Finite Group Databasebeta

← #30636657

Subgroups — #30636657

Order 16256.

294 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 37 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#22549731C211yesmaximal, characteristic
2#22549600C211yesmaximal, characteristic, complemented
3#323×C127C211yesmaximal, characteristic, complemented
4C24×D127C411yescharacteristic
5C2×C22⋊Dic127C411yescharacteristic
6C2×(C2×D127)⋊C4—21no—
7C2×(C2×D127)⋊C4—21no—
8C22⋊(C127⋊C8)—21no—
9C2×C22⋊C4×C127C2211yescharacteristic
10C22⋊C8×C127—21no—
11C23×D127—21no—
12C23×D127—21no—
13C23×D127—41no—
14C23×D127—41no—
15C23×D127—42no—
16C23×D127—42no—
17C23×D127C812yescomplemented
18C22×Dic127—21no—
19C2×C127⋊C8—41no—
20C22⋊Dic127—41no—
21(C2×D127)⋊C4—42no—
22(C2×D127)⋊C4—42no—
23C24×C127C2×C411yescharacteristic
24C22×C4×C127D411yescharacteristic
25C22×C4×C127D411yescharacteristic
26C2×C8×C127—41no—
27C22⋊C4×C127—41no—
28C22×D127—22no—
29C22×D127—22no—
30C22×D127—22no—
31C22×D127—44no—
32C22×D127—44no—
33C22×D127—41no—
34C22×D127—41no—
35C22×D127—42no—
36C22×D127—48no—
37C22×D127—42no—
38C22×D127—42no—
39C22×D127—41no—
40C22×D127—41no—
41C22×D127—42no—
42C22×D127—42no—
43C22×D127—42no—
44C2×Dic127—41no—
45C2×Dic127—41no—
46C2×Dic127—41no—
47C127⋊C8—81no—
48C23×C127—21no—
49C23×C127—21no—
50C23×C127—41no—
51C23×C127—41no—
52C23×C127C22⋊C411yescharacteristic
53C23×C127C2×C811yescharacteristic
54C23×C127M1611yescharacteristic
55C2×C4×C127—21no—
56C2×C4×C127—21no—
57C2×C4×C127—41no—
58C2×C4×C127—41no—
59C8×C127—81no—
60C22×C127—21no—
61C22×C127—21no—
62C22×C127—21no—
63C22×C127—21no—
64C22×C127—41no—
65C22×C127—41no—
66C22×C127—42no—
67C22×C127—41no—
68C22×C127—41no—
69C22×C127C22⋊C811yescharacteristic
70C22×C127#9911yescharacteristic
71C22×C127#10011yescharacteristic
72C2×D127—24no—
73C2×D127—42no—
74C2×D127—42no—
75C2×D127—42no—
76C2×D127—44no—
77C2×D127—44no—
78C2×D127—44no—
79C2×D127—42no—
80C2×D127—42no—
81C2×D127—42no—
82C2×D127—44no—
83C4×C127—41no—
84C4×C127—41no—
85Dic127—81no—
86C2×C127—21no—
87C2×C127—21no—
88C2×C127—41no—
89C2×C127—41no—
90C2×C127#32311yescharacteristic
91C2×C127AΣL(1,𝔽16)11yescharacteristic
92C2×C127AΣL(1,𝔽16)11yescharacteristic
93D127—44no—
94#1316—1271nomaximal, Sylow
95C127#131611yescharacteristic, complemented, Sylow
96#379—1271no—
97#323—1271no—
98#323D12711yescharacteristic, complemented
99C25—1271no—
100C2×C22⋊C4—1271no—
101C2×C22⋊C4—2541no—
102C2×C22⋊C4—2541no—
103C2×C22⋊C4C2×D12711yescharacteristic
104C22⋊C8—21no—
105C22⋊C8—2541no—
106C24—1272no—
107C24—5081no—
108C24—5081no—
109C24—5082no—
110C24—5082no—
111C24—2541no—
112C24—2541no—
113C24C4×D12711yescharacteristic
114C22×C4—2541no—
115C22×C4C22⋊D12711yescharacteristic
116C22×C4D50811yescharacteristic
117C2×C8—41no—
118C2×C8—5081no—
119C22⋊C4—41no—
120C22⋊C4—5082no—
121C22⋊C4—5082no—
122C22⋊C4—5081no—
123C23—21no—
124C23—21no—
125C23—41no—
126C23—41no—
127C23—5082no—
128C23—5081no—
129C23—5081no—
130C23—5082no—
131C23—5082no—
132C23—5082no—
133C23—5088no—
134C23—5082no—
135C23—5082no—
136C23—5084no—
137C23—5084no—
138C23—5081no—
139C23—5081no—
140C23—2542no—
141C23—2542no—
142C23—2542no—
143C23(C2×D127)⋊C411yescharacteristic
144C23C8×D12711yescharacteristic
145C23#1452323111yescharacteristic
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—5081no—
151C2×C4—5081no—
152C2×C4—5081no—
153C8—81no—
154C8—10161no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22—21no—
159C22—41no—
160C22—41no—
161C22—42no—
162C22—41no—
163C22—41no—
164C22—5082no—
165C22—5082no—
166C22—5082no—
167C22—5084no—
168C22—5084no—
169C22—5084no—
170C22—5084no—
171C22—5082no—
172C22—5082no—
173C22—5082no—
174C22—2544no—
175C22(C2×D127)⋊C811yescharacteristic
176C22#1686112311yescharacteristic
177C22#1686112511yescharacteristic
178C4—41no—
179C4—41no—
180C4—10161no—
181C2—21no—
182C2—21no—
183C2—41no—
184C2—41no—
185C2—5084no—
186C2#2254959911yescharacteristic
187C2#2254966211yescharacteristic
188C2#2254966411yescharacteristic
subgroupG/H
centerC22#16861123
commutatorC23×C127C2×C8
FrattiniC2×C22⋊C4C2×D127
Fitting#323×C127C2
socleC22×C127#99
prime radicalC24×D127C4
cosocleC2×C22⋊C4×C127C22
hypercenter#323D127
hypocenterC127#1316
LCSG ▷ C23×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #22549600C2semidirect
3. #323×C127C2semidirect
17. C23×D127C8semidirect
95. C127#1316semidirect
98. #323D127semidirect

Maximal subgroups

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

subgroupindex
#225497312
#225496002
#323×C1272
#1316127