Finite Group Databasebeta

← C31⋊C3×#6899 #36382348

Subgroups — C31⋊C3×#6899 #36382348

Order 22599.

788 conjugacy classes of subgroups; only the 292 normal subgroups are shown.

normal subgroups only

This group has 788 conjugacy classes of subgroups — too many to draw. Only its 292 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×C31⋊C3C311yesmaximal, characteristic
2#21419358C312yesmaximal, complemented
3#21419357C312yesmaximal, complemented
4#21419356C316yesmaximal, complemented
5#21419355C316yesmaximal, complemented
6#21419354C311yesmaximal, characteristic, complemented
7#21419353C311yesmaximal, characteristic, complemented
8#21419352C312yesmaximal, complemented
9#21419351C316yesmaximal, complemented
10C9⋊C9×C31⋊C3C316yesmaximal, complemented
11C9⋊C9×C31⋊C3C312yesmaximal, complemented
12C32⋊C9×C31⋊C3C313yesmaximal
13C92×C31⋊C3C311yesmaximal, characteristic, complemented
14C31×#6899C311yesmaximal, characteristic, complemented
15C33×C31⋊C3C3211yescharacteristic
16C3×#11719689C3212yes—
17C3×#11719686C3211yescharacteristic
18C3×#11719686C3212yes—
19C3×#11719687C3211yescharacteristic
20C3×#11719687C3212yes—
213−1+2×C31⋊C3—39no—
22C3×C9×C31⋊C3C3213yes—
23C3×C9×C31⋊C3C3216yes—
24C3×C9×C31⋊C3C3212yes—
25C3×C9×C31⋊C3C3211yescharacteristic
26#14578887C3216yescomplemented
27#14578887C3212yescomplemented
28#14578886C3216yescomplemented
29#14578886C3212yescomplemented
30(C31⋊C9)⋊C9C3216yescomplemented
31(C31⋊C9)⋊C9C3216yescomplemented
32(C31⋊C9)⋊C9C32112yescomplemented
33(C31⋊C9)⋊C9C3216yescomplemented
34(C31⋊C9)⋊C9C3216yescomplemented
35(C31⋊C9)⋊C9C32112yescomplemented
36C32⋊(C31⋊C9)C3216yes—
37(C3×C31⋊C3)⋊C9C3213yes—
38(C3×C31⋊C3)⋊C9C3213yes—
39(C3×C31⋊C3)⋊C9C3216yes—
40(C3×C31⋊C3)⋊C9C3216yes—
41C9×C31⋊C9C3216yescomplemented
42C9×C31⋊C9C3212yescomplemented
43C3×3−1+2×C31C3211yescharacteristic
44C31×C9⋊C9C3216yescomplemented
45C31×C9⋊C9C3212yescomplemented
46C31×C32⋊C9C3213yes—
47C92×C31C3211yescharacteristic, complemented
48C33×C31C3311yescharacteristic
49C32×C31⋊C3—33no—
50C32×C31⋊C3—39no—
51C32×C31⋊C3C3311yescharacteristic
52C32×C31⋊C3C3311yescharacteristic
53C32×C31⋊C3C3311yescharacteristic
543−1+2×C31—33no—
55C3×C9×C31C3313yes—
56C3×C9×C31C3316yes—
57C3×C9×C31C3312yes—
58C3×C9×C31C3311yescharacteristic
59#11719689—36no—
60C3×C31⋊C9C3316yes—
61C3×C31⋊C9C3316yes—
62C3×C31⋊C9C3316yes—
63C3×C31⋊C9C3312yes—
64C3×C31⋊C9C3312yes—
65C3×C31⋊C9C3312yes—
66#11719687—33no—
67#11719687—36no—
68#11719686—33no—
69#11719686—36no—
70C9×C31⋊C3—36no—
71C9×C31⋊C3—39no—
72C9×C31⋊C3—318no—
73C9×C31⋊C33−1+219yes—
74C3×#6899—311nomaximal, Sylow
75C32×C31—31no—
76C32×C31—33no—
77C32×C31C3411yescharacteristic
78C3×C31⋊C3—31no—
79C3×C31⋊C3—31no—
80C3×C31⋊C3—33no—
81C3×C31⋊C3—33no—
82C3×C31⋊C3—99no—
83C3×C31⋊C3C3×3−1+213yes—
84C3×C31⋊C3#73719yes—
85C9×C31—32no—
86C9×C31—33no—
87C9×C31—36no—
88C9×C31C3×3−1+213yes—
89C31⋊C9—32no—
90C31⋊C9—32no—
91C31⋊C9—36no—
92C31⋊C9—36no—
93C31⋊C9—36no—
94C31⋊C9C3×3−1+216yes—
95C32×3−1+2—311no—
96#6899—312no—
97#6899—316no—
98#6899—316no—
99#6899—312no—
100#6899—312no—
101#6899—316no—
102#6899—311no—
103#6899—311no—
104#6899C31⋊C311yescharacteristic, direct product factor
105C3×C9⋊C9—312no—
106C3×C9⋊C9—316no—
107C3×C32⋊C9—313no—
108C3×C92—311no—
109C3×C31—91no—
110C3×C31C32×3−1+211yescharacteristic
111C3×C31C3×#73713yes—
112C31⋊C3—91no—
113C31⋊C3—91no—
114C31⋊C3#689919yesdirect product factor
115C34—311no—
116C3×3−1+2—312no—
117C3×3−1+2—312no—
118C3×3−1+2—311no—
119C3×3−1+2—312no—
120C3×3−1+2—311no—
121C3×3−1+2—939no—
122C3×3−1+2C3×C31⋊C311yescharacteristic
123C32×C9—311no—
124C32×C9—312no—
125C32×C9—313no—
126C32×C9—316no—
127C9⋊C9—3112no—
128C9⋊C9—3112no—
129C9⋊C9—316no—
130C9⋊C9—316no—
131C9⋊C9—316no—
132C9⋊C9—316no—
133C9⋊C9—316no—
134C9⋊C9—312no—
135C9⋊C9—316no—
136C9⋊C9—312no—
137C9⋊C9C3×C31⋊C316yescomplemented
138C9⋊C9C3×C31⋊C312yescomplemented
139C32⋊C9—316no—
140C32⋊C9—316no—
141C32⋊C9—316no—
142C32⋊C9—313no—
143C32⋊C9—313no—
144C32⋊C9C3×C31⋊C313yes—
145C92—316no—
146C92—312no—
147C92C3×C31⋊C311yescharacteristic, complemented
148C31C3×#689911yescharacteristic, complemented, Sylow
149C33—311no—
150C33—311no—
151C33—311no—
152C33—933no—
153C33—939no—
154C33C32×C31⋊C311yescharacteristic
1553−1+2—33no—
1563−1+2—936no—
1573−1+2—936no—
1583−1+2—936no—
1593−1+2—933no—
1603−1+2—933no—
161C3×C9—312no—
162C3×C9—319no—
163C3×C9—316no—
164C3×C9—316no—
165C3×C9—316no—
166C3×C9—312no—
167C3×C9—312no—
168C3×C9—939no—
169C3×C9—9318no—
170C3×C9—936no—
171C3×C9C32×C31⋊C313yes—
172C3×C9C32×C31⋊C316yes—
173C3×C9C32×C31⋊C312yes—
174C3×C9C32×C31⋊C311yescharacteristic
175C32—31no—
176C32—33no—
177C32—2799no—
178C32—313no—
179C32—319no—
180C32—931no—
181C32—931no—
182C32—933no—
183C32—933no—
184C32C33×C31⋊C311yescharacteristic
185C9—32no—
186C9—33no—
187C9—36no—
188C9—316no—
189C9—932no—
190C9—932no—
191C9—936no—
192C9—936no—
193C9—936no—
194C93−1+2×C31⋊C313yes—
195C3—91no—
196C3—2791no—
197C3—2791no—
198C3—319no—
199C3C3×3−1+2×C31⋊C311yescharacteristic
200C3#737×C31⋊C313yes—
subgroupG/H
center, FrattiniC32C33×C31⋊C3
commutator, socle, cosocleC32×C31C34
FittingC31×#6899C3
prime radicalC33×C31⋊C3C32
hypercenter#6899C31⋊C3
hypocenterC31C3×#6899
LCSG ▷ C32×C31 ▷ C31
UCSC1 ◁ C32 ◁ #6899

Complements

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

NH ≅ G/Ntype
2. #21419358C3semidirect
3. #21419357C3semidirect
4. #21419356C3semidirect
5. #21419355C3semidirect
6. #21419354C3semidirect
7. #21419353C3semidirect
8. #21419352C3semidirect
9. #21419351C3semidirect
10. C9⋊C9×C31⋊C3C3semidirect
11. C9⋊C9×C31⋊C3C3semidirect
13. C92×C31⋊C3C3semidirect
14. C31×#6899C3semidirect
26. #14578887C32semidirect
27. #14578887C32semidirect
28. #14578886C32semidirect
29. #14578886C32semidirect
30. (C31⋊C9)⋊C9C32semidirect
31. (C31⋊C9)⋊C9C32semidirect
32. (C31⋊C9)⋊C9C32semidirect
33. (C31⋊C9)⋊C9C32semidirect
34. (C31⋊C9)⋊C9C32semidirect
35. (C31⋊C9)⋊C9C32semidirect
41. C9×C31⋊C9C32semidirect
42. C9×C31⋊C9C32semidirect
44. C31×C9⋊C9C32semidirect
45. C31×C9⋊C9C32semidirect
47. C92×C31C32semidirect
104. #6899C31⋊C3direct (N × H)
114. C31⋊C3#6899direct (N × H)
137. C9⋊C9C3×C31⋊C3semidirect
138. C9⋊C9C3×C31⋊C3semidirect
147. C92C3×C31⋊C3semidirect
148. C31C3×#6899semidirect

Maximal subgroups

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

subgroupindex
C3×3−1+2×C31⋊C33
#214193583
#214193573
#214193563
#214193553
#214193543
#214193533
#214193523
#214193513
C9⋊C9×C31⋊C33
C32⋊C9×C31⋊C33
C92×C31⋊C33
C31×#68993
C3×#689931