Finite Group Databasebeta

← #24919680

Subgroups — #24919680

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#19128386C211yesmaximal, characteristic, complemented
2#19128339C211yesmaximal, characteristic
3#19128057C211yesmaximal, characteristic, complemented
4#19128041C211yesmaximal, characteristic, complemented
5#19128019C211yesmaximal, characteristic, complemented
6#19127999C211yesmaximal, characteristic, complemented
7#19127958C211yesmaximal, characteristic, complemented
8#19127947C211yesmaximal, characteristic, complemented
9#19127926C211yesmaximal, characteristic, complemented
10#19127863C211yesmaximal, characteristic, complemented
11#19127853C211yesmaximal, characteristic, complemented
12#19127792C211yesmaximal, characteristic, complemented
13#19127777C211yesmaximal, characteristic, complemented
14#19127704C211yesmaximal, characteristic, complemented
15#417×C83C211yesmaximal, characteristic, complemented
16C2×D4×D83C2211yescharacteristic
17C2×#13050580C2211yescharacteristic, complemented
18C2×#13050579C2211yescharacteristic
19C2×#13050578C2211yescharacteristic
20C2×#13050573C2211yescharacteristic
21#14581787—21no—
22#14581783C2211yescharacteristic, complemented
23#14581782—21no—
24#14581781C2211yescharacteristic
25#14581775—21no—
26#14581775—21no—
27#14581772C2211yescharacteristic
28#14581771C2211yescharacteristic, complemented
29#14581768C2211yescharacteristic
30#14581766C2211yescharacteristic
31#14581764—21no—
32#14581760—21no—
33#14581757C2211yescharacteristic, complemented
34D4×Dic83C2211yescharacteristic
35C4×C22⋊D83C2211yescharacteristic
36#14581751—21no—
37#14581750C2211yescharacteristic
38C2×(C2×D83)⋊C4—21no—
39#14581726C2211yescharacteristic, complemented
40#14581723C2211yescharacteristic, complemented
41#14581723C2211yescharacteristic, complemented
42Dic83⋊C8C2211yescharacteristic, complemented
43#14581717C2211yescharacteristic, complemented
44#14581716C2211yescharacteristic, complemented
45#14581714C2211yescharacteristic
46#14581713C2211yescharacteristic
47#14581712C2211yescharacteristic
48#14581712C2211yescharacteristic
49C22⋊(C83⋊C8)C2211yescharacteristic, complemented
50(C2×D83)⋊C8C2211yescharacteristic
51C8×Dic83C2211yescharacteristic, complemented
52C2×#41×C83C2211yescharacteristic
53C2×M16×C83C2211yescharacteristic, complemented
54C2×C4⋊C4×C83C2211yescharacteristic, complemented
55#103×C83C2211yescharacteristic, complemented
56#103×C83C2211yescharacteristic, complemented
57#102×C83C2211yescharacteristic
58#102×C83C2211yescharacteristic
59C23×D83—21no—
60C2×Dic166C2311yescharacteristic
61C2×D332C2311yescharacteristic
62C2×C22⋊D83—21no—
63C2×C22⋊D83—21no—
64C2×C22⋊D83D411yescharacteristic
65D4×D83—22no—
66D4×D83—41no—
67C22×Dic83—21no—
68C2×C4×D83C2311yescharacteristic
69#13050580—41no—
70#13050579—41no—
71#13050578—22no—
72#13050573—41no—
73C2×C83⋊C8C2311yescharacteristic
74C2×C83⋊C8C2311yescharacteristic
75C4⋊Dic83—21no—
76C4⋊Dic83—21no—
77C4⋊Dic83C2311yescharacteristic
78C4⋊Dic83C2311yescharacteristic
79Dic83⋊C4—21no—
80Dic83⋊C4—21no—
81Dic83⋊C4—21no—
82Dic83⋊C4—21no—
83Dic83⋊C4D411yescharacteristic
84C22⋊Dic83—21no—
85C22⋊Dic83—21no—
86C22⋊Dic83D411yescharacteristic
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)⋊C4D411yescharacteristic
94C4×Dic83—21no—
95C4×Dic83—21no—
96C4×Dic83C2311yescharacteristic
97#41×C83—22no—
98#41×C83—41no—
99C2×Q8×C83C2311yescharacteristic
100C2×D4×C83—21no—
101C2×D4×C83C2311yescharacteristic
102C22×C4×C83—21no—
103C22×C4×C83—21no—
104C22×C4×C83C2311yescharacteristic
105M16×C83—41no—
106C2×C8×C83C2311yescharacteristic
107C2×C8×C83C2311yescharacteristic
108C4⋊C4×C83—21no—
109C4⋊C4×C83C2311yescharacteristic
110C4⋊C4×C83C2311yescharacteristic
111C23×C83—21no—
112C23×C83C2×D411yescharacteristic
113C22×D83—21no—
114C22×D83—42no—
115C22×D83—42no—
116C22×D83—81no—
117C22×D83C2×D411yescharacteristic
118Q8×C83—21no—
119Q8×C83—21no—
120D4×C83—21no—
121D4×C83—42no—
122D4×C83#4112yes—
123C2×C4×C83—21no—
124C2×C4×C83—21no—
125C2×C4×C83—21no—
126C2×C4×C83—21no—
127C2×C4×C83—21no—
128C2×C4×C83—42no—
129C2×C4×C83—42no—
130C2×C4×C83C2×D411yescharacteristic
131C2×C4×C83C2411yescharacteristic
132C22⋊D83—41no—
133C22⋊D83—42no—
134C22⋊D83—81no—
135C2×Dic83—21no—
136C2×Dic83—21no—
137C2×Dic83—21no—
138C2×Dic83—81no—
139C2×Dic83C2×D411yescharacteristic
140C2×Dic83C2×D411yescharacteristic
141C2×Dic83C2×D411yescharacteristic
142D332—21no—
143D332—21no—
144C4×D83—21no—
145C4×D83—21no—
146Dic166—21no—
147Dic166—21no—
148C8×C83—21no—
149C8×C83—21no—
150C83⋊C8—21no—
151C83⋊C8—21no—
152C22×C83—22no—
153C22×C83—21no—
154C22×C83—21no—
155C22×C83—21no—
156C22×C83—41no—
157C22×C83—41no—
158C22×C83C22×D411yescharacteristic
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—41no—
168C4×C83—41no—
169C4×C83—41no—
170C4×C83C2×#4111yescharacteristic
171C4×C83H(5,𝔽2)11yescharacteristic
172Dic83—21no—
173Dic83—21no—
174Dic83—21no—
175Dic83—41no—
176Dic83—41no—
177Dic83—41no—
178C2×C83—42no—
179C2×C83—41no—
180C2×C83#54611yescharacteristic
181C2×C83#57511yescharacteristic
182C2×C83#57711yescharacteristic
183D83—41no—
184D83—81no—
185#3323—831nomaximal, Sylow
186C83#332311yescharacteristic, complemented, Sylow
187#552—831no—
188#546—831no—
189#488—831no—
190#485—831no—
191#481—831no—
192#478—831no—
193#470—831no—
194#468—831no—
195#464—831no—
196#452—831no—
197#450—831no—
198C4×SD16—831no—
199#436—831no—
200#418—831no—
201#417D8311yescharacteristic, complemented
202C2×#41C2×D8311yescharacteristic
203C22×D4—831no—
204C2×Q16—831no—
205C2×SD16—831no—
206C2×SD16—831no—
207C2×M16—831no—
208C2×M16C2×D8311yescharacteristic, complemented
209#126—1661no—
210#125—831no—
211#125—1661no—
212#124—831no—
213#123—1661no—
214#123—1661no—
215C22⋊Q8—831no—
216C22⋊Q8—831no—
217C22⋊Q8—831no—
218#121—831no—
219#121—1661no—
220#120—1661no—
221C4×Q8—831no—
222C4×D4—831no—
223C4×D4—831no—
224#117—831no—
225#117—1661no—
226C2×C4⋊C4C2×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#103C2×D8311yescharacteristic, complemented
235#103C2×D8311yescharacteristic, complemented
236#102—831no—
237#102—831no—
238#102—831no—
239#102—831no—
240#102C2×D8311yescharacteristic
241#102C2×D8311yescharacteristic
242C22⋊C8—831no—
243C22⋊C8—831no—
244C4×C8—831no—
245C24—1661no—
246#41—22no—
247#41—41no—
248C2×Q8—831no—
249C2×Q8C22×D8311yescharacteristic
250C2×D4—21no—
251C2×D4—3321no—
252C2×D4—831no—
253C2×D4—831no—
254C2×D4—1661no—
255C2×D4—1662no—
256C2×D4—1661no—
257C2×D4C22×D8311yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—831no—
261C22×C4—1661no—
262C22×C4C22×D8311yescharacteristic
263Q16—3321no—
264SD16—3321no—
265SD16—1662no—
266M16—41no—
267M16—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⋊C4—1661no—
281C4⋊C4—1661no—
282C4⋊C4C22×D8311yescharacteristic
283C4⋊C4C22×D8311yescharacteristic
284C22⋊C4—831no—
285C22⋊C4—831no—
286C22⋊C4—1661no—
287C22⋊C4—1661no—
288C22⋊C4—1661no—
289C22⋊C4—1661no—
290C22⋊C4—1661no—
291C22⋊C4—1661no—
292C22⋊C4—1661no—
293C22⋊C4—1661no—
294C42—831no—
295C42—1661no—
296C42—1661no—
297C23—21no—
298C23—3322no—
299C23—3322no—
300C23—831no—
301C23—6641no—
302C23—1661no—
303C23D4×D8311yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—1661no—
307Q8—1661no—
308D4—21no—
309D4—42no—
310D4—3322no—
311D4—3321no—
312D4—6641no—
313D4—1661no—
314D4—1661no—
315D4#1305059012yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—831no—
324C2×C4—831no—
325C2×C4—831no—
326C2×C4—6641no—
327C2×C4—1661no—
328C2×C4—1661no—
329C2×C4—1661no—
330C2×C4—1661no—
331C2×C4—1661no—
332C2×C4D4×D8311yescharacteristic
333C2×C4C23×D8311yescharacteristic
334C8—21no—
335C8—21no—
336C8—1661no—
337C8—1661no—
338C22—21no—
339C22—22no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—3321no—
345C22—3321no—
346C22—3321no—
347C22—6644no—
348C22—1661no—
349C22—1661no—
350C22—1661no—
351C22C2×D4×D8311yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—3321no—
357C4—3321no—
358C4—3321no—
359C4—1661no—
360C4—1661no—
361C4—1661no—
362C4C2×#1305059011yescharacteristic
363C4#1458183711yescharacteristic
364C2—41no—
365C2—42no—
366C2—3321no—
367C2—6641no—
368C2#1912834111yescharacteristic
369C2#1912849711yescharacteristic
370C2#1912851011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#417×C83C2
socleC22×C83C22×D4
prime radical#19128339C2
hypercenter#417D83
hypocenterC83#3323
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #19128386C2semidirect
3. #19128057C2semidirect
4. #19128041C2semidirect
5. #19128019C2semidirect
6. #19127999C2semidirect
7. #19127958C2semidirect
8. #19127947C2semidirect
9. #19127926C2semidirect
10. #19127863C2semidirect
11. #19127853C2semidirect
12. #19127792C2semidirect
13. #19127777C2semidirect
14. #19127704C2semidirect
15. #417×C83C2semidirect
17. C2×#13050580C22semidirect
22. #14581783C22semidirect
28. #14581771C22semidirect
33. #14581757C22semidirect
39. #14581726C22semidirect
40. #14581723C22semidirect
41. #14581723C22semidirect
42. Dic83⋊C8C22semidirect
43. #14581717C22semidirect
44. #14581716C22semidirect
49. C22⋊(C83⋊C8)C22semidirect
51. C8×Dic83C22semidirect
53. C2×M16×C83C22semidirect
54. C2×C4⋊C4×C83C22semidirect
55. #103×C83C22semidirect
56. #103×C83C22semidirect
186. C83#3323semidirect
201. #417D83semidirect
208. C2×M16C2×D83semidirect
226. C2×C4⋊C4C2×D83semidirect
234. #103C2×D83semidirect
235. #103C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191283862
#191283392
#191280572
#191280412
#191280192
#191279992
#191279582
#191279472
#191279262
#191278632
#191278532
#191277922
#191277772
#191277042
#417×C832
#332383