Finite Group Databasebeta

← #24917099

Subgroups — #24917099

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×#13050578C211yesmaximal, characteristic, complemented
2C2×#14581799C211yesmaximal, characteristic, complemented
3#19128272C211yesmaximal, characteristic, complemented
4#19127945C214yesmaximal, complemented
5#19127928C214yesmaximal, complemented
6#19127737C211yesmaximal, characteristic, complemented
7#19127709C211yesmaximal, characteristic, complemented
8#19127706C211yesmaximal, characteristic, complemented
9#535×C83C211yesmaximal, characteristic
10C22×Dic166C2211yescharacteristic
11C2×#13050578—24no—
12C2×#13050578C2212yes—
13C2×#13050578C2212yes—
14#14581799—24no—
15C2×#13050574C2211yescharacteristic
16C2×#13050574C2212yescomplemented
17C22×C83⋊C8C2211yescharacteristic
18C22×C83⋊C8C2212yescomplemented
19#14581790—21no—
20#14581779—21no—
21#14581773—22no—
22#14581771C2214yescomplemented
23#14581751C2211yescharacteristic
24#14581725C2212yescomplemented
25#14581723C2212yescomplemented
26#14581715C2214yescomplemented
27#14581713C2214yes—
28C2×#41×C83C2211yescharacteristic
29C22×D4×C83C2211yescharacteristic
30#127×C83—21no—
31#124×C83—21no—
32#121×C83C2214yes—
33#120×C83—22no—
34#117×C83C2211yescharacteristic
35C2×Dic166—22no—
36C2×Dic166—22no—
37C2×Dic166—21no—
38C2×Dic166C2312yes—
39C22×Dic83—21no—
40#13050578—24no—
41#13050578—24no—
42#13050574—22no—
43#13050574—24no—
44C2×C83⋊C8—24no—
45C2×C83⋊C8D414yescomplemented
46C2×C83⋊C8C2312yes—
47C2×C83⋊C8C2311yescharacteristic
48C2×C83⋊C8C2311yescharacteristic
49C4⋊Dic83C2312yes—
50Dic83⋊C4—24no—
51C22⋊Dic83—24no—
52C22⋊Dic83—21no—
53C4×Dic83—21no—
54C24×C83—21no—
55#41×C83—22no—
56#41×C83—22no—
57C2×Q8×C83D411yescharacteristic
58C2×D4×C83—24no—
59C2×D4×C83—22no—
60C2×D4×C83—21no—
61C2×D4×C83—22no—
62C2×D4×C83D412yes—
63C2×D4×C83D411yescharacteristic
64C2×D4×C83C2312yes—
65C22×C4×C83C2312yes—
66C22×C4×C83C2311yescharacteristic
67C4⋊C4×C83C2312yes—
68C22⋊C4×C83—21no—
69C22⋊C4×C83—24no—
70C42×C83—21no—
71C23×C83—22no—
72C23×C83—42no—
73C23×C83—82no—
74C23×C83C2×D412yes—
75C23×C83C2×D411yescharacteristic
76Q8×C83—22no—
77D4×C83—24no—
78D4×C83—22no—
79D4×C83—22no—
80D4×C83—22no—
81D4×C83—42no—
82D4×C83—84no—
83C2×C4×C83—22no—
84C2×C4×C83—22no—
85C2×C4×C83—22no—
86C2×C4×C83C2×D412yes—
87C2×C4×C83C2×D412yes—
88C2×C4×C83C2×D411yescharacteristic
89C2×C4×C83#4112yes—
90C2×C4×C83C2411yescharacteristic
91C2×Dic83—22no—
92C2×Dic83—22no—
93C2×Dic83—42no—
94Dic166—22no—
95Dic166—22no—
96Dic166—42no—
97C83⋊C8—22no—
98C83⋊C8C2×D412yes—
99C83⋊C8C2×D412yes—
100C22×C83—22no—
101C22×C83—22no—
102C22×C83—24no—
103C22×C83—42no—
104C22×C83—44no—
105C22×C83—84no—
106C22×C83#12112yes—
107C22×C83C22×D411yescharacteristic
108C4×C83—22no—
109C4×C83—42no—
110C4×C83#12112yes—
111C4×C83C22×D411yescharacteristic
112C4×C83C2×#4111yescharacteristic
113Dic83—42no—
114Dic83—42no—
115C2×C83—22no—
116C2×C83—42no—
117C2×C83—82no—
118C2×C83C2×#12111yescharacteristic
119C2×C83#57712yes—
120#3057—831nomaximal, Sylow
121C83#305711yescharacteristic, complemented, Sylow
122C22×SD16—831no—
123C2×#131—831no—
124#536—831no—
125#535D8311yescharacteristic
126#468—834no—
127#465—834no—
128#427—831no—
129#419—831no—
130#418—831no—
131C2×#41C2×D8311yescharacteristic
132C22×Q8—831no—
133C22×D4C2×D8311yescharacteristic
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—1661no—
143#127—21no—
144#124—21no—
145#124—1661no—
146#123—1662no—
147C22⋊Q8—834no—
148#121C2×D8314yes—
149#120—22no—
150#117—831no—
151#117C2×D8311yescharacteristic
152#107—832no—
153#106—832no—
154#103—834no—
155#102—834no—
156C24—21no—
157#41—22no—
158#41—22no—
159C2×Q8—832no—
160C2×Q8—1662no—
161C2×Q8—1662no—
162C2×Q8—1661no—
163C2×Q8C22⋊D8311yescharacteristic
164C2×D4—24no—
165C2×D4—22no—
166C2×D4—21no—
167C2×D4—22no—
168C2×D4C22⋊D8312yes—
169C2×D4C22⋊D8311yescharacteristic
170C2×D4C22×D8312yes—
171C22×C4—1661no—
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—832no—
184C4⋊C4—1664no—
185C4⋊C4C22×D8312yes—
186C22⋊C4—21no—
187C22⋊C4—24no—
188C22⋊C4—1661no—
189C22⋊C4—1664no—
190C42—21no—
191C42—1661no—
192C23—22no—
193C23—42no—
194C23—82no—
195C23C2×C22⋊D8312yes—
196C23C2×C22⋊D8311yescharacteristic
197Q8—22no—
198Q8—3322no—
199Q8—1662no—
200Q8—1662no—
201D4—24no—
202D4—22no—
203D4—22no—
204D4—22no—
205D4—42no—
206D4—84no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—3322no—
211C2×C4—1662no—
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—42no—
225C22—44no—
226C22—84no—
227C22#1458176412yes—
228C22C22×C22⋊D8311yescharacteristic
229C4—22no—
230C4—42no—
231C4—3322no—
232C4—3322no—
233C4#1458176412yes—
234C4C2×D4×D8311yescharacteristic
235C4C2×#1305058911yescharacteristic
236C2—22no—
237C2—42no—
238C2—82no—
239C2C2×#1458176411yescharacteristic
240C2#1912851212yes—
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting, prime radical#535×C83C2
socleC22×C83C22×D4
hypercenter#535D83
hypocenterC83#3057
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C22×#13050578C2semidirect
2. C2×#14581799C2semidirect
3. #19128272C2semidirect
4. #19127945C2semidirect
5. #19127928C2semidirect
6. #19127737C2semidirect
7. #19127709C2semidirect
8. #19127706C2semidirect
16. C2×#13050574C22semidirect
18. C22×C83⋊C8C22semidirect
22. #14581771C22semidirect
24. #14581725C22semidirect
25. #14581723C22semidirect
26. #14581715C22semidirect
45. C2×C83⋊C8D4semidirect
121. C83#3057semidirect

Maximal subgroups

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

subgroupindex
C22×#130505782
C2×#145817992
#191282722
#191279452
#191279282
#191277372
#191277092
#191277062
#535×C832
#305783