Finite Group Databasebeta

← #24919504

Subgroups — #24919504

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#19128424C211yesmaximal, characteristic, complemented
2#19128346C211yesmaximal, characteristic
3#19128061C211yesmaximal, characteristic, complemented
4#19128044C211yesmaximal, characteristic, complemented
5#19128028C211yesmaximal, characteristic, complemented
6#19127991C211yesmaximal, characteristic, complemented
7#19127958C211yesmaximal, characteristic, complemented
8#19127933C211yesmaximal, characteristic, complemented
9#19127923C211yesmaximal, characteristic, complemented
10#19127860C211yesmaximal, characteristic, complemented
11#19127850C211yesmaximal, characteristic, complemented
12#19127774C211yesmaximal, characteristic, complemented
13#19127704C211yesmaximal, characteristic, complemented
14#19127694C211yesmaximal, characteristic, complemented
15#440×C83C211yesmaximal, characteristic, complemented
16C2×#13050589C2211yescharacteristic
17C2×D4×D83C2211yescharacteristic
18C2×#13050580C2211yescharacteristic, complemented
19C2×#13050577C2211yescharacteristic
20C2×#13050573C2211yescharacteristic
21C2×C8×D83C2211yescharacteristic, complemented
22#14581792C2211yescharacteristic, complemented
23#14581787—21no—
24#14581787—21no—
25#14581780—21no—
26#14581780C2211yescharacteristic
27#14581776—21no—
28#14581772C2211yescharacteristic
29#14581768C2211yescharacteristic
30#14581768C2211yescharacteristic, complemented
31#14581767C2211yescharacteristic
32#14581766—21no—
33#14581762—21no—
34C4×Dic166—21no—
35C4×D332C2211yescharacteristic
36#14581750C2211yescharacteristic
37#14581750C2211yescharacteristic, complemented
38C2×(C2×D83)⋊C4—21no—
39#14581723C2211yescharacteristic, complemented
40#14581723C2211yescharacteristic, complemented
41C4⋊(C83⋊C8)C2211yescharacteristic, complemented
42#14581717C2211yescharacteristic, complemented
43#14581717C2211yescharacteristic, complemented
44#14581715C2211yescharacteristic, complemented
45#14581714C2211yescharacteristic
46#14581714C2211yescharacteristic
47#14581713C2211yescharacteristic
48#14581713C2211yescharacteristic
49#14581712C2211yescharacteristic
50(C2×D83)⋊C8C2211yescharacteristic
51(C2×D83)⋊C8C2211yescharacteristic, complemented
52C2×SD16×C83C2211yescharacteristic
53C4×Q8×C83C2211yescharacteristic, complemented
54C4×D4×C83C2211yescharacteristic
55#107×C83C2211yescharacteristic, complemented
56#103×C83C2211yescharacteristic, complemented
57#102×C83C2211yescharacteristic
58#97×C83C2211yescharacteristic, complemented
59C23×D83—21no—
60#13050589—22no—
61#13050589—41no—
62C2×Dic166—21no—
63C2×Dic166C2311yescharacteristic
64C2×D332—21no—
65C2×D332D411yescharacteristic
66C2×D332C2311yescharacteristic
67C2×C22⋊D83—21no—
68C2×C22⋊D83—21no—
69D4×D83—22no—
70D4×D83—41no—
71C22×Dic83—21no—
72C2×C4×D83C2311yescharacteristic
73C2×C4×D83C2311yescharacteristic
74#13050580—41no—
75#13050577—41no—
76#13050573—41no—
77C2×C83⋊C8C2311yescharacteristic
78C2×C83⋊C8C2311yescharacteristic
79C8×D83—41no—
80C4⋊Dic83—21no—
81C4⋊Dic83—21no—
82C4⋊Dic83D411yescharacteristic
83C4⋊Dic83C2311yescharacteristic
84C4⋊Dic83C2311yescharacteristic
85Dic83⋊C4—21no—
86Dic83⋊C4—21no—
87(C2×D83)⋊C4—21no—
88(C2×D83)⋊C4—21no—
89(C2×D83)⋊C4—21no—
90(C2×D83)⋊C4—21no—
91(C2×D83)⋊C4—21no—
92(C2×D83)⋊C4—21no—
93(C2×D83)⋊C4—21no—
94(C2×D83)⋊C4—21no—
95(C2×D83)⋊C4D411yescharacteristic
96(C2×D83)⋊C4D411yescharacteristic
97C4×Dic83—21no—
98C4×Dic83—21no—
99C2×Q8×C83C2311yescharacteristic
100C2×D4×C83C2311yescharacteristic
101C22×C4×C83—21no—
102SD16×C83—22no—
103C2×C8×C83C2311yescharacteristic
104C2×C8×C83C2311yescharacteristic
105C4⋊C4×C83—21no—
106C4⋊C4×C83C2311yescharacteristic
107C4⋊C4×C83C2311yescharacteristic
108C22⋊C4×C83—21no—
109C42×C83—21no—
110C42×C83C2311yescharacteristic
111C23×C83—21no—
112C22×D83—21no—
113C22×D83—42no—
114C22×D83—42no—
115C22×D83—81no—
116C22×D83C2×D411yescharacteristic
117C22×D83C2×D411yescharacteristic
118Q8×C83—21no—
119Q8×C83—21no—
120D4×C83—21no—
121D4×C83#4112yes—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×C4×C83—21no—
125C2×C4×C83—81no—
126C2×C4×C83C2×D411yescharacteristic
127C2×C4×C83C2×D411yescharacteristic
128C2×C4×C83C2411yescharacteristic
129C22⋊D83—42no—
130C22⋊D83—42no—
131C2×Dic83—21no—
132C2×Dic83—21no—
133C2×Dic83—21no—
134C2×Dic83—42no—
135C2×Dic83C2×D411yescharacteristic
136C2×Dic83C2×D411yescharacteristic
137D332—21no—
138D332—21no—
139D332—41no—
140D332—81no—
141C4×D83—21no—
142C4×D83—21no—
143C4×D83—21no—
144C4×D83—21no—
145Dic166—21no—
146Dic166—21no—
147Dic166—42no—
148C8×C83—21no—
149C8×C83—21no—
150C83⋊C8—21no—
151C83⋊C8—21no—
152C22×C83—22no—
153C22×C83—41no—
154C22×C83—41no—
155C22×C83C22×D411yescharacteristic
156C2×D83—21no—
157C2×D83—21no—
158C2×D83—21no—
159C2×D83—21no—
160C2×D83—21no—
161C2×D83—21no—
162C2×D83—41no—
163C2×D83—41no—
164C2×D83—41no—
165C2×D83—84no—
166C4×C83—21no—
167C4×C83—21no—
168C4×C83—41no—
169C4×C83—41no—
170C4×C83—41no—
171C4×C83C2×#4111yescharacteristic
172C4×C83H(5,𝔽2)11yescharacteristic
173Dic83—21no—
174Dic83—21no—
175Dic83—41no—
176Dic83—41no—
177Dic83—41no—
178C2×C83—42no—
179C2×C83#54611yescharacteristic
180C2×C83#57511yescharacteristic
181C2×C83#57711yescharacteristic
182D83—41no—
183D83—41no—
184D83—81no—
185#3311—831nomaximal, Sylow
186C83#331111yescharacteristic, complemented, Sylow
187#556—831no—
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#440D8311yescharacteristic, complemented
199#435—831no—
200#418—831no—
201#416—831no—
202C2×#41—831no—
203C22×D4—831no—
204C2×Q16—831no—
205C2×SD16—831no—
206C2×SD16C2×D8311yescharacteristic
207C2×M16—831no—
208C22×C8—831no—
209C4⋊Q8—831no—
210#126—1661no—
211#126—1661no—
212#124—831no—
213#124—1661no—
214#123—1661no—
215C22⋊Q8—831no—
216C22⋊Q8—831no—
217C22⋊Q8—831no—
218#121—831no—
219#121—1661no—
220#120—1661no—
221C4×Q8—1661no—
222C4×Q8C2×D8311yescharacteristic, complemented
223C4×D4—831no—
224C4×D4C2×D8311yescharacteristic
225#117—831no—
226#117—831no—
227C2×C22⋊C4—1661no—
228#107C2×D8311yescharacteristic, complemented
229#106—831no—
230#106—831no—
231C4⋊C8—831no—
232#103—831no—
233#103—831no—
234#103—831no—
235#103C2×D8311yescharacteristic, complemented
236#102—831no—
237#102—831no—
238#102—831no—
239#102—831no—
240#102—831no—
241#102C2×D8311yescharacteristic
242C22⋊C8—831no—
243C22⋊C8—831no—
244#97C2×D8311yescharacteristic, complemented
245C24—1661no—
246#41—3321no—
247#41—1662no—
248C2×Q8—831no—
249C2×Q8—1661no—
250C2×Q8C22×D8311yescharacteristic
251C2×D4—3321no—
252C2×D4—831no—
253C2×D4—831no—
254C2×D4—1661no—
255C2×D4—1661no—
256C2×D4—1661no—
257C2×D4—1662no—
258C2×D4C22×D8311yescharacteristic
259C22×C4—21no—
260C22×C4—831no—
261C22×C4—831no—
262C22×C4—1661no—
263Q16—3321no—
264SD16—22no—
265SD16—3321no—
266M16—3321no—
267C2×C8—3321no—
268C2×C8—831no—
269C2×C8—831no—
270C2×C8C22×D8311yescharacteristic
271C2×C8C22×D8311yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—831no—
274C4⋊C4—831no—
275C4⋊C4—831no—
276C4⋊C4—1661no—
277C4⋊C4—1661no—
278C4⋊C4—1661no—
279C4⋊C4—1661no—
280C4⋊C4C22×D8311yescharacteristic
281C4⋊C4C22×D8311yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—831no—
284C22⋊C4—831no—
285C22⋊C4—1661no—
286C22⋊C4—1661no—
287C22⋊C4—1661no—
288C22⋊C4—1661no—
289C22⋊C4—1661no—
290C22⋊C4—1661no—
291C22⋊C4—1661no—
292C22⋊C4—1661no—
293C42—21no—
294C42—1661no—
295C42—1661no—
296C42C22×D8311yescharacteristic
297C23—21no—
298C23—3322no—
299C23—3322no—
300C23—831no—
301C23—831no—
302C23—6641no—
303C23—1661no—
304Q8—21no—
305Q8—21no—
306Q8—3322no—
307Q8—1661no—
308Q8—1661no—
309D4—21no—
310D4—3321no—
311D4—3322no—
312D4—3322no—
313D4—6641no—
314D4—1661no—
315D4—1661no—
316D4#1305059012yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—81no—
321C2×C4—3322no—
322C2×C4—831no—
323C2×C4—831no—
324C2×C4—1661no—
325C2×C4—1661no—
326C2×C4—1661no—
327C2×C4—1661no—
328C2×C4—1661no—
329C2×C4—1661no—
330C2×C4—1661no—
331C2×C4D4×D8311yescharacteristic
332C2×C4D4×D8311yescharacteristic
333C2×C4C23×D8311yescharacteristic
334C8—21no—
335C8—21no—
336C8—1661no—
337C8—1661no—
338C22—22no—
339C22—41no—
340C22—41no—
341C22—3321no—
342C22—3321no—
343C22—3321no—
344C22—6644no—
345C22—1661no—
346C22—1661no—
347C22—1661no—
348C22—1661no—
349C22—1661no—
350C22—1661no—
351C22C2×D4×D8311yescharacteristic
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—3321no—
358C4—3321no—
359C4—3321no—
360C4—1661no—
361C4—1661no—
362C4C2×#1305059011yescharacteristic
363C4#1458183811yescharacteristic
364C2—42no—
365C2—3321no—
366C2—3321no—
367C2—6641no—
368C2#1912834711yescharacteristic
369C2#1912850011yescharacteristic
370C2#1912851111yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#440×C83C2
socleC22×C83C22×D4
prime radical#19128346C2
hypercenter#440D83
hypocenterC83#3311
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #19128424C2semidirect
3. #19128061C2semidirect
4. #19128044C2semidirect
5. #19128028C2semidirect
6. #19127991C2semidirect
7. #19127958C2semidirect
8. #19127933C2semidirect
9. #19127923C2semidirect
10. #19127860C2semidirect
11. #19127850C2semidirect
12. #19127774C2semidirect
13. #19127704C2semidirect
14. #19127694C2semidirect
15. #440×C83C2semidirect
18. C2×#13050580C22semidirect
21. C2×C8×D83C22semidirect
22. #14581792C22semidirect
30. #14581768C22semidirect
37. #14581750C22semidirect
39. #14581723C22semidirect
40. #14581723C22semidirect
41. C4⋊(C83⋊C8)C22semidirect
42. #14581717C22semidirect
43. #14581717C22semidirect
44. #14581715C22semidirect
51. (C2×D83)⋊C8C22semidirect
53. C4×Q8×C83C22semidirect
55. #107×C83C22semidirect
56. #103×C83C22semidirect
58. #97×C83C22semidirect
186. C83#3311semidirect
198. #440D83semidirect
222. C4×Q8C2×D83semidirect
228. #107C2×D83semidirect
235. #103C2×D83semidirect
244. #97C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191284242
#191283462
#191280612
#191280442
#191280282
#191279912
#191279582
#191279332
#191279232
#191278602
#191278502
#191277742
#191277042
#191276942
#440×C832
#331183