Finite Group Databasebeta

← #30637320

Subgroups — #30637320

Order 16256.

270 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 63 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#22550425C211yesmaximal, characteristic
2#22549828C212yesmaximal, complemented
3#22549616C211yesmaximal, characteristic, complemented
4#22549615C212yesmaximal, complemented
5#329×C127C211yesmaximal, characteristic, complemented
6C2×#14523249C411yescharacteristic
7C2×#14523247C411yescharacteristic
8C2×#14523231—22no—
9#16861207C411yescharacteristic
10#16861205C411yescharacteristic
11#16861197C2211yescharacteristic
12#16861182—22no—
13(C22⋊D127)⋊C4—22no—
14C42×D127C2211yescharacteristic
15#16861120C2212yes—
16C8×Dic127—22no—
17#124×C127C2211yescharacteristic
18#97×C127C2212yes—
19#14523249—42no—
20#14523247—42no—
21C2×Dic254C2×C411yescharacteristic
22C2×D508—21no—
23C2×D508C2×C411yescharacteristic
24C2×C22⋊D127—21no—
25C2×C22⋊D127—22no—
26C22×Dic127—21no—
27C2×C4×D127—21no—
28C2×C4×D127C2×C411yescharacteristic
29C2×C4×D127D412yes—
30#14523231—28no—
31C2×C127⋊C8—22no—
32Dic127⋊C4—22no—
33(C2×D127)⋊C4—22no—
34C4×Dic127C2×C411yescharacteristic
35C4×Dic127D412yes—
36C2×Q8×C127C2×C411yescharacteristic
37C2×D4×C127C2×C411yescharacteristic
38C2×C8×C127—22no—
39C22⋊C4×C127—22no—
40C42×C127C2311yescharacteristic
41C22×D127—21no—
42C22×D127C22⋊C411yescharacteristic
43Dic254—41no—
44D508—41no—
45D508—42no—
46C22⋊D127—42no—
47C22⋊D127—44no—
48C2×Dic127—21no—
49C2×Dic127—42no—
50C2×Dic127C22⋊C411yescharacteristic
51C2×Dic127C22⋊C412yes—
52C4×D127—22no—
53C4×D127—24no—
54C4×D127—42no—
55C127⋊C8—24no—
56C23×C127—21no—
57Q8×C127—41no—
58D4×C127—41no—
59C2×C4×C127—21no—
60C2×C4×C127C22×C411yescharacteristic
61C2×C4×C127C2×D412yes—
62C8×C127—24no—
63C22×C127—41no—
64C22×C127—42no—
65C22×C127C2×C22⋊C411yescharacteristic
66C2×D127—21no—
67C2×D127—22no—
68C2×D127—41no—
69C2×D127—42no—
70C4×C127—21no—
71C4×C127—22no—
72C4×C127—41no—
73Dic127—41no—
74Dic127#10012yes—
75Dic127#10414yes—
76C2×C127—81no—
77C2×C127C2×#10011yescharacteristic
78C2×C127C2×#10412yes—
79D127—41no—
80D127—81no—
81#1525—1271nomaximal, Sylow
82C127#152511yescharacteristic, complemented, Sylow
83#532—1271no—
84C4×M16—1272no—
85#329—1271no—
86#329—1272no—
87#329D12711yescharacteristic, complemented
88C2×#41—1271no—
89C2×#41—1271no—
90C2×M16—2542no—
91C4⋊Q8—1271no—
92#127—1271no—
93#124—1271no—
94#124C2×D12711yescharacteristic
95#121—2542no—
96C4×D4—2542no—
97C2×C42—1271no—
98#97—1272no—
99#97C2×D12712yes—
100C4×C8—2542no—
101#41—5082no—
102#41—5082no—
103C2×Q8—1271no—
104C2×Q8C4×D12711yescharacteristic
105C2×D4—1271no—
106C2×D4—2541no—
107C2×D4—2542no—
108C2×D4—2541no—
109C2×D4C4×D12711yescharacteristic
110C22×C4—1271no—
111C22×C4—1272no—
112C22×C4—2541no—
113C22×C4—2541no—
114M16—2548no—
115C2×C8—22no—
116C2×C8—2542no—
117C4⋊C4—2542no—
118C22⋊C4—22no—
119C22⋊C4—2542no—
120C42—1271no—
121C42—1272no—
122C42C22×D12711yescharacteristic
123C23—21no—
124C23—1271no—
125C23—2541no—
126Q8—41no—
127Q8—5081no—
128D4—41no—
129D4—5081no—
130D4—5082no—
131D4—5084no—
132D4—5082no—
133C2×C4—21no—
134C2×C4—1272no—
135C2×C4—1271no—
136C2×C4—5082no—
137C2×C4—5082no—
138C2×C4—2544no—
139C2×C4—2542no—
140C2×C4—2541no—
141C2×C4C2×C4×D12711yescharacteristic
142C2×C4D4×D12712yes—
143C8—24no—
144C8—2544no—
145C22—41no—
146C22—42no—
147C22—5081no—
148C22—5082no—
149C22—2541no—
150C22—2542no—
151C22C22⋊C4×D12711yescharacteristic
152C4—21no—
153C4—22no—
154C4—41no—
155C4—1272no—
156C4—1274no—
157C4—5081no—
158C2—81no—
159C2—10161no—
160C2—5081no—
161C2#100×D12711yescharacteristic
162C2#104×D12712yes—
subgroupG/H
centerC22C22⋊C4×D127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
Fitting#329×C127C2
socleC22×C127C2×C22⋊C4
prime radical#22550425C2
cosocleC42×C127C23
hypercenter#329D127
hypocenterC127#1525
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #22549828C2semidirect
3. #22549616C2semidirect
4. #22549615C2semidirect
5. #329×C127C2semidirect
82. C127#1525semidirect
87. #329D127semidirect

Maximal subgroups

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

subgroupindex
#225504252
#225498282
#225496162
#225496152
#329×C1272
#1525127