Finite Group Databasebeta

← #30639258

Subgroups — #30639258

Order 16256.

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

normal subgroups only

This group has 344 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
1#22550643C211yesmaximal, characteristic, complemented
2#22549881C212yesmaximal, complemented
3#22549850C211yesmaximal, characteristic, complemented
4#22549632C212yesmaximal, complemented
5#535×C127C211yesmaximal, characteristic
6#16861216—24no—
7C2×#14523232—22no—
8C2×#14523232C2212yes—
9#16861168C2212yes—
10#16861130—24no—
11#16861124—22no—
12C2×#41×C127C2211yescharacteristic
13C22×D4×C127C2211yescharacteristic
14#127×C127C412yescomplemented
15#124×C127C412yescomplemented
16#121×C127—22no—
17#120×C127—22no—
18#117×C127C2211yescharacteristic
19#14523232—24no—
20#14523232—22no—
21#14523232—22no—
22C2×C127⋊C8—24no—
23C2×C127⋊C8D414yescomplemented
24C4⋊Dic127—22no—
25C22⋊Dic127—22no—
26C4×Dic127—22no—
27C24×C127—21no—
28#41×C127—22no—
29#41×C127—22no—
30C2×Q8×C127C2×C411yescharacteristic
31C2×D4×C127—24no—
32C2×D4×C127—22no—
33C2×D4×C127—21no—
34C2×D4×C127—22no—
35C2×D4×C127C2×C412yes—
36C2×D4×C127C2×C411yescharacteristic
37C2×D4×C127D412yes—
38C22×C4×C127D412yes—
39C22×C4×C127C2311yescharacteristic
40C4⋊C4×C127—21no—
41C22⋊C4×C127—21no—
42C22⋊C4×C127—24no—
43C42×C127C2×C412yes—
44C2×Dic127—42no—
45C127⋊C8—24no—
46C23×C127—22no—
47C23×C127—42no—
48C23×C127—82no—
49C23×C127C22⋊C412yes—
50C23×C127C2×D411yescharacteristic
51Q8×C127—22no—
52D4×C127—24no—
53D4×C127—22no—
54D4×C127—24no—
55D4×C127—41no—
56D4×C127—41no—
57D4×C127—84no—
58C2×C4×C127—22no—
59C2×C4×C127—22no—
60C2×C4×C127—22no—
61C2×C4×C127C22⋊C412yes—
62C2×C4×C127C22×C411yescharacteristic
63C2×C4×C127C2×D412yes—
64C2×C4×C127C2×D411yescharacteristic
65C2×C4×C127#4112yes—
66C22×C127—22no—
67C22×C127—41no—
68C22×C127—42no—
69C22×C127—44no—
70C22×C127—42no—
71C22×C127—84no—
72C22×C127C2×C22⋊C411yescharacteristic
73C22×C127#12112yes—
74C4×C127—22no—
75C4×C127—42no—
76C4×C127C4×D412yes—
77C4×C127#12011yescharacteristic
78C4×C127#12311yescharacteristic
79Dic127—82no—
80C2×C127—22no—
81C2×C127—21no—
82C2×C127—42no—
83C2×C127—82no—
84C2×C127#38611yescharacteristic
85#1910—1271nomaximal, Sylow
86C127#191011yescharacteristic, complemented, Sylow
87#568—1271no—
88#535D12711yescharacteristic
89#418—1272no—
90#410—1271no—
91H(3,ℤ/4)—1272no—
92C2×#41C2×D12711yescharacteristic
93C22×D4C2×D12711yescharacteristic
94#131—2544no—
95C2×M16—1272no—
96C2×M16—2542no—
97#127Dic12712yescomplemented
98#124Dic12712yescomplemented
99#121—22no—
100#120—22no—
101#117—1272no—
102#117C2×D12711yescharacteristic
103#102—2544no—
104#99—2542no—
105C24—21no—
106#41—22no—
107#41—22no—
108C2×Q8C2×Dic12711yescharacteristic
109C2×D4—24no—
110C2×D4—22no—
111C2×D4—21no—
112C2×D4—22no—
113C2×D4C2×Dic12712yes—
114C2×D4C2×Dic12711yescharacteristic
115C2×D4C22⋊D12712yes—
116C22×C4C22⋊D12712yes—
117C22×C4C22×D12711yescharacteristic
118M16—2544no—
119M16—2542no—
120M16—2542no—
121C2×C8—1274no—
122C2×C8—2544no—
123C4⋊C4—21no—
124C4⋊C4—2542no—
125C22⋊C4—21no—
126C22⋊C4—24no—
127C22⋊C4—2542no—
128C42—2542no—
129C42C2×Dic12712yes—
130C23—22no—
131C23—42no—
132C23—82no—
133C23C22⋊Dic12712yes—
134C23C2×C22⋊D12711yescharacteristic
135Q8—22no—
136D4—24no—
137D4—22no—
138D4—24no—
139D4—41no—
140D4—41no—
141D4—84no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—5082no—
146C2×C4C22⋊Dic12712yes—
147C2×C4C22×Dic12711yescharacteristic
148C2×C4D4×D12712yes—
149C2×C4C2×C22⋊D12711yescharacteristic
150C2×C4#1452324712yes—
151C8—2544no—
152C22—22no—
153C22—41no—
154C22—42no—
155C22—44no—
156C22—42no—
157C22—84no—
158C22C2×C22⋊Dic12711yescharacteristic
159C22#1686118112yes—
160C4—22no—
161C4—42no—
162C4—10162no—
163C4D4×Dic12712yes—
164C4#1686117711yescharacteristic
165C4#1686119011yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—82no—
170C2#2254975611yescharacteristic
subgroupG/H
centerC2#22549756
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting, prime radical#535×C127C2
socleC2×C127#386
cosocleC22×C4×C127C23
hypercenter#535D127
hypocenterC127#1910
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #22550643C2semidirect
2. #22549881C2semidirect
3. #22549850C2semidirect
4. #22549632C2semidirect
14. #127×C127C4semidirect
15. #124×C127C4semidirect
23. C2×C127⋊C8D4semidirect
86. C127#1910semidirect
97. #127Dic127semidirect
98. #124Dic127semidirect

Maximal subgroups

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

subgroupindex
#225506432
#225498812
#225498502
#225496322
#535×C1272
#1910127