Finite Group Databasebeta

← #30639797

Subgroups — #30639797

Order 16256.

394 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 69 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#22550387C212yesmaximal, complemented
3#22549892C212yesmaximal, complemented
4#22549633C211yesmaximal, characteristic, complemented
5#407×C127C211yesmaximal, characteristic, complemented
6C2×#14523248C2212yes—
7#16861242—22no—
8#16861237—22no—
9#16861231—22no—
10#16861192—22no—
11#16861188—22no—
12#16861183—22no—
13#16861178—22no—
14C4×C22⋊D127—24no—
15#16861168C2212yes—
16#16861135—24no—
17C23×C4×C127C2211yescharacteristic
18C2×M16×C127C2211yescharacteristic
19C2×M16×C127C2211yescharacteristic
20C22⋊C8×C127—22no—
21#14523248—22no—
22#14523248—22no—
23#14523248—22no—
24#14523248—22no—
25C2×Dic254D412yescomplemented
26C2×D508D412yescomplemented
27C2×C22⋊D127—24no—
28C2×C22⋊D127—22no—
29C2×C4×D127—22no—
30Dic508—22no—
31#14523237—24no—
32D1016—22no—
33C4⋊Dic127—22no—
34Dic127⋊C4—24no—
35C22⋊Dic127—24no—
36C22⋊Dic127—22no—
37(C2×D127)⋊C4—24no—
38C4×Dic127D414yescomplemented
39C24×C127D411yescharacteristic
40C22×C4×C127—22no—
41C22×C4×C127—21no—
42C22×C4×C127—22no—
43C22×C4×C127—21no—
44C22×C4×C127D411yescharacteristic
45C22×C4×C127C2311yescharacteristic
46M16×C127—22no—
47M16×C127—22no—
48C2×C8×C127—21no—
49C2×C8×C127D412yescomplemented
50C22×D127—22no—
51Dic254—22no—
52Dic254—22no—
53D508—22no—
54D508—22no—
55C22⋊D127—48no—
56C22⋊D127—42no—
57C22⋊D127—42no—
58C2×Dic127—24no—
59C2×Dic127—22no—
60C4×D127—42no—
61C4×D127—42no—
62C23×C127—22no—
63C23×C127—21no—
64C23×C127—42no—
65C23×C127C2×D411yescharacteristic
66C2×C4×C127—22no—
67C2×C4×C127—21no—
68C2×C4×C127—41no—
69C2×C4×C127—42no—
70C2×C4×C127—41no—
71C2×C4×C127C2×D412yes—
72C2×C4×C127C2×D412yes—
73C2×C4×C127C2×D411yescharacteristic
74C2×C4×C127#4111yescharacteristic
75C8×C127—22no—
76C8×C127—41no—
77C22×C127—22no—
78C22×C127—24no—
79C22×C127—41no—
80C22×C127—42no—
81C22×C127—41no—
82C22×C127—41no—
83C22×C127#12012yes—
84C22×C127#12111yescharacteristic
85C2×D127—44no—
86C2×D127—42no—
87C4×C127—41no—
88C4×C127#12011yescharacteristic
89C4×C127#12112yes—
90C4×C127#12711yescharacteristic
91Dic127—44no—
92Dic127—42no—
93C2×C127—22no—
94C2×C127—21no—
95C2×C127—42no—
96C2×C127#39211yescharacteristic
97D127—82no—
98#2010—1271nomaximal, Sylow
99C127#201011yescharacteristic, complemented, Sylow
100#575—1271no—
101#525—1272no—
102#421—1272no—
103#407D12711yescharacteristic, complemented
104H(3,ℤ/4)—1271no—
105C2×#41—1272no—
106C23×C4C2×D12711yescharacteristic
107#137—2542no—
108AGL(1,ℤ/8)—2542no—
109#135—2542no—
110C2×M16C2×D12711yescharacteristic
111C2×M16C2×D12711yescharacteristic
112#123—2542no—
113C22⋊Q8—2542no—
114#121—2542no—
115#120—2542no—
116C4×D4—2544no—
117#117—1272no—
118#104—2544no—
119C22⋊C8—22no—
120C24D50811yescharacteristic
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—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—2542no—
134C22×C4D50811yescharacteristic
135C22×C4C22×D12711yescharacteristic
136Q16—2542no—
137SD16—2544no—
138D8—2542no—
139M16—22no—
140M16—22no—
141C2×C8—21no—
142C2×C8C22⋊D12712yescomplemented
143C4⋊C4—2544no—
144C4⋊C4—2542no—
145C22⋊C4—2544no—
146C22⋊C4—2544no—
147C22⋊C4—2542no—
148C42—1274no—
149C23—22no—
150C23—21no—
151C23—42no—
152C23—2542no—
153C23C2×D50811yescharacteristic
154Q8—2542no—
155Q8—2542no—
156D4—5088no—
157D4—5082no—
158D4—5082no—
159D4—2542no—
160D4—2542no—
161C2×C4—22no—
162C2×C4—21no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4—5082no—
167C2×C4—5082no—
168C2×C4—2544no—
169C2×C4—2542no—
170C2×C4D4×D12712yes—
171C2×C4D4×D12712yes—
172C2×C4C2×C22⋊D12711yescharacteristic
173C2×C4#1452324811yescharacteristic
174C8—22no—
175C8—41no—
176C22—24no—
177C22—22no—
178C22—41no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—5084no—
183C22—5082no—
184C22#1686117912yes—
185C22#1686118311yescharacteristic
186C4—41no—
187C4—5084no—
188C4—5082no—
189C4#1686117711yescharacteristic
190C4#1686118212yes—
191C4#1686120511yescharacteristic
192C2—22no—
193C2—21no—
194C2—42no—
195C2—10162no—
196C2#2254978511yescharacteristic
subgroupG/H
centerC4#16861205
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#407×C127C2
socleC2×C127#392
hypercenter#407D127
hypocenterC127#2010
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C22×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. #22550676C2semidirect
2. #22550387C2semidirect
3. #22549892C2semidirect
4. #22549633C2semidirect
5. #407×C127C2semidirect
25. C2×Dic254D4semidirect
26. C2×D508D4semidirect
38. C4×Dic127D4semidirect
49. C2×C8×C127D4semidirect
99. C127#2010semidirect
103. #407D127semidirect
142. C2×C8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225506762
#225503872
#225498922
#225496332
#407×C1272
#2010127