Finite Group Databasebeta

← #28349599

Subgroups — #28349599

Order 13696.

370 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 113 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#20293429C212yesmaximal, complemented
2#20293064C214yesmaximal, complemented
3#20293062C212yesmaximal, complemented
4#20293060C212yesmaximal, complemented
5C2×D1712C211yesmaximal, characteristic, complemented
6D16×D107C212yesmaximal, complemented
7#20293043C211yesmaximal, characteristic, complemented
8#508×C107C211yesmaximal, characteristic, complemented
9#16812155—22no—
10#16812136—24no—
11#16812134—22no—
12#16812132C2214yescomplemented
13C2×D856C2212yescomplemented
14D8×D107C2214yescomplemented
15#16812115C2211yescharacteristic
16#16812052C2214yescomplemented
17D1712C2212yescomplemented
18D1712C2212yescomplemented
19#16812050C2214yescomplemented
20#16812049C2211yescharacteristic, complemented
21#16812048C2212yescomplemented
22C16×D107C2212yescomplemented
23#135×C107C2212yescomplemented
24Q32×C107C2211yescharacteristic, complemented
25SD32×C107C2212yescomplemented
26D16×C107C2211yescharacteristic, complemented
27C2×C16×C107C2211yescharacteristic, complemented
28#13133819—24no—
29#13133818—42no—
30#13133818D411yescharacteristic
31C2×D428—22no—
32C2×D428—42no—
33D4×D107—24no—
34D4×D107—44no—
35#13133805—24no—
36D856—22no—
37D856C2314yes—
38#13133803—24no—
39#13133802D411yescharacteristic
40#13133801D412yes—
41C8×D107C2312yes—
42C107⋊C16C2312yes—
43#41×C107—22no—
44Q16×C107C2312yes—
45SD16×C107—22no—
46D8×C107C2312yes—
47C2×C8×C107C2311yescharacteristic
48C16×C107C2311yescharacteristic
49C16×C107C2311yescharacteristic
50C22×D107—42no—
51C22×D107—44no—
52Q8×C107—22no—
53D4×C107—22no—
54D4×C107—42no—
55C2×C4×C107—42no—
56C2×C4×C107C2×D411yescharacteristic
57C22⋊D107—44no—
58C22⋊D107D812yes—
59D428—24no—
60D428—44no—
61D428—42no—
62D428D811yescharacteristic
63C4×D107—44no—
64C4×D107C2×D412yes—
65Dic214D811yescharacteristic
66C8×C107C2×D411yescharacteristic
67C8×C107C2411yescharacteristic
68C107⋊C8C2×D412yes—
69C22×C107—42no—
70C22×C107C2×D811yescharacteristic
71C2×D107—44no—
72C2×D107—164no—
73C2×D107—162no—
74C2×D107C2×D812yes—
75C4×C107—42no—
76C4×C107C2×D811yescharacteristic
77C4×C107C22×D411yescharacteristic
78Dic107C2×D812yes—
79C2×C107—21no—
80C2×C107—82no—
81C2×C107C22×D811yescharacteristic
82D107—22no—
83D107—84no—
84#3415—1071nomaximal, Sylow
85C107#341511yescharacteristic, complemented, Sylow
86#576—1072no—
87#509—1072no—
88#509—1074no—
89#508—1072no—
90#508D10711yescharacteristic, complemented
91C2×D16—1072no—
92C2×D16—1071no—
93#504—1071no—
94H(5,𝔽2)—2142no—
95AGL(1,ℤ/8)—2144no—
96AGL(1,ℤ/8)—2142no—
97#135—1074no—
98#135C2×D10712yescomplemented
99C2×D8—1074no—
100C2×D8—1072no—
101#131—1071no—
102Q32C2×D10711yescharacteristic, complemented
103SD32—1074no—
104SD32C2×D10712yescomplemented
105D16—1074no—
106D16—1072no—
107D16—1072no—
108D16C2×D10711yescharacteristic, complemented
109M32—1071no—
110M32—1072no—
111C2×C16—1072no—
112C2×C16C2×D10711yescharacteristic, complemented
113#41—22no—
114#41—1071no—
115#41—4282no—
116#41—2144no—
117C2×D4—4282no—
118C2×D4—4284no—
119C2×D4—2144no—
120C2×D4—2142no—
121Q16C22×D10712yes—
122SD16—22no—
123SD16—2144no—
124D8—1074no—
125D8—2144no—
126D8—2142no—
127D8C22×D10712yes—
128M16—1071no—
129M16—1072no—
130C2×C8—1072no—
131C2×C8C22×D10711yescharacteristic
132C16—1072no—
133C16C22×D10711yescharacteristic
134C16C22×D10711yescharacteristic
135C23—4284no—
136C23—4282no—
137Q8—22no—
138Q8—1071no—
139D4—22no—
140D4—42no—
141D4—1072no—
142D4—1071no—
143D4—4284no—
144D4—4284no—
145D4—4282no—
146D4—2144no—
147C2×C4—42no—
148C2×C4—1072no—
149C2×C4—4284no—
150C2×C4D4×D10711yescharacteristic
151C8—1072no—
152C8D4×D10711yescharacteristic
153C8C23×D10711yescharacteristic
154C22—42no—
155C22—1072no—
156C22—4284no—
157C22—17122no—
158C22—17124no—
159C22D8×D10711yescharacteristic
160C4—42no—
161C4—1072no—
162C4D8×D10711yescharacteristic
163C4C2×D4×D10711yescharacteristic
164C2—21no—
165C2—82no—
166C2—8564no—
167C2—2142no—
168C2C2×D8×D10711yescharacteristic
subgroupG/H
centerC2C2×D8×D107
commutator, cosocleC8×C107C24
FrattiniC8C23×D107
Fitting#508×C107C2
socleC2×C107C22×D8
hypercenter#508D107
hypocenterC107#3415
LCSG ▷ C8×C107 ▷ C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #20293429C2semidirect
2. #20293064C2semidirect
3. #20293062C2semidirect
4. #20293060C2semidirect
5. C2×D1712C2semidirect
6. D16×D107C2semidirect
7. #20293043C2semidirect
8. #508×C107C2semidirect
12. #16812132C22semidirect
13. C2×D856C22semidirect
14. D8×D107C22semidirect
16. #16812052C22semidirect
17. D1712C22semidirect
18. D1712C22semidirect
19. #16812050C22semidirect
20. #16812049C22semidirect
21. #16812048C22semidirect
22. C16×D107C22semidirect
23. #135×C107C22semidirect
24. Q32×C107C22semidirect
25. SD32×C107C22semidirect
26. D16×C107C22semidirect
27. C2×C16×C107C22semidirect
85. C107#3415semidirect
90. #508D107semidirect
98. #135C2×D107semidirect
102. Q32C2×D107semidirect
104. SD32C2×D107semidirect
108. D16C2×D107semidirect
112. C2×C16C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202934292
#202930642
#202930622
#202930602
C2×D17122
D16×D1072
#202930432
#508×C1072
#3415107