Finite Group Databasebeta

← C3×#10604560 #14306039

Subgroups — C3×#10604560 #14306039

Order 1920.

640 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66002C211yesmaximal, characteristic
2C2×C3×#4010C212yesmaximal, complemented
3C2×C3×#4008C212yesmaximal, complemented
4C3×#65686C211yesmaximal, characteristic, complemented
5C3×#65681C211yesmaximal, characteristic, complemented
6#10604560C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×#713C2211yescharacteristic
8C2×C3×#711C2211yescharacteristic
9C2×C3×C4×AGL(1,𝔽5)C2212yes—
10C2×C3×#700C2212yes—
11C3×C5×#124C411yescharacteristic, complemented
12C3×#4090C411yescharacteristic, complemented
13C3×#4086C411yescharacteristic, complemented
14C3×#4030—22no—
15C3×#4027—22no—
16C3×#4021C411yescharacteristic, complemented
17C3×C42×D5C2211yescharacteristic
18C3×#4010—24no—
19C3×#4008—24no—
20C3×(C2×AGL(1,𝔽5))⋊C4—22no—
21C3×C4⋊#170—22no—
22C3×#1963—51nomaximal, Hall
23#66002C2×C311yescharacteristic
24C2×#4010C2×C312yescomplemented
25C2×#4008C2×C312yescomplemented
26#65686C2×C311yescharacteristic, complemented
27#65681C2×C311yescharacteristic, complemented
28C22×C3×AGL(1,𝔽5)—22no—
29C2×C3×C5×Q8C2×C411yescharacteristic
30C2×C3×C5×D4C2×C411yescharacteristic
31C2×C3×C22⋊D5—21no—
32C2×C3×C22⋊D5—22no—
33C22×C3×Dic5—21no—
34C3×#713—22no—
35C3×#713—22no—
36C3×#711—22no—
37C3×#711—22no—
38C2×C3×D20—21no—
39C2×C3×D20C2×C411yescharacteristic
40C2×C3×C4×D5—21no—
41C2×C3×C4×D5D412yes—
42C2×C3×C4×D5C2311yescharacteristic
43C2×C3×Dic10C2×C411yescharacteristic
44C2×C3×#170—22no—
45C3×C4×AGL(1,𝔽5)—22no—
46C3×C4×AGL(1,𝔽5)D414yescomplemented
47C3×#700—22no—
48C3×#700—22no—
49C3×C5×C22⋊C4—22no—
50C3×C42×C5C2×C411yescharacteristic
51C3×#685—22no—
52C3×#683—22no—
53C3×C4×Dic5C2×C411yescharacteristic
54C3×C4×Dic5D412yes—
55C3×#532—51no—
56C3×#423—51no—
57C2×C3×#104—52no—
58C2×C3×#104—52no—
59C3×#376—51no—
60C2×#713C22×C311yescharacteristic
61C2×#711C22×C311yescharacteristic
62C2×C4×AGL(1,𝔽5)C22×C312yes—
63C2×#700C22×C312yes—
64C5×#124C3×C411yescharacteristic, complemented
65#4090C3×C411yescharacteristic, complemented
66#4086C3×C411yescharacteristic, complemented
67#4030—22no—
68#4027—22no—
69#4021C3×C411yescharacteristic, complemented
70C42×D5C22×C311yescharacteristic
71#4010—24no—
72#4008—24no—
73(C2×AGL(1,𝔽5))⋊C4—22no—
74C4⋊#170—22no—
75#1963—51noSylow
76C23×C3×C5—21no—
77C22×C3×D5—21no—
78C22×C3×D5C2×D411yescharacteristic
79C2×C3×AGL(1,𝔽5)—24no—
80C2×C3×AGL(1,𝔽5)—44no—
81C3×C5×Q8—22no—
82C3×C5×D4—22no—
83C2×C3×C4×C5—21no—
84C2×C3×C4×C5C22⋊C412yes—
85C2×C3×C4×C5C22×C411yescharacteristic
86C3×C22⋊D5—44no—
87C3×C22⋊D5—42no—
88C2×C3×Dic5—21no—
89C2×C3×Dic5—41no—
90C2×C3×Dic5—41no—
91C2×C3×Dic5C22⋊C412yes—
92C2×C3×Dic5C2×D411yescharacteristic
93C3×D20—22no—
94C3×D20—42no—
95C3×C4×D5—42no—
96C3×C4×D5—41no—
97C3×C4×D5—41no—
98C3×C4×D5C2×D412yes—
99C3×C4×D5#4112yes—
100C3×Dic10—22no—
101C3×#170—42no—
102C2×C3×#41—51no—
103C2×C3×#41—51no—
104C2×C3×M16—52no—
105C3×C4⋊Q8—51no—
106C3×#127—51no—
107C3×#124—51no—
108C3×#124AGL(1,𝔽5)11yescharacteristic, complemented
109C3×#121—102no—
110C3×C4×D4—102no—
111C2×C3×C42—51no—
112C2×C3×C42—52no—
113C3×C4⋊C8—102no—
114C3×#104—104no—
115C3×#104—104no—
116C3×#95—102no—
117C22×AGL(1,𝔽5)—22no—
118C2×C5×Q8C2×C3×C411yescharacteristic
119C2×C5×D4C2×C3×C411yescharacteristic
120C2×C22⋊D5—21no—
121C2×C22⋊D5—22no—
122C22×Dic5—21no—
123#713—22no—
124#713—22no—
125#711—22no—
126#711—22no—
127C2×D20—21no—
128C2×D20C2×C3×C411yescharacteristic
129C2×C4×D5—21no—
130C2×C4×D5C3×D412yes—
131C2×C4×D5C23×C311yescharacteristic
132C2×Dic10C2×C3×C411yescharacteristic
133C2×#170—22no—
134C4×AGL(1,𝔽5)—22no—
135C4×AGL(1,𝔽5)C3×D414yescomplemented
136#700—22no—
137#700—22no—
138C5×C22⋊C4—22no—
139C42×C5C2×C3×C411yescharacteristic
140#685—22no—
141#683—22no—
142C4×Dic5C2×C3×C411yescharacteristic
143C4×Dic5C3×D412yes—
144#532—51no—
145#423—51no—
146C2×#104—52no—
147C2×#104—52no—
148#376—51no—
149C22×C3×C5—41no—
150C22×C3×C5—42no—
151C22×C3×C5C2×C22⋊C411yescharacteristic
152C2×C3×D5—22no—
153C2×C3×D5—41no—
154C2×C3×D5—42no—
155C2×C3×D5#12112yes—
156C3×AGL(1,𝔽5)—44no—
157C3×C4×C5—22no—
158C3×C4×C5—41no—
159C3×C4×C5C4×D412yes—
160C3×Dic5—22no—
161C3×Dic5—41no—
162C3×Dic5#12411yescharacteristic
163C3×Dic5#12711yescharacteristic
164C3×#41—102no—
165C3×#41—102no—
166C3×#41—102no—
167C3×#41—102no—
168C2×C3×Q8—51no—
169C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
170C2×C3×D4—51no—
171C2×C3×D4—101no—
172C2×C3×D4—102no—
173C2×C3×D4—101no—
174C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
175C22×C3×C4—51no—
176C22×C3×C4—52no—
177C22×C3×C4—101no—
178C22×C3×C4—101no—
179C22×C3×C4—102no—
180C3×M16—102no—
181C3×M16—102no—
182C2×C3×C8—102no—
183C3×C4⋊C4—102no—
184C3×C22⋊C4—22no—
185C3×C22⋊C4—102no—
186C3×C42—52no—
187C3×C42—51no—
188C3×C42—54no—
189C3×C42—102no—
190C3×C42C2×AGL(1,𝔽5)11yescharacteristic
191C23×C5—21no—
192C22×D5—21no—
193C22×D5C2×C3×D411yescharacteristic
194C2×AGL(1,𝔽5)—24no—
195C2×AGL(1,𝔽5)—44no—
196C5×Q8—22no—
197C5×D4—22no—
198C2×C4×C5—21no—
199C2×C4×C5C3×C22⋊C412yes—
200C2×C4×C5C22×C3×C411yescharacteristic
201C22⋊D5—44no—
202C22⋊D5—42no—
203C2×Dic5—21no—
204C2×Dic5—41no—
205C2×Dic5—41no—
206C2×Dic5C3×C22⋊C412yes—
207C2×Dic5C2×C3×D411yescharacteristic
208D20—22no—
209D20—42no—
210C4×D5—42no—
211C4×D5—41no—
212C4×D5—41no—
213C4×D5C2×C3×D412yes—
214C4×D5C3×#4112yes—
215Dic10—22no—
216#170—42no—
217C2×#41—51no—
218C2×#41—51no—
219C2×M16—52no—
220C4⋊Q8—51no—
221#127—51no—
222#124—51no—
223#124C3×AGL(1,𝔽5)11yescharacteristic, complemented
224#121—102no—
225C4×D4—102no—
226C2×C42—51no—
227C2×C42—52no—
228C4⋊C8—102no—
229#104—104no—
230#104—104no—
231#95—102no—
232C2×C3×C5—81no—
233C2×C3×C5#39011yescharacteristic
234C2×C3×C5#42112yes—
235C3×D5—22no—
236C3×D5—81no—
237C23×C3—21no—
238C23×C3—51no—
239C23×C3—101no—
240C3×Q8—22no—
241C3×Q8—102no—
242C3×D4—22no—
243C3×D4—102no—
244C3×D4—202no—
245C3×D4—204no—
246C3×D4—202no—
247C2×C3×C4—21no—
248C2×C3×C4—51no—
249C2×C3×C4—52no—
250C2×C3×C4—52no—
251C2×C3×C4—52no—
252C2×C3×C4—104no—
253C2×C3×C4—101no—
254C2×C3×C4—202no—
255C2×C3×C4—201no—
256C2×C3×C4—201no—
257C2×C3×C4—201no—
258C2×C3×C4—201no—
259C2×C3×C4—204no—
260C2×C3×C4C22⋊AGL(1,𝔽5)12yes—
261C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
262C3×C8—202no—
263C22×C5—41no—
264C22×C5—42no—
265C22×C5C2×C3×C22⋊C411yescharacteristic
266C2×D5—22no—
267C2×D5—41no—
268C2×D5—42no—
269C2×D5C3×#12112yes—
270AGL(1,𝔽5)—44no—
271C4×C5—22no—
272C4×C5—41no—
273C4×C5C3×C4×D412yes—
274Dic5—22no—
275Dic5—41no—
276Dic5C3×#12411yescharacteristic
277Dic5C3×#12711yescharacteristic
278#41—102no—
279#41—102no—
280#41—102no—
281#41—102no—
282C2×Q8—51no—
283C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
284C2×D4—51no—
285C2×D4—101no—
286C2×D4—102no—
287C2×D4—101no—
288C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
289C22×C4—51no—
290C22×C4—52no—
291C22×C4—101no—
292C22×C4—101no—
293C22×C4—102no—
294M16—102no—
295M16—102no—
296C2×C8—102no—
297C4⋊C4—102no—
298C22⋊C4—22no—
299C22⋊C4—102no—
300C42—52no—
301C42—51no—
302C42—54no—
303C42—102no—
304C42C2×C3×AGL(1,𝔽5)11yescharacteristic
305C3×C5#196311yescharacteristic, complemented, Hall
306C22×C3—42no—
307C22×C3—41no—
308C22×C3—52no—
309C22×C3—102no—
310C22×C3—202no—
311C22×C3—201no—
312C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
313C3×C4—22no—
314C3×C4—41no—
315C3×C4—51no—
316C3×C4—51no—
317C3×C4—102no—
318C3×C4—204no—
319C3×C4—201no—
320C3×C4D4×AGL(1,𝔽5)12yes—
321C2×C5—81no—
322C2×C5C3×#39011yescharacteristic
323C2×C5C3×#42112yes—
324D5—22no—
325D5—81no—
326C23—21no—
327C23—51no—
328C23—101no—
329Q8—22no—
330Q8—102no—
331D4—22no—
332D4—102no—
333D4—202no—
334D4—204no—
335D4—202no—
336C2×C4—21no—
337C2×C4—51no—
338C2×C4—52no—
339C2×C4—52no—
340C2×C4—52no—
341C2×C4—104no—
342C2×C4—101no—
343C2×C4—202no—
344C2×C4—201no—
345C2×C4—201no—
346C2×C4—201no—
347C2×C4—201no—
348C2×C4—204no—
349C2×C4C3×C22⋊AGL(1,𝔽5)12yes—
350C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
351C8—202no—
352C2×C3—81no—
353C2×C3—102no—
354C2×C3—401no—
355C2×C3#6578011yescharacteristic
356C2×C3#6579512yes—
357C5C3×#196311yescharacteristic, complemented, Sylow
358C22—41no—
359C22—42no—
360C22—52no—
361C22—102no—
362C22—201no—
363C22—202no—
364C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
365C4—22no—
366C4—41no—
367C4—51no—
368C4—51no—
369C4—102no—
370C4—204no—
371C4—201no—
372C4C3×D4×AGL(1,𝔽5)12yes—
373C3#1060456011yescharacteristic, direct product factor, Sylow
374C2—81no—
375C2—102no—
376C2—401no—
377C2C3×#6578011yescharacteristic
378C2C3×#6579512yes—
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×#124C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC3×#66002C2
cosocleC2×C4×D5C23×C3
hypercenterC3×#124AGL(1,𝔽5)
hypocenterC5C3×#1963
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#124

Complements

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

NH ≅ G/Ntype
2. C2×C3×#4010C2semidirect
3. C2×C3×#4008C2semidirect
4. C3×#65686C2semidirect
5. C3×#65681C2semidirect
6. #10604560C3direct (N × H)
11. C3×C5×#124C4semidirect
12. C3×#4090C4semidirect
13. C3×#4086C4semidirect
16. C3×#4021C4semidirect
24. C2×#4010C2×C3semidirect
25. C2×#4008C2×C3semidirect
26. #65686C2×C3semidirect
27. #65681C2×C3semidirect
46. C3×C4×AGL(1,𝔽5)D4semidirect
64. C5×#124C3×C4semidirect
65. #4090C3×C4semidirect
66. #4086C3×C4semidirect
69. #4021C3×C4semidirect
108. C3×#124AGL(1,𝔽5)semidirect
135. C4×AGL(1,𝔽5)C3×D4semidirect
223. #124C3×AGL(1,𝔽5)semidirect
305. C3×C5#1963semidirect
357. C5C3×#1963semidirect
373. C3#10604560direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#660022
C2×C3×#40102
C2×C3×#40082
C3×#656862
C3×#656812
#106045603
C3×#19635