Finite Group Databasebeta

← #44534976

Subgroups — #44534976

Order 31347.

836 conjugacy classes of subgroups; only the 298 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 298 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
1C32×#11923957C311yesmaximal, characteristic
2C3×3−1+2×C43⋊C3C311yesmaximal, characteristic
3#23782849C313yesmaximal, complemented
4#23782849C316yesmaximal, complemented
5#23782828C313yesmaximal, complemented
6#23782826C316yesmaximal, complemented
7#23782817C312yesmaximal, complemented
8#23782816C311yesmaximal, characteristic, complemented
9#23782814C316yesmaximal, complemented
10#23782812C312yesmaximal, complemented
11#23782797C312yesmaximal
12#23782792C312yesmaximal, complemented
13C9×#11923958C311yesmaximal, characteristic, complemented
14#11923957⋊C9C313yesmaximal, complemented
15C43×#6894C311yesmaximal, characteristic, complemented
16C33×C43⋊C3C3211yescharacteristic
17C3×#11923958C3211yescharacteristic
18C3×#11923958C3211yescharacteristic
19C3×#11923957—39no—
20C3×#11923957C3213yes—
21C3×#11923957C3212yes—
22C3×#11923957C3212yes—
23C3×#11923957C3211yescharacteristic
24C3×#11923957C3212yes—
25C3×#11923957C3211yescharacteristic
26C3×#11923957C3216yes—
27C3×#11923957C3211yescharacteristic
28C3×#11923956C3213yes—
293−1+2×C43⋊C3—39no—
30C32×C43⋊C9C3212yes—
31C3×C9×C43⋊C3C3213yes—
32#16812500C3213yescomplemented
33#16812500C3216yescomplemented
34(C43⋊C9)⋊C9C3216yescomplemented
35(C43⋊C9)⋊C9C3216yescomplemented
36(C43⋊C9)⋊C9C3216yescomplemented
37(C43⋊C9)⋊C9C3216yescomplemented
38(C43⋊C9)⋊C9C3216yescomplemented
39(C43⋊C9)⋊C9C3216yescomplemented
40(C43⋊C9)⋊C9C3216yescomplemented
41(C43⋊C9)⋊C9C3216yescomplemented
42(C43⋊C9)⋊C9C3216yescomplemented
43C32⋊(C43⋊C9)C3212yes—
44C32⋊(C43⋊C9)C3212yes—
45(C3×C43⋊C3)⋊C9C3216yes—
46(C3×C43⋊C3)⋊C9C3216yes—
47C9×C43⋊C9C3213yescomplemented
48C9×C43⋊C9C3216yescomplemented
49C3×3−1+2×C43C3211yescharacteristic
50C32×C9×C43C3211yescharacteristic
51C43×C9⋊C9C3213yescomplemented
52C43×C9⋊C9C3216yescomplemented
53C43×C32⋊C9C3212yes—
54C32×C43⋊C3—33no—
55C32×C43⋊C3—33no—
56C32×C43⋊C3—92no—
57C32×C43⋊C3C3311yescharacteristic
58C32×C43⋊C3C3311yescharacteristic
59C32×C43⋊C3C3311yescharacteristic
60#11923958—33no—
61#119239583−1+219yescomplemented
62#11923957—36no—
63#11923957—33no—
64#11923957—33no—
65#11923957—39no—
66#11923957—36no—
67#11923957—36no—
68#11923957—96no—
69#119239573−1+219yescomplemented
70#11923956—39no—
71C3×C43⋊C9—36no—
72C3×C43⋊C9C3316yes—
73C3×C43⋊C9C3316yes—
74C3×C43⋊C9C3312yes—
75C3×C43⋊C9C3313yes—
76C3×C43⋊C9C3312yes—
77C3×C43⋊C9C3313yes—
78C3×C43⋊C9C3312yes—
79C9×C43⋊C3—39no—
80C33×C43C3311yescharacteristic
813−1+2×C43—33no—
82C3×C9×C43—33no—
83C3×C9×C43C3316yes—
84C3×C9×C43C3312yes—
85C3×C9×C43C3313yes—
86C3×C9×C43C3311yescharacteristic
873−1+22—431nomaximal, Sylow
88C32×C43—31no—
89C32×C43—32no—
90C32×C43C3×3−1+213yes—
91C32×C43C3411yescharacteristic
92C3×C43⋊C3—31no—
93C3×C43⋊C3—32no—
94C3×C43⋊C3—31no—
95C3×C43⋊C3—31no—
96C3×C43⋊C3—31no—
97C3×C43⋊C3—31no—
98C3×C43⋊C3—32no—
99C3×C43⋊C3—32no—
100C3×C43⋊C3—99no—
101C3×C43⋊C3C3×3−1+213yes—
102C9×C43—32no—
103C9×C43—33no—
104C9×C43—36no—
105C9×C43C3×3−1+213yescomplemented
106C43⋊C9—33no—
107C43⋊C9—32no—
108C43⋊C9—32no—
109C43⋊C9—36no—
110C43⋊C9—36no—
111C43⋊C9C3×3−1+219yescomplemented
112C43⋊C9C3×3−1+216yescomplemented
113C32×3−1+2—431no—
114C32×3−1+2—431no—
115#6900—436no—
116#6900—433no—
117#6896—436no—
118#6896—433no—
119#6894—432no—
120#6894—431no—
121#6894—436no—
122#6894—432no—
123#6894C43⋊C311yescharacteristic, complemented
124#6891—432no—
125C9×3−1+2—431no—
126C9×3−1+2—433no—
127C9×3−1+2—432no—
128C3×C43—33no—
129C3×C43C32×3−1+211yescharacteristic
130C3×C43C32×3−1+211yescharacteristic
131C3×C433−1+412yes—
132C43⋊C3—33no—
133C43⋊C3—91no—
134C43⋊C3—91no—
135C34—431no—
136C3×3−1+2—431no—
137C3×3−1+2—431no—
138C3×3−1+2—431no—
139C3×3−1+2—433no—
140C3×3−1+2—432no—
141C3×3−1+2—432no—
142C3×3−1+2—431no—
143C3×3−1+2—433no—
144C3×3−1+2—432no—
145C3×3−1+2—431no—
146C3×3−1+2—436no—
147C3×3−1+2—1299no—
148C3×3−1+2—1299no—
149C3×3−1+2C3×C43⋊C311yescharacteristic
150C32×C9—432no—
151C32×C9—433no—
152C32×C9C3×C43⋊C311yescharacteristic
153C9⋊C9—436no—
154C9⋊C9—436no—
155C9⋊C9—436no—
156C9⋊C9—436no—
157C9⋊C9—436no—
158C9⋊C9—436no—
159C9⋊C9—436no—
160C9⋊C9—436no—
161C9⋊C9—436no—
162C9⋊C9—433no—
163C9⋊C9—436no—
164C9⋊C9C3×C43⋊C313yescomplemented
165C9⋊C9C3×C43⋊C316yescomplemented
166C32⋊C9—432no—
167C32⋊C9—432no—
168C32⋊C9—436no—
169C32⋊C9—436no—
170C32⋊C9C3×C43⋊C312yes—
171C92—436no—
172C92—433no—
173C433−1+2211yescharacteristic, complemented, Sylow
174C33—431no—
175C33—431no—
176C33—431no—
177C33—1293no—
178C33—1293no—
179C33—3872no—
180C33C32×C43⋊C311yescharacteristic
1813−1+2—33no—
1823−1+2—439no—
1833−1+2—439no—
1843−1+2—1299no—
1853−1+2—1296no—
1863−1+2—1296no—
1873−1+2—1299no—
1883−1+2—1296no—
1893−1+2—1293no—
1903−1+2—1293no—
1913−1+2—1293no—
1923−1+2—3876no—
193C3×C9—33no—
194C3×C9—432no—
195C3×C9—433no—
196C3×C9—436no—
197C3×C9—436no—
198C3×C9—432no—
199C3×C9—432no—
200C3×C9—433no—
201C3×C9—1299no—
202C3×C9—1296no—
203C3×C9C32×C43⋊C316yes—
204C3×C9C32×C43⋊C313yes—
205C3×C9C32×C43⋊C312yes—
206C3×C9C32×C43⋊C311yescharacteristic
207C32—31no—
208C32—32no—
209C32—433no—
210C32—1291no—
211C32—1292no—
212C32—1291no—
213C32—1291no—
214C32—1291no—
215C32—1291no—
216C32—1292no—
217C32—1292no—
218C32—3879no—
219C323−1+2×C43⋊C313yes—
220C32C33×C43⋊C311yescharacteristic
221C9—32no—
222C9—33no—
223C9—36no—
224C9—439no—
225C9—436no—
226C9—1293no—
227C9—1292no—
228C9—1292no—
229C9—1296no—
230C9—1296no—
231C93−1+2×C43⋊C313yescomplemented
232C3—33no—
233C3—1293no—
234C3—3871no—
235C3—3871no—
236C3C3×3−1+2×C43⋊C311yescharacteristic
237C3C3×3−1+2×C43⋊C311yescharacteristic
238C3#2378292212yes—
subgroupG/H
center, FrattiniC32C33×C43⋊C3
commutator, socle, cosocleC32×C43C34
FittingC43×#6894C3
prime radicalC33×C43⋊C3C32
hypercenter#6894C43⋊C3
hypocenterC433−1+22
LCSG ▷ C32×C43 ▷ C43
UCSC1 ◁ C32 ◁ #6894

Complements

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

NH ≅ G/Ntype
3. #23782849C3semidirect
4. #23782849C3semidirect
5. #23782828C3semidirect
6. #23782826C3semidirect
7. #23782817C3semidirect
8. #23782816C3semidirect
9. #23782814C3semidirect
10. #23782812C3semidirect
12. #23782792C3semidirect
13. C9×#11923958C3semidirect
14. #11923957⋊C9C3semidirect
15. C43×#6894C3semidirect
32. #16812500C32semidirect
33. #16812500C32semidirect
34. (C43⋊C9)⋊C9C32semidirect
35. (C43⋊C9)⋊C9C32semidirect
36. (C43⋊C9)⋊C9C32semidirect
37. (C43⋊C9)⋊C9C32semidirect
38. (C43⋊C9)⋊C9C32semidirect
39. (C43⋊C9)⋊C9C32semidirect
40. (C43⋊C9)⋊C9C32semidirect
41. (C43⋊C9)⋊C9C32semidirect
42. (C43⋊C9)⋊C9C32semidirect
47. C9×C43⋊C9C32semidirect
48. C9×C43⋊C9C32semidirect
51. C43×C9⋊C9C32semidirect
52. C43×C9⋊C9C32semidirect
61. #119239583−1+2semidirect
69. #119239573−1+2semidirect
105. C9×C43C3×3−1+2semidirect
111. C43⋊C9C3×3−1+2semidirect
112. C43⋊C9C3×3−1+2semidirect
123. #6894C43⋊C3semidirect
164. C9⋊C9C3×C43⋊C3semidirect
165. C9⋊C9C3×C43⋊C3semidirect
173. C433−1+22semidirect
231. C93−1+2×C43⋊C3semidirect

Maximal subgroups

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

subgroupindex
C32×#119239573
C3×3−1+2×C43⋊C33
#237828493
#237828283
#237828263
#237828173
#237828163
#237828143
#237828123
#237827973
#237827923
C9×#119239583
#11923957⋊C93
C43×#68943
3−1+2243