Finite Group Databasebeta

← #17051922

Subgroups — #17051922

Order 4352.

692 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14537444C211yesmaximal, characteristic
2#14537442C211yesmaximal, characteristic
3C2×#11762845C211yesmaximal, characteristic
4#14529536C212yesmaximal, complemented
5#14529386C212yesmaximal, complemented
6#14529381C212yesmaximal, complemented
7#14528884C211yesmaximal, characteristic
8#14528820C211yesmaximal, characteristic
9#14528667C212yesmaximal, complemented
10#14528562C211yesmaximal, characteristic
11C17×#1826C211yesmaximal, characteristic
12C23×#3622C2211yescharacteristic
13C2×#10587612C2211yescharacteristic
14#11762870—22no—
15#11762867—22no—
16#11762866—21no—
17#11762863—21no—
18#11762862—21no—
19#11762859—21no—
20C2×#10587493C2212yes—
21#11762845C411yescharacteristic, complemented
22#11762845C411yescharacteristic, complemented
23#11762845C411yescharacteristic, complemented
24#11762845C411yescharacteristic, complemented
25#11762845C2211yescharacteristic
26#11762845C2211yescharacteristic
27#11762845C2211yescharacteristic
28#11762845C2211yescharacteristic
29C2×#10587478C411yescharacteristic, complemented
30C2×#10587478C411yescharacteristic, complemented
31C2×#10587479C411yescharacteristic, complemented
32C2×#10587479C411yescharacteristic, complemented
33#11762837C2212yes—
34#11762836C2212yes—
35#11762773C2211yescharacteristic
36C2×C22⋊#3622C2211yescharacteristic
37#11762767C2212yes—
38#11762765C2212yes—
39C2×#10587474C2212yes—
40#11762573C2212yes—
41#11762569C2212yes—
42#11762569C2212yes—
43#11762564C2212yes—
44C2×C17×#117C2211yescharacteristic
45C17×#407C2211yescharacteristic
46C4×C22⋊C4×C17C2211yescharacteristic
47C17×#339C2212yes—
48C17×#339C2212yes—
49C23×C4×C17C2311yescharacteristic
50C2×M16×C17C2312yes—
51C17×#117—21no—
52C17×#117—21no—
53C17×#117C2×C411yescharacteristic
54C17×#117C2×C411yescharacteristic
55C17×#117C2311yescharacteristic
56C17×#117C2311yescharacteristic
57C2×C4⋊C4×C17C2×C411yescharacteristic
58C2×C4⋊C4×C17C2×C411yescharacteristic
59C2×C22⋊C4×C17—21no—
60C2×C22⋊C4×C17—21no—
61C2×C42×C17—21no—
62C2×C42×C17C2312yes—
63C2×C22⋊Dic17—21no—
64#10587612—21no—
65#10587612—21no—
66#10587612C2×C411yescharacteristic
67#10587612C2×C411yescharacteristic
68#10587612C2311yescharacteristic
69#10587612C2311yescharacteristic
70C2×C4⋊Dic17C2×C411yescharacteristic
71C2×C4⋊Dic17C2×C411yescharacteristic
72C2×C4×Dic17—21no—
73C2×#63223C2312yes—
74C22×#3622—24no—
75C22×#3622C2×C411yescharacteristic
76C22×#3622C2×C411yescharacteristic
77C22×#3622C2×C411yescharacteristic
78C22×#3622C2×C411yescharacteristic
79C22×#3622C2311yescharacteristic
80C22×#3622C2312yes—
81C22×#3622C2311yescharacteristic
82C17×C22⋊C8—21no—
83C17×C22⋊C8—21no—
84C17×#95—21no—
85C22⋊#3622—21no—
86C22⋊#3622—21no—
87C22⋊#3622D411yescharacteristic
88C22⋊#3622D411yescharacteristic
89C22⋊#3622D412yes—
90#10587493—24no—
91#10587486—22no—
92#10587485—22no—
93#10587485—22no—
94C8×Dic17—22no—
95#10587479C2×C411yescharacteristic
96#10587479C2×C411yescharacteristic
97#10587479C2×C411yescharacteristic
98#10587479C2×C411yescharacteristic
99#10587479C2×C411yescharacteristic
100#10587479C2×C411yescharacteristic
101#10587479C2×C411yescharacteristic
102#10587479C2×C411yescharacteristic
103#10587478C2×C411yescharacteristic
104#10587478C2×C411yescharacteristic
105#10587478C2×C411yescharacteristic
106#10587478C2×C411yescharacteristic
107#10587478C2×C411yescharacteristic
108#10587478C2×C411yescharacteristic
109#10587478C2×C411yescharacteristic
110#10587478C2×C411yescharacteristic
111C4⋊#3622—22no—
112C4⋊#3622—22no—
113#10587474—22no—
114#10587474—24no—
115C4×#3622—22no—
116C24×C17#4111yescharacteristic
117C22×C4×C17—22no—
118C22×C4×C17—22no—
119C22×C4×C17—21no—
120C22×C4×C17—21no—
121C22×C4×C17—22no—
122C22×C4×C17C22×C411yescharacteristic
123C22×C4×C17C22×C411yescharacteristic
124C22×C4×C17C2×D411yescharacteristic
125C22×C4×C17C2×D411yescharacteristic
126C22×C4×C17#4111yescharacteristic
127C22×C4×C17C2411yescharacteristic
128C22×Dic17—21no—
129C22×Dic17—21no—
130M16×C17—22no—
131M16×C17—22no—
132C2×C8×C17—22no—
133C4⋊C4×C17—21no—
134C4⋊C4×C17—21no—
135C4⋊C4×C17C22×C411yescharacteristic
136C4⋊C4×C17C22×C411yescharacteristic
137C4⋊C4×C17C22×C411yescharacteristic
138C4⋊C4×C17C22×C411yescharacteristic
139C22⋊C4×C17—21no—
140C22⋊C4×C17—21no—
141C22⋊C4×C17—21no—
142C22⋊C4×C17—21no—
143C22⋊C4×C17—42no—
144C42×C17—21no—
145C42×C17—21no—
146C42×C17—21no—
147C42×C17—21no—
148C42×C17—22no—
149C42×C17—22no—
150C22⋊Dic17—21no—
151C22⋊Dic17—21no—
152C22⋊Dic17—21no—
153C22⋊Dic17—21no—
154C4⋊Dic17—21no—
155C4⋊Dic17—21no—
156C4⋊Dic17C22×C411yescharacteristic
157C4⋊Dic17C22×C411yescharacteristic
158C4⋊Dic17C22×C411yescharacteristic
159C4⋊Dic17C22×C411yescharacteristic
160C4×Dic17—21no—
161C4×Dic17—21no—
162C4×Dic17—21no—
163C4×Dic17—21no—
164#63223—22no—
165#63223—22no—
166C2×#3622—22no—
167C2×#3622—22no—
168C2×#3622—22no—
169C2×#3622—22no—
170C2×#3622—22no—
171C2×#3622—22no—
172C2×#3622—22no—
173C2×#3622—22no—
174C2×#3622C22×C411yescharacteristic
175C2×#3622C22×C411yescharacteristic
176C2×#3622C22×C411yescharacteristic
177C2×#3622C22×C411yescharacteristic
178C2×#3622C2×D412yes—
179C2×#3622C2×D412yes—
180C2×#3622#4114yes—
181#22607—171nomaximal, Sylow
182C23×C17—22no—
183C23×C17—22no—
184C23×C17—41no—
185C23×C17C4×D411yescharacteristic
186C23×C17C4×D411yescharacteristic
187C23×C17C2×#4111yescharacteristic
188C2×C4×C17—22no—
189C2×C4×C17—21no—
190C2×C4×C17—21no—
191C2×C4×C17—21no—
192C2×C4×C17—21no—
193C2×C4×C17—22no—
194C2×C4×C17—22no—
195C2×C4×C17—21no—
196C2×C4×C17—21no—
197C2×C4×C17—21no—
198C2×C4×C17—21no—
199C2×C4×C17—42no—
200C2×C4×C17—42no—
201C2×C4×C17—42no—
202C2×C4×C17—42no—
203C2×C4×C17—81no—
204C2×C4×C17C4×D411yescharacteristic
205C2×C4×C17C4×D411yescharacteristic
206C2×C4×C17C23×C411yescharacteristic
207C2×C4×C17C22×D411yescharacteristic
208C2×C4×C17C2×#4112yes—
209C2×C4×C172−1+412yes—
210C2×Dic17—21no—
211C2×Dic17—21no—
212C2×Dic17—21no—
213C2×Dic17—21no—
214C2×Dic17—42no—
215C2×Dic17—81no—
216C8×C17—42no—
217#3622—21no—
218#3622—21no—
219#3622—22no—
220#3622—22no—
221#3622—42no—
222#3622#11712yes—
223#3622#11712yes—
224#2924—171no—
225#2924—171no—
226C2×#427—171no—
227#1946—172no—
228#1922—172no—
229#1922—172no—
230#1826—171no—
231#1826D1711yescharacteristic
232#1810—171no—
233#1773—172no—
234#1752—171no—
235C22×C17—21no—
236C22×C17—21no—
237C22×C17—21no—
238C22×C17—21no—
239C22×C17—21no—
240C22×C17—21no—
241C22×C17—21no—
242C22×C17—44no—
243C22×C17#44412yes—
244C22×C17C2×C4×D411yescharacteristic
245C22×C17#55212yes—
246C4×C17—21no—
247C4×C17—21no—
248C4×C17—41no—
249C4×C17—41no—
250C4×C17—41no—
251C4×C17—41no—
252C4×C17—82no—
253C4×C17C2×#11711yescharacteristic
254C4×C17#51911yescharacteristic
255C4×C17#54712yes—
256Dic17—41no—
257Dic17—41no—
258Dic17—41no—
259Dic17—41no—
260C23×C8—171no—
261C2×#117—171no—
262C2×#117C2×D1711yescharacteristic
263#433—342no—
264#433—342no—
265#432—341no—
266#432—341no—
267#431—341no—
268#431—341no—
269C2×#108—172no—
270#427—171no—
271#427—171no—
272#427—171no—
273#427—171no—
274#427—171no—
275#427—171no—
276#427—171no—
277#427—171no—
278C2×#107—171no—
279C2×#107—171no—
280C2×#106—171no—
281C2×#106—171no—
282#424—172no—
283#424—172no—
284#407—171no—
285#407C2×D1711yescharacteristic
286C2×C22⋊C8—171no—
287#405—172no—
288#405—172no—
289C2×#97—172no—
290C4×C22⋊C4C2×D1711yescharacteristic
291#341—172no—
292#339—172no—
293#339—172no—
294#339C2×D1712yes—
295#339C2×D1712yes—
296#336—172no—
297C2×C17—22no—
298C2×C17—22no—
299C2×C17—41no—
300C2×C17#234411yescharacteristic
301C2×C17C2×#44411yescharacteristic
302C2×C17#295711yescharacteristic
303C23×C4C22×D1711yescharacteristic
304C2×M16—172no—
305C2×M16C22×D1712yes—
306C22×C8—171no—
307C22×C8—171no—
308C22×C8—171no—
309C22×C8—171no—
310C22×C8—171no—
311C22×C8—171no—
312C22×C8—172no—
313C22×C8—344no—
314#117—21no—
315#117—21no—
316#117—171no—
317#117—171no—
318#117—171no—
319#117—171no—
320#117—341no—
321#117—341no—
322#117C4×D1711yescharacteristic
323#117C4×D1711yescharacteristic
324#117C22×D1711yescharacteristic
325#117C22×D1711yescharacteristic
326C2×C4⋊C4—171no—
327C2×C4⋊C4—171no—
328C2×C4⋊C4C4×D1711yescharacteristic
329C2×C4⋊C4C4×D1711yescharacteristic
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—341no—
333C2×C42—21no—
334C2×C42—341no—
335C2×C42C22×D1712yes—
336#108—344no—
337#107—171no—
338#107—171no—
339#107—171no—
340#107—171no—
341#107—171no—
342#107—171no—
343#107—171no—
344#107—171no—
345#106—171no—
346#106—171no—
347#106—171no—
348#106—171no—
349#106—171no—
350#106—171no—
351#106—171no—
352#106—171no—
353C4⋊C8—342no—
354C4⋊C8—342no—
355C4⋊C8—342no—
356C4⋊C8—342no—
357C22⋊C8—21no—
358C22⋊C8—21no—
359C22⋊C8—171no—
360C22⋊C8—171no—
361C22⋊C8—172no—
362C22⋊C8—341no—
363C22⋊C8—341no—
364#97—344no—
365#97—342no—
366#97—342no—
367C4×C8—342no—
368C4×C8—342no—
369#95—21no—
370C17#2260711yescharacteristic, complemented, Sylow
371C24#6325411yescharacteristic
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—341no—
378C22×C4—341no—
379C22×C4C2×C4×D1711yescharacteristic
380C22×C4C2×C4×D1711yescharacteristic
381C22×C4D4×D1711yescharacteristic
382C22×C4D4×D1711yescharacteristic
383C22×C4#6325411yescharacteristic
384C22×C4C23×D1711yescharacteristic
385M16—22no—
386M16—22no—
387M16—342no—
388M16—342no—
389C2×C8—22no—
390C2×C8—171no—
391C2×C8—171no—
392C2×C8—174no—
393C2×C8—171no—
394C2×C8—171no—
395C2×C8—172no—
396C2×C8—172no—
397C2×C8—342no—
398C2×C8—342no—
399C2×C8—342no—
400C2×C8—342no—
401C2×C8—342no—
402C2×C8—342no—
403C2×C8—342no—
404C2×C8—342no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—171no—
408C4⋊C4—171no—
409C4⋊C4—171no—
410C4⋊C4—171no—
411C4⋊C4—341no—
412C4⋊C4—341no—
413C4⋊C4C2×C4×D1711yescharacteristic
414C4⋊C4C2×C4×D1711yescharacteristic
415C4⋊C4C2×C4×D1711yescharacteristic
416C4⋊C4C2×C4×D1711yescharacteristic
417C22⋊C4—21no—
418C22⋊C4—21no—
419C22⋊C4—21no—
420C22⋊C4—21no—
421C22⋊C4—42no—
422C22⋊C4—341no—
423C22⋊C4—341no—
424C22⋊C4—341no—
425C22⋊C4—341no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—22no—
431C42—22no—
432C42—341no—
433C42—341no—
434C42—341no—
435C42—341no—
436C23—22no—
437C23—22no—
438C23—41no—
439C23#1058756711yescharacteristic
440C23#1058756711yescharacteristic
441C23C2×#6325411yescharacteristic
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—81no—
458C2×C4—341no—
459C2×C4—341no—
460C2×C4—341no—
461C2×C4—341no—
462C2×C4—682no—
463C2×C4—1361no—
464C2×C4#1058756711yescharacteristic
465C2×C4#1058756711yescharacteristic
466C2×C4C22×C4×D1711yescharacteristic
467C2×C4C2×D4×D1711yescharacteristic
468C2×C4#41×D1712yes—
469C2×C4#1058769712yes—
470C8—42no—
471C8—172no—
472C8—172no—
473C8—342no—
474C8—342no—
475C8—341no—
476C8—341no—
477C8—682no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—44no—
486C22#1176292812yes—
487C22C2×#1058756711yescharacteristic
488C22#1176349612yes—
489C4—21no—
490C4—21no—
491C4—41no—
492C4—41no—
493C4—41no—
494C4—41no—
495C4—82no—
496C4—681no—
497C4—681no—
498C4—681no—
499C4—681no—
500C4#117×D1711yescharacteristic
501C4#1176328611yescharacteristic
502C4#1176345712yes—
503C2—22no—
504C2—22no—
505C2—41no—
506C2#1453164811yescharacteristic
507C2C2×#1176292811yescharacteristic
508C2#1453769711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#1826C2
socleC22×C17C2×C4×D4
prime radicalC24×C17#41
cosocleC22×C4×C17C24
hypercenter#1826D17
hypocenterC17#22607
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1826

Complements

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

NH ≅ G/Ntype
4. #14529536C2semidirect
5. #14529386C2semidirect
6. #14529381C2semidirect
9. #14528667C2semidirect
21. #11762845C4semidirect
22. #11762845C4semidirect
23. #11762845C4semidirect
24. #11762845C4semidirect
29. C2×#10587478C4semidirect
30. C2×#10587478C4semidirect
31. C2×#10587479C4semidirect
32. C2×#10587479C4semidirect
370. C17#22607semidirect

Maximal subgroups

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

subgroupindex
#145374442
#145374422
C2×#117628452
#145295362
#145293862
#145293812
#145288842
#145288202
#145286672
#145285622
C17×#18262
#2260717