Finite Group Databasebeta

← #30639955

Subgroups — #30639955

Order 16256.

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

normal subgroups only

This group has 306 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#22549745C211yesmaximal, characteristic, complemented
5#343×C127C211yesmaximal, characteristic, complemented
6C2×#14523248C2212yes—
7#16861189—22no—
8#16861184—22no—
9#16861183—22no—
10C2×C4⋊Dic127C2211yescharacteristic
11C2×Dic127⋊C4—22no—
12C2×C4×Dic127C2211yescharacteristic
13#16861135—24no—
14#16861134—22no—
15#16861131—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—
31Dic127⋊C4—44no—
32(C2×D127)⋊C4—24no—
33C4×Dic127—21no—
34C4×Dic127D412yescomplemented
35C22×C4×C127D412yes—
36C22×C4×C127C2311yescharacteristic
37M16×C127—22no—
38M16×C127—22no—
39C2×C8×C127—22no—
40C4⋊C4×C127—22no—
41C22×D127—22no—
42Dic254—22no—
43Dic254—22no—
44D508—22no—
45D508—22no—
46D508—82no—
47C22⋊D127—42no—
48C22⋊D127—42no—
49C2×Dic127—22no—
50C2×Dic127—22no—
51C2×Dic127—22no—
52C2×Dic127—22no—
53C2×Dic127—42no—
54C2×Dic127—41no—
55C4×D127—42no—
56C4×D127—42no—
57C23×C127C2×D411yescharacteristic
58C2×C4×C127—22no—
59C2×C4×C127—42no—
60C2×C4×C127C2×D411yescharacteristic
61C2×C4×C127C2×D412yes—
62C2×C4×C127#4112yes—
63C2×C4×C127#4111yescharacteristic
64C8×C127—42no—
65C22×C127—21no—
66C22×C127—21no—
67C22×C127#12011yescharacteristic
68C22×C127#12111yescharacteristic
69C22×C127#12411yescharacteristic
70C2×D127—42no—
71C2×D127—42no—
72C2×D127—42no—
73C4×C127—42no—
74C4×C127#12112yes—
75C4×C127#12312yes—
76Dic127—24no—
77Dic127—42no—
78Dic127—81no—
79C2×C127—21no—
80C2×C127—21no—
81C2×C127#39411yescharacteristic
82C2×C127#45311yescharacteristic
83C2×C127#45611yescharacteristic
84D127—82no—
85#2039—1271nomaximal, Sylow
86C127#203911yescharacteristic, complemented, Sylow
87#537—1271no—
88C2×#104—1272no—
89#417—1272no—
90#384—1271no—
91#343D12711yescharacteristic, complemented
92C2×#41—1272no—
93C2×M16C2×D12712yes—
94C22⋊Q8—2542no—
95#121—2542no—
96#121—2542no—
97C2×C4⋊C4—1271no—
98C2×C4⋊C4—2542no—
99C2×C4⋊C4C2×D12711yescharacteristic
100C2×C42—1271no—
101#104—2544no—
102#103—2542no—
103#102—2542no—
104#41—2542no—
105#41—2542no—
106#41—2542no—
107#41—2542no—
108C2×Q8—1272no—
109C2×D4—1272no—
110C2×D4—2542no—
111C2×D4—2542no—
112C22×C4—2541no—
113C22×C4—2542no—
114C22×C4—2541no—
115C22×C4D50812yes—
116C22×C4C22×D12711yescharacteristic
117M16—22no—
118M16—22no—
119C2×C8—22no—
120C4⋊C4—22no—
121C4⋊C4—5084no—
122C4⋊C4—2542no—
123C4⋊C4—2542no—
124C22⋊C4—2544no—
125C42—1272no—
126C42—2541no—
127C23—2542no—
128C23C2×D50811yescharacteristic
129Q8—2542no—
130Q8—2542no—
131D4—10162no—
132D4—5082no—
133D4—5082no—
134D4—2542no—
135D4—2542no—
136C2×C4—22no—
137C2×C4—42no—
138C2×C4—5082no—
139C2×C4—5081no—
140C2×C4—5082no—
141C2×C4—5082no—
142C2×C4—2542no—
143C2×C4—2542no—
144C2×C4—2542no—
145C2×C4—2542no—
146C2×C4D4×D12712yes—
147C2×C4D4×D12711yescharacteristic
148C2×C4#1452324812yes—
149C2×C4#1452324911yescharacteristic
150C8—42no—
151C22—21no—
152C22—21no—
153C22—5082no—
154C22—5082no—
155C22—5082no—
156C22#1686117911yescharacteristic
157C22#1686118411yescharacteristic
158C22#1686119711yescharacteristic
159C4—42no—
160C4—10161no—
161C4—5082no—
162C4—2544no—
163C4#1686118212yes—
164C4#1686119412yes—
165C2—21no—
166C2—21no—
167C2—10162no—
168C2#2254979411yescharacteristic
169C2#2255004611yescharacteristic
170C2#2255006111yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#343×C127C2
socleC22×C127#120
prime radical#22550457C2
hypercenter#343D127
hypocenterC127#2039
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. #22549745C2semidirect
5. #343×C127C2semidirect
34. C4×Dic127D4semidirect
86. C127#2039semidirect
91. #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
#225497452
#343×C1272
#2039127