Finite Group Databasebeta

← #16936637

Subgroups — #16936637

Order 4352.

728 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763011C211yesmaximal, characteristic, complemented
2#14533697C211yesmaximal, characteristic
3#14529746C211yesmaximal, characteristic, complemented
4C2×D4⋊#3622C211yesmaximal, characteristic, complemented
5C2×#11762536C211yesmaximal, characteristic, complemented
6#14526668C211yesmaximal, characteristic, complemented
7C2×C17×#474C211yesmaximal, characteristic, complemented
8C22×#63230—21no—
9C2×#10587616—21no—
10C2×#10587624—21no—
11C2×C4×D68C2211yescharacteristic
12#11763011—22no—
13#11763011—22no—
14C2×C4⋊#3622C2211yescharacteristic
15C2×#10587506—21no—
16C2×C4×#3622C2211yescharacteristic
17#11762725—22no—
18#11762679C2211yescharacteristic
19#11762669—21no—
20#11762666—21no—
21#11762571—22no—
22D4⋊#3622—22no—
23#11762536—22no—
24C2×C17×C4⋊Q8C2211yescharacteristic
25C2×C4×D4×C17C2211yescharacteristic
26C17×#474—22no—
27C17×#474—22no—
28C2×C17×#107—21no—
29C2×C17×C4⋊C8C2211yescharacteristic
30C2×C17×#102—21no—
31C22×Q8×C17—21no—
32C22×D4×C17D411yescharacteristic
33C23×C4×C17—21no—
34C22×C22⋊D17—22no—
35C22×D68D411yescharacteristic
36C22×C4×D17—21no—
37C22×C8×C17—21no—
38C17×C4⋊Q8—21no—
39C17×C4⋊Q8—21no—
40C17×C4⋊Q8—21no—
41C17×C4⋊Q8D412yescomplemented
42C4×D4×C17—21no—
43C4×D4×C17—21no—
44C4×D4×C17—21no—
45C4×D4×C17—21no—
46C2×C4⋊C4×C17—21no—
47C2×C4⋊C4×C17—21no—
48C2×C4⋊C4×C17D411yescharacteristic
49C2×C22⋊C4×C17—21no—
50C2×C42×C17C2311yescharacteristic
51C2×C22⋊Dic17—22no—
52C2×#63230—24no—
53C2×#63230—42no—
54C2×#63230—42no—
55#10587624—42no—
56#10587624—42no—
57#10587616—42no—
58#10587616—42no—
59C2×#63227—22no—
60C2×#63227—21no—
61C2×C4⋊Dic17—21no—
62C2×C4⋊Dic17—21no—
63C2×C4⋊Dic17D411yescharacteristic
64C2×C4⋊Dic17D412yes—
65C22×#3622—21no—
66C22×#3622—21no—
67C4×D68—21no—
68C4×D68—21no—
69C4×D68—21no—
70C4×D68—21no—
71C17×#107—24no—
72C17×C4⋊C8—21no—
73C17×C4⋊C8D412yescomplemented
74C17×#102—24no—
75#10587506—24no—
76#10587502—22no—
77C4⋊#3622—21no—
78C4⋊#3622D412yescomplemented
79C4×#3622—21no—
80C4×#3622—21no—
81C24×C17—21no—
82C23×D17—21no—
83C2×Q8×C17—22no—
84C2×Q8×C17—41no—
85C2×Q8×C17—41no—
86C2×Q8×C17—41no—
87C2×Q8×C17—41no—
88C2×Q8×C17—41no—
89C2×D4×C17—21no—
90C2×D4×C17—21no—
91C2×D4×C17—21no—
92C2×D4×C17—21no—
93C2×D4×C17—21no—
94C2×D4×C17SD1612yescomplemented
95C22×C4×C17—21no—
96C22×C4×C17—41no—
97C22×C4×C17—41no—
98C22×C4×C17—41no—
99C22×C4×C17—41no—
100C22×C4×C17—41no—
101C22×C4×C17—41no—
102C22×C4×C17—41no—
103C22×C4×C17C2×D411yescharacteristic
104C22×C4×C17C2×D411yescharacteristic
105C22×C4×C17C2×D411yescharacteristic
106C2×C22⋊D17—44no—
107C2×C22⋊D17—44no—
108C2×C22⋊D17—82no—
109C22×Dic17—21no—
110C22×Dic17—22no—
111C2×D68—21no—
112C2×D68—21no—
113C2×D68—21no—
114C2×D68—21no—
115C2×D68—21no—
116C2×D68D812yes—
117C2×C4×D17—41no—
118C2×C4×D17—41no—
119C2×C4×D17—41no—
120C2×C4×D17—41no—
121C2×C4×D17—41no—
122C2×C4×D17—41no—
123C2×C8×C17—22no—
124C2×C8×C17—41no—
125C2×C8×C17—41no—
126C4⋊C4×C17—22no—
127C4⋊C4×C17—21no—
128C4⋊C4×C17—22no—
129C4⋊C4×C17—21no—
130C4⋊C4×C17—41no—
131C4⋊C4×C17—41no—
132C22⋊C4×C17—41no—
133C22⋊C4×C17—41no—
134C42×C17C2×D411yescharacteristic
135C42×C17C2×D411yescharacteristic
136C42×C17C2×D411yescharacteristic
137C42×C17#4111yescharacteristic
138C22⋊Dic17—44no—
139#63230—48no—
140#63227—44no—
141#63227—41no—
142#63227—41no—
143C4⋊Dic17—21no—
144C4⋊Dic17—21no—
145C4⋊Dic17—41no—
146C4⋊Dic17—41no—
147C4⋊Dic17—41no—
148C4⋊Dic17—41no—
149C4⋊Dic17—42no—
150C2×#3622—22no—
151C2×#3622—22no—
152C2×#3622—41no—
153C2×#3622—41no—
154C2×#3622—41no—
155C2×#3622—41no—
156#10231—171nomaximal, Sylow
157C23×C17—22no—
158C23×C17—41no—
159C23×C17—41no—
160C23×C17—41no—
161C23×C17—41no—
162C23×C17—41no—
163C23×C17—41no—
164C23×C17#12011yescharacteristic
165C22×D17—22no—
166C22×D17—41no—
167C22×D17—41no—
168C22×D17—41no—
169C22×D17—41no—
170C22×D17—41no—
171C22×D17—41no—
172Q8×C17—42no—
173Q8×C17—42no—
174Q8×C17—81no—
175Q8×C17—81no—
176D4×C17—22no—
177D4×C17—22no—
178D4×C17—41no—
179D4×C17—41no—
180C2×C4×C17—21no—
181C2×C4×C17—21no—
182C2×C4×C17—41no—
183C2×C4×C17—42no—
184C2×C4×C17—41no—
185C2×C4×C17—41no—
186C2×C4×C17—81no—
187C2×C4×C17—81no—
188C2×C4×C17—81no—
189C2×C4×C17—81no—
190C2×C4×C17—81no—
191C2×C4×C17—81no—
192C2×C4×C17#12011yescharacteristic
193C2×C4×C17#12011yescharacteristic
194C2×C4×C17#12111yescharacteristic
195C2×C4×C17#12111yescharacteristic
196C2×C4×C17#12111yescharacteristic
197C2×C4×C17#12711yescharacteristic
198C2×C4×C17C2×D811yescharacteristic
199C2×C4×C17C2×SD1611yescharacteristic
200C2×C4×C17#13511yescharacteristic
201C2×C4×C17#13511yescharacteristic
202C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
203C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
204C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
205C2×C4×C17#13711yescharacteristic
206C22⋊D17—88no—
207C2×Dic17—41no—
208C2×Dic17—42no—
209C2×Dic17—41no—
210C2×Dic17—42no—
211C2×Dic17—41no—
212C2×Dic17—42no—
213D68—22no—
214D68—22no—
215D68—41no—
216D68—41no—
217C4×D17—81no—
218C4×D17—81no—
219C4×D17—81no—
220C4×D17—81no—
221C8×C17—42no—
222#3622—42no—
223#3622—42no—
224C2×#474D1711yescharacteristic, complemented
225C2×#461—171no—
226#2537—171no—
227#1990—171no—
228C2×D4⋊C8—171no—
229C2×D4⋊C8—171no—
230#1296—171no—
231C22×C17—42no—
232C22×C17—42no—
233C22×C17—42no—
234C22×C17—81no—
235C22×C17—81no—
236C22×C17—81no—
237C22×C17—81no—
238C22×C17#39211yescharacteristic
239C22×C17#44711yescharacteristic
240C22×C17#44911yescharacteristic
241C22×C17#45011yescharacteristic
242C22×C17#45211yescharacteristic
243C22×C17#45511yescharacteristic
244C22×C17#45611yescharacteristic
245C2×D17—42no—
246C2×D17—42no—
247C2×D17—42no—
248C2×D17—81no—
249C2×D17—81no—
250C2×D17—81no—
251C2×D17—81no—
252C4×C17—21no—
253C4×C17—21no—
254C4×C17—81no—
255C4×C17—82no—
256C4×C17#44711yescharacteristic
257C4×C17#44811yescharacteristic
258C4×C17#44911yescharacteristic
259C4×C17#44911yescharacteristic
260C4×C17#46511yescharacteristic
261C4×C17#46611yescharacteristic
262C4×C17#46811yescharacteristic
263C4×C17#46911yescharacteristic
264Dic17—82no—
265Dic17—81no—
266C22×SD16—341no—
267C2×C4⋊Q8C2×D1711yescharacteristic
268C2×#121—341no—
269C2×#121—341no—
270C2×C4×D4—171no—
271C2×C4×D4C2×D1711yescharacteristic
272#474—22no—
273#474—22no—
274#461—342no—
275#461—342no—
276C2×#107—21no—
277C2×C4⋊C8—171no—
278C2×C4⋊C8C2×D1711yescharacteristic
279C2×#103—341no—
280C2×#102—21no—
281C2×C4×C8—171no—
282#394—342no—
283#384—171no—
284#380—341no—
285#380—341no—
286#340—342no—
287D4⋊C8—342no—
288D4⋊C8—342no—
289C2×C17—82no—
290C2×C17#163911yescharacteristic
291C2×C17#164611yescharacteristic
292C2×C17#164711yescharacteristic
293C2×C17#165411yescharacteristic
294C2×C17#199911yescharacteristic
295C2×C17#200011yescharacteristic
296C2×C17#200711yescharacteristic
297D17—82no—
298C22×Q8—21no—
299C22×D4—171no—
300C22×D4—342no—
301C22×D4C22⋊D1711yescharacteristic
302C23×C4—21no—
303C23×C4—341no—
304C2×SD16—344no—
305C2×SD16—682no—
306C2×SD16—682no—
307C22×C8—21no—
308C22×C8—341no—
309C22×C8—341no—
310C4⋊Q8—21no—
311C4⋊Q8—21no—
312C4⋊Q8—21no—
313C4⋊Q8C22⋊D1712yescomplemented
314#121—682no—
315#121—682no—
316#121—682no—
317#121—682no—
318C4×D4—21no—
319C4×D4—21no—
320C4×D4—21no—
321C4×D4—21no—
322C4×D4—341no—
323C4×D4—341no—
324C4×D4—341no—
325C4×D4—341no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—171no—
329C2×C4⋊C4—172no—
330C2×C4⋊C4—341no—
331C2×C4⋊C4—341no—
332C2×C4⋊C4C22⋊D1711yescharacteristic
333C2×C22⋊C4—21no—
334C2×C22⋊C4—342no—
335C2×C22⋊C4—342no—
336C2×C22⋊C4—341no—
337C2×C42C22×D1711yescharacteristic
338#107—24no—
339C4⋊C8—21no—
340C4⋊C8—172no—
341C4⋊C8—341no—
342C4⋊C8C22⋊D1712yescomplemented
343#103—344no—
344#102—24no—
345C4×C8—341no—
346C4×C8—341no—
347#95—342no—
348C17#1023111yescharacteristic, complemented, Sylow
349C24—21no—
350C24—341no—
351C2×Q8—22no—
352C2×Q8—41no—
353C2×Q8—41no—
354C2×Q8—41no—
355C2×Q8—41no—
356C2×Q8—41no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—172no—
363C2×D4—341no—
364C2×D4—341no—
365C2×D4—341no—
366C2×D4—341no—
367C2×D4—341no—
368C2×D4—684no—
369C2×D4—684no—
370C2×D4—1362no—
371C2×D4#6323012yescomplemented
372C22×C4—21no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—341no—
381C22×C4—342no—
382C22×C4—681no—
383C22×C4—681no—
384C22×C4—681no—
385C22×C4—681no—
386C22×C4—681no—
387C22×C4—681no—
388C22×C4D4×D1711yescharacteristic
389C22×C4D4×D1711yescharacteristic
390C22×C4C2×C22⋊D1711yescharacteristic
391SD16—688no—
392C2×C8—22no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—342no—
396C2×C8—342no—
397C2×C8—681no—
398C2×C8—681no—
399C2×C8—681no—
400C2×C8—681no—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—341no—
408C4⋊C4—341no—
409C4⋊C4—682no—
410C4⋊C4—681no—
411C4⋊C4—681no—
412C4⋊C4—681no—
413C4⋊C4—681no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—684no—
417C22⋊C4—684no—
418C22⋊C4—681no—
419C22⋊C4—681no—
420C42D4×D1711yescharacteristic
421C42#6325411yescharacteristic
422C42C2×C22⋊D1711yescharacteristic
423C42C2×C22⋊D1711yescharacteristic
424C23—22no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—342no—
432C23—681no—
433C23—681no—
434C23—681no—
435C23—681no—
436C23—681no—
437C23—681no—
438C23#1058762311yescharacteristic
439Q8—42no—
440Q8—42no—
441Q8—81no—
442Q8—81no—
443D4—22no—
444D4—22no—
445D4—41no—
446D4—41no—
447D4—342no—
448D4—342no—
449D4—681no—
450D4—681no—
451D4—1368no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—681no—
465C2×C4—682no—
466C2×C4—681no—
467C2×C4—681no—
468C2×C4—682no—
469C2×C4—682no—
470C2×C4—1361no—
471C2×C4—1361no—
472C2×C4—1361no—
473C2×C4—1361no—
474C2×C4D8×D1711yescharacteristic
475C2×C4#1058759711yescharacteristic
476C2×C4#1058759711yescharacteristic
477C2×C4#1058759811yescharacteristic
478C2×C4#1058762311yescharacteristic
479C2×C4#1058762411yescharacteristic
480C2×C4#1058762511yescharacteristic
481C2×C4#1058762511yescharacteristic
482C2×C4#1058762611yescharacteristic
483C2×C4C2×#6323011yescharacteristic
484C2×C4#1058762811yescharacteristic
485C2×C4#1058763511yescharacteristic
486C2×C4#1058763611yescharacteristic
487C2×C4#1058763911yescharacteristic
488C8—42no—
489C8—682no—
490C8—682no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—682no—
499C22—682no—
500C22—682no—
501C22—1361no—
502C22—1361no—
503C22—1361no—
504C22—1361no—
505C22#1176271711yescharacteristic
506C22#1176294111yescharacteristic
507C22#1176295111yescharacteristic
508C22#1176295911yescharacteristic
509C22#1176296911yescharacteristic
510C22#1176298511yescharacteristic
511C22#1176299011yescharacteristic
512C4—21no—
513C4—21no—
514C4—81no—
515C4—82no—
516C4—1362no—
517C4—1361no—
518C4#1176294111yescharacteristic
519C4#1176294811yescharacteristic
520C4#1176295111yescharacteristic
521C4#1176295411yescharacteristic
522C4#1176304011yescharacteristic
523C4#1176304211yescharacteristic
524C4#1176305711yescharacteristic
525C4#1176305911yescharacteristic
526C2—82no—
527C2—1362no—
528C2#1452796211yescharacteristic
529C2#1452801111yescharacteristic
530C2#1452801811yescharacteristic
531C2#1452806711yescharacteristic
532C2#1452977911yescharacteristic
533C2#1452978511yescharacteristic
534C2#1452982911yescharacteristic
subgroupG/H
centerC23#10587623
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC2×C17×#474C2
socleC23×C17#120
prime radical#14533697C2
hypercenterC2×#474D17
hypocenterC17#10231
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. C2×#11763011C2semidirect
3. #14529746C2semidirect
4. C2×D4⋊#3622C2semidirect
5. C2×#11762536C2semidirect
6. #14526668C2semidirect
7. C2×C17×#474C2semidirect
41. C17×C4⋊Q8D4semidirect
73. C17×C4⋊C8D4semidirect
78. C4⋊#3622D4semidirect
94. C2×D4×C17SD16semidirect
224. C2×#474D17semidirect
313. C4⋊Q8C22⋊D17semidirect
342. C4⋊C8C22⋊D17semidirect
348. C17#10231semidirect
371. C2×D4#63230semidirect

Maximal subgroups

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

subgroupindex
C2×#117630112
#145336972
#145297462
C2×D4⋊#36222
C2×#117625362
#145266682
C2×C17×#4742
#1023117