Finite Group Databasebeta

← #41007654

Subgroups — #41007654

Order 26973.

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

normal subgroups only

This group has 926 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×H(3,𝔽3)×C37⋊C3C311yesmaximal, characteristic
2#22582561C312yesmaximal, complemented
3#22582560C312yesmaximal, complemented
4#22582559C311yesmaximal, characteristic, complemented
5#22582558C311yesmaximal, characteristic, complemented
6#22582557C316yesmaximal, complemented
7#22582557C316yesmaximal, complemented
8#22582556C312yesmaximal, complemented
9#22582555C316yesmaximal, complemented
10#22582553C313yesmaximal, complemented
11#22582552C313yesmaximal, complemented
12#22582551C313yesmaximal, complemented
13C3×#15688502C311yesmaximal, characteristic, complemented
14C32⋊C9×C37⋊C3C312yesmaximal, complemented
15C37×C9⋊H(3,𝔽3)C311yesmaximal, characteristic, complemented
16C33×C37⋊C3C3213yes—
17C33×C37⋊C3C3211yescharacteristic
18C3×#11758608C3216yescomplemented
19C3×#11758605C3216yescomplemented
20C3×#11758605C3213yescomplemented
21C3×#11758606C3216yescomplemented
22C3×#11758606C3213yescomplemented
23C3×C32⋊(C37⋊C3)C3216yes—
24C3×C32⋊(C37⋊C3)C3211yescharacteristic
25C3×C32⋊(C37⋊C3)C3211yescharacteristic
26H(3,𝔽3)×C37⋊C3—39no—
27C32×#66524C3212yescomplemented
28C3×C9×C37⋊C3C3216yescomplemented
29C32⋊#66524C3212yescomplemented
30C32⋊#66524C3212yescomplemented
31C32⋊#66524C3216yescomplemented
32C32⋊#66524C3216yescomplemented
33#15688502C3216yes—
34#15688502C3211yescharacteristic
35#15688502C3211yescharacteristic
36#15688502C3211yescharacteristic
37#15688502C3216yescomplemented
38#15688502C3212yescomplemented
39#15688502C3212yescomplemented
40#15688502C3212yescomplemented
41#15688502C3213yescomplemented
42#15688502C3213yescomplemented
43#15688502C3216yescomplemented
44#15688502C3216yescomplemented
45#15688502C3216yescomplemented
46#15688502C3213yescomplemented
47#15688502C3216yescomplemented
48#15688502C3213yescomplemented
49C3×3−1+2×C37C3213yescomplemented
50C3×H(3,𝔽3)×C37C3211yescharacteristic
51C32×C9×C37C3211yescharacteristic, complemented
52C37×C32⋊C9C3212yescomplemented
53C37×C32⋊C9C3213yescomplemented
54C37×C32⋊C9C3213yescomplemented
55C33×C37C3313yes—
56C33×C37C3311yescharacteristic
57C32×C37⋊C3—39no—
58C32×C37⋊C3—91no—
59C32×C37⋊C3—93no—
60C32×C37⋊C3—91no—
61C32×C37⋊C3—91no—
62C32×C37⋊C3—93no—
63C32×C37⋊C3—93no—
64C32×C37⋊C33−1+219yescomplemented
65C32×C37⋊C3C3313yes—
66C32×C37⋊C3C3313yes—
67C32×C37⋊C3C3311yescharacteristic
68C32×C37⋊C3C3312yes—
693−1+2×C37—93no—
70H(3,𝔽3)×C37—33no—
71C3×C9×C37—91no—
72C3×C9×C37C3311yescharacteristic
73C3×C9×C37C3311yescharacteristic
74C3×C9×C37C3316yes—
75C3×C9×C37C3311yescharacteristic
76C32⋊(C37⋊C3)—33no—
77C32⋊(C37⋊C3)—33no—
78C32⋊(C37⋊C3)—318no—
79#11758608—96no—
80C3×#66524—92no—
81C3×#66524C3312yes—
82C3×#66524C3316yes—
83C3×#66524C3316yes—
84C3×#66524C3312yes—
85C3×#66524C3312yes—
86#11758606—93no—
87#11758606—96no—
88#11758605—93no—
89#11758605—96no—
90C9×C37⋊C3—96no—
91#10624883—371nomaximal, Sylow
92C32×C37—31no—
93C32×C37—31no—
94C32×C37—31no—
95C32×C37—33no—
96C32×C37—33no—
97C32×C37—33no—
98C32×C37—33no—
99C32×C37C3×3−1+213yes—
100C32×C37C3411yescharacteristic
101C3×C37⋊C3—32no—
102C3×C37⋊C3—31no—
103C3×C37⋊C3—31no—
104C3×C37⋊C3—31no—
105C3×C37⋊C3—32no—
106C3×C37⋊C3—32no—
107C3×C37⋊C3—33no—
108C3×C37⋊C3—33no—
109C3×C37⋊C3—33no—
110C3×C37⋊C3—33no—
111C3×C37⋊C3—33no—
112C3×C37⋊C3—33no—
113C3×C37⋊C3—33no—
114C3×C37⋊C3—33no—
115C3×C37⋊C3—99no—
116C3×C37⋊C3—927no—
117C3×C37⋊C3C3×3−1+216yes—
118C3×C37⋊C3C3×3−1+213yes—
119C9×C37—36no—
120C9×C37—31no—
121C9×C37—31no—
122C9×C37—31no—
123#66524—36no—
124#66524—36no—
125#66524—32no—
126#66524—32no—
127#66524—32no—
128C32×H(3,𝔽3)—371no—
129C9⋊H(3,𝔽3)—376no—
130C9⋊H(3,𝔽3)—372no—
131C9⋊H(3,𝔽3)—371no—
132C9⋊H(3,𝔽3)—371no—
133C9⋊H(3,𝔽3)—376no—
134C9⋊H(3,𝔽3)—376no—
135C9⋊H(3,𝔽3)—372no—
136C9⋊H(3,𝔽3)—372no—
137C9⋊H(3,𝔽3)C37⋊C311yescharacteristic, complemented
138#6891—373no—
139#6891—373no—
140#6891—373no—
141C3×C32⋊C9—371no—
142C3×C32⋊C9—372no—
143C3×C37—33no—
144C3×C37—93no—
145C3×C37C32×3−1+211yescharacteristic
146C3×C37H(5,𝔽3)11yescharacteristic
147C3×C373−1+411yescharacteristic
148C3×C373−1+411yescharacteristic
149C37⋊C3—33no—
150C37⋊C3—36no—
151C37⋊C3—93no—
152C37⋊C3—93no—
153C34—371no—
154C34—373no—
155C3×3−1+2—376no—
156C3×3−1+2—376no—
157C3×3−1+2—373no—
158C3×3−1+2—376no—
159C3×3−1+2—373no—
160C3×3−1+2C3×C37⋊C313yescomplemented
161C3×H(3,𝔽3)—376no—
162C3×H(3,𝔽3)—371no—
163C3×H(3,𝔽3)—371no—
164C3×H(3,𝔽3)—1119no—
165C3×H(3,𝔽3)C3×C37⋊C311yescharacteristic
166C32×C9—372no—
167C32×C9—376no—
168C32×C9C3×C37⋊C311yescharacteristic, complemented
169C32⋊C9—372no—
170C32⋊C9—372no—
171C32⋊C9—376no—
172C32⋊C9—376no—
173C32⋊C9—371no—
174C32⋊C9—376no—
175C32⋊C9—371no—
176C32⋊C9—371no—
177C32⋊C9—372no—
178C32⋊C9—376no—
179C32⋊C9—372no—
180C32⋊C9—372no—
181C32⋊C9—373no—
182C32⋊C9—373no—
183C32⋊C9—376no—
184C32⋊C9—376no—
185C32⋊C9—376no—
186C32⋊C9—376no—
187C32⋊C9—373no—
188C32⋊C9—373no—
189C32⋊C9C3×C37⋊C312yescomplemented
190C32⋊C9C3×C37⋊C313yescomplemented
191C32⋊C9C3×C37⋊C313yescomplemented
192C37#1062488311yescharacteristic, complemented, Sylow
193C33—379no—
194C33—371no—
195C33—372no—
196C33—373no—
197C33—373no—
198C33—3331no—
199C33—3331no—
200C33—3331no—
201C33—3333no—
202C33—3333no—
203C33—3333no—
204C33—1119no—
205C33C32×C37⋊C313yes—
206C33C32×C37⋊C311yescharacteristic
2073−1+2—93no—
2083−1+2—3333no—
2093−1+2—3336no—
2103−1+2—3333no—
2113−1+2—3336no—
2123−1+2—3336no—
213H(3,𝔽3)—33no—
214H(3,𝔽3)—11118no—
215H(3,𝔽3)—1113no—
216H(3,𝔽3)—1113no—
217C3×C9—91no—
218C3×C9—372no—
219C3×C9—376no—
220C3×C9—376no—
221C3×C9—372no—
222C3×C9—372no—
223C3×C9—3336no—
224C3×C9—3332no—
225C3×C9C32×C37⋊C311yescharacteristic
226C3×C9C32×C37⋊C316yes—
227C3×C9C32×C37⋊C311yescharacteristic
228C3×C9C32×C37⋊C311yescharacteristic
229C32—31no—
230C32—31no—
231C32—31no—
232C32—33no—
233C32—33no—
234C32—33no—
235C32—33no—
236C32—376no—
237C32—373no—
238C32—3339no—
239C32—33327no—
240C32—1113no—
241C32—1113no—
242C32—1113no—
243C32—1113no—
244C32—1113no—
245C32—1113no—
246C32—1113no—
247C32—1113no—
248C32—1112no—
249C32—1111no—
250C32—1111no—
251C32—1112no—
252C32—1111no—
253C32—1112no—
254C323−1+2×C37⋊C313yes—
255C32C33×C37⋊C311yescharacteristic
256C9—31no—
257C9—31no—
258C9—31no—
259C9—36no—
260C9—1116no—
261C9—1116no—
262C9—1112no—
263C9—1112no—
264C9—1112no—
265C3—33no—
266C3—93no—
267C3—3333no—
268C3—3333no—
269C3—1113no—
270C3—1116no—
271C3C3×3−1+2×C37⋊C311yescharacteristic
272C3C37⋊H(5,𝔽3)11yescharacteristic
273C3#2258267711yescharacteristic
274C3#2258267811yescharacteristic
subgroupG/H
center, FrattiniC32C33×C37⋊C3
commutator, socle, cosocleC32×C37C34
FittingC37×C9⋊H(3,𝔽3)C3
prime radicalC3×H(3,𝔽3)×C37⋊C3C3
hypercenterC9⋊H(3,𝔽3)C37⋊C3
hypocenterC37#10624883
LCSG ▷ C32×C37 ▷ C37
UCSC1 ◁ C32 ◁ C9⋊H(3,𝔽3)

Complements

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

NH ≅ G/Ntype
2. #22582561C3semidirect
3. #22582560C3semidirect
4. #22582559C3semidirect
5. #22582558C3semidirect
6. #22582557C3semidirect
7. #22582557C3semidirect
8. #22582556C3semidirect
9. #22582555C3semidirect
10. #22582553C3semidirect
11. #22582552C3semidirect
12. #22582551C3semidirect
13. C3×#15688502C3semidirect
14. C32⋊C9×C37⋊C3C3semidirect
15. C37×C9⋊H(3,𝔽3)C3semidirect
18. C3×#11758608C32semidirect
19. C3×#11758605C32semidirect
20. C3×#11758605C32semidirect
21. C3×#11758606C32semidirect
22. C3×#11758606C32semidirect
27. C32×#66524C32semidirect
28. C3×C9×C37⋊C3C32semidirect
29. C32⋊#66524C32semidirect
30. C32⋊#66524C32semidirect
31. C32⋊#66524C32semidirect
32. C32⋊#66524C32semidirect
37. #15688502C32semidirect
38. #15688502C32semidirect
39. #15688502C32semidirect
40. #15688502C32semidirect
41. #15688502C32semidirect
42. #15688502C32semidirect
43. #15688502C32semidirect
44. #15688502C32semidirect
45. #15688502C32semidirect
46. #15688502C32semidirect
47. #15688502C32semidirect
48. #15688502C32semidirect
49. C3×3−1+2×C37C32semidirect
51. C32×C9×C37C32semidirect
52. C37×C32⋊C9C32semidirect
53. C37×C32⋊C9C32semidirect
54. C37×C32⋊C9C32semidirect
64. C32×C37⋊C33−1+2semidirect
137. C9⋊H(3,𝔽3)C37⋊C3semidirect
160. C3×3−1+2C3×C37⋊C3semidirect
168. C32×C9C3×C37⋊C3semidirect
189. C32⋊C9C3×C37⋊C3semidirect
190. C32⋊C9C3×C37⋊C3semidirect
191. C32⋊C9C3×C37⋊C3semidirect
192. C37#10624883semidirect

Maximal subgroups

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

subgroupindex
C3×H(3,𝔽3)×C37⋊C33
#225825613
#225825603
#225825593
#225825583
#225825573
#225825563
#225825553
#225825533
#225825523
#225825513
C3×#156885023
C32⋊C9×C37⋊C33
C37×C9⋊H(3,𝔽3)3
#1062488337