Finite Group Databasebeta

← #41007798

Subgroups — #41007798

Order 26973.

1148 conjugacy classes of subgroups; only the 277 normal subgroups are shown.

normal subgroups only

This group has 1148 conjugacy classes of subgroups — too many to draw. Only its 277 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×#15688532C312yesmaximal, complemented
2C3×#15688531C311yesmaximal, characteristic, complemented
3C3×H(3,𝔽3)×C37⋊C3C311yesmaximal, characteristic
4#22582565C312yesmaximal, complemented
5#22582564C311yesmaximal, characteristic, complemented
6#22582563C316yesmaximal, complemented
7#22582563C316yesmaximal, complemented
8#22582563C316yesmaximal, complemented
9#22582562C312yesmaximal, complemented
10#22582553C313yesmaximal, complemented
11#22582552C313yesmaximal, complemented
12#22582551C312yesmaximal, complemented
13C3×#15688502C311yesmaximal, characteristic, complemented
14C32⋊C9×C37⋊C3C313yesmaximal, complemented
15C37×#6893C311yesmaximal, characteristic, complemented
16C33×C37⋊C3C3213yes—
17C33×C37⋊C3C3211yescharacteristic
18#15688532—318no—
19#15688531—39no—
20C3×#11758608C3212yescomplemented
21C3×#11758608C3212yescomplemented
22C3×#11758605C3213yescomplemented
23C3×#11758605C3216yescomplemented
24C3×#11758606C3213yescomplemented
25C3×#11758606C3216yescomplemented
26C3×C32⋊(C37⋊C3)C3216yes—
27C3×C32⋊(C37⋊C3)C3211yescharacteristic
28C3×C32⋊(C37⋊C3)C3211yescharacteristic
29H(3,𝔽3)×C37⋊C3—39no—
30C32×#66524C3212yescomplemented
31C3×C9×C37⋊C3C3213yescomplemented
32C3×C9×C37⋊C3C3216yescomplemented
33C32⋊#66524C3216yescomplemented
34C32⋊#66524C3216yescomplemented
35C32⋊#66524C3216yescomplemented
36#15688502C3211yescharacteristic
37#15688502C3216yes—
38#15688502C3211yescharacteristic
39#15688502C3211yescharacteristic
40#15688502C3213yescomplemented
41#15688502C3216yescomplemented
42#15688502C3212yescomplemented
43#15688502C3212yescomplemented
44#15688502C3212yescomplemented
45#15688502C32112yescomplemented
46#15688502C3213yescomplemented
47#15688502C3216yescomplemented
48#15688502C3213yescomplemented
49#15688502C3216yescomplemented
50C3×3−1+2×C37C3212yescomplemented
51C3×H(3,𝔽3)×C37C3211yescharacteristic
52C32×C9×C37C3211yescharacteristic, complemented
53C37×C32⋊C9C3213yescomplemented
54C37×C32⋊C9C3213yescomplemented
55C37×C32⋊C9C3213yescomplemented
56C33×C37C3313yes—
57C33×C37C3311yescharacteristic
58C32×C37⋊C3—33no—
59C32×C37⋊C3—39no—
60C32×C37⋊C3—39no—
61C32×C37⋊C3—33no—
62C32×C37⋊C3—93no—
63C32×C37⋊C3—93no—
64C32×C37⋊C3—91no—
65C32×C37⋊C3—91no—
66C32×C37⋊C3C3313yes—
67C32×C37⋊C3C3312yes—
68C32×C37⋊C3C3313yes—
69C32×C37⋊C3C3311yescharacteristic
703−1+2×C37—36no—
71H(3,𝔽3)×C37—33no—
72C3×C9×C37—33no—
73C3×C9×C37C3311yescharacteristic
74C3×C9×C37C3311yescharacteristic
75C3×C9×C37C3311yescharacteristic
76C3×C9×C37C3316yes—
77C32⋊(C37⋊C3)—33no—
78C32⋊(C37⋊C3)—318no—
79C32⋊(C37⋊C3)H(3,𝔽3)19yes—
80#11758608—36no—
81#11758608—36no—
82C3×#66524—36no—
83C3×#66524C3312yes—
84C3×#66524C3316yes—
85C3×#66524C3316yes—
86C3×#66524C3312yes—
87C3×#66524C3312yes—
88#11758606—39no—
89#11758606—318no—
90#11758605—39no—
91#11758605—318no—
92C9×C37⋊C3—39no—
93C9×C37⋊C3—318no—
94#10624894—371nomaximal, Sylow
95C32×C37—31no—
96C32×C37—31no—
97C32×C37—31no—
98C32×C37—33no—
99C32×C37—33no—
100C32×C37—33no—
101C32×C37—33no—
102C32×C37C3×H(3,𝔽3)13yes—
103C32×C37C3411yescharacteristic
104C3×C37⋊C3—31no—
105C3×C37⋊C3—33no—
106C3×C37⋊C3—31no—
107C3×C37⋊C3—31no—
108C3×C37⋊C3—33no—
109C3×C37⋊C3—33no—
110C3×C37⋊C3—33no—
111C3×C37⋊C3—32no—
112C3×C37⋊C3—33no—
113C3×C37⋊C3—32no—
114C3×C37⋊C3—33no—
115C3×C37⋊C3—32no—
116C3×C37⋊C3—33no—
117C3×C37⋊C3—32no—
118C3×C37⋊C3—927no—
119C3×C37⋊C3—99no—
120C3×C37⋊C3C3×H(3,𝔽3)19yes—
121C3×C37⋊C3#73713yescomplemented
122C9×C37—31no—
123C9×C37—36no—
124C9×C37—31no—
125C9×C37—31no—
126#66524—32no—
127#66524—36no—
128#66524—36no—
129#66524—32no—
130#66524—32no—
131C3×#737—372no—
132C3×#737—371no—
133C32×H(3,𝔽3)—371no—
134#6893—372no—
135#6893—372no—
136#6893—371no—
137#6893—376no—
138#6893—376no—
139#6893—376no—
140#6893C37⋊C311yescharacteristic, complemented
141#6891—372no—
142#6891—373no—
143#6891—373no—
144C3×C32⋊C9—371no—
145C3×C32⋊C9—373no—
146C3×C37—33no—
147C3×C37—93no—
148C3×C37C32×H(3,𝔽3)11yescharacteristic
149C3×C37C3×#73711yescharacteristic
150C3×C373−1+411yescharacteristic
151C3×C373−1+411yescharacteristic
152C37⋊C3—33no—
153C37⋊C3—39no—
154C37⋊C3—93no—
155C37⋊C3—92no—
156C34—373no—
157C34—371no—
158#737—1119no—
159#737—11118no—
160C3×3−1+2—372no—
161C3×3−1+2—372no—
162C3×3−1+2—376no—
163C3×3−1+2—373no—
164C3×3−1+2—376no—
165C3×3−1+2—373no—
166C3×3−1+2C3×C37⋊C312yescomplemented
167C3×H(3,𝔽3)—371no—
168C3×H(3,𝔽3)—376no—
169C3×H(3,𝔽3)—371no—
170C3×H(3,𝔽3)—1119no—
171C3×H(3,𝔽3)C3×C37⋊C311yescharacteristic
172C32×C9—372no—
173C32×C9—376no—
174C32×C9—373no—
175C32×C9C3×C37⋊C311yescharacteristic, complemented
176C32⋊C9—376no—
177C32⋊C9—376no—
178C32⋊C9—376no—
179C32⋊C9—371no—
180C32⋊C9—376no—
181C32⋊C9—371no—
182C32⋊C9—371no—
183C32⋊C9—376no—
184C32⋊C9—372no—
185C32⋊C9—372no—
186C32⋊C9—373no—
187C32⋊C9—372no—
188C32⋊C9—3712no—
189C32⋊C9—376no—
190C32⋊C9—376no—
191C32⋊C9—373no—
192C32⋊C9—373no—
193C32⋊C9C3×C37⋊C313yescomplemented
194C32⋊C9C3×C37⋊C313yescomplemented
195C32⋊C9C3×C37⋊C313yescomplemented
196C37#1062489411yescharacteristic, complemented, Sylow
197C33—371no—
198C33—373no—
199C33—372no—
200C33—373no—
201C33—3331no—
202C33—3331no—
203C33—3333no—
204C33—3333no—
205C33—1119no—
206C33—1113no—
207C33—1119no—
208C33—1113no—
209C33C32×C37⋊C313yes—
210C33C32×C37⋊C311yescharacteristic
2113−1+2—36no—
2123−1+2—1119no—
2133−1+2—11118no—
2143−1+2—1119no—
2153−1+2—11118no—
2163−1+2—1116no—
2173−1+2—1116no—
218H(3,𝔽3)—33no—
219H(3,𝔽3)—379no—
220H(3,𝔽3)—11118no—
221H(3,𝔽3)—1113no—
222C3×C9—33no—
223C3×C9—372no—
224C3×C9—376no—
225C3×C9—376no—
226C3×C9—372no—
227C3×C9—372no—
228C3×C9—1119no—
229C3×C9—11118no—
230C3×C9—1116no—
231C3×C9C32×C37⋊C311yescharacteristic
232C3×C9C32×C37⋊C316yes—
233C3×C9C32×C37⋊C311yescharacteristic
234C3×C9C32×C37⋊C311yescharacteristic
235C32—31no—
236C32—31no—
237C32—31no—
238C32—33no—
239C32—33no—
240C32—33no—
241C32—33no—
242C32—379no—
243C32—373no—
244C32—3339no—
245C32—33327no—
246C32—1112no—
247C32—1113no—
248C32—1112no—
249C32—1113no—
250C32—1112no—
251C32—1113no—
252C32—1112no—
253C32—1113no—
254C32—1111no—
255C32—1113no—
256C32—1111no—
257C32—1113no—
258C32—1111no—
259C32—1113no—
260C32H(3,𝔽3)×C37⋊C313yes—
261C32C33×C37⋊C311yescharacteristic
262C9—31no—
263C9—31no—
264C9—31no—
265C9—36no—
266C9—1112no—
267C9—1116no—
268C9—1116no—
269C9—1112no—
270C9—1112no—
271C3—33no—
272C3—93no—
273C3—3332no—
274C3—3333no—
275C3—1113no—
276C3—1119no—
277C3C3×H(3,𝔽3)×C37⋊C311yescharacteristic
278C3#737×C37⋊C311yescharacteristic
279C3#2258267711yescharacteristic
280C3#2258267811yescharacteristic
subgroupG/H
center, FrattiniC32C33×C37⋊C3
commutator, socle, cosocleC32×C37C34
FittingC37×#6893C3
prime radicalC3×H(3,𝔽3)×C37⋊C3C3
hypercenter#6893C37⋊C3
hypocenterC37#10624894
LCSG ▷ C32×C37 ▷ C37
UCSC1 ◁ C32 ◁ #6893

Complements

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

NH ≅ G/Ntype
1. C3×#15688532C3semidirect
2. C3×#15688531C3semidirect
4. #22582565C3semidirect
5. #22582564C3semidirect
6. #22582563C3semidirect
7. #22582563C3semidirect
8. #22582563C3semidirect
9. #22582562C3semidirect
10. #22582553C3semidirect
11. #22582552C3semidirect
12. #22582551C3semidirect
13. C3×#15688502C3semidirect
14. C32⋊C9×C37⋊C3C3semidirect
15. C37×#6893C3semidirect
20. C3×#11758608C32semidirect
21. C3×#11758608C32semidirect
22. C3×#11758605C32semidirect
23. C3×#11758605C32semidirect
24. C3×#11758606C32semidirect
25. C3×#11758606C32semidirect
30. C32×#66524C32semidirect
31. C3×C9×C37⋊C3C32semidirect
32. C3×C9×C37⋊C3C32semidirect
33. C32⋊#66524C32semidirect
34. C32⋊#66524C32semidirect
35. C32⋊#66524C32semidirect
40. #15688502C32semidirect
41. #15688502C32semidirect
42. #15688502C32semidirect
43. #15688502C32semidirect
44. #15688502C32semidirect
45. #15688502C32semidirect
46. #15688502C32semidirect
47. #15688502C32semidirect
48. #15688502C32semidirect
49. #15688502C32semidirect
50. C3×3−1+2×C37C32semidirect
52. C32×C9×C37C32semidirect
53. C37×C32⋊C9C32semidirect
54. C37×C32⋊C9C32semidirect
55. C37×C32⋊C9C32semidirect
121. C3×C37⋊C3#737semidirect
140. #6893C37⋊C3semidirect
166. C3×3−1+2C3×C37⋊C3semidirect
175. C32×C9C3×C37⋊C3semidirect
193. C32⋊C9C3×C37⋊C3semidirect
194. C32⋊C9C3×C37⋊C3semidirect
195. C32⋊C9C3×C37⋊C3semidirect
196. C37#10624894semidirect

Maximal subgroups

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

subgroupindex
C3×#156885323
C3×#156885313
C3×H(3,𝔽3)×C37⋊C33
#225825653
#225825643
#225825633
#225825623
#225825533
#225825523
#225825513
C3×#156885023
C32⋊C9×C37⋊C33
C37×#68933
#1062489437