Finite Group Databasebeta

← #30640076

Subgroups — #30640076

Order 16256.

246 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 75 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#22550507C211yesmaximal, characteristic
2#22549918C212yesmaximal
3#22549745C211yesmaximal, characteristic
4#22549637C212yesmaximal
5#343×C127C211yesmaximal, characteristic
6#16861208—22no—
7#16861200—22no—
8#16861188—22no—
9#16861168C2212yes—
10C2×C4⋊Dic127C2211yescharacteristic
11C2×Dic127⋊C4—22no—
12C2×C4×Dic127C2211yescharacteristic
13#16861143—22no—
14#16861141—22no—
15C2×M16×C127C2212yes—
16C2×C4⋊C4×C127C2211yescharacteristic
17C2×Dic254—22no—
18C22×Dic127—21no—
19C22×Dic127—21no—
20C4⋊Dic127—22no—
21C4⋊Dic127—22no—
22C4⋊Dic127D412yes—
23C4⋊Dic127D412yes—
24Dic127⋊C4—24no—
25Dic127⋊C4—44no—
26C22⋊Dic127—22no—
27C4×Dic127—22no—
28C4×Dic127—21no—
29C4×Dic127Q812yescomplemented
30C22×C4×C127D412yes—
31C22×C4×C127C2311yescharacteristic
32M16×C127—22no—
33M16×C127Q814yescomplemented
34C2×C8×C127—22no—
35C4⋊C4×C127—22no—
36Dic254—82no—
37C2×Dic127—22no—
38C2×Dic127—22no—
39C2×Dic127—22no—
40C2×Dic127—22no—
41C2×Dic127—22no—
42C2×Dic127—42no—
43C2×Dic127—41no—
44C23×C127C2×D411yescharacteristic
45C2×C4×C127—22no—
46C2×C4×C127—42no—
47C2×C4×C127C2×D412yes—
48C2×C4×C127C2×Q811yescharacteristic
49C2×C4×C127C2×Q812yes—
50C2×C4×C127#4111yescharacteristic
51C8×C127—24no—
52C22×C127—21no—
53C22×C127—21no—
54C22×C127#12011yescharacteristic
55C22×C127C22⋊Q811yescharacteristic
56C22×C127C4⋊Q811yescharacteristic
57C4×C127—42no—
58C4×C127C22⋊Q812yes—
59C4×C127C4⋊Q812yes—
60Dic127—24no—
61Dic127—42no—
62Dic127—42no—
63Dic127—81no—
64C2×C127—21no—
65C2×C127—21no—
66C2×C127#39511yescharacteristic
67C2×C127#45311yescharacteristic
68C2×C127#45611yescharacteristic
69#2060—1271nomaximal, Sylow
70C127#206011yescharacteristic, complemented, Sylow
71#544—1271no—
72#428—1272no—
73#384—1271no—
74#343D12711yescharacteristic
75#339—1272no—
76C2×M16C2×D12712yes—
77C4⋊Q8—2542no—
78#125—2542no—
79C22⋊Q8—2542no—
80#117—1272no—
81C2×C4⋊C4—1271no—
82C2×C4⋊C4—2542no—
83C2×C4⋊C4C2×D12711yescharacteristic
84C2×C42—1271no—
85#107—2542no—
86#106—2542no—
87C2×Q8—2542no—
88C22×C4—2541no—
89C22×C4—2541no—
90C22×C4D50812yes—
91C22×C4C22×D12711yescharacteristic
92M16—22no—
93M16Dic25414yescomplemented
94C2×C8—22no—
95C4⋊C4—22no—
96C4⋊C4—1272no—
97C4⋊C4—1272no—
98C4⋊C4—5084no—
99C4⋊C4—2544no—
100C4⋊C4—2542no—
101C4⋊C4—2542no—
102C22⋊C4—2542no—
103C42—1272no—
104C42—2542no—
105C42—2541no—
106C23C2×D50811yescharacteristic
107Q8—10162no—
108C2×C4—22no—
109C2×C4—42no—
110C2×C4—5082no—
111C2×C4—5081no—
112C2×C4—2542no—
113C2×C4—2542no—
114C2×C4—2542no—
115C2×C4—2542no—
116C2×C4—2542no—
117C2×C4D4×D12712yes—
118C2×C4Q8×D12711yescharacteristic
119C2×C4C2×Dic25412yes—
120C2×C4#1452324711yescharacteristic
121C8—24no—
122C22—21no—
123C22—21no—
124C22#1686117911yescharacteristic
125C22#1686118911yescharacteristic
126C22#1686120911yescharacteristic
127C4—42no—
128C4—10161no—
129C4—5082no—
130C4—5082no—
131C4—2544no—
132C4#1686118612yes—
133C4#1686120812yes—
134C2—21no—
135C2—21no—
136C2#2254979711yescharacteristic
137C2#2255004611yescharacteristic
138C2#2255006111yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#343×C127C2
socleC22×C127#120
prime radicalC23×C127C2×D4
hypercenter#343D127
hypocenterC127#2060
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
29. C4×Dic127Q8semidirect
33. M16×C127Q8semidirect
70. C127#2060semidirect
93. M16Dic254semidirect

Maximal subgroups

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

subgroupindex
#225505072
#225499182
#225497452
#225496372
#343×C1272
#2060127