—
623C2×C4—341no—
624C2×C4—688no—
625C2×C4—681no—
626C2×C4—681no—
627C2×C4—681no—
628C2×C4—681no—
629C2×C4—681no—
630C2×C4—681no—
631C2×C4—681no—
632C2×C4—1364no—
633C2×C4#1058758212yes—
634C2×C4C2×#6325211yescharacteristic
635C2×C4C2×D4×D1711yescharacteristic
636C2×C4#1058769111yescharacteristic
637C2×C4C2×Q8×D1711yescharacteristic
638C2×C4#41×D1711yescharacteristic
639C2×C4#1058769611yescharacteristic
640C8—22no—
641C8—41no—
642C8—681no—
643C8—681no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22#1176309314yescomplemented
651C22C2×#1058758211yescharacteristic
652C22#1176344311yescharacteristic
653C22#1176344811yescharacteristic
654C22C2×SD16×D1711yescharacteristic
655C22C2×#1058760111yescharacteristic
656C22#1176361611yescharacteristic
657C22#1176362311yescharacteristic
658C4—22no—
659C4—81no—
660C4—81no—
661C4—81no—
662C4—81no—
663C4—344no—
664C4—342no—
665C4—342no—
666C4—681no—
667C4—682no—
668C4—681no—
669C4—682no—
670C4—1361no—
671C4C22⋊Q8×D1711yescharacteristic
672C4#1176343411yescharacteristic
673C4#1176344911yescharacteristic
674C4#1176348311yescharacteristic
675C2—24no—
676C2#1453245211yescharacteristic
677C2C2×#1176309311yescharacteristic
678C2#1453862911yescharacteristic
679C2#1454035111yescharacteristic
680C2#1454047711yescharacteristic
681C2#1454064411yescharacteristic
682C2#1454090111yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutator, cosocleC22×C4×C17C24
FrattiniC22×C4C23×D17
FittingC17×#1807C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
hypercenter#1807D17
hypocenterC17#25955
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×#11763093C2semidirect
5. #14530324C2semidirect
6. #14530166C2semidirect
7. #14530160C2semidirect
8. #14529924C2semidirect
9. #14529053C2semidirect
10. #14529049C2semidirect
13. #14528788C2semidirect
71. C22×Dic34Q8semidirect
298. Dic34C22⋊Q8semidirect
324. C22×C17#475semidirect
478. C17#25955semidirect
650. C22#11763093semidirect

Maximal subgroups

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

subgroupindex
C2×#117630932
C22×#105874872
#145324872
#145316162
#145303242
#145301662
#145301602
#145299242
#145290532
#145290492
#145289762
#145289622
#145287882
#145287182
C17×#18072
#2595517