Finite Group Databasebeta

← #30637437

Subgroups — #30637437

Order 16256.

198 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 65 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#22550431C211yesmaximal, characteristic
2#22549932C211yesmaximal, characteristic
3#22549928C211yesmaximal, characteristic
4#22549624C211yesmaximal, characteristic, complemented
5#22549624C211yesmaximal, characteristic, complemented
6#22549622C211yesmaximal, characteristic, complemented
7#332×C127C211yesmaximal, characteristic, complemented
8#16861209C2211yescharacteristic
9#16861200—21no—
10#16861189C412yes—
11#16861185—21no—
12Q8×Dic127—21no—
13#16861174C412yes—
14#16861167—21no—
15C42×D127C2211yescharacteristic
16C4⋊(C127⋊C8)C2211yescharacteristic
17C4⋊(C127⋊C8)C2211yescharacteristic
18(C2×D127)⋊C8—21no—
19(C2×D127)⋊C8—21no—
20#16861119—21no—
21C8×Dic127—21no—
22C4⋊Q8×C127C2211yescharacteristic
23C4⋊C8×C127C2211yescharacteristic
24C4⋊C8×C127C2211yescharacteristic
25C2×Dic254C2×C412yes—
26C2×C4×D127C2×C411yescharacteristic
27C2×C4×D127D411yescharacteristic
28C2×C4×D127D411yescharacteristic
29C2×C127⋊C8—21no—
30C2×C127⋊C8—21no—
31C4⋊Dic127—22no—
32C4⋊Dic127—21no—
33C4⋊Dic127—21no—
34Dic127⋊C4—22no—
35(C2×D127)⋊C4—22no—
36C4×Dic127—22no—
37C4×Dic127C2×C411yescharacteristic
38C4×Dic127D411yescharacteristic
39C4×Dic127D411yescharacteristic
40C2×Q8×C127—21no—
41C2×C8×C127—21no—
42C2×C8×C127—21no—
43C4⋊C4×C127—21no—
44C4⋊C4×C127C2×C412yes—
45C42×C127C2311yescharacteristic
46C22×D127C22⋊C411yescharacteristic
47Dic254—42no—
48C2×Dic127—22no—
49C2×Dic127C22⋊C411yescharacteristic
50C2×Dic127C22⋊C411yescharacteristic
51C2×Dic127C22⋊C411yescharacteristic
52C4×D127—21no—
53C4×D127—21no—
54C4×D127—21no—
55C4×D127—21no—
56C4×D127—41no—
57C127⋊C8—41no—
58C127⋊C8—41no—
59Q8×C127—41no—
60Q8×C127—41no—
61C2×C4×C127—22no—
62C2×C4×C127C22×C411yescharacteristic
63C2×C4×C127C2×D411yescharacteristic
64C2×C4×C127C2×D411yescharacteristic
65C8×C127—41no—
66C8×C127—41no—
67C22×C127C2×C22⋊C411yescharacteristic
68C2×D127—21no—
69C2×D127—21no—
70C2×D127—21no—
71C4×C127—21no—
72C4×C127—42no—
73C4×C127#13511yescharacteristic
74C4×C127#13511yescharacteristic
75C4×C127#13711yescharacteristic
76C4×C127#13711yescharacteristic
77Dic127—21no—
78Dic127—21no—
79Dic127—21no—
80Dic127—42no—
81C2×C127#41311yescharacteristic
82C2×C127#41611yescharacteristic
83C2×C127#41911yescharacteristic
84D127—41no—
85#1554—1271nomaximal, Sylow
86C127#155411yescharacteristic, complemented, Sylow
87#533—1271no—
88#432—1271no—
89#431—1271no—
90#333—1271no—
91#333—1271no—
92#332—1271no—
93#332D12711yescharacteristic, complemented
94C4⋊Q8—1271no—
95C4⋊Q8C2×D12711yescharacteristic
96#125—2541no—
97C22⋊Q8—1272no—
98C22⋊Q8—2541no—
99C4×Q8—1272no—
100C4×Q8—2541no—
101#117—2541no—
102C2×C42—1271no—
103C4⋊C8—1271no—
104C4⋊C8—1271no—
105C4⋊C8C2×D12711yescharacteristic
106C4⋊C8C2×D12711yescharacteristic
107C22⋊C8—2541no—
108C22⋊C8—2541no—
109#97—2541no—
110C4×C8—2541no—
111C2×Q8—21no—
112C2×Q8—1272no—
113C22×C4—1271no—
114C22×C4—1271no—
115C22×C4—1271no—
116C2×C8—21no—
117C2×C8—21no—
118C2×C8—2541no—
119C2×C8—2541no—
120C4⋊C4—21no—
121C4⋊C4—2542no—
122C4⋊C4—2542no—
123C4⋊C4—2541no—
124C4⋊C4—2541no—
125C4⋊C4C4×D12712yes—
126C22⋊C4—2542no—
127C42—1271no—
128C42—1271no—
129C42—1271no—
130C42—2542no—
131C42C22×D12711yescharacteristic
132C23—1271no—
133Q8—41no—
134Q8—41no—
135Q8—5082no—
136C2×C4—22no—
137C2×C4—1271no—
138C2×C4—1271no—
139C2×C4—1271no—
140C2×C4—5081no—
141C2×C4—2542no—
142C2×C4—2541no—
143C2×C4—2541no—
144C2×C4—2541no—
145C2×C4—2541no—
146C2×C4C2×C4×D12711yescharacteristic
147C2×C4D4×D12711yescharacteristic
148C2×C4D4×D12711yescharacteristic
149C8—41no—
150C8—41no—
151C8—5081no—
152C8—5081no—
153C22—2541no—
154C22—2541no—
155C22—2541no—
156C22C22⋊C4×D12711yescharacteristic
157C4—21no—
158C4—42no—
159C4—5082no—
160C4—2541no—
161C4—2541no—
162C4—2541no—
163C4#1686122811yescharacteristic
164C4#1686123011yescharacteristic
165C4#1686123811yescharacteristic
166C4#1686124111yescharacteristic
167C2—5081no—
168C2#2254985911yescharacteristic
169C2#2254986911yescharacteristic
170C2#2254988311yescharacteristic
subgroupG/H
centerC22C22⋊C4×D127
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
Fitting#332×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC22×D127C22⋊C4
cosocleC42×C127C23
hypercenter#332D127
hypocenterC127#1554
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
4. #22549624C2semidirect
5. #22549624C2semidirect
6. #22549622C2semidirect
7. #332×C127C2semidirect
86. C127#1554semidirect
93. #332D127semidirect

Maximal subgroups

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

subgroupindex
#225504312
#225499322
#225499282
#225496242
#225496222
#332×C1272
#1554127