Finite Group Databasebeta

← D31×#585 #16860517

Subgroups — D31×#585 #16860517

Order 3968.

5956 conjugacy classes of subgroups; only the 3203 normal subgroups are shown.

normal subgroups only

This group has 5956 conjugacy classes of subgroups — too many to draw. Only its 3203 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#14521886C216yesmaximal, complemented
2#14521885C2115yesmaximal, complemented
3#14521884C2110yesmaximal, complemented
42−1+4×D31C216yesmaximal, complemented
5H(5,𝔽2)×D31C2110yesmaximal, complemented
6C2×#41×D31C2115yesmaximal, complemented
7C31×#585C211yesmaximal, characteristic, complemented
8C31×2−1+4C2216yescomplemented
9C31×H(5,𝔽2)C22110yescomplemented
10C2×#41×C31C22115yescomplemented
11#11758567C22160yescomplemented
12#11758566C22160yescomplemented
13#41×D31C22120yescomplemented
14#41×D31C22160yescomplemented
15#41×D31C22160yescomplemented
16#41×D31C22160yescomplemented
17#11758564C22130yescomplemented
18C2×#92499C22115yescomplemented
19C2×Q8×D31C22115yescomplemented
20#11758561C22190yescomplemented
21C2×#92497C22145yescomplemented
22C2×D4×D31C22145yescomplemented
23C2×#92495C22145yescomplemented
24C22×C4×D31C22115yescomplemented
25C23×D31C23115yes—
26#41×C31C23120yescomplemented
27#41×C31C23160yescomplemented
28C2×Q8×C31C23115yescomplemented
29C2×D4×C31C23145yescomplemented
30C22×C4×C31C23115yescomplemented
31C2×C22⋊D31C23190yescomplemented
32C22×Dic31C23115yescomplemented
33#92499C23120yescomplemented
34#92499C23160yescomplemented
35Q8×D31C23120yes—
36Q8×D31C23160yescomplemented
37#92497C23160yescomplemented
38#92497C231180yescomplemented
39D4×D31C23160yes—
40D4×D31C231180yescomplemented
41#92495C23160yescomplemented
42#92495C231180yescomplemented
43C2×D124C23145yescomplemented
44C2×C4×D31C23115yes—
45C2×C4×D31C23145yes—
46C2×C4×D31C23145yescomplemented
47C2×C4×D31C23145yescomplemented
48C2×Dic62C23145yescomplemented
49C23×C31C24115yes—
50C22×D31—430no—
51C22×D31C24115yes—
52C22×D31C24145yes—
53Q8×C31C24120yes—
54D4×C31C24160yes—
55C2×C4×C31C24145yes—
56C2×C4×C31C24115yes—
57C22⋊D31C241120yes—
58C2×Dic31C24145yes—
59C2×Dic31C24115yes—
60D124C24160yes—
61C4×D31C2411yescharacteristic
62C4×D31C24115yes—
63C4×D31C24115yes—
64C4×D31C24115yes—
65C4×D31C24190yes—
66Dic62C24160yes—
67C2×#585—311nomaximal, Sylow
68C22×C31—415no—
69C22×C31C25115yes—
70C2×D31—230no—
71C2×D31—445no—
72C2×D31C25115yes—
73C2×D31C2511yescharacteristic
74C4×C31C25115yes—
75C4×C31C2511yescharacteristic
76Dic31C25115yes—
77Dic31C2511yescharacteristic
78#585—316no—
79#585—3110no—
80#585—3115no—
81#585D3111yescharacteristic, direct product factor
82C2×2−1+4—316no—
83C2×H(5,𝔽2)—3110no—
84C22×#41—3115no—
85C2×C31—215no—
86C2×C31C2611yescharacteristic
87D31—215no—
88D31#58512yesdirect product factor
892−1+4—3160no—
902−1+4—3130no—
912−1+4C2×D3116yescomplemented
92H(5,𝔽2)—3160no—
93H(5,𝔽2)—3190no—
94H(5,𝔽2)C2×D31110yescomplemented
95C2×#41—3120no—
96C2×#41—3160no—
97C2×#41—3160no—
98C2×#41—3160no—
99C2×#41—3115no—
100C2×#41—3145no—
101C2×#41—3145no—
102C2×#41C2×D31115yescomplemented
103C22×Q8—3115no—
104C22×D4—3145no—
105C23×C4—3115no—
106C31C2×#58511yescharacteristic, complemented, Sylow
107C24—3115no—
108#41—3120no—
109#41—3160no—
110#41—3160no—
111#41—31180no—
112#41—3160no—
113#41—31180no—
114#41C22×D31120yescomplemented
115#41C22×D31160yescomplemented
116C2×Q8—3120no—
117C2×Q8—3160no—
118C2×Q8—3145no—
119C2×Q8C22×D31115yescomplemented
120C2×D4—3190no—
121C2×D4—3160no—
122C2×D4—31180no—
123C2×D4—3145no—
124C2×D4C22×D31145yescomplemented
125C22×C4—3115no—
126C22×C4—3115no—
127C22×C4—3145no—
128C22×C4—3145no—
129C22×C4—3145no—
130C22×C4C22×D31115yescomplemented
131C23—3115no—
132C23—3145no—
133C23—12430no—
134C23C23×D31115yes—
135Q8—3160no—
136Q8C23×D31120yes—
137D4—31120no—
138D4—3160no—
139D4C23×D31160yes—
140C2×C4—3115no—
141C2×C4—3145no—
142C2×C4—3115no—
143C2×C4—3115no—
144C2×C4—3190no—
145C2×C4—311no—
146C2×C4—3115no—
147C2×C4C23×D31145yes—
148C2×C4C23×D31115yes—
149C22—415no—
150C22—311no—
151C22—3115no—
152C22—6230no—
153C22—12445no—
154C22C24×D31115yes—
155C4—311no—
156C4—3115no—
157C4C24×D31115yes—
158C4C24×D3111yescharacteristic
159C2—215no—
160C2—312no—
161C2—6215no—
162C2C25×D3111yescharacteristic
subgroupG/H
centerC4C24×D31
commutator, socle, cosocleC2×C31C26
FrattiniC2C25×D31
FittingC31×#585C2
hypercenter#585D31
hypocenterC31C2×#585
LCSG ▷ C2×C31 ▷ C31
UCSC1 ◁ C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #14521886C2semidirect
2. #14521885C2semidirect
3. #14521884C2semidirect
4. 2−1+4×D31C2semidirect
5. H(5,𝔽2)×D31C2semidirect
6. C2×#41×D31C2semidirect
7. C31×#585C2semidirect
8. C31×2−1+4C22semidirect
9. C31×H(5,𝔽2)C22semidirect
10. C2×#41×C31C22semidirect
11. #11758567C22semidirect
12. #11758566C22semidirect
13. #41×D31C22semidirect
14. #41×D31C22semidirect
15. #41×D31C22semidirect
16. #41×D31C22semidirect
17. #11758564C22semidirect
18. C2×#92499C22semidirect
19. C2×Q8×D31C22semidirect
20. #11758561C22semidirect
21. C2×#92497C22semidirect
22. C2×D4×D31C22semidirect
23. C2×#92495C22semidirect
24. C22×C4×D31C22semidirect
26. #41×C31C23semidirect
27. #41×C31C23semidirect
28. C2×Q8×C31C23semidirect
29. C2×D4×C31C23semidirect
30. C22×C4×C31C23semidirect
31. C2×C22⋊D31C23semidirect
32. C22×Dic31C23semidirect
33. #92499C23semidirect
34. #92499C23semidirect
36. Q8×D31C23semidirect
37. #92497C23semidirect
38. #92497C23semidirect
40. D4×D31C23semidirect
41. #92495C23semidirect
42. #92495C23semidirect
43. C2×D124C23semidirect
46. C2×C4×D31C23semidirect
47. C2×C4×D31C23semidirect
48. C2×Dic62C23semidirect
81. #585D31direct (N × H)
88. D31#585direct (N × H)
91. 2−1+4C2×D31semidirect
94. H(5,𝔽2)C2×D31semidirect
102. C2×#41C2×D31semidirect
106. C31C2×#585semidirect
114. #41C22×D31semidirect
115. #41C22×D31semidirect
119. C2×Q8C22×D31semidirect
124. C2×D4C22×D31semidirect
130. C22×C4C22×D31semidirect

Maximal subgroups

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

subgroupindex
#145218862
#145218852
#145218842
2−1+4×D312
H(5,𝔽2)×D312
C2×#41×D312
C31×#5852
C2×#58531