Finite Group Databasebeta

← #30640072

Subgroups — #30640072

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#22549919C212yesmaximal
2#22549745C211yesmaximal, characteristic
3#22549645C211yesmaximal, characteristic
4#22549635C212yesmaximal
5#544×C127C211yesmaximal, characteristic
6C2×#14523232C2212yes—
7C2×C4⋊Dic127C2211yescharacteristic
8C2×Dic127⋊C4—22no—
9C2×C4×Dic127C2211yescharacteristic
10#16861142—22no—
11#16861140—22no—
12C4⋊Q8×C127—22no—
13#125×C127—22no—
14C22⋊Q8×C127—22no—
15#117×C127C2212yes—
16C2×C4⋊C4×C127C2211yescharacteristic
17C22×Dic127—21no—
18C22×Dic127—21no—
19#14523232—22no—
20#14523232Q814yescomplemented
21C2×C127⋊C8—22no—
22C4⋊Dic127—22no—
23Dic127⋊C4—44no—
24C4×Dic127—21no—
25C4×Dic127Q812yescomplemented
26C2×Q8×C127—22no—
27C22×C4×C127D412yes—
28C22×C4×C127C2311yescharacteristic
29C4⋊C4×C127—22no—
30C4⋊C4×C127—24no—
31C4⋊C4×C127—22no—
32C4⋊C4×C127D412yes—
33C4⋊C4×C127D412yes—
34C22⋊C4×C127—22no—
35C42×C127—22no—
36C2×Dic127—22no—
37C2×Dic127—22no—
38C2×Dic127—22no—
39C2×Dic127—42no—
40C2×Dic127—41no—
41C127⋊C8—24no—
42C23×C127C2×D411yescharacteristic
43Q8×C127—82no—
44C2×C4×C127—22no—
45C2×C4×C127—22no—
46C2×C4×C127—22no—
47C2×C4×C127—42no—
48C2×C4×C127C2×D412yes—
49C2×C4×C127C2×Q812yes—
50C2×C4×C127C2×Q811yescharacteristic
51C2×C4×C127#4111yescharacteristic
52C22×C127—21no—
53C22×C127—21no—
54C22×C127#12011yescharacteristic
55C22×C127C22⋊Q811yescharacteristic
56C22×C127C4⋊Q811yescharacteristic
57C4×C127—42no—
58C4×C127—42no—
59C4×C127—42no—
60C4×C127C22⋊Q812yes—
61C4×C127C4⋊Q812yes—
62Dic127—24no—
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#544D12711yescharacteristic
72#428—1272no—
73#384—1271no—
74#343—1271no—
75#339—1272no—
76C2×M16—1272no—
77C4⋊Q8—22no—
78#125—22no—
79C22⋊Q8—22no—
80#117C2×D12712yes—
81C2×C4⋊C4—1271no—
82C2×C4⋊C4—2542no—
83C2×C4⋊C4C2×D12711yescharacteristic
84C2×C42—1271no—
85#107—2542no—
86#106—2542no—
87C2×Q8—22no—
88C22×C4—2541no—
89C22×C4—2541no—
90C22×C4C22⋊D12712yes—
91C22×C4C22×D12711yescharacteristic
92M16—1274no—
93M16—2542no—
94C2×C8—2542no—
95C4⋊C4—22no—
96C4⋊C4—24no—
97C4⋊C4—22no—
98C4⋊C4—5084no—
99C4⋊C4—2542no—
100C4⋊C4C22⋊D12712yes—
101C4⋊C4C22⋊D12712yes—
102C22⋊C4—22no—
103C42—22no—
104C42—1272no—
105C42—2541no—
106C23C2×C22⋊D12711yescharacteristic
107Q8—82no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—22no—
111C2×C4—42no—
112C2×C4—5082no—
113C2×C4—5081no—
114C2×C4—2542no—
115C2×C4—2542no—
116C2×C4—2542no—
117C2×C4C2×C22⋊D12712yes—
118C2×C4Q8×D12712yes—
119C2×C4Q8×D12711yescharacteristic
120C2×C4#1452324911yescharacteristic
121C8—2544no—
122C22—21no—
123C22—21no—
124C22#1686117811yescharacteristic
125C22#1686118511yescharacteristic
126C22#1686120711yescharacteristic
127C4—42no—
128C4—42no—
129C4—42no—
130C4—10161no—
131C4—2544no—
132C4#1686118512yes—
133C4#1686120712yes—
134C2—21no—
135C2—21no—
136C2#2254979511yescharacteristic
137C2#2255004411yescharacteristic
138C2#2255005911yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#544×C127C2
socleC22×C127#120
prime radicalC23×C127C2×D4
hypercenter#544D127
hypocenterC127#2060
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #544

Complements

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

NH ≅ G/Ntype
20. #14523232Q8semidirect
25. C4×Dic127Q8semidirect
70. C127#2060semidirect

Maximal subgroups

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

subgroupindex
#225499192
#225497452
#225496452
#225496352
#544×C1272
#2060127