Finite Group Databasebeta

← #30636849

Subgroups — #30636849

Order 16256.

234 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 57 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×#16861124C211yesmaximal, characteristic
2#22549756C211yesmaximal, characteristic
3C2×#100×C127C211yesmaximal, characteristic
4C23×Dic127C411yescharacteristic, complemented
5C2×C22⋊Dic127—21no—
6C2×C22⋊Dic127—21no—
7C2×C22⋊Dic127C411yescharacteristic, complemented
8#16861124—21no—
9#16861124C412yescomplemented
10C127⋊#95—21no—
11C22×D4×C127C2211yescharacteristic
12C2×M16×C127—21no—
13#100×C127—21no—
14#100×C127C412yescomplemented
15C22×Dic127—21no—
16C22×Dic127—21no—
17C22×Dic127—41no—
18C22×Dic127—41no—
19C22×Dic127—42no—
20C22⋊Dic127—22no—
21C22⋊Dic127—41no—
22C22⋊Dic127—41no—
23C22⋊Dic127—42no—
24C24×C127C2×C411yescharacteristic
25C24×C127D411yescharacteristic
26C2×D4×C127—41no—
27C2×D4×C127—41no—
28C2×D4×C127C2×C411yescharacteristic
29C2×D4×C127C2×C411yescharacteristic
30C2×D4×C127D411yescharacteristic
31C2×D4×C127Q811yescharacteristic
32C22×C4×C127D411yescharacteristic
33M16×C127—22no—
34M16×C127—41no—
35C2×C8×C127—41no—
36C2×Dic127—22no—
37C2×Dic127—41no—
38C2×Dic127—42no—
39C2×Dic127—41no—
40C2×Dic127—42no—
41C2×Dic127—42no—
42C2×Dic127—41no—
43C2×Dic127—41no—
44C2×Dic127—41no—
45C2×Dic127—81no—
46C2×Dic127—81no—
47C23×C127—21no—
48C23×C127—21no—
49C23×C127—21no—
50C23×C127—21no—
51C23×C127—41no—
52C23×C127—41no—
53C23×C127—41no—
54C23×C127—41no—
55C23×C127C4211yescharacteristic
56C23×C127C22⋊C411yescharacteristic
57C23×C127C22⋊C411yescharacteristic
58C23×C127C4⋊C411yescharacteristic
59C23×C127C4⋊C411yescharacteristic
60D4×C127—41no—
61D4×C127—41no—
62D4×C127—41no—
63D4×C127—41no—
64C2×C4×C127—21no—
65C2×C4×C127—21no—
66C2×C4×C127C22⋊C411yescharacteristic
67C2×C4×C127C4⋊C411yescharacteristic
68C8×C127—42no—
69C22×C127—21no—
70C22×C127—21no—
71C22×C127—21no—
72C22×C127—21no—
73C22×C127—21no—
74C22×C127—21no—
75C22×C127—41no—
76C22×C127—41no—
77C22×C127—41no—
78C22×C127—41no—
79C22×C127—42no—
80C22×C127—41no—
81C22×C127—41no—
82C22×C127—41no—
83C22×C127—41no—
84C22×C127—81no—
85C22×C127#9511yescharacteristic
86C22×C127#9911yescharacteristic
87C22×C127#9911yescharacteristic
88C4×C127—21no—
89C4×C127—21no—
90Dic127—42no—
91Dic127—81no—
92Dic127—82no—
93C2×C127—21no—
94C2×C127—21no—
95C2×C127—41no—
96C2×C127—41no—
97C2×C127—41no—
98C2×C127—41no—
99C2×C127#34211yescharacteristic
100C2×C127AΣL(1,𝔽16)11yescharacteristic
101C2×C127#35211yescharacteristic
102#1390—1271nomaximal, Sylow
103C127#139011yescharacteristic, complemented, Sylow
104C2×#100D12711yescharacteristic
105C2×#99—1271no—
106#386—1271no—
107C22×D4C2×D12711yescharacteristic
108C23×C4—1271no—
109C2×M16—21no—
110C2×C22⋊C4—1271no—
111C2×C22⋊C4—2541no—
112C2×C22⋊C4—2541no—
113#100—21no—
114#100Dic12712yescomplemented
115#99—1272no—
116#99—2541no—
117#95—2541no—
118C24C4×D12711yescharacteristic
119C24D50811yescharacteristic
120C2×D4—41no—
121C2×D4—41no—
122C2×D4C4×D12711yescharacteristic
123C2×D4C2×Dic12711yescharacteristic
124C2×D4C22⋊D12711yescharacteristic
125C2×D4Dic25411yescharacteristic
126C22×C4—5081no—
127C22×C4—5081no—
128C22×C4—5082no—
129C22×C4—2541no—
130C22×C4—2541no—
131C22×C4C22⋊D12711yescharacteristic
132M16—22no—
133M16—41no—
134C2×C8—41no—
135C22⋊C4—5081no—
136C22⋊C4—5082no—
137C22⋊C4—5081no—
138C22⋊C4—2542no—
139C23—21no—
140C23—21no—
141C23—21no—
142C23—21no—
143C23—41no—
144C23—41no—
145C23—41no—
146C23—41no—
147C23C4×Dic12711yescharacteristic
148C23(C2×D127)⋊C411yescharacteristic
149C23(C2×D127)⋊C411yescharacteristic
150C23Dic127⋊C411yescharacteristic
151C23C4⋊Dic12711yescharacteristic
152D4—41no—
153D4—41no—
154D4—41no—
155D4—41no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—10161no—
159C2×C4—10161no—
160C2×C4—5081no—
161C2×C4—5082no—
162C2×C4—5081no—
163C2×C4—5081no—
164C2×C4—5081no—
165C2×C4—5082no—
166C2×C4—5081no—
167C2×C4—5082no—
168C2×C4—2542no—
169C2×C4C22⋊Dic12711yescharacteristic
170C2×C4Dic127⋊C411yescharacteristic
171C8—42no—
172C22—21no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—42no—
183C22—41no—
184C22—41no—
185C22—41no—
186C22—41no—
187C22—81no—
188C22C127⋊#9511yescharacteristic
189C22#1686112311yescharacteristic
190C22#1686112311yescharacteristic
191C4—21no—
192C4—21no—
193C4—10161no—
194C4—10162no—
195C4—5082no—
196C2—21no—
197C2—21no—
198C2—41no—
199C2—41no—
200C2—41no—
201C2—41no—
202C2#2254964211yescharacteristic
203C2#2254966411yescharacteristic
204C2#2254966711yescharacteristic
subgroupG/H
centerC22#16861123
commutatorC23×C127C42
FrattiniC22×D4C2×D127
FittingC2×#100×C127C2
socleC22×C127#99
prime radical, cosocleC22×D4×C127C22
hypercenterC2×#100D127
hypocenterC127#1390
LCSG ▷ C23×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
4. C23×Dic127C4semidirect
7. C2×C22⋊Dic127C4semidirect
9. #16861124C4semidirect
14. #100×C127C4semidirect
103. C127#1390semidirect
114. #100Dic127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611242
#225497562
C2×#100×C1272
#1390127