Finite Group Databasebeta

← #30639159

Subgroups — #30639159

Order 16256.

432 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 129 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×#16861130C212yesmaximal, complemented
3#22549736C211yesmaximal, characteristic, complemented
4#22549638C212yesmaximal, complemented
5C2×#121×C127C211yesmaximal, characteristic
6C22×C127⋊C8—24no—
7C22×C127⋊C8—22no—
8C22×C127⋊C8C2212yes—
9C2×C4⋊Dic127C2212yes—
10C2×C22⋊Dic127—21no—
11#16861130—24no—
12C127⋊#95—21no—
13C22×D4×C127—21no—
14C22×D4×C127C2211yescharacteristic
15C23×C4×C127C2211yescharacteristic
16#121×C127—22no—
17#121×C127C414yescomplemented
18C2×C4⋊C4×C127C2211yescharacteristic
19C2×C22⋊C4×C127—21no—
20C22×Dic127—22no—
21C2×C127⋊C8—28no—
22C2×C127⋊C8—22no—
23C2×C127⋊C8—22no—
24C2×C127⋊C8D414yescomplemented
25C4⋊Dic127—22no—
26C4⋊Dic127—22no—
27C22⋊Dic127—42no—
28C24×C127—21no—
29C24×C127D411yescharacteristic
30C2×D4×C127—24no—
31C2×D4×C127—22no—
32C2×D4×C127—21no—
33C2×D4×C127—22no—
34C2×D4×C127—42no—
35C2×D4×C127—42no—
36C2×D4×C127C2×C412yes—
37C22×C4×C127—22no—
38C22×C4×C127—22no—
39C22×C4×C127—21no—
40C22×C4×C127C2×C412yes—
41C22×C4×C127D411yescharacteristic
42C22×C4×C127D412yes—
43C22×C4×C127C2311yescharacteristic
44C4⋊C4×C127—21no—
45C4⋊C4×C127C2×C412yes—
46C22⋊C4×C127—24no—
47C2×Dic127—42no—
48C2×Dic127—42no—
49C2×Dic127—42no—
50C127⋊C8—24no—
51C23×C127—22no—
52C23×C127—22no—
53C23×C127—22no—
54C23×C127—42no—
55C23×C127—41no—
56C23×C127—41no—
57C23×C127—42no—
58C23×C127C22⋊C412yes—
59C23×C127D812yes—
60C23×C127SD1612yes—
61C23×C127C2×D411yescharacteristic
62D4×C127—24no—
63D4×C127—48no—
64D4×C127—41no—
65D4×C127—41no—
66C2×C4×C127—22no—
67C2×C4×C127—24no—
68C2×C4×C127—22no—
69C2×C4×C127—22no—
70C2×C4×C127—22no—
71C2×C4×C127—41no—
72C2×C4×C127—41no—
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—42no—
83C22×C127—42no—
84C22×C127—42no—
85C22×C127—82no—
86C22×C127—82no—
87C22×C127#10214yescomplemented
88C22×C127C2×C22⋊C411yescharacteristic
89C22×C127#12112yes—
90C22×C127C2×D811yescharacteristic
91C22×C127C2×SD1611yescharacteristic
92C22×C127#13512yes—
93C4×C127—22no—
94C4×C127—42no—
95C4×C127C4×D412yes—
96C4×C127#12011yescharacteristic
97C4×C127#12311yescharacteristic
98Dic127—82no—
99C2×C127—24no—
100C2×C127—82no—
101C2×C127#38611yescharacteristic
102C2×C127C2×#10211yescharacteristic
103C2×C127#41611yescharacteristic
104C2×C127#46512yes—
105C2×C127#46612yes—
106#1893—1271nomaximal, Sylow
107C127#189311yescharacteristic, complemented, Sylow
108C23×C8—1271no—
109C2×#121D12711yescharacteristic
110C2×#102—1272no—
111#380—1271no—
112#340—1272no—
113C22×D4—21no—
114C22×D4C2×D12711yescharacteristic
115C23×C4C2×D12711yescharacteristic
116C22×C8—1272no—
117C22×C8—2544no—
118C22×C8—2542no—
119#121—22no—
120#121Dic12714yescomplemented
121C2×C4⋊C4—1272no—
122C2×C4⋊C4C2×D12711yescharacteristic
123C2×C22⋊C4—21no—
124C2×C22⋊C4—2541no—
125#102—2544no—
126#95—2541no—
127C24—21no—
128C24C22⋊D12711yescharacteristic
129C2×D4—24no—
130C2×D4—22no—
131C2×D4—21no—
132C2×D4—22no—
133C2×D4—42no—
134C2×D4—42no—
135C2×D4C2×Dic12712yes—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—2542no—
140C22×C4C2×Dic12712yes—
141C22×C4C22⋊D12711yescharacteristic
142C22×C4C22⋊D12712yes—
143C22×C4C22×D12711yescharacteristic
144C2×C8—1274no—
145C2×C8—2548no—
146C2×C8—2542no—
147C2×C8—2542no—
148C4⋊C4—21no—
149C4⋊C4—2542no—
150C4⋊C4—2542no—
151C4⋊C4C2×Dic12712yes—
152C22⋊C4—24no—
153C22⋊C4—5082no—
154C23—22no—
155C23—22no—
156C23—22no—
157C23—41no—
158C23—41no—
159C23—42no—
160C23—42no—
161C23C22⋊Dic12712yes—
162C23#1452323312yes—
163C23#1452323612yes—
164C23C2×C22⋊D12711yescharacteristic
165D4—24no—
166D4—48no—
167D4—41no—
168D4—41no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—24no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—5082no—
177C2×C4—5082no—
178C2×C4—5082no—
179C2×C4C22⋊Dic12712yes—
180C2×C4C22×Dic12711yescharacteristic
181C2×C4D4×D12712yes—
182C2×C4C2×C22⋊D12711yescharacteristic
183C2×C4#1452324712yes—
184C8—2544no—
185C22—24no—
186C22—24no—
187C22—22no—
188C22—22no—
189C22—42no—
190C22—42no—
191C22—42no—
192C22—82no—
193C22—82no—
194C22#1686113014yescomplemented
195C22C2×C22⋊Dic12711yescharacteristic
196C22#1686118112yes—
197C22C2×#1452323311yescharacteristic
198C22C2×#1452323611yescharacteristic
199C22#1686122912yes—
200C4—22no—
201C4—42no—
202C4—10162no—
203C4D4×Dic12712yes—
204C4#1686117711yescharacteristic
205C4#1686119011yescharacteristic
206C2—24no—
207C2—82no—
208C2#2254975611yescharacteristic
209C2C2×#1686113011yescharacteristic
210C2#2254987211yescharacteristic
211C2#2255010312yes—
212C2#2255011012yes—
subgroupG/H
centerC23C2×C22⋊D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting, prime radicalC2×#121×C127C2
socleC23×C127C2×D4
cosocleC22×C4×C127C23
hypercenterC2×#121D127
hypocenterC127#1893
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C23×C127⋊C8C2semidirect
2. C2×#16861130C2semidirect
3. #22549736C2semidirect
4. #22549638C2semidirect
17. #121×C127C4semidirect
24. C2×C127⋊C8D4semidirect
87. C22×C127#102semidirect
107. C127#1893semidirect
120. #121Dic127semidirect
194. C22#16861130semidirect

Maximal subgroups

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

subgroupindex
C23×C127⋊C82
C2×#168611302
#225497362
#225496382
C2×#121×C1272
#1893127