Finite Group Databasebeta

← #24919493

Subgroups — #24919493

Order 10624.

402 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128347C211yesmaximal, characteristic
2#19128063C211yesmaximal, characteristic, complemented
3#19128042C211yesmaximal, characteristic, complemented
4#19128025C211yesmaximal, characteristic, complemented
5#19127988C211yesmaximal, characteristic, complemented
6#19127957C211yesmaximal, characteristic, complemented
7#19127932C211yesmaximal, characteristic, complemented
8#19127923C211yesmaximal, characteristic, complemented
9#19127861C211yesmaximal, characteristic, complemented
10#19127851C211yesmaximal, characteristic, complemented
11#19127799C211yesmaximal, characteristic, complemented
12#19127773C211yesmaximal, characteristic, complemented
13#19127705C211yesmaximal, characteristic, complemented
14#19127695C211yesmaximal, characteristic, complemented
15#556×C83C211yesmaximal, characteristic, complemented
16C2×#13050590C2211yescharacteristic
17C22×D332C2211yescharacteristic
18C2×#13050580C2211yescharacteristic, complemented
19C2×#13050577C2211yescharacteristic
20C2×#13050577C2211yescharacteristic
21C2×#13050574C2211yescharacteristic
22C22×C83⋊C8C2211yescharacteristic, complemented
23#14581780C2211yescharacteristic
24#14581777—21no—
25#14581771C2211yescharacteristic
26#14581766C2211yescharacteristic
27#14581765—21no—
28#14581760—21no—
29C4×D332C2211yescharacteristic
30C22⋊C4×D83—21no—
31#14581725C2211yescharacteristic, complemented
32#14581723C2211yescharacteristic, complemented
33#14581723C2211yescharacteristic, complemented
34C4⋊(C83⋊C8)C2211yescharacteristic, complemented
35#14581716C2211yescharacteristic, complemented
36#14581715C2211yescharacteristic, complemented
37#14581715C2211yescharacteristic, complemented
38#14581715C2211yescharacteristic, complemented
39#14581712C2211yescharacteristic
40#14581712C2211yescharacteristic
41#14581712C2211yescharacteristic
42#14581712C2211yescharacteristic
43#14581712C2211yescharacteristic
44#14581712C2211yescharacteristic
45C22⋊(C83⋊C8)C2211yescharacteristic
46C22⋊(C83⋊C8)C2211yescharacteristic, complemented
47#14581703C2211yescharacteristic, complemented
48C4⋊Q8×C83C2211yescharacteristic, complemented
49#126×C83—21no—
50#126×C83—21no—
51#124×C83—21no—
52C22⋊Q8×C83C2211yescharacteristic
53C22⋊Q8×C83C2211yescharacteristic, complemented
54C4×Q8×C83—21no—
55C4×Q8×C83C2211yescharacteristic, complemented
56C4×D4×C83C2211yescharacteristic
57#117×C83C2211yescharacteristic
58#117×C83C2211yescharacteristic, complemented
59C23×D83—21no—
60#13050590—22no—
61#13050590—41no—
62C2×Dic166C2311yescharacteristic
63C2×D332—22no—
64C2×D332—21no—
65C2×D332—41no—
66C2×D332C2311yescharacteristic
67C2×D332C2311yescharacteristic
68C2×C22⋊D83—21no—
69C2×C22⋊D83—21no—
70C2×C4×D83—21no—
71C2×C4×D83—21no—
72#13050580—41no—
73#13050577—22no—
74#13050577—41no—
75#13050574—41no—
76C2×C83⋊C8—41no—
77C2×C83⋊C8C2311yescharacteristic
78C2×C83⋊C8C2311yescharacteristic
79C2×C83⋊C8C2311yescharacteristic
80C2×C83⋊C8C2311yescharacteristic
81C4⋊Dic83C2311yescharacteristic
82Dic83⋊C4—21no—
83C22⋊Dic83—21no—
84(C2×D83)⋊C4—21no—
85(C2×D83)⋊C4—21no—
86(C2×D83)⋊C4—21no—
87(C2×D83)⋊C4—21no—
88C2×Q8×C83—21no—
89C2×Q8×C83C2311yescharacteristic
90C2×D4×C83D411yescharacteristic
91C22×C4×C83C2311yescharacteristic
92C22×C4×C83C2311yescharacteristic
93C4⋊C4×C83—21no—
94C4⋊C4×C83—21no—
95C4⋊C4×C83—21no—
96C4⋊C4×C83—21no—
97C4⋊C4×C83D411yescharacteristic
98C4⋊C4×C83C2311yescharacteristic
99C4⋊C4×C83C2311yescharacteristic
100C4⋊C4×C83C2311yescharacteristic
101C22⋊C4×C83—21no—
102C22⋊C4×C83—21no—
103C22⋊C4×C83—21no—
104C22⋊C4×C83—21no—
105C22⋊C4×C83D411yescharacteristic
106C22⋊C4×C83D411yescharacteristic
107C42×C83—21no—
108C42×C83—21no—
109C42×C83—21no—
110C42×C83C2311yescharacteristic
111C23×C83C2×D411yescharacteristic
112C23×C83C2×D411yescharacteristic
113C22×D83—21no—
114C22×D83—21no—
115C22×D83—42no—
116C22×D83—42no—
117C22×D83—81no—
118Q8×C83—21no—
119Q8×C83—21no—
120Q8×C83—42no—
121D4×C83—41no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×C4×C83—21no—
125C2×C4×C83—21no—
126C2×C4×C83—21no—
127C2×C4×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83C2×D411yescharacteristic
131C2×C4×C83C2×D411yescharacteristic
132C2×C4×C83C2×D411yescharacteristic
133C2×C4×C83C2×D411yescharacteristic
134C2×C4×C83C2411yescharacteristic
135C22⋊D83—42no—
136C22⋊D83—81no—
137C2×Dic83—21no—
138C2×Dic83—21no—
139D332—21no—
140D332—21no—
141D332—21no—
142D332—42no—
143D332#4112yes—
144C4×D83—42no—
145C4×D83—81no—
146Dic166—21no—
147Dic166—21no—
148C83⋊C8—21no—
149C83⋊C8—21no—
150C83⋊C8—21no—
151C83⋊C8—21no—
152C22×C83—21no—
153C22×C83—21no—
154C22×C83—21no—
155C22×C83—21no—
156C22×C83—21no—
157C22×C83—21no—
158C22×C83C22×D411yescharacteristic
159C2×D83—22no—
160C2×D83—41no—
161C2×D83—41no—
162C2×D83—41no—
163C2×D83—41no—
164C2×D83—41no—
165C2×D83—84no—
166C4×C83—21no—
167C4×C83—21no—
168C4×C83—21no—
169C4×C83—21no—
170C4×C83—41no—
171C4×C83—41no—
172C4×C83—41no—
173C4×C83—41no—
174C4×C83C2×#4111yescharacteristic
175C4×C83H(5,𝔽2)11yescharacteristic
176Dic83—41no—
177Dic83—41no—
178C2×C83—41no—
179C2×C83—41no—
180C2×C83#54611yescharacteristic
181C2×C83#57511yescharacteristic
182C2×C83#57711yescharacteristic
183D83—42no—
184D83—81no—
185#3311—831nomaximal, Sylow
186C83#331111yescharacteristic, complemented, Sylow
187#556D8311yescharacteristic, complemented
188#546—831no—
189#489—831no—
190#485—831no—
191#482—831no—
192#476—831no—
193#470—831no—
194#465—831no—
195#464—831no—
196#452—831no—
197#450—831no—
198#440—831no—
199#435—831no—
200#418—831no—
201#416—831no—
202C2×#41—831no—
203C22×D4—831no—
204C2×Q16—831no—
205C2×SD16—831no—
206C2×SD16—831no—
207C2×M16—831no—
208C22×C8—831no—
209C4⋊Q8C2×D8311yescharacteristic, complemented
210#126—21no—
211#126—21no—
212#124—21no—
213#124—831no—
214#123—1661no—
215C22⋊Q8—831no—
216C22⋊Q8C2×D8311yescharacteristic
217C22⋊Q8C2×D8311yescharacteristic, complemented
218#121—831no—
219#121—1661no—
220#120—1661no—
221C4×Q8—21no—
222C4×Q8C2×D8311yescharacteristic, complemented
223C4×D4—831no—
224C4×D4C2×D8311yescharacteristic
225#117C2×D8311yescharacteristic
226#117C2×D8311yescharacteristic, complemented
227C2×C22⋊C4—1661no—
228#107—831no—
229#106—831no—
230#106—831no—
231C4⋊C8—831no—
232#103—831no—
233#103—831no—
234#103—831no—
235#103—831no—
236#102—831no—
237#102—831no—
238#102—831no—
239#102—831no—
240#102—831no—
241#102—831no—
242C22⋊C8—831no—
243C22⋊C8—831no—
244#97—831no—
245C24—1661no—
246#41—3321no—
247#41—1662no—
248C2×Q8—21no—
249C2×Q8—831no—
250C2×Q8C22×D8311yescharacteristic
251C2×D4—3321no—
252C2×D4—831no—
253C2×D4—831no—
254C2×D4—1662no—
255C2×D4—1661no—
256C2×D4—1661no—
257C2×D4—1661no—
258C2×D4C22⋊D8311yescharacteristic
259C22×C4—1661no—
260C22×C4—1661no—
261C22×C4C22×D8311yescharacteristic
262C22×C4C22×D8311yescharacteristic
263Q16—3321no—
264SD16—3321no—
265SD16—1662no—
266M16—3321no—
267C2×C8—3321no—
268C2×C8—831no—
269C2×C8—831no—
270C2×C8—831no—
271C2×C8—831no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—831no—
277C4⋊C4—1661no—
278C4⋊C4C22⋊D8311yescharacteristic
279C4⋊C4C22×D8311yescharacteristic
280C4⋊C4C22×D8311yescharacteristic
281C4⋊C4C22×D8311yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—1661no—
287C22⋊C4—1661no—
288C22⋊C4—1661no—
289C22⋊C4—1661no—
290C22⋊C4—1661no—
291C22⋊C4C22⋊D8311yescharacteristic
292C22⋊C4C22⋊D8311yescharacteristic
293C42—21no—
294C42—21no—
295C42—21no—
296C42C22×D8311yescharacteristic
297C23—3322no—
298C23—3322no—
299C23—6641no—
300C23—1661no—
301C23—1661no—
302C23C2×C22⋊D8311yescharacteristic
303C23C2×C22⋊D8311yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—42no—
307Q8—1661no—
308Q8—1661no—
309D4—41no—
310D4—3322no—
311D4—3322no—
312D4—832no—
313D4—6641no—
314D4—1661no—
315D4—1661no—
316D4—1661no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—3322no—
326C2×C4—6641no—
327C2×C4—1661no—
328C2×C4—1661no—
329C2×C4C2×C22⋊D8311yescharacteristic
330C2×C4C2×C22⋊D8311yescharacteristic
331C2×C4C2×C22⋊D8311yescharacteristic
332C2×C4C2×C22⋊D8311yescharacteristic
333C2×C4C23×D8311yescharacteristic
334C8—1661no—
335C8—1661no—
336C8—1661no—
337C8—1661no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—3321no—
345C22—3321no—
346C22—3321no—
347C22—3321no—
348C22—3321no—
349C22—6644no—
350C22—1662no—
351C22C22×C22⋊D8311yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—3321no—
361C4—3321no—
362C4#41×D8311yescharacteristic
363C4#1458183811yescharacteristic
364C2—41no—
365C2—41no—
366C2—3322no—
367C2—6641no—
368C2#1912833911yescharacteristic
369C2#1912849911yescharacteristic
370C2#1912850911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#556×C83C2
socleC22×C83C22×D4
prime radical#19128347C2
hypercenter#556D83
hypocenterC83#3311
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ #556

Complements

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

NH ≅ G/Ntype
2. #19128063C2semidirect
3. #19128042C2semidirect
4. #19128025C2semidirect
5. #19127988C2semidirect
6. #19127957C2semidirect
7. #19127932C2semidirect
8. #19127923C2semidirect
9. #19127861C2semidirect
10. #19127851C2semidirect
11. #19127799C2semidirect
12. #19127773C2semidirect
13. #19127705C2semidirect
14. #19127695C2semidirect
15. #556×C83C2semidirect
18. C2×#13050580C22semidirect
22. C22×C83⋊C8C22semidirect
31. #14581725C22semidirect
32. #14581723C22semidirect
33. #14581723C22semidirect
34. C4⋊(C83⋊C8)C22semidirect
35. #14581716C22semidirect
36. #14581715C22semidirect
37. #14581715C22semidirect
38. #14581715C22semidirect
46. C22⋊(C83⋊C8)C22semidirect
47. #14581703C22semidirect
48. C4⋊Q8×C83C22semidirect
53. C22⋊Q8×C83C22semidirect
55. C4×Q8×C83C22semidirect
58. #117×C83C22semidirect
186. C83#3311semidirect
187. #556D83semidirect
209. C4⋊Q8C2×D83semidirect
217. C22⋊Q8C2×D83semidirect
222. C4×Q8C2×D83semidirect
226. #117C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191283472
#191280632
#191280422
#191280252
#191279882
#191279572
#191279322
#191279232
#191278612
#191278512
#191277992
#191277732
#191277052
#191276952
#556×C832
#331183