Finite Group Databasebeta

← #30637366

Subgroups — #30637366

Order 16256.

194 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 59 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#22550432C211yesmaximal, characteristic
2#22549931C212yesmaximal
3#22549618C211yesmaximal, characteristic, complemented
4#22549617C212yesmaximal, complemented
5#330×C127C211yesmaximal, characteristic, complemented
6#16861208C412yes—
7#16861201C2211yescharacteristic
8#16861187—22no—
9#16861174—22no—
10#16861167C412yes—
11C42×D127C2211yescharacteristic
12Dic127⋊C8—22no—
13(C2×D127)⋊C8—22no—
14#16861120C2212yes—
15#125×C127C2211yescharacteristic
16#97×C127C2212yes—
17C2×Dic254—22no—
18C2×C4×D127C2×C411yescharacteristic
19C2×C4×D127D412yes—
20C2×C127⋊C8—22no—
21C4⋊Dic127C2×C412yes—
22Dic127⋊C4—22no—
23Dic127⋊C4—22no—
24(C2×D127)⋊C4—22no—
25C4×Dic127—22no—
26C4×Dic127C2×C411yescharacteristic
27C4×Dic127D412yes—
28C2×C8×C127—22no—
29C4⋊C4×C127—22no—
30C4⋊C4×C127C2×C412yes—
31C42×C127C2311yescharacteristic
32C22×D127C22⋊C411yescharacteristic
33Dic254—44no—
34C2×Dic127—22no—
35C2×Dic127C22⋊C412yes—
36C2×Dic127C22⋊C411yescharacteristic
37C4×D127—22no—
38C4×D127—42no—
39C127⋊C8—42no—
40C2×C4×C127—22no—
41C2×C4×C127C22×C411yescharacteristic
42C2×C4×C127C2×D412yes—
43C8×C127—42no—
44C22×C127C2×C22⋊C411yescharacteristic
45C2×D127—21no—
46C2×D127—22no—
47C4×C127—21no—
48C4×C127—22no—
49C4×C127—42no—
50Dic127—22no—
51Dic127—42no—
52Dic127#10112yes—
53C2×C127C2×#10111yescharacteristic
54C2×C127#42112yes—
55D127—41no—
56#1537—1271nomaximal, Sylow
57C127#153711yescharacteristic, complemented, Sylow
58#533—1271no—
59#432—1272no—
60#330—1271no—
61#330—1272no—
62#330D12711yescharacteristic, complemented
63C4⋊Q8—1272no—
64#125—1271no—
65#125C2×D12711yescharacteristic
66C22⋊Q8—2542no—
67C4×Q8—2542no—
68#117—1272no—
69C2×C42—1271no—
70C4⋊C8—2542no—
71C22⋊C8—2542no—
72#97—1272no—
73#97C2×D12712yes—
74C2×Q8—2542no—
75C22×C4—1272no—
76C22×C4—1271no—
77C2×C8—22no—
78C2×C8—2542no—
79C4⋊C4—22no—
80C4⋊C4—1272no—
81C4⋊C4—2542no—
82C4⋊C4—2542no—
83C4⋊C4C4×D12712yes—
84C22⋊C4—2542no—
85C42—1271no—
86C42—1272no—
87C42—2542no—
88C42C22×D12711yescharacteristic
89C23—1271no—
90Q8—5084no—
91C2×C4—22no—
92C2×C4—1271no—
93C2×C4—1272no—
94C2×C4—5082no—
95C2×C4—2542no—
96C2×C4—2542no—
97C2×C4C2×C4×D12711yescharacteristic
98C2×C4D4×D12712yes—
99C8—42no—
100C8—5082no—
101C22—2541no—
102C22—2542no—
103C22C22⋊C4×D12711yescharacteristic
104C4—21no—
105C4—22no—
106C4—42no—
107C4—1272no—
108C4—5082no—
109C4—2542no—
110C2—5081no—
111C2#101×D12711yescharacteristic
112C2#2254989112yes—
subgroupG/H
centerC22C22⋊C4×D127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
Fitting#330×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC22×D127C22⋊C4
cosocleC42×C127C23
hypercenter#330D127
hypocenterC127#1537
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
3. #22549618C2semidirect
4. #22549617C2semidirect
5. #330×C127C2semidirect
57. C127#1537semidirect
62. #330D127semidirect

Maximal subgroups

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

subgroupindex
#225504322
#225499312
#225496182
#225496172
#330×C1272
#1537127