Finite Group Databasebeta

← #17045540

Subgroups — #17045540

Order 4352.

1080 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762948C211yesmaximal, characteristic
2C4×#10587616C211yesmaximal, characteristic
3#14529668C212yesmaximal, complemented
4#14529477C212yesmaximal, complemented
5#14529422C212yesmaximal, complemented
6#14529121C212yesmaximal, complemented
7#14529101C211yesmaximal, characteristic
8#14528972C211yesmaximal, characteristic, complemented
9#14528719C211yesmaximal, characteristic, complemented
10#14528546C211yesmaximal, characteristic, complemented
11C2×C17×#391C211yesmaximal, characteristic, complemented
12C22×#63230C2212yes—
13C2×#10587616C2211yescharacteristic
14C2×C4×D68C2211yescharacteristic
15C2×C4×C22⋊D17—21no—
16#11762948C418yescomplemented
17C2×#10587478C2212yes—
18C2×#10587506C2212yes—
19C2×#10587506C2212yes—
20C2×#10587506C2212yes—
21C2×#10587480C2212yes—
22C2×C22⋊#3622C2211yescharacteristic
23C2×C22⋊#3622C2211yescharacteristic
24C2×#10587474C2212yes—
25#11762711C2211yescharacteristic
26#11762707C2211yescharacteristic
27#11762700—21no—
28#11762688—21no—
29#11762669C2211yescharacteristic
30C4×C4⋊Dic17C2211yescharacteristic
31C4×#63227—21no—
32#11762571C2212yes—
33#11762571C2212yes—
34#11762564C2212yes—
35C23×Q8×C17C2211yescharacteristic
36C22×C4⋊C4×C17C2211yescharacteristic
37C22×C42×C17C2211yescharacteristic
38C17×#391—21no—
39C17×#391—21no—
40C17×#391C2212yes—
41C17×#391C2212yes—
42C2×C17×#95—21no—
43C2×C17×#95—21no—
44C22×Q8×C17—21no—
45C22×Q8×C17—22no—
46C22×Q8×C17—42no—
47C22×Q8×C17—42no—
48C22×Q8×C17C2×C414yescomplemented
49C22×Q8×C17C2312yes—
50C23×C4×C17—21no—
51C23×C4×C17—21no—
52C23×C4×C17D411yescharacteristic
53C23×C4×C17D411yescharacteristic
54C23×C4×C17C2311yescharacteristic
55C22×C22⋊D17—21no—
56C22×D68C2311yescharacteristic
57C22×C4×D17—21no—
58C2×C4⋊C4×C17—21no—
59C2×C4⋊C4×C17—42no—
60C2×C4⋊C4×C17C2312yes—
61C2×C42×C17—42no—
62C2×C42×C17D411yescharacteristic
63C2×C42×C17D411yescharacteristic
64C2×C42×C17C2311yescharacteristic
65C2×C42×C17C2311yescharacteristic
66C2×C22⋊Dic17—21no—
67C2×#63230—42no—
68C2×#63230—42no—
69C2×#63230C2×C418yescomplemented
70#10587616—22no—
71#10587616C2×C414yes—
72C4×C22⋊D17—44no—
73C2×#63227—21no—
74C2×#63227—21no—
75C2×#63227—21no—
76C2×#63227—21no—
77C2×C4⋊Dic17C2311yescharacteristic
78C2×C4⋊Dic17C2311yescharacteristic
79C2×C4⋊Dic17C2311yescharacteristic
80C2×#63225—21no—
81C2×C4×Dic17—21no—
82C22×#3622C2312yes—
83C22×#3622C2312yes—
84C4×D68—42no—
85C17×#95—22no—
86C17×#95—22no—
87C17×#95—22no—
88C17×#95—22no—
89#10587506—42no—
90#10587506—42no—
91#10587506C2×C418yescomplemented
92#10587502—21no—
93#10587502—21no—
94#10587502—21no—
95C22⋊#3622—22no—
96C22⋊#3622C2×C414yescomplemented
97#10587480—42no—
98#10587478—42no—
99#10587474—42no—
100C24×C17C2×D411yescharacteristic
101C23×D17—21no—
102C2×Q8×C17—24no—
103C2×Q8×C17—22no—
104C2×Q8×C17—22no—
105C2×Q8×C17—22no—
106C2×Q8×C17—22no—
107C2×Q8×C17—22no—
108C2×Q8×C17—48no—
109C2×Q8×C17—42no—
110C2×Q8×C17—42no—
111C2×Q8×C17—48no—
112C2×Q8×C17C22×C414yes—
113C22×C4×C17—22no—
114C22×C4×C17—22no—
115C22×C4×C17—24no—
116C22×C4×C17—22no—
117C22×C4×C17—22no—
118C22×C4×C17—22no—
119C22×C4×C17—22no—
120C22×C4×C17—22no—
121C22×C4×C17—41no—
122C22×C4×C17—41no—
123C22×C4×C17—41no—
124C22×C4×C17—42no—
125C22×C4×C17—42no—
126C22×C4×C17—42no—
127C22×C4×C17—42no—
128C22×C4×C17—42no—
129C22×C4×C17C22×C412yes—
130C22×C4×C17C2×D411yescharacteristic
131C22×C4×C17C2×D411yescharacteristic
132C22×C4×C17C2×D411yescharacteristic
133C22×C4×C17C2×D411yescharacteristic
134C22×C4×C17C2×D411yescharacteristic
135C22×C4×C17#4112yes—
136C22×C4×C17C2411yescharacteristic
137C2×C22⋊D17—24no—
138C2×C22⋊D17—42no—
139C2×C22⋊D17—42no—
140C22×Dic17—21no—
141C22×Dic17—21no—
142C22×Dic17—21no—
143C2×D68—22no—
144C2×D68—22no—
145C2×D68—21no—
146C2×D68C22×C412yes—
147C2×C4×D17—42no—
148C2×C4×D17—42no—
149C2×C4×D17—42no—
150C4⋊C4×C17—22no—
151C4⋊C4×C17—22no—
152C4⋊C4×C17—42no—
153C42×C17—22no—
154C42×C17—22no—
155C42×C17—41no—
156C42×C17—41no—
157C22⋊Dic17—42no—
158#63230—48no—
159#63227—24no—
160#63227—42no—
161#63227—42no—
162#63227—42no—
163C4⋊Dic17—21no—
164C4⋊Dic17—22no—
165C4⋊Dic17—22no—
166C4⋊Dic17C22×C412yes—
167#63225—42no—
168C4×Dic17—42no—
169C2×#3622—22no—
170C2×#3622—22no—
171C2×#3622—22no—
172C2×#3622—22no—
173C2×#3622—22no—
174C2×#3622C22×C414yes—
175#22047—171nomaximal, Sylow
176C23×C17—21no—
177C23×C17—21no—
178C23×C17—21no—
179C23×C17—21no—
180C23×C17C4×D412yes—
181C23×C17AGL(1,ℤ/8)12yes—
182C23×C17#13712yes—
183C23×C17C22×D411yescharacteristic
184C22×D17—22no—
185C22×D17—41no—
186C22×D17—41no—
187C22×D17—41no—
188C22×D17—41no—
189C22×D17—41no—
190C22×D17—41no—
191Q8×C17—24no—
192Q8×C17—24no—
193Q8×C17—48no—
194Q8×C17—42no—
195Q8×C17—42no—
196C2×C4×C17—21no—
197C2×C4×C17—21no—
198C2×C4×C17—21no—
199C2×C4×C17—21no—
200C2×C4×C17—24no—
201C2×C4×C17—24no—
202C2×C4×C17—24no—
203C2×C4×C17—21no—
204C2×C4×C17—21no—
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—44no—
213C2×C4×C17—44no—
214C2×C4×C17—48no—
215C2×C4×C17—42no—
216C2×C4×C17—42no—
217C2×C4×C17—42no—
218C2×C4×C17—42no—
219C2×C4×C17—42no—
220C2×C4×C17—84no—
221C2×C4×C17C4×D412yes—
222C2×C4×C17#13712yes—
223C2×C4×C17C23×C411yescharacteristic
224C2×C4×C17C2×#4111yescharacteristic
225C2×C4×C17H(5,𝔽2)12yes—
226C2×C4×C17H(5,𝔽2)12yes—
227C22⋊D17—48no—
228C2×Dic17—22no—
229C2×Dic17—41no—
230C2×Dic17—41no—
231C2×Dic17—41no—
232C2×Dic17—41no—
233C2×Dic17—41no—
234C2×Dic17—41no—
235C2×Dic17—41no—
236C2×Dic17—41no—
237D68—24no—
238D68—42no—
239C4×D17—84no—
240#3622—24no—
241#3622—42no—
242C2×#448—171no—
243C4×#121—171no—
244C2×#391D1711yescharacteristic, complemented
245#1973—172no—
246#1935—172no—
247#1928—172no—
248#1877—172no—
249#1874—171no—
250#1852—171no—
251#1787—171no—
252#1749—171no—
253C22×C17—22no—
254C22×C17—22no—
255C22×C17—22no—
256C22×C17—22no—
257C22×C17—22no—
258C22×C17—22no—
259C22×C17#44014yes—
260C22×C17C2×C4×D411yescharacteristic
261C22×C17#53411yescharacteristic
262C22×C17#53511yescharacteristic
263C22×C17C2×AGL(1,ℤ/8)11yescharacteristic
264C22×C17C2×#13711yescharacteristic
265C22×C17C2×#13711yescharacteristic
266C22×C17#57511yescharacteristic
267C2×D17—42no—
268C2×D17—42no—
269C2×D17—42no—
270C2×D17—81no—
271C2×D17—81no—
272C2×D17—81no—
273C2×D17—81no—
274C4×C17—22no—
275C4×C17—22no—
276C4×C17—44no—
277C4×C17—41no—
278C4×C17—41no—
279C4×C17—41no—
280C4×C17—82no—
281C4×C17#51812yes—
282C4×C17#53812yes—
283Dic17—42no—
284Dic17—81no—
285Dic17—81no—
286C23×Q8C2×D1711yescharacteristic
287C22×SD16—172no—
288C2×#121—171no—
289C2×C4×D4—171no—
290C2×C4×D4—341no—
291C22×C4⋊C4C2×D1711yescharacteristic
292C22×C42C2×D1711yescharacteristic
293#448—178no—
294C2×#107—172no—
295C2×#103—172no—
296C2×#103—172no—
297C2×#103—172no—
298C2×#102—172no—
299C2×C22⋊C8—171no—
300C2×C22⋊C8—171no—
301C2×#97—172no—
302#391—21no—
303#391—21no—
304#391C2×D1712yes—
305#391C2×D1712yes—
306#390—171no—
307#389—171no—
308#388—341no—
309#386—341no—
310#380—171no—
311C4×C4⋊C4—171no—
312C4×C22⋊C4—341no—
313C2×#95—21no—
314C2×#95—21no—
315#340—172no—
316#340—172no—
317#336—172no—
318C2×C17—24no—
319C2×C17#232111yescharacteristic
320C2×C17C2×#44011yescharacteristic
321C2×C17C4×#13711yescharacteristic
322C2×C17#319311yescharacteristic
323C2×C17#319611yescharacteristic
324C2×C17#321311yescharacteristic
325C2×C17#321811yescharacteristic
326D17—82no—
327C22×Q8—22no—
328C22×Q8—21no—
329C22×Q8—42no—
330C22×Q8—42no—
331C22×Q8C4×D1714yescomplemented
332C22×Q8C22×D1712yes—
333C22×D4—171no—
334C22×D4—341no—
335C23×C4—21no—
336C23×C4—21no—
337C23×C4—341no—
338C23×C4C22⋊D1711yescharacteristic
339C23×C4C22⋊D1711yescharacteristic
340C23×C4C22×D1711yescharacteristic
341C2×SD16—178no—
342C2×SD16—682no—
343C2×SD16—682no—
344C22×C8—172no—
345C22×C8—172no—
346#121—174no—
347#121—342no—
348C4×D4—682no—
349C4×D4—684no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—42no—
352C2×C4⋊C4—171no—
353C2×C4⋊C4—171no—
354C2×C4⋊C4—171no—
355C2×C4⋊C4—341no—
356C2×C4⋊C4C22×D1712yes—
357C2×C22⋊C4—341no—
358C2×C22⋊C4—341no—
359C2×C22⋊C4—341no—
360C2×C22⋊C4—341no—
361C2×C22⋊C4—341no—
362C2×C42—42no—
363C2×C42—341no—
364C2×C42C22⋊D1711yescharacteristic
365C2×C42C22⋊D1711yescharacteristic
366C2×C42C22×D1711yescharacteristic
367C2×C42C22×D1711yescharacteristic
368#107—682no—
369#103—178no—
370#103—682no—
371#103—682no—
372#102—682no—
373C22⋊C8—174no—
374C22⋊C8—342no—
375#97—682no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—341no—
381#95—341no—
382#95—341no—
383C17#2204711yescharacteristic, complemented, Sylow
384C24—341no—
385C24C2×C22⋊D1711yescharacteristic
386C2×Q8—22no—
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—48no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—48no—
396C2×Q8C2×C4×D1714yes—
397C2×D4—172no—
398C2×D4—344no—
399C2×D4—341no—
400C2×D4—342no—
401C2×D4—342no—
402C2×D4—682no—
403C2×D4—682no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—341no—
421C22×C4—341no—
422C22×C4—341no—
423C22×C4—682no—
424C22×C4—682no—
425C22×C4—682no—
426C22×C4C2×C4×D1712yes—
427C22×C4#6325212yes—
428C22×C4C2×C22⋊D1711yescharacteristic
429C22×C4C2×C22⋊D1711yescharacteristic
430C22×C4C2×C22⋊D1711yescharacteristic
431C22×C4C2×C22⋊D1711yescharacteristic
432C22×C4C2×C22⋊D1711yescharacteristic
433C22×C4C23×D1711yescharacteristic
434SD16—688no—
435C2×C8—174no—
436C2×C8—342no—
437C2×C8—342no—
438C2×C8—342no—
439C2×C8—342no—
440C2×C8—342no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—42no—
444C4⋊C4—172no—
445C4⋊C4—341no—
446C4⋊C4—342no—
447C4⋊C4—342no—
448C4⋊C4—682no—
449C22⋊C4—344no—
450C22⋊C4—682no—
451C22⋊C4—682no—
452C22⋊C4—682no—
453C22⋊C4—682no—
454C42—22no—
455C42—22no—
456C42—41no—
457C42—41no—
458C42—682no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—342no—
464C23—681no—
465C23—681no—
466C23—681no—
467C23—681no—
468C23—681no—
469C23—681no—
470C23C4×C22⋊D1712yes—
471C23#1058762812yes—
472C23#1058763512yes—
473C23C22×C22⋊D1711yescharacteristic
474Q8—24no—
475Q8—24no—
476Q8—48no—
477Q8—42no—
478Q8—42no—
479D4—344no—
480D4—688no—
481D4—682no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—44no—
499C2×C4—48no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—84no—
507C2×C4—342no—
508C2×C4—681no—
509C2×C4—681no—
510C2×C4—681no—
511C2×C4—681no—
512C2×C4—681no—
513C2×C4—681no—
514C2×C4—681no—
515C2×C4—681no—
516C2×C4—1364no—
517C2×C4C4×C22⋊D1712yes—
518C2×C4#1058762812yes—
519C2×C4C22×C4×D1711yescharacteristic
520C2×C4C2×#6325211yescharacteristic
521C2×C4#1058769112yes—
522C2×C4#1058769112yes—
523C8—344no—
524C8—682no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—682no—
532C22—682no—
533C22—682no—
534C22—1361no—
535C22—1361no—
536C22—1361no—
537C22—1361no—
538C22#1176290714yes—
539C22C2×C4×C22⋊D1711yescharacteristic
540C22#1176336811yescharacteristic
541C22#1176337611yescharacteristic
542C22C2×#1058763511yescharacteristic
543C22C2×#1058762811yescharacteristic
544C22C2×#1058762811yescharacteristic
545C22#1176360711yescharacteristic
546C4—22no—
547C4—22no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—44no—
552C4—82no—
553C4—682no—
554C4—1361no—
555C4—1361no—
556C4#1176328212yes—
557C4#1176339712yes—
558C2—24no—
559C2—1362no—
560C2#1453144211yescharacteristic
561C2C2×#1176290711yescharacteristic
562C2C4×#1058762811yescharacteristic
563C2#1453958611yescharacteristic
564C2#1453961411yescharacteristic
565C2#1453977111yescharacteristic
566C2#1453981111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC2×C17×#391C2
socleC23×C17C22×D4
prime radicalC2×#10587616C22
cosocleC22×C4×C17C24
hypercenterC2×#391D17
hypocenterC17#22047
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
3. #14529668C2semidirect
4. #14529477C2semidirect
5. #14529422C2semidirect
6. #14529121C2semidirect
8. #14528972C2semidirect
9. #14528719C2semidirect
10. #14528546C2semidirect
11. C2×C17×#391C2semidirect
16. #11762948C4semidirect
48. C22×Q8×C17C2×C4semidirect
69. C2×#63230C2×C4semidirect
91. #10587506C2×C4semidirect
96. C22⋊#3622C2×C4semidirect
244. C2×#391D17semidirect
331. C22×Q8C4×D17semidirect
383. C17#22047semidirect

Maximal subgroups

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

subgroupindex
C2×#117629482
C4×#105876162
#145296682
#145294772
#145294222
#145291212
#145291012
#145289722
#145287192
#145285462
C2×C17×#3912
#2204717