Finite Group Databasebeta

← #30637340

Subgroups — #30637340

Order 16256.

256 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 73 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#22550426C211yesmaximal, characteristic
2#22549933C211yesmaximal, characteristic, complemented
3#22549615C214yesmaximal, complemented
4C2×#97×C127C211yesmaximal, characteristic, complemented
5C2×#14523248C412yes—
6#16861209C411yescharacteristic
7#16861206C411yescharacteristic
8#16861197C2212yes—
9#16861183—21no—
10#16861183—21no—
11C4×D508—21no—
12C4×D508—21no—
13C4⋊(C127⋊C8)—21no—
14C22⋊(C127⋊C8)—21no—
15#16861120C2212yes—
16C22×C8×C127—21no—
17C2×C42×C127C2211yescharacteristic
18#97×C127—21no—
19#97×C127C2212yes—
20#14523248—44no—
21C2×Dic254C2×C412yes—
22C2×D508—21no—
23C2×D508—21no—
24C2×D508C2×C412yes—
25C2×C22⋊D127—22no—
26C2×C4×D127—22no—
27C2×C127⋊C8—22no—
28C4⋊Dic127—21no—
29C4⋊Dic127—21no—
30(C2×D127)⋊C4—22no—
31(C2×D127)⋊C4—22no—
32C22×C4×C127C2×C411yescharacteristic
33C22×C4×C127D411yescharacteristic
34C22×C4×C127D411yescharacteristic
35C2×C8×C127—24no—
36C2×C8×C127—22no—
37C42×C127C2×C411yescharacteristic
38C42×C127D411yescharacteristic
39C42×C127D411yescharacteristic
40C42×C127C2311yescharacteristic
41C22×D127—22no—
42Dic254—42no—
43D508—42no—
44D508—42no—
45D508—42no—
46C22⋊D127—44no—
47C2×Dic127—22no—
48C4×D127—44no—
49C127⋊C8—42no—
50C23×C127C22⋊C411yescharacteristic
51C2×C4×C127—22no—
52C2×C4×C127—22no—
53C2×C4×C127—41no—
54C2×C4×C127C22⋊C411yescharacteristic
55C2×C4×C127C22⋊C411yescharacteristic
56C2×C4×C127C22⋊C411yescharacteristic
57C2×C4×C127C22×C411yescharacteristic
58C2×C4×C127C2×D411yescharacteristic
59C2×C4×C127C2×D411yescharacteristic
60C8×C127—24no—
61C22×C127—21no—
62C22×C127—21no—
63C22×C127#10012yes—
64C22×C127C2×C22⋊C411yescharacteristic
65C2×D127—42no—
66C2×D127—42no—
67C2×D127—42no—
68C4×C127—21no—
69C4×C127—21no—
70C4×C127—21no—
71C4×C127—21no—
72C4×C127—21no—
73C4×C127#10412yes—
74Dic127—42no—
75C2×C127—22no—
76C2×C127C2×#10011yescharacteristic
77C2×C127C2×#10411yescharacteristic
78C2×C127#42111yescharacteristic
79D127—82no—
80#1530—1271nomaximal, Sylow
81C127#153011yescharacteristic, complemented, Sylow
82#532—1271no—
83#432—1271no—
84C2×#97D12711yescharacteristic, complemented
85#329—1274no—
86C2×#41—1272no—
87C22×C8—21no—
88C4⋊Q8—1271no—
89#127—1271no—
90#124—1272no—
91#121—2541no—
92#121—2541no—
93C4×D4—2541no—
94C4×D4—2541no—
95C2×C42C2×D12711yescharacteristic
96C4⋊C8—2541no—
97C22⋊C8—2541no—
98#97—21no—
99#97—1272no—
100#97C2×D12712yes—
101#41—5084no—
102C2×Q8—1272no—
103C2×D4—1272no—
104C2×D4—2542no—
105C2×D4—2541no—
106C2×D4—2541no—
107C22×C4—2542no—
108C22×C4C4×D12711yescharacteristic
109C22×C4C22⋊D12711yescharacteristic
110C22×C4D50811yescharacteristic
111C2×C8—24no—
112C2×C8—22no—
113C2×C8—2542no—
114C4⋊C4—2541no—
115C4⋊C4—2541no—
116C22⋊C4—2542no—
117C22⋊C4—2542no—
118C42C4×D12711yescharacteristic
119C42C22⋊D12711yescharacteristic
120C42D50811yescharacteristic
121C42C22×D12711yescharacteristic
122C23—2542no—
123C23(C2×D127)⋊C411yescharacteristic
124Q8—5082no—
125D4—5082no—
126D4—5082no—
127D4—5084no—
128D4—5082no—
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—41no—
132C2×C4—5084no—
133C2×C4—2542no—
134C2×C4(C2×D127)⋊C411yescharacteristic
135C2×C4(C2×D127)⋊C411yescharacteristic
136C2×C4(C2×D127)⋊C411yescharacteristic
137C2×C4C2×C4×D12711yescharacteristic
138C2×C4C2×C22⋊D12711yescharacteristic
139C2×C4C2×D50811yescharacteristic
140C8—24no—
141C8—5082no—
142C22—21no—
143C22—21no—
144C22—5082no—
145C22—5082no—
146C22—5082no—
147C22#1686112512yes—
148C22C2×(C2×D127)⋊C411yescharacteristic
149C4—21no—
150C4—21no—
151C4—21no—
152C4—21no—
153C4—21no—
154C4—5082no—
155C4#1686113712yes—
156C2—22no—
157C2—10162no—
158C2C2×#1686112511yescharacteristic
159C2C2×#1686113711yescharacteristic
160C2#2254989211yescharacteristic
subgroupG/H
centerC22C2×(C2×D127)⋊C4
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
FittingC2×#97×C127C2
socleC22×C127C2×C22⋊C4
prime radical#22550426C2
cosocleC42×C127C23
hypercenterC2×#97D127
hypocenterC127#1530
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#97

Complements

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

NH ≅ G/Ntype
2. #22549933C2semidirect
3. #22549615C2semidirect
4. C2×#97×C127C2semidirect
81. C127#1530semidirect
84. C2×#97D127semidirect

Maximal subgroups

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

subgroupindex
#225504262
#225499332
#225496152
C2×#97×C1272
#1530127