Finite Group Databasebeta

← #30641327

Subgroups — #30641327

Order 16256.

672 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 189 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
1C22×(C2×D127)⋊C4C211yesmaximal, characteristic
2#22549768C214yesmaximal, complemented
3#22549766C214yesmaximal, complemented
4#22549736C211yesmaximal, characteristic, complemented
5#22549731C212yesmaximal
6C22⋊C4×Dic127C212yesmaximal, complemented
7#380×C127C211yesmaximal, characteristic, complemented
8C24×D127C2211yescharacteristic
9C23×Dic127C2211yescharacteristic
10C2×C4⋊Dic127C2212yescomplemented
11C2×C22⋊Dic127C2212yes—
12C2×(C2×D127)⋊C4C418yescomplemented
13C2×(C2×D127)⋊C4C2214yes—
14C2×(C2×D127)⋊C4C2214yes—
15C2×(C2×D127)⋊C4C2212yes—
16C2×(C2×D127)⋊C4C2212yes—
17C2×C4×Dic127C2214yescomplemented
18C127⋊#95C2212yescomplemented
19C127⋊#95C2214yescomplemented
20C23×C4×C127C2211yescharacteristic
21C2×C4⋊C4×C127C2212yescomplemented
22C2×C22⋊C4×C127C2212yes—
23#95×C127C2212yescomplemented
24C23×D127—22no—
25C23×D127—22no—
26C23×D127—42no—
27C23×D127—42no—
28C23×D127C2×C414yes—
29C23×D127C2312yes—
30C22×Dic127—42no—
31C22×Dic127C2×C414yes—
32C22×Dic127C2314yes—
33C22×Dic127C2312yes—
34C4⋊Dic127—42no—
35C22⋊Dic127—24no—
36(C2×D127)⋊C4—44no—
37(C2×D127)⋊C4—44no—
38(C2×D127)⋊C4C2×C418yes—
39(C2×D127)⋊C4C2×C418yes—
40C4×Dic127—44no—
41C24×C127C2311yescharacteristic
42C22×C4×C127—42no—
43C22×C4×C127C2×C414yes—
44C22×C4×C127C2314yes—
45C22×C4×C127C2311yescharacteristic
46C22×C4×C127C2311yescharacteristic
47C4⋊C4×C127—42no—
48C22⋊C4×C127—24no—
49C22×D127—28no—
50C22×D127—22no—
51C22×D127—22no—
52C22×D127—24no—
53C22×D127—24no—
54C22×D127—44no—
55C22×D127—44no—
56C22×D127—48no—
57C22×D127—48no—
58C22×D127C22×C414yes—
59C2×Dic127—24no—
60C2×Dic127—24no—
61C2×Dic127—24no—
62C2×Dic127—24no—
63C2×Dic127—44no—
64C2×Dic127C22×C414yes—
65C23×C127—22no—
66C23×C127—22no—
67C23×C127C22×C412yes—
68C23×C127#4112yes—
69C23×C127#4112yes—
70C23×C127C2411yescharacteristic
71C2×C4×C127—22no—
72C2×C4×C127—22no—
73C2×C4×C127—24no—
74C2×C4×C127—24no—
75C2×C4×C127—24no—
76C2×C4×C127—44no—
77C2×C4×C127C22×C412yes—
78C2×C4×C127C22×C412yes—
79C22×C127—22no—
80C22×C127—22no—
81C22×C127—24no—
82C22×C127—24no—
83C22×C127#11714yes—
84C22×C127C23×C411yescharacteristic
85C22×C127C2×#4111yescharacteristic
86C22×C127C2×#4111yescharacteristic
87C22×C127H(5,𝔽2)12yes—
88C22×C127H(5,𝔽2)12yes—
89C2×D127—24no—
90C2×D127—24no—
91C2×D127—24no—
92C2×D127—416no—
93C2×D127—44no—
94C2×D127—44no—
95C4×C127—22no—
96C4×C127—22no—
97C4×C127—44no—
98Dic127—24no—
99Dic127—44no—
100C2×C127—24no—
101C2×C127C2×#11711yescharacteristic
102C2×C127#51811yescharacteristic
103C2×C127#51811yescharacteristic
104C2×C127#53812yes—
105C2×C127#53812yes—
106D127—44no—
107#2312—1271nomaximal, Sylow
108C127#231211yescharacteristic, complemented, Sylow
109C22×C22⋊C4—1271no—
110#388—1274no—
111#388—1274no—
112#380—1271no—
113#380D12711yescharacteristic, complemented
114#379—1272no—
115C4×C22⋊C4—1272no—
116C25—1271no—
117C23×C4—1271no—
118C23×C4C2×D12711yescharacteristic
119C2×C4⋊C4—1272no—
120C2×C4⋊C4C2×D12712yescomplemented
121C2×C22⋊C4—1278no—
122C2×C22⋊C4—1272no—
123C2×C22⋊C4—1272no—
124C2×C22⋊C4—1274no—
125C2×C22⋊C4—1274no—
126C2×C22⋊C4—1272no—
127C2×C22⋊C4C2×D12712yes—
128C2×C42—1274no—
129#95—1274no—
130#95—1272no—
131#95C2×D12712yescomplemented
132C24—1274no—
133C24—1272no—
134C24—5082no—
135C24—5082no—
136C24—2542no—
137C24—2542no—
138C24C22×D12711yescharacteristic
139C22×C4—42no—
140C22×C4—1274no—
141C22×C4—1272no—
142C22×C4—1274no—
143C22×C4—5082no—
144C22×C4C4×D12714yes—
145C22×C4C22×D12714yes—
146C22×C4C22×D12711yescharacteristic
147C22×C4C22×D12711yescharacteristic
148C4⋊C4—42no—
149C4⋊C4—5082no—
150C22⋊C4—24no—
151C22⋊C4—1278no—
152C22⋊C4—1278no—
153C22⋊C4—5084no—
154C22⋊C4—5084no—
155C22⋊C4—2544no—
156C42—5084no—
157C23—22no—
158C23—22no—
159C23—1274no—
160C23—5088no—
161C23—5088no—
162C23—5084no—
163C23—5084no—
164C23—2542no—
165C23—2542no—
166C23—2544no—
167C23—2544no—
168C23—2548no—
169C23C2×C4×D12712yes—
170C23#1452324712yes—
171C23#1452324912yes—
172C23C23×D12711yescharacteristic
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—24no—
176C2×C4—24no—
177C2×C4—24no—
178C2×C4—44no—
179C2×C4—1274no—
180C2×C4—5084no—
181C2×C4—2544no—
182C2×C4—2544no—
183C2×C4—2544no—
184C2×C4—2544no—
185C2×C4C2×C4×D12712yes—
186C2×C4C2×C4×D12712yes—
187C22—22no—
188C22—22no—
189C22—24no—
190C22—24no—
191C22—50816no—
192C22—5084no—
193C22—5084no—
194C22—2544no—
195C22—2544no—
196C22—2544no—
197C22#1686116714yes—
198C22C22×C4×D12711yescharacteristic
199C22C2×#1452324711yescharacteristic
200C22C2×#1452324911yescharacteristic
201C22#1686125412yes—
202C22#1686125512yes—
203C4—22no—
204C4—22no—
205C4—44no—
206C4—5084no—
207C4—2544no—
208C2—24no—
209C2—5084no—
210C2C2×#1686116711yescharacteristic
211C2#2255034911yescharacteristic
212C2#2255035011yescharacteristic
213C2#2255046312yes—
214C2#2255046612yes—
subgroupG/H
center, FrattiniC23C23×D127
commutatorC22×C127C23×C4
Fitting#380×C127C2
socle, cosocleC23×C127C24
prime radicalC24×D127C22
hypercenter#380D127
hypocenterC127#2312
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. #22549768C2semidirect
3. #22549766C2semidirect
4. #22549736C2semidirect
6. C22⋊C4×Dic127C2semidirect
7. #380×C127C2semidirect
10. C2×C4⋊Dic127C22semidirect
12. C2×(C2×D127)⋊C4C4semidirect
17. C2×C4×Dic127C22semidirect
18. C127⋊#95C22semidirect
19. C127⋊#95C22semidirect
21. C2×C4⋊C4×C127C22semidirect
23. #95×C127C22semidirect
108. C127#2312semidirect
113. #380D127semidirect
120. C2×C4⋊C4C2×D127semidirect
131. #95C2×D127semidirect

Maximal subgroups

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

subgroupindex
C22×(C2×D127)⋊C42
#225497682
#225497662
#225497362
#225497312
C22⋊C4×Dic1272
#380×C1272
#2312127