Finite Group Databasebeta

← #30639115

Subgroups — #30639115

Order 16256.

360 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 83 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×#16861198C211yesmaximal, characteristic
2#22550350C211yesmaximal, characteristic
3#22549885C211yesmaximal, characteristic, complemented
4C2×#16861125C211yesmaximal, characteristic
5#22549637C211yesmaximal, characteristic, complemented
6#22549635C211yesmaximal, characteristic, complemented
7#419×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C2×#14523232C2211yescharacteristic
10#16861198—22no—
11#16861198C414yescomplemented
12C4×D508—22no—
13#16861171—22no—
14#16861168C2211yescharacteristic
15C2×(C2×D127)⋊C4—22no—
16C22⋊C4×D127—22no—
17C2×C4×Dic127C2211yescharacteristic
18#16861133—22no—
19#16861125—22no—
20C22×Q8×C127C2211yescharacteristic
21C2×M16×C127C2211yescharacteristic
22#117×C127C2211yescharacteristic
23#103×C127—22no—
24C23×D127D412yes—
25C2×D508—21no—
26C2×D508—21no—
27C2×D508—21no—
28C2×D508—21no—
29C2×D508C2×C412yes—
30C2×D508D412yes—
31C22×Dic127—21no—
32C2×C4×D127—24no—
33#14523232—21no—
34#14523232—21no—
35C2×C127⋊C8—21no—
36C4⋊Dic127D412yes—
37C22⋊Dic127—21no—
38(C2×D127)⋊C4—24no—
39(C2×D127)⋊C4—28no—
40C4×Dic127—21no—
41C4×Dic127—21no—
42C4×Dic127C2×C412yescomplemented
43C2×Q8×C127—21no—
44C2×Q8×C127—21no—
45C2×Q8×C127—21no—
46C2×Q8×C127—21no—
47C2×Q8×C127—21no—
48C2×Q8×C127C2×C412yescomplemented
49C22×C4×C127—21no—
50C22×C4×C127C2311yescharacteristic
51M16×C127—21no—
52M16×C127—21no—
53C2×C8×C127—21no—
54C4⋊C4×C127D412yes—
55C22⋊C4×C127—21no—
56C42×C127—21no—
57C22×D127—22no—
58C22×D127—22no—
59C22×D127—44no—
60C22×D127C22⋊C414yes—
61D508—22no—
62D508—22no—
63D508—22no—
64D508—22no—
65C2×Dic127—22no—
66C2×Dic127—22no—
67C2×Dic127—42no—
68C4×D127—84no—
69C127⋊C8—41no—
70C23×C127C2×D411yescharacteristic
71Q8×C127—22no—
72Q8×C127—22no—
73Q8×C127—41no—
74Q8×C127—41no—
75C2×C4×C127—22no—
76C2×C4×C127—22no—
77C2×C4×C127—41no—
78C2×C4×C127—41no—
79C2×C4×C127C22×C411yescharacteristic
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127C2×D411yescharacteristic
82C2×C4×C127C2×D411yescharacteristic
83C2×C4×C127#4111yescharacteristic
84C2×C4×C127#4111yescharacteristic
85C8×C127—41no—
86C22×C127—22no—
87C22×C127C2×C22⋊C411yescharacteristic
88C22×C127#12011yescharacteristic
89C22×C127#12311yescharacteristic
90C2×D127—24no—
91C2×D127—44no—
92C2×D127—44no—
93C2×D127—82no—
94C4×C127—42no—
95C4×C127—42no—
96C4×C127C4×D411yescharacteristic
97C4×C127C4×D411yescharacteristic
98C4×C127#12111yescharacteristic
99C4×C127#12111yescharacteristic
100Dic127—42no—
101Dic127—42no—
102C2×C127—21no—
103C2×C127—21no—
104C2×C127#38611yescharacteristic
105C2×C127#45512yes—
106D127—44no—
107#1885—1271nomaximal, Sylow
108C127#188511yescharacteristic, complemented, Sylow
109C2×#124—1271no—
110#518—1271no—
111#419—1271no—
112#419D12711yescharacteristic, complemented
113C2×#100—1271no—
114#339—1271no—
115#339—1271no—
116C22×Q8C2×D12711yescharacteristic
117C22×D4—1271no—
118C2×M16—1271no—
119C2×M16C2×D12711yescharacteristic
120#124—1274no—
121#124—2542no—
122C4×D4—2542no—
123C4×D4—2542no—
124#117—1271no—
125#117C2×D12711yescharacteristic
126C2×C22⋊C4—2542no—
127C2×C22⋊C4—2542no—
128C2×C42—1271no—
129#103—22no—
130#103—2542no—
131#100—2542no—
132C24—1272no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8C4×D12712yescomplemented
139C2×D4—1272no—
140C2×D4—1272no—
141C2×D4—2541no—
142C2×D4—2541no—
143C2×D4—2541no—
144C2×D4—2541no—
145C22×C4—21no—
146C22×C4—2544no—
147C22×C4—2541no—
148C22×C4C22×D12711yescharacteristic
149M16—21no—
150M16—21no—
151M16—2541no—
152M16—2541no—
153C2×C8—21no—
154C2×C8—2541no—
155C4⋊C4—1272no—
156C4⋊C4C22⋊D12712yes—
157C22⋊C4—21no—
158C22⋊C4—2548no—
159C22⋊C4—2544no—
160C22⋊C4—2541no—
161C42—21no—
162C42—1272no—
163C42—2541no—
164C42—2541no—
165C23—1274no—
166C23—5084no—
167C23—2542no—
168C23—2542no—
169C23D4×D12711yescharacteristic
170Q8—22no—
171Q8—22no—
172Q8—41no—
173Q8—41no—
174D4—2542no—
175D4—2542no—
176D4—2542no—
177D4—2542no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—10164no—
183C2×C4—5082no—
184C2×C4—2542no—
185C2×C4—2542no—
186C2×C4C2×C4×D12711yescharacteristic
187C2×C4D4×D12711yescharacteristic
188C2×C4D4×D12711yescharacteristic
189C2×C4C2×C22⋊D12711yescharacteristic
190C2×C4#1452324711yescharacteristic
191C2×C4#1452324811yescharacteristic
192C8—41no—
193C8—5081no—
194C22—22no—
195C22—10162no—
196C22—5084no—
197C22—5084no—
198C22—2544no—
199C22C22⋊C4×D12711yescharacteristic
200C22#1686117711yescharacteristic
201C22#1686119111yescharacteristic
202C4—42no—
203C4—42no—
204C4—5082no—
205C4—5082no—
206C4(C22⋊D127)⋊C411yescharacteristic
207C4C4×C22⋊D12711yescharacteristic
208C4#1686118011yescharacteristic
209C4#1686118211yescharacteristic
210C2—21no—
211C2—21no—
212C2—5084no—
213C2#2254975211yescharacteristic
214C2#2255005212yes—
subgroupG/H
centerC22#16861177
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#419×C127C2
socleC22×C127#120
prime radicalC22×D508C22
cosocleC22×C4×C127C23
hypercenter#419D127
hypocenterC127#1885
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. #22549885C2semidirect
5. #22549637C2semidirect
6. #22549635C2semidirect
7. #419×C127C2semidirect
11. #16861198C4semidirect
42. C4×Dic127C2×C4semidirect
48. C2×Q8×C127C2×C4semidirect
108. C127#1885semidirect
112. #419D127semidirect
138. C2×Q8C4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611982
#225503502
#225498852
C2×#168611252
#225496372
#225496352
#419×C1272
#1885127