Finite Group Databasebeta

← #30640396

Subgroups — #30640396

Order 16256.

286 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 55 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#22550439C211yesmaximal, characteristic
2#99×D127C212yesmaximal, complemented
3#22549670C212yesmaximal, complemented
4#22549669C211yesmaximal, characteristic, complemented
5#353×C127C211yesmaximal, characteristic, complemented
6C2×#14523247C411yescharacteristic, complemented
7C2×D4×D127C2211yescharacteristic
8#16861205C411yescharacteristic, complemented
9#16861197C411yescharacteristic, complemented
10#16861196C2211yescharacteristic
11#16861181—22no—
12#16861177—22no—
13#16861166C411yescharacteristic, complemented
14C22⋊C4×D127—22no—
15#16861124C2212yes—
16#16861123—22no—
17#127×C127C2211yescharacteristic
18#99×C127C2212yes—
19C23×D127D412yes—
20#14523247—42no—
21C2×Dic254C2×C411yescharacteristic, complemented
22C2×D508C2×C411yescharacteristic
23C2×C22⋊D127—21no—
24C2×C22⋊D127—22no—
25C2×C22⋊D127D412yes—
26D4×D127—42no—
27C22×Dic127—21no—
28C2×C4×D127—42no—
29C2×C4×D127C2×C411yescharacteristic
30Dic127⋊C4—21no—
31C22⋊Dic127—22no—
32C22⋊Dic127—22no—
33(C2×D127)⋊C4—22no—
34(C2×D127)⋊C4—21no—
35(C2×D127)⋊C4—42no—
36C4×Dic127C2×C411yescharacteristic, complemented
37C2×D4×C127—22no—
38C2×D4×C127C2×C411yescharacteristic, complemented
39C2×D4×C127C2311yescharacteristic
40C22⋊C4×C127—22no—
41C42×C127C2×C411yescharacteristic, complemented
42C22×D127—22no—
43C22×D127—22no—
44C22×D127—82no—
45C22×D127C22⋊C412yes—
46C22×D127C22⋊C411yescharacteristic
47Dic254—41no—
48D508—41no—
49C22⋊D127—42no—
50C22⋊D127—42no—
51C22⋊D127—82no—
52C2×Dic127—21no—
53C2×Dic127—21no—
54C2×Dic127—42no—
55C2×Dic127—42no—
56C2×Dic127C22⋊C411yescharacteristic
57C4×D127—22no—
58C4×D127—84no—
59C23×C127—21no—
60C23×C127C2×D412yes—
61D4×C127—41no—
62D4×C127—41no—
63D4×C127—82no—
64C2×C4×C127—21no—
65C2×C4×C127—42no—
66C2×C4×C127C22×C411yescharacteristic
67C22×C127—22no—
68C22×C127—42no—
69C22×C127—42no—
70C22×C127—41no—
71C22×C127C2×C22⋊C411yescharacteristic
72C2×D127—22no—
73C2×D127—41no—
74C2×D127—44no—
75C2×D127—42no—
76C2×D127—82no—
77C2×D127#9912yes—
78C4×C127—21no—
79C4×C127—41no—
80C4×C127—82no—
81Dic127—21no—
82Dic127—41no—
83Dic127—41no—
84Dic127—82no—
85C2×C127—21no—
86C2×C127—42no—
87C2×C127—81no—
88C2×C127C2×#9911yescharacteristic
89D127—22no—
90D127—42no—
91#2127—1271nomaximal, Sylow
92C127#212711yescharacteristic, complemented, Sylow
93#535—1271no—
94C2×#99—1272no—
95#354—1272no—
96#353—1271no—
97#353D12711yescharacteristic, complemented
98C2×#41—1271no—
99C22×D4—1271no—
100#127—1271no—
101#127C2×D12711yescharacteristic
102#124—1271no—
103#124—1271no—
104#121—2542no—
105#120—2542no—
106#117—1271no—
107C2×C22⋊C4—2542no—
108#99—1272no—
109#99—2542no—
110#99C2×D12712yes—
111C24—1272no—
112#41—5082no—
113C2×Q8—1271no—
114C2×D4—22no—
115C2×D4—1272no—
116C2×D4—1271no—
117C2×D4—5082no—
118C2×D4—2542no—
119C2×D4—2541no—
120C2×D4C4×D12711yescharacteristic, complemented
121C2×D4C22×D12711yescharacteristic
122C22×C4—1271no—
123C22×C4—5082no—
124C22×C4—2541no—
125C4⋊C4—2541no—
126C22⋊C4—22no—
127C22⋊C4—5082no—
128C22⋊C4—2541no—
129C22⋊C4—2542no—
130C22⋊C4—2542no—
131C22⋊C4—2542no—
132C42—1271no—
133C42C4×D12711yescharacteristic, complemented
134C23—21no—
135C23—1271no—
136C23—1272no—
137C23—10162no—
138C23—2542no—
139C23—2542no—
140C23D4×D12712yes—
141Q8—5081no—
142D4—41no—
143D4—41no—
144D4—82no—
145D4—10162no—
146D4—5081no—
147D4—5082no—
148D4—5082no—
149C2×C4—21no—
150C2×C4—42no—
151C2×C4—1271no—
152C2×C4—10164no—
153C2×C4—5082no—
154C2×C4—5082no—
155C2×C4—2542no—
156C2×C4—2541no—
157C2×C4—2541no—
158C2×C4C2×C4×D12711yescharacteristic
159C22—22no—
160C22—42no—
161C22—42no—
162C22—41no—
163C22—1272no—
164C22—10162no—
165C22—5082no—
166C22—5081no—
167C22—5084no—
168C22—2542no—
169C22C22⋊C4×D12711yescharacteristic
170C4—21no—
171C4—41no—
172C4—82no—
173C4—10162no—
174C4—5081no—
175C4—5081no—
176C4—2541no—
177C2—21no—
178C2—42no—
179C2—81no—
180C2—5082no—
181C2—2542no—
182C2#99×D12711yescharacteristic
subgroupG/H
centerC2#99×D127
commutatorC2×C4×C127C22×C4
FrattiniC2×D4C22×D127
Fitting#353×C127C2
socleC2×C127C2×#99
prime radical#22550439C2
cosocleC2×D4×C127C23
hypercenter#353D127
hypocenterC127#2127
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #99×D127C2semidirect
3. #22549670C2semidirect
4. #22549669C2semidirect
5. #353×C127C2semidirect
6. C2×#14523247C4semidirect
8. #16861205C4semidirect
9. #16861197C4semidirect
13. #16861166C4semidirect
21. C2×Dic254C2×C4semidirect
36. C4×Dic127C2×C4semidirect
38. C2×D4×C127C2×C4semidirect
41. C42×C127C2×C4semidirect
92. C127#2127semidirect
97. #353D127semidirect
120. C2×D4C4×D127semidirect
133. C42C4×D127semidirect

Maximal subgroups

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

subgroupindex
#225504392
#99×D1272
#225496702
#225496692
#353×C1272
#2127127