Finite Group Databasebeta

← #24916740

Subgroups — #24916740

Order 10624.

752 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14581822C211yesmaximal, characteristic, complemented
2C22×#13050578C211yesmaximal, characteristic, complemented
3#19128272C211yesmaximal, characteristic, complemented
4#19127863C214yesmaximal, complemented
5#19127852C214yesmaximal, complemented
6#19127706C211yesmaximal, characteristic, complemented
7#19127697C211yesmaximal, characteristic, complemented
8#19127668C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C83C211yesmaximal, characteristic
10C22×Dic166C2211yescharacteristic, complemented
11#14581822—24no—
12C2×#13050580C2212yescomplemented
13C2×#13050578—24no—
14C2×#13050578C2212yescomplemented
15C2×#13050578C2214yescomplemented
16C2×#13050574C2211yescharacteristic, complemented
17C22×C83⋊C8C2211yescharacteristic, complemented
18#14581790—21no—
19#14581779—21no—
20#14581773—22no—
21#14581771C2214yescomplemented
22#14581751C2211yescharacteristic, complemented
23#14581716C2212yescomplemented
24#14581713C2212yescomplemented
25#14581713C2214yescomplemented
26C22⋊(C83⋊C8)C2214yescomplemented
27H(5,𝔽2)×C83—24no—
28H(5,𝔽2)×C83—24no—
29C2×#41×C83—21no—
30C2×#41×C83C2212yes—
31C2×#41×C83C2211yescharacteristic
32C2×#41×C83C2211yescharacteristic
33C22×D4×C83—22no—
34C22×D4×C83—21no—
35C22×D4×C83C2212yes—
36C22×D4×C83C2211yescharacteristic
37C2×Dic166—22no—
38C2×Dic166—22no—
39C2×Dic166—21no—
40C2×Dic166C2312yes—
41C22×Dic83—21no—
42#13050580—24no—
43#13050578—28no—
44#13050578—24no—
45#13050574—22no—
46C2×C83⋊C8—22no—
47C2×C83⋊C8C2312yes—
48C2×C83⋊C8C2312yes—
49C4⋊Dic83C2312yes—
50Dic83⋊C4—24no—
51C22⋊Dic83—24no—
52C22⋊Dic83—21no—
53C4×Dic83—21no—
54C24×C83—22no—
55C24×C83—21no—
56#41×C83—28no—
57#41×C83—24no—
58#41×C83—24no—
59#41×C83—22no—
60#41×C83—22no—
61#41×C83—22no—
62#41×C83D414yes—
63C2×Q8×C83D411yescharacteristic
64C2×Q8×C83C2311yescharacteristic
65C2×D4×C83—216no—
66C2×D4×C83—28no—
67C2×D4×C83—22no—
68C2×D4×C83—21no—
69C2×D4×C83—24no—
70C2×D4×C83—24no—
71C2×D4×C83—22no—
72C2×D4×C83—21no—
73C2×D4×C83—22no—
74C2×D4×C83D412yes—
75C2×D4×C83D411yescharacteristic
76C2×D4×C83D414yes—
77C2×D4×C83C2311yescharacteristic
78C2×D4×C83C2312yes—
79C22×C4×C83—22no—
80C22×C4×C83—21no—
81C22×C4×C83C2312yes—
82C22×C4×C83C2311yescharacteristic
83C23×C83—28no—
84C23×C83—22no—
85C23×C83—21no—
86C23×C83—24no—
87C23×C83—82no—
88C23×C83—84no—
89C23×C83C2×D412yes—
90C23×C83C2×D411yescharacteristic
91Q8×C83—22no—
92Q8×C83—21no—
93Q8×C83C2×D412yes—
94D4×C83—28no—
95D4×C83—28no—
96D4×C83—24no—
97D4×C83—24no—
98D4×C83—22no—
99D4×C83—24no—
100D4×C83—22no—
101D4×C83—21no—
102D4×C83C2×D412yes—
103D4×C83C2×D414yes—
104C2×C4×C83—28no—
105C2×C4×C83—21no—
106C2×C4×C83—24no—
107C2×C4×C83—22no—
108C2×C4×C83—22no—
109C2×C4×C83C2×D412yes—
110C2×C4×C83C2×D411yescharacteristic
111C2×C4×C83C2×D412yes—
112C2×C4×C83C2×D412yes—
113C2×C4×C83C2411yescharacteristic
114C2×Dic83—22no—
115C2×Dic83—22no—
116C2×Dic83—42no—
117Dic166—22no—
118Dic166—22no—
119Dic166—42no—
120C83⋊C8—22no—
121C83⋊C8—22no—
122C22×C83—22no—
123C22×C83—22no—
124C22×C83—24no—
125C22×C83—22no—
126C22×C83—24no—
127C22×C83—44no—
128C22×C83—42no—
129C22×C83—84no—
130C22×C83—88no—
131C22×C83#12012yes—
132C22×C83C22×D411yescharacteristic
133C4×C83—22no—
134C4×C83—22no—
135C4×C83#12012yes—
136C4×C83C22×D411yescharacteristic
137C4×C83C22×D411yescharacteristic
138Dic83—42no—
139Dic83—42no—
140C2×C83—22no—
141C2×C83—42no—
142C2×C83—44no—
143C2×C83—42no—
144C2×C83C2×#12011yescharacteristic
145C2×C83#57712yes—
146#3009—831nomaximal, Sylow
147C83#300911yescharacteristic, complemented, Sylow
148C2×H(5,𝔽2)D8311yescharacteristic
149C2×#137—831no—
150C22×SD16—831no—
151#536—831no—
152#452—834no—
153#450—834no—
154#418—831no—
155#416—831no—
156#408—831no—
157H(5,𝔽2)—24no—
158H(5,𝔽2)—24no—
159C2×#41—21no—
160C2×#41C2×D8312yes—
161C2×#41C2×D8311yescharacteristic
162C2×#41C2×D8311yescharacteristic
163C22×Q8—831no—
164C22×D4—22no—
165C22×D4—21no—
166C22×D4C2×D8312yes—
167C22×D4C2×D8311yescharacteristic
168#137—1664no—
169C2×Q16—832no—
170C2×SD16—834no—
171C2×SD16—832no—
172C2×SD16—1664no—
173C2×M16—831no—
174C22×C8—831no—
175C4⋊Q8—1661no—
176#124—1661no—
177#123—1662no—
178C22⋊Q8—834no—
179#117—831no—
180#103—832no—
181#102—832no—
182#102—834no—
183C22⋊C8—834no—
184C24—22no—
185C24—21no—
186#41—28no—
187#41—24no—
188#41—24no—
189#41—22no—
190#41—22no—
191#41—22no—
192#41C22⋊D8314yes—
193C2×Q8—832no—
194C2×Q8—1662no—
195C2×Q8—1662no—
196C2×Q8—1661no—
197C2×Q8C22⋊D8311yescharacteristic
198C2×Q8C22×D8311yescharacteristic
199C2×D4—216no—
200C2×D4—28no—
201C2×D4—22no—
202C2×D4—21no—
203C2×D4—24no—
204C2×D4—24no—
205C2×D4—22no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4C22⋊D8312yes—
209C2×D4C22⋊D8311yescharacteristic
210C2×D4C22⋊D8314yes—
211C2×D4C22×D8311yescharacteristic
212C2×D4C22×D8312yes—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—1661no—
216C22×C4C22×D8312yes—
217C22×C4C22×D8311yescharacteristic
218Q16—1664no—
219SD16—1668no—
220SD16—1664no—
221M16—1662no—
222C2×C8—832no—
223C2×C8—832no—
224C2×C8—1662no—
225C4⋊C4—832no—
226C4⋊C4—1664no—
227C22⋊C4—1661no—
228C22⋊C4—1664no—
229C42—1661no—
230C23—28no—
231C23—22no—
232C23—21no—
233C23—24no—
234C23—82no—
235C23—84no—
236C23C2×C22⋊D8312yes—
237C23C2×C22⋊D8311yescharacteristic
238Q8—22no—
239Q8—21no—
240Q8—3322no—
241Q8—1662no—
242Q8—1662no—
243Q8C2×C22⋊D8312yes—
244D4—28no—
245D4—28no—
246D4—24no—
247D4—24no—
248D4—22no—
249D4—24no—
250D4—22no—
251D4—21no—
252D4C2×C22⋊D8312yes—
253D4C2×C22⋊D8314yes—
254C2×C4—28no—
255C2×C4—21no—
256C2×C4—24no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—3322no—
260C2×C4—1662no—
261C2×C4—1662no—
262C2×C4C2×C22⋊D8312yes—
263C2×C4C2×C22⋊D8311yescharacteristic
264C2×C4C2×C22⋊D8312yes—
265C2×C4C2×C22⋊D8312yes—
266C2×C4C23×D8311yescharacteristic
267C8—1662no—
268C8—1662no—
269C22—22no—
270C22—22no—
271C22—24no—
272C22—22no—
273C22—24no—
274C22—44no—
275C22—42no—
276C22—84no—
277C22—88no—
278C22#1458176112yes—
279C22C22×C22⋊D8311yescharacteristic
280C4—22no—
281C4—22no—
282C4—3322no—
283C4—3322no—
284C4#1458176112yes—
285C4C22×C22⋊D8311yescharacteristic
286C4C22×C22⋊D8311yescharacteristic
287C2—22no—
288C2—42no—
289C2—44no—
290C2—42no—
291C2C2×#1458176111yescharacteristic
292C2#1912851212yes—
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting, prime radicalC2×H(5,𝔽2)×C83C2
socleC22×C83C22×D4
hypercenterC2×H(5,𝔽2)D83
hypocenterC83#3009
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#14581822C2semidirect
2. C22×#13050578C2semidirect
3. #19128272C2semidirect
4. #19127863C2semidirect
5. #19127852C2semidirect
6. #19127706C2semidirect
7. #19127697C2semidirect
8. #19127668C2semidirect
10. C22×Dic166C22semidirect
12. C2×#13050580C22semidirect
14. C2×#13050578C22semidirect
15. C2×#13050578C22semidirect
16. C2×#13050574C22semidirect
17. C22×C83⋊C8C22semidirect
21. #14581771C22semidirect
22. #14581751C22semidirect
23. #14581716C22semidirect
24. #14581713C22semidirect
25. #14581713C22semidirect
26. C22⋊(C83⋊C8)C22semidirect
147. C83#3009semidirect

Maximal subgroups

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

subgroupindex
C2×#145818222
C22×#130505782
#191282722
#191278632
#191278522
#191277062
#191276972
#191276682
C2×H(5,𝔽2)×C832
#300983