Finite Group Databasebeta

← C3×#67⋊AGL(1,𝔽5) #13088659

Subgroups — C3×#67⋊AGL(1,𝔽5) #13088659

Order 1440.

410 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×ΓL(1,𝔽16)C211yesmaximal, characteristic, complemented
2C2×C3×D3×AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
3C3×D5×#67C211yesmaximal, characteristic
4C32×C22⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
5C3×Dic3⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
6C3×(C2×D3)⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
7C3×Dic3×AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
8C3×D4×AGL(1,𝔽5)—31nomaximal
9#67⋊AGL(1,𝔽5)C311yesmaximal, characteristic, direct product factor
10C22×C32×D5C2211yescharacteristic
11C2×C3×D3×D5C2211yescharacteristic
12C2×C3×ΓL(1,𝔽16)—22no—
13C2×C3×ΓL(1,𝔽16)C2211yescharacteristic
14C2×C3×ΓL(1,𝔽16)C2211yescharacteristic
15C2×C32×AGL(1,𝔽5)C2211yescharacteristic
16C2×C32×AGL(1,𝔽5)C2211yescharacteristic
17C3×D3×AGL(1,𝔽5)—22no—
18C3×#1220C411yescharacteristic, complemented
19C3×C5×#67C411yescharacteristic, complemented
20C3×#1202C411yescharacteristic, complemented
21C3×#1201C411yescharacteristic, complemented
22C3×D5×Dic3C2211yescharacteristic
23C3×#67⋊C4—51nomaximal, Hall
24C22×ΓL(1,𝔽16)C2×C311yescharacteristic, complemented
25C22×C3×AGL(1,𝔽5)—31no—
26C22×C3×AGL(1,𝔽5)—31no—
27C2×D3×AGL(1,𝔽5)C2×C311yescharacteristic, complemented
28C3×D4×D5—31no—
29D5×#67C2×C311yescharacteristic
30C3×C22⋊AGL(1,𝔽5)—21no—
31C3×C22⋊AGL(1,𝔽5)—31no—
32C3×C22⋊AGL(1,𝔽5)D311yescharacteristic, complemented
33C3×C22⋊AGL(1,𝔽5)C2×C311yescharacteristic, complemented
34C3×C4⋊AGL(1,𝔽5)—31no—
35C3×C4×AGL(1,𝔽5)—31no—
36Dic3⋊AGL(1,𝔽5)C2×C311yescharacteristic, complemented
37(C2×D3)⋊AGL(1,𝔽5)C2×C311yescharacteristic, complemented
38Dic3×AGL(1,𝔽5)C2×C311yescharacteristic, complemented
39C22×C32×C5C2×C411yescharacteristic, complemented
40C2×C3×D15C2×C411yescharacteristic, complemented
41C2×C3×C5×D3C2×C411yescharacteristic, complemented
42C2×C32×D5—21no—
43C2×C32×D5—21no—
44C2×C32×D5C2×C411yescharacteristic, complemented
45C2×C32×D5C2311yescharacteristic
46C3×D3×D5—21no—
47C3×D3×D5—21no—
48C3×ΓL(1,𝔽16)—21no—
49C3×ΓL(1,𝔽16)D412yescomplemented
50C32×AGL(1,𝔽5)—21no—
51C32×AGL(1,𝔽5)—21no—
52C3×Dic15C2×C411yescharacteristic, complemented
53C3×C5×Dic3C2×C411yescharacteristic, complemented
54D4×AGL(1,𝔽5)—31noHall
55C2×C3×#67—51no—
56C22×C3×Dic3—51no—
57C2×C3×C4×D3—51no—
58C32×C22⋊C4—51no—
59C3×(C2×D3)⋊C4—51no—
60C3×Dic3⋊C4—51no—
61C3×C4×Dic3—51no—
62C22×C3×D5—21no—
63C22×C3×D5—31no—
64C22×C3×D5C2×D311yescharacteristic
65C22×C3×D5C22×C311yescharacteristic
66C2×D3×D5C22×C311yescharacteristic
67C2×ΓL(1,𝔽16)—22no—
68C2×ΓL(1,𝔽16)C22×C311yescharacteristic
69C2×ΓL(1,𝔽16)C22×C311yescharacteristic
70C2×C3×AGL(1,𝔽5)—21no—
71C2×C3×AGL(1,𝔽5)—21no—
72C2×C3×AGL(1,𝔽5)—31no—
73C2×C3×AGL(1,𝔽5)—31no—
74C2×C3×AGL(1,𝔽5)—62no—
75C2×C3×AGL(1,𝔽5)—62no—
76C2×C3×AGL(1,𝔽5)C2×D311yescharacteristic
77C2×C3×AGL(1,𝔽5)C2×D311yescharacteristic
78C2×C3×AGL(1,𝔽5)C22×C311yescharacteristic
79C2×C3×AGL(1,𝔽5)C22×C311yescharacteristic
80D3×AGL(1,𝔽5)—22no—
81C3×C5×D4—31no—
82#1220C3×C411yescharacteristic, complemented
83C5×#67C3×C411yescharacteristic, complemented
84C3×C22⋊D5—31no—
85C3×C22⋊D5—31no—
86C3×D20—31no—
87C3×C4×D5—31no—
88#1202C3×C411yescharacteristic, complemented
89#1201C3×C411yescharacteristic, complemented
90D5×Dic3C22×C311yescharacteristic
91C3×C4×D4—151no—
92#67⋊C4—51no—
93C2×C32×C5—21no—
94C2×C32×C5C22×C411yescharacteristic
95C3×D15—21no—
96C3×C5×D3—21no—
97C32×D5—21no—
98C32×D5C2×D411yescharacteristic
99C32×D5#4111yescharacteristic
100C22×AGL(1,𝔽5)—31no—
101C22×AGL(1,𝔽5)—31no—
102D4×D5—31no—
103C22⋊AGL(1,𝔽5)—31no—
104C22⋊AGL(1,𝔽5)C3×D311yescharacteristic, complemented
105C4⋊AGL(1,𝔽5)—31no—
106C4×AGL(1,𝔽5)—31no—
107C23×C32—51no—
108C22×C3×D3—51no—
109C2×C32×C4—51no—
110C2×C32×C4—51no—
111C3×#67—51no—
112C3×#67—51no—
113C3×#67—51no—
114C3×#67AGL(1,𝔽5)11yescharacteristic, complemented
115C2×C3×Dic3—51no—
116C2×C3×Dic3—51no—
117C2×C3×Dic3—51no—
118C2×C3×Dic3—102no—
119C3×C4×D3—102no—
120C22×C3×C5—21no—
121C22×C3×C5—31no—
122C22×C3×C5C4×D311yescharacteristic, complemented
123C22×C3×C5C2×C3×C411yescharacteristic, complemented
124C2×D15C2×C3×C411yescharacteristic, complemented
125C2×C5×D3C2×C3×C411yescharacteristic, complemented
126C2×C3×D5—21no—
127C2×C3×D5—21no—
128C2×C3×D5—21no—
129C2×C3×D5—21no—
130C2×C3×D5—21no—
131C2×C3×D5—21no—
132C2×C3×D5—31no—
133C2×C3×D5—41no—
134C2×C3×D5—41no—
135C2×C3×D5—61no—
136C2×C3×D5—61no—
137C2×C3×D5C4×D311yescharacteristic, complemented
138C2×C3×D5C2×C3×C411yescharacteristic, complemented
139C2×C3×D5C22×D311yescharacteristic
140C2×C3×D5C23×C311yescharacteristic
141D3×D5—21no—
142D3×D5—21no—
143ΓL(1,𝔽16)—21no—
144ΓL(1,𝔽16)C3×D412yescomplemented
145C3×AGL(1,𝔽5)—21no—
146C3×AGL(1,𝔽5)—21no—
147C3×AGL(1,𝔽5)—21no—
148C3×AGL(1,𝔽5)—21no—
149C3×AGL(1,𝔽5)—32no—
150C3×AGL(1,𝔽5)—41no—
151C3×AGL(1,𝔽5)—41no—
152C3×AGL(1,𝔽5)—61no—
153C3×C4×C5—31no—
154Dic15C2×C3×C411yescharacteristic, complemented
155C3×Dic5—31no—
156C5×Dic3C2×C3×C411yescharacteristic, complemented
157C2×C3×D4—151no—
158C22×C3×C4—151no—
159C22×C3×C4—151no—
160C2×#67—51no—
161C22×Dic3—51no—
162C2×C4×D3—51no—
163C3×C4⋊C4—151no—
164C3×C22⋊C4—51no—
165C3×C22⋊C4—51no—
166C3×C22⋊C4—101no—
167C3×C22⋊C4—151no—
168C3×C42—151no—
169(C2×D3)⋊C4—51no—
170Dic3⋊C4—51no—
171C4×Dic3—51no—
172C32×C5C4×D411yescharacteristic, complemented, Hall
173C22×D5—31no—
174C22×D5C2×C3×D311yescharacteristic
175C2×AGL(1,𝔽5)—31no—
176C2×AGL(1,𝔽5)—31no—
177C2×AGL(1,𝔽5)—62no—
178C2×AGL(1,𝔽5)—62no—
179C2×AGL(1,𝔽5)C2×C3×D311yescharacteristic
180C2×AGL(1,𝔽5)C2×C3×D311yescharacteristic
181C5×D4—31no—
182C22⋊D5—31no—
183C22⋊D5—31no—
184D20—31no—
185C4×D5—31no—
186C22×C32—51no—
187C22×C32—51no—
188C22×C32—101no—
189C22×C32—101no—
190C22×C32C2×AGL(1,𝔽5)11yescharacteristic, complemented
191C2×C3×D3—51no—
192C2×C3×D3—101no—
193C2×C3×D3—101no—
194C2×C3×D3C2×AGL(1,𝔽5)11yescharacteristic, complemented
195C32×C4—101no—
196C32×C4—101no—
197C3×Dic3—52no—
198C3×Dic3—51no—
199C3×Dic3—101no—
200C3×Dic3C2×AGL(1,𝔽5)11yescharacteristic, complemented
201C4×D4—151noSylow
202C2×C3×C5—21no—
203C2×C3×C5—21no—
204C2×C3×C5—21no—
205C2×C3×C5—41no—
206C2×C3×C5—61no—
207C2×C3×C5C2×C4×D311yescharacteristic
208C2×C3×C5C22×C3×C411yescharacteristic
209D15—21no—
210C3×D5—21no—
211C3×D5—21no—
212C3×D5—21no—
213C3×D5—21no—
214C3×D5—41no—
215C3×D5—61no—
216C3×D5D3×D411yescharacteristic
217C3×D5#23711yescharacteristic
218C3×D5C2×C3×D411yescharacteristic
219C3×D5C3×#4111yescharacteristic
220C5×D3—21no—
221C23×C3—51no—
222C23×C3—51no—
223C23×C3—101no—
224C23×C3—151no—
225C22×D3—51no—
226C3×D4—31no—
227C3×D4—151no—
228C3×D4—151no—
229C3×D4—151no—
230C2×C3×C4—51no—
231C2×C3×C4—51no—
232C2×C3×C4—51no—
233C2×C3×C4—51no—
234C2×C3×C4—101no—
235C2×C3×C4—101no—
236C2×C3×C4—151no—
237C2×C3×C4—151no—
238C2×C3×C4—151no—
239C2×C3×C4—302no—
240C2×C3×C4—302no—
241#67—51no—
242#67—51no—
243#67—51no—
244#67C3×AGL(1,𝔽5)11yescharacteristic, complemented
245C2×Dic3—51no—
246C2×Dic3—51no—
247C2×Dic3—51no—
248C2×Dic3—102no—
249C4×D3—102no—
250C22×C5—31no—
251C22×C5C3×C4×D311yescharacteristic, complemented
252C2×D5—21no—
253C2×D5—21no—
254C2×D5—31no—
255C2×D5—61no—
256C2×D5—61no—
257C2×D5C3×C4×D311yescharacteristic, complemented
258C2×D5C22×C3×D311yescharacteristic
259AGL(1,𝔽5)—21no—
260AGL(1,𝔽5)—21no—
261AGL(1,𝔽5)—32no—
262AGL(1,𝔽5)—61no—
263C4×C5—31no—
264Dic5—31no—
265C2×C32—21no—
266C2×C32—51no—
267C2×C32—51no—
268C2×C32—101no—
269C2×C32C22×AGL(1,𝔽5)11yescharacteristic
270C3×D3—21no—
271C3×D3—101no—
272C2×D4—151no—
273C22×C4—151no—
274C22×C4—151no—
275C4⋊C4—151no—
276C22⋊C4—51no—
277C22⋊C4—151no—
278C42—151no—
279C3×C5—21no—
280C3×C5#67⋊C411yescharacteristic, complemented
281C3×C5C3×C4×D411yescharacteristic, complemented
282C22×C3—21no—
283C22×C3—31no—
284C22×C3—51no—
285C22×C3—51no—
286C22×C3—51no—
287C22×C3—51no—
288C22×C3—101no—
289C22×C3—101no—
290C22×C3—101no—
291C22×C3—101no—
292C22×C3—101no—
293C22×C3—101no—
294C22×C3—151no—
295C22×C3—201no—
296C22×C3—201no—
297C22×C3—301no—
298C22×C3—301no—
299C22×C3D3×AGL(1,𝔽5)11yescharacteristic, complemented
300C22×C3C2×C3×AGL(1,𝔽5)11yescharacteristic, complemented
301C2×D3—51no—
302C2×D3—101no—
303C2×D3—101no—
304C2×D3C2×C3×AGL(1,𝔽5)11yescharacteristic, complemented
305C3×C4—31no—
306C3×C4—101no—
307C3×C4—101no—
308C3×C4—101no—
309C3×C4—101no—
310C3×C4—152no—
311C3×C4—151no—
312C3×C4—201no—
313C3×C4—201no—
314C3×C4—301no—
315Dic3—52no—
316Dic3—51no—
317Dic3—101no—
318Dic3C2×C3×AGL(1,𝔽5)11yescharacteristic, complemented
319C2×C5—21no—
320C2×C5—61no—
321C2×C5C2×C3×C4×D311yescharacteristic
322D5—21no—
323D5—61no—
324D5C3×D3×D411yescharacteristic
325D5C3×#23711yescharacteristic
326C32D4×AGL(1,𝔽5)11yescharacteristic, complemented, Sylow
327C23—51no—
328C23—151no—
329D4—31no—
330D4—151no—
331D4—151no—
332D4—151no—
333C2×C4—51no—
334C2×C4—51no—
335C2×C4—151no—
336C2×C4—151no—
337C2×C4—151no—
338C2×C4—302no—
339C2×C4—302no—
340C2×C3—21no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—41no—
344C2×C3—51no—
345C2×C3—51no—
346C2×C3—51no—
347C2×C3—51no—
348C2×C3—61no—
349C2×C3—101no—
350C2×C3—101no—
351C2×C3—101no—
352C2×C3—101no—
353C2×C3—201no—
354C2×C3—301no—
355C2×C3C2×D3×AGL(1,𝔽5)11yescharacteristic
356C2×C3C22×C3×AGL(1,𝔽5)11yescharacteristic
357D3—21no—
358D3—101no—
359C5C3×#67⋊C411yescharacteristic, complemented, Sylow
360C22—31no—
361C22—51no—
362C22—51no—
363C22—101no—
364C22—101no—
365C22—151no—
366C22—301no—
367C22—301no—
368C22C3×D3×AGL(1,𝔽5)11yescharacteristic, complemented
369C4—31no—
370C4—101no—
371C4—101no—
372C4—152no—
373C4—151no—
374C4—301no—
375C3—21no—
376C3#67⋊AGL(1,𝔽5)11yescharacteristic, direct product factor
377C3C3×D4×AGL(1,𝔽5)11yescharacteristic, complemented
378C2—21no—
379C2—51no—
380C2—51no—
381C2—61no—
382C2—101no—
383C2—301no—
384C2C2×C3×D3×AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC2×C3C2×D3×AGL(1,𝔽5)
commutatorC2×C3×C5C22×C3×C4
FrattiniC2C2×C3×D3×AGL(1,𝔽5)
FittingC22×C32×C5C2×C4
socleC2×C32×C5C22×C4
prime radicalC3×D5×#67C2
cosocleC2×C3×D5C23×C3
hypercenterC22×C3D3×AGL(1,𝔽5)
hypocenterC3×C5C3×C4×D4
LCSG ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C3 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C22×C3×ΓL(1,𝔽16)C2semidirect
2. C2×C3×D3×AGL(1,𝔽5)C2semidirect
4. C32×C22⋊AGL(1,𝔽5)C2semidirect
5. C3×Dic3⋊AGL(1,𝔽5)C2semidirect
6. C3×(C2×D3)⋊AGL(1,𝔽5)C2semidirect
7. C3×Dic3×AGL(1,𝔽5)C2semidirect
9. #67⋊AGL(1,𝔽5)C3direct (N × H)
18. C3×#1220C4semidirect
19. C3×C5×#67C4semidirect
20. C3×#1202C4semidirect
21. C3×#1201C4semidirect
24. C22×ΓL(1,𝔽16)C2×C3semidirect
27. C2×D3×AGL(1,𝔽5)C2×C3semidirect
32. C3×C22⋊AGL(1,𝔽5)D3semidirect
33. C3×C22⋊AGL(1,𝔽5)C2×C3semidirect
36. Dic3⋊AGL(1,𝔽5)C2×C3semidirect
37. (C2×D3)⋊AGL(1,𝔽5)C2×C3semidirect
38. Dic3×AGL(1,𝔽5)C2×C3semidirect
39. C22×C32×C5C2×C4semidirect
40. C2×C3×D15C2×C4semidirect
41. C2×C3×C5×D3C2×C4semidirect
44. C2×C32×D5C2×C4semidirect
49. C3×ΓL(1,𝔽16)D4semidirect
52. C3×Dic15C2×C4semidirect
53. C3×C5×Dic3C2×C4semidirect
82. #1220C3×C4semidirect
83. C5×#67C3×C4semidirect
88. #1202C3×C4semidirect
89. #1201C3×C4semidirect
104. C22⋊AGL(1,𝔽5)C3×D3semidirect
114. C3×#67AGL(1,𝔽5)semidirect
122. C22×C3×C5C4×D3semidirect
123. C22×C3×C5C2×C3×C4semidirect
124. C2×D15C2×C3×C4semidirect
125. C2×C5×D3C2×C3×C4semidirect
137. C2×C3×D5C4×D3semidirect
138. C2×C3×D5C2×C3×C4semidirect
144. ΓL(1,𝔽16)C3×D4semidirect
154. Dic15C2×C3×C4semidirect
156. C5×Dic3C2×C3×C4semidirect
172. C32×C5C4×D4semidirect
190. C22×C32C2×AGL(1,𝔽5)semidirect
194. C2×C3×D3C2×AGL(1,𝔽5)semidirect
200. C3×Dic3C2×AGL(1,𝔽5)semidirect
244. #67C3×AGL(1,𝔽5)semidirect
251. C22×C5C3×C4×D3semidirect
257. C2×D5C3×C4×D3semidirect
280. C3×C5#67⋊C4semidirect
281. C3×C5C3×C4×D4semidirect
299. C22×C3D3×AGL(1,𝔽5)semidirect
300. C22×C3C2×C3×AGL(1,𝔽5)semidirect
304. C2×D3C2×C3×AGL(1,𝔽5)semidirect
318. Dic3C2×C3×AGL(1,𝔽5)semidirect
326. C32D4×AGL(1,𝔽5)semidirect
359. C5C3×#67⋊C4semidirect
368. C22C3×D3×AGL(1,𝔽5)semidirect
376. C3#67⋊AGL(1,𝔽5)direct (N × H)
377. C3C3×D4×AGL(1,𝔽5)semidirect

Maximal subgroups

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

subgroupindex
C22×C3×ΓL(1,𝔽16)2
C2×C3×D3×AGL(1,𝔽5)2
C3×D5×#672
C32×C22⋊AGL(1,𝔽5)2
C3×Dic3⋊AGL(1,𝔽5)2
C3×(C2×D3)⋊AGL(1,𝔽5)2
C3×Dic3×AGL(1,𝔽5)2
C3×D4×AGL(1,𝔽5)3
#67⋊AGL(1,𝔽5)3
C3×#67⋊C45