Finite Group Databasebeta

← #17052693

Subgroups — #17052693

Order 4352.

982 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762909C211yesmaximal, characteristic
2C2×#11762906C211yesmaximal, characteristic
3C2×#11762900C211yesmaximal, characteristic
4#14531696C211yesmaximal, characteristic, complemented
5#14531694C211yesmaximal, characteristic
6#14529521C211yesmaximal, characteristic, complemented
7#14529516C211yesmaximal, characteristic, complemented
8#14529053C211yesmaximal, characteristic, complemented
9#14529043C211yesmaximal, characteristic
10#14529039C211yesmaximal, characteristic, complemented
11#14529029C211yesmaximal, characteristic
12#14528950C211yesmaximal, characteristic, complemented
13#14528800C211yesmaximal, characteristic, complemented
14#14528661C211yesmaximal, characteristic, complemented
15C17×#1802C211yesmaximal, characteristic
16C22×#63229C2211yescharacteristic
17C2×C4×Dic34C2211yescharacteristic
18C2×#10587573C2211yescharacteristic
19C2×#10587573C2211yescharacteristic
20C2×D4×Dic17C2211yescharacteristic
21C2×D4×Dic17C2211yescharacteristic
22#11762909C418yescomplemented
23#11762906—24no—
24#11762900—24no—
25C2×#10587478C2211yescharacteristic
26C2×#10587478C2211yescharacteristic
27C2×#10587478C2211yescharacteristic
28C2×#10587479C2211yescharacteristic
29C2×#10587479C2211yescharacteristic
30C2×#10587487C2211yescharacteristic
31C2×#10587487C2211yescharacteristic
32C2×#10587481C2211yescharacteristic
33C2×#10587503C2211yescharacteristic
34C2×#10587474C2211yescharacteristic
35C2×#10587486C2211yescharacteristic
36C2×C8×Dic17C2211yescharacteristic
37#11762708C2211yescharacteristic
38#11762708C2211yescharacteristic
39#11762705C2211yescharacteristic
40#11762705C2211yescharacteristic
41#11762704—21no—
42#11762704—21no—
43#11762684—21no—
44#11762684—21no—
45#11762678—21no—
46#11762675C2211yescharacteristic
47#11762669—21no—
48C4⋊C4×Dic17—21no—
49C4×C4⋊Dic17C2211yescharacteristic
50C2×#10587502—21no—
51#11762572C2211yescharacteristic
52#11762572C2211yescharacteristic
53#11762563C2211yescharacteristic
54#11762563C2211yescharacteristic
55C2×C4×D4×C17C2211yescharacteristic
56C2×C17×C4⋊C8C2211yescharacteristic
57C2×C17×#102C2211yescharacteristic
58C2×C17×#102C2211yescharacteristic
59C17×#384C2211yescharacteristic
60C17×#340C2211yescharacteristic
61C17×#340C2211yescharacteristic
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×C17C2×C414yes—
71C2×C4⋊C4×C17—21no—
72C2×C4⋊C4×C17—21no—
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C22⋊C4×C17—21no—
77C2×C42×C17C2311yescharacteristic
78C2×C22⋊Dic17—21no—
79C2×C22⋊Dic17—21no—
80D4×Dic17—22no—
81D4×Dic17—22no—
82D4×Dic17—41no—
83D4×Dic17—41no—
84C2×#63229C2×C414yes—
85C2×#63229D414yes—
86C2×#63229Q814yes—
87C2×C4⋊Dic17—21no—
88C2×C4⋊Dic17—21no—
89C2×C4⋊Dic17D411yescharacteristic
90C2×C4⋊Dic17D411yescharacteristic
91C2×C4⋊Dic17C2311yescharacteristic
92C2×C4⋊Dic17C2311yescharacteristic
93C2×C4⋊Dic17C2311yescharacteristic
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×Dic17—21no—
102C2×C4×Dic17C2311yescharacteristic
103C2×C4×Dic17C2311yescharacteristic
104C22×#3622C2311yescharacteristic
105C22×#3622C2311yescharacteristic
106#10587573—22no—
107#10587573—22no—
108#10587573—41no—
109#10587573—41no—
110C4×Dic34—22no—
111C4×Dic34C2×C414yescomplemented
112C17×C4⋊C8—22no—
113C17×#102—22no—
114C17×#102—22no—
115#10587503C2×C414yes—
116#10587502—21no—
117#10587502—21no—
118#10587502—21no—
119#10587502—21no—
120#10587502—21no—
121#10587502—21no—
122#10587502D411yescharacteristic
123#10587502D411yescharacteristic
124#10587487—22no—
125#10587487—22no—
126#10587486—22no—
127C8×Dic17—22no—
128#10587481C2×C414yescomplemented
129#10587479—22no—
130#10587479—41no—
131#10587478—22no—
132#10587478—41no—
133#10587478C2×C414yescomplemented
134#10587474C2×C414yescomplemented
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—24no—
144C22×C4×C17—21no—
145C22×C4×C17—21no—
146C22×C4×C17—81no—
147C22×C4×C17—81no—
148C22×C4×C17C2×D411yescharacteristic
149C22×C4×C17C2×D411yescharacteristic
150C22×C4×C17C2411yescharacteristic
151C22×Dic17—21no—
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×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×C17—21no—
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×C17—81no—
184C4⋊C4×C17C22×C412yes—
185C22⋊C4×C17—24no—
186C42×C17C22×C412yes—
187C42×C17#4112yes—
188C22⋊Dic17—42no—
189C22⋊Dic17—42no—
190#63229C4⋊C4116yescomplemented
191C4⋊Dic17—21no—
192C4⋊Dic17—22no—
193C4⋊Dic17—21no—
194C4⋊Dic17—21no—
195C4⋊Dic17—21no—
196C4⋊Dic17—21no—
197C4⋊Dic17—41no—
198C4⋊Dic17—81no—
199C4⋊Dic17—81no—
200C4⋊Dic17C22×C412yes—
201#63225—24no—
202#63225—42no—
203#63225—42no—
204#63225—81no—
205#63225—81no—
206C4×Dic17—24no—
207C4×Dic17—21no—
208C4×Dic17—21no—
209C4×Dic17—21no—
210C4×Dic17—21no—
211C4×Dic17—42no—
212C4×Dic17—42no—
213C2×#3622—21no—
214C2×#3622—21no—
215C2×#3622C22×C412yes—
216C2×#3622C22×C412yes—
217C2×#3622C2×D412yes—
218C2×#3622C2×Q812yes—
219#22678—171nomaximal, Sylow
220C23×C17—22no—
221C23×C17—22no—
222C23×C17—22no—
223C23×C17—41no—
224C23×C17—41no—
225C23×C17—41no—
226C23×C17—41no—
227C23×C17C22×D411yescharacteristic
228D4×C17—22no—
229D4×C17—22no—
230D4×C17—22no—
231D4×C17C2×C4⋊C414yes—
232C2×C4×C17—21no—
233C2×C4×C17—21no—
234C2×C4×C17—21no—
235C2×C4×C17—21no—
236C2×C4×C17—22no—
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—41no—
245C2×C4×C17—84no—
246C2×C4×C17C4×D412yes—
247C2×C4×C17C4×D412yes—
248C2×C4×C17C23×C411yescharacteristic
249C2×C4×C17C22×D411yescharacteristic
250C2×C4×C17C22×Q811yescharacteristic
251C2×C4×C17C2×#4111yescharacteristic
252C2×C4×C17H(5,𝔽2)11yescharacteristic
253C2×C4×C172−1+411yescharacteristic
254C2×Dic17—22no—
255C2×Dic17—22no—
256C2×Dic17—21no—
257C2×Dic17—22no—
258C2×Dic17—21no—
259C2×Dic17—22no—
260C2×Dic17—21no—
261C2×Dic17—21no—
262C2×Dic17—21no—
263C2×Dic17—21no—
264C2×Dic17—21no—
265C2×Dic17—21no—
266C2×Dic17—21no—
267C2×Dic17—21no—
268C2×Dic17—21no—
269C2×Dic17—41no—
270C2×Dic17—41no—
271C2×Dic17—41no—
272C2×Dic17—41no—
273C2×Dic17—41no—
274C2×Dic17—41no—
275C2×Dic17—41no—
276C2×Dic17—41no—
277C2×Dic17—84no—
278C2×Dic17—84no—
279C2×Dic17#13712yes—
280Dic34—22no—
281Dic34—22no—
282Dic34—22no—
283Dic34C2×C4⋊C414yes—
284C8×C17—41no—
285C8×C17—41no—
286#3622—22no—
287#3622C2×C4⋊C414yes—
288C2×#440—171no—
289C2×#440—171no—
290C2×C4×SD16—171no—
291Q8×C4⋊C4—171no—
292#2349—171no—
293#1943—171no—
294#1942—171no—
295#1865—171no—
296#1864—171no—
297#1863—171no—
298#1862—171no—
299#1847—171no—
300#1805—171no—
301#1802D1711yescharacteristic
302#1771—171no—
303C22×C17—24no—
304C22×C17—42no—
305C22×C17—42no—
306C22×C17—42no—
307C22×C17—42no—
308C22×C17—42no—
309C22×C17—42no—
310C22×C17C2×C4×D411yescharacteristic
311C22×C17D4211yescharacteristic
312C22×C17D4×Q811yescharacteristic
313C22×C17C2×#13711yescharacteristic
314C22×C17#57511yescharacteristic
315C22×C17#57611yescharacteristic
316C22×C17#57711yescharacteristic
317C4×C17—22no—
318C4×C17—22no—
319C4×C17—42no—
320C4×C17—81no—
321C4×C17—81no—
322C4×C17C22×C4⋊C411yescharacteristic
323C4×C17#52011yescharacteristic
324C4×C17#54711yescharacteristic
325C4×C17#55411yescharacteristic
326Dic17—24no—
327Dic17—22no—
328Dic17—42no—
329Dic17—41no—
330Dic17—41no—
331Dic17—41no—
332Dic17—81no—
333Dic17—81no—
334C22×SD16—171no—
335C2×C4×Q8—171no—
336C2×C4×Q8—171no—
337C2×C4×Q8—171no—
338C2×C4×D4—171no—
339C2×C4×D4—171no—
340C2×C4×D4C2×D1711yescharacteristic
341#440—178no—
342#440—344no—
343C4×SD16—344no—
344C2×#107—171no—
345C2×#107—171no—
346C2×#107—171no—
347C2×#106—171no—
348C2×#106—171no—
349C2×C4⋊C8C2×D1711yescharacteristic
350C2×#103—171no—
351C2×#103—171no—
352C2×#103—171no—
353C2×#102—171no—
354C2×#102C2×D1711yescharacteristic
355C2×#102C2×D1711yescharacteristic
356C2×#97—171no—
357C2×#97—171no—
358C2×C4×C8—171no—
359#389—171no—
360#389—171no—
361#389—171no—
362#389—171no—
363#389—341no—
364#389—341no—
365#385—341no—
366#385—341no—
367#384—341no—
368#384C2×D1711yescharacteristic
369#382—171no—
370#380—341no—
371C4×C4⋊C4—171no—
372C4×C4⋊C4—341no—
373C2×#95—341no—
374#340—171no—
375#340—171no—
376#340C2×D1711yescharacteristic
377#340C2×D1711yescharacteristic
378#336—171no—
379#336—171no—
380C2×C17—44no—
381C2×C17D4×C4⋊C411yescharacteristic
382C2×C17C4×#13711yescharacteristic
383C2×C17#294611yescharacteristic
384C2×C17#326611yescharacteristic
385C2×C17#326811yescharacteristic
386C2×C17#338511yescharacteristic
387C2×C17#338811yescharacteristic
388C22×Q8—171no—
389C22×D4C22×D1711yescharacteristic
390C23×C4—21no—
391C23×C4—341no—
392C23×C4—341no—
393C2×SD16—174no—
394C2×SD16—174no—
395C2×SD16—174no—
396C22×C8—171no—
397C22×C8—171no—
398C22×C8C22×D1711yescharacteristic
399C22×C8C22×D1711yescharacteristic
400C4×Q8—174no—
401C4×Q8—342no—
402C4×Q8—342no—
403C4×Q8—342no—
404C4×Q8—681no—
405C4×Q8—681no—
406C4×D4—22no—
407C4×D4—342no—
408C4×D4—342no—
409C4×D4—681no—
410C4×D4—681no—
411C4×D4C4×D1714yes—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—171no—
415C2×C4⋊C4—171no—
416C2×C4⋊C4—171no—
417C2×C4⋊C4—171no—
418C2×C4⋊C4—171no—
419C2×C4⋊C4—341no—
420C2×C4⋊C4—341no—
421C2×C4⋊C4—341no—
422C2×C4⋊C4—341no—
423C2×C4⋊C4—341no—
424C2×C4⋊C4—341no—
425C2×C4⋊C4—341no—
426C2×C4⋊C4C22×D1711yescharacteristic
427C2×C4⋊C4C22×D1711yescharacteristic
428C2×C4⋊C4C22×D1711yescharacteristic
429C2×C22⋊C4—21no—
430C2×C22⋊C4—341no—
431C2×C22⋊C4—341no—
432C2×C42—171no—
433C2×C42—171no—
434C2×C42—341no—
435C2×C42—341no—
436C2×C42—341no—
437C2×C42C22×D1711yescharacteristic
438#107—174no—
439#107—342no—
440#107—681no—
441#106—342no—
442#106—681no—
443C4⋊C8—22no—
444#103—174no—
445#103—342no—
446#103—342no—
447#102—22no—
448#102—22no—
449#102—174no—
450#97—174no—
451#97—342no—
452C4×C8—342no—
453#95—171no—
454#95—171no—
455#95—341no—
456#95—341no—
457#95—341no—
458#95—341no—
459#95—341no—
460#95—341no—
461C17#2267811yescharacteristic, complemented, Sylow
462C24—21no—
463C2×Q8—172no—
464C2×Q8—172no—
465C2×Q8—172no—
466C2×Q8—341no—
467C2×Q8—341no—
468C2×Q8—341no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4C2×C4×D1712yes—
473C2×D4D4×D1712yes—
474C2×D4Q8×D1712yes—
475C22×C4—21no—
476C22×C4—24no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—81no—
480C22×C4—81no—
481C22×C4—171no—
482C22×C4—171no—
483C22×C4—171no—
484C22×C4—171no—
485C22×C4—341no—
486C22×C4—341no—
487C22×C4—341no—
488C22×C4—341no—
489C22×C4—682no—
490C22×C4—682no—
491C22×C4—682no—
492C22×C4—682no—
493C22×C4—1361no—
494C22×C4—1361no—
495C22×C4D4×D1711yescharacteristic
496C22×C4D4×D1711yescharacteristic
497C22×C4C23×D1711yescharacteristic
498SD16—1716no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—172no—
506C2×C8—172no—
507C2×C8—172no—
508C2×C8—172no—
509C2×C8—341no—
510C2×C8—341no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—42no—
517C4⋊C4—81no—
518C4⋊C4—172no—
519C4⋊C4—342no—
520C4⋊C4—341no—
521C4⋊C4—344no—
522C4⋊C4—341no—
523C4⋊C4—341no—
524C4⋊C4—341no—
525C4⋊C4—341no—
526C4⋊C4—681no—
527C4⋊C4—682no—
528C4⋊C4—682no—
529C4⋊C4—1361no—
530C4⋊C4—1361no—
531C4⋊C4—1361no—
532C4⋊C4—1361no—
533C4⋊C4C2×C4×D1712yes—
534C22⋊C4—24no—
535C22⋊C4—682no—
536C22⋊C4—682no—
537C42—341no—
538C42—341no—
539C42—341no—
540C42—341no—
541C42—344no—
542C42—682no—
543C42—682no—
544C42C2×C4×D1712yes—
545C42#6325412yes—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23C2×D4×D1711yescharacteristic
554Q8—174no—
555Q8—342no—
556Q8—342no—
557Q8—342no—
558D4—22no—
559D4—22no—
560D4—22no—
561D4C4⋊C4×D1714yes—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—84no—
576C2×C4—172no—
577C2×C4—342no—
578C2×C4—342no—
579C2×C4—341no—
580C2×C4—341no—
581C2×C4—342no—
582C2×C4—342no—
583C2×C4—341no—
584C2×C4—341no—
585C2×C4—341no—
586C2×C4—341no—
587C2×C4—341no—
588C2×C4—341no—
589C2×C4—341no—
590C2×C4—341no—
591C2×C4—341no—
592C2×C4—681no—
593C2×C4—681no—
594C2×C4—681no—
595C2×C4—681no—
596C2×C4—681no—
597C2×C4—681no—
598C2×C4—681no—
599C2×C4—681no—
600C2×C4—1364no—
601C2×C4—1364no—
602C2×C4#1058756712yes—
603C2×C4#1058756712yes—
604C2×C4C22×C4×D1711yescharacteristic
605C2×C4C2×D4×D1711yescharacteristic
606C2×C4C2×#6325411yescharacteristic
607C2×C4C2×Q8×D1711yescharacteristic
608C2×C4#1058769611yescharacteristic
609C2×C4#1058769711yescharacteristic
610C8—41no—
611C8—41no—
612C8—174no—
613C8—342no—
614C22—24no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22C2×#1058756711yescharacteristic
622C22#1176344311yescharacteristic
623C22#1176347211yescharacteristic
624C22#137×D1711yescharacteristic
625C22#1176360511yescharacteristic
626C22#1176361011yescharacteristic
627C22#1176361511yescharacteristic
628C4—22no—
629C4—22no—
630C4—42no—
631C4—81no—
632C4—81no—
633C4—342no—
634C4—344no—
635C4—681no—
636C4—681no—
637C4—681no—
638C4—682no—
639C4—1361no—
640C4—1361no—
641C4C2×C4⋊C4×D1711yescharacteristic
642C4#1176328911yescharacteristic
643C4#1176345911yescharacteristic
644C4#1176351711yescharacteristic
645C2—44no—
646C2#1453168011yescharacteristic
647C2#1453760811yescharacteristic
648C2#1453761011yescharacteristic
649C2#1454030011yescharacteristic
650C2#1454033011yescharacteristic
651C2#1454168911yescharacteristic
652C2#1454172211yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1802C2
socleC23×C17C22×D4
prime radicalC22×D4×C17C23
cosocleC22×C4×C17C24
hypercenter#1802D17
hypocenterC17#22678
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
4. #14531696C2semidirect
6. #14529521C2semidirect
7. #14529516C2semidirect
8. #14529053C2semidirect
10. #14529039C2semidirect
12. #14528950C2semidirect
13. #14528800C2semidirect
14. #14528661C2semidirect
22. #11762909C4semidirect
111. C4×Dic34C2×C4semidirect
128. #10587481C2×C4semidirect
133. #10587478C2×C4semidirect
134. #10587474C2×C4semidirect
190. #63229C4⋊C4semidirect
461. C17#22678semidirect

Maximal subgroups

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

subgroupindex
C2×#117629092
C2×#117629062
C2×#117629002
#145316962
#145316942
#145295212
#145295162
#145290532
#145290432
#145290392
#145290292
#145289502
#145288002
#145286612
C17×#18022
#2267817