Finite Group Databasebeta

← #31010310

Subgroups — #31010310

Order 17152.

738 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863831C211yesmaximal, characteristic, complemented
2C22×#14523983C211yesmaximal, characteristic, complemented
3C2×#16863632C211yesmaximal, characteristic, complemented
4#22562324C212yesmaximal, complemented
5#22562310C212yesmaximal, complemented
6#22562307C212yesmaximal, complemented
7#22562293C212yesmaximal, complemented
8#22561998C211yesmaximal, characteristic, complemented
9#22561924C211yesmaximal, characteristic, complemented
10#22561921C211yesmaximal, characteristic, complemented
11C67×#3050C211yesmaximal, characteristic, complemented
12C22×D536C2211yescharacteristic
13C2×#14524017—21no—
14#16863831—24no—
15C2×#14523986C2212yescomplemented
16C2×#14523984C2212yescomplemented
17C2×#14523983—24no—
18C2×#14523983C2211yescharacteristic
19C2×#14523983C2211yescharacteristic
20C2×#14523983C2212yescomplemented
21C22×C67⋊C16C2212yescomplemented
22#16863632—21no—
23#16863632—21no—
24#16863632C2212yes—
25#16863632C2212yes—
26C2×#14523977C2211yescharacteristic
27C2×#14523965—21no—
28#16863223C2212yescomplemented
29#16863221C2212yescomplemented
30#16863200C2212yescomplemented
31#16863197C2212yescomplemented
32#16863196C2212yescomplemented
33#16863196C2212yescomplemented
34C2×#135×C67C2211yescharacteristic
35C23×C8×C67C2211yescharacteristic
36#525×C67—21no—
37#467×C67C2211yescharacteristic, complemented
38#467×C67C2211yescharacteristic, complemented
39#466×C67C2211yescharacteristic, complemented
40#466×C67C2211yescharacteristic, complemented
41#465×C67—21no—
42#465×C67—21no—
43#427×C67C2211yescharacteristic
44#416×C67—21no—
45C22×D268—21no—
46C22×C22⋊D67—41no—
47C2×D536—22no—
48C2×D536—21no—
49C2×D536—22no—
50C2×D536C2312yes—
51#14524017—22no—
52#14524017—22no—
53#14524017—41no—
54#14524017—41no—
55C2×C4⋊Dic67—21no—
56C2×(C2×D67)⋊C4—41no—
57#14523986—24no—
58#14523984—24no—
59#14523983—24no—
60#14523983—22no—
61#14523983D414yescomplemented
62C2×C67⋊C16—24no—
63C2×C67⋊C16C2312yes—
64C2×C67⋊C16C2312yes—
65#14523977—21no—
66#14523977C2312yes—
67#14523965—22no—
68#14523965—22no—
69C2×#41×C67—21no—
70C23×C4×C67D411yescharacteristic
71#135×C67—22no—
72#135×C67—22no—
73C2×Q16×C67C2311yescharacteristic
74C2×SD16×C67—21no—
75C2×D8×C67C2311yescharacteristic
76C22×C8×C67—22no—
77C22×C8×C67—22no—
78C22×C8×C67D411yescharacteristic
79C22×C8×C67D411yescharacteristic
80C22×C8×C67D411yescharacteristic
81C22×C8×C67C2311yescharacteristic
82C22×C8×C67C2311yescharacteristic
83C22×C8×C67C2311yescharacteristic
84#123×C67—41no—
85C22⋊Q8×C67—21no—
86C22⋊Q8×C67—21no—
87#121×C67—21no—
88#121×C67—21no—
89#120×C67—41no—
90C4×D4×C67—41no—
91C4×D4×C67—41no—
92#117×C67—21no—
93#107×C67C2311yescharacteristic
94#107×C67C2311yescharacteristic
95#106×C67—21no—
96#103×C67—21no—
97#103×C67—21no—
98#102×C67—21no—
99#102×C67—21no—
100C23×D67—41no—
101C2×D268—22no—
102C2×D268—42no—
103C2×D268—42no—
104C2×D268—41no—
105C2×C22⋊D67—42no—
106C2×C22⋊D67—42no—
107C2×C22⋊D67—84no—
108C22×Dic67—41no—
109D536—22no—
110D536—22no—
111D536—42no—
112C4⋊Dic67—22no—
113C4⋊Dic67—41no—
114(C2×D67)⋊C4—42no—
115(C2×D67)⋊C4—42no—
116C67⋊C16—22no—
117C67⋊C16C2×D414yescomplemented
118C24×C67D811yescharacteristic
119#41×C67—42no—
120#41×C67—42no—
121C2×Q8×C67—21no—
122C2×D4×C67—21no—
123C2×D4×C67—41no—
124C2×D4×C67—41no—
125C2×D4×C67—41no—
126C22×C4×C67—22no—
127C22×C4×C67—22no—
128C22×C4×C67—41no—
129C22×C4×C67D811yescharacteristic
130C22×C4×C67D811yescharacteristic
131C22×C4×C67D811yescharacteristic
132C22×C4×C67C2×D411yescharacteristic
133C22×C4×C67C2×D411yescharacteristic
134C22×C4×C67C2×D411yescharacteristic
135Q16×C67—21no—
136Q16×C67—21no—
137SD16×C67—42no—
138D8×C67—21no—
139D8×C67—21no—
140C2×C8×C67—22no—
141C2×C8×C67—22no—
142C2×C8×C67—22no—
143C2×C8×C67—21no—
144C2×C8×C67—21no—
145C2×C8×C67—21no—
146C2×C8×C67—21no—
147C2×C8×C67C2×D412yes—
148C2×C8×C67C2×D411yescharacteristic
149C2×C8×C67C2×D411yescharacteristic
150C2×C8×C67C2×D411yescharacteristic
151C2×C8×C67#4112yes—
152C2×C8×C67C2411yescharacteristic
153C4⋊C4×C67—21no—
154C4⋊C4×C67—21no—
155C4⋊C4×C67—41no—
156C4⋊C4×C67—41no—
157C22⋊C4×C67—41no—
158C22⋊C4×C67—41no—
159C22⋊C4×C67—41no—
160C22⋊C4×C67—41no—
161C22⋊C4×C67—41no—
162C42×C67—41no—
163C23×C67—22no—
164C23×C67—22no—
165C23×C67—41no—
166C23×C67—41no—
167C23×C67C2×D811yescharacteristic
168C23×C67C2×D811yescharacteristic
169C23×C67C2×D811yescharacteristic
170C22×D67—42no—
171C22×D67—82no—
172C22×D67—82no—
173C22×D67—82no—
174Q8×C67—41no—
175Q8×C67—41no—
176D4×C67—41no—
177D4×C67—41no—
178D4×C67—82no—
179D4×C67—82no—
180D4×C67—82no—
181C2×C4×C67—22no—
182C2×C4×C67—21no—
183C2×C4×C67—21no—
184C2×C4×C67—21no—
185C2×C4×C67—21no—
186C2×C4×C67—22no—
187C2×C4×C67—22no—
188C2×C4×C67—41no—
189C2×C4×C67—41no—
190C2×C4×C67—41no—
191C2×C4×C67—82no—
192C2×C4×C67#12112yes—
193C2×C4×C67C2×D811yescharacteristic
194C2×C4×C67C2×D811yescharacteristic
195C2×C4×C67C2×D811yescharacteristic
196C2×C4×C67#13512yes—
197C2×C4×C67C22×D411yescharacteristic
198C22⋊D67—84no—
199C22⋊D67—84no—
200C2×Dic67—42no—
201C2×Dic67—82no—
202D268—42no—
203D268—42no—
204D268—82no—
205C8×C67—21no—
206C8×C67—21no—
207C8×C67#12112yes—
208C8×C67C22×D411yescharacteristic
209C8×C67C2×#4111yescharacteristic
210C22×C67—21no—
211C22×C67—21no—
212C22×C67—21no—
213C22×C67—21no—
214C22×C67—21no—
215C22×C67—21no—
216C22×C67—21no—
217C22×C67—44no—
218C22×C67—81no—
219C22×C67—81no—
220C22×C67—81no—
221C22×C67#46612yes—
222C22×C67#50912yescomplemented
223C22×C67C22×D811yescharacteristic
224C2×D67—82no—
225C2×D67—82no—
226C2×D67—82no—
227C2×D67—164no—
228C4×C67—21no—
229C4×C67—21no—
230C4×C67—81no—
231C4×C67—81no—
232C4×C67—81no—
233C4×C67#46612yes—
234C4×C67C2×#12111yescharacteristic
235C4×C67C2×#13511yescharacteristic
236Dic67—82no—
237#33605—671nomaximal, Sylow
238C2×C67—22no—
239C2×C67—22no—
240C2×C67—41no—
241C2×C67—161no—
242C2×C67C2×#46611yescharacteristic
243C2×C67C2×#50911yescharacteristic
244C2×C67#341411yescharacteristic
245D67—162no—
246C2×#509—671no—
247C22×M32—671no—
248#3050D6711yescharacteristic, complemented
249C2×#466—671no—
250#2220—672no—
251#2218—672no—
252#2217—672no—
253#2215—672no—
254#2159—671no—
255#2140—671no—
256#2139—671no—
257C67#3360511yescharacteristic, complemented, Sylow
258C2×#135C2×D6711yescharacteristic
259C22×D8—671no—
260C23×C8C2×D6711yescharacteristic
261#525—21no—
262C2×#121—1341no—
263#509—1344no—
264C2×SD32—672no—
265C2×D16—672no—
266C2×M32—671no—
267C2×M32—671no—
268C2×M32—672no—
269C2×M32—1344no—
270C22×C16—672no—
271#467C2×D6711yescharacteristic, complemented
272#467C2×D6711yescharacteristic, complemented
273#466—672no—
274#466—672no—
275#466—1341no—
276#466—1341no—
277#466C2×D6711yescharacteristic, complemented
278#466C2×D6711yescharacteristic, complemented
279#465—21no—
280#465—21no—
281#427C2×D6711yescharacteristic
282C2×#107—671no—
283#416—21no—
284C2×#102—1341no—
285#367—672no—
286#366—672no—
287#358—672no—
288#357—672no—
289#357—672no—
290#357—672no—
291C2×#41—21no—
292C22×D4—2681no—
293C22×D4—1341no—
294C23×C4C22⋊D6711yescharacteristic
295#135—22no—
296#135—22no—
297C2×Q16C22×D6711yescharacteristic
298C2×SD16—21no—
299C2×D8—672no—
300C2×D8—1342no—
301C2×D8—1341no—
302C2×D8—1342no—
303C2×D8C22×D6711yescharacteristic
304C22×C8—22no—
305C22×C8—22no—
306C22×C8C22⋊D6711yescharacteristic
307C22×C8C22⋊D6711yescharacteristic
308C22×C8C22⋊D6711yescharacteristic
309C22×C8C22×D6711yescharacteristic
310C22×C8C22×D6711yescharacteristic
311C22×C8C22×D6711yescharacteristic
312#123—41no—
313C22⋊Q8—21no—
314C22⋊Q8—21no—
315#121—21no—
316#121—21no—
317#121—2681no—
318#121—2681no—
319#121—1342no—
320#121—1342no—
321#120—41no—
322C4×D4—41no—
323C4×D4—41no—
324#117—21no—
325C2×C4⋊C4—1341no—
326C2×C22⋊C4—2681no—
327SD32—1344no—
328D16—1344no—
329M32—674no—
330M32—1344no—
331M32—1342no—
332C2×C16—672no—
333C2×C16—672no—
334C2×C16—1344no—
335#107—672no—
336#107—1341no—
337#107C22×D6711yescharacteristic
338#107C22×D6711yescharacteristic
339#106—21no—
340#103—21no—
341#103—21no—
342#102—21no—
343#102—21no—
344#102—1342no—
345#102—1342no—
346C24—2681no—
347C24#1176161311yescharacteristic
348#41—42no—
349#41—42no—
350C2×Q8—21no—
351C2×D4—21no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—2682no—
356C2×D4—2682no—
357C2×D4—2682no—
358C2×D4—2681no—
359C2×D4—2682no—
360C2×D4—5364no—
361C2×D4—1342no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—41no—
365C22×C4—2681no—
366C22×C4#1176161311yescharacteristic
367C22×C4#1176161311yescharacteristic
368C22×C4#1176161311yescharacteristic
369C22×C4C2×C22⋊D6711yescharacteristic
370C22×C4C2×C22⋊D6711yescharacteristic
371C22×C4C2×C22⋊D6711yescharacteristic
372Q16—21no—
373Q16—21no—
374SD16—42no—
375D8—21no—
376D8—21no—
377D8—2682no—
378D8—1342no—
379D8—1342no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8D4×D6712yes—
388C2×C8C2×C22⋊D6711yescharacteristic
389C2×C8C2×C22⋊D6711yescharacteristic
390C2×C8C2×C22⋊D6711yescharacteristic
391C2×C8#1176162712yes—
392C2×C8C23×D6711yescharacteristic
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—2681no—
398C4⋊C4—1342no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—2682no—
405C22⋊C4—2682no—
406C42—41no—
407C16—674no—
408C16—1342no—
409C23—22no—
410C23—22no—
411C23—41no—
412C23—41no—
413C23—2682no—
414C23—5362no—
415C23—5362no—
416C23—5362no—
417C23C2×#1176161311yescharacteristic
418C23C2×#1176161311yescharacteristic
419C23C2×#1176161311yescharacteristic
420Q8—41no—
421Q8—41no—
422D4—41no—
423D4—41no—
424D4—82no—
425D4—82no—
426D4—82no—
427D4—2682no—
428D4—2682no—
429D4—5364no—
430D4—5364no—
431D4—5362no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—82no—
443C2×C4—2682no—
444C2×C4—5362no—
445C2×C4#1452401512yes—
446C2×C4C2×#1176161311yescharacteristic
447C2×C4C2×#1176161311yescharacteristic
448C2×C4C2×#1176161311yescharacteristic
449C2×C4#1452406312yes—
450C2×C4C22×C22⋊D6711yescharacteristic
451C8—21no—
452C8—21no—
453C8#1452401512yes—
454C8C2×D4×D6711yescharacteristic
455C8C2×#1176162711yescharacteristic
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—44no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22—5362no—
468C22—5362no—
469C22—5362no—
470C22—10724no—
471C22#1686363012yes—
472C22#1686383112yescomplemented
473C22C22×#1176161311yescharacteristic
474C4—21no—
475C4—21no—
476C4—81no—
477C4—81no—
478C4—81no—
479C4—5362no—
480C4#1686363012yes—
481C4C2×#1452401511yescharacteristic
482C4C2×#1452406311yescharacteristic
483C2—22no—
484C2—22no—
485C2—41no—
486C2—161no—
487C2—10722no—
488C2C2×#1686363011yescharacteristic
489C2C2×#1686383111yescharacteristic
490C2#2257334311yescharacteristic
subgroupG/H
centerC22C22×#11761613
commutator, cosocleC2×C8×C67C24
FrattiniC2×C8C23×D67
FittingC67×#3050C2
socleC22×C67C22×D8
hypercenter#3050D67
hypocenterC67#33605
LCSG ▷ C2×C8×C67 ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #3050

Complements

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

NH ≅ G/Ntype
1. C2×#16863831C2semidirect
2. C22×#14523983C2semidirect
3. C2×#16863632C2semidirect
4. #22562324C2semidirect
5. #22562310C2semidirect
6. #22562307C2semidirect
7. #22562293C2semidirect
8. #22561998C2semidirect
9. #22561924C2semidirect
10. #22561921C2semidirect
11. C67×#3050C2semidirect
15. C2×#14523986C22semidirect
16. C2×#14523984C22semidirect
20. C2×#14523983C22semidirect
21. C22×C67⋊C16C22semidirect
28. #16863223C22semidirect
29. #16863221C22semidirect
30. #16863200C22semidirect
31. #16863197C22semidirect
32. #16863196C22semidirect
33. #16863196C22semidirect
37. #467×C67C22semidirect
38. #467×C67C22semidirect
39. #466×C67C22semidirect
40. #466×C67C22semidirect
61. #14523983D4semidirect
117. C67⋊C16C2×D4semidirect
222. C22×C67#509semidirect
248. #3050D67semidirect
257. C67#33605semidirect
271. #467C2×D67semidirect
272. #467C2×D67semidirect
277. #466C2×D67semidirect
278. #466C2×D67semidirect
472. C22#16863831semidirect

Maximal subgroups

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

subgroupindex
C2×#168638312
C22×#145239832
C2×#168636322
#225623242
#225623102
#225623072
#225622932
#225619982
#225619242
#225619212
C67×#30502
#3360567