Finite Group Databasebeta

← #41008210

Subgroups — #41008210

Order 26973.

980 conjugacy classes of subgroups; only the 268 normal subgroups are shown.

normal subgroups only

This group has 980 conjugacy classes of subgroups — too many to draw. Only its 268 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#22582677C313yesmaximal, complemented
2C32×#11758606C311yesmaximal, characteristic, complemented
3#22582663C314yesmaximal, complemented
4#22582659C316yesmaximal, complemented
5#22582658C312yesmaximal, complemented
6#22582656C316yesmaximal, complemented
7#22582650C312yesmaximal, complemented
8#22582646C311yesmaximal, characteristic, complemented
9#22582640C313yesmaximal, complemented
10#22582633C312yesmaximal, complemented
11#22582629C316yesmaximal, complemented
12AΣL(1,𝔽27)×C37⋊C3C313yesmaximal, complemented
13C37×#6908C311yesmaximal, characteristic, complemented
14C33×C37⋊C3C3211yescharacteristic
15#15688532—312no—
16#15688532C32112yescomplemented
17#15688531—36no—
18#15688531C3216yescomplemented
19#15688530C3216yescomplemented
20C3×#11758606—36no—
21C3×#11758606—33no—
22C3×#11758606—33no—
23C3×#11758606C3211yescharacteristic
24C3×#11758606C3211yescharacteristic
25C3×#11758606C3211yescharacteristic
26C3×#11758606C3212yescomplemented
27C3×#11758606C3212yescomplemented
28C3×#11758606C3212yescomplemented
293−1+2×C37⋊C3—36no—
303−1+2×C37⋊C3C3216yes—
31H(3,𝔽3)×C37⋊C3C3213yes—
32C32×#66524C3212yescomplemented
33#15688521C3214yescomplemented
34#15688520C32112yescomplemented
35#15688519C3212yescomplemented
36#15688518C3216yescomplemented
37#15688517C32112yescomplemented
38#15688516C3216yescomplemented
39#15688515C3216yescomplemented
40#15688514C32112yescomplemented
41#15688513C3216yescomplemented
42#15688512C3216yescomplemented
43C37×#737C3213yescomplemented
44C32×C9×C37C3211yescharacteristic, complemented
45C37×#733C3212yescomplemented
46C37×#732C3211yescharacteristic, complemented
47C37×#731C3213yescomplemented
48C37×AΣL(1,𝔽27)C3213yescomplemented
49C33×C37C3311yescharacteristic
50C32×C37⋊C3—33no—
51C32×C37⋊C3—33no—
52C32×C37⋊C3—93no—
53C32×C37⋊C3C3312yes—
54C32×C37⋊C3C3311yescharacteristic
553−1+2×C37—36no—
563−1+2×C37C3316yes—
57H(3,𝔽3)×C37C3313yes—
58C3×C9×C37—32no—
59C3×C9×C37—31no—
60C3×C9×C37—33no—
61C3×C9×C37C3311yescharacteristic
62C3×C9×C37C3311yescharacteristic
63C3×C9×C37C3311yescharacteristic
64C32⋊(C37⋊C3)—36no—
65C32⋊(C37⋊C3)C3316yes—
66#11758608—312no—
67#11758608C33112yes—
68C3×#66524—36no—
69C3×#66524—36no—
70C3×#66524C3312yes—
71C3×#66524C3312yes—
72C3×#66524C3312yes—
73#11758606—36no—
74#11758606—33no—
75#11758606—33no—
76#11758606—34no—
77#11758606—32no—
78#11758606—36no—
79#11758606—36no—
80#11758606—36no—
81#11758606—312no—
82#11758606H(3,𝔽3)16yes—
83#11758606H(3,𝔽3)13yes—
84#11758605—36no—
85#11758605—36no—
86#11758605—312no—
87C9×C37⋊C3—36no—
88C9×C37⋊C3—36no—
89C9×C37⋊C3—312no—
90#10624934—371nomaximal, Sylow
91C32×C37—33no—
92C32×C37—31no—
93C32×C37—33no—
94C32×C37C3411yescharacteristic
95C3×C37⋊C3—36no—
96C3×C37⋊C3—32no—
97C3×C37⋊C3—91no—
98C3×C37⋊C3—99no—
99C3×C37⋊C3—92no—
100C3×C37⋊C3—273no—
101C3×C37⋊C3C3×H(3,𝔽3)13yes—
102C9×C37—36no—
103C9×C37—31no—
104C9×C37—31no—
105C9×C37C3×H(3,𝔽3)12yes—
106C9×C37C3×H(3,𝔽3)11yescharacteristic
107#66524—312no—
108#66524—32no—
109#66524—32no—
110#66524C3×H(3,𝔽3)16yes—
1113−1+4—373no—
112C32×3−1+2—371no—
113#6913—376no—
114#6913—374no—
115#6913—372no—
116#6913—376no—
117#6912—372no—
118#6911—371no—
119#6910—373no—
120#6908—372no—
121#6908—376no—
122#6908C37⋊C311yescharacteristic, complemented
123C3×AΣL(1,𝔽27)—373no—
124C3×C37—33no—
125C3×C37—31no—
126C3×C37—93no—
127C3×C37C32×H(3,𝔽3)11yescharacteristic
128C37⋊C3—33no—
129C37⋊C3—92no—
130C37⋊C3—96no—
131C34—371no—
132#737—3712no—
133#737—376no—
134#737—376no—
135#737—1116no—
136#737—11112no—
137#737C3×C37⋊C313yescomplemented
138C3×3−1+2—371no—
139C3×3−1+2—372no—
140C3×3−1+2—372no—
141C3×3−1+2—371no—
142C3×3−1+2—371no—
143C3×3−1+2—372no—
144C3×3−1+2—376no—
145C3×3−1+2—1113no—
146C3×3−1+2—1116no—
147C3×3−1+2—1113no—
148C3×3−1+2—1116no—
149C3×H(3,𝔽3)—373no—
150C32×C9—372no—
151C32×C9C3×C37⋊C311yescharacteristic, complemented
152#733—3712no—
153#733—374no—
154#733C3×C37⋊C312yescomplemented
155#732—376no—
156#732—372no—
157#732C3×C37⋊C311yescharacteristic, complemented
158#731—3712no—
159#731—376no—
160#731—376no—
161#731C3×C37⋊C313yescomplemented
162AΣL(1,𝔽27)—376no—
163AΣL(1,𝔽27)—3712no—
164AΣL(1,𝔽27)—376no—
165AΣL(1,𝔽27)C3×C37⋊C313yescomplemented
166C37#1062493411yescharacteristic, complemented, Sylow
167C33—371no—
168C33—372no—
169C33—3333no—
170C33—1113no—
171C33—1113no—
172C33C32×C37⋊C311yescharacteristic
1733−1+2—36no—
1743−1+2—373no—
1753−1+2—376no—
1763−1+2—3712no—
1773−1+2—11112no—
1783−1+2—1116no—
1793−1+2—1116no—
1803−1+2—11112no—
1813−1+2—1116no—
1823−1+2—1113no—
1833−1+2—1113no—
1843−1+2—1116no—
1853−1+2—11112no—
1863−1+2—1116no—
1873−1+2—1112no—
1883−1+2—1114no—
1893−1+2—1116no—
1903−1+2C32×C37⋊C316yes—
191H(3,𝔽3)—376no—
192H(3,𝔽3)—1116no—
193H(3,𝔽3)C32×C37⋊C313yes—
194C3×C9—31no—
195C3×C9—32no—
196C3×C9—33no—
197C3×C9—372no—
198C3×C9—372no—
199C3×C9—372no—
200C3×C9—11112no—
201C3×C9—1116no—
202C3×C9—1116no—
203C3×C9—1116no—
204C3×C9—1116no—
205C3×C9C32×C37⋊C311yescharacteristic
206C3×C9C32×C37⋊C311yescharacteristic
207C3×C9C32×C37⋊C311yescharacteristic
208C32—33no—
209C32—31no—
210C32—33no—
211C32—373no—
212C32—3332no—
213C32—3339no—
214C32—3331no—
215C32—1116no—
216C32—1112no—
217C32—9993no—
218C32C33×C37⋊C311yescharacteristic
219C9—31no—
220C9—31no—
221C9—36no—
222C9—376no—
223C9—1112no—
224C9—1112no—
225C9—11112no—
226C9H(3,𝔽3)×C37⋊C311yescharacteristic
227C9H(3,𝔽3)×C37⋊C312yes—
228C3—31no—
229C3—33no—
230C3—93no—
231C3—3332no—
232C3—3336no—
233C3—1113no—
234C3C3×H(3,𝔽3)×C37⋊C311yescharacteristic
subgroupG/H
centerC3C3×H(3,𝔽3)×C37⋊C3
commutator, cosocleC32×C37C34
FrattiniC32C33×C37⋊C3
FittingC37×#6908C3
socleC3×C37C32×H(3,𝔽3)
hypercenter#6908C37⋊C3
hypocenterC37#10624934
LCSG ▷ C32×C37 ▷ C3×C37 ▷ C37
UCSC1 ◁ C3 ◁ C3×C9 ◁ #6908

Complements

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

NH ≅ G/Ntype
1. #22582677C3semidirect
2. C32×#11758606C3semidirect
3. #22582663C3semidirect
4. #22582659C3semidirect
5. #22582658C3semidirect
6. #22582656C3semidirect
7. #22582650C3semidirect
8. #22582646C3semidirect
9. #22582640C3semidirect
10. #22582633C3semidirect
11. #22582629C3semidirect
12. AΣL(1,𝔽27)×C37⋊C3C3semidirect
13. C37×#6908C3semidirect
16. #15688532C32semidirect
18. #15688531C32semidirect
19. #15688530C32semidirect
26. C3×#11758606C32semidirect
27. C3×#11758606C32semidirect
28. C3×#11758606C32semidirect
32. C32×#66524C32semidirect
33. #15688521C32semidirect
34. #15688520C32semidirect
35. #15688519C32semidirect
36. #15688518C32semidirect
37. #15688517C32semidirect
38. #15688516C32semidirect
39. #15688515C32semidirect
40. #15688514C32semidirect
41. #15688513C32semidirect
42. #15688512C32semidirect
43. C37×#737C32semidirect
44. C32×C9×C37C32semidirect
45. C37×#733C32semidirect
46. C37×#732C32semidirect
47. C37×#731C32semidirect
48. C37×AΣL(1,𝔽27)C32semidirect
122. #6908C37⋊C3semidirect
137. #737C3×C37⋊C3semidirect
151. C32×C9C3×C37⋊C3semidirect
154. #733C3×C37⋊C3semidirect
157. #732C3×C37⋊C3semidirect
161. #731C3×C37⋊C3semidirect
165. AΣL(1,𝔽27)C3×C37⋊C3semidirect
166. C37#10624934semidirect

Maximal subgroups

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

subgroupindex
#225826773
C32×#117586063
#225826633
#225826593
#225826583
#225826563
#225826503
#225826463
#225826403
#225826333
#225826293
AΣL(1,𝔽27)×C37⋊C33
C37×#69083
#1062493437