Finite Group Databasebeta

← #30640246

Subgroups — #30640246

Order 16256.

222 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 65 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#22549925C211yesmaximal, characteristic, complemented
2#22549924C212yesmaximal, complemented
3#22549840C212yesmaximal
4C127⋊#338C211yesmaximal, characteristic, complemented
5#407×C127C211yesmaximal, characteristic
6C2×#14523232C2212yes—
7C2×#14523232C2212yes—
8#16861145—22no—
9#16861144—24no—
10C22⋊(C127⋊C8)—24no—
11C23×C4×C127C2211yescharacteristic
12C2×M16×C127C2211yescharacteristic
13C2×M16×C127C2211yescharacteristic
14C22⋊C8×C127—22no—
15#14523232—22no—
16#14523232—22no—
17#14523232—22no—
18#14523232—22no—
19C2×C127⋊C8—22no—
20C2×C127⋊C8D414yes—
21C24×C127Q811yescharacteristic
22C22×C4×C127—21no—
23C22×C4×C127—22no—
24C22×C4×C127—22no—
25C22×C4×C127—21no—
26C22×C4×C127Q811yescharacteristic
27C22×C4×C127C2311yescharacteristic
28M16×C127—22no—
29M16×C127—22no—
30C2×C8×C127—21no—
31C2×C8×C127D412yes—
32C127⋊C8—24no—
33C127⋊C8—42no—
34C23×C127—22no—
35C23×C127—21no—
36C23×C127—42no—
37C23×C127C2×Q811yescharacteristic
38C2×C4×C127—22no—
39C2×C4×C127—21no—
40C2×C4×C127—41no—
41C2×C4×C127—42no—
42C2×C4×C127—41no—
43C2×C4×C127C2×D412yes—
44C2×C4×C127C2×D411yescharacteristic
45C2×C4×C127#4112yes—
46C2×C4×C127#4111yescharacteristic
47C8×C127—22no—
48C8×C127—41no—
49C22×C127—24no—
50C22×C127—22no—
51C22×C127—41no—
52C22×C127—42no—
53C22×C127—41no—
54C22×C127—41no—
55C22×C127C22⋊Q812yes—
56C22×C127C22⋊Q811yescharacteristic
57C4×C127—41no—
58C4×C127#12311yescharacteristic
59C4×C127#12312yes—
60C4×C127#12711yescharacteristic
61C2×C127—22no—
62C2×C127—21no—
63C2×C127—42no—
64C2×C127#39911yescharacteristic
65#2091—1271nomaximal, Sylow
66C127#209111yescharacteristic, complemented, Sylow
67#430—1272no—
68#430—1271no—
69#407—1272no—
70#407D12711yescharacteristic
71#338—1271no—
72C23×C4C2×D12711yescharacteristic
73C2×M16—1272no—
74C2×M16—1272no—
75C2×M16C2×D12711yescharacteristic
76C2×M16C2×D12711yescharacteristic
77#108—2544no—
78#108—2542no—
79C22⋊C8—22no—
80C22⋊C8—2544no—
81C24Dic25411yescharacteristic
82C22×C4—22no—
83C22×C4—21no—
84C22×C4—22no—
85C22×C4—21no—
86C22×C4Dic25411yescharacteristic
87C22×C4C22×D12711yescharacteristic
88M16—22no—
89M16—22no—
90M16—2542no—
91M16—2542no—
92M16—2542no—
93M16—2542no—
94C2×C8—21no—
95C2×C8—1274no—
96C2×C8—2542no—
97C2×C8C22⋊D12712yes—
98C23—22no—
99C23—21no—
100C23—42no—
101C23C2×Dic25411yescharacteristic
102C2×C4—22no—
103C2×C4—21no—
104C2×C4—41no—
105C2×C4—42no—
106C2×C4—41no—
107C2×C4D4×D12712yes—
108C2×C4C2×C22⋊D12711yescharacteristic
109C2×C4#1452324712yes—
110C2×C4#1452324811yescharacteristic
111C8—22no—
112C8—41no—
113C8—5082no—
114C8—2544no—
115C22—24no—
116C22—22no—
117C22—41no—
118C22—42no—
119C22—41no—
120C22—41no—
121C22#1686118612yes—
122C22#1686118811yescharacteristic
123C4—41no—
124C4#1686119011yescharacteristic
125C4#1686119112yes—
126C4#1686120511yescharacteristic
127C2—22no—
128C2—21no—
129C2—42no—
130C2#2254981211yescharacteristic
subgroupG/H
centerC4#16861205
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#407×C127C2
socleC2×C127#399
prime radicalC24×C127Q8
hypercenter#407D127
hypocenterC127#2091
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. #22549925C2semidirect
2. #22549924C2semidirect
4. C127⋊#338C2semidirect
66. C127#2091semidirect

Maximal subgroups

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

subgroupindex
#225499252
#225499242
#225498402
C127⋊#3382
#407×C1272
#2091127