Finite Group Databasebeta

← #24918304

Subgroups — #24918304

Order 10624.

418 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128214C211yesmaximal, characteristic, complemented
2#19128208C211yesmaximal, characteristic, complemented
3#19128041C211yesmaximal, characteristic, complemented
4#19128039C211yesmaximal, characteristic, complemented
5#19128026C211yesmaximal, characteristic, complemented
6#19128024C211yesmaximal, characteristic, complemented
7#19127956C211yesmaximal, characteristic, complemented
8#19127953C211yesmaximal, characteristic, complemented
9#19127950C211yesmaximal, characteristic, complemented
10#19127944C211yesmaximal, characteristic, complemented
11#19127862C211yesmaximal, characteristic, complemented
12#19127846C211yesmaximal, characteristic, complemented
13#19127845C211yesmaximal, characteristic, complemented
14#19127663C211yesmaximal, characteristic, complemented
15#452×C83C211yesmaximal, characteristic, complemented
16C2×#13050591C2211yescharacteristic
17C22×C4×D83C2211yescharacteristic
18C2×#13050577C2211yescharacteristic, complemented
19C2×#13050575C2211yescharacteristic, complemented
20C2×#13050573C2211yescharacteristic
21C2×#13050573C2211yescharacteristic
22#14581777—21no—
23#14581773—21no—
24#14581768C2211yescharacteristic
25#14581768C2211yescharacteristic
26#14581768C2211yescharacteristic
27#14581767C2211yescharacteristic
28#14581766C2211yescharacteristic
29#14581763C2211yescharacteristic
30#14581763C2211yescharacteristic
31#14581762—21no—
32#14581760—21no—
33#14581754—21no—
34C4×C22⋊D83—21no—
35C4×C22⋊D83—21no—
36(C22⋊D83)⋊C4—21no—
37#14581751C2211yescharacteristic
38#14581750C2211yescharacteristic
39#14581726C2211yescharacteristic, complemented
40#14581725C2211yescharacteristic, complemented
41#14581724C2211yescharacteristic, complemented
42#14581723C2211yescharacteristic, complemented
43#14581716C2211yescharacteristic, complemented
44#14581716C2211yescharacteristic, complemented
45#14581716C2211yescharacteristic, complemented
46#14581714C2211yescharacteristic, complemented
47#14581714C2211yescharacteristic, complemented
48#14581713C2211yescharacteristic, complemented
49C22⋊(C83⋊C8)C2211yescharacteristic
50(C2×D83)⋊C8C2211yescharacteristic
51(C2×D83)⋊C8C2211yescharacteristic
52C2×#41×C83C2211yescharacteristic
53C2×Q16×C83C2211yescharacteristic, complemented
54C2×SD16×C83C2211yescharacteristic, complemented
55C22⋊Q8×C83C2211yescharacteristic
56#103×C83C2211yescharacteristic, complemented
57#102×C83C2211yescharacteristic, complemented
58C22⋊C8×C83C2211yescharacteristic
59C23×D83D411yescharacteristic
60#13050591—41no—
61#13050591—41no—
62C2×D332—21no—
63C2×D332—21no—
64C2×D332C2311yescharacteristic
65C2×C22⋊D83—21no—
66C2×C22⋊D83—21no—
67C2×C22⋊D83—21no—
68C22×Dic83D411yescharacteristic
69C2×C4×D83—21no—
70C2×C4×D83—41no—
71C2×C4×D83—41no—
72C2×C4×D83D411yescharacteristic
73C2×C4×D83D411yescharacteristic
74C2×C4×D83C2311yescharacteristic
75C2×C4×D83C2311yescharacteristic
76#13050577—41no—
77#13050575—41no—
78#13050573—41no—
79#13050573—41no—
80C2×C83⋊C8C2311yescharacteristic
81C2×C83⋊C8C2311yescharacteristic
82C4⋊Dic83C2311yescharacteristic
83C4⋊Dic83C2311yescharacteristic
84C4⋊Dic83C2311yescharacteristic
85Dic83⋊C4—21no—
86Dic83⋊C4—21no—
87Dic83⋊C4—21no—
88C22⋊Dic83—21no—
89C22⋊Dic83—21no—
90C22⋊Dic83—21no—
91(C2×D83)⋊C4—21no—
92(C2×D83)⋊C4—21no—
93(C2×D83)⋊C4—21no—
94(C2×D83)⋊C4—21no—
95(C2×D83)⋊C4—21no—
96(C2×D83)⋊C4—21no—
97C4×Dic83—21no—
98C4×Dic83—21no—
99#41×C83—41no—
100#41×C83—41no—
101C2×Q8×C83C2311yescharacteristic
102C2×Q8×C83C2311yescharacteristic
103C2×D4×C83—21no—
104C2×D4×C83C2311yescharacteristic
105C22×C4×C83—21no—
106C22×C4×C83C2311yescharacteristic
107Q16×C83—41no—
108SD16×C83—41no—
109C2×C8×C83C2311yescharacteristic
110C2×C8×C83C2311yescharacteristic
111C4⋊C4×C83—21no—
112C4⋊C4×C83C2311yescharacteristic
113C22⋊C4×C83—21no—
114C23×C83—21no—
115C23×C83C2×D411yescharacteristic
116C22×D83—21no—
117C22×D83—41no—
118C22×D83—41no—
119C22×D83—41no—
120C22×D83C2×D411yescharacteristic
121C22×D83C2×D411yescharacteristic
122Q8×C83—21no—
123Q8×C83—21no—
124Q8×C83—21no—
125Q8×C83—21no—
126D4×C83—21no—
127D4×C83—21no—
128D4×C83—42no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—21no—
133C2×C4×C83—21no—
134C2×C4×C83—42no—
135C2×C4×C83C2×D411yescharacteristic
136C2×C4×C83C2411yescharacteristic
137C22⋊D83—42no—
138C22⋊D83—42no—
139C22⋊D83—42no—
140C2×Dic83—21no—
141C2×Dic83—21no—
142C2×Dic83—21no—
143C2×Dic83—41no—
144C2×Dic83C2×D411yescharacteristic
145C2×Dic83C2×D411yescharacteristic
146D332—21no—
147D332—21no—
148D332—42no—
149D332—42no—
150C4×D83—21no—
151C4×D83—21no—
152C4×D83—21no—
153C4×D83—21no—
154C4×D83—41no—
155C4×D83—41no—
156C4×D83—42no—
157C8×C83—21no—
158C8×C83—21no—
159C83⋊C8—21no—
160C83⋊C8—21no—
161C22×C83—21no—
162C22×C83—21no—
163C22×C83—21no—
164C22×C83—41no—
165C22×C83—41no—
166C22×C83—41no—
167C22×C83C22×D411yescharacteristic
168C2×D83—21no—
169C2×D83—21no—
170C2×D83—21no—
171C2×D83—21no—
172C2×D83—21no—
173C2×D83—21no—
174C2×D83—44no—
175C2×D83—41no—
176C2×D83—41no—
177C2×D83—41no—
178C4×C83—21no—
179C4×C83—41no—
180C4×C83—41no—
181C4×C83—41no—
182C4×C83H(5,𝔽2)11yescharacteristic
183C4×C83H(5,𝔽2)11yescharacteristic
184Dic83—21no—
185Dic83—21no—
186Dic83—41no—
187Dic83—41no—
188Dic83—41no—
189C2×C83—41no—
190C2×C83—81no—
191C2×C83#53411yescharacteristic
192C2×C83#57511yescharacteristic
193C2×C83#57511yescharacteristic
194D83—41no—
195D83—41no—
196D83—81no—
197#3197—831nomaximal, Sylow
198C83#319711yescharacteristic, complemented, Sylow
199#525—831no—
200#525—831no—
201#485—831no—
202#485—831no—
203#482—831no—
204#482—831no—
205#470—831no—
206#469—831no—
207#468—831no—
208#468—831no—
209#452—831no—
210#452D8311yescharacteristic, complemented
211#449—831no—
212#449—831no—
213#407—831no—
214C2×#41—831no—
215C2×#41C2×D8311yescharacteristic
216C23×C4—831no—
217C2×Q16C2×D8311yescharacteristic, complemented
218C2×SD16—831no—
219C2×SD16C2×D8311yescharacteristic, complemented
220C2×D8—831no—
221C2×M16—831no—
222C2×M16—831no—
223#123—1661no—
224#123—1661no—
225C22⋊Q8—831no—
226C22⋊Q8—831no—
227C22⋊Q8—831no—
228C22⋊Q8C2×D8311yescharacteristic
229#121—831no—
230#121—831no—
231#121—831no—
232#121—831no—
233#120—1661no—
234#120—1661no—
235C4×D4—1661no—
236C4×D4—1661no—
237C4×D4—1661no—
238C4×D4—1661no—
239#117—831no—
240#117—831no—
241#107—831no—
242#107—831no—
243#106—831no—
244#106—831no—
245#103—831no—
246#103—831no—
247#103—831no—
248#103C2×D8311yescharacteristic, complemented
249#102—831no—
250#102—831no—
251#102—831no—
252#102C2×D8311yescharacteristic, complemented
253C22⋊C8—831no—
254C22⋊C8—831no—
255C22⋊C8—831no—
256C22⋊C8C2×D8311yescharacteristic
257C24—831no—
258#41—41no—
259#41—41no—
260#41—3321no—
261#41—3321no—
262C2×Q8C22×D8311yescharacteristic
263C2×Q8C22×D8311yescharacteristic
264C2×D4—21no—
265C2×D4—831no—
266C2×D4—1661no—
267C2×D4—1661no—
268C2×D4—1661no—
269C2×D4—1661no—
270C2×D4—1661no—
271C2×D4C22×D8311yescharacteristic
272C22×C4—21no—
273C22×C4—3321no—
274C22×C4—3321no—
275C22×C4—831no—
276C22×C4—831no—
277C22×C4—831no—
278C22×C4—831no—
279C22×C4—831no—
280C22×C4—1661no—
281C22×C4C22×D8311yescharacteristic
282Q16—41no—
283SD16—41no—
284SD16—3321no—
285D8—3321no—
286M16—3321no—
287M16—3321no—
288C2×C8—831no—
289C2×C8—831no—
290C2×C8C22×D8311yescharacteristic
291C2×C8C22×D8311yescharacteristic
292C4⋊C4—21no—
293C4⋊C4—831no—
294C4⋊C4—831no—
295C4⋊C4—831no—
296C4⋊C4—1661no—
297C4⋊C4—1661no—
298C4⋊C4—1661no—
299C4⋊C4C22×D8311yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—1661no—
302C22⋊C4—1661no—
303C22⋊C4—1661no—
304C22⋊C4—1661no—
305C22⋊C4—1661no—
306C22⋊C4—1661no—
307C22⋊C4—1661no—
308C22⋊C4—1661no—
309C22⋊C4—1661no—
310C42—1661no—
311C42—1661no—
312C23—21no—
313C23—3321no—
314C23—3321no—
315C23—3321no—
316C23—831no—
317C23—831no—
318C23—1661no—
319C23D4×D8311yescharacteristic
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—21no—
324D4—21no—
325D4—21no—
326D4—42no—
327D4—3322no—
328D4—3322no—
329D4—3322no—
330D4—3322no—
331D4—3322no—
332D4—1661no—
333D4—1661no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—42no—
340C2×C4—3321no—
341C2×C4—3321no—
342C2×C4—3322no—
343C2×C4—3321no—
344C2×C4—831no—
345C2×C4—831no—
346C2×C4—1661no—
347C2×C4—1661no—
348C2×C4—1661no—
349C2×C4—1661no—
350C2×C4—1661no—
351C2×C4—1661no—
352C2×C4—1661no—
353C2×C4D4×D8311yescharacteristic
354C2×C4C23×D8311yescharacteristic
355C8—21no—
356C8—21no—
357C8—1661no—
358C8—1661no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—3324no—
366C22—3321no—
367C22—3321no—
368C22—3321no—
369C22—1661no—
370C22—1661no—
371C22—1661no—
372C22—1661no—
373C22—1661no—
374C22—1661no—
375C22C2×D4×D8311yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—3321no—
381C4—3321no—
382C4—3321no—
383C4—1661no—
384C4—1661no—
385C4#1458183711yescharacteristic
386C4#1458183711yescharacteristic
387C2—41no—
388C2—81no—
389C2—3321no—
390C2—3321no—
391C2—6641no—
392C2#1912826111yescharacteristic
393C2#1912849811yescharacteristic
394C2#1912850011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#452×C83C2
socleC22×C83C22×D4
hypercenter#452D83
hypocenterC83#3197
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #19128214C2semidirect
2. #19128208C2semidirect
3. #19128041C2semidirect
4. #19128039C2semidirect
5. #19128026C2semidirect
6. #19128024C2semidirect
7. #19127956C2semidirect
8. #19127953C2semidirect
9. #19127950C2semidirect
10. #19127944C2semidirect
11. #19127862C2semidirect
12. #19127846C2semidirect
13. #19127845C2semidirect
14. #19127663C2semidirect
15. #452×C83C2semidirect
18. C2×#13050577C22semidirect
19. C2×#13050575C22semidirect
39. #14581726C22semidirect
40. #14581725C22semidirect
41. #14581724C22semidirect
42. #14581723C22semidirect
43. #14581716C22semidirect
44. #14581716C22semidirect
45. #14581716C22semidirect
46. #14581714C22semidirect
47. #14581714C22semidirect
48. #14581713C22semidirect
53. C2×Q16×C83C22semidirect
54. C2×SD16×C83C22semidirect
56. #103×C83C22semidirect
57. #102×C83C22semidirect
198. C83#3197semidirect
210. #452D83semidirect
217. C2×Q16C2×D83semidirect
219. C2×SD16C2×D83semidirect
248. #103C2×D83semidirect
252. #102C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282142
#191282082
#191280412
#191280392
#191280262
#191280242
#191279562
#191279532
#191279502
#191279442
#191278622
#191278462
#191278452
#191276632
#452×C832
#319783