Finite Group Databasebeta

← #27170425

Subgroups — #27170425

Order 12928.

230 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 49 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#20265567C212yesmaximal, complemented
2#20265543C211yesmaximal, characteristic, complemented
3#20264919C211yesmaximal, characteristic, complemented
4#20264917C212yesmaximal, complemented
5#562×C101C211yesmaximal, characteristic, complemented
6C2×#13108061—22no—
7C2×#13108059—22no—
8#15692252C2211yescharacteristic
9#15692175—22no—
10#15692166C2211yescharacteristic
11#15692166C2212yes—
12#125×C101C2211yescharacteristic
13#124×C101C2212yes—
14#123×C101—22no—
15C22⋊Q8×C101—22no—
16#121×C101—22no—
17C2×D404—21no—
18C2×D404—22no—
19#13108061—44no—
20#13108059—44no—
21C2×#11717579—22no—
22C2×#11717579—21no—
23C2×Q8×C101D412yes—
24C2×D4×C101—22no—
25C2×D4×C101D412yes—
26C22×C4×C101—22no—
27C4⋊C4×C101—22no—
28C4⋊C4×C101—22no—
29C4⋊C4×C101D412yes—
30C22⋊C4×C101—22no—
31C22⋊C4×C101—22no—
32C22⋊C4×C101—22no—
33C42×C101C2311yescharacteristic
34C23×C101—22no—
35C22×D101—41no—
36Q8×C101—42no—
37D4×C101—42no—
38D4×C101—82no—
39C2×C4×C101—22no—
40C2×C4×C101—22no—
41C2×C4×C101—82no—
42C2×C4×C101C2×D411yescharacteristic
43C2×C4×C101C2×D412yes—
44D404—42no—
45D404—42no—
46D404—42no—
47#11717579—24no—
48#11717579—41no—
49C22×C101—42no—
50C22×C101—42no—
51C22×C101—42no—
52C22×C101#12011yescharacteristic
53C2×D101—82no—
54C2×D101—81no—
55C4×C101—22no—
56C4×C101—21no—
57C4×C101—42no—
58C4×C101—42no—
59C2×C101—82no—
60C2×C101#45312yes—
61C2×C101#45411yescharacteristic
62D101—161no—
63#1656—1011nomaximal, Sylow
64C101#165611yescharacteristic, complemented, Sylow
65#562D10111yescharacteristic, complemented
66#496—1012no—
67#490—1011no—
68#330—1011no—
69#329—1012no—
70C2×SD16—2022no—
71C2×D8—2022no—
72#127—1011no—
73#125C2×D10111yescharacteristic
74#124C2×D10112yes—
75#123—22no—
76C22⋊Q8—22no—
77#121—22no—
78#102—2022no—
79#97—1011no—
80#97—1012no—
81C2×Q8C22⋊D10112yes—
82C2×D4—22no—
83C2×D4—2021no—
84C2×D4—2022no—
85C2×D4C22⋊D10112yes—
86C22×C4—22no—
87SD16—4044no—
88D8—4044no—
89C2×C8—2021no—
90C2×C8—2022no—
91C4⋊C4—22no—
92C4⋊C4—22no—
93C4⋊C4C22⋊D10112yes—
94C22⋊C4—22no—
95C22⋊C4—22no—
96C22⋊C4—22no—
97C42C22×D10111yescharacteristic
98C23—22no—
99C23—4041no—
100Q8—42no—
101D4—42no—
102D4—82no—
103D4—4042no—
104D4—4042no—
105D4—4042no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—82no—
109C2×C4C2×C22⋊D10111yescharacteristic
110C2×C4C2×C22⋊D10112yes—
111C8—4041no—
112C8—2024no—
113C22—42no—
114C22—42no—
115C22—42no—
116C22—8082no—
117C22—8081no—
118C22#1569222411yescharacteristic
119C4—22no—
120C4—21no—
121C4—42no—
122C4—42no—
123C2—82no—
124C2—16161no—
125C2#2026534512yes—
126C2#2026535011yescharacteristic
subgroupG/H
centerC22#15692224
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#562×C101C2
socleC22×C101#120
hypercenter#562D101
hypocenterC101#1656
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #562

Complements

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

NH ≅ G/Ntype
1. #20265567C2semidirect
2. #20265543C2semidirect
3. #20264919C2semidirect
4. #20264917C2semidirect
5. #562×C101C2semidirect
64. C101#1656semidirect
65. #562D101semidirect

Maximal subgroups

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

subgroupindex
#202655672
#202655432
#202649192
#202649172
#562×C1012
#1656101