Finite Group Databasebeta

← #30637291

Subgroups — #30637291

Order 16256.

264 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 85 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#22550426C211yesmaximal, characteristic
2#22549742C211yesmaximal, characteristic, complemented
3#22549611C214yesmaximal, complemented
4#431×C127C211yesmaximal, characteristic, complemented
5C2×#14523248—21no—
6#16861209C411yescharacteristic, complemented
7#16861206C411yescharacteristic, complemented
8#16861197—21no—
9#16861183—21no—
10#16861183C2212yes—
11C4×D508—21no—
12C4×D508C412yescomplemented
13C2×C4⋊Dic127—21no—
14C127⋊#95—21no—
15C127⋊#95C2212yes—
16C2×C42×C127C2211yescharacteristic
17C4⋊C8×C127—21no—
18C22⋊C8×C127C2212yes—
19C4×C8×C127—21no—
20#14523248—42no—
21#14523248—42no—
22C2×Dic254—21no—
23C2×D508—21no—
24C2×D508—21no—
25C2×D508C2×C412yes—
26C2×C22⋊D127—22no—
27C22×Dic127—22no—
28C2×C4×D127—22no—
29C4⋊Dic127—21no—
30C4⋊Dic127—42no—
31C4⋊Dic127C2×C412yes—
32(C2×D127)⋊C4—22no—
33(C2×D127)⋊C4—22no—
34C22×C4×C127D411yescharacteristic
35C22×C4×C127D411yescharacteristic
36C22×C4×C127C2311yescharacteristic
37C2×C8×C127—22no—
38C42×C127C2×C411yescharacteristic
39C42×C127C2×C411yescharacteristic
40C42×C127D411yescharacteristic
41C42×C127D411yescharacteristic
42C22×D127—22no—
43Dic254—42no—
44D508—42no—
45D508—42no—
46D508—42no—
47C22⋊D127—42no—
48C22⋊D127—42no—
49C2×Dic127—22no—
50C2×Dic127—42no—
51C2×Dic127—42no—
52C2×Dic127—42no—
53C4×D127—44no—
54C23×C127C2×D411yescharacteristic
55C2×C4×C127—41no—
56C2×C4×C127—41no—
57C2×C4×C127C22⋊C411yescharacteristic
58C2×C4×C127C22⋊C411yescharacteristic
59C2×C4×C127C22⋊C411yescharacteristic
60C2×C4×C127C22⋊C411yescharacteristic
61C2×C4×C127D812yes—
62C2×C4×C127SD1612yes—
63C2×C4×C127C22×C411yescharacteristic
64C2×C4×C127C2×D411yescharacteristic
65C8×C127—42no—
66C22×C127—21no—
67C22×C127—21no—
68C22×C127C2×C22⋊C411yescharacteristic
69C22×C127C2×D811yescharacteristic
70C22×C127C2×SD1611yescharacteristic
71C2×D127—42no—
72C2×D127—42no—
73C2×D127—42no—
74C4×C127—21no—
75C4×C127—21no—
76C4×C127—21no—
77C4×C127—21no—
78C4×C127#10212yes—
79C4×C127#10212yes—
80Dic127—42no—
81Dic127—82no—
82C2×C127—21no—
83C2×C127—21no—
84C2×C127#41011yescharacteristic
85C2×C127C2×#10211yescharacteristic
86C2×C127#42111yescharacteristic
87D127—82no—
88#1517—1271nomaximal, Sylow
89C127#151711yescharacteristic, complemented, Sylow
90#532—1271no—
91#431D12711yescharacteristic, complemented
92#382—1271no—
93#327—1274no—
94C2×#41—2541no—
95C4⋊Q8—1271no—
96#127—1271no—
97#124—2541no—
98#121—1272no—
99#121—2541no—
100C4×D4—1272no—
101C4×D4—2541no—
102C2×C4⋊C4—2541no—
103C2×C42C2×D12711yescharacteristic
104C4⋊C8—21no—
105C22⋊C8C2×D12712yes—
106C4×C8—21no—
107#95—1272no—
108#95—2541no—
109#41—5082no—
110#41—5082no—
111C2×Q8—2541no—
112C2×D4—1272no—
113C2×D4—2542no—
114C2×D4—2541no—
115C2×D4—2541no—
116C22×C4—2542no—
117C22×C4—2542no—
118C22×C4D50811yescharacteristic
119C22×C4D50811yescharacteristic
120C22×C4C22×D12711yescharacteristic
121C2×C8—22no—
122C4⋊C4—1272no—
123C4⋊C4—5082no—
124C4⋊C4—2541no—
125C22⋊C4—2542no—
126C22⋊C4—2542no—
127C42C4×D12711yescharacteristic
128C42C4×D12711yescharacteristic
129C42C22⋊D12711yescharacteristic
130C42C22⋊D12711yescharacteristic
131C23—2542no—
132C23C2×D50811yescharacteristic
133Q8—5082no—
134D4—5082no—
135D4—5082no—
136D4—5082no—
137D4—5082no—
138D4—5082no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—5082no—
142C2×C4—5082no—
143C2×C4—5082no—
144C2×C4—5084no—
145C2×C4—2542no—
146C2×C4(C2×D127)⋊C411yescharacteristic
147C2×C4(C2×D127)⋊C411yescharacteristic
148C2×C4(C2×D127)⋊C411yescharacteristic
149C2×C4(C2×D127)⋊C411yescharacteristic
150C2×C4D101612yes—
151C2×C4#1452323712yes—
152C2×C4C2×C4×D12711yescharacteristic
153C2×C4C2×C22⋊D12711yescharacteristic
154C8—42no—
155C22—21no—
156C22—21no—
157C22—5082no—
158C22—5082no—
159C22—5082no—
160C22C2×(C2×D127)⋊C411yescharacteristic
161C22C2×D101611yescharacteristic
162C22C2×#1452323711yescharacteristic
163C4—21no—
164C4—21no—
165C4—21no—
166C4—21no—
167C4—10162no—
168C4—5082no—
169C4#1686113112yes—
170C4#1686113112yes—
171C2—21no—
172C2—21no—
173C2—10162no—
174C2#2254984911yescharacteristic
175C2C2×#1686113111yescharacteristic
176C2#2254989211yescharacteristic
subgroupG/H
centerC22C2×(C2×D127)⋊C4
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#431×C127C2
socleC22×C127C2×C22⋊C4
prime radical#22550426C2
cosocleC22×C4×C127C23
hypercenter#431D127
hypocenterC127#1517
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
2. #22549742C2semidirect
3. #22549611C2semidirect
4. #431×C127C2semidirect
6. #16861209C4semidirect
7. #16861206C4semidirect
12. C4×D508C4semidirect
89. C127#1517semidirect
91. #431D127semidirect

Maximal subgroups

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

subgroupindex
#225504262
#225497422
#225496112
#431×C1272
#1517127