Finite Group Databasebeta

← #30639156

Subgroups — #30639156

Order 16256.

432 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C23×C127⋊C8C211yesmaximal, characteristic, complemented
2C2×#16861183C211yesmaximal, characteristic
3C2×#16861129C212yesmaximal, complemented
4#22549638C212yesmaximal, complemented
5#380×C127C211yesmaximal, characteristic, complemented
6C22×D508C2211yescharacteristic
7C22×C22⋊D127—21no—
8C22×C127⋊C8—24no—
9C22×C127⋊C8—22no—
10C22×C127⋊C8C2212yes—
11#16861183—22no—
12#16861183C414yescomplemented
13C2×C4⋊Dic127C2211yescharacteristic
14C2×(C2×D127)⋊C4—21no—
15#16861129—24no—
16C23×C4×C127C2211yescharacteristic
17C2×C4⋊C4×C127C2212yes—
18C2×C22⋊C4×C127—21no—
19#95×C127—21no—
20C23×D127—21no—
21C2×D508—22no—
22C2×D508—21no—
23C2×D508—22no—
24C2×D508C2×C412yes—
25C2×C22⋊D127—24no—
26C2×C22⋊D127—42no—
27C2×C22⋊D127—42no—
28C22×Dic127—21no—
29C2×C127⋊C8—28no—
30C2×C127⋊C8—22no—
31C2×C127⋊C8—22no—
32C2×C127⋊C8D414yescomplemented
33C4⋊Dic127—21no—
34C4⋊Dic127C2×C412yes—
35(C2×D127)⋊C4—24no—
36C24×C127D411yescharacteristic
37C22×C4×C127—22no—
38C22×C4×C127—22no—
39C22×C4×C127—22no—
40C22×C4×C127C2×C412yes—
41C22×C4×C127D411yescharacteristic
42C22×C4×C127D412yes—
43C22×C4×C127C2311yescharacteristic
44C4⋊C4×C127—22no—
45C4⋊C4×C127—22no—
46C22⋊C4×C127—42no—
47C22×D127—22no—
48C22×D127—42no—
49C22×D127—41no—
50C22×D127—41no—
51C22×D127—42no—
52D508—24no—
53D508—41no—
54D508—41no—
55C22⋊D127—48no—
56C2×Dic127—22no—
57C2×Dic127—41no—
58C2×Dic127—41no—
59C127⋊C8—24no—
60C23×C127—22no—
61C23×C127—22no—
62C23×C127C22⋊C412yes—
63C23×C127D812yes—
64C23×C127SD1612yes—
65C23×C127C2×D411yescharacteristic
66C2×C4×C127—22no—
67C2×C4×C127—24no—
68C2×C4×C127—22no—
69C2×C4×C127—22no—
70C2×C4×C127—42no—
71C2×C4×C127—42no—
72C2×C4×C127—42no—
73C2×C4×C127C22⋊C412yes—
74C2×C4×C127C22×C411yescharacteristic
75C2×C4×C127C2×D412yes—
76C2×C4×C127C2×D411yescharacteristic
77C2×C4×C127#4112yes—
78C22×C127—22no—
79C22×C127—22no—
80C22×C127—24no—
81C22×C127—24no—
82C22×C127#10214yescomplemented
83C22×C127C2×C22⋊C411yescharacteristic
84C22×C127#12112yes—
85C22×C127C2×D811yescharacteristic
86C22×C127C2×SD1611yescharacteristic
87C22×C127#13512yes—
88C2×D127—42no—
89C2×D127—42no—
90C2×D127—42no—
91C2×D127—82no—
92C2×D127—82no—
93C4×C127—22no—
94C4×C127—82no—
95C4×C127C4×D412yes—
96C4×C127#12011yescharacteristic
97C4×C127#12311yescharacteristic
98Dic127—42no—
99C2×C127—24no—
100C2×C127#38611yescharacteristic
101C2×C127C2×#10211yescharacteristic
102C2×C127#41611yescharacteristic
103C2×C127#46512yes—
104C2×C127#46612yes—
105D127—82no—
106#1893—1271nomaximal, Sylow
107C127#189311yescharacteristic, complemented, Sylow
108C23×C8—1271no—
109C2×#121—1271no—
110C2×#102—1272no—
111#380D12711yescharacteristic, complemented
112#340—1272no—
113C22×D4—1271no—
114C22×D4—2541no—
115C23×C4C2×D12711yescharacteristic
116C22×C8—1272no—
117C22×C8—2544no—
118C22×C8—2542no—
119#121—1274no—
120#121—2542no—
121C2×C4⋊C4—1271no—
122C2×C4⋊C4C2×D12712yes—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—2541no—
125#102—2544no—
126#95—21no—
127C24—2541no—
128C24C22⋊D12711yescharacteristic
129C2×D4—1272no—
130C2×D4—5082no—
131C2×D4—5082no—
132C2×D4—2544no—
133C2×D4—2542no—
134C2×D4—2541no—
135C2×D4—2542no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—2541no—
140C22×C4C4×D12712yes—
141C22×C4C22⋊D12711yescharacteristic
142C22×C4D50812yes—
143C22×C4C22×D12711yescharacteristic
144C2×C8—1274no—
145C2×C8—2548no—
146C2×C8—2542no—
147C2×C8—2542no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—1272no—
151C4⋊C4—2541no—
152C22⋊C4—42no—
153C22⋊C4—2544no—
154C23—22no—
155C23—22no—
156C23—5081no—
157C23—5081no—
158C23—5082no—
159C23—5082no—
160C23—2542no—
161C23(C2×D127)⋊C412yes—
162C23#1452323312yes—
163C23#1452323512yes—
164C23C2×C22⋊D12711yescharacteristic
165D4—5088no—
166D4—5081no—
167D4—5081no—
168D4—2544no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—24no—
172C2×C4—22no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—5081no—
177C2×C4—5081no—
178C2×C4—2542no—
179C2×C4(C2×D127)⋊C412yes—
180C2×C4C2×C4×D12711yescharacteristic
181C2×C4D4×D12712yes—
182C2×C4C2×D50811yescharacteristic
183C2×C4#1452324712yes—
184C8—2544no—
185C22—24no—
186C22—24no—
187C22—22no—
188C22—22no—
189C22—10162no—
190C22—10162no—
191C22—5082no—
192C22—5082no—
193C22—5082no—
194C22#1686112914yescomplemented
195C22C2×(C2×D127)⋊C411yescharacteristic
196C22#1686118112yes—
197C22C2×#1452323311yescharacteristic
198C22C2×#1452323511yescharacteristic
199C22#1686122912yes—
200C4—22no—
201C4—82no—
202C4—5082no—
203C4(C22⋊D127)⋊C412yes—
204C4#1686117911yescharacteristic
205C4#1686119311yescharacteristic
206C2—24no—
207C2—10162no—
208C2#2254975311yescharacteristic
209C2C2×#1686112911yescharacteristic
210C2#2254987011yescharacteristic
211C2#2255010712yes—
212C2#2255011012yes—
subgroupG/H
centerC23C2×C22⋊D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#380×C127C2
socleC23×C127C2×D4
prime radicalC2×#16861183C2
cosocleC22×C4×C127C23
hypercenter#380D127
hypocenterC127#1893
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
1. C23×C127⋊C8C2semidirect
3. C2×#16861129C2semidirect
4. #22549638C2semidirect
5. #380×C127C2semidirect
12. #16861183C4semidirect
32. C2×C127⋊C8D4semidirect
82. C22×C127#102semidirect
107. C127#1893semidirect
111. #380D127semidirect
194. C22#16861129semidirect

Maximal subgroups

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

subgroupindex
C23×C127⋊C82
C2×#168611832
C2×#168611292
#225496382
#380×C1272
#1893127