Finite Group Databasebeta

← #30649054

Subgroups — #30649054

Order 16256.

400 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#22550431C211yesmaximal, characteristic, complemented
2#22550427C211yesmaximal, characteristic
3#22550232C214yesmaximal, complemented
4#22550226C211yesmaximal, characteristic, complemented
5#22550224C211yesmaximal, characteristic, complemented
6#22550223C211yesmaximal, characteristic, complemented
7#22549871C214yesmaximal, complemented
8C4×C8×D127C211yesmaximal, characteristic, complemented
9#487×C127C211yesmaximal, characteristic, complemented
10C2×#14523249C2212yes—
11C2×C8×D127C2212yes—
12#16861209C2211yescharacteristic
13#16861206C2211yescharacteristic
14#16861200C2212yes—
15#16861198C2212yes—
16#16861189—21no—
17#16861185—21no—
18#16861184—21no—
19#16861184—21no—
20Q8×Dic127—21no—
21#16861174—21no—
22#16861171—21no—
23#16861171—21no—
24#16861167C2212yes—
25C42×D127C2211yescharacteristic
26#16861133C2214yes—
27#16861131C2214yes—
28#16861129C2214yes—
29C4×C127⋊C8C2211yescharacteristic
30C8×Dic127C2212yes—
31C4⋊Q8×C127C2211yescharacteristic
32C4⋊Q8×C127C2211yescharacteristic
33#103×C127C2214yes—
34C4×C8×C127C2211yescharacteristic
35#14523249—28no—
36C2×Dic254—21no—
37C2×D508—22no—
38C2×D508—21no—
39C2×D508—21no—
40C2×D508C2312yes—
41C2×C4×D127—22no—
42C2×C4×D127D411yescharacteristic
43C2×C4×D127D411yescharacteristic
44C2×C4×D127C2311yescharacteristic
45C2×C127⋊C8C2312yes—
46C8×D127—24no—
47C4⋊Dic127—22no—
48C4⋊Dic127—21no—
49C4⋊Dic127C2312yes—
50Dic127⋊C4—22no—
51(C2×D127)⋊C4—22no—
52(C2×D127)⋊C4—22no—
53(C2×D127)⋊C4—22no—
54C4×Dic127—22no—
55C4×Dic127D411yescharacteristic
56C4×Dic127D411yescharacteristic
57C4×Dic127C2311yescharacteristic
58C2×Q8×C127—21no—
59C2×Q8×C127C2312yes—
60C2×C8×C127C2312yes—
61C4⋊C4×C127—21no—
62C4⋊C4×C127—21no—
63C4⋊C4×C127—21no—
64C4⋊C4×C127C2312yes—
65C42×C127C2311yescharacteristic
66C22×D127—22no—
67C22×D127C2×D411yescharacteristic
68Dic254—42no—
69D508—24no—
70D508—44no—
71D508—42no—
72D508—42no—
73C2×Dic127—22no—
74C2×Dic127C2×D411yescharacteristic
75C2×Dic127C2×D411yescharacteristic
76C2×Dic127C2×D411yescharacteristic
77C4×D127—22no—
78C4×D127—22no—
79C4×D127—44no—
80C4×D127#4114yes—
81C127⋊C8—22no—
82Q8×C127—24no—
83Q8×C127—42no—
84C2×C4×C127—22no—
85C2×C4×C127—22no—
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127C2411yescharacteristic
89C8×C127—22no—
90C22×C127C22×D411yescharacteristic
91C2×D127—21no—
92C2×D127—21no—
93C2×D127—42no—
94C2×D127—42no—
95C2×D127—42no—
96C2×D127#12412yes—
97C4×C127—42no—
98C4×C127—42no—
99C4×C127#13512yes—
100C4×C127#13512yes—
101C4×C127C2×#4112yes—
102Dic127—21no—
103Dic127—21no—
104Dic127—42no—
105Dic127#12412yes—
106C2×C127C2×#12411yescharacteristic
107C2×C127C2×#13511yescharacteristic
108C2×C127C2×#13511yescharacteristic
109D127—22no—
110D127—82no—
111#3131—1271nomaximal, Sylow
112C127#313111yescharacteristic, complemented, Sylow
113#533—1271no—
114#532—1271no—
115#488—1274no—
116#487—1271no—
117#487D12711yescharacteristic, complemented
118#486—1271no—
119#486—1271no—
120#416—1274no—
121C2×C4×C8—1271no—
122C2×#41—1272no—
123C22×C8—1272no—
124C4⋊Q8—1271no—
125C4⋊Q8C2×D12711yescharacteristic
126C4⋊Q8C2×D12711yescharacteristic
127#127—1271no—
128#125—1272no—
129#124—1272no—
130C22⋊Q8—2541no—
131C22⋊Q8—2541no—
132#121—2541no—
133#121—2541no—
134C4×Q8—2541no—
135C4×Q8—2541no—
136C4×D4—2541no—
137C4×D4—2541no—
138#117—1272no—
139C2×C42—1271no—
140#103—1274no—
141#103C2×D12714yes—
142#102—1274no—
143#102—1274no—
144C4×C8—1272no—
145C4×C8—1271no—
146C4×C8C2×D12711yescharacteristic
147#41—2548no—
148C2×Q8—21no—
149C2×Q8—2541no—
150C2×Q8C22×D12712yes—
151C2×D4—1272no—
152C2×D4—2542no—
153C2×D4—2541no—
154C2×D4—2541no—
155C22×C4—1271no—
156C22×C4—1271no—
157C22×C4—1271no—
158C22×C4—2542no—
159C2×C8—1272no—
160C2×C8—2544no—
161C2×C8C22×D12712yes—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—1272no—
166C4⋊C4—2542no—
167C4⋊C4—2542no—
168C4⋊C4—2541no—
169C4⋊C4C22×D12712yes—
170C22⋊C4—2542no—
171C22⋊C4—2542no—
172C22⋊C4—2542no—
173C42—1271no—
174C42—1271no—
175C42—1271no—
176C42—2542no—
177C42C22×D12711yescharacteristic
178C23—1271no—
179C23—2542no—
180Q8—24no—
181Q8—42no—
182Q8—5082no—
183D4—5084no—
184D4—5082no—
185D4—5082no—
186D4—2544no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—1271no—
190C2×C4—1274no—
191C2×C4—1271no—
192C2×C4—1271no—
193C2×C4—5084no—
194C2×C4—2542no—
195C2×C4—2542no—
196C2×C4—2542no—
197C2×C4D4×D12711yescharacteristic
198C2×C4D4×D12711yescharacteristic
199C2×C4C23×D12711yescharacteristic
200C8—22no—
201C8—2542no—
202C22—1272no—
203C22—5082no—
204C22—5082no—
205C22—5082no—
206C22—2541no—
207C22—2541no—
208C22C2×D4×D12711yescharacteristic
209C4—42no—
210C4—42no—
211C4—1272no—
212C4—5082no—
213C4—2541no—
214C4—2541no—
215C4#1686123012yes—
216C4#1686123212yes—
217C4#41×D12712yes—
218C2—10162no—
219C2—2542no—
220C2#124×D12711yescharacteristic
221C2C2×#1686123011yescharacteristic
222C2C2×#1686123211yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#487×C127C2
socleC22×C127C22×D4
prime radical#22550427C2
hypercenter#487D127
hypocenterC127#3131
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #22550431C2semidirect
3. #22550232C2semidirect
4. #22550226C2semidirect
5. #22550224C2semidirect
6. #22550223C2semidirect
7. #22549871C2semidirect
8. C4×C8×D127C2semidirect
9. #487×C127C2semidirect
112. C127#3131semidirect
117. #487D127semidirect

Maximal subgroups

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

subgroupindex
#225504312
#225504272
#225502322
#225502262
#225502242
#225502232
#225498712
C4×C8×D1272
#487×C1272
#3131127