Finite Group Databasebeta

← #24916987

Subgroups — #24916987

Order 10624.

490 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 133 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×#13050579C211yesmaximal, characteristic, complemented
3#19128265C211yesmaximal, characteristic
4#19128178C211yesmaximal, characteristic, complemented
5#19127926C214yesmaximal, complemented
6#19127900C214yesmaximal, complemented
7#19127727C211yesmaximal, characteristic, complemented
8#19127705C211yesmaximal, characteristic, complemented
9#416×C83C211yesmaximal, characteristic, complemented
10C22×Dic166C2211yescharacteristic
11C22×D332C2211yescharacteristic
12#14581822—24no—
13C2×#13050580C2212yescomplemented
14C2×#13050579—24no—
15C2×#13050579C2214yes—
16C2×#13050578C2212yescomplemented
17C2×#13050574C2211yescharacteristic
18#14581788C2212yes—
19#14581781C2212yes—
20#14581766—22no—
21#14581760—22no—
22C4×Dic166—22no—
23#14581757C2214yescomplemented
24#14581751C2211yescharacteristic
25#14581748—22no—
26Dic83⋊C8C2214yescomplemented
27#14581712C2214yes—
28C2×#41×C83C2211yescharacteristic
29C22×C8×C83C2211yescharacteristic
30#117×C83C2211yescharacteristic
31#103×C83C2212yescomplemented
32#102×C83C2212yescomplemented
33C23×D83—21no—
34C2×Dic166—21no—
35C2×Dic166—22no—
36C2×Dic166D414yescomplemented
37C2×Dic166C2312yes—
38C2×D332—22no—
39C2×D332—22no—
40C2×D332—21no—
41C2×D332C2312yes—
42C2×C22⋊D83—24no—
43C22×Dic83—21no—
44#13050580—24no—
45#13050579—28no—
46#13050578—24no—
47#13050574—22no—
48C2×C83⋊C8C2312yes—
49C4⋊Dic83D412yes—
50Dic83⋊C4—24no—
51C22⋊Dic83D412yes—
52(C2×D83)⋊C4—24no—
53C4×Dic83—24no—
54C4×Dic83C2312yes—
55#41×C83—22no—
56#41×C83—22no—
57C2×Q8×C83C2311yescharacteristic
58C2×D4×C83—21no—
59C2×D4×C83C2311yescharacteristic
60C22×C4×C83—21no—
61C22×C4×C83C2311yescharacteristic
62C2×C8×C83—22no—
63C2×C8×C83C2312yes—
64C4⋊C4×C83C2312yes—
65C22⋊C4×C83—21no—
66C42×C83—21no—
67C23×C83—21no—
68C23×C83C2×D411yescharacteristic
69C22×D83—22no—
70C22×D83—42no—
71C22×D83—82no—
72Q8×C83—21no—
73Q8×C83—21no—
74D4×C83—21no—
75D4×C83—21no—
76D4×C83—42no—
77C2×C4×C83—22no—
78C2×C4×C83—21no—
79C2×C4×C83—42no—
80C2×C4×C83C2×D411yescharacteristic
81C2×C4×C83C2×D412yes—
82C2×C4×C83#4112yes—
83C2×C4×C83C2411yescharacteristic
84C22⋊D83—84no—
85C2×Dic83—24no—
86C2×Dic83—22no—
87C2×Dic83C2×D414yes—
88D332—22no—
89D332—22no—
90D332—42no—
91Dic166—24no—
92Dic166—22no—
93Dic166C2×D414yes—
94C8×C83—22no—
95C83⋊C8—22no—
96C22×C83—22no—
97C22×C83—41no—
98C22×C83—42no—
99C22×C83#12112yes—
100C22×C83C22×D411yescharacteristic
101C2×D83—42no—
102C2×D83—44no—
103C2×D83—84no—
104C4×C83—41no—
105C4×C83—42no—
106C4×C83#12112yes—
107C4×C83C22×D411yescharacteristic
108C4×C83C2×#4111yescharacteristic
109Dic83—24no—
110Dic83—24no—
111C2×C83—22no—
112C2×C83—81no—
113C2×C83C2×#12111yescharacteristic
114C2×C83#57712yes—
115D83—82no—
116#3043—831nomaximal, Sylow
117C83#304311yescharacteristic, complemented, Sylow
118C2×#137—831no—
119C22×SD16—831no—
120#535—831no—
121#519—831no—
122#464—834no—
123#460—834no—
124#424—831no—
125#418—831no—
126#416D8311yescharacteristic, complemented
127C2×#41C2×D8311yescharacteristic
128C22×Q8—831no—
129C22×D4—831no—
130#137—1664no—
131C2×Q16—832no—
132C2×SD16—832no—
133C2×SD16—834no—
134C2×SD16—1664no—
135C2×M16—831no—
136C22×C8C2×D8311yescharacteristic
137#127—832no—
138#124—832no—
139#121—1662no—
140#120—1662no—
141C4×Q8—834no—
142C4×Q8—1662no—
143#117—831no—
144#117—1662no—
145#117C2×D8311yescharacteristic
146C4⋊C8—834no—
147#103C2×D8312yescomplemented
148#102—834no—
149#102C2×D8312yescomplemented
150C24—1661no—
151#41—22no—
152#41—22no—
153C2×Q8—834no—
154C2×Q8—832no—
155C2×Q8—1661no—
156C2×Q8—1662no—
157C2×Q8C22×D8311yescharacteristic
158C2×D4—21no—
159C2×D4—832no—
160C2×D4—1664no—
161C2×D4—1661no—
162C2×D4—1662no—
163C2×D4—1662no—
164C2×D4C22×D8311yescharacteristic
165C22×C4—21no—
166C22×C4—1661no—
167C22×C4C22×D8311yescharacteristic
168Q16—1664no—
169SD16—1668no—
170SD16—1664no—
171M16—1662no—
172C2×C8—22no—
173C2×C8—832no—
174C2×C8C22×D8312yes—
175C4⋊C4—832no—
176C4⋊C4—1664no—
177C4⋊C4C22×D8312yes—
178C22⋊C4—21no—
179C22⋊C4—832no—
180C22⋊C4—1664no—
181C42—21no—
182C42—832no—
183C42—1664no—
184C23—21no—
185C23—3322no—
186C23—6642no—
187C23—1662no—
188C23D4×D8311yescharacteristic
189Q8—21no—
190Q8—21no—
191Q8—834no—
192Q8—1664no—
193Q8—1662no—
194D4—21no—
195D4—21no—
196D4—42no—
197D4—3322no—
198D4—6644no—
199D4—1662no—
200D4—1662no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—42no—
204C2×C4—834no—
205C2×C4—1664no—
206C2×C4—1662no—
207C2×C4D4×D8311yescharacteristic
208C2×C4D4×D8312yes—
209C2×C4#1305059012yes—
210C2×C4C23×D8311yescharacteristic
211C8—22no—
212C8—1662no—
213C22—22no—
214C22—41no—
215C22—42no—
216C22—3322no—
217C22—3324no—
218C22—6644no—
219C22#1458176512yes—
220C22C2×D4×D8311yescharacteristic
221C4—41no—
222C4—42no—
223C4—1664no—
224C4—1664no—
225C4#1458176512yes—
226C4C2×D4×D8311yescharacteristic
227C4C2×#1305059011yescharacteristic
228C2—22no—
229C2—81no—
230C2—6642no—
231C2C2×#1458176511yescharacteristic
232C2#1912850812yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#416×C83C2
socleC22×C83C22×D4
prime radical#19128265C2
hypercenter#416D83
hypocenterC83#3043
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#14581822C2semidirect
2. C22×#13050579C2semidirect
4. #19128178C2semidirect
5. #19127926C2semidirect
6. #19127900C2semidirect
7. #19127727C2semidirect
8. #19127705C2semidirect
9. #416×C83C2semidirect
13. C2×#13050580C22semidirect
16. C2×#13050578C22semidirect
23. #14581757C22semidirect
26. Dic83⋊C8C22semidirect
31. #103×C83C22semidirect
32. #102×C83C22semidirect
36. C2×Dic166D4semidirect
117. C83#3043semidirect
126. #416D83semidirect
147. #103C2×D83semidirect
149. #102C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818222
C22×#130505792
#191282652
#191281782
#191279262
#191279002
#191277272
#191277052
#416×C832
#304383