Finite Group Databasebeta

← #30639960

Subgroups — #30639960

Order 16256.

290 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 67 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×#16861135C212yesmaximal, complemented
3#22549878C212yesmaximal, complemented
4#22549741C211yesmaximal, characteristic, complemented
5#343×C127C211yesmaximal, characteristic, complemented
6C2×#14523248C2212yes—
7#16861189—22no—
8#16861184—22no—
9#16861183—22no—
10C2×C4⋊Dic127C2211yescharacteristic
11C2×C4×Dic127C2211yescharacteristic
12#16861135—24no—
13#16861134—22no—
14#16861131—22no—
15C127⋊#95—22no—
16C2×M16×C127C2212yes—
17C2×C4⋊C4×C127C2211yescharacteristic
18#14523248—22no—
19#14523248—22no—
20#14523248—22no—
21#14523248—22no—
22C2×Dic254D412yes—
23C2×D508—22no—
24C2×D508D412yes—
25C2×C22⋊D127—22no—
26C22×Dic127—21no—
27C22×Dic127—21no—
28C2×C4×D127—22no—
29C4⋊Dic127—22no—
30C4⋊Dic127—22no—
31(C2×D127)⋊C4—24no—
32C4×Dic127—21no—
33C4×Dic127D412yescomplemented
34C22×C4×C127D412yes—
35C22×C4×C127C2311yescharacteristic
36M16×C127—22no—
37M16×C127—22no—
38C2×C8×C127—22no—
39C4⋊C4×C127—22no—
40C22×D127—22no—
41Dic254—22no—
42Dic254—22no—
43D508—22no—
44D508—22no—
45D508—82no—
46C22⋊D127—42no—
47C22⋊D127—42no—
48C2×Dic127—22no—
49C2×Dic127—22no—
50C2×Dic127—41no—
51C2×Dic127—41no—
52C2×Dic127—42no—
53C2×Dic127—41no—
54C4×D127—42no—
55C4×D127—42no—
56C23×C127C2×D411yescharacteristic
57C2×C4×C127—22no—
58C2×C4×C127—42no—
59C2×C4×C127C2×D411yescharacteristic
60C2×C4×C127C2×D412yes—
61C2×C4×C127#4112yes—
62C2×C4×C127#4111yescharacteristic
63C8×C127—42no—
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#12112yes—
74C4×C127#12312yes—
75Dic127—42no—
76Dic127—42no—
77Dic127—81no—
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#382—1271no—
90#343D12711yescharacteristic, complemented
91C2×#41—1272no—
92C2×M16C2×D12712yes—
93C22⋊Q8—2542no—
94#121—2542no—
95#121—2542no—
96C2×C4⋊C4—1271no—
97C2×C4⋊C4C2×D12711yescharacteristic
98C2×C42—1271no—
99#104—2544no—
100#103—2542no—
101#102—2542no—
102#95—2542no—
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—2541no—
112C22×C4—2541no—
113C22×C4—2542no—
114C22×C4D50812yes—
115C22×C4C22×D12711yescharacteristic
116M16—22no—
117M16—22no—
118C2×C8—22no—
119C4⋊C4—22no—
120C4⋊C4—2542no—
121C4⋊C4—2542no—
122C22⋊C4—2544no—
123C42—1272no—
124C42—2541no—
125C23—2542no—
126C23C2×D50811yescharacteristic
127Q8—2542no—
128Q8—2542no—
129D4—10162no—
130D4—5082no—
131D4—5082no—
132D4—2542no—
133D4—2542no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—5082no—
137C2×C4—5081no—
138C2×C4—5081no—
139C2×C4—5081no—
140C2×C4—5082no—
141C2×C4—5082no—
142C2×C4—2542no—
143C2×C4—2542no—
144C2×C4D4×D12712yes—
145C2×C4D4×D12711yescharacteristic
146C2×C4#1452324812yes—
147C2×C4#1452324911yescharacteristic
148C8—42no—
149C22—21no—
150C22—21no—
151C22—5082no—
152C22—5082no—
153C22—5082no—
154C22#1686117911yescharacteristic
155C22#1686118411yescharacteristic
156C22#1686119711yescharacteristic
157C4—42no—
158C4—10161no—
159C4—5082no—
160C4—5082no—
161C4#1686118212yes—
162C4#1686119412yes—
163C2—21no—
164C2—21no—
165C2—10162no—
166C2#2254979411yescharacteristic
167C2#2255005111yescharacteristic
168C2#2255005611yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#343×C127C2
socleC22×C127#120
prime radical#22550457C2
hypercenter#343D127
hypocenterC127#2040
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
2. C2×#16861135C2semidirect
3. #22549878C2semidirect
4. #22549741C2semidirect
5. #343×C127C2semidirect
33. C4×Dic127D4semidirect
85. C127#2040semidirect
90. #343D127semidirect

Maximal subgroups

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

subgroupindex
#225504572
C2×#168611352
#225498782
#225497412
#343×C1272
#2040127