Finite Group Databasebeta

← #72247271

Subgroups — #72247271

Order 4860.

316 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 53 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
1D5×#6909C211yesmaximal, characteristic
2C3×3−1+2⋊AGL(1,𝔽5)—31nomaximal
3C3×3−1+2⋊AGL(1,𝔽5)—31nomaximal
4C3×(C3⋊ΓL(1,𝔽16))⋊C3—31nomaximal
5C3×(C3⋊ΓL(1,𝔽16))⋊C3—31nomaximal
6#13108191C311yesmaximal, characteristic, complemented
7C5×#6909C411yescharacteristic, complemented, Hall
8#66330⋊C3—51nomaximal, Hall
9C3×D5×3−1+2D311yescharacteristic
10C3×D5×3−1+2D311yescharacteristic
11C3×D5×H(3,𝔽3)D311yescharacteristic
12C3×D5×H(3,𝔽3)D311yescharacteristic
13D5×AΣL(1,𝔽27)—21no—
14D5×AΣL(1,𝔽27)—21no—
15D5×AΣL(1,𝔽27)—21no—
16D5×AΣL(1,𝔽27)—21no—
17D5×AΣL(1,𝔽27)C2×C311yescharacteristic
18C3×C3⋊ΓL(1,𝔽16)—31no—
19C3×C3⋊ΓL(1,𝔽16)—31no—
20C32×ΓL(1,𝔽16)—91no—
213−1+2⋊AGL(1,𝔽5)—31no—
223−1+2⋊AGL(1,𝔽5)—31no—
233−1+2⋊AGL(1,𝔽5)—31no—
243−1+2⋊AGL(1,𝔽5)—31no—
253−1+2⋊AGL(1,𝔽5)—31no—
263−1+2⋊AGL(1,𝔽5)—31no—
27(C3⋊ΓL(1,𝔽16))⋊C3—31no—
28(C3⋊ΓL(1,𝔽16))⋊C3—31no—
29(C3⋊ΓL(1,𝔽16))⋊C3—31no—
30(C3⋊ΓL(1,𝔽16))⋊C3—31no—
31(C3⋊ΓL(1,𝔽16))⋊C3—31no—
32(C3⋊ΓL(1,𝔽16))⋊C3—31no—
33C3×C9⋊AGL(1,𝔽5)—31no—
34C3×C9⋊AGL(1,𝔽5)—31no—
35C2×#6909—51no—
36C3×C5×3−1+2Dic311yescharacteristic, complemented
37C3×C5×3−1+2Dic311yescharacteristic, complemented
38C3×C5×H(3,𝔽3)Dic311yescharacteristic, complemented
39C3×C5×H(3,𝔽3)Dic311yescharacteristic, complemented
40C5×AΣL(1,𝔽27)—21no—
41C5×AΣL(1,𝔽27)—21no—
42C5×AΣL(1,𝔽27)—21no—
43C5×AΣL(1,𝔽27)—21no—
44C5×AΣL(1,𝔽27)C3×C411yescharacteristic, complemented
45C3×3−1+2⋊C4—151no—
46C3×3−1+2⋊C4—151no—
47C3×(C3⋊Dic3)⋊C3—151no—
48C3×(C3⋊Dic3)⋊C3—151no—
49#66330—51no—
50C33×D5—21no—
51C33×D5—61no—
52C33×D5C3×D311yescharacteristic
53C33×D5C3⋊D311yescharacteristic
54D5×3−1+2—21no—
55D5×3−1+2—21no—
56D5×3−1+2—31no—
57D5×3−1+2—31no—
58D5×3−1+2—61no—
59D5×3−1+2—61no—
60D5×3−1+2C3×D311yescharacteristic
61D5×3−1+2C3×D311yescharacteristic
62D5×H(3,𝔽3)—21no—
63D5×H(3,𝔽3)—31no—
64D5×H(3,𝔽3)—61no—
65D5×H(3,𝔽3)—61no—
66D5×H(3,𝔽3)—61no—
67D5×H(3,𝔽3)—61no—
68D5×H(3,𝔽3)C3×D311yescharacteristic
69C3×C9×D5—61no—
70C3×C9×D5—61no—
71#6909AGL(1,𝔽5)11yescharacteristic, complemented, Sylow
72C3⋊ΓL(1,𝔽16)—31no—
73C3⋊ΓL(1,𝔽16)—31no—
74C3×ΓL(1,𝔽16)—91no—
75C3×ΓL(1,𝔽16)—91no—
76C3×ΓL(1,𝔽16)—91no—
77C3×ΓL(1,𝔽16)—91no—
78C3×ΓL(1,𝔽16)—271no—
79C3×ΓL(1,𝔽16)—271no—
80C32×AGL(1,𝔽5)—271no—
81C9⋊AGL(1,𝔽5)—31no—
82C9⋊AGL(1,𝔽5)—31no—
83C2×C3×3−1+2—51no—
84C2×C3×3−1+2—51no—
85C2×C3×H(3,𝔽3)—51no—
86C2×C3×H(3,𝔽3)—51no—
87C2×AΣL(1,𝔽27)—51no—
88C2×AΣL(1,𝔽27)—101no—
89C2×AΣL(1,𝔽27)—101no—
90C2×AΣL(1,𝔽27)—101no—
91C2×AΣL(1,𝔽27)—101no—
92C33×C5—21no—
93C33×C5—61no—
94C33×C5C3×Dic311yescharacteristic, complemented
95C33×C5C3⋊Dic311yescharacteristic
96C5×3−1+2—21no—
97C5×3−1+2—21no—
98C5×3−1+2—31no—
99C5×3−1+2—31no—
100C5×3−1+2—61no—
101C5×3−1+2—61no—
102C5×3−1+2C3×Dic311yescharacteristic, complemented
103C5×3−1+2C3×Dic311yescharacteristic, complemented
104C5×H(3,𝔽3)—21no—
105C5×H(3,𝔽3)—31no—
106C5×H(3,𝔽3)—61no—
107C5×H(3,𝔽3)—61no—
108C5×H(3,𝔽3)—61no—
109C5×H(3,𝔽3)—61no—
110C5×H(3,𝔽3)C3×Dic311yescharacteristic
111C3×C5×C9—61no—
112C3×C5×C9—61no—
113C3×C3⋊Dic3—151no—
114C3×C3⋊Dic3—151no—
115C32×Dic3—451no—
1163−1+2⋊C4—151no—
1173−1+2⋊C4—151no—
1183−1+2⋊C4—151no—
1193−1+2⋊C4—151no—
1203−1+2⋊C4—151no—
1213−1+2⋊C4—151no—
122(C3⋊Dic3)⋊C3—151no—
123(C3⋊Dic3)⋊C3—151no—
124(C3⋊Dic3)⋊C3—151no—
125(C3⋊Dic3)⋊C3—151no—
126(C3⋊Dic3)⋊C3—151no—
127(C3⋊Dic3)⋊C3—151no—
128C3×Dic9—151no—
129C3×Dic9—151no—
130C32×D5—31no—
131C32×D5—31no—
132C32×D5—61no—
133C32×D5—61no—
134C32×D5—181no—
135C32×D5—181no—
136C32×D5—181no—
137C32×D5—181no—
138C32×D5—181no—
139C32×D5#27211yescharacteristic
140C32×D5#27211yescharacteristic
141C32×D5#27211yescharacteristic
142C32×D5C3×C3⋊D311yescharacteristic
143C9×D5—61no—
144C9×D5—61no—
145C9×D5—61no—
146C9×D5—61no—
147C3×3−1+2ΓL(1,𝔽16)11yescharacteristic, complemented
148C3×3−1+2ΓL(1,𝔽16)11yescharacteristic, complemented
149C3×H(3,𝔽3)ΓL(1,𝔽16)11yescharacteristic, complemented
150C3×H(3,𝔽3)ΓL(1,𝔽16)11yescharacteristic, complemented
151AΣL(1,𝔽27)—21no—
152AΣL(1,𝔽27)—21no—
153AΣL(1,𝔽27)—21no—
154AΣL(1,𝔽27)—21no—
155AΣL(1,𝔽27)C3×AGL(1,𝔽5)11yescharacteristic, complemented
156ΓL(1,𝔽16)—91no—
157ΓL(1,𝔽16)—271no—
158ΓL(1,𝔽16)—271no—
159C3×AGL(1,𝔽5)—271no—
160C3×AGL(1,𝔽5)—271no—
161C3×AGL(1,𝔽5)—271no—
162C3×AGL(1,𝔽5)—271no—
163C2×C33—51no—
164C2×C33—51no—
165C2×C33—101no—
166C2×C33—301no—
167C2×3−1+2—51no—
168C2×3−1+2—51no—
169C2×3−1+2—101no—
170C2×3−1+2—101no—
171C2×3−1+2—151no—
172C2×3−1+2—301no—
173C2×3−1+2—301no—
174C2×3−1+2—301no—
175C2×H(3,𝔽3)—51no—
176C2×H(3,𝔽3)—101no—
177C2×H(3,𝔽3)—151no—
178C2×H(3,𝔽3)—301no—
179C2×H(3,𝔽3)—301no—
180C2×H(3,𝔽3)—301no—
181C2×H(3,𝔽3)—301no—
182C2×C3×C9—151no—
183C2×C3×C9—151no—
184C32×C5—31no—
185C32×C5—31no—
186C32×C5—61no—
187C32×C5—61no—
188C32×C5—181no—
189C32×C5—181no—
190C32×C5—181no—
191C32×C5—181no—
192C32×C5—181no—
193C32×C5#108411yescharacteristic
194C32×C5#108411yescharacteristic
195C32×C5#108411yescharacteristic
196C32×C5C3×C3⋊Dic311yescharacteristic
197C5×C9—61no—
198C5×C9—61no—
199C5×C9—61no—
200C5×C9—61no—
201C32×C4—1351no—
202C3⋊Dic3—151no—
203C3⋊Dic3—151no—
204C3×Dic3—451no—
205C3×Dic3—451no—
206C3×Dic3—451no—
207C3×Dic3—451no—
208C3×Dic3—1351no—
209C3×Dic3—1351no—
210Dic9—151no—
211Dic9—151no—
212C3×D5—31no—
213C3×D5—31no—
214C3×D5—61no—
215C3×D5—61no—
216C3×D5—181no—
217C3×D5—181no—
218C3×D5—181no—
219C3×D5—181no—
220C3×D5C3×#27211yescharacteristic
221C33—21no—
222C33—61no—
223C33C3×ΓL(1,𝔽16)11yescharacteristic, complemented
224C33C3⋊ΓL(1,𝔽16)11yescharacteristic
2253−1+2—21no—
2263−1+2—21no—
2273−1+2—31no—
2283−1+2—61no—
2293−1+2—61no—
2303−1+2—61no—
2313−1+2C3×ΓL(1,𝔽16)11yescharacteristic, complemented
2323−1+2C3×ΓL(1,𝔽16)11yescharacteristic, complemented
233H(3,𝔽3)—21no—
234H(3,𝔽3)—31no—
235H(3,𝔽3)—61no—
236H(3,𝔽3)—61no—
237H(3,𝔽3)—61no—
238H(3,𝔽3)—61no—
239H(3,𝔽3)C3×ΓL(1,𝔽16)11yescharacteristic
240C3×C9—31no—
241C3×C9—31no—
242AGL(1,𝔽5)—271noHall
243C2×C32—51no—
244C2×C32—51no—
245C2×C32—51no—
246C2×C32—51no—
247C2×C32—151no—
248C2×C32—301no—
249C2×C32—301no—
250C2×C32—301no—
251C2×C32—451no—
252C2×C32—901no—
253C2×C32—901no—
254C2×C32—901no—
255C2×C32—901no—
256C2×C9—151no—
257C2×C9—301no—
258C2×C9—301no—
259C2×C9—301no—
260C3×C5—31no—
261C3×C5—31no—
262C3×C5—61no—
263C3×C5—61no—
264C3×C5—181no—
265C3×C5—181no—
266C3×C5—181no—
267C3×C5—181no—
268C3×C5C3×#108411yescharacteristic
269C3×C4—1351no—
270C3×C4—1351no—
271C3×C4—1351no—
272C3×C4—1351no—
273Dic3—451no—
274Dic3—1351no—
275Dic3—1351no—
276D5#4183⋊C311yescharacteristic
277C32—31no—
278C32—61no—
279C32—61no—
280C32—61no—
281C32—91no—
282C32—181no—
283C32—181no—
284C32—181no—
285C32—181no—
286C32#1058738011yescharacteristic
287C32#1058738011yescharacteristic
288C32#1058738011yescharacteristic
289C32C3×C3⋊ΓL(1,𝔽16)11yescharacteristic
290C9—31no—
291C9—61no—
292C9—61no—
293C9—61no—
294C2×C3—51no—
295C2×C3—301no—
296C2×C3—301no—
297C2×C3—301no—
298C2×C3—301no—
299C2×C3—451no—
300C2×C3—901no—
301C2×C3—901no—
302C2×C3—901no—
303C5#66330⋊C311yescharacteristic, complemented, Sylow
304C4—1351noSylow
305C3—61no—
306C3—61no—
307C3—61no—
308C3—61no—
309C3—91no—
310C3—181no—
311C3—181no—
312C3—181no—
313C3C3×#1058738011yescharacteristic
314C2—51no—
subgroupG/H
center, hypercenterC1G
commutator, hypocenterC5×AΣL(1,𝔽27)C3×C4
FrattiniC32C3×C3⋊ΓL(1,𝔽16)
FittingC5×#6909C4
socleC3×C5C3×#1084
prime radicalD5×#6909C2
cosocleD5×AΣL(1,𝔽27)C2×C3
LCSG ▷ C5×AΣL(1,𝔽27)
UCSC1

Complements

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

NH ≅ G/Ntype
6. #13108191C3semidirect
7. C5×#6909C4semidirect
36. C3×C5×3−1+2Dic3semidirect
37. C3×C5×3−1+2Dic3semidirect
38. C3×C5×H(3,𝔽3)Dic3semidirect
39. C3×C5×H(3,𝔽3)Dic3semidirect
44. C5×AΣL(1,𝔽27)C3×C4semidirect
71. #6909AGL(1,𝔽5)semidirect
94. C33×C5C3×Dic3semidirect
102. C5×3−1+2C3×Dic3semidirect
103. C5×3−1+2C3×Dic3semidirect
147. C3×3−1+2ΓL(1,𝔽16)semidirect
148. C3×3−1+2ΓL(1,𝔽16)semidirect
149. C3×H(3,𝔽3)ΓL(1,𝔽16)semidirect
150. C3×H(3,𝔽3)ΓL(1,𝔽16)semidirect
155. AΣL(1,𝔽27)C3×AGL(1,𝔽5)semidirect
223. C33C3×ΓL(1,𝔽16)semidirect
231. 3−1+2C3×ΓL(1,𝔽16)semidirect
232. 3−1+2C3×ΓL(1,𝔽16)semidirect
303. C5#66330⋊C3semidirect

Maximal subgroups

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

subgroupindex
D5×#69092
C3×3−1+2⋊AGL(1,𝔽5)3
C3×(C3⋊ΓL(1,𝔽16))⋊C33
#131081913
#66330⋊C35