Finite Group Databasebeta

← C3×#22582557 #41007541

Subgroups — C3×#22582557 #41007541

Order 26973.

1328 conjugacy classes of subgroups; only the 328 normal subgroups are shown.

normal subgroups only

This group has 1328 conjugacy classes of subgroups — too many to draw. Only its 328 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×#11758608C311yesmaximal, characteristic, complemented
2C32×#11758605C311yesmaximal, characteristic, complemented
3C32×#11758606C311yesmaximal, characteristic, complemented
4C32×C32⋊(C37⋊C3)C311yesmaximal, characteristic
5C32×C9×C37⋊C3C311yesmaximal, characteristic, complemented
6#22582557C3127yesmaximal, direct product factor
7C3×C32⋊#66524C311yesmaximal, characteristic, complemented
8C3×#15688502C311yesmaximal, characteristic, complemented
9C3×#15688502C311yesmaximal, characteristic, complemented
10C3×#15688502C311yesmaximal, characteristic, complemented
11C3×#15688502C311yesmaximal, characteristic, complemented
12C3×#15688502C311yesmaximal, characteristic, complemented
13C3×#15688502C311yesmaximal, characteristic, complemented
14C3×C37×C32⋊C9C311yesmaximal, characteristic, complemented
15C33×C37⋊C3C3211yescharacteristic
16C33×C37⋊C3C3211yescharacteristic
17C33×C37⋊C3C3211yescharacteristic
18C3×#11758608—39no—
19C3×#11758608C3219yescomplemented
20C3×#11758605—39no—
21C3×#11758605C3219yescomplemented
22C3×#11758606—39no—
23C3×#11758606C3219yescomplemented
24C3×C32⋊(C37⋊C3)—39no—
25C3×C32⋊(C37⋊C3)C3219yes—
26C32×#66524C3211yescharacteristic
27C32×#66524C3211yescharacteristic
28C32×#66524C3211yescharacteristic
29C32×#66524C3211yescharacteristic
30C32×#66524C3211yescharacteristic
31C32×#66524C3211yescharacteristic
32C3×C9×C37⋊C3—39no—
33C3×C9×C37⋊C3C3219yescomplemented
34C32⋊#66524C3219yescomplemented
35#15688502C3219yescomplemented
36#15688502C3219yescomplemented
37#15688502C3219yescomplemented
38#15688502C3219yescomplemented
39#15688502C3219yescomplemented
40#15688502C3219yescomplemented
41C34×C37C3211yescharacteristic
42C32×C9×C37C3211yescharacteristic
43C32×C9×C37C3211yescharacteristic
44C32×C9×C37C3211yescharacteristic
45C37×C32⋊C9C3219yescomplemented
46C33×C37—33no—
47C33×C37—33no—
48C33×C37—33no—
49C33×C37—33no—
50C33×C37C3313yes—
51C33×C37C3311yescharacteristic
52C32×C37⋊C3—33no—
53C32×C37⋊C3—33no—
54C32×C37⋊C3—33no—
55C32×C37⋊C3—33no—
56C32×C37⋊C3—33no—
57C32×C37⋊C3—33no—
58C32×C37⋊C3—33no—
59C32×C37⋊C3—33no—
60C32×C37⋊C3—33no—
61C32×C37⋊C3—33no—
62C32×C37⋊C3—33no—
63C32×C37⋊C33−1+219yescomplemented
64C32×C37⋊C3C3313yes—
65C32×C37⋊C3C3313yes—
66C32×C37⋊C3C3313yes—
67C3×C9×C37—33no—
68C3×C9×C37—33no—
69C3×C9×C37—33no—
70C3×C9×C37C3313yes—
71C3×C9×C37C3313yes—
72C3×C9×C37C3313yes—
73C32⋊(C37⋊C3)—327no—
74#11758608—327no—
75C3×#66524—33no—
76C3×#66524—33no—
77C3×#66524—33no—
78C3×#66524—33no—
79C3×#66524—33no—
80C3×#66524H(3,𝔽3)19yescomplemented
81C3×#66524C3313yes—
82C3×#66524C3313yes—
83C3×#66524C3313yes—
84C3×#66524C3313yes—
85C3×#66524C3313yes—
86C3×#66524C3313yes—
87#11758606—327no—
88#11758605—327no—
89C9×C37⋊C3—327no—
90C3×C9⋊H(3,𝔽3)—371nomaximal, Sylow
91C32×C37—33no—
92C32×C37—33no—
93C32×C37—33no—
94C32×C37—33no—
95C32×C37—99no—
96C32×C37C3×H(3,𝔽3)13yes—
97C32×C37C3×3−1+213yes—
98C32×C37#73713yes—
99C32×C37#73713yes—
100C32×C37C3411yescharacteristic
101C3×C37⋊C3—33no—
102C3×C37⋊C3—33no—
103C3×C37⋊C3—33no—
104C3×C37⋊C3—327no—
105C3×C37⋊C3—33no—
106C3×C37⋊C3—33no—
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—99no—
114C3×C37⋊C3—99no—
115C3×C37⋊C3C3×3−1+219yescomplemented
116C9×C37—33no—
117C9×C37—33no—
118C9×C37—33no—
119#66524—33no—
120#66524—33no—
121#66524—33no—
122#66524—33no—
123#66524—33no—
124#66524C3×H(3,𝔽3)19yescomplemented
125C32×3−1+2—371no—
126C32×3−1+2—371no—
127C32×3−1+2—371no—
128C32×H(3,𝔽3)—371no—
129C33×C9—371no—
130C9⋊H(3,𝔽3)—3727no—
131C3×C32⋊C9—371no—
132C3×C32⋊C9—371no—
133C3×C32⋊C9—371no—
134C3×C32⋊C9—371no—
135C3×C32⋊C9—371no—
136C3×C32⋊C9—371no—
137C3×C32⋊C9—371no—
138C3×C32⋊C9C37⋊C311yescharacteristic, complemented
139C3×C37—93no—
140C3×C37C9⋊H(3,𝔽3)19yescomplemented
141C3×C37C32×H(3,𝔽3)11yescharacteristic
142C3×C37C32×3−1+211yescharacteristic
143C3×C37C3×#73711yescharacteristic
144C3×C37C3×#73711yescharacteristic
145C37⋊C3—39no—
146C37⋊C3—93no—
147C37⋊C3—93no—
148C34—371no—
149C34—371no—
150C34—371no—
151C34C3×C37⋊C311yescharacteristic
152C3×3−1+2—379no—
153C3×3−1+2—379no—
154C3×3−1+2—379no—
155C3×3−1+2—1119no—
156C3×3−1+2—1119no—
157C3×3−1+2—1119no—
158C3×H(3,𝔽3)—379no—
159C3×H(3,𝔽3)—1119no—
160C32×C9—371no—
161C32×C9—371no—
162C32×C9—371no—
163C32×C9—371no—
164C32×C9—371no—
165C32×C9—371no—
166C32×C9—379no—
167C32×C9—1119no—
168C32×C9C3×C37⋊C311yescharacteristic
169C32×C9C3×C37⋊C311yescharacteristic
170C32×C9C3×C37⋊C311yescharacteristic
171C32⋊C9—379no—
172C32⋊C9—379no—
173C32⋊C9—379no—
174C32⋊C9—379no—
175C32⋊C9—379no—
176C32⋊C9—379no—
177C32⋊C9—379no—
178C32⋊C9C3×C37⋊C319yescomplemented
179C37C3×C9⋊H(3,𝔽3)11yescharacteristic, complemented, Sylow
180C33—33no—
181C33—33no—
182C33—33no—
183C33—33no—
184C33—379no—
185C33—373no—
186C33—373no—
187C33—373no—
188C33—1113no—
189C33—1113no—
190C33—1113no—
191C33—1113no—
192C33—1113no—
193C33—1113no—
194C33—1113no—
195C33—1113no—
196C33—1113no—
197C33—1113no—
198C33—1113no—
199C33C32×C37⋊C313yes—
200C33C32×C37⋊C311yescharacteristic
2013−1+2—11127no—
2023−1+2—11127no—
2033−1+2—11127no—
204H(3,𝔽3)—11127no—
205C3×C9—33no—
206C3×C9—33no—
207C3×C9—33no—
208C3×C9—379no—
209C3×C9—373no—
210C3×C9—373no—
211C3×C9—373no—
212C3×C9—373no—
213C3×C9—373no—
214C3×C9—373no—
215C3×C9—11127no—
216C3×C9—1113no—
217C3×C9—1113no—
218C3×C9—1113no—
219C3×C9—1113no—
220C3×C9—1113no—
221C3×C9C32×C37⋊C313yes—
222C3×C9C32×C37⋊C313yes—
223C3×C9C32×C37⋊C313yes—
224C32—33no—
225C32—33no—
226C32—33no—
227C32—33no—
228C32—99no—
229C32—379no—
230C32—3339no—
231C32—3339no—
232C32—1113no—
233C32—1113no—
234C32—1113no—
235C32—1113no—
236C32—1113no—
237C32—1113no—
238C32—1113no—
239C32—1113no—
240C32—11127no—
241C32—1113no—
242C32—1113no—
243C32—1113no—
244C32H(3,𝔽3)×C37⋊C313yes—
245C323−1+2×C37⋊C313yes—
246C32#1568853113yes—
247C32#1568853213yes—
248C32C33×C37⋊C311yescharacteristic
249C9—33no—
250C9—33no—
251C9—33no—
252C9—379no—
253C9—1113no—
254C9—1113no—
255C9—1113no—
256C9—1113no—
257C9—1113no—
258C3—93no—
259C3—3333no—
260C3—3333no—
261C3—1119no—
262C3#2258255719yesdirect product factor
263C3C3×H(3,𝔽3)×C37⋊C311yescharacteristic
264C3C3×3−1+2×C37⋊C311yescharacteristic
265C3C3×#1568853111yescharacteristic
266C3C3×#1568853211yescharacteristic
subgroupG/H
centerC33C32×C37⋊C3
commutator, cosocleC32×C37C34
FrattiniC32C33×C37⋊C3
FittingC3×C37×C32⋊C9C3
socleC33×C37C33
prime radicalC32×C32⋊(C37⋊C3)C3
hypercenterC3×C32⋊C9C37⋊C3
hypocenterC37C3×C9⋊H(3,𝔽3)
LCSG ▷ C32×C37 ▷ C37
UCSC1 ◁ C33 ◁ C3×C32⋊C9

Complements

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

NH ≅ G/Ntype
1. C32×#11758608C3semidirect
2. C32×#11758605C3semidirect
3. C32×#11758606C3semidirect
5. C32×C9×C37⋊C3C3semidirect
6. #22582557C3direct (N × H)
7. C3×C32⋊#66524C3semidirect
8. C3×#15688502C3semidirect
9. C3×#15688502C3semidirect
10. C3×#15688502C3semidirect
11. C3×#15688502C3semidirect
12. C3×#15688502C3semidirect
13. C3×#15688502C3semidirect
14. C3×C37×C32⋊C9C3semidirect
19. C3×#11758608C32semidirect
21. C3×#11758605C32semidirect
23. C3×#11758606C32semidirect
33. C3×C9×C37⋊C3C32semidirect
34. C32⋊#66524C32semidirect
35. #15688502C32semidirect
36. #15688502C32semidirect
37. #15688502C32semidirect
38. #15688502C32semidirect
39. #15688502C32semidirect
40. #15688502C32semidirect
45. C37×C32⋊C9C32semidirect
63. C32×C37⋊C33−1+2semidirect
80. C3×#66524H(3,𝔽3)semidirect
115. C3×C37⋊C3C3×3−1+2semidirect
124. #66524C3×H(3,𝔽3)semidirect
138. C3×C32⋊C9C37⋊C3semidirect
140. C3×C37C9⋊H(3,𝔽3)semidirect
178. C32⋊C9C3×C37⋊C3semidirect
179. C37C3×C9⋊H(3,𝔽3)semidirect
262. C3#22582557direct (N × H)

Maximal subgroups

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

subgroupindex
C32×#117586083
C32×#117586053
C32×#117586063
C32×C32⋊(C37⋊C3)3
C32×C9×C37⋊C33
#225825573
C3×C32⋊#665243
C3×#156885023
C3×C37×C32⋊C93
C3×C9⋊H(3,𝔽3)37