Finite Group Databasebeta

← #24919546

Subgroups — #24919546

Order 10624.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128408C211yesmaximal, characteristic, complemented
2#19128347C211yesmaximal, characteristic
3#19128061C211yesmaximal, characteristic, complemented
4#19128037C211yesmaximal, characteristic, complemented
5#19128015C211yesmaximal, characteristic, complemented
6#19127982C211yesmaximal, characteristic, complemented
7#19127947C211yesmaximal, characteristic, complemented
8#19127942C211yesmaximal, characteristic, complemented
9#19127924C211yesmaximal, characteristic, complemented
10#19127865C211yesmaximal, characteristic, complemented
11#19127854C211yesmaximal, characteristic, complemented
12#19127803C211yesmaximal, characteristic, complemented
13#19127703C211yesmaximal, characteristic, complemented
14C2×#14581714C211yesmaximal, characteristic, complemented
15#435×C83C211yesmaximal, characteristic, complemented
16C2×#13050590C2211yescharacteristic
17C22×D332C2211yescharacteristic
18C2×Dic332C2211yescharacteristic, complemented
19C2×#13050579C2211yescharacteristic
20C2×#13050579C2211yescharacteristic
21#14581792C2211yescharacteristic, complemented
22#14581783—21no—
23#14581780C2211yescharacteristic
24#14581777—21no—
25#14581771C2211yescharacteristic
26#14581771C2211yescharacteristic
27#14581771C2211yescharacteristic, complemented
28#14581769—21no—
29#14581769—21no—
30#14581766C2211yescharacteristic
31#14581765—21no—
32#14581760—21no—
33C4×Dic166C2211yescharacteristic, complemented
34C4×D332C2211yescharacteristic
35D4×Dic83—21no—
36C2×C4⋊Dic83C2211yescharacteristic
37C2×C4⋊Dic83C2211yescharacteristic, complemented
38C22⋊C4×D83—21no—
39#14581726C2211yescharacteristic, complemented
40#14581726C2211yescharacteristic, complemented
41#14581726C2211yescharacteristic, complemented
42#14581717C2211yescharacteristic, complemented
43#14581717C2211yescharacteristic, complemented
44#14581717C2211yescharacteristic, complemented
45#14581717C2211yescharacteristic, complemented
46#14581714C2211yescharacteristic
47#14581714C2211yescharacteristic
48#14581714C2211yescharacteristic
49#14581714C2211yescharacteristic
50#14581714C2211yescharacteristic
51#14581714C2211yescharacteristic
52C2×M16×C83C2211yescharacteristic, complemented
53C22×C8×C83C2211yescharacteristic
54C4×D4×C83C2211yescharacteristic
55C4⋊C8×C83C2211yescharacteristic, complemented
56C22⋊C8×C83C2211yescharacteristic
57C22⋊C8×C83C2211yescharacteristic, complemented
58#97×C83C2211yescharacteristic, complemented
59C23×D83—21no—
60#13050590—22no—
61#13050590—41no—
62C2×Dic166—21no—
63C2×Dic166C2311yescharacteristic
64C2×Dic166C2311yescharacteristic
65C2×D332—22no—
66C2×D332—21no—
67C2×D332—41no—
68C2×D332C2311yescharacteristic
69C2×D332C2311yescharacteristic
70C2×C22⋊D83—21no—
71C2×C22⋊D83—21no—
72C22×Dic83—21no—
73C22×Dic83—21no—
74C2×C4×D83—21no—
75C2×C4×D83—21no—
76Dic332—41no—
77#13050579—22no—
78#13050579—41no—
79C4⋊Dic83—21no—
80C4⋊Dic83—21no—
81C4⋊Dic83—21no—
82C4⋊Dic83C2311yescharacteristic
83C4⋊Dic83C2311yescharacteristic
84C4⋊Dic83C2311yescharacteristic
85C4⋊Dic83C2311yescharacteristic
86Dic83⋊C4—21no—
87Dic83⋊C4—21no—
88Dic83⋊C4—21no—
89Dic83⋊C4—21no—
90C22⋊Dic83—21no—
91C22⋊Dic83—21no—
92C22⋊Dic83—21no—
93(C2×D83)⋊C4—21no—
94(C2×D83)⋊C4—21no—
95(C2×D83)⋊C4—21no—
96(C2×D83)⋊C4—21no—
97C4×Dic83—21no—
98C2×D4×C83D411yescharacteristic
99C22×C4×C83C2311yescharacteristic
100C22×C4×C83C2311yescharacteristic
101M16×C83—41no—
102C2×C8×C83—41no—
103C2×C8×C83C2311yescharacteristic
104C2×C8×C83C2311yescharacteristic
105C2×C8×C83C2311yescharacteristic
106C2×C8×C83C2311yescharacteristic
107C4⋊C4×C83D411yescharacteristic
108C22⋊C4×C83D411yescharacteristic
109C22⋊C4×C83D411yescharacteristic
110C42×C83C2311yescharacteristic
111C23×C83C2×D411yescharacteristic
112C23×C83C2×D411yescharacteristic
113C22×D83—21no—
114C22×D83—21no—
115C22×D83—42no—
116C22×D83—42no—
117C22×D83—81no—
118D4×C83—22no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83C2×D411yescharacteristic
124C2×C4×C83C2×D411yescharacteristic
125C2×C4×C83C2×D411yescharacteristic
126C2×C4×C83C2×D411yescharacteristic
127C2×C4×C83C2411yescharacteristic
128C22⋊D83—42no—
129C22⋊D83—81no—
130C2×Dic83—21no—
131C2×Dic83—21no—
132C2×Dic83—21no—
133C2×Dic83—21no—
134C2×Dic83—21no—
135C2×Dic83—21no—
136C2×Dic83—81no—
137C2×Dic83—81no—
138D332—21no—
139D332—21no—
140D332—21no—
141D332—42no—
142D332#4112yes—
143C4×D83—42no—
144C4×D83—81no—
145Dic166—21no—
146Dic166—21no—
147Dic166—21no—
148Dic166—21no—
149Dic166—81no—
150C8×C83—21no—
151C8×C83—21no—
152C8×C83—21no—
153C8×C83—21no—
154C22×C83—21no—
155C22×C83—21no—
156C22×C83—21no—
157C22×C83—21no—
158C22×C83—21no—
159C22×C83#13712yes—
160C22×C83C22×D411yescharacteristic
161C2×D83—22no—
162C2×D83—41no—
163C2×D83—41no—
164C2×D83—41no—
165C2×D83—41no—
166C2×D83—41no—
167C2×D83—84no—
168C4×C83—21no—
169C4×C83—21no—
170C4×C83—21no—
171C4×C83—21no—
172C4×C83C2×#4111yescharacteristic
173C4×C83H(5,𝔽2)11yescharacteristic
174Dic83—41no—
175Dic83—41no—
176Dic83—41no—
177Dic83—41no—
178Dic83—41no—
179Dic83—41no—
180C2×C83—22no—
181C2×C83—41no—
182C2×C83#54611yescharacteristic
183C2×C83C2×#13711yescharacteristic
184C2×C83#57611yescharacteristic
185D83—42no—
186D83—81no—
187#3314—831nomaximal, Sylow
188C83#331411yescharacteristic, complemented, Sylow
189#554—831no—
190#546—831no—
191#489—831no—
192#484—831no—
193#480—831no—
194#474—831no—
195#468—831no—
196#467—831no—
197#464—831no—
198#452—831no—
199#450—831no—
200#440—831no—
201#435D8311yescharacteristic, complemented
202#417—831no—
203C2×#102—831no—
204C2×#41—831no—
205C22×D4—831no—
206C2×Q16—831no—
207C2×SD16—831no—
208C2×SD16—831no—
209C2×M16C2×D8311yescharacteristic, complemented
210C22×C8C2×D8311yescharacteristic
211C4⋊Q8—831no—
212#125—1661no—
213#124—831no—
214#123—1661no—
215C22⋊Q8—831no—
216C22⋊Q8—831no—
217C22⋊Q8—831no—
218C22⋊Q8—1661no—
219C22⋊Q8—1661no—
220#121—831no—
221#121—1661no—
222#120—1661no—
223C4×Q8—831no—
224C4×D4—831no—
225C4×D4—1661no—
226C4×D4C2×D8311yescharacteristic
227C2×C4⋊C4—831no—
228C2×C4⋊C4—831no—
229C2×C22⋊C4—1661no—
230#107—831no—
231#107—831no—
232#107—831no—
233C4⋊C8C2×D8311yescharacteristic, complemented
234#103—831no—
235#103—831no—
236#103—831no—
237#103—831no—
238#102—831no—
239#102—831no—
240#102—831no—
241#102—831no—
242#102—831no—
243#102—831no—
244C22⋊C8C2×D8311yescharacteristic
245C22⋊C8C2×D8311yescharacteristic, complemented
246#97C2×D8311yescharacteristic, complemented
247C24—1661no—
248#41—3321no—
249#41—1662no—
250C2×Q8—831no—
251C2×Q8—831no—
252C2×Q8—1661no—
253C2×D4—3321no—
254C2×D4—831no—
255C2×D4—831no—
256C2×D4—1662no—
257C2×D4—1661no—
258C2×D4—1661no—
259C2×D4—1661no—
260C2×D4D33211yescharacteristic
261C22×C4—1661no—
262C22×C4—1661no—
263C22×C4—1661no—
264C22×C4—1661no—
265C22×C4C22×D8311yescharacteristic
266C22×C4C22×D8311yescharacteristic
267Q16—3321no—
268SD16—3321no—
269SD16—1662no—
270M16—41no—
271C2×C8—41no—
272C2×C8C22×D8311yescharacteristic
273C2×C8C22×D8311yescharacteristic
274C2×C8C22×D8311yescharacteristic
275C2×C8C22×D8311yescharacteristic
276C4⋊C4—831no—
277C4⋊C4—831no—
278C4⋊C4—831no—
279C4⋊C4—831no—
280C4⋊C4—1661no—
281C4⋊C4—1661no—
282C4⋊C4—1661no—
283C4⋊C4—1661no—
284C4⋊C4—1661no—
285C4⋊C4—1661no—
286C4⋊C4—1661no—
287C4⋊C4D33211yescharacteristic
288C22⋊C4—1661no—
289C22⋊C4—1661no—
290C22⋊C4—1661no—
291C22⋊C4—1661no—
292C22⋊C4—1661no—
293C22⋊C4—1661no—
294C22⋊C4—1661no—
295C22⋊C4D33211yescharacteristic
296C22⋊C4D33211yescharacteristic
297C42—1661no—
298C42C22×D8311yescharacteristic
299C23—3322no—
300C23—3322no—
301C23—6641no—
302C23—1661no—
303C23—1661no—
304C23C2×D33211yescharacteristic
305C23C2×D33211yescharacteristic
306Q8—6641no—
307Q8—1661no—
308Q8—1661no—
309Q8—1661no—
310Q8—1661no—
311D4—22no—
312D4—3322no—
313D4—3322no—
314D4—832no—
315D4—6641no—
316D4—1661no—
317D4—1661no—
318D4—1661no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—3322no—
324C2×C4—6641no—
325C2×C4—6641no—
326C2×C4—6641no—
327C2×C4—1661no—
328C2×C4—1661no—
329C2×C4—1661no—
330C2×C4—1661no—
331C2×C4—1661no—
332C2×C4—1661no—
333C2×C4C2×D33211yescharacteristic
334C2×C4C2×D33211yescharacteristic
335C2×C4C2×D33211yescharacteristic
336C2×C4C2×D33211yescharacteristic
337C2×C4C23×D8311yescharacteristic
338C8—21no—
339C8—21no—
340C8—21no—
341C8—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—3321no—
348C22—3321no—
349C22—3321no—
350C22—3321no—
351C22—3321no—
352C22—6644no—
353C22—1662no—
354C22#1458182512yes—
355C22C22×D33211yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—3321no—
361C4—3321no—
362C4—3321no—
363C4—3321no—
364C4—3321no—
365C4—3321no—
366C4#41×D8311yescharacteristic
367C4#1458183711yescharacteristic
368C2—22no—
369C2—41no—
370C2—3322no—
371C2—6641no—
372C2#1912834611yescharacteristic
373C2C2×#1458182511yescharacteristic
374C2#1912850611yescharacteristic
subgroupG/H
centerC22C22×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#435×C83C2
socleC22×C83C22×D4
prime radical#19128347C2
hypercenter#435D83
hypocenterC83#3314
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #19128408C2semidirect
3. #19128061C2semidirect
4. #19128037C2semidirect
5. #19128015C2semidirect
6. #19127982C2semidirect
7. #19127947C2semidirect
8. #19127942C2semidirect
9. #19127924C2semidirect
10. #19127865C2semidirect
11. #19127854C2semidirect
12. #19127803C2semidirect
13. #19127703C2semidirect
14. C2×#14581714C2semidirect
15. #435×C83C2semidirect
18. C2×Dic332C22semidirect
21. #14581792C22semidirect
27. #14581771C22semidirect
33. C4×Dic166C22semidirect
37. C2×C4⋊Dic83C22semidirect
39. #14581726C22semidirect
40. #14581726C22semidirect
41. #14581726C22semidirect
42. #14581717C22semidirect
43. #14581717C22semidirect
44. #14581717C22semidirect
45. #14581717C22semidirect
52. C2×M16×C83C22semidirect
55. C4⋊C8×C83C22semidirect
57. C22⋊C8×C83C22semidirect
58. #97×C83C22semidirect
188. C83#3314semidirect
201. #435D83semidirect
209. C2×M16C2×D83semidirect
233. C4⋊C8C2×D83semidirect
245. C22⋊C8C2×D83semidirect
246. #97C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191284082
#191283472
#191280612
#191280372
#191280152
#191279822
#191279472
#191279422
#191279242
#191278652
#191278542
#191278032
#191277032
C2×#145817142
#435×C832
#331483