Finite Group Databasebeta

← #30640400

Subgroups — #30640400

Order 16256.

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

normal subgroups only

This group has 270 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#22549848C212yesmaximal, complemented
3#22549669C211yesmaximal, characteristic, complemented
4#22549668C212yesmaximal, complemented
5#353×C127C211yesmaximal, characteristic, complemented
6C2×#14523247C2211yescharacteristic
7C2×D4×D127C411yescharacteristic, complemented
8#16861205C2211yescharacteristic
9#16861197C411yescharacteristic, complemented
10#16861196C411yescharacteristic, complemented
11#16861181—22no—
12#16861177—22no—
13#16861166C411yescharacteristic, complemented
14#16861165—22no—
15#16861124C2212yescomplemented
16#16861123—22no—
17#127×C127C2211yescharacteristic
18#99×C127C2212yescomplemented
19C23×D127—21no—
20#14523247—42no—
21C2×Dic254C2×C411yescharacteristic
22C2×D508C2×C411yescharacteristic
23C2×C22⋊D127—21no—
24C2×C22⋊D127—22no—
25C2×C22⋊D127D412yes—
26D4×D127—42no—
27C22×Dic127D412yes—
28C2×C4×D127C2×C411yescharacteristic
29Dic127⋊C4—21no—
30Dic127⋊C4—42no—
31C22⋊Dic127—22no—
32C22⋊Dic127—22no—
33(C2×D127)⋊C4—21no—
34(C2×D127)⋊C4—22no—
35C4×Dic127—42no—
36C4×Dic127C2×C411yescharacteristic
37C2×D4×C127—22no—
38C2×D4×C127C2×C411yescharacteristic
39C2×D4×C127C2311yescharacteristic
40C22⋊C4×C127—22no—
41C42×C127C2×C411yescharacteristic
42C22×D127—21no—
43C22×D127—42no—
44C22×D127—82no—
45C22×D127C22⋊C411yescharacteristic
46Dic254—41no—
47D508—41no—
48C22⋊D127—42no—
49C22⋊D127—42no—
50C22⋊D127—82no—
51C2×Dic127—22no—
52C2×Dic127—22no—
53C2×Dic127—21no—
54C2×Dic127—42no—
55C2×Dic127C22⋊C412yes—
56C2×Dic127C22⋊C411yescharacteristic
57C4×D127—22no—
58C23×C127—21no—
59C23×C127C2×D412yes—
60D4×C127—41no—
61D4×C127—41no—
62D4×C127—82no—
63C2×C4×C127—21no—
64C2×C4×C127—42no—
65C2×C4×C127C22×C411yescharacteristic
66C22×C127—22no—
67C22×C127—42no—
68C22×C127—42no—
69C22×C127—41no—
70C22×C127C2×C22⋊C411yescharacteristic
71C2×D127—21no—
72C2×D127—41no—
73C2×D127—42no—
74C2×D127—41no—
75C2×D127—84no—
76C4×C127—21no—
77C4×C127—41no—
78C4×C127—82no—
79Dic127—22no—
80Dic127—41no—
81Dic127—82no—
82Dic127#9912yes—
83C2×C127—21no—
84C2×C127—42no—
85C2×C127—81no—
86C2×C127C2×#9911yescharacteristic
87D127—41no—
88D127—81no—
89#2128—1271nomaximal, Sylow
90C127#212811yescharacteristic, complemented, Sylow
91#535—1271no—
92#410—1272no—
93#353—1271no—
94#353—1272no—
95#353D12711yescharacteristic, complemented
96C2×#41—1271no—
97C22×D4—1271no—
98#127—1271no—
99#127C2×D12711yescharacteristic
100#124—1271no—
101#124—1271no—
102#121—2542no—
103#120—2542no—
104#117—1271no—
105#117—2542no—
106#99—1272no—
107#99—2542no—
108#99C2×D12712yescomplemented
109C24—2541no—
110#41—5082no—
111C2×Q8—1271no—
112C2×D4—22no—
113C2×D4—1272no—
114C2×D4—1271no—
115C2×D4—5082no—
116C2×D4—2541no—
117C2×D4—2542no—
118C2×D4C4×D12711yescharacteristic
119C2×D4C22×D12711yescharacteristic
120C22×C4—1272no—
121C22×C4—1271no—
122C4⋊C4—5082no—
123C4⋊C4—2541no—
124C22⋊C4—22no—
125C22⋊C4—2542no—
126C22⋊C4—2541no—
127C22⋊C4—2542no—
128C22⋊C4—2542no—
129C42—1271no—
130C42—5082no—
131C42C4×D12711yescharacteristic
132C23—21no—
133C23—1271no—
134C23—10162no—
135C23—5082no—
136C23—2541no—
137C23D4×D12712yes—
138Q8—5081no—
139D4—41no—
140D4—41no—
141D4—82no—
142D4—10162no—
143D4—5081no—
144D4—5082no—
145D4—5082no—
146C2×C4—21no—
147C2×C4—42no—
148C2×C4—1271no—
149C2×C4—1272no—
150C2×C4—5082no—
151C2×C4—2542no—
152C2×C4—2542no—
153C2×C4—2542no—
154C2×C4—2541no—
155C2×C4C2×C4×D12711yescharacteristic
156C22—22no—
157C22—42no—
158C22—42no—
159C22—41no—
160C22—10164no—
161C22—5082no—
162C22—5081no—
163C22—5081no—
164C22—2541no—
165C22C22⋊C4×D12711yescharacteristic
166C4—21no—
167C4—41no—
168C4—82no—
169C4—1272no—
170C4—10162no—
171C4—5081no—
172C4—2542no—
173C2—21no—
174C2—42no—
175C2—81no—
176C2—10161no—
177C2—5081no—
178C2#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#2128
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. #22549848C2semidirect
3. #22549669C2semidirect
4. #22549668C2semidirect
5. #353×C127C2semidirect
7. C2×D4×D127C4semidirect
9. #16861197C4semidirect
10. #16861196C4semidirect
13. #16861166C4semidirect
15. #16861124C22semidirect
18. #99×C127C22semidirect
90. C127#2128semidirect
95. #353D127semidirect
108. #99C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504392
#225498482
#225496692
#225496682
#353×C1272
#2128127