Finite Group Databasebeta

← #24917404

Subgroups — #24917404

Order 10624.

462 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 137 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×#14581766C211yesmaximal, characteristic
2C2×#14581751C211yesmaximal, characteristic, complemented
3#19128026C214yesmaximal, complemented
4#19128020C212yesmaximal, complemented
5#19128015C212yesmaximal, complemented
6#19127743C212yesmaximal, complemented
7#19127707C212yesmaximal, complemented
8#407×C83C211yesmaximal, characteristic, complemented
9C22×D332C2211yescharacteristic
10C22×C22⋊D83—21no—
11#14581766—21no—
12#14581766—21no—
13#14581766C2212yes—
14#14581766C2212yes—
15#14581751—21no—
16#14581751—21no—
17#14581751C2212yes—
18#14581751C2212yescomplemented
19C2×C4⋊Dic83C2211yescharacteristic
20C2×C4⋊Dic83C2211yescharacteristic, complemented
21C2×C4⋊Dic83C2211yescharacteristic, complemented
22C2×C22⋊Dic83—21no—
23C2×(C2×D83)⋊C4—21no—
24C2×C4×Dic83—21no—
25#14581726C2214yescomplemented
26#14581724C2214yescomplemented
27#14581714C2214yescomplemented
28#14581714C2214yescomplemented
29C23×C4×C83C2211yescharacteristic
30C2×M16×C83C2212yes—
31C22⋊C8×C83C2212yescomplemented
32C22⋊C8×C83C2212yescomplemented
33C23×D83—21no—
34C2×D332—24no—
35C2×D332—21no—
36C2×D332C2312yes—
37C2×C22⋊D83—22no—
38C2×C22⋊D83—22no—
39C2×C22⋊D83—44no—
40C22×Dic83—21no—
41C22×Dic83—21no—
42C22×Dic83—21no—
43C4⋊Dic83—21no—
44C4⋊Dic83—21no—
45C4⋊Dic83—21no—
46C4⋊Dic83C2312yes—
47C4⋊Dic83C2312yes—
48C4⋊Dic83C2312yes—
49C22⋊Dic83—22no—
50C22⋊Dic83—22no—
51(C2×D83)⋊C4—22no—
52(C2×D83)⋊C4—22no—
53C4×Dic83—22no—
54C4×Dic83—22no—
55C24×C83D411yescharacteristic
56C22×C4×C83—24no—
57C22×C4×C83D411yescharacteristic
58C22×C4×C83D411yescharacteristic
59C22×C4×C83D411yescharacteristic
60C22×C4×C83C2311yescharacteristic
61C22×C4×C83C2311yescharacteristic
62C22×C4×C83C2311yescharacteristic
63M16×C83—24no—
64C2×C8×C83C2314yes—
65C23×C83—22no—
66C23×C83—22no—
67C23×C83—41no—
68C23×C83C2×D411yescharacteristic
69C23×C83C2×D411yescharacteristic
70C23×C83C2×D411yescharacteristic
71C22×D83—22no—
72C22×D83—42no—
73C22×D83—42no—
74C22×D83—42no—
75C2×C4×C83—22no—
76C2×C4×C83—22no—
77C2×C4×C83—22no—
78C2×C4×C83—22no—
79C2×C4×C83—22no—
80C2×C4×C83C2×D411yescharacteristic
81C2×C4×C83C2×D411yescharacteristic
82C2×C4×C83C2×D411yescharacteristic
83C2×C4×C83#4114yes—
84C2×C4×C83C2411yescharacteristic
85C22⋊D83—44no—
86C22⋊D83—44no—
87C2×Dic83—22no—
88C2×Dic83—22no—
89C2×Dic83—22no—
90C2×Dic83—42no—
91C2×Dic83—42no—
92C2×Dic83—42no—
93D332—24no—
94D332—42no—
95C8×C83—24no—
96C22×C83—21no—
97C22×C83—21no—
98C22×C83—21no—
99C22×C83—21no—
100C22×C83—21no—
101C22×C83—21no—
102C22×C83—21no—
103C22×C83—44no—
104C22×C83#12312yes—
105C22×C83AGL(1,ℤ/8)12yes—
106C22×C83C22×D411yescharacteristic
107C2×D83—42no—
108C2×D83—42no—
109C2×D83—42no—
110C2×D83—84no—
111C4×C83—21no—
112C4×C83—21no—
113C4×C83#12312yes—
114C4×C83C2×#4112yes—
115Dic83—42no—
116Dic83—42no—
117Dic83—42no—
118C2×C83—22no—
119C2×C83—22no—
120C2×C83—41no—
121C2×C83C2×#12311yescharacteristic
122C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
123C2×C83#57511yescharacteristic
124D83—82no—
125#3094—831nomaximal, Sylow
126C83#309411yescharacteristic, complemented, Sylow
127C2×#121—831no—
128C2×#117—831no—
129#482—834no—
130#481—832no—
131#480—832no—
132#428—832no—
133#418—832no—
134#407D8311yescharacteristic, complemented
135C22×D4—831no—
136C22×D4—1661no—
137C23×C4C2×D8311yescharacteristic
138C2×M16C2×D8312yes—
139#121—832no—
140#121—832no—
141#121—1661no—
142#121—1661no—
143#117—832no—
144#117—832no—
145#117—1661no—
146#117—1661no—
147C2×C4⋊C4—831no—
148C2×C4⋊C4—831no—
149C2×C4⋊C4—831no—
150C2×C22⋊C4—1661no—
151C2×C22⋊C4—1661no—
152C2×C42—1661no—
153#107—834no—
154#106—834no—
155#102—834no—
156#102—834no—
157C22⋊C8C2×D8312yescomplemented
158C22⋊C8C2×D8312yescomplemented
159C24—1661no—
160C24D33211yescharacteristic
161C2×D4—3324no—
162C2×D4—832no—
163C2×D4—1662no—
164C2×D4—1662no—
165C2×D4—1661no—
166C2×D4—1664no—
167C22×C4—24no—
168C22×C4—1661no—
169C22×C4—1661no—
170C22×C4—1661no—
171C22×C4D33211yescharacteristic
172C22×C4D33211yescharacteristic
173C22×C4D33211yescharacteristic
174C22×C4C22×D8311yescharacteristic
175C22×C4C22×D8311yescharacteristic
176C22×C4C22×D8311yescharacteristic
177M16—24no—
178C2×C8C22×D8314yes—
179C4⋊C4—832no—
180C4⋊C4—832no—
181C4⋊C4—832no—
182C4⋊C4—1661no—
183C4⋊C4—1661no—
184C4⋊C4—1661no—
185C22⋊C4—1662no—
186C22⋊C4—1662no—
187C22⋊C4—1662no—
188C22⋊C4—1662no—
189C42—1662no—
190C42—1662no—
191C23—22no—
192C23—22no—
193C23—41no—
194C23—3322no—
195C23—3322no—
196C23—3322no—
197C23—1662no—
198C23C2×D33211yescharacteristic
199C23C2×D33211yescharacteristic
200C23C2×D33211yescharacteristic
201D4—3324no—
202D4—3324no—
203D4—3322no—
204D4—1664no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—3322no—
211C2×C4—3322no—
212C2×C4—3322no—
213C2×C4—1662no—
214C2×C4—1662no—
215C2×C4—1662no—
216C2×C4C2×D33211yescharacteristic
217C2×C4C2×D33211yescharacteristic
218C2×C4C2×D33211yescharacteristic
219C2×C4#1305058914yes—
220C2×C4C23×D8311yescharacteristic
221C8—24no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—44no—
230C22—3322no—
231C22—3322no—
232C22—3322no—
233C22—6644no—
234C22#1458177612yes—
235C22#1458182012yes—
236C22C22×D33211yescharacteristic
237C4—21no—
238C4—21no—
239C4—3322no—
240C4—3322no—
241C4—3322no—
242C4#1458177612yes—
243C4C2×#1305058912yes—
244C2—22no—
245C2—22no—
246C2—41no—
247C2—6642no—
248C2C2×#1458177611yescharacteristic
249C2C2×#1458182011yescharacteristic
250C2#1912850111yescharacteristic
subgroupG/H
centerC22C22×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#407×C83C2
socleC22×C83C22×D4
prime radicalC2×#14581766C2
hypercenter#407D83
hypocenterC83#3094
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#14581751C2semidirect
3. #19128026C2semidirect
4. #19128020C2semidirect
5. #19128015C2semidirect
6. #19127743C2semidirect
7. #19127707C2semidirect
8. #407×C83C2semidirect
18. #14581751C22semidirect
20. C2×C4⋊Dic83C22semidirect
21. C2×C4⋊Dic83C22semidirect
25. #14581726C22semidirect
26. #14581724C22semidirect
27. #14581714C22semidirect
28. #14581714C22semidirect
31. C22⋊C8×C83C22semidirect
32. C22⋊C8×C83C22semidirect
126. C83#3094semidirect
134. #407D83semidirect
157. C22⋊C8C2×D83semidirect
158. C22⋊C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145817662
C2×#145817512
#191280262
#191280202
#191280152
#191277432
#191277072
#407×C832
#309483