Finite Group Databasebeta

← #30639128

Subgroups — #30639128

Order 16256.

444 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C2×#16861205C211yesmaximal, characteristic
2#22549880C212yesmaximal, complemented
3C2×#16861126C211yesmaximal, characteristic, complemented
4#22549635C212yesmaximal, complemented
5#518×C127C211yesmaximal, characteristic, complemented
6C22×D508C2211yescharacteristic
7C22×C22⋊D127—22no—
8C2×#14523232C2212yes—
9#16861205—22no—
10#16861205C414yescomplemented
11C2×C4×Dic127C2211yescharacteristic
12#16861129—24no—
13#16861126—22no—
14C22×D4×C127C2211yescharacteristic
15C4×D4×C127—24no—
16#117×C127C2212yes—
17C2×C22⋊C4×C127—22no—
18C23×D127—21no—
19C2×D508—22no—
20C2×D508—21no—
21C2×D508—22no—
22C2×D508C2×C412yescomplemented
23C2×C22⋊D127—28no—
24C2×C22⋊D127—48no—
25C22×Dic127—21no—
26#14523232—22no—
27#14523232—22no—
28C2×C127⋊C8—22no—
29C4×Dic127—21no—
30C4×Dic127C2×C412yescomplemented
31C24×C127D412yes—
32C2×D4×C127—22no—
33C2×D4×C127—22no—
34C2×D4×C127C2×C412yes—
35C2×D4×C127D412yes—
36C22×C4×C127—24no—
37C22×C4×C127C2311yescharacteristic
38C4⋊C4×C127D414yescomplemented
39C22⋊C4×C127—24no—
40C22⋊C4×C127—22no—
41C42×C127—22no—
42C22×D127—22no—
43C22×D127—44no—
44C22×D127—41no—
45C22×D127—41no—
46D508—24no—
47D508—42no—
48C22⋊D127—416no—
49C2×Dic127—24no—
50C2×Dic127—22no—
51C127⋊C8—42no—
52C23×C127—22no—
53C23×C127—22no—
54C23×C127—44no—
55C23×C127C22⋊C414yes—
56C23×C127C2×D411yescharacteristic
57D4×C127—24no—
58D4×C127—24no—
59C2×C4×C127—24no—
60C2×C4×C127—84no—
61C2×C4×C127C22×C411yescharacteristic
62C2×C4×C127C2×D412yes—
63C2×C4×C127C2×D411yescharacteristic
64C2×C4×C127#4112yes—
65C22×C127—22no—
66C22×C127—24no—
67C22×C127—44no—
68C22×C127—44no—
69C22×C127—82no—
70C22×C127C2×C22⋊C411yescharacteristic
71C22×C127#12011yescharacteristic
72C22×C127#12311yescharacteristic
73C2×D127—44no—
74C2×D127—42no—
75C2×D127—82no—
76C2×D127—82no—
77C4×C127—44no—
78C4×C127C4×D412yes—
79C4×C127#12112yes—
80Dic127—24no—
81C2×C127—21no—
82C2×C127—21no—
83C2×C127—44no—
84C2×C127#38611yescharacteristic
85C2×C127#45312yes—
86D127—82no—
87#1887—1271nomaximal, Sylow
88C127#188711yescharacteristic, complemented, Sylow
89C2×#127—1271no—
90#518D12711yescharacteristic, complemented
91#418—1272no—
92C2×#100—1271no—
93#339—1272no—
94C22×D4—1271no—
95C22×D4—2542no—
96C22×D4C2×D12711yescharacteristic
97C2×M16—1272no—
98#127—1274no—
99#127—2542no—
100C4×D4—24no—
101#117C2×D12712yes—
102C2×C22⋊C4—22no—
103C2×C42—1271no—
104#102—2544no—
105#100—2542no—
106C24—2541no—
107C24C22⋊D12712yes—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4—1272no—
111C2×D4—5088no—
112C2×D4—2548no—
113C2×D4—2542no—
114C2×D4—2541no—
115C2×D4—2542no—
116C2×D4C4×D12712yes—
117C2×D4D50812yes—
118C22×C4—24no—
119C22×C4—2541no—
120C22×C4C22×D12711yescharacteristic
121M16—2542no—
122M16—2542no—
123C2×C8—2542no—
124C4⋊C4C22⋊D12714yescomplemented
125C22⋊C4—24no—
126C22⋊C4—22no—
127C42—22no—
128C42—1272no—
129C42—2541no—
130C23—22no—
131C23—22no—
132C23—44no—
133C23—5084no—
134C23—5081no—
135C23—5081no—
136C23—2542no—
137C23(C2×D127)⋊C414yes—
138C23C2×C22⋊D12711yescharacteristic
139D4—24no—
140D4—24no—
141D4—50816no—
142D4—5082no—
143D4—2544no—
144C2×C4—24no—
145C2×C4—84no—
146C2×C4—2544no—
147C2×C4—2542no—
148C2×C4C2×C4×D12711yescharacteristic
149C2×C4C2×C22⋊D12712yes—
150C2×C4C2×D50811yescharacteristic
151C2×C4#1452324812yes—
152C8—5082no—
153C22—22no—
154C22—24no—
155C22—44no—
156C22—44no—
157C22—82no—
158C22—10162no—
159C22—10162no—
160C22—5084no—
161C22—5082no—
162C22C2×(C2×D127)⋊C411yescharacteristic
163C22#1686117811yescharacteristic
164C22#1686119211yescharacteristic
165C4—44no—
166C4—2544no—
167C4C4×C22⋊D12712yes—
168C4#1686118312yes—
169C2—21no—
170C2—21no—
171C2—44no—
172C2—10162no—
173C2#2254975511yescharacteristic
174C2#2255004412yes—
subgroupG/H
centerC22#16861178
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#518×C127C2
socleC22×C127#120
prime radicalC2×#16861205C2
cosocleC22×C4×C127C23
hypercenter#518D127
hypocenterC127#1887
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #518

Complements

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

NH ≅ G/Ntype
2. #22549880C2semidirect
3. C2×#16861126C2semidirect
4. #22549635C2semidirect
5. #518×C127C2semidirect
10. #16861205C4semidirect
22. C2×D508C2×C4semidirect
30. C4×Dic127C2×C4semidirect
38. C4⋊C4×C127D4semidirect
88. C127#1887semidirect
90. #518D127semidirect
124. C4⋊C4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612052
#225498802
C2×#168611262
#225496352
#518×C1272
#1887127