Finite Group Databasebeta

← #27180042

Subgroups — #27180042

Order 12928.

458 conjugacy classes of subgroups; only the 217 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 217 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
1#20265944C212yesmaximal, complemented
2#20265299C214yesmaximal, complemented
3#20265294C214yesmaximal, complemented
4#20265226C211yesmaximal, characteristic, complemented
5C2×#15692182C212yesmaximal, complemented
6M16×Dic101C211yesmaximal, characteristic, complemented
7#575×C101C211yesmaximal, characteristic
8#15692262—24no—
9#15692262C2218yescomplemented
10C2×#13108058—24no—
11C2×#13108058C2212yes—
12C2×#13108058C2211yescharacteristic
13C2×C4×Dic101C2211yescharacteristic
14#15692191C2214yescomplemented
15#15692182C2214yescomplemented
16#15692182C2214yescomplemented
17#15692165C2212yescomplemented
18C8×Dic101C2212yescomplemented
19C2×#41×C101C2212yes—
20#137×C101C414yescomplemented
21AGL(1,ℤ/8)×C101C414yescomplemented
22#135×C101C2214yes—
23C2×M16×C101C2211yescharacteristic
24C22×Dic101—21no—
25#13108058—22no—
26#13108058—28no—
27#13108058D414yes—
28#13108058C2314yes—
29C2×#11717579—22no—
30C2×#11717579—28no—
31C2×#11717579C2312yes—
32C4×Dic101—21no—
33C4×Dic101C2311yescharacteristic
34C4×Dic101C2311yescharacteristic
35#41×C101—22no—
36#41×C101—22no—
37#41×C101C2×C414yes—
38#41×C101C2314yes—
39C2×Q8×C101C2×C412yes—
40C2×D4×C101—22no—
41C2×D4×C101C2×C412yes—
42C22×C4×C101—22no—
43C22×C4×C101C2311yescharacteristic
44Q16×C101C2×C414yescomplemented
45SD16×C101C2×C418yescomplemented
46D8×C101C2×C414yescomplemented
47M16×C101C2×C414yes—
48C2×C8×C101C2312yes—
49C23×C101—22no—
50C23×C101#4111yescharacteristic
51Q8×C101—22no—
52Q8×C101C22×C414yes—
53D4×C101—24no—
54D4×C101—24no—
55D4×C101—22no—
56D4×C101C22×C414yes—
57C2×C4×C101—22no—
58C2×C4×C101—24no—
59C2×C4×C101—24no—
60C2×C4×C101C22×C412yes—
61C2×C4×C101C2×D412yes—
62C2×C4×C101#4111yescharacteristic
63C2×C4×C101C2411yescharacteristic
64C2×Dic101—21no—
65C2×Dic101—21no—
66C2×Dic101—42no—
67C8×C101C22×C414yes—
68#11717579—24no—
69#11717579C2×D414yes—
70C22×C101—24no—
71C22×C101—41no—
72C22×C101—82no—
73C22×C101C4×D412yes—
74C22×C101C2×#4111yescharacteristic
75C4×C101—24no—
76C4×C101C4×D412yes—
77C4×C101C23×C411yescharacteristic
78C4×C101C22×D411yescharacteristic
79Dic101—22no—
80Dic101—41no—
81C2×C101—21no—
82C2×C101—22no—
83C2×C101—44no—
84C2×C101C2×C4×D411yescharacteristic
85#2956—1011nomaximal, Sylow
86C101#295611yescharacteristic, complemented, Sylow
87#575D10111yescharacteristic
88#568—1012no—
89#444—1014no—
90#443—1014no—
91#430—1011no—
92C2×#104—1012no—
93C4×M16—1011no—
94C2×#41C2×D10112yes—
95#137Dic10114yescomplemented
96AGL(1,ℤ/8)Dic10114yescomplemented
97#135C2×D10114yes—
98#131—1018no—
99#131—2024no—
100C2×M16—1011no—
101C2×M16—1012no—
102C2×M16—2024no—
103C2×M16C2×D10111yescharacteristic
104C2×C42—1011no—
105#108—1014no—
106#104—1014no—
107#104—1014no—
108#97—1012no—
109C4×C8—1012no—
110#41—22no—
111#41—22no—
112#41C2×Dic10114yes—
113#41C22×D10114yes—
114C2×Q8C2×Dic10112yes—
115C2×D4—22no—
116C2×D4C2×Dic10112yes—
117C22×C4—22no—
118C22×C4—2021no—
119C22×C4C22×D10111yescharacteristic
120Q16C2×Dic10114yescomplemented
121SD16C2×Dic10118yescomplemented
122D8C2×Dic10114yescomplemented
123M16—1014no—
124M16—1014no—
125M16—2028no—
126M16—2022no—
127M16C2×Dic10114yes—
128C2×C8—1012no—
129C2×C8—2028no—
130C2×C8—2022no—
131C2×C8C22×D10112yes—
132C42—1011no—
133C42—1011no—
134C42—2021no—
135C23—22no—
136C23#1310807311yescharacteristic
137Q8—22no—
138Q8C22×Dic10114yes—
139D4—24no—
140D4—24no—
141D4—22no—
142D4C22×Dic10114yes—
143C2×C4—22no—
144C2×C4—24no—
145C2×C4—24no—
146C2×C4—4042no—
147C2×C4—2021no—
148C2×C4—2021no—
149C2×C4C22×Dic10112yes—
150C2×C4D4×D10112yes—
151C2×C4#1310807311yescharacteristic
152C2×C4C23×D10111yescharacteristic
153C8—1014no—
154C8—2024no—
155C8C22×Dic10114yes—
156C22—24no—
157C22—41no—
158C22—82no—
159C22D4×Dic10112yes—
160C22C2×#1310807311yescharacteristic
161C4—24no—
162C4—4041no—
163C4—2022no—
164C4D4×Dic10112yes—
165C4C23×Dic10111yescharacteristic
166C4C2×D4×D10111yescharacteristic
167C2—21no—
168C2—22no—
169C2—44no—
170C2C2×D4×Dic10111yescharacteristic
subgroupG/H
centerC4C2×D4×D101
commutatorC4×C101C23×C4
FrattiniC2×C4C23×D101
Fitting, prime radical#575×C101C2
socleC2×C101C2×C4×D4
cosocleC2×C4×C101C24
hypercenter#575D101
hypocenterC101#2956
LCSG ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #20265944C2semidirect
2. #20265299C2semidirect
3. #20265294C2semidirect
4. #20265226C2semidirect
5. C2×#15692182C2semidirect
6. M16×Dic101C2semidirect
9. #15692262C22semidirect
14. #15692191C22semidirect
15. #15692182C22semidirect
16. #15692182C22semidirect
17. #15692165C22semidirect
18. C8×Dic101C22semidirect
20. #137×C101C4semidirect
21. AGL(1,ℤ/8)×C101C4semidirect
44. Q16×C101C2×C4semidirect
45. SD16×C101C2×C4semidirect
46. D8×C101C2×C4semidirect
86. C101#2956semidirect
95. #137Dic101semidirect
96. AGL(1,ℤ/8)Dic101semidirect
120. Q16C2×Dic101semidirect
121. SD16C2×Dic101semidirect
122. D8C2×Dic101semidirect

Maximal subgroups

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

subgroupindex
#202659442
#202652992
#202652942
#202652262
C2×#156921822
M16×Dic1012
#575×C1012
#2956101