Finite Group Databasebeta

← #17052744

Subgroups — #17052744

Order 4352.

986 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762900C211yesmaximal, characteristic
2C2×#11762900C211yesmaximal, characteristic
3C2×C4×#63229C211yesmaximal, characteristic
4#14531703C211yesmaximal, characteristic, complemented
5#14531689C211yesmaximal, characteristic
6#14529511C211yesmaximal, characteristic, complemented
7#14529509C211yesmaximal, characteristic, complemented
8#14529041C211yesmaximal, characteristic, complemented
9#14529035C211yesmaximal, characteristic, complemented
10#14529033C211yesmaximal, characteristic
11#14528947C211yesmaximal, characteristic, complemented
12#14528798C211yesmaximal, characteristic, complemented
13#14528792C211yesmaximal, characteristic
14#14528645C211yesmaximal, characteristic, complemented
15C17×#1862C211yesmaximal, characteristic
16C22×#63229C2211yescharacteristic
17C2×C4×Dic34C2211yescharacteristic
18C2×#10587573C2211yescharacteristic
19C2×#10587573C2211yescharacteristic
20C2×D4×Dic17C2211yescharacteristic
21C2×D4×Dic17C2211yescharacteristic
22#11762900—24no—
23#11762900C418yescomplemented
24C4×#63229—24no—
25C2×#10587489C2211yescharacteristic
26C2×#10587479C2211yescharacteristic
27C2×#10587479C2211yescharacteristic
28C2×#10587479C2211yescharacteristic
29C2×#10587485C2211yescharacteristic
30C2×#10587487C2211yescharacteristic
31C2×#10587487C2211yescharacteristic
32C2×#10587481C2211yescharacteristic
33C2×#10587503C2211yescharacteristic
34C2×C4×#3622C2211yescharacteristic
35C2×C8×Dic17C2211yescharacteristic
36C2×C8×Dic17C2211yescharacteristic
37#11762708C2211yescharacteristic
38#11762707C2211yescharacteristic
39#11762706—21no—
40#11762705C2211yescharacteristic
41#11762704—21no—
42#11762682—21no—
43#11762681—21no—
44#11762674C2211yescharacteristic
45#11762674C2211yescharacteristic
46#11762673—21no—
47#11762667—21no—
48C4⋊C4×Dic17—21no—
49C4×#63225C2211yescharacteristic
50C2×#10587502—21no—
51#11762572C2211yescharacteristic
52#11762571C2211yescharacteristic
53#11762571C2211yescharacteristic
54#11762563C2211yescharacteristic
55C2×C4×D4×C17C2211yescharacteristic
56C2×C17×#107C2211yescharacteristic
57C2×C17×#102C2211yescharacteristic
58C2×C17×#102C2211yescharacteristic
59C17×#389C2211yescharacteristic
60C17×#340C2211yescharacteristic
61C17×#336C2211yescharacteristic
62C22×D4×C17C2311yescharacteristic
63C23×C4×C17—21no—
64C23×Dic17—21no—
65C23×Dic17—21no—
66C22×Dic34C2311yescharacteristic
67C22×C8×C17C2311yescharacteristic
68C22×C8×C17C2311yescharacteristic
69C4×D4×C17—22no—
70C4×D4×C17—41no—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C22⋊C4×C17—21no—
76C2×C42×C17C2311yescharacteristic
77C2×C22⋊Dic17—21no—
78C2×C22⋊Dic17—21no—
79D4×Dic17—22no—
80D4×Dic17—22no—
81D4×Dic17—41no—
82D4×Dic17C2×C414yes—
83C2×#63229C2×C414yes—
84C2×#63229D414yes—
85C2×#63229Q814yes—
86C2×C4⋊Dic17—21no—
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4⋊Dic17C2311yescharacteristic
90C2×#63225—21no—
91C2×#63225—21no—
92C2×#63225—21no—
93C2×#63225—21no—
94C2×#63225—21no—
95C2×#63225D411yescharacteristic
96C2×#63225D411yescharacteristic
97C2×C4×Dic17—21no—
98C2×C4×Dic17—21no—
99C2×C4×Dic17—21no—
100C2×C4×Dic17C2311yescharacteristic
101C2×C4×Dic17C2311yescharacteristic
102C22×#3622C2311yescharacteristic
103C22×#3622C2311yescharacteristic
104#10587573—22no—
105#10587573—22no—
106#10587573—41no—
107#10587573C2×C414yescomplemented
108C4×Dic34—22no—
109C4×Dic34—41no—
110C17×#107—41no—
111C17×#102—22no—
112C17×#102C2×C414yes—
113C17×#95—21no—
114#10587503—22no—
115#10587502—21no—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502D411yescharacteristic
123#10587502D411yescharacteristic
124#10587489—41no—
125#10587487—22no—
126#10587487C2×C414yescomplemented
127#10587485—22no—
128C8×Dic17—22no—
129C8×Dic17C2×C414yescomplemented
130#10587481—22no—
131#10587479—22no—
132#10587479—22no—
133#10587479C2×C414yescomplemented
134C4×#3622—22no—
135C24×C17—21no—
136C2×D4×C17—21no—
137C2×D4×C17—21no—
138C2×D4×C17—21no—
139C2×D4×C17C22×C412yes—
140C2×D4×C17C2×D412yes—
141C2×D4×C17C2×Q812yes—
142C22×C4×C17—21no—
143C22×C4×C17—21no—
144C22×C4×C17—21no—
145C22×C4×C17—42no—
146C22×C4×C17—42no—
147C22×C4×C17—81no—
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2411yescharacteristic
151C22×Dic17—21no—
152C22×Dic17—21no—
153C22×Dic17—24no—
154C22×Dic17—21no—
155C22×Dic17—21no—
156C22×Dic17—42no—
157C22×Dic17—42no—
158C22×Dic17—81no—
159C22×Dic17—81no—
160C22×Dic17—81no—
161C22×Dic17C2×D411yescharacteristic
162C22×Dic17C2×D411yescharacteristic
163C22×Dic17C2×D411yescharacteristic
164C22×Dic17C2×D411yescharacteristic
165C2×Dic34—21no—
166C2×Dic34—21no—
167C2×Dic34—21no—
168C2×Dic34C22×C412yes—
169C2×Dic34C2×D412yes—
170C2×Dic34C2×Q812yes—
171C2×C8×C17—21no—
172C2×C8×C17—21no—
173C2×C8×C17—21no—
174C2×C8×C17—21no—
175C2×C8×C17—21no—
176C2×C8×C17C22×C412yes—
177C4⋊C4×C17—21no—
178C4⋊C4×C17—21no—
179C4⋊C4×C17—21no—
180C4⋊C4×C17—21no—
181C4⋊C4×C17—21no—
182C4⋊C4×C17—42no—
183C4⋊C4×C17C22×C412yes—
184C22⋊C4×C17—42no—
185C42×C17—21no—
186C42×C17—21no—
187C22⋊Dic17—24no—
188C22⋊Dic17—42no—
189#63229C4⋊C4116yescomplemented
190C4⋊Dic17—21no—
191C4⋊Dic17—21no—
192C4⋊Dic17—21no—
193C4⋊Dic17—21no—
194C4⋊Dic17—21no—
195C4⋊Dic17—42no—
196C4⋊Dic17C22×C412yes—
197#63225—22no—
198#63225—24no—
199#63225—22no—
200#63225—42no—
201#63225—42no—
202#63225—81no—
203#63225—81no—
204C4×Dic17—21no—
205C4×Dic17—21no—
206C4×Dic17—24no—
207C4×Dic17—42no—
208C4×Dic17—42no—
209C4×Dic17C22×C412yes—
210C4×Dic17#4112yes—
211C2×#3622—21no—
212C2×#3622—21no—
213C2×#3622—21no—
214C2×#3622C22×C412yes—
215C2×#3622C2×D412yes—
216C2×#3622C2×Q812yes—
217#22683—171nomaximal, Sylow
218C23×C17—22no—
219C23×C17—22no—
220C23×C17—22no—
221C23×C17—41no—
222C23×C17—41no—
223C23×C17—41no—
224C23×C17—41no—
225C23×C17C22×D411yescharacteristic
226D4×C17—22no—
227D4×C17—22no—
228D4×C17—22no—
229D4×C17C2×C4⋊C414yes—
230C2×C4×C17—21no—
231C2×C4×C17—21no—
232C2×C4×C17—22no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—41no—
237C2×C4×C17—41no—
238C2×C4×C17—41no—
239C2×C4×C17—41no—
240C2×C4×C17—41no—
241C2×C4×C17—41no—
242C2×C4×C17—41no—
243C2×C4×C17—41no—
244C2×C4×C17—84no—
245C2×C4×C17#13512yes—
246C2×C4×C17C23×C411yescharacteristic
247C2×C4×C17C22×D411yescharacteristic
248C2×C4×C17C22×Q811yescharacteristic
249C2×C4×C17C2×#4111yescharacteristic
250C2×C4×C17H(5,𝔽2)11yescharacteristic
251C2×C4×C172−1+411yescharacteristic
252C2×Dic17—22no—
253C2×Dic17—22no—
254C2×Dic17—21no—
255C2×Dic17—22no—
256C2×Dic17—21no—
257C2×Dic17—21no—
258C2×Dic17—21no—
259C2×Dic17—21no—
260C2×Dic17—21no—
261C2×Dic17—22no—
262C2×Dic17—21no—
263C2×Dic17—21no—
264C2×Dic17—21no—
265C2×Dic17—41no—
266C2×Dic17—41no—
267C2×Dic17—41no—
268C2×Dic17—41no—
269C2×Dic17—41no—
270C2×Dic17—41no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273C2×Dic17—84no—
274C2×Dic17—84no—
275C2×Dic17C4×D412yes—
276C2×Dic17C4×D412yes—
277C2×Dic17#13512yes—
278Dic34—22no—
279Dic34—22no—
280Dic34—22no—
281Dic34C2×C4⋊C414yes—
282C8×C17—22no—
283C8×C17—41no—
284#3622—41no—
285#3622C2×C4⋊C414yes—
286C2×C4×SD16—171no—
287C2×C4×SD16—171no—
288C2×C4×SD16—171no—
289#2351—171no—
290#2349—171no—
291#1941—171no—
292#1941—171no—
293#1863—171no—
294#1863—171no—
295#1862—171no—
296#1862D1711yescharacteristic
297#1846—171no—
298#1805—171no—
299#1804—171no—
300C8×C4⋊C4—171no—
301C22×C17—24no—
302C22×C17—42no—
303C22×C17—42no—
304C22×C17—42no—
305C22×C17—42no—
306C22×C17—42no—
307C22×C17—42no—
308C22×C17C2×C4×D411yescharacteristic
309C22×C17D4211yescharacteristic
310C22×C17D4×Q811yescharacteristic
311C22×C17C2×#13511yescharacteristic
312C22×C17C2×#13511yescharacteristic
313C22×C17#57711yescharacteristic
314C22×C17#57711yescharacteristic
315C4×C17—22no—
316C4×C17—42no—
317C4×C17—41no—
318C4×C17—81no—
319C4×C17—81no—
320C4×C17C22×C4⋊C411yescharacteristic
321C4×C17#52011yescharacteristic
322C4×C17#54711yescharacteristic
323C4×C17#55411yescharacteristic
324Dic17—24no—
325Dic17—22no—
326Dic17—22no—
327Dic17—22no—
328Dic17—42no—
329Dic17—41no—
330Dic17—81no—
331Dic17—81no—
332C22×SD16—171no—
333C2×C4×Q8—171no—
334C2×C4×Q8—171no—
335C2×C4×Q8—171no—
336C2×C4×D4—171no—
337C2×C4×D4—171no—
338C2×C4×D4C2×D1711yescharacteristic
339C4×SD16—178no—
340C4×SD16—344no—
341C4×SD16—344no—
342C2×#107—171no—
343C2×#107C2×D1711yescharacteristic
344C2×#106—171no—
345C2×#106—171no—
346C2×#106—171no—
347C2×C4⋊C8—171no—
348C2×#103—171no—
349C2×#103—171no—
350C2×#103—171no—
351C2×#102—171no—
352C2×#102C2×D1711yescharacteristic
353C2×#102C2×D1711yescharacteristic
354C2×C4×C8—171no—
355C2×C4×C8—171no—
356C2×C4×C8—171no—
357#389—171no—
358#389—171no—
359#389—171no—
360#389—341no—
361#389—341no—
362#389C2×D1711yescharacteristic
363#385—341no—
364#385—341no—
365#382—171no—
366#382—171no—
367#382—341no—
368#380—341no—
369C4×C4⋊C4—171no—
370C4×C4⋊C4—341no—
371C2×#95—341no—
372#340—171no—
373#340—171no—
374#340—171no—
375#340C2×D1711yescharacteristic
376#336—171no—
377#336C2×D1711yescharacteristic
378C2×C17—44no—
379C2×C17D4×C4⋊C411yescharacteristic
380C2×C17C4×#13511yescharacteristic
381C2×C17#294911yescharacteristic
382C2×C17#328211yescharacteristic
383C2×C17#328211yescharacteristic
384C2×C17#338111yescharacteristic
385C2×C17#338111yescharacteristic
386C22×Q8—171no—
387C22×D4C22×D1711yescharacteristic
388C23×C4—21no—
389C23×C4—341no—
390C23×C4—341no—
391C2×SD16—174no—
392C2×SD16—174no—
393C2×SD16—174no—
394C22×C8—171no—
395C22×C8—171no—
396C22×C8C22×D1711yescharacteristic
397C22×C8C22×D1711yescharacteristic
398C4×Q8—174no—
399C4×Q8—342no—
400C4×Q8—342no—
401C4×Q8—342no—
402C4×Q8—681no—
403C4×Q8—681no—
404C4×D4—22no—
405C4×D4—41no—
406C4×D4—174no—
407C4×D4—342no—
408C4×D4—342no—
409C4×D4—681no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—171no—
412C2×C4⋊C4—171no—
413C2×C4⋊C4—171no—
414C2×C4⋊C4—171no—
415C2×C4⋊C4—171no—
416C2×C4⋊C4—341no—
417C2×C4⋊C4—341no—
418C2×C4⋊C4—341no—
419C2×C4⋊C4—341no—
420C2×C4⋊C4—341no—
421C2×C4⋊C4—341no—
422C2×C4⋊C4C22×D1711yescharacteristic
423C2×C4⋊C4C22×D1711yescharacteristic
424C2×C4⋊C4C22×D1711yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—341no—
427C2×C22⋊C4—341no—
428C2×C42—171no—
429C2×C42—171no—
430C2×C42—341no—
431C2×C42—341no—
432C2×C42—341no—
433C2×C42C22×D1711yescharacteristic
434#107—41no—
435#107—681no—
436#106—174no—
437#106—342no—
438#106—342no—
439C4⋊C8—342no—
440#103—174no—
441#103—342no—
442#103—342no—
443#102—22no—
444#102—342no—
445#102C4×D1714yes—
446C4×C8—174no—
447C4×C8—342no—
448C4×C8—342no—
449#95—21no—
450#95—171no—
451#95—171no—
452#95—341no—
453#95—341no—
454#95—341no—
455#95—341no—
456#95—341no—
457#95—341no—
458#95—341no—
459C17#2268311yescharacteristic, complemented, Sylow
460C24—21no—
461C2×Q8—172no—
462C2×Q8—172no—
463C2×Q8—172no—
464C2×Q8—341no—
465C2×Q8—341no—
466C2×Q8—341no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4C2×C4×D1712yes—
471C2×D4D4×D1712yes—
472C2×D4Q8×D1712yes—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—81no—
479C22×C4—171no—
480C22×C4—171no—
481C22×C4—171no—
482C22×C4—171no—
483C22×C4—344no—
484C22×C4—341no—
485C22×C4—341no—
486C22×C4—341no—
487C22×C4—341no—
488C22×C4—682no—
489C22×C4—682no—
490C22×C4—1361no—
491C22×C4—1361no—
492C22×C4—1361no—
493C22×C4D4×D1711yescharacteristic
494C22×C4D4×D1711yescharacteristic
495C22×C4C23×D1711yescharacteristic
496SD16—1716no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—172no—
503C2×C8—172no—
504C2×C8—172no—
505C2×C8—341no—
506C2×C8—341no—
507C2×C8—341no—
508C2×C8C2×C4×D1712yes—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—42no—
515C4⋊C4—172no—
516C4⋊C4—342no—
517C4⋊C4—344no—
518C4⋊C4—341no—
519C4⋊C4—342no—
520C4⋊C4—341no—
521C4⋊C4—341no—
522C4⋊C4—341no—
523C4⋊C4—341no—
524C4⋊C4—682no—
525C4⋊C4—682no—
526C4⋊C4—682no—
527C4⋊C4—1361no—
528C4⋊C4—1361no—
529C4⋊C4C2×C4×D1712yes—
530C22⋊C4—42no—
531C22⋊C4—344no—
532C22⋊C4—682no—
533C42—21no—
534C42—21no—
535C42—172no—
536C42—172no—
537C42—344no—
538C42—341no—
539C42—341no—
540C42—682no—
541C42—682no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23C2×D4×D1711yescharacteristic
550Q8—174no—
551Q8—342no—
552Q8—342no—
553Q8—342no—
554D4—22no—
555D4—22no—
556D4—22no—
557D4C4⋊C4×D1714yes—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—84no—
573C2×C4—172no—
574C2×C4—172no—
575C2×C4—172no—
576C2×C4—342no—
577C2×C4—342no—
578C2×C4—341no—
579C2×C4—341no—
580C2×C4—341no—
581C2×C4—341no—
582C2×C4—341no—
583C2×C4—341no—
584C2×C4—342no—
585C2×C4—342no—
586C2×C4—341no—
587C2×C4—341no—
588C2×C4—341no—
589C2×C4—681no—
590C2×C4—681no—
591C2×C4—681no—
592C2×C4—681no—
593C2×C4—681no—
594C2×C4—681no—
595C2×C4—681no—
596C2×C4—681no—
597C2×C4—1364no—
598C2×C4—1364no—
599C2×C4#1058759812yes—
600C2×C4C22×C4×D1711yescharacteristic
601C2×C4C2×D4×D1711yescharacteristic
602C2×C4#1058769111yescharacteristic
603C2×C4C2×Q8×D1711yescharacteristic
604C2×C4#41×D1711yescharacteristic
605C2×C4#1058769711yescharacteristic
606C8—22no—
607C8—41no—
608C8—174no—
609C8—681no—
610C22—24no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22C4×D4×D1711yescharacteristic
618C22#1176344111yescharacteristic
619C22#1176347211yescharacteristic
620C22#135×D1711yescharacteristic
621C22C2×#1058759811yescharacteristic
622C22#1176361811yescharacteristic
623C22#1176361911yescharacteristic
624C4—22no—
625C4—42no—
626C4—41no—
627C4—81no—
628C4—81no—
629C4—342no—
630C4—344no—
631C4—342no—
632C4—342no—
633C4—682no—
634C4—681no—
635C4—1361no—
636C4—1361no—
637C4C2×C4⋊C4×D1711yescharacteristic
638C4#1176328811yescharacteristic
639C4#1176345711yescharacteristic
640C4#1176351311yescharacteristic
641C2—44no—
642C2#1453168211yescharacteristic
643C2C4×#1058759811yescharacteristic
644C2#1453764011yescharacteristic
645C2#1454048911yescharacteristic
646C2#1454049511yescharacteristic
647C2#1454165411yescharacteristic
648C2#1454166311yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1862C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenter#1862D17
hypocenterC17#22683
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1862

Complements

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

NH ≅ G/Ntype
4. #14531703C2semidirect
6. #14529511C2semidirect
7. #14529509C2semidirect
8. #14529041C2semidirect
9. #14529035C2semidirect
11. #14528947C2semidirect
12. #14528798C2semidirect
14. #14528645C2semidirect
23. #11762900C4semidirect
107. #10587573C2×C4semidirect
126. #10587487C2×C4semidirect
129. C8×Dic17C2×C4semidirect
133. #10587479C2×C4semidirect
189. #63229C4⋊C4semidirect
459. C17#22683semidirect

Maximal subgroups

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

subgroupindex
C2×#117629002
C2×C4×#632292
#145317032
#145316892
#145295112
#145295092
#145290412
#145290352
#145290332
#145289472
#145287982
#145287922
#145286452
C17×#18622
#2268317