Finite Group Databasebeta

← #27170424

Subgroups — #27170424

Order 12928.

230 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 45 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#20265925C211yesmaximal, characteristic, complemented
2#20265564C212yesmaximal, complemented
3#20264919C211yesmaximal, characteristic, complemented
4#20264916C212yesmaximal, complemented
5#490×C101C211yesmaximal, characteristic, complemented
6C2×#13108062—22no—
7C2×#13108059—22no—
8#15692243C2212yes—
9#15692240—22no—
10#15692233—22no—
11#15692230—22no—
12#15692166C2212yes—
13#127×C101C2211yescharacteristic
14#125×C101C2211yescharacteristic
15#102×C101—22no—
16#97×C101C2211yescharacteristic
17C2×Dic202D412yes—
18C2×D404—22no—
19C2×D404D412yes—
20C2×C4×D101—22no—
21#13108062—44no—
22#13108059—44no—
23C2×#11717579—22no—
24#13108053—22no—
25#13108051—22no—
26#13108051—22no—
27#13108051—22no—
28C2×D4×C101—21no—
29C2×D4×C101—22no—
30C2×C8×C101—21no—
31C4⋊C4×C101—22no—
32C4⋊C4×C101D412yes—
33C42×C101C2311yescharacteristic
34C23×C101—41no—
35C22×D101—22no—
36D4×C101—42no—
37D4×C101—42no—
38D4×C101—42no—
39C2×C4×C101—22no—
40C2×C4×C101C2×D412yes—
41C2×C4×C101C2×D411yescharacteristic
42C2×Dic101—22no—
43D404—42no—
44D404—82no—
45C4×D101—82no—
46Dic202—42no—
47C8×C101—41no—
48#11717579—24no—
49C22×C101—82no—
50C22×C101—81no—
51C22×C101#12011yescharacteristic
52C2×D101—42no—
53C2×D101—42no—
54C2×D101—42no—
55C4×C101—22no—
56C4×C101—21no—
57C4×C101—42no—
58Dic101—42no—
59C2×C101—161no—
60C2×C101#45312yes—
61C2×C101#45411yescharacteristic
62D101—82no—
63#1656—1011nomaximal, Sylow
64C101#165611yescharacteristic, complemented, Sylow
65#562—1011no—
66#496—1012no—
67#490D10111yescharacteristic, complemented
68#330—1011no—
69#329—1012no—
70C2×SD16—2022no—
71C2×D8—2022no—
72#127C2×D10111yescharacteristic
73#125C2×D10111yescharacteristic
74#124—1012no—
75#123—2022no—
76C22⋊Q8—2022no—
77#121—2022no—
78#102—22no—
79#97—1012no—
80#97C2×D10111yescharacteristic
81C2×Q8—1012no—
82C2×D4—22no—
83C2×D4—21no—
84C2×D4—1012no—
85C2×D4—2022no—
86C22×C4—2022no—
87SD16—4044no—
88D8—4044no—
89C2×C8—21no—
90C2×C8—2022no—
91C4⋊C4—22no—
92C4⋊C4—2022no—
93C4⋊C4C22⋊D10112yes—
94C22⋊C4—2022no—
95C22⋊C4—2022no—
96C22⋊C4—2022no—
97C42C22×D10111yescharacteristic
98C23—41no—
99C23—2022no—
100Q8—4042no—
101D4—42no—
102D4—42no—
103D4—42no—
104D4—8082no—
105D4—4042no—
106C2×C4—22no—
107C2×C4—8082no—
108C2×C4—2022no—
109C2×C4D4×D10112yes—
110C2×C4C2×C22⋊D10111yescharacteristic
111C8—41no—
112C8—2024no—
113C22—82no—
114C22—81no—
115C22—4042no—
116C22—4042no—
117C22—4042no—
118C22#1569222311yescharacteristic
119C4—22no—
120C4—21no—
121C4—42no—
122C4—4042no—
123C2—161no—
124C2—8082no—
125C2#2026534312yes—
126C2#2026534911yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#490×C101C2
socleC22×C101#120
hypercenter#490D101
hypocenterC101#1656
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #20265925C2semidirect
2. #20265564C2semidirect
3. #20264919C2semidirect
4. #20264916C2semidirect
5. #490×C101C2semidirect
64. C101#1656semidirect
67. #490D101semidirect

Maximal subgroups

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

subgroupindex
#202659252
#202655642
#202649192
#202649162
#490×C1012
#1656101