Finite Group Databasebeta

← #30637319

Subgroups — #30637319

Order 16256.

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

normal subgroups only

This group has 234 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
1#22550410C211yesmaximal, characteristic
2C2×#16861120C211yesmaximal, characteristic, complemented
3#22549617C212yesmaximal, complemented
4#22549615C212yesmaximal, complemented
5C4×M16×C127C211yesmaximal, characteristic, complemented
6C22×C127⋊C8—21no—
7#16861203—21no—
8#16861201C2211yescharacteristic
9#16861197C2211yescharacteristic
10#16861192—21no—
11#16861188C411yescharacteristic
12#16861183C411yescharacteristic
13C4×Dic254C411yescharacteristic
14C4×D508C411yescharacteristic
15C4×C22⋊D127—21no—
16#16861166—21no—
17#16861120—21no—
18#16861120C2212yes—
19C2×M16×C127—21no—
20C2×C42×C127C2211yescharacteristic
21#97×C127C2212yes—
22C4×C8×C127—21no—
23C2×Dic254C2×C411yescharacteristic
24C2×D508C2×C411yescharacteristic
25C2×C22⋊D127—21no—
26C2×C4×D127—21no—
27C2×C127⋊C8—24no—
28C2×C127⋊C8—22no—
29C4⋊Dic127C2×C411yescharacteristic
30C4⋊Dic127C2×C411yescharacteristic
31Dic127⋊C4—21no—
32Dic127⋊C4—21no—
33Dic127⋊C4—21no—
34Dic127⋊C4—21no—
35C22⋊Dic127—21no—
36(C2×D127)⋊C4—21no—
37(C2×D127)⋊C4—21no—
38(C2×D127)⋊C4—21no—
39(C2×D127)⋊C4—21no—
40C4×Dic127—21no—
41C22×C4×C127C2×C411yescharacteristic
42C22×C4×C127D411yescharacteristic
43C22×C4×C127D411yescharacteristic
44M16×C127—24no—
45C2×C8×C127—22no—
46C42×C127C2×C411yescharacteristic
47C42×C127D411yescharacteristic
48C42×C127D411yescharacteristic
49C42×C127C2311yescharacteristic
50C22×D127—21no—
51Dic254—41no—
52D508—41no—
53C22⋊D127—42no—
54C2×Dic127—21no—
55C2×Dic127—21no—
56C2×Dic127—21no—
57C4×D127—42no—
58C127⋊C8—24no—
59C23×C127C22⋊C411yescharacteristic
60C2×C4×C127—22no—
61C2×C4×C127—22no—
62C2×C4×C127—41no—
63C2×C4×C127C22⋊C411yescharacteristic
64C2×C4×C127C22⋊C411yescharacteristic
65C2×C4×C127C22⋊C411yescharacteristic
66C2×C4×C127C22×C411yescharacteristic
67C2×C4×C127C2×D411yescharacteristic
68C2×C4×C127C2×D411yescharacteristic
69C8×C127—24no—
70C22×C127—21no—
71C22×C127—21no—
72C22×C127#10412yes—
73C22×C127C2×C22⋊C411yescharacteristic
74C2×D127—41no—
75C2×D127—41no—
76C2×D127—41no—
77C4×C127—21no—
78C4×C127—21no—
79C4×C127—21no—
80C4×C127—21no—
81C4×C127—21no—
82C4×C127#10412yes—
83Dic127—41no—
84Dic127—41no—
85Dic127—41no—
86C2×C127—22no—
87C2×C127#41311yescharacteristic
88C2×C127C2×#10411yescharacteristic
89C2×C127C2×#10411yescharacteristic
90D127—81no—
91#1524—1271nomaximal, Sylow
92C127#152411yescharacteristic, complemented, Sylow
93#529—1271no—
94C4×M16D12711yescharacteristic, complemented
95C2×#97—1271no—
96#330—1272no—
97#329—1272no—
98C2×M16—21no—
99C22×C8—2541no—
100#126—2541no—
101#125—1271no—
102#124—1271no—
103#123—2541no—
104C22⋊Q8—1271no—
105#121—1271no—
106C4×Q8—1271no—
107C4×D4—1271no—
108C4×D4—2541no—
109#117—2541no—
110C2×C42C2×D12711yescharacteristic
111#97—1272no—
112#97—2541no—
113#97C2×D12712yes—
114C4×C8—21no—
115C2×Q8—1271no—
116C2×D4—1271no—
117C2×D4—2541no—
118C22×C4—2541no—
119C22×C4C4×D12711yescharacteristic
120C22×C4C22⋊D12711yescharacteristic
121C22×C4D50811yescharacteristic
122M16—24no—
123C2×C8—22no—
124C2×C8—2544no—
125C2×C8—2542no—
126C4⋊C4—1271no—
127C4⋊C4—1271no—
128C4⋊C4—2541no—
129C4⋊C4—2541no—
130C4⋊C4—2541no—
131C4⋊C4—2541no—
132C22⋊C4—2541no—
133C22⋊C4—2541no—
134C22⋊C4—2541no—
135C22⋊C4—2541no—
136C22⋊C4—2541no—
137C42—2541no—
138C42C4×D12711yescharacteristic
139C42C22⋊D12711yescharacteristic
140C42D50811yescharacteristic
141C42C22×D12711yescharacteristic
142C23—2541no—
143C23(C2×D127)⋊C411yescharacteristic
144Q8—5081no—
145D4—5081no—
146D4—5082no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—41no—
150C2×C4—5082no—
151C2×C4—2541no—
152C2×C4—2541no—
153C2×C4—2541no—
154C2×C4(C2×D127)⋊C411yescharacteristic
155C2×C4(C2×D127)⋊C411yescharacteristic
156C2×C4(C2×D127)⋊C411yescharacteristic
157C2×C4C2×C4×D12711yescharacteristic
158C2×C4C2×C22⋊D12711yescharacteristic
159C2×C4C2×D50811yescharacteristic
160C8—24no—
161C8—2544no—
162C22—21no—
163C22—21no—
164C22—5081no—
165C22—5081no—
166C22—5081no—
167C22#1686113712yes—
168C22C2×(C2×D127)⋊C411yescharacteristic
169C4—21no—
170C4—21no—
171C4—21no—
172C4—21no—
173C4—21no—
174C4—5081no—
175C4—5081no—
176C4—5081no—
177C4#1686113512yes—
178C2—22no—
179C2—10161no—
180C2#2254985811yescharacteristic
181C2C2×#1686113511yescharacteristic
182C2C2×#1686113711yescharacteristic
subgroupG/H
centerC2×C4(C2×D127)⋊C4
commutatorC2×C4×C127C22×C4
FrattiniC42C22×D127
FittingC4×M16×C127C2
socleC22×C127C2×C22⋊C4
prime radical#16861183C4
cosocleC42×C127C23
hypercenterC4×M16D127
hypocenterC127#1524
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
2. C2×#16861120C2semidirect
3. #22549617C2semidirect
4. #22549615C2semidirect
5. C4×M16×C127C2semidirect
92. C127#1524semidirect
94. C4×M16D127semidirect

Maximal subgroups

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

subgroupindex
#225504102
C2×#168611202
#225496172
#225496152
C4×M16×C1272
#1524127