Finite Group Databasebeta

← #28345372

Subgroups — #28345372

Order 13696.

458 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 185 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#20293427C211yesmaximal, characteristic
2#20293393C212yesmaximal, complemented
3#20292750C214yesmaximal, complemented
4#20292745C214yesmaximal, complemented
5#20292676C211yesmaximal, characteristic, complemented
6C2×#16812037C212yesmaximal, complemented
7C4×M16×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2212yes—
9#16812142C414yescomplemented
10#16812137C414yescomplemented
11#16812131C2214yes—
12#16812117—24no—
13#16812115C2218yescomplemented
14C2×#13133802C2212yes—
15M16×D107—24no—
16#16812045C2214yescomplemented
17#16812037C2214yescomplemented
18#16812037C2214yescomplemented
19C2×M16×C107C2211yescharacteristic
20C2×M16×C107C2211yescharacteristic
21C2×C42×C107C2211yescharacteristic
22#97×C107C2212yescomplemented
23C4×C8×C107C2212yescomplemented
24#13133818—22no—
25#13133818—22no—
26#13133818C2×C414yes—
27#13133818C2314yes—
28C2×Dic214C2×C412yes—
29C2×D428C2×C412yes—
30C2×C22⋊D107—22no—
31C2×C4×D107—22no—
32Dic428C2×C414yescomplemented
33#13133807C2×C418yescomplemented
34D856C2×C414yescomplemented
35#13133802—22no—
36#13133802C2314yes—
37#13133801—28no—
38C2×C107⋊C8—22no—
39C8×D107—28no—
40C22×C4×C107—21no—
41C22×C4×C107C2311yescharacteristic
42M16×C107C2×C414yes—
43M16×C107D414yes—
44C2×C8×C107C2312yes—
45C2×C8×C107C2312yes—
46C42×C107—21no—
47C42×C107C2311yescharacteristic
48C42×C107C2311yescharacteristic
49C23×C107#4111yescharacteristic
50C22×D107—22no—
51C2×C4×C107—21no—
52C2×C4×C107—21no—
53C2×C4×C107—42no—
54C2×C4×C107C22×C412yes—
55C2×C4×C107C2×D412yes—
56C2×C4×C107#4111yescharacteristic
57C2×C4×C107C2411yescharacteristic
58C22⋊D107—24no—
59C22⋊D107—24no—
60C2×Dic107—22no—
61D428—22no—
62D428C22×C414yes—
63C4×D107—24no—
64C4×D107—24no—
65Dic214—22no—
66Dic214C22×C414yes—
67C8×C107C22×C414yes—
68C8×C107C2×D414yes—
69C107⋊C8—24no—
70C22×C107—41no—
71C22×C107C4×D412yes—
72C22×C107C2×#4111yescharacteristic
73C2×D107—24no—
74C2×D107—82no—
75C4×C107—22no—
76C4×C107—41no—
77C4×C107C4×D412yes—
78C4×C107C23×C411yescharacteristic
79C4×C107C22×D411yescharacteristic
80Dic107—24no—
81C2×C107—21no—
82C2×C107—22no—
83C2×C107C2×C4×D411yescharacteristic
84D107—44no—
85#2956—1071nomaximal, Sylow
86C107#295611yescharacteristic, complemented, Sylow
87#575—1071no—
88#568—1072no—
89#444—1074no—
90#443—1074no—
91#430—1071no—
92C2×#104—1072no—
93C4×M16D10711yescharacteristic, complemented
94C2×#41—1072no—
95#137—1074no—
96AGL(1,ℤ/8)—1074no—
97#135—1074no—
98#131—1078no—
99#131—2144no—
100C2×M16—1072no—
101C2×M16—2144no—
102C2×M16C2×D10711yescharacteristic
103C2×M16C2×D10711yescharacteristic
104C2×C42C2×D10711yescharacteristic
105#108—1074no—
106#104—1074no—
107#104—1074no—
108#97C2×D10712yescomplemented
109C4×C8C2×D10712yescomplemented
110#41—1074no—
111#41—1074no—
112#41—2142no—
113#41—2142no—
114C2×Q8—1072no—
115C2×D4—1072no—
116C2×D4—2142no—
117C22×C4—21no—
118C22×C4—2142no—
119C22×C4C22×D10711yescharacteristic
120Q16—1074no—
121SD16—1078no—
122D8—1074no—
123M16—1074no—
124M16—2142no—
125M16—2148no—
126M16C4×D10714yes—
127M16D42814yes—
128C2×C8—2142no—
129C2×C8—2148no—
130C2×C8C22×D10712yes—
131C2×C8C22×D10712yes—
132C42—21no—
133C42C22×D10711yescharacteristic
134C42C22×D10711yescharacteristic
135C23—2142no—
136C23#1313381811yescharacteristic
137Q8—1074no—
138Q8—2142no—
139D4—1074no—
140D4—2144no—
141D4—2144no—
142D4—2142no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—42no—
146C2×C4—2142no—
147C2×C4—2144no—
148C2×C4—2144no—
149C2×C4C2×C4×D10712yes—
150C2×C4C2×D42812yes—
151C2×C4#1313381811yescharacteristic
152C2×C4C23×D10711yescharacteristic
153C8—2144no—
154C8C2×C4×D10714yes—
155C8C2×D42814yes—
156C22—41no—
157C22—8562no—
158C22—2144no—
159C22C4×D42812yes—
160C22C2×#1313381811yescharacteristic
161C4—22no—
162C4—41no—
163C4—2144no—
164C4C4×D42812yes—
165C4C22×C4×D10711yescharacteristic
166C4C22×D42811yescharacteristic
167C2—21no—
168C2—22no—
169C2—4284no—
170C2C2×C4×D42811yescharacteristic
subgroupG/H
centerC4C22×D428
commutatorC4×C107C23×C4
FrattiniC2×C4C23×D107
FittingC4×M16×C107C2
socleC2×C107C2×C4×D4
prime radical#20293427C2
cosocleC2×C4×C107C24
hypercenterC4×M16D107
hypocenterC107#2956
LCSG ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C4 ◁ C2×M16 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
2. #20293393C2semidirect
3. #20292750C2semidirect
4. #20292745C2semidirect
5. #20292676C2semidirect
6. C2×#16812037C2semidirect
7. C4×M16×C107C2semidirect
9. #16812142C4semidirect
10. #16812137C4semidirect
13. #16812115C22semidirect
16. #16812045C22semidirect
17. #16812037C22semidirect
18. #16812037C22semidirect
22. #97×C107C22semidirect
23. C4×C8×C107C22semidirect
32. Dic428C2×C4semidirect
33. #13133807C2×C4semidirect
34. D856C2×C4semidirect
86. C107#2956semidirect
93. C4×M16D107semidirect
108. #97C2×D107semidirect
109. C4×C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934272
#202933932
#202927502
#202927452
#202926762
C2×#168120372
C4×M16×C1072
#2956107