Finite Group Databasebeta

← #30636857

Subgroups — #30636857

Order 16256.

172 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 57 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
1C2×#16861128C212yesmaximal
2#391×C127C211yesmaximal, characteristic
3C2×#14523232—22no—
4#16861128—22no—
5#16861128C414yescomplemented
6C22×Q8×C127C2211yescharacteristic
7C2×C4⋊C4×C127C411yescharacteristic
8C2×C42×C127C411yescharacteristic
9#95×C127—22no—
10#14523232—24no—
11#14523232—42no—
12C2×C127⋊C8—42no—
13C2×Q8×C127—41no—
14C2×Q8×C127—41no—
15C2×Q8×C127C2×C412yes—
16C2×Q8×C127D411yescharacteristic
17C2×Q8×C127Q811yescharacteristic
18C22×C4×C127—21no—
19C22×C4×C127—21no—
20C22×C4×C127C2×C411yescharacteristic
21C22×C4×C127D412yes—
22C4⋊C4×C127—41no—
23C42×C127—41no—
24C127⋊C8—44no—
25C23×C127C22⋊C411yescharacteristic
26Q8×C127—42no—
27Q8×C127—41no—
28Q8×C127—41no—
29C2×C4×C127—21no—
30C2×C4×C127—21no—
31C2×C4×C127—22no—
32C2×C4×C127—22no—
33C2×C4×C127—22no—
34C2×C4×C127—41no—
35C2×C4×C127—41no—
36C2×C4×C127—42no—
37C2×C4×C127—41no—
38C2×C4×C127C4211yescharacteristic
39C2×C4×C127C22⋊C412yes—
40C2×C4×C127C4⋊C412yes—
41C2×C4×C127C4⋊C411yescharacteristic
42C22×C127—21no—
43C22×C127—21no—
44C22×C127#9511yescharacteristic
45C22×C127#9911yescharacteristic
46C22×C127#9911yescharacteristic
47C4×C127—21no—
48C4×C127—22no—
49C4×C127—22no—
50C4×C127—21no—
51C4×C127—42no—
52C4×C127—81no—
53C2×C127—21no—
54C2×C127—21no—
55C2×C127#34211yescharacteristic
56C2×C127#35511yescharacteristic
57C2×C127#35611yescharacteristic
58#1394—1271nomaximal, Sylow
59C127#139411yescharacteristic, complemented, Sylow
60C2×#101—1272no—
61#391D12711yescharacteristic
62C22×Q8C2×D12711yescharacteristic
63C2×M16—2542no—
64C2×C4⋊C4Dic12711yescharacteristic
65C2×C42Dic12711yescharacteristic
66#101—1274no—
67#101—2542no—
68#95—22no—
69C2×Q8—41no—
70C2×Q8—41no—
71C2×Q8C4×D12712yes—
72C2×Q8D50811yescharacteristic
73C2×Q8Dic25411yescharacteristic
74C22×C4—21no—
75C22×C4—21no—
76C22×C4C2×Dic12711yescharacteristic
77C22×C4C22⋊D12712yes—
78M16—5082no—
79M16—2544no—
80C2×C8—5082no—
81C4⋊C4—41no—
82C42—41no—
83C23C22⋊Dic12711yescharacteristic
84Q8—42no—
85Q8—41no—
86Q8—41no—
87C2×C4—21no—
88C2×C4—22no—
89C2×C4—21no—
90C2×C4—22no—
91C2×C4—22no—
92C2×C4—41no—
93C2×C4—41no—
94C2×C4—42no—
95C2×C4—41no—
96C2×C4C4×Dic12711yescharacteristic
97C2×C4(C2×D127)⋊C412yes—
98C2×C4Dic127⋊C412yes—
99C2×C4C4⋊Dic12711yescharacteristic
100C8—5084no—
101C22—21no—
102C22—21no—
103C22C127⋊#9511yescharacteristic
104C22#1686112411yescharacteristic
105C22#1686112411yescharacteristic
106C4—22no—
107C4—21no—
108C4—22no—
109C4—21no—
110C4—42no—
111C4—81no—
112C2—21no—
113C2—21no—
114C2#2254964311yescharacteristic
115C2#2254967311yescharacteristic
116C2#2254967511yescharacteristic
subgroupG/H
centerC22#16861124
commutatorC2×C4×C127C42
FrattiniC22×Q8C2×D127
Fitting#391×C127C2
socleC22×C127#99
prime radicalC23×C127C22⋊C4
cosocleC22×Q8×C127C22
hypercenter#391D127
hypocenterC127#1394
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #391

Complements

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

NH ≅ G/Ntype
5. #16861128C4semidirect
59. C127#1394semidirect

Maximal subgroups

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

subgroupindex
C2×#168611282
#391×C1272
#1394127