Finite Group Databasebeta

← #17013162

Subgroups — #17013162

Order 4352.

1170 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14539935C211yesmaximal, characteristic, complemented
2#14538792C211yesmaximal, characteristic, complemented
3#14538581C211yesmaximal, characteristic, complemented
4#14528019C214yesmaximal, complemented
5#14527935C214yesmaximal, complemented
6#14527469C211yesmaximal, characteristic, complemented
7#14527209C211yesmaximal, characteristic, complemented
8#14527175C211yesmaximal, characteristic, complemented
9C2×C17×#547C211yesmaximal, characteristic
10#11763366C2211yescharacteristic
11#11763212C2212yescomplemented
12#11763161C2212yescomplemented
13#11763144—22no—
14#11763139—21no—
15#11763109—22no—
16#11763101—22no—
17#11763099C2212yescomplemented
18#11763099C2212yescomplemented
19#11763089C2212yescomplemented
20#11763014C2212yescomplemented
21#11762960—21no—
22C4×#63229—22no—
23#11762862C2211yescharacteristic
24#11762862C2211yescharacteristic
25#11762849—21no—
26#11762833C2211yescharacteristic
27#11762814—21no—
28#11762555C2214yescomplemented
29D4⋊#3622C2214yescomplemented
30D4⋊#3622C2214yescomplemented
31C22×#41×C17—21no—
32C17×#547—22no—
33C17×#547—22no—
34C17×#547—22no—
35C17×#547C2214yes—
36C2×C17×C4⋊Q8C2211yescharacteristic
37C2×C17×#123—22no—
38C2×C17×C22⋊Q8—21no—
39C2×C17×#121—21no—
40C2×C4×D4×C17C2211yescharacteristic
41C2×C4×D4×C17C2211yescharacteristic
42C22×C4⋊C4×C17—21no—
43C2×#41×C17—24no—
44C2×#41×C17—24no—
45C2×#41×C17—48no—
46C22×Q8×C17—21no—
47C22×D4×C17—22no—
48C22×D4×C17D411yescharacteristic
49C22×D4×C17D411yescharacteristic
50C23×C4×C17—22no—
51C23×C4×C17—21no—
52C23×C4×C17—21no—
53C17×C4⋊Q8—21no—
54C17×C4⋊Q8—21no—
55C17×C4⋊Q8D412yes—
56C17×C4⋊Q8C2312yes—
57C17×#123—28no—
58C17×#123—28no—
59C17×C22⋊Q8—24no—
60C17×C22⋊Q8—24no—
61C17×#121—24no—
62C17×#121—24no—
63C4×D4×C17—21no—
64C4×D4×C17—21no—
65C4×D4×C17—21no—
66C4×D4×C17—21no—
67C4×D4×C17D412yes—
68C4×D4×C17D412yes—
69C4×D4×C17C2312yes—
70C4×D4×C17C2312yes—
71C2×C4⋊C4×C17—22no—
72C2×C4⋊C4×C17—24no—
73C2×C4⋊C4×C17—21no—
74C2×C4⋊C4×C17—24no—
75C2×C4⋊C4×C17D411yescharacteristic
76C2×C4⋊C4×C17D411yescharacteristic
77C2×C4⋊C4×C17D412yes—
78C2×C22⋊C4×C17—22no—
79C2×C22⋊C4×C17—21no—
80C2×C22⋊C4×C17—21no—
81C2×C42×C17C2311yescharacteristic
82C2×#63229—22no—
83#10587612—21no—
84#10587610—21no—
85#10587610—21no—
86C2×#63223—21no—
87#10587556—21no—
88#10587555C2311yescharacteristic
89#10587555C2311yescharacteristic
90C4×Dic34—21no—
91C4×Dic34—21no—
92#10587503—22no—
93#10587503—22no—
94#10587503—22no—
95C22⋊#3622—21no—
96C22⋊#3622—21no—
97#10587481—22no—
98#10587481—24no—
99#10587479—22no—
100#10587479—22no—
101#10587478—24no—
102#10587478—22no—
103C4⋊#3622—21no—
104C4⋊#3622—21no—
105C4⋊#3622C2312yes—
106C4⋊#3622C2312yes—
107C4×#3622—21no—
108C4×#3622C2312yes—
109C24×C17—21no—
110C24×C17—21no—
111#41×C17—416no—
112#41×C17—48no—
113#41×C17—48no—
114C2×Q8×C17—22no—
115C2×Q8×C17—22no—
116C2×Q8×C17—44no—
117C2×D4×C17—24no—
118C2×D4×C17—28no—
119C2×D4×C17—21no—
120C2×D4×C17—21no—
121C2×D4×C17—22no—
122C2×D4×C17—22no—
123C2×D4×C17—48no—
124C2×D4×C17C2×D412yes—
125C2×D4×C17C2×D412yes—
126C22×C4×C17—28no—
127C22×C4×C17—21no—
128C22×C4×C17—22no—
129C22×C4×C17—21no—
130C22×C4×C17—22no—
131C22×C4×C17—22no—
132C22×C4×C17—22no—
133C22×C4×C17—22no—
134C22×C4×C17—44no—
135C22×C4×C17—48no—
136C22×C4×C17—42no—
137C22×C4×C17—42no—
138C22×C4×C17C2×D411yescharacteristic
139C22×C4×C17C2×D411yescharacteristic
140C22×C4×C17C2×D411yescharacteristic
141C2×Dic34—21no—
142C2×Dic34—21no—
143C4⋊C4×C17—28no—
144C4⋊C4×C17—21no—
145C4⋊C4×C17—22no—
146C4⋊C4×C17—22no—
147C4⋊C4×C17—21no—
148C4⋊C4×C17—22no—
149C4⋊C4×C17C2×D412yes—
150C4⋊C4×C17C2×D414yes—
151C4⋊C4×C17C2×D412yes—
152C22⋊C4×C17—28no—
153C22⋊C4×C17—22no—
154C22⋊C4×C17—22no—
155C22⋊C4×C17—22no—
156C22⋊C4×C17—22no—
157C42×C17C2×D411yescharacteristic
158C42×C17C2×D411yescharacteristic
159C42×C17C2×D411yescharacteristic
160C42×C17C2411yescharacteristic
161C22⋊Dic17—41no—
162#63229—44no—
163C4⋊Dic17—21no—
164C4⋊Dic17—21no—
165C4⋊Dic17—21no—
166C4⋊Dic17—21no—
167#63225—41no—
168#63225—41no—
169C4×Dic17—41no—
170#63223—42no—
171C2×#3622—22no—
172C2×#3622—22no—
173C2×#3622—22no—
174#18783—171nomaximal, Sylow
175C23×C17—24no—
176C23×C17—22no—
177C23×C17—22no—
178C23×C17—42no—
179C23×C17—42no—
180C23×C17—42no—
181C23×C17—42no—
182C23×C17—42no—
183C23×C17#12011yescharacteristic
184Q8×C17—44no—
185Q8×C17—44no—
186D4×C17—24no—
187D4×C17—22no—
188D4×C17—22no—
189D4×C17—22no—
190D4×C17—416no—
191D4×C17—42no—
192D4×C17AGL(1,ℤ/8)14yes—
193C2×C4×C17—22no—
194C2×C4×C17—22no—
195C2×C4×C17—22no—
196C2×C4×C17—24no—
197C2×C4×C17—41no—
198C2×C4×C17—44no—
199C2×C4×C17—42no—
200C2×C4×C17—44no—
201C2×C4×C17—416no—
202C2×C4×C17—42no—
203C2×C4×C17—42no—
204C2×C4×C17—44no—
205C2×C4×C17—42no—
206C2×C4×C17—44no—
207C2×C4×C17#12011yescharacteristic
208C2×C4×C17#12011yescharacteristic
209C2×C4×C17#12011yescharacteristic
210C2×C4×C17#13512yes—
211C2×C4×C17#13712yes—
212C2×C4×C17C22×D411yescharacteristic
213C2×C4×C17C22×D411yescharacteristic
214C2×C4×C17C22×D411yescharacteristic
215C2×Dic17—41no—
216C2×Dic17—41no—
217Dic34—42no—
218Dic34—42no—
219#3622—42no—
220#3622—42no—
221#3622—42no—
222C2×#547D1711yescharacteristic
223#3230—171no—
224#3101—171no—
225#3077—171no—
226#1647—174no—
227#1635—174no—
228#1549—171no—
229#1486—171no—
230#1478—171no—
231C22×C17—21no—
232C22×C17—21no—
233C22×C17—24no—
234C22×C17—48no—
235C22×C17—42no—
236C22×C17—42no—
237C22×C17—42no—
238C22×C17—42no—
239C22×C17—42no—
240C22×C17—84no—
241C22×C17#45212yes—
242C22×C17C2×#12011yescharacteristic
243C4×C17—21no—
244C4×C17—21no—
245C4×C17—21no—
246C4×C17—42no—
247C4×C17—42no—
248C4×C17—44no—
249C4×C17#45212yes—
250C4×C17C2×#13511yescharacteristic
251C4×C17C2×AGL(1,ℤ/8)12yes—
252C4×C17C2×#13711yescharacteristic
253Dic17—81no—
254Dic17—81no—
255C22×#41—21no—
256#547—22no—
257#547—22no—
258#547—22no—
259#547C2×D1714yes—
260#533—171no—
261C2×C4⋊Q8C2×D1711yescharacteristic
262C2×#123—22no—
263C2×C22⋊Q8—21no—
264C2×#121—21no—
265C2×C4×D4C2×D1711yescharacteristic
266C2×C4×D4C2×D1711yescharacteristic
267C22×C4⋊C4—21no—
268#500—172no—
269#487—172no—
270#484—342no—
271#483—341no—
272#478—342no—
273#477—342no—
274#476—172no—
275#476—172no—
276#474—172no—
277#461—172no—
278#450—341no—
279C4×SD16—342no—
280#431—171no—
281#431—171no—
282#428—341no—
283#423—171no—
284#418—341no—
285#332—174no—
286D4⋊C8—174no—
287D4⋊C8—174no—
288C2×C17—22no—
289C2×C17—44no—
290C2×C17—82no—
291C2×C17C2×#45211yescharacteristic
292C2×C17#300311yescharacteristic
293C2×C17#302011yescharacteristic
294C2×#41—24no—
295C2×#41—24no—
296C2×#41—48no—
297C22×Q8—21no—
298C22×D4—22no—
299C22×D4C22⋊D1711yescharacteristic
300C22×D4C22⋊D1711yescharacteristic
301C23×C4—22no—
302C23×C4—21no—
303C23×C4—21no—
304C2×SD16—342no—
305C2×M16—341no—
306C4⋊Q8—21no—
307C4⋊Q8—21no—
308C4⋊Q8—171no—
309C4⋊Q8—171no—
310C4⋊Q8C22⋊D1712yes—
311C4⋊Q8C22×D1712yes—
312#125—341no—
313#123—28no—
314#123—28no—
315C22⋊Q8—24no—
316C22⋊Q8—24no—
317C22⋊Q8—341no—
318C22⋊Q8—341no—
319#121—24no—
320#121—24no—
321C4×Q8—341no—
322C4×Q8—341no—
323C4×D4—21no—
324C4×D4—21no—
325C4×D4—21no—
326C4×D4—21no—
327C4×D4C22⋊D1712yes—
328C4×D4C22⋊D1712yes—
329C4×D4C22×D1712yes—
330C4×D4C22×D1712yes—
331#117—341no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4C22⋊D1711yescharacteristic
337C2×C4⋊C4C22⋊D1711yescharacteristic
338C2×C4⋊C4C22⋊D1712yes—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C42C22×D1711yescharacteristic
343#107—342no—
344#107—344no—
345#106—342no—
346#106—342no—
347C4⋊C8—172no—
348C4⋊C8—172no—
349C4⋊C8—341no—
350C4⋊C8—341no—
351#103—342no—
352#103—344no—
353#102—342no—
354#102—342no—
355#102—342no—
356C22⋊C8—341no—
357C22⋊C8—341no—
358C4×C8—172no—
359C4×C8—341no—
360C17#1878311yescharacteristic, complemented, Sylow
361C24—21no—
362C24—21no—
363#41—416no—
364#41—48no—
365#41—48no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—44no—
369C2×Q8—341no—
370C2×Q8—341no—
371C2×D4—24no—
372C2×D4—28no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—48no—
378C2×D4C2×C22⋊D1712yes—
379C2×D4C2×C22⋊D1712yes—
380C22×C4—28no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—44no—
389C22×C4—48no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4C2×C22⋊D1711yescharacteristic
393C22×C4C2×C22⋊D1711yescharacteristic
394C22×C4C2×C22⋊D1711yescharacteristic
395SD16—684no—
396M16—682no—
397C2×C8—342no—
398C2×C8—342no—
399C2×C8—342no—
400C4⋊C4—28no—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—341no—
407C4⋊C4—341no—
408C4⋊C4—341no—
409C4⋊C4—341no—
410C4⋊C4—681no—
411C4⋊C4—681no—
412C4⋊C4C2×C22⋊D1712yes—
413C4⋊C4C2×C22⋊D1714yes—
414C4⋊C4C2×C22⋊D1712yes—
415C22⋊C4—28no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—681no—
421C42—681no—
422C42C2×C22⋊D1711yescharacteristic
423C42C2×C22⋊D1711yescharacteristic
424C42C2×C22⋊D1711yescharacteristic
425C42C23×D1711yescharacteristic
426C23—24no—
427C23—22no—
428C23—22no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23#1058763911yescharacteristic
435Q8—44no—
436Q8—44no—
437Q8—682no—
438Q8—682no—
439D4—24no—
440D4—22no—
441D4—22no—
442D4—22no—
443D4—416no—
444D4—42no—
445D4#1058761814yes—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—41no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—416no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—681no—
461C2×C4—681no—
462C2×C4#1058763612yes—
463C2×C4#1058763712yes—
464C2×C4#1058763911yescharacteristic
465C2×C4#1058763911yescharacteristic
466C2×C4#1058763911yescharacteristic
467C2×C4C22×C22⋊D1711yescharacteristic
468C2×C4C22×C22⋊D1711yescharacteristic
469C2×C4C22×C22⋊D1711yescharacteristic
470C8—682no—
471C8—682no—
472C8—682no—
473C22—21no—
474C22—21no—
475C22—24no—
476C22—48no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—84no—
483C22#1176297112yes—
484C22C2×#1058763911yescharacteristic
485C4—21no—
486C4—21no—
487C4—21no—
488C4—42no—
489C4—42no—
490C4—44no—
491C4—1361no—
492C4—1361no—
493C4#1176297112yes—
494C4C2×#1058763611yescharacteristic
495C4C2×#1058761812yes—
496C4C2×#1058763711yescharacteristic
497C2—22no—
498C2—44no—
499C2—82no—
500C2C2×#1176297111yescharacteristic
501C2#1453801111yescharacteristic
502C2#1453814811yescharacteristic
subgroupG/H
centerC22C2×#10587639
commutator, cosocleC42×C17C24
FrattiniC42C23×D17
Fitting, prime radicalC2×C17×#547C2
socleC22×C17C2×#120
hypercenterC2×#547D17
hypocenterC17#18783
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. #14539935C2semidirect
2. #14538792C2semidirect
3. #14538581C2semidirect
4. #14528019C2semidirect
5. #14527935C2semidirect
6. #14527469C2semidirect
7. #14527209C2semidirect
8. #14527175C2semidirect
11. #11763212C22semidirect
12. #11763161C22semidirect
17. #11763099C22semidirect
18. #11763099C22semidirect
19. #11763089C22semidirect
20. #11763014C22semidirect
28. #11762555C22semidirect
29. D4⋊#3622C22semidirect
30. D4⋊#3622C22semidirect
360. C17#18783semidirect

Maximal subgroups

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

subgroupindex
#145399352
#145387922
#145385812
#145280192
#145279352
#145274692
#145272092
#145271752
C2×C17×#5472
#1878317