Finite Group Databasebeta

← #17052071

Subgroups — #17052071

Order 4352.

810 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531677C211yesmaximal, characteristic
2#14531653C211yesmaximal, characteristic
3#14529523C211yesmaximal, characteristic, complemented
4#14529518C211yesmaximal, characteristic, complemented
5#14529378C211yesmaximal, characteristic, complemented
6#14529371C211yesmaximal, characteristic, complemented
7#14529364C211yesmaximal, characteristic, complemented
8#14529357C211yesmaximal, characteristic, complemented
9#14528879C211yesmaximal, characteristic
10#14528874C211yesmaximal, characteristic
11#14528817C211yesmaximal, characteristic
12#14528684C211yesmaximal, characteristic, complemented
13#106×Dic17C211yesmaximal, characteristic, complemented
14#14528565C211yesmaximal, characteristic
15C2×C17×#428C211yesmaximal, characteristic
16C2×#10587612—21no—
17C2×#10587489C2211yescharacteristic
18C2×#10587488C2211yescharacteristic
19C2×C22⋊#3622C2211yescharacteristic
20C2×#10587486C2211yescharacteristic
21C2×C8×Dic17C2211yescharacteristic
22#11762708C2211yescharacteristic
23#11762708C2211yescharacteristic
24#11762708C2211yescharacteristic
25#11762708C2211yescharacteristic
26#11762702—21no—
27#11762702—21no—
28#11762695—21no—
29#11762695—21no—
30#11762677—21no—
31#11762672—21no—
32#11762669C2211yescharacteristic
33#11762669C2211yescharacteristic
34C4⋊C4×Dic17C2211yescharacteristic
35C4⋊C4×Dic17C2211yescharacteristic
36C4⋊C4×Dic17C2211yescharacteristic
37C4⋊C4×Dic17C2211yescharacteristic
38C4×C4⋊Dic17—21no—
39C4×C22⋊Dic17C2211yescharacteristic
40#11762572C2211yescharacteristic
41#11762572C2211yescharacteristic
42#11762571C2211yescharacteristic
43#11762571C2211yescharacteristic
44#11762571C2211yescharacteristic
45#11762571C2211yescharacteristic
46#11762571C2211yescharacteristic
47#11762571C2211yescharacteristic
48#11762571C2211yescharacteristic
49#11762571C2211yescharacteristic
50#11762563C2211yescharacteristic
51#11762563C2211yescharacteristic
52C22×M16×C17C2211yescharacteristic
53C2×C17×#117C2211yescharacteristic
54C22×C4⋊C4×C17C2211yescharacteristic
55C17×#428C418yescomplemented
56C2×C17×#107C2211yescharacteristic
57C2×C17×#107C2211yescharacteristic
58C2×C17×#106C2211yescharacteristic
59C2×C17×#106C2211yescharacteristic
60C23×C4×C17—21no—
61C23×C4×C17C2311yescharacteristic
62C2×M16×C17—22no—
63C2×M16×C17—22no—
64C2×M16×C17C2×C414yes—
65C22×C8×C17C2311yescharacteristic
66C22×C8×C17C2311yescharacteristic
67C17×#117—22no—
68C17×#117C2×C414yes—
69C2×C4⋊C4×C17—21no—
70C2×C4⋊C4×C17—22no—
71C2×C4⋊C4×C17C2×C414yes—
72C2×C4⋊C4×C17C2311yescharacteristic
73C2×C4⋊C4×C17C2311yescharacteristic
74C2×C4⋊C4×C17C2311yescharacteristic
75C2×C4⋊C4×C17C2311yescharacteristic
76C2×C22⋊C4×C17—21no—
77C2×C42×C17—21no—
78C2×C22⋊Dic17—21no—
79C2×C22⋊Dic17—21no—
80C2×C22⋊Dic17D411yescharacteristic
81C2×C22⋊Dic17D411yescharacteristic
82#10587612—44no—
83C2×C4⋊Dic17—21no—
84C2×C4⋊Dic17—21no—
85C2×C4⋊Dic17—21no—
86C2×C4⋊Dic17—21no—
87C2×C4⋊Dic17C2311yescharacteristic
88C2×C4⋊Dic17C2311yescharacteristic
89C2×C4⋊Dic17C2311yescharacteristic
90C2×C4⋊Dic17C2311yescharacteristic
91C2×C4×Dic17—21no—
92C2×C4×Dic17—21no—
93C2×C4×Dic17—21no—
94C2×C4×Dic17—21no—
95C2×C4×Dic17C2311yescharacteristic
96C2×C4×Dic17C2311yescharacteristic
97C22×#3622C2311yescharacteristic
98C22×#3622C2311yescharacteristic
99C17×#107C2×C414yes—
100C17×#107C2×C414yes—
101C17×#106C2×C414yes—
102C17×#106C2×C414yes—
103#10587502—21no—
104#10587502—21no—
105#10587502—21no—
106#10587502—21no—
107#10587502—21no—
108#10587502—21no—
109#10587502—21no—
110#10587502—21no—
111#10587502—21no—
112#10587502—21no—
113#10587502D411yescharacteristic
114#10587502D411yescharacteristic
115C22⋊#3622—22no—
116#10587489—22no—
117#10587488—22no—
118#10587486—22no—
119C8×Dic17—22no—
120C24×C17C2×D411yescharacteristic
121C22×C4×C17—21no—
122C22×C4×C17—21no—
123C22×C4×C17—21no—
124C22×C4×C17—21no—
125C22×C4×C17—24no—
126C22×C4×C17—21no—
127C22×C4×C17—21no—
128C22×C4×C17—21no—
129C22×C4×C17—21no—
130C22×C4×C17—42no—
131C22×C4×C17—42no—
132C22×C4×C17C22×C412yes—
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17#4112yes—
135C22×C4×C17C2411yescharacteristic
136C22×Dic17—21no—
137C22×Dic17—21no—
138C22×Dic17—21no—
139C22×Dic17—21no—
140C22×Dic17C2×D411yescharacteristic
141C22×Dic17C2×D411yescharacteristic
142C22×Dic17C2×D411yescharacteristic
143C22×Dic17C2×D411yescharacteristic
144M16×C17—28no—
145C2×C8×C17—21no—
146C2×C8×C17—21no—
147C2×C8×C17C22×C412yes—
148C2×C8×C17C22×C412yes—
149C2×C8×C17#4112yes—
150C2×C8×C17#4112yes—
151C4⋊C4×C17—24no—
152C4⋊C4×C17—21no—
153C4⋊C4×C17—21no—
154C4⋊C4×C17—21no—
155C4⋊C4×C17—21no—
156C4⋊C4×C17C22×C412yes—
157C4⋊C4×C17C22×C412yes—
158C4⋊C4×C17C22×C412yes—
159C4⋊C4×C17C22×C412yes—
160C22⋊C4×C17—24no—
161C42×C17—24no—
162C22⋊Dic17—22no—
163C22⋊Dic17—41no—
164C22⋊Dic17—42no—
165C22⋊Dic17—81no—
166C4⋊Dic17—21no—
167C4⋊Dic17—21no—
168C4⋊Dic17—21no—
169C4⋊Dic17—21no—
170C4⋊Dic17—21no—
171C4⋊Dic17—21no—
172C4⋊Dic17—21no—
173C4⋊Dic17—21no—
174C4⋊Dic17—42no—
175C4⋊Dic17—42no—
176C4⋊Dic17—42no—
177C4⋊Dic17—42no—
178C4×Dic17—21no—
179C4×Dic17—21no—
180C4×Dic17—21no—
181C4×Dic17—21no—
182C4×Dic17—42no—
183C4×Dic17—42no—
184C4×Dic17—81no—
185C4×Dic17—81no—
186C2×#3622—21no—
187C2×#3622—21no—
188C2×#3622—21no—
189C2×#3622—21no—
190C2×#3622—21no—
191C2×#3622—21no—
192#22622—171nomaximal, Sylow
193C23×C17—21no—
194C23×C17—21no—
195C23×C17—21no—
196C23×C17—21no—
197C23×C17—21no—
198C23×C17—21no—
199C23×C17C4×D412yes—
200C23×C17C22×D411yescharacteristic
201C2×C4×C17—21no—
202C2×C4×C17—21no—
203C2×C4×C17—22no—
204C2×C4×C17—22no—
205C2×C4×C17—22no—
206C2×C4×C17—22no—
207C2×C4×C17—22no—
208C2×C4×C17—22no—
209C2×C4×C17—22no—
210C2×C4×C17—22no—
211C2×C4×C17—48no—
212C2×C4×C17—41no—
213C2×C4×C17—41no—
214C2×C4×C17—41no—
215C2×C4×C17—41no—
216C2×C4×C17—41no—
217C2×C4×C17—41no—
218C2×C4×C17—41no—
219C2×C4×C17—41no—
220C2×C4×C17C4×D412yes—
221C2×C4×C17C23×C411yescharacteristic
222C2×C4×C17C2×#4111yescharacteristic
223C2×C4×C17C2×#4111yescharacteristic
224C2×C4×C17C2×#4111yescharacteristic
225C2×C4×C172−1+411yescharacteristic
226C2×C4×C172−1+411yescharacteristic
227C2×Dic17—21no—
228C2×Dic17—21no—
229C2×Dic17—21no—
230C2×Dic17—21no—
231C2×Dic17—21no—
232C2×Dic17—21no—
233C2×Dic17—21no—
234C2×Dic17—21no—
235C2×Dic17—21no—
236C2×Dic17—21no—
237C2×Dic17—21no—
238C2×Dic17—41no—
239C2×Dic17—41no—
240C2×Dic17—41no—
241C2×Dic17—41no—
242C2×Dic17—41no—
243C2×Dic17—41no—
244C2×Dic17—41no—
245C2×Dic17—41no—
246C2×Dic17—41no—
247C2×Dic17—41no—
248C2×Dic17—41no—
249C2×Dic17—41no—
250C2×Dic17AGL(1,ℤ/8)12yes—
251C8×C17—22no—
252C8×C17#11714yescomplemented
253#3622—41no—
254#3622—41no—
255C2×#428D1711yescharacteristic
256#2347—171no—
257#2344—171no—
258#1943—171no—
259#1942—171no—
260#1921—171no—
261#1920—171no—
262#1919—171no—
263#1918—171no—
264#1825—171no—
265#1824—171no—
266#1809—171no—
267#1776—171no—
268C4×#106—171no—
269#1753—171no—
270C22×C17—22no—
271C22×C17—22no—
272C22×C17—22no—
273C22×C17—41no—
274C22×C17—41no—
275C22×C17—41no—
276C22×C17—41no—
277C22×C17C2×C4×D411yescharacteristic
278C22×C17#54711yescharacteristic
279C22×C17#54711yescharacteristic
280C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C17#57511yescharacteristic
282C22×C17#57711yescharacteristic
283C22×C17#57811yescharacteristic
284C4×C17—22no—
285C4×C17—42no—
286C4×C17—42no—
287C4×C17—42no—
288C4×C17—42no—
289C4×C17C2×#11711yescharacteristic
290C4×C17#51911yescharacteristic
291C4×C17#55211yescharacteristic
292C4×C17#55211yescharacteristic
293Dic17—22no—
294Dic17—41no—
295Dic17—41no—
296Dic17—41no—
297Dic17—81no—
298Dic17—81no—
299Dic17—81no—
300Dic17—81no—
301C22×M16C2×D1711yescharacteristic
302C2×#117—341no—
303C2×#117C2×D1711yescharacteristic
304C22×C4⋊C4C2×D1711yescharacteristic
305#428Dic1718yescomplemented
306C2×#107—171no—
307C2×#107C2×D1711yescharacteristic
308C2×#107C2×D1711yescharacteristic
309C2×#106—171no—
310C2×#106C2×D1711yescharacteristic
311C2×#106C2×D1711yescharacteristic
312C2×C22⋊C8—171no—
313C2×#97—171no—
314C2×C4×C8—171no—
315#389—171no—
316#389—171no—
317#389—171no—
318#389—171no—
319#388—341no—
320#388—341no—
321#387—341no—
322#387—341no—
323#383—341no—
324#381—341no—
325#380—171no—
326#380—171no—
327C4×C4⋊C4—171no—
328C4×C4⋊C4—171no—
329C4×C4⋊C4—171no—
330C4×C4⋊C4—171no—
331C4×C4⋊C4—341no—
332C4×C22⋊C4—171no—
333#340—171no—
334#340—171no—
335#340—171no—
336#340—171no—
337#340—171no—
338#340—171no—
339#340—171no—
340#340—171no—
341#340—171no—
342#340—171no—
343#336—171no—
344#336—171no—
345C2×C17—42no—
346C2×C17#234411yescharacteristic
347C2×C17C4×AGL(1,ℤ/8)11yescharacteristic
348C2×C17#294711yescharacteristic
349C2×C17#334511yescharacteristic
350C2×C17#334911yescharacteristic
351C2×C17#335111yescharacteristic
352C2×C17#335311yescharacteristic
353C23×C4—21no—
354C23×C4C22×D1711yescharacteristic
355C2×M16—22no—
356C2×M16—22no—
357C2×M16C2×Dic1714yes—
358C22×C8—171no—
359C22×C8—171no—
360C22×C8C22×D1711yescharacteristic
361C22×C8C22×D1711yescharacteristic
362#117—22no—
363#117—684no—
364#117C2×Dic1714yes—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—171no—
368C2×C4⋊C4—171no—
369C2×C4⋊C4—171no—
370C2×C4⋊C4—171no—
371C2×C4⋊C4—341no—
372C2×C4⋊C4—341no—
373C2×C4⋊C4—341no—
374C2×C4⋊C4—341no—
375C2×C4⋊C4C2×Dic1714yes—
376C2×C4⋊C4C22×D1711yescharacteristic
377C2×C4⋊C4C22×D1711yescharacteristic
378C2×C4⋊C4C22×D1711yescharacteristic
379C2×C4⋊C4C22×D1711yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—171no—
382C2×C22⋊C4—171no—
383C2×C22⋊C4—341no—
384C2×C22⋊C4—341no—
385C2×C42—21no—
386C2×C42—171no—
387C2×C42—171no—
388C2×C42—341no—
389C2×C42—341no—
390C2×C42—341no—
391C2×C42—341no—
392#107—342no—
393#107C2×Dic1714yes—
394#107C2×Dic1714yes—
395#106—342no—
396#106C2×Dic1714yes—
397#106C2×Dic1714yes—
398C22⋊C8—342no—
399#97—342no—
400C4×C8—342no—
401#95—171no—
402#95—171no—
403#95—341no—
404#95—341no—
405#95—341no—
406#95—341no—
407#95—341no—
408#95—341no—
409#95—341no—
410#95—341no—
411#95—341no—
412#95—341no—
413C17#2262211yescharacteristic, complemented, Sylow
414C24D4×D1711yescharacteristic
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—171no—
427C22×C4—171no—
428C22×C4—171no—
429C22×C4—171no—
430C22×C4—341no—
431C22×C4—341no—
432C22×C4—341no—
433C22×C4—341no—
434C22×C4D4×D1711yescharacteristic
435C22×C4#6325412yes—
436C22×C4C22×Dic1712yes—
437C22×C4C23×D1711yescharacteristic
438M16—28no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—341no—
442C2×C8—341no—
443C2×C8—341no—
444C2×C8—341no—
445C2×C8—341no—
446C2×C8—341no—
447C2×C8#6325212yes—
448C2×C8#6325212yes—
449C2×C8C22×Dic1712yes—
450C2×C8C22×Dic1712yes—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—24no—
456C4⋊C4—341no—
457C4⋊C4—341no—
458C4⋊C4—341no—
459C4⋊C4—341no—
460C4⋊C4—341no—
461C4⋊C4—341no—
462C4⋊C4—341no—
463C4⋊C4—341no—
464C4⋊C4—682no—
465C4⋊C4—682no—
466C4⋊C4—682no—
467C4⋊C4—682no—
468C4⋊C4C22×Dic1712yes—
469C4⋊C4C22×Dic1712yes—
470C4⋊C4C22×Dic1712yes—
471C4⋊C4C22×Dic1712yes—
472C22⋊C4—24no—
473C22⋊C4—342no—
474C22⋊C4—681no—
475C22⋊C4—682no—
476C22⋊C4—1361no—
477C42—24no—
478C42—341no—
479C42—341no—
480C42—341no—
481C42—341no—
482C42—682no—
483C42—682no—
484C42—1361no—
485C42—1361no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23D4×Dic1712yes—
493C23C2×D4×D1711yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—48no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—172no—
514C2×C4—341no—
515C2×C4—341no—
516C2×C4—341no—
517C2×C4—341no—
518C2×C4—341no—
519C2×C4—341no—
520C2×C4—341no—
521C2×C4—341no—
522C2×C4—341no—
523C2×C4—341no—
524C2×C4—341no—
525C2×C4—681no—
526C2×C4—681no—
527C2×C4—681no—
528C2×C4—681no—
529C2×C4—681no—
530C2×C4—681no—
531C2×C4—681no—
532C2×C4—681no—
533C2×C4—681no—
534C2×C4—681no—
535C2×C4—681no—
536C2×C4—681no—
537C2×C4D4×Dic1712yes—
538C2×C4C2×#6325211yescharacteristic
539C2×C4C2×#6325211yescharacteristic
540C2×C4C2×#6325411yescharacteristic
541C2×C4#1058769411yescharacteristic
542C2×C4#1058769411yescharacteristic
543C2×C4C23×Dic1711yescharacteristic
544C8—22no—
545C8—681no—
546C8—681no—
547C8#1058761214yescomplemented
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×D4×Dic1711yescharacteristic
556C22#1176346011yescharacteristic
557C22#1176346011yescharacteristic
558C22AGL(1,ℤ/8)×D1711yescharacteristic
559C22#1176360511yescharacteristic
560C22#1176361811yescharacteristic
561C22#1176362511yescharacteristic
562C4—22no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—342no—
568C4—681no—
569C4—681no—
570C4—681no—
571C4—1361no—
572C4—1361no—
573C4—1361no—
574C4—1361no—
575C4C2×#1058761211yescharacteristic
576C4#1176328711yescharacteristic
577C4#1176349411yescharacteristic
578C4#1176349411yescharacteristic
579C2—42no—
580C2#1453165311yescharacteristic
581C2AGL(1,ℤ/8)×Dic1711yescharacteristic
582C2#1453762811yescharacteristic
583C2#1454125511yescharacteristic
584C2#1454131511yescharacteristic
585C2#1454134011yescharacteristic
586C2#1454136411yescharacteristic
subgroupG/H
centerC23C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C17×#428C2
socleC23×C17C22×D4
prime radicalC24×C17C2×D4
cosocleC22×C4×C17C24
hypercenterC2×#428D17
hypocenterC17#22622
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
3. #14529523C2semidirect
4. #14529518C2semidirect
5. #14529378C2semidirect
6. #14529371C2semidirect
7. #14529364C2semidirect
8. #14529357C2semidirect
12. #14528684C2semidirect
13. #106×Dic17C2semidirect
55. C17×#428C4semidirect
252. C8×C17#117semidirect
305. #428Dic17semidirect
413. C17#22622semidirect
547. C8#10587612semidirect

Maximal subgroups

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

subgroupindex
#145316772
#145316532
#145295232
#145295182
#145293782
#145293712
#145293642
#145293572
#145288792
#145288742
#145288172
#145286842
#106×Dic172
#145285652
C2×C17×#4282
#2262217