Finite Group Databasebeta

← #41007802

Subgroups — #41007802

Order 26973.

1244 conjugacy classes of subgroups; only the 298 normal subgroups are shown.

normal subgroups only

This group has 1244 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×#11758605C311yesmaximal, characteristic, complemented
2C3×3−1+2×C37⋊C3C313yesmaximal, complemented
3C3×H(3,𝔽3)×C37⋊C3C311yesmaximal, characteristic
4#22582560C312yesmaximal, complemented
5#22582559C311yesmaximal, characteristic, complemented
6#22582557C3112yesmaximal, complemented
7#22582556C312yesmaximal, complemented
8#22582555C316yesmaximal, complemented
9#22582553C312yesmaximal, complemented
10#22582553C316yesmaximal, complemented
11C9×C32⋊(C37⋊C3)C311yesmaximal, characteristic, complemented
12#22582539C312yesmaximal, complemented
13C37×C9⋊H(3,𝔽3)C311yesmaximal, characteristic, complemented
14C33×C37⋊C3C3213yes—
15C33×C37⋊C3C3211yescharacteristic
16C3×#11758608C3216yescomplemented
17C3×#11758605—39no—
18C3×#11758605C3216yes—
19C3×#11758605C3212yes—
20C3×#11758605C3211yescharacteristic
21C3×#11758605C3212yescomplemented
22C3×#11758605C3214yescomplemented
23C3×#11758606C3213yescomplemented
24C3×#11758606C3216yescomplemented
253−1+2×C37⋊C3—327no—
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⋊C3C3216yescomplemented
32C3×C9×C37⋊C3C3213yescomplemented
33C32⋊#66524C3212yescomplemented
34C32⋊#66524C3212yescomplemented
35C32⋊#66524C32112yescomplemented
36#15688502C32112yescomplemented
37#15688502C3216yescomplemented
38#15688502C32112yescomplemented
39#15688502C32112yescomplemented
40#15688502C3216yescomplemented
41C3×3−1+2×C37C3213yescomplemented
42C3×H(3,𝔽3)×C37C3211yescharacteristic
43C32×C9×C37C3211yescharacteristic, complemented
44C37×C32⋊C9C3212yescomplemented
45C37×C32⋊C9C3216yescomplemented
46C33×C37C3313yes—
47C33×C37C3311yescharacteristic
48C32×C37⋊C3—33no—
49C32×C37⋊C3—39no—
50C32×C37⋊C3—33no—
51C32×C37⋊C3—39no—
52C32×C37⋊C3—96no—
53C32×C37⋊C3—92no—
54C32×C37⋊C3C3313yes—
55C32×C37⋊C3C3312yes—
56C32×C37⋊C3C3311yescharacteristic
57C32×C37⋊C3C3313yes—
583−1+2×C37—39no—
59H(3,𝔽3)×C37—33no—
60C3×C9×C37—33no—
61C3×C9×C37C3312yes—
62C3×C9×C37C3316yes—
63C3×C9×C37C3311yescharacteristic
64C32⋊(C37⋊C3)—33no—
65C32⋊(C37⋊C3)—318no—
66C32⋊(C37⋊C3)3−1+219yescomplemented
67#11758608—318no—
68C3×#66524—36no—
69C3×#66524C3316yes—
70C3×#66524C3316yes—
71C3×#66524C3314yes—
72C3×#66524C3312yes—
73#11758606—39no—
74#11758606—318no—
75#11758605—318no—
76#11758605—36no—
77#11758605—36no—
78#11758605—312no—
79#11758605H(3,𝔽3)19yescomplemented
80C9×C37⋊C3—39no—
81C9×C37⋊C3—318no—
82H(3,𝔽3)×3−1+2—371nomaximal, Sylow
83C32×C37—31no—
84C32×C37—32no—
85C32×C37—33no—
86C32×C37—33no—
87C32×C37—36no—
88C32×C37C3×3−1+213yescomplemented
89C32×C37C3411yescharacteristic
90C3×C37⋊C3—33no—
91C3×C37⋊C3—31no—
92C3×C37⋊C3—32no—
93C3×C37⋊C3—36no—
94C3×C37⋊C3—32no—
95C3×C37⋊C3—33no—
96C3×C37⋊C3—33no—
97C3×C37⋊C3—32no—
98C3×C37⋊C3—36no—
99C3×C37⋊C3—34no—
100C3×C37⋊C3—927no—
101C3×C37⋊C3—99no—
102C3×C37⋊C3C3×H(3,𝔽3)13yes—
103C3×C37⋊C3C3×3−1+219yescomplemented
104C9×C37—36no—
105C9×C37—32no—
106C9×C37C3×H(3,𝔽3)13yescomplemented
107#66524—36no—
108#66524—36no—
109#66524—34no—
110#66524C3×H(3,𝔽3)16yescomplemented
111C32×3−1+2—373no—
112C32×3−1+2—371no—
113C32×H(3,𝔽3)—371no—
114C9⋊H(3,𝔽3)—376no—
115C9⋊H(3,𝔽3)—372no—
116C9⋊H(3,𝔽3)—371no—
117C9⋊H(3,𝔽3)—3712no—
118C9⋊H(3,𝔽3)—372no—
119C9⋊H(3,𝔽3)C37⋊C311yescharacteristic, complemented
120#6891—372no—
121#6891—376no—
122C9×H(3,𝔽3)—371no—
123C9×H(3,𝔽3)—372no—
124C3×C37—33no—
125C3×C37—93no—
126C3×C37C32×H(3,𝔽3)11yescharacteristic
127C3×C37C32×3−1+211yescharacteristic
128C3×C373−1+412yes—
129C37⋊C3—33no—
130C37⋊C3—39no—
131C37⋊C3—92no—
132C37⋊C3—93no—
133C34—373no—
134C34—371no—
135C3×3−1+2—376no—
136C3×3−1+2—371no—
137C3×3−1+2—376no—
138C3×3−1+2—373no—
139C3×3−1+2—376no—
140C3×3−1+2—372no—
141C3×3−1+2—374no—
142C3×3−1+2—372no—
143C3×3−1+2—1119no—
144C3×3−1+2—11127no—
145C3×3−1+2C3×C37⋊C313yescomplemented
146C3×H(3,𝔽3)—371no—
147C3×H(3,𝔽3)—376no—
148C3×H(3,𝔽3)—371no—
149C3×H(3,𝔽3)—1119no—
150C3×H(3,𝔽3)C3×C37⋊C311yescharacteristic
151C32×C9—372no—
152C32×C9—376no—
153C32×C9—373no—
154C32×C9C3×C37⋊C311yescharacteristic, complemented
155C32⋊C9—372no—
156C32⋊C9—372no—
157C32⋊C9—3712no—
158C32⋊C9—3712no—
159C32⋊C9—376no—
160C32⋊C9—3712no—
161C32⋊C9—3712no—
162C32⋊C9—376no—
163C32⋊C9C3×C37⋊C312yescomplemented
164C32⋊C9C3×C37⋊C316yescomplemented
165C37H(3,𝔽3)×3−1+211yescharacteristic, complemented, Sylow
166C33—373no—
167C33—371no—
168C33—373no—
169C33—372no—
170C33—3332no—
171C33—3336no—
172C33—1119no—
173C33—1113no—
174C33—1119no—
175C33—1113no—
176C33C32×C37⋊C313yes—
177C33C32×C37⋊C311yescharacteristic
1783−1+2—39no—
1793−1+2—379no—
1803−1+2—11112no—
1813−1+2—11118no—
1823−1+2—1116no—
1833−1+2—11118no—
1843−1+2—11118no—
1853−1+2—1116no—
1863−1+2—1119no—
187H(3,𝔽3)—33no—
188H(3,𝔽3)—379no—
189H(3,𝔽3)—11118no—
190H(3,𝔽3)—1113no—
191C3×C9—33no—
192C3×C9—372no—
193C3×C9—376no—
194C3×C9—376no—
195C3×C9—374no—
196C3×C9—11118no—
197C3×C9—1116no—
198C3×C9—1119no—
199C3×C9C32×C37⋊C316yes—
200C3×C9C32×C37⋊C312yes—
201C3×C9C32×C37⋊C311yescharacteristic
202C32—31no—
203C32—32no—
204C32—33no—
205C32—33no—
206C32—36no—
207C32—373no—
208C32—379no—
209C32—3339no—
210C32—33327no—
211C32—1112no—
212C32—1113no—
213C32—1112no—
214C32—1113no—
215C32—1114no—
216C32—1116no—
217C32—1113no—
218C32—1111no—
219C32—1112no—
220C32—1116no—
221C323−1+2×C37⋊C313yescomplemented
222C32C33×C37⋊C311yescharacteristic
223C9—32no—
224C9—36no—
225C9—376no—
226C9—1116no—
227C9—1114no—
228C9—1116no—
229C9H(3,𝔽3)×C37⋊C313yescomplemented
230C3—33no—
231C3—93no—
232C3—3332no—
233C3—3333no—
234C3—1113no—
235C3—1119no—
236C3C3×H(3,𝔽3)×C37⋊C311yescharacteristic
237C3C3×3−1+2×C37⋊C311yescharacteristic
238C3#2258267812yes—
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
hypocenterC37H(3,𝔽3)×3−1+2
LCSG ▷ C32×C37 ▷ C37
UCSC1 ◁ C32 ◁ C9⋊H(3,𝔽3)

Complements

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

NH ≅ G/Ntype
1. C32×#11758605C3semidirect
2. C3×3−1+2×C37⋊C3C3semidirect
4. #22582560C3semidirect
5. #22582559C3semidirect
6. #22582557C3semidirect
7. #22582556C3semidirect
8. #22582555C3semidirect
9. #22582553C3semidirect
10. #22582553C3semidirect
11. C9×C32⋊(C37⋊C3)C3semidirect
12. #22582539C3semidirect
13. C37×C9⋊H(3,𝔽3)C3semidirect
16. C3×#11758608C32semidirect
21. C3×#11758605C32semidirect
22. C3×#11758605C32semidirect
23. C3×#11758606C32semidirect
24. C3×#11758606C32semidirect
30. C32×#66524C32semidirect
31. C3×C9×C37⋊C3C32semidirect
32. C3×C9×C37⋊C3C32semidirect
33. C32⋊#66524C32semidirect
34. C32⋊#66524C32semidirect
35. C32⋊#66524C32semidirect
36. #15688502C32semidirect
37. #15688502C32semidirect
38. #15688502C32semidirect
39. #15688502C32semidirect
40. #15688502C32semidirect
41. C3×3−1+2×C37C32semidirect
43. C32×C9×C37C32semidirect
44. C37×C32⋊C9C32semidirect
45. C37×C32⋊C9C32semidirect
66. C32⋊(C37⋊C3)3−1+2semidirect
79. #11758605H(3,𝔽3)semidirect
88. C32×C37C3×3−1+2semidirect
103. C3×C37⋊C3C3×3−1+2semidirect
106. C9×C37C3×H(3,𝔽3)semidirect
110. #66524C3×H(3,𝔽3)semidirect
119. C9⋊H(3,𝔽3)C37⋊C3semidirect
145. C3×3−1+2C3×C37⋊C3semidirect
154. C32×C9C3×C37⋊C3semidirect
163. C32⋊C9C3×C37⋊C3semidirect
164. C32⋊C9C3×C37⋊C3semidirect
165. C37H(3,𝔽3)×3−1+2semidirect
221. C323−1+2×C37⋊C3semidirect
229. C9H(3,𝔽3)×C37⋊C3semidirect

Maximal subgroups

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

subgroupindex
C32×#117586053
C3×3−1+2×C37⋊C33
C3×H(3,𝔽3)×C37⋊C33
#225825603
#225825593
#225825573
#225825563
#225825553
#225825533
C9×C32⋊(C37⋊C3)3
#225825393
C37×C9⋊H(3,𝔽3)3
H(3,𝔽3)×3−1+237