Finite Group Databasebeta

← #17039227

Subgroups — #17039227

Order 4352.

826 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762870C211yesmaximal, characteristic
2#14531140C211yesmaximal, characteristic
3#14529550C211yesmaximal, characteristic, complemented
4#14529550C211yesmaximal, characteristic, complemented
5#14529545C211yesmaximal, characteristic, complemented
6#14529540C211yesmaximal, characteristic, complemented
7#14528981C211yesmaximal, characteristic
8#14528981C211yesmaximal, characteristic
9#14528976C211yesmaximal, characteristic
10#14528971C211yesmaximal, characteristic
11#14528682C211yesmaximal, characteristic, complemented
12#14528682C211yesmaximal, characteristic, complemented
13C4×#10587478C211yesmaximal, characteristic, complemented
14C4×#10587479C211yesmaximal, characteristic, complemented
15C17×#2314C211yesmaximal, characteristic
16C2×#10587612C2211yescharacteristic
17C22×C4⋊Dic17C2211yescharacteristic
18#11762870C418yescomplemented
19C2×#10587478C2211yescharacteristic
20C2×#10587478C2211yescharacteristic
21C2×#10587478C2211yescharacteristic
22C2×#10587478C2211yescharacteristic
23C2×#10587479C2211yescharacteristic
24C2×#10587479C2211yescharacteristic
25C2×#10587479C2211yescharacteristic
26C2×#10587479C2211yescharacteristic
27C2×C4⋊#3622C2211yescharacteristic
28C2×C4⋊#3622C2211yescharacteristic
29C2×C22⋊#3622C2211yescharacteristic
30C2×C22⋊#3622C2211yescharacteristic
31C2×#10587474C2211yescharacteristic
32C2×C4×#3622C2211yescharacteristic
33C4⋊C4×Dic17—21no—
34C4⋊C4×Dic17—21no—
35C4×C4⋊Dic17C2211yescharacteristic
36C4×C4⋊Dic17C2211yescharacteristic
37C4×C4⋊Dic17C2211yescharacteristic
38C4×C4⋊Dic17C2211yescharacteristic
39C22⋊C4×Dic17—21no—
40C22⋊C4×Dic17—21no—
41#11762571C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49C2×C17×#117C2211yescharacteristic
50C17×#388—21no—
51C17×#388—21no—
52C17×#387—21no—
53C17×#387—21no—
54C17×#384C2211yescharacteristic
55C17×#382C2211yescharacteristic
56C17×#380C2211yescharacteristic
57C17×#380C2211yescharacteristic
58C4×C4⋊C4×C17C2211yescharacteristic
59C4×C4⋊C4×C17C2211yescharacteristic
60C23×C4×C17C2311yescharacteristic
61C23×Dic17—21no—
62C17×#117C2×C414yes—
63C17×#117Q814yes—
64C2×C4⋊C4×C17—21no—
65C2×C4⋊C4×C17—21no—
66C2×C4⋊C4×C17D411yescharacteristic
67C2×C4⋊C4×C17D411yescharacteristic
68C2×C4⋊C4×C17C2311yescharacteristic
69C2×C4⋊C4×C17C2311yescharacteristic
70C2×C4⋊C4×C17C2311yescharacteristic
71C2×C4⋊C4×C17C2311yescharacteristic
72C2×C22⋊C4×C17—21no—
73C2×C22⋊C4×C17—21no—
74C2×C22⋊C4×C17D411yescharacteristic
75C2×C22⋊C4×C17D411yescharacteristic
76C2×C42×C17—21no—
77C2×C42×C17—21no—
78C2×C42×C17C2311yescharacteristic
79C2×C42×C17C2311yescharacteristic
80C2×C22⋊Dic17—21no—
81#10587612—22no—
82#10587612—22no—
83C2×C4⋊Dic17—22no—
84C2×C4⋊Dic17—22no—
85C2×C4⋊Dic17—21no—
86C2×C4⋊Dic17C2311yescharacteristic
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4×Dic17—21no—
91C2×C4×Dic17—21no—
92C2×C4×Dic17—21no—
93C2×C4×Dic17—21no—
94C2×C4×Dic17—21no—
95C22×#3622C2311yescharacteristic
96C22×#3622C2311yescharacteristic
97C22×#3622C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99C17×#95—21no—
100C17×#95—21no—
101C17×#95—21no—
102C17×#95—21no—
103C17×#95—21no—
104C17×#95—21no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109C22⋊#3622C2×C414yes—
110C22⋊#3622C2×C414yes—
111#10587479—22no—
112#10587479—22no—
113#10587479—22no—
114#10587479—22no—
115#10587478—22no—
116#10587478—22no—
117#10587478—22no—
118#10587478—22no—
119C4⋊#3622C2×C414yescomplemented
120C4⋊#3622C2×C414yescomplemented
121#10587474C2×C414yescomplemented
122C4×#3622C2×C414yescomplemented
123C24×C17C2×D411yescharacteristic
124C22×C4×C17—21no—
125C22×C4×C17—21no—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—21no—
131C22×C4×C17—21no—
132C22×C4×C17C22×C412yes—
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17C2×D411yescharacteristic
136C22×C4×C17C2×D411yescharacteristic
137C22×C4×C17C2×D411yescharacteristic
138C22×C4×C17C2×Q812yes—
139C22×C4×C17C2411yescharacteristic
140C22×Dic17—21no—
141C22×Dic17—21no—
142C22×Dic17—21no—
143C22×Dic17—21no—
144C22×Dic17—24no—
145C22×Dic17—81no—
146C22×Dic17—81no—
147C4⋊C4×C17—21no—
148C4⋊C4×C17—21no—
149C4⋊C4×C17—21no—
150C4⋊C4×C17—21no—
151C4⋊C4×C17—21no—
152C4⋊C4×C17—21no—
153C4⋊C4×C17—21no—
154C4⋊C4×C17—21no—
155C4⋊C4×C17—81no—
156C4⋊C4×C17—81no—
157C4⋊C4×C17C4⋊C414yes—
158C4⋊C4×C17C4⋊C414yes—
159C22⋊C4×C17—81no—
160C22⋊C4×C17—81no—
161C22⋊C4×C17C4⋊C414yes—
162C22⋊C4×C17C4⋊C414yes—
163C42×C17—81no—
164C42×C17—81no—
165C42×C17C22×C412yes—
166C42×C17C22×C412yes—
167C42×C17C2×Q812yes—
168C42×C17C2×Q812yes—
169C22⋊Dic17—24no—
170C4⋊Dic17—24no—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17—21no—
175C4⋊Dic17#4112yes—
176C4⋊Dic17#4112yes—
177C4⋊Dic17#4112yes—
178C4⋊Dic17#4112yes—
179C4×Dic17—24no—
180C4×Dic17—81no—
181C4×Dic17—81no—
182C4×Dic17—81no—
183C4×Dic17—81no—
184C2×#3622—21no—
185C2×#3622—21no—
186C2×#3622—21no—
187C2×#3622—21no—
188C2×#3622—21no—
189C2×#3622—21no—
190C2×#3622—21no—
191C2×#3622—21no—
192C2×#3622C22×C412yes—
193C2×#3622C22×C412yes—
194C2×#3622C22×C412yes—
195C2×#3622C22×C412yes—
196#21367—171nomaximal, Sylow
197C23×C17—21no—
198C23×C17—21no—
199C23×C17—21no—
200C23×C17—21no—
201C23×C17—21no—
202C23×C17—21no—
203C23×C17C2×C4⋊C412yes—
204C23×C17C22×D411yescharacteristic
205C2×C4×C17—21no—
206C2×C4×C17—21no—
207C2×C4×C17—21no—
208C2×C4×C17—21no—
209C2×C4×C17—21no—
210C2×C4×C17—21no—
211C2×C4×C17—21no—
212C2×C4×C17—21no—
213C2×C4×C17—21no—
214C2×C4×C17—21no—
215C2×C4×C17—22no—
216C2×C4×C17—22no—
217C2×C4×C17—22no—
218C2×C4×C17—22no—
219C2×C4×C17—41no—
220C2×C4×C17—41no—
221C2×C4×C17—41no—
222C2×C4×C17—41no—
223C2×C4×C17—41no—
224C2×C4×C17—41no—
225C2×C4×C17—41no—
226C2×C4×C17—41no—
227C2×C4×C17—41no—
228C2×C4×C17—41no—
229C2×C4×C17—41no—
230C2×C4×C17—41no—
231C2×C4×C17C2×C4⋊C412yes—
232C2×C4×C17C2×C4⋊C412yes—
233C2×C4×C17C2×C4⋊C412yes—
234C2×C4×C17C2×C4⋊C412yes—
235C2×C4×C17C2×C4⋊C412yes—
236C2×C4×C17C23×C411yescharacteristic
237C2×C4×C17C22×Q811yescharacteristic
238C2×C4×C17C2×#4111yescharacteristic
239C2×C4×C17C2×#4111yescharacteristic
240C2×C4×C17C2×#4111yescharacteristic
241C2×C4×C17C2×#4111yescharacteristic
242C2×Dic17—22no—
243C2×Dic17—22no—
244C2×Dic17—22no—
245C2×Dic17—22no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17—41no—
251C2×Dic17—41no—
252C2×Dic17—41no—
253C2×Dic17—41no—
254C2×Dic17—84no—
255#3622—22no—
256#3622—22no—
257#3622—22no—
258#3622—22no—
259C2×#433—171no—
260#2314D1711yescharacteristic
261#2274—171no—
262#1949—171no—
263#1949—171no—
264#1948—171no—
265#1947—171no—
266#1854—171no—
267#1854—171no—
268#1853—171no—
269#1852—171no—
270#1776—171no—
271#1776—171no—
272C4×#107—171no—
273C4×#106—171no—
274C22×C17—22no—
275C22×C17—22no—
276C22×C17—22no—
277C22×C17—41no—
278C22×C17—41no—
279C22×C17—41no—
280C22×C17—41no—
281C22×C17C22×C4⋊C411yescharacteristic
282C22×C17#52511yescharacteristic
283C22×C17#52511yescharacteristic
284C22×C17#57611yescharacteristic
285C22×C17#57711yescharacteristic
286C22×C17#57711yescharacteristic
287C22×C17#57811yescharacteristic
288C4×C17—22no—
289C4×C17—22no—
290C4×C17—22no—
291C4×C17—22no—
292C4×C17—22no—
293C4×C17—81no—
294C4×C17—81no—
295C4×C17—81no—
296C4×C17—81no—
297C4×C17C4×#4111yescharacteristic
298C4×C17C4×#4111yescharacteristic
299C4×C17#53311yescharacteristic
300C4×C17#53311yescharacteristic
301Dic17—42no—
302Dic17—42no—
303Dic17—42no—
304Dic17—42no—
305C2×#117—171no—
306C2×#117C2×D1711yescharacteristic
307C22×C4⋊C4—171no—
308#433—178no—
309C2×#107—171no—
310C2×#107—171no—
311C2×#107—171no—
312C2×#107—171no—
313C2×#106—171no—
314C2×#106—171no—
315C2×#106—171no—
316C2×#106—171no—
317C2×C4⋊C8—171no—
318C2×C4⋊C8—171no—
319C2×C22⋊C8—171no—
320C2×C22⋊C8—171no—
321C2×#97—171no—
322C2×C4×C8—171no—
323#388—21no—
324#388—21no—
325#387—21no—
326#387—21no—
327#384C2×D1711yescharacteristic
328#382C2×D1711yescharacteristic
329#380C2×D1711yescharacteristic
330#380C2×D1711yescharacteristic
331C4×C4⋊C4—171no—
332C4×C4⋊C4—171no—
333C4×C4⋊C4—171no—
334C4×C4⋊C4—171no—
335C4×C4⋊C4—341no—
336C4×C4⋊C4—341no—
337C4×C4⋊C4C2×D1711yescharacteristic
338C4×C4⋊C4C2×D1711yescharacteristic
339C4×C22⋊C4—341no—
340C4×C22⋊C4—341no—
341#340—171no—
342#340—171no—
343#340—171no—
344#340—171no—
345#340—171no—
346#340—171no—
347#340—171no—
348#340—171no—
349C2×C17—42no—
350C2×C17#229611yescharacteristic
351C2×C17#291211yescharacteristic
352C2×C17#291311yescharacteristic
353C2×C17#311811yescharacteristic
354C2×C17#311911yescharacteristic
355C2×C17#312011yescharacteristic
356C2×C17#312111yescharacteristic
357C23×C4—341no—
358C23×C4C22×D1711yescharacteristic
359C22×C8—171no—
360C22×C8—171no—
361C22×C8—171no—
362C22×C8—171no—
363#117—342no—
364#117—342no—
365#117Dic3414yes—
366#117C4×D1714yes—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—171no—
370C2×C4⋊C4—171no—
371C2×C4⋊C4—171no—
372C2×C4⋊C4—171no—
373C2×C4⋊C4—342no—
374C2×C4⋊C4—342no—
375C2×C4⋊C4—341no—
376C2×C4⋊C4C22⋊D1711yescharacteristic
377C2×C4⋊C4C22⋊D1711yescharacteristic
378C2×C4⋊C4C22×D1711yescharacteristic
379C2×C4⋊C4C22×D1711yescharacteristic
380C2×C4⋊C4C22×D1711yescharacteristic
381C2×C4⋊C4C22×D1711yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—341no—
385C2×C22⋊C4C22⋊D1711yescharacteristic
386C2×C22⋊C4C22⋊D1711yescharacteristic
387C2×C42—21no—
388C2×C42—21no—
389C2×C42—341no—
390C2×C42—341no—
391C2×C42—341no—
392C2×C42—341no—
393C2×C42—341no—
394C2×C42C22×D1711yescharacteristic
395C2×C42C22×D1711yescharacteristic
396#107—342no—
397#107—342no—
398#107—342no—
399#107—342no—
400#106—342no—
401#106—342no—
402#106—342no—
403#106—342no—
404C4⋊C8—174no—
405C4⋊C8—174no—
406C22⋊C8—174no—
407C22⋊C8—174no—
408#97—174no—
409C4×C8—174no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95—21no—
415#95—21no—
416#95—341no—
417#95—341no—
418#95—341no—
419#95—341no—
420C17#2136711yescharacteristic, complemented, Sylow
421C24C2×C22⋊D1711yescharacteristic
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—341no—
431C22×C4—341no—
432C22×C4—341no—
433C22×C4—341no—
434C22×C4—344no—
435C22×C4—1361no—
436C22×C4—1361no—
437C22×C4C2×Dic3412yes—
438C22×C4C2×C4×D1712yes—
439C22×C4C2×C22⋊D1711yescharacteristic
440C22×C4C2×C22⋊D1711yescharacteristic
441C22×C4C2×C22⋊D1711yescharacteristic
442C22×C4C2×C22⋊D1711yescharacteristic
443C22×C4C2×C22⋊D1711yescharacteristic
444C22×C4C23×D1711yescharacteristic
445C2×C8—172no—
446C2×C8—172no—
447C2×C8—172no—
448C2×C8—172no—
449C2×C8—341no—
450C2×C8—341no—
451C2×C8—341no—
452C2×C8—341no—
453C2×C8—341no—
454C2×C8—341no—
455C2×C8—341no—
456C2×C8—341no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—81no—
466C4⋊C4—81no—
467C4⋊C4—172no—
468C4⋊C4—172no—
469C4⋊C4—172no—
470C4⋊C4—172no—
471C4⋊C4—344no—
472C4⋊C4—341no—
473C4⋊C4—341no—
474C4⋊C4—341no—
475C4⋊C4—341no—
476C4⋊C4#6322514yes—
477C4⋊C4#6322514yes—
478C22⋊C4—81no—
479C22⋊C4—81no—
480C22⋊C4—344no—
481C22⋊C4#6322514yes—
482C22⋊C4#6322514yes—
483C42—81no—
484C42—81no—
485C42—344no—
486C42—1361no—
487C42—1361no—
488C42—1361no—
489C42—1361no—
490C42C2×Dic3412yes—
491C42C2×Dic3412yes—
492C42C2×C4×D1712yes—
493C42C2×C4×D1712yes—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C2×#6322512yes—
501C23C22×C22⋊D1711yescharacteristic
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—342no—
529C2×C4—342no—
530C2×C4—342no—
531C2×C4—342no—
532C2×C4—681no—
533C2×C4—681no—
534C2×C4—681no—
535C2×C4—681no—
536C2×C4—681no—
537C2×C4—681no—
538C2×C4—681no—
539C2×C4—681no—
540C2×C4—1364no—
541C2×C4C2×#6322512yes—
542C2×C4C2×#6322512yes—
543C2×C4C2×#6322512yes—
544C2×C4C2×#6322512yes—
545C2×C4C2×#6322512yes—
546C2×C4C22×Dic3411yescharacteristic
547C2×C4C22×C4×D1711yescharacteristic
548C2×C4#41×D1711yescharacteristic
549C2×C4#41×D1711yescharacteristic
550C2×C4#41×D1711yescharacteristic
551C2×C4#41×D1711yescharacteristic
552C8—342no—
553C8—342no—
554C8—342no—
555C8—342no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22C22×#6322511yescharacteristic
564C22#1176331611yescharacteristic
565C22#1176331611yescharacteristic
566C22#1176361211yescharacteristic
567C22#1176361711yescharacteristic
568C22#1176362011yescharacteristic
569C22#1176362411yescharacteristic
570C4—22no—
571C4—22no—
572C4—22no—
573C4—22no—
574C4—22no—
575C4—81no—
576C4—81no—
577C4—81no—
578C4—81no—
579C4—682no—
580C4—682no—
581C4—682no—
582C4—682no—
583C4#1176327711yescharacteristic
584C4#1176327711yescharacteristic
585C4#1176336211yescharacteristic
586C4#1176336211yescharacteristic
587C2—42no—
588C2#1453126611yescharacteristic
589C2#1453737711yescharacteristic
590C2#1453738211yescharacteristic
591C2#1453896211yescharacteristic
592C2#1453897711yescharacteristic
593C2#1453899211yescharacteristic
594C2#1453900711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#2314C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
cosocleC22×C4×C17C24
hypercenter#2314D17
hypocenterC17#21367
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2314

Complements

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

NH ≅ G/Ntype
3. #14529550C2semidirect
4. #14529550C2semidirect
5. #14529545C2semidirect
6. #14529540C2semidirect
11. #14528682C2semidirect
12. #14528682C2semidirect
13. C4×#10587478C2semidirect
14. C4×#10587479C2semidirect
18. #11762870C4semidirect
119. C4⋊#3622C2×C4semidirect
120. C4⋊#3622C2×C4semidirect
121. #10587474C2×C4semidirect
122. C4×#3622C2×C4semidirect
420. C17#21367semidirect

Maximal subgroups

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

subgroupindex
C2×#117628702
#145311402
#145295502
#145295452
#145295402
#145289812
#145289762
#145289712
#145286822
C4×#105874782
C4×#105874792
C17×#23142
#2136717