Finite Group Databasebeta

← #36381340

Subgroups — #36381340

Order 22599.

488 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 61 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#21419438C311yesmaximal, characteristic, complemented
2#21419405C311yesmaximal, characteristic, complemented
3#21419402C311yesmaximal, characteristic, complemented
4#21419402C311yesmaximal, characteristic, complemented
5#21419401C311yesmaximal, characteristic, complemented
6#21419397C311yesmaximal, characteristic, complemented
7#21419396C311yesmaximal, characteristic, complemented
8#21419396C311yesmaximal, characteristic, complemented
9C3×#14578895C311yesmaximal, characteristic, complemented
10C3×#14578893C311yesmaximal, characteristic, complemented
11#21419309C311yesmaximal, characteristic, complemented
12(C3×C31⋊C3)⋊H(3,𝔽3)C311yesmaximal, characteristic, complemented
13C31×#6909C311yesmaximal, characteristic, complemented
14C33×C31⋊C3C3211yescharacteristic
15#14578911—33no—
16#14578911—33no—
17#14578911—33no—
18C3×#11719689—33no—
19C3×#11719689C3211yescharacteristic
20C3×#11719689C3211yescharacteristic
21C3×#11719689C3211yescharacteristic
22C3×#11719689C3211yescharacteristic
23C3×#11719689C3211yescharacteristic
24C3×C32⋊(C31⋊C3)—33no—
25C3×C32⋊(C31⋊C3)C3211yescharacteristic
26C3×C32⋊(C31⋊C3)C3211yescharacteristic
27C3×C32⋊(C31⋊C3)C3211yescharacteristic
28#14578899—33no—
29#14578898—33no—
30#14578898—33no—
31#14578897—33no—
32#14578895—33no—
33#14578895—33no—
34#14578894—33no—
35#14578894—33no—
36#14578893—33no—
37(C3×C31⋊C3)⋊C9—33no—
38C3×3−1+2×C31C3211yescharacteristic
39C3×3−1+2×C31C3211yescharacteristic
40C3×H(3,𝔽3)×C31C3211yescharacteristic
41C3×H(3,𝔽3)×C31C3211yescharacteristic
42C31×AΣL(1,𝔽27)—33no—
43C33×C31—31no—
44C33×C31H(3,𝔽3)13yes—
45C33×C31C3311yescharacteristic
46C32×C31⋊C3—31no—
47C32×C31⋊C3—31no—
48C32×C31⋊C3—31no—
49C32×C31⋊C3—31no—
50C32×C31⋊C3—31no—
51C32×C31⋊C3—31no—
52C32×C31⋊C3—31no—
53C32×C31⋊C3—93no—
543−1+2×C31—31no—
553−1+2×C31—31no—
563−1+2×C31—31no—
573−1+2×C31—31no—
583−1+2×C31—31no—
593−1+2×C31—31no—
60H(3,𝔽3)×C31—31no—
61H(3,𝔽3)×C31—31no—
62H(3,𝔽3)×C31—31no—
63H(3,𝔽3)×C31—31no—
64H(3,𝔽3)×C31—31no—
65H(3,𝔽3)×C31—31no—
66C3×C9×C31—31no—
67C3×C9×C31—31no—
68C32⋊(C31⋊C3)—31no—
69C32⋊(C31⋊C3)—31no—
70C32⋊(C31⋊C3)—31no—
71C32⋊(C31⋊C3)—31no—
72C32⋊(C31⋊C3)—31no—
73C32⋊(C31⋊C3)—31no—
74C32⋊(C31⋊C3)—31no—
75C32⋊(C31⋊C3)—31no—
76C32⋊(C31⋊C3)—31no—
77C32⋊(C31⋊C3)—99no—
78#11719689—318no—
79#11719689—31no—
80#11719689—31no—
81#11719689—31no—
82#11719689—31no—
83#11719689—31no—
84#11719689—31no—
85#11719689—31no—
86#11719689—31no—
87#11719689—31no—
88#11719689—31no—
89#11719689—31no—
90#11719689—31no—
91#11719689—31no—
92#11719689H(3,𝔽3)13yescomplemented
93#11719689H(3,𝔽3)13yescomplemented
94C3×C31⋊C9—39no—
95C3×C31⋊C9—31no—
96C3×C31⋊C9—31no—
97C3×C31⋊C9—31no—
98C3×C31⋊C9—31no—
99C3×C31⋊C9H(3,𝔽3)13yescomplemented
100#10624749—311nomaximal, Sylow
101C32×C31—33no—
102C32×C31—91no—
103C32×C31—93no—
104C32×C31—91no—
105C32×C31—271no—
106C32×C31C3×H(3,𝔽3)11yescharacteristic
107C32×C31C3×H(3,𝔽3)11yescharacteristic
108C32×C31C3×H(3,𝔽3)11yescharacteristic
109C32×C31C3×H(3,𝔽3)11yescharacteristic
110C3×C31⋊C3—33no—
111C3×C31⋊C3—33no—
112C3×C31⋊C3—92no—
113C3×C31⋊C3—92no—
114C3×C31⋊C3—91no—
115C3×C31⋊C3—93no—
116C3×C31⋊C3—93no—
117C3×C31⋊C3—93no—
118C3×C31⋊C3—93no—
119C3×C31⋊C3—91no—
120C3×C31⋊C3—91no—
121C3×C31⋊C3—91no—
122C3×C31⋊C3—271no—
123C9×C31—91no—
124C9×C31—91no—
125C31⋊C9—33no—
126C31⋊C9—33no—
127C31⋊C9—33no—
128C31⋊C9—91no—
129C31⋊C9—91no—
1303−1+4—311no—
131#6911—311no—
132#6910—311no—
133#6910—311no—
134#6910—311no—
135#6909—311no—
136#6909—311no—
137#6909—311no—
138#6909C31⋊C311yescharacteristic, complemented
139C3×AΣL(1,𝔽27)—311no—
140C3×AΣL(1,𝔽27)—311no—
141#6891—311no—
142C32⋊H(3,𝔽3)—311no—
143C3×C31—31no—
144C3×C31—31no—
145C3×C31—31no—
146C3×C31—31no—
147C3×C31—93no—
148C3×C31—271no—
149C3×C31C32⋊H(3,𝔽3)11yescharacteristic
150C31⋊C3—93no—
151C31⋊C3—271no—
152C31⋊C3—271no—
153C31⋊C3—271no—
154C34—311no—
155#737—933no—
156#737—933no—
157#737—933no—
158C3×3−1+2—311no—
159C3×3−1+2—311no—
160C3×3−1+2—311no—
161C3×3−1+2—311no—
162C3×3−1+2—311no—
163C3×3−1+2—933no—
164C3×3−1+2C3×C31⋊C311yescharacteristic
165C3×3−1+2C3×C31⋊C311yescharacteristic
166C3×H(3,𝔽3)—311no—
167C3×H(3,𝔽3)—311no—
168C3×H(3,𝔽3)—311no—
169C3×H(3,𝔽3)—933no—
170C3×H(3,𝔽3)C3×C31⋊C311yescharacteristic
171C3×H(3,𝔽3)C3×C31⋊C311yescharacteristic
172#732—933no—
173#731—933no—
174#731—933no—
175#731—933no—
176AΣL(1,𝔽27)—33no—
177AΣL(1,𝔽27)—933no—
178AΣL(1,𝔽27)—933no—
179AΣL(1,𝔽27)—933no—
180AΣL(1,𝔽27)—933no—
181AΣL(1,𝔽27)—933no—
182C32⋊C9—933no—
183C31#1062474911yescharacteristic, complemented, Sylow
184C33—31no—
185C33—2793no—
186C33—931no—
187C33—931no—
188C33—931no—
189C33—931no—
190C33—931no—
191C33—931no—
192C33—931no—
193C33C32⋊(C31⋊C3)13yes—
194C33C32×C31⋊C311yescharacteristic
1953−1+2—31no—
1963−1+2—31no—
1973−1+2—31no—
1983−1+2—31no—
1993−1+2—31no—
2003−1+2—31no—
2013−1+2—313no—
2023−1+2—313no—
2033−1+2—931no—
2043−1+2—931no—
2053−1+2—931no—
2063−1+2—931no—
2073−1+2—931no—
2083−1+2—931no—
2093−1+2—9318no—
2103−1+2—931no—
2113−1+2—931no—
2123−1+2—931no—
2133−1+2—931no—
2143−1+2—931no—
2153−1+2—931no—
2163−1+2—931no—
217H(3,𝔽3)—31no—
218H(3,𝔽3)—31no—
219H(3,𝔽3)—31no—
220H(3,𝔽3)—31no—
221H(3,𝔽3)—31no—
222H(3,𝔽3)—31no—
223H(3,𝔽3)—2799no—
224H(3,𝔽3)—931no—
225H(3,𝔽3)—931no—
226H(3,𝔽3)—931no—
227H(3,𝔽3)—931no—
228H(3,𝔽3)—931no—
229H(3,𝔽3)—931no—
230H(3,𝔽3)—931no—
231H(3,𝔽3)—931no—
232H(3,𝔽3)—931no—
233C3×C9—31no—
234C3×C9—31no—
235C3×C9—313no—
236C3×C9—931no—
237C3×C9—931no—
238C3×C9—939no—
239C3×C9—931no—
240C3×C9—931no—
241C32—33no—
242C32—91no—
243C32—93no—
244C32—91no—
245C32—2793no—
246C32—2793no—
247C32—2792no—
248C32—2792no—
249C32—2793no—
250C32—2793no—
251C32—2791no—
252C32—2791no—
253C32—2791no—
254C32—2791no—
255C32—271no—
256C32—8371no—
257C32—933no—
258C32—933no—
259C32H(3,𝔽3)×C31⋊C311yescharacteristic
260C32H(3,𝔽3)×C31⋊C311yescharacteristic
261C32H(3,𝔽3)×C31⋊C311yescharacteristic
262C32C3×C32⋊(C31⋊C3)11yescharacteristic
263C9—91no—
264C9—91no—
265C9—2791no—
266C9—2791no—
267C9—933no—
268C9—933no—
269C9—933no—
270C3—31no—
271C3—31no—
272C3—31no—
273C3—31no—
274C3—93no—
275C3—2793no—
276C3—271no—
277C3—8371no—
278C3—8371no—
279C3—8371no—
280C3(C3×C31⋊C3)⋊H(3,𝔽3)11yescharacteristic
subgroupG/H
centerC3(C3×C31⋊C3)⋊H(3,𝔽3)
commutator, cosocleC33×C31C33
FrattiniC33C32×C31⋊C3
FittingC31×#6909C3
socleC3×C31C32⋊H(3,𝔽3)
hypercenter#6909C31⋊C3
hypocenterC31#10624749
LCSG ▷ C33×C31 ▷ C3×C31 ▷ C31
UCSC1 ◁ C3 ◁ C33 ◁ #6909

Complements

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

NH ≅ G/Ntype
1. #21419438C3semidirect
2. #21419405C3semidirect
3. #21419402C3semidirect
4. #21419402C3semidirect
5. #21419401C3semidirect
6. #21419397C3semidirect
7. #21419396C3semidirect
8. #21419396C3semidirect
9. C3×#14578895C3semidirect
10. C3×#14578893C3semidirect
11. #21419309C3semidirect
12. (C3×C31⋊C3)⋊H(3,𝔽3)C3semidirect
13. C31×#6909C3semidirect
92. #11719689H(3,𝔽3)semidirect
93. #11719689H(3,𝔽3)semidirect
99. C3×C31⋊C9H(3,𝔽3)semidirect
138. #6909C31⋊C3semidirect
183. C31#10624749semidirect

Maximal subgroups

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

subgroupindex
#214194383
#214194053
#214194023
#214194013
#214193973
#214193963
C3×#145788953
C3×#145788933
#214193093
(C3×C31⋊C3)⋊H(3,𝔽3)3
C31×#69093
#1062474931