Finite Group Databasebeta

← #30637062

Subgroups — #30637062

Order 16256.

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

normal subgroups only

This group has 252 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
1C2×#16861166C211yesmaximal, characteristic
2#22549933C211yesmaximal, characteristic, complemented
3#22549601C212yesmaximal, complemented
4#22549599C212yesmaximal
5#432×C127C211yesmaximal, characteristic, complemented
6C22×C4×D127C411yescharacteristic
7#16861166—21no—
8#16861166—21no—
9#16861166—21no—
10#16861166—21no—
11C2×Dic127⋊C4C411yescharacteristic
12C2×Dic127⋊C4C2211yescharacteristic
13C2×(C2×D127)⋊C4C411yescharacteristic
14C2×(C2×D127)⋊C4C2211yescharacteristic
15C2×C4×Dic127C411yescharacteristic
16C4⋊(C127⋊C8)—21no—
17C22⋊(C127⋊C8)C2212yes—
18#16861120—21no—
19C2×C42×C127C2211yescharacteristic
20C4⋊C8×C127—21no—
21C22⋊C8×C127C2212yes—
22#97×C127—21no—
23C23×D127C2×C411yescharacteristic
24C22×Dic127C2×C411yescharacteristic
25C22×Dic127C2×C411yescharacteristic
26C22×Dic127C2×C411yescharacteristic
27C2×C4×D127—22no—
28C2×C4×D127—22no—
29C2×C4×D127—21no—
30C2×C4×D127—21no—
31C2×C127⋊C8—22no—
32Dic127⋊C4—21no—
33Dic127⋊C4—21no—
34Dic127⋊C4—21no—
35Dic127⋊C4—21no—
36(C2×D127)⋊C4—21no—
37(C2×D127)⋊C4—21no—
38(C2×D127)⋊C4—21no—
39(C2×D127)⋊C4—21no—
40C4×Dic127—21no—
41C4×Dic127—21no—
42C22×C4×C127C2×C411yescharacteristic
43C22×C4×C127C2×C411yescharacteristic
44C22×C4×C127C2311yescharacteristic
45C2×C8×C127—22no—
46C42×C127D411yescharacteristic
47C42×C127D411yescharacteristic
48C42×C127D411yescharacteristic
49C42×C127D411yescharacteristic
50C22×D127—21no—
51C22×D127—21no—
52C22×D127—41no—
53C22×D127—41no—
54C22×D127M1612yes—
55C2×Dic127—21no—
56C2×Dic127—21no—
57C2×Dic127—21no—
58C2×Dic127—21no—
59C2×Dic127—41no—
60C2×Dic127—41no—
61C2×Dic127M1612yes—
62C4×D127—24no—
63C4×D127—24no—
64C127⋊C8—42no—
65C23×C127C22×C411yescharacteristic
66C2×C4×C127—21no—
67C2×C4×C127—21no—
68C2×C4×C127—41no—
69C2×C4×C127—41no—
70C2×C4×C127C22⋊C411yescharacteristic
71C2×C4×C127C22⋊C411yescharacteristic
72C2×C4×C127C22⋊C411yescharacteristic
73C2×C4×C127C22⋊C411yescharacteristic
74C2×C4×C127C2×D411yescharacteristic
75C2×C4×C127C2×D411yescharacteristic
76C8×C127—42no—
77C22×C127—21no—
78C22×C127—21no—
79C22×C127C2×C22⋊C411yescharacteristic
80C22×C127C2×M1611yescharacteristic
81C22×C127C2×M1611yescharacteristic
82C2×D127—22no—
83C2×D127—42no—
84C2×D127—41no—
85C2×D127—41no—
86C2×D127—42no—
87C4×C127—21no—
88C4×C127—21no—
89C4×C127—21no—
90C4×C127—21no—
91C4×C127—21no—
92C4×C127—21no—
93Dic127—22no—
94Dic127—41no—
95Dic127—41no—
96C2×C127—21no—
97C2×C127—21no—
98C2×C127#40711yescharacteristic
99C2×C127#41011yescharacteristic
100C2×C127#41311yescharacteristic
101D127—42no—
102#1464—1271nomaximal, Sylow
103C127#146411yescharacteristic, complemented, Sylow
104C2×#117—1271no—
105#432—1271no—
106#432D12711yescharacteristic, complemented
107#324—1272no—
108#323—1272no—
109C23×C4—1271no—
110#117—2541no—
111#117—2541no—
112#117—2541no—
113#117—2541no—
114C2×C4⋊C4—1271no—
115C2×C4⋊C4—1271no—
116C2×C22⋊C4—1271no—
117C2×C22⋊C4—1271no—
118C2×C42—1271no—
119C2×C42C2×D12711yescharacteristic
120C4⋊C8—21no—
121C4⋊C8—2541no—
122C22⋊C8—1272no—
123C22⋊C8C2×D12712yes—
124#97—21no—
125#97—2541no—
126C24—1271no—
127C22×C4—1271no—
128C22×C4—1271no—
129C22×C4—1271no—
130C22×C4—2542no—
131C22×C4—2542no—
132C22×C4—2541no—
133C22×C4—2541no—
134C22×C4C4×D12711yescharacteristic
135C22×C4C4×D12711yescharacteristic
136C22×C4C22×D12711yescharacteristic
137C2×C8—22no—
138C2×C8—2542no—
139C4⋊C4—2541no—
140C4⋊C4—2541no—
141C4⋊C4—2541no—
142C4⋊C4—2541no—
143C22⋊C4—2541no—
144C22⋊C4—2541no—
145C22⋊C4—2541no—
146C22⋊C4—2541no—
147C42—2541no—
148C42—2541no—
149C42C22⋊D12711yescharacteristic
150C42C22⋊D12711yescharacteristic
151C42D50811yescharacteristic
152C42D50811yescharacteristic
153C23—1272no—
154C23—5081no—
155C23—5081no—
156C23—2541no—
157C23—2541no—
158C23C2×C4×D12711yescharacteristic
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—1272no—
164C2×C4—5081no—
165C2×C4—5081no—
166C2×C4—2544no—
167C2×C4—2544no—
168C2×C4—2541no—
169C2×C4—2541no—
170C2×C4—2541no—
171C2×C4—2541no—
172C2×C4(C2×D127)⋊C411yescharacteristic
173C2×C4(C2×D127)⋊C411yescharacteristic
174C2×C4(C2×D127)⋊C411yescharacteristic
175C2×C4(C2×D127)⋊C411yescharacteristic
176C2×C4C2×C22⋊D12711yescharacteristic
177C2×C4C2×D50811yescharacteristic
178C8—42no—
179C8—5082no—
180C22—21no—
181C22—21no—
182C22—5082no—
183C22—5082no—
184C22—5081no—
185C22—5081no—
186C22—2542no—
187C22C2×(C2×D127)⋊C411yescharacteristic
188C22M16×D12711yescharacteristic
189C22M16×D12711yescharacteristic
190C4—21no—
191C4—21no—
192C4—21no—
193C4—21no—
194C4—21no—
195C4—21no—
196C4—5081no—
197C4—5081no—
198C4—2542no—
199C2—21no—
200C2—21no—
201C2—5082no—
202C2#2254983911yescharacteristic
203C2#2254984911yescharacteristic
204C2#2254985811yescharacteristic
subgroupG/H
centerC22C2×(C2×D127)⋊C4
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#432×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC23×D127C2×C4
cosocleC22×C4×C127C23
hypercenter#432D127
hypocenterC127#1464
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. #22549933C2semidirect
3. #22549601C2semidirect
5. #432×C127C2semidirect
103. C127#1464semidirect
106. #432D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611662
#225499332
#225496012
#225495992
#432×C1272
#1464127