Finite Group Databasebeta

← #16936624

Subgroups — #16936624

Order 4352.

744 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763001C211yesmaximal, characteristic
2#14533702C211yesmaximal, characteristic, complemented
3#14529591C211yesmaximal, characteristic, complemented
4C2×Q8⋊#3622C211yesmaximal, characteristic, complemented
5C2×#11762536C211yesmaximal, characteristic, complemented
6#14526668C211yesmaximal, characteristic, complemented
7C2×C17×#460C211yesmaximal, characteristic, complemented
8C22×#63228—21no—
9C2×#10587632—21no—
10C2×C4×D68C2211yescharacteristic
11#11763001—22no—
12#11763001—22no—
13C2×C4⋊#3622C2211yescharacteristic
14C2×#10587503—21no—
15C2×#10587503—22no—
16C2×C4×#3622C2211yescharacteristic
17#11762731—22no—
18#11762711—21no—
19#11762707—21no—
20#11762679C2211yescharacteristic
21#11762666—21no—
22Q8⋊#3622—22no—
23#11762536—22no—
24C22×SD16×C17—21no—
25C2×C17×#127C2211yescharacteristic
26C2×C4×Q8×C17C2211yescharacteristic
27C17×#460—22no—
28C17×#460—22no—
29C2×C17×C4⋊C8C2211yescharacteristic
30C2×C17×#102—21no—
31C22×Q8×C17D411yescharacteristic
32C22×D4×C17—21no—
33C22×D4×C17—21no—
34C22×D4×C17—21no—
35C22×D68D411yescharacteristic
36C22×C4×D17—21no—
37C2×SD16×C17—24no—
38C2×SD16×C17—42no—
39C2×SD16×C17—42no—
40C22×C8×C17—21no—
41C17×#127—21no—
42C17×#127—21no—
43C17×#127—21no—
44C17×#127D412yes—
45C4×Q8×C17—21no—
46C4×Q8×C17—21no—
47C4×Q8×C17—21no—
48C4×Q8×C17—21no—
49C2×C4⋊C4×C17—21no—
50C2×C4⋊C4×C17D411yescharacteristic
51C2×C42×C17—21no—
52C2×C42×C17C2311yescharacteristic
53#10587632—42no—
54#10587632—42no—
55C2×#63228—24no—
56C2×#63228—42no—
57C2×#63228—42no—
58C2×#63227—22no—
59C2×#63227—22no—
60C2×#63227—21no—
61C2×C4⋊Dic17—21no—
62C2×C4⋊Dic17—21no—
63C2×C4⋊Dic17D411yescharacteristic
64C2×C4⋊Dic17D412yes—
65C2×#63225—22no—
66C22×#3622—21no—
67C22×#3622—21no—
68C4×D68—21no—
69C4×D68—21no—
70C4×D68—21no—
71C4×D68—21no—
72C17×C4⋊C8—21no—
73C17×C4⋊C8D412yes—
74C17×#102—24no—
75#10587503—24no—
76#10587503—44no—
77#10587502—22no—
78C4⋊#3622—21no—
79C4⋊#3622D412yes—
80C4×#3622—21no—
81C4×#3622—21no—
82C24×C17—41no—
83C23×D17—21no—
84C2×Q8×C17—21no—
85C2×Q8×C17—21no—
86C2×Q8×C17—21no—
87C2×Q8×C17—21no—
88C2×Q8×C17—21no—
89C2×Q8×C17D812yescomplemented
90C2×D4×C17—22no—
91C2×D4×C17—22no—
92C2×D4×C17—22no—
93C2×D4×C17—42no—
94C2×D4×C17—42no—
95C2×D4×C17—41no—
96C2×D4×C17—41no—
97C2×D4×C17—41no—
98C2×D4×C17—41no—
99C2×D4×C17—41no—
100C2×D4×C17—41no—
101C2×D4×C17—41no—
102C2×D4×C17—41no—
103C2×D4×C17—41no—
104C2×D4×C17—41no—
105C2×D4×C17—41no—
106C22×C4×C17—21no—
107C22×C4×C17—21no—
108C22×C4×C17C2×D411yescharacteristic
109C22×C4×C17C2×D411yescharacteristic
110C22×C4×C17C2×D411yescharacteristic
111C22×Dic17—21no—
112C22×Dic17—22no—
113C2×D68—21no—
114C2×D68—21no—
115C2×D68—21no—
116C2×D68—21no—
117C2×D68—21no—
118C2×D68SD1612yescomplemented
119C2×C4×D17—41no—
120C2×C4×D17—41no—
121C2×C4×D17—41no—
122C2×C4×D17—41no—
123C2×C4×D17—41no—
124C2×C4×D17—41no—
125SD16×C17—48no—
126C2×C8×C17—22no—
127C2×C8×C17—41no—
128C2×C8×C17—41no—
129C4⋊C4×C17—21no—
130C4⋊C4×C17—21no—
131C4⋊C4×C17—41no—
132C4⋊C4×C17—41no—
133C42×C17—41no—
134C42×C17—41no—
135C42×C17C2×D411yescharacteristic
136C42×C17C2×D411yescharacteristic
137C42×C17C2×D411yescharacteristic
138C42×C17#4111yescharacteristic
139#63228—48no—
140#63227—44no—
141#63227—41no—
142#63227—41no—
143#63227—82no—
144C4⋊Dic17—21no—
145C4⋊Dic17—21no—
146C4⋊Dic17—41no—
147C4⋊Dic17—41no—
148C4⋊Dic17—41no—
149C4⋊Dic17—41no—
150C4⋊Dic17—42no—
151#63225—44no—
152C2×#3622—22no—
153C2×#3622—22no—
154C2×#3622—41no—
155C2×#3622—41no—
156C2×#3622—41no—
157C2×#3622—41no—
158#10229—171nomaximal, Sylow
159C23×C17—42no—
160C23×C17—81no—
161C23×C17—81no—
162C23×C17—81no—
163C23×C17—81no—
164C23×C17—81no—
165C23×C17—81no—
166C23×C17#12011yescharacteristic
167C22×D17—22no—
168C22×D17—41no—
169C22×D17—41no—
170C22×D17—41no—
171C22×D17—41no—
172C22×D17—41no—
173C22×D17—41no—
174Q8×C17—22no—
175Q8×C17—22no—
176Q8×C17—41no—
177Q8×C17—41no—
178D4×C17—44no—
179D4×C17—42no—
180D4×C17—42no—
181D4×C17—42no—
182D4×C17—42no—
183D4×C17—82no—
184D4×C17—81no—
185D4×C17—81no—
186D4×C17—81no—
187D4×C17—81no—
188C2×C4×C17—21no—
189C2×C4×C17—21no—
190C2×C4×C17—22no—
191C2×C4×C17—41no—
192C2×C4×C17—41no—
193C2×C4×C17—41no—
194C2×C4×C17—41no—
195C2×C4×C17—41no—
196C2×C4×C17#12011yescharacteristic
197C2×C4×C17#12011yescharacteristic
198C2×C4×C17#12111yescharacteristic
199C2×C4×C17#12111yescharacteristic
200C2×C4×C17#12111yescharacteristic
201C2×C4×C17#12711yescharacteristic
202C2×C4×C17C2×D811yescharacteristic
203C2×C4×C17C2×SD1611yescharacteristic
204C2×C4×C17#13511yescharacteristic
205C2×C4×C17#13511yescharacteristic
206C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
207C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
208C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
209C2×C4×C17#13711yescharacteristic
210C2×Dic17—41no—
211C2×Dic17—41no—
212C2×Dic17—42no—
213C2×Dic17—41no—
214C2×Dic17—42no—
215C2×Dic17—42no—
216D68—22no—
217D68—22no—
218D68—41no—
219D68—41no—
220C4×D17—81no—
221C4×D17—81no—
222C4×D17—81no—
223C4×D17—81no—
224C8×C17—42no—
225#3622—42no—
226#3622—42no—
227C2×#460D1711yescharacteristic, complemented
228C2×#459—171no—
229#2538—171no—
230#1957—171no—
231C2×Q8⋊C8—171no—
232C2×D4⋊C8—171no—
233#1296—171no—
234C22×C17—82no—
235C22×C17—82no—
236C22×C17—82no—
237C22×C17—161no—
238C22×C17—161no—
239C22×C17—161no—
240C22×C17—161no—
241C22×C17#39211yescharacteristic
242C22×C17#44711yescharacteristic
243C22×C17#44811yescharacteristic
244C22×C17#44911yescharacteristic
245C22×C17#44911yescharacteristic
246C22×C17#45311yescharacteristic
247C22×C17#45411yescharacteristic
248C2×D17—42no—
249C2×D17—42no—
250C2×D17—42no—
251C2×D17—81no—
252C2×D17—81no—
253C2×D17—81no—
254C2×D17—81no—
255C4×C17—21no—
256C4×C17—21no—
257C4×C17—42no—
258C4×C17—81no—
259C4×C17#44711yescharacteristic
260C4×C17#44911yescharacteristic
261C4×C17#45011yescharacteristic
262C4×C17#45211yescharacteristic
263C4×C17#46511yescharacteristic
264C4×C17#46611yescharacteristic
265C4×C17#46811yescharacteristic
266C4×C17#46911yescharacteristic
267Dic17—82no—
268Dic17—81no—
269C22×SD16—21no—
270C22×D8—341no—
271C2×#127C2×D1711yescharacteristic
272C2×C22⋊Q8—341no—
273C2×C4×Q8C2×D1711yescharacteristic
274C2×C4×D4—171no—
275#460—22no—
276#460—22no—
277#459—342no—
278#459—342no—
279C2×C4⋊C8—171no—
280C2×C4⋊C8C2×D1711yescharacteristic
281C2×#102—21no—
282C2×#102—341no—
283C2×#102—342no—
284C2×C4×C8—171no—
285#396—342no—
286#390—341no—
287#389—341no—
288#384—171no—
289#380—341no—
290Q8⋊C8—342no—
291D4⋊C8—342no—
292C2×C17—162no—
293C2×C17#162111yescharacteristic
294C2×C17#162211yescharacteristic
295C2×C17#162711yescharacteristic
296C2×C17#162811yescharacteristic
297C2×C17#199911yescharacteristic
298C2×C17#200011yescharacteristic
299C2×C17#200611yescharacteristic
300D17—82no—
301C22×Q8C22⋊D1711yescharacteristic
302C22×D4—21no—
303C22×D4—21no—
304C22×D4—21no—
305C22×D4—171no—
306C23×C4—341no—
307C2×SD16—24no—
308C2×SD16—42no—
309C2×SD16—42no—
310C2×D8—344no—
311C2×D8—682no—
312C2×D8—682no—
313C22×C8—21no—
314C22×C8—341no—
315C22×C8—341no—
316#127—21no—
317#127—21no—
318#127—21no—
319#127C22⋊D1712yes—
320C22⋊Q8—682no—
321C22⋊Q8—682no—
322C4×Q8—21no—
323C4×Q8—21no—
324C4×Q8—21no—
325C4×Q8—21no—
326C4×D4—341no—
327C4×D4—341no—
328C4×D4—341no—
329C4×D4—341no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—171no—
332C2×C4⋊C4—172no—
333C2×C4⋊C4—342no—
334C2×C4⋊C4—341no—
335C2×C4⋊C4—341no—
336C2×C4⋊C4C22⋊D1711yescharacteristic
337C2×C22⋊C4—342no—
338C2×C22⋊C4—341no—
339C2×C22⋊C4—342no—
340C2×C42—21no—
341C2×C42C22×D1711yescharacteristic
342C4⋊C8—21no—
343C4⋊C8—172no—
344C4⋊C8—341no—
345C4⋊C8C22⋊D1712yes—
346#102—24no—
347#102—344no—
348#102—684no—
349C4×C8—341no—
350C4×C8—341no—
351#95—342no—
352C17#1022911yescharacteristic, complemented, Sylow
353C24—41no—
354C24—341no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8#6322812yescomplemented
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—172no—
378C2×D4—341no—
379C2×D4—341no—
380C2×D4—341no—
381C2×D4—341no—
382C2×D4—341no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—341no—
386C22×C4—342no—
387C22×C4—681no—
388C22×C4—681no—
389C22×C4—681no—
390C22×C4—681no—
391C22×C4—681no—
392C22×C4—681no—
393C22×C4D4×D1711yescharacteristic
394C22×C4D4×D1711yescharacteristic
395C22×C4C2×C22⋊D1711yescharacteristic
396SD16—48no—
397D8—688no—
398C2×C8—22no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—342no—
402C2×C8—342no—
403C2×C8—681no—
404C2×C8—681no—
405C2×C8—681no—
406C2×C8—681no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—341no—
412C4⋊C4—341no—
413C4⋊C4—682no—
414C4⋊C4—684no—
415C4⋊C4—681no—
416C4⋊C4—681no—
417C4⋊C4—681no—
418C4⋊C4—681no—
419C22⋊C4—684no—
420C22⋊C4—681no—
421C22⋊C4—681no—
422C22⋊C4—1362no—
423C42—41no—
424C42—41no—
425C42D4×D1711yescharacteristic
426C42#6325411yescharacteristic
427C42C2×C22⋊D1711yescharacteristic
428C42C2×C22⋊D1711yescharacteristic
429C23—42no—
430C23—81no—
431C23—81no—
432C23—81no—
433C23—81no—
434C23—81no—
435C23—81no—
436C23—342no—
437C23—681no—
438C23—681no—
439C23—681no—
440C23—681no—
441C23—681no—
442C23—681no—
443C23#1058762311yescharacteristic
444Q8—22no—
445Q8—22no—
446Q8—41no—
447Q8—41no—
448D4—44no—
449D4—42no—
450D4—42no—
451D4—42no—
452D4—42no—
453D4—82no—
454D4—81no—
455D4—81no—
456D4—81no—
457D4—81no—
458D4—342no—
459D4—342no—
460D4—681no—
461D4—681no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—681no—
471C2×C4—682no—
472C2×C4—681no—
473C2×C4—682no—
474C2×C4—682no—
475C2×C4—681no—
476C2×C4—1361no—
477C2×C4—1361no—
478C2×C4—1361no—
479C2×C4—1361no—
480C2×C4SD16×D1711yescharacteristic
481C2×C4#1058760011yescharacteristic
482C2×C4#1058760111yescharacteristic
483C2×C4#1058760211yescharacteristic
484C2×C4C2×#6322811yescharacteristic
485C2×C4#1058761811yescharacteristic
486C2×C4#1058762311yescharacteristic
487C2×C4#1058762411yescharacteristic
488C2×C4#1058762511yescharacteristic
489C2×C4#1058762511yescharacteristic
490C2×C4#1058762611yescharacteristic
491C2×C4#1058763511yescharacteristic
492C2×C4#1058763611yescharacteristic
493C2×C4#1058763911yescharacteristic
494C8—42no—
495C8—682no—
496C8—682no—
497C22—82no—
498C22—82no—
499C22—82no—
500C22—161no—
501C22—161no—
502C22—161no—
503C22—161no—
504C22—682no—
505C22—682no—
506C22—682no—
507C22—1361no—
508C22—1361no—
509C22—1361no—
510C22—1361no—
511C22#1176271711yescharacteristic
512C22#1176294211yescharacteristic
513C22#1176294611yescharacteristic
514C22#1176295211yescharacteristic
515C22#1176295311yescharacteristic
516C22#1176297511yescharacteristic
517C22#1176298011yescharacteristic
518C4—21no—
519C4—21no—
520C4—42no—
521C4—81no—
522C4—1362no—
523C4—1361no—
524C4#1176294311yescharacteristic
525C4#1176295411yescharacteristic
526C4#1176295811yescharacteristic
527C4#1176296811yescharacteristic
528C4#1176303511yescharacteristic
529C4#1176304311yescharacteristic
530C4#1176305211yescharacteristic
531C4#1176306011yescharacteristic
532C2—162no—
533C2—1362no—
534C2#1452784711yescharacteristic
535C2#1452785511yescharacteristic
536C2#1452788511yescharacteristic
537C2#1452789311yescharacteristic
538C2#1452978011yescharacteristic
539C2#1452978411yescharacteristic
540C2#1452982411yescharacteristic
subgroupG/H
centerC23#10587623
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC2×C17×#460C2
socleC23×C17#120
prime radicalC2×#11763001C2
hypercenterC2×#460D17
hypocenterC17#10229
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
2. #14533702C2semidirect
3. #14529591C2semidirect
4. C2×Q8⋊#3622C2semidirect
5. C2×#11762536C2semidirect
6. #14526668C2semidirect
7. C2×C17×#460C2semidirect
89. C2×Q8×C17D8semidirect
118. C2×D68SD16semidirect
227. C2×#460D17semidirect
352. C17#10229semidirect
360. C2×Q8#63228semidirect

Maximal subgroups

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

subgroupindex
C2×#117630012
#145337022
#145295912
C2×Q8⋊#36222
C2×#117625362
#145266682
C2×C17×#4602
#1022917