Finite Group Databasebeta

← #24918555

Subgroups — #24918555

Order 10624.

408 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128318C211yesmaximal, characteristic, complemented
2#19128265C211yesmaximal, characteristic, complemented
3#19128089C211yesmaximal, characteristic, complemented
4#19128082C211yesmaximal, characteristic, complemented
5#19128062C211yesmaximal, characteristic, complemented
6#19128046C211yesmaximal, characteristic, complemented
7#19127926C211yesmaximal, characteristic, complemented
8#19127918C211yesmaximal, characteristic, complemented
9#19127900C211yesmaximal, characteristic, complemented
10#19127894C211yesmaximal, characteristic, complemented
11#19127863C211yesmaximal, characteristic, complemented
12#19127854C211yesmaximal, characteristic, complemented
13#19127834C211yesmaximal, characteristic, complemented
14#19127759C211yesmaximal, characteristic, complemented
15#449×C83C211yesmaximal, characteristic, complemented
16C22×D332C2211yescharacteristic
17C2×#13050580C2211yescharacteristic, complemented
18C2×#13050579C2211yescharacteristic
19C2×#13050579C2211yescharacteristic
20C2×#13050578C2211yescharacteristic, complemented
21C2×#13050575C2211yescharacteristic
22C2×#13050575C2211yescharacteristic
23#14581791C2211yescharacteristic, complemented
24#14581788C2211yescharacteristic
25#14581788C2211yescharacteristic
26#14581785—21no—
27#14581781C2211yescharacteristic
28#14581781C2211yescharacteristic
29#14581779—21no—
30#14581779C2211yescharacteristic, complemented
31#14581773—21no—
32#14581771C2211yescharacteristic
33#14581769—21no—
34#14581766—21no—
35#14581766—21no—
36#14581760—21no—
37#14581760—21no—
38#14581757C2211yescharacteristic, complemented
39D4×Dic83C2211yescharacteristic, complemented
40#14581751C2211yescharacteristic
41Dic83⋊C8C2211yescharacteristic, complemented
42Dic83⋊C8C2211yescharacteristic, complemented
43#14581716C2211yescharacteristic, complemented
44#14581714C2211yescharacteristic
45#14581714C2211yescharacteristic
46#14581713C2211yescharacteristic, complemented
47#14581712C2211yescharacteristic
48#14581712C2211yescharacteristic
49C22⋊(C83⋊C8)C2211yescharacteristic
50#14581702C2211yescharacteristic, complemented
51C8×Dic83C2211yescharacteristic, complemented
52C2×#41×C83C2211yescharacteristic
53C2×SD16×C83C2211yescharacteristic, complemented
54C2×D8×C83C2211yescharacteristic, complemented
55#121×C83C2211yescharacteristic
56#103×C83C2211yescharacteristic, complemented
57#102×C83C2211yescharacteristic, complemented
58C22⋊C8×C83C2211yescharacteristic
59C23×D83—21no—
60C2×Dic166—21no—
61C2×Dic166C2311yescharacteristic
62C2×D332—21no—
63C2×D332—41no—
64C2×D332—41no—
65C2×D332C2311yescharacteristic
66C2×D332C2311yescharacteristic
67C2×C22⋊D83—21no—
68C2×C22⋊D83—21no—
69C2×C22⋊D83—21no—
70C2×C22⋊D83—21no—
71C22×Dic83—21no—
72#13050580—41no—
73#13050579—41no—
74#13050579—41no—
75#13050578—41no—
76#13050575—41no—
77#13050575—41no—
78C2×C83⋊C8C2311yescharacteristic
79C2×C83⋊C8C2311yescharacteristic
80C4⋊Dic83D411yescharacteristic
81C4⋊Dic83D411yescharacteristic
82C4⋊Dic83C2311yescharacteristic
83Dic83⋊C4—21no—
84Dic83⋊C4—21no—
85Dic83⋊C4—21no—
86C22⋊Dic83—21no—
87C22⋊Dic83—21no—
88C22⋊Dic83—21no—
89C22⋊Dic83—21no—
90C22⋊Dic83D411yescharacteristic
91C22⋊Dic83D411yescharacteristic
92(C2×D83)⋊C4—21no—
93(C2×D83)⋊C4—21no—
94(C2×D83)⋊C4—21no—
95(C2×D83)⋊C4—21no—
96C4×Dic83—21no—
97C4×Dic83C2311yescharacteristic
98C4×Dic83C2311yescharacteristic
99#41×C83—41no—
100#41×C83—41no—
101C2×Q8×C83C2311yescharacteristic
102C2×D4×C83—21no—
103C2×D4×C83—21no—
104C2×D4×C83C2311yescharacteristic
105C2×D4×C83C2311yescharacteristic
106C22×C4×C83—21no—
107C22×C4×C83C2311yescharacteristic
108SD16×C83—41no—
109D8×C83—41no—
110C2×C8×C83C2311yescharacteristic
111C2×C8×C83C2311yescharacteristic
112C4⋊C4×C83C2311yescharacteristic
113C22⋊C4×C83—21no—
114C23×C83—21no—
115C23×C83—21no—
116C23×C83C2×D411yescharacteristic
117C22×D83—21no—
118C22×D83—21no—
119C22×D83—42no—
120C22×D83—81no—
121C22×D83—81no—
122Q8×C83—21no—
123Q8×C83—21no—
124D4×C83—21no—
125D4×C83—21no—
126D4×C83—21no—
127D4×C83—21no—
128D4×C83—42no—
129D4×C83—81no—
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—81no—
138C22⋊D83—81no—
139C22⋊D83—81no—
140C22⋊D83—81no—
141C2×Dic83—21no—
142C2×Dic83—21no—
143C2×Dic83—81no—
144C2×Dic83C2×D411yescharacteristic
145C2×Dic83C2×D411yescharacteristic
146C2×Dic83C2×D411yescharacteristic
147C2×Dic83C2×D411yescharacteristic
148D332—21no—
149D332—21no—
150D332—21no—
151D332—21no—
152D332—42no—
153Dic166—21no—
154Dic166—21no—
155Dic166—81no—
156C8×C83—21no—
157C8×C83—21no—
158C83⋊C8—21no—
159C83⋊C8—21no—
160C22×C83—21no—
161C22×C83—21no—
162C22×C83—21no—
163C22×C83—41no—
164C22×C83—41no—
165C22×C83—41no—
166C22×C83—41no—
167C22×C83—41no—
168C22×C83—41no—
169C22×C83C22×D411yescharacteristic
170C2×D83—41no—
171C2×D83—41no—
172C2×D83—41no—
173C2×D83—41no—
174C2×D83—41no—
175C2×D83—41no—
176C2×D83—84no—
177C4×C83—21no—
178C4×C83—41no—
179C4×C83—41no—
180C4×C83H(5,𝔽2)11yescharacteristic
181C4×C83H(5,𝔽2)11yescharacteristic
182Dic83—21no—
183Dic83—21no—
184Dic83—21no—
185Dic83—21no—
186Dic83—41no—
187Dic83—41no—
188C2×C83—41no—
189C2×C83—81no—
190C2×C83—81no—
191C2×C83#53511yescharacteristic
192C2×C83#57611yescharacteristic
193C2×C83#57711yescharacteristic
194D83—81no—
195D83—81no—
196#3224—831nomaximal, Sylow
197C83#322411yescharacteristic, complemented, Sylow
198#542—831no—
199#535—831no—
200#496—831no—
201#495—831no—
202#489—831no—
203#486—831no—
204#464—831no—
205#463—831no—
206#460—831no—
207#459—831no—
208#452—831no—
209#450—831no—
210#449D8311yescharacteristic, complemented
211#447—831no—
212#433—831no—
213C2×#41C2×D8311yescharacteristic
214C22×D4—831no—
215C2×Q16—831no—
216C2×SD16—831no—
217C2×SD16—831no—
218C2×SD16—831no—
219C2×SD16C2×D8311yescharacteristic, complemented
220C2×D8—831no—
221C2×D8—831no—
222C2×D8C2×D8311yescharacteristic, complemented
223C4⋊Q8—831no—
224#127—831no—
225#127—831no—
226#126—1661no—
227#124—831no—
228#124—831no—
229#124—831no—
230#124—1661no—
231#123—1661no—
232C22⋊Q8—831no—
233C22⋊Q8—1661no—
234#121—1661no—
235#121—1661no—
236#121C2×D8311yescharacteristic
237#120—1661no—
238#120—1661no—
239C4×Q8—831no—
240C4×D4—831no—
241#117—831no—
242C4⋊C8—831no—
243C4⋊C8—831no—
244#103—831no—
245#103C2×D8311yescharacteristic, complemented
246#102—831no—
247#102—831no—
248#102—831no—
249#102—831no—
250#102—831no—
251#102C2×D8311yescharacteristic, complemented
252C22⋊C8—831no—
253C22⋊C8C2×D8311yescharacteristic
254#97—831no—
255C4×C8—831no—
256C24—1661no—
257#41—41no—
258#41—41no—
259C2×Q8—831no—
260C2×Q8—1661no—
261C2×Q8C22×D8311yescharacteristic
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—3321no—
265C2×D4—3321no—
266C2×D4—831no—
267C2×D4—831no—
268C2×D4—1661no—
269C2×D4—1661no—
270C2×D4—1661no—
271C2×D4—1661no—
272C2×D4—1661no—
273C2×D4C22×D8311yescharacteristic
274C2×D4C22×D8311yescharacteristic
275C22×C4—21no—
276C22×C4—1661no—
277C22×C4C22×D8311yescharacteristic
278Q16—3321no—
279SD16—41no—
280SD16—3321no—
281SD16—3321no—
282SD16—3321no—
283D8—41no—
284D8—3321no—
285D8—3321no—
286C2×C8—831no—
287C2×C8—831no—
288C2×C8C22×D8311yescharacteristic
289C2×C8C22×D8311yescharacteristic
290C4⋊C4—831no—
291C4⋊C4—831no—
292C4⋊C4—831no—
293C4⋊C4—1661no—
294C4⋊C4—1661no—
295C4⋊C4—1661no—
296C4⋊C4C22×D8311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—831no—
299C22⋊C4—831no—
300C22⋊C4—1661no—
301C22⋊C4—1661no—
302C22⋊C4—1661no—
303C22⋊C4—1661no—
304C22⋊C4—1661no—
305C22⋊C4—1661no—
306C22⋊C4—1661no—
307C22⋊C4—1661no—
308C42—831no—
309C42—831no—
310C42—1661no—
311C23—21no—
312C23—21no—
313C23—3322no—
314C23—6641no—
315C23—6641no—
316C23—1661no—
317C23—1661no—
318C23D4×D8311yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—6641no—
322Q8—1661no—
323Q8—1661no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—21no—
328D4—42no—
329D4—81no—
330D4—3322no—
331D4—6641no—
332D4—6641no—
333D4—6641no—
334D4—6641no—
335D4—1661no—
336D4—1661no—
337D4—1661no—
338D4—1661no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—42no—
344C2×C4—831no—
345C2×C4—831no—
346C2×C4—831no—
347C2×C4—831no—
348C2×C4—6641no—
349C2×C4—1661no—
350C2×C4—1661no—
351C2×C4D4×D8311yescharacteristic
352C2×C4C23×D8311yescharacteristic
353C8—21no—
354C8—21no—
355C8—1661no—
356C8—1661no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—3321no—
367C22—3321no—
368C22—3321no—
369C22—3321no—
370C22—3321no—
371C22—3321no—
372C22—6644no—
373C22C2×D4×D8311yescharacteristic
374C4—21no—
375C4—41no—
376C4—41no—
377C4—3321no—
378C4—3321no—
379C4—1661no—
380C4—1661no—
381C4—1661no—
382C4—1661no—
383C4#1458183711yescharacteristic
384C4#1458183711yescharacteristic
385C2—41no—
386C2—81no—
387C2—81no—
388C2—6641no—
389C2—6641no—
390C2#1912826611yescharacteristic
391C2#1912850511yescharacteristic
392C2#1912850811yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#449×C83C2
socleC22×C83C22×D4
hypercenter#449D83
hypocenterC83#3224
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #19128318C2semidirect
2. #19128265C2semidirect
3. #19128089C2semidirect
4. #19128082C2semidirect
5. #19128062C2semidirect
6. #19128046C2semidirect
7. #19127926C2semidirect
8. #19127918C2semidirect
9. #19127900C2semidirect
10. #19127894C2semidirect
11. #19127863C2semidirect
12. #19127854C2semidirect
13. #19127834C2semidirect
14. #19127759C2semidirect
15. #449×C83C2semidirect
17. C2×#13050580C22semidirect
20. C2×#13050578C22semidirect
23. #14581791C22semidirect
30. #14581779C22semidirect
38. #14581757C22semidirect
39. D4×Dic83C22semidirect
41. Dic83⋊C8C22semidirect
42. Dic83⋊C8C22semidirect
43. #14581716C22semidirect
46. #14581713C22semidirect
50. #14581702C22semidirect
51. C8×Dic83C22semidirect
53. C2×SD16×C83C22semidirect
54. C2×D8×C83C22semidirect
56. #103×C83C22semidirect
57. #102×C83C22semidirect
197. C83#3224semidirect
210. #449D83semidirect
219. C2×SD16C2×D83semidirect
222. C2×D8C2×D83semidirect
245. #103C2×D83semidirect
251. #102C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191283182
#191282652
#191280892
#191280822
#191280622
#191280462
#191279262
#191279182
#191279002
#191278942
#191278632
#191278542
#191278342
#191277592
#449×C832
#322483