Finite Group Databasebeta

← #30638058

Subgroups — #30638058

Order 16256.

172 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 49 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#22550286C212yesmaximal
2#22550246C211yesmaximal, characteristic
3#22549618C212yesmaximal
4#22549618C211yesmaximal, characteristic
5#564×C127C211yesmaximal, characteristic
6#16861209C2211yescharacteristic
7#16861142—22no—
8#16861140—22no—
9#16861132—22no—
10#16861120C2212yes—
11#16861120C2211yescharacteristic
12#125×C127—22no—
13#125×C127—22no—
14#125×C127—22no—
15#125×C127C2212yes—
16#125×C127C2211yescharacteristic
17C2×Dic254—21no—
18C2×C127⋊C8—22no—
19C2×C127⋊C8—21no—
20C4⋊Dic127—22no—
21C4⋊C4×C127—22no—
22C4⋊C4×C127—22no—
23C4⋊C4×C127—22no—
24C4⋊C4×C127—22no—
25C4⋊C4×C127—22no—
26C4⋊C4×C127—22no—
27C4⋊C4×C127D412yes—
28C4⋊C4×C127D412yes—
29C4⋊C4×C127D412yes—
30C42×C127—22no—
31C42×C127C2311yescharacteristic
32Dic254—42no—
33C2×Dic127—41no—
34C127⋊C8—24no—
35C127⋊C8—41no—
36C2×C4×C127—22no—
37C2×C4×C127—22no—
38C2×C4×C127—22no—
39C2×C4×C127C2×D412yes—
40C2×C4×C127C2×D411yescharacteristic
41C22×C127#12011yescharacteristic
42C4×C127—21no—
43C4×C127—22no—
44C4×C127—42no—
45C4×C127—42no—
46C4×C127—42no—
47Dic127—81no—
48C2×C127#45411yescharacteristic
49C2×C127#45612yes—
50#1663—1271nomaximal, Sylow
51C127#166311yescharacteristic, complemented, Sylow
52#564D12711yescharacteristic
53#501—1272no—
54#491—1271no—
55#330—1272no—
56#330—1271no—
57C4⋊Q8—1271no—
58#125—22no—
59#125—22no—
60#125—22no—
61#125C2×D12711yescharacteristic
62#125C2×D12712yes—
63#107—2542no—
64#106—2542no—
65#103—2542no—
66#97—1271no—
67#97—1272no—
68C2×Q8—2541no—
69C2×C8—2541no—
70C2×C8—2542no—
71C4⋊C4—22no—
72C4⋊C4—22no—
73C4⋊C4—22no—
74C4⋊C4—22no—
75C4⋊C4—22no—
76C4⋊C4—22no—
77C4⋊C4—2542no—
78C4⋊C4C22⋊D12712yes—
79C4⋊C4C22⋊D12712yes—
80C4⋊C4C22⋊D12712yes—
81C42—22no—
82C42C22×D12711yescharacteristic
83Q8—5082no—
84C2×C4—22no—
85C2×C4—22no—
86C2×C4—22no—
87C2×C4—5081no—
88C2×C4C2×C22⋊D12712yes—
89C2×C4C2×C22⋊D12711yescharacteristic
90C8—5081no—
91C8—2544no—
92C22#1686117811yescharacteristic
93C4—21no—
94C4—22no—
95C4—42no—
96C4—42no—
97C4—42no—
98C4—10161no—
99C2#2255004911yescharacteristic
100C2#2255005912yes—
subgroupG/H
centerC22#16861178
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#564×C127C2
socle, prime radicalC22×C127#120
hypercenter#564D127
hypocenterC127#1663
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #564

Complements

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

NH ≅ G/Ntype
51. C127#1663semidirect

Maximal subgroups

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

subgroupindex
#225502862
#225502462
#225496182
#564×C1272
#1663127