Finite Group Databasebeta

← C3×#10604424 #14282927

Subgroups — C3×#10604424 #14282927

Order 1920.

480 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65912C211yesmaximal, characteristic
2C3×C4×C4⋊AGL(1,𝔽5)C211yesmaximal, characteristic
3C3×#65678C211yesmaximal, characteristic
4C3×#64926C212yesmaximal
5C3×#64869C212yesmaximal
6#10604424C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×C4⋊AGL(1,𝔽5)—21no—
8C2×C3×C4×AGL(1,𝔽5)—21no—
9C3×C4×C5×Q8C411yescharacteristic, complemented
10C3×#4092C2211yescharacteristic
11C3×#4090C411yescharacteristic, complemented
12C3×#4043C2211yescharacteristic
13C3×#4042—21no—
14C3×#4035—21no—
15C3×#4034C411yescharacteristic, complemented
16C3×#4018—21no—
17C3×C42×D5C2211yescharacteristic
18C3×C4×Dic10—21no—
19C3×C4×Dic10C411yescharacteristic, complemented
20C3×(C2×AGL(1,𝔽5))⋊C4C2212yes—
21C3×#4004—21no—
22C3×#4003C2212yes—
23C3×C4×#170—21no—
24C3×#1503—51nomaximal, Hall
25#65912C2×C311yescharacteristic
26C4×C4⋊AGL(1,𝔽5)C2×C311yescharacteristic
27#65678C2×C311yescharacteristic
28#64926C2×C312yes—
29#64869C2×C312yes—
30C22×C3×AGL(1,𝔽5)—22no—
31C2×C3×C5×Q8C2×C411yescharacteristic
32C2×C3×C4×D5D411yescharacteristic
33C2×C3×C4×D5D411yescharacteristic
34C2×C3×C4×D5C2311yescharacteristic
35C2×C3×Dic10—21no—
36C2×C3×Dic10C2×C411yescharacteristic
37C2×C3×#170—22no—
38C3×C4⋊AGL(1,𝔽5)—24no—
39C3×C4×AGL(1,𝔽5)—42no—
40C3×C5×C4⋊C4—21no—
41C3×C5×C4⋊C4C2×C411yescharacteristic
42C3×C42×C5—21no—
43C3×C42×C5C2×C411yescharacteristic
44C3×#685—21no—
45C3×#685—21no—
46C3×C4⋊Dic5—21no—
47C3×C4⋊Dic5C2×C411yescharacteristic
48C3×#683—21no—
49C3×#683—21no—
50C3×#683—21no—
51C3×#683—21no—
52C3×#683—21no—
53C3×C4×Dic5—21no—
54C3×C4×Dic5C2×C411yescharacteristic
55C3×C4×Dic5D411yescharacteristic
56C3×C4×Dic5D411yescharacteristic
57C3×#533—51no—
58C3×#431—51no—
59C3×C4×C4⋊C4—51no—
60C3×#328—52no—
61C3×#328—52no—
62C2×C4⋊AGL(1,𝔽5)—21no—
63C2×C4×AGL(1,𝔽5)—21no—
64C4×C5×Q8C3×C411yescharacteristic, complemented
65#4092C22×C311yescharacteristic
66#4090C3×C411yescharacteristic, complemented
67#4043C22×C311yescharacteristic
68#4042—21no—
69#4035—21no—
70#4034C3×C411yescharacteristic, complemented
71#4018—21no—
72C42×D5C22×C311yescharacteristic
73C4×Dic10—21no—
74C4×Dic10C3×C411yescharacteristic, complemented
75(C2×AGL(1,𝔽5))⋊C4C22×C312yes—
76#4004—21no—
77#4003C22×C312yes—
78C4×#170—21no—
79#1503—51noSylow
80C22×C3×D5C2×D411yescharacteristic
81C2×C3×AGL(1,𝔽5)—24no—
82C2×C3×AGL(1,𝔽5)—42no—
83C2×C3×AGL(1,𝔽5)—42no—
84C3×C5×Q8—41no—
85C2×C3×C4×C5—21no—
86C2×C3×C4×C5—21no—
87C2×C3×C4×C5C22⋊C411yescharacteristic
88C2×C3×C4×C5C22⋊C411yescharacteristic
89C2×C3×C4×C5C22×C411yescharacteristic
90C2×C3×Dic5—21no—
91C2×C3×Dic5—21no—
92C2×C3×Dic5C22⋊C411yescharacteristic
93C2×C3×Dic5C22⋊C411yescharacteristic
94C2×C3×Dic5C2×D411yescharacteristic
95C3×C4×D5—22no—
96C3×C4×D5—41no—
97C3×C4×D5SD1612yes—
98C3×C4×D5Q1612yes—
99C3×Dic10—42no—
100C3×Dic10—41no—
101C3×#170—42no—
102C3×C4⋊Q8—51no—
103C3×C4⋊Q8—51no—
104C3×#125—101no—
105C3×C22⋊Q8—51no—
106C3×C22⋊Q8—51no—
107C3×C22⋊Q8—101no—
108C3×C4×Q8—51no—
109C3×C4×Q8—101no—
110C3×C4×Q8AGL(1,𝔽5)11yescharacteristic, complemented
111C3×#117—101no—
112C2×C3×C4⋊C4—101no—
113C2×C3×C42—51no—
114C2×C3×C42—101no—
115C3×C4⋊C8—101no—
116C3×C22⋊C8—52no—
117C3×C4×C8—101no—
118C3×#95—52no—
119C22×AGL(1,𝔽5)—22no—
120C2×C5×Q8C2×C3×C411yescharacteristic
121C2×C4×D5C3×D411yescharacteristic
122C2×C4×D5C3×D411yescharacteristic
123C2×C4×D5C23×C311yescharacteristic
124C2×Dic10—21no—
125C2×Dic10C2×C3×C411yescharacteristic
126C2×#170—22no—
127C4⋊AGL(1,𝔽5)—24no—
128C4×AGL(1,𝔽5)—42no—
129C5×C4⋊C4—21no—
130C5×C4⋊C4C2×C3×C411yescharacteristic
131C42×C5—21no—
132C42×C5C2×C3×C411yescharacteristic
133#685—21no—
134#685—21no—
135C4⋊Dic5—21no—
136C4⋊Dic5C2×C3×C411yescharacteristic
137#683—21no—
138#683—21no—
139#683—21no—
140#683—21no—
141#683—21no—
142C4×Dic5—21no—
143C4×Dic5C2×C3×C411yescharacteristic
144C4×Dic5C3×D411yescharacteristic
145C4×Dic5C3×D411yescharacteristic
146#533—51no—
147#431—51no—
148C4×C4⋊C4—51no—
149#328—52no—
150#328—52no—
151C22×C3×C5C2×C22⋊C411yescharacteristic
152C2×C3×D5—21no—
153C2×C3×D5—21no—
154C2×C3×D5C2×SD1611yescharacteristic
155C2×C3×D5C2×Q1611yescharacteristic
156C3×AGL(1,𝔽5)—44no—
157C3×C4×C5—21no—
158C3×C4×C5—21no—
159C3×C4×C5—41no—
160C3×C4×C5—41no—
161C3×C4×C5#10312yes—
162C3×Dic5—21no—
163C3×Dic5—41no—
164C3×Dic5—41no—
165C3×Dic5#10312yes—
166C3×Dic5#10412yes—
167C2×C3×Q8—51no—
168C2×C3×Q8—101no—
169C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
170C22×C3×C4—51no—
171C22×C3×C4—51no—
172C22×C3×C4—51no—
173C22×C3×C4—102no—
174C2×C3×C8—102no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—51no—
177C3×C4⋊C4—101no—
178C3×C4⋊C4—101no—
179C3×C4⋊C4—101no—
180C3×C4⋊C4—101no—
181C3×C4⋊C4—101no—
182C3×C4⋊C4—104no—
183C3×C4⋊C4—101no—
184C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
185C3×C22⋊C4—101no—
186C3×C22⋊C4—101no—
187C3×C42—21no—
188C3×C42—51no—
189C3×C42—51no—
190C3×C42—51no—
191C3×C42—101no—
192C3×C42—202no—
193C3×C42C2×AGL(1,𝔽5)11yescharacteristic
194C22×D5C2×C3×D411yescharacteristic
195C2×AGL(1,𝔽5)—24no—
196C2×AGL(1,𝔽5)—42no—
197C2×AGL(1,𝔽5)—42no—
198C5×Q8—41no—
199C2×C4×C5—21no—
200C2×C4×C5—21no—
201C2×C4×C5C3×C22⋊C411yescharacteristic
202C2×C4×C5C3×C22⋊C411yescharacteristic
203C2×C4×C5C22×C3×C411yescharacteristic
204C2×Dic5—21no—
205C2×Dic5—21no—
206C2×Dic5C3×C22⋊C411yescharacteristic
207C2×Dic5C3×C22⋊C411yescharacteristic
208C2×Dic5C2×C3×D411yescharacteristic
209C4×D5—22no—
210C4×D5—41no—
211C4×D5C3×SD1612yes—
212C4×D5C3×Q1612yes—
213Dic10—42no—
214Dic10—41no—
215#170—42no—
216C4⋊Q8—51no—
217C4⋊Q8—51no—
218#125—101no—
219C22⋊Q8—51no—
220C22⋊Q8—51no—
221C22⋊Q8—101no—
222C4×Q8—51no—
223C4×Q8—101no—
224C4×Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
225#117—101no—
226C2×C4⋊C4—101no—
227C2×C42—51no—
228C2×C42—101no—
229C4⋊C8—101no—
230C22⋊C8—52no—
231C4×C8—101no—
232#95—52no—
233C2×C3×C5#41011yescharacteristic
234C2×C3×C5C2×#10311yescharacteristic
235C2×C3×C5C2×#10411yescharacteristic
236C3×D5—21no—
237C3×D5—21no—
238C23×C3—51no—
239C3×Q8—41no—
240C3×Q8—201no—
241C3×Q8—202no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—51no—
245C2×C3×C4—51no—
246C2×C3×C4—52no—
247C2×C3×C4—52no—
248C2×C3×C4—51no—
249C2×C3×C4—102no—
250C2×C3×C4—104no—
251C2×C3×C4—101no—
252C2×C3×C4—101no—
253C2×C3×C4—201no—
254C2×C3×C4—202no—
255C2×C3×C4—202no—
256C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
257C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
258C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
259C3×C8—202no—
260C22×C5C2×C3×C22⋊C411yescharacteristic
261C2×D5—21no—
262C2×D5—21no—
263C2×D5C2×C3×SD1611yescharacteristic
264C2×D5C2×C3×Q1611yescharacteristic
265AGL(1,𝔽5)—44no—
266C4×C5—21no—
267C4×C5—21no—
268C4×C5—41no—
269C4×C5—41no—
270C4×C5C3×#10312yes—
271Dic5—21no—
272Dic5—41no—
273Dic5—41no—
274Dic5C3×#10312yes—
275Dic5C3×#10412yes—
276C2×Q8—51no—
277C2×Q8—101no—
278C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
279C22×C4—51no—
280C22×C4—51no—
281C22×C4—51no—
282C22×C4—102no—
283C2×C8—102no—
284C4⋊C4—21no—
285C4⋊C4—51no—
286C4⋊C4—101no—
287C4⋊C4—101no—
288C4⋊C4—101no—
289C4⋊C4—101no—
290C4⋊C4—101no—
291C4⋊C4—104no—
292C4⋊C4—101no—
293C4⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic
294C22⋊C4—101no—
295C22⋊C4—101no—
296C42—21no—
297C42—51no—
298C42—51no—
299C42—51no—
300C42—101no—
301C42—202no—
302C42C2×C3×AGL(1,𝔽5)11yescharacteristic
303C3×C5#150311yescharacteristic, complemented, Hall
304C22×C3—51no—
305C22×C3—51no—
306C22×C3—101no—
307C22×C3—101no—
308C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—41no—
312C3×C4—41no—
313C3×C4—52no—
314C3×C4—52no—
315C3×C4—101no—
316C3×C4—204no—
317C3×C4—201no—
318C3×C4—201no—
319C3×C4#400912yes—
320C2×C5C3×#41011yescharacteristic
321C2×C5C2×C3×#10311yescharacteristic
322C2×C5C2×C3×#10411yescharacteristic
323D5—21no—
324D5—21no—
325C23—51no—
326Q8—41no—
327Q8—201no—
328Q8—202no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—51no—
332C2×C4—51no—
333C2×C4—52no—
334C2×C4—52no—
335C2×C4—51no—
336C2×C4—102no—
337C2×C4—104no—
338C2×C4—101no—
339C2×C4—101no—
340C2×C4—201no—
341C2×C4—202no—
342C2×C4—202no—
343C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
344C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
345C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
346C8—202no—
347C2×C3—101no—
348C2×C3—101no—
349C2×C3#6574511yescharacteristic
350C2×C3C2×#400911yescharacteristic
351C2×C3D5⋊#10411yescharacteristic
352C5C3×#150311yescharacteristic, complemented, Sylow
353C22—51no—
354C22—51no—
355C22—101no—
356C22—101no—
357C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—52no—
363C4—52no—
364C4—101no—
365C4—204no—
366C4—201no—
367C4—201no—
368C4C3×#400912yes—
369C3#1060442411yescharacteristic, direct product factor, Sylow
370C2—101no—
371C2—101no—
372C2C3×#6574511yescharacteristic
373C2C2×C3×#400911yescharacteristic
374C2C3×D5⋊#10411yescharacteristic
subgroupG/H
centerC2×C3×C4C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C4×C5×Q8C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC22×C3×D5C2×D4
cosocleC2×C4×D5C23×C3
hypercenterC3×C4×Q8AGL(1,𝔽5)
hypocenterC5C3×#1503
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C2×C3×C4 ◁ C3×C42 ◁ C3×C4×Q8

Complements

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

NH ≅ G/Ntype
6. #10604424C3direct (N × H)
9. C3×C4×C5×Q8C4semidirect
11. C3×#4090C4semidirect
15. C3×#4034C4semidirect
19. C3×C4×Dic10C4semidirect
64. C4×C5×Q8C3×C4semidirect
66. #4090C3×C4semidirect
70. #4034C3×C4semidirect
74. C4×Dic10C3×C4semidirect
110. C3×C4×Q8AGL(1,𝔽5)semidirect
224. C4×Q8C3×AGL(1,𝔽5)semidirect
303. C3×C5#1503semidirect
352. C5C3×#1503semidirect
369. C3#10604424direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#659122
C3×C4×C4⋊AGL(1,𝔽5)2
C3×#656782
C3×#649262
C3×#648692
#106044243
C3×#15035