Finite Group Databasebeta

← C3×#10604538 #14283208

Subgroups — C3×#10604538 #14283208

Order 1920.

484 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66014C211yesmaximal, characteristic
2C3×C4×C4⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
3C3×#65684C211yesmaximal, characteristic, complemented
4C3×#64926C212yesmaximal, complemented
5C3×#64920C212yesmaximal
6#10604538C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×C4⋊AGL(1,𝔽5)—21no—
8C2×C3×C4×AGL(1,𝔽5)—21no—
9C3×C5×#124C411yescharacteristic, complemented
10C3×#4092C2211yescharacteristic
11C3×Q8×Dic5C411yescharacteristic, complemented
12C3×#4084C2211yescharacteristic
13C3×D4×Dic5C411yescharacteristic, complemented
14C3×#4029—21no—
15C3×#4027—21no—
16C3×#4025—21no—
17C3×#4023—21no—
18C3×C42×D5C2211yescharacteristic
19C3×#4016C411yescharacteristic, complemented
20C3×(C2×AGL(1,𝔽5))⋊C4C2212yes—
21C3×#4004—21no—
22C3×#4003C2212yes—
23C3×#4002—21no—
24C3×#1509—51nomaximal, Hall
25#66014C2×C311yescharacteristic
26C4×C4⋊AGL(1,𝔽5)C2×C311yescharacteristic, complemented
27#65684C2×C311yescharacteristic, complemented
28#64926C2×C312yescomplemented
29#64920C2×C312yes—
30C22×C3×AGL(1,𝔽5)—22no—
31C2×C3×C5×Q8C2×C411yescharacteristic
32C2×C3×C5×D4C2×C411yescharacteristic
33C2×C3×C22⋊D5—21no—
34C22×C3×Dic5—21no—
35C2×C3×C4×D5D411yescharacteristic
36C2×C3×C4×D5D411yescharacteristic
37C2×C3×C4×D5C2311yescharacteristic
38C2×C3×#170—22no—
39C3×C4⋊AGL(1,𝔽5)—24no—
40C3×C4×AGL(1,𝔽5)—42no—
41C3×C5×C22⋊C4—21no—
42C3×C5×C22⋊C4—21no—
43C3×C42×C5C2×C411yescharacteristic
44C3×C22⋊Dic5—21no—
45C3×C22⋊Dic5—21no—
46C3×#685—21no—
47C3×C4⋊Dic5—21no—
48C3×C4⋊Dic5C2×C411yescharacteristic
49C3×C4⋊Dic5C2×C411yescharacteristic
50C3×#683—21no—
51C3×#683—21no—
52C3×#683—21no—
53C3×C4×Dic5—21no—
54C3×C4×Dic5C2×C411yescharacteristic
55C3×C4×Dic5D411yescharacteristic
56C3×C4×Dic5D411yescharacteristic
57C3×#529—51no—
58C3×#432—51no—
59C3×C4×C4⋊C4—51no—
60C3×#328—52no—
61C3×#327—52no—
62C2×C4⋊AGL(1,𝔽5)—21no—
63C2×C4×AGL(1,𝔽5)—21no—
64C5×#124C3×C411yescharacteristic, complemented
65#4092C22×C311yescharacteristic
66Q8×Dic5C3×C411yescharacteristic, complemented
67#4084C22×C311yescharacteristic
68D4×Dic5C3×C411yescharacteristic, complemented
69#4029—21no—
70#4027—21no—
71#4025—21no—
72#4023—21no—
73C42×D5C22×C311yescharacteristic
74#4016C3×C411yescharacteristic, complemented
75(C2×AGL(1,𝔽5))⋊C4C22×C312yes—
76#4004—21no—
77#4003C22×C312yes—
78#4002—21no—
79#1509—51noSylow
80C23×C3×C5—21no—
81C22×C3×D5C2×D411yescharacteristic
82C2×C3×AGL(1,𝔽5)—24no—
83C2×C3×AGL(1,𝔽5)—42no—
84C2×C3×AGL(1,𝔽5)—42no—
85C3×C5×Q8—41no—
86C3×C5×D4—41no—
87C2×C3×C4×C5—21no—
88C2×C3×C4×C5C22⋊C411yescharacteristic
89C2×C3×C4×C5C22⋊C411yescharacteristic
90C2×C3×C4×C5C22×C411yescharacteristic
91C3×C22⋊D5—41no—
92C3×C22⋊D5—41no—
93C2×C3×Dic5—21no—
94C2×C3×Dic5—21no—
95C2×C3×Dic5—42no—
96C2×C3×Dic5C22⋊C411yescharacteristic
97C2×C3×Dic5C22⋊C411yescharacteristic
98C2×C3×Dic5C2×D411yescharacteristic
99C3×C4×D5—21no—
100C3×C4×D5—21no—
101C3×C4×D5—22no—
102C3×C4×D5—41no—
103C3×#170—42no—
104C3×#126—101no—
105C3×#125—51no—
106C3×#124AGL(1,𝔽5)11yescharacteristic, complemented
107C3×#123—101no—
108C3×C22⋊Q8—51no—
109C3×#121—51no—
110C3×C4×Q8—51no—
111C3×C4×D4—51no—
112C3×C4×D4—101no—
113C3×#117—101no—
114C2×C3×C4⋊C4—101no—
115C2×C3×C42—51no—
116C2×C3×C42—101no—
117C3×C4⋊C8—101no—
118C3×C22⋊C8—52no—
119C3×#97—101no—
120C3×#95—52no—
121C22×AGL(1,𝔽5)—22no—
122C2×C5×Q8C2×C3×C411yescharacteristic
123C2×C5×D4C2×C3×C411yescharacteristic
124C2×C22⋊D5—21no—
125C22×Dic5—21no—
126C2×C4×D5C3×D411yescharacteristic
127C2×C4×D5C3×D411yescharacteristic
128C2×C4×D5C23×C311yescharacteristic
129C2×#170—22no—
130C4⋊AGL(1,𝔽5)—24no—
131C4×AGL(1,𝔽5)—42no—
132C5×C22⋊C4—21no—
133C5×C22⋊C4—21no—
134C42×C5C2×C3×C411yescharacteristic
135C22⋊Dic5—21no—
136C22⋊Dic5—21no—
137#685—21no—
138C4⋊Dic5—21no—
139C4⋊Dic5C2×C3×C411yescharacteristic
140C4⋊Dic5C2×C3×C411yescharacteristic
141#683—21no—
142#683—21no—
143#683—21no—
144C4×Dic5—21no—
145C4×Dic5C2×C3×C411yescharacteristic
146C4×Dic5C3×D411yescharacteristic
147C4×Dic5C3×D411yescharacteristic
148#529—51no—
149#432—51no—
150C4×C4⋊C4—51no—
151#328—52no—
152#327—52no—
153C22×C3×C5—41no—
154C22×C3×C5—41no—
155C22×C3×C5—41no—
156C22×C3×C5C2×C22⋊C411yescharacteristic
157C2×C3×D5—21no—
158C2×C3×D5—21no—
159C2×C3×D5AGL(1,ℤ/8)11yescharacteristic
160C2×C3×D5#13711yescharacteristic
161C3×AGL(1,𝔽5)—44no—
162C3×C4×C5—21no—
163C3×C4×C5—21no—
164C3×C4×C5—21no—
165C3×C4×C5—41no—
166C3×Dic5—21no—
167C3×Dic5—21no—
168C3×Dic5—41no—
169C3×Dic5—41no—
170C3×Dic5#10412yes—
171C2×C3×Q8C2×AGL(1,𝔽5)11yescharacteristic
172C2×C3×D4—101no—
173C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
174C22×C3×C4—51no—
175C22×C3×C4—51no—
176C22×C3×C4—51no—
177C22×C3×C4—101no—
178C22×C3×C4—102no—
179C2×C3×C8—102no—
180C3×C4⋊C4—51no—
181C3×C4⋊C4—51no—
182C3×C4⋊C4—101no—
183C3×C4⋊C4—101no—
184C3×C4⋊C4—104no—
185C3×C4⋊C4—101no—
186C3×C4⋊C4—101no—
187C3×C22⋊C4—21no—
188C3×C22⋊C4—21no—
189C3×C22⋊C4—101no—
190C3×C22⋊C4—101no—
191C3×C22⋊C4—101no—
192C3×C42—51no—
193C3×C42—51no—
194C3×C42—51no—
195C3×C42—101no—
196C3×C42—202no—
197C3×C42C2×AGL(1,𝔽5)11yescharacteristic
198C23×C5—21no—
199C22×D5C2×C3×D411yescharacteristic
200C2×AGL(1,𝔽5)—24no—
201C2×AGL(1,𝔽5)—42no—
202C2×AGL(1,𝔽5)—42no—
203C5×Q8—41no—
204C5×D4—41no—
205C2×C4×C5—21no—
206C2×C4×C5C3×C22⋊C411yescharacteristic
207C2×C4×C5C3×C22⋊C411yescharacteristic
208C2×C4×C5C22×C3×C411yescharacteristic
209C22⋊D5—41no—
210C22⋊D5—41no—
211C2×Dic5—21no—
212C2×Dic5—21no—
213C2×Dic5—42no—
214C2×Dic5C3×C22⋊C411yescharacteristic
215C2×Dic5C3×C22⋊C411yescharacteristic
216C2×Dic5C2×C3×D411yescharacteristic
217C4×D5—21no—
218C4×D5—21no—
219C4×D5—22no—
220C4×D5—41no—
221#170—42no—
222#126—101no—
223#125—51no—
224#124C3×AGL(1,𝔽5)11yescharacteristic, complemented
225#123—101no—
226C22⋊Q8—51no—
227#121—51no—
228C4×Q8—51no—
229C4×D4—51no—
230C4×D4—101no—
231#117—101no—
232C2×C4⋊C4—101no—
233C2×C42—51no—
234C2×C42—101no—
235C4⋊C8—101no—
236C22⋊C8—52no—
237#97—101no—
238#95—52no—
239C2×C3×C5—81no—
240C2×C3×C5#41011yescharacteristic
241C2×C3×C5#41711yescharacteristic
242C2×C3×C5C2×#10411yescharacteristic
243C3×D5—21no—
244C3×D5—21no—
245C23×C3—21no—
246C23×C3—51no—
247C3×Q8—41no—
248C3×D4—41no—
249C3×D4—201no—
250C3×D4—201no—
251C2×C3×C4—21no—
252C2×C3×C4—51no—
253C2×C3×C4—51no—
254C2×C3×C4—51no—
255C2×C3×C4—102no—
256C2×C3×C4—101no—
257C2×C3×C4—101no—
258C2×C3×C4—104no—
259C2×C3×C4—101no—
260C2×C3×C4—101no—
261C2×C3×C4—201no—
262C2×C3×C4—202no—
263C2×C3×C4—202no—
264C2×C3×C4—202no—
265C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
266C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
267C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
268C3×C8—202no—
269C22×C5—41no—
270C22×C5—41no—
271C22×C5—41no—
272C22×C5C2×C3×C22⋊C411yescharacteristic
273C2×D5—21no—
274C2×D5—21no—
275C2×D5C3×AGL(1,ℤ/8)11yescharacteristic
276C2×D5C3×#13711yescharacteristic
277AGL(1,𝔽5)—44no—
278C4×C5—21no—
279C4×C5—21no—
280C4×C5—21no—
281C4×C5—41no—
282Dic5—21no—
283Dic5—21no—
284Dic5—41no—
285Dic5—41no—
286Dic5C3×#10412yes—
287C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
288C2×D4—101no—
289C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
290C22×C4—51no—
291C22×C4—51no—
292C22×C4—51no—
293C22×C4—101no—
294C22×C4—102no—
295C2×C8—102no—
296C4⋊C4—51no—
297C4⋊C4—51no—
298C4⋊C4—101no—
299C4⋊C4—101no—
300C4⋊C4—104no—
301C4⋊C4—101no—
302C4⋊C4—101no—
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—101no—
306C22⋊C4—101no—
307C22⋊C4—101no—
308C42—51no—
309C42—51no—
310C42—51no—
311C42—101no—
312C42—202no—
313C42C2×C3×AGL(1,𝔽5)11yescharacteristic
314C3×C5#150911yescharacteristic, complemented, Hall
315C22×C3—41no—
316C22×C3—41no—
317C22×C3—41no—
318C22×C3—51no—
319C22×C3—51no—
320C22×C3—101no—
321C22×C3—101no—
322C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—41no—
327C3×C4—52no—
328C3×C4—101no—
329C3×C4—101no—
330C3×C4—204no—
331C3×C4—201no—
332C3×C4—201no—
333C2×C5—81no—
334C2×C5C3×#41011yescharacteristic
335C2×C5C3×#41711yescharacteristic
336C2×C5C2×C3×#10411yescharacteristic
337D5—21no—
338D5—21no—
339C23—21no—
340C23—51no—
341Q8—41no—
342D4—41no—
343D4—201no—
344D4—201no—
345C2×C4—21no—
346C2×C4—51no—
347C2×C4—51no—
348C2×C4—51no—
349C2×C4—102no—
350C2×C4—101no—
351C2×C4—101no—
352C2×C4—104no—
353C2×C4—101no—
354C2×C4—101no—
355C2×C4—201no—
356C2×C4—202no—
357C2×C4—202no—
358C2×C4—202no—
359C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
360C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
361C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
362C8—202no—
363C2×C3—81no—
364C2×C3—101no—
365C2×C3—101no—
366C2×C3#6577111yescharacteristic
367C2×C3D5⋊#10411yescharacteristic
368C2×C3#6579311yescharacteristic
369C5C3×#150911yescharacteristic, complemented, Sylow
370C22—41no—
371C22—41no—
372C22—41no—
373C22—51no—
374C22—51no—
375C22—101no—
376C22—101no—
377C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—41no—
382C4—52no—
383C4—101no—
384C4—101no—
385C4—204no—
386C4—201no—
387C4—201no—
388C3#1060453811yescharacteristic, direct product factor, Sylow
389C2—81no—
390C2—101no—
391C2—101no—
392C2C3×#6577111yescharacteristic
393C2C3×D5⋊#10411yescharacteristic
394C2C3×#6579311yescharacteristic
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×#4084C22
cosocleC2×C4×D5C23×C3
hypercenterC3×#124AGL(1,𝔽5)
hypocenterC5C3×#1509
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#124

Complements

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

NH ≅ G/Ntype
2. C3×C4×C4⋊AGL(1,𝔽5)C2semidirect
3. C3×#65684C2semidirect
4. C3×#64926C2semidirect
6. #10604538C3direct (N × H)
9. C3×C5×#124C4semidirect
11. C3×Q8×Dic5C4semidirect
13. C3×D4×Dic5C4semidirect
19. C3×#4016C4semidirect
26. C4×C4⋊AGL(1,𝔽5)C2×C3semidirect
27. #65684C2×C3semidirect
28. #64926C2×C3semidirect
64. C5×#124C3×C4semidirect
66. Q8×Dic5C3×C4semidirect
68. D4×Dic5C3×C4semidirect
74. #4016C3×C4semidirect
106. C3×#124AGL(1,𝔽5)semidirect
224. #124C3×AGL(1,𝔽5)semidirect
314. C3×C5#1509semidirect
369. C5C3×#1509semidirect
388. C3#10604538direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#660142
C3×C4×C4⋊AGL(1,𝔽5)2
C3×#656842
C3×#649262
C3×#649202
#106045383
C3×#15095