Finite Group Databasebeta

← #30639798

Subgroups — #30639798

Order 16256.

394 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 71 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#22550676C211yesmaximal, characteristic, complemented
2#22550388C212yesmaximal, complemented
3#22549894C212yesmaximal, complemented
4#22549839C211yesmaximal, characteristic, complemented
5H(3,ℤ/4)×C127C211yesmaximal, characteristic, complemented
6C2×#14523248C2212yes—
7C22×C4×D127C2211yescharacteristic
8#16861242—22no—
9#16861237—22no—
10#16861231—22no—
11C2×#14523232C2211yescharacteristic
12#16861193—22no—
13#16861189—22no—
14#16861184—22no—
15#16861179—22no—
16C4×D508—24no—
17#16861137—24no—
18(C2×D127)⋊C8—22no—
19C2×M16×C127C2211yescharacteristic
20#117×C127C2212yes—
21C23×D127D411yescharacteristic
22#14523248—22no—
23#14523248—22no—
24#14523248—22no—
25#14523248—22no—
26C2×Dic254D412yescomplemented
27C2×D508—24no—
28C2×D508D412yescomplemented
29C2×C22⋊D127—22no—
30C22×Dic127D411yescharacteristic
31C2×C4×D127—22no—
32C2×C4×D127—21no—
33C2×C4×D127—22no—
34C2×C4×D127—22no—
35C2×C4×D127—21no—
36Dic508—22no—
37#14523237—24no—
38D1016—22no—
39#14523232—22no—
40C2×C127⋊C8—21no—
41C4⋊Dic127—24no—
42(C2×D127)⋊C4—24no—
43(C2×D127)⋊C4—24no—
44C22×C4×C127C2311yescharacteristic
45M16×C127—22no—
46C2×C8×C127D412yescomplemented
47C4⋊C4×C127—22no—
48C22⋊C4×C127—22no—
49C42×C127D414yescomplemented
50C22×D127—22no—
51C22×D127—21no—
52C22×D127—22no—
53C22×D127—42no—
54Dic254—22no—
55Dic254—22no—
56D508—22no—
57D508—22no—
58D508—48no—
59C22⋊D127—42no—
60C22⋊D127—42no—
61C2×Dic127—22no—
62C2×Dic127—22no—
63C2×Dic127—21no—
64C4×D127—41no—
65C4×D127—42no—
66C4×D127—42no—
67C4×D127—41no—
68C4×D127—42no—
69C127⋊C8—41no—
70C23×C127C2×D411yescharacteristic
71C2×C4×C127—24no—
72C2×C4×C127C2×D412yes—
73C2×C4×C127C2×D411yescharacteristic
74C2×C4×C127C2×D412yes—
75C2×C4×C127#4111yescharacteristic
76C8×C127—22no—
77C22×C127—41no—
78C22×C127#12012yes—
79C22×C127#12111yescharacteristic
80C2×D127—24no—
81C2×D127—22no—
82C2×D127—42no—
83C2×D127—41no—
84C2×D127—41no—
85C2×D127—44no—
86C2×D127—42no—
87C4×C127—44no—
88C4×C127#12011yescharacteristic
89C4×C127#12112yes—
90C4×C127#12711yescharacteristic
91Dic127—41no—
92Dic127—42no—
93C2×C127—22no—
94C2×C127—21no—
95C2×C127#39211yescharacteristic
96D127—42no—
97D127—82no—
98#2010—1271nomaximal, Sylow
99C127#201011yescharacteristic, complemented, Sylow
100#575—1271no—
101#525—1272no—
102#421—1272no—
103#407—1271no—
104H(3,ℤ/4)D12711yescharacteristic, complemented
105C2×#41—1272no—
106C23×C4—1271no—
107#137—2542no—
108AGL(1,ℤ/8)—2542no—
109#135—2542no—
110C2×M16—1271no—
111C2×M16C2×D12711yescharacteristic
112#123—2542no—
113C22⋊Q8—2542no—
114#121—2542no—
115#120—2542no—
116C4×D4—2544no—
117#117C2×D12712yes—
118#104—2544no—
119C22⋊C8—2542no—
120C24—1271no—
121#41—2542no—
122#41—2542no—
123#41—2542no—
124#41—2542no—
125C2×Q8—1272no—
126C2×D4—1272no—
127C2×D4—2544no—
128C2×D4—2542no—
129C22×C4—1271no—
130C22×C4—2542no—
131C22×C4—2541no—
132C22×C4—2542no—
133C22×C4—2542no—
134C22×C4—2541no—
135C22×C4C22×D12711yescharacteristic
136Q16—2542no—
137SD16—2544no—
138D8—2542no—
139M16—22no—
140M16—2542no—
141C2×C8—2541no—
142C2×C8D50812yescomplemented
143C4⋊C4—22no—
144C4⋊C4—2544no—
145C22⋊C4—22no—
146C22⋊C4—2544no—
147C22⋊C4—2544no—
148C42D50814yescomplemented
149C23—5082no—
150C23—2542no—
151C23—2541no—
152C23—2542no—
153C23D4×D12711yescharacteristic
154Q8—2542no—
155Q8—2542no—
156D4—5088no—
157D4—5082no—
158D4—5082no—
159D4—2542no—
160D4—2542no—
161C2×C4—24no—
162C2×C4—5082no—
163C2×C4—5082no—
164C2×C4—5081no—
165C2×C4—5082no—
166C2×C4—5081no—
167C2×C4—2541no—
168C2×C4—2542no—
169C2×C4—2542no—
170C2×C4D4×D12712yes—
171C2×C4C2×D50812yes—
172C2×C4C2×D50811yescharacteristic
173C2×C4#1452324911yescharacteristic
174C8—22no—
175C8—5081no—
176C22—41no—
177C22—5084no—
178C22—5082no—
179C22—5081no—
180C22—5082no—
181C22—5081no—
182C22—2544no—
183C22—2542no—
184C22#1686117912yes—
185C22#1686118411yescharacteristic
186C4—44no—
187C4—5081no—
188C4—5082no—
189C4#1686117911yescharacteristic
190C4#1686118412yes—
191C4#1686120611yescharacteristic
192C2—22no—
193C2—21no—
194C2—10162no—
195C2—5082no—
196C2#2254978611yescharacteristic
subgroupG/H
centerC4#16861206
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingH(3,ℤ/4)×C127C2
socleC2×C127#392
hypercenterH(3,ℤ/4)D127
hypocenterC127#2010
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ H(3,ℤ/4)

Complements

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

NH ≅ G/Ntype
1. #22550676C2semidirect
2. #22550388C2semidirect
3. #22549894C2semidirect
4. #22549839C2semidirect
5. H(3,ℤ/4)×C127C2semidirect
26. C2×Dic254D4semidirect
28. C2×D508D4semidirect
46. C2×C8×C127D4semidirect
49. C42×C127D4semidirect
99. C127#2010semidirect
104. H(3,ℤ/4)D127semidirect
142. C2×C8D508semidirect
148. C42D508semidirect

Maximal subgroups

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

subgroupindex
#225506762
#225503882
#225498942
#225498392
H(3,ℤ/4)×C1272
#2010127