Finite Group Databasebeta

← #30640399

Subgroups — #30640399

Order 16256.

286 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 73 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×#16861124C212yesmaximal, complemented
2#22549671C212yesmaximal, complemented
3#22549669C212yesmaximal, complemented
4#535×C127C211yesmaximal, characteristic
5C2×C22⋊Dic127—22no—
6#16861124—22no—
7#16861124C2214yes—
8C2×#41×C127C411yescharacteristic, complemented
9C22×D4×C127C2211yescharacteristic
10#127×C127C411yescharacteristic, complemented
11#127×C127C2211yescharacteristic
12#124×C127C411yescharacteristic, complemented
13#124×C127C2211yescharacteristic
14#121×C127—22no—
15#120×C127—22no—
16#117×C127C411yescharacteristic, complemented
17C22×Dic127—42no—
18C22⋊Dic127—24no—
19C22⋊Dic127—42no—
20C24×C127D412yes—
21#41×C127—42no—
22C2×Q8×C127C2×C411yescharacteristic, complemented
23C2×D4×C127—21no—
24C2×D4×C127—22no—
25C2×D4×C127—22no—
26C2×D4×C127—42no—
27C2×D4×C127C2×C411yescharacteristic
28C2×D4×C127C2×C411yescharacteristic, complemented
29C2×D4×C127D412yes—
30C2×D4×C127C2311yescharacteristic
31C22×C4×C127—21no—
32C22×C4×C127C2×C411yescharacteristic
33C4⋊C4×C127—21no—
34C22⋊C4×C127—21no—
35C22⋊C4×C127—22no—
36C22⋊C4×C127—22no—
37C42×C127C2×C411yescharacteristic, complemented
38C42×C127C2×C411yescharacteristic, complemented
39C2×Dic127—44no—
40C2×Dic127—84no—
41C23×C127—22no—
42C23×C127—22no—
43C23×C127—21no—
44C23×C127—82no—
45C23×C127C22⋊C412yes—
46C23×C127C22⋊C411yescharacteristic
47C23×C127C2×D412yes—
48Q8×C127—41no—
49D4×C127—41no—
50D4×C127—42no—
51D4×C127—41no—
52D4×C127—41no—
53D4×C127—42no—
54D4×C127—82no—
55D4×C127—82no—
56C2×C4×C127—22no—
57C2×C4×C127—21no—
58C2×C4×C127—21no—
59C2×C4×C127—21no—
60C2×C4×C127—42no—
61C2×C4×C127C22⋊C411yescharacteristic
62C2×C4×C127C22×C411yescharacteristic
63C22×C127—22no—
64C22×C127—22no—
65C22×C127—41no—
66C22×C127—42no—
67C22×C127—42no—
68C22×C127—44no—
69C22×C127—42no—
70C22×C127—41no—
71C22×C127—82no—
72C22×C127#9912yes—
73C22×C127C2×C22⋊C411yescharacteristic
74C4×C127—21no—
75C4×C127—21no—
76C4×C127—41no—
77C4×C127—41no—
78C4×C127—41no—
79Dic127—84no—
80C2×C127—21no—
81C2×C127—22no—
82C2×C127—42no—
83C2×C127—42no—
84C2×C127—81no—
85C2×C127C2×#9911yescharacteristic
86#2127—1271nomaximal, Sylow
87C127#212711yescharacteristic, complemented, Sylow
88#535D12711yescharacteristic
89C2×#99—1272no—
90#354—1272no—
91#353—1272no—
92C2×#41Dic12711yescharacteristic, complemented
93C22×D4C2×D12711yescharacteristic
94#127Dic12711yescharacteristic, complemented
95#127C2×D12711yescharacteristic
96#124Dic12711yescharacteristic, complemented
97#124C2×D12711yescharacteristic
98#121—22no—
99#120—22no—
100#117Dic12711yescharacteristic, complemented
101C2×C22⋊C4—2542no—
102#99—1274no—
103#99—2542no—
104C24C22⋊D12712yes—
105#41—42no—
106C2×Q8C2×Dic12711yescharacteristic, complemented
107C2×D4—21no—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4—42no—
111C2×D4C2×Dic12711yescharacteristic
112C2×D4C2×Dic12711yescharacteristic, complemented
113C2×D4C22⋊D12712yes—
114C2×D4C22×D12711yescharacteristic
115C22×C4—21no—
116C22×C4—5082no—
117C22×C4C2×Dic12711yescharacteristic
118C4⋊C4—21no—
119C22⋊C4—22no—
120C22⋊C4—21no—
121C22⋊C4—22no—
122C22⋊C4—5082no—
123C22⋊C4—2544no—
124C42C2×Dic12711yescharacteristic, complemented
125C42C2×Dic12711yescharacteristic, complemented
126C23—22no—
127C23—22no—
128C23—21no—
129C23—82no—
130C23C22⋊Dic12712yes—
131C23C22⋊Dic12711yescharacteristic
132C23C2×C22⋊D12712yes—
133Q8—41no—
134D4—41no—
135D4—42no—
136D4—41no—
137D4—41no—
138D4—42no—
139D4—82no—
140D4—82no—
141C2×C4—22no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—42no—
146C2×C4—10164no—
147C2×C4—5084no—
148C2×C4C22⋊Dic12711yescharacteristic
149C2×C4C22×Dic12711yescharacteristic
150C22—22no—
151C22—22no—
152C22—41no—
153C22—42no—
154C22—42no—
155C22—44no—
156C22—42no—
157C22—41no—
158C22—82no—
159C22#1686112412yes—
160C22C2×C22⋊Dic12711yescharacteristic
161C4—21no—
162C4—21no—
163C4—41no—
164C4—41no—
165C4—41no—
166C4—10164no—
167C2—21no—
168C2—22no—
169C2—42no—
170C2—42no—
171C2—81no—
172C2C2×#1686112411yescharacteristic
subgroupG/H
centerC2C2×#16861124
commutatorC2×C4×C127C22×C4
FrattiniC2×D4C22×D127
Fitting, prime radical#535×C127C2
socleC2×C127C2×#99
cosocleC2×D4×C127C23
hypercenter#535D127
hypocenterC127#2127
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C23 ◁ C22×D4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#16861124C2semidirect
2. #22549671C2semidirect
3. #22549669C2semidirect
8. C2×#41×C127C4semidirect
10. #127×C127C4semidirect
12. #124×C127C4semidirect
16. #117×C127C4semidirect
22. C2×Q8×C127C2×C4semidirect
28. C2×D4×C127C2×C4semidirect
37. C42×C127C2×C4semidirect
38. C42×C127C2×C4semidirect
87. C127#2127semidirect
92. C2×#41Dic127semidirect
94. #127Dic127semidirect
96. #124Dic127semidirect
100. #117Dic127semidirect
106. C2×Q8C2×Dic127semidirect
112. C2×D4C2×Dic127semidirect
124. C42C2×Dic127semidirect
125. C42C2×Dic127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611242
#225496712
#225496692
#535×C1272
#2127127