Finite Group Databasebeta

← #24919549

Subgroups — #24919549

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#19128411C211yesmaximal, characteristic, complemented
2#19128346C211yesmaximal, characteristic
3#19128061C211yesmaximal, characteristic, complemented
4#19128038C211yesmaximal, characteristic, complemented
5#19128016C211yesmaximal, characteristic, complemented
6#19127981C211yesmaximal, characteristic, complemented
7#19127950C211yesmaximal, characteristic, complemented
8#19127941C211yesmaximal, characteristic, complemented
9#19127923C211yesmaximal, characteristic, complemented
10#19127860C211yesmaximal, characteristic, complemented
11#19127850C211yesmaximal, characteristic, complemented
12#19127774C211yesmaximal, characteristic, complemented
13#19127699C211yesmaximal, characteristic, complemented
14#102×D83C211yesmaximal, characteristic, complemented
15#440×C83C211yesmaximal, characteristic, complemented
16C2×#13050589C2211yescharacteristic
17C2×D4×D83C2211yescharacteristic
18C2×#13050580C2211yescharacteristic, complemented
19C2×#13050577C2211yescharacteristic
20C2×#13050573C2211yescharacteristic, complemented
21C2×C8×D83C2211yescharacteristic
22#14581792C2211yescharacteristic, complemented
23#14581784—21no—
24#14581780C2211yescharacteristic
25#14581776—21no—
26#14581772C2211yescharacteristic
27#14581770—21no—
28#14581770—21no—
29#14581768C2211yescharacteristic
30#14581768C2211yescharacteristic, complemented
31#14581767C2211yescharacteristic
32#14581766—21no—
33#14581762—21no—
34C4×D332—21no—
35C4×D332C2211yescharacteristic
36C4⋊C4×D83C2211yescharacteristic
37C4⋊C4×D83C2211yescharacteristic, complemented
38C2×(C2×D83)⋊C4—21no—
39#14581725C2211yescharacteristic, complemented
40#14581725C2211yescharacteristic, 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×D83—21no—
73C2×C4×D83—21no—
74C2×C4×D83C2311yescharacteristic
75C2×C4×D83C2311yescharacteristic
76#13050580—41no—
77#13050577—41no—
78#13050573—41no—
79C2×C83⋊C8C2311yescharacteristic
80C2×C83⋊C8C2311yescharacteristic
81C8×D83—41no—
82C4⋊Dic83—21no—
83C4⋊Dic83—21no—
84C4⋊Dic83D411yescharacteristic
85C4⋊Dic83C2311yescharacteristic
86C4⋊Dic83C2311yescharacteristic
87Dic83⋊C4—21no—
88Dic83⋊C4—21no—
89Dic83⋊C4—21no—
90Dic83⋊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)⋊C4—21no—
96(C2×D83)⋊C4—21no—
97(C2×D83)⋊C4D411yescharacteristic
98(C2×D83)⋊C4D411yescharacteristic
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—22no—
138D332—21no—
139D332—21no—
140D332—81no—
141C4×D83—21no—
142C4×D83—21no—
143C4×D83—21no—
144C4×D83—21no—
145C4×D83—81no—
146C4×D83—81no—
147Dic166—21no—
148Dic166—21no—
149Dic166—81no—
150C8×C83—21no—
151C8×C83—21no—
152C83⋊C8—21no—
153C83⋊C8—21no—
154C22×C83—22no—
155C22×C83—41no—
156C22×C83—41no—
157C22×C83C22×D411yescharacteristic
158C2×D83—21no—
159C2×D83—21no—
160C2×D83—21no—
161C2×D83—21no—
162C2×D83—21no—
163C2×D83—41no—
164C2×D83—41no—
165C2×D83—41no—
166C2×D83—84no—
167C2×D83#13712yes—
168C4×C83—21no—
169C4×C83—21no—
170C4×C83—41no—
171C4×C83—41no—
172C4×C83—41no—
173C4×C83C2×#4111yescharacteristic
174C4×C83H(5,𝔽2)11yescharacteristic
175Dic83—21no—
176Dic83—21no—
177Dic83—41no—
178Dic83—41no—
179Dic83—41no—
180C2×C83—42no—
181C2×C83#54611yescharacteristic
182C2×C83C2×#13711yescharacteristic
183C2×C83#57611yescharacteristic
184D83—22no—
185D83—41no—
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#440D8311yescharacteristic, complemented
201#435—831no—
202#417—831no—
203C2×#102—831no—
204C2×#41—831no—
205C22×D4—831no—
206C2×Q16—831no—
207C2×SD16—831no—
208C2×SD16C2×D8311yescharacteristic
209C2×M16—831no—
210C22×C8—831no—
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×Q8C2×D8311yescharacteristic, complemented
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#107C2×D8311yescharacteristic, complemented
233C4⋊C8—831no—
234#103—831no—
235#103—831no—
236#103—831no—
237#103C2×D8311yescharacteristic, complemented
238#102—831no—
239#102—831no—
240#102—831no—
241#102—831no—
242#102—831no—
243#102C2×D8311yescharacteristic
244C22⋊C8—831no—
245C22⋊C8—831no—
246#97C2×D8311yescharacteristic, complemented
247C24—1661no—
248#41—3321no—
249#41—1662no—
250C2×Q8—831no—
251C2×Q8—1661no—
252C2×Q8C22×D8311yescharacteristic
253C2×D4—3321no—
254C2×D4—831no—
255C2×D4—831no—
256C2×D4—1661no—
257C2×D4—1661no—
258C2×D4—1661no—
259C2×D4—1662no—
260C2×D4C22×D8311yescharacteristic
261C22×C4—21no—
262C22×C4—831no—
263C22×C4—831no—
264C22×C4—1661no—
265C22×C4—1661no—
266C22×C4—1661no—
267Q16—3321no—
268SD16—22no—
269SD16—3321no—
270M16—3321no—
271C2×C8—3321no—
272C2×C8—831no—
273C2×C8—831no—
274C2×C8C22×D8311yescharacteristic
275C2×C8C22×D8311yescharacteristic
276C4⋊C4—21no—
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⋊C4C22×D8311yescharacteristic
287C4⋊C4C22×D8311yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—831no—
290C22⋊C4—831no—
291C22⋊C4—1661no—
292C22⋊C4—1661no—
293C22⋊C4—1661no—
294C22⋊C4—1661no—
295C22⋊C4—1661no—
296C22⋊C4—1661no—
297C42—21no—
298C42C22×D8311yescharacteristic
299C23—21no—
300C23—3322no—
301C23—3322no—
302C23—831no—
303C23—831no—
304C23—6641no—
305C23—1661no—
306Q8—21no—
307Q8—21no—
308Q8—6641no—
309Q8—1661no—
310Q8—1661no—
311D4—21no—
312D4—3322no—
313D4—3322no—
314D4—6641no—
315D4—1662no—
316D4—1661no—
317D4—1661no—
318D4#1305059012yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—81no—
323C2×C4—3322no—
324C2×C4—831no—
325C2×C4—831no—
326C2×C4—6641no—
327C2×C4—6641no—
328C2×C4—1661no—
329C2×C4—1661no—
330C2×C4—1661no—
331C2×C4—1661no—
332C2×C4—1661no—
333C2×C4—1661no—
334C2×C4—1661no—
335C2×C4D4×D8311yescharacteristic
336C2×C4D4×D8311yescharacteristic
337C2×C4C23×D8311yescharacteristic
338C8—21no—
339C8—21no—
340C8—1661no—
341C8—1661no—
342C22—22no—
343C22—41no—
344C22—41no—
345C22—3321no—
346C22—3321no—
347C22—3321no—
348C22—832no—
349C22—6644no—
350C22—1661no—
351C22—1661no—
352C22—1661no—
353C22—1661no—
354C22—1661no—
355C22C2×D4×D8311yescharacteristic
356C4—21no—
357C4—21no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—3321no—
362C4—3321no—
363C4—3321no—
364C4—1661no—
365C4—1661no—
366C4C2×#1305059011yescharacteristic
367C4#1458183811yescharacteristic
368C2—42no—
369C2—3321no—
370C2—6641no—
371C2—1662no—
372C2#1912834711yescharacteristic
373C2#137×D8311yescharacteristic
374C2#1912850511yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#440×C83C2
socleC22×C83C22×D4
prime radical#19128346C2
hypercenter#440D83
hypocenterC83#3314
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. #19128411C2semidirect
3. #19128061C2semidirect
4. #19128038C2semidirect
5. #19128016C2semidirect
6. #19127981C2semidirect
7. #19127950C2semidirect
8. #19127941C2semidirect
9. #19127923C2semidirect
10. #19127860C2semidirect
11. #19127850C2semidirect
12. #19127774C2semidirect
13. #19127699C2semidirect
14. #102×D83C2semidirect
15. #440×C83C2semidirect
18. C2×#13050580C22semidirect
20. C2×#13050573C22semidirect
22. #14581792C22semidirect
30. #14581768C22semidirect
37. C4⋊C4×D83C22semidirect
39. #14581725C22semidirect
40. #14581725C22semidirect
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
188. C83#3314semidirect
200. #440D83semidirect
223. C4×Q8C2×D83semidirect
232. #107C2×D83semidirect
237. #103C2×D83semidirect
246. #97C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191284112
#191283462
#191280612
#191280382
#191280162
#191279812
#191279502
#191279412
#191279232
#191278602
#191278502
#191277742
#191276992
#102×D832
#440×C832
#331483