Finite Group Databasebeta

← #24916986

Subgroups — #24916986

Order 10624.

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

normal subgroups only

This group has 490 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
2#19128265C211yesmaximal, characteristic
3#19128179C211yesmaximal, characteristic, complemented
4#19127925C214yesmaximal, complemented
5#19127899C214yesmaximal, complemented
6#19127729C211yesmaximal, characteristic, complemented
7#19127707C211yesmaximal, characteristic, complemented
8#19127697C211yesmaximal, characteristic, complemented
9C2×#137×C83C211yesmaximal, characteristic, complemented
10C22×D332C2211yescharacteristic
11C2×#13050577—24no—
12C2×#13050577C2214yes—
13C22×C83⋊C8C2211yescharacteristic
14#14581788C2212yes—
15#14581781C2212yes—
16#14581766—22no—
17#14581760—22no—
18Q8×Dic83—22no—
19Q8×Dic83C2214yescomplemented
20#14581751—22no—
21#14581751C2211yescharacteristic
22#14581751C2211yescharacteristic
23Dic83⋊C8C2214yescomplemented
24#14581716C2212yescomplemented
25#14581714C2214yes—
26#14581713C2212yescomplemented
27C2×#41×C83C2211yescharacteristic
28C22×Q8×C83C2211yescharacteristic
29#137×C83—24no—
30C2×Q16×C83C2212yescomplemented
31C2×SD16×C83C2212yescomplemented
32C2×M16×C83C2211yescharacteristic
33C23×D83—21no—
34C2×D332—22no—
35C2×D332—22no—
36C2×D332—21no—
37C2×D332C2312yes—
38C2×C22⋊D83—24no—
39#13050577—28no—
40C2×C83⋊C8—22no—
41C2×C83⋊C8C2312yes—
42C4⋊Dic83—24no—
43C4⋊Dic83D412yes—
44C4⋊Dic83C2312yes—
45C22⋊Dic83—21no—
46C22⋊Dic83D412yes—
47(C2×D83)⋊C4—24no—
48C4×Dic83—21no—
49C4×Dic83—24no—
50C4×Dic83C2312yes—
51#41×C83—22no—
52#41×C83—22no—
53C2×Q8×C83—21no—
54C2×Q8×C83—22no—
55C2×Q8×C83D414yescomplemented
56C2×Q8×C83C2311yescharacteristic
57C2×Q8×C83C2312yes—
58C2×D4×C83—21no—
59C2×D4×C83C2311yescharacteristic
60C22×C4×C83—21no—
61C22×C4×C83—21no—
62C22×C4×C83C2311yescharacteristic
63Q16×C83—24no—
64SD16×C83—24no—
65M16×C83—22no—
66C2×C8×C83C2312yes—
67C23×C83—21no—
68C23×C83C2×D411yescharacteristic
69C22×D83—22no—
70C22×D83—42no—
71C22×D83—82no—
72Q8×C83—24no—
73Q8×C83—22no—
74Q8×C83—21no—
75Q8×C83—21no—
76Q8×C83C2×D414yes—
77D4×C83—21no—
78D4×C83—21no—
79D4×C83—42no—
80C2×C4×C83—24no—
81C2×C4×C83—22no—
82C2×C4×C83—21no—
83C2×C4×C83—42no—
84C2×C4×C83C2×D411yescharacteristic
85C2×C4×C83C2×D412yes—
86C2×C4×C83#4112yes—
87C2×C4×C83C2411yescharacteristic
88C22⋊D83—84no—
89C2×Dic83—22no—
90C2×Dic83C2×D414yes—
91D332—22no—
92D332—22no—
93D332—42no—
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—24no—
105C4×C83—41no—
106C4×C83#12112yes—
107C4×C83C22×D411yescharacteristic
108C4×C83C2×#4111yescharacteristic
109Dic83—24no—
110Dic83—42no—
111C2×C83—22no—
112C2×C83—81no—
113C2×C83C2×#12111yescharacteristic
114C2×C83#57712yes—
115D83—82no—
116#3043—831nomaximal, Sylow
117C83#304311yescharacteristic, complemented, Sylow
118C2×#137D8311yescharacteristic, complemented
119C22×SD16—831no—
120#535—831no—
121#519—831no—
122#464—834no—
123#460—834no—
124#424—831no—
125#418—831no—
126#416—831no—
127C2×#41C2×D8311yescharacteristic
128C22×Q8C2×D8311yescharacteristic
129C22×D4—831no—
130#137—24no—
131C2×Q16C2×D8312yescomplemented
132C2×SD16—834no—
133C2×SD16—1664no—
134C2×SD16C2×D8312yescomplemented
135C2×M16C2×D8311yescharacteristic
136C22×C8—831no—
137#127—832no—
138#124—832no—
139#121—1662no—
140#120—1662no—
141C4×Q8—834no—
142C4×Q8—1662no—
143#117—831no—
144#117—831no—
145#117—1662no—
146C4⋊C8—834no—
147#103—832no—
148#102—832no—
149#102—834no—
150C24—1661no—
151#41—22no—
152#41—22no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8C22⋊D8314yescomplemented
156C2×Q8C22×D8311yescharacteristic
157C2×Q8C22×D8312yes—
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—21no—
167C22×C4C22×D8311yescharacteristic
168Q16—24no—
169SD16—24no—
170SD16—1668no—
171M16—22no—
172C2×C8—832no—
173C2×C8—1662no—
174C2×C8C22×D8312yes—
175C4⋊C4—832no—
176C4⋊C4—832no—
177C4⋊C4—1664no—
178C22⋊C4—832no—
179C22⋊C4—1664no—
180C22⋊C4—1661no—
181C42—832no—
182C42—1664no—
183C42—1661no—
184C23—21no—
185C23—3322no—
186C23—6642no—
187C23—1662no—
188C23D4×D8311yescharacteristic
189Q8—24no—
190Q8—22no—
191Q8—21no—
192Q8—21no—
193Q8C2×C22⋊D8314yes—
194D4—21no—
195D4—21no—
196D4—42no—
197D4—3322no—
198D4—6644no—
199D4—1662no—
200D4—1662no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—42no—
205C2×C4—834no—
206C2×C4—1662no—
207C2×C4D4×D8311yescharacteristic
208C2×C4C2×C22⋊D8312yes—
209C2×C4#1305058912yes—
210C2×C4C23×D8311yescharacteristic
211C8—22no—
212C8—1662no—
213C22—22no—
214C22—41no—
215C22—42no—
216C22—3322no—
217C22—3324no—
218C22—6644no—
219C22#1458176412yes—
220C22C2×D4×D8311yescharacteristic
221C4—24no—
222C4—41no—
223C4—3322no—
224C4—1664no—
225C4#1458176412yes—
226C4C22×C22⋊D8311yescharacteristic
227C4C2×#1305058911yescharacteristic
228C2—22no—
229C2—81no—
230C2—6642no—
231C2C2×#1458176411yescharacteristic
232C2#1912851012yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
FittingC2×#137×C83C2
socleC22×C83C22×D4
prime radical#19128265C2
hypercenterC2×#137D83
hypocenterC83#3043
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C22×#13050577C2semidirect
3. #19128179C2semidirect
4. #19127925C2semidirect
5. #19127899C2semidirect
6. #19127729C2semidirect
7. #19127707C2semidirect
8. #19127697C2semidirect
9. C2×#137×C83C2semidirect
19. Q8×Dic83C22semidirect
23. Dic83⋊C8C22semidirect
24. #14581716C22semidirect
26. #14581713C22semidirect
30. C2×Q16×C83C22semidirect
31. C2×SD16×C83C22semidirect
55. C2×Q8×C83D4semidirect
117. C83#3043semidirect
118. C2×#137D83semidirect
131. C2×Q16C2×D83semidirect
134. C2×SD16C2×D83semidirect
155. C2×Q8C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C22×#130505772
#191282652
#191281792
#191279252
#191278992
#191277292
#191277072
#191276972
C2×#137×C832
#304383