Finite Group Databasebeta

← #41008220

Subgroups — #41008220

Order 26973.

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

normal subgroups only

This group has 948 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
2C3×#15688531C311yesmaximal, characteristic, complemented
3#22582662C314yesmaximal, complemented
4#22582657C312yesmaximal, complemented
5#22582656C316yesmaximal, complemented
6#22582654C316yesmaximal, complemented
7#22582649C312yesmaximal, complemented
8#22582645C311yesmaximal, characteristic, complemented
9#22582636C313yesmaximal, complemented
10#22582633C312yesmaximal, complemented
11#22582632C316yesmaximal, complemented
12#731×C37⋊C3C313yesmaximal, complemented
13C37×#6908C311yesmaximal, characteristic, complemented
14#15688532—312no—
15#15688532C32112yescomplemented
16#15688531—36no—
17#15688531—33no—
18#15688531—36no—
19#15688531C3216yescomplemented
20#15688530C3216yescomplemented
21C3×#11758608C3212yescomplemented
22C3×#11758605C3211yescharacteristic
23C3×#11758605C3212yescomplemented
24C3×#11758606—33no—
25C3×#11758606C3211yescharacteristic
26C3×#11758606C3212yescomplemented
273−1+2×C37⋊C3—36no—
283−1+2×C37⋊C3C3216yes—
29C3×C32⋊(C37⋊C3)C3211yescharacteristic
30H(3,𝔽3)×C37⋊C3C3213yes—
31C3×C9×C37⋊C3C3211yescharacteristic
32C3×C9×C37⋊C3C3212yescomplemented
33#15688521C3214yescomplemented
34#15688520C32112yescomplemented
35#15688519C3212yes—
36#15688518C3216yes—
37#15688517C32112yescomplemented
38#15688516C3216yescomplemented
39#15688515C3216yescomplemented
40#15688514C32112yescomplemented
41#15688513C3216yescomplemented
42#15688512C3216yescomplemented
43C37×#737C3213yescomplemented
44C32×C9×C37C3211yescharacteristic, complemented
45C37×#733C3212yescomplemented
46C37×#732C3211yescharacteristic
47C37×#731C3213yescomplemented
48C37×AΣL(1,𝔽27)C3213yescomplemented
49C33×C37C3311yescharacteristic
50C32×C37⋊C3—33no—
51C32×C37⋊C3C3312yes—
52C32×C37⋊C3C3311yescharacteristic
533−1+2×C37—36no—
543−1+2×C37C3316yes—
55H(3,𝔽3)×C37C3313yes—
56C3×C9×C37—32no—
57C3×C9×C37—31no—
58C3×C9×C37—33no—
59C3×C9×C37C3311yescharacteristic
60C3×C9×C37C3311yescharacteristic
61C3×C9×C37C3311yescharacteristic
62C32⋊(C37⋊C3)—36no—
63C32⋊(C37⋊C3)—33no—
64C32⋊(C37⋊C3)C3316yes—
65#11758608—312no—
66#11758608—36no—
67#11758608C33112yes—
68C3×#66524—36no—
69C3×#66524C3312yes—
70C3×#66524C3312yes—
71C3×#66524C3312yes—
72#11758606—36no—
73#11758606—36no—
74#11758606—34no—
75#11758606—32no—
76#11758606—312no—
77#11758606H(3,𝔽3)16yes—
78#11758606H(3,𝔽3)13yes—
79#11758605—36no—
80#11758605—33no—
81#11758605—36no—
82#11758605—312no—
83#11758605—36no—
84C9×C37⋊C3—33no—
85C9×C37⋊C3—36no—
86C9×C37⋊C3—36no—
87C9×C37⋊C3—36no—
88C9×C37⋊C3—312no—
89#10624935—371nomaximal, Sylow
90C32×C37—33no—
91C32×C37—31no—
92C32×C37—91no—
93C32×C37C3411yescharacteristic
94C3×C37⋊C3—32no—
95C3×C37⋊C3—36no—
96C3×C37⋊C3—91no—
97C3×C37⋊C3—92no—
98C3×C37⋊C3—273no—
99C3×C37⋊C3C3×H(3,𝔽3)13yes—
100C9×C37—31no—
101C9×C37—36no—
102C9×C37—31no—
103C9×C37C3×H(3,𝔽3)12yes—
104C9×C37C3×H(3,𝔽3)11yescharacteristic
105#66524—312no—
106#66524—32no—
107#66524—32no—
108#66524C3×H(3,𝔽3)16yes—
1093−1+4—373no—
110C3×#737—371no—
111#6913—376no—
112#6913—374no—
113#6913—372no—
114#6913—376no—
115#6912—372no—
116#6911—371no—
117#6909—373no—
118#6908—372no—
119#6908—376no—
120#6908C37⋊C311yescharacteristic, complemented
121C3×#731—373no—
122C3×C37—31no—
123C3×C37—93no—
124C3×C37—91no—
125C3×C37C32×H(3,𝔽3)11yescharacteristic
126C37⋊C3—33no—
127C37⋊C3—92no—
128C37⋊C3—96no—
129#737—3712no—
130#737—376no—
131#737—376no—
132#737—1113no—
133#737—1116no—
134#737—1116no—
135#737—11112no—
136#737C3×C37⋊C313yescomplemented
137C3×3−1+2—372no—
138C3×3−1+2—371no—
139C3×3−1+2—372no—
140C3×3−1+2—371no—
141C3×3−1+2—372no—
142C3×3−1+2—376no—
143C3×3−1+2—1113no—
144C3×3−1+2—1116no—
145C3×H(3,𝔽3)—373no—
146C3×H(3,𝔽3)—371no—
147C32×C9—371no—
148C32×C9—372no—
149C32×C9C3×C37⋊C311yescharacteristic, complemented
150#733—3712no—
151#733—374no—
152#733C3×C37⋊C312yescomplemented
153#732—376no—
154#732—372no—
155#732C3×C37⋊C311yescharacteristic
156#731—3712no—
157#731—376no—
158#731—376no—
159#731C3×C37⋊C313yescomplemented
160AΣL(1,𝔽27)—376no—
161AΣL(1,𝔽27)—3712no—
162AΣL(1,𝔽27)—376no—
163AΣL(1,𝔽27)C3×C37⋊C313yescomplemented
164C37#1062493511yescharacteristic, complemented, Sylow
165C33—371no—
166C33—372no—
167C33—1113no—
168C33C32×C37⋊C311yescharacteristic
1693−1+2—36no—
1703−1+2—373no—
1713−1+2—376no—
1723−1+2—3712no—
1733−1+2—1116no—
1743−1+2—11112no—
1753−1+2—11112no—
1763−1+2—1113no—
1773−1+2—1116no—
1783−1+2—1116no—
1793−1+2—1116no—
1803−1+2—11112no—
1813−1+2—1116no—
1823−1+2—1112no—
1833−1+2—1114no—
1843−1+2—1116no—
1853−1+2C32×C37⋊C316yes—
186H(3,𝔽3)—376no—
187H(3,𝔽3)—1113no—
188H(3,𝔽3)—1116no—
189H(3,𝔽3)C32×C37⋊C313yes—
190C3×C9—31no—
191C3×C9—32no—
192C3×C9—33no—
193C3×C9—372no—
194C3×C9—372no—
195C3×C9—372no—
196C3×C9—1116no—
197C3×C9—11112no—
198C3×C9—1113no—
199C3×C9—1116no—
200C3×C9—1116no—
201C3×C9—1116no—
202C3×C9C32×C37⋊C311yescharacteristic
203C3×C9C32×C37⋊C311yescharacteristic
204C3×C9C32×C37⋊C311yescharacteristic
205C32—31no—
206C32—33no—
207C32—91no—
208C32—373no—
209C32—3332no—
210C32—3331no—
211C32—1116no—
212C32—1112no—
213C32—9993no—
214C32C33×C37⋊C311yescharacteristic
215C9—31no—
216C9—31no—
217C9—36no—
218C9—376no—
219C9—1112no—
220C9—1112no—
221C9—11112no—
222C9H(3,𝔽3)×C37⋊C311yescharacteristic
223C9H(3,𝔽3)×C37⋊C312yes—
224C3—31no—
225C3—91no—
226C3—93no—
227C3—3332no—
228C3—3336no—
229C3—1113no—
230C3C3×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#10624935
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. C3×#15688531C3semidirect
3. #22582662C3semidirect
4. #22582657C3semidirect
5. #22582656C3semidirect
6. #22582654C3semidirect
7. #22582649C3semidirect
8. #22582645C3semidirect
9. #22582636C3semidirect
10. #22582633C3semidirect
11. #22582632C3semidirect
12. #731×C37⋊C3C3semidirect
13. C37×#6908C3semidirect
15. #15688532C32semidirect
19. #15688531C32semidirect
20. #15688530C32semidirect
21. C3×#11758608C32semidirect
23. C3×#11758605C32semidirect
26. C3×#11758606C32semidirect
32. C3×C9×C37⋊C3C32semidirect
33. #15688521C32semidirect
34. #15688520C32semidirect
37. #15688517C32semidirect
38. #15688516C32semidirect
39. #15688515C32semidirect
40. #15688514C32semidirect
41. #15688513C32semidirect
42. #15688512C32semidirect
43. C37×#737C32semidirect
44. C32×C9×C37C32semidirect
45. C37×#733C32semidirect
47. C37×#731C32semidirect
48. C37×AΣL(1,𝔽27)C32semidirect
120. #6908C37⋊C3semidirect
136. #737C3×C37⋊C3semidirect
149. C32×C9C3×C37⋊C3semidirect
152. #733C3×C37⋊C3semidirect
159. #731C3×C37⋊C3semidirect
163. AΣL(1,𝔽27)C3×C37⋊C3semidirect
164. C37#10624935semidirect

Maximal subgroups

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

subgroupindex
#225826773
C3×#156885313
#225826623
#225826573
#225826563
#225826543
#225826493
#225826453
#225826363
#225826333
#225826323
#731×C37⋊C33
C37×#69083
#1062493537