Finite Group Databasebeta

← #30638683

Subgroups — #30638683

Order 16256.

620 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 101 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×#16861254C211yesmaximal, characteristic
2C2×#16861137C212yesmaximal, complemented
3#22549875C212yesmaximal, complemented
4C2×#16861126C211yesmaximal, characteristic, complemented
5#390×C127C211yesmaximal, characteristic, complemented
6#16861254—24no—
7#16861254—22no—
8#16861254C414yescomplemented
9C2×#14523248C2212yes—
10C2×#14523247—22no—
11C22×C22⋊D127—22no—
12C2×D4×D127—22no—
13C2×#14523232C2212yescomplemented
14#16861137—24no—
15#16861132—22no—
16#16861129—22no—
17#16861126—22no—
18C22×D4×C127C2211yescharacteristic
19C2×C4⋊C4×C127C2211yescharacteristic, complemented
20C2×C22⋊C4×C127—22no—
21C2×C42×C127C2211yescharacteristic, complemented
22C23×D127—22no—
23#14523248—22no—
24#14523248—22no—
25#14523248C2×C414yes—
26#14523248D414yes—
27#14523247—28no—
28#14523247—28no—
29C2×Dic254D412yes—
30C2×D508D412yes—
31C2×C22⋊D127—24no—
32C2×C22⋊D127—28no—
33C2×C22⋊D127—28no—
34C2×C22⋊D127—22no—
35D4×D127—28no—
36D4×D127—28no—
37C22×Dic127—22no—
38C2×C4×D127—22no—
39#14523232—22no—
40#14523232—22no—
41C2×C127⋊C8—22no—
42C24×C127D412yes—
43C2×D4×C127—22no—
44C2×D4×C127—22no—
45C2×D4×C127C2×C412yes—
46C2×D4×C127D412yes—
47C22×C4×C127—21no—
48C22×C4×C127—21no—
49C22×C4×C127C2311yescharacteristic
50C4⋊C4×C127—22no—
51C22⋊C4×C127—44no—
52C42×C127—22no—
53C22×D127—22no—
54C22×D127—28no—
55C22×D127—82no—
56C22×D127—82no—
57Dic254—22no—
58Dic254C22⋊C414yescomplemented
59D508—22no—
60D508C22⋊C414yescomplemented
61C22⋊D127—216no—
62C22⋊D127—24no—
63C22⋊D127—24no—
64C2×Dic127—28no—
65C2×Dic127—22no—
66C4×D127—24no—
67C4×D127—24no—
68C127⋊C8—42no—
69C23×C127—22no—
70C23×C127—22no—
71C23×C127—42no—
72C23×C127—42no—
73C23×C127C22⋊C414yes—
74C23×C127C2×D411yescharacteristic
75D4×C127—24no—
76D4×C127—24no—
77C2×C4×C127—22no—
78C2×C4×C127—41no—
79C2×C4×C127—41no—
80C2×C4×C127—42no—
81C2×C4×C127—41no—
82C2×C4×C127C22×C411yescharacteristic
83C2×C4×C127C2×D412yes—
84C2×C4×C127C2×D412yes—
85C2×C4×C127C2×D411yescharacteristic
86C22×C127—21no—
87C22×C127—21no—
88C22×C127—24no—
89C22×C127—44no—
90C22×C127—42no—
91C22×C127—42no—
92C22×C127—82no—
93C22×C127C2×C22⋊C411yescharacteristic
94C22×C127#12011yescharacteristic
95C22×C127#12011yescharacteristic
96C2×D127—24no—
97C2×D127—42no—
98C2×D127—42no—
99C2×D127—88no—
100C4×C127—42no—
101C4×C127—81no—
102C4×C127C2×C22⋊C412yes—
103C4×C127#12012yes—
104Dic127—24no—
105C2×C127—21no—
106C2×C127—21no—
107C2×C127—44no—
108C2×C127#37911yescharacteristic
109C2×C127#45311yescharacteristic
110C2×C127#45511yescharacteristic
111D127—44no—
112#1792—1271nomaximal, Sylow
113C127#179211yescharacteristic, complemented, Sylow
114C2×H(5,𝔽2)—1271no—
115C2×#104—1272no—
116#417—1272no—
117C2×#100—1271no—
118#390D12711yescharacteristic, complemented
119H(5,𝔽2)—1274no—
120H(5,𝔽2)—2544no—
121H(5,𝔽2)—2542no—
122C2×#41—1272no—
123C2×#41—2542no—
124C22×D4—2542no—
125C22×D4—2542no—
126C22×D4C2×D12711yescharacteristic
127C2×M16—1272no—
128C2×C4⋊C4C2×D12711yescharacteristic, complemented
129C2×C22⋊C4—22no—
130C2×C42C2×D12711yescharacteristic, complemented
131#104—2544no—
132#103—2542no—
133#102—2542no—
134#100—2542no—
135C24—2542no—
136C24C22⋊D12712yes—
137#41—1274no—
138#41—1274no—
139#41—2548no—
140#41—2548no—
141#41—2542no—
142#41—2542no—
143C2×Q8—1272no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—1272no—
147C2×D4—2548no—
148C2×D4—2548no—
149C2×D4—2544no—
150C2×D4—2548no—
151C2×D4—2548no—
152C2×D4—2542no—
153C2×D4C4×D12712yes—
154C2×D4D50812yes—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—2542no—
158C22×C4—2542no—
159C22×C4C22×D12711yescharacteristic
160M16—2542no—
161M16—2542no—
162C2×C8—2542no—
163C4⋊C4—22no—
164C22⋊C4—44no—
165C42—22no—
166C23—22no—
167C23—22no—
168C23—42no—
169C23—42no—
170C23—10162no—
171C23—10162no—
172C23—2548no—
173C23—2542no—
174C23(C2×D127)⋊C414yes—
175C23C2×C22⋊D12711yescharacteristic
176Q8—1274no—
177Q8—2542no—
178D4—24no—
179D4—24no—
180D4—1274no—
181D4—25416no—
182D4—2544no—
183D4—2544no—
184D4—2542no—
185C2×C4—22no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—2548no—
191C2×C4—2542no—
192C2×C4—2544no—
193C2×C4—2544no—
194C2×C4C2×C4×D12711yescharacteristic
195C2×C4D4×D12712yes—
196C2×C4D4×D12712yes—
197C2×C4C2×D50811yescharacteristic
198C8—5082no—
199C22—21no—
200C22—21no—
201C22—24no—
202C22—44no—
203C22—42no—
204C22—42no—
205C22—82no—
206C22—10168no—
207C22—5082no—
208C22—5082no—
209C22—2544no—
210C22C2×(C2×D127)⋊C411yescharacteristic
211C22#1686117711yescharacteristic
212C22#1686117711yescharacteristic
213C4—42no—
214C4—81no—
215C4—2544no—
216C4C22⋊C4×D12712yes—
217C4#1686117912yes—
218C2—21no—
219C2—21no—
220C2—44no—
221C2—5084no—
222C2#2254973111yescharacteristic
223C2#2255004311yescharacteristic
224C2#2255005311yescharacteristic
subgroupG/H
centerC22#16861177
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#390×C127C2
socleC22×C127#120
prime radicalC2×#16861254C2
cosocleC22×C4×C127C23
hypercenter#390D127
hypocenterC127#1792
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#16861137C2semidirect
3. #22549875C2semidirect
4. C2×#16861126C2semidirect
5. #390×C127C2semidirect
8. #16861254C4semidirect
13. C2×#14523232C22semidirect
19. C2×C4⋊C4×C127C22semidirect
21. C2×C42×C127C22semidirect
58. Dic254C22⋊C4semidirect
60. D508C22⋊C4semidirect
113. C127#1792semidirect
118. #390D127semidirect
128. C2×C4⋊C4C2×D127semidirect
130. C2×C42C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612542
C2×#168611372
#225498752
C2×#168611262
#390×C1272
#1792127