Finite Group Databasebeta

← #30639959

Subgroups — #30639959

Order 16256.

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

normal subgroups only

This group has 290 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#22550457C211yesmaximal, characteristic
2C2×#16861137C212yesmaximal, complemented
3#22549875C212yesmaximal, complemented
4#22549645C211yesmaximal, characteristic, complemented
5#382×C127C211yesmaximal, characteristic, complemented
6C2×#14523248C2212yes—
7C2×#14523232C2212yes—
8#16861189—22no—
9#16861184—22no—
10#16861183—22no—
11#16861137—24no—
12#16861132—22no—
13#16861129—22no—
14C2×C4⋊C4×C127C2211yescharacteristic
15C2×C4⋊C4×C127C2211yescharacteristic
16C2×C42×C127C2211yescharacteristic
17#95×C127—22no—
18#14523248—22no—
19#14523248—22no—
20#14523248—22no—
21#14523248—22no—
22C2×Dic254D412yes—
23C2×D508—22no—
24C2×D508D412yes—
25C2×C22⋊D127—22no—
26C2×C4×D127—22no—
27#14523232—22no—
28#14523232—22no—
29C2×C127⋊C8—22no—
30C4⋊Dic127—22no—
31(C2×D127)⋊C4—24no—
32C22×C4×C127—21no—
33C22×C4×C127—21no—
34C22×C4×C127D412yes—
35C22×C4×C127C2311yescharacteristic
36C4⋊C4×C127—22no—
37C4⋊C4×C127—22no—
38C42×C127—21no—
39C42×C127D412yescomplemented
40C22×D127—22no—
41Dic254—22no—
42Dic254—22no—
43D508—22no—
44D508—22no—
45D508—82no—
46C22⋊D127—42no—
47C22⋊D127—42no—
48C2×Dic127—22no—
49C4×D127—42no—
50C4×D127—42no—
51C127⋊C8—42no—
52C23×C127C2×D411yescharacteristic
53C2×C4×C127—22no—
54C2×C4×C127—22no—
55C2×C4×C127—41no—
56C2×C4×C127—41no—
57C2×C4×C127—42no—
58C2×C4×C127—42no—
59C2×C4×C127—41no—
60C2×C4×C127C2×D412yes—
61C2×C4×C127C2×D411yescharacteristic
62C2×C4×C127#4111yescharacteristic
63C2×C4×C127#4112yes—
64C22×C127—21no—
65C22×C127—21no—
66C22×C127#12011yescharacteristic
67C22×C127#12111yescharacteristic
68C22×C127#12411yescharacteristic
69C2×D127—42no—
70C2×D127—42no—
71C2×D127—42no—
72C4×C127—42no—
73C4×C127—42no—
74C4×C127—81no—
75C4×C127#12112yes—
76C4×C127#12312yes—
77Dic127—42no—
78C2×C127—21no—
79C2×C127—21no—
80C2×C127#39411yescharacteristic
81C2×C127#45411yescharacteristic
82C2×C127#45511yescharacteristic
83D127—82no—
84#2040—1271nomaximal, Sylow
85C127#204011yescharacteristic, complemented, Sylow
86#537—1271no—
87C2×#104—1272no—
88#417—1272no—
89#382D12711yescharacteristic, complemented
90#343—1271no—
91C2×#41—1272no—
92C2×M16—1272no—
93C22⋊Q8—2542no—
94#121—2542no—
95#121—2542no—
96C2×C4⋊C4C2×D12711yescharacteristic
97C2×C4⋊C4C2×D12711yescharacteristic
98C2×C42C2×D12711yescharacteristic
99#104—2544no—
100#103—2542no—
101#102—2542no—
102#95—22no—
103#41—2542no—
104#41—2542no—
105#41—2542no—
106#41—2542no—
107C2×Q8—1272no—
108C2×D4—1272no—
109C2×D4—2542no—
110C2×D4—2542no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—2542no—
114C22×C4C22⋊D12712yes—
115C22×C4C22×D12711yescharacteristic
116M16—2542no—
117M16—2542no—
118C2×C8—2542no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—2542no—
122C22⋊C4—2544no—
123C42—21no—
124C42D50812yescomplemented
125C23—2542no—
126C23C2×C22⋊D12711yescharacteristic
127Q8—2542no—
128Q8—2542no—
129D4—10162no—
130D4—5082no—
131D4—5082no—
132D4—2542no—
133D4—2542no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—41no—
141C2×C4—5082no—
142C2×C4—5082no—
143C2×C4—2542no—
144C2×C4D4×D12712yes—
145C2×C4C2×D50811yescharacteristic
146C2×C4#1452324811yescharacteristic
147C2×C4#1452324912yes—
148C8—5082no—
149C22—21no—
150C22—21no—
151C22—5082no—
152C22—5082no—
153C22—5082no—
154C22#1686117711yescharacteristic
155C22#1686118311yescharacteristic
156C22#1686119811yescharacteristic
157C4—42no—
158C4—42no—
159C4—81no—
160C4—5082no—
161C4#1686118412yes—
162C4#1686119412yes—
163C2—21no—
164C2—21no—
165C2—10162no—
166C2#2254979311yescharacteristic
167C2#2255004811yescharacteristic
168C2#2255005311yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#382×C127C2
socleC22×C127#120
prime radical#22550457C2
hypercenter#382D127
hypocenterC127#2040
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #382

Complements

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

NH ≅ G/Ntype
2. C2×#16861137C2semidirect
3. #22549875C2semidirect
4. #22549645C2semidirect
5. #382×C127C2semidirect
39. C42×C127D4semidirect
85. C127#2040semidirect
89. #382D127semidirect
124. C42D508semidirect

Maximal subgroups

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

subgroupindex
#225504572
C2×#168611372
#225498752
#225496452
#382×C1272
#2040127