Finite Group Databasebeta

← #30649132

Subgroups — #30649132

Order 16256.

392 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 117 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#22550427C211yesmaximal, characteristic
2#125×D127C211yesmaximal, characteristic, complemented
3#22550244C211yesmaximal, characteristic, complemented
4#22550242C211yesmaximal, characteristic, complemented
5#22550241C211yesmaximal, characteristic, complemented
6#22550232C214yesmaximal, complemented
7#22549876C214yesmaximal, complemented
8#22549828C211yesmaximal, characteristic, complemented
9#491×C127C211yesmaximal, characteristic, complemented
10C2×#14523249C2212yes—
11C2×#14523231C2212yes—
12#16861206C2211yescharacteristic
13#16861201C2211yescharacteristic
14#16861200C2212yes—
15#16861199—22no—
16#16861198C2212yes—
17#16861184—22no—
18#16861171—22no—
19C4⋊C4×D127—22no—
20C4⋊C4×D127C2212yes—
21C42×D127C2211yescharacteristic
22#16861133C2214yes—
23#16861131C2214yes—
24#16861129C2214yes—
25#16861120C2211yescharacteristic
26C8×Dic127C2212yes—
27C4⋊Q8×C127C2211yescharacteristic
28#125×C127C2211yescharacteristic
29#103×C127C2214yes—
30#97×C127C2211yescharacteristic
31#14523249—28no—
32C2×D508—22no—
33C2×D508—22no—
34C2×D508C2312yes—
35C2×C4×D127—22no—
36C2×C4×D127—22no—
37C2×C4×D127D412yes—
38C2×C4×D127C2311yescharacteristic
39#14523231—24no—
40C2×C127⋊C8C2312yes—
41C4⋊Dic127—22no—
42C4⋊Dic127C2312yes—
43Dic127⋊C4—22no—
44Dic127⋊C4—22no—
45Dic127⋊C4—22no—
46(C2×D127)⋊C4—24no—
47C4×Dic127D412yes—
48C4×Dic127C2311yescharacteristic
49C2×Q8×C127C2312yes—
50C2×C8×C127C2312yes—
51C4⋊C4×C127—22no—
52C4⋊C4×C127—22no—
53C4⋊C4×C127C2312yes—
54C42×C127C2311yescharacteristic
55C22×D127—22no—
56C22×D127C2×D411yescharacteristic
57D508—24no—
58D508—44no—
59D508—44no—
60C2×Dic127—22no—
61C2×Dic127C2×D412yes—
62C2×Dic127C2×D411yescharacteristic
63C4×D127—42no—
64C4×D127—44no—
65C4×D127—44no—
66C4×D127#4114yes—
67C127⋊C8—22no—
68Q8×C127—24no—
69C2×C4×C127—22no—
70C2×C4×C127—22no—
71C2×C4×C127C2×D412yes—
72C2×C4×C127C2411yescharacteristic
73C8×C127—22no—
74C22×C127C22×D411yescharacteristic
75C2×D127—22no—
76C2×D127—42no—
77C2×D127—44no—
78C2×D127#12412yes—
79C4×C127—22no—
80C4×C127—42no—
81C4×C127—42no—
82C4×C127C2×#4112yes—
83Dic127—22no—
84Dic127—42no—
85Dic127#12412yes—
86C2×C127C2×#12411yescharacteristic
87C2×C127#57512yes—
88D127—22no—
89D127—82no—
90#3142—1271nomaximal, Sylow
91C127#314211yescharacteristic, complemented, Sylow
92#532—1271no—
93C2×#125—1271no—
94#491—1271no—
95#491D12711yescharacteristic, complemented
96#490—1271no—
97#490—1271no—
98#488—1274no—
99#417—1274no—
100C4×M16—1271no—
101C2×#41—1272no—
102C2×M16—1272no—
103C4⋊Q8C2×D12711yescharacteristic
104#127—1271no—
105#125—1272no—
106#125—1271no—
107#125—2542no—
108#125C2×D12711yescharacteristic
109#124—1272no—
110#121—2542no—
111C4×D4—2542no—
112C2×C4⋊C4—1272no—
113C2×C4⋊C4—2542no—
114C2×C42—1271no—
115#103—1274no—
116#103C2×D12714yes—
117#102—1274no—
118#102—1274no—
119#97—1271no—
120#97C2×D12711yescharacteristic
121C4×C8—1272no—
122#41—2548no—
123C2×Q8C22×D12712yes—
124C2×D4—1272no—
125C2×D4—2542no—
126C2×D4—2542no—
127C22×C4—1271no—
128C22×C4—1272no—
129C22×C4—2542no—
130C22×C4—2542no—
131M16—2544no—
132C2×C8—1272no—
133C2×C8C22×D12712yes—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—1272no—
137C4⋊C4—2542no—
138C4⋊C4—2542no—
139C4⋊C4—2542no—
140C4⋊C4—2542no—
141C4⋊C4C22×D12712yes—
142C22⋊C4—2544no—
143C42—1272no—
144C42—1271no—
145C42C22×D12711yescharacteristic
146C23—1271no—
147C23—2542no—
148Q8—24no—
149D4—5084no—
150D4—5084no—
151D4—2544no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—1271no—
155C2×C4—1274no—
156C2×C4—1272no—
157C2×C4—5082no—
158C2×C4—5084no—
159C2×C4—5084no—
160C2×C4—2542no—
161C2×C4D4×D12712yes—
162C2×C4C23×D12711yescharacteristic
163C8—22no—
164C8—2542no—
165C22—1272no—
166C22—5082no—
167C22—5084no—
168C22—2542no—
169C22C2×D4×D12711yescharacteristic
170C4—22no—
171C4—42no—
172C4—42no—
173C4—1272no—
174C4—5082no—
175C4—2542no—
176C4#41×D12712yes—
177C2—10162no—
178C2—2542no—
179C2#124×D12711yescharacteristic
180C2#2255067512yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#491×C127C2
socleC22×C127C22×D4
prime radical#22550427C2
hypercenter#491D127
hypocenterC127#3142
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
2. #125×D127C2semidirect
3. #22550244C2semidirect
4. #22550242C2semidirect
5. #22550241C2semidirect
6. #22550232C2semidirect
7. #22549876C2semidirect
8. #22549828C2semidirect
9. #491×C127C2semidirect
91. C127#3142semidirect
95. #491D127semidirect

Maximal subgroups

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

subgroupindex
#225504272
#125×D1272
#225502442
#225502422
#225502412
#225502322
#225498762
#225498282
#491×C1272
#3142127