Finite Group Databasebeta

← #44534807

Subgroups — #44534807

Order 31347.

716 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 274 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
1C3×3−1+2×C43⋊C3C311yesmaximal, characteristic
2#23782845C312yesmaximal, complemented
3#23782845C316yesmaximal, complemented
4#23782844C312yesmaximal, complemented
5#23782843C312yesmaximal, complemented
6#23782842C316yesmaximal, complemented
7#23782841C316yesmaximal, complemented
8#23782840C311yesmaximal, characteristic, complemented
9#23782839C311yesmaximal, characteristic, complemented
10#23782821C316yesmaximal, complemented
11#23782795C313yesmaximal
12#23782781C312yesmaximal, complemented
13#23782780C311yesmaximal, characteristic, complemented
14C43×#6900C311yesmaximal, characteristic, complemented
15C33×C43⋊C3C3211yescharacteristic
16C3×#11923958C3212yes—
17C3×#11923958C3216yes—
18C3×#11923957C3211yescharacteristic
19C3×#11923957C3212yes—
20C3×#11923956C3211yescharacteristic
21C3×#11923956C3212yes—
223−1+2×C43⋊C3—39no—
23C3×C9×C43⋊C3C3212yes—
24C3×C9×C43⋊C3C3211yescharacteristic
25#16812500C3212yescomplemented
26#16812500C3216yescomplemented
27#16812499C3212yescomplemented
28#16812499C3216yescomplemented
29(C43⋊C9)⋊C9C3216yescomplemented
30(C43⋊C9)⋊C9C3216yescomplemented
31(C43⋊C9)⋊C9C3216yescomplemented
32(C43⋊C9)⋊C9C3216yescomplemented
33(C43⋊C9)⋊C9C32112yescomplemented
34(C43⋊C9)⋊C9C3216yescomplemented
35(C43⋊C9)⋊C9C3216yescomplemented
36(C3×C43⋊C3)⋊C9C3213yes—
37(C3×C43⋊C3)⋊C9C3216yes—
38(C3×C43⋊C3)⋊C9C3213yes—
39(C3×C43⋊C3)⋊C9C3213yes—
40(C3×C43⋊C3)⋊C9C3216yes—
41(C3×C43⋊C3)⋊C9C3216yes—
42C9×C43⋊C9C3212yescomplemented
43C9×C43⋊C9C3216yescomplemented
44C3×3−1+2×C43C3211yescharacteristic
45C3×3−1+2×C43C3213yes—
46C43×C9⋊C9C3212yescomplemented
47C43×C9⋊C9C3216yescomplemented
48C92×C43C3211yescharacteristic, complemented
49C32×C43⋊C3—33no—
50C32×C43⋊C3—93no—
51C32×C43⋊C3C3311yescharacteristic
52C32×C43⋊C3C3311yescharacteristic
53C32×C43⋊C3C3311yescharacteristic
54#11923958—36no—
55#11923958—96no—
56#11923957—33no—
57#11923957—36no—
58#11923956—33no—
59#11923956—36no—
60C3×C43⋊C9C3316yes—
61C3×C43⋊C9C3316yes—
62C3×C43⋊C9C3316yes—
63C3×C43⋊C9C3312yes—
64C3×C43⋊C9C3312yes—
65C3×C43⋊C9C3312yes—
66C9×C43⋊C3—36no—
67C9×C43⋊C33−1+219yescomplemented
68C33×C43C3311yescharacteristic
693−1+2×C43—33no—
703−1+2×C43—93no—
71C3×C9×C43C3313yes—
72C3×C9×C43C3316yes—
73C3×C9×C43C3312yes—
74C3×C9×C43C3311yescharacteristic
75#10624884—431nomaximal, Sylow
76C32×C43—31no—
77C32×C43—33no—
78C32×C43C3411yescharacteristic
79C3×C43⋊C3—33no—
80C3×C43⋊C3—31no—
81C3×C43⋊C3—31no—
82C3×C43⋊C3—33no—
83C3×C43⋊C3—33no—
84C3×C43⋊C3—99no—
85C3×C43⋊C3C3×3−1+213yes—
86C9×C43—32no—
87C9×C43—33no—
88C9×C43—36no—
89C9×C43C3×3−1+213yes—
90C43⋊C9—32no—
91C43⋊C9—32no—
92C43⋊C9—36no—
93C43⋊C9—36no—
94C43⋊C9—36no—
95C43⋊C9C3×3−1+216yes—
96C32×3−1+2—431no—
97#6900—436no—
98#6900—432no—
99#6900C43⋊C311yescharacteristic, complemented
100#6899—436no—
101#6899—436no—
102#6899—432no—
103#6899—432no—
104#6899—431no—
105#6899—431no—
106#6895—436no—
107#6891—433no—
108#6887—432no—
109#6887—431no—
110C3×C43—91no—
111C3×C43C32×3−1+211yescharacteristic
112C3×C433−1+413yes—
113C43⋊C3—33no—
114C43⋊C3—91no—
115C43⋊C3—91no—
116C34—431no—
117C3×3−1+2—432no—
118C3×3−1+2—436no—
119C3×3−1+2—432no—
120C3×3−1+2—431no—
121C3×3−1+2—432no—
122C3×3−1+2—431no—
123C3×3−1+2—1299no—
124C3×3−1+2C3×C43⋊C311yescharacteristic
125C3×3−1+2C3×C43⋊C313yes—
126C32×C9—431no—
127C32×C9—432no—
128C9⋊C9—436no—
129C9⋊C9—436no—
130C9⋊C9—4312no—
131C9⋊C9—436no—
132C9⋊C9—436no—
133C9⋊C9—436no—
134C9⋊C9—436no—
135C9⋊C9—432no—
136C9⋊C9—436no—
137C9⋊C9—432no—
138C9⋊C9—436no—
139C9⋊C9C3×C43⋊C312yescomplemented
140C9⋊C9C3×C43⋊C316yescomplemented
141C32⋊C9—436no—
142C32⋊C9—433no—
143C32⋊C9—436no—
144C32⋊C9—436no—
145C32⋊C9—433no—
146C32⋊C9—433no—
147C92—436no—
148C92—432no—
149C92C3×C43⋊C311yescharacteristic, complemented
150C43#1062488411yescharacteristic, complemented, Sylow
151C33—431no—
152C33—431no—
153C33—431no—
154C33—1293no—
155C33—3873no—
156C33C32×C43⋊C311yescharacteristic
1573−1+2—33no—
1583−1+2—93no—
1593−1+2—1296no—
1603−1+2—1296no—
1613−1+2—1296no—
1623−1+2—1293no—
1633−1+2—1293no—
1643−1+2—3876no—
165C3×C9—432no—
166C3×C9—439no—
167C3×C9—436no—
168C3×C9—436no—
169C3×C9—436no—
170C3×C9—432no—
171C3×C9—432no—
172C3×C9—1296no—
173C3×C9C32×C43⋊C316yes—
174C3×C9C32×C43⋊C313yes—
175C3×C9C32×C43⋊C312yes—
176C3×C9C32×C43⋊C311yescharacteristic
177C32—31no—
178C32—33no—
179C32—433no—
180C32—1293no—
181C32—1291no—
182C32—1291no—
183C32—1293no—
184C32—1293no—
185C32—3879no—
186C32C33×C43⋊C311yescharacteristic
187C9—32no—
188C9—36no—
189C9—33no—
190C9—436no—
191C9—1292no—
192C9—1292no—
193C9—1296no—
194C9—1296no—
195C9—1296no—
196C93−1+2×C43⋊C313yes—
197C3—91no—
198C3—1293no—
199C3—3871no—
200C3—3871no—
201C3C3×3−1+2×C43⋊C311yescharacteristic
202C3#2378292313yes—
subgroupG/H
center, FrattiniC32C33×C43⋊C3
commutator, socle, cosocleC32×C43C34
FittingC43×#6900C3
prime radicalC33×C43⋊C3C32
hypercenter#6900C43⋊C3
hypocenterC43#10624884
LCSG ▷ C32×C43 ▷ C43
UCSC1 ◁ C32 ◁ #6900

Complements

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

NH ≅ G/Ntype
2. #23782845C3semidirect
3. #23782845C3semidirect
4. #23782844C3semidirect
5. #23782843C3semidirect
6. #23782842C3semidirect
7. #23782841C3semidirect
8. #23782840C3semidirect
9. #23782839C3semidirect
10. #23782821C3semidirect
12. #23782781C3semidirect
13. #23782780C3semidirect
14. C43×#6900C3semidirect
25. #16812500C32semidirect
26. #16812500C32semidirect
27. #16812499C32semidirect
28. #16812499C32semidirect
29. (C43⋊C9)⋊C9C32semidirect
30. (C43⋊C9)⋊C9C32semidirect
31. (C43⋊C9)⋊C9C32semidirect
32. (C43⋊C9)⋊C9C32semidirect
33. (C43⋊C9)⋊C9C32semidirect
34. (C43⋊C9)⋊C9C32semidirect
35. (C43⋊C9)⋊C9C32semidirect
42. C9×C43⋊C9C32semidirect
43. C9×C43⋊C9C32semidirect
46. C43×C9⋊C9C32semidirect
47. C43×C9⋊C9C32semidirect
48. C92×C43C32semidirect
67. C9×C43⋊C33−1+2semidirect
99. #6900C43⋊C3semidirect
139. C9⋊C9C3×C43⋊C3semidirect
140. C9⋊C9C3×C43⋊C3semidirect
149. C92C3×C43⋊C3semidirect
150. C43#10624884semidirect

Maximal subgroups

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

subgroupindex
C3×3−1+2×C43⋊C33
#237828453
#237828443
#237828433
#237828423
#237828413
#237828403
#237828393
#237828213
#237827953
#237827813
#237827803
C43×#69003
#1062488443