Finite Group Databasebeta

← #28348254

Subgroups — #28348254

Order 13696.

460 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 117 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#20293433C214yesmaximal, complemented
2#20292887C214yesmaximal, complemented
3#20292809C212yesmaximal, complemented
4#20292794C212yesmaximal, complemented
5#20292745C212yesmaximal, complemented
6#492×C107C211yesmaximal, characteristic, complemented
7#16812157C2212yes—
8#16812154C2212yescomplemented
9#16812140C2214yescomplemented
10#16812138—24no—
11#16812136—24no—
12#16812135C2214yes—
13#16812131C2212yes—
14#16812130—24no—
15SD16×D107—24no—
16#16812115C2214yescomplemented
17#16812045C2212yescomplemented
18#16812037C2214yescomplemented
19#16812028C2212yescomplemented
20#16812026C2212yescomplemented
21C2×SD16×C107C2214yescomplemented
22C4⋊Q8×C107C2211yescharacteristic, complemented
23#127×C107C2211yescharacteristic
24C4×C8×C107C2211yescharacteristic, complemented
25#13133819—24no—
26#13133818—42no—
27#13133818C2314yes—
28#13133817—24no—
29Q8×D107—24no—
30C2×C22⋊D107—42no—
31D4×D107—24no—
32Dic428D412yescomplemented
33#13133808—24no—
34#13133807D414yescomplemented
35#13133806—24no—
36#13133805—24no—
37D856D412yescomplemented
38#13133803—24no—
39#13133802C2314yes—
40#13133801—24no—
41C8×D107—24no—
42C2×Q8×C107C2312yes—
43C2×D4×C107—41no—
44C2×D4×C107C2312yes—
45SD16×C107—24no—
46SD16×C107—24no—
47C2×C8×C107C2312yes—
48C4⋊C4×C107—41no—
49C42×C107C2311yescharacteristic
50C23×C107—22no—
51C22×D107—42no—
52Q8×C107—24no—
53D4×C107—24no—
54D4×C107—82no—
55C2×C4×C107—21no—
56C2×C4×C107—22no—
57C2×C4×C107C2411yescharacteristic
58C22⋊D107—24no—
59C22⋊D107—44no—
60C2×Dic107—42no—
61D428—42no—
62D428C2×D414yes—
63C4×D107—24no—
64C4×D107—44no—
65Dic214—42no—
66Dic214C2×D414yes—
67C8×C107C2×D412yes—
68C8×C107C2×D412yes—
69C107⋊C8—24no—
70C22×C107—42no—
71C22×C107—82no—
72C22×C107H(5,𝔽2)11yescharacteristic
73C2×D107—24no—
74C2×D107—162no—
75C4×C107—22no—
76C4×C107—42no—
77C4×C107C22×D412yes—
78Dic107—24no—
79C2×C107—21no—
80C2×C107—82no—
81C2×C107D4211yescharacteristic
82D107—44no—
83#3291—1071nomaximal, Sylow
84C107#329111yescharacteristic, complemented, Sylow
85#577—1074no—
86#492D10711yescharacteristic, complemented
87#471—1074no—
88#456—1072no—
89#453—1072no—
90#443—1072no—
912−1+4—1072no—
92H(5,𝔽2)—1072no—
93#137—1074no—
94#137—2144no—
95AGL(1,ℤ/8)—1074no—
96AGL(1,ℤ/8)—2144no—
97#135—1072no—
98#135—2144no—
99C2×SD16—2144no—
100C2×SD16C2×D10714yescomplemented
101#131—1074no—
102C4⋊Q8C2×D10711yescharacteristic, complemented
103#127C2×D10711yescharacteristic
104#108—1072no—
105#104—1074no—
106#101—1072no—
107#100—1072no—
108C4×C8C2×D10711yescharacteristic, complemented
109#41—1074no—
110#41—4282no—
111#41—2144no—
112#41—2144no—
113C2×Q8—2144no—
114C2×Q8C22×D10712yes—
115C2×D4—41no—
116C2×D4—4282no—
117C2×D4—2144no—
118C2×D4C22×D10712yes—
119Q16—1072no—
120Q16—2144no—
121SD16—24no—
122SD16—24no—
123SD16—1074no—
124SD16—2144no—
125SD16—2144no—
126D8—1072no—
127D8—2144no—
128M16—1074no—
129M16—2144no—
130C2×C8—2144no—
131C2×C8C22×D10712yes—
132C4⋊C4—41no—
133C42C22×D10711yescharacteristic
134C23—22no—
135C23—4282no—
136Q8—24no—
137Q8—1074no—
138Q8—4282no—
139D4—24no—
140D4—82no—
141D4—1074no—
142D4—4282no—
143D4—4284no—
144D4—2144no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—4284no—
148C2×C4—4282no—
149C2×C4—2144no—
150C2×C4C23×D10711yescharacteristic
151C8—2144no—
152C8D4×D10712yes—
153C8D4×D10712yes—
154C22—42no—
155C22—82no—
156C22—17122no—
157C22—2144no—
158C22#1681215411yescharacteristic
159C4—22no—
160C4—42no—
161C4—2144no—
162C4C2×D4×D10712yes—
163C2—21no—
164C2—82no—
165C2—4284no—
166C2#2029326211yescharacteristic
subgroupG/H
centerC2#20293262
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#492×C107C2
socleC2×C107D42
hypercenter#492D107
hypocenterC107#3291
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #20293433C2semidirect
2. #20292887C2semidirect
3. #20292809C2semidirect
4. #20292794C2semidirect
5. #20292745C2semidirect
6. #492×C107C2semidirect
8. #16812154C22semidirect
9. #16812140C22semidirect
16. #16812115C22semidirect
17. #16812045C22semidirect
18. #16812037C22semidirect
19. #16812028C22semidirect
20. #16812026C22semidirect
21. C2×SD16×C107C22semidirect
22. C4⋊Q8×C107C22semidirect
24. C4×C8×C107C22semidirect
32. Dic428D4semidirect
34. #13133807D4semidirect
37. D856D4semidirect
84. C107#3291semidirect
86. #492D107semidirect
100. C2×SD16C2×D107semidirect
102. C4⋊Q8C2×D107semidirect
108. C4×C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934332
#202928872
#202928092
#202927942
#202927452
#492×C1072
#3291107