Finite Group Databasebeta

← #16936667

Subgroups — #16936667

Order 4352.

712 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763100C211yesmaximal, characteristic, complemented
2C2×#11763099C211yesmaximal, characteristic, complemented
3#14533697C211yesmaximal, characteristic
4#14529586C211yesmaximal, characteristic, complemented
5C2×#11762538C211yesmaximal, characteristic, complemented
6#14526668C211yesmaximal, characteristic, complemented
7C2×C17×D4⋊C8C211yesmaximal, characteristic, complemented
8C2×#10587555C2211yescharacteristic
9C2×#10587616—21no—
10C2×#10587624—21no—
11C2×C4×D68C2211yescharacteristic
12#11763100—22no—
13#11763100—22no—
14#11763099—22no—
15#11763099—22no—
16C2×#10587488—21no—
17C2×#10587479—21no—
18C2×C4⋊#3622C2211yescharacteristic
19C2×#10587487—22no—
20C2×#10587492—21no—
21C2×#10587503—21no—
22#11762725—22no—
23#11762679C2211yescharacteristic
24#11762669—21no—
25#11762666—21no—
26#11762538—22no—
27C2×C4×D4×C17C2211yescharacteristic
28C2×C17×C4⋊C8C2211yescharacteristic
29C2×C4×C8×C17C2211yescharacteristic
30C17×D4⋊C8—22no—
31C22×D4×C17D411yescharacteristic
32C23×C4×C17—21no—
33C22×C22⋊D17—22no—
34C22×D68D411yescharacteristic
35C22×C4×D17—21no—
36C22×Dic34—21no—
37C22×C8×C17—21no—
38C22×C8×C17—21no—
39C4×D4×C17—21no—
40C4×D4×C17—21no—
41C4×D4×C17—21no—
42C4×D4×C17—21no—
43C2×C4⋊C4×C17D411yescharacteristic
44C2×C22⋊C4×C17—21no—
45C2×C42×C17C2311yescharacteristic
46C2×C22⋊Dic17—22no—
47#10587624—42no—
48#10587624—42no—
49#10587616—42no—
50#10587616—42no—
51C2×#63227—22no—
52C2×#63227—21no—
53C2×C4⋊Dic17—21no—
54C2×C4⋊Dic17—21no—
55C2×C4⋊Dic17—21no—
56C2×C4⋊Dic17—21no—
57C2×C4⋊Dic17D411yescharacteristic
58C2×C4⋊Dic17D412yes—
59C22×#3622—21no—
60C4×D68—21no—
61C4×D68—21no—
62C4×D68—21no—
63C4×D68—21no—
64#10587555—21no—
65#10587555—21no—
66#10587555—21no—
67#10587555D412yescomplemented
68C17×C4⋊C8—21no—
69C17×C4⋊C8D412yescomplemented
70C4×C8×C17—21no—
71C4×C8×C17—21no—
72#10587503—24no—
73#10587502—22no—
74#10587492—24no—
75#10587488—24no—
76#10587487—44no—
77#10587479—24no—
78C4⋊#3622—21no—
79C4⋊#3622D412yescomplemented
80C24×C17—21no—
81C23×D17—21no—
82C2×D4×C17—21no—
83C2×D4×C17—21no—
84C2×D4×C17—21no—
85C2×D4×C17—21no—
86C2×D4×C17—21no—
87C2×D4×C17SD1612yes—
88C22×C4×C17—21no—
89C22×C4×C17—41no—
90C22×C4×C17—41no—
91C22×C4×C17—41no—
92C22×C4×C17—41no—
93C22×C4×C17—41no—
94C22×C4×C17—41no—
95C22×C4×C17C2×D411yescharacteristic
96C22×C4×C17C2×D411yescharacteristic
97C22×C4×C17C2×D411yescharacteristic
98C2×C22⋊D17—44no—
99C2×C22⋊D17—44no—
100C2×C22⋊D17—82no—
101C22×Dic17—21no—
102C22×Dic17—22no—
103C22×Dic17—41no—
104C2×D68—21no—
105C2×D68—21no—
106C2×D68—21no—
107C2×D68—21no—
108C2×D68—21no—
109C2×D68SD1612yes—
110C2×C4×D17—41no—
111C2×C4×D17—41no—
112C2×C4×D17—41no—
113C2×C4×D17—41no—
114C2×C4×D17—41no—
115C2×C4×D17—41no—
116C2×Dic34—22no—
117C2×Dic34—42no—
118C2×Dic34—42no—
119C2×Dic34—41no—
120C2×C8×C17—22no—
121C2×C8×C17—22no—
122C2×C8×C17—41no—
123C2×C8×C17—41no—
124C2×C8×C17—41no—
125C2×C8×C17—41no—
126C4⋊C4×C17—21no—
127C4⋊C4×C17—21no—
128C22⋊C4×C17—41no—
129C22⋊C4×C17—41no—
130C42×C17C2×D411yescharacteristic
131C42×C17C2×D411yescharacteristic
132C42×C17C2×D411yescharacteristic
133C42×C17#4111yescharacteristic
134C22⋊Dic17—44no—
135#63227—44no—
136#63227—41no—
137#63227—41no—
138C4⋊Dic17—22no—
139C4⋊Dic17—21no—
140C4⋊Dic17—22no—
141C4⋊Dic17—21no—
142C4⋊Dic17—41no—
143C4⋊Dic17—41no—
144C4⋊Dic17—41no—
145C4⋊Dic17—41no—
146C4⋊Dic17—41no—
147C4⋊Dic17—41no—
148C4⋊Dic17—42no—
149C2×#3622—22no—
150C2×#3622—41no—
151C2×#3622—41no—
152#10235—171nomaximal, Sylow
153C23×C17—22no—
154C23×C17—41no—
155C23×C17—41no—
156C23×C17—41no—
157C23×C17—41no—
158C23×C17—41no—
159C23×C17—41no—
160C23×C17#12011yescharacteristic
161C22×D17—22no—
162C22×D17—41no—
163C22×D17—41no—
164C22×D17—41no—
165C22×D17—41no—
166C22×D17—41no—
167C22×D17—41no—
168D4×C17—22no—
169D4×C17—22no—
170D4×C17—41no—
171D4×C17—41no—
172C2×C4×C17—21no—
173C2×C4×C17—21no—
174C2×C4×C17—41no—
175C2×C4×C17—41no—
176C2×C4×C17—41no—
177C2×C4×C17—81no—
178C2×C4×C17—81no—
179C2×C4×C17—81no—
180C2×C4×C17—81no—
181C2×C4×C17#12011yescharacteristic
182C2×C4×C17#12011yescharacteristic
183C2×C4×C17#12111yescharacteristic
184C2×C4×C17#12111yescharacteristic
185C2×C4×C17#12111yescharacteristic
186C2×C4×C17#12711yescharacteristic
187C2×C4×C17C2×SD1611yescharacteristic
188C2×C4×C17C2×SD1611yescharacteristic
189C2×C4×C17#13511yescharacteristic
190C2×C4×C17#13511yescharacteristic
191C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
192C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
193C2×C4×C17#13711yescharacteristic
194C2×C4×C17#13711yescharacteristic
195C22⋊D17—88no—
196C2×Dic17—42no—
197C2×Dic17—41no—
198C2×Dic17—41no—
199C2×Dic17—42no—
200C2×Dic17—42no—
201C2×Dic17—41no—
202C2×Dic17—42no—
203C2×Dic17—81no—
204C2×Dic17—81no—
205D68—22no—
206D68—22no—
207D68—41no—
208D68—41no—
209C4×D17—81no—
210C4×D17—81no—
211C4×D17—81no—
212C4×D17—81no—
213Dic34—44no—
214Dic34—82no—
215C8×C17—42no—
216C8×C17—42no—
217#3622—42no—
218C2×#476—171no—
219C2×#476—171no—
220#2537—171no—
221#1956—171no—
222C2×D4⋊C8—171no—
223C2×D4⋊C8D1711yescharacteristic, complemented
224#1296—171no—
225C22×C17—42no—
226C22×C17—42no—
227C22×C17—42no—
228C22×C17—81no—
229C22×C17—81no—
230C22×C17—81no—
231C22×C17—81no—
232C22×C17#39211yescharacteristic
233C22×C17#45011yescharacteristic
234C22×C17#45011yescharacteristic
235C22×C17#45211yescharacteristic
236C22×C17#45211yescharacteristic
237C22×C17#45311yescharacteristic
238C22×C17#45411yescharacteristic
239C2×D17—42no—
240C2×D17—42no—
241C2×D17—42no—
242C2×D17—81no—
243C2×D17—81no—
244C2×D17—81no—
245C2×D17—81no—
246C4×C17—21no—
247C4×C17—21no—
248C4×C17—81no—
249C4×C17#44811yescharacteristic
250C4×C17#44811yescharacteristic
251C4×C17#44911yescharacteristic
252C4×C17#44911yescharacteristic
253C4×C17#46511yescharacteristic
254C4×C17#46511yescharacteristic
255C4×C17#46811yescharacteristic
256C4×C17#46811yescharacteristic
257Dic17—82no—
258Dic17—81no—
259Dic17—82no—
260C2×C4⋊Q8—171no—
261C2×#121—341no—
262C2×#121—341no—
263C2×C4×D4—171no—
264C2×C4×D4C2×D1711yescharacteristic
265#476—342no—
266#476—342no—
267#476—342no—
268#476—342no—
269C2×#106—341no—
270C2×#106—341no—
271C2×C4⋊C8—171no—
272C2×C4⋊C8C2×D1711yescharacteristic
273C2×#103—342no—
274C2×#102—341no—
275C2×#102—341no—
276C2×C4×C8C2×D1711yescharacteristic
277#394—342no—
278#384—171no—
279#380—341no—
280#380—341no—
281D4⋊C8—22no—
282D4⋊C8—342no—
283C2×C17—82no—
284C2×C17#162911yescharacteristic
285C2×C17#163411yescharacteristic
286C2×C17#163511yescharacteristic
287C2×C17#163511yescharacteristic
288C2×C17#200011yescharacteristic
289C2×C17#200011yescharacteristic
290C2×C17#200611yescharacteristic
291D17—82no—
292C22×Q8—341no—
293C22×D4—171no—
294C22×D4—342no—
295C22×D4D6811yescharacteristic
296C23×C4—21no—
297C23×C4—341no—
298C22×C8—21no—
299C22×C8—21no—
300C22×C8—341no—
301C4⋊Q8—172no—
302C4⋊Q8—341no—
303C4⋊Q8—341no—
304C4⋊Q8—341no—
305#121—682no—
306#121—682no—
307#121—682no—
308#121—682no—
309C4×D4—21no—
310C4×D4—21no—
311C4×D4—21no—
312C4×D4—21no—
313C4×D4—341no—
314C4×D4—341no—
315C4×D4—341no—
316C4×D4—341no—
317C2×C4⋊C4—171no—
318C2×C4⋊C4—172no—
319C2×C4⋊C4—341no—
320C2×C4⋊C4—341no—
321C2×C4⋊C4—341no—
322C2×C4⋊C4—341no—
323C2×C4⋊C4D6811yescharacteristic
324C2×C22⋊C4—21no—
325C2×C22⋊C4—342no—
326C2×C22⋊C4—342no—
327C2×C22⋊C4—341no—
328C2×C42C22×D1711yescharacteristic
329#106—344no—
330#106—344no—
331C4⋊C8—21no—
332C4⋊C8—172no—
333C4⋊C8—341no—
334C4⋊C8C22⋊D1712yescomplemented
335#103—684no—
336#102—344no—
337#102—344no—
338C4×C8—21no—
339C4×C8—21no—
340#95—342no—
341C17#1023511yescharacteristic, complemented, Sylow
342C24—21no—
343C24—341no—
344C2×Q8—342no—
345C2×Q8—682no—
346C2×Q8—682no—
347C2×Q8—681no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—172no—
354C2×D4—341no—
355C2×D4—341no—
356C2×D4—341no—
357C2×D4—341no—
358C2×D4—341no—
359C2×D4—684no—
360C2×D4—684no—
361C2×D4—1362no—
362C2×D4#6321912yes—
363C22×C4—21no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—341no—
371C22×C4—342no—
372C22×C4—681no—
373C22×C4—681no—
374C22×C4—681no—
375C22×C4—681no—
376C22×C4—681no—
377C22×C4—681no—
378C22×C4—681no—
379C22×C4C2×D6811yescharacteristic
380C22×C4D4×D1711yescharacteristic
381C22×C4D4×D1711yescharacteristic
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—342no—
389C2×C8—681no—
390C2×C8—681no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—342no—
394C4⋊C4—341no—
395C4⋊C4—342no—
396C4⋊C4—341no—
397C4⋊C4—682no—
398C4⋊C4—681no—
399C4⋊C4—681no—
400C4⋊C4—681no—
401C4⋊C4—681no—
402C4⋊C4—681no—
403C4⋊C4—681no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—684no—
407C22⋊C4—684no—
408C22⋊C4—681no—
409C22⋊C4—681no—
410C42#6325211yescharacteristic
411C42D4×D1711yescharacteristic
412C42D4×D1711yescharacteristic
413C42C2×C22⋊D1711yescharacteristic
414C23—22no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—342no—
422C23—681no—
423C23—681no—
424C23—681no—
425C23—681no—
426C23—681no—
427C23—681no—
428C23#1058756811yescharacteristic
429Q8—684no—
430Q8—1362no—
431D4—22no—
432D4—22no—
433D4—41no—
434D4—41no—
435D4—342no—
436D4—342no—
437D4—681no—
438D4—681no—
439D4—1368no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—682no—
450C2×C4—681no—
451C2×C4—682no—
452C2×C4—681no—
453C2×C4—681no—
454C2×C4—682no—
455C2×C4—682no—
456C2×C4—1361no—
457C2×C4—1361no—
458C2×C4—1361no—
459C2×C4—1361no—
460C2×C4—1361no—
461C2×C4—1361no—
462C2×C4#1058756811yescharacteristic
463C2×C4#1058757011yescharacteristic
464C2×C4#1058757011yescharacteristic
465C2×C4C2×#6321911yescharacteristic
466C2×C4#1058759011yescharacteristic
467C2×C4#1058759411yescharacteristic
468C2×C4#1058759511yescharacteristic
469C2×C4#1058759711yescharacteristic
470C2×C4#1058759811yescharacteristic
471C2×C4SD16×D1711yescharacteristic
472C2×C4#1058760111yescharacteristic
473C2×C4#1058761611yescharacteristic
474C2×C4#1058762311yescharacteristic
475C2×C4#1058762611yescharacteristic
476C8—42no—
477C8—42no—
478C8—682no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22—682no—
487C22—682no—
488C22—682no—
489C22—1361no—
490C22—1361no—
491C22—1361no—
492C22—1361no—
493C22#1176271811yescharacteristic
494C22#1176296111yescharacteristic
495C22#1176296211yescharacteristic
496C22#1176297211yescharacteristic
497C22#1176297311yescharacteristic
498C22#1176297811yescharacteristic
499C22#1176298311yescharacteristic
500C4—21no—
501C4—21no—
502C4—81no—
503C4—1362no—
504C4—1362no—
505C4—1361no—
506C4#1176294611yescharacteristic
507C4#1176295011yescharacteristic
508C4#1176295111yescharacteristic
509C4#1176295611yescharacteristic
510C4#1176303711yescharacteristic
511C4#1176303811yescharacteristic
512C4#1176305411yescharacteristic
513C4#1176305511yescharacteristic
514C2—82no—
515C2—1362no—
516C2#1452790211yescharacteristic
517C2#1452793111yescharacteristic
518C2#1452793711yescharacteristic
519C2#1452793811yescharacteristic
520C2#1452978811yescharacteristic
521C2#1452978911yescharacteristic
522C2#1452982711yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC2×C17×D4⋊C8C2
socleC23×C17#120
prime radical#14533697C2
hypercenterC2×D4⋊C8D17
hypocenterC17#10235
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×D4⋊C8

Complements

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

NH ≅ G/Ntype
1. C2×#11763100C2semidirect
2. C2×#11763099C2semidirect
4. #14529586C2semidirect
5. C2×#11762538C2semidirect
6. #14526668C2semidirect
7. C2×C17×D4⋊C8C2semidirect
67. #10587555D4semidirect
69. C17×C4⋊C8D4semidirect
79. C4⋊#3622D4semidirect
223. C2×D4⋊C8D17semidirect
334. C4⋊C8C22⋊D17semidirect
341. C17#10235semidirect

Maximal subgroups

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

subgroupindex
C2×#117631002
C2×#117630992
#145336972
#145295862
C2×#117625382
#145266682
C2×C17×D4⋊C82
#1023517