Finite Group Databasebeta

← #30638033

Subgroups — #30638033

Order 16256.

182 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 45 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
1C127⋊#564C211yesmaximal, characteristic, complemented
2#22550242C211yesmaximal, characteristic
3#22550241C212yesmaximal
4#22549617C212yesmaximal, complemented
5#330×C127C211yesmaximal, characteristic, complemented
6#16861206C2211yescharacteristic
7#16861201C2212yes—
8#16861200—22no—
9#16861199—22no—
10#16861199—22no—
11#16861131—22no—
12#16861131—22no—
13#16861129—22no—
14#16861120C2211yescharacteristic
15#125×C127C2211yescharacteristic
16#97×C127C2212yes—
17C2×D508—21no—
18C2×D508—22no—
19C2×C127⋊C8—21no—
20C4⋊Dic127—22no—
21C4⋊Dic127D412yes—
22C4⋊Dic127D412yes—
23Dic127⋊C4—22no—
24Dic127⋊C4—22no—
25Dic127⋊C4—22no—
26Dic127⋊C4—22no—
27C4×Dic127—22no—
28C2×C8×C127—22no—
29C4⋊C4×C127—22no—
30C4⋊C4×C127D412yes—
31C42×C127C2311yescharacteristic
32C22×D127—41no—
33D508—44no—
34D508—42no—
35C2×Dic127—22no—
36C2×Dic127—22no—
37C127⋊C8—41no—
38C2×C4×C127—22no—
39C2×C4×C127C2×D412yes—
40C2×C4×C127C2×D411yescharacteristic
41C8×C127—42no—
42C22×C127#12011yescharacteristic
43C2×D127—82no—
44C2×D127—81no—
45C4×C127—22no—
46C4×C127—21no—
47C4×C127—42no—
48Dic127—42no—
49Dic127—42no—
50C2×C127#45412yes—
51C2×C127#45411yescharacteristic
52D127—161no—
53#1658—1271nomaximal, Sylow
54C127#165811yescharacteristic, complemented, Sylow
55#564—1271no—
56#490—1271no—
57#490—1272no—
58#330—1272no—
59#330D12711yescharacteristic, complemented
60#127—1271no—
61#125—1272no—
62#125—2542no—
63#125—2542no—
64#125—2542no—
65#125C2×D12711yescharacteristic
66#102—2542no—
67#102—2542no—
68#102—2542no—
69#97—1271no—
70#97C2×D12712yes—
71C2×D4—2542no—
72C2×D4—2541no—
73C2×C8—22no—
74C2×C8—2541no—
75C4⋊C4—22no—
76C4⋊C4—1272no—
77C4⋊C4—1272no—
78C4⋊C4—2542no—
79C4⋊C4—2542no—
80C4⋊C4—2542no—
81C4⋊C4—2542no—
82C4⋊C4—2542no—
83C4⋊C4D50812yes—
84C42—2542no—
85C42C22×D12711yescharacteristic
86C23—5081no—
87D4—5084no—
88D4—5082no—
89C2×C4—22no—
90C2×C4—2542no—
91C2×C4—2542no—
92C2×C4D4×D12712yes—
93C2×C4C2×D50811yescharacteristic
94C8—42no—
95C8—5081no—
96C22—10162no—
97C22—10161no—
98C22#1686117911yescharacteristic
99C4—22no—
100C4—21no—
101C4—42no—
102C4—5082no—
103C4—5082no—
104C2—20321no—
105C2#2255005012yes—
106C2#2255005111yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#330×C127C2
socleC22×C127#120
prime radical#16861206C22
hypercenter#330D127
hypocenterC127#1658
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
1. C127⋊#564C2semidirect
4. #22549617C2semidirect
5. #330×C127C2semidirect
54. C127#1658semidirect
59. #330D127semidirect

Maximal subgroups

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

subgroupindex
C127⋊#5642
#225502422
#225502412
#225496172
#330×C1272
#1658127