Finite Group Databasebeta

← #24917104

Subgroups — #24917104

Order 10624.

492 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#13050577C211yesmaximal, characteristic, complemented
2C2×#14581799C211yesmaximal, characteristic, complemented
3#19128265C211yesmaximal, characteristic
4#19127949C214yesmaximal, complemented
5#19127932C214yesmaximal, complemented
6#19127737C211yesmaximal, characteristic, complemented
7#19127710C211yesmaximal, characteristic, complemented
8#19127705C211yesmaximal, characteristic, complemented
9#536×C83C211yesmaximal, characteristic, complemented
10C22×D332C2211yescharacteristic
11C2×#13050577—24no—
12C2×#13050577C2212yes—
13C2×#13050577C2212yes—
14#14581799—24no—
15C2×#13050574C2211yescharacteristic
16C2×#13050574C2212yescomplemented
17C22×C83⋊C8C2211yescharacteristic
18C22×C83⋊C8C2212yescomplemented
19#14581788—21no—
20#14581781—21no—
21#14581766C2214yes—
22#14581760—22no—
23#14581751C2211yescharacteristic
24#14581725C2212yescomplemented
25#14581723C2212yescomplemented
26#14581716C2214yescomplemented
27#14581712C2214yes—
28C2×#41×C83C2211yescharacteristic
29C22×Q8×C83C2211yescharacteristic
30C4⋊Q8×C83—21no—
31#124×C83—21no—
32#123×C83—22no—
33C22⋊Q8×C83C2214yescomplemented
34#117×C83C2211yescharacteristic
35C23×D83—21no—
36C2×D332—22no—
37C2×D332—22no—
38C2×D332—21no—
39C2×D332C2312yes—
40C2×C22⋊D83—24no—
41#13050577—24no—
42#13050577—24no—
43#13050574—22no—
44#13050574—24no—
45C2×C83⋊C8—24no—
46C2×C83⋊C8D414yescomplemented
47C2×C83⋊C8C2312yes—
48C2×C83⋊C8C2311yescharacteristic
49C2×C83⋊C8C2311yescharacteristic
50C4⋊Dic83C2312yes—
51C22⋊Dic83—21no—
52(C2×D83)⋊C4—24no—
53C4×Dic83—21no—
54#41×C83—22no—
55#41×C83—22no—
56C2×Q8×C83—21no—
57C2×Q8×C83—22no—
58C2×Q8×C83—22no—
59C2×Q8×C83D411yescharacteristic
60C2×Q8×C83C2312yes—
61C2×D4×C83D412yes—
62C2×D4×C83D411yescharacteristic
63C22×C4×C83—21no—
64C22×C4×C83C2312yes—
65C22×C4×C83C2311yescharacteristic
66C4⋊C4×C83—24no—
67C4⋊C4×C83C2312yes—
68C22⋊C4×C83—21no—
69C22⋊C4×C83—24no—
70C42×C83—21no—
71C23×C83C2×D412yes—
72C23×C83C2×D411yescharacteristic
73C22×D83—22no—
74C22×D83—42no—
75C22×D83—82no—
76Q8×C83—22no—
77Q8×C83—22no—
78Q8×C83—22no—
79Q8×C83—42no—
80D4×C83—24no—
81D4×C83—22no—
82C2×C4×C83—22no—
83C2×C4×C83—22no—
84C2×C4×C83—22no—
85C2×C4×C83—22no—
86C2×C4×C83—42no—
87C2×C4×C83C2×D412yes—
88C2×C4×C83C2×D412yes—
89C2×C4×C83C2×D411yescharacteristic
90C2×C4×C83#4112yes—
91C2×C4×C83C2411yescharacteristic
92C22⋊D83—84no—
93C2×Dic83—22no—
94D332—22no—
95D332—22no—
96D332—42no—
97C83⋊C8—22no—
98C83⋊C8C2×D412yes—
99C83⋊C8C2×D412yes—
100C22×C83—22no—
101C22×C83—22no—
102C22×C83—24no—
103C22×C83#12112yes—
104C22×C83C22×D411yescharacteristic
105C2×D83—42no—
106C2×D83—44no—
107C2×D83—84no—
108C4×C83—22no—
109C4×C83—42no—
110C4×C83—42no—
111C4×C83#12112yes—
112C4×C83C22×D411yescharacteristic
113C4×C83C2×#4111yescharacteristic
114Dic83—42no—
115C2×C83—22no—
116C2×C83—42no—
117C2×C83C2×#12111yescharacteristic
118C2×C83#57712yes—
119D83—82no—
120#3057—831nomaximal, Sylow
121C83#305711yescharacteristic, complemented, Sylow
122C22×SD16—831no—
123C2×#131—831no—
124#536D8311yescharacteristic, complemented
125#535—831no—
126#468—834no—
127#465—834no—
128#427—831no—
129#419—831no—
130#418—831no—
131C2×#41C2×D8311yescharacteristic
132C22×Q8C2×D8311yescharacteristic
133C22×D4—831no—
134C2×SD16—832no—
135C2×SD16—832no—
136C2×SD16—1664no—
137#131—1664no—
138C2×M16—831no—
139C2×M16—832no—
140C22×C8—831no—
141C22×C8—832no—
142C4⋊Q8—21no—
143#127—1661no—
144#124—21no—
145#124—1661no—
146#123—22no—
147C22⋊Q8C2×D8314yescomplemented
148#121—834no—
149#120—1662no—
150#117—831no—
151#117C2×D8311yescharacteristic
152#107—832no—
153#106—832no—
154#103—834no—
155#102—834no—
156C24—1661no—
157#41—22no—
158#41—22no—
159C2×Q8—21no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8C22⋊D8311yescharacteristic
163C2×Q8C22×D8312yes—
164C2×D4—832no—
165C2×D4—1664no—
166C2×D4—1662no—
167C2×D4—1661no—
168C2×D4—1662no—
169C2×D4C22⋊D8312yes—
170C2×D4C22⋊D8311yescharacteristic
171C22×C4—21no—
172C22×C4C22×D8312yes—
173C22×C4C22×D8311yescharacteristic
174SD16—1664no—
175SD16—1664no—
176M16—1664no—
177M16—1662no—
178C2×C8—834no—
179C2×C8—831no—
180C2×C8—831no—
181C2×C8—832no—
182C2×C8—1664no—
183C4⋊C4—24no—
184C4⋊C4—832no—
185C4⋊C4C22×D8312yes—
186C22⋊C4—21no—
187C22⋊C4—24no—
188C22⋊C4—1661no—
189C22⋊C4—1664no—
190C42—21no—
191C42—1661no—
192C23—3322no—
193C23—6642no—
194C23—1662no—
195C23C2×C22⋊D8312yes—
196C23C2×C22⋊D8311yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—22no—
200Q8—42no—
201D4—24no—
202D4—22no—
203D4—3322no—
204D4—6644no—
205D4—1662no—
206D4—1662no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—1662no—
213C2×C4D4×D8312yes—
214C2×C4C2×C22⋊D8312yes—
215C2×C4C2×C22⋊D8311yescharacteristic
216C2×C4#1305058912yes—
217C2×C4C23×D8311yescharacteristic
218C8—832no—
219C8—832no—
220C8—1662no—
221C22—22no—
222C22—22no—
223C22—24no—
224C22—3322no—
225C22—3324no—
226C22—6644no—
227C22#1458176412yes—
228C22C22×C22⋊D8311yescharacteristic
229C4—22no—
230C4—42no—
231C4—42no—
232C4—3322no—
233C4#1458176412yes—
234C4C2×D4×D8311yescharacteristic
235C4C2×#1305058911yescharacteristic
236C2—22no—
237C2—42no—
238C2—6642no—
239C2C2×#1458176411yescharacteristic
240C2#1912850912yes—
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#536×C83C2
socleC22×C83C22×D4
prime radical#19128265C2
hypercenter#536D83
hypocenterC83#3057
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C22×#13050577C2semidirect
2. C2×#14581799C2semidirect
4. #19127949C2semidirect
5. #19127932C2semidirect
6. #19127737C2semidirect
7. #19127710C2semidirect
8. #19127705C2semidirect
9. #536×C83C2semidirect
16. C2×#13050574C22semidirect
18. C22×C83⋊C8C22semidirect
24. #14581725C22semidirect
25. #14581723C22semidirect
26. #14581716C22semidirect
33. C22⋊Q8×C83C22semidirect
46. C2×C83⋊C8D4semidirect
121. C83#3057semidirect
124. #536D83semidirect
147. C22⋊Q8C2×D83semidirect

Maximal subgroups

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

subgroupindex
C22×#130505772
C2×#145817992
#191282652
#191279492
#191279322
#191277372
#191277102
#191277052
#536×C832
#305783