Finite Group Databasebeta

← #30637787

Subgroups — #30637787

Order 16256.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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#22550612C211yesmaximal, characteristic, complemented
2#22550073C212yesmaximal, complemented
3#22549620C211yesmaximal, characteristic, complemented
4#22549608C212yesmaximal, complemented
5#493×C127C211yesmaximal, characteristic
6C2×#14523236—22no—
7#16861196—22no—
8#16861190—22no—
9C4×Dic254C2212yes—
10D4×Dic127—22no—
11C4⋊(C127⋊C8)C2212yes—
12#16861130—22no—
13C2×D8×C127—22no—
14#127×C127C2211yescharacteristic
15#127×C127C2211yescharacteristic
16C4×C8×C127C2211yescharacteristic
17C2×Dic254D412yes—
18C22×Dic127—22no—
19#14523236—44no—
20C2×C127⋊C8—22no—
21C4⋊Dic127D412yes—
22Dic127⋊C4—22no—
23C22⋊Dic127—22no—
24C22⋊Dic127—24no—
25C4×Dic127—22no—
26C2×D4×C127—22no—
27C2×D4×C127—22no—
28C2×D4×C127—21no—
29C2×D4×C127D412yes—
30D8×C127—44no—
31C2×C8×C127—21no—
32C42×C127C2311yescharacteristic
33Dic254—22no—
34Dic254—22no—
35C2×Dic127—22no—
36C2×Dic127—22no—
37C2×Dic127—82no—
38C127⋊C8—42no—
39C23×C127—22no—
40C23×C127—41no—
41D4×C127—42no—
42D4×C127—42no—
43D4×C127—42no—
44D4×C127—42no—
45D4×C127—42no—
46D4×C127—42no—
47C2×C4×C127C2×D412yes—
48C2×C4×C127C2×D411yescharacteristic
49C8×C127—22no—
50C22×C127—44no—
51C22×C127—42no—
52C22×C127—82no—
53C22×C127—81no—
54C22×C127#12011yescharacteristic
55C4×C127—21no—
56C4×C127#13512yes—
57C4×C127AGL(1,ℤ/8)12yes—
58Dic127—42no—
59Dic127—42no—
60C2×C127—82no—
61C2×C127—161no—
62C2×C127#44912yes—
63C2×C127#45311yescharacteristic
64#1620—1271nomaximal, Sylow
65C127#162011yescharacteristic, complemented, Sylow
66#559—1271no—
67#493D12711yescharacteristic
68#460—1272no—
69#331—1271no—
70Q8⋊C8—1272no—
71C2×SD16—2542no—
72C2×D8—22no—
73#127C2×D12711yescharacteristic
74#127C2×D12711yescharacteristic
75#124—2542no—
76#123—2542no—
77C4×Q8—1272no—
78C4×D4—2542no—
79C4⋊C8—1272no—
80#102—2542no—
81C4×C8C2×D12711yescharacteristic
82C2×Q8—1272no—
83C2×D4—22no—
84C2×D4—22no—
85C2×D4—21no—
86C2×D4C22⋊D12712yes—
87C22×C4—2542no—
88SD16—5084no—
89D8—44no—
90C2×C8—21no—
91C2×C8—2542no—
92C4⋊C4—1272no—
93C4⋊C4—2542no—
94C22⋊C4—2544no—
95C22⋊C4—2542no—
96C42—2542no—
97C42C22×D12711yescharacteristic
98C23—22no—
99C23—41no—
100Q8—2542no—
101Q8—2542no—
102D4—42no—
103D4—42no—
104D4—42no—
105D4—42no—
106D4—42no—
107D4—42no—
108C2×C4—10162no—
109C2×C4—2542no—
110C2×C4—2542no—
111C2×C4D4×D12712yes—
112C2×C4C2×C22⋊D12711yescharacteristic
113C8—22no—
114C8—5082no—
115C22—44no—
116C22—42no—
117C22—82no—
118C22—81no—
119C22#1686117711yescharacteristic
120C4—21no—
121C4—5082no—
122C4—5082no—
123C4#1686122812yes—
124C4#1686123312yes—
125C2—82no—
126C2—161no—
127C2#2255001812yes—
128C2#2255004311yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting, prime radical#493×C127C2
socleC22×C127#120
hypercenter#493D127
hypocenterC127#1620
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. #22550612C2semidirect
2. #22550073C2semidirect
3. #22549620C2semidirect
4. #22549608C2semidirect
65. C127#1620semidirect

Maximal subgroups

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

subgroupindex
#225506122
#225500732
#225496202
#225496082
#493×C1272
#1620127